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abstract: 'We determine the quark-hadron transition line in the whole region of temperature ($T$) and baryon-number chemical potential ($\mu_{\rm B}$) from lattice QCD results and neutron-star mass measurements, making the quark-hadron hybrid model that is consistent with the two solid constraints. The quark part of the hybrid model is the Polyakov-loop extended Nambu-Jona-Lasinio (PNJL) model with entanglement vertex that reproduces lattice QCD results at $\mu_{\rm B}/T=0$, while the hadron part is the hadron resonance gas model with volume-exclusion effect that reproduces neutron-star mass measurements and the neutron-matter equation of state calculated from two- and three-nucleon forces based on the chiral effective field theory. The lower bound of the critical $\mu_{\rm B}$ of the quark-hadron transition at zero $T$ is $\mu_{\rm B}\sim 1.6$ GeV. The interplay between the heavy-ion collision physics around $\mu_{\rm B}/T =6$ and the neutron-star physics at $\mu_{\rm B}/T =\infty$ is discussed.'
author:
- Takahiro Sasaki
- Nobutoshi Yasutake
- Michio Kohno
- Hiroaki Kouno
- Masanobu Yahiro
title: ' Determination of quark-hadron transition from lattice QCD and neutron-star observation '
---
Introduction
============
The phase diagram of quantum chromodynamics (QCD) is a key to understanding not only natural phenomena such as compact stars and the early Universe but also laboratory experiments such as relativistic heavy-ion collisions[@KL1994; @Aoki; @Fukushima-review]. The first-principle lattice QCD (LQCD) simulation as a quantitative analysis of the phase diagram[@KL1994; @Aoki], however, has the severe sign problem at middle and large $\mu_{\rm B}/T$, where $T$ is temperature and $\mu_{\rm B}$ is baryon-number chemical potential. Therefore the QCD phase diagram is still unknown particularly at $\mu_{\rm B}/T \gsim 1$, although many possibilities are proposed by effective models there. A steady way of approaching the middle and large $\mu_{\rm B}/T$ regions is gathering solid information from different regions and extracting a consistent picture from the information.
LQCD simulations are quite successful at $\mu_{\rm B}/T \lsim 1$[@KL1994; @Aoki; @AliKhan; @Bazavov; @Borsanyi]. They are providing high-precision results for the realistic 2+1 flavor system at the present day, for example the transition temperature, the equation of state (EoS), and fluctuations of conserved charges [@Bazavov; @Borsanyi]. As a way of extending the understanding to the $\mu_{\rm B}/T \gsim 1$ region, we can consider effective models such as the Polyakov-loop extended Nambu-Jona-Lasinio (PNJL) model [@Sakai-EPNJL; @Sasaki-Columbia; @PNJL8; @Fukushima-Kashiwa; @nlPNJL; @Rossner; @Sakai-extendedZ3; @Gatto; @Sakai-signproblem; @Sakai-vector2008; @Sakai-EOS; @Meisinger; @Fukushima; @Megias; @Ratti; @Bratovic; @Lourenco]. Actually, some improved versions of the PNJL model yield desirable results consistent with LQCD simulations at $\mu_{\rm B}/T \lsim 1$ [@Sakai-EPNJL; @Sasaki-Columbia; @nlPNJL; @Fukushima-Kashiwa; @PNJL8]. However, the model approach has still various ambiguity at large $\mu_{\rm B}/T$.
A key issue in the large $\mu_{\rm B}/T$ limit, i.e. at finite $\mu_{\rm B}$ but vanishing $T$, is the EoS of nuclear matter. It is one of the most important subjects in nuclear physics to understand properties of symmetric nuclear matter and neutron matter microscopically from realistic baryon-baryon interactions. Various theoretical frameworks have been developed to study the subject. The results seem to be reliable because most of them are now converging a common result, but the common result cannot reproduce empirical saturation properties properly if one starts with realistic two-nucleon forces (2NF). This insufficiency is probably due to the lack of including three-nucleon forces (3NF). Recent development of the chiral effective field theory (Ch-EFT) [@WE79; @CDB] provides a way of determining 2NF and 3NF systematically from symmetries of underlying QCD. Although the Ch-EFT interaction is, by construction, to be applied at low and normal nuclear densities, the standard many-nucleon calculation using the Ch-EFT 2NF and 3NF at these densities should provide the predictive base for considering the neutron-matter EoS at higher densities. The combination of this new constraint and the experimental constraint [@Danielewicz] evaluated from the heavy-ion collision measurements is considered to be useful to determine the nuclear-matter EoS solidly.
The mass-radius (MR) relation of neutron star (NS) is sensitive to the nuclear-matter EoS [@Lattimer]. In this sense, astrophysical observation is another valuable source of information to provide a strong constraint on the EoS. Recent observations suggest the existence of massive NSs ($\sim 2 M_\odot $), which seems to exclude the possibility of soft EoS [@Demorest; @Antoniadis]. However, there exists uncertainties on the radius of NSs from varying observations. Steiner [*et al.*]{} have adopted the statistical approach to constrain this uncertainty, and have provided the best fitting against various observations on the MR relation [@Steiner].
There is a possibility that the quark-hadron phase transition occurs in NSs. The observations on the MR relation yield a strong constraint on both the quark and hadron phases, while the nuclear-matter EoS determined from the Ch-EFT 2NF and 3NF and the heavy-ion collision measurements does on the hadron phase. Therefore, the combination of the solid constraints may answer an important question, whether the quark-hadron phase transition occurs in NSs and further what is the critical chemical potential of the transition if it occurs. This is nothing but to clarify the QCD phase diagram in the large $\mu_{\rm B}/T$ limit.
In this paper, we determine the QCD phase diagram in the whole region from $\mu_{\rm B}/T=0$ to infinity, constructing a reliable quark-hadron hybrid model. The quark part of the hybrid model is the Polyakov-loop extended Nambu-Jona-Lasinio (PNJL) model with entanglement vertex that reproduces LQCD data at finite imaginary $\mu_{\rm B}$, finite real- and imaginary-isospin chemical potentials, small real $\mu_{\rm B}$[@Sakai-EPNJL; @Sasaki-Columbia], and strong magnetic field[@Gatto]. The hadron part of the hybrid model is the hadron resonance gas (HRG) model with volume-exclusion effect that reproduces the NS observations and the nuclear-matter EoS evaluated from the Ch-EFT 2NF and 3NF and the heavy-ion collision measurements. The volume-exclusion effect is necessary to reproduce the repulsive nature of the nuclear-matter EoS. The EoS provided by the hybrid model preserves the causality even at high $\mu_{\rm B}$. In order to construct the nuclear-matter EoS from the Ch-EFT 2NF and 3NF, we employ the lowest-order Brueckner theory (LOBT) in pure neutron matter with the Jülich N$^3$LO interaction [@EHM09]. The lower bound of the critical $\mu_{\rm B}$ of the quark-hadron transition at $T=0$ is found to be $\mu_{\rm B}\sim 1.6$ GeV. We also investigate the interplay between the heavy-ion collision physics around $\mu_{\rm B}/T =6$ and the neutron-star physics at $\mu_{\rm B}/T =\infty$.
This paper is organized as follows. In Sec. \[Modelsetting\], we present the quark-hadron hybrid model and evaluate the nuclear-matter EoS from the Ch-EFT 2NF and 3NF by using the LOBT. Numerical results are shown in Sec. \[Numericalresults\]. Section \[Summary\] is devoted to a summary.
Model setting {#Modelsetting}
=============
We consider a two-phase model to treat the quark-hadron phase transition by assuming that the transition is the first order[@Shao; @Blaschke; @Bonanno:2011ch; @Ippolito:2007hn]. For the quark phase, we use the entanglement PNJL (EPNJL) model [@Sakai-EPNJL; @Sakai-EOS; @Sasaki-Columbia]. This is an extension of the PNJL model and yields consistent results with LQCD data for finite imaginary $\mu_{\rm B}$, finite real- and imaginary-isospin chemical potentials, small real $\mu_{\rm B}$[@Sakai-EPNJL; @Sasaki-Columbia], and strong magnetic field[@Gatto]. For the hadron phase, we use the HRG model. The model is successful in reproducing the QCD EoS below the transition temperature at $\mu_{\rm B}/T=0$ [@Bazavov; @Borsanyi]. This model is extended for the baryon part to include the volume-exclusion effect. The effect is necessary to reproduce the repulsive nature of the nuclear-matter EoS. The volume-exclusion radius is fitted to reproduce the nuclear-matter EoS determined from the Ch-EFT 2NF and 3NF and the heavy-ion collision measurements.
In this work, we consider the 2-flavor system and do not take into account the existence of hyperons[@Schulze]. Even with hyperons, the fraction of hyperons is suppressed by the existence of quarks in NS [@Maruyama2007]. Hence, the possibility of the appearance of quarks is first discussed in this paper. The possibility of the appearance of hyperons will be discussed in a forthcoming paper.
Quark phase
-----------
We first consider the quark phase with the two-flavor EPNJL model. The Lagrangian density is obtained in Euclidean spacetime by $$\begin{aligned}
{\cal L}_{\rm EPNJL}
&=&
\bar{q}(\gamma_\nu D_\nu + {\hat m_0} - \gamma_4 {\hat \mu} )q
-G(\Phi )[(\bar{q}q )^2 +(\bar{q}i\gamma_5\vec{\tau}q )^2]
\nonumber\\
&&
+{\cal U}(\Phi [A],\Phi^* [A],T) ,
\label{EPNJLmodel}\end{aligned}$$ where $D_\nu=\partial_\nu - i\delta_{\nu 4}A_{4}^{a}{\lambda_a /2}$ with the Gell-Mann matrices $\lambda_a$. The two-flavor quark fields $q=(q_{\rm u},q_{\rm d})$ have masses ${\hat m_0}={\rm diag}(m_{\rm u},m_{\rm d})$, and the quark-number chemical potential matrix ${\hat \mu}$ is defined by ${\hat \mu}={\rm diag}(\mu_{\rm u},\mu_{\rm d})$. Baryon-number chemical potential is obtained by $\mu_{\rm B}=3(\mu_{\rm u}+\mu_{\rm d})/2$.
The gauge field $A_\mu$ is treated as a homogeneous and static background field. The Polyakov-loop $\Phi$ and its conjugate $\Phi ^*$ are determined in the Euclidean space by $$\Phi = \frac{1}{3}{\rm tr}_{\rm c}(L),~~~~~
\Phi^* = \frac{1}{3}{\rm tr}_{\rm c}({\bar L}),
\label{Polyakov}$$ where $L = \exp(i A_4/T)$ with $A_4/T={\rm diag}(\phi_r,\phi_g,\phi_b)$ in the Polyakov-gauge; note that $\lambda_a$ is traceless and hence $\phi_r+\phi_g+\phi_b=0$. Therefore we obtain $$\begin{aligned}
\Phi &=&{1\over{3}}(e^{i\phi_r}+e^{i\phi_g}+e^{i\phi_b})
\nonumber\\
&=&{1\over{3}}(e^{i\phi_r}+e^{i\phi_g}+e^{-i(\phi_r+\phi_g)}),
\notag\\
\Phi^* &=&{1\over{3}}(e^{-i\phi_r}+e^{-i\phi_g}+e^{-i\phi_b})
\nonumber\\
&=&{1\over{3}}(e^{-i\phi_r}+e^{-i\phi_g}+e^{i(\phi_r+\phi_g)}) .
\label{Polyakov_explict}\end{aligned}$$ We use the Polyakov-loop potential $\mathcal{U}$ of Ref. [@Rossner]: $$\begin{aligned}
{\cal U}
&=&
T^4\Bigl[
-\frac{a(T)}{2} {\Phi}^*\Phi
\nonumber\\
&&
+ b(T)\ln(1 - 6{\Phi\Phi^*} + 4(\Phi^3+{\Phi^*}^3) - 3(\Phi\Phi^*)^2 )
\Bigr]\end{aligned}$$ with $$a(T)
=
a_0
+ a_1\left(\frac{T_0}{T}\right)
+ a_2\left(\frac{T_0}{T}\right)^2,~~~~
b(T)
=
b_3\left(\frac{T_0}{T}\right)^3.$$ The parameter set in $\mathcal{U}$ is fitted to LQCD data at finite $T$ in the pure gauge limit. The parameters except $T_0$ are summarized in Table \[table-para\]. The Polyakov potential yields a first-order deconfinement phase transition at $T=T_0$ in the pure gauge theory. The original value of $T_0$ is $270$ MeV determined from the pure gauge LQCD data, but the EPNJL model with this value of $T_0$ yields a larger value of the pseudocritical temperature $T_\mathrm{c}$ of the deconfinement transition at zero chemical potential than $T_{\rm c}\approx 173\pm 8$ MeV predicted by full LQCD [@Karsch1994; @Kaczmarek; @Karsch2001]. Therefore we rescale $T_0$ to 190 MeV so that the EPNJL model can reproduce $T_{\rm c}=174$ MeV [@Sakai-EPNJL].
$a_0$ $a_1$ $a_2$ $b_3$
------------- ------------- ------------- -------------
$3.51$ $-2.47$ $15.2$ $-1.75$
: Summary of the parameter set in the Polyakov-loop potential sector determined in Ref. [@Rossner]. All parameters are dimensionless. []{data-label="table-para"}
The four-quark vertex originates from the one-gluon exchange between quarks and its higher-order diagrams. If the gluon field $A_{\nu}$ has a vacuum expectation value $\langle A_{0} \rangle$ in its time component, $A_{\nu}$ is coupled to $\langle A_{0} \rangle$ and then to $\Phi$ through $L$. Hence the effective four-quark vertex can depend on $\Phi$ [@Kondo]. In this paper, we use the following form for $G(\Phi)$[@Sakai-EPNJL]: $$\begin{aligned}
G(\Phi)=G_{\rm S}[1-\alpha_1\Phi\Phi^*-\alpha_2(\Phi^3+\Phi^{*3})].
\label{entanglement-vertex}\end{aligned}$$ This form preserves the chiral symmetry, the charge conjugation ($C$) symmetry and the extended $\mathbb{Z}_3$ symmetry [@Sakai-extendedZ3]. We take the parameters $(\alpha_1,\alpha_2)=(0.2,0.2)$ to reproduce LQCD data at imaginary $\mu_{\rm B}$ [@Sakai-EPNJL]. It is expected that $\Phi$ dependence of $G(\Phi )$ will be determined in future by the accurate method such as the exact renormalization group method [@Braun; @Kondo; @Wetterich].
Performing the mean-field approximation and the path integral over the quark field, one can obtain the thermodynamic potential $\Omega$ (per volume): $$\begin{aligned}
\frac{\Omega}{V}
&=&
G(\Phi )\sigma^2 + \mathcal{U}
- 2N_{\rm c}\sum_{f={\rm u,d}}
\int_{\Lambda}
\frac{d^3p}{(2\pi )^3}
E_f
\nonumber
\\
&&
-
\frac{2N_{\rm c}}{\beta}\sum_{f={\rm u,d}}
\int
\frac{d^3p}{(2\pi )^3}
\Bigl\{
\ln
\Bigl[
1+3\Phi e^{-\beta (E_f-\mu_f)}
\nonumber\\
&&
+3\Phi^*e^{-2\beta (E_f-\mu_f)}+e^{-3\beta (E_f-\mu_f)}
\Bigr]
\nonumber\\
&&
+
\ln
\Bigl[
1+
3\Phi^*e^{-\beta (E_f+\mu_f)}+3\Phi e^{-2\beta (E_f+\mu_f)}
\nonumber\\
&&
+e^{-3\beta (E_f+\mu_f)}
\Bigr]
\Bigr\}
\label{EPNJL-Omega}\end{aligned}$$ with $$\begin{aligned}
E_f = \sqrt{\vec{p}^{~2}+M_f^2}
,~~
M_f = m_0-2G(\Phi )\sigma
,~~
\sigma\equiv\braket{\bar{q}q}.\end{aligned}$$ The quark-number densities $n_{\rm u}$ and $n_{\rm d}$ are obtained by $$\begin{aligned}
n_f
=
-
\frac{\partial}{\partial\mu_f}\left(\frac{\Omega}{V}\right)\end{aligned}$$ for $f={\rm u},{\rm d}$ and the pressure $P$ is defined as $P=-\Omega + \Omega_0$, where $\Omega_0$ is thermodynamic potential at $T=\mu_{\rm u}=\mu_{\rm d}=0$. The three-dimensional cutoff is introduced for the momentum integration, since this model is nonrenormalizable; this regularization is denoted by $\int_{\Lambda}$ in Eq. . For simplicity, we assume isospin symmetry for ${\rm u}$ and ${\rm d}$ masses: $m_{l} \equiv m_{\rm u}=m_{\rm d}$. At $T=0$, the EPNJL model agrees with the NJL model that has three parameters; $G_{\rm S}$, $m_l$, and $\Lambda$. One of the typical parameter sets is shown in Table \[Table\_NJL\] [@Kashiwa]. These parameters are fitted to empirical values of pion mass and decay constant at vacuum.
$m_l({\rm MeV})$ $\Lambda({\rm MeV})$ $G_{\rm S} ({\rm GeV}^{-2})$
---------------------- -------------------------- ----------------------------------
$5.5$ $631.5$ $5.498$
: Summary of the parameter set in the NJL sector taken from Ref. [@Kashiwa]. \[Table\_NJL\]
The classical variables $X=\Phi$, ${\Phi}^*$ and $\sigma$ are determined by the stationary conditions $$\frac{\partial\Omega}{\partial X}=0.$$ The solutions to the stationary conditions do not give the global minimum of $\Omega$ necessarily. They may yield a local minimum or even a maximum. We then have checked that the solutions yield the global minimum when the solutions $X(T, \mu_{\rm u},\mu_{\rm d})$ are inserted into Eq. (\[EPNJL-Omega\]). In this work, we employ an approximation $\Phi = \Phi^*$ for numerical simplicity, because the approximation is good and hence sufficient for the present analysis[@Sakai-signproblem].
Repulsive forces among quarks are crucial to account for the 2$M_{\odot}$ NS observation [@Masuda; @Shao], since they harden the EoS of quark matter. We then introduce the vector-type four-body interaction to the EPNJL model [@Sakai-vector2008], $${\cal L}_{\rm EPNJL}
~\rightarrow~
{\cal L}_{\rm EPNJL}
+G_{\rm V}(\bar{q}\gamma_{\mu}q)^2.$$ The corresponding thermodynamic potential is obtained by the replacement, $$\begin{aligned}
\mu_f
&\rightarrow&
\mu_f - 2G_{\rm V}n_{\rm q},
\\
G(\Phi )\sigma^2
&\rightarrow&
G(\Phi )\sigma^2 - G_{\rm V}n_{\rm q}^2\end{aligned}$$ with $n_{\rm q}\equiv\braket{q^{\dagger}q}$. Here, $n_{\rm q}$ is determined in a self-consistent manner to satisfy the thermodynamic relation, $$-
\frac{\partial}{\partial \mu_{\rm q}}
\left(\frac{\Omega}{V}\right)
=
n_{\rm q},$$ where $\mu_{\rm q}=\mu_{\rm B}/3=(\mu_{\rm u}+\mu_{\rm d})/2$. The parameter $G_{\rm V}$ is treated as a free parameter in this paper. $G_{\rm V}$ dependence of the quark-hadron phase transition will be discussed in Sec. \[Numericalresults\].
Hadron phase
------------
Now we consider the hadron phase by using the HRG model and its extension. The pressure of the HRG model is composed of meson and baryon parts, $$P_{\rm H}
=
P_{\rm M}
+
P_{\rm B}$$ where $P_{\rm H}, P_{\rm M}$ and $P_{\rm B}$ are pressures of hadronic, mesonic and baryonic matters, respectively. For the meson part, we use the HRG model with no extension: $$\begin{aligned}
P_{\rm M}
&=&
\sum_i
d_iT
\int
\frac{d^3p}{(2\pi )^3}
\ln
\left( 1-e^{-\beta E_i}\right)
\\
E_i
&=&
\sqrt{\vec{p}^{~2}+M_i^2},\end{aligned}$$ where the summation is taken over all meson species and $M_i$ and $d_i$ are mass and degeneracy of $i$th meson, respectively.
For the baryon sector, the volume-exclusion effect [@Sakai-EOS; @Rischke; @Steinheimer] is introduced to reproduce the repulsive nature of the nuclear-matter EoS determined from the Ch-EFT 2NF and 3NF and the heavy-ion collision measurements that will be shown later in Sec. \[LOBT\].
We consider the system of particles having a finite volume $v$, characterized by thermodynamic variables ($T,V,\mu$). Following Refs. [@Sakai-EOS; @Rischke; @Steinheimer], we approximate the system of finite-volume particles by the mimic system of point particles with ($T,\tilde{V},\tilde{\mu}$) defined by $$\begin{aligned}
\tilde{V}&=&V-vN_{\rm B},\\
\tilde{\mu}&=&\mu - vP, \end{aligned}$$ where $N_{\rm B}$ is the total baryon number. The $P$ and $N_{\rm B}$ should be the same between the original and mimic systems. The chemical potential $\tilde{\mu}$ of the mimic system is determined to preserve the thermodynamic consistency. The procedure can be extended to the multi-species system composed of proton (p) and neutron (n), and the pressure of the mimic system is obtained by $$\begin{aligned}
P_{\rm B}
&=&
\frac{2}{\beta}\sum_{i=p,n}
\int\frac{d^3p}{(2\pi )^3}
\Bigl[
\ln
\left(
1+e^{-\beta (E_i-\tilde{\mu}_i)}
\right)
\nonumber\\
&&
+
\ln
\left(
1+e^{-\beta (E_i+\tilde{\mu}_i)}
\right)
\Bigl]\end{aligned}$$ with $E=\sqrt{p^2+M_i^2}$, $M_{\rm p}=938$ MeV, and $M_{\rm n}=940$ MeV[@PDG]. The entropy density ($s$) and the number densities ($n_{\rm p},n_{\rm n}$) of the original system are obtained from those of mimic system by $$\begin{aligned}
s
&=&
\frac{\tilde{s}}{1+v\tilde{n}_{\rm B}},
\\
n_i
&=&
\frac{\tilde{n}_i}{1+v\tilde{n}_{\rm B}},\end{aligned}$$ with $i={\rm p},{\rm n}$ and $\tilde{n}_{\rm B}=\tilde{n}_{\rm p}+\tilde{n}_{\rm n}$.
LOBT calculation with Ch-EFT interactions {#LOBT}
-----------------------------------------
The Brueckner theory is a standard framework to describe nuclear matter starting from realistic 2N interactions. The reaction matrix $G$, defined by the $G$-matrix equation $$G_{12}=v_{12}+v_{12}\frac{Q}{\omega -(t_1+U_1+t_2+U_2)}G_{12},$$ properly deals with short range (high momentum) singularities of the 2N potential $v_{12}$. The self-consistent determination of the single-particle (s.p.) potential $U$, $$\langle i|U|i\rangle \equiv \sum_j^{\rm occupied} \langle ij|G_{12}|ij-ji \rangle$$ corresponds to the inclusion of a certain class of higher-order correlations. In the above expression, $Q$ stands for the Pauli exclusion, $t_i$ is a kinetic energy operator, and $\omega$ is a sum of the initial two-nucleon s.p. energies. The reliability of the lowest-order calculation in the Brueckner theory has been demonstrated by the estimation of the smallness of the contribution of higher-order correlations on the one hand and by the consistency with the results from other methods such as variational framework[@Baldo].
The Ch-EFT provides a systematic determination of 2NF and 3NF. It is prohibitively hard, at present, to do full many-body calculations for infinite matter with including 3NF. The effects can be estimated by introducing a density-dependent effective 2N force $v_{12(3)}$ obtained by folding the third nucleon in infinite matter considered: $$\begin{aligned}
\langle \bk_1' \sigma_1' \tau_1', \bk_2'\sigma_2'\tau_2'|v_{12(3)}
|\bk_1 \sigma_1 \tau_1, \bk_2\sigma_2\tau_2\rangle_A \nonumber \\
=\sum_{\bk_3\sigma_3\tau_3} \langle \bk_1'\sigma_1'\tau_1',
\bk_2'\sigma_2'\tau_2', \bk_3\sigma_3\tau_3|v_{123} \nonumber \\
|\bk_1\sigma_1\tau_1,
\bk_2\sigma_2\tau_2, \bk_3\sigma_3\tau_3\rangle_A, \label{eq:efv}\end{aligned}$$ where $\sigma$ and $\tau$ stand for the spin and isospin indices, and two-remaining nucleons are assumed to be in the center-of-mass frame, namely $\bk_1'+\bk_2'=\bk_1+\bk_2$. The suffix $A$ denotes an antisymmetrized matrix element. The $G$-matrix equation is set up for the two-body interaction $v_{12}+\frac{1}{3}v_{12(3)}$. The factor $\frac{1}{3}$ is necessary for properly taking into account the combinatorial factor in evaluating the total energy. The LOBT $G$-matrix calculation in this approximation turns out to give quantitatively satisfactory description for the fundamental properties of nucleon many-body systems, namely saturation and strong spin-orbit field: the latter is essential for accounting for nuclear shell structure. These results were briefly reported in Ref. [@MK12]. Detailed accounts will be given in a separate paper.
In neutron matter, the contact $c_E$ term of the Ch-EFT 3NF vanishes and the $c_D$ term contributes negligibly. This means that the 3NF contributions in neutron matter are determined by the parameters that are fixed in the 2NF sector. Thus ambiguities concerning the 3NF contributions are minimal with the use of the Ch-EFT, in contrast to past studies in which phenomenological regulations were often applied. Because many-body correlation effects are expected not to be large because of the absence of strong tensor-force correlations in the $^3$E channel, the LOBT energies should be reliable in neutron matter.
Calculated energies of neutron matter with and without 3NF are shown in Fig. \[chiPT\], where the cutoff energy $\Lambda_{\rm EFT}$ of the Ch-EFT 2NF and 3NF is 550 MeV. The solid and dashed curves are results using the Ch-EFT interactions with and without 3NF, respectively. The energy curve without 3NF is very close to that of the standard modern 2NF, AV18 [@AV18]. For comparison, energies from the variational calculation by Illinois group [@APR] are included, which are frequently referred to as the standard EoS for discussing NS properties although their 3NF is phenomenological to some extent. It is interesting that the present prediction based on the Ch-EFT shows good correspondence to those energies.
In the application of the Ch-EFT, an estimation of theoretical uncertainties due to the uncertainties of the low-energy constants is customarily presented. As for the neutron-matter EoS, it is instructive to consult the estimation by Krüger [*et al.*]{} [@KTHS]. They show, in their Hatree-Fock type calculations that the neutron-matter energy at saturation density is in a range of $-14 \sim -17$ MeV for the Ch-EFT potential of the Jülich group [@EHM09] with the cutoff parameter of 450/700 MeV from uncertainties of coupling constants and cutoff parameters as well as many-body theoretical treatment. Following this estimation, we add the shaded are to indicate possible uncertainties, simply assuming the $\pm 8$ % of the potential contribution, which is $-18.6$ MeV at saturation density.
![ Neutron-matter energies as a function of the density $n_{\rm B}$. The solid and dashed curves are results of the Ch-EFT interactions with and without 3NF, respectively. The dotted curve shows results of the AV18 2NF [@AV18]. The typical result of the variational method by the Illinois group [@APR] is include by a dot-dashed curve, in which the Urbana 3NF is used together with the AV18. []{data-label="chiPT"}](./nmate13jun.eps){width="45.00000%"}
Numerical results {#Numericalresults}
=================
Zero temperature {#Zero temperature}
----------------
At zero temperature, the present hybrid model becomes simpler. Mesons do not contribute to the pressure, and the quark phase is described by the NJL model, since the EPNJL model is reduced to the NJL model there. In this section, we discuss the MR relation of NS, assuming that the hadron phase is a neutron-matter system.
The NJL model for the quark phase is solved under the condition $$2n_{\rm u} = n_{\rm d},$$ and the neutron-number density ($n_{\rm n}$) and its chemical potential ($\mu_{\rm n}$) are given by $$\begin{aligned}
n_{\rm n}
&=&
\frac{2n_{\rm d}-n_{\rm u}}{3},
\\
\mu_{\rm n}
&=&
\mu_{\rm u}+2\mu_{\rm d}.\end{aligned}$$ In the HRG model for the hadron phase, neutrons are assumed to have the exclusion volume $v$ which depends on $\tilde{\mu}_{\rm B}$. The dependence is parameterized as $$\begin{aligned}
v
&=&
\frac{4}{3}\pi r_{\rm excl}^3,
\\
r_{\rm excl}(\tilde{\mu}_{\rm B})
&=&
r_0+r_1\tilde{\mu}_{\rm B}+r_2\tilde{\mu}_{\rm B}^2.\end{aligned}$$ Figure \[fitting\] shows $n_{\rm B}$ dependence of the neutron-matter pressure; note that $n_{\rm B}=n_{\rm n}$ in neutron matter and it is normalized by the normal nuclear density $\rho_0=0.17$ (${\rm fm}^{-3}$). Closed squares denote the results of LOBT calculations with the Ch-EFT 2NF and 3NF. The results are plotted in the region of $n_{\rm B} < 2\rho_0$, since the Fermi energy becomes larger than the cutoff energy $\Lambda_{\rm ERT}$ beyond $n_{\rm B} = 2\rho_0$. As shown in panel (a), the result (solid line) of the HRG model with the volume-exclusion effect well reproduces the results of LOBT calculations at $\rho_0 \lsim n_{\rm B} \lsim 2\rho_0$, when $$\begin{aligned}
r_0&=&0.50 (\rm fm),\\
r_1&=&0.50 (\rm fm/GeV),\\
r_2&=&- 0.34 (\rm fm/GeV^2) .\end{aligned}$$ More precisely, the difference between the two results is at most $2 ({\rm MeV/fm^3})$, but the deviation is smaller than the theoretical uncertainty of the Ch-EFT EoS estimated in Sec. \[LOBT\]. For $n_{\rm B} < \rho_0$, the agreement of the extended HRG model with the Ch-EFT EoS is not perfect, so the Ch-EFT EoS itself is used there whenever the MR relation is evaluated.
In panel (b), the neutron-matter pressure is plotted at higher $n_{\rm B}$. The hatching area shows the empirical EoS [@Danielewicz] evaluated from heavy-ion collisions in which the uncertainty coming from the symmetry energy is taken into account. The present HRG model is also consistent with this empirical result.
![ Baryon-number density ($n_{\rm B}$) dependence of pressure ($P$) for neutron matter. $n_{\rm B}$ is normalized by the normal nuclear density $\rho_0=0.17$ (${\rm fm}^{-3}$). In the panel (b), experimental data is taken from Ref.[@Danielewicz]. []{data-label="fitting"}](./fit-v3.eps "fig:"){width="45.00000%"} ![ Baryon-number density ($n_{\rm B}$) dependence of pressure ($P$) for neutron matter. $n_{\rm B}$ is normalized by the normal nuclear density $\rho_0=0.17$ (${\rm fm}^{-3}$). In the panel (b), experimental data is taken from Ref.[@Danielewicz]. []{data-label="fitting"}](./fit2-v3.eps "fig:"){width="45.00000%"}
The speed of sound ($c_{\rm S}$) relative to the speed of light ($c$) is obtained by $$\frac{c_{\rm S}}{c}
=
\sqrt{\frac{dP}{d\varepsilon}}$$ with the energy density $\varepsilon$. The ratio $c_{\rm S}/c$ should be smaller than 1 to preserve the causality. As shown in Fig. \[sound\] that shows $n_{\rm B}$ dependence of $c_{\rm S}/c$, the present HRG model satisfies the causality even in the high-density region.
![ Baryon-number density ($n_{\rm B}$) dependence of the speed of sound ($c_{\rm S}$) in neutron matter. []{data-label="sound"}](./sound-v3.eps){width="45.00000%"}
Figure \[ex-radius\] shows $n_{\rm B}$ dependence of the neutron exclusion radius $r_{\rm excl}$. The resulting $r_{\rm excl}$ determined from the Ch-EFT and the empirical EoS has weak $n_{\rm B}$ dependence and the value is around 0.6 fm that is not far from the proton charge radius 0.877 fm[@PDG]. This fact implies that the present model is reasonable as an effective model.
![ Baryon-number density ($n_{\rm B}$) dependence of neutron exclusion radius ($r_{\rm excl}$). []{data-label="ex-radius"}](./rn-v3.eps){width="45.00000%"}
The MR relation of NS is obtained by solving the static and spherically symmetric Einstein equation, i.e., the Tolman-Oppenheimer-Volkoff (TOV) equation, $$\begin{aligned}
\frac{dP}{dr}
&=&
-G_{\rm N}\frac{\varepsilon m}{r^2}
\left(1+\frac{P}{\varepsilon}\right)
\left(1+\frac{4\pi Pr^3}{m}\right)
\left(1-\frac{2G_{\rm N}m}{r}\right)^{-1} ,
\nonumber\\
\frac{dm}{dr}
&=&
4\pi r^2\varepsilon \end{aligned}$$ with $G_{\rm N}$ being the gravitational constant [@Shapiro], where $$\begin{aligned}
m(r)=\int^r_04\pi r^{\prime 2}\varepsilon (r')dr'\end{aligned}$$ corresponds to the gravitational mass of the sphere of radius $r$. The solutions, $m(r)$ and $P(r)$, can be obtained by integrating the TOV equations numerically, when the EoS, $P=P(\varepsilon)$, is given. The integration stops at $r=R$ where $P(R)=0$, and the maximum value $R$ is the radius of NS and the mass is given by $M=m(R)$. Here, we adopt the Baym-Pethick-Sutherland (BPS) EoS for the outer crust [@Baym]. Although, for the inner crust, we should consider the non-uniform structures, namely the pasta structures [@Maruyama2010], we just connect the outer crust EoS to the Ch-EFT EoS at the subnuclear density smoothly, since this simplification does not affect on the MR relation. Similarly the Ch-EFT EoS is connected to the HRG-model EoS at $n_{\rm B}\sim \rho_0$.
Figure \[MR-full\] shows the MR relation obtained by the hadron model mentioned above. The model result (dashed line) is compared with two observation data. The first one obtained by A. W. Steiner [*et al.*]{} is the best fitting against various observations on the MR relation [@Steiner]. This is not a strong constraint because of the uncertainty of the analysis particularly on X-ray burst phenomena. The second one has been obtained by P. B. Demorest [*et al.*]{} from measurements of pulsar J1614-2230 [@Demorest]. This yields the lower bound of maximum NS mass, $M=(1.97\pm 0.04)M_{\odot}$ and is a strong constraint. The present hadron model yields a consistent result with both the observations.
![ The mass-radius relation obtained by the neutron matter with quark-hadron transition. The two observation data are taken from Ref. [@Demorest; @Steiner]. []{data-label="MR-full"}](./MR-v3.eps){width="45.00000%"}
Next, we consider the quark-hadron transition with the Maxwell construction by assuming that the transition is the first-order. The transition occurs, when the two phases satisfy the conditions $$\begin{aligned}
\mu_{\rm u} + 2\mu_{\rm d}&=&\mu_{\rm n},\\
P_{\rm Q}(\mu_{\rm u},\mu_{\rm d})&=&P_{\rm H}(\mu_{\rm n}).\end{aligned}$$ Here we do not consider the finite-size effects due to the Coulomb interaction and the surface tension [@Yasutake-review]. We will study these effects on the EoS in the future.
Once the quark phase appears as a consequence of the quark-hadron phase transition, it softens the EoS. The quark-matter part of the EoS depends on the strength of $G_{\rm V}$; more precisely, it becomes hard as $G_{\rm V}$ increases. Hence, the lower bound of $G_{\rm V}$ is determined from the 2$M_{\odot}$ NS observation. The lower bound of such $G_{\rm V}$ is $0.03G_{\rm S}$, as shown below. Figure \[MR-full\] shows the MR-relation determined by the present hybrid model. The solid line shows the result of the hybrid model with $G_{\rm V}=0.03G_{\rm S}$, while the dashed line represents the result of the hadron model that corresponds to the hybrid model with $G_{\rm V}=\infty$. Thus the hybrid model is consistent with the 2$M_{\odot}$ NS observation, when $G_{\rm V} \ge 0.03G_{\rm S}$.
Finite temperature
------------------
In this section, we consider the symmetric matter by setting $\mu_{\rm p}=\mu_{\rm n}=\mu_{\rm B}$ and $\mu_{\rm u}=\mu_{\rm d}=\mu_{\rm B}/3$. Understanding of the symmetric matter at finite $T$ is important to elucidate early universe or heavy-ion collisions.
Figure \[EoS-LQCD\] shows $T$ dependence of (a) the pressure and (b) the energy density obtained by the hybrid model in comparison with LQCD results [@AliKhan], where $T$ is normalized by the deconfinement transition temperature $T_c$. The deconfinement transition is crossover at $\mu_{\rm B}=0$ in both of LQCD simulations and the EPNJL model. The transition temperature defined by the peak of susceptibility is $T_c=174$ MeV for both the results [@Sakai-EPNJL]. The hybrid model (solid line) shows the first-order quark-hadron transition, whereas the LQCD simulations (closed squares) do the crossover transition. Except for the transition temperature $T \approx 1.1T_c$ of the first-order quark-hadron transition, the model results almost reproduce the LQCD results.
![ T dependence of (a) the pressure and (b) the energy density obtained by the hybrid model. The result are normalized by their Stefan-Boltzmann limits. LQCD data is taken from Ref. [@AliKhan]. []{data-label="EoS-LQCD"}](./pres-mu0-v3.eps "fig:"){width="45.00000%"} ![ T dependence of (a) the pressure and (b) the energy density obtained by the hybrid model. The result are normalized by their Stefan-Boltzmann limits. LQCD data is taken from Ref. [@AliKhan]. []{data-label="EoS-LQCD"}](./ener-mu0-v3.eps "fig:"){width="45.00000%"}
Figure \[phase-diagram\] is the phase diagram in the $\mu_{\rm B}$-$T$ plane. The thick solid line is the quark-hadron transition line obtained by the hybrid model with $G_{\rm V}=0.03G_{\rm S}$. The transition is the first order everywhere. In this sense, this is an approximate result at least at $\mu_{\rm B}/T < 1$, since LQCD simulations show that the deconfinement (quark-hadron) transition is crossover there. As an important result, the first-order quark-hadron transition line is close to the crossover deconfinement transition line (dot-dashed line) obtained by the EPNJL model at $\mu_{\rm B}/T < 1$, where the deconfinement transition line is simply defined as a line satisfying $\Phi = 0.5$. Noting that the EPNJL model well simulates LQCD results at $\mu_{\rm B}/T < 1$, one can see that the present hybrid model is a rather good effective model even at small $\mu_{\rm B}/T$. The dashed and dotted lines correspond to the first-order and crossover chiral transition lines, whereas the closed square is the critical endpoint (CEP) of the chiral transition.
As already mentioned in Sec. \[Zero temperature\], the present hybrid model is consistent with the NS observations at $T=0$, when $G_{\rm V} \ge 0.03G_{\rm S}$. In the hybrid model with $G_{\rm V}=0.03G_{\rm S}$, the critical baryon-number chemical potential $\mu_{\rm B}^{\rm (c)}$ of the first-order quark-hadron transition at $T=0$ is $1.6$ GeV, as shown in Fig. \[phase-diagram\]. This is the lower bound of $\mu_{\rm B}^{\rm (c)}$, since $G_{\rm V}$ can vary from $0.03G_{\rm S}$ to $\infty$; actually, $\mu_{\rm B}^{\rm (c)}$ is shifted to higher $\mu_{\rm B}$ as $G_{\rm V}$ increases, as shown later in Fig. \[phase-diagram2\]. This is the primary result of the present work. In the EPNJL model, meanwhile, the critical baryon-number chemical potential of the chiral transition at $T=0$ is 1 GeV. The point belongs to the hadron phase in the hybrid model. Thus, we do not have any conclusive result on the chiral transition at $T=0$. This is an important problem to be solved in future.
![ Phase diagram in the $\mu_{\rm B}$-$T$ plane. The solid line represents a quark-hadron transition line given by the hybrid model. The other lines and symbol are obtained by the EPNJL model. The dashed (dotted) line correspond to the first-order (crossover) chiral transition line, and the dot-dashed line is a contour line corresponds to $\Phi = 0.5$. The closed square is the critical endpoint (CEP). []{data-label="phase-diagram"}](./phase-v3.eps){width="45.00000%"}
![ Phase diagram in the $\mu_{\rm B}$-$T$ plane. The dashed line is the result of the hybrid model with $G_{\rm V}=0.03 G_{\rm S}$; the line corresponds to the thick solid line in Fig. \[phase-diagram\]. The thick-solid line corresponds to the case of $G_{\rm V}=0.2 G_{\rm S}$. Two thin-solid lines mean lines of $\mu_{\rm B}/T=3$ and $6$, respectively. []{data-label="phase-diagram2"}](./phase-4-v3.eps){width="45.00000%"}
In principle one can determine the strength of $G_{\rm V}$ from LQCD simulations present at $\mu_{\rm B} /T<3$, but in practice the strength thus determined has large ambiguity[@Bratovic; @Lourenco]. Figure \[phase-diagram2\] shows the phase diagram in the $\mu_{\rm B}$-$T$ plane predicted by the hybrid model with different values of $G_{\rm V}$. The dashed and solid lines correspond to the cases of $G_{\rm V}=0.03 G_{\rm S}$ and $0.2 G_{\rm S}$, respectively. The phase transition line is insensitive to the variance of $G_{\rm V}$ at $\mu_{\rm B} /T<3$, but rather sensitive at $\mu_{\rm B}/T \approx 6$. Thus the physics at $\mu_{\rm B}/T \approx 6$ is strongly related to the NS physics at $\mu_{\rm B}/T = \infty$. If the quark-hadron transition line at $\mu_{\rm B}/T \approx 6$ is determined by LQCD simulations or heavy-ion collision experiments, it will also determine $\mu_{\rm B}^{\rm (c)}$ more strictly.
Summary {#Summary}
=======
We have studied the QCD phase diagram in the whole region from $\mu_{\rm B}/T=0$ to infinity, constructing the quark-hadron hybrid model that is consistent with LQCD results at $\mu_{\rm B}/T=0$ and at $\mu_{\rm B}/T=\infty$ with NS observations and the neutron-matter EoS evaluated from the Ch-EFT 2NF and 3NF and the heavy-ion collision measurements. The EoS provided by the model preserves the causality even at high $n_{\rm B}$. At $n_{\rm B} < 2\rho_0$ the baryon part of the EoS agrees with the neutron-matter EoS constructed from the Ch-EFT 2NF and 3NF with the lowest-order Brueckner theory (LOBT). The Ch-EFT provides a systematic framework of constructing 2NF and 3NF, and the 3NF yields a significant effect on the EoS at $n_{\rm B} > \rho_0$. In this sense, the use of the Ch-EFT, which respects symmetries of QCD, is inevitable to construct the neutron-matter EoS with no ambiguity.
We have determined the lower bound of the critical chemical potential of the quark-hadron transition at $T=0$: $$\mu_{\rm B}^{\rm (c)}\sim 1.6~{\rm GeV}.$$ This is the primary result of this work. In the NJL model, the first-order chiral transition occurs at $\mu_{\rm B}^{\rm (c)}=1$ GeV, when $T=0$. The point is located in the hadron phase in the hybrid model. Thus, the critical chemical potential of the chiral transition at $T=0$ is unknown. In this sense, the NJL model is not good enough at $T=0$. It is then highly required to introduce baryon degrees of freedom in the effective model.
We have also shown the interplay between the heavy-ion collision physics at $\mu_{\rm B}/T \approx 6$ and the NS physics at $\mu_{\rm B}/T = \infty$. If the vector coupling $G_{\rm V}$ is determined at $\mu_{\rm B}/T \approx 6$ from heavy-ion collision measurements, the information determines the critical chemical potential of the quark-hadron transition at $T=0$ and hence properties of NS in the inner core. This fact strongly suggests that these two regions should be studied simultaneously.
The authors thank A. Nakamura and K. Nagano for useful discussions. T.S. is supported by JSPS KAKENHI Grant No. 23-2790. N.Y. is supported by JSPS KAKENHI Grant No. 2510-5510. M.K. is supported by a Grant-in-Aid for Scientific Research (C) from the Japan Society for the Promotion of Science (Grant Nos. 22540288 and 25400266).
[19]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , , ****, (); , Lect. Notes Phys. [**583**]{}, 209 (2002). , , , , , ****, (). , , ****, (); , ****, (); , , (). , ****, (). , , ****, (); ****, (). , , , , , , , , ****, (); , , , , , , ****, (). , , ****, (). , ****, (); ****, (). , , , ****, (). , , , ****, (). , , , ****, (). , , , , ****, (). , , , , , ****, (). , , , , ****, (); , , , , ****, (). , , , , ****, (). , , , , ****, (). , ****, (). , , , , ****, (). , , , , ****, (). , , , , , ****, (). , (). , , , (). , , , ****, (). , ****, (). , ****, (). , , , ****, (). , ****, (). , , , , , ****, (). , , ****, (). , , , ****, (). , , , ****, (). , , , , , , , ****, (); , , , , , (). , , , , , ****, (). L. Bonanno and A. Sedrakian, Astron. Astrophys. [**539**]{}, A16 (2012) N. Ippolito, M. Ruggieri, D. Rischke, A. Sedrakian and F. Weber, Phys. Rev. D [**77**]{}, 023004 (2008) , ****, (); and references therein. , , , , ****, (). , , ****, (); , ****, (). , , , , ****, (). , , , ****, (). , ****, (). , , , , ****, (); , , (). , ****, (). , , , , , ****, (). , , , ****, (). , , , , ****, (). , , , ****, (). , , ****, (). , ****, (). , ****, (). , , , ****, (). , , , ****, (). , , , , (). S.L. Shapiro and S.A. Teukolsky, [*Black Holes, White Dwarfs, and Neutron Stars*]{} (John Wiley & Sons, New York, 1983). , , , ****, (). , , , ****, (). Nobutoshi. Yasutake, Tsuneo Noda, Hajime Sotani, Toshiki Maruyama, and Toshitaka Tatsumi, [*Recent Advances in Quarks Research*]{}, Nova, (2013), Chap.4, pp.63, ISBN 9781622579709, arXiv:1208.0427\[astro-ph\].
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title: 'Zero-Shot Feature Selection'
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abstract: 'The Bloch sphere is a familiar and useful geometrical picture of the time evolution of a single spin or a quantal two-level system. The analogous geometrical picture for three-level systems is presented, with several applications. The relevant SU(3) group and su(3) algebra are eight-dimensional objects and are realized in our picture as two four-dimensional manifolds that describe the time evolution operator. The first, called the base manifold, is the counterpart of the S$^2$ Bloch sphere, whereas the second, called the fiber, generalizes the single U(1) phase of a single spin. Now four-dimensional, it breaks down further into smaller objects depending on alternative representations that we discuss. Geometrical phases are also developed and presented for specific applications. Arbitrary time-dependent couplings between three levels or between two spins (qubits) with SU(3) Hamiltonians can be conveniently handled through these geometrical objects.'
address: ' Dept of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803-4001,USA\'
author:
- Sai Vinjanampathy and A R P Rau
title: 'Bloch sphere like construction of SU(3) Hamiltonians using Unitary Integration'
---
\[intro\]Introduction
=====================
Three-level systems are of fundamental importance to many branches of physics. While two levels give the simplest model for the dynamics of discrete systems, three levels illustrate the role that an intermediate state can play in inducing transitions between the other two. Canonical examples of this include applications in quantum optics that use three-level atoms to control quantum state evolution [@eit]. Such laser control is used, for instance, to transfer population between two states using stimulated Raman adiabatic passage (STIRAP) [@stirap; @stirap2] and chirped adiabatic passage (CARP) [@Chelkowski]. In some of these systems, the interaction of the radiation with the atom is represented as a time-dependent Hamiltonian inducing an energy separation between the two states that varies with time. For a non-zero sweep rate, it can be shown that there is finite transition probability between the states [@LZ; @LZ2; @LZ3]. The study of Landau-Zener transitions in multilevel systems is of interest to understand the interplay between various level crossings [@vitanov]. Particle physics represents another example where three-level systems play a central role as, for example, the oscillations of neutrino flavor eigenstates [@neutrino].
The general Hamiltonian of a three-level system involves 8 independent operators. Such a set can also naturally arise as a subgroup of higher level systems where there is some degeneracy involved. Thus, several important two-qubit problems in quantum computing and quantum information can be so written in terms of eight operators that form a subalgebra of the full fifteen operators that describe two spins. The Hamiltonian describing anisotropic spin exchange is an example of one such important physical problem. While isotropic spin exchange has been explored to design two-qubit gates in quantum computing, anisotropic spin exchange has been studied as a possible impediment to two-qubit gate operations [@divincenzo; @divincenz2]. Such a SU(3) Hamiltonian is given by $$\begin{aligned}
\label{DV}
\mathbf{H}(t)=J(t)(\vec{\bm{\sigma}}.\vec{\bm{\tau}} +
\vec{\beta}(t).(\vec{\bm{\sigma}}
\times \vec{\bm{\tau}}) + \vec{\bm{\sigma}}.\mathbf{\Gamma}(t).\vec{\bm{\tau}}),\end{aligned}$$ when written in terms of a scalar, a vector and a symmetric tensor operator expressed in terms of two Pauli spins. Here, $\vec{\beta}(t)$ is the Dzyaloshinksii-Moriya vector [@dz1; @dz2] and $\mathbf{\Gamma}(t)$ is the (traceless) symmetric interaction term. While the first term is the familiar Ising interaction Hamiltonian [@Chandler], the last two terms are due to spin-orbit coupling.
Given this wide applicability, a geometrical picture of the dynamics of three-level systems can be useful. For a two-level system, the geometry of the evolution operator is well known. Any density matrix can be written as $\bm{\rho}=(I^{(2)}+\vec{n}.\vec{\bm{\sigma}})/2$, where $\vec{\bm{\sigma}}$ are the Pauli matrices. Unitary evolution of $\bm{\rho}$ is represented as the vector $\vec{n}$ rotating on the surface of a three dimensional unit sphere called the Bloch sphere [@poincare]. This vector, along with a phase, accounts for the three parameters describing the time evolution operator of a two-level system. The vector $\vec{n}$, along with the phase factor, is shown in Fig. (\[Fig1\]). The vector $\vec{n}$ shown traces out the “base manifold” and together with the global phase factor or “fiber” at each point on that manifold is referred to as a “fiber bundle” [@Bengtsson]. While the density matrix is independent of it, the complete description of the system requires this phase as well. The aim of this paper is to provide an analogous geometrical picture for a three-level system with appropriate generalizations of the base and fiber.
Some work already exists regarding the geometry of SU(3). Following Wei and Norman [@wei], Dattoli and Torre have constructed the “Rabi matrix" for a general SU(3) unitary evolution in [@dattoli]. Mosseri and Dandoloff in [@remy] described the generalization of the Bloch sphere construction of single qubits to two qubits via the Hopf fibration description. This method relies upon the homomorphism between the SU(2) and SO(3) groups and likewise between the SU(4) and SO(6) groups. In [@tilma], the authors propose a generalized Euler angle parameterization for SU(4). This decomposition is similar to the work in [@oldui; @oldui2; @oldui3; @oldui4; @restui; @restui2; @restui3] into which fits our treatment of SU(3) in this paper.
Another well known choice of the $(N^{2}-1)$ generators $\mathbf{s}_{j}$ of the SU(N) group was studied in [@hioe; @dattoli2]. Consider $\mathbf{s}_{j}$, chosen to be traceless and Hermitian such that $[\mathbf{s}_{i},\mathbf{s}_{j}]=2i f_{ijk}\mathbf{s}_{k}$ and $Tr \{
\mathbf{s}_{i}\mathbf{s}_{j} \} = 2 \delta_{jk} $. Here, $f_{ijk}$ is the completely antisymmetric symbol which for a two-level system is the Levi-Civita symbol $\epsilon_{ijk}$, and a repeated index is summed over. In this basis, the Hamiltonian is written as $
\mathbf{H}(t) = \Gamma_{i} \mathbf{s}_{i} $. With this choice, the Liouville-Von Neumann equation for the density matrix $\bm{\rho}=\mathbf{I}/N+S_{j}\mathbf{s}_{j}/2$ becomes $\dot{S}_{i}=f_{ijk}\Gamma_{j}S_{k}$. Note that for the N=2 case, this is the familiar Bloch sphere representation. But, for SU(3), this representation differs from the one we present in two aspects. Firstly, the “coherence vector”, whose elements are real and are given by $S_{j}$, experiences rotations in a $(N^{2}-1)$ dimensional space. For instance, for SU(3), the coherence vector undergoes rotations in an eight-dimensional space. Arbitrary rotations in eight dimensions are characterized by 28 parameters. But since a three-level Hamiltonian is only characterized by 8 real quantities, this means that the coherence vector is not permitted arbitrary rotations and is instead constrained. Secondly, the coherence vector representation does not differentiate between local and non-local operations. Our decomposition of the time evolution operator into a diagonal and an off-diagonal term in this paper is more suited for this differentiation. Such a parameterization of the time evolution operator in terms of local and non-local operations can be useful in understanding entanglement. The aim of this paper is to discuss the geometry of two-qubit time evolution operators in terms of such a decomposition. The authors in [@englert] discuss an alternative decomposition of two-qubit states in terms of two three-vectors and a $3 \times
3$ dyadic to discuss entanglement.
A series of papers presented a systematic approach to studying N-level systems using a program of unitary integration [@oldui; @oldui2; @oldui3; @oldui4; @newui; @newui2; @restui; @restui2; @restui3]. Continuing this program, we present a complete analytical solution to the three-level problem that generalizes the Bloch sphere approach to three levels. Below, we define the fiber bundle via two different decompositions which allows us to extract the geometric phases associated with a three-level system (for a discussion on the quantum phases of three-level systems, see [@gpqutrit; @berry2and3]). These fiber bundles are $\{$SU(3)/SU(2)$\times$ U(1)$\}\times\{$SU(2)$\times$U(1)$\}$ and $\{$SU(4)/\[SU(2)$\times$SU(2)\]$\}
\times\{$SU(2)$\times$SU(2)$\}$.
![Bloch or Poincare sphere representation for SU(2). The base manifold is the $S^{2}$ sphere while the fiber is given by the U(1) phase at each point on that sphere. Together, we have the fiber bundle SU(2) $\simeq$ S$^2\times$U(1).[]{data-label="Fig1"}](Fig1){height="2" width="3"}
The structure of this paper is as follows: Section \[ui\] outlines the unitary integration program to solve time-dependent operator equations. Section \[su3\] uses this technique for the solution of a general time-dependent SU(3) Hamiltonian completely analytically. Section \[discussion\] presents the geometry of the time evolution operator for SU(3) with some applications. Section \[GP\] presents a coordinate description that is useful to define the geometric phase for three-level systems, and Section \[conclusions\] presents the conclusions. The appendix will present an alternative analytical solution to the three-level problem by exploiting the natural embedding of SU(3) in SU(4).
\[ui\]Unitary Integration
=========================
Many important applications in physics involve time dependence in the Hamiltonian. For such systems, the time evolution operator is not given by the simple exponentiation of the Hamiltonian [@sakurai]. To handle the time evolution for such Hamiltonians iteratively, “Unitary Integration" was proposed in [@oldui; @oldui2; @oldui3; @oldui4]. Earlier work with this technique is presented in [@wei; @dattoli2]. Later, the technique was presented as generalizing the SU(2) example to solve iteratively for the time evolution operator $\mathbf{U}^{(N)}(t)$ of N-level systems [@newui; @newui2]. Consider the N-dimensional Hamiltonian $\mathbf{H}^{(N)}$ given by $$\label{Hamiltonian}
\mathbf{H}^{(N)}=\left(\begin{array}{cc}\mathbf{H}^{(N-n)}&\mathbf{V
}\\ \mathbf{V}^{\dagger}&
\mathbf{H}^{(n)}\end{array} \right).$$ The diagonal blocks are (N$\--$n)- and (n)-dimensional square matrices, respectively, while $\mathbf{V}$ is an $(N-n) \times (n)$-dimensional matrix.
The evolution operator $\mathbf{U}^{(N)}(t)$ for such a $\mathbf{H}^{(N)}$ is written as a product of two operators $\mathbf{U}^{(N)}(t)=\widetilde{U}_{1}\widetilde{U}_{2}$, where $$\begin{aligned}
\label{nonunit_u}
\widetilde{U}_{1}=\left(\begin{array}{cc}\mathbf{I}^{(N-n)}& \mathbf{z}(t)\\
\mathbf{0}^{\dagger}&
\mathbf{I}^{(n)} \end{array} \right)\left(\begin{array}{cc}\mathbf{I}^{(N-n)}&
\mathbf{0}\\
\mathbf{w}^{\dagger}(t)& \mathbf{I}^{(n)} \end{array} \right)
,\\
\widetilde{U}_{2}=\left(\begin{array}{cc}\widetilde{\mathbf{U}}^{(N-n)}&
\mathbf{0}\\ \mathbf{0}^{\dagger}&
\widetilde{\mathbf{U}}^{n} \end{array} \right).\nonumber\end{aligned}$$ For any $N$, $n$ is arbitrary with $1 \leq n < N$, and tilde denotes that the matrices need not be unitary. The product of three factors parallels the product of exponentials in three Pauli matrices. Equations defining the rectangular matrices $\mathbf{z}(t)$ and $\mathbf{w}^{\dagger}(t)$ are developed and the problem is reduced to the two residual $(N-n)$- and $(n)$ dimensional evolution problems sitting as diagonal blocks of $\widetilde{U}_{2}$. $\mathbf{z}(t)$ and $\mathbf{w}^{\dagger}(t)$ are related to each other through the unitarity of $\mathbf{U}^{(N)}(t)$ [@newui; @newui2]: $$\begin{aligned}
\mathbf{z}=-\bm{\gamma}_{1}\mathbf{w}=-\mathbf{w}\bm{\gamma}_{2},\end{aligned}$$ with $\bm{\gamma}_{1}=\hat{\mathbf{I}}^{(N-n)}+\mathbf{z}.\mathbf{z}^{\dagger}$ and $\bm{\gamma}_{2}=\hat{\mathbf{I}}^{(n)}+\mathbf{z}^{\dagger}.\mathbf{z}$.
With $\mathbf{U}^{(N)}(t)$ in such a product form, the Schrödinger equation is written as $$\begin{aligned}
\label{effeqn}
i \dot{\widetilde{U}}_{2}(t)=\mathbf{H}_{\mathtt{eff}}\widetilde{U}_{2},\\
\nonumber
\mathbf{H}_{\mathtt{eff}}=\widetilde{U}^{-1}_{1}\mathbf{H}^{(N)}\widetilde{U}_{1
} \-- i \widetilde{U}^{-1}_{1}\dot{\widetilde{U}}_{1}.\end{aligned}$$ Here, overdot denotes differentiation with respect to time. Since $\widetilde{U}_{2}$ is block diagonal, the off-diagonal blocks of equation (\[effeqn\]) define the equation satisfied by $\mathbf{z}$ $$\label{zdot}
i\dot{\mathbf{z}}=\mathbf{H}^{(N-n)}\mathbf{z} + \mathbf{V}
\--\mathbf{z}(\mathbf{V}^{\dagger}\mathbf{z}+\mathbf{H}^{(n)}).$$ Note that the initial condition $U^{N}(0)=\mathbf{I}^{N}$ implies that $\widetilde{U}_{1}(0)=\mathbf{I}^{(N-n)}$, $\widetilde{U}_{2}(0)=\mathbf{I}^{(n)}$ and $\mathbf{z}(0)=\mathbf{0}^{(N-n)}$. equation (\[zdot\]), along with the initial condition can be solved to determine $\mathbf{z}$ and thereby $\widetilde{U}_{1}$ and $\mathbf{H}_{\mathtt{eff}}$ for subsequent solution of equation (\[effeqn\]) for $\widetilde{U}_{2}$. In this manner, the procedure iteratively determines $U^{(N)}(t)$.
Before discussing the geometry of the time evolution operators for this unitary case, we briefly mention the procedure to deal with non-Hermitian Hamiltonians. For such a non-Hermitian Hamiltonian, $$\label{Hamiltonian_non_Hermitian}
\mathbf{H}^{(N)}=\left(\begin{array}{cc}\widetilde{\mathbf{H}}^{(N-n)}&\mathbf{V
}\\ \mathbf{Y}^{\dagger}&
\widetilde{\mathbf{H}}^{(n)}\end{array} \right),$$ where tilde denotes possibly non-Hermitian character, and the off-diagonal components $\mathbf{V}$ and $\mathbf{Y}$ are independent. In this case, equation (\[zdot\]) is replaced by $$\label{zdot_non_Hermitian}
i\dot{\mathbf{z}}=\widetilde{\mathbf{H}}^{(N-n)}\mathbf{z} + \mathbf{V}
\--\mathbf{z}(\mathbf{Y}^{\dagger}\mathbf{z}+\widetilde{\mathbf{H}}^{(n)}),$$ and there is a separate equation governing the evolution of $\mathbf{w}$ given by $$\label{wdot_non_Hermitian}
i\dot{\mathbf{w}}^{\dagger}=\mathbf{w}^{\dagger}(\mathbf{z}\mathbf{Y}^{\dagger}
-\widetilde{\mathbf{H}}^{(N-n)})+(\widetilde{\mathbf{H}}^{(n)} +
\mathbf{Y}^{\dagger}\mathbf{z})\mathbf{w}^{\dagger} +\mathbf{Y}^{\dagger}.$$ The diagonal terms of the time-evolution operators are governed by $$\begin{aligned}
\label{effeqn_non_Hermitian}
i \dot{\widetilde{U}}_{2}(t)=\left(\begin{array}{cc}
\widetilde{\mathbf{H}}^{(N-n)}-\mathbf{z}\mathbf{Y}^{\dagger}&0\\
0&\widetilde{\mathbf{H}}^{(n)}+\mathbf{Y}^{\dagger}\mathbf{z}
\end{array}
\right)\widetilde{U}_{2}.\end{aligned}$$
Returning to the case where the Hamiltonian is Hermitian, it is convenient to render the two matrices $\widetilde{U}_{1}$ and $\widetilde{U}_{2}$ themselves unitary [@newui; @newui2]. For this purpose, a “gauge factor" $b$ is chosen such that the unitary counterparts of $\widetilde{U}_{1}$ and $\widetilde{U}_{2}$ are defined via $U_{1}=\widetilde{U}_{1}b$ and $U_{2}=b^{-1}\widetilde{U}_{1}$. Since $\widetilde{U}^{\dagger}_{1}\widetilde{U}_{1}=diag(\gamma^{(-1)}_{1},\gamma_{2}
)$, this would imply that b is the “Hermitian square-root" of $diag(\gamma^{(-1)}_{1},\gamma_{2})$. This “Hermitian square-root" is defined by the relation $(b^{(-1)})^{\dagger}b^{(-1)}=diag(\gamma^{(-1)}_{1},\gamma_{2})$. Inspection of the power series expansion of $\gamma_{1}^{(\pm\frac{1}{2})}=(\hat{\mathbf{I}}+\mathbf{z}.\mathbf{z}^{\dagger}
)^{(\pm\frac{1}{2})}$ and $\gamma_{2}^{(\pm\frac{1}{2})}=(\hat{\mathbf{I}}+\mathbf{z}^{\dagger}.\mathbf{z}
)^{(\pm\frac{1}{2})}$ show that since each term in the expansion is Hermitian, matrices $\gamma^{\pm\frac{1}{2}}_{1}$ and $\gamma^{\pm\frac{1}{2}}_{2}$ are Hermitian and have non-negative eigenvalues. Because of this, it is sufficient to define $b$ as the inverse square root via $b^{(-2)}=diag(\gamma^{(-1)}_{1},\gamma_{2})$.
Furthermore, $H_{\mathtt{eff}}$ in equation (\[effeqn\]) is Hermitian for the unitary counterpart $U_{1}$. The upper diagonal block of this Hermitian Hamiltonian accompanying the decomposition $U=U_{1}U_{2}$ is given by $$\label{heffup}
\frac{i}{2}[\frac{d(\gamma_{1}^{-\frac{1}{2}})}{dt},\gamma_{1}^{\frac{1}{2}}]
+
\frac{1}{2}\left(\gamma_{1}^{-\frac{1}{2}}(\widetilde{\mathbf{H}}^{(N-n)}
-\mathbf { z }
\mathbf{V}^{\dagger})\gamma_{1}^{\frac{1}{2}}
+ H.c.\right),$$ where \[,\] represents the commutator and *H.c.* stands for the Hermitian conjugate. The lower diagonal block is similarly given by $$\label{heffdown}
\frac{i}{2}[\frac{d(\gamma_{2}^{-\frac{1}{2}})}{dt},\gamma_{2}^{\frac{1}{2}}]
+
\frac{1}{2}\left(\gamma_{2}^{-\frac{1}{2}}(\widetilde{\mathbf{H}}^{(N-n)}
+\mathbf { z } ^
{\dagger}\mathbf{V})\gamma_{2}^{\frac{1}{2}}
+ H.c.\right).$$ For $N=3$, $n=1$, these diagonal blocks define an SU(2)- and a U(1) Hamiltonian and $\mathbf{z}$ is a pair of complex numbers. The SU(2) Hamiltonian is in turn rendered in terms of its fiber bundle in Fig. (\[Fig1\]) and the U(1) Hamiltonian corresponds to a phase. Together, they describe a four-dimensional fiber for SU(3) over the base manifold, also four dimensional, of $\mathbf{z}$.
Alternatively, $N=3$ SU(3) problems may be conveniently seen as a part of $N=4$ SU(4) problems, making contact with two qubit systems that are extensively studied. In this case, for $N=4$, $n=2$, these diagonal blocks define two SU(2) Hamiltonians and $\mathbf{z}$ is a $2\times2$ matrix representable in terms of Pauli spinors. Generally, it is 8-dimensional while the fiber has seven dimensions (two SU(2) and a mutual phase) but for the SU(3) subgroup of SU(4),both the base and manifold again reduce to four dimensions each. With $\textbf{z}$ a pair of complex numbers, the non-trivial part of geometrizing SU(3) is thereby reduced to describing this four-dimensional manifold. Exploring this for the $N=3$, $n=1$ decomposition will be the content of the next section whereas the Appendix gives the alternative SU(4) rendering.
\[su3\]Geometry of general SU(3) time evolution operator
========================================================
A general time-dependent three-level Hamiltonian may be written in terms of eight linearly independent operators of a three-level system. Such a Hamiltonian can also be written in terms of a subgroup of 15 operators of a four-level system. Before the time evolution operator is presented in the SU(3) basis in terms of a $N=3$, $n=1$ decomposition, we will note that it can be rendered in a few alternative ways.
First, a general time-dependent four-level Hamiltonian may be written as $H(t)=\sum_i c_{i}\mathbf{O}_{i}$. Here $c_{i}$ are time-dependent and $\mathbf{O}_{i}$ are the unit matrix and 15 linearly independent operators of a 4-level system that may be chosen in a variety of matrix representations. One choice used in particle physics are the so called Greiner matrices [@greiner; @oldui; @oldui2; @oldui3; @oldui4]. Another choice consists of using $\vec{\bm{\sigma}}$, $\vec{\bm{\tau}}$, $\vec{\bm{\sigma}}\otimes\vec{\bm{\tau}}$ and the $4\times4$ unit matrix. Such a choice was discussed in [@restui; @restui2] and will be used throughout this paper. As it stands, the above Hamiltonian describes a general four-level atom with 4 energies and 6 complex couplings. Note that only the three differences in energies are important. Restricting the 15 coefficients $c_{i}$ to a smaller number allows this Hamiltonian to describe various physical Hamiltonians, forming different subalgebras of the su(4) algebra [@restui]. For example, if two of the six complex couplings are zero (levels 1 and 4 and levels 2 and 3 of a four-level atom not coupled), then the Hamiltonian may be recast such that the operators involved belong to an so(5) subalgebra [@restui]. On the other hand, if levels 2 and 3 are degenerate and level 4 is uncoupled from the rest, then the problem may be recast in terms of only eight operators belonging to the su(3) subalgebra of su(4). This is illustrated in Fig. \[Fig\_energylevels\] and is one of the systems of interest in this paper.
![Levels $\vert2\rangle$ and $\vert3\rangle$ couple equally to $\vert1\rangle$ and to $\vert4\rangle$, which are themselves coupled. The three complex coupling matrix elements and two energy positions define such an SU(3) system.[]{data-label="Fig_energylevels"}](Fig2){height="2" width="2"}
Alternatively, after one arrives at the linear equation for the $N=4$, $n=2$ decomposition, one can represent the resulting vector in terms of six homogeneous coordinates. This is the so-called “Plücker coordinate” representation for the SU(3) Hamiltonian. These coordinates as well as the alternative derivation are presented in the appendix. The $N=3$, $n=1$ decomposition will be the content of the rest of this section.
Consider the Hamiltonian in the basis of the Gell-Mann lambda matrices [@georgi] $H(t)=\sum_{i}a_{i}\bm{\lambda}_{i}$. The $N=3$, $n=1$ decomposition consists of writing the time evolution operator in terms of a product of two matrices $U=\widetilde{U}_{1} \widetilde{U}_{2}$ where $\widetilde{U}_{1}$ is composed of a (2$\times$1)-dimensional **z**, as explained in Sec. II. The equation that governs the evolution of **z**, equation (\[zdot\]), can be written in this case as $$\label{su3_z_eqn}
\dot{z}_{\mu}=-iV_{\mu}-iF_{\mu\nu}z_{\nu}+iV^{*}_{\nu}z_{\nu}z_{\mu}
;\; \mu ,\nu=1,2.$$ Here, the symbols used in defining $\dot{\mathbf{z}}$ are defined as $V=(a_{4}-ia_{5},a_{6}-ia_{7})$, and $$F=\left(\begin{array}{cc}
a_{3}+\sqrt{3} a_{8}&a_{1}-i a_{2}\\
a_{1}+i a_{2}&-a_{3}+\sqrt{3} a_{8}\\
\end{array} \right).$$ Using the transformation equations $m_{1,2}=-z_{1,2}(De^{i\phi})^{-1}$, $m_{3}=(De^{i\phi})^{-1}$ and $\vert m_{1} \vert^{2}+\vert m_{2}\vert^{2}+\vert
m_{3} \vert^{2}=1$ leads to the evolution equation for $\vec{m}=(m_{1r},m_{2r},m_{3r},m_{1i},m_{2i},m_{3i})^{T}$: $$\label{su3_Rotation}
\fl
\dot{\vec{m}}=\left(\begin{array}{cccccc}
0&-a_{2}&a_{5}&a_{3}+\sqrt{3}a_{8}&a_{1}&-a_{4}\\
a_{2}&0&a_{7}&a_{1}&-a_{3}+\sqrt{3}a_{8}&-a_{6}\\
-a_{5}&-a_{7}&0&-a_{4}&-a_{6}&0\\
-a_{3}-\sqrt{3}a_{8}&-a_{1}&a_{4}&0&-a_{2}&a_{5}\\
-a_{1}&a_{3}-\sqrt{3}a_{8}&a_{6}&a_{2}&0&a_{7}\\
a_{4}&a_{6}&0&-a_{5}&-a_{7}&0\\
\end{array} \right)
\vec{m},$$ which describes the rotation of a unit vector in a six dimensional space of the real and imaginary parts of $\vec{m}$ defined by $m_{\mu}=m_{\mu r}+im_{\mu
i}$. In the above equations, $D=(1+\vert z_{1}\vert^{2}+\vert z_{2}\vert^{2})^{1/2}$ and $i\dot{\phi}=i(V^{*}_{\nu}z_{\nu}+V_{\nu}z^{*}_{\nu})$. The phase $\phi$ is real and determined only up to a constant factor. Since the real and imaginary parts of $m_{3}$ are not independently defined, the geometrical description of the base manifold for the $N=3$, $n=1$ decomposition may be thought of as a point on the surface of a constrained six-dimensional unit sphere.
The two constraints, namely $\vert m_{1}
\vert^{2}+\vert m_{2}
\vert^{2}+\vert m_{3} \vert^{2}=1$ and the “phase arbitrariness" of $\phi$, reduce the 6-dimensional manifold of the three-dimensional complex vector $\vec{m}$ to a four-dimensional manifold in agreement with there being only four independent parameters in $\textbf{z}$.The first condition defines the base as a vector on an $S^{5}$ sphere while the phase arbitrariness serves as an additional constraint. The fiber, on the other hand, is an SU(2) block, evolving as a vector on $S^{2}$ Poincare-like sphere with a phase at each point, and a U(1) block that amounts to an extra phase.This is presented schematically in Fig.(\[Bloch3\_SU3\]), as the product of three matrices of the evolution operator.
![The base and fiber for the SU(3) group. The first two factors give the base manifold, an $S^{5}$ sphere with a phase arbitrariness defined in the text. The fiber, described by the third matrix, is composed of a Bloch sphere and a phase associated with each of its points, and the second an extra phase represented by a vertical line.[]{data-label="Bloch3_SU3"}](Fig3){height="2" width="4"}
The alternative $N=4$, $n=2$ decomposition in the appendix yields the equation of motion for $m_{\mu}=-z_{\mu}/De^{i \phi}$ in equation (\[rotation\_SU4\]). Following equation (\[heffup\]) and equation (\[heffdown\]), we see that for this case, the two remaining blocks of the time evolution operator, namely $\widetilde{U}^{(4-2)}$ and $\widetilde{U}^{(2)}$, can be transformed into unitary matrices for SU(2). The fiber evolves as vectors on two identical $S^{2}$ Bloch-like spheres with a mutual phase, whose evolution is coupled to the base that evolves as a vector on an $S^{5}$ sphere. This is illustrated in Fig. (\[Bloch3\_SU4\]).
![The base and fiber for the SU(3) group via the $N=4$, $n=2$ decomposition. The base again is given by an $S^{5}$ sphere as in Fig. (\[Bloch3\_SU3\]). The fiber is composed of two identical SU(2) Bloch spheres plus phase, and an extra mutual phase between them. The four parameters each of base and fiber again account for all eight parameters of the SU(3).[]{data-label="Bloch3_SU4"}](Fig4){height="2" width="4"}
Either decomposition can be used to study various physical processes as will be discussed in the next section.
\[discussion\]Applications
==========================
It is often desirable to control the time evolution of quantum states to manipulate an input state into a desirable output state. In [@mitra; @mitra2], the authors considered a Hamiltonian of the form $\mathbf{H}_{0}-\mu
\mathcal{E}(t)$, where $\mathbf{H}_{0}$ is a free-field Hamiltonian and $\mu
\mathcal{E}(t)$ is a control field. To illustrate the “Hamiltonian encoding” scheme to control quantum systems, the authors considered a three-level system and studied stimulated Raman adiabatic passage (STIRAP), an atomic coherence effect that employs interference between quantum states to transfer population completely from a given initial state to a specific final state. This is done through a “counterintuitive” pulse sequence. Consider the Hamiltonian $$\begin{aligned}
\label{Ham_mitra}
\mathbf{H}(t)=\left(\begin{array}{ccc}
0&G_{1}(t)&0\\
G_{1}(t)&2 \Delta&G_{2}(t)\\
0&G_{2}(t)&0
\end{array}\right).\end{aligned}$$ Here $G_{1,2}(t)=2.5$exp$[-(t-t_{1,2})^{2}/\tau^{2}]$ and $\Delta=0.1$. The initial population is in the upper state. For $t_{1}=\tau$, $t_{2}=0$ and $\tau=3$, it is seen that the two empty states are coupled first via $G_{2}(t)$ and then the levels $\vert 1 \rangle$ and $\vert 2 \rangle$ are coupled through $G_{1}$. The dynamics of the populations reveal complete population transfer. A complete solution as per Section \[su3\] was constructed for this model and the results are presented in Fig. \[Fig\_Mitra\] in total agreement with the results of [@mitra].
![Population $P_{1j}=\vert \langle1\vert j \rangle \vert^{2}$ plotted as a function of time. The initial population in state $\vert 1 \rangle$ is completely transferred to $\vert 3 \rangle$. Both the unitary integration solution and the direct numerical solution [@mitra] are plotted and they coincide at all times.[]{data-label="Fig_Mitra"}](Fig5){height="1.7" width="2.75"}
Quantum control can also be achieved by understanding the nature of tunneling. The famous Landau-Zener formula [@LZ; @LZ2; @LZ3] predicts the transition probability of the ground state of a two-level system when the energy levels adiabatically undergo a crossing. The study of level crossings has since been extended to multi-level systems. For example, in [@agarwal], the authors considered a three-level atom to study population trapping by manipulating the phase acquired as a three-level system evolves under the influence of frequency modulated fields [@agarwal2]. Such a frequency modulated field is given by $$\begin{aligned}
\mathbf{E}(t)=
\mathbf{E}_{1}e^{-i[\omega_{1}t+\varphi_{1}(t)]}+\mathbf{E}_{2}e^{-i[\omega_{2}
t+\varphi_{2}(t)]}+c.c.\\
\varphi_{i}(t)=M_{i}\sin{\Omega_{i}t}.\end{aligned}$$ Here, $c.c.$ stands for complex conjugation. The phase $\varphi_{i}(t)$ in the exponent can be written in terms of Bessel functions as [@Abramowitz] $$e^{M_{j}\sin{\Omega_{j}t}}=\sum^{\infty}_{k=-\infty}J_{k}(M_{j})e^{ik\Omega_{j}
t}.$$ For large values of $\Omega_{j}$, the leading contribution for slow time scales would come from $J_{0}(M_{j})$. Hence, for large $\Omega_{j}$, the interaction Hamiltonian can be written as $$\begin{aligned}
\mathbf{H}_{int}(t)=
-\mathbf{d}.(\mathbf{E}_{1}J_{0}(M_{1})+\mathbf{E}_{2}J_{0}(M_{2}).\end{aligned}$$ Hence, for values of $M_{1,2}$ that are zeros of the zeroth-order Bessel functions, the interaction Hamiltonian is zero and population trapping is observed. Under this assumption, consider the full Hamiltonian under the rotating wave approximation, $$\begin{aligned}
\mathbf{H}(t)=\left(\begin{array}{ccc}
E_{1}(t)&G_{1}(t)&0\\
G^{*}_{1}(t)&0&G_{2}(t)\\
0&G^{*}_{2}(t)&E_{3}(t)
\end{array}\right).\end{aligned}$$ Here, $E_{1}(t)=\Delta_{1}-M_{1}\Omega_{1}\cos(\Omega_{1} t+\theta)$ and $E_{3}(t)=-\Delta_{2}+M_{2}\Omega_{2}\cos(\Omega_{2}t)$. Results are presented in Fig. \[Fig\_Agarwal\], and for the parameter values $\Omega_{1,2}=1$, $\Delta_{1}=-\Delta_{2}=10$, $\theta=0$ and $G_{1,2}=6$, demonstrate the phenomenon of population localization discussed in [@agarwal].
![(a) For $M_{1,2}=7$ and the other parameter values given in the text, there is no population trapping observed. (b) The energy landscape for $M_{1,2}=30.6346$ showing energy level crossing. (c) Population trapping is observed with $M_{1,2}=30.6346$ which corresponds to the tenth zero of the zeroth-order Bessel function. Note that the thick line is $P_{11}$ and the thin line corresponds to $P_{12}$. The results agree completely with [@agarwal].[]{data-label="Fig_Agarwal"}](Fig6){height="4.8" width="3.2"}
As a final illustration of the unitary integration technique applied to three-level systems, let us consider the example discussed in [@kancheva]. Here, a three-level system is subject to strong fields and the correlation between the scattered light spectrum and the atom dynamics is discussed. The authors consider the Hamiltonian $$\begin{aligned}
\label{Ham_kancheva}
\mathbf{H}(t)=\left(\begin{array}{ccc}
0&0&G_{1}(t)\\
0&0&G_{2}(t)\\
G^{*}_{1}(t)&G^{*}_{2}(t)&0
\end{array}\right).\end{aligned}$$ Here, $G_{1,2}(t)=-V_{1,2}e^{-i \delta t}$. The time evolution of the states calculated as per our procedure in Section \[su3\] is plotted in Fig. \[Fig\_kancheva\] for different values of the parameters. All of these results agree with those given in [@kancheva]. Further features of the base and fiber will be presented at the end of the next section.
![(a) Populations $P_{1j}=\vert \langle 1 \vert j \rangle \vert^{2}$ for $\delta=5$, $V_{1}=2$ and $V_{2}=1$. $P_{11}$ is given by the solid line and $P_{12}$ is given by the thin line. (b) Same as (a), for $\delta=12$. Note that $P_{13}$ oscillates close to zero at all times. (c) $P_{1j}$ for $\delta=12$, $V_{1}=1$ and $V_{2}=2$.[]{data-label="Fig_kancheva"}](Fig7){height="4.8" width="3.2"}
\[GP\]Geometric phase for SU(3) group
=====================================
Many physical systems give rise to a measurable phase that does not depend directly on the dynamical equations that govern the evolution of the system, but depends only on the geometry of the path traversed by vectors characterizing the state of the system. This geometric phase is denoted by $\gamma_{g}$ and is given by the integral [@berry], $$\begin{aligned}
\gamma_{g}=\int d\mathbf{R}\;.\langle n(\mathbf{R}(t)) \vert
i\nabla_{\mathbf{R}} \vert n(\mathbf{R}(t)) \rangle,\end{aligned}$$ where the state evolution is governed by a set of internal coordinates that parameterize the Hamiltonian $\mathbf{R}(t)$, and $\nabla_{\mathbf{R}}$ is the gradient in the space of these internal coordinates. This phase has been generalized to non-cyclic non-adiabatic evolution of quantum systems [@wilczek; @wilczek2; @wilczek3]. The purpose of this section is to present this phase in terms of coordinates on the Bloch sphere for two-level systems and extend it to three-level systems.
In two-level systems, the time evolution operator is described by three parameters as described in Section \[intro\]. Two of these parameters describe a point on the Bloch sphere. Traversing closed loops on this Bloch sphere returns the quantum system to its initial state as described by the two parameters on the Bloch sphere but not the third parameter of an overall phase. Hence, general closed loops on the Bloch sphere do not correspond to closed loops in the space of the full unitary operator. This discrepancy in the phase between the initial and final state corresponds to the geometric phase given above and amounts to changes along the fiber at each point on the sphere. To formalize this, consider $U_{1}$, given by equation (\[nonunit\_u\]) as unitarized through the matrix $b$ in Section \[ui\], which for $N=2$, $n=1$ takes the form $$\begin{aligned}
U_{1}=\frac{1}{\sqrt{1+\vert
\mathbf{z}\vert^{2}}}\left(\begin{array}{cc}1&\mathbf{z}\\ -\mathbf{z}^{*}&
1 \end{array} \right).\end{aligned}$$
By identifying $\cos{\frac{\theta}{2}}=(1+\vert
\mathbf{z}\vert^{2})^{-\frac{1}{2}}$ and $\sin{\frac{\theta}{2}}e^{-i \epsilon}=-\mathbf{z} (1+\vert
\mathbf{z}\vert^{2})^{-\frac{1}{2}}$, we get the usual description of the base manifold in terms of the angles $0\leq\theta<\pi$ and $0\leq\epsilon<2\pi$ that are associated with the Bloch sphere, namely, $$\begin{aligned}
U_{1}=\left(\begin{array}{cc}\cos{\frac{\theta}{2}}&-\sin{\frac{\theta}{2}}e^{
-i\epsilon}\\\sin{\frac{\theta}{2}}e^{i\epsilon}&
\cos{\frac{\theta}{2}} \end{array} \right).\end{aligned}$$ In terms of the parameters $\theta$ and $\epsilon$, the Hamiltonian $H(t)=-\vec{a}.\vec{\bm{\sigma}}$ is given by $$\begin{aligned}
H(t)=\left(\begin{array}{cc}-\cos{\theta}&-\sin{\theta}e^{-i\epsilon}\\-\sin{
\theta}e^{i\epsilon}&
\cos{\theta} \end{array} \right).\end{aligned}$$ equation (\[effeqn\]) governing the evolution of the fiber $U_{2}$ has two terms. The first term is evaluated as $$\begin{aligned}
U^{\dagger}_{1}H(t)U_{1}=\left(\begin{array}{cc}-1&0\\0&
1 \end{array} \right),\end{aligned}$$ which corresponds to the eigenvalues of the Hamiltonian. To evaluate the second term, consider the case whereby the vector on the Bloch sphere traverses a closed path defined by a constant $\theta$. The second term is then given by $$\begin{aligned}
U^{\dagger}_{1}\frac{\partial{U}_{1}}{\partial(-i\epsilon)}=\left(\begin{array}{
cc}-\sin^{2}{\frac{\theta}{2}}&-\frac{1}{2}\sin{\theta}e^{-i\epsilon}\\-\frac{1}
{
2}\sin{\theta}e^{i\epsilon}&\sin^{2}{\frac{\theta}{2}}
\end{array} \right).\end{aligned}$$ Integrating $\epsilon$ from 0 to $2\pi$ yields $$\begin{aligned}
\int^{2\pi}_{0}{d\epsilon}U^{\dagger}_{1}\frac{\partial{U}_{1}}{
\partial(-i\epsilon)}=\left(\begin{array}{cc}\pi(1-\cos{\theta}
)&0\\0&-\pi(1-\cos
{\theta})
\end{array} \right),\end{aligned}$$ which is the correct formula for the geometric phase of a two-level system [@berry].
To extend this analysis to three-level systems, we consider the $N=3$, $n=1$ decomposition. The matrix $U_{1}=\widetilde{U}_{1}.b$ is now given by $$\begin{aligned}
U_{1}=\left(\begin{array}{cc}I^{(2)}-\frac{1}{D(D+1)}\mathbf{z}\mathbf{z}^{
\dagger}&\frac{\mathbf{z}}{D}\\-\frac{\mathbf{z}^{\dagger}}{D}&
\frac{1}{D} \end{array} \right),\end{aligned}$$ where $\mathbf{z}$ is a complex column vector $(z_{1},z_{2})^{T}$ and $D=\sqrt{1+\vert\mathbf{z}\vert^{2}}$. Care has to be taken in assigning angles to elements of this matrix such that the transformation satisfies two conditions: the $U_{1}$ matrix should not depend on $\phi$ and the transformation must be commensurate with the definition of $\vec{m}$. To this effect, we transform $\mathbf{z}$ into polar coordinates: $z_{1}=-\tan{\frac{\theta_{1}}{2}}\cos{\frac{\theta_{2}}{2}}e^{i\epsilon_{1}}$, $z_{2}=-\tan{\frac{\theta_{1}}{2}}\sin{\frac{\theta_{2}}{2}}e^{i\epsilon_{2}}$. These transformation equations imply that $D=\sqrt{1+\vert\mathbf{z}\vert^{2}}=\sec{\frac{\theta_{1}}{2}}$, $m_{1}=\sin{\frac{\theta_{1}}{2}}\cos{\frac{\theta_{2}}{2}}e^{i(\epsilon_{1}
-\phi)}$, $m_{2}=\sin{\frac{\theta_{1}}{2}}\sin{\frac{\theta_{2}}{2}}e^{i(\epsilon_{2}
-\phi)}$ and $m_{3}=\cos{\frac{\theta_{2}}{2}}e^{-i\phi}$. The $U_{1}$ matrix is given by $$\begin{aligned}
\fl
U_{1}=\left(\begin{array}{ccc}
1-2\sin^{2}{\frac{\theta_{1}}{4}}\cos^{2}{\frac{\theta_{2}}{2}}
&-\sin^{2}{\frac{\theta_{1}}{4}}\sin{\theta_{2}}e^{i(\epsilon_{1}-\epsilon_{2})}
&-\sin{\frac{\theta_{1}}{2}}\cos{\frac{\theta_{2}}{2}}e^{i\epsilon_{1}} \\
-\sin^{2}{\frac{\theta_{1}}{4}}\sin{\theta_{2}}e^{-i(\epsilon_{1}-\epsilon_{2})}
&1-2\sin^{2}{\frac{\theta_{1}}{4}}\sin^{2}{\frac{\theta_{2}}{2}}&
-\sin{\frac{\theta_{1}}{2}}\sin{\frac{\theta_{2}}{2}}e^{i\epsilon_{2}}\\
\sin{\frac{\theta_{1}}{2}}\cos{\frac{\theta_{2}}{2}}e^{-i\epsilon_{1}}&\sin{
\frac{\theta_{1}}{2}}\sin{\frac{\theta_{2}}{2}}e^{-i\epsilon_{2}}&\cos{\frac{
\theta_{1}}{2}}\\
\end{array}\right).\end{aligned}$$ In the above equation, the range on the angles $0\leq\theta_{i}<\pi$ and $0\leq\epsilon_{i}<2\pi$ are chosen so that the absolute value of each element of the time-evolution operator is positive [@aravind]. Hence $U_{1}$ can be represented as two vectors on a sphere, at angles $(\theta_{1},\epsilon_{1})$ and $(\theta_{2},\epsilon_{2})$ respectively. This is represented in Fig. (\[SU3\_GP\_Blochfig\]).
![The base manifold $U_{1}$ is characterized by two sets of angles $0\leq\theta_{i}<\pi$, $0\leq\epsilon_{i}<2\pi$ which can be represented as two vectors with polar angles $(\theta_{1},\epsilon_{1})$ and $(\theta_{2},\epsilon_{2})$.[]{data-label="SU3_GP_Blochfig"}](Fig8){height="3" width="3.5"}
Since the columns of a unitary operator correspond to normalized eigenvectors, we can consider the last column of the matrix above, $\vert\psi\rangle=(-\sin{\frac{\theta_{1}}{2}}\cos{\frac{\theta_{2}}{2}}e^{
i\epsilon_{1}},-\sin{\frac{\theta_{1}}{2}}\sin{\frac{\theta_{2}}{2}}e^{
i\epsilon_{2}},\cos{\frac{\theta_{1}}{2}})^{T}$, and evaluate the so-called connection 1-form given by [@arno; @bohms; @book] $$\begin{aligned}
\mathcal{A}=-i\langle\psi\vert d\vert\psi\rangle.\end{aligned}$$ The Abelian geometric phase, given by $\gamma_{g}=\int{\mathcal{A}}$ is evaluated to be $$\begin{aligned}
\gamma_{g}=-\frac{1}{2}\int{\sin^{2}{\frac{\theta_{1}}{2}}\left((d\epsilon_{1}
+d\epsilon_{2})+\cos{\theta_{2}}(d\epsilon_{1}-d\epsilon_{2})\right)}.\end{aligned}$$ If the various angles are relabelled $\epsilon_{1}\rightarrow-\gamma-\alpha$, $\epsilon_{2}\rightarrow-\gamma+\alpha$, $\theta_{1}\rightarrow2\theta$ and $\theta_{2}\rightarrow2\beta$, the formula above agrees with [@byrd] and [@aravind]. The time-evolution operator above can now be used as in the case of SU(2) to evaluate the dynamic contribution $\int U^{\dagger}_{1}H(t)U_{1}$ and the geometric contribution to the time evolution operator which is given by $-i\int
U^{\dagger}_{1}dU_{1}$, where $dU_{1}=\frac{dU_{1}}{d\theta_{i}}
d\theta_{i}+\frac{dU_{1}}{d\epsilon_{i}}d\epsilon_{i}$, $i=1,2$.
This description of the base manifold in terms of $(\theta_{i},\epsilon_{i})$ can now be used to describe the dynamics of various physical processes. Fig. (\[Kancheva2vec\]) represents the base manifold corresponding to the results in Fig. (\[Fig\_kancheva\]). $(\theta_{1},\epsilon_{1})$ depend on all the parameters that define the system while $(\theta_{2},\epsilon_{2})$ depend only on the ratio $V_{1}/V_{2}$. Also note that the maximum value of $\epsilon_{2}$, corresponding to the maximum latitude traversed by the black curve, is inversely proportional to $\delta$. Such observations can be used to control the dynamics of this system.
![ The base manifold corresponding to the results in Fig. (\[Fig\_kancheva\]) for the three-level system of $\cite{kancheva}$. For the first column, $V_{1}=1$, $V_{2}=2$. The second column corresponds to $V_{1}=2$, $V_{2}=2$ and the third to $V_{1}=2$, $V_{2}=1$. The rows correspond to $\delta=1$, $\delta=5$ and $\delta=50$. The thin black curve describes $(\theta_{1},\epsilon_{1})$ and the thick red curve the set $(\theta_{2},\epsilon_{2})$.[]{data-label="Kancheva2vec"}](Fig9){height="4" width="4"}
\[conclusions\]Conclusions
==========================
The ability to decouple the time dependence of operator equations from the non-commuting nature of the operators is the central feature of unitary integration and also characterizes the Bloch sphere representation for the evolution of a single spin. By doing so, the quantum mechanical evolution is rendered a “classical" picture of a rotating unit vector. For a two-level atom, the Bloch sphere representation along with a phase completely determines the time evolution operator. In this paper, we have extended this program to deal with the time evolution operator belonging to the SU(3) group. This complements the work in [@newui] for SU(4) Hamiltonians of two qubit systems. We have also extended the analysis of geometric phase to three-level systems by providing an explicit coordinate representation for the SU(3) time evolution operator.
\[appendix\]Alternative derivations for a general SU(3) Hamiltonian.
====================================================================
Consider a three-level Hamiltonian written in terms of the Gell-Mann matrices [@georgi] as $H(t)=\sum_{i=1}^{8}a_{i}\bm{\lambda}_{i}$. To exploit the fact that this Hamiltonian is a subgroup of four-level problems, it is represented in terms of the O matrices [@restui] as $$\begin{aligned}
\label{SU(3) derivation}
\fl
2\frac{a_{8}}{\sqrt{3}}\mathbf{O}_{2}+(a_{3}-\frac{a_{8}}{\sqrt{3}})\mathbf
{O}_{3}+(2a_{3}+2\frac{a_{8}}{\sqrt{3}})\mathbf{O}_{4}
+a_{4}\mathbf{O}_{5}+a_{5}\mathbf{O}_{6}+2a_{4}\mathbf{O}_{7}+2a_{5}\mathbf{O}_{
8}+\nonumber\\
\fl\qquad
a_{1}\mathbf{O}_{9}+
a_{2}\mathbf{O}_{10}+2a_{1}\mathbf{O}_{11}+2a_{2}\mathbf{O}_{12}+2a_{6}\mathbf{O
}_{13}+2a_{6}\mathbf{O}_{14}-2a_{7}\mathbf{O}_{15}+2a_{7}\mathbf{O}_{16}.\end{aligned}$$ This embeds the Hamiltonian $H(t)=\sum_{i}a_{i}\bm{\lambda}_{i}$ as a 4$\times$4 matrix with zeros along the last row and column. In such a representation, the various entries of the Hamiltonian equation (\[Hamiltonian\]) are given by $$\begin{aligned}
H^{(4-2)}=\frac{1}{\sqrt{3}}a_{8}\mathbf{I}^{(2)}+a_{1}\bm{\sigma}_{1}+a_{2}
\bm{\sigma}_{2}+a_{3}\bm{\sigma}_{3},\\
H^{(2)}=-\frac{1}{\sqrt{3}}a_{8}\mathbf{I}^{(2)}-\frac{1}{\sqrt{3}}a_{8}\bm{
\sigma}_{1},\\
\mathbf{V}=\frac{1}{2}(a_{4}-ia_{5})\mathbf{I}^{(2)}+\frac{1}{2}(a_{6}-ia_{7}
)\bm{\sigma}_{1}\\\nonumber
-i\frac{1}{2}(a_{6}-ia_{7})\bm{\sigma}_{2}+\frac{1}{2}(a_{4}-ia_{5})\bm{
\sigma}_{3}.\end{aligned}$$ Writing $\mathbf{z}$ in the standard Clifford basis as $\mathbf{z}=\frac{1}{2}z_{4}\bm{I}^{(2)}-\frac{i}{2}\sum_{i}z_{i}\bm{\sigma}_{i}
$, it follows from equation (\[zdot\]) that $z_{1}=iz_{2}$ and $z_{3}=iz_{4}$ and the equation reduces precisely to equation (\[su3\_z\_eqn\]). The geometry described in Section \[su3\] can thus be derived from either of these decompositions of the time evolution operator.
The SU(3) subgroup in equation (\[SU(3) derivation\]) is one among many SU(3) subgroups embedded in SU(4). Another choice corresponds to the Dzyaloshinskii-Moriya interaction Hamiltonian [@dz1; @dz2] and is also of interest because the 4$\times$4 matrices now do not have a trivial row and column of zeros. In the two-spin basis, this Hamiltonian is given by $$\begin{aligned}
\label{expandedDV}
\fl
H(t)=\sum_{i}c_{i}\mathbf{O}_{i}=a_{1}(\mathbf{O}_{2}+\mathbf{O}_{3})+2
a_{2}(\mathbf{O}_{15}+\mathbf{O}_{16})+2 a_{3}(\mathbf{O}_{14}\--
\mathbf{O}_{13})+2 a_{4}(\mathbf{O}_{7}+\mathbf{O}_{11})\nonumber\\
\fl\qquad
+a_{5}(\mathbf{O}_{6}+\mathbf{O}_{10})+a_{6}(\mathbf{O}_{5}+\mathbf{O}_{9})+2
a_{7}(\mathbf{O}_{8}+\mathbf{O}_{12})+\frac{2 a_{8}}{\sqrt{3}}(2
\mathbf{O}_{4}\--\mathbf{O}_{13}\--\mathbf{O}_{14}).\end{aligned}$$ The correspondence between the coefficients in terms of $\mathbf{O}$ and in terms of the $\bm{\lambda}$ matrices is : $c_{1}=0$, $c_{2}=a_{1}$, $c_{3}=a_{1}$, $c_{4}=4a_{8}/\sqrt{3}$, $c_{5}=a_{6}$, $c_{6}=a_{5}$, $c_{7}=2a_{4}$, $c_{8}=2a_{7}$, $c_{9}=a_{6}$, $c_{10}=a_{5}$, $c_{11}=2a_{4}$, $c_{12}=2a_{7}$, $c_{13}=-2a_{3}-2a_{8}/\sqrt{3}$, $c_{14}=2a_{3}-2a_{8}/\sqrt{3}$, $c_{15}=2a_{2}$ and $c_{16}=2a_{2}$. Relabeling of the states $1\rightarrow2$, $2\rightarrow3$, $3\rightarrow4$ and $4\rightarrow1$ expresses the Hamiltonian as $$\begin{aligned}
H^{(4-2)}=\frac{1}{\sqrt{3}}a_{8}\mathbf{I}^{(2)}-a_{3}\bm{\sigma}_{1}-a_{2}
\bm{\sigma}_{2}-a_{1}\bm{\sigma}_{3},\\
H^{(2)}=-\frac{1}{\sqrt{3}}a_{8}\mathbf{I}^{(2)}-\frac{1}{\sqrt{3}}a_{8}\bm{
\sigma}_{1},\\
\mathbf{V}=\frac{1}{2}(a_{6}-ia_{7})\mathbf{I}^{(2)}+\frac{1}{2}(a_{6}-ia_{7}
)\bm{\sigma}_{1}\\\nonumber
-\frac{1}{2}(a_{5}+ia_{4})\bm{\sigma}_{2}-\frac{1}{2}(a_{4}-ia_{5})\bm{
\sigma}_{3}.\end{aligned}$$ If $\mathbf{z}$ is written in terms of the standard Clifford basis $(\hat{\mathbf{I}},-i\vec{\bm{\sigma}})$ as $\mathbf{z}=\frac{1}{2}z_{4}\mathbf{I}^{(2)}-\frac{i}{2}\sum_{i=1}^{3}z_{i}\bm{
\sigma}_{i} $ , it follows from equation (\[zdot\]) that $z_{1}=i z_{4}$ and $z_{2}=i z_{3}$. This is consistent with the parameter count that since the inhomogeneity $\mathbf{V}$ has only two free complex parameters (namely $V_{1}=a_{6}-ia_{7}$ and $V_{2}=a_{4}-ia_{5}$), the complex $\mathbf{z}$ matrix should be composed only of two independent complex parameters, $z_{1}$ and $z_{2}$. With the above analysis, equation (\[zdot\]) becomes for the pair of complex numbers $$\frac{1}{2}\dot{z}_{\mu}=\frac{1}{2}X_{\mu}-iF_{\mu\nu}z_{\nu}+2G_{\nu}z_{\nu}z_
{ \mu}
;\;\mu,\nu=1,2.$$ Here $X=(V_{1}/2,-i V_{2}/2)$, $G=(2V_{1}^{*},2i V_{2}^{*})$ and $$-i F=\left(\begin{array}{cc}
i a_{3}-\sqrt{3} i a_{8}&a_{1}+i a_{2}\\
-a_{1}+i a_{2}&-i a_{3}-\sqrt{3}i a_{8}\\
\end{array} \right).$$
Paralleling the technique employed to solve an SO(5) Hamiltonian in [@newui; @newui2], we transform $\mathbf{z}$ into a complex vector $\vec{m}$: $m_{\mu}=\frac{-2z_{\mu}e^{i\phi}}{D}$ and $m_{3}=\frac{e^{i\phi}}{D}$ such that $\vert m_{1} \vert^{2}+\vert
m_{2} \vert^{2}+\vert m_{3} \vert^{2}=1$,with $D=(1+4(\vert z_{1} \vert^{2}+\vert z_{2}
\vert^{2}))^{1/2}$. This leads to the new set of evolution equations
$$\dot{\vec{m}}=\left(\begin{array}{ccc}
i a_{3}-\sqrt{3} i a_{8}&a_{1}+i a_{2}&-a_{6}+i a_{7}\\
-a_{1}+i a_{2}&-i a_{3}-\sqrt{3}i a_{8}&a_{5}+i a_{4}\\
a_{6}+i a_{7}&-a_{5}+i a_{4}&0
\end{array} \right)\vec{m}.$$
This can be written as an equation describing the rotation of the real and imaginary components of the vector $\vec{m}=(m_{1r},m_{2r},m_{3r},m_{1i},m_{2i},m_{3i})^{T}$, $$\begin{aligned}
\label{rotation_SU4}
\fl
\dot{\vec{m}}=\left(\begin{array}{cccccc}
0&a_{1}&-a_{6}&-a_{3}+\sqrt{3}a_{8}&-a_{2}&-a_{7}\\
-a_{1}&0&a_{5}&-a_{2}&a_{3}+\sqrt{3}a_{8}&-a_{4}\\
a_{6}&-a_{5}&0&-a_{7}&-a_{4}&0\\
a_{3}-\sqrt{3}a_{8}&a_{2}&a_{7}&0&a_{1}&-a_{6}\\
a_{2}&-a_{3}-\sqrt{3}a_{8}&a_{4}&-a_{1}&0&a_{5}\\
a_{7}&a_{4}&0&a_{6}&-a_{5}&0\\
\end{array} \right)\vec{m}.\end{aligned}$$ Here, the coefficients $c_{i}$ are written in terms of the coefficients $a_{i}$, whose correspondence was given earlier in this section. Also note that $m_{\mu}=m_{\mu
r}+im_{\mu i}$, $D=(1+\vert z_{1}\vert^{2}+\vert
z_{2}\vert^{2})^{\frac{1}{2}}$ and $\dot{\phi}=(V^{*}_{\nu}z_{\nu}+V_{\nu}z^{*}_{\nu})$. Simplifying this leads to the equation $i\dot{\phi}=-2(X_{\mu}z_{\mu}^{*}-X_{\mu}^{*}z_{\mu})$ for the evolution of $\phi$ which is clearly real but determined only to within a constant. A little algebra yields for the effective Hamiltonian given by equation (\[heffup\]),
$$\begin{aligned}
H^{(4-2)}-\frac{1}{(D+1)}(\mathbf{z}\mathbf{V}^{\dagger}+\mathbf{V}\mathbf{z}^{
\dagger})-
\frac{1}{2(D+1)^{2}}(\mathbf{z}\mathbf{V}^{\dagger}\mathbf{z}\mathbf{z}^{\dagger
} \- +\mathbf{z}\mathbf{z}^{\dagger}\mathbf{V}\mathbf{z}^{\dagger}),\end{aligned}$$
and for the effective Hamiltonian given by equation (\[heffdown\]), the expression $H^{(2)}+(\mathbf{z}^{\dagger}\mathbf{V}+\mathbf{V}^{\dagger}\mathbf{z})/2.$
Another representation of the SU(3) subgroup of SU(4) Hamiltonians is given by the so called “Plücker coordinate” representation of the SU(4) group discussed in [@newui; @newui2]. For an arbitrary SU(4) matrix, the Plücker coordinates are defined as a set of six parameters $(P_{12},P_{13},P_{14},P_{23},P_{24},P_{34})$ such that $P_{12}P_{34}-P_{13}P_{24}+P_{14}P_{23}=0$ and $\sum \vert
P_{ij}\vert^{2}=1$. They can be written in terms of the unit vector $\vec{m}$ and are given by $$\left(\begin{array}{c}
P_{12}\\
P_{13}\\
P_{14}\\
P_{23}\\
P_{24}\\
P_{34}\\
\end{array} \right)=\frac{1}{2}
\left(\begin{array}{c}
im_{6}-m_{5}\\
im_{1}+m_{2}\\
-im_{3}+m_{4}\\
-im_{3}-m_{4}\\
-im_{1}+m_{2}\\
im_{6}+m_{5}\\
\end{array} \right).$$ The linear equation of motion for $\vec{m}$ translates into an evolution equation for $\mathbf{P}=(P_{12},-P_{13},P_{14},P_{23},P_{24},P_{34})$ of the form $i\dot{\mathbf{P}}=\mathbf{H}_{P}\mathbf{P}$. Here, $\mathbf{H}_{P}$ is given by $$\mathbf{H}_{P}=\left(\begin{array}{cc}
\mathbf{H}_{P1}&\mathbf{V}_{P}\\
\mathbf{V}^{\dagger}_{P}&\mathbf{H}_{P2}\\
\end{array}\right),$$ where $$\begin{aligned}
\fl
\mathbf{H}_{P1}=\left(\begin{array}{ccc}
2a_{8}/\sqrt{3}&a_{64-}+ia_{75-}&a_{64-}+ia_{75-}\\
a_{64-}-ia_{75-}&-a_{1}&a_{8}/\sqrt{3}\\
a_{64-}-ia_{75-}&a_{8}/\sqrt{3}&-a_{1}
\end{array}
\right),\\
\fl
\mathbf{H}_{P2}=\left(\begin{array}{ccc}
a_{1}&-a_{8}/\sqrt{3}&-a_{64-}-ia_{75-}\\
-a_{8}/\sqrt{3}&a_{1}&-a_{64-}-ia_{75-}\\
-a_{64-}+ia_{75-}&-a_{64-}+ia_{75-}&-2a_{8}/\sqrt{3}
\end{array}
\right),\\
\fl
\mathbf{V}_{P}=\left(\begin{array}{ccc}
-a_{64+}-ia_{75+}&a_{64+}+ia_{75+}&0\\
a_{32-}&0&-a_{64+}-ia_{75+}\\
0&-a_{32-}&a_{64+}-ia_{75+}
\end{array}
\right).\end{aligned}$$ In the above equation, $a_{ij\pm}$ denotes $a_{i}\pm a_{j}$.
References {#references .unnumbered}
==========
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|
---
abstract: 'Motivated by recent STM experiments, we explore the magnetic field induced Kondo effect that takes place at symmetry protected level crossings in finite Co adatom chains. We argue that the effective two-level system realized at a level crossing acts as an extended impurity coupled to the conduction electrons of the substrate by a distribution of Kondo couplings at the sites of the chain. Using auxiliary-field quantum Monte Carlo simulations, which quantitatively reproduce the field dependence of the zero-bias signal, we show that a proper Kondo resonance is present at the sites where the effective Kondo coupling dominates. Our modeling and numerical simulations provide a theoretical basis for the interpretation of the STM spectrum in terms of level crossings of the Co adatom chains.'
author:
- Bimla Danu
- Fakher Assaad
- Frédéric Mila
bibliography:
- 'Kondo\_ref.bib'
title: Exploring the Kondo effect of an extended impurity with chains of Co adatoms in a magnetic field
---
The Kondo effect is one of the most extensively studied and adequately addressed many-body process occurring due to the screening of a local moment by a conduction electron cloud [@hewson1993; @Kondo1964; @Wilson1975]. On the experimental side, recent advances in scanning tunneling microscopy (STM) open greater opportunities to realise and investigate the Kondo effect in various Kondo nanostructures [@Madhavan1998; @Cronenwett1999; @Park2002; @Herrero2005; @Wahl2005; @Neel2011; @Spinelli2015; @Toskovic2016]. For instance, the recent STM experiments on finite atomic spin-chain realizations of Co adatoms in the presence of an external magnetic field have revealed an interesting interplay between the Kondo problem and the physics of quantum spin chains in a field [@Toskovic2016]. Co adatoms on a Cu$_2$N/Cu(100) surface carry a spin-3/2 with a strong uniaxial hard-axis anisotropy($D$) [@Otte2008; @Spinelli2015], and applying an external magnetic field perpendicular to the surface, the Co adatom chain effectively behaves like a spin-1/2 XXZ chain in transverse field. The magnetic field induced level crossings of finite XXZ and SU(2) chains are similar so that the experiments of Ref. \[\] can be discussed in the context of the SU(2) invariant version of the model: $$\begin{aligned}
\Hhat & = & -t\sum_{\langle \i,\j\rangle, \sigma}(\hat c^{\dagger}_{\i,\sigma} \hat c_{\j,\sigma}+h.c)+J_k \sum^L_{{\boldsymbol{l}} =1} \hat{{\boldsymbol{S}}}^c_{{\boldsymbol{l}}} \cdot \hat{{\boldsymbol{S}}}_{{\boldsymbol{l}}} \nonumber \\
& & +J_h\sum^{L-1}_{{\boldsymbol{l}}=1} \hat{{\boldsymbol{S}}}_{{\boldsymbol{l}}} \cdot \hat{{\boldsymbol{S}}}_{{\boldsymbol{l}}+\Delta{\boldsymbol{l}}} -g \mu_B h^{\bf z} \sum^{L}_{{\boldsymbol{l}}=1} \hat{S}^z_{{\boldsymbol{l}}}.
\label{model_ham}\end{aligned}$$ Here, $t$ is the hopping parameter of the conduction electrons, $J_k$ the antiferromagnetic Kondo coupling between a Co adatom and the conduction electrons, $J_h$ the Heisenberg antiferromagnetic coupling, $h^{\bf z}$ an external magnetic field in the $z$ direction, $L$ the length of the Heisenberg chain, $\hat{{\boldsymbol{S}}}_{{\boldsymbol{l}}}$ spin-1/2 operators and $\hat{{\boldsymbol{S}}}^c_{{\boldsymbol{l}}}=\frac{1}{2}\sum_{\sigma,\sigma^\prime} \hat{c}^{\dagger}_{{\boldsymbol{l}}, \sigma} {\boldsymbol{\sigma}}_{\sigma,\sigma^\prime} \hat{c}^{}_{{\boldsymbol{l}},\sigma^\prime}$ denotes the spin of conduction electrons. Throughout the calculation we set $t,\mu_B=1$ and $g=2$.
When the Kondo coupling is switched off ($J_k=0$), the chain undergoes a series of level crossings that lead to steps in the magnetization curve. In the geometry used in the experiment of Ref. \[\], $J_h$ and the Kondo energy $\epsilon_k=k_B T_k$ are both of the order of 0.2 meV, with two important consequences: There is a competition between the Heisenberg coupling and the Kondo effect, and one can reach the saturation field of the isolated chain.
The main result of the STM experiments of Ref. \[\] is to demonstrate that the differential conductance exhibits a series of anomalies as a function of the field, and that these anomalies coincide with the fields at which the isolated chain is expected to undergo level crossings. These anomalies are strongly site dependent however. Occasionally they take the typical V-shape of the Kondo resonance of a single impurity split by a magnetic field, but in most cases they are less pronounced if at all.
In this Letter, our goal is to provide a theory of STM that is valid at low temperature and that puts the measurements in the appropriate Kondo context. As we shall see, a Kondo effect is indeed present at each level crossing, but it corresponds to that of an extended impurity of size the length of the chain. The Hilbert space of this extended impurity corresponds to the twofold degenerate ground state of the spin chain at the level crossing. The presence or absence of a Kondo resonance at a given Co site where the STM signal is recorded depends i) on a matrix element encoding the fact that the ground state of the spin chain is addressed at a given Co site, and ii) on the magnitude of the effective Kondo coupling between the extended impurity and the substrate at the considered Co location.
[*Method.*]{} At high temperatures, the differential conductance in the presence of the Kondo coupling can be calculated using perturbation theory. While the results reproduce the gross features of the experimental data, they are limited to a regime above the Kondo temperature, and cannot account for the details of the low temperature data of Ref. \[\]. For a particle-hole symmetric conduction band, our model can be simulated with the auxiliary field quantum Monte Carlo (QMC) algorithm without encountering the negative sign problem. We have used the finite temperature algorithm [@Blankenbecler81; @White89; @Assaad08_rev] of the ALF-project [@ALF_v1] and followed Refs. \[\] for the implementation of our Kondo model. In the QMC calculation we consider a $20\times20$ square lattice with unit lattice constant and hopping matrix element $t$ and consider a linear arrangement of magnetic adatoms at distance $\Delta {\boldsymbol{l}} =(0,3)$ or $ \Delta {\boldsymbol{l}} = (3,2) $ from each other (up to $L=7$). To overcome the finite size effects we included an orbital magnetic field corresponding to a single flux quantum traversing the whole lattice [@Assaad2002], and a rather large value of the Kondo interaction $J_k/t=2$ so as to ensure that the Kondo scale of the single impurity problem remains larger than the finite size level spacing of the conduction electrons. Finally we consider $J_h/t=1.8$.
In the case of a single adatom ($L=1$), the problem reduces to that of a single impurity. The low temperature STM signal observed in Ref. \[\] consists of a single peak, the Kondo resonance, consistent with a tunneling process from sample to tip that goes through the localized d-orbital of the Co adatoms. To account for this in the realm of the Kondo model we compute co-tunneling processes [@Figgins2010; @Ternes2015; @Morr2017] given by: $ A_{{\boldsymbol{l}}}(\omega) = - \text{Im} G_{{\boldsymbol{l}}}^{\text{ret}} (\omega ) $ with $ G_{{\boldsymbol{l}}}^{\text{ret}} (\omega) = - i \int_{0}^{\infty} d\tau e^{i \omega \tau} \sum_{\sigma } \big< \big\{ \tilde{d}_{{\boldsymbol{l}},\sigma}^{}(\tau), \tilde{d}_{{\boldsymbol{l}},\sigma}^{\dagger} (0) \big\} \big> $ and $ \tilde{d}_{{\boldsymbol{l}},\sigma}^{\dagger} = \hat{c}^{\dagger}_{{\boldsymbol{l}}, -\sigma} \hat{S}^{\sigma}_{{\boldsymbol{l}}} + \sigma \hat{c}_{{\boldsymbol{l}},\sigma}^{\dagger} \hat{S}^{z}_{{\boldsymbol{l}}}$. Here $\sigma = \pm$ runs over the two spin polarization and $ \hat{S}^{\pm}_{{\boldsymbol{l}}} = \hat{S}^{x}_{{\boldsymbol{l}}} \pm i \hat{S}^{y}_{{\boldsymbol{l}}} $. This form can be obtained by starting from the single impurity Anderson model and applying the canonical Schrieffer-Wolf transformation (see supplemental material of Ref. \[\]) and agrees with the expression given in Ref. \[\].
Let us start by showing examples of spectral functions at level crossings obtained from QMC by stochastic analytic continuation [@KBeach2004]. As apparent from Fig. \[fig:Aomega\_vs\_omega\_L1234\]a), for a single impurity we observe the characteristic temperature dependence of a Kondo resonance at zero field.
![The spectral function computed using stochastic analytical continuation algorithm [@KBeach2004] at a given level crossings up to $L=4$. For $L>1$ we choose $J_h/t=1.8$ and $J_k/t=2$. The corresponding Kondo scale is extracted in Fig. \[fig:Tchi\_vs\_T\_L1234\].[]{data-label="fig:Aomega_vs_omega_L1234"}](Aomega_vs_omega_L1234){width="47.00000%"}
Fig. \[fig:Aomega\_vs\_omega\_L1234\] also shows the magnetic field induced Kondo resonances. For the two site chain there is a single level crossing between the singlet and the triplet at $g\mu_Bh^z=\Delta_1^{0,1}$. In the generic Kondo problem, time reversal symmetry protects the two-fold degeneracy of the impurity state. Here parity protects the level crossings and a Kondo resonance is apparent on both adatom sites, see Fig. \[fig:Aomega\_vs\_omega\_L1234\]b). For the three site chain two level crossings occur before saturation. The ground state is a spin-1/2 doublet in zero field and resonances are seen on the first and third adatoms, see Fig. \[fig:Aomega\_vs\_omega\_L1234\]c). At the second level crossing, the resonance is seen only on the central adatom, see Fig. \[fig:Aomega\_vs\_omega\_L1234\]d). For $L=4$, Kondo resonances emerge on outer adatoms at the first level crossing, see Fig.\[fig:Aomega\_vs\_omega\_L1234\]e), and on the central adatoms for the second level crossing, see Fig. \[fig:Aomega\_vs\_omega\_L1234\].f).
These results have been obtained at temperatures already representative of the low temperature regime, and they reproduce the main features of the experimental results (see Supplemental Material, Ref. \[\], Fig. \[fig:dIbydV\_vs\_V\_STM\]). However, to make a quantitative comparison with the experiments, which correspond to much lower temperature, we will concentrate on the zero bias differential conductance measured in the STM experiment [@Spinelli2015; @Toskovic2016; @Otte2008] as $$\begin{aligned}
dI_{{\boldsymbol{l}}}/dV(V=0) =2\frac{e^2}{\hbar} \int^{\infty}_{-\infty} d \omega \Big(-\frac{df(\omega)}{d\omega}\Big) A_{{\boldsymbol{l}}}(\omega)
\end{aligned}$$ where $f(\omega)$ is a Fermi function. In the low temperature limit the above maps onto: $$dI_{{\boldsymbol{l}}}/dV(V=0) \simeq 2\frac{e^2}{\hbar} A_{{\boldsymbol{l}}}(\omega=0) \simeq 2\frac{e^2}{\pi\hbar} \beta G_{{\boldsymbol{l}}}(\tau = \beta/2)$$ where $G_{{\boldsymbol{l}}} (\tau) = \sum_{\sigma}\langle \tilde{d}_{{\boldsymbol{l}},\sigma}^{}(\tau) \tilde{d}_{{\boldsymbol{l}},\sigma}^{\dagger}(0) \rangle $ is the imaginary time Green function which can be directly computed in the auxiliary field QMC. This approach avoids analytical continuation and our discussion will be based on the field dependence of this quantity. In the zero temperature limit the above equation is exact, and a more precise account of the zero bias differential conductance at finite temperature without using the analytical continuation can be obtained following Refs. \[\].
![The $\tilde d$-spectral function at $\omega=0$ computed by QMC for a Heisenberg chain in a field together with the zero bias conductance measured in the STM experiment (in atomic units) for an XXZ chain in transverse field [@Toskovic2016]. The magnetic field axis is normalised by the maximum values in both cases. The continuous grey vertical lines and the dashed grey lines denote the expected exact positions of the level crossings for a Heisenberg chain and for an XXZ one [@Toskovic2016], respectively.[]{data-label="comp_Greenf"}](L1234_Aomega0_vs_field_QMC_STM.pdf){width="45.00000%"}
The QMC results of the local spectral function at zero frequency for $k_BT/t=1/30$ are compared to the zero bias conductance reported in Ref. \[\] as a function of external magnetic field in Fig. \[comp\_Greenf\]. Noticeably, up to four atoms the zero frequency spectral function shows excellent agreement with the corresponding zero bias conductance measured in the experiment. The temperature scales in the QMC and STM are comparable: the data presented in Fig. \[comp\_Greenf\] are computed below $k_BT^l_k/8t$, where $T^l_k$ is an estimate of Kondo temperature from scaling of local spin susceptibility [@Hirsch1986; @Assaad2002] at each level crossings (see below), while the STM data are taken at $330$ mK $\sim T^{Co}_k/8$ ($k_BT/t\sim1/35$ in QMC).
To associate an adatom dependent Kondo temperature to each level crossing, we compute the local transverse susceptibility, $\chi_{{\boldsymbol{l}}}=\int^{\beta}_{0} d \tau\langle \hat S^{+}_{{\boldsymbol{l}}}(\tau)\hat S^{-}_{{\boldsymbol{l}}}(0)+h.c \rangle$, where $\hat S^{\pm}_{{\boldsymbol{l}}}(\tau)=e^{\tau \hat H} \hat S^\pm_{{\boldsymbol{l}}} e^{-\tau \hat H}$. Interestingly we observe in Fig. \[fig:Tchi\_vs\_T\_L1234\] that when the STM data at an adatom site shows a resonance, the local susceptiblity follows the expected universal behaviour, $T\chi_l = f(T/T^l_k)$, where $T^l_k$ corresponds to the adatom and level crossing resolved Kondo temperature.
![Local probe transverse susceptibility as a function of temperature computed for $J_h/t=1.8$ up to $L=4$ at sites where the level crossing leads to a Kondo resonance. The sites not shown do not follow the typical temperature dependence on Kondo couplings up to $k_BT/t\approx1/40$.[]{data-label="fig:Tchi_vs_T_L1234"}](Tchi_vs_TbyTk_L1234){width="45.00000%"}
Fig. \[fig:peakheight\_vs\_T\] plots the temperature dependence of the zero frequency spectral function as estimated by $A_{{\boldsymbol{l}}}(\omega=0) \simeq
\frac{1}{\pi} \beta G_{{\boldsymbol{l}}}(\tau = \beta/2) $. For $L=3$ and $L=4$ strong site dependence of the signal emerges below the Kondo temperature. We first concentrate on cases where we observe a dominant resonance. For these cases, the temperature dependence of the zero-bias conductance at the various sites (see Fig. \[fig:peakheight\_vs\_T\]) shows logarithmic increase at the Kondo scale upon reducing the temperature. At the other sites no such increase is observed. Quite remarkably, depending on the site, level crossings can show up as a peak, a dip, or a change of slope as a function of field. Accordingly, in the experimental results of Ref. \[\], the frequency dependence at a critical field may or may not show a Kondo resonance.
![Exact zero-bias conductance as a function of temperature normalised by the corresponding Kondo scale at each level crossings up to $L=4$.[]{data-label="fig:peakheight_vs_T"}](Aomega0_vs_TbyTk_L1234.pdf){width="45.00000%"}
*Effective model.* To provide an effective model for a given level crossing, $p$, we project the Hamiltonian on the two fold degenerate Hilbert space of the level crossing. Clearly such a strategy is valid only if the gap separating the next excited states of the spin chain is large compared to the effective Kondo scale. For our SU(2) model, this approximation will necessarily fail in the large $L$ limit, but as we will see below it provides an accurate account of the QMC data for small $L$. Let $\{|\mbf\rangle=|\Sbf_\mbf\Sbf^z_\mbf\rangle_p,|\mmbf \rangle = | \Sbf_\mmbf\Sbf^z_\mmbf\rangle_p\}$ be the eigenstate of the Heisenberg chain with energies $e_{m_1,p} $ , $e_{m_2,p} $ that span the Hilbert space of the level crossing and $\hat{P} $ the projector onto this space. Defining a vector of Pauli spin matrices ${\boldsymbol{\tau}}$ that act on this Hilbert space, the projected Hamiltonian, up to a constant, reads: $$\begin{aligned}
&&\Hhat^{eff}_p=-t\sum_{\langle \i,\j\rangle, \sigma}(\hat c^{\dagger}_{\i,\sigma} \hat c_{\j,\sigma}+h.c) + \nonumber
J_k \sum_{{\boldsymbol{l}}}\Big\{j^{\perp}_{{\boldsymbol{l}},p}(\hat S^{x,c}_{{\boldsymbol{l}}} \hat{\tau}^x+ \\ &&\hat{S}^{y,c}_{{\boldsymbol{l}}} \hat \tau^y)+j^{z}_{{\boldsymbol{l}},p}\hat S^{z,c}_{{\boldsymbol{l}}} \hat{\tau}^z+\mu_{{\boldsymbol{l}},p} S^{z,c}_{{\boldsymbol{l}}}+(\Delta^{\mbf,\mmbf}_p-g\mu_Bh^{\bf z})\hat n\Big\}
\label{effe_ham}\end{aligned}$$ The energy difference between the two local states is given by $\Delta^{\mbf,\mmbf}_p=e_{\mmbf,p}-e_{\mbf,p}$, $\hat{n}=\frac{1}{2}(\identity+ \hat {\tau}^z)$, and the site dependent effective couplings and the effective magnetic field are defined as; $j^{\perp}_{{\boldsymbol{l}},p}=\langle \mmbf|\hat{S}^{x,y}_{{\boldsymbol{l}}} |\mbf\rangle$, $2j^{z}_{{\boldsymbol{l}},p}=-\langle \mbf|\hat{S}^z_{{\boldsymbol{l}}} |\mbf \rangle+\langle \mmbf|\hat{S}^z_{{\boldsymbol{l}}} |\mmbf \rangle$ and $2\mu_{{\boldsymbol{l}},p}=\langle \mbf|\hat{S}^z_{{\boldsymbol{l}}} |\mbf \rangle+\langle \mmbf| \hat{S}^z_{{\boldsymbol{l}}} |\mmbf \rangle$.
![A chain of Co adatoms on a substrate implement, at a level crossing, the Kondo model of an extended impurity.[]{data-label="Coadatoms_skech"}](Co_adatoms_sketch.pdf){width="48.00000%"}
------- ---------------------------------------------------------------------------------------------------------------------------------------------------------
$L=2$ $j^{\perp}_{1(2),1}= \frac{1}{2\sqrt{2}}$, $j^{z}_{1(2),1}=\frac{1}{4}$
$L=3$ $j^{\perp}_{1(3),1}=\frac{1}{3}$,$j^{\perp}_{2,1}=\frac{1}{6}$, $j^{z}_{1(3),1}=\frac{1}{3}$, $j^{z}_{2,1}=\frac{1}{6}$
$L=3$ $j^{\perp}_{1(3),2}=\frac{1}{2\sqrt{6}}$, $j^{\perp}_{2,2}=\frac{1}{\sqrt{6}}$, $j^{z}_{1(3),2}=\frac{1}{12}$,$j^{z}_{2,2}=\frac{1}{3}$
$L=4$ $j^{\perp}_{1(4),1}=\frac {1}{4 \sqrt{6}{\sqrt{2+\sqrt{2}}}}\Big[ \frac{1+\sqrt{2}}{1+\sqrt{3}}+1+\sqrt{1+\frac{\sqrt{3}}{2}}\big(1+\sqrt{2}\big)\Big]$
$j^{\perp}_{2(3),1}= \frac {1}{4 \sqrt{6}{\sqrt{2+\sqrt{2}}}}\Big[ \frac{1}{1+\sqrt{3}}+(1+\sqrt{2})+\sqrt{1+\frac{\sqrt{3}}{2}}\Big]$
$j^{z}_{1(4),1}=\frac{1}{16} (2+\sqrt{2})$, $j^{z}_{2(3),1}=\frac{1}{16} (2-\sqrt{2})$
$L=4$ $j^{\perp}_{1(4),2}=\frac {1}{4{\sqrt{2+\sqrt{2}}}} $, $j^{\perp}_{2(3),2}= \frac {1+\sqrt{2}}{4{\sqrt{2+\sqrt{2}}}}$
$j^{z}_{1(4),2}=\frac{1}{16} (2-\sqrt{2})$, $j^{z}_{2(3),2}=\frac{1}{16} (2+\sqrt{2})$
------- ---------------------------------------------------------------------------------------------------------------------------------------------------------
: \[tab:j\_x,z\]Effective couplings $j^{\perp,z}_{{\boldsymbol{l}}({\boldsymbol{l}}^\prime),p}$ at the level crossings($p=1, \cdots,\frac{L}{2}(\frac{L+1}{2})$ for a fixed even(odd) $L$.)
This model can be interpreted as the Kondo model of an extended impurity that is coupled to the conduction electrons at $L$ points (see Fig. \[Coadatoms\_skech\]). The projection does not affect the U(1) spin symmetry of the model, but as shown in Table \[tab:j\_x,z\] it yields a strong site dependence of the effective couplings $j^{\perp,z}_{{\boldsymbol{l}},p}$. To compute the co-tunneling within the effective model, we still have to project the spin operator onto the level-crossing Hilbert space $ \hat{P} \tilde{d}_{{\boldsymbol{l}},\sigma}^{\dagger} \hat{P} = \hat{c}^{\dagger}_{{\boldsymbol{l}}, -\sigma} \hat{P} \hat{S}^{\sigma}_{{\boldsymbol{l}}} \hat{P} + \sigma \hat{c}_{{\boldsymbol{l}},\sigma}^{\dagger} \hat{P} S^{z}_{{\boldsymbol{l}}} \hat{P} $ so that it acquires a site dependence when written in terms of ${\boldsymbol{\tau}}$ operators. This reflects the fact that in the experiment the extended states $| \mbf \rangle $ and $ | \mmbf \rangle $ are addressed via manipulation of one of the constituent Co d-spins. A detailed numerical analysis of the effective model along these lines is left for future studies. However, it is already clear that it provides a qualitative interpretation of the data in terms of a projection induced hierarchy of Kondo scales (see Ref. \[\], Table \[tab:epsilonk\_L1234567\]). Consider for instance the four-site chain at the first, $p=1$, level crossing for which $j^{z}_{1(4),1} / j^{z}_{2(3),1} = 5.83$ and $j^{\perp}_{1(4),1}/j^{\perp}_{2(3),1} = 1.25 $. Thereby, the Kondo singlet will be predominantly formed by an entangled state of the degenerate two level system, $ \left\{ | \mbf \rangle, | \mmbf \rangle \right\} $ and a symmetric combination of the conduction electron spins on sites one and four (see Ref. \[\], Section \[Effec\_Kondo\_scale\].). This provides an understanding of the observed Kondo like temperature dependence of the zero-bias conductance at sites one and four.
In Fig. \[comp\_567\] we consider chains up to seven spins. Here, an accurate comparison of the zero bias conductance is hard due to the asymmetric line shape of the Kondo resonances that arise in the experiment due to the potential scattering between tip and sample [@Figgins2010; @von2015; @Ternes2015; @Morr2017], but a reasonable agreement can be achieved slightly away from the zero bias around $V\sim0.4-0.8$ mV as shown in Fig. \[comp\_567\].
![QMC results for five, six and seven atom spin-chains together with the STM data around $V\sim0.6$ mV [@Toskovic2016] as a function of magnetic field (normalised by the maximum values).[]{data-label="comp_567"}](L567_Aomega0_vs_field_STM_QMC.pdf){width="45.00000%"}
As the chain size grows, the spectrum will collapse, and our understanding in terms of the projection onto the two-fold level crossing Hilbert space fails. In fact in this limit we expect a crossover to Kondo lattice behavior characterized at the mean field level by hybridized heavy and light bands in a magnetic field (see Ref. \[\], Section \[LargeN\_meanfield\]). For an infinite Heisenberg chain and within this approximation, we expect the conductance to reflect the local spinon density of states.
To summarize, we have shown that the STM measurements on Co adatom chains agree remarkably well with QMC simulations of a model of spin-1/2 chains in a field coupled to a conducting substrate via local Kondo couplings. To interpret the strong site dependence of the signal, we have performed a projection onto the symmetry protected level crossing Hilbert space of the spin chain that leads to the notion of an extended impurity with site dependent Kondo couplings. Consequently, screening happens via the dominant channel such that site and level crossing dependent Kondo resonances are observed in the STM co-tunneling conductance as well as in the Monte Carlo simulations. As a function of chain length, the above construction will progressively fail and we expect a crossover to Kondo lattice physics, in which the STM signal will pick up the two spinon continuum of the spin chain. Further work at understanding the details of the extended impurity Kondo model, as well as the crossover to the lattice limit is presently under progress.
We thank S. Otte, M. Raczkowski, and M. Ternes for very useful discussions and S. Otte and M. Ternes for providing the STM data. FFA thanks the DFG collaborative research centre SFB1170 To CoTronics (project C01) for financial support as well as the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter – *ct.qmat* (EXC 2147, project-id 39085490). The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for funding this project by providing computing time on the GCS Supercomputer SUPERMUC-NG at Leibniz Supercomputing Centre (www.lrz.de). FM and BD acknowledge the Swiss National Science Foundation t and its SINERGIA network “Mott physics beyond the Heisenberg model” for financial support. This work was also supported by EPFL through the use of the facilities of its Scientific IT and Application Support Center.
Effective models at level crossings {#H_effL1234}
-----------------------------------
To derive the effective model at level crossing $p$ (see Fig. \[fig:eigenspectrum\_vs\_hz\] and Fig. \[fig:gapSU2\_vs\_hz\]) we project the Hamiltonian of Eq. (\[model\_ham\]) onto the Hilbert space spanned by the states $\big\{|\mbf\rangle, |\mmbf\rangle\big\}$ of the level crossing of a finite Heisenberg chain at a given critical magnetic field. $$\begin{aligned}
\hat H^{eff}_p=-t\sum_{\langle \i,\j\rangle, \sigma}(\hat c^{\dagger}_{\i,\sigma} \hat c_{\j,\sigma}+h.c)+ J_k\sum^L_{{\boldsymbol{l}}=1} \sum_{\mibf \mjbf=\mbf,\mmbf} \Sbf^c_{{\boldsymbol{l}}} \cdot |\mjbf\rangle \langle \mjbf |\Sbf_{{\boldsymbol{l}}}|\mibf\rangle \langle \mibf|\nonumber\\+(e_{\mbf,p}-\Sbf^z_{\mbf,p} h^z) |\mbf\rangle\langle \mbf|+(e_{\mmbf,p}-\Sbf^z_{\mmbf,p} h^z) |\mmbf\rangle\langle \mmbf|.
\label{HL1234P}\end{aligned}$$ The states are explicitly given in Table \[tab:exactm1m2\].
[l | l]{} $L=2$& $|0\rangle=\frac{1}{\sqrt{2}}\left(\big|\uparrow\downarrow\rangle-|\downarrow\uparrow\big\rangle\right)$,$\big|1\big\rangle=\big|\uparrow \uparrow\big\rangle$\
& $e_{0,1}=-\frac{3}{4}J_h$, $e_{1,1}=\frac{J_h}{4}$, $\Delta^{0,1}_1=J_h$, $\mu_{1(2),1}=\frac{1}{4}$\
\
$L=3$& $\big|\frac{1}{2}\big\rangle=\frac{1}{\sqrt{6}}\big(|\downarrow\uparrow\uparrow\rangle-2|\uparrow\downarrow\uparrow\rangle+\uparrow\uparrow\downarrow\rangle\big)$, $\big|-\frac{1}{2}\big\rangle=\frac{1}{\sqrt{6}}\left(|\downarrow\downarrow\uparrow\rangle-2|\downarrow\uparrow\downarrow\rangle+\uparrow\downarrow\downarrow\rangle\right)$\
& $e_{\frac{1}{2},1}=-J_h$, $e_{-\frac{1}{2},1}=-J_h$, $\Delta^{-\frac{1}{2},\frac{1}{2}}_1=0$, $\mu_{1(3),1}=0$, $\mu_{2,1}=0$\
\
$L=3$& $\big|\frac{1}{2}\big\rangle=\frac{1}{\sqrt{6}}\big(\big|\downarrow\uparrow\uparrow\big\rangle-2\big|\uparrow\downarrow\uparrow\big\rangle+\big|\uparrow\uparrow\downarrow\big\rangle\big)$,$\big|\frac{3}{2}\big\rangle=\big|\uparrow \uparrow\uparrow\big\rangle$\
& $e_{\frac{1}{2},2}=-J_h$, $e_{\frac{3}{2},2}=\frac{J_h}{2}$, $\Delta^{\frac{1}{2},\frac{3}{2}}_2=\frac{3}{2}J_h$, $\mu_{1(3),2}=\frac{5}{12}$, $\mu_{2,2}=\frac{1}{6}$\
\
$L=4$ & $\big|0\big\rangle=\frac{1}{\sqrt{6}}\Big[\frac{1}{(1+\sqrt{3})}\big(\big|\downarrow\downarrow\uparrow\uparrow\big\rangle+\big|\uparrow\uparrow\downarrow\downarrow\big\rangle\big)+\big|\downarrow\uparrow\uparrow\downarrow\big\rangle+\big|\uparrow\downarrow\downarrow\uparrow\big\rangle-\sqrt{1+\frac{\sqrt{3}}{2}}\big(\big|\downarrow\uparrow\downarrow\uparrow\big\rangle+\big|\uparrow\downarrow\uparrow\downarrow\big\rangle\big)\Big]$\
&$\big|1\big\rangle=\frac{1}{2\sqrt{2+\sqrt{2}}}\Big[-\big|\downarrow\uparrow\uparrow\uparrow\big\rangle+\big|\uparrow\uparrow\uparrow\downarrow\big\rangle+(1+\sqrt{2})\big(\big|\uparrow\downarrow\uparrow\uparrow\big\rangle-|\uparrow\uparrow\downarrow\uparrow\big\rangle\big)\Big]$\
&$e_{0,1}=\frac{J_h}{4}\big(-3-2\sqrt{3}\big)$,$e_{1,1}=\frac{J_h}{4}\big(-1-2\sqrt{2}\big)$, $\Delta^{0,1}_1=\frac{J_h}{2} (1-\sqrt{2}+\sqrt{3})$,$\mu_{1(4),1}=\frac{(2+\sqrt{2})}{16}$,$\mu_{2(3),1}=\frac{(2-\sqrt{2})}{16}$\
\
$L=4$&$\big|1\big\rangle=\frac{1}{2\sqrt{2+\sqrt{2}}}\Big[-\big|\downarrow\uparrow\uparrow\uparrow\big\rangle+\big|\uparrow\uparrow\uparrow\downarrow\big\rangle+(1+\sqrt{2})\big(\big|\uparrow\downarrow\uparrow\uparrow\big\rangle-\big|\uparrow\uparrow\downarrow\uparrow\big\rangle\big)\Big]$, $\big|2\big\rangle=\big|\uparrow\uparrow\uparrow\uparrow\big\rangle$\
& $e_{1,2}=\frac{J_h}{4}\big(-1-2\sqrt{2}\big)$,$e_{2,2}=\frac{3J_h}{4}$, $\Delta^{1,2}_2=J_h\big(1+\frac{1}{\sqrt{2}}\big)$, $\mu_{1(4),2}=\frac{(6+\sqrt{2})}{16}$,$\mu_{2(3),2}=\frac{(6-\sqrt{2})}{16}$
We further introduce pseudo spin-1/2 operators; $\hat{\tau}^x=|\mmbf\rangle\langle \mbf|+|\mbf\rangle\langle \mmbf|$, $\hat {\tau}^y=-i(|\mmbf\rangle\langle \mbf|-|\mbf\rangle\langle \mmbf|)$ and $\hat{\tau}^z=|\mmbf\rangle \langle \mmbf|-|\mbf\rangle \langle \mbf|$ and impose the constraint $\identity=|\mbf\rangle \langle \mbf|+|\mmbf\rangle \langle \mmbf|$. In terms of the ${\boldsymbol{\tau}}$ operators Eq. (\[HL1234P\]) takes the form given in Eq. (\[effe\_ham\]) with a set of site dependent effective couplings given in Table \[tab:j\_x,z\]. The other effective parameters of Eq. (\[effe\_ham\]) are given in Table \[tab:exactm1m2\] up to L=4. When calculating the local matrix elements, we obtain an alternative $\pm$ sign of $j^{\perp}_{{\boldsymbol{l}},p}$ which does not affect Kondo physics [@Anderson11970; @Anderson1970]. For simplicity we omitted this sign $(-1)^{{\boldsymbol{l}}}$ in Table \[tab:j\_x,z\] of the main text. Noticeably, for a Heisenberg chain ($J_h\sum^{L-1}_{{\boldsymbol{l}}=1} \hat{{\boldsymbol{S}}}_{{\boldsymbol{l}}} \cdot \hat{{\boldsymbol{S}}}_{{\boldsymbol{l}}+1}$) the effective couplings $(j^{\perp,z}_{{\boldsymbol{l}}({\boldsymbol{l}}^\prime),p})$ are independent of $J_h$ (see Table \[tab:j\_x,z\]). However, for an XXZ chain ($J_{xy}\sum^{L-1}_{{\boldsymbol{l}}=1} (\hat{S^x}_{{\boldsymbol{l}}}\hat{S^x}_{{\boldsymbol{l}}+1}+\hat{S^y}_{{\boldsymbol{l}}}\hat{S^y}_{{\boldsymbol{l}}+1})+J_{zz}\sum^{L-1}_{{\boldsymbol{l}}=1} \hat{S^z}_{{\boldsymbol{l}}}\hat{S^z}_{{\boldsymbol{l}}+1}$) the ratio of the exchange parameters $J_{zz}/J_{xy}$ appears in the expression of the effective couplings. Since they do not have a simple form, we choose to plot them as a function of $J_{zz}/J_{xy}$ in Fig.\[fig:XY\_SU2\_Kondoscale\_L3\] and Fig.\[fig:XY\_SU2\_Kondoscale\_L4\] for chains of three and four atoms respectively. As for the Heisenberg chain ($J_{zz}/J_{xy}=1$), there is always a strong site dependence of the effective couplings when varying $J_{zz}/J_{xy}=0, \cdots,1$. Hence, the magnetic field induced level crossings in finite XXZ and Heisenberg chains is expected to show a similar site dependent Kondo physics.
![Top: Site dependent effective couplings as a function of $J_{zz}/J_{xy}$ at level crossings for $L=4$. Bottom: Corresponding effective Kondo scale ($\epsilon^k_{l(l^\prime),p}\sim e^{-\frac{1} {j_{l,p}}}$) as a function of $J_{zz}/J_{xy}$.[]{data-label="fig:XY_SU2_Kondoscale_L4"}](Kondo_temp_L3_XY_XXZ_SU2.pdf){width="11.5cm"}
![Top: Site dependent effective couplings as a function of $J_{zz}/J_{xy}$ at level crossings for $L=4$. Bottom: Corresponding effective Kondo scale ($\epsilon^k_{l(l^\prime),p}\sim e^{-\frac{1} {j_{l,p}}}$) as a function of $J_{zz}/J_{xy}$.[]{data-label="fig:XY_SU2_Kondoscale_L4"}](Kondo_temp_L4_XY_XXZ_SU2.pdf){width="12.cm"}
![Lowest energy excitations as a function of magnetic field (in $z$-direction in units of $g \mu_B$) for a spin-1/2 Heisenberg chain.[]{data-label="fig:gapSU2_vs_hz"}](EigL24567_SU2){width="\textwidth"}
![Lowest energy excitations as a function of magnetic field (in $z$-direction in units of $g \mu_B$) for a spin-1/2 Heisenberg chain.[]{data-label="fig:gapSU2_vs_hz"}](GapL24567_SU2){width="\textwidth"}
Effective Kondo Scale at level crossings (Anderson Poor man scaling approach) {#Effec_Kondo_scale}
-----------------------------------------------------------------------------
Starting from the effective Kondo Hamiltonian for a chain of two atoms at singlet-triplet level crossing, which reads $$\begin{aligned}
\Hhat^{eff}_1 & = &e_{0,1}-t\sum_{\langle \i,\j\rangle, \sigma}(\hat c^{\dagger}_{\i,\sigma} \hat c_{\j,\sigma}+h.c)+J_k\Big\{j^{\perp} \big[(\hat S^{x,c}_{{{\boldsymbol{l}}}_{1}}+\hat S^{x,c}_{{{\boldsymbol{l}}}_{2}})\hat \tau^x+(\hat S^{y,c}_{{{\boldsymbol{l}}}_{1}}+
\hat S^{y,c}_{{{\boldsymbol{l}}}_{2}})\hat \tau^y\big] \nonumber\\ && +j^{z}(\hat S^{z,c}_{{{\boldsymbol{l}}}_{1}}+ \hat S^{z,c}_{{{\boldsymbol{l}}}_{2}}) \hat{\tau}^z +\mu (\hat S^{z,c}_{{{\boldsymbol{l}}}_{1}}+
\hat S^{z,c}_{{{\boldsymbol{l}}}_{2}}) \Big\} + \frac{1}{2}\big(\Delta^{0,1}_1-g\mu_Bh^z\big)\big (\identity+\hat \tau^z\big)
\label{HameffL2AS}\end{aligned}$$ where, $j^{\perp}=\frac{1}{2\sqrt{2}}$, $j^z=\frac{1}{4}$ and $\mu =\frac{1}{4}$, we use the following unitary transformation for conduction electrons in Eq. (\[HameffL2AS\]), $$\begin{aligned}
\hat \S^c_B=\frac{1}{\sqrt{2}}\Big(\hat{\S}^c_{{{\boldsymbol{l}}}_{1}}+\hat{\S}^c_{{{\boldsymbol{l}}}_{2}}\Big), \quad \hat \S^c_A=\frac{1}{\sqrt{2}}\Big(\hat{\S}^c_{{{\boldsymbol{l}}}_{1}}-\hat{\S}^c_{{{\boldsymbol{l}}}_{2}}\Big)
\label{UT_AS}
\end{aligned}$$ to rewrite it as $$\begin{aligned}
\Hhat^{eff}_1 =e_{0,1} -t\sum_{\langle \i,\j\rangle, \sigma}(\hat c^{\dagger}_{\i,\sigma} \hat c_{\j,\sigma}+h.c)+ \tilde{j}^{\perp}( \hat {S}^{x,c}_{B}\hat{\tau}^x+\hat{S}^{y,c}_{B}\hat{\tau}^y)+\tilde{j}^{z} \hat{S}^{z,c}_{B} \hat \tau^z +\tilde\mu \hat{S}^{z,c}_{B}+\frac{1}{2}\big(\Delta^{0,1}_1-g\mu_Bh^z\big)\big (\identity+\hat \tau^z\big)
\label{HameffAL2}
\end{aligned}$$ where, $\tilde{j}^\perp=\sqrt{2}J_k j^{\perp}$ and $\tilde{j}^z=\sqrt{2}J_k j^{z}$. The effective Hamiltonian given in Eq.(\[HameffAL2\]) corresponds to an anisotropic single impurity Kondo Hamiltonian. Following Anderson’s poor man scaling approach [@Anderson11970; @Anderson1970] the effective Kondo scale can be estimated by integrating and solving two differential equations (see below) obtained from $T$-matrix scattering process of the conduction electron scattering off the pseudo spin-1/2 degree of freedom, ${\boldsymbol{\tau}}$: $$\begin{aligned}
\frac {d \tilde {j}^{z}}{d \ln D} =2\rho (\tilde {j}^{\perp})^2
\label{dbydjz}\end{aligned}$$ and $$\begin{aligned}
\frac {d \tilde {j}^{\perp}}{d \ln D}=2\rho \tilde {j}^{\perp} \tilde {j}^{z}
\label{dbydjxy}
\end{aligned}$$ where $D$ is the half bandwidth cutoff and $\rho$ the density of states at the Fermi level. In the isotropic case $\tilde {j}^{\perp}= \tilde {j}^{z}={j}$, the two differential Eqs. are identical and yield the Kondo scale; $\epsilon^k \sim e^{-\frac{1}{\rho j}}$. In the anisotropic case $\tilde {j}^{\perp}\ne\tilde {j}^{z}$ the two Eqs. (\[dbydjz\]) and. (\[dbydjxy\]) give a scaling trajectory $(\tilde{j}^z)^2-(\tilde j^{\perp})^2 =const$, and the effective Kondo scale can be obtained by the flow of the renormalised couplings along the trajectory. For a chain of length $L$ one can write the effective Hamiltonian in terms of $\hat {\S}^c_{B(A)}$ as: $$\begin{aligned}
\Hhat^{eff}_p &=-t\sum_{\langle \i,\j\rangle, \sigma}(\hat c^{\dagger}_{\i,\sigma} \hat c_{\j,\sigma}+h.c)+cJ_k \sum_{{\boldsymbol{l}}} \Big\{ j^{\perp}_{{\boldsymbol{l}},p}( \hat {S}^{x,c}_{{\boldsymbol{l}},B(A)}\hat{\tau}^x+\hat{S}^{y,c}_{{\boldsymbol{l}},B(A)}\hat{\tau}^y)+j^{z}_{{\boldsymbol{l}},p} \hat{S}^{z,c}_{{\boldsymbol{l}},B(A)} \hat \tau^z +\mu_{{\boldsymbol{l}},p} \hat{S}^{z,c}_{{\boldsymbol{l}},B(A)}\Big\}\\ & +\frac{1}{2}\big(\Delta^{\mbf,\mmbf}_p-g\mu_Bh^z\big)\big (\identity+\hat \tau^z\big).
\label{HameffASL}
\end{aligned}$$ Here, the summation $\sum_{{\boldsymbol{l}}}$ goes over ${\boldsymbol{l}}=1, \cdots, \frac{L}{2} (\frac{L+1}{2})$ for even(odd) $L$, and, $c$ is a normalisation factor arising from the bonding or antibonding ($B(A)$) selections of conduction electrons involved in the Kondo effect. This selection rule stems from the inversion symmetry present in the Heisenberg chain and involves pairs of sites ${\boldsymbol{l}}(=1, \cdots, \frac{L}{2})$ and ${\boldsymbol{l}}^\prime(=L-({\boldsymbol{l}}-1), \cdots,\frac{L}{2}+1$). One can define a site dependent effective Kondo scale of Eq. (\[HameffASL\]) by considering the flow of renormalised couplings along the scaling trajectories, $$\begin{aligned}
(\tilde {j}^z_{{\boldsymbol{l}},p})^2-(\tilde j^{\perp}_{{\boldsymbol{l}},p})^2 =const
\label{scaling_jxjzL}\end{aligned}$$ where $\tilde{j}^\perp_{{\boldsymbol{l}},p}=cJ_k j^{\perp}_{{\boldsymbol{l}},p}$ and $\tilde{j}^z_{{\boldsymbol{l}},p}=cJ_k j^{z}_{{\boldsymbol{l}},p}$. To estimate the Kondo scale we use $c=\sqrt{2}$ if two atoms at the position ${\boldsymbol{l}}$ and ${\boldsymbol{l}}^\prime$ are symmetrically involved in the Kondo resonance and $c=1$ if only one atom shows a Kondo resonance. The latter case corresponds for example to the central atom of an odd sized chain. Furthermore, we use a constant density of states $\rho=1$ in all cases. At a level crossing $p$ and depending on the sign of $\tilde{j}^{z}_{{\boldsymbol{l}},p}$ [@Anderson11970; @Anderson1970; @Yosida1991; @Romeike2006; @Rok2008] (see Table \[tab:epsilonk\_signjz\]) a relative estimate of Kondo scale ($\epsilon^k_{{\boldsymbol{l}},p}\sim e^{-\frac{1} { \rho \tilde j_{{\boldsymbol{l}},p}}}$) is given in Table \[tab:epsilonk\_L1234567\].
$\tilde{j}^z_{{\boldsymbol{l}},p}$ $>$ 0 $\tilde{j}^z_{{\boldsymbol{l}},p}$ $<$ 0
------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --
$|\tilde{j}^{\perp}_{{\boldsymbol{l}},p}|>|\tilde{j}^z_{{\boldsymbol{l}},p}|$ $\frac{1}{\tilde{j}_{{\boldsymbol{l}},p}}=\frac{1}{\sqrt {({\tilde{j}^{\perp}_{{\boldsymbol{l}},p})^2-(\tilde{j}^{z}_{{\boldsymbol{l}},p})^2 }}} \times \tan^{-1} \Big (\frac {\sqrt{ (\tilde{j}^{\perp}_{{\boldsymbol{l}},p})^2-(\tilde{j}^{z}_{{\boldsymbol{l}},p})^2}}{\tilde{j}^{z}_{{\boldsymbol{l}},p}} \Big)$ $\frac{1}{\tilde{j}_{{\boldsymbol{l}},p}}=\frac{1}{\sqrt {({\tilde{j}^{\perp}_{{\boldsymbol{l}},p})^2-(\tilde{j}^{z}_{{\boldsymbol{l}},p})^2}}} \times \Big[ \pi+ \tan^{-1} \Big (\frac {\sqrt{ (\tilde{j}^{\perp}_{{\boldsymbol{l}},p})^2-(\tilde{j}^{z}_{{\boldsymbol{l}},p})^2}}{\tilde{j}^{z}_{{\boldsymbol{l}},p}} \Big)\Big]$
$|\tilde{j}^{\perp}_{{\boldsymbol{l}},p}|<|\tilde{j}^z_{{\boldsymbol{l}},p}|$ $\frac{1}{\tilde{j}_{{\boldsymbol{l}},p}}=\frac{1}{\sqrt {({\tilde{j}^{z}_{{\boldsymbol{l}},p})^2-(\tilde{j}^{\perp}_{{\boldsymbol{l}},p})^2 }}} \times\tanh^{-1} \Big(\frac {\sqrt{(\tilde{j}^{z}_{{\boldsymbol{l}},p})^2-(\tilde{j}^{\perp}_{{\boldsymbol{l}},p})^2}}{\tilde{j}^{z}_{{\boldsymbol{l}},p}} \Big)$ 0
: \[tab:epsilonk\_signjz\] Kondo scale ($\epsilon^k_{{\boldsymbol{l}},p}\sim e^{-\frac{1} { \rho \tilde j_{{\boldsymbol{l}},p}}}$) according to sign of $\tilde{j}^{z}_{{\boldsymbol{l}},p}$ for an anisotropic Kondo Hamiltonian.
------- --------------------------------------------- ----------------------------------------- ----------------------------------------- ---------------------------------------
$L=2$ $\epsilon^{k}_{1(2),1}\sim0.329(0.227)$
$L=3$ $\epsilon^{k}_{1(3),1}\sim 0.346(0.243)$ $\epsilon^{k}_{2,1}\sim0.049(0.018)$
$L=3$ $\epsilon^{k}_{1(3),2}\sim0.113(0.055)$ $\epsilon^{k}_{2,2}\sim0.271(0.176)$
$L=4$ $\epsilon^{k}_{1(4),1}\sim0.259(0.165)$ $\epsilon^{k}_{2(3),1}\sim0.113(0.055)$
$L=4$ $\epsilon^{k}_{1(4),2}\sim0.029(0.009)$ $\epsilon^{k}_{2(3),2}\sim0.293(0.194)$
$L=5$ $\epsilon^{k}_{1(5),1}\sim0.251(0.159)$ $\epsilon^{k}_{2(4),1}\sim0.091(0.041)$ $\epsilon^k_{3,1}\sim0.171(0.095)$
$L=5$ $\epsilon^{k}_{1(5),2}\sim0.124(0.061)$ $\epsilon^{k}_{2(4),2}\sim0.249(0.157)$ $\epsilon^{k}_{3,2}\sim0.0004(0.00003)$
$L=5$ $\epsilon^{k}_{1(5),3}\sim0.007(0.001)$ $\epsilon^{k}_{2(4),3}\sim0.189(0.109)$ $\epsilon^{k}_{3,3}\sim0.164(0.089)$
$L=6$ $\epsilon^{k}_{1(6),1}\sim0.196(0.114)$ $\epsilon^{k}_{2(5),1}\sim0.026(0.008)$ $\epsilon^{k}_{3(4),1}\sim0.151(0.079)$
$L=6$ $\epsilon^{k}_{1(6),2}\sim0.054(0.021)$ $\epsilon^{k}_{2(5),2}\sim0.251(0.159)$ $\epsilon^{k}_{3(4),2}\sim0.051(0.019)$
$L=6$ $\epsilon^{k}_{1(6),3}\sim0.002(0.0002)$ $\epsilon^{k}_{2(5),3}\sim0.113(0.055)$ $\epsilon^{k}_{3(4),3}\sim0.224(0.136)$
$L=7$ $\epsilon^{k}_{1(7),1}\sim0.186(0.106)$ $\epsilon^{k}_{2(6),1}\sim0.062(0.025)$ $\epsilon^{k}_{3(5),1}\sim0.231(0.142)$ $\epsilon^{k}_{4,1}\sim0.035(0.012)$
$L=7$ $\epsilon^{k}_{1(7),2}\sim0.103(0.048)$ $\epsilon^{k}_{2(6),2}\sim0.119(0.059)$ $\epsilon^{k}_{3(5),2}\sim0.012(0.003)$ $\epsilon^{k}_{4,2}\sim0.151(0.081)$
$L=7$ $\epsilon^{k}_{1(7),3}\sim0.022(0.006)$ $\epsilon^{k}_{2(6),3}\sim0.214(0.128)$ $\epsilon^{k}_{3(5),3}\sim0.123(0.062)$ $\epsilon^{k}_{4,3}\sim0.003(0.0004)$
$L=7$ $\epsilon^{k}_{1(7),4}\sim0.00017(10^{-6})$ $\epsilon^{k}_{2(6),4}\sim0.063(0.025)$ $\epsilon^{k}_{3(5),4}\sim0.168(0.093)$ $\epsilon^{k}_{4,4}\sim0.108(0.052)$
------- --------------------------------------------- ----------------------------------------- ----------------------------------------- ---------------------------------------
: \[tab:epsilonk\_L1234567\] Effective Kondo scale $(\epsilon^k_{{\boldsymbol{l}}({\boldsymbol{l}}^\prime),p}\sim e^{-\frac{1} {\tilde{j}_{{\boldsymbol{l}},p}}})$ at the level crossings for two $J_k=2(1.5)$ values up to L=7 using Anderson poor man scaling approach. Correspondingly, the Fig.\[fig:spectral\_temp\] shows that proper Kondo resonances appears in the QMC simulation at sites where $\epsilon^k_{{\boldsymbol{l}}({\boldsymbol{l}}^\prime),p}$ dominates.
A detailed temperature dependence of $A_{{\boldsymbol{l}}}(\omega=0) \simeq \frac{1}{\pi} \beta G_{{\boldsymbol{l}}}(\tau = \beta/2) $ as a function of magnetic field and chain length is given in Fig. \[fig:spectral\_temp\]. Upon inspection, one will see that at a given critical magnetic field corresponding to a level crossing, a peak occurs at the site where the local Kondo scale dominates.
![QMC results for spectral function at $\omega=0$ as a function of an external magnetic field for $J_k/t=2$, $J_h/t=1.8$ and for different values of inverse temperature $\beta=t/k_BT$ up to $L=7$. This figure is directly comparable with Fig.3 reported in Ref. \[\] along a cut on zero bias conductance data around 330 mK.[]{data-label="fig:spectral_temp"}](Spectralomega0_atom1234567.pdf){width="\textwidth"}
Large-N mean field for the infinite Heisenberg chain of adatoms {#LargeN_meanfield}
---------------------------------------------------------------
We consider an infinite Heisenberg chain of adatoms with periodic boundary conditions. The unit cell, ${\boldsymbol{l}}$, contains $n \in \left[ 1 \cdots N_c \right] $ conduction electrons $ \hat{c}_{{\boldsymbol{l}},n,\sigma} $ and a single spin degree of freedom. In this case, we can use the identity $J_h {\boldsymbol{S}}_{{\boldsymbol{l}}} \cdot {\boldsymbol{S}}_{{\boldsymbol{l}}+1} = -\frac{J_h}{4} \left( D^{}_{{\boldsymbol{l}},{\boldsymbol{l}}+1} D^{\dagger}_{{\boldsymbol{l}},{\boldsymbol{l}}+1} + D^{\dagger}_{{\boldsymbol{l}},{\boldsymbol{l}}+1} D^{}_{{\boldsymbol{l}},{\boldsymbol{l}}+1} \right) $ with ${\boldsymbol{S}}_l = \frac{1}{2} {\boldsymbol{d}}^{\dagger}_l {\boldsymbol{\sigma}} {\boldsymbol{d}}_{{\boldsymbol{l}}}$, $D^{}_{{\boldsymbol{l}},{\boldsymbol{l}}+1} = {\boldsymbol{d}}^{\dagger}_{{\boldsymbol{l}}} {\boldsymbol{d}}^{}_{{\boldsymbol{l}}+1} $ and constraint $ {\boldsymbol{d}}^{\dagger}_{{\boldsymbol{l}}} {\boldsymbol{d}}^{}_{{\boldsymbol{l}}} =1$ to define the large-N mean-field saddle point of Eq. (\[model\_ham\]) $$\begin{aligned}
\hat{H}_{MF} =& & \sum_{n,n',k,\sigma} \hat{c}^{\dagger}_{k,n,\sigma} T(k)_{n,n'} \hat{c}^{}_{k,n,\sigma}
-\frac{J_h\chi} {4} \sum_{{\boldsymbol{l}},\sigma} ( \hat{d}^{\dagger}_{{\boldsymbol{l}},\sigma} \hat{d}^{}_{{{\boldsymbol{l}}}+1,\sigma}+ h.c.) - \frac{1}{2}g \mu_B h^z
\sum_{l,\sigma} \sigma \hat{d}^{\dagger}_{{\boldsymbol{l}},\sigma} \hat{d}^{}_{{\boldsymbol{l}},\sigma} \nonumber \\
& & -\frac{J_kV}{4} \sum_{{{\boldsymbol{l}}},\sigma} \left( \hat{c}^{\dagger}_{{{\boldsymbol{l}}},0,\sigma} \hat{d}^{}_{{\boldsymbol{l}},0,\sigma} + h.c. \right) - \lambda \sum_{{{\boldsymbol{l}}},\sigma} \hat{d}^{\dagger}_{{{\boldsymbol{l}}},\sigma}\hat{d}^{}_{{{\boldsymbol{l}}},\sigma} \nonumber \end{aligned}$$ describing the hybridization of a band of spinons with the conduction electron. Here the mean field order parameters $ V = \langle \hat{c}^{\dagger}_{{\boldsymbol{l}},0,\sigma} \hat{d}^{}_{{\boldsymbol{l}},0,\sigma} \rangle $ and $\chi = \langle \hat{d}^{\dagger}_{{\boldsymbol{l}},\sigma} \hat{d}^{}_{{\boldsymbol{l}}+1,\sigma} \rangle $ have to be determined self-consistently and the Lagrange multiplier $\lambda$ enforces the constraint on average.
![Differential conductance (in atomic units) as a function of voltage measured in STM experiment of Ref. \[\].[]{data-label="fig:dIbydV_vs_V_STM"}](dIbydV_vs_V_STM_L1234.pdf){width="\textwidth"}
|
---
abstract: 'We consider the design of self-testers for quantum gates. A self-tester for the gates ${\boldsymbol{F}}_1,\ldots, {\boldsymbol{F}}_m$ is a classical procedure that, given any gates ${\boldsymbol{G}}_1, \ldots, {\boldsymbol{G}}_m$, decides with high probability if each ${\boldsymbol{G}}_i$ is close to ${\boldsymbol{F}}_i$. This decision has to rely only on measuring in the computational basis the effect of iterating the gates on the classical states. It turns out that instead of individual gates, we can only design procedures for families of gates. To achieve our goal we borrow some elegant ideas of the theory of program testing: we characterize the gate families by specific properties, we develop a theory of robustness for them, and show that they lead to self-testers. In particular we prove that the universal and fault-tolerant set of gates consisting of a Hadamard gate, a ${{\mathrm{c\text{-}NOT}}}$ gate, and a phase rotation gate of angle $\pi/4$ is self-testable.'
author:
- '*Wim van Dam*[^1]'
- '*Frédéric Magniez*[^2]'
- '*Michele Mosca*[^3]'
- '*Miklos Santha*[^4]'
title: |
**Self-Testing of Universal and Fault-Tolerant\
Sets of Quantum Gates**
---
Introduction
============
In the last decade quantum computing has become an extremely active research area. The initial idea that the simulation of quantum physical systems might be out of reach for classical devices goes back to Feynman[@Fey82]. He raised the possibility that computational devices based on the principles of quantum mechanics might be more powerful than classical ones. This challenge to the quantitative version of the Church-Turing thesis, which asserts that all physically realisable computational devices can be simulated with only a polynomial overhead by a probabilistic Turing machine, is the driving force behind the study of quantum computers and algorithms.
The first formal models of quantum computing, the quantum Turing machine and quantum circuits were defined by Deutsch[@Deu85; @Deu89]. Yao has shown[@Yao93] that these two models have polynomially equivalent computational power when the circuits are uniform. In a sequence of papers oracles have been exhibited [@DJ92; @BB92; @BV97; @Sim97], relative to which quantum Turing machines are more powerful than classical (probabilistic or non-deterministic) ones. These results culminated in the seminal paper of Shor[@Sho97] where he gave polynomial time quantum algorithms for the factoring and the discrete logarithm problems.
A quantum circuit operates on $n$ quantum bits (qubits), where $n$ is some integer. The actual computation takes place in the Hilbert space ${\mathbb{C}}^{\{0,1\}^n}$ whose computational basis consists of the $2^n$ orthonormal vectors ${| i \rangle}$ for $i \in {\{0,1\}}^n.$ According to the standard model, during the computation the state of the system is a unit length linear superposition of the basis states. The computational steps of the system are done by quantum gates which perform unitary operations and are local in the sense that they involve only a constant number of qubits. At the end of the computation a measurement takes place on one of the qubits. This is a probabilistic experiment whose outcome can be $0$ or $1$, and the probability of measuring the bit $b$ is the squared length of the projection of the superposition to the subspace spanned by the basis states that are compatible with the outcome. As a result of a measurement, the state of the system becomes this projected state.
The most convenient way to describe all possible operations on a quantum register is in the formalism of ‘density matrices’. In this approach, which differs from the Dirac notation, the quantum operations are described by completely positive superoperators (CPSOs) that act on matrices. These density matrices describe mixed states (that is, classical probability distributions over pure quantum states), and the CPSOs correspond exactly to all the physically allowed transformations on them. Such a model of quantum circuits with mixed states was described by Aharonov, Kitaev and Nisan[@AKN98], and we will adopt it here. The unitary quantum gates of the standard model and measurements are special CPSOs. CPSOs can be simulated by unitary quantum gates on a larger number of qubits, and in [@AKN98] it was shown that the computational powers of the two models are polynomially equivalent.
Unitary quantum gates for small number of qubits have been extensively studied. One reason is that although quantum gates for up to three qubits have already been built, constructing gates for large numbers seems to be elusive. Another reason is that universal sets of gates can be built from them, which means that they can simulate (approximately) any unitary transformation on an arbitrary number of qubits. The first universal quantum gate which operates on three qubits was identified by Deutsch[@Deu89]. After a long sequence of work on universal quantum gates [@DiV95; @Bar95; @DBE95; @Llo95; @BBC95; @Sho96; @KLZ96; @Kit97], Boykin et al.[@bmprv99] have recently shown that the set consisting of a Hadamard gate, a ${{\mathrm{c\text{-}NOT}}}$ gate, and a phase rotation gate of angle $\pi/4$ is universal. In order to form a practical basis for quantum computation, a universal set must also be able to operate in a noisy environment, and therefore has to be fault-tolerant[@Sho96; @AB97; @Kit97; @KLZ98]. The above set of three gates has the additional advantage of also being fault-tolerant.
Experimental procedures for determining the properties of quantum “black boxes” were given by Chuang and Nielsen[@CN97] and Poyatos, Cirac and Zoller[@PCZ97], however these procedures implicitly require apparatus that has already been tested and characterized.
The idea of self-testing in quantum devices is implicit in the work of Adleman, Demarrais and Huang[@ADH97]. They have developed a procedure by which a quantum Turing machine is able to estimate its internal angle by its own means under the hypothesis that the machine is unitary. In the context of quantum cryptography Mayers and Yao[@MY98] have designed tests for deciding if a photon source is perfect. These tests guarantee that if source passes them then it is adequate for the security of the Bennett-Brassard[@BB84] quantum key distribution protocol.
In this paper we develop the theory of self-testing of quantum gates by classical procedures. Given a CPSO ${\boldsymbol{G}}$ for $n$ qubits, and a family $\mathcal{F}$ of unitary CPSOs, we would like to decide if ${\boldsymbol{G}}$ belongs to $\mathcal{F}$. Intuitively, a self-tester is a procedure that answers the question “${\boldsymbol{G}}\in \mathcal{F}$ ?” by interacting with the CPSO ${\boldsymbol{G}}$ in a purely classical way. More precisely, it will be a probabilistic algorithm that is able to access ${\boldsymbol{G}}$ as a black box in the following sense: it can prepare the classical states $w\in\{0,1\}^n$, iterate ${\boldsymbol{G}}$ on these states, and afterwards, measure in the computational basis. The access must be seen as a whole, performed by a specific, experimental oracle for ${\boldsymbol{G}}$: once the basis state ${w}$ and the number of iterations $k$ have been specified, the program in one step gets back one of the possible probabilistic outcomes of measuring the state of the system after ${\boldsymbol{G}}$ is iterated $k$-times on ${w}$. The intermediate quantum states of this process cannot be used by the program, which cannot perform any other quantum operations either. For $0
\leq \delta_1 \leq
\delta_2$, such an algorithm will be a $(\delta_1,
\delta_2)$-tester for $\mathcal{F}$ if for every CPSO ${\boldsymbol{G}}$, whenever the distance of ${\boldsymbol{G}}$ and $\mathcal{F}$ is at most $\delta_1$ (in some norm), it accepts with high probability, and whenever the same distance is greater than $\delta_2$, it rejects with high probability, where the probability is taken over the measurements performed by the oracle and by the internal coin tosses of the algorithm. Finally we will say that $\mathcal{F}$ is [*testable*]{} if for every $\delta_2 > 0$, there exists $0 <
\delta_1 \leq \delta_2$ such that there exists a $(\delta_1,
\delta_2)$-tester for $\mathcal{F}$. These definitions can be extended to several classes of CPSOs.
The study of self-testing programs is a well-established research area which was initiated by the work of Blum, Luby and Rubinfeld[@BLR93], Rubinfeld[@Rub90], Lipton[@Lip91] and Gemmel [@GLR91]. The purpose of a self-tester for a function family is to detect by simple means if a program which is accessible as an oracle computes a function from the given family. This clearly inspired the definition of our self-testers which have the particularity that they should test quantum objects that they can access only in some particular way. The analogy with self-testing does not stop with the definition. One of the main tools in self-testing of function families is the characterization of these families by robust properties. Informally, a property is robust if whenever a function satisfies the property approximately, then it is close to a function which satisfies it exactly. The concept of robustness was introduced and its implication for self-testing was first studied by Rubinfeld and Sudan[@RS96] and by Rubinfeld[@Rub94]. It will play a crucial role in our case.
We note in the Preliminaries that for any real ${\varphi}$ the states ${| 1 \rangle}$ and $e^{i{\varphi}}{| 1 \rangle}$ are experimentally indistinguishable. This implies that if we start by only distinguishing the classical states $0$ and $1$ then there are families of CPSOs which are indistinguishable as well. For example, let ${\boldsymbol{H}}$ be the well-known Hadamard gate, and let ${\boldsymbol{H}}_{{\varphi}}$ be the same gate expressed in the basis $({| 0 \rangle}, e^{i {\varphi}}{| 1 \rangle})$, for ${\varphi}\in [0,2\pi)$. Any experiment that starts in state $0$ or $1$ and uses only ${\boldsymbol{H}}$ will produce outcomes $0$ and $1$ with the same probabilities as the same experiment with ${\boldsymbol{H_{{\varphi}}}}$. Thus no experiment that uses this quantum gate alone can distinguish it from all the other Hadamard gates. Indeed, as stated later in Fact \[conjugate\_basis\], a family $\mathcal{F}$ containing ${\boldsymbol{H}}$ can only be tested if the entire Hadamard family $\mathcal{H}=\{{\boldsymbol{H}}_{\varphi}:{\varphi}\in
[0,2\pi)\}$ is included in $\mathcal{F}$.
The main result of this paper is **[Theorem [\[maintheo\]]{}]{}** which states that for several sets of unitary CPSOs, in particular, the Hadamard gates family, Hadamard gates together with ${{\mathrm{c\text{-}NOT}}}$ gates, and Hadamard gates with ${{\mathrm{c\text{-}NOT}}}$ and phase rotation gates of angle $\pm\pi/4$, are testable. This last family is of particular importance since every triplet in the family forms a universal and fault-tolerant set of gates for quantum computation[@bmprv99].
For the proof we will define the notion of experimental equations which are functional equations for CPSOs corresponding to the properties of the quantum gate that a self-tester can approximately test. These tests are done via the interaction with the experimental oracle. The proof itself contains three parts. In **Theorems \[general\]**, **\[couple\]**, and **\[cnot\]** we will exhibit experimental equations for the families of unitary CPSOs we want to characterize. In **[Theorem [\[robustness\]]{}]{}** we will show that actually all experimental equations are robust; in fact, the distance of a CPSO from the target family is polynomially related to the error tolerated in the experimental equations. Finally **[Theorem [\[robtotester\]]{}]{}** gives self-testers for CPSO families which are characterized by a finite set of robust experimental equations.
In some cases, we are able to calculate explicitly the polynomial bound in the robustness of experimental equations. Such a result will be illustrated in **[Theorem [\[hadamardrob\]]{}]{}** for the equations characterizing the Hadamard family $\mathcal{H}$.
Technically, these results will be based on the representation of one-qubit states and CPSOs in ${\mathbb{R}}^3$, where they are respectively vectors in the unit ball of ${\mathbb{R}}^3$, and particular affine transformations. This correspondence is known as the Bloch Ball representation.
Preliminaries
=============
The quantum state
-----------------
A [*pure state*]{} in a quantum physical system is described by a unit vector in a Hilbert space. In the [*Dirac*]{} notation it is denoted by ${| \psi \rangle}$. In particular a [*qubit*]{} (a quantum two-state system) is an element of the Hilbert space ${{\mathbb{C}}^{\{0,1\}}}$. The orthonormal basis containing ${| 0 \rangle}$ and ${| 1 \rangle}$ is called the [*computational basis*]{} of ${{\mathbb{C}}^{\{0,1\}}}$. Therefore a pure state ${| \psi \rangle}\in{{\mathbb{C}}^{\{0,1\}}}$ is a [*superposition*]{} of the computational basis states, that is, ${| \psi \rangle}=c_{0}{| 0 \rangle}+c_{1}{| 1 \rangle}$, with ${| c_{0} |}^{2}+{| c_{1} |}^{2}=1$. A physical system which deals with $n$ qubits is described mathematically by the $2^n$-dimensional Hilbert space which is by definition ${{\mathbb{C}}^{\{0,1\}}}\otimes\cdots\otimes{{\mathbb{C}}^{\{0,1\}}}$, that is, the $n^{\mbox{\footnotesize
th}}$ tensor power of ${{\mathbb{C}}^{\{0,1\}}}$. Let $N=2^n$. The computational basis of this space consists of the $N$ orthonormal states ${| i \rangle}$ for $0\leq i<N$. If $i$ is in binary notation $i_1 i_2\ldots i_n$, then ${| i_1\ldots i_n \rangle}={| i_1 \rangle}\ldots{| i_n \rangle}$, where this is a short notation for ${| i_1 \rangle}\otimes\cdots\otimes{| i_n \rangle}$. All vectors and matrices will be expressed in the computational basis. The transposed complex conjugate ${| \psi \rangle}^\dag$ of ${| \psi \rangle}$ is denoted by ${\langle \psi |}$. The inner product between ${| \psi \rangle}$ and ${| \psi' \rangle}$ is denoted by ${\langle \psi | \psi' \rangle}$, and their outer product by ${| \psi \rangle\langle \psi' |}$.
Quantum systems can also be in more general states than what can be described by pure states. The most general states are *mixed states*, described by a probability distribution over pure states. Such a mixture can be denoted by $\{(p_{k},{| \psi_{k} \rangle}) : k\in{\mathbb{N}}\}$, where the system is in the pure state ${| \psi_{k} \rangle}$ with probability $p_{k}$.
Different mixtures (even different pure states ${| \psi \rangle}$) can represent the same physical system. This notational redundancy can be avoided if we use the formalism of the density matrices. A *density matrix* that represents an $n$-qubit state is an $N\times N$ Hermitian semi-positive matrix with trace $1$. The pure state ${| \psi \rangle}$ in this representation is described by the density matrix $\psi={| \psi \rangle\langle \psi |}$, and a mixture $\{(p_{k},{| \psi_{k} \rangle})
: \ k\in{\mathbb{N}}\}$ by the density matrix $\psi=\sum_{k\in{\mathbb{N}}}p_{k}{| \psi_{k} \rangle\langle \psi_{k} |}$. For example, the pure states $e^{i\gamma}{| \psi \rangle}$, for $\gamma\in[0,2\pi)$, or the mixtures $\{({\mbox{$\frac{1}{2}$}},{| 0 \rangle}),({\mbox{$\frac{1}{2}$}},{| 1 \rangle})\}$ and $\{({\mbox{$\frac{1}{2}$}},\frac{{| 0 \rangle}+{| 1 \rangle}}{\sqrt{2}}),
({\mbox{$\frac{1}{2}$}},\frac{{| 0 \rangle}-{| 1 \rangle}}{\sqrt{2}})\}$ have respectively the same density matrix.
Since a density matrix is Hermitian semi-positive, its eigenvectors are orthogonal and its eigenvalues are non-negative. Because the density matrix has trace $1$, its eigenvalues sum to $1$. Therefore a density matrix represents the mixture of its orthonormal eigenvectors, where the probabilities are the respective eigenvalues. Note that diagonal density matrices correspond to a mixture over pure states ${| i \rangle}$, for $0\leq i<N$. Density matrices that represent pure states have a simple algebraic characterization: $\rho$ is a pure state if and only if it has two eigenvalues, $0$ with multiplicity $N-1$ and $1$ with multiplicity $1$, equivalently $\rho$ is a pure state exactly when $\rho^2=\rho$.
A $2\times 2$ Hermitian matrix of unit trace is semi-positive if and only if its determinant is between $0$ and $1/4$. Therefore in the case of one qubit, any density matrix $\rho$ can be written as $\rho = p{| 0 \rangle\langle 0 |}+(1-p){| 1 \rangle\langle 1 |}
+\alpha{| 1 \rangle\langle 0 |}+{\alpha^*}{| 0 \rangle\langle 1 |}$, where $p\in[0,1]$, and $\alpha$ is a complex number such that ${| \alpha |}^2\leq
p(1-p)$. This density matrix will be denoted by $\rho(p,\alpha)$. Remark that $\rho(p,\alpha)$ is a pure state exactly when ${| \alpha |}^2= p(1-p)$, that is, its determinant is $0$.
Superoperators
--------------
The evolution of physical systems is described by specific transformations on density matrices, that is, on operators. A [*superoperator*]{} for $n$ qubits is a linear transformation on ${\mathbb{C}}^{N\times N}$. A [*positive*]{} superoperator (PSO) is a superoperator that sends density matrices to density matrices. A [*completely positive*]{} superoperator (CPSO) ${\boldsymbol{G}}$ is a PSO such that for all positive integers $M$, ${\boldsymbol{G}}\otimes{\boldsymbol{I}}_{M}$ is also a PSO, where ${\boldsymbol{I}}_{M}$ is the identity on ${\mathbb{C}}^{M\times M}$. CPSOs are exactly the physically allowed transformations on density matrices. An example of a PSO for one qubit that is not a CPSO is the transpose superoperator ${\boldsymbol{T}}$ defined by ${\boldsymbol{T}}({| i \rangle\langle j |})={| j \rangle\langle i |}$, for $0\leq i,j\leq 1$.
Quantum computation is based on the possibility of constructing some particular CPSOs, [*unitary*]{} superoperators, which preserve the set of pure states. These operators are characterized by transformations from ${\mathrm{U}}(N)$, the set of $N\times N$ unitary matrices. For any $A\in {\mathrm{U}}(N)$, we define a CPSO which maps a density matrix $\rho$ into $A\rho A^\dag$. When the underlying unitary transformation $A$ is clear from the context, by somewhat abusing the notation, we will denote this CPSO simply by ${\boldsymbol{A}}$. If ${| \psi' \rangle}$ denotes $A{| \psi \rangle}$, then the unitary superoperator ${\boldsymbol{A}}$ maps the pure state $\psi$ to the pure state $\psi'$. As was the case in the Dirac representation of states, there is the same phase redundancy in the set of unitary transformations ${\mathrm{U}}(N)$. If $A\in{\mathrm{U}}(N)$, then for all $\gamma\in
[0,2\pi)$, the transformations $e^{i\gamma} A$ are different, however the corresponding superoperators are identical. We will therefore focus on ${\mathrm{U}}(N)/{\mathrm{U}}(1)$.
Measurements
------------
Measurements form another important class of (non-unitary) CPSOs. They describe physical transformations corresponding to the observation of the system. We will define now formally one of the simplest classes of measurements which correspond to the projections to elements of the computational basis.
A [*Von Neumann measurement in the computational basis*]{} of $n$ qubits is the $n$-qubit CPSO ${\boldsymbol{M}}$ that, for every density matrix $\rho$, satisfies ${\boldsymbol{M}}(\rho)_{i,i}=\rho_{i,i}$ and ${\boldsymbol{M}}(\rho)_{i,j}=0$, for $i\neq j$.
In the case of one qubit, the Von Neumann measurement in the computational basis maps the density matrix $\rho(p,\alpha)$ into $\rho(p,0)$. We will say that $p={\langle 0 |}\rho{| 0 \rangle}$ is the [*probability of measuring*]{} ${| 0 \rangle\langle 0 |}$, and we will denote it by ${\mathrm{Pr}^{0}[{\rho}]}$.
In general, a [*Von Neumann measurement*]{} of $n$ qubits in any basis can be viewed as the Von Neumann measurement in the computational basis preceded by some unitary superoperator.
The Bloch Ball representation
-----------------------------
Specific for the one-qubit case, there is an isomorphism between the group ${\mathrm{U}}(2)/{\mathrm{U}}(1)$ and the special rotation group ${\mathrm{SO}}(3)$, the set of $3\times 3$ orthogonal matrices with determinant $1$. This allows us to represent one-qubit states as vectors in the unit ball of ${\mathbb{R}}^3$, and unitary superoperators as rotations on ${\mathbb{R}}^3$. We will now describe exactly this correspondence.
The [*Bloch Ball*]{} ${\mathcal{B}}$ (respectively [*Bloch Sphere*]{} ${\mathcal{S}}$) is the unit ball (respectively unit sphere) of the Euclidean affine space ${\mathbb{R}}^{3}$. Any point ${\overline{u}}\in{\mathbb{R}}^{3}$ determines a vector with the same coordinates which we will also denote by ${\overline{u}}$. The inner product of ${\overline{u}}$ and ${\overline{v}}$ will be denoted by $({\overline{u}},{\overline{v}})$, and their Euclidean norm by ${|\! | {\overline{u}} |\! |}$.
Each point ${\overline{u}}\in{\mathbb{R}}^{3}$ can be also characterized by its norm $r\geq 0$, its latitude $\theta\in[0,\pi]$, and its longitude ${\varphi}\in[0,2\pi)$. The [*latitude*]{} is the angle between the $z$-axis and the vector ${\overline{u}}$, and the [*longitude*]{} is the angle between the $x$-axis and the orthogonal projection of ${\overline{u}}$ in the plane defined by $z=0$. If ${\overline{u}}=(x,y,z)$, then these parameters satisfy $x=r\sin\theta\cos{\varphi}$, $y=r\sin\theta\sin{\varphi}$ and $z=r\cos\theta$.
For every density matrix $\rho$ for one qubit there exists a unique point ${\overline{\rho}}=(x,y,z) \in{\mathcal{B}}$ such that $$\begin{aligned}
\rho
&=&\frac{1}{2}{\left({
\begin{array}{cc}
1+z & x-i y \\
x+i y & 1-z \\
\end{array}
}\right)}.\end{aligned}$$ This mapping is a bijection that also obeys $$\begin{aligned}
{\overline{\rho(p,\alpha)}} & = & (2{\mathrm{Re}}(\alpha),2{\mathrm{Im}}(\alpha),2p-1).\end{aligned}$$
In this formalism, the pure states are nicely characterized in ${\mathcal{B}}$ by their norm.
A density matrix $\rho$ represents a pure state if and only if ${\overline{\rho}}\in{\mathcal{S}}$, that is, ${|\! | {\overline{\rho}} |\! |}=1$.
Also, if $\theta\in[0,\pi]$ and ${\varphi}\in[0,2\pi)$ are respectively the latitude and the longitude of ${\overline{\psi}}\in{\mathcal{S}}$, then the corresponding density matrix represents a pure state and satisfies ${| \psi \rangle} =
\cos(\theta/2){| 0 \rangle}+\sin(\theta/2)e^{i{\varphi}}{| 1 \rangle}$. Observe that the pure states ${| \psi \rangle}$ and ${| \psi^{\perp} \rangle}$ are orthogonal if and only if ${\overline{\psi}}=-{\overline{\psi^\perp}}$. We will use the following notation for the six pure states along the $x$, $y$ and $z$ axes: ${| \zeta_x^\pm \rangle} =
{\frac{1}{\sqrt{2}}}({| 0 \rangle}\pm{| 1 \rangle})$, ${| \zeta_y^\pm \rangle} =
{\frac{1}{\sqrt{2}}}({| 0 \rangle}\pm i {| 1 \rangle})$, ${| \zeta_z^+ \rangle} = {| 0 \rangle}$, and ${| \zeta_z^- \rangle} = {| 1 \rangle}$, with the respective coordinates $(\pm
1,0,0)$, $(0,\pm 1,0)$ and $(0,0,\pm 1)$ in ${\mathbb{R}}^3$.
For each CPSO ${\boldsymbol{G}}$, there exists a unique affine transformation ${{\boldsymbol{{\overline{G}}}}}$ over ${\mathbb{R}}^{3}$, which maps the ball ${\mathcal{B}}$ into ${\mathcal{B}}$ and is such that, for all density matrices $\rho$, ${{\boldsymbol{{\overline{G}}}}}({\overline{\rho}})={\overline{{\boldsymbol{G}}(\rho)}}$. Unitary superoperators have a nice characterization in ${\mathcal{B}}$.
The map between ${\mathrm{U}}(2)/{\mathrm{U}}(1)$ and ${\mathrm{SO}}(3)$, which sends $A$ to ${{\boldsymbol{{\overline{A}}}}}$, is an isomorphism.
For $\alpha\in(-\pi,\pi]$, $\theta\in[0,\frac{\pi}{2}]$, and ${\varphi}\in[0,2\pi)$, we will define the unitary transformation $R_{\alpha,\theta,{\varphi}}$ over ${\mathbb{C}}^{2}$. If ${| \psi \rangle}=\cos(\theta/2){| 0 \rangle} +e^{i{\varphi}}\sin(\theta/2){| 1 \rangle}$ and ${| \psi^\perp \rangle}=\sin(\theta/2){| 0 \rangle}
-e^{i{\varphi}}\cos(\theta/2){| 1 \rangle}$ then by definition $R_{\alpha,\theta,{\varphi}}{| \psi \rangle}={| \psi \rangle}$ and $R_{\alpha,\theta,{\varphi}}{| \psi^\perp \rangle}=e^{i\alpha}{| \psi^\perp \rangle}$. If ${\boldsymbol{A}}$ is a unitary superoperator then we have ${\boldsymbol{A}}={\boldsymbol{R}}_{\alpha,\theta,{\varphi}}$ for some $\alpha$, $\theta$, and ${\varphi}$. In ${\mathbb{R}}^3$ the transformation ${{\boldsymbol{{\overline{R}}}}}_{\alpha,\theta,{\varphi}}$ is the rotation of angle $\alpha$ whose axis cuts the sphere ${\mathcal{S}}$ in the points ${\overline{\psi}}$ and ${\overline{\psi^\perp}}$. Note that for $\theta=0$ the CPSO ${\boldsymbol{R}}_{\alpha,0,{\varphi}}$ does not depend on ${\varphi}$. We will denote this phase rotation by ${\boldsymbol{R}}_{\alpha}$.
The affine transformation in ${\mathcal{B}}$ which corresponds to the Von Neumann measurement in the computational basis is the orthogonal projection to the $z$-axis. Therefore it maps ${\overline{\rho}}=(x,y,z)$ into $(0,0,z)$, the point which corresponds to the density matrix $\frac{1+z}{2}{| 0 \rangle\langle 0 |}+\frac{1-z}{2}{| 1 \rangle\langle 1 |}$. Thus ${\mathrm{Pr}^{0}[{\rho}]}=\frac{1+z}{2}$.
Norm and distance
-----------------
Let $N=2^n$. We will consider the [*trace norm*]{} on ${\mathbb{C}}^{N\times
N}$ which is defined as follows: for all $V\in{\mathbb{C}}^{N\times
N}$, ${{|\! | V |\! |}_{1}}={\mathrm{Tr}}\sqrt{V^{\dag}V}$. This norm has several advantages when we consider the difference of density matrices. Given a Von Neumann measurement, a density matrix induces a probability distribution over the basis of the measurement. The trace norm of the difference of two density matrices is the maximal variation distance between the two induced probability distributions, over all Von Neumann measurements. It also satisfies the following properties.
For all density matrices $\rho(p,\alpha)$ and $\rho(q,\beta)$ for one qubit we have: $$\begin{array}{rcccl}
{{|\! | \rho(p,\alpha)-\rho(q,\beta) |\! |}_{1}}&=&
{|\! | {\overline{\rho(p,\alpha)}}-{\overline{\rho(q,\beta)}} |\! |}
&=&2\sqrt{(p-q)^2+{| \alpha-\beta |}^2}.
\end{array}$$
For all $V \in{\mathbb{C}}^{N \times N}$ and $W\in{\mathbb{C}}^{M\times M}$ we have ${{|\! | V{\otimes}W |\! |}_{1}}={{|\! | V |\! |}_{1}}{{|\! | W |\! |}_{1}}$ and ${| {\mathrm{Tr}}(V) |}\leq{{|\! | V |\! |}_{1}}$. For density matrices $\rho$ it holds that ${{|\! | \rho |\! |}_{1}}=1$.
For $n$-qubit superoperators, the superoperator norm associated to the trace norm is defined as $$\begin{aligned}
{{|\! | {\boldsymbol{G}} |\! |}_{\infty}}& = &
\sup\{
{{|\! | {\boldsymbol{G}}(V) |\! |}_{1}} : {{|\! | V |\! |}_{1}}=1\}.\end{aligned}$$ This norm is always $1$ when ${\boldsymbol{G}}$ is a CPSO. The norm ${{|\! | \, |\! |}_{\infty}}$ can be easily generalized for $k$-tuples of superoperators by ${{|\! | ({\boldsymbol{G}}_1,\ldots,{\boldsymbol{G}}_k) |\! |}_{\infty}}=
\max({{|\! | {\boldsymbol{G}}_1 |\! |}_{\infty}},\ldots,{{|\! | {\boldsymbol{G}}_k |\! |}_{\infty}}).$ We will denote by ${{\mathrm{dist}}_{\infty}}$ the natural induced distance by the norm ${{|\! | \, |\! |}_{\infty}}$.
Properties of CPSOs
===================
Here we will establish the properties of CPSOs that we will need for the characterization of our CPSO families. In this extended abstract we will omit the proof of Lemma \[lemma1\], and the proof of Lemma \[lemma2\] will be in [Appendix [\[APL2\]]{}]{}.
\[lemma1\] Let ${\boldsymbol{G}}$ be a CPSO for one qubit, and let $\rho$ and $\tau$ be density matrices for one qubit.
1. ${{|\! | {\boldsymbol{G}}(\rho)-{\boldsymbol{G}}(\tau) |\! |}_{1}}\leq{{|\! | \rho-\tau |\! |}_{1}}$.
2. If ${\boldsymbol{G}}$ is not constant and ${\boldsymbol{G}}(\rho)$ is a pure state then $\rho$ is a pure state.
An affine transformation of ${\mathbb{R}}^3$ is uniquely defined by the images of four non-coplanar points. Surprisingly, if the transformation is a CPSO for one qubit, the images of three points are sometimes sufficient. The following will make this precise more generally for $n$ qubits.
\[lemma2\] Let $n\geq 1$ be an integer, and let $\rho_1$, $\rho_2$, and $\rho_3$ be three distinct one-qubit density matrices representing pure states, such that the plane in ${\mathbb{R}}^3$ containing the points ${\overline{\rho_1}},{\overline{\rho_2}},{\overline{\rho_3}}$ goes through the center of ${\mathcal{B}}$. If ${\boldsymbol{G}}$ is a CPSO for $n$ qubits which acts as the identity on the set $\{\rho_1,\rho_2,\rho_3\}^{\otimes n}$, then ${\boldsymbol{G}}$ is the identity mapping.
We also use the property that for CPSOs unitarity and invertibility are equivalent (see e.g. [@Pre98 Ch. 3, Sec. 8]).
\[lemma1c\] Let ${\boldsymbol{G}}$ be a CPSO for $n$ qubits. If there exists a CPSO ${\boldsymbol{H}}$ for $n$ qubits such that ${\boldsymbol{H}}\circ{\boldsymbol{G}}$ is the identity mapping, then ${\boldsymbol{G}}$ is a unitary superoperator.
Characterization
================
One-Qubit CPSO Families
-----------------------
In this section, every CPSO will be for one qubit. First we define the notion of experimental equations, and then we show that several important CPSO families are characterizable by them.
An [*experimental equation*]{} in one variable is a CPSO equation of the form $$\begin{aligned}
\label{expCPSOeq}
{\mathrm{Pr}^{0}[{{\boldsymbol{G}}^{k}({| b \rangle\langle b |})}]} & = & r,\end{aligned}$$ where $k$ is a non-negative integer, $b\in\{0,1\}$, and $0\leq
r\leq 1$. We will call the left-hand side of the equation the [*probability term*]{}, and the right-hand side the [*constant term*]{}. The [*size*]{} of this equation is $k$. A CPSO ${\boldsymbol{G}}$ will “almost” satisfy the equations if, for example, it is the result of adding small systematic and random errors (independent of time) to a CPSO that does. For ${\varepsilon}\geq 0$, the CPSO ${\boldsymbol{G}}$ [*${\varepsilon}$-satisfies*]{} [(\[expCPSOeq\])]{} if ${| {\mathrm{Pr}^{0}[{{\boldsymbol{G}}^{k} ({| b \rangle\langle b |})}]}-r |}\leq{\varepsilon},$ and when ${\varepsilon}=0$ we will just say that ${\boldsymbol{G}}$ [*satisfies*]{} [(\[expCPSOeq\])]{}. Let $(E)$ be a finite set of experimental equations. If ${\boldsymbol{G}}$ ${\varepsilon}$-satisfies all equations in $(E)$ we say that ${\boldsymbol{G}}$ ${\varepsilon}$-satisfies $(E)$. If some ${\boldsymbol{G}}$ satisfies $(E)$ then $(E)$ is [*satisfiable*]{}. The set $\{{\boldsymbol{G}} :
{\boldsymbol{G}}\mbox{ satisfies $(E)$}\}$ will be denoted by $\mathcal{F}_{(E)}$. A family $\mathcal{F}$ of CPSOs is [*characterizable*]{} if it is $\mathcal{F}_{(E)}$ for some finite set $(E)$ of experimental equations. In this case we say that $(E)$ [*characterizes*]{} $\mathcal{F}$.
All these definitions generalize naturally for $m$-tuples of CPSOs for $m\geq 2$. In what follows we will need only the case $m=2$. An [*experimental equation*]{} in two CPSO variables is an equation of the form $$\begin{aligned}
\label{expCPSOeq2}
{\mathrm{Pr}^{0}[{{\boldsymbol{F}}^{k_{1}}\circ{\boldsymbol{G}}^{l_1}\circ\cdots\circ
{\boldsymbol{F}}^{k_{t}}\circ{\boldsymbol{G}}^{l_{t}}({| b \rangle\langle b |})}]} & = & r,\end{aligned}$$ where $k_1,\ldots,k_t,l_1,\ldots,l_t$ are non-negative integers, $b\in\{0,1\}$, and $0\leq r\leq 1$.
We discuss now the existence of finite sets of experimental equations in one variable that characterize unitary superoperators, that is, the operators ${\boldsymbol{R}}_{\alpha,\theta,{\varphi}}$, for $\alpha\in(-\pi,\pi]$, $\theta\in[0,\pi/2]$, and ${\varphi}\in[0,2\pi)$. First observe that due to the restrictions of experimental equations, there are unitary superoperators that they cannot distinguish.
\[conjugate\_basis\] Let $\alpha\in[0,\pi]$, $\theta\in[0,\pi/2]$, and ${\varphi}_1,{\varphi}_2\in[0,2\pi)$ such that ${\varphi}_1\neq{\varphi}_2$. Let $(E)$ be a finite set of experimental equations in $m$ variables. If $({\boldsymbol{R}}_{\alpha,\theta,{\varphi}_1},
{\boldsymbol{G}}_2,\ldots,{\boldsymbol{G}}_m)$ satisfies $(E)$ then there exist ${\boldsymbol{G}}'_2,\ldots,{\boldsymbol{G}}'_m$ and ${\boldsymbol{G}}''_2,\ldots,{\boldsymbol{G}}''_m$ such that $({\boldsymbol{R}}_{-\alpha,\theta,{\varphi}_1},
{\boldsymbol{G}}'_2,\ldots,{\boldsymbol{G}}'_m)$ and $({\boldsymbol{R}}_{\alpha,\theta,{\varphi}_2},
{\boldsymbol{G}}''_2,\ldots, {\boldsymbol{G}}''_m)$ both satisfy $(E)$.
In the Bloch Ball formalism this corresponds to the following degrees of freedom in the choice of the orthonormal basis of ${\mathbb{R}}^3$. Since experimental equations contain exactly the states ${| 0 \rangle\langle 0 |}$ and ${| 1 \rangle\langle 1 |}$ there is no freedom in the choice of the $z$-axis, but there is complete freedom in the choice of the $x$ and $y$ axes. The indistinguishability of the latitude ${\varphi}$ corresponds to the freedom of choosing the oriented $x$-axis, and the indistinguishability of the sign of $\alpha$ corresponds to the freedom of choosing the orientation of the $y$-axis.
We introduce the following notations. Let $\mathcal{R}_{\alpha,\theta}$ denote the superoperator family $\{{\boldsymbol{R}}_{\pm\alpha,\theta,{\varphi}} : {\varphi}\in[0,2\pi)\}$. For ${\varphi}\in[0,2\pi)$, let the ${{\mathrm{NOT}}}_{\varphi}$ transformation be defined by ${{\mathrm{NOT}}}_{\varphi}{| 0 \rangle} = e^{i{\varphi}}{| 1 \rangle}$ and ${{\mathrm{NOT}}}_{\varphi}(e^{i{\varphi}}{| 1 \rangle})={| 0 \rangle}$, and recall that the Hadamard transformation $H_{\varphi}$ obeys $H_{{\varphi}}{| 0 \rangle}=({| 0 \rangle}+e^{i{\varphi}}{| 1 \rangle})/\sqrt{2}$ and $H_{{\varphi}}(e^{i{\varphi}}{| 1 \rangle})=({| 0 \rangle}-e^{i{\varphi}}{| 1 \rangle})/\sqrt{2}$. Observe that ${\boldsymbol{H}}_{\varphi}={\boldsymbol{R}}_{\pi,\pi/4,{\varphi}}$ and ${\boldsymbol{{{\mathrm{NOT}}}}}_{\varphi}={\boldsymbol{R}}_{\pi,\pi/2,{\varphi}}$, for ${\varphi}\in[0,2\pi)$. Finally let $\mathcal{H}=
\{{\boldsymbol{H}}_{\varphi}: {\varphi}\in[0,2\pi)\}$, and $\mathcal{N}=\{{\boldsymbol{{{\mathrm{NOT}}}}}_{\varphi}: {\varphi}\in[0,2\pi)\}$.
Since the sign of $\alpha$ cannot be determined, we will assume that $\alpha$ is in the interval $[0,\pi]$. We will also consider only unitary superoperators such that $\alpha/\pi$ is rational. This is a reasonable choice since these superoperators form a dense subset of all unitary superoperators. For such a unitary superoperator, let $n_\alpha$ be the smallest positive integer $n$ for which $n\alpha=0 \mod{2\pi}$. Then either $n_\alpha=1$, or $n_\alpha\geq 2$ and there exists $t\geq 1$ which is coprime with $n_\alpha$ such that $\alpha=(t/n_\alpha)2\pi$. Observe that the case $n_\alpha=1$ corresponds to the identity superoperator.
Our first theorem shows that almost all families $\mathcal{R}_{\alpha,\theta}$ are characterizable by some finite set of experimental equations. In particular $\mathcal{H}$ is characterizable.
\[general\] Let $(\alpha,\theta)\in(0,\pi]\times(0,\pi/2]\backslash\{(\pi,\pi/2)\}$ be such that $\alpha/\pi$ is rational. Let $z_k(\alpha,\theta)=\cos^{2}\theta+\sin^{2}\theta\cos(k\alpha)$. Then the following experimental equations characterize $\mathcal{R}_{\alpha,\theta}$: $$\label{generaltest1}
\begin{array}{rcll}
{\mathrm{Pr}^{0}[{{\boldsymbol{G}}^{n_\alpha}({| 1 \rangle\langle 1 |})}]}= 0
& \textrm{ and } &
{\mathrm{Pr}^{0}[{{\boldsymbol{G}}^{k}({| 0 \rangle\langle 0 |})}]} = {\mbox{$\frac{1}{2}$}}+{\mbox{$\frac{1}{2}$}}z_{k}(\alpha,\theta),
& k\in{\{1,2,\ldots,n_\alpha\}}.
\end{array}$$
[[**Proof:** ]{}]{}First observe that every CPSO in $\mathcal{R}_{\alpha,\theta}$ satisfies the equations of the theorem since the $z$-coordinate of ${\overline{{\boldsymbol{R}}_{\alpha,\theta,{\varphi}}^k({| 0 \rangle\langle 0 |})}}$ is $z_k(\alpha,\theta)$ for every ${\varphi}\in[0,2\pi)$. Let ${\boldsymbol{G}}$ be a CPSO which satisfies these equations. We will prove that ${\boldsymbol{G}}$ is a unitary superoperator. Then, [Fact [\[unicityofangles\]]{}]{} implies that ${\boldsymbol{G}}\in\mathcal{R}_{\alpha,\theta}$.
Since $z_1(\alpha,\theta)\neq \pm 1$, we know ${\boldsymbol{G}}({| 0 \rangle\langle 0 |})\not\in\{{| 0 \rangle\langle 0 |},{| 1 \rangle\langle 1 |}\}$. Observing that ${\boldsymbol{G}}^{n_\alpha}({| 0 \rangle\langle 0 |})={| 0 \rangle\langle 0 |}$, [Lemma [\[lemma1\]]{}]{}(b) implies that ${\boldsymbol{G}}({| 0 \rangle\langle 0 |})$ is a pure state. Thus ${| 0 \rangle\langle 0 |}$, ${| 1 \rangle\langle 1 |}$, and ${\boldsymbol{G}}({| 0 \rangle\langle 0 |})$ are distinct pure states, and since ${\boldsymbol{G}}^{n_\alpha}$ acts as the identity on them, by [Lemma [\[lemma2\]]{}]{} it is the identity mapping. Hence by [Lemma [\[lemma1c\]]{}]{} ${\boldsymbol{G}}$ is a unitary superoperator.
------------------------------------------------------------------------
\[unicityofangles\] Let $\alpha\in(0,\pi]$, $\theta\in(0,\pi/2]$, $\alpha'\in(-\pi,\pi]$, $\theta'\in(0,\pi/2]$ be such that $\alpha/\pi$ is rational. If $z_k(\alpha,\theta)=
z_k(\alpha',\theta')$, for $k\in{\{1,2,\ldots,n_\alpha\}}$, then ${| \alpha' |}=\alpha$ and $\theta'=\theta$.
The remaining families $\mathcal{R}_{\alpha,\theta}$ for which $\alpha/\pi$ is rational are $\{{\boldsymbol{R}}_{-\alpha},{\boldsymbol{R}}_\alpha\}$, for $\alpha\in[0,\pi]$, and $\mathcal{N}$. Let us recall that ${\boldsymbol{M}}$ is the CPSO which represents the Von Neumann measurement in the computational basis. Since ${\boldsymbol{M}}$ satisfies exactly the same equations as ${\boldsymbol{R}}_{\pm\alpha}$, and ${\boldsymbol{{{\mathrm{NOT}}}}}_0\circ{\boldsymbol{M}}$ satisfies exactly the same equations as ${\boldsymbol{{{\mathrm{NOT}}}}}_{\varphi}$, for any ${\varphi}\in[0,2\pi)$, these families are not characterizable by experimental equations in one variable. Nevertheless it turns out that together with the family $\mathcal{H}$ they become characterizable. This is stated in the following theorem whose proof is omitted.
\[couple\] The family $\{({\boldsymbol{H}}_{\varphi},{\boldsymbol{{{\mathrm{NOT}}}}}_{{\varphi}}) : {\varphi}\in[0,2\pi)\}
\subset \mathcal{H}\times\mathcal{N}$ is characterized by the experimental equations in two variables $({\boldsymbol{F}},{\boldsymbol{G}})$: $$\left\{{\begin{array}{lll}
{\mathrm{Pr}^{0}[{{\boldsymbol{F}}({| 0 \rangle\langle 0 |})}]}={\mbox{$\frac{1}{2}$}},\!\!&
{\mathrm{Pr}^{0}[{{\boldsymbol{F}}^2({| 0 \rangle\langle 0 |})}]}=1,\!\!&
{\mathrm{Pr}^{0}[{{\boldsymbol{F}}^2({| 1 \rangle\langle 1 |})}]}=0,\!\!\\
\vspace{-.1cm}
& & \\
{\mathrm{Pr}^{0}[{{\boldsymbol{G}}({| 0 \rangle\langle 0 |})}]}=0,\!\!&
{\mathrm{Pr}^{0}[{{\boldsymbol{G}}({| 1 \rangle\langle 1 |})}]}=1,\!\!& \\
\vspace{-.1cm}
& & \\
{\mathrm{Pr}^{0}[{{\boldsymbol{F}}\circ {\boldsymbol{G}}^2\circ {\boldsymbol{F}}({| 0 \rangle\langle 0 |})}]}=1,\!\!&
{\mathrm{Pr}^{0}[{{\boldsymbol{F}}\circ {\boldsymbol{G}}\circ {\boldsymbol{F}}({| 0 \rangle\langle 0 |})}]}=1.&\!\!\\
\end{array}}\right.$$ If $\alpha/\pi$ is rational, then the family $\mathcal{H}\times\{ {\boldsymbol{R}}_{\pm\alpha}\}$ is characterized by the experimental equations in two variables $({\boldsymbol{F}},{\boldsymbol{G}})$: $$\left\{{\begin{array}{lll}
{\mathrm{Pr}^{0}[{{\boldsymbol{F}}({| 0 \rangle\langle 0 |})}]}={\mbox{$\frac{1}{2}$}},\!\!&
{\mathrm{Pr}^{0}[{{\boldsymbol{F}}^2({| 0 \rangle\langle 0 |})}]}=1,\!\!&
{\mathrm{Pr}^{0}[{{\boldsymbol{F}}^2({| 1 \rangle\langle 1 |})}]}=0,\!\!\\
\vspace{-.1cm}
& & \\
{\mathrm{Pr}^{0}[{{\boldsymbol{G}}({| 0 \rangle\langle 0 |})}]}=1,\!\!&
{\mathrm{Pr}^{0}[{{\boldsymbol{G}}({| 1 \rangle\langle 1 |})}]}=0,\!\!& \\
\vspace{-.1cm}
& & \\
{\mathrm{Pr}^{0}[{{\boldsymbol{F}}\circ {\boldsymbol{G}}^{n_\alpha}\circ
{\boldsymbol{F}}({| 0 \rangle\langle 0 |})}]}=1,\!\!&
{\mathrm{Pr}^{0}[{{\boldsymbol{F}}\circ {\boldsymbol{G}}\circ
{\boldsymbol{F}}({| 0 \rangle\langle 0 |})}]}={\mbox{$\frac{1}{2}$}}+{\mbox{$\frac{1}{2}$}}\cos\alpha.\!\!& \\
\end{array}}\right.$$
Characterization of ${\boldsymbol{{{\mathrm{c\text{-}NOT}}}}}$ gates
--------------------------------------------------------------------
In this section we will extend our theory of characterization of CPSO families for several qubits. In particular, we will show that the family of ${\boldsymbol{{{\mathrm{c\text{-}NOT}}}}}$ gates together with the family $\mathcal{H}$ is characterizable. First we need some definitions.
For every ${\varphi}\in[0,2\pi)$, we define ${{\mathrm{c\text{-}NOT}}}_{\varphi}$ as the only unitary transformation over ${\mathbb{C}}^4$ satisfying ${{\mathrm{c\text{-}NOT}}}_{{\varphi}}({| 0 \rangle}{| \psi \rangle})={| 0 \rangle}{| \psi \rangle}$ and ${{\mathrm{c\text{-}NOT}}}_{{\varphi}}{| 1 \rangle}{| \psi \rangle}={| 1 \rangle}{{\mathrm{NOT}}}_{\varphi}{| \psi \rangle}$, for all ${| \psi \rangle}\in{\mathbb{C}}^2$.
We extend the definition of the experimental equation for CPSOs given in [(\[expCPSOeq2\])]{} for $n$ qubits. It is an equation of the form $$\begin{aligned}
\label{expCPSOeq3}
{\mathrm{Pr}^{v}[{{\boldsymbol{F}}^{k_{1}}\circ{\boldsymbol{G}}^{l_1}\circ\cdots\circ
{\boldsymbol{F}}^{k_{t}}\circ{\boldsymbol{G}}^{l_{t}}({| w \rangle\langle w |})}]}& = & r,\end{aligned}$$ where in addition to the notation of [(\[expCPSOeq2\])]{} $v,w\in\{0,1\}^n$, and $\mathrm{Pr}^{v}$ is the probability of measuring ${| v \rangle\langle v |}$. For the variables ${\boldsymbol{F}}$ and ${\boldsymbol{G}}$ of [(\[expCPSOeq3\])]{}, we also allow both the tensor product of two CPSO variables and the tensor product of a CPSO variable with the identity. We now state the characterization.
\[cnot\] The family $\{({\boldsymbol{H}}_{\varphi},{\boldsymbol{{{\mathrm{c\text{-}NOT}}}}}_{{\varphi}}) : {\varphi}\in[0,2\pi)\}$ is characterized by the experimental equations in two variables $({\boldsymbol{F}},{\boldsymbol{G}})$: $$\left\{{\begin{array}{llll}
{\mathrm{Pr}^{0}[{{\boldsymbol{F}}({| 0 \rangle\langle 0 |})}]}={\mbox{$\frac{1}{2}$}},\!\!&
{\mathrm{Pr}^{0}[{{\boldsymbol{F}}^2({| 0 \rangle\langle 0 |})}]}=1,\!\!&
{\mathrm{Pr}^{0}[{{\boldsymbol{F}}^2({| 1 \rangle\langle 1 |})}]}=0,\!\!& \\
\vspace{-.1cm}
&&& \\
{\mathrm{Pr}^{00}[{{\boldsymbol{G}}({| 00 \rangle\langle 00 |})}]}=1,\!\!&
{\mathrm{Pr}^{01}[{{\boldsymbol{G}}({| 01 \rangle\langle 01 |})}]}=1,\!\!&
{\mathrm{Pr}^{11}[{{\boldsymbol{G}}({| 10 \rangle\langle 10 |})}]}=1,\!\!&
{\mathrm{Pr}^{10}[{{\boldsymbol{G}}({| 11 \rangle\langle 11 |})}]}=1,\!\!\\
\vspace{-.1cm}
&&& \\
\multicolumn{2}{l}{
{\mathrm{Pr}^{00}[{({\boldsymbol{I}}_2\otimes {\boldsymbol{F}})\circ {\boldsymbol{G}}\circ
({\boldsymbol{I}}_2\otimes {\boldsymbol{F}})({| 00 \rangle\langle 00 |})}]}=1,\!\!}&
\multicolumn{2}{l}{
{\mathrm{Pr}^{10}[{({\boldsymbol{I}}_2\otimes {\boldsymbol{F}})\circ {\boldsymbol{G}}\circ
({\boldsymbol{I}}_2\otimes {\boldsymbol{F}})({| 10 \rangle\langle 10 |})}]}=1,\!\!}\\
\vspace{-.1cm}
&&& \\
\multicolumn{2}{l}{
{\mathrm{Pr}^{00}[{({\boldsymbol{F}}\otimes{\boldsymbol{I}}_2)\circ {\boldsymbol{G}}^2\circ
({\boldsymbol{F}}\otimes{\boldsymbol{I}}_2)({| 00 \rangle\langle 00 |})}]}=1,\!\!}&
\multicolumn{2}{l}{
{\mathrm{Pr}^{01}[{({\boldsymbol{F}}\otimes{\boldsymbol{I}}_2)\circ {\boldsymbol{G}}^2\circ
({\boldsymbol{F}}\otimes{\boldsymbol{I}}_2)({| 01 \rangle\langle 01 |})}]}=1,\!\!}\\
\vspace{-.1cm}
&&& \\
\multicolumn{2}{l}{
{\mathrm{Pr}^{00}[{({\boldsymbol{F}}\otimes{\boldsymbol{F}})\circ {\boldsymbol{G}}\circ
({\boldsymbol{F}}\otimes{\boldsymbol{F}})({| 00 \rangle\langle 00 |})}]}=1.\!\!}&&
\end{array}}\right.$$
[[**Proof:** ]{}]{}Let ${\boldsymbol{F}}$ and ${\boldsymbol{G}}$ satisfy these equations. By [Theorem [\[general\]]{}]{}, with $\alpha=\pi$ and $\theta=\pi/4$, the first three equations imply that ${\boldsymbol{F}}={\boldsymbol{H}}_{\varphi}$, for some ${\varphi}\in[0,2\pi)$. Using [Lemma [\[lemma2\]]{}]{}, the remaining equations imply that ${\boldsymbol{G}}^2={\boldsymbol{I}}_4$, and it follows from [Lemma [\[lemma1c\]]{}]{} that ${\boldsymbol{G}}$ is a unitary CPSO. A straightforward verification then shows that indeed ${\boldsymbol{G}}={\boldsymbol{{{\mathrm{c\text{-}NOT}}}_{\varphi}}}$.
------------------------------------------------------------------------
Robustness
==========
In this section we introduce the notion of robustness for experimental equations which will be the crucial ingredient for proving self-testability. In this extended abstract we will deal only with the case of experimental equations for one qubit and in one variable. [From]{} now on $(E)$ will always denote a set of such equations. Similar results can be obtained for several qubits and several variables.
Let ${\varepsilon},\delta\geq 0$, and let $(E)$ be a finite satisfiable set of experimental equations. We say that $(E)$ is [*(${\varepsilon},\delta$)-robust*]{} if whenever a CPSO ${\boldsymbol{G}}$ ${\varepsilon}$-satisfies $(E)$, we have ${{\mathrm{dist}}_{\infty}}({\boldsymbol{G}},\mathcal{F}_{(E)})\leq\delta$.
When a CPSO family is characterized by a finite set of experimental equations $(E)$, one would like to prove that $(E)$ is robust. The next theorem shows that this is always the case.
\[robustness\] Let $(E)$ be a finite satisfiable set of experimental equations. Then there exists an integer $k\geq 1$ and a real $C>0$ such that for all ${\varepsilon}\geq 0$, $(E)$ is (${\varepsilon},C{\varepsilon}^{1/k}$)-robust.
[[**Proof:** ]{}]{}We will use basic notions from algebraic geometry for which we refer the reader for example to [@br90]. In the proof, ${\mathbb{C}}$ is identified with ${\mathbb{R}}^2$. Then the set $K$ of CPSOs for one qubit is a real compact semi-algebraic set. Suppose that in $(E)$ there are $d$ equations. Let $f:K\rightarrow{\mathbb{R}}$ be the function that maps the CPSO ${\boldsymbol{G}}$ to the maximum of the magnitudes of the difference between the probability term and the constant term of the $i^{\mbox{\scriptsize th}}$ equation in $(E)$, for $i=1,\ldots,d$. By definition of $f$, we get $f^{-1}(0)=\mathcal{F}_{(E)}$. Moreover, $f$ is a continuous semi-algebraic function, since it is the maximum of the magnitudes of polynomial functions in the (real) coefficients of ${\boldsymbol{G}}$.
Let $g:K\rightarrow{\mathbb{R}}$ defined in ${\boldsymbol{G}}$ by $g({\boldsymbol{G}})={{\mathrm{dist}}_{\infty}}({\boldsymbol{G}},\mathcal{F}_{(E)})$. Since $K$ is a compact semi-algebraic set, $g$ is a continuous semi-algebraic function. Moreover, for all ${\boldsymbol{G}}\in K$, we have $f({\boldsymbol{G}})=0$ if and only if $g({\boldsymbol{G}})=0$. Then [Fact [\[semi-alg\]]{}]{} concludes the proof.
------------------------------------------------------------------------
For a proof of the following fact, see for example [@br90 Prop. 2.3.11].
\[semi-alg\] Let $X\subseteq{\mathbb{R}}^m$ be a compact semi-algebraic set. Let $f,g:X\rightarrow{\mathbb{R}}$ be continuous semi-algebraic functions. Assume that for all $x\in X$, if $f(x)=0$ then $g(x)=0$. Then there exists an integer $k\geq 1$ and a real $C>0$ such that, for all $x\in X$, ${| g(x) |}^k\leq C{| f(x) |}$.
In some cases we can explicitly compute the constants $C$ and $k$ of [Theorem [\[robustness\]]{}]{}. We will illustrate these techniques with the equations in [Theorem [\[general\]]{}]{} for the case $\alpha=\pi$ and $\theta=\pi/4$. Let us recall that these equations characterize the set $\mathcal{H}$.
\[hadamardrob\] For every $0 \leq {\varepsilon}\leq 1$, the following equations are (${\varepsilon},4579\sqrt{{\varepsilon}}$)-robust: $$\begin{aligned}
\label{approxhadamard}
{\mathrm{Pr}^{0}[{{\boldsymbol{G}}({| 0 \rangle\langle 0 |})}]}={\mbox{$\frac{1}{2}$}},&
{\mathrm{Pr}^{0}[{{\boldsymbol{G}}^2({| 0 \rangle\langle 0 |})}]}=1,
\textrm{ and} &
{\mathrm{Pr}^{0}[{{\boldsymbol{G}}^2({| 1 \rangle\langle 1 |})}]}=0.\end{aligned}$$
The proof of this theorem will be given in [Appendix [\[APTH\]]{}]{}.
Quantum Self-Testers
====================
In this final section we define formally our testers and establish the relationship between robust equations and testability. Again, we will do it here only for the case of one qubit and one variable. Let ${\boldsymbol{G}}$ be a CPSO. The [*experimental oracle*]{} ${\mathcal{O}[{\boldsymbol{G}}]}$ for ${\boldsymbol{G}}$ is a probabilistic procedure. It takes inputs from $\{0,1\}\times{\mathbb{N}}$ and generates outcomes from the set $\{0,1\}$ such that for every $k\in{\mathbb{N}}$, $$\begin{aligned}
\Pr[{\mathcal{O}[{\boldsymbol{G}}]}(b,k)=0]
& = &
{\mathrm{Pr}^{0}[{{\boldsymbol{G}}^{k}({| b \rangle\langle b |})}]}.\end{aligned}$$ An oracle program $T$ with an experimental oracle ${\mathcal{O}[{\boldsymbol{G}}]}$ is a program denoted by $T^{{\mathcal{O}[{\boldsymbol{G}}]}}$ which can ask queries from the experimental oracle in the following sense: when it presents a query $(b,k)$ to the oracle, in one computational step it receives the probabilistic outcome of ${\mathcal{O}[{\boldsymbol{G}}]}$ on it.
Let $\mathcal{F}$ be a family of CPSOs, and let $0\leq\delta_1\leq\delta_2<1$. A [*$(\delta_1,\delta_2)$-tester for*]{} $\mathcal{F}$ is a probabilistic oracle program $T$ such that for every CPSO ${\boldsymbol{G}}$,
- if ${{\mathrm{dist}}_{\infty}}({\boldsymbol{G}},\mathcal{F})\leq\delta_1$ then $\Pr[T^{{\mathcal{O}[{\boldsymbol{G}}]}}\mbox{ says { \tt PASS}}]\geq 2/3$,
- if ${{\mathrm{dist}}_{\infty}}({\boldsymbol{G}},\mathcal{F})>\delta_2$ then $\Pr[T^{{\mathcal{O}[{\boldsymbol{G}}]}}\mbox{ says { \tt FAIL}}]\geq 2/3$,
where the probability is taken over the probability distribution of the outcomes of the experimental oracle and the internal coin tosses of the program.
\[robtotester\] Let ${\varepsilon},\delta>0$, and let $(E)$ be a satisfiable set of $d$ experimental equations such that the size of every equation is at most $k$. If $(E)$ is (${\varepsilon},\delta$)-robust then there exists an $({\varepsilon}/(3k),\delta)$-tester for $\mathcal{F}_{(E)}$ which makes $O(d\ln (d)/{\varepsilon}^2)$ queries.
<span style="font-variant:small-caps;">Sketch of proof.</span> We will describe a probabilistic oracle program $T$. Let ${\boldsymbol{G}}$ be a CPSO. We can suppose that for every equation in $(E)$, $T$ has a rational number $\tilde{r}$ such that ${| \tilde{r}-r |}\leq{\varepsilon}/6$, where $r$ is the constant term of the equation. By sampling the oracle ${\mathcal{O}[{\boldsymbol{G}}]}$, for every equation in $(E)$, $T$ obtains a value $\tilde{p}$ such that ${| \tilde{p}-p |}\leq{\varepsilon}/6$ with probability at least $1-1/(3d)$, where $p$ is the probability term of the equation. A standard Chernoff bound argument shows that this is feasible with $O(\ln(d)/{\varepsilon}^2)$ queries for each equation. If for every equation ${| \tilde{p}-\tilde{r} |}\leq 2{\varepsilon}/3$, then $T$ says [PASS]{}, otherwise $T$ says [FAIL]{}. Using the robustness of $(E)$ and [Lemma [\[lemmar\]]{}]{}, one can verify that $T$ is a $({\varepsilon}/(3k),\delta)$-tester for $\mathcal{F}_{(E)}$.
------------------------------------------------------------------------
\[lemmar\] Let $(E)$ be a finite satisfiable set of experimental equations such that the size of every equation is at most $k$, and let ${\boldsymbol{G}}$ be a CPSO. For every ${\varepsilon}\geq 0$, if ${{\mathrm{dist}}_{\infty}}({\boldsymbol{G}},\mathcal{F}_{(E)})\leq{\varepsilon}$ then ${\boldsymbol{G}}$ ($k{\varepsilon}$)-satisfies $(E)$.
Our main result is the consequence of Theorems \[general\], \[couple\], \[cnot\], \[robustness\], \[hadamardrob\], \[robtotester\], and the many-qubit generalizations of them.
\[maintheo\] Let $\mathcal{F}$ be one of the following families :
- $\mathcal{R}_{\alpha,\theta}\ $ for $(\alpha,\theta)\in(0,\pi]\times(0,\pi/2]\backslash\{(\pi,\pi/2)\}$ where $\alpha/\pi$ is rational,
- $\{({\boldsymbol{H}}_{\varphi},{\boldsymbol{{{\mathrm{NOT}}}}}_{{\varphi}}) : {\varphi}\in[0,2\pi)\}$,
- $\mathcal{H}\times\{{\boldsymbol{R}}_{\pm\alpha}\}\ $ for $\alpha/\pi$ rational,
- $\{({\boldsymbol{H}}_{\varphi},{\boldsymbol{{{\mathrm{c\text{-}NOT}}}}}_{{\varphi}})
: {\varphi}\in[0,2\pi)\}$,
- $\{({\boldsymbol{H}}_{\varphi},{\boldsymbol{R}}_{s\pi/4},{\boldsymbol{{{\mathrm{c\text{-}NOT}}}}}_{{\varphi}})
: {\varphi}\in[0,2\pi),s=\pm 1\}$.
Then there exists an integer $k\geq 1$ and a real $C>0$ such that, for all ${\varepsilon}>0$, $\mathcal{F}$ has an $({\varepsilon},C{\varepsilon}^{1/k})$-tester which makes $O(1/{\varepsilon}^2)$ queries. Moreover, for every $0< {\varepsilon}\leq 1$, $\mathcal{H}$ has an $({\varepsilon}/6,4579\sqrt{{\varepsilon}})$-tester which makes $O(1/{\varepsilon}^2)$ queries.
Note that each triplet of the last family forms a universal and fault-tolerant set of quantum gates[@bmprv99].
Acknowledgements
================
We would like to thank Jean-Benoit Bost, Stéphane Boucheron, Charles Delorme, Stéphane Gonnord, Lucien Hardy, Richard Jozsa, and Vlatko Vedral for several useful discussions and advice.
This work has been supported by C.E.S.G., Wolfson College Oxford, Hewlett-Packard, European TMR Research Network ERP-4061PL95-1412, the Institute for Logic, Language and Computation in Amsterdam, ESPRIT Working Group RAND2 no. 21726, NSERC, British-French Bilateral Project ALLIANCE no. 98101, and the Quantum Information Theory programme of the European Science Foundation.
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Appendix: Proof of [Lemma \[lemma2\]]{}[\[APL2\]]{}
===================================================
<span style="font-variant:small-caps;">Proof of [Lemma [\[lemma2\]]{}]{}.</span> Let $P$ be the plane defined in ${\mathbb{R}}^3$ by ${\overline{\rho_1}}$, ${\overline{\rho_2}}$ and ${\overline{\rho_3}}$. To simplify the discussion, we suppose w.l.o.g. that ${\overline{\zeta^{\pm}_z}}$ and ${\overline{\zeta^{\pm}_x}}$ are in $P$. Every one-qubit $\rho$ satisfying ${\overline{\rho}}\in P$ is a linear combination of $\rho_1$, $\rho_2$ and $\rho_3$. Therefore by linearity of ${\boldsymbol{G}}$ we get that it acts as the identity on $\{\rho : {\overline{\rho}}\in
P\}^{\otimes n}$. Moreover it is sufficient to show that ${\boldsymbol{G}}$ is the identity on density matrices representing non-entangled pure states, since they form a basis for all density matrices.
For this, for every $k$, let $A_k$ be the set of density matrices representing $k$-qubit non-entangled pure states, and let $B_k=\{\zeta_x^{\pm},\zeta_z^{\pm}\}^{\otimes n}$. We will show by induction on $k$ that, for every $0\leq k\leq n$, the CPSO ${\boldsymbol{G}}$ acts as the identity on $A_k\otimes B_{n-k}$. The case $k=0$ follows by the hypothesis of the lemma.
Suppose the statement is true for some $k$. Fix $\sigma\in A_k$ and $\tau\in B_{n-k-1}$. For every one-qubit density matrix $\rho$ let $\tilde{\rho}$ denote the $n$-qubit density matrix $\sigma\otimes\rho\otimes\tau$.
We now prove that $G(\tilde{\rho})=\tilde{\rho}$, for every $\rho\in A_1$. For this, we use the fact that the density matrix $\Psi^{+}$ representing the entangled EPR state $({| 00 \rangle}+{| 11 \rangle})/\sqrt{2}$, can be written in terms of tensor products of the $\zeta$ states: $$\begin{aligned}
\Psi^+ & = &
{\mbox{$\frac{1}{2}$}}(
\zeta_x^+ {\otimes}\zeta_x^+ +
\zeta_x^- {\otimes}\zeta_x^- +
\zeta_z^+ {\otimes}\zeta_z^+ +
\zeta_z^- {\otimes}\zeta_z^-)
-{\mbox{$\frac{1}{2}$}}(\zeta_y^+ {\otimes}\zeta_y^+ +
\zeta_y^- {\otimes}\zeta_y^-).\end{aligned}$$ This can be generalized for the pure state ${| \mu \rangle}=({| \tilde{0} \rangle}{| \tilde{0} \rangle}
+{| \tilde{1} \rangle}{| \tilde{1} \rangle})/\sqrt{2}$: $$\begin{aligned}
\mu & = &
{\mbox{$\frac{1}{2}$}}(
\tilde{\zeta}_x^+ {\otimes}\tilde{\zeta}_x^+ +
\tilde{\zeta}_x^- {\otimes}\tilde{\zeta}_x^- +
\tilde{\zeta}_z^+ {\otimes}\tilde{\zeta}_z^+ +
\tilde{\zeta}_z^- {\otimes}\tilde{\zeta}_z^-)
-{\mbox{$\frac{1}{2}$}}(\tilde{\zeta}_y^+ {\otimes}\tilde{\zeta}_y^+ +
\tilde{\zeta}_y^- {\otimes}\tilde{\zeta}_y^-).\end{aligned}$$ If we apply the CPSO ${\boldsymbol{I}}_{2^n} {\otimes}{\boldsymbol{G}}$ to the state $\mu$ we get: $$\begin{aligned}
({\boldsymbol{I}}_{2^n} {\otimes}{\boldsymbol{G}})(\mu) & = &
{\mbox{$\frac{1}{2}$}}(
\tilde{\zeta}_x^{+} {\otimes}\tilde{\zeta}_x^{+} +
\tilde{\zeta}_x^{-} {\otimes}\tilde{\zeta}_x^{-} +
\tilde{\zeta}_z^{+} {\otimes}\tilde{\zeta}_z^{+} +
\tilde{\zeta}_z^{-} {\otimes}\tilde{\zeta}_z^{-}
-\tilde{\zeta}_y^{+} {\otimes}{\boldsymbol{G}}(\tilde{\zeta}_y^{+})
-\tilde{\zeta}_y^{-} {\otimes}{\boldsymbol{G}}(\tilde{\zeta}_y^{-})).\end{aligned}$$ If ${| {\varphi}\rangle}$ and ${| {\varphi}' \rangle}$ are orthogonal $n$-qubit pure states, then let $\Phi^-_{{\varphi}{\varphi}'}=({| {\varphi}\rangle}{| {\varphi}' \rangle}-
{| {\varphi}' \rangle}{| {\varphi}\rangle})/\sqrt{2}$. Since $\Phi^-_{{\varphi}{\varphi}'}$ is orthogonal to all symmetric $2n$-qubit pure states of the form $\psi {\otimes}\psi$, by projecting $({\boldsymbol{I}}_{2^n} {\otimes}{\boldsymbol{G}})(\mu)$ to $\Phi^-_{{\varphi}{\varphi}'}$ we obtain: $$\begin{aligned}
{\langle \Phi^-_{{\varphi}{\varphi}'} |}
({\boldsymbol{I}}_{2^n} {\otimes}{\boldsymbol{G}})(\mu)
{| \Phi^-_{{\varphi}{\varphi}'} \rangle}
& = &
-{\mbox{$\frac{1}{2}$}}{\langle \Phi^-_{{\varphi}{\varphi}'} |}\tilde{\zeta}_y^+
{\otimes}{\boldsymbol{G}}(\tilde{\zeta}_y^+){| \Phi^-_{{\varphi}{\varphi}'} \rangle}
-{\mbox{$\frac{1}{2}$}}{\langle \Phi^-_{{\varphi}{\varphi}'} |}\tilde{\zeta}_y^-
{\otimes}{\boldsymbol{G}}(\tilde{\zeta}_y^-){| \Phi^-_{{\varphi}{\varphi}'} \rangle}.\end{aligned}$$ Since ${\boldsymbol{G}}$ is a CPSO, the left-hand side of this equality is non-negative and in the right-hand side both terms are non-positive. Therefore for every orthogonal $n$-qubit pure states ${| {\varphi}\rangle}$ and ${| {\varphi}' \rangle}$, we get $$\begin{array}{rcccl}
{\langle \Phi^-_{{\varphi}{\varphi}'} |}\tilde{\zeta}_y^+
{\otimes}{\boldsymbol{G}}(\tilde{\zeta_y}^+){| \Phi^-_{{\varphi}{\varphi}'} \rangle} & = &
{\langle \Phi^-_{{\varphi}{\varphi}'} |}\tilde{\zeta}_y^-
{\otimes}{\boldsymbol{G}}(\tilde{\zeta_y}^-){| \Phi^-_{{\varphi}{\varphi}'} \rangle} & = & 0.
\end{array}$$
A straightforward calculation then shows that ${\boldsymbol{G}}(\tilde{\zeta_y}^{\pm})=\tilde{\zeta}_y^{\pm}$. Therefore ${\boldsymbol{G}}$ acts as the identity on density matrices $\tilde{\zeta}_z^{\pm}$, $\tilde{\zeta}_x^+$ and $\tilde{\zeta}_y^+$, which generate all density matrices, and thus ${\boldsymbol{G}}(\tilde{\rho})=\tilde{\rho}$.
------------------------------------------------------------------------
Appendix: Proof of [Theorem \[hadamardrob\]]{}\[APTH\]
======================================================
\[cnxnormso\] Let ${\boldsymbol{G}}$ be a superoperator on ${\mathbb{C}}^{2\times 2}$. Let $0\leq{\varepsilon}\leq 1$ be such that ${{|\! | {\boldsymbol{G}}(\zeta_x^\pm)-\zeta_x^\pm |\! |}_{1}}$, ${{|\! | {\boldsymbol{G}}(\zeta_y^\pm)-\zeta_y^\pm |\! |}_{1}}$, ${{|\! | {\boldsymbol{G}}(\zeta_z^\pm)-\zeta_z^\pm |\! |}_{1}}\leq{\varepsilon}$; then ${{|\! | {\boldsymbol{G}}-{\boldsymbol{I}}_2 |\! |}_{\infty}}\leq 8{\varepsilon}$.
\[lemma5\] Let ${\overline{u}}$ and ${\overline{v}}$ be two orthonormal vectors in ${\mathbb{R}}^3$, and $0\leq{\varepsilon}\leq 1$ a constant. If ${\boldsymbol{G}}$ is a CPSO for one qubit such that ${|\! | {{\boldsymbol{{\overline{G}}}}}(\pm{\overline{u}})-\pm{\overline{u}} |\! |}\leq{\varepsilon}$ and ${|\! | {{\boldsymbol{{\overline{G}}}}}(\pm{\overline{v}})-\pm{\overline{v}} |\! |}\leq{\varepsilon}$, then ${{|\! | {\boldsymbol{G}}-{\boldsymbol{I}}_{2} |\! |}_{\infty}}\leq 241{\varepsilon}$.
[[**Proof:** ]{}]{}We can suppose w.l.o.g. that $u=\zeta_x^+$ and $v=\zeta_z^+$. Let $\rho={\boldsymbol{G}}(\zeta_y^+)$, where ${\overline{\rho}}=(x,y,z)$. [From]{} [Lemma [\[lemma1\]]{}]{} it follows that ${{|\! | {\boldsymbol{G}}(\zeta_z^+)-\rho |\! |}_{1}}
\leq {{|\! | \zeta_z^+-\zeta_y^+ |\! |}_{1}} = \sqrt{2}$. By the assumption of this lemma we have that ${{|\! | {\boldsymbol{G}}(\zeta_z^+)-\zeta_z^+ |\! |}_{1}}\leq {\varepsilon}$, and hence ${{|\! | \zeta_z^+-\rho |\! |}_{1}}\leq \sqrt{2}+{\varepsilon}$. The same relation holds also for the other three fixed points $\zeta_z^-$, $\zeta_x^+$, and $\zeta_x^-$. As a result, the three coordinates of ${\overline{\rho}}$ have to obey the four inequalities $$\begin{array}{rcccl}\label{eq:xyzrestriction}
x^2+y^2 + (z\pm 1)^2
\textrm{ and }
(x\pm 1)^2+y^2 + z^2 &
\leq &
(\sqrt{2}+{\varepsilon})^2
& \leq &
2+4{\varepsilon}\end{array}$$
A second set of restrictions on $(x,y,z)$ comes from the complete positivity of ${\boldsymbol{G}}$. Again we use the decomposition of the EPR state $\Psi^+$, to analyze the two-qubit state: $$\begin{aligned}
({\boldsymbol{I}}_2 {\otimes}{\boldsymbol{G}})(\Psi^+)
& = &
{\mbox{$\frac{1}{2}$}}(
\zeta_x^+ {\otimes}{\boldsymbol{G}}(\zeta_x^+) +
\zeta_x^- {\otimes}{\boldsymbol{G}}(\zeta_x^-))
+{\mbox{$\frac{1}{2}$}}(
\zeta_z^+ {\otimes}{\boldsymbol{G}}(\zeta_z^+) +
\zeta_z^- {\otimes}{\boldsymbol{G}}(\zeta_z^-)) \\
&&
-{\mbox{$\frac{1}{2}$}}({\zeta_y^+ {\otimes}{\boldsymbol{G}}(\zeta_y^+)
+\zeta_y^- {\otimes}{\boldsymbol{G}}(\zeta_y^-)}).\end{aligned}$$ Using the hypothesis, the projection of this state onto the anti-symmetrical entangled qubit pair ${| \Phi^- \rangle} =
({| 01 \rangle}-{| 10 \rangle})/\sqrt{2}$ yields $$\begin{aligned}
{\langle \Phi^- |}
({\boldsymbol{I}}_2 {\otimes}{\boldsymbol{G}})(\Psi^+)
{| \Phi^- \rangle}
& \leq & 2{\varepsilon}-{\mbox{$\frac{1}{2}$}}{\langle \Phi^- |}\zeta_y^+
{\otimes}{\boldsymbol{G}}(\zeta_y^+){| \Phi^- \rangle}
-{\mbox{$\frac{1}{2}$}}{\langle \Phi^- |}\zeta_y^-
{\otimes}{\boldsymbol{G}}(\zeta_y^-){| \Phi^- \rangle}.\end{aligned}$$ Since ${\boldsymbol{G}}$ is a CPSO, as in [Lemma [\[lemma2\]]{}]{} we get ${\langle \Phi^- |}\zeta_y^+{\otimes}\rho{| \Phi^- \rangle} \leq 4{\varepsilon}$. A straightforward calculation shows that this last relation is equivalent with a restriction on the $y$ coordinate: $y \geq 1-16{\varepsilon}$.
This last inequality implies $y^2\geq 1-32{\varepsilon}$, which combined with the restrictions of [(\[eq:xyzrestriction\])]{}, leads to the conclusion that $\left(x\pm 1\right)^2 \leq 2+4{\varepsilon}-y^2-z^2 \leq
1+36{\varepsilon}$, and similarly $\left(z\pm 1\right)^2 \leq 1+36{\varepsilon}$. The $x$ and $z$ coordinates of ${\overline{\rho}}$ satisfy $|x|,|z|
\leq 18{\varepsilon}$.
These bounds imply $$\begin{aligned}
&\begin{array}{rcccl}
{{|\! | {\boldsymbol{G}}(\zeta_y^+)-\zeta_y^+ |\! |}_{1}}
& = &
\sqrt{x^2+(y-1)^2+z^2}
& \leq &
\sqrt{904}{\varepsilon}.
\end{array}&\end{aligned}$$ The same result can be proved for $\zeta_y^-$. Therefore by [Fact [\[cnxnormso\]]{}]{} we can conclude the proof.
------------------------------------------------------------------------
<span style="font-variant:small-caps;">Proof of [Theorem [\[hadamardrob\]]{}]{}.</span> Let ${\boldsymbol{G}}$ be a CPSO which ${\varepsilon}$-satisfies the equations. First we will show there is a point ${\overline{\rho}}\in{\mathcal{S}}$ with $z$-coordinate $0$ whose distance from ${\overline{{\boldsymbol{G}}({| 0 \rangle\langle 0 |})}}$ is at most $10\sqrt{{\varepsilon}}$. The last two equations imply that ${{|\! | {\boldsymbol{G}}^2({| b \rangle\langle b |})-{| b \rangle\langle b |} |\! |}_{1}}\leq 3\sqrt{{\varepsilon}}$, for $b=0,1$. Therefore ${{|\! | {\boldsymbol{G}}^2({| 0 \rangle\langle 0 |})-
{\boldsymbol{G}}^2({| 1 \rangle\langle 1 |}) |\! |}_{1}}\geq 2 - 6\sqrt{{\varepsilon}}$, and by [Lemma [\[lemma1\]]{}]{}(a) we have ${{|\! | {\boldsymbol{G}}({| 0 \rangle\langle 0 |})-
{\boldsymbol{G}}({| 1 \rangle\langle 1 |}) |\! |}_{1}}\geq 2 - 6\sqrt{{\varepsilon}}$. Thus ${|\! | {\overline{{\boldsymbol{G}}({| b \rangle\langle b |})}} |\! |}\geq 1-6\sqrt{{\varepsilon}}$, for $b=0,1$. Let $\tau=\rho({\mbox{$\frac{1}{2}$}},\alpha)$, where ${\boldsymbol{G}}({| 0 \rangle\langle 0 |})=\rho(p,\alpha)$. The first equation implies that ${|\! | {\overline{\tau}}-{\overline{{\boldsymbol{G}}({| 0 \rangle\langle 0 |})}} |\! |}\leq 2{\varepsilon}$. Therefore for ${\overline{\rho}}={\overline{\tau}}/{|\! | {\overline{\tau}} |\! |}$ we get ${{|\! | {\boldsymbol{G}}({| 0 \rangle\langle 0 |})-\rho |\! |}_{1}}\leq 10\sqrt{{\varepsilon}}$.
The point ${\overline{\rho}}$ on ${\mathcal{S}}$ uniquely defines ${\varphi}\in[0,2\pi)$ such that ${\overline{{\boldsymbol{H}}_{\varphi}({| 0 \rangle\langle 0 |})}}={\overline{\rho}}$. One can verify that ${{\boldsymbol{H}}_{{\varphi}}^{-1}\circ{\boldsymbol{G}}}$ acts as the identity with error at most $19\sqrt{{\varepsilon}}$ on the four density matrices ${{| 0 \rangle\langle 0 |}}$, ${{| 1 \rangle\langle 1 |}}$, ${{\boldsymbol{H}}_{\varphi}({| 0 \rangle\langle 0 |})}$, and ${{\boldsymbol{H}}_{\varphi}({| 1 \rangle\langle 1 |})}$. [From]{} [Lemma [\[lemma5\]]{}]{} we conclude that ${{|\! | {\boldsymbol{G}}-{\boldsymbol{H}}_{{\varphi}} |\! |}_{\infty}}\leq 4579\sqrt{{\varepsilon}}$.
------------------------------------------------------------------------
[^1]: C.W.I. Amsterdam; Centre for Quantum Computation, University of Oxford. [wimvdam@qubit.org]{}
[^2]: Université Paris Sud, LRI. [magniez@lri.fr]{}
[^3]: University of Waterloo; Centre for Quantum Computation, University of Oxford. [mosca@qubit.org]{}
[^4]: CNRS, Université Paris Sud, LRI. [santha@lri.fr]{}
|
---
author:
- Grace Dupuis
bibliography:
- 'hpscx.bib'
title: Collider Constraints and Prospects of a Scalar Singlet Extension to Higgs Portal Dark Matter
---
Introduction {#scn:intro}
============
The discovery of the Higgs boson at the Large Hadron Collider (LHC) is one of the most significant scientific achievements of late. By finally providing long-awaited evidence of the previously undiscovered scalar sector of the Standard Model (SM), this groundbreaking result has sparked much excitement. Not only does this discovery verify the well-established fundamental theory of particle physics, it also presents new possibilities to probe physics beyond the Standard Model. Although this result is a major achievement in the field of particle physics, the Higgs sector still remains largely unexplored. Specifically, it has yet to be determined whether the discovered scalar is indeed the Higgs boson of the Standard Model, or a piece of some extended theory. Such a question lies under the realm of Higgs precision measurements. Future experiments, reaching higher energies and employing advancing techniques will shed light on precision aspects such as the Higgs CP nature, its self-couplings and couplings to electroweak vector bosons, and possible deviations from the Standard Model that may indicate a larger Higgs sector.
As successful as the Standard Model has proven to be, there are still missing pieces, one of which is the failure to account for the identity of dark matter (DM) as a particle species. Extensions of the SM scalar sector lend themselves to models of DM, through potential DM candidates and/or new mediators bridging the dark and visible sectors. The study presented here considers one of the simplest such extensions — the addition of a scalar singlet field, which couples to a fermionic DM candidate. The scalar singlet extension is attractive in its simplicity, hence, similar models have been studied extensively in the literature. The implications on the Higgs sector of a pure scalar singlet extension have been investigated [@Robens:2015gla; @Pruna:2013bma; @Godunov:2015nea; @Martin-Lozano:2015dja; @Falkowski:2015iwa], as well as the viability of such a model in a dark matter context [@Berlin:2015wwa; @Kouvaris:2014uoa; @Buckley:2014fba; @Ghorbani:2014qpa], with subtle distinctions. There is a wide range of models which contain a new scalar singlet, including similar studies of the implications of an additional Higgs-like scalar, with varying instances of the scalar potential and Higgs-scalar interaction terms [@Basso:2013nza; @Basso:2012nh].
Often such models, which are typically referred to as Higgs portal, couple the SM field directly to a DM candidate [@Fedderke:2014wda; @Baek:2011aa]. Studies of scalar singlet extensions also exist which take the additional scalar as the DM candidate, as it is protected under a $Z_{2}$ symmetry [@Queiroz:2014pra; @Feng:2014vea; @Martin:2014sxa; @Cline:2013gha]. The new scalar here acquires a vacuum expectation value (vev) and mixes with the SM Higgs field, giving two scalar mediators that act as the portal to the dark fermion. The case of a dark fermion with its SM interactions mediated by a new scalar or pseudoscalar has also been considered, but in the context of a bottom-up, effective field theory (EFT) approach, without considering the mixing of the scalar and SM Higgs [@Izaguirre:2014vva], or embedded in a more complex model with additional extensions [@Varzielas:2015sno] .
In the following study, a scalar singlet extension is investigated, with applicability to a model of dark matter. Focus is placed on the collider implications of the mixing in the Higgs sector, particularly the prospective influence of the additional scalar in Higgs precision measurements at the proposed International Linear Collider (ILC). The applicability of the model in a dark matter context is addressed in associated invisible signatures, and compatibility with direct detection (DD) limits. As current LHC data is consistent with a SM Higgs, it is vital to further consider the scenario in the context of future experiments, in the case where an additional Higgs-like scalar has evaded detection. Under this assumption, the discovery potential at a proposed linear collider is explored. A lepton collider is advantageous in possessing greater sensitivity to Higgs precision measurements, to which the LHC may not be sensitive. It is discussed how a precision environment may probe the scalar parameters of the model, and what distinguishing features may be observable. In particular, the study focuses on the Higgs self-coupling through double Higgs production processes.
This work is structured as follows. The model and its parametrization are presented in section \[scn:model\]. As the implications of new decay modes are examined, expressions for the new decay widths are given in section \[scn:widths\]. The LHC constraints are presented in section \[scn:LHC\], including both a review of Higgs exclusion searches and measured signals, as well as specific invisible searches. The topic of discovery reach and projected limits at a future linear collider is explored in section \[scn:ILC\]. The compatibility with direct detection data is addressed in section \[scn:DD\]. A concluding discussion is then given on the viability and prospects of the model.
Model {#scn:model}
=====
The scalar sector of the SM is supplemented with an additional real scalar field, which is a singlet under the SM gauge group and couples to a fermionic dark matter candidate via a Yukawa term. The additional scalar acquires a vev in the same way as the familiar SM Higgs field, thereby inducing mixing between the new scalar and the Higgs. The model is parametrized as follows.\
\
The scalar potential is modified, according to $$V(\Phi_1, \phi_2) = \lambda_1 (\Phi_1^{\dagger}\Phi_1 - v_1^2/2)^2 +
\frac{\lambda_2}{4} (\phi_2^2 - v_2^2)^2 + \frac{\lambda_{12}}{2} (\Phi_1^{\dagger}\Phi_1 - v_1^2/2)
(\phi_2^2 - v_2^2),
\label{eqn:potl}$$ where $\Phi_1$ is the SM Higgs doublet and $v_1$ is equivalent to the SM Higgs vev, specifically $v_1 = 246 $ GeV; $\phi_2$ is the additional field. As the new field is a singlet under the SM gauge, the familiar symmetry breaking mechanism, with respect to the electroweak sector, is unchanged. Hence, the SM Higgs vev and other electroweak parameters are the same as in the SM case.
A Yukawa interaction is introduced, coupling a dark, vector-like fermion to the scalar singlet field, as described by the following interaction Lagrangian: $$\mathcal{L}_{\mathrm{dark}} \supset -\frac{1}{2} \phi_2 \left( g_{L} \bar{\psi}_{L}\psi_{L}^{c} +
g_{R}\bar{\psi}_{R} \psi_{R}^{c} \right) + \mathrm{h.c.}
\label{eqn:Ldark}$$ There is a discrete symmetry in this case by which the fields transform as $$\begin{aligned}
\psi_{L} &\rightarrow i \psi_{L} \\
\psi_{R} &\rightarrow -i \psi_{R} \\
\phi_2 &\rightarrow -\phi_2.\end{aligned}$$ This symmetry forbids both a bare Dirac mass term, as well as any terms of odd order in $\phi_2$ in the scalar potential. Eq. \[eqn:potl\] represents the most general scalar potential consistent with the symmetry. Both scalar fields are allowed to acquire vevs, $$\Phi_1 \rightarrow \frac{1}{\sqrt{2}} {0 \choose v_1 + h_1(x)},
\quad \phi_2 \rightarrow v_2 + h_2(x).
\label{eqn:vevs}$$ Beginning with the scalar sector, after diagonalizing the mass terms, one finds the following mass eigenvalues:
$$m_{H,S}^2 = (\lambda_1 v_1^2 + \lambda_2 v_2^2) \pm \sqrt{(\lambda_1 v_1^2 -
\lambda_2 v_2^2)^2 +(\lambda_{12} v_1 v_2)^2},
\label{eqn:massvals1}$$
or $$m_{S}^2 = m_{H}^2 \pm \delta m^2; \quad \delta m^2 = 2 \sqrt{(\lambda_1 v_1^2 -
\lambda_2 v_2^2)^2 +(\lambda_{12} v_1 v_2)^2}.
\label{eqn:massvals2}$$ \[eqn:massvals\]
with mass eigenstates given by $$\left( \begin{array}{c} H \\ S \end{array} \right) = \begin{pmatrix} c_{\theta_h} & -
s_{\theta_h} \\ s_{\theta_h} & c_{\theta_h} \end{pmatrix}
\left( \begin{array}{c} h_1 \\ h_2\end{array} \right).
\label{eqn:mixing}$$ Here $c_{\theta_h}$ and $s_{\theta_h}$ denote $\cos\theta_h$ and $\sin\theta_h$ respectively. The mixing angle is given by the following expression: $$\tan 2\theta_h = \frac{-\lambda_{12} v_1 v_2}{\lambda_1v_1^2 - \lambda_2v_2^2}$$ with $\theta_h \in \left[ -\pi/2, \pi/2 \right]$. $H$ is taken to be the recently discovered Higgs boson, with mass fixed at $m_H = 125.09$ GeV [@Agashe:2014kda], and the mass of the additional scalar is allowed to be either lighter or heavier than $H$.
The scalar sector is described by three additional parameters — the vev of the singlet field, the mass of the new scalar, and the scalar-Higgs mixing angle, $\{ v_2, m_S, \theta_h \}$. In terms of these parameters, the potential coefficients are
$$\begin{aligned}
\lambda_1 &= \frac{1}{2v_1^2} \left( c_{\theta_h}^2 m_H^2 + s_{\theta_h}^2 m_S^2 \right)
\label{eqn:lambda1} \\
\lambda_2 &= \frac{1}{2 v_2^2} \left( s_{\theta_h}^2 m_H^2 + c_{\theta_h}^2 m_S^2 \right)
\label{eqn:lambda2} \\
\lambda_{12} & = \frac{(m_S^2 - m_H^2)}{2 v_1 v_2} \sin 2\theta_h
\label{eqn:lambda12}.\end{aligned}$$
\[eqn:lambdas\]
The couplings of $H$ and $S$ to SM particles are simply given by those of the SM Higgs, scaled respectively by $\cos\theta_h$ and $\sin\theta_h$. The extension of the scalar potential results in modified expressions for the scalar self-couplings. The triple scalar self-coupling coefficients for $H$ and $S$, denoted respectively by $g_H$ and $g_S$, are given by
$$\begin{aligned}
g_H = \frac{m_H^2}{2 v_1 v_2}\left( v_2 c_{\theta_h}^3 - v_1 s_{\theta_h}^3 \right) \label{eqn:gH} \\
g_S = \frac{m_S^2}{2 v_1 v_2}\left( v_1 c_{\theta_h}^3 + v_2 s_{\theta_h}^3 \right) \label{eqn:gS}.\end{aligned}$$
\[eqn:tricouplings\]
Furthermore, scalar mixing gives rise to additional interaction vertices between the two scalars, $H$-$H$-$S$ and $H$-$S$-$S$, with respective coupling strengths $$\begin{aligned}
\mu &= \frac{\sin 2\theta_h }{2 v_1 v_2} \left( v_1 s_{\theta_h} +
v_2 c_{\theta_h} \right) \left( m_H^2 + \frac{m_S^2}{2} \right) \label{eqn:mucoup} \\
\eta &= \frac{\sin 2\theta_h }{2 v_1 v_2} \left( -v_1 c_{\theta_h} +
v_2 s_{\theta_h} \right) \left( m_S^2 + \frac{m_H^2}{2} \right). \label{eqn:etacoup}\end{aligned}$$
The triple scalar interaction terms, both self-interactions and mixed scalar interactions are summarized in the following Lagrangian: $$-\mathcal{L} \supset g_H H^3 + g_S S^3 + \mu H^2 S + \eta H S^2.$$
Returning to the interaction term which describes the dark sector of the model, as given in eq. \[eqn:Ldark\], as $S$ acquires a vev, one generically obtains two Majorana mass states, $$\chi_1 = \begin{pmatrix} \psi_L \\ \psi_L^c \end{pmatrix}, \qquad
\chi_2 = \begin{pmatrix} \psi_R^c \\ \psi_R \end{pmatrix}$$ with masses $m_1 = g_L v_2$ and $m_2 = g_R v_2$. In this analysis the degenerate case is considered, in which the left and right couplings are related by $g_L = -g_R \equiv g$. This represents the simplest extension of this scalar extension to a dark matter scenario, giving a single dark matter candidate; the more general model described by an extended parameter space is left for future study. In the case of two mass-degenerate Majorana states, one can equivalently describe the picture in terms of a single Dirac fermion, $\chi = \left( \xi \quad \eta^c \right)^{T}$, where $$\xi = \frac{1}{\sqrt{2}} \left( \psi_L + i \psi_R^c \right) \quad
\eta= \frac{1}{\sqrt{2}} \left( \psi_L - i \psi_R^c \right),$$ with mass $m_{\chi} = g v_2$. The dark sector is then described by $$\bar{\chi} ( i\slashed{\partial} - m_{\chi}) \chi - \frac{m_{\chi}}{v_2} h_2 \bar{\chi} \chi.$$ With the addition of the dark matter mass, the model is then completely described by the set of four parameters, $$\{ v_2, m_S, \theta_h, m_{\chi} \}.$$
New Decay Channels {#scn:widths}
==================
Partial Widths
--------------
A dark matter candidate that couples to the Higgs introduces a new contribution to the Higgs invisible width, $\Gamma_{\mathrm{inv}}$. The associated experimental signature is a channel resulting in missing energy, denoted $\slashed{E}_{T}$. For $m_{\chi} < m_H/2$, the channel $H \rightarrow \chi \overline{\chi}$ gives a contribution $$\Gamma_{\mathrm{inv}} = \frac{s_{\theta_{h}}^2 m_H}{8\pi} \left( \frac{m_{\chi}}{v_2} \right)^2
\left (1 - \frac{4m_{\chi}^2}{m_H^2}\right)^{3/2}.
\label{eqn:inviswidth}$$ Moreover, additional non-SM decay modes arise from the scalar mixing terms. The total Higgs width receives new contributions from the decay to an $S$ pair. If $S$ is sufficiently light, the decay $H \rightarrow S S$ is kinematically allowed, with a partial width given by $$\Gamma_{HSS} = \frac{\eta^2}{8 \pi m_H} \left ( 1 - \frac{4m_S^2}{m_H^2} \right )^{1/2}.
\label{eqn:WdthHSS}$$
Scalar mass hierarchy Accessible decay modes
----------------------- ---------------------------
$m_S < m_H/2$ $H \rightarrow SS$
$m_H/2 < m_S < m_H$ $H \rightarrow Sf\bar{f}$
$m_H < m_S < 2m_H$ $S \rightarrow Hf\bar{f}$
$2m_H < m_S$ $S \rightarrow HH$
: Summary of the available scalar-to-scalar decay modes for various regions of the additional scalar mass, $m_S$, relative to the SM Higgs mass. It is implicit that for each case the invisible decay to $\chi\bar{\chi}$ is included for each scalar if its mass is greater than $2m_{\chi}$. For the intermediate mass ranges, $f$ may denote either a SM fermion, or $\chi$, if kinematically permitted.[]{data-label="tab:wdthtab"}
In light of experimental exclusion limits on a light Higgs, perhaps the other mass regions are the more interesting cases — that is either the case in which $m_S \sim m_H$, or the scenario in which the new scalar is much heavier and has not yet been discovered at the energies currently probed by colliders. If $S$ is very heavy, the dominant new scalar decay mode is its decay to an $H$ pair; the expression for the width is analogous to eq. \[eqn:WdthHSS\] upon the exchanges $m_H \leftrightarrow m_S$ and $\eta \rightarrow \mu$. For the intermediate regions, one may consider the three-body decays to one real and one virtual scalar, with the latter decaying to a fermion pair. More specifically, for the case in which $m_H/2 < m_S < m_H$, the accessible decay is $H \rightarrow SS^{*} \rightarrow Sf\bar{f}$, where $f$ may be any SM fermion (excluding the top quark), or $\chi$, if the DM is sufficiently light. The expressions for the relevant three-body widths (distinguishing between $f_{SM}$ and $\chi$) are given by the following:
$$\begin{aligned}
\Gamma(H \rightarrow Sf\overline{f}) =& \frac{s_{\theta_h}^2 \eta^2}{8\pi^3}
\frac{m_S^2}{m_H^3} \left( \frac{m_f}{v_1}\right)^2 I(2m_f/m_S, m_H/m_S) ,
\\
\Gamma(H \rightarrow S\chi\overline{\chi}) =& \frac{c_{\theta_h}^2 \eta^2}{8\pi^3}
\frac{m_S^2}{m_H^3}\left( \frac{m_{\chi}}{v_2}\right)^2 I(2m_{\chi}/m_S, m_H/m_S).\end{aligned}$$
Here $$I(y,z) = \int_{1}^{x_{max}} \mathrm{d}x \frac{\sqrt{x^2 - 1}}{(z - 2x)^2}
\frac{(1 + z^2 - y^2 - 2zx)^{3/2}}{(1 + z^2 - 2zx)^{1/2}},$$
where $x_{max} = (1 + z^2 - y^2)/(2z)$. The $y$ term is neglected for $y = 2m_f/m_S$. As would be expected, the expressions for the intermediate case with $S$ heavier than $H$ are obtained by exchanging $m_H \leftrightarrow m_S$, $\eta \rightarrow \mu$, and $s_{\theta_h} \leftrightarrow c_{\theta_h}$. The mass regions and accessible decay channels are summarized in table \[tab:wdthtab\].
Scalar Couplings {#sscn:couplings}
----------------
[0.45]{} ![Value of the $H$-$H$-$S$ coupling constant, $\mu$, over various regions of scalar parameter space.The left figure gives the dependence in $(m_S, v_2)$ space for a discrete value of the mixing angle, while the right shows the dependence over $(m_S, s_{\theta_h})$ space for chosen value of $v_2$.[]{data-label="fig:mucoup"}](couplings1a.png "fig:"){width="\linewidth"}
[0.45]{} ![Value of the $H$-$H$-$S$ coupling constant, $\mu$, over various regions of scalar parameter space.The left figure gives the dependence in $(m_S, v_2)$ space for a discrete value of the mixing angle, while the right shows the dependence over $(m_S, s_{\theta_h})$ space for chosen value of $v_2$.[]{data-label="fig:mucoup"}](couplings1b.png "fig:"){width="\linewidth"}
[0.45]{} ![Dependence of $H$-$S$-$S$ coupling, $\eta$, on the scalar parameter space. The description of subfigures is as given in figure \[fig:mucoup\].[]{data-label="fig:etacoup"}](couplings2a.png "fig:"){width="\linewidth"}
[0.45]{} ![Dependence of $H$-$S$-$S$ coupling, $\eta$, on the scalar parameter space. The description of subfigures is as given in figure \[fig:mucoup\].[]{data-label="fig:etacoup"}](couplings2b.png "fig:"){width="\linewidth"}
The interaction vertices between $H$ and $S$, arising from the mixed cubic terms in the scalar potential, provide possibly significant new contributions to decay widths and production processes. [^1] It can be seen from eqs. \[eqn:mucoup\] and \[eqn:etacoup\] that these couplings may become large for certain values of the mass and vev of the additional scalar. Physical quantities which would otherwise be suppressed — either by a small mixing angle or by nature of being a higher order effect — may be non-negligible. Figures \[fig:mucoup\] and \[fig:etacoup\] show the dependence of the scalar couplings, $\mu$ and $\eta$ respectively, on the scalar parameters.
Three-Body Decays {#sscn:3wdths}
-----------------
[0.49]{} ![Relative magnitude of the three-body decay widths to a scalar and SM fermion pair. A simplified branching ratio is shown, taken as the ratio of the three-body width to the total visible width. The width $H \rightarrow S f \bar{f}$ is shown in the left figure, and $S \rightarrow H f \bar{f}$ on the right. In both cases, the scalar vev is fixed to be $v_2 = 200$ GeV, and the mixing angle is varied discretely.[]{data-label="fig:3wdthsff"}](Hwidth_3body_ff.png "fig:"){width="\linewidth"}
[0.49]{} ![Relative magnitude of the three-body decay widths to a scalar and SM fermion pair. A simplified branching ratio is shown, taken as the ratio of the three-body width to the total visible width. The width $H \rightarrow S f \bar{f}$ is shown in the left figure, and $S \rightarrow H f \bar{f}$ on the right. In both cases, the scalar vev is fixed to be $v_2 = 200$ GeV, and the mixing angle is varied discretely.[]{data-label="fig:3wdthsff"}](Swidth_3body_ff.png "fig:"){width="\linewidth"}
[0.45]{} ![Branching ratio for the three-body decays to a scalar and dark matter pair. The dark matter mass is fixed at $m_{\chi} = 10$ GeV, and the scalar mixing angle is set to $\sin\theta_h = 0.5$. Branching ratios for $H \rightarrow S \chi \bar{\chi}$ and $S \rightarrow H \chi \bar{\chi}$ are shown in the left and right figures respectively.[]{data-label="fig:3wdthsXX"}](Hwidth_3body_xx.png "fig:"){width="\linewidth"}
[0.45]{} ![Branching ratio for the three-body decays to a scalar and dark matter pair. The dark matter mass is fixed at $m_{\chi} = 10$ GeV, and the scalar mixing angle is set to $\sin\theta_h = 0.5$. Branching ratios for $H \rightarrow S \chi \bar{\chi}$ and $S \rightarrow H \chi \bar{\chi}$ are shown in the left and right figures respectively.[]{data-label="fig:3wdthsXX"}](Swidth_3body_xx.png "fig:"){width="\linewidth"}
One typically expects the contribution of the three-body widths to be subdominant, due to suppression by the phase space integration. As previously noted, for certain regions of parameter space, the tri-scalar couplings may be large enough to partially offset these effects. In the intermediate mass regions, where the three-body decays are kinematically relevant, the magnitudes of these couplings grow very large only for extremal values of the mixing angle or the scalar vev, however it warrants further consideration to verify that the three-body decay widths remain subdominant, and may be neglected. A more detailed exploration of the relative size of the three-body contribution is presented in the following. The partial width of the three-body decay is calculated and compared with the more dominant contributions.
The mass regions of interest relevant for the three-body decays of $H$ and $S$ are the two intermediate ranges — that is, $m_H/2 < m_S < m_H$ and $m_H < m_S < 2 m_H$. The different choices of final-state fermion pair – either a SM fermion, or $\chi$ — are presented separately. The decay $H(S)$ to $S(H)$ and a SM fermion pair is shown in figure \[fig:3wdthsff\]. In this case, the reference point for comparison is chosen to be the total width for decay to SM particles, $$\begin{aligned}
\Gamma_{SM}^{H} &= c_{\theta_h}^2 \Gamma_{h}^{SM}(m_H) \\
\Gamma_{SM}^{S} &= s_{\theta_h}^2 \Gamma_{h}^{SM}(m_S),
\label{eqn:viswidths}\end{aligned}$$ i.e., the total Higgs width under the SM, scaled by the appropriate factor of the mixing angle. This effectively gives the branching ratio under a few simplifications. The other new contributions to the total width are the two body decays to a scalar pair, which is kinematically forbidden in the intermediate mass regions, and to $\chi\bar{\chi}$, which is neglected for simplicity; the inclusion of the invisible decay to the total width would simply result in a further overall reduction of the branching ratio. The result is presented as a function of $m_S$, discretely varying the mixing angle. It can be seen in figures \[fig:mucoup\] and \[fig:etacoup\], that over the intermediate mass intervals, both couplings $\mu$ and $\eta$ show little dependence on the scalar vev, except for a small increase towards very small values of $v_2$. Hence, its value is fixed to $v_2 = 200$ GeV. Several values of $\sin\theta_h$ are shown for the $H$ decay width, given in the left figure. Despite the non-trivial functional dependence of the coupling, the primary effect of varying the mixing angle is an overall scaling of the width, $\propto \sin^2\theta_h$. This dependence vanishes in the case of the corresponding $S$ decay, due to the factor of $\sin^2\theta_h$ in the normalizing visible width, as seen in eq. \[eqn:viswidths\]. A very minimal effect was seen upon varying the mixing angle, and so only one value is shown.
When considering the analogous decay to $\chi$ rather than a SM fermion, the dark matter mass enters into the three-body width; the invisible decay is thus included in the analysis as the parameter space is already extended. The normalizing quantity is chosen as the total width, giving the branching ratio for $H(S) \rightarrow S(H) \chi\bar{\chi}$ . The results are shown in figure \[fig:3wdthsXX\]. In the given result, the value of $m_{\chi}$ was chosen to be $10$ GeV. The primary influence of the $\chi$ mass on the magnitude of the three-body width is through its contribution to the coupling, through the ratio $m_{\chi}/v_2$, and results primarily in an overall scaling. For most of the scalar mass range of interest however, the values of the dark matter mass for which the relevant decays are kinematically permitted, are at most of order $\sim 10$ GeV, i.e. $m_{\chi} \lesssim 50$ GeV. Hence, as the order of $m_{\chi}$ does not change drastically in comparison to the variation of the scalar vev, its value is fixed such that it is near the larger end of the allowed range, and the three-body decays of both $H$ and $S$ are allowed over most of the $m_S$ range considered. In keeping with the aim of investigating the possible extremal values of these partial widths, a further simplification is made with respect to the mixing angle. The most apparent effect of the mixing angle on the overall magnitude is an overall scaling by $\sim \sin\theta_h$; therefore it is set to roughly what will be later shown to be the maximum value allowed by experimental constraints, with $\sin\theta_h=0.5$.
Over most of parameter space, the three-body contributions to the $H$ and $S$ widths are considerably small; except for large values of the mixing angle or small values of the vev, and values of $m_S \simeq m_H/2$ or $\simeq 2 m_H$, in most cases $BR \lesssim 10^{-4}$. For most parameter values, one is justified in neglecting these contributions in calculating the total width. In the case of the decay involving a $\chi\bar{\chi}$ pair, the branching ratio does become more significant for very small values of the new scalar vev, i.e. $\mathcal{O}(10)$ GeV, or more specifically, when $v_2 \sim m_{\chi}$. However, as will be discussed later, such small values of $v_2$ are disfavoured by other experimental and theoretical limits, and the remaining analysis focuses on vevs of order $100$ GeV and larger, neglecting these three-body contributions. The couplings $\mu$ and $\eta$ do assume larger values for regions of parameter space over which the three-body decays are relevant, but they only become very large (i.e. orders of magnitude larger) in regions where these decays are kinematically forbidden, and the quantity of interest is not the total widths. This may have relevance towards off-shell scalar production processes, a topic which will be further discussed in later sections.
LHC Limits {#scn:LHC}
==========
Current LHC Higgs data have been found to be consistent with the SM Higgs sector. Such a result places limits on a scalar extension, both with respect to the modification of production cross sections of the SM-like Higgs, as well as the fact that a second scalar has evaded discovery as yet. This scenario, in which a new scalar has escaped detection by hadron colliders at the currently achieved energies and luminosities, generally constrains either the mixing parameter to be small or the scalar mass to be heavy. The limits posed by LHC Higgs data are reviewed here, and limits resulting from new contributions to the Higgs widths are determined.
Higgs Signal Strength and Exclusion Searches {#sscn:HBHS}
--------------------------------------------
[0.46]{} ![Limits on scalar parameter space, $(m_S, |\sin \theta_h|)$ from current collider data. Left: Regions excluded by LEP and Tevatron/LHC Higgs exclusion searches, at $95\%$ c.l. Right: Preferred regions compatible with measured LHC Higgs signal strength. The regions allowed at $68\%$, $95\%$, and $99\%$ c.l. are shown.[]{data-label="fig:HBHSlim"}](HB_excl_single.png "fig:"){width="\linewidth"}
[0.47]{} ![Limits on scalar parameter space, $(m_S, |\sin \theta_h|)$ from current collider data. Left: Regions excluded by LEP and Tevatron/LHC Higgs exclusion searches, at $95\%$ c.l. Right: Preferred regions compatible with measured LHC Higgs signal strength. The regions allowed at $68\%$, $95\%$, and $99\%$ c.l. are shown.[]{data-label="fig:HBHSlim"}](HS_excl_single.png "fig:"){width="\linewidth"}
In the following section, the compatibility of a mixed two-Higgs scenario with current hadronic collider data is reviewed. The scalar parameter space is strictly constrained by Higgs searches, with respect to both past exclusion searches, and measured signals, particularly in light of the recent Higgs discovery. The precise limits on a scalar singlet extension posed by LHC Higgs data have been determined by ref. [@Robens:2015gla]. Since publication of this work, updated LHC Higgs results have been released, as well as a version of the publicly available code used throughout the analysis, which incorporates these new data. There is also a slight distinction from the present analysis, in the convention chosen to define the mass eigenstates, as well as the inclusion here of the nonzero $H \rightarrow SS$ and $S \rightarrow HH$ branching ratios. [^2] In light of this, the limits are reproduced here for completion, under these modifications, following the procedure of [@Robens:2015gla]. [^3] The publicly available code <span style="font-variant:small-caps;">HiggsBoundsv4.3.1</span> [@Bechtle:2015pma; @arXiv:1311.0055; @arXiv:1301.2345; @arXiv:1102.1898; @arXiv:0811.4169] is used to determine the limiting region in the space defined by the scalar mass and mixing angle, by exclusion from LEP, Tevatron and LHC data. In particular, LEP searches from refs. [@Searches:2001aa; @Abbiendi:2002qp; @Searches:2001ab; @Abdallah:2003ry; @Abbiendi:2007ac; @Searches:2001ac; @Abbiendi:2001kp; @Achard:2004cf; @Schael:2006cr; @Abdallah:2003wd; @Abdallah:2004wy] are used, and experimental LHC/Tevatron results are those in refs. [@ATLASnotes; @LHWGnotes; @Aad:2014xva; @Chatrchyan:2012sn; @Chatrchyan:2014tja; @Aad:2012tfa; @Aad:2012tj; @Chatrchyan:2012tx; @Aad:2011ec; @ATLAS:2012ac; @ATLAS:2011af; @Khachatryan:2014wca; @ATLAS:2011ae; @Chatrchyan:2012dg; @Aad:2014fia; @Aad:2014yja; @Aad:2014iia; @Aad:2011rv; @ATLAS:2011aa; @ATLAS:2012ad; @Chatrchyan:2013vaa; @ATLAS:2012ae; @Aad:2014ioa; @Chatrchyan:2012ft; @CDFnotes; @D0notes; @Abazov:2010ci; @Aaltonen:2009vf; @Abazov:2010zk; @Aaltonen:2011nh; @Aaltonen:2010cm; @Benjamin:2011sv; @Group:2012zca; @Aaltonen:2009ke; @Abazov:2011qz; @Abazov:2008wg; @Aaltonen:2012qt; @Abazov:2009yi; @Benjamin:2010xb; @Abazov:2011jh; @Abazov:2011ed; @TEVNPHWorking:2011aa; @Abazov:2010hn; @Aaltonen:2008ec; @Aaltonen:2009dh; @Abazov:2009aa; @Abazov:2010ct; @Abazov:2009kq], with more recent LHC results from Refs. [@Aad:2014vgg; @arXiv:1509.04670; @arXiv:1506.02301; @arXiv:1504.00936; @arXiv:1507.05930; @arXiv:1509.00389] . The region allowed at $95\%$ c.l. is shown in figure \[fig:HBHSlim\], with regions excluded by LEP and Tevatron/LHC presented separately.
While the limits posed by exclusion searches constrain the model in the absence of a signal, the recent Higgs discovery necessitates the complementary limits from compatibility with this observed signal. Results are determined using <span style="font-variant:small-caps;">HiggsSignalsv1.4.0</span> [@Stal:2013hwa; @Bechtle:2013xfa]. The experimental results used to obtain the constraints are given in refs. [@Aad:2015gba; @Aad:2014eha; @Aad:2015vsa; @ATLAS:2014aga; @ATLAS-CONF-2015-005; @Aad:2014eva; @Aad:2014xzb; @Khachatryan:2014ira; @Chatrchyan:2014vua; @Chatrchyan:2013iaa; @Chatrchyan:2013mxa; @CMS-PAS-HIG-13-005]. For light scalars, $m_S < 80$ GeV, the strictest limit is found by LEP exclusion, giving an upper bound of $|\sin\theta_h| \lesssim 0.15$, while the upper limit from LHC exclusion searches is between $|\sin\theta_h | \lesssim 0.3$ at its most stringent, in the region $m_S \sim 170$–$250$ GeV, and is as large as 0.4–0.6 for higher masses. The constraint posed by the measured signal strength in Higgs production at LHC, which places an upper bound (at $95\%$ c.l.) on the mixing angle of $| \sin\theta_h | \lesssim 0.5$, for scalar masses $m_S < 100$ GeV or $m_S > 150$ GeV, and $| \sin\theta_h | \lesssim 0.4$ for scalar masses within a $25$ GeV window of $125$ GeV. When $m_S$ approaches $m_H$, within the detector mass resolution, this represents the degenerate limit which simply reproduces the SM signal — in other words, the signal strength for a $125$ GeV scalar with SM couplings scaled by $\cos^2 \theta_h + \sin^2 \theta_h = 1$ — hence the mixing angle is unconstrained in this narrow region. Figure \[fig:HBHSlim\] shows the regions preferred by LHC Higgs signal compatibility.
Additional limits arise from electroweak precision data (EWPD), and theoretical constraints such as perturbativity and renormalization group (RG) evolution of the couplings, which may tighten the mixing angle upper bound, to $|\sin\theta_h | \lesssim 0.3$ in the heavier $S$ region — $m_S \gtrsim 600$ GeV [@Robens:2015gla]. Such considerations also place further lower bounds on the $S$ vev in some cases, again disfavouring small $v_2$; the restrictions were not in conflict with any of the present results. Similar analyses in refs. [@Martin-Lozano:2015dja; @Falkowski:2015iwa] also include such theoretical considerations, obtaining similar bounds.
Invisible Signals {#sscn:invis}
-----------------
[0.36]{} ![Limits on $(m_{\chi}, v_2)$ parameter space, from ATLAS upper bounds on signal strength in Higgs invisible decays, for discrete values of the mixing angle, below the upper bound set in section \[sscn:HBHS\]. The regions below the curves are excluded at $95\%$ c.l.[]{data-label="fig:invisWdth"}](invisible_ATLAS_1.png "fig:"){width="\linewidth"}
[0.3]{} ![Limits on $(m_{\chi}, v_2)$ parameter space, from ATLAS upper bounds on signal strength in Higgs invisible decays, for discrete values of the mixing angle, below the upper bound set in section \[sscn:HBHS\]. The regions below the curves are excluded at $95\%$ c.l.[]{data-label="fig:invisWdth"}](invisible_ATLAS_2.png "fig:"){width="\linewidth"}
[0.3]{} ![Limits on $(m_{\chi}, v_2)$ parameter space, from ATLAS upper bounds on signal strength in Higgs invisible decays, for discrete values of the mixing angle, below the upper bound set in section \[sscn:HBHS\]. The regions below the curves are excluded at $95\%$ c.l.[]{data-label="fig:invisWdth"}](invisible_ATLAS_3.png "fig:"){width="\linewidth"}
The extension to the SM scalar sector results in a strict constraint posed by Higgs production searches and measurements, due to scaling of Higgs couplings to SM particles and the allowed mass of a new Higgs-like scalar; the allowed values of the scalar parameters are described in the previous subsection. The remaining parameter — the dark matter mass — becomes significant when focusing on searches with invisible signatures. A direct limit on the dark matter mass, as well as the additional scalar vev, is found in Higgs production with subsequent invisible decay channels.
A recent ATLAS search [@Aad:2015pla] for invisible Higgs decays, in both vector boson fusion (VBF), and associated vector boson production modes, gives an upper limit on the Higgs invisible branching ratio. ATLAS obtains limits from three production channels — vector boson fusion, associated Z production with subsequent leptonic Z decays, and associated vector boson, $V$ ($W$ or $Z$), with hadronic $W$/$Z$ decay, as well the combined limit from the three. The search corresponds to $4.7$ fb$^{-1}$ of data at centre of mass energy $7$ TeV, and $20.3$ fb$^{-1}$ at $8$ TeV. Upper limits are given at $95\%$ c.l. on the production cross section times invisible branching ratio signal strength. This signal strength, denoted here by $\zeta$ is simply the production cross section times branching ratio, normalized by the SM production cross section $$\zeta = \frac{\sigma}{\sigma_{SM}} \times \mathcal{B}(h \rightarrow \mathrm{inv}).
\label{eqn:zetadefn}$$ More specifically, for $H$, this becomes $$\zeta_{H} = c_{\theta_h}^2 \frac{\Gamma_{\mathrm{inv}}}{\Gamma_{\mathrm{tot}}}
= \frac{c_{\theta_h}^2 s_{\theta_h}^2 \Gamma_{\mathrm{inv}}}{c_{\theta_h}^2 \Gamma_{SM}
+ s_{\theta_h}^2 \Gamma_{\mathrm{inv}}}.
\label{eqn:zeta}$$ Here, $\Gamma_{\mathrm{inv}}(m_{\chi}, v_2)$ is as given in eq. \[eqn:inviswidth\], with the factor $s_{\theta_h}$ written explicitly. The corresponding observable for $S$ production and decay is obtained under exchange $m_H \rightarrow m_{S}$ and $s_{\theta_h} \leftrightarrow c_{\theta_h}$.
ATLAS sets upper limits on the normalized signal strength of $0.28$, $0.75$, and $0.78$ in the VBF, leptonic $ZH$, and hadronic $VH$ channels, with a combined limit of $0.25$. The resulting bound on $(m_{\chi}, v_{2})$ is given in figure \[fig:invisWdth\]. The $95\%$ c.l. exclusion is shown, using the individual channels, as well the combined results. The mixing angle is varied discretely, with values below the upper limit obtained in section \[sscn:HBHS\]. The constraint represents the experimental bound to the invisible $H$ width, at mass $125$ GeV. A similar search by CMS [@Chatrchyan:2014tja] gives an upper bound on the invisible branching ratio as a function of Higgs mass, although the limit at $m_H=125$ GeV is looser than that given by ref. [@Aad:2015pla]. As this gives a bound for a general scalar mass, this is applied to the invisible $S$ width. For mixing angle values which remain consistent with the Higgs production signal strength however, the corresponding limit applied to the invisible branching ratio of $S$ does not significantly constrain the parameter space.
Total Higgs Width {#sscn:totalwidth}
-----------------
An analysis by CMS obtains a limit on the width of the Higgs boson, by a method which uses the ratio between on-shell and off-shell cross section measurements [@Khachatryan:2015mma]. CMS obtains an upper limit on the total Higgs width of $26$ MeV, at $95 \%$ c.l.
[0.36]{} ![Limit on dark matter parameters resulting from invisible contribution to total Higgs width. Results are obtained using the CMS upper bound on the total Higgs width, assuming the absence of additional two-body scalar decays, i.e. in the heavy $S$ regime. The shaded regions are excluded at $95\%$ c.l.[]{data-label="fig:totwidthinv"}](totalwidth_inv_1.png "fig:"){width="\linewidth"}
[0.3]{} ![Limit on dark matter parameters resulting from invisible contribution to total Higgs width. Results are obtained using the CMS upper bound on the total Higgs width, assuming the absence of additional two-body scalar decays, i.e. in the heavy $S$ regime. The shaded regions are excluded at $95\%$ c.l.[]{data-label="fig:totwidthinv"}](totalwidth_inv_2.png "fig:"){width="\linewidth"}
[0.3]{} ![Limit on dark matter parameters resulting from invisible contribution to total Higgs width. Results are obtained using the CMS upper bound on the total Higgs width, assuming the absence of additional two-body scalar decays, i.e. in the heavy $S$ regime. The shaded regions are excluded at $95\%$ c.l.[]{data-label="fig:totwidthinv"}](totalwidth_inv_3.png "fig:"){width="\linewidth"}
For most of the scalar mass range, this provides an additional constraint on the dark matter mass and coupling; if $m_{S} > m_{H}/2 \sim 63$ GeV, the decay of $H$ to an $S$ pair is kinematically forbidden, and as seen in section \[sscn:3wdths\], the three-body contributions are negligible. Assuming that the only non-SM contribution to the Higgs width is the invisible decay $H \rightarrow \chi \bar{\chi}$, the resulting limit on the dark matter mass and scalar vev is given in figure \[fig:totwidthinv\]. The constraint posed by measured invisible decays however, gives the stricter limit.
[0.37]{} ![Limit resulting from contribution of $H \rightarrow S S$ decay to total Higgs width. The shaded regions represent the region excluded at $95\%$ c.l. by the CMS limit. From the top row down, $\sin \theta_h$, $v_2$, $m_S$ are each varied discretely in turn. The blue curve/region corresponds to the case for which the decay to $SS$ is the only non-SM contribution, and red to the case which also includes the invisible channel.[]{data-label="fig:totwidth"}](totalwidth_mS-v2_1.png "fig:"){width="\linewidth"}
[0.3]{} ![Limit resulting from contribution of $H \rightarrow S S$ decay to total Higgs width. The shaded regions represent the region excluded at $95\%$ c.l. by the CMS limit. From the top row down, $\sin \theta_h$, $v_2$, $m_S$ are each varied discretely in turn. The blue curve/region corresponds to the case for which the decay to $SS$ is the only non-SM contribution, and red to the case which also includes the invisible channel.[]{data-label="fig:totwidth"}](totalwidth_mS-v2_2.png "fig:"){width="\linewidth"}
[0.3]{} ![Limit resulting from contribution of $H \rightarrow S S$ decay to total Higgs width. The shaded regions represent the region excluded at $95\%$ c.l. by the CMS limit. From the top row down, $\sin \theta_h$, $v_2$, $m_S$ are each varied discretely in turn. The blue curve/region corresponds to the case for which the decay to $SS$ is the only non-SM contribution, and red to the case which also includes the invisible channel.[]{data-label="fig:totwidth"}](totalwidth_mS-v2_3.png "fig:"){width="\linewidth"}
[0.37]{} ![Limit resulting from contribution of $H \rightarrow S S$ decay to total Higgs width. The shaded regions represent the region excluded at $95\%$ c.l. by the CMS limit. From the top row down, $\sin \theta_h$, $v_2$, $m_S$ are each varied discretely in turn. The blue curve/region corresponds to the case for which the decay to $SS$ is the only non-SM contribution, and red to the case which also includes the invisible channel.[]{data-label="fig:totwidth"}](totalwidth_mS-sh_1.png "fig:"){width="\linewidth"}
[0.3]{} ![Limit resulting from contribution of $H \rightarrow S S$ decay to total Higgs width. The shaded regions represent the region excluded at $95\%$ c.l. by the CMS limit. From the top row down, $\sin \theta_h$, $v_2$, $m_S$ are each varied discretely in turn. The blue curve/region corresponds to the case for which the decay to $SS$ is the only non-SM contribution, and red to the case which also includes the invisible channel.[]{data-label="fig:totwidth"}](totalwidth_mS-sh_2.png "fig:"){width="\linewidth"}
[0.3]{} ![Limit resulting from contribution of $H \rightarrow S S$ decay to total Higgs width. The shaded regions represent the region excluded at $95\%$ c.l. by the CMS limit. From the top row down, $\sin \theta_h$, $v_2$, $m_S$ are each varied discretely in turn. The blue curve/region corresponds to the case for which the decay to $SS$ is the only non-SM contribution, and red to the case which also includes the invisible channel.[]{data-label="fig:totwidth"}](totalwidth_mS-sh_3.png "fig:"){width="\linewidth"}
[0.37]{} ![Limit resulting from contribution of $H \rightarrow S S$ decay to total Higgs width. The shaded regions represent the region excluded at $95\%$ c.l. by the CMS limit. From the top row down, $\sin \theta_h$, $v_2$, $m_S$ are each varied discretely in turn. The blue curve/region corresponds to the case for which the decay to $SS$ is the only non-SM contribution, and red to the case which also includes the invisible channel.[]{data-label="fig:totwidth"}](totalwidth_v2-sh_1.png "fig:"){width="\linewidth"}
[0.3]{} ![Limit resulting from contribution of $H \rightarrow S S$ decay to total Higgs width. The shaded regions represent the region excluded at $95\%$ c.l. by the CMS limit. From the top row down, $\sin \theta_h$, $v_2$, $m_S$ are each varied discretely in turn. The blue curve/region corresponds to the case for which the decay to $SS$ is the only non-SM contribution, and red to the case which also includes the invisible channel.[]{data-label="fig:totwidth"}](totalwidth_v2-sh_2.png "fig:"){width="\linewidth"}
[0.3]{} ![Limit resulting from contribution of $H \rightarrow S S$ decay to total Higgs width. The shaded regions represent the region excluded at $95\%$ c.l. by the CMS limit. From the top row down, $\sin \theta_h$, $v_2$, $m_S$ are each varied discretely in turn. The blue curve/region corresponds to the case for which the decay to $SS$ is the only non-SM contribution, and red to the case which also includes the invisible channel.[]{data-label="fig:totwidth"}](totalwidth_v2-sh_3.png "fig:"){width="\linewidth"}
In the light $S$ regime — that is, in the subregion of parameter space for which $m_S < m_H/2$ — the total width also receives a contribution from the decay channel $H \rightarrow S S$. This provides additional direct experimental sensitivity on the allowed values of the scalar vev, as it enters into the mixed scalar coupling, which depends additionally on the mixing angle and scalar mass. Each of these three parameters, $(\sin\theta_h, v_2, m_S)$, is varied discretely in turn, giving the exclusion region in the other two. The results are given in figure \[fig:totwidth\]. Both the case for which the scalar decay channel is the only new contribution (which is the case for heavy dark matter), as well as that for which both scalar and invisible decays are present. In the latter case, $m_{\chi}$ is chosen such that the invisible contribution is below the upper bound over most of the $v_2$ range considered, as shown in figures \[fig:invisWdth\] and \[fig:totwidthinv\]; the value is taken to be $m_{\chi} = 10$ GeV.
The resulting exclusion regions in figure \[fig:totwidth\] show the extremal allowed values of the mixing angle and scalar vev, which are more or less proportionate as each is varied. The figures in the top row show a lower bound on $v_2$, which increases with the mixing angle, and remains constant with $m_S$, in the kinematically allowed range. Varying $v_2$ discretely (shown in the middle row), one sees an upper bound on $|\sin \theta_h|$ which relaxes with increasing $v_2$. The regions in $(v_2, s_h)$ (bottom row) illustrate this behaviour. Smaller values of the scalar vev, $v_2 \lesssim 100$ GeV are disfavoured, except for vary small values of the mixing angle; the allowed range of $\sin \theta_h$ is relaxed with increasing $v_2$, and for $v_2 > 200$ GeV the upper bound on $\sin \theta_h$ is well above that imposed by the LHC signal strength.
ILC Prospects {#scn:ILC}
=============
The current experimental results impose that an additional Higgs-like scalar, or any general extended Higgs sector, has evaded detection at the current collider energies. A possible next step which suggests itself, is to move toward a precision environment and search for signatures of a scalar extension, through its influence on Higgs precision measurements. In the following discussion, the potential discovery reach, and possible signatures of a mixed Higgs scenario are considered at the ILC.
By nature of being a lepton collider the ILC will offer substantially cleaner signals, and is therefore much better suited to precision measurements; such a task is not possible at the LHC due to the significant QCD background. The cleaner environment, as well as detectors and instrumentation associated with a lepton collider, allow for the possibility to reconstruct Higgses more cleanly and in additional decay channels. Furthermore, the capability for polarized beams adds greater sensitivity to spin effects, by allowing greater control on the initial helicity states. This is advantageous in that beam polarization provides the ability to fully characterize a process with respect to the differences in interactions and couplings between left and right-handed particles. Beam polarization is particularly important in electroweak processes, which are sensitive to spin. There are several key motivations for using polarized beams at ILC, which vary with the particular type of process under study. Firstly, oppositely polarized beams enhance luminosity in electron-positron annihilation processes, as an electron annihilates a positron of opposite helicity. Beam polarization asymmetry is also an informative variable in precision electroweak measurements, via the $e^+ e^- \rightarrow f \bar{f}$ process. Another advantage is the ability to increase the signal to background ratio, in processes that occur predominantly through a specific initial helicity configuration, thereby optimizing either certain SM signals or new physics searches. In light of its proposed features and capabilities, Higgs parameters such as the top Yukawa coupling, Higgs branching ratios, couplings to vector bosons, and the Higgs self coupling (of particular focus here), are perhaps well within the reach of the ILC.
[0.4]{} ![Feynman diagrams for leptonic Higgs production. From left to right, and top to bottom, subprocesses are: vector boson fusion (VBF), Higgs-strahlung, and for higher energies, associated top production ($t\bar{t} H$).[]{data-label="fig:leptHiggsprod"}](leptVBF.png "fig:"){width="0.9\linewidth"}
[0.4]{} ![Feynman diagrams for leptonic Higgs production. From left to right, and top to bottom, subprocesses are: vector boson fusion (VBF), Higgs-strahlung, and for higher energies, associated top production ($t\bar{t} H$).[]{data-label="fig:leptHiggsprod"}](leptHiggsstrahlung.png "fig:"){width="\linewidth"}
[0.45]{} ![Feynman diagrams for leptonic Higgs production. From left to right, and top to bottom, subprocesses are: vector boson fusion (VBF), Higgs-strahlung, and for higher energies, associated top production ($t\bar{t} H$).[]{data-label="fig:leptHiggsprod"}](leptttH.png "fig:"){width="\linewidth"}
In lepton collisions, one loses the gluonic contributions that dominate LHC Higgs production, and the remaining processes are the electroweak modes. The leading modes in leptonic Higgs production are the Higgs-strahlung process, and $W$ boson fusion. Additional contributions are present in $Z$ boson fusion, and associated heavy quark production at higher energies. The corresponding Feynman diagrams are shown in figure \[fig:leptHiggsprod\]. Diagrams correspond interchangeably to either $H$ or $S$ production, with the couplings scaled by the appropriate factor of the mixing angle. Naturally, the Higgs production cross section is larger in hadron collisions than in lepton collisions, due to the leading QCD processes; by experimental design the LHC optimizes Higgs production, with discovery being the priority. Although production cross sections are smaller at a lepton collider, higher order effects are much less significant in the case of electroweak processes, offering greater ease of calculability and less theoretical uncertainty associated with higher order effects.
![Leptonic Higgs production cross section as a function of centre of mass energy, for beam polarization $P(e^{+}, e^{-}) = (30, -80)$. The Higgs mass is taken to be $125$ GeV.[]{data-label="fig:leptXSCM"}](higgsprod_xs_lept_vEcm_Pol.png){width="0.75\linewidth"}
Cross sections for Higgs production in $e^- e^+$ collisions are determined at LO, using <span style="font-variant:small-caps;">MadGraph5</span> [@Alwall:2014hca]. The cross sections as a function of centre of mass energy are shown in figure \[fig:leptXSCM\], for beam polarization $(P_{e^+}, P_{e^-}) = (30, -80)$. More specifically, this denotes an electron beam which is $80\%$ left-polarized, and a positron beam that is $30\%$ right-polarized. The formal definition of polarization is $$P = \frac{N_R - N_L}{N_R + N_L},$$ where $N_R$ and $N_L$ are the number of particles with spin parallel or antiparallel to the direction of motion. An equivalent interpretation of this number, is the percentage of events for which the helicity is known, with positive or negative signifying right or left.
A possible experimental program includes running at centre of mass energies $250$ GeV, $500$ GeV, and $1$ TeV, with ability to obtain beam polarizations of $(P_{e^+}, P_{e^-}) = (30, -80)$ at the lower two energies, and $(20, -80)$ at $1$ TeV [@TDRvol1]. The displayed cross sections correspond to production of the SM-like Higgs, i.e. $m_H=125$ GeV, and omitting the mixing angle factor $\cos^2\theta_h$.
[0.4]{} ![Higgs production cross section in electron-positron collisions, for centre of mass energies, $\sqrt{s} = 250, 500$ GeV, and $1$ TeV. Cross sections are given at leading order, for electron polarization $-80$, and positron polarization $30$ for $250$ and $500$ GeV, and $20$ for $1$ TeV.[]{data-label="fig:leptXSmass"}](higgsprod_xs_lept_vMh_Pol_250GeV.png "fig:"){width="\linewidth"}
[0.4]{} ![Higgs production cross section in electron-positron collisions, for centre of mass energies, $\sqrt{s} = 250, 500$ GeV, and $1$ TeV. Cross sections are given at leading order, for electron polarization $-80$, and positron polarization $30$ for $250$ and $500$ GeV, and $20$ for $1$ TeV.[]{data-label="fig:leptXSmass"}](higgsprod_xs_lept_vMh_Pol_500GeV.png "fig:"){width="\linewidth"}
[0.4]{} ![Higgs production cross section in electron-positron collisions, for centre of mass energies, $\sqrt{s} = 250, 500$ GeV, and $1$ TeV. Cross sections are given at leading order, for electron polarization $-80$, and positron polarization $30$ for $250$ and $500$ GeV, and $20$ for $1$ TeV.[]{data-label="fig:leptXSmass"}](higgsprod_xs_lept_vMh_Pol_1000GeV.png "fig:"){width="\linewidth"}
Depending on the mass of the additional scalar, it may also be possible to observe direct production of the new particle. For each centre of mass energy, and the particular range of $m_S$, there are various possible modes by which the additional scalar may be observed. In addition to direct $S$ production, the new scalar may also be observed by Higgs production and subsequent decay to a scalar pair — for the light $S$ case — or through double scalar production processes. Cross sections for single $S$ production are presented in Figure \[fig:leptXSmass\] as a function of $S$ mass, for proposed operational energies of the ILC and respective beam polarizations corresponding to each centre of mass energy. Maintaining convention, $S$ production cross sections have been normalized by the mixing factor, omitting the scaling by $\sin^2\theta_h$ in this figure.
An important feature of the ILC is the sensitivity to processes in which two Higgs (or two scalars) are produced. Such processes could be a distinguishing signature of a mixed Higgs model, and would allow the possibility to probe scalar parameters which are uniquely modified, an effect which is not measurable at the LHC. In the following section, the potential observation of the di-Higgs processes is further explored.
Di-Higgs Processes {#sscn:diHiggs}
------------------
It is interesting to note that the potentially large couplings of the trilinear scalar terms, and modification to the SM value of the Higgs self-couplings, could result in distinguishing effects in measurements of the double Higgs processes. Precision measurements required to probe the Higgs self-couplings and other parameters of the scalar potential, are beyond the scope of current hadron colliders; such measurements fall under the domain of the ILC. The scalar interaction vertices give rise to double scalar production processes which may produce $H$, $S$, or $H$-$S$ pairs. The following will focus on measurement of the double Higgs ($H$ pair) process.
An example Feynman diagram for di-Higgs production is given in figure \[fig:diHiggsFeyn\], showing the contributions of the self-coupling and mixed scalar coupling, to the double Higgs-strahlung process.
[0.425]{} ![Diagrams showing contribution of Higgs self coupling (left) and mixed scalar coupling (right) to di-Higgs production.[]{data-label="fig:diHiggsFeyn"}](HHZ1.png "fig:"){width="\linewidth"}
$ \; + \;$
[0.425]{} ![Diagrams showing contribution of Higgs self coupling (left) and mixed scalar coupling (right) to di-Higgs production.[]{data-label="fig:diHiggsFeyn"}](HHZ2.png "fig:"){width="\linewidth"}
Additional diagrams contributing to this process are given in appendix \[app:diag\]. These are referred to as ‘background’ subprocesses, in the sense that the contributions are not proportional to either the self-coupling, or mixed scalar coupling. These contributions arise from the Higgs-vector boson interactions. Hence, the dependence of the di-Higgs cross section on these scalar couplings is not straightforward; translating measurement of the cross section to a measurement of either one of the couplings is therefore more complex than in the SM case. For this reason, the quantitative effect on measurement of the cross section is analyzed, rather than that of the self-coupling. At higher energies, approaching $1$ TeV, the $W$ fusion mode becomes more significant. Analogous diagrams exist for double Higgs production in this channel, which are also given in appendix \[app:diag\].
[0.45]{} ![Cross sections for scalar pair production processes, presented both as a function of centre of mass energy (left) and scalar mass, $m_S$, at $\sqrt{s}=1$ TeV (right). In the former case, the scalar mass and vev are both taken to be $200$ GeV, and the mixing angle is fixed at $s_{\theta_h}=0.4$.[]{data-label="fig:diHiggsProdxs"}](Di-higgs_prodxs_vEcm_Pol.png "fig:"){width="\linewidth"}
[0.45]{} ![Cross sections for scalar pair production processes, presented both as a function of centre of mass energy (left) and scalar mass, $m_S$, at $\sqrt{s}=1$ TeV (right). In the former case, the scalar mass and vev are both taken to be $200$ GeV, and the mixing angle is fixed at $s_{\theta_h}=0.4$.[]{data-label="fig:diHiggsProdxs"}](Di-higgs_prodxs_vMS_Pol_1000GeV.png "fig:"){width="\linewidth"}
Under this model, the cross section for di-Higgs production differs from the SM value in several aspects:
- the triple Higgs self-coupling is modified, according to $$\frac{m_H^2}{2v_1} \rightarrow
\frac{m_H^2}{2 v_1 v_2}\left( v_2 c_{\theta_h}^3 - v_1 s_{\theta_h}^3 \right)$$
- the additional contribution from the mixed trilinear scalar vertex, proportional to $$\mu = \frac{\sin 2\theta_h }{2 v_1 v_2} \left( v_1 s_{\theta_h} +
v_2 c_{\theta_h} \right) \left( m_H^2 + \frac{m_S^2}{2} \right)$$
- scaling by various factors of $\cos \theta_h$ and $\sin \theta_h$ in the Higgs’ couplings to vector bosons
{width="0.5\linewidth"}
$\longrightarrow \quad \cos\theta_h, \: \sin\theta_h$
{width="0.5\linewidth"}
$\longrightarrow \quad \cos^2\theta_h, \: \sin^2\theta_h$
The cross section for the di-Higgs process is given in figure \[fig:diHiggsProdxs\], including also the $H$-$S$, and double $S$ processes, showing both the $HHZ$ and $HH\nu_e\bar{\nu}_e$ production channels. The dependence on both the centre of mass energy, and the $S$ mass is shown. In both cases, the mixing angle is fixed near its maximal allowed value, $\sin\theta_h = 0.4$, and in the latter case the $S$ mass and vev are set to $m_S = 200$ GeV and $v_2 = 200$ GeV. Cross sections are calculated using <span style="font-variant:small-caps;">MadGraph5</span>, and the model as described in section \[scn:model\] is implemented via <span style="font-variant:small-caps;">FeynRulesv2.0</span> [@Alloul:2013bka].
![Measurable region in $(m_S, \sin\theta_h)$ space, based on fractional change in the $HHZ$ production cross section, using expected ILC precision in $\Delta\sigma/\sigma$. Value of $v_2$ is varied discretely, as indicated. Cross sections are calculated at $500$ GeV, and $(P_{e^-}, P_{e^+}) = (-80, 30)$.[]{data-label="fig:dsigmaHHZ"}](DiHiggs_hhz_pol30_varv2_1.png){width="\linewidth"}
![Measurable region in $(m_S, \sin\theta_h)$ space, based on fractional change in the $HHZ$ production cross section, using expected ILC precision in $\Delta\sigma/\sigma$. Value of $v_2$ is varied discretely, as indicated. Cross sections are calculated at $500$ GeV, and $(P_{e^-}, P_{e^+}) = (-80, 30)$.[]{data-label="fig:dsigmaHHZ"}](DiHiggs_hhz_pol30_varv2_2.png){width="\linewidth"}
![Measurable region in $(m_S, \sin\theta_h)$ space, based on fractional change in the $HHZ$ production cross section, using expected ILC precision in $\Delta\sigma/\sigma$. Value of $v_2$ is varied discretely, as indicated. Cross sections are calculated at $500$ GeV, and $(P_{e^-}, P_{e^+}) = (-80, 30)$.[]{data-label="fig:dsigmaHHZ"}](DiHiggs_hhz_pol30_varv2_3.png){width="\linewidth"}
![Measurable region in $(m_S, \sin\theta_h)$ space, based on fractional change in the $HHZ$ production cross section, using expected ILC precision in $\Delta\sigma/\sigma$. Value of $v_2$ is varied discretely, as indicated. Cross sections are calculated at $500$ GeV, and $(P_{e^-}, P_{e^+}) = (-80, 30)$.[]{data-label="fig:dsigmaHHZ"}](DiHiggs_hhz_pol30_varv2_4.png){width="\linewidth"}
![Measurable region in $(m_S, \sin\theta_h)$ space, based on fractional change in the $HH\nu_e\bar{\nu}_e$ production cross section, using expected ILC precision in $\Delta\sigma/\sigma$. Value of $v_2$ is varied discretely, as indicated. Cross sections are calculated at $1$ TeV, and $(P_{e^-}, P_{e^+}) = (-80, 20)$.[]{data-label="fig:dsigmaHHvv"}](DiHiggs_hhvv_pol20_varv2_1.png){width="\linewidth"}
![Measurable region in $(m_S, \sin\theta_h)$ space, based on fractional change in the $HH\nu_e\bar{\nu}_e$ production cross section, using expected ILC precision in $\Delta\sigma/\sigma$. Value of $v_2$ is varied discretely, as indicated. Cross sections are calculated at $1$ TeV, and $(P_{e^-}, P_{e^+}) = (-80, 20)$.[]{data-label="fig:dsigmaHHvv"}](DiHiggs_hhvv_pol20_varv2_2.png){width="\linewidth"}
![Measurable region in $(m_S, \sin\theta_h)$ space, based on fractional change in the $HH\nu_e\bar{\nu}_e$ production cross section, using expected ILC precision in $\Delta\sigma/\sigma$. Value of $v_2$ is varied discretely, as indicated. Cross sections are calculated at $1$ TeV, and $(P_{e^-}, P_{e^+}) = (-80, 20)$.[]{data-label="fig:dsigmaHHvv"}](DiHiggs_hhvv_pol20_varv2_3.png){width="\linewidth"}
![Measurable region in $(m_S, \sin\theta_h)$ space, based on fractional change in the $HH\nu_e\bar{\nu}_e$ production cross section, using expected ILC precision in $\Delta\sigma/\sigma$. Value of $v_2$ is varied discretely, as indicated. Cross sections are calculated at $1$ TeV, and $(P_{e^-}, P_{e^+}) = (-80, 20)$.[]{data-label="fig:dsigmaHHvv"}](DiHiggs_hhvv_pol20_varv2_4.png){width="\linewidth"}
The expected precision in measurement of the cross section for di-Higgs production has been determined by ILC projections obtained by full detector simulations of SM processes. The technical design report gives an expected precision of $\Delta\sigma/\sigma = 0.27$ for the $HHZ$ process [@TDRvol2], and $\Delta\sigma/\sigma = 0.23$ for the $HH\nu_e\bar{\nu}_e$ is obtained in ref. [@selfcoupHvv]. The fractional change in the di-Higgs production cross sections for both the $HHZ$ and $HH\nu_e\bar{\nu}_e$ processes is determined, and the parameter regions over which the effect of the mixed scalar model would be measurable is established. The fractional change in cross section, compared with the SM value, for both the $HHZ$ and $HH\nu_e\bar{\nu}_e$ processes is calculated and plotted over the remaining scalar parameter space, allowed by existing LHC constraints. Specifically, the quantity $$\frac{\lvert \sigma(m_S, \sin\theta_h, v_2) - \sigma_{SM} \rvert}{\sigma_{SM}}$$ is determined.
The region over which the effect of the scalar mixing is within the expected ILC precision of the measurement is presented, showing the region of scalar parameter space which is within the reach of measurement via di-Higgs production. Results are presented in figures \[fig:dsigmaHHZ\] and \[fig:dsigmaHHvv\], in the $HHZ$ and $HH\nu_e\bar{\nu}_e$ channels respectively. The strongest dependence is found to be in $(m_S, \sin\theta_h)$, while varying the value of $v_2$ has a minimal effect on the bounds of the measurable region. Cross sections are calculated for $HHZ$ production at $\sqrt{s} = 500$ GeV and $(P_{e^-}, P_{e^+}) = (-80, 30)$, and for $HH\nu_e\bar{\nu}_e$ at $\sqrt{s} = 1$ TeV and $(P_{e^-}, P_{e^+}) = (-80, 20)$ — consistent with the values corresponding to the expected precision in the cross section measurements. The greatest change in the cross section is found to be for larger values of $m_S$, due to the exchanged $S$ going on shell and decaying to an $HH$ pair, in the contribution $\propto \mu$. For light $S$, there is still a possible measurable effect for $\sin\theta_h > 0.25-0.4$ that is within the region allowed by existing LHC constraints.
Direct Detection {#scn:DD}
================
![Feynman diagrams for DM-quark scattering.[]{data-label="fig:feynDD"}](DD_H.png){width="0.8\linewidth"}
$+$
![Feynman diagrams for DM-quark scattering.[]{data-label="fig:feynDD"}](DD_S.png){width="0.8\linewidth"}
In any model that includes a generic weakly interacting massive particle (WIMP), limits inevitably arise from DM direct detection searches. In this case, the spin-independent cross section for DM scattering by nucleons receives a contribution from the Higgs and scalar couplings to the DM candidate $\chi$. The scattering process occurs by $t$-channel exchange of either $H$ or $S$. The corresponding Feynman diagrams for the parton subprocesses are shown in figure \[fig:feynDD\].
The spin-independent cross section for $\chi$-nucleon scattering is then given by $$\sigma_{SI} = \left( \frac{s_{\theta_h} c_{\theta_h}}{v_2}\right)^2
m_{\chi}^2 \left( \frac{f_N m_N}{v_1} \right)^2
\frac{\mu^2}{\pi} \left( \frac{1}{m_H^2} - \frac{1}{m_S^2} \right)^2.$$ The Higgs coupling to nucleons (rather than partons) is accounted for in the factor $f_N m_N/v_1$, with $f_N = 0.303$ [@Cline:2013gha]. Here, $\mu$ is the $\chi$-nucleon reduced mass, $\mu = m_{\chi} m_N/(m_{\chi} + m_N)$ and the difference in proton and neutron mass is assumed negligible, taking $m_N=0.946$ GeV.
Current results from LUX [@LUX2015] give the most stringent limit on direct detection of WIMP-nucleon scattering. They present an upper limit at $90 \%$ c.l. on the spin-independent cross section, as a function of the WIMP mass.
![LUX upper limit on the effective coupling $g_{\mathrm{eff}} = s_{\theta_h}
c_{\theta_h}/v_2$. The dependence on both the dark matter mass (left) and scalar mass (right) are shown.[]{data-label="fig:LUXlimit"}](LUXlimit_varmS.png){width="\linewidth"}
![LUX upper limit on the effective coupling $g_{\mathrm{eff}} = s_{\theta_h}
c_{\theta_h}/v_2$. The dependence on both the dark matter mass (left) and scalar mass (right) are shown.[]{data-label="fig:LUXlimit"}](LUXlimit_varmX.png){width="\linewidth"}
The resulting limit on the effective coupling, denoted $g_{\mathrm{eff}} = s_{\theta_h} c_{\theta_h}/v_2$, and its dependence on both the scalar and DM masses is determined, as shown in figure \[fig:LUXlimit\]. The upper limit on $g_{\mathrm{eff}}$, based on the LUX $90 \%$ c.l. bound, is determined as a function of $m_{\chi}$, while discretely varying $m_{S}$, and vice versa. Discrete values of $m_S = \{ 50, 130, 500 \}$ GeV, and $m_{\chi} = \{ 5, 10, 30 \}$ GeV are chosen.
Noting first the $m_{\chi}$ dependence of the effective coupling limit, the limit is most stringent for heavier $\chi$, due to the additional factor of $m_{\chi}^2$ in the scalar-dark matter coupling. The limit relaxes for light $\chi$, roughly less than $10$ GeV. Considering the dependence of this limit on the scalar mass, the coupling is more tightly constrained for $m_S \lesssim 100$ GeV, while the limit weakens slightly for heavier $m_S > m_H$. A special case is seen as $S$ approaches the degenerate limit, $m_{S} \sim m_H$, due to destructive interference of the $H$ and $S$ contributions. In this narrow region, the upper bound on the coupling is substantially relaxed. For lighter $\chi$, the upper limit on the effective coupling is between $0.01 - 0.1$ GeV$^{-1}$, and can be as large as $\sim 1$ GeV$^{-1}$ for values of $m_S$ near $m_H$. For heavier $\chi$, this coupling is restricted to be $\mathcal{O}(10^{-3})$ GeV$^{-1}$, or smaller. For illustrative purposes, the dependence of the minimum allowed value of the scalar vev on the value of $\sin \theta_h$ is shown in figure \[fig:v2min\], for several maximum values of $g_{\mathrm{eff}}$.
![Dependence of the minimum allowed value of $v_{2}$ on $\sin \theta_h$, for several values of the upper bound on $g_{\mathrm{eff}}$.[]{data-label="fig:v2min"}](v2min.png){width="57.50000%"}
Conclusion {#scn:conc}
==========
Although the current LHC Higgs data are consistent with the Standard Model, there is still the possibility that the observed Higgs could be part of an extended theory, in particular, that a Higgs sector with multiple scalars could be avoiding detection at the current levels of precision. A mixed two-Higgs scenario has been considered, under the addition of a real singlet scalar field to the Standard Model scalar sector. Strict limits from the measured signal strength in LHC Higgs production, and exclusion searches constrain the mixing angle such that the upper bound on $|\sin\theta_h|$ is between 0.3 and 0.5 for various ranges of the scalar mass, above $m_S \sim 80$ GeV, while LEP exclusion place a stricter bound for smaller masses, of $|\sin\theta_h| \lesssim 0.15$. For a subset of the parameter space, additional limits arise on the scalar parameters, including the vev of the additional scalar field, from new contributions to the total width of the Higgs, due to decays to a scalar pair. For the case of a light $S$, the region of scalar parameter space allowed by measurements of the total Higgs width is determined, giving the allowed values of the mixing angle and scalar vev. Smaller values of $v_2$, i.e. $v_2 \sim 10$ GeV tend to be disfavoured in this case.
This analysis primarily considered scalar masses of approximately $100$ GeV or larger, but masses as low as $1-10$ GeV were briefly considered in some cases — specifically the contribution to the Higgs width from decays to a scalar pair, and in the region excluded by LEP data. Although $\sim 1$ GeV scalar masses were not the focus here, it should be noted that in the region of $m_S \lesssim 4$ GeV, there is potential for the mixing angle to be strictly constrained by recent LHCb data corresponding to a search for hidden-sector bosons in $B^0$ decays [@Aaij:2015tna].
The model is also extended to include a possible application to dark matter, via the addition of a chiral Yukawa coupling of the scalar singlet to a dark fermion. Two Majorana mass states generally result as the scalar acquires a vev, which are equivalently expressed as a single Dirac fermion in the degenerate case which is considered here. The initial implications of the simpler case of a single dark fermion in this extended Higgs portal framework are investigated, with the extended case of two non-degenerate dark matter species left for future work. Bounds on the Higgs invisible width place limits on the dark sector of the model. Specifically, such measurements constrain the scalar vev to be approximately greater than $100$ or $500$ GeV, for $\sin\theta_h = 0.1, 0.4$ respectively, for values of $m_{\chi}$ in the region kinematically allowed by the $H \rightarrow \chi \bar{\chi}$ decay. The lower bound on $v_2$ is relaxed for very light $\chi$, $m_{\chi} \sim 1$ GeV. Alternatively, the constraint on $v_2$ is avoided for $m_{\chi} > m_H/2$. Additional limits on the parameters describing the dark sector interaction are obtained by direct detection results, from $H$ and $S$ mediated contributions to the spin-independent $\chi$-nucleon scattering cross section. The maximally allowed value of the effective coupling, $g_{\mathrm{eff}} = \sin\theta_h \cos\theta_h/v_2$, is determined as a function of the dark matter mass, with additional dependence on the mass of the new scalar. The resulting constraint places a lower bound on the scalar vev, which varies with the values of the mixing angle and masses of $S$ and $\chi$, and disfavours heavier $\chi$. It is noted than possible tension with the LUX result is avoided under three possibilities: either lighter dark matter with $m_{\chi} \lesssim 10$ GeV, $v_2$ values of order $1$ TeV, or values of $m_S$ which are near the SM Higgs mass. Under the scenario in which an additional scalar has evaded detection at current hadron colliders, the remaining parameter space is probed in a Higgs precision environment, investigating the discovery potential at ILC. The presence of the additional scalar, and induced mixing, results in unique contributions to the di-Higgs production process. Based on expected precision in measurement of the di-Higgs cross section, it is shown that there exists a measurable region in the remaining allowed parameter space, over which the effect of scalar mixing could be observed. Future experiments in Higgs precision will offer interesting results, and the opportunity to further constrain, or perhaps detect, new physics in the Higgs sector.
Di-Higgs Production Diagrams {#app:diag}
============================
Additional Feynman diagrams contributing to di-Higgs production processes at ILC are given here. Included, are the background diagrams for $HHZ$ production, and the diagrams for $HH\nu_e\bar{\nu}_e$ production, via $WW$ fusion, including both background, and the scalar trilinear and self coupling contributions. It should be noted, that $HH\nu_e\bar{\nu}_e$ production also includes the corresponding Higgs-strahlung diagrams in which the final-state $Z$ decays to $\nu_e \bar{\nu}_e$.
{width="\linewidth"}
{width="\linewidth"}
{width="\linewidth"}
{width="\linewidth"}
{width="\linewidth"}
{width="\linewidth"}
{width="\linewidth"}
{width="\linewidth"}
The author would like to thank Jim Cline for valuable comments, guidance, and supervision of this project, as well as Brigitte Vachon and Guy Moore for helpful discussions. Research is supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada.
[^1]: The contribution of this vertex to Higgs production at a linear collider will be further discussed in section \[sscn:diHiggs\]
[^2]: Ref. [@Robens:2015gla] takes the scalar defined by $c_{\theta_h} h_1- s_{\theta_h}h_2$ to be the lighter of the two, rather than fixing it to be the $125$ GeV boson, as is done here, effectively switching $\sin\theta_h \rightarrow \cos\theta_h$, for $m_S < 125$ GeV.
[^3]: Ref. [@Martin-Lozano:2015dja] conducts a similar analysis, using an alternative approach to the signal strength limit.
|
---
address: |
Institut f" ur Physik, Technische Universität Dortmund\
D-44221 Dortmund, Germany
author:
- MARTIN JUNG
bibliography:
- 'EDM\_proceedings.bib'
title: 'THE ELECTRON EDM AND EDMS IN TWO-HIGGS-DOUBLET MODELS'
---
Introduction
============
Electric dipole moments (EDMs) provide a competitive means to search for new physics (NP), complementary to strategies like direct searches at hadron colliders, but also to other indirect searches like the flavour-changing processes investigated at the flavour factories. The exceptional sensitivity is in two ways related to the very specific connection between flavour and CP violation[^1] in the SM, embodied by the Kobayashi-Maskawa mechanism:[@Kobayashi] firstly, it is very effective in suppressing flavour-changing neutral currents (FCNCs), and even more so flavour-conserving ones involving CP violation. An exception is provided by the gluonic operator $\mathcal{O}_{G\tilde G}\propto\epsilon_{\mu\nu\rho\sigma}G^{\mu\nu}G^{\rho\sigma}$: its potentially very large contribution to hadronic EDMs is, however, strongly bounded experimentally. In this work it is implicitly assumed that it is effectively removed by Peccei-Quinn symmetry[@Peccei:1977hh] or a similar mechanism. The remaining SM contributions then lead to EDMs many orders of magnitude below the present limits, *e.g.*[@Khriplovich:1981ca; @Gavela:1981sk; @McKellar:1987tf; @Mannel:2012qk] $d_n^{\rm SM,CKM}\lesssim(10^{-32}-10^{-31})\,e\,{\rm cm}$. Importantly, for leptonic EDMs no assumption regarding $\mathcal{O}_{G\tilde G}$ is necessary; the SM contribution to the electron EDM is estimated to be[@Pospelov:1991zt; @Booth:1993af; @Pospelov:2013sca] $d_e^{\rm SM}\lesssim10^{-38}\,e\,{\rm cm}$. The observation of an EDM with the present experimental precision would therefore clearly constitute a NP signal, especially in the leptonic sector. The second way the Kobayashi-Maskawa mechanism plays a role is that in a generic NP scenario, the absence of such a powerful suppression typically yields contributions that are large compared to experimental limits. On the other hand, Sakharov’s conditions [@Sakharov:1967dj] require the presence of new sources of CP violation to explain the observed baryon asymmetry in the universe; while this does not necessarily imply sizable EDMs, it yields a strong motivation to search for such sources. This combination of tiny SM “background” and comparatively large expected NP contributions renders EDMs a precision laboratory for NP searches.
A potential discovery of a non-vanishing EDM would therefore indicate a NP signal several orders of magnitude above the SM expectation, rather independent of theoretical uncertainties or the specific source. However, when casting existing experimental limits (and also potential signals) into bounds on model parameters, both issues need to be addressed. Specifically, since experiments are typically carried out using composite systems, that is, nucleons, atoms or molecules, the relation to more fundamental quantities like the electron EDM requires the evaluation of complicated matrix elements which constitute the main source of theoretical uncertainty. Furthermore, there are different sources for EDMs in theories beyond the SM, which can exhibit cancellations. For heavy paramagnetic systems, potential cancellations can be taken into account, leading to a more reliable, model-independent limit on the electron EDM,[@Jung:2013mg] discussed in the next section. This is then used together with other limits in section \[sec::2HDMs\] to constrain the CP-violating parameters in Two-Higgs-Doublet models (2HDMs). Specifically, models with new CP-violating phases in the Yukawa interactions used to be discarded because of potentially huge EDMs. While the present experimental limits impose strong bounds on the corresponding parameters, we show that in models with an appropriate flavour structure they have not yet to be unnaturally small.[@Jung:2013hka] However, large enhancements in other CP-violating observables are strongly restricted by these bounds. Furthermore, the generic size for EDMs lies well within reach of planned and ongoing next-generation experiments. These will therefore provide critical tests for this class of models. We conclude in section \[sec::conclusions\].
Framework
=========
Relating experimental data to fundamental parameters proceeds in a series of effective theories. The available competitive observables, that is, the EDMs of thorium monoxide and ytterbium fluoride molecules[@Baron:2013eja; @Hudson:2011zz], thallium and mercury atoms[@Regan:2002ta; @Griffith:2009zz] and the neutron[@Baker:2006ts] (see also [@Serebrov:2013tba]), are related by atomic, nuclear and QCD calculations to the coefficients of an effective theory on a hadronic scale (see, *e.g.*, Ref.): $$\begin{aligned}
\label{eq::Leff}
\mathcal{L}^{\rm EDM}_{\rm eff} = -\!\!\sum_{f}\frac{d_f^\gamma}{2}\mathcal{O}_f^\gamma-\sum_q\frac{d_q^C}{2}\mathcal{O}_q^C+C_W \mathcal{O}_W+\sum_{f,f'}C_{ff'}\mathcal{O}_{ff'}\,.\end{aligned}$$ The operator basis consists of (colour-)EDM operators $\mathcal{O}^{\gamma,C}_f$ ($f=e,q$, $q=u,d,s$), the Weinberg operator $\mathcal{O}_W$ and T- and P-violating four-fermion operators $\mathcal{O}_{ff'}$ without derivatives (see,*e.g.*, Ref.). Since these calculations do not depend on the NP model under consideration, this is used as the interface between the experimental side and the high-energy calculations: the latter provide the model-specific expressions for the Wilson coefficients in Eq. , with at least one more intermediate effective theory at the electroweak scale.
Model-independent extraction of the electron EDM\[sec::eEDMMI\]
===============================================================
Within a given model, typically different operators from Eq. dominate in different regions of the parameter space. Heavy paramagnetic systems are an exception in this respect: their EDMs receive two contributions scaling at least like $d\sim Z^3$,[@Sandars:1965xx; @Sandars:1966xx; @Flambaum:1976vg] which therefore dominate the others; one term is directly proportional to the electron EDM $d_e$, the other stems from electron-nucleon interactions, parametrized by a dimensionless parameter $\tilde C_S$. The energy shift $\Delta E=\hbar\omega$ of molecules $M$ in an external field, as measured in,[@Baron:2013eja; @Hudson:2011zz] is given in terms of these contributions as well: $$\omega = 2\pi \left(\frac{W_d^M}{2} d_e+\frac{W_c^M}{2}\tilde C_S\right)\,.$$ The necessary constants $W_{d,c}^M$, as estimated in [@Jung:2013mg; @Jung:2013hka], are $W_d^{YbF}=-(1.3\pm0.1)10^{25}\,{\rm Hz}/e~{\rm cm}$, $W_c^{YbF}=-(92\pm9)\,{\rm kHz}$, $W_d^{ThO}=-(3.67\pm0.18)10^{25}\,{\rm Hz}/e~{\rm cm}$, $W_c^{ThO}=-(598\pm90)\,{\rm kHz}$. In the literature it is common, however, to extract the electron EDM by setting $\tilde C_S\to0$ (and neglecting theory uncertainties). While this is reasonable in some models, it is not a model-independent procedure. Actually, a single measurement cannot be translated into a limit on $d_e$ without an assumption on $\tilde C_S$. In principle, however, clearly both contributions can be extracted, once more than one measurement is available. Using additionally the measurement for mercury, this has been done in Ref.[@Jung:2013mg], leading to a bound on $d_e$ competitive with the naive extraction at the time. The present situation, illustrated in Fig. \[fig::eEDM\][@Jung:2013mg; @Jung:2013hka] on the left, is that the in principle much stronger limit[@Baron:2013eja] cannot easily be translated into a much better constraint on the electron EDM (comparing the projections on the $d_e$ axis of the blue ellipsis versus the one of its overlap with the dark green fan yields and improvement from $|d_e|\leq 1.4\times 10^{-27}\,e~{\rm cm}$ to $|d_e|\leq 1.0\times 10^{-27}\,e~{\rm cm}$, only), since no second competitive measurement is available.
This will change in the future, as illustrated in the same figure on the right. Note that it is of special importance to have measurements with significantly differing values of $W_d^M/W_c^M$,[@Jung:2013mg] that is, measurements with atoms/molecules of different weight.[@PhysRevA.85.029901] In the meantime we propose to use a fine-tuning argument instead of neglecting the $\tilde C_S$ contribution completely: allowing the latter by itself to saturate the experimental limit at most $n=1,2,3,\ldots$ times, *i.e.* excluding very large cancellations, we obtain a still conservative upper limit on $d_e$, *e.g.*[@Jung:2013hka] $$|d_e|\leq 0.25\times 10^{-27}e\,{\rm cm}\quad (n=2)\,.$$
EDMs in Two-Higgs-Doublet Models\[sec::2HDMs\]
==============================================
We calculate the Wilson coefficients in Eq. for 2HDMs with new CP-violating phases. To that aim, we use a general parametrization for the charged current Yukawa couplings in the Higgs basis, $$\label{eq::LHcharged}
\mathcal L_Y^{H^\pm} \! =\! - \frac{\sqrt{2}}{v}\, H^+ \left\{ \bar{u} \left[ V \varsigma_d M_d \mathcal P_R - \varsigma_u\, M_u^\dagger V \mathcal P_L \right] d\, +\, \bar{\nu} \varsigma_l M_l \mathcal P_R l\right\} + \;\mathrm{h.c.} \, ,$$ where the $M_i$ are diagonal mass matrices, $V$ denotes the CKM matrix, and the $\varsigma_f$ in principle arbitrary complex matrices. We give below the constraints in terms of elements of these matrices, which can be translated into the parameters of any given 2HDM model. To be specific and able to relate the resulting bounds also to those from other observables, we will furthermore consider the Aligned 2HDM (A2HDM),[@Pich:2009sp; @Jung:2010ik] where the $\varsigma_i$ are complex numbers, thereby avoiding FCNCs on tree level while still allowing for a rich phenomenology including additional CP-violating phases. For the couplings of the neutral Higgs states, we obtain similarly $$\label{eq::LHneutral}
\mathcal L_Y^{\varphi_i^0} \! =\! -\frac{1}{v}\; \sum_{\varphi, f}\, \varphi^0_i \; \bar{f}\,y^{\varphi^0_i}_f\, M_f \mathcal P_R f\; + \;\mathrm{h.c.} \, ,$$ with the fields $\varphi_i^0=\{h,H,A\}$ denoting the neutral scalar mass eigenstates. Denoting the fermion species as $F(f)$, *e.g.* $F(u)=F(c)=F(t)=u$, we can write the appearing couplings as *e.g.* $y_{f}^{\varphi^0_i} = {\mathcal{R}}_{i1} + ({\mathcal{R}}_{i2} + i\,{\mathcal{R}}_{i3})\,\left(\varsigma_{F(f)}\right)_{ff}$ (for $F(f)=d,l$) to allow for the general form of $\varsigma_{u,d,l}$. ${\mathcal{R}}$ denotes the rotation defined by $\mathcal{M}^2_{\rm diag}=\mathcal{R} \mathcal{M}^2\mathcal{R}^T$, relating the mass eigenstates to the neutral scalar fields in the Higgs basis. In a general 2HDM, the $\varsigma_{u,d,l}$ are the matrices introduced in Eq. , only the diagonal elements of which are relevant here. This expression for the couplings $y_{f}$ reflects the fact that in the neutral Higgs couplings CP violation may enter from the Yukawa couplings as well as the scalar potential, rendering the phenomenological discussion very complicated. However, orthogonality of the matrix ${\mathcal{R}}$ implies the relation $$\label{eq::ycancellation}
\sum_i {\rm Re}\left(y_f^{\varphi^0_i}\right){\rm Im}\left(y_{f'}^{\varphi^0_i}\right)=\pm{\rm Im}\left[(\varsigma_{F(f)}^*)_{ff}(\varsigma_{F(f')})_{f'f'}\right]\,,$$ which vanishes for real $\varsigma_i$ (as *e.g.* the case for $\mathcal Z_2$ models) and for $f=f'$ (fermions of the same family if the $\varsigma_i$ are family-universal as *e.g.* in the A2HDM). Importantly, the right-hand side is independent of the parameters of the scalar potential. While in practical calculations there are mass-dependent weight factors in the sum on the left, the relation still holds exactly in two limits: trivially so when the neutral scalars are degenerate, but also in the decoupling limit.[@Jung:2013hka] Therefore, in general cancellations can be expected for any mass spectrum and the influence of CP violation in the potential is reduced. Clearly, this observation provides a protection against large EDMs for models which exhibit new CP-violating parameters in the potential, only. Below we will assume this relation to hold and evaluate the right-hand side with a common weight factor at an intermediate effective neutral Higgs mass $\overline{M}_\varphi$.
The fact that so far no significant NP signals have been observed directly implies a highly non-trivial flavour structure of the theory. In models fulfilling this requirement, the main contributions then stem typically from two-loop diagrams, namely the Weinberg operator and so-called Barr-Zee diagrams[@Weinberg:1989dx; @Barr:1990vd]; additionally, enhanced four-fermion operators can be important for heavy atoms/molecules, as emphasized above. We refer the reader to[@Jung:2013hka] for the relevant expressions in 2HDMs and show below directly the resulting constraints.
The electron EDM receives contributions mostly from Barr-Zee diagrams.[@BowserChao:1997bb] The resulting constraints on ${\rm Im}(\varsigma_{u,33}\varsigma_{l,11}^*)$ are shown in Fig. \[fig::A2HDMeEDM\], demonstrating the strength of this observable.
For the A2HDM, this becomes even more obvious when comparing with the bound on the absolute value of this parameter combination obtained from leptonic and semileptonic decays,[@Jung:2010ik; @Celis:2012dk] which is about a factor 1000 weaker.
For the neutron, the constraint induced in the charged-Higgs sector via the Weinberg operator is shown in Fig. \[fig::nEDMWzetaud\] on the left.
Again no fine-tuning is necessary to avoid this bound. On the other hand it prohibits large CP-violating effects in other observables. On the right, the maximally allowed band is shown together with the constraint from the branching ratio in $b\to s\gamma$, again for the A2HDM; from the discussion in Refs.[@Jung:2010ab; @Jung:2012vu] follows for the NP contribution that $|A_{\rm CP}(b\to s\gamma)|\lesssim1\%$.
Conclusions\[sec::conclusions\]
===============================
EDMs provide unique constraints for the CP-violating sectors of NP models. We discussed the model-independent extraction of the electron EDM from measurements in heavy paramagnetic systems and the application of EDM constraints to general 2HDMs. While so far no severe fine-tuning is necessary to avoid the resulting bounds, they prohibit large effects in other CP-violating observables in concrete models like the A2HDM. Given the present strength of the constraints, forthcoming experiments will test a crucial part of the parameter space and might turn existing bounds into observations.
Acknowledgments {#acknowledgments .unnumbered}
===============
I would like to thank Toni Pich for a fruitful and enjoyable collaboration, as well as the organizers and participants of Moriond EW 2014 for a very pleasant conference. This work is supported in part by the Bundesministerium für Bildung und Forschung (BMBF).
References {#references .unnumbered}
==========
[^1]: EDMs are T,P-odd, implying also CP violation when assuming CPT to conserved as we will in this article.
|
---
author:
- 'D. Elbaz'
- 'M. Dickinson'
- 'H.S. Hwang'
- 'T. Díaz-Santos'
- 'G. Magdis'
- 'B. Magnelli'
- 'D. Le Borgne'
- 'F. Galliano'
- 'M. Pannella'
- 'P. Chanial'
- 'L. Armus'
- 'V. Charmandaris'
- 'E. Daddi'
- 'H. Aussel'
- 'P. Popesso'
- 'J. Kartaltepe'
- 'B. Altieri'
- 'I. Valtchanov'
- 'D. Coia'
- 'H. Dannerbauer'
- 'K. Dasyra'
- 'R. Leiton'
- 'J. Mazzarella'
- 'D.M. Alexander'
- 'V. Buat'
- 'D. Burgarella'
- 'R.-R. Chary'
- 'R. Gilli'
- 'R.J. Ivison'
- 'S. Juneau'
- 'E. Le Floc’h'
- 'D. Lutz'
- 'G.E. Morrison'
- 'J.R. Mullaney'
- 'E. Murphy'
- 'A. Pope'
- 'D. Scott'
- 'M. Brodwin'
- 'D. Calzetti'
- 'C. Cesarsky'
- 'S. Charlot'
- 'H. Dole'
- 'P. Eisenhardt'
- 'H.C. Ferguson'
- 'N. F[ö]{}rster Schreiber'
- 'D. Frayer'
- 'M. Giavalisco'
- 'M. Huynh'
- 'A.M. Koekemoer'
- 'C. Papovich'
- 'N. Reddy'
- 'C. Surace'
- 'H. Teplitz'
- 'M.S. Yun'
- 'G. Wilson'
date: 'Received 11 May 2011; accepted 3 August 2011'
title: 'GOODS–[*Herschel*]{}: an infrared main sequence for star-forming galaxies[^1]'
---
Introduction {#SEC:intro}
============
It is now well established that $\sim$85% of the baryon mass contained in present-day stars formed at 0$<$$z$$<$2.5 (see, e.g., Marchesini et al. 2009 and references therein) and that most energy radiated during this epoch by newly formed stars was heavily obscured by dust. To understand how present-day galaxies were made, it is therefore imperative to accurately determine the bolometric output of dust, hence the total IR luminosity, $L_{\rm IR}^{\rm tot}$, integrated from 8 to 1000$\mu$m. In the past, this key information on the actual star formation rate (SFR) experienced by distant galaxies was determined by extrapolating observations in the mid-IR and sub-millimeter (sub-mm) or by correcting their UV luminosities for extinction. These extrapolations implied that the number density per unit comoving volume of luminous IR galaxies (LIRGs, 10$^{11}$$\leq$$L_{\rm IR}$/L$_{\sun}$$<$10$^{12}$) was 70 times larger at $z$$\sim$1, i.e., $\sim$ 8 Gyr ago, when LIRGs were responsible for most of the cosmic SFR density per unit co-moving volume (see e.g., Chary & Elbaz 2001 – hereafter CE01, Le Floch et al. 2005, Magnelli et al. 2009). Earlier in the past, at $z$$\sim$2, sub-mm and [*Spitzer*]{} observations revealed that the contribution to the cosmic SFR density of even more active objects, the ultraluminous IR galaxies (ULIRGs, $L_{\rm IR}$$\geq$10$^{12}$ L$_{\sun}$), was as important as for LIRGs (Chapman et al. 2005, Papovich et al. 2007, Caputi et al. 2007, Daddi et al. 2007a, Magnelli et al. 2009, 2011). However, none of these studies used rest-frame far-IR measurements of individual galaxies at wavelengths where the IR spectral energy distribution (SED) of star-forming galaxies is known to peak. At best, they relied on stacking of far-IR data from individually undetected sources.
With the launch of the [*Herschel*]{} Space Observatory (Pilbratt et al. 2010), it has now become possible to measure the total IR luminosity of distant galaxies directly. Using shallower [*Herschel*]{} data than the present study, Elbaz et al. (2010) showed that extrapolations of $L_{\rm IR}^{\rm tot}$ from the mid-IR (24$\mu$m passband), which was done under the assumption that the IR SEDs of star-forming galaxies remained the same at all epochs, were correct below $z$$\lesssim$1.3, with an uncertainty of only 0.15 dex. However, the extension of this assumption to (U)LIRGs at $z$$\gtrsim$1.3, in large part relying on stacking, failed by a factor 3-5 typically (Elbaz et al. 2010, Nordon et al. 2010). This finding confirmed the past discovery of a so-called “mid-IR excess” population of galaxies (Daddi et al. 2007a, Papovich et al. 2007, Magnelli et al. 2011): the 8$\mu$m rest-frame emission of $z$$\sim$2 (U)LIRGs was excessively strong compared to the IR SED of local galaxies with equivalent luminosities when deriving $L_{\rm IR}^{\rm tot}$ from the radio continuum at 1.4 GHz, from stacked measurements from [*Spitzer*]{}-MIPS 70$\mu$m, or from the UV luminosity corrected for extinction.
Various causes have been invoked to explain this “mid-IR excess” population: *(i)* an evolution of the IR SEDs of galaxies; *(ii)* the presence of an active galactic nucleus (AGN) heating dust to temperatures of a few 100 K; or *(iii)* limitations in local libraries of template SEDs, i.e., the $k$-correction effect on distant galaxies probing regimes where the SEDs were not accurately calibrated. Evidence pointing toward an important role played by obscured AGN to explain these discrepancies (point *(ii)*) came from the stacking of [*Chandra*]{} X-ray images at the positions of the most luminous $z$$\sim$2 BzK galaxies (Daddi et al. 2007a). The most luminous of these distant galaxies were detected in both the soft (0.5–2 keV) and hard (2–7 keV) X-ray channels of [*Chandra*]{} and exhibited a flux ratio typical of heavily obscured ($N_H$$\geq$10$^{23}$ cm$^{-2}$) or even Compton thick AGN ($N_H$$\geq$10$^{24}$ cm$^{-2}$). Surprisingly, however, a high fraction of the same objects, when observed in mid-IR spectroscopy with the [*Spitzer*]{} IR spectrograph (IRS), were found to possess intense polycyclic aromatic hydrocarbon (PAH, Léger $\&$ Puget 1984, Puget $\&$ Léger 1989, Allamandola et al. 1989) broad lines with equivalent widths strongly dominating over the hot to warm dust continuum (Rigby et al. 2008, Farrah et al. 2008, Murphy et al. 2009, Fadda et al. 2010, Takagi et al. 2010). Deeper Chandra observations have since showed that only $\sim$25% of the $z$$\sim$2 BzK-selected mid-IR galaxies hosted heavily obscured AGN, the rest being otherwise composed of relatively unobscured AGNs and star-forming galaxies (Alexander et al. 2011). This would instead favor points *(i)* or *(iii)* above.
In this paper, we present the deepest 100 to 500$\mu$m far-IR observations obtained with the [*Herschel*]{} Space Observatory as part of the GOODS–[*Herschel*]{} Open Time Key Program with the PACS (Poglitsch et al. 2010) and SPIRE (Griffin et al. 2010) instruments. Thanks to the unique power of [*Herschel*]{} to determine the bolometric output of star-forming galaxies, we demonstrate that incorrect extrapolations of $L_{\rm IR}^{\rm tot}$ from 24$\mu$m observations at $z$$\gtrsim$1.5, and the associated claim for a “mid-IR excess” population, do not indicate a drastic evolution of infrared SEDs, nor the ubiquity of warm AGN-heated dust dominating the mid-IR emission. Instead, we show that the 8$\mu$m bolometric correction factor ($IR8$$\equiv$$L_{\rm IR}^{\rm tot}$/$L_8$) is universal in the range 0$<$$z$$\leq$2.5, hence defining an IR “main sequence” (MS). We show that past incorrect extrapolations resulted from the confusion between galaxies with extended star formation and those with compact starbursts, which exhibit notably different infrared SEDs.
We present evidence that this IR main sequence is directly related to the redshift dependent SFR – M\* relation (Noeske et al. 2007, Elbaz et al. 2007, Daddi et al. 2007a, 2009, Pannella et al. 2009, Magdis et al. 2010a, Gonzalez et al. 2011) and is able to separate galaxies between those experiencing a “normal” mode of extended star formation and starbursts with compact projected star formation densities. This distinction between a majority of “main sequence” (MS) galaxies and a minority of compact “starbursts” (SB) is analogous to the recent finding of two regimes of star formation in the Schmidt-Kennicutt (SK) law, with MS galaxies following the classical SK relation while the SFR of SB galaxies is an order of magnitude greater than expected from their projected gas surface density (Daddi et al. 2010, Genzel et al. 2010). To separate these two star-formation modes, the GOODS–[*Herschel*]{} observations of distant galaxies are supplemented by a reference sample of local galaxies using a compilation of data from [*IRAS*]{}, [*AKARI*]{}, [*Spitzer*]{}, SDSS and radio observations.
The GOODS–[*Herschel*]{} observations and catalogs are presented in Section \[SEC:data\]. The main limitation of the [*Herschel*]{} catalogs, the confusion limit, and a “clean index” identifying sources with robust photometry are discussed in Sect. \[SEC:clean\]. The high- and low-redshift galaxy samples are introduced in Section \[SEC:highlowz\] together with a description of the method used to compute total IR luminosities, stellar masses and photometric redshifts. The IR main sequence is presented in Section \[SEC:IR8\] where the so-called “mid-IR excess problem” is addressed and a solution proposed using the $IR8$ bolometric correction factor. This parameter, which relies on the same rest-frame wavelengths independent of galaxy redshift, is used to separate star-forming galaxies in two modes: a main sequence and a starburst mode. In the following sections, $IR8$ is shown to correlate closely with the IR surface brightness, hence with the projected star formation density, and with the starburst intensity, that we quantify here with a parameter named “starburstiness”, for local (Section \[SEC:compactness\]) and distant (Section \[SEC:MSSB\]) galaxies. It is shown that galaxies exhibiting enhanced $IR8$ values are undergoing a compact starburst phase. The universality of $IR8$ among main sequence star-forming galaxies is used to produce a prototypical IR SED for galaxies in the main sequence mode of star formation in Section \[SEC:sed\]. We combine [*Spitzer*]{} and [*Herschel*]{} photometry in many passbands for galaxies at $0 < z < 2.5$ to derive composite SEDs for both main sequence and starburst galaxies. Finally, galaxies exhibiting an AGN signature are discussed in Section \[SEC:AGN\], where we present a technique to identify obscured AGN candidates that would be unrecognized by previous methods.
We use below a cosmology with $H_0$=70 $\mathrm{km s^{-1} Mpc^{-1}}$, $\Omega_M=0.3$, $\Omega_\Lambda=0.7$ and we assume a Salpeter initial mass function (IMF, Salpeter 1955) when deriving SFRs and stellar masses.
{width="3.2cm"} {width="3.2cm"} {width="3.2cm"} {width="3.2cm"} {width="3.2cm"} {width="3.2cm"} {width="3.2cm"} {width="3.2cm"} {width="3.2cm"} {width="3.2cm"}
![Composite three color image of the GOODS–north field (10$\arcmin$$\times$15$\arcmin$) at 100$\mu$m (blue), 160$\mu$m (green) and 250$\mu$m (red). North is up and east is left.[]{data-label="FIG:GN"}](elbaz_fig2_gr.eps){width="9.0cm"}
![Composite three color image of the GOODS–south field (10$\arcmin$$\times$10$\arcmin$) at 24$\mu$m (blue), 100$\mu$m (green) and 160$\mu$m (red). North is up and east is left.[]{data-label="FIG:GS"}](elbaz_fig3_gr.eps){width="8.5cm"}
GOODS–[*Herschel*]{} data and catalogs {#SEC:data}
======================================
Observations
------------
The sample of high-redshift galaxies analyzed here consists of galaxies observed in the two Great Observatories Origins Deep Survey (GOODS) fields in the Northern and Southern hemispheres. Observations with the [*Herschel*]{} Space Observatory were obtained as part of the open time key program GOODS–[*Herschel*]{} (PI D.Elbaz), for a total time of 361.3 hours. PACS observations at 100 and 160$\mu$m cover the whole GOODS–north field of 10$\arcmin$$\times$16$\arcmin$ and part of GOODS–south, i.e., 10$\arcmin$$\times$10$\arcmin$ (but reaching the largest depths over $\sim$64 arcmin$^2$). When considering the total observing times of 124 hours in GOODS–N and 206.3 hours in GOODS–S (including 2.6 and 5 hours of overheads), the PACS GOODS–[*Herschel*]{} observations reach a total integration time per sky position of 2.4 hours in GOODS–N and of 15.1 hours in GOODS–S, i.e., 6.3 times longer. Due to the larger beam size and observing configuration, the SPIRE observations of GOODS–N cover a field of 900 arcmin$^2$, hence largely encompassing the central 10$\arcmin$$\times$16$\arcmin$, for a total observing time of 31.1 hours and an integration time per sky position of 16.8 hours.
Fig. \[FIG:stamps\] shows a montage of images (each 5$\arcmin$$\times$5$\arcmin$) from [*Spitzer*]{}–IRAC at 3.6$\mu$m to SPIRE at 500$\mu$m. This illustrates the impact of the increasing beam size as a function of wavelength: the number of sources that are clearly visible at each wavelength increases when going from the longest to the shortest wavelengths (with the exception of the 70$\mu$m image, which comes from [*Spitzer*]{} and not [*Herschel*]{}). Composite three color images of GOODS–N at 100–160–250$\mu$m and GOODS–S at 24–100–160$\mu$m are shown in Figs. \[FIG:GN\] and \[FIG:GS\].
Catalogs
--------
### Source extraction
Flux densities and their associated uncertainties were obtained from point source fitting using 24$\mu$m prior positions. For the largest passbands of SPIRE (i.e., 350 and 500$\mu$m), the 24$\mu$m priors are much too numerous and would lead to an over-deblending of the actual sources. Hence, we defined priors with the following procedure. For PACS-100$\mu$m, we used MIPS-24$\mu$m priors down to the 3$\sigma$ limit and imposing a minimum flux density of 20$\mu$Jy. For PACS-160$\mu$m and SPIRE-250$\mu$m, we restricted the 24$\mu$m priors to the 5$\sigma$ (30$\mu$Jy) limit (reducing the number of priors by about 35%). For SPIRE-350 and 500$\mu$m, we kept only the 24$\mu$m priors for sources with a S/N ratio greater than 2 at 250$\mu$m. These criteria were chosen from Monte Carlo simulations (see Sect. \[SEC:simulations\]) to avoid using too many priors that would result in subdividing flux densities artificially, while producing residual maps (after PSF subtracting the sources brighter than the detection limit) with no obvious sources remaining.
![GOODS–[*Herschel*]{} detection limits (from Table \[TAB:depths\]) and total IR luminosities of the [*Herschel*]{} sources as a function of redshift (filled dots: spectroscopic, open dots: photometric). The right axis is the SFR derived from SFR\[M$_{\sun}$yr$^{-1}$\]=1.72$\times$10$^{-10}$$\times$$L_{\rm IR}^{\rm tot}$\[L$_{\sun}$\] (Kennicutt 1998a). These limits were computed assuming the local library of template SEDs of CE01. For comparison, the [*Spitzer*]{} MIPS-24$\mu$m detection limit is represented as well as the knee of the total IR luminosity function, as derived by Magnelli et al. (2009, 2010).[]{data-label="FIG:limits"}](elbaz_fig4.ps){width="9cm"}
Fig. \[FIG:limits\] can be used to infer the reliability of the [*Spitzer*]{}-MIPS 24$\mu$m images of the two GOODS fields for identifying potential blending issues with [*Herschel*]{}. It shows that the 20$\mu$Jy depth at 24$\mu$m (3$\sigma$) reaches fainter sources than any of the [*Herschel*]{} bands, down to the confusion level and up to a redshift of $z$$\sim$3. The technique used to estimate total IR luminosities for the [*Herschel*]{} sources is discussed in Sect. \[SEC:method\]. Hence the positions of 24$\mu$m sources can be used to perform robust PSF fitting source detection and flux measurements on the [*Herschel*]{} maps. We validated the efficiency of this technique by checking that no sources remain in the residual images after subtracting the detected sources or by independently extracting sources using a blind source extraction technique (Starfinder: Diolaiti et al. 2000). Although it is the case that most [*Herschel*]{} sources have a 24$\mu$m counterpart, a few 24$\mu$m-dropout galaxies were found, i.e., galaxies detected by [*Herschel*]{} but not at 24$\mu$m. This will be the subject of a companion paper (Magdis et al. 2011). But these objects represent less than 1% of the [*Herschel*]{} sources.
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$\lambda$ FWHM Depth Depth
($\mu$m) () Nb N$_{z}^{spec}$ % $z_{spec}$ % $z_{all}$ Nb % $z_{sp}$ Nb N$_{z}^{spec}$ % $z_{spec}$ % $z_{all}$ Nb % $z_{sp}$
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16)
3.6 1.6$^{\mathrm{(a)}}$ 17$\mu$Jy 24476 3489 14 55 – – 20$\mu$Jy 13791 2933 21 99 – –
4.5 1.6$^{\mathrm{(a)}}$ 24$\mu$Jy 19957 3445 17 63 – – 35$\mu$Jy 10203 2808 28 99 – –
5.8 1.7$^{\mathrm{(a)}}$ 129$\mu$Jy 8182 2648 32 76 – – 137$\mu$Jy 4005 1943 49 100 – –
8 2.0 150$\mu$Jy 6020 2324 39 84 – – 134$\mu$Jy 3519 1760 50 100 – –
16 4.0 32$\mu$Jy 1297 870 67 90 – – 52$\mu$Jy 883 571 65 83 – –
24 5.7 21$\mu$Jy 2575 1284 50 91 – – 20$\mu$Jy 2063 1054 51 81 – –
70 18.0 2.4 mJy 150 118 79 83 94 98 3.1 mJy 456 77 17 100 50 17
100 6.7 1.1 mJy 1095 693 63 93 959 72 0.8 mJy 531 375 71 91 485 77
160 11.0 2.7 mJy 781 517 66 94 355 77 2.4 mJy 296 216 73 90 170 84
250 18.1 5.7 mJy$^{\mathrm{(b)}}$ 374 251 67 94 194 80 – – – – – – –
350 24.9 7.2 mJy$^{\mathrm{(b)}}$ 173 114 66 94 91 78 – – – – – – –
500 36.6 9 mJy$^{\mathrm{(b)}}$ 24 11 46 96 11 73 – – – – – – –
All – – 1263 776 61 89 990 72 – 555 385 69 91 498 41
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**Notes.** *Column definitions:* Col.(1) central wavelength of the passband; Col.(2) full width half maximum (FWHM) of the point spread function (PSF) in the passband. In the shortest bands, the FWHM is limited by the under-sampling of the PSF; Col.(3) depth of the image at that wavelength, i.e., flux density of the faintest sources of the catalog. The depths listed in the table correspond to the 3$\sigma$ limit. Due to local noise variations in the maps, some sources with slightly fainter flux densities may lie above this signal-to-noise threshold; Col.(4) total number of point sources above the 3-$\sigma$ limit. For the [*Herschel*]{} PACS and SPIRE passbands, we list the number of sources identified using a PSF-fitting based on [*Spitzer*]{}-MIPS 24$\mu$m prior positions, themselves resulting from [*Spitzer*]{}-IRAC 3.6$\mu$m priors; Col.(5) number of sources identified with an optical counterpart having a spectroscopic redshift; Col.(6) fraction of sources with an optical counterpart having a spectroscopic redshift; Col.(7) fraction of sources with an optical counterpart having either a spectroscopic or a photometric redshift; Col.(8) number of sources used in the present study, i.e., sources which are both “clean” (not polluted by bright neighbors as discussed in Sect. \[SEC:clean\]) and for which a redshift either photometric or spectroscopic was measured; Col.(9) fraction of the sources listed in column (8) for which a spectroscopic redshift was determined. Cols.(10) to (16) for GOODS–south are defined as Cols.(3) to (9) for GOODS–north.\
$^{\mathrm{(a)}}$ The characteristics of the GOODS–[*Herschel*]{} PACS 100, 160$\mu$m and SPIRE 250, 350 and 500$\mu$m images and catalogs are given in the bottom part of the table. The SPIRE images extend over 30$\times$30 but we restricted the present analysis, hence also the number of sources given in the table, to the same field as the one covered with the PACS bands and to the sources with a 24$\mu$m counterpart. The upper part of the table lists the characteristics of the [*Spitzer*]{} IRAC 3.6, 4.5, 5.8, 8$\mu$m, IRS peakup array 16$\mu$m and MIPS 24, 70$\mu$m images and catalogs.\
$^{\mathrm{(b)}}$ The depth of the SPIRE catalogs given here applies only to the sub-sample of “clean” galaxies, located in isolated areas as probed by the density maps obtained from the shorter wavelengths starting at 24$\mu$m (see Sect. \[SEC:clean\]). This explains why they are much lower than the statistical confusion limits as listed by Nguyen et al. (2010) of 29, 31 and 34 mJy (5$\sigma_{\rm conf}$ confusion limits). \[TAB:depths\]
### Limiting depths of the catalogs and flux uncertainties {#SEC:simulations}
The noise in the [*Herschel*]{} catalogs results from the combined effects of (1) instrumental effects $+$ photon noise, (2) background fluctuations due to the presence of sources below the detection threshold (photometric confusion noise, see Dole et al. 2004), (3) blending due to neighboring sources, above the detection threshold (source density contribution to the confusion noise). In both PACS and SPIRE images (except at 100$\mu$m in GOODS–N), the depths of the GOODS–[*Herschel*]{} observations are always limited by confusion, i.e., (2) and (3) are always stronger than (1). Global confusion limits have been determined for PACS (Berta et al. 2011) and SPIRE (Nguyen et al. 2010). However these global definitions assume no a priori knowledge on the local projected densities of sources, as if e.g., 500$\mu$m sources were distributed in an independent manner with respect to shorter wavelengths such as the 250 and 350$\mu$m ones, or even down to 24$\mu$m. Moreover, the flux limit associated to source blending, (3), is often artificially set to be the flux density above which 10% of the sources are blended, even though statistical studies, such as the present one, could afford higher fractions as long as the photometric uncertainty is well controlled. Actual observations instead demonstrate that shorter wavelengths do provide a good proxy for the density field of longer wavelengths (see Fig. \[FIG:stamps\]). Hence we define the 3$\sigma$ (or 5$\sigma$) sensitivity limits of the GOODS–[*Herschel*]{} catalogs as the flux densities above which at least 68% of the sources can be extracted with a photometric accuracy better than 33% on the basis of Monte Carlo simulations and we use the positions of 24$\mu$m sources as priors to extract sources from PSF fitting. Individual sources are attributed a “clean” flag depending on the underlying density field as defined in Sect. \[SEC:clean\].
Flux uncertainties were derived in two independent ways. First *(i)*, we added artificial sources into the real [*Herschel*]{} images and applied the source extraction procedure. This process was repeated a large number of times (Monte Carlo – MC – simulations). Second *(ii)*, we measured the local noise level at the position of each source on the residual images produced after subtracting sources detected above the detection threshold. The first technique gives a noise level for a given flux density averaged over the whole map, while the second one provides a local noise estimate. In the MC simulations, we define the 3$\sigma$ (or 5$\sigma$) sensitivity limits in all bands as the flux densities above which a photometric accuracy better than 33% (or 20%) is achieved for at least 68% of the sources in the faintest flux density bin (as in Magnelli et al. 2009, 2011).
Technique *(i)* provides a statistical noise level attributed for a given flux density which accounts for all three noise components but is independent of local variations of the noise. The histogram of the output - input flux densities of the MC simulations follows a Gaussian shape whose $rms$ was used to define the typical limiting depths of the [*Herschel*]{} catalogs listed in Col.(3) of Table \[TAB:depths\]. All GOODS–[*Herschel*]{} images (except the PACS–100$\mu$m image in GOODS–N) reach the 3$\sigma$ confusion level, i.e., the flux density for which the photometric accuracy is better than 33% for at least 68% of the sources is more than three times higher than the instrumental noise level.
In technique *(ii)*, only the noise components (1) and (2) are taken into account, since the objects participating in the third component (source blending) have been subtracted to produce the residual images. However, imperfect subtraction of sources, due to local blending, may inflate the local residuals in the maps after source subtraction. In the PACS images and catalogs, both techniques result in very similar noise levels. A statistical limiting depth was computed by convolving the residual images with the PACS beam at each wavelength and measuring the $rms$ of the distribution of individual pixels. This method resulted in the same depths as in technique *(i)* and listed in Col.(3) of Table \[TAB:depths\]. Instead, for the SPIRE data, local noise estimates in the residual maps were found to be systematically lower than those measured with technique *(i)*. On average, sources with a SPIRE flux density corresponding to the detection threshold of 3$\sigma$ in the MC simulations are found to present a local signal-to-noise ratio of 5 in the residual maps. For SPIRE sources, this implies that we consider only sources above the 5$\sigma$ limit in the residual maps, to be consistent with the 3$\sigma$ limit resulting from the MC simulations.
Due to local noise variations in the maps, there can be small numbers of sources with flux densities slightly fainter than the nominal detection limits, which explains the presence of sources below the horizontal lines in Fig. \[FIG:fluxes\].
{width="18cm"}
Local confusion limit and “clean index” {#SEC:clean}
---------------------------------------
The main source of uncertainty, in the SPIRE images in particular, comes from the high source density relative to the beam size, i.e., the so-called confusion limit (see Condon 1974). Assuming that this limit applies equally at all positions of the sky, Nguyen et al. (2010) estimated that the floor below which SPIRE sources may not be extracted is $\sim$ 30 mJy, corresponding to 5$\sigma_{\rm conf}$ confusion limits of 29, 31 and 34 mJy/beam for beams of 18.1, 24.9and 36.6 FWHM at 250, 350 and 500$\mu$m respectively.
However, this “global confusion limit” is defined assuming no a priori knowledge on the projected density map of the underlying galaxy population. If one instead assumes that shorter wavelengths, at a higher spatial resolution, can be used to define the local galaxy density at a given galaxy position, then a “local confusion limit” can be defined. In practice, this means that not all SPIRE sources are located at a place where several bright PACS or MIPS–24$\mu$m fall in the SPIRE beam. Following this recipe, Hwang et al. (2010a) defined a “clean index” that was attributed to all individual [*Herschel*]{} detections under the following conditions: a 500$\mu$m source is flagged as “clean” if its 24$\mu$m prior has at most one bright neighbor in the [*Spitzer*]{}-MIPS 24$\mu$m band (where “bright” means an F$_{24}$$>$50% of the central 24$\mu$m source) within 20$\arcsec$ (1.1$\times$FWHM of [*Herschel*]{} at 250$\mu$m) and no bright neighbor in each one of the shorter [*Herschel*]{} passbands, i.e., at 100, 160, 250, 350 and 500$\mu$m within 1.1$\times$FWHM of [*Herschel*]{} in these passbands (see Table \[TAB:depths\]). As a result, we only kept 11 clean sources at 500$\mu$m for which we consider that the photometry is reliable. The criterion becomes less critical for the shorter bands, since we only consider the presence of bright neighbors at shorter wavelengths. As a result, the number of 350$\mu$m detections is an order of magnitude larger than at 500$\mu$m. This “local confusion limit” was empirically defined after visually inspecting the data for all individual sources but a more detailed investigation of this quality flag using simulations of the actual GOODS sources both spatially and in redshift confirms its robustness (Leiton et al. 2011, in prep.). For galaxies for which this “clean index” condition is not met in some bands, unphysical jumps in the IR SED are observed. This may lead to wrong estimates of the dust temperature for example, systematically shifting it to colder values, since source blending affects preferentially the longest wavelengths.
With the sensitivity limits of GOODS–[*Herschel*]{}, Fig. \[FIG:limits\] shows that below a redshift of $z$$\sim$3 the shortest wavelengths are always deeper than the longest ones, hence one can take advantage of these higher resolution images to better constrain the confusion limit at local, instead of global, scales. Moreover, we note that the fluxes in e.g., the SPIRE bands are not independent of those measured in the 24$\mu$m and 100$\mu$m passbands. They even follow a tight correlation (see Elbaz et al. 2010 and the present analysis), again in the redshift range of interest here, i.e., $z$$\sim$0–2.5. Hence it is possible to map the density of IR sources and to flag sources in relatively isolated areas, with respect to similarly bright or brighter IR sources. If the “clean index” did not reject efficiently problematic measurements, this would result in an increase of the dispersion in the figures presented in this paper. Since we will show that these dispersions are quite small already, if this effect was corrected, it would only reinforce our results. Typically, half of the [*Herschel*]{} sources detected at $\lambda$$>$160$\mu$m survive this criterion (see Table \[TAB:depths\]).
High- and low-redshift galaxy samples {#SEC:highlowz}
=====================================
The GOODS–[*Herschel*]{} galaxy sample {#SEC:GH}
--------------------------------------
Both GOODS fields have been subject to intensive follow-up campaigns, resulting in a spectroscopic redshift completeness greater than 70% for the [*Herschel*]{} sources (Table \[TAB:depths\]). We use a compilation of 3630 and 3018 spectroscopic redshifts for GOODS–N (Cohen et al. 2000, Wirth et al. 2004, Barger, Cowie & Wang 2008, and Stern et al. in prep.) and GOODS–S (Le Fèvre et al. 2004, Mignoli et al. 2005, Vanzella et al. 2008, Popesso et al. 2009, Balestra et al. 2010, Silverman et al. 2010, and Xia et al. 2010) respectively. Photometric redshifts and stellar masses are computed in both fields from U-band to IRAC 4.5$\mu$m photometric data using Z-PEG (Le Borgne & Rocca-Volmerange 2002). The templates used for both photometric redshifts and stellar mass estimates are determined from PEGASE.2 (Fioc & Rocca-Volmerange 1999) are were produced using nine scenarios for the star formation history (see Le Borgne & Rocca-Volmerange 2002) with various star-formation efficiencies and infall timescales, ranging from a pure starburst to an almost continuous star-formation rate, aged between 1 Myr and 13 Gyr (200 ages). There is no constraint on the formation redshift. The templates are required to be younger than the age of the Universe at any redshift.
The redshift distributions of the sources individually detected in each of the [*Herschel*]{} bands, as well as at 24$\mu$m with [*Spitzer*]{}, are presented in Fig. \[FIG:fluxes\] for both fields. This illustrates the relative power of these bands to detect sources as a function of redshift. While the 500$\mu$m band samples sources at all redshifts from $z$=0 to 4, it only provides a handful of objects: 24 galaxies in total within the 10$\times$15 size of the GOODS–N field, with only 11 flagged as clean, 73% of which have a spectroscopic redshift determination. In comparison, more than a thousand sources are detected in the 100$\mu$m band, the vast majority being flagged as clean and 72% having a spectroscopic redshift.
Table \[TAB:depths\] also lists the characteristics of the other IR catalogs that we use in the present study. The GOODS [*Spitzer*]{} IRAC catalogs were created using SExtractor (Bertin & Arnouts 1996), detecting sources in a weighted combination of the 3.6 and 4.5$\mu$m images, with matched-aperture photometry in the four IRAC bands, using appropriate aperture corrections to total flux. The [*Spitzer*]{} 24$\mu$m and 70$\mu$m catalogs (Magnelli et al. 2011) use data from the [*Spitzer*]{} GOODS and FIDEL programs (PI: M. Dickinson). Sources detected in the IRAC images are used as priors to extract the 24$\mu$m fluxes, and then in turn a subset of those 24$\mu$m sources are used as priors to extract fluxes at 70$\mu$m. The 16$\mu$m data comes from [*Spitzer*]{} IRS peak-up array imaging (Teplitz et al. 2011); here again, 16$\mu$m catalog fluxes are extracted using IRAC priors. In this study, we make particular use of the [*Spitzer*]{} data to quantify the redshift dependence of the IR SEDs while minimizing mid-infrared k-corrections by measuring the rest-frame 8$\mu$m emission of galaxies at $z$$\sim$0, 1 and 2 from their observed fluxes in the IRAC-8$\mu$m, IRS-16$\mu$m and MIPS-24$\mu$m passbands. Table \[TAB:depths\] also gives the spectroscopic (%$z_{spec}$), and photometric $+$ spectroscopic (%$z_{all}$) completeness of the IR catalogs from 3.6 to 500$\mu$m within the fiducial GOODS area. As noted previously, the SPIRE images of GOODS–N cover a wider field, but here we do not count the sources detected outside the regular GOODS area.
Known AGN were excluded from the sample and will be discussed separately in Section \[SEC:AGN\]. X-ray/optical AGN were identified from one of the following criteria: $L_{\rm X}$\[0.5-8.0 keV\] $>$ 3$\times$10$^{42}$ ergs s$^{-1}$, a hardness ratio (ratio of the counts in the 2-8 keV to 0.5-2 keV passbands) higher than 0.8, N$_{\rm H}\geq$10$^{22}$ cm$^{-2}$, or broad/high-ionization AGN emission lines (Bauer et al. 2004). We also excluded power-law AGN, i.e., galaxies showing a rising continuum emission in the IRAC bands due to hot dust radiation (see definition in Sect. \[SEC:AGN\]).
Total infrared luminosities {#SEC:method}
---------------------------
Total IR luminosities, $L_{\rm IR}^{\rm Herschel}$, for GOODS–[*Herschel*]{} galaxies were determined by allowing the normalization of the CE01 template SEDs to vary and choosing the one that minimizes the $\chi^2$ fit to the [*Herschel*]{} measured flux densities. At the highest redshifts considered in the present analysis ($z \approx 2.5$), the [*Herschel*]{} 100$\mu$m passband samples rest-frame mid-IR wavelengths. Hence, to avoid mixing galaxies with and without direct far-IR detections, we require at least one photometric measurement at wavelengths longer than 30$\mu$m in the rest-frame. This excludes a few high redshift galaxies detected only at 100$\mu$m. Total IR luminosities, $L_{\rm IR}^{\rm Herschel}$, were integrated from 8 to 1000$\mu$m on the best-fitting normalized CE01 SED. When only one or two [*Herschel*]{} measurements are available above 30$\mu$m, the degeneracy of the fit being large, we use the standard CE01 technique, i.e., we use the SED with the closest luminosity from the CE01 library without allowing any renormalization.
In order to quantify the impact of the choice of a given set of SEDs to fit the [*Herschel*]{} measurements and determine $L_{\rm IR}^{\rm Herschel}$, we have repeated the same exercise with another SED library from Dale & Helou (2002, DH02). The ratio of the $L_{\rm IR}^{\rm Herschel}$ values derived with one or the other family of SEDs has a median of 1 and a dispersion of 12%–rms. The uncertainty in the determination of $L_{\rm IR}^{\rm Herschel}$ is therefore dominated by the actual error bars on the [*Herschel*]{} flux measurements rather than by the choice of the SED library. In order to account for the latter source of uncertainty, we have generated a series of 100 realizations of the [*Herschel*]{} flux measurements assuming a Gaussian distribution within their error bars and determined 100 values of $L_{\rm IR}^{\rm Herschel}$ by fitting those realizations independently. The final $L_{\rm IR}^{\rm Herschel}$ associated to a given galaxy is the median of the 100 Monte Carlo estimates and its error bar is the rms around the median. This procedure was repeated for each individual galaxy.
Since we will compare the distant GOODS–[*Herschel*]{} galaxies to a reference sample of local galaxies for which $L_{\rm IR}^{\rm tot}$ is estimated from [*IRAS*]{} measurements alone, as a consistency check we computed the total IR luminosity that we would obtain for the GOODS–[*Herschel*]{} galaxies if we had used Eq. \[EQ:sanders\] (taken from Sanders & Mirabel 1996), $$\begin{array}{l}
L_{\rm IR}/L_{\odot} = 4 \pi D_{lum}^2[{\rm m}] \left[1.8\times10^{-14} ( FIR [{\rm W m}^{-2}] ) \right] / 3.826\times10^{26} \\
{\rm where~} FIR = 13.48 F_{12\mu m} + 5.16 F_{25\mu m} + 2.58 F_{60\mu m} + F_{100\mu m}~,
\end{array}
\label{EQ:sanders}$$ as a proxy for the derivation of the 8 – 1000$\mu$m luminosity, instead of the actual integral over the IR SED. The [*IRAS*]{} flux densities $F_{12\mu m}$, $F_{25\mu m}$, $F_{60\mu m}$ and $F_{100\mu m}$ in Eq. \[EQ:sanders\] are in Jy. Both techniques give equivalent total IR luminosities within 5%, hence again the dominant cause of discrepancy in the comparison is related to flux uncertainties.
--------------- ----- ----------------- ----- ------- ------- --------- ----------
Local (2) (3) (4)
samples Nb R$_{\rm radio}$ FEE $<$10 10–11 11 – 12 $\geq$12
[*ISO*]{} 150 11 0 36 56 14 45
[*AKARI*]{} 287 47 0 63 164 55 9
[*Spitzer*]{} 211 58 211 0 44 154 13
Total 648 116 211 99 264 223 67
--------------- ----- ----------------- ----- ------- ------- --------- ----------
: Number of galaxies and total IR luminosity range of the local galaxy samples.
**Notes**. Column (2) total number of objects for each local sample; Col.(3) number of galaxies with a radio continuum (1.4 GHz) size estimate; Col.(4) number of galaxies for which a fraction of extended emission (FEE) was measured (Díaz-Santos et al. 2010), i.e., fraction of the mid-IR continuum at 13.2$\mu$m more extended than the *Spitzer*/IRS resolution of $3.6\,\arcsec$ (see Sect. \[SEC:compactness\]). \[TAB:local\]
Local galaxy reference sample {#SEC:local}
-----------------------------
The local galaxy reference sample that we use in this paper consists of galaxies detected with the [*Infrared Space Observatory*]{} ([*ISO*]{}), [*AKARI*]{}, and [*Spitzer*]{}. Their rest-frame 8$\mu$m luminosities and total IR luminosities are compared to those of the GOODS–[*Herschel*]{} galaxies. Galaxies with direct IRAC–8$\mu$m measurements from [*Spitzer*]{} are supplemented with galaxies with [*ISO*]{} 6.75$\mu$m and [*AKARI*]{} 9$\mu$m photometry, for which pseudo-IRAC 8$\mu$m luminosities, $L_8$, were computed using the IR SED of M82 (Förster Schreiber et al. 2001, Elbaz et al. 2002). The [*ISO*]{} and [*AKARI*]{} samples span a wide range of relatively low luminosity galaxies, together with a sample of ULIRGs, while the [*Spitzer*]{} sample contains a quite complete sample of local LIRGs (see Table \[TAB:local\]).
### Local [*ISO*]{} galaxy sample {#SEC:ISOlocal}
The mid-IR luminosities of this sample of 150 galaxies described in CE01 and Elbaz et al. (2002) were obtained from measurements taken with [*ISO*]{}. The sample includes 110 galaxies closer than 300 Mpc and spanning a wide range of mid-IR luminosities estimated from ISOCAM-LW2 (5–8.5$\mu$m, centered at 6.75$\mu$m) and 41 ULIRGs, at distances 80 to 900 Mpc, with mid-IR luminosities determined with the PHOT-S spectrograph of ISOPHOT (Rigopoulou et al. 1999). We refer to CE01 for a discussion of the conversion of the PHOT-S spectra into broadband luminosities equivalent to the LW2 filter. Pseudo-IRAC 8$\mu$m luminosities, $L_8$, were estimated by first convolving the ISOCAM CVF spectrum of M82 (Förster Schreiber et al. 2001, Elbaz et al. 2002) to the ISOCAM-LW2 and IRAC-8$\mu$m bandpasses and then normalizing the resulting luminosities to the observed luminosity for each of the 150 galaxies, in order to derive their $L_8$. Since both filters are wide and largely overlapping, the conversion depends very little on the exact shape of the spectrum used for the conversion and we checked that indeed using the CE01 SEDs (for example) instead of that for M82 would make negligible differences with respect to the actual dispersion of galaxies in the $L_{\rm IR}^{\rm tot}$ – $L_8$ diagram. Total IR luminosities, $L_{\rm IR}^{\rm tot}$, were derived from the four [*IRAS*]{} band measurements using Eq. \[EQ:sanders\].
### Local [*AKARI*]{} galaxy sample {#SEC:AKARIlocal}
Galaxies with mid-infrared (9$\mu$m) measurements from [*AKARI*]{} were cross-matched with the [*IRAS*]{} Faint Sources Catalog ver. 2 (FSC-2; Moshir, Kopman & Conrow 1992) and with spectroscopic redshifts from the Sloan Digital Sky Survey Data Release 7 (SDSS DR7; Abazajian et al. 2009) supplemented by a photometric sample of galaxies with redshifts available in the literature (Hwang et al. 2010b). For both [*IRAS*]{} and [*AKARI*]{}, we consider only the sources with reliable flux densities. A total of 287 galaxies have 9$\mu$m flux densities from the [*AKARI*]{}/Infrared Camera (IRC, Onaka et al. 2007) Point Source Catalog (PSC ver. 1.0, Ishihara et al. 2010) reaching a detection limit of 50 mJy (5$\sigma$) with a uniform distribution over the whole sky and closer than $\sim$450 Mpc ($z$$<$0.1). As in Sect. \[SEC:ISOlocal\], pseudo-IRAC 8$\mu$m luminosities, $L_8$, were computed by convolving the ISOCAM CVF spectrum of M82 with the [*AKARI*]{}-IRC 9$\mu$m bandpass to estimate the conversion factor between the IRC–9$\mu$m and IRAC–8$\mu$m luminosities assuming the same IR SED for all galaxies. The effective wavelength of the [*AKARI*]{} 9$\mu$m passband is 8.6$\mu$m (Ishihara et al. 2010), not far from that of the IRAC-8$\mu$m filter (7.9$\mu$m, Fazio et al. 2004).
Total IR luminosities were computed from the four [*IRAS*]{} bands using Eq. \[EQ:sanders\]. The IRC–9$\mu$m measurements were not used in the computation of $L_{\rm IR}^{\rm tot}$. Far-IR measurements were supplemented with the [*AKARI*]{}/Far-Infrared Surveyor (FIS; Kawada et al. 2007) all-sky survey Bright Source Catalogue (BSC ver. 1.0) that contains 427 071 sources, with measured flux densities at 65, 90, 140 and 160$\mu$m. We used the supplementary far-IR measurements for 16% of the sample for which there is no 12$\mu$m nor 25$\mu$m reliable measurement from IRAS. We checked the consistency of these IR estimates from [*AKARI*]{} with those obtained from [*IRAS*]{} alone and found that [*AKARI*]{} luminosities were systematically lower by 10%. We corrected those 16% galaxies by this factor.
### Local [*Spitzer*]{} galaxy sample {#SEC:SPITZERlocal}
A sample of 202 [*IRAS*]{} sources, consisting of 291 individual galaxies (some blended at [*IRAS*]{} resolution), were observed with the IR spectrograph (IRS) on-board *Spitzer* as part of the Great Observatories All-sky LIRG Survey project (GOALS; Armus et al. 2009). The sources were drawn from the *IRAS* Revised Bright Galaxy Sample (RBGS; Sanders et al. 2003) and represent a complete sub-sample of systems ($z\,<\,0.088$) with IR luminosities originally defined to be in the range of $10^{11}$L$_{\odot}$$\leq$$L_{\rm IR}$$\leq$10$^{13}$L$_{\odot}$. The GOALS sample includes 200 LIRGs and 22 ULIRGs. The total IR luminosities of the systems were derived using their *IRAS* measurements and Eq. 1 (see Armus et al. 2009 for further details on this calculation).
Using the spectral images obtained with the short-low module of IRS, Díaz-Santos et al. (2010) measured the spatial extent of the light radiated in the mid-IR continuum at 13.2$\mu$m of a sub-sample of 211 individual galaxies (closer than 350 Mpc) for which data were available at the time of publication and sources could be detected. We use these size estimates in our analysis regarding the link between star formation compactness and the $IR8$ ratio. This fraction of extended emission (FEE) is directly related to the spatial distribution of the star formation regions and presents the advantage of being measured in a wavelength range not affected by the presence/absence of emission lines such as PAHs. For the multiple systems unresolved by IRAS, Díaz-Santos et al. (2010) distributed the total IR luminosity between galaxies proportionally to their *Spitzer*/MIPS–24$\mu$m fluxes. Due to this redistribution of the luminosity, there are now 44 galaxies with IR luminosities less than 10$^{11}$ L$_{\odot}$ in our sample. Added to these normal star-forming galaxies, the present sample finally includes 154 LIRGs and 13 ULIRGs (with 10$^{12}$$\leq$$L_{\rm IR}^{\rm tot}$/L$_{\odot}$$<$4$\times$10$^{12}$). IRAC-8$\mu$m luminosities for these galaxies are from Mazzarella et al. (in prep). Stellar masses were derived by cross-matching the GOALS sample with 2MASS and converting the Ks luminosities into stellar masses (excluding remnants) using a using a mass-to-light ratio $M_*$/$L_{Ks}$=0.7 M$_{\odot}$/L$_{K,\odot}$ computed from PEGASE 2 (Fioc & Rocca-Volmerange 1997, 1999) assuming a Salpeter IMF and an age of 12 Gyr.
Universality of $IR8$ (=$L_{\rm IR}$/$L_{8}$): an IR main sequence {#SEC:IR8}
==================================================================
{width="9cm"} {width="9cm"}
The mid-infrared excess problem {#SEC:MIRexcess}
-------------------------------
Before the launch of [*Herschel*]{}, the derivation of $L_{\rm IR}^{\rm tot}$, hence also of the SFR, of distant galaxies had to rely on extrapolations from either mid-IR or sub-mm photometry. While there are many reasons why extrapolations from the mid-IR could be wrong (evolution in metallicity, geometry of star formation regions, evolution of the relative contributions of broad emission lines and continuum), it was instead found that they work relatively well up to $z$$\sim$1.5. Using shallower [*Herschel*]{} data than the present study, Elbaz et al. (2010) compared $L_{\rm IR}^{\rm tot}$, estimated from [*Herschel*]{} PACS and SPIRE, to $L_{\rm IR}^{24}$ – the total IR luminosity extrapolated from the observed [*Spitzer*]{} mid-IR 24$\mu$m flux density – and found that they agreed within a dispersion of only 0.15 dex. The CE01 technique used to extrapolate $L_{\rm IR}^{24}$ attributes a single IR SED per total IR luminosity. Hence a given 24$\mu$m flux density is attributed the $L_{\rm IR}^{\rm tot}$ of the SED that would yield the same flux 24$\mu$m flux density at that redshift.
Stacking [*Spitzer*]{} MIPS-70$\mu$m measurements at prior positions defined by 24$\mu$m sources in specific redshift intervals, Magnelli et al. (2009) found that the rest-frame 24$\mu$m/(1+$z$) and 70$\mu$m/(1+$z$) luminosities were perfectly consistent with those derived using the CE01 technique for galaxies at $z$$\leq$1.3. Although the 70$\mu$m passband probes the mid-IR regime for redshifts $z$$\gtrsim$0.8, it presents the advantage of sampling the continuum IR emission of distant galaxies without being affected by the potentially uncertain contribution of PAHs, contrary to that at 24$\mu$m. At $z$$\geq$1.5 however, extrapolations from 24$\mu$m measurements using local SED templates were found to systematically overestimate the 70$\mu$m measurements (Magnelli et al. 2011). This mid-IR excess, first identified by comparing $L_{\rm IR}^{24}$ with radio, MIPS-70$\mu$m and 160$\mu$m stacking (Daddi et al. 2007a, Papovich et al. 2007, Magnelli et al. 2011) has recently been confirmed with [*Herschel*]{} by Nordon et al. (2010) on a small sample of $z$$\sim$2 galaxies detected with PACS and by stacking PACS images on 24$\mu$m priors (Elbaz et al. 2010, Nordon et al. 2010).
Here, thanks to the unique depth of the GOODS–[*Herschel*]{} images, we are able to compare $L_{\rm IR}^{24}$ to $L_{\rm IR}^{\rm tot}$ for a much larger number of galaxies than in Elbaz et al. (2010) and, more importantly, for direct detections at $z$$>$1.5. In the left-hand part of Fig. \[FIG:IR8\], we show that the mid-IR excess problem is not artificially produced by imperfections that could result from the indirect stacking measurements, but instead takes place for individually detected galaxies at $z$$>$1.5 and at high 24$\mu$m flux densities, corresponding to $L_{\rm IR}^{24}$$>$10$^{12}$ L$_{\odot}$. Although known AGN were not included in the sample, unknown AGN may still remain. Indeed it has been proposed that the mid-IR excess problem could be due to the presence of unidentified AGN affected by strong extinction, possibly Compton thick (Daddi et al. 2007b, see also Papovich et al. 2007). At these high redshifts, the re-processed radiation of a buried AGN may dominate the mid-IR light measured in the 24$\mu$m passband, while the far-IR emission probed by [*Herschel*]{} would be dominated by dust-reprocessed stellar light. Indeed, studies of local dusty AGN have demonstrated that their contribution to the IR emission of a galaxy drops rapidly above 20$\mu$m in the rest-frame (Netzer et al. 2007). However, this explanation for the mid-IR excess problem was recently called into question by mid-IR spectroscopy of $z$$\sim$2 galaxies obtained using the [*Spitzer*]{} IRS spectrograph showing the presence of strong PAH emission lines where one would expect hot dust continuum emission to dominate if this regime were dominated by a buried AGN (Murphy et al. 2009, Fadda et al. 2010) and by deeper Chandra observations (Alexander et al. 2011).
Resolving the mid-IR excess problem: universality of $IR8$ {#SEC:univIR8}
----------------------------------------------------------
We have seen that extrapolations of $L_{\rm IR}^{\rm tot}$ from 24$\mu$m measurements using the CE01 technique fail at $z$$>$1.5. We also find that using the same technique with another set of template SEDs, such as the DH02 ones, fails in a similar way.
We wish to test the main hypothesis on which the CE01 technique relies, namely, that IR SEDs do not evolve with redshift. If that was the case, then a single SED could be used to derive the $L_{\rm IR}^{\rm tot}$ of any galaxy whatever the rest-frame wavelength probed, as long as it falls in the dust reprocessed stellar light wavelength range. Indeed, local galaxies are observed to follow tight correlations between their mid-IR luminosities at 6.75, 12, 15, 25$\mu$m and $L_{\rm IR}^{\rm tot}$ (see CE01, Elbaz et al. 2002) as well as with their SFR as derived from the Pa$\alpha$ line (Calzetti et al. 2007) for the [*Spitzer*]{} passbands at 8 and 24$\mu$m. This technique fails at $z$$>$1.5, which has until now been interpreted as evidence that distant IR SEDs are different from local ones. However, in order to properly test the redshift evolution of the IR SEDs, it is necessary to compare measurements in the same wavelength range for galaxies at all redshifts. For that purpose, we now compute the same rest-frame mid-IR luminosity, $L_8$ (=$\nu$$L_{\nu}$\[8$\mu$m\]), defined as the luminosity that would be measured in the IRAC–8$\mu$m passband in the rest-frame. We choose this particular wavelength range because it can be computed from $z$$\sim$0 to 2.5 with minimum extrapolations using the IRAC-8$\mu$m filter for nearby galaxies ($z$$<$0.5), the IRS-16$\mu$m peak-up array for intermediate redshifts around $z$$\sim$1 (0.5$\leq$z$<$1.5) and the MIPS-24$\mu$m passband at $z$$\sim$2 (1.5$\leq$$z$$\leq$2.5). Even in these conditions, small k-corrections need to be applied in order to calculate $L_8$ for the same rest-frame passband. This was done using the mid-IR SED of M82 for all galaxies. We verified that using other SEDs, such as the CE01 or DH02 templates, would alter $L_8$ by factors that are small when compared with the dispersion of the observed $L_{\rm IR}^{\rm tot}$ – $L_8$ relation. The results are shown in the right-hand part of Fig. \[FIG:IR8\]. Surprisingly, when plotting galaxies at all redshifts and luminosities in the same wavelength range, we no longer see a discrepancy between galaxies above and below $z$$\sim$1.5. The sliding median of the $IR8$ ratio, defined as $IR8$=$L_{\rm IR}^{\rm tot}/L_8$, – illustrated by white points connected with a solid grey line in the right-hand part of Fig. \[FIG:IR8\] – remains flat and equal to $IR8$=4.9 \[-2.2,+2.9\] (solid and dashed lines in Fig. \[FIG:IR8\]-right) from $L_8$=10$^9$ to 5$\times$10$^{11}$ L$_{\odot}$ or equivalently from $L_{\rm IR}^{\rm tot}$=5$\times$10$^{9}$ to 3$\times$10$^{12}$ L$_{\odot}$. The 68% dispersion around the median is only $\pm$0.2 dex.
![***Upper panel:*** Distribution of total IR luminosities ($L_{\rm IR}^{\rm {\it Herschel}}$) of the GOODS–[*Herschel*]{} galaxies classified as “clean” as a function of redshift. The solid and dashed red lines are the detection limits of the GOODS–S and GOODS–N images respectively. Spectroscopic and photometric redshifts are shown with filled and open dots respectively. ***Bottom panel:*** $IR8$ ratio (=$L_{\rm IR}^{\rm {\it Herschel}}$/$L_8$) as a function of redshift. The solid and dashed horizontal black lines are the median and 16th and 84th percentiles of the distribution (Eq. \[EQ:IR8\]), i.e., $IR8$=$4.9$ $[-2.2,+2.9]$. The solid and dashed red lines show the detection limits of the GOODS–S and GOODS–N images above $z$=1.5. The sliding median of the sources detected by [*Herschel*]{} is shown with black open circles connected with a solid line. A weighted combination of detections with stacked measurements (as in Fig. \[FIG:IR8\] and as described in Sect. \[SEC:univIR8\]) is shown with open yellow triangles (both fields combined). []{data-label="FIG:IR8z"}](elbaz_fig7.ps){width="9cm"}
![$L_{\rm IR}^{\rm tot}$\[IRAS\] versus $L_8$ for local galaxies including only the [*ISO*]{} sample of galaxies used to build the CE01 library of template SEDs and converted from 6.75$\mu$m to 8$\mu$m using the SED of M82. The light blue line shows the position of the CE01 SED templates, built to follow two power laws in the $L_{\rm IR}^{\rm tot}$ – $L_8$ relation.[]{data-label="FIG:IR8_iso"}](elbaz_fig8.ps){width="9cm"}
In order to test possible selection effects on the galaxies used to determine the $IR8$ ratio, we combined [*Herschel*]{} detections with stacked measurements on 24$\mu$m prior positions. This was done by defining intervals of luminosity in $L_8$, e.g., from the 16$\mu$m band for sources around $z$$\sim$1 or 24$\mu$m for sources around $z$$\sim$2. In a given $L_8$ interval, we determined the median of the $L_{\rm IR}^{\rm tot}$ obtained for detections on one hand (white dots connected with a solid line in the right-hand part of Fig. \[FIG:IR8\]) and on the other hand measured average PACS 100$\mu$m and 160$\mu$m flux densities for the sources with no [*Herschel*]{} detection by stacking sub-images of 60on a side at their 24$\mu$m prior positions. These sub-images were extracted from the residual images to avoid contamination by detections. The average stacked PACS-100$\mu$m and 160$\mu$m flux densities were converted into total IR luminosities using the CE01 library of template SEDs, selected based on luminosity at the median redshift of the galaxies in that $L_8$ luminosity interval. We found no systematic difference when deriving $L_{\mathrm IR}^{\mathrm tot}$ from the PACS 100 $\mu$m or 160 $\mu$m data when using the CE01 templates for the extrapolation (see also Elbaz et al. 2010) . Both PACS bands gave consistent values for $L_{\rm IR}^{\rm tot}$. The two values obtained for $L_{\rm IR}^{\rm tot}$ from detected and stacked undetected sources were then combined according to a weight depending on the number of sources in each group within this $L_8$ interval and on the signal-to-noise ratio of these measurements (quadratically), in order to avoid giving the same weight to both measurements if they have the same number of sources but very different S/N ratios. The resulting $L_{\rm IR}^{\rm tot}$ – $L_8$ relation is shown with yellow open triangles separately for each GOODS field. Since the 100$\mu$m and 160$\mu$m gave similar results, we only present in the right-hand part of Fig. \[FIG:IR8\] the result obtained from the 100$\mu$m band. Again, the typical $IR8$ ratio appears to be flat, independent of both luminosity and redshift. The range of luminosities probed by GOODS–[*Herschel*]{} varies as a function of redshift as shown in the upper panel of Fig. \[FIG:IR8z\], where we represent the distribution of total IR luminosities measured with [*Herschel*]{} as a function of redshift for the galaxies classified as “clean” (Sect. \[SEC:data\]). This is due to the combination of limited volume at low redshifts – limiting the ability to detect rare luminous objects – and depth at high redshifts – limiting the ability to detect distant low luminosity objects. In the bottom panel of Fig. \[FIG:IR8z\], we show the redshift evolution of the $IR8$ ratio. It is flat up to $z$$\sim$2 and then, due to the shallower detection limit of [*Herschel*]{} compared to [*Spitzer*]{}–24$\mu$m, it is slightly larger than the typical value, since only galaxies with high $L_{\rm IR}^{\rm tot}$/$L_8$ can be detected by [*Herschel*]{}.
Hence, we do not see a mid-IR excess when comparing systematically $L_{\rm IR}^{\rm tot}$ to 8$\mu$m rest-frame data. In particular, if AGN were playing a more important role at $z$$>$1.5 than at lower redshifts, we would expect to see a change in $IR8$ at this redshift cut-off contrary to what is actually observed. The cause for the mid-IR discrepancy is therefore not specific to galaxies at $z > 1.5$, but is instead due to the templates used to represent $z \sim 0$ ULIRGs. Locally, galaxies with $L_{\rm IR}^{\rm tot}$ $>$ 10$^{12}$ L$_{\odot}$ are very rare, most probably because galaxies today are relatively gas-poor compared to those at high redshift. Moreover, they have infrared SEDs that are not typical of star-forming galaxies in general, including those of most distant ULIRGs. The majority of high-redshift galaxies, even ultraluminous ones, share the same IR properties as do local, normal, star-forming galaxies with lower total luminosities. Galaxies with SEDs like those of local ULIRGs do exist at high redshift, but they do not dominate high redshift ULIRGs by number as they do in the present day.
Origin of the “mid-IR excess” discrepancy {#SEC:localIR8}
-----------------------------------------
{width="8.5cm"} {width="8.5cm"}
Fig. \[FIG:IR8\_iso\] shows the original $L_{\rm IR}^{\rm tot}$–$L_8$ data that were used to build the CE01 library of template SEDs. The solid line in the figure shows the relation traced by the SED templates. Originally, the mid-IR luminosity was computed from the ISOCAM–LW2 filter at 6.75$\mu$m, $L_{6.7}$, which we convert here to $L_8$ using the SED of M82. The conversion was validated by a sub-sample of galaxies for which we have measurements with both ISOCAM–LW2 and IRAC–8$\mu$m. While the trend followed by the CE01 templates is consistent with the GOODS–[*Herschel*]{} galaxies below $L_8\sim$10$^{10}$ L$_{\odot}$, there is a break above this luminosity threshold that was required to fit the local ULIRGs in this diagram.
In Fig. \[FIG:IR8\_local\]-left, we supplement the original local [*ISO*]{} sample with the 287 [*AKARI*]{} galaxies introduced in Sect. \[SEC:AKARIlocal\]. With this larger sample, we see galaxies extending the low luminosity trend beyond the threshold of $L_8\sim$10$^{10}$ L$_{\odot}$, with a flat $IR8$ ratio. This trend is similar to the one found for the GOODS–[*Herschel*]{} galaxies (background larger orange symbols as in Fig. \[FIG:IR8\]) and the extended local sample is well contained within the 16th and 84th percentiles around the median of the GOODS–[*Herschel*]{} sample (solid and dashed lines in Fig. \[FIG:IR8\]).
The median of both samples are very similar (see Eqs. \[EQ:IR8\],\[EQ:IR8\_local\]), $$IR8^{\rm local} = 4.8~~~[-1.7,+6.4]
\label{EQ:IR8_local}$$ $$IR8^{\rm GOODS-{\it Herschel}} = 4.9~~~[-2.2,+2.9]
\label{EQ:IR8}$$ Note, however, the large upper limit of the 68% dispersion in Eq. \[EQ:IR8\_local\], which is mainly due to the elevated $IR8$ values of the local ULIRGs, as seen in the left-hand panel of Fig. \[FIG:IR8\_local\]. The medians of both samples are shifted to higher values because of the asymmetric tails of galaxies with large values of $IR8$, as shown in the right-hand part of Fig. \[FIG:IR8\_local\] where we compare the $IR8$ distribution for the local [*ISO*]{}$+$[*AKARI*]{} galaxies (upper panel) with that of the GOODS–[*Herschel*]{} galaxies (lower panel). Both distributions present the same properties: they can be fitted by a Gaussian and a tail of high–$IR8$ values. The central values and widths $\sigma$ of the Gaussian distributions are very similar for both samples (Eqs. \[EQ:IR8local\],\[EQ:IR8gh\]), $$IR8^{\rm local} ({\rm center~Gaussian})= 3.9~~~[\sigma=1.25]
\label{EQ:IR8local}$$ $$IR8^{\rm GOODS-{\it Herschel}} ({\rm center~Gaussian})= 4.0~~~[\sigma=1.6]~,
\label{EQ:IR8gh}$$ again reinforcing the interpretation that the distant galaxies behave very similarly to local galaxies. If the IR SED of galaxies were different at low and high redshift, then one would not expect them to have the same distributions in $IR8$.
Hence we do not find evidence for different IR SEDs in distant galaxies. Instead, we find that local and distant galaxies are both distributed in two quite well-defined regimes: a Gaussian distribution containing nearly 80% of the galaxies, which share a universal $IR8$ ratio of $\sim$4, and a sub-population of $\sim$20% of galaxies with larger $IR8$ values. The exact proportion of this sub-population is not absolutely determined from this analysis, since it depends on the flux limit used to define the local reference sample, while the distant sample mixes together galaxies spanning a large range of redshifts and luminosities. Nevertheless, the objects in the high-$IR8$ tail remain a minority at both low and high redshift compared with those in the Gaussian distribution.
In the following, we call the dominant population “main sequence” galaxies, since they follow a Universal trend in $L_{\rm IR}^{\rm tot}$–$L_8$ valid at all redshifts and luminosities. We also justify this choice in the next sections by showing that this population also follows a main sequence in SFR – $M_*$, while galaxies with an excess $IR8$ ratio systematically exhibit an excess sSFR (=SFR/$M_*$). In the local sample, ULIRGs are clearly members of the second population whereas $z$$\sim$2 ULIRGs mostly belong to the Gaussian distribution, hence are main sequence galaxies. It is therefore the weight of both populations that has changed with time and that is at the origin of the mid-IR excess problem. The CE01 SED library, illustrated by a blue line in Figs. \[FIG:IR8\_iso\] and \[FIG:IR8\_local\], reaches values of $IR8$ that are more than five times larger than the typical value for main sequence galaxies. This leads to an overestimate of $L_{\rm IR}^{\rm tot}$ when the SED templates for local ULIRGs are used to extrapolate from 24$\mu$m photometry for main sequence galaxies at $z$$\sim$2. Note, however, that it is not necessary to call for a new physics for the IR SED of these galaxies that would justify, e.g., stronger PAH equivalent widths, since most of the distant LIRGs and ULIRGs belong to the same main sequence as local normal star-forming galaxies. It is well-known that local (U)LIRGs are experiencing a starburst phase, with compact star formation regions, triggered in most cases by major mergers (see e.g., Armus et al. 1987, Sanders et al. 1988, Murphy et al. 1996, Veilleux, Kim & Sanders 2002 for ULIRGs and Ishida 2004 for LIRGs). This leads us to the investigation of the role of compactness presented in the next section. Indeed, if local ULIRGs are known to form stars in compact regions and are found to be atypical in terms of $IR8$, then it would be logical to expect that distant ULIRGs instead are less compact, perhaps as a result of their higher gas fractions. Note also that galaxies with an excess $IR8$ ratio are found at all luminosities and redshifts and are not only a characteristic of ULIRGs.
$IR8$ as a tracer of star formation compactness and “starburstiness” in local galaxies {#SEC:compactness}
======================================================================================
The size and compactness of the star formation regions in galaxies is a key parameter that can affect the IR SED of galaxies. Chanial et al. (2007) showed that the dust temperature (T$_{\rm dust}$) estimated from the [*IRAS*]{} 60 over 100$\mu$m flux ratio, R(60/100), is very sensitive to the spatial scale over which most of the IR light is produced. It is known that there is a rough correlation of R(60/100), hence T$_{\rm dust}$, with $L_{\rm IR}^{\rm tot}$ (Soifer et al. 1987): locally, the most luminous galaxies are warmer. This relation has recently been established with [*AKARI*]{} and [*Herschel*]{} in the local and distant Universe (Hwang et al. 2010a). Locally, where galaxies can be spatially resolved in the far-IR or radio, Chanial et al. (2007) showed that the dispersion in the $L_{\rm IR}$ – T$_{\rm dust}$ relation was significantly reduced by replacing $L_{\rm IR}$ by the IR surface brightness, $\Sigma_{\rm IR}$. We extend this analysis to the relation between this star formation compactness indicator, $\Sigma_{\rm IR}$, and $IR8$, the far-IR over mid-IR luminosity ratio. In the present study, the term “compactness” is used to refer to the overall size of the starburst and not to the local clumpiness of the various star formation regions, which we cannot measure in most cases.
An extension of the Chanial et al. analysis to the brighter IR luminosity range of (U)LIRGS has become possible thanks to the work of Díaz-Santos et al. (2010). They used *Spitzer*/IRS data to derive the fraction of extended emission of the mid-IR continuum of the GOALS galaxy sample (Sect. \[SEC:SPITZERlocal\]) at 13.2$\,\mu$m.
Determination of the projected star formation density {#SEC:proj_density}
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### Radio/Far-IR projected surface brightness {#SEC:RADIOsizes}
Due to the limited angular resolution of far-IR data, we first estimate the sizes of star formation regions from radio imaging by cross-matching the local galaxy sample with existing radio continuum surveys and then convert them into far-IR sizes using a correlation determined from a small sample of galaxies resolved in both wavelength domains as in Chanial et al. (2007).
The IRAS-60$\mu$m and VLA radio continuum (RC, 20 cm) azimuthally averaged surface brightness profiles of a sample of 22 nearby spiral galaxies was fitted by a combination of exponential and Gaussian functions by Mayya & Rengarayan (1997). The angular resolutions of the 60$\mu$m and 20 cm maps used in that study was about 1, so we deconvolved their synthesized profiles by a 1beam and derived the intrinsic half-light radii r$_{\rm IR}$ (at 60$\mu$m) and r$_{\rm RC}$. The half-light radii estimates at both wavelengths are strongly correlated (Fig. \[FIG:rir\_radio\]); a logarithmic bisector fit to the data is given in Eq. \[EQ:radio\_size\]: $$r_{\rm IR} = (0.86 \pm 0.05)~~ r_{\rm RC}
\label{EQ:radio_size}$$ Hence in the following, we estimate the far-IR sizes of the star formation regions of our local galaxy sample from their radio continuum half-light radius using Eq. \[EQ:radio\_size\]. The existence of such correlation is not surprising, since the radio and far-IR emission of star-forming galaxies are known to present a tight correlation (Yun et al. 2001, de Jong et al. 1985, Helou, Soifer & Rowan-Robinson 1985): the radio emission is predominantly produced by the synchrotron radiation of supernova remnants and the bulk of the far-IR emission is due to UV light from young and massive stars reprocessed by interstellar dust. Hence, we consider this size estimate to be a good proxy for the global size of the star formation regions of galaxies. This is obviously an approximation, since this does not account for the clumpiness or granularity of the region, but this is the best that we can do with existing datasets.
![IRAS-60$\mu$m versus VLA 20 cm radio continuum half-light radius correlation.[]{data-label="FIG:rir_radio"}](elbaz_fig10.ps){width="9cm"}
Our local galaxy sample was cross-matched with the NRAO VLA Sky Survey (NVSS, Condon et al. 1998) and the Faint Images of the Radio Sky at Twenty-cm (FIRST, Becker, White & Helfand 1995), both obtained with the VLA at 20 cm. A total of 11, 47 and 58 galaxies have radio sizes in our [*ISO*]{}, [*AKARI*]{} and [*Spitzer*]{} local galaxy samples (see Table \[TAB:local\]).
We computed the IR surface brightness using Eq. \[EQ:surfIR\], $$\Sigma_{\rm IR} = \frac{L_{\rm IR}/2}{\pi r_{IR}^{2}}~,
\label{EQ:surfIR}$$ where the IR luminosity is divided by 2 since r$_{\rm IR}$ is the far-IR (60$\mu$m) half-light radius, which is derived from the 20 cm radio measurements using Eq. \[EQ:radio\_size\].
### Mid-IR compactness {#SEC:MIRsizes}
Using the low spectral resolution staring mode of the *Spitzer*/IRS, Díaz-Santos et al. (2010) measured the spatial extent of the mid-IR continuum emission at 13.2$\,\mu$m for 211 local (U)LIRGs of the GOALS sample (see section 2.2.3). The 13.2$\,\mu$m emission probes the warm dust (very small grains, VSGs) heated by the UV continuum of young and massive stars, and hence traces regions of dust-obscured star formation. Instead of measuring the half-light radius of the sources at this wavelength, Díaz-Santos et al. (2010) calculated their fraction of extended emission, or FEE, which they defined as the fraction of light in a galaxy that does not arise from its spatially unresolved central component. Conversely, the compactness of a source can be defined as the percentage of light that is unresolved, that is, 100$\times$(1$-$FEE). The angular resolution of *Spitzer*/IRS at 13.2$\,\mu$m is $\sim\,3.6\,\arcsec$ which, at the median distance of the sample used in this work, 91Mpc, results in a spatial resolution of 1.7kpc.
In the following, we consider galaxies as “compact” if their 13.2$\,\mu$m compactness is greater than 60%. With this definition, we find that 55% (117/211) of the GOALS galaxies are compact. Interestingly, while it is true that the fraction of galaxies showing compact star formation (i.e., compact hot dust emission) increases with increasing $L_{\rm IR}^{\rm tot}$ (hence also with SFR), the compact population is not systematically associated with the most luminous sources. On the contrary, galaxies with compact star formation can be found at all luminosities (see Figure 4 of Díaz-Santos et al. 2010).
![Comparison of the two compactness indicators: $\Sigma_{\rm IR}$ (=$L_{\rm IR}^{\rm tot}$/(2$\times$$\pi$$R_{\rm IR}^2$)), the IR surface brightness, and 13.2$\mu$m compactness (percentage of unresolved [*Spitzer*]{}/IRS light at 13.2$\mu$m, Díaz-Santos et al. 2010).[]{data-label="FIG:compactness"}](elbaz_fig11.ps){width="9cm"}
![Distribution of IR sizes (half-light radius) of 119 local galaxies as derived from their 1.4 GHz radio continuum using Eq. \[EQ:radio\_size\]. Galaxies with an IR surface brightness greater than $\Sigma_{\rm IR}$=3$\times$10$^{10}$ L$_{\odot}$kpc$^{-2}$, i.e., compact galaxies, are in blue, while extended galaxies are in red. The vertical dashed line indicates the typical resolution of $\sim$1.7 kpc of the mid-IR compactness at 13.2$\mu$m estimated by Díaz-Santos et al. (2010).[]{data-label="FIG:IRsize"}](elbaz_fig12.ps){width="9cm"}
### Identification of the galaxies with compact star formation {#SEC:compactSF}
In order to check whether both star formation compactness indicators are consistent, we used the 58 galaxies from the GOALS sample for which we can determine both $\Sigma_{\rm IR}$ from the radio sizes (Sect. \[SEC:RADIOsizes\]) and a 13.2$\mu$m compactness (Sect. \[SEC:MIRsizes\]). The comparison of both compactness indicators shows a correlation with a dispersion of $\sim$0.45 dex (Fig. \[FIG:compactness\]). The critical threshold of 60% in the 13.2$\mu$m compactness above which we classify galaxies as compact corresponds to $\Sigma_{\rm IR}$$\sim$3$\times$10$^{10}$ L$_{\odot}$ kpc$^{-2}$. Hence, we hereafter classify as compact the galaxies for which $\Sigma_{\rm IR}$$\ge$3$\times$10$^{10}$ L$_{\odot}$ kpc$^{-2}$. This threshold is more than two orders of magnitude lower than typical upper limits for star formation on small (kpc) scales (see Soifer et al. 2001). We note that if it were not averaged on large scales, the local star formation surface density could be much higher in many of these sources.
In Fig. \[FIG:IRsize\], we present the distribution of far-IR sizes of extended (red) and compact (blue) galaxies, estimated from radio 20 cm imaging using Eq. \[EQ:radio\_size\]. The median far-IR sizes of compact and extended galaxies are 0.5 kpc and 1.8 kpc respectively. The typical spatial resolution reached at 13.2$\mu$m, i.e., 1.7 kpc, is close to the typical size of extended galaxies and is significantly larger than the median size for compact galaxies. This contributes to the relatively high dispersion seen in Fig. \[FIG:compactness\]. With a linear resolution of $\sim$0.2 kpc at the average distance of the GOALS sample, the radio estimator is therefore a finer discriminant of compact galaxies when good quality radio data exist.
In the following, we use both compactness indicators, i.e., radio and 13.2$\mu$m, with no distinction to define the projected IR surface brightness, $\Sigma_{\rm IR}$ (=$L_{\rm IR}^{\rm tot}$/(2$\times$$\pi$$R_{\rm IR}^2$)). For galaxies with a measured radio size, $R_{\rm IR}$ is computed using Eq. \[EQ:radio\_size\], while for galaxies with a 13.2$\mu$m compactness estimate but no radio size we use the relation presented in Fig. \[FIG:compactness\].
![Dependence of $IR8$ (=$L_{\rm IR}^{\rm tot}$/$L$\[8$\mu$m\]) with $\Sigma_{\rm IR}$ (=$L_{\rm IR}^{\rm tot}$/(2$\times$$\pi$$R_{\rm IR}^2$)), the IR surface brightness. Galaxies to the right of the vertical dotted line are considered to be compact star-forming galaxies ($\Sigma_{\rm IR}$$\ge$3$\times$10$^{10}$ L$_{\odot}$ kpc$^{-2}$). Galaxies above the horizontal dotted line exhibit $IR8$ ratio that is two times larger than that of main sequence galaxies, which follow the Gaussian $IR8$ distribution shown in Fig. \[FIG:IR8\_local\]). The sliding median is shown with open blue dots. It is fitted by the solid line in red and its 16th and 84th percentiles are fitted with the dashed red lines (see Eq. \[EQ:IR8\_sigma\]). []{data-label="FIG:radio_compact"}](elbaz_fig13.ps){width="8.5cm"}
![$L_{\rm IR}^{\rm tot}$ (top) and $IR8$ ratio (bottom) versus $L$\[8$\mu$m\] for the local galaxy sample (ISO, [*AKARI*]{}, [*Spitzer*]{}-GOALS). Galaxies with compact mid-IR (more than 60% of the light emitted at 13.2$\mu$m within the resolution element of 3.6) and radio ($\Sigma_{\rm IR}$$\geq$3$\times$10$^{10}$ L$_{\odot}$) light distributions are marked with large filled red dots. Here the solid and dashed lines present the median and 68% dispersion around the Gaussian distribution as defined in Eq \[EQ:IR8local\]. []{data-label="FIG:setMSlocal"}](elbaz_fig14.ps){width="9cm"}
$IR8$, a star formation compactness indicator {#SEC:IR8compact}
---------------------------------------------
The $IR8$ ratio is compared to the IR surface brightness, $\Sigma_{\rm IR}$, in Fig. \[FIG:radio\_compact\]. The number of galaxies presented in this figure is larger than in Fig. \[FIG:compactness\] because we include sources with no radio size estimate as well. We find that $IR8$ is correlated with $\Sigma_{\rm IR}$ for local galaxies following Eq. \[EQ:IR8\_sigma\], $$IR8 = 0.22~[-0.05,+0.06] \times \Sigma_{\rm IR}^{0.15}~,
\label{EQ:IR8_sigma}$$ where $\Sigma_{\rm IR}$ is in L$_{\odot}$ kpc$^{-2}$. Hence $IR8$ is a good proxy for the projected IR surface brightness of local galaxies. Galaxies with strong $IR8$ ratios are also those which harbor the highest star-formation compactness.
We showed in Fig. \[FIG:compactness\] that galaxies having more than 60% of their 13.2$\mu$m emission unresolved by [*Spitzer*]{}–IRS, defined as compact star-forming galaxies by Díaz-Santos et al. (2010), presented an IR surface brightness of $\Sigma_{\rm IR}$$\geq$3$\times$10$^{10}$ L$_{\odot}$kpc$^{-2}$. This threshold is illustrated in Fig. \[FIG:radio\_compact\] by a vertical dotted line. It crosses the best-fitting relation of Eq. \[EQ:IR8\_sigma\] at $IR8$=8, i.e., twice the central value of the Gaussian distribution of main sequence galaxies (Fig. \[FIG:IR8\_local\] and Eq. \[EQ:IR8local\]). As a result, compact star-forming galaxies, with $\Sigma_{\rm IR}$$\geq$3$\times$10$^{10}$ L$_{\odot}$kpc$^{-2}$, systematically present an excess in $IR8$, whereas nearly all galaxies with extended star formation exhibit a ’normal’ $IR8$, i.e., within the Gaussian distribution of Fig. \[FIG:IR8\_local\]. This is illustrated in Fig. \[FIG:setMSlocal\], reproducing the $IR8$–$L8$ diagram for local galaxies of Fig. \[FIG:IR8\_local\], this time including the GOALS sample. The sub-sample of galaxies with measured IR surface brightnesses are represented with large symbols, with blue and red marking galaxies with extended and compact star-formation respectively, i.e., $\Sigma_{\rm IR}$ lower and greater than 3$\times$10$^{10}$ L$_{\odot}$kpc$^{-2}$. Galaxies with compact star formation systematically lie above the typical range of $IR8$ values. The trend can be extended to the local ULIRGs with no size measurement, since they are known to experience compact starbursts driven by major mergers (Armus et al. 1987, Sanders et al. 1988, Murphy et al. 1996, Veilleux, Kim & Sanders 2002).
![Fractions of compact local galaxies (black line), local galaxies with $IR8$$>$8 (blue dashed line) and GOODS–[*Herschel*]{} galaxies with $IR8$$>$8 (red dash-dotted line) as a function of total IR luminosity. The median redshift of the GOODS–[*Herschel*]{} galaxies varies with $L_{\rm IR}^{\rm tot}$, and is indicated by labels in grey along the red dash-dotted line. []{data-label="FIG:pcIR8"}](elbaz_fig15.ps){width="8.5cm"}
Very interestingly, the proportion of galaxies with compact star formation rises with $L_{\rm IR}^{\rm tot}$ following a path very similar to the proportion of galaxies with $IR8$$>$8 (Fig. \[FIG:pcIR8\]), the 68% upper limit of the GOODS–[*Herschel*]{} galaxies (Eq. \[EQ:IR8\]). Hence, $IR8$ can be considered as a good proxy of the star formation compactness of local galaxies. This can be very useful for galaxies with no radio size measurement.
In comparison, the fraction of excess $IR8$ sources within the GOODS–[*Herschel*]{} sample remains low (around 20%, see Fig. \[FIG:IR8\]-right and Fig. \[FIG:IR8\_local\]) and never reaches such high proportions as seen in local ULIRGs. Due to the [*Herschel*]{} detection limit, however, only ULIRGs are individually detected at $z$$>$2. The $IR8$ parameter is found to be biased towards high values in these galaxies which are responsible for the increase in the compactness fraction in the [*Herschel*]{} sample at the highest redshifts from 20 to 40%.
Globally, this analysis suggests that compact sources have been a minor fraction of star-forming galaxies at all epochs, but locally, due to the low gas content of galaxies, compact sources make the dominant population of ULIRGs. Extending the analysis of local galaxies to the distant ones, this also suggests that compact star formation takes place at all luminosities but does not dominate the majority of distant ULIRGs. Conversely, knowing the compactness and mid-IR luminosity of a galaxy, one may optimize the determination of its total IR luminosity from mid-IR observations alone. This is discussed in Sect. \[SEC:sed\].
Finally, we have assumed in this section that the compactness measured either from the radio or from the mid-IR continuum is associated with star formation. We discuss the role of AGN in Sect. \[SEC:AGN\], but we can already note that when an AGN contributes to the IR emission of a galaxy, it does so mainly at wavelengths shorter than 20$\mu$m (Netzer et al. 2007, Mullaney et al. 2011a). If AGN were contributing to the infrared emission, they would tend to boost $L_8$ relative to $L_{\rm IR}^{\rm tot}$, therefore reducing $IR8$. Instead, we see that an increasing compactness corresponds to an increase in $IR8$ as well. This reinforces the idea that we are dealing here with star formation compactness and not an effect produced by the presence of an active nucleus. In the next section, we show that compact star-forming galaxies are generally experiencing a starburst phase.
{width="9.cm"} {width="9cm"}
$IR8$, a starburst indicator {#SEC:IR8starburst}
----------------------------
In the previous section, we have seen that high $IR8$ values were systematically found in galaxies with compact star formation regions. We now show that these galaxies are experiencing a starburst phase. In the following, a star-forming galaxy is considered to be experiencing a starburst phase if its “current SFR” is twice or more stronger than its “averaged past SFR” ($<$SFR$>$), i.e., if its birthrate parameter $b$=SFR/$<$SFR$>$ (Kennicutt 1983) is greater than 2. Here $<$SFR$>$=$M_*$/$t_{\rm gal}$, where $t_{\rm gal}$ is the age of the galaxy. Alternatively, a star-forming galaxy may be defined as a starburst if the time it would take to produce its current stellar mass, hence its stellar mass doubling timescale, $\tau$, defined in Eq. \[EQ:t2\], $$\tau~[{\rm Gyr}]=M_{\star} ~[{\rm M}_{\odot}]~/~SFR~[{\rm M}_{\odot} ~{\rm Gyr}^{-1}] = 1 / sSFR~[{\rm Gyr}^{-1}]~,
\label{EQ:t2}$$ is small when compared to its age. Both definitions are equivalent if one assumes that galaxies at a given epoch have similar ages.
In recent years, a tight correlation between SFR and $M_*$ has been discovered which defines a typical specific SFR, (sSFR = SFR/$M_*$), for “normal star-forming galaxies” as opposed to “starburst galaxies”. This relation evolves with redshift but a tight correlation between SFR and $M_*$ is observed at all redshifts from $z$$\sim$0 to 7 (Brinchmann et al. 2004, Noeske et al. 2007, Elbaz et al. 2007, Daddi et al. 2007a, 2009, Pannella et al. 2009, Magdis et al. 2010a, Gonzalez et al. 2011). Hence, we use the sSFR definition of a starburst since it can be applied at all lookback times. In the present section, we consider only local star-forming galaxies. The SFR – $M_*$ relation for local [*AKARI*]{} galaxies is shown in the left-hand part of Fig. \[FIG:sfrAKARI\]. The best fit to this relation is a one-to-one correlation (0.26 dex–rms), hence a constant sSFR$\sim$0.25 Gyr$^{-1}$ or $\tau$$\sim$4 Gyr (Eq. \[EQ:t2\]). Local galaxies with compact star formation (large red dots), as defined in the previous section, are systematically found to have higher sSFR, i.e., $\tau$$<$ 1 Gyr, than that of normal star-forming galaxies.
Since there is a continuous distribution of galaxies ranging from the normal mode of star formation, with $\tau$$\sim$4 Gyr, to extreme starbursts, that can double their stellar masses in $\tau$$\sim$50 Myr, we quantify the intensity of a starburst by the parameter $R_{\rm SB}$, which measures the excess in sSFR of a star-forming galaxy (which we label its “starburstiness”), as defined in Eq. \[EQ:starburstiness\]:
$$R_{\rm SB} = sSFR / sSFR_{\rm MS} = \tau_{\rm MS} / \tau~~~~[>~2~~{\rm for~starbursts}]~,
\label{EQ:starburstiness}$$
where the subscript MS indicates the typical value for main sequence galaxies at the redshift of the galaxy in question. A starburst is defined to be a galaxy with $R_{\rm SB}$$\geq$2. 75% of the galaxies with compact star formation ($\Sigma_{\rm IR}$$\geq$3$\times$10$^{10}$ L$_{\odot}$ kpc$^{-2}$) have $R_{\rm SB}$$>$2, hence are also in a starburst mode, and 93% of them have $R_{\rm SB}$$>$1. Conversely, 79% of the starburst galaxies are “compact”. Globally, starburst galaxies with sSFR $>$ 2 $\times$ $<$sSFR$>$ have a median $\Sigma_{\rm IR}$$\sim$1.6$\times$10$^{11}$ L$_{\odot}$ kpc$^{-2}$, hence more than 5 times higher than the critical IR surface brightness above which galaxies are compact. The size of their star-forming regions is typically 2.3 times smaller than that of galaxies with sSFR$_{\rm MS}$.
![$R_{\rm SB}$=sSFR/sSFR$_{MS}$ versus $IR8$ (=$L_{\rm IR}^{\rm tot}$/$L_8$) for $z$$\sim$0 galaxies ([*AKARI*]{} and GOALS samples). The red line is the best fit (plain) and its 0.3 dex dispersion (dashed). []{data-label="FIG:sSFR_lirl8"}](elbaz_fig17.ps){width="8.5cm"}
The sSFR and $\Sigma_{\rm IR}$ are correlated with a 0.2 dex dispersion (Fig. \[FIG:sfrAKARI\]-right) following Eq. \[EQ:sSFR\_Sir\], where sSFR is in Gyr$^{-1}$ and $\Sigma_{\rm IR}$ in L$_{\odot}$kpc$^{-2}$: $$sSFR = 1.81~[-0.66,+1.05]\times10^{-4}~\times~\Sigma_{\rm IR}^{0.33}
\label{EQ:sSFR_Sir}$$ Both parameters measure specific quantities related to the SFR: the sSFR is measured per unit stellar mass, while $\Sigma_{\rm IR}$ is related to the SFR (derived from $L_{\rm IR}^{\rm tot}$) per unit area. Note, however, that it was not obvious [*a priori*]{} that these quantities should be correlated, since the stellar mass of most galaxies is dominated by the old stellar population, whereas the IR (or radio) size used to derive $\Sigma_{\rm IR}$ measures the spatial distribution of young and massive stars.
Because the starburstiness and the $IR8$ ratio are both enhanced in compact star-forming galaxies, they are also correlated as shown by Fig. \[FIG:sSFR\_lirl8\]. The fit to this correlation is given in Eq. \[EQ:R\_IR8\]: $$R_{\rm SB} = (IR8/4)^{1.2}
\label{EQ:R_IR8}$$ The dispersion in this relation is 0.3 dex. Hence we find that it is mostly compact starbursting galaxies that present atypically strong $IR8$ bolometric correction factors, although there is not a sharp separation of both regimes, but instead a continuum of values.
$IR8$ as a tracer of star formation compactness and “starburstiness” in distant galaxies {#SEC:MSSB}
========================================================================================
In the previous section, we have defined two modes of star formation:
- a normal mode that we called the infrared main sequence, in which galaxies present a universal $IR8$ bolometric correction factor and a moderate star formation compactness, $\Sigma_{\rm IR}$, and
- a starburst mode, identified by an excess SFR per unit stellar mass, hence sSFR, as compared to the typical sSFR of most local galaxies.
Galaxies with an enhanced $IR8$ ratio were systematically found to be forming their stars in the starburst mode and to show a strong star formation compactness ($\Sigma_{\rm IR}$$>$3$\times$10$^{10}$ L$_{\odot}$kpc$^{-2}$). In order to separate these two modes of star formation in distant galaxies as well, we first need to define the typical sSFR of star-forming galaxies in a given redshift domain. This definition has become possible since the recent discovery that star-forming galaxies follow a tight correlation between their SFR and $M_*$ with a typical dispersion of 0.3 dex over a large range of redshifts: $z$$\sim$0 (Brinchmann et al. 2004), $z$$\sim$1 (Noeske et al. 2007, Elbaz et al. 2007), $z$$\sim$2 (Daddi et al. 2007a, Pannella et al. 2009), $z$$\sim$3 (Magdis et al. 2010a), $z$$\sim$4 (Daddi et al. 2009, Lee et al. 2011) and even up to $z$$\sim$7 (Gonzalez et al. 2011).
Evolution of the specific SFR with cosmic time and definition of main sequence versus starburst galaxies
--------------------------------------------------------------------------------------------------------
![Redshift evolution of the median specific SFR (sSFR=SFR/$M_*$) of star-forming galaxies. Values for individual GOODS–[*Herschel*]{} galaxies are shown as grey points. Median sSFR values in redshift bins are shown with open circles (blue for GOODS–N and black for GOODS–S). Values combining individual detections and stacking measurements for undetected sources are shown with filled triangles (blue upward for GOODS–N and black downward for GOODS–S). The red solid line is the fit shown in Eq. \[EQ:sSFRz\], and the dashed lines are a factor 2 above and below this fit. Starbursts are defined as galaxies with a sSFR$>$2$\times$sSFR$_{\rm MS}$ (blue zone). The yellow zone shows the galaxies with significantly lower sSFR values. []{data-label="FIG:sSFRz"}](elbaz_fig18.ps){width="9.2cm"}
In the following, we assume that the slope of the SFR – $M_*$ relation is equal to 1 at all redshifts, hence that the specific SFR, sSFR (=SFR/$M_*$), is independent of stellar mass at fixed redshift. A small departure from this value would not strongly affect our conclusions and the same logic may be applied for a different slope. At $z$$\sim$0, our local reference sample is well fitted by a constant sSFR (see Fig. \[FIG:sfrAKARI\]-left), although the best-fitting slope is 0.77 (Elbaz et al. 2007). At $z$$\sim$1$\pm$0.3, Elbaz et al. (2007) find a slope of 0.9 but we checked that the dispersion of the data allows a nearly equally good fit with a slope of 1. At $z$$\sim$2, Pannella et al. (2009) find a slope of 0.95 consistent with the value obtained by Daddi et al. (2007a) in the same redshift range. Lyman-break galaxies at $z$$\sim$3 (Magdis et al. 2010a) and $z$$\sim$4 (Daddi et al. 2009) are also consistent with a slope of unity. From a different perspective, Peng et al. (2010a) argue that a slope of unity is required to keep an invariant Schechter function for the stellar mass function of star-forming galaxies from $z$$\sim$0 to 1 as observed from COSMOS data, while non-zero values would result in a change of the faint-end slope of the mass function that would be inconsistent with the observations.
However, the slope of the SFR – $M_*$ relation is sensitive to the technique used to select the sample of star-forming galaxies. Karim et al. (2011) find two different slopes depending on the selection of their sample: a slope lower than 1 for a mildly star-forming sample, and a slope of unity when selecting more actively star-forming galaxies (see their Fig.13). Using shallower [*Herschel*]{} data than the present observations, Rodighiero et al. (2010) found a slope lower than unity.
Assuming that the slope of the SFR – $M_*$ relation remains equal to 1 at all redshifts, a main sequence mode of star formation can be defined by the median sSFR in a given redshift interval, sSFR$_{\rm MS}(z)$. The starburstiness, described in Eq. \[EQ:starburstiness\], measures the offset relative to this typical sSFR. Since at any redshift – at least in the redshift range of interest here, i.e., $z$$<$3 – most galaxies belong to the main sequence in SFR – $M_*$, we assume that the median sSFR measured within a given redshift interval is a good proxy to the sSFR$_{\rm MS}(z)$ defining the MS. Galaxies detected with [*Herschel*]{} follow the trend shown with open circles in Fig. \[FIG:sSFRz\] (blue for GOODS–N and black for GOODS–S). We have performed the analysis independently for both GOODS fields in order to check the impact of cosmic variance on our result. To correct for incompleteness, we performed stacking measurements as for Fig. \[FIG:IR8\] but in redshift intervals. The stacking was done on the PACS-100$\mu$m images using the 24$\mu$m sources as a list of prior positions. The resulting values (blue upward triangles for GOODS–N and black downward triangles for GOODS–S) were computed by weighting detections and stacking measurements by the number of sources used in both samples per redshift interval. The SFR was derived from $L_{\rm IR}^{\rm tot}$ extrapolated from the PACS-100$\mu$m band photometry using the CE01 technique. The CE01 method works well for 100$\mu$m measurements up to $z$$\sim$3 as noted already in Elbaz et al. (2010), and we confirm this agreement with the extended sample of detected sources in the present analysis (Sect. \[SEC:IRtot\]). The trends found for both fields are in good agreement. The stacking $+$ detection measurements for GOODS–N are slightly lower than those obtained for GOODS–S which may result from a combination of cosmic variance and the fact that the GOODS–S image is deeper.
The redshift evolution of sSFR$_{\rm MS}(z)$ (Fig. \[FIG:sSFRz\]), accounting for both detections and stacked measurements, is well fitted by Eq. \[EQ:sSFRz\], $$sSFR_{\rm MS}~[{\rm Gyr}^{-1}]=26 \times t_{\rm cosmic}^{-2.2}~,
\label{EQ:sSFRz}$$ where $t_{\rm cosmic}$ is the cosmic time elapsed since the Big Bang in Gyr. A starburst can be defined by its sSFR following Eq. \[EQ:sSFRz\_SB\], $$sSFR_{\rm SB}~[{\rm Gyr}^{-1}]>52 \times t_{\rm cosmic}^{-2.2}~.
\label{EQ:sSFRz_SB}$$ The intensity of such starbursts, or “starburstiness”, is then defined by the excess sSFR: $R_{\rm SB}$=sSFR$_{\rm SB}$/sSFR$_{\rm MS}$. Due to the evolution observed with cosmic time, a galaxy with a sSFR twice as large as the local MS value would be considered a starburst today, but a galaxy with the same sSFR at $z$$\sim$1 would be part of the main sequence.
![$R_{\rm SB}$=sSFR/sSFR$_{MS}$ versus $IR8$ (=$L_{\rm IR}^{\rm tot}$/$L_8$) for the distant GOODS–[*Herschel*]{} galaxies. The green lines show the range of values occupied by main sequence galaxies (68% dispersion) in sSFR (horizontal lines; $R_{\rm SB}$=1$\pm$1) and $IR8$ (vertical lines; $IR8_{MS}$=4$\pm$1.6, see Eq. \[EQ:IR8gh\]). The thick grey line show the sliding median for the GOODS–[*Herschel*]{} galaxies. The diagonal blue lines are the best fit (solid) and 68% dispersion (dashed) for local galaxies as in Fig. \[FIG:sSFR\_lirl8\]. Large open dots show the position of sub-mm galaxies from Menendez-Delmestre et al. (2009) and Pope et al. (2008a). []{data-label="FIG:sSFR_lirl8_distant"}](elbaz_fig19.ps){width="9cm"}
We have seen that for local galaxies, the starburstiness and $IR8$ are correlated (see Fig. \[FIG:sSFR\_lirl8\]). The same exercise for distant GOODS–[*Herschel*]{} galaxies, mixing galaxies of all luminosities and redshifts, is shown in Fig. \[FIG:sSFR\_lirl8\_distant\]. Distant galaxies exhibit a non negligible dispersion, but their sliding median, shown by a thick grey line in Fig. \[FIG:sSFR\_lirl8\_distant\], is coincident with the best fit relation for local galaxies (solid and dashed blue lines).
We find that 80% of the galaxies which belong to the SFR – $M_*$ main sequence – with 0.5$\leq$$R_{\rm SB}$$\leq$2 – also belong to the main sequence in $IR8$ – $IR8_{\rm MS}$=4$\pm$1.6 (Eq. \[EQ:IR8gh\]). Hence we confirm that the two definitions of “main sequence galaxies” are similar and that on average they represent the same galaxy population. We note also that even though there is a tail toward stronger starburstiness and compactness, i.e., increased $R_{\rm SB}$ and $IR8$, this regime of parameter space is only sparsely populated in the GOODS–[*Herschel*]{} sample, which suggests that analogs to the local compact starbursts predominantly produced by major mergers remain a minority among the distant galaxy population. Finally, we see that sub-mm galaxies (large open circles in Fig. \[FIG:sSFR\_lirl8\_distant\]) also follow the same trend.
Star formation compactness of distant galaxies {#SEC:distant_compactness}
----------------------------------------------
We have shown that local galaxies with high $\Sigma_{\rm IR}$ values also exhibit high $IR8$ ratios. We do not have IR or radio size estimates for the distant galaxy population, but we can use the high resolution [*HST*]{}–ACS images to study the spatial distribution of the rest-frame UV light in the populations of MS and SB galaxies. It has been suggested that distant (U)LIRGs at 1.5$<$$z$$<$2.5 (Daddi et al. 2007a) and at $z$$\sim$3 (Magdis et al. 2010b) are not optically thick since the SFR derived from the UV after correcting for extinction using the Calzetti et al. (2000) law is consistent with the SFR derived from radio stacking measurements at these redshifts (see also Nordon et al. 2010).
![Stacked images (5 on a side) centered on main sequence (*left column*) and starburst (*right column*) galaxy positions. Typical MS galaxies are selected to have $R_{\rm SB}$$=$1$\pm$0.1 and SB galaxies $R_{\rm SB}$$\geq$3. Each image results from the stacking of [*HST*]{}–ACS images in the B (4350Å), V (6060Å) and I (7750Å) bands corresponding to the rest-frame UV at $\sim$2700Å at $z$=0.7 (first line), $z$=1.2 (second line) and $z$=1.8 (third line) respectively. []{data-label="FIG:stacksUV"}](elbaz_fig20.eps){width="7.5cm"}
---------- --------------- ------------------ ------------------
Redshift Main Sequence
$R_{\rm SB}$$>$2 $R_{\rm SB}$$>$3
0.7 5.2 kpc 3.9 kpc 2.5 kpc
1.2 4.4 kpc 3.3 kpc 2.5 kpc
1.8 3.0 kpc 2.5 kpc 2.0 kpc
---------- --------------- ------------------ ------------------
: UV – 2700Å half-light radii of distant main sequence and starburst galaxies.
\[TAB:uvsizes\]
We use [*HST*]{}–ACS images in the $B$ (4350Å), $V$ (6060Å) and $I$ (7750Å) bands to sample the same rest-frame UV wavelength of $\sim$2700Å at $z$=0.7, 1.2 and 1.8 respectively. MS galaxies are selected to have $R_{\rm SB}$$=$1$\pm$0.1 (Eq. \[EQ:starburstiness\]) whereas SB galaxies are defined as galaxies with $R_{\rm SB}$$\geq$2. We also tested a stricter definition for starbursts, $R_{\rm SB}$$\geq$3 to avoid contamination from MS galaxies (Table \[TAB:uvsizes\]). The result of the stacking of [*HST*]{}–ACS sub-images is shown in Fig. \[FIG:stacksUV\] for MS (left column) and SB galaxies with $R_{\rm SB}$$\geq$3 (right column). It is clear that the sizes of the starbursts are more compact than those of the main sequence galaxies. The half-light radius of each stacked image was measured with GALFIT (Peng et al. 2010b) and is listed in Table \[TAB:uvsizes\].
These sizes are consistent with those obtained by Ferguson et al. (2004). SB galaxies typically exhibit half-light radii that are two times smaller than those of MS galaxies, implying projected star formation densities that are 4 times larger. We verified that this was not due to a mass selection effect by matching the stellar masses in both samples and obtained similar results, although with larger uncertainties. These sizes are larger than the radio-derived IR half-light radii of the local sample of MS (1.8 kpc) and SB (0.5 kpc) galaxies. However, since the distant galaxy sample has a different mass and luminosity selection than that of the local reference sample, we cannot directly compare their sizes. However, the difference in the [*relative*]{} sizes among the high-redshift galaxies confirms that star formation in distant starbursts is more concentrated than that in distant main sequence galaxies. This, again, is strong evidence for a greater concentration of star formation in galaxies with higher specific SFRs. Since we have seen that sSFR and $IR8$ are correlated (Fig. \[FIG:sSFR\_lirl8\_distant\]), this implies that in distant galaxies, like in local ones, galaxies with strong $IR8$ ratios are likely to be compact starbursts.
This result is consistent with the work of Rujopakarn et al. (2011), who measured IR luminosity surface densities for distant (U)LIRGs similar to those found in local normal star-forming galaxies. However, we find that this is the case for most but not all high redshift (U)LIRGs. Compact starbursts do exist in the distant Universe, even among (U)LIRGs, but they are not the dominant population.
{width="9cm"} {width="9cm"}
Toward a universal IR SED for Main Sequence and Starburst galaxies {#SEC:sed}
==================================================================
Medium resolution IR SED for main sequence ($IR8$$\sim4$) and starburst ($IR8$$>8$) galaxies
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At $z$$<$2.5 – where we can estimate the rest-frame $L_8$ from [*Spitzer*]{} IRAC, IRS and MIPS photometry as well as reliable $L_{\rm IR}$ from [*Herschel*]{} measurements at rest-frame $\lambda$$>$30$\mu$m – the $IR8$ (=$L_{\rm IR}$/$L_8$) ratio follows a Gaussian distribution centered on $IR8$$\sim$4 (Eq. \[EQ:IR8gh\], Fig. \[FIG:IR8\_local\]), with a tail skewed toward higher values for compact starbursts. This defines two populations of star-forming galaxies or, more precisely, two modes of star formation: the MS and SB modes. Galaxies in the MS mode form the Gaussian part of the $IR8$ distribution and present typical sSFR values (i.e., $R_{\rm SB}$$\sim$1) while SB exhibit stronger $IR8$ values (see Fig. \[FIG:IR8\_local\]) and a stronger “starburstiness” ($R_{\rm SB}$$>$2).
$IR8$ is universal among MS galaxies of all luminosities and redshifts. This suggests that these galaxies share a common IR SED. In the local Universe, the rest-frame $L_{12}$, $L_{25}$, $L_{60}$, $L_{100}$ from [*IRAS*]{} and $L_{15}$ from ISOCAM were also found to be nearly directly proportional to $L_{\rm IR}^{\rm tot}$ (see CE01 and Elbaz et al. 2002), hence reinforcing this idea. To produce the typical IR SED of MS and SB galaxies, we use $k$-correction as a spectroscopic tool. We separate MS and SB galaxies by their $IR8$ ratios: $IR8$=4$\pm$2 for MS galaxies (as in Eq. \[EQ:IR8gh\]) and $IR8$$>$8 (hence $>$2$\sigma$ away from the MS) for SB galaxies. We then normalize the individual IR SEDs by a factor 10$^{11}$/$L_{\rm IR}^{\rm tot}$ so that all galaxies are normalized to the same reference luminosity of $L_{\rm IR}^{\rm tot}$=10$^{11}$ L$_{\odot}$. The result is shown with light grey dots in the left-hand part of Fig. \[FIG:SEDs\] for MS galaxies and in the right-hand part of Fig. \[FIG:SEDs\] for SB galaxies. A sliding median was computed in wavelength intervals which always encompass 25$\pm$5 galaxies (blue points for MS in Fig. \[FIG:SEDs\]-left and red points for SB in Fig. \[FIG:SEDs\]-right). As a result, the typical MS and SB IR SEDs have an effective resolution of $\lambda$/$\Delta \lambda$=25 and 10 respectively, nearly homogeneously distributed in wavelength from 3 to 350$\mu$m.
The typical MS IR SED in the left-hand part of Fig. \[FIG:SEDs\] has a broad far-IR bump centered around 90$\mu$m, suggesting a wide range of dust temperatures around an effective value of $\sim$30 K, and strong PAH features in emission. Instead, the typical IR SED for SB galaxies (Fig. \[FIG:SEDs\]-right) presents a narrower far-IR bump peaking around $\lambda$$\sim$70–80$\mu$m, corresponding to an effective dust temperature of $\sim$40 K, and weak PAH emission lines. We note however, that these prototypical IR SEDs result from the combination of 267 and 111 galaxies for the MS and SB modes, respectively. They therefore should be considered as average SEDs, acknowledging that there is a continuous transition from one to the other with increasing $IR8$ or star-formation compactness. In the next Section, we provide a model fit to these SEDs to better describe their properties.
SED decomposition of main sequence and starburst galaxies {#SEC:decomp}
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In order to interpret the physical nature of the MS and SB SEDs derived in the previous section, we adopt a simple phenomenological approach. We decompose the two classes of SEDs with the linear combination of two templates, shown in Fig. \[FIG:SED\_decomposition\]: (1) a “[*star-forming region*]{}” component including H$\,$[ii]{} regions and the surrounding photo-dissociation region (labeled SF), and (2) a “[*diffuse ISM*]{}” (interstellar medium) component accounting for the quiescent regions (labeled ISM). The luminosity ratio of the two components controls the [*IR8*]{} parameter. This SED decomposition is not unique and the two components used here are not rigorously associated with physical regions of the galaxies.
The SED of each sub-component is given by the model of Galliano et al. (2011, in prep.; also presented by Galametz et al. 2009). This model adopts the Galactic dust properties of Zubko, Dwek & Arendt (2004). To account for the diversity of physical conditions within a galaxy, we combine the emission of grains exposed to different starlight intensities, $U$ (normalized to the solar neighborhood value of $2.2\times10^{-5}\;\rm W\,m^{-2}$). We assume, following Dale et al. (2001), that the mass fraction of dust exposed to a given starlight intensity follows a power-law (index $\alpha$): $dM_{\rm dust}/dU\propto U^{-\alpha}$. The two cutoffs are $U_{min}$ and $U_{min}+\Delta U$. We fit the two SEDs simultaneously, varying only the luminosity ratio of the two components. We add a stellar continuum to fit the short wavelengths (see Galametz et al. 2009 for a description). This component is a minor correction. In summary, the free parameters for the fit are:
- the starlight intensity distribution parameters ($\alpha$, $U_{min}$ and $\Delta U$) of each sub-component;
- the PAH mass fraction and charge of each sub-component;
- the luminosity ratio of the two components for the main sequence and for the starburst;
- the contribution of the stellar continuum (negligible here)
The fits are shown with solid black lines in Fig. \[FIG:SEDs\] while the derived templates that we used for the decomposition are shown in blue and red lines in Fig. \[FIG:SED\_decomposition\]. The most relevant parameters are summarized in Table \[tab:decomp\]. The “[*diffuse ISM*]{}” component has colder dust and a larger PAH mass fraction than the “[*star-forming region*]{}” SED.
![[*Components used in the fit of Fig. \[FIG:SEDs\]*]{}. These two components have been constrained by the simultaneous fit of the two SEDs (Fig. \[FIG:SEDs\]). The main sequence and starburst SEDs are the linear composition of these two components. The luminosity ratio of these components controls IR8. []{data-label="FIG:SED_decomposition"}](elbaz_fig22.eps){width="9cm"}
The main differences between galaxies in the MS and SB modes are:
- the effective T$_{\rm dust}$ of galaxies in the SB mode is warmer than that of MS galaxies, i.e., $\sim$40 K versus $\sim$31 K;
- the contribution of diffuse ISM emission to the SB SED is negligible (8%), consistent with the strong compactness seen both for local starbursts (in radio and mid-IR imaging, see Sect. \[SEC:compactness\]) and for high redshift analogs (in the rest-frame UV, see Sect. \[SEC:distant\_compactness\]);
- the MS SED requires a wider distribution of dust temperatures, typically ranging from 15 to 50 K;
- the stronger contribution of PAH lines to the broadband mid-IR emission in the MS SED is the main cause for the difference in $IR8$ ratios between the two populations.
Galaxies are distributed continuously between the MS and various degrees of SB strength, hence this decomposition technique can be used in the future to produce SEDs suitable for ranges of $IR8$ or sSFR values, in the form of a new library of template SEDs. We note, however, that this decomposition of the typical MS and SB SEDs is not unique. For example, the SB SED is very similar to the CE01 template for a local galaxy with $L_{\rm IR} = 6 \times 10^{11}$ L$_{\odot}$ galaxy in the local Universe, which turns out to be close to the observed median luminosity of the starbursts. Instead, the MS SED is closer to the CE01 SED for a 4$\times$10$^{9}$ L$_{\odot}$ galaxy in the local Universe.
We note also that a direct fit of the Rayleigh-Jeans portions of both SEDs would favor an effective emissivity index of $\beta$=1.5 for the MS and $\beta$=2 for the SB. However, this is a degenerate problem. Indeed, the effective emissivity index $\beta$ is not necessarily equal to the intrinsic $\beta$ of the grains. A temperature distribution of grains having an intrinsic $\beta = 2$ would flatten the sub-mm SED and can give an effective $\beta$ of $\simeq1.5$, as it is the case for our star-forming region. Finally, it is also not possible to disentangle some potential contribution from an AGN, particularly for the SB SED. Indeed, AGN are known to be ubiquitous in LIRGs (Iwasawa et al. 2011) and ULIRGs (Nardini et al. 2010), and they may contribute in part to the mid-IR continuum, mostly in SB SEDs, since those are both more compact and exhibit lower PAH equivalent widths than they do MS galaxies. However, even if AGN may contribute to some fraction of the light in these galaxies, they cannot dominate both in the mid and far-IR regimes since we find evidence that PAHs dominate around 8$\mu$m in both MS and SB galaxy types, even if they are stronger in the MS SED. The high $IR8$ values measured for SBs also suggest that star formation dominates the IR emission in these galaxies. In Sect. \[SEC:AGN\] we present a technique to search for hidden AGN activity in the GOODS–[*Herschel*]{} galaxies.
[lrrr]{}\
& *ISM & *Star Forming & \[units\]\
$\langle U\rangle$ & 1.8 & 757 & $[2.2\times10^{-5}\;\rm W\,m^{-2}]$\
$T_{\rm eff}$ & 19 & 53 & \[K\]\
$f_{\rm PAH}$ & 1.9 & 0.3 & $[4.6\,\%]$\
[*IR8*]{} & 15 & 3 & …\
\
& Main Sequence & Starburst &\
$\phi$ & 62 & 92 & \[$\%$\]\
[*IR8*]{} & 5 & 11 & …\
$T_{\rm eff}^{\rm peak}$ & 31 & 40 & \[K\]\
**
**Notes.** $\langle U\rangle$ is the luminosity averaged starlight intensity; $T_{\rm eff}\simeq \langle U\rangle^{1/6}\times 17.5$ K is the corresponding effective temperature of the grains; $f_{\rm PAH}$ is the PAH-to-total-dust mass fraction; $\phi=L_{\rm SF}/(L_{\rm ISM}+L_{\rm SF})$ is the luminosity fraction of the star-forming component. $T_{\rm eff}^{\rm peak}$ is the effective dust temperature corresponding to the peak of the far-IR bump using Wien’s law. \[tab:decomp\]
![Ratio of the extrapolated ($L_{\rm IR}^{\lambda}$) over [*Herschel*]{} total IR luminosity as a function of redshift, for all clean GOODS–[*Herschel*]{} galaxies. $L_{\rm IR}^{\lambda}$ is computed by normalizing the main sequence SED to the broadband photometric measurement at $\lambda$. The 5 passbands used for the extrapolation are, from the top to bottom, [*Spitzer*]{} MIPS–24$\mu$m, [*Herschel*]{} PACS–100$\mu$m & 160$\mu$m, [*Herschel*]{} SPIRE–250$\mu$m & 350$\mu$m. $L_{\rm IR}$=$L_{\rm IR}^{\rm Herschel}$ is measured using the full set of [*Herschel*]{} measurements at rest-frame wavelengths $\lambda$$>$30$\mu$m to normalize the main sequence SED and by integrating over 8–1000$\mu$m. Black triangles: AGN. The solid lines with error bars are the sliding median and the 16th and 84th percentiles around it. Upper panel: SED of M82 and MIPS 24$\mu$m filter at $z$=0.25, 0.9, 1.4, 2.[]{data-label="FIG:IRtot"}](elbaz_fig23.ps){width="9.25cm"}
Derivation of total IR luminosities from monochromatic measurements {#SEC:IRtot}
-------------------------------------------------------------------
Now that we have defined a typical IR SED for main sequence galaxies, this SED may be used to extrapolate the total IR luminosity of galaxies for which only one measurement exists. Ideally, one would need to know the value of $IR8$ or equivalently the starburstiness, $R_{\rm SB}$, of a galaxy, to know whether to use the MS or SB SED. But this would require to already know the actual SFR of a galaxy, which is what we are looking for. An alternative technique would consist in using the star formation compactness of a galaxy, or $\Sigma_{\rm IR}$, to determine if it is in the main sequence or starburst mode.
Assuming that all galaxies share the same MS SED, in Fig. \[FIG:IRtot\] we compare the total IR luminosity that can be extrapolated from a single passband using the main sequence IR SED ($L_{\rm IR}^{\lambda}$) with the value ($L_{\rm IR}^{\rm tot}$) measured from an SED fit to galaxies with “clean” [*Herschel*]{} detections in several bandpasses (see Sect. \[SEC:clean\]). Both luminosities agree with an average uncertainty of $\sim$35% when $L_{\rm IR}^{\lambda}$ is extrapolated from 24$\mu$m and $\sim$20% when $L_{\rm IR}^{\lambda}$ is derived from one of the 100 to 350$\mu$m wavelengths. This is remarkable since we only used a single IR SED to extrapolate $L_{\rm IR}^{\lambda}$ for all galaxies. The MS SED does a better job than the CE01 technique (see Fig.3 in Elbaz et al. 2010), which overestimates $L_{\rm IR}^{\rm tot}$ from 24$\mu$m measurements at $z$$>$1.5 as well as from SPIRE 250 and 350$\mu$m measurements at $z$$<$1.3. Note however that individual galaxies do present a wide range of IR SEDs with different dust temperatures.
Finally, we note that these extrapolations work nearly equally well for X-ray AGN (black open triangles in Fig. \[FIG:IRtot\]) on average, although the dispersion is slightly larger for these galaxies. This suggests that star formation dominates the IR emission in the hosts of typical AGN in deep-field X-ray surveys. We discuss the properties of AGN in detail in Sect. \[SEC:AGN\].
Interpretation of the connection between compactness, starburstiness and $IR8$
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At fixed redshift, a normal galaxy forms stars at a rate proportional to its gas mass divided by the free-fall time, as expressed in Eq. \[EQ:SFRinterp\], $$\begin{array}{l}
SFR \propto M_{\rm gas} / \tau_{\rm free-fall} \propto M_{\rm gas} ~~ \rho_{\rm gas + stars}^{0.5} \\
{\rm since}~\tau_{\rm free-fall} \propto 1 / \sqrt{G \rho_{\rm gas + stars}}~,
\end{array}
\label{EQ:SFRinterp}$$ If one assumes that the free-fall time is dominated by the gas density, then it follows that the right term of the equation is proportional to $\rho_{\rm gas}^{1.5}$, as in the Schmidt-Kennicutt relation and close to the value of 1.4 found by Kennicutt (1998b) for projected gas and SFR densities. However the role of stars (in the free-fall time) may not be negligible in some conditions and may partly explain why including them in the relation may reduce the observed dispersion, as proposed by Shi et al. (2011).
If, instead, we consider separately the roles played by the density and gas mass in Eq. \[EQ:SFRinterp\], we may interpret that SB galaxies form stars more efficiently as a result of a greater $\rho_{\rm gas + stars}$, hence shorter $\tau_{\rm free-fall}$, possibly due to a merger. In the case of more distant galaxies, a galaxy with a similar stellar mass will naturally possess a higher gas fraction, and hence gas mass for its stellar mass, which will result in a greater sSFR. In this framework, where the difference of sSFR with redshift comes from a greater gas mass content in the past, but with similar gas densities, it is natural that MS galaxies exhibit similar $IR8$ values and share a common prototypical IR SED. In the case of a SB, where the density is increased (e.g., by a merger), the $IR8$ ratio is very sensitive to the geometry of the young stellar population (see, e.g., Galliano, Dwek & Chanial 2008, in particular their Fig.6). Increasing the compactness of the young stellar population, hence $\Sigma_{\rm IR}$, in the case of SB would increase the radiation field and push the photo-dissociation region farther away where molecules such as PAHs can survive and emit their light. The equivalent width of PAHs would then be reduced and the contribution of continuum emission increased, resulting in greater $IR8$ values.
{width="8.5cm"} {width="8.5cm"}
Unveiling dusty AGN within starburst galaxies {#SEC:AGN}
=============================================
The $IR8$ bolometric correction factor for X-ray and power-law AGN
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It is known that AGN can heat the dust that surrounds them to temperatures of several hundreds of Kelvin and produce a spectrum that can dominate the mid-IR emission of a galaxy, but which typically falls off steeply at wavelengths longer than 20$\mu$m (Netzer et al. 2007; see also Mullaney et al. 2011a). AGN can therefore make quite a significant contribution to the emission around 8$\mu$m. In this study, we have identified AGN using all available criteria and have excluded them from most aspects of the analysis up to this point. Only highly obscured and unidentified AGN may still be present in the sample that we used to carry out the $IR8$ analysis.
Before trying to define techniques to identify those hidden AGN, we should examine the properties of known and well-recognized AGN in the GOODS–[*Herschel*]{} data. We divide the known AGN into two populations (defined in Sect. \[SEC:GH\]): the X-ray/optical AGN and the infrared power-law AGN. The reason for this separation is that infrared power-law sources already show evidence that part of their mid-IR emission is powered by an AGN, since they have been identified as galaxies with a rising mid-infrared continuum from 3.6 to 8$\mu$m. We used the criteria given in Eq. \[EQ:PLagn\], taken from Le Floc’h et al. (in prep.; technique similar to that from Ivison et al. 2004 and Pope et al. 2008b for sub-mm galaxies): $$\begin{array}{l}
S_{\nu}[4.5\,\mu m] > S_{\nu}[3.6\,\mu m] \\
log_{10}\left(\frac{S_{\nu}[24\,\mu m]}{S_{\nu}[8\,\mu m]}\right) < 3.64 \times log_{10}\left(\frac{S_{\nu}[8\,\mu m]}{S_{\nu}[4.5\,\mu m]} \right) + 0.15
\end{array}
\label{EQ:PLagn}$$ X-ray/optical AGN that also meet the infrared power-law criteria are counted as power-law sources in the following discussion.
Surprisingly, AGN of both types exhibit $IR8$ ratios that are centered on the median of star-forming galaxies (upper panels of Fig. \[FIG:AGN\]), i.e., $IR8$$\sim$4.9 (Eq. \[EQ:IR8\]). This behavior may be understood for the non-power-law X-ray/optical AGN, since there is no evidence that high amounts of radiation from the AGN heats the surrounding dust in these galaxies. Hence for those galaxies, it is the star formation that is most probably responsible for both the 8$\mu$m and far-IR emission.
The situation is different for power-law AGN, however. In this case, we know, by definition, that a hot dust continuum is present that exceeds the stellar continuum emission at wavelengths shorter than 8$\mu$m. Still, the $IR8$ ratios observed for these galaxies remains similar to that found for star-forming galaxies (upper panel of Fig. \[FIG:AGN\]). This is consistent with Fig. \[FIG:IRtot\], where we showed that extrapolations of the total IR luminosity from any single photometric measurement between the observed 24 to 350$\mu$m passbands were nearly as accurate for AGN as for star-forming galaxies with no X-ray or optical AGN signatures.
However, we have seen in Sect. \[SEC:MSSB\] that compact starbursts have larger $IR8$ ratios than do normal star-forming galaxies. Hence it is possible that two mechanisms act in opposite ways: some contribution from the hot dust heated by an AGN to the mid-IR light may be counterbalanced by the presence of a starburst that increases the far-IR over mid-IR ratio. To test this possibility, we correct for the effect of starbursts on $IR8$ in the next section.
Correction of the effect of starbursts on $IR8$ {#SEC:SBcorrection}
-----------------------------------------------
We showed that a starburst induces an enhancement of the far-IR emission at fixed 8$\mu$m luminosity, whereas an AGN may induce an increase of the 8$\mu$m luminosity at fixed far-IR luminosity. The enhancement of $IR8$ in the presence of a starburst is proportional to its intensity, as measured by the starburstiness, $R_{\rm SB}$ (see Fig. \[FIG:sSFR\_lirl8\] and Fig. \[FIG:sSFR\_lirl8\_distant\]). Hence it can be corrected by normalizing $IR8$ by $R_{\rm SB}$, i.e., replacing $IR8$ by $IR8$/$R_{\rm SB}$.
The de-boosted $IR8$ ratios galaxies with known AGN are shown in the lower panels of Fig. \[FIG:AGN\] (open triangles). Interestingly, we find that the two AGN populations behave differently. The $IR8$ ratios of X-ray/optical AGN (from which we have excluded power-law AGN) remain centered on the region of star-forming galaxies. The fraction of AGN falling below the lower limit of main sequence star-forming galaxies increases from 11% to 22% but the same happens above the upper limit, which illustrates that this is just a result of the enhanced dispersion produced when correcting by $R_{\rm SB}$. This suggests that the IR emission of X-ray AGN is predominantly powered by star formation (as also confirmed by Mullaney et al. 2001b).
The case of the power-law AGN is very different. The fraction of galaxies falling below the lower limit in $IR8$ of MS star-forming galaxies rises from 33% (already three times higher than for the X-ray AGN) to 70% after dividing by $R_{\rm SB}$. Hence the majority of the power-law AGN show evidence for an 8$\mu$m excess, but this excess mid-infrared emission was disguised by the presence of a concurrent starburst.
Two important conclusions can be derived from this observation:
1. most of the IR emission from non-power-law X-ray/optical AGN appears to be powered by dust heated by stars. Hence their IR luminosities may be used to derive SFRs;
2. the bulk of power-law AGN host both an obscured AGN and a compact starburst.
The second point suggests that there is a physical link between both activities, the obscured AGN and the starburst, since they take place at the same time. It makes sense that infrared power-law AGN are associated with both compact starbursts and obscured active nuclei since compactness is required both to explain the excess sSFR of these galaxies as well as their dust obscuration. We note that the power-law criterion has not been demonstrated to be a perfect tracer of dusty AGNs, hence some of the galaxies selected by this criterion may just be purely star-forming galaxies.
Searching for unknown obscured AGN
----------------------------------
![$IR8$=$L_{\rm IR}^{\rm tot}$/$L_{8 \mu m}$ ratio as a function of $L_{8 \mu m}$ for star-forming galaxies, excluding recognized X-ray/optical AGN and infrared power-law AGN. The plain and dashed horizontal lines show the center and width of the Gaussian distribution of main sequence galaxies (Fig. \[FIG:IR8\_local\] and Eq. \[EQ:IR8gh\]). ***Upper panel:*** position of the star-forming galaxies as measured. Most fall on the same sequence as local star-forming galaxies. ***Bottom panel:*** position of the same galaxies after correcting their $IR8$ ratio for starburstiness (unless 0.5$\leq$$R_{\rm SB}$$\leq$2, where $R_{\rm SB}$ is consistent with unity within the error bars). []{data-label="FIG:deboostdistgals"}](elbaz_fig25.ps){width="9cm"}
We have seen in Sect. \[SEC:SBcorrection\] that the presence of a starburst could hide the signature of a dusty AGN on $IR8$, and that this could be corrected by normalizing $IR8$ by $R_{\rm SB}$. In Sect. \[SEC:SBcorrection\], we applied this correction only to known AGNs. We now consider the possibility that galaxies with neither X-ray nor optical evidence of an AGN may harbor a dust-obscured AGN whose signature is masked by the co-existence of a starburst.
In the upper panel of Fig. \[FIG:deboostdistgals\] we present the $IR8$ ratios of GOODS–[*Herschel*]{} galaxies as in the bottom panel of the right-hand part of Fig. \[FIG:IR8\]. The central value and width of the Gaussian distribution of main sequence galaxies (Fig. \[FIG:IR8\_local\] and Eq. \[EQ:IR8gh\]) is shown with plain and dashed lines respectively. Only 2% of the galaxies fall below the lower limit of the IR main sequence. After correcting $IR8$ by $R_{\rm SB}$, as in Sect. \[SEC:SBcorrection\], we find that the fraction of galaxies falling below the $IR8$=2.4 lower limit of the main sequence is increased by a factor of 8, reaching 17%, whereas the number of sources above the main sequence was reduced by 15%. The effect is stronger in (U)LIRGs, i.e., above $L_8$$\sim$3$\times$10$^{10}$ L$_{\odot}$, where the fraction of galaxies below $IR8$=2.4 reaches 25%. These galaxies present a starburstiness coefficient that would normally put them in the high-$IR8$ tail of the distribution. Instead, they exhibit normal values of $IR8$, and fall down to low values of $IR8$/$R_{\rm SB}$. This suggests that part of their 8$\mu$m rest-frame radiation is powered by a dust-obscured AGN that was not identified from X-ray or optical signatures. These candidate obscured AGN behave similarly to power-law AGN, but they were not identified as such because of the presence of a starburst. They are very good candidates for the missing Compton Thick AGN needed to explain the peak emission of the cosmic X-ray background around 30 keV (Gilli, Comastri & Hasinger 2007).
Finally, we applied this technique to a sample of sub-millimeter galaxies (SMGs) from Menendez-Delmestre et al. (2009) and Pope et al. (2008a). The $IR8$ and $R_{\rm SB}$ values of these galaxies are shown in Fig. \[FIG:sSFR\_lirl8\_distant\]. While most objects follow the trend defined for local galaxies (albeit with a wide dispersion), 11 out of a total of 28 SMGs exhibit a starburstiness ($R_{\rm SB}$) higher than expected for their $IR8$. Hence if we correct $IR8$ for starburstiness in these galaxies, we find evidence for the presence of hidden AGN activity. The most extreme cases are the SMGs known as “C1”, “GN39a” and “GN39b” from Pope et al. (2008a). C1, or SMM J123600+621047, is a $z$$\sim$2.002 SMG which has the strongest mid-IR continuum and weakest PAH emission lines in a sample of SMGs with [*Spitzer*]{} IRS spectroscopy analyzed by Pope et al. (2008a). Those authors interpret this as evidence that 80% of the mid-IR emission from this object arises from an AGN. It is undetected in the 2 Ms CDF-N data (Alexander et al. 2003), and its X-ray to 6$\mu$m luminosity ratio indicates that it hosts a Compton-thick AGN (Alexander et al. 2008). GN39a and GN39b are both classified as obscured AGN based on their strong X-ray hardness ratios, although their mid-IR spectra only show 10-40% AGN contribution. Hence the technique appears to be efficient even in the case of extreme systems such as distant SMGs.
Discussion and conclusion
=========================
The mode in which galaxies form their stars seems to follow some fairly simple scaling laws. The Schmidt-Kennicutt law, which connects the surface densities of gas and star formation in the local Universe (Kennicutt 1998b), has been recently extended to the study of distant galaxies. Two star formation modes have thus been identified: so-called “normal” star formation, and an accelerated mode, where the star formation efficiency (SFE) is increased, probably due to the merger of two galaxies (Daddi et al. 2010, Genzel et al. 2010). The different SFEs of these two modes are difficult to recognize because of the observational challenges associated with measuring the mass and density of molecular gas at high redshift. Indeed, the CO luminosity to H$_2$ conversion factor is poorly known and is based on $^{12}$CO $J$=1–0 emission locally, while observations of distant galaxies rely on higher-$J$ transitions (see, e.g., Ivison et al. 2011). Similarly, star-forming galaxies follow another scaling law: the SFR – $M_*$ relation, which measures the characteristic time to double the stellar mass of a galaxy. At each cosmic epoch, one can identify a typical sSFR for star-forming galaxies. This distinguishes a main sequence of “normal” star-forming galaxies from a minority population of starburst galaxies with elevated sSFR.
Our analysis of the deep surveys carried out in the open time key program GOODS–[*Herschel*]{} allowed us to establish a third scaling law for star-forming galaxies relating the total IR luminosity of galaxies, $L_{\rm IR}$, hence their SFR, to the broadband 8$\mu$m luminosity, $L_8$. We showed that the 8$\mu$m bolometric correction factor, $IR8$$\equiv$$L_{\rm IR}$/$L_8$, exhibits a Gaussian distribution containing the vast majority of star-forming galaxies both locally and up to $z$$\sim$2.5, centered on $IR8$$\sim$4. This defines an IR main sequence for star-forming galaxies. Outliers from this main sequence produce a tail skewed toward higher values of $IR8$. We find that this sub-population ($<$20%) is due to galaxies experiencing compact star-formation in a starburst mode.
The projected star-formation densities of present-day galaxies were estimated from their IR surface brightnesses, $\Sigma_{\rm IR}$, as measured from the size of their radio and/or 13.2$\mu$m continuum emission. For distant galaxies, we stacked rest-frame UV–2700Å images from [*HST*]{}–ACS in the $B$, $V$ and $I$ filters for galaxies located at $z$$\sim$ 0.7, 1.2 and 1.8 respectively. We find that at all times the projected star-formation density of galaxies in the high-$IR8$ tail is more compact ($\Sigma_{\rm IR}$$>$3$\times$10$^{10}$ L$_{\odot}$kpc$^{-2}$ at $z$$\sim$0) than in galaxies belonging to the IR main sequence, which includes distant (U)LIRGs as well.
Using the more accurate SFRs derived from [*Herschel*]{} data for galaxies at 0$<$$z$$<$3, we established the evolution of the typical specific SFR (sSFR=SFR/$M_*$) for star-forming galaxies. This allowed us to separate main sequence and starbursting galaxies thanks to a new parameter, labeled “starburstiness” (R$_{\rm SB}$), which measures the excess sSFR with respect to the SFR – $M_*$ main sequence, i.e., R$_{\rm SB}$=sSFR/sSFR$_{\rm MS}$($z$). We find that galaxies belonging to this main sequence (R$_{\rm SB}$=1$\pm$2) also belong to the one defined by the Gaussian distribution of $IR8$, and that the compact, star-forming galaxies that make up the high-$IR8$ tail fall systematically above the SFR – $M_*$ relation, with strong starburstiness (R$_{\rm SB}$$>$2–3). Indeed, we find that $IR8$ is strongly correlated with R$_{\rm SB}$ in general. Hence $IR8$ appears to be a good proxy for identifying compact starbursts, most probably triggered by merger events. In the present-day Universe, most (U)LIRGs are found to be experiencing compact star-formation during a starburst phase, which is not the case for most distant (U)LIRGs. Most probably, the very high SFRs of local (U)LIRGs can only be achieved during mergers, whereas distant galaxies are more gas-rich and can sustain these large SFRs in other ways. As a result of this difference, previous studies that have used local (U)LIRG SED templates, with their large, starbursting $IR8$ ratios, to extrapolate from MIPS 24$\mu$m photometry of galaxies at $z > 1.5$ have overestimated their total infrared luminosities and SFRs, resulting in the so-called “mid-IR excess” issue.
Using $k$-correction as a spectrophotometric tool for converting broadband photometric measurements at various redshifts into a medium resolution IR SED, we were able to determine the prototypical IR SED of MS and SB galaxies with a resolution of $\lambda$/$\Delta \lambda$=25 and 10 respectively. The SED of MS galaxies presents strong PAH emission line features, a broad far-IR bump resulting from a combination of emission from dust at different temperatures ranging typically from 15 to 50 K, and an effective dust temperature of 31 K, as derived from the peak wavelength of the IR SED. Galaxies that inhabit the SB regime instead exhibit weak PAH equivalent widths and a sharper far-IR bump with an effective dust temperature of $\sim$40 K. Although PAHs are stronger in MS than in SB galaxies, they are found to be present in the SEDs of both, implying that the IR emission in both populations is primarily powered by star formation and not AGN activity.
Finally, we present evidence that the mid-to-far IR emission of X-ray active galactic nuclei is dominantly produced by star formation, and that power-law AGNs systematically occur in compact, dusty starbursts. After correcting for the excess $IR8$ due to star formation – estimated from the starburstiness R$_{\rm SB}$ which we showed correlates with $IR8$ – we identify candidate members of for a sub-population of extremely obscured AGN that have not been identified as such by any other method.
Future studies will be dedicated to understanding the origin of the increase of $IR8$ with compactness and starburstiness, by separating the relative contributions of PAH lines and continuum emission, relating the starburstiness with the local environment of galaxies as well as with their dust temperature. ALMA and eMERLIN will soon provide powerful tools to measure the spatial distribution of star formation in distant galaxies at high angular resolution, making it possible not only to understand the compactness but also the clumpiness of the star-forming regions.
Spectro-imaging with integral field instruments like VLT/SINFONI will also be essential for measuring kinematic signatures to assess whether galaxy interactions play an important role in this process (see, e.g., Förster Schreiber et al. 2009, Shapiro et al. 2009). Most star formation takes place among main sequence galaxies, which suggests that internal, secular processes dominate over the role of mergers. This also accounts more naturally for the long duty cycle of their star-forming phase (Noeske et al. 2007, Daddi et al. 2007a, 2010). Finally, we note that this technique of separating MS and SB galaxies based on their IR star formation compactness will be extremely useful to extrapolate accurate total IR luminosities and SFRs of distant galaxies when using the mid-IR camera MIRI on board the [*James Webb Space Telescope*]{}.
We wish to thank R.Gobat for generating the three color images of the GOODS fields and our referee Kai Noeske for his constructive comments that helped improving the paper. D.Elbaz and H.S.Hwang thank the Centre National d’Etudes Spatiales (CNES) for their support. D.Elbaz wishes to thank the French National Agency for Research (ANR) for their support (ANR-09-BLAN-0224). VC would like to acknowledge partial support from the EU ToK grant 39965 and FP7-REGPOT 206469. Support for this work was also provided by NASA through an award issued by JPL/Caltech. PACS has been developed by a consortium of institutes led by MPE (Germany) and including UVIE (Austria); KU Leuven, CSL, IMEC (Belgium); CEA, LAM (France); MPIA (Germany); INAFIFSI/OAA/OAP/OAT, LENS, SISSA (Italy); IAC (Spain). This development has been supported by the funding agencies BMVIT (Austria), ESA-PRODEX (Belgium), CEA/CNES (France), DLR (Germany), ASI/INAF (Italy), and CICYT/MCYT (Spain). SPIRE has been developed by a consortium of institutes led by Cardiff University (UK) and including Univ. Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, Univ. Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); Imperial College London, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK); and Caltech, JPL, NHSC, Univ. Colorado (USA). This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); Stockholm Observatory (Sweden); STFC (UK); and NASA (USA).
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[^1]: [*Herschel*]{} is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.
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abstract: 'We investigate the dynamics of granular columns of point particles that interact via long-ranged hydrodynamic interactions and that fall under the action of gravity. We investigate the influence of inertia using the Green function for the Oseen Equation. The initial conditions (density and aspect ratio) are systematically varied. Our results suggest that universal self similar laws may be sufficient to characterize the temporal and structural evolution of the granular columns. A characteristic time above which an instability is triggered (that may enable the formation of clusters) is also retrieved and discussed.'
author:
- 'Hamza Chraïbi\*$^{1}$ and Yacine Amarouchene\*$^{1}$'
title: Sedimentation of granular columns in the viscous and weakly inertial regimes
---
Granular materials that are assemblies of discrete macroscopic solid particles with sizes large enough that Brownian motion is irrelevant, have been a subject of intensive research during the past few year [@Andreotti]. They are ubiquitous in our everyday lives and remain at the heart of several geophysical (sand dunes, coastal geomorphology, avalanches...) and industrial processes (chemical, pharmaceutical, food, agricultural...) [@Andreotti]. The variety of these fields make these granular materials subject to very different flow and stress conditions. In particular, when the particles are suspended in a fluid, one may expect that subtle hydrodynamic effects should play a leading role [@Guazelli]. This must be contrasted with the case of dry granular materials for which the influence of the carrying fluid is negligible. In that case both the inelasticity of the collisions and/or the friction between the grains are crucial [@Andreotti]. While the falling of a single or couple of particles in purely viscous and weakly inertial regimes was well described by Stokes and Oseen [@Brenner], understanding the interactions of a cloud of particles remains a challenge, as complex collective dynamics emerge due to the multiple long ranged interactions (see fluidized beds [@Caflisch; @GuazelliHinch]). Similar difficulties exist also for n-body gravitational problems. Therefore, many investigations were led in order to better understand the behavior of these particle laden flows, presenting a large panel of geometries like jets, streams, drops, spherical clouds. The sedimentation of spherical clouds of particles, in an external fluid of variable viscosity, has been recently investigated experimentally and numerically [@pignatel11]. At the exception of the experimental work of Nicolas [@nicolas02], investigations related to jets or column of particles focused mainly on highly viscous fluids (i.e. zero Reynolds numbers limit) [@pignatel09; @crosby12], air and moderate vacuum (Large Reynolds numbers limit) [@amarouchene2008; @prado11; @prado13] or other kinds of interactions : capillary bridges, Wan Der Waals forces [@mobius06; @royer09; @waitukaitis11; @ulrich12]... \
In this letter, we present an investigation that fully characterizes, using point-particle simulations, the dynamics of freely falling granular columns in different flow regimes, clarifying the dependence to the Reynolds number, the aspect ratio and the particle density.\
The main characteristics of the present system are: (i) solid particles suspended in a viscous fluid, and interacting by virtue of the fluid, (ii) particles heavier than the fluid, thus sedimenting on account of gravity. (iii) No continuous supply of particles in the granular cylinder.\
We will first describe the model used for the numerical simulations, defining the characteristic quantities of the problem and its relevant parameters before presenting and discussing our results.At the beginning of the simulation, we randomly initialize the positions of $N_{0}$ particles in a cylindrical column of radius $R_{0}$ and length $H_{0}$, such as the dimensionless particle density $n_{0}=N_{0}/(\pi h^{\ast})$ is homogeneous ($h^{\ast}=H_{0}/R_{0}$). In addition to their settling velocity $U_{\eta}=F/(6\pi\eta a)$ in the fluid of viscosity $\eta$ under the action of the gravitational force $F$, the point particles of radius $a$ are subject to the hydrodynamic pairwise interactions modeled by the dimensionless Green function of the Oseen Equation [@Brenner; @Guazelli; @pignatel11] which represents the additional velocity induced on a point particle by another point particle distant by $\mathbf{d}=(d_{x},d_{y},d_{z})$ : $$u_{k}^{\ast}=\frac{3}{4}a^{\ast}\left( \frac{d_{k}}{d^{2}}\left[ \frac{2l^{\ast}}{d}(1-E)-E\right] +\frac{E}{d}\delta_{kz}\right) ~;~k=x,y,z\label{oseen}$$$$E=\exp\left( -(1+\frac{d_{z}}{d})\frac{d}{2l^{\ast}}\right) ~;~a^{\ast
}=a/R_{0}~;~l^{\ast}=\eta/(U_{\eta}\rho_{f}R_{0})$$ In equation (\[oseen\]), all lengths and velocities were made dimensionless using $U_{\eta}$ as a reference velocity and $R_{0}$ as a reference length. A reference time $\tau_{\eta}=R_{0}/U_{\eta}$ was also defined. $\rho_{f}$ is the mass density of the external fluid and $l^{\ast}$ represents the importance of the viscous effects. Note that the velocity given by equation (\[oseen\]) is solution to the corrected Navier-Stokes Equation, which models the weakly inertial regime: $$\rho_{f}(\mathbf{U_{\eta}}\cdot\mathbf{grad})\mathbf{u}=-\mathbf{grad}p+\eta\triangle\mathbf{u}~~~$$ where $p$ is the fluid pressure.By choosing the frame of reference moving with the terminal settling velocity of an isolated particle, we compute all the $N_{0}-1$ interactions on each particle and obtain a set of equations describing the motions of the particles, of the form : $$\frac{dM_{ki}}{dt}=\sum_{j\neq i}u_{k}^{\ast}~~~~~k=x,y,z~~i=1,N_{0}~~j=1,N_{0}$$ This equation is integrated using an Adams-Bashford time-marching algorithm and at each iteration, we obtain the Cartesian position $M=(x,y,z)$ of each particle. The detection of the interface of a granular column is performed by dividing axially the domain into $h^{\ast}$ overlapping volumes. For each volume, the radial position of the interface is calculated by calculating the mean radial position of the farthest particles. The parameters for the simulations are the aspect ratio $h^{\ast}$, the particle density $n_{0}$, $l^{\ast}$ and $a^{\ast}$. However, one can notice that equation (\[oseen\]) is linear with respect to $a^{\ast}$ and therefore the dynamics of the problem will vary linearly with it. As a consequence, we set $a^{\ast}=0.05$ for all simulations. As our simulations neglect particle-particle collisions they are only applicable to dilute regimes. It is interesting to write the particle Reynolds number such as $Re_{p}=a^{\ast}/l^{\ast}=aU_{\eta}\rho_{f}/\eta$ which variation in this problem is performed by varying $\eta$, however as we are investigating the behavior of a macroscopic object, we have to define a macroscopic Reynolds number $Re=R_{0}U_{col}\rho_{f}/\eta$, where $U_{col}$ is the characteristic velocity of a cylindrical column. $U_{col}$ can be defined using the settling velocity of a vertical cylinder of aspect ratio $h^{\ast}$ in a viscous fluid, hence $U_{col}=P\ln(h^{\ast}/2)/(2\pi\eta H_{0})$, $P$ being the macroscopic gravitational force. Calculating the equivalent mass of the column from its volume fraction, one can find that $U_{col}=3\pi
n_{0}a^{\ast}\ln(h^{\ast}/2)U_{\eta}$ and therefore $Re=3\pi n_{0}\ln(h^{\ast
}/2)Re_{p}$.
{width="1.\columnwidth"}\[columns\]
\
An example of simulations performed varying $Re$ are shown in figure 1. We can first observe that while they fall, all the columns stretch and thin. We can also observe that a leading mushroom shaped plume forms at the front [@pignatel09] while a particle leakage can be observed at the rear. For the last instant, we can see that the columns lose their cohesion and eventually detach into shorter columns and droplets, which means that a varicose instability grows in time. Considering now the effect of the Reynolds number in relative frames, we can see that increasing $Re$ relatively slows down the falling of the columns and increases their effective cohesion. We can also observe that size of the leading mushroom is larger for a same instant. It is important to understand that, in order to compare them, columns for different $Re$ were represented in different frames. As the viscosity of the fluid for $Re=0.05$ is much larger than the viscosity for $Re=50$, columns at large $Re$ in the absolute frame will experience a faster dynamics. These results are qualitatively comparable to those of Pignatel [@pignatel11], which showed that increasing $Re$ enhances effective cohesion and slows the falling of spherical clouds of particles.
{width="1.\columnwidth"}
\
Figure 2 shows the time variation of the center of mass $V_{mass}$ reduced by a corrected characteristic velocity of the column $U_{cyl}$. Indeed, when varying $Re_{p}$ at fixed $h^{\ast}$ and $n_{0}$, we observed a correction in $V_{mass}$ which was not taken into account in $U_{col}$. This correction is shown in the inset of figure 2, where $V_{mass}/U_{\eta}$ decreases with $Re_{p}$ following a logarithmic behavior. We successfully retrieved this behavior by the function $f(Re_{p})=1/(1+\ln(1+l^{\ast-1}))=1/(1+\ln
(1+Re_{p}/a^{\ast}))$ and in order to take into account the $Re_{p}$ dependence of the dynamics, we defined a new characteristic velocity $U_{cyl}=3\pi U_{\eta}n_{0}a^{\ast}\ln(h^{\ast}/2)f(Re_{p})$ along with a macroscopic characteristic time $\tau_{cyl}=H_{0}/U_{cyl}$. Finally, we can observe in figure 2 that for a large set of different parameters, all the temporal evolution of $V_{mass}/U_{cyl}$ collapse in a single universal curve. It shows that for $t\ll\tau_{cyl}$, the column fall with a constant velocity $U_{cyl}$ before decreasing with time following a logarithmic behavior when $t\gg\tau_{cyl}$. This confirms that $U_{cyl}$ and $\tau_{cyl}$ are the adequate velocity and characteristic time that describe the falling of the columns.
{width="1.\columnwidth"}
\
The variation of particle mean density $<n(t)>/n_{0}$ versus dimensionless time $t/\tau_{\eta}$ is presented in figure 3. For different Reynolds number and initial particle densities, we observe a first incompressible regime where $<n(t)>$ is almost constant followed by a weakly compressible regime where the mean particle density experiences a slow time decay. While in the main figure, there seems to be different dynamics, the inset of figure 3 shows that using the characteristic time $\tau_{cyl}$ provides a better collapse of the data. In addition, we can see that $t\ll
\tau_{cyl}$ corresponds to an incompressible regime while $t\gg\tau_{cyl}$ corresponds to a weakly compressible flow.
{width="1.\columnwidth"}
\
The deformations of the columns are displayed in figure 4. It provides an adequate description of the dynamics of both the reduced length $H(t)/H_{0}$ and of the reduced mean radius $R(t)/R_{0}$ (calculated excluding the extremities of the column). Once again, the dynamics of the column deformation for a large set of different parameters $h^{\ast}$, $n_{0}$ and $Re_{p}$ are represented by universal curves. When $t\ll\tau_{cyl}$ the columns remain undeformed while for $\tau_{cyl}<t<10\tau_{cyl}$, the length increases following a universal scaling such as $H(t)\sim H_{0}(t/\tau_{cyl})^{2/3}$ and the mean radius decreases such as $R(t)\sim R_{0}(t/\tau_{cyl})^{-1/3}$. Assuming a weakly compressible flow for $t>\tau_{cyl}$ (in agreement with figure 3), the volume of the column has to remain constant, i.e. $\pi
R(t)^{2}H(t)\sim\pi R_{0}^{2}H_{0}$ which is well recovered by the previous scaling laws. Now let us focus on the strain rate $dV_{z}/dz$ selected at a local scale. It is an important parameter to describe the elongational stretching applied to the columns. It is known that stretching stabilizes liquid columns and prevent instabilities from growing and forming satellite drops [@eggers97]. Figure 5 provides a universal curve representing the variation of the reduced axial velocity gradient $(dV_{z}/dz)/(n^{\ast}U_{\eta}/H_{0})$ versus reduced time $t/\tau_{cyl}$ where different set of parameters present a good collapse. The elongation rate, or axial velocity gradient (which extent grows with time along the column), is deduced from the axial velocities of the particles at the rear of the columns, as shown in the inset. We can clearly see that in the incompressible regime ($t<\tau_{cyl}$), the elongational rate remains constant and scales like $n^{\ast}U_{\eta}/H_{0}$. This scaling comes from the fact that a single particle is on average surrounded by $2N_{0}/h^{\ast}=n^{\ast}$ particles (i.e. the particles contained in a sphere of diameter $2R_{0}$), therefore its characteristic velocity is $n^{\ast
}U_{\eta}$ while its characteristic axial length is $H_{0}$. In the weakly compressible regime ($t>\tau_{cyl}$), $dV_{z}/dz$ decays like $t^{-1}$. This is consistent with an incompressible self similar decay of the column radius $R(t)\sim R_0 t^{\alpha}$ that gives $(dV_{z}/dz)=-\frac{2}{R}\frac{dR}{dt}\sim
t^{-1}$ independently of the thinning exponent $\alpha$. Although not strictly comparable as they perform event-driven simulations of streams of particles interacting via collisions and cohesive forces and not via hydrodynamic interactions, Ulrich and Zippelius [@ulrich12] showed a similar result in the case of particles that fall in vacuum under the action of gravity. In that case the elongational rate is simply retrieved from the incompressibility condition and the velocity field imposed by the free fall. Finally, figure 5 also displays snapshots of columns at different Reynolds number. We observe that for $t\ll\tau_{cyl}$, the column are cohesive and destabilization has not yet occurred, while the columns for $t>\tau_{cyl}$ show a clear destabilization due to the development of a varicose instability.
{width="1.\columnwidth"}
\
Finally, let us focus on the description of the instability that leads to the destabilization of the columns. The main features of the instability are shown in figure 6. In the main panel of figure 6, we can note that the value of the most unstable wavelength $\lambda$ is almost constant and shows no clear dependence on the Reynolds number ($\lambda$ was deduced from the interface profile). In the regime $Re<<1$, we found $\lambda
\sim15R_{0}$ for $n_{0}=5$ and $\lambda\sim12R_{0}$ for $n_{0}=10$ which are both consistent with the values found in earlier investigations [@pignatel09; @crosby12] dedicated to the effect of $n_{0}$. The inset of figure 6 shows the temporal evolution of the standard deviation of the reduced radial variation (excluding the front drop) $\sigma n_{0}^{1/2}$ for $n_{0}=5$ and $10$ and for $Re=0.08, 0.8, 8$. We can see that the data collapse for $Re\ll1$, in good agreement with Crosby and Lister who suggested that the growth of the standard deviation of the reduced radial variation are mainly due to fluctuations in the average number density of particles along the axial distance about its mean value [@crosby12] . However, $\sigma$ seems to present larger values for $Re\gg1$. This means that increasing the Reynolds number may have a noticeable effect on the varicose instability. This induces a stronger effective cohesion and leads to a more efficient destabilization. These observations provide another route to the instability of granular jets along with the recently observed clustering due to cohesion and liquid bridges between grains [@royer09; @waitukaitis11]. Furthermore, our results suggest that the sedimentation of particle-laden jets may eventually furnish an interesting system to study the compressible Rayleigh -Plateau instability as suggested recently [@Miyamoto].\
{width="1.\columnwidth"}
\
To conclude, we have shown that universal scaling laws fully characterize the dynamics of free falling granular columns in viscous fluids. The characteristic velocity $U_{cyl}$ scales linearly with the particle density, while it shows a logarithmic increase with the aspect ratio and a decreasing logarithmic correction with the particle Reynolds number. A universal characteristic time $\tau_{cyl}$ based on $U_{cyl}$ and the column length $H_{0}$ has also been retrieved. When $t<\tau_{cyl}$, the flow could be considered as incompressible, and the columns deform only slightly and are subjected to a constant strain rate $n^{\ast}U_{\eta}/H_{0}$ while falling at a constant velocity $U_{cyl}$. For $t>\tau_{cyl}$, we showed that the flow was weakly compressible, and that the columns were subjected to an elongational rate decaying like $t^{-1}$, while they stretched like $t^{2/3}$ and thinned like $t^{-1/3}$ before the development of a varicose instability leading to a long wavelength destabilization. Finally, we found that the most unstable wavelength of the instability of the order of $\sim10R_{0}$ is almost independent of inertia corrections while the growth rate of the most unstable mode shows a clear increase with the Reynolds number.\
[**Acknowledgement**]{}\
We thank Pierre Navarot for the preliminary investigation . This research is supported by CR Aquitaine Grants no. 2006111101035, no. 20091101004 and by ANR Grant no. ANR-09-JCJC-0092.\
[ Corresponding authors’ e-mails :\
h.chraibi@loma.u-bordeaux1.fr\
y.amarouchene@loma.u-bordeaux1.fr]{}\
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J. R. Royer, D. J. Evans, L. Oyarte, Q. Guo, E. Kapit & M. E. Möbius and S. R. Waitukaitis and H. M. Jaeger, Nature. **459**, 11010 (2009).
S. R. Waitukaitis, H. F. Grütjen, J. R. Royer & H. M. Jaeger, Phys. Rev. E. **83** 051302 (2011).
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---
abstract: 'The unique fluctuation-dissipation theorem for equilibrium stands in contrast with the wide variety of nonequilibrium linear response formul[æ]{}. Their most traditional approach is “analytic”, which, in the absence of detailed balance, introduces the logarithm of the stationary probability density as observable. The theory of dynamical systems offers an alternative with a formula that continues to work when the stationary distribution is not smooth. We show that this method works equally well for stochastic dynamics, and we illustrate it with a numerical example for the perturbation of circadian cycles. A second “probabilistic” approach starts from dynamical ensembles and expands the probability weights on path space. This line suggests new physical questions, as we meet the [*frenetic*]{} contribution to linear response, and the relevance of the change in dynamical activity in the relaxation to a (new) nonequilibrium condition.'
address:
- '$^1$ Dipartimento di Fisica ed Astronomia, Università di Padova, Via Marzolo 8, I-35131 Padova, Italy'
- '$^2$ INFN, Sezione di Padova, Via Marzolo 8, I-35131 Padova, Italy'
- '$^3$ Instituut voor Theoretische Fysica, K.U. Leuven, Celestijnenlaad 400D, Leuven, Belgium'
author:
- 'M Baiesi$^{1,2}$ and C Maes$^3$'
title: An update on nonequilibrium linear response
---
Introduction {#intro}
============
An open system in contact with a large environment is in stationary equilibrium for a given reduced scale of description when at each moment its condition realizes minimal free energy. The total entropy production is then zero and the evolution is time-reversal invariant. Small disturbances break the stationary behavior and after some time an eventually new equilibrium is established mainly via dissipative effects. The comparison with the original equilibrium is the domain of linear response: since about 60 years now [@cal51; @kub66], for classical, quantum, open, and closed systems basically the same formula relates the response to a small perturbation with an equilibrium correlation function. This formula (later called Kubo formula) exactly picks up the physical interpretation in terms of dissipation, hence the name fluctuation-dissipation theorem [@tod92].
Response theory continues making sense for stimuli to nonequilibrium systems, where entropy is being produced already before the perturbation. Over the years various types of linear response formul[æ]{} have indeed been obtained for nonequilibria, including rather diverse but specific models such as in climatology and for glassy or coarsening dynamics. For better orientation the present paper identifies some common traits between the various approaches to put them in a more unified framework. Suppose for example that one wants a numerical code for predicting the response of an out-of-equilibrium systems, i.e., without actually perturbing the system. Without a clear picture of the available theoretical results, the choice of the method, if any, would be limited and possibly suboptimal. To answer requests like these, we aim at classifying the features of several “extended fluctuation-dissipation theorems” for nonequilibrium, highlight briefly their strengths and weaknesses, and the eventual relations between them.
Reviews and classifications of nonequilibrium response already exist. The paper by Marini Bettolo Marconi, Puglisi, Rondoni, and Vulpiani has reviewed response in statistical physics in the light of recent fluctuation relations [@mar08]. Seifert and Speck have introduced a classification of some fluctuation-dissipation theorems into three classes [@sei10]. Chetrite and Gupta present a more mathematical view [@che11]. The present work includes the case of deterministic dynamics and attempts a concise classification of linear response for nonequilibrium. We will conclude that there are three main classes. The first class of formul[æ]{} can be derived from the Kubo-Agarwal formula, which itself starts from a Dyson-expansion of the perturbed Markov semi-group. A second approach originates in the theory of dynamical systems and can be applied when the stationary distribution is not smooth; it gives rise to a numerical algorithm which in fact also applies for certain stochastic dynamics, as we show. The third class is much more probabilistic and treats noise as an important observable in the linear response. The synthesis is there provided by introducing the excess in dynamical activity in a second “frenetic” contribution to the traditional Kubo fluctuation-dissipation formula. In that sense, the name fluctuation–[*dissipation*]{} theorem (even “extended”) is not fully suitable for nonequilibrium systems, as e.g. their return to stationary nonequilibrium is not uniquely characterized by dissipation [@mae11]. On the mathematical side, our classification is to work either with generators (for Markov evolutions) or to work with weights on path space (mostly still limited to classical dynamical ensembles). Referring to those, we call it the analytic approach (Section \[per\]) versus the probabilistic approach (Section \[pro\]). Remarks complementing those in previous sections, and conclusions, are found in Section \[rema\].
Analytic approach {#per}
=================
The framework we consider is that of Markov dynamics (no memory) on regions of ${{\mathbb R}}^d$. That includes deterministic dynamical systems, being essentially first order in time. It also includes jump processes as described with Master equations, but we use here the language of diffusion because it reduces naturally to that of deterministic dynamical systems in the limit of zero noise, realized here with a state-independent diffusion constant $D \to 0$. For even much greater simplicity we choose overdamped diffusions where the velocity field $$\begin{aligned}
\label{lan}
\dot{x}_t = &v(x_t)& \quad\mbox{is given by}\\
&v(x_t)& = F(x_t) + \sqrt{2 D }\, \xi_t\nonumber\end{aligned}$$ for standard white noise $\xi_t$ [@ris89]. In Section \[path\] we also treat (underdamped or inertial) Langevin processes. In all these cases it makes sense to speak about the so called backward generator $L$, working on observables (Heisenberg picture). The expectation of an observable $A$ at time $t\geq 0$ is then given by $$\label{gena}
{{\left< A(x_t) \right>}} = \int \rho(dx) \left( e^{t L} A\right)(x)$$ when at time zero the states are distributed with probability $\rho$, possibly singular with respect to the reference volume element $dx$ on state space. We can also abbreviate that as ${\textrm{d}}{{\left< A(t) \right>}} / {\textrm{d}}t = {{\left< LA\,(t) \right>}}$. When no confusion can arise, we continue to write $A(t) =A(x_t)$ for the (most often random) value of the observable at time $t\geq 0$, and $\langle B(0) A(t)\rangle$ for the time-correlation between observable $B$ at time zero and observable $A$ at time $t$.
For the overdamped diffusion (\[lan\]) the generator is $$\label{gene}
(LA)\,(x) = (F \cdot \nabla A)\,(x) + D \,\Delta A\,(x)$$ A perturbation changes the drift $F \to F^h\equiv F+h F_1$ where, for simplicity, we avoid inserting a time-dependence in the small amplitude $h$, applied at all times $t > 0$. This leads to a perturbed backward generator $L^h \equiv L + h L_1$ with $L_1 A= F_1 \cdot \nabla A$. The change in expectations at times $t$ with respect to what we had for the unperturbed dynamics follows from : $$\langle A(t)\rangle^h - \langle A(t)\rangle = \int \rho({\textrm{d}}x)\,
\,\big(e^{tL^h} - e^{tL}\big)A\,(x)$$
To linear order in $h$, $$e^{tL^h} - e^{tL} = \int_0^t e^{sL}\,(L^h-L)\,e^{(t-s)L} {\textrm{d}}s + {{\mathcal O}}(h^2)$$ yielding a linear response $$\begin{aligned}
\frac{{\textrm{d}}\langle A(t)\rangle^h}{{\textrm{d}}h}{\bigg|}_{h=0}
&\equiv& \lim_{h\to 0} \frac{\langle A(t)\rangle^h - \langle A(t)\rangle}{h} \nonumber\\
&\equiv&
\chi(t) = \int_0^t {\textrm{d}}s \,R(t,s)\end{aligned}$$ including a susceptibility $\chi(t)$ as the integration of a response function of the form $$\label{dys}
\fbox {$ \displaystyle
R(t,s) = \int \rho({\textrm{d}}x)\;\big(e^{sL}\,L_1\,e^{(t-s)L}\,A\big)\,(x)
$}$$ More generally, when applying a time-dependent perturbation $h_s$ at time $s>0$ we also have $$\frac{\delta \langle A(t)\rangle^h}{\delta h_s}{\bigg|}_{h=0} = R(t,s),\quad s<t$$ We mostly restrict ourselves to the case where $\rho$ is stationary: the response depends only on the time difference $\tau=t-s$ and we can write $$\fbox {$ \displaystyle
R(\tau) = \int\, \rho({\textrm{d}}x)\,L_1 e^{\tau L}A\,(x) \label{dys2a}
$}$$ as the central object of study for the linear response of stationary Markov evolutions within an analytic approach[^1].
The direct reading of the right-hand side of , further discussed under Section \[sec:det\], is that $e^{\tau L}A$ evolves the observable $A$ for a time $\tau$ and then $L_1$ acts on the result to evaluate it in state $x$. However, we start in the next section with the more frequent approach of acting on $\rho$.
Acting on probabilities {#apr}
-----------------------
In this section we focus on manipulations with the stationary probability distribution $\rho$. The basic step from is partial integration, which means that it is assumed here that $\rho$ has a smooth density with respect to the reference volume element, $\rho({\textrm{d}}x) = \rho(x) \,{\textrm{d}}x$. In many cases that appears to be a reasonable physical assumption when the level of description is mesoscopic to macroscopic, independent of whether the system is driven or not (see [@col12]).
### Kubo–Agarwal formula
Assuming a smooth density $\rho$ we have that can be rewritten as $$\label{agar0}
R(\tau) = \int\, {\textrm{d}}x\, \rho(x)\,\frac{{{\mathcal L}}_1\rho}{\rho}(x)\, e^{\tau L}A\,(x)$$ where the adjoint ${{\mathcal L}}_1$ is defined by $\int {\textrm{d}}x ({{\mathcal L}}_1\rho)(x)\,A(x) = \int {\textrm{d}}x \rho(x)\,(L_1 A)(x)$. Adjoints are forward generators of the time-evolution on densities, as appears e.g. in Master equations. For the diffusion the adjoint of $L$ is the operator of the Fokker-Planck equation ($\partial_t\rho = {{\mathcal L}}\rho$, see [@ris89]) $$\label{FPL}
{\cal L} \rho = -\nabla\cdot(F\rho)+\Delta(D \rho)$$ and ${\cal L}_1 \rho = -\nabla\cdot(F_1\rho)$ so that takes the form $$\label{difag}
R(\tau) = -{{\left< [\nabla \cdot F_1(0) + F_1(0)\cdot\nabla \log \rho(0)]\,A(\tau) \right>}}$$ which is a specific realization of $$\label{agar}
\fbox {$ \displaystyle R(\tau) = {{\left< \frac{{{\mathcal L}}_1\rho}{\rho}(0)\,A(\tau) \right>}} $}
= \langle B(0)\,A(\tau)\rangle$$ with [*observable*]{} $B(x) = \frac{{{\mathcal L}}_1\rho}{\rho}(x)$. Note that in general the stationary expectation $\langle B\rangle =0$ because $\int {\textrm{d}}x\, {{\mathcal L}}_1 \rho(x) = 0$ from the normalization of $\rho$. Applications of that Agarwal formula [@aga72] in practice meet the difficulty of needing to know the density $\rho$ (which is usually not available) and the details of the dynamics for ${{\mathcal L}}_1$. It is thus a result (from partial integration) on a formal level.
Formula is associated to equilibrium, see formula (2.5) in [@rue09]. Of course we have only used that $\rho$ is smooth. It is easy to verify that in the case of detailed balance with conservative forces $F=-\nabla U$ at temperature $D=1/\beta$ we obtain the Kubo formula [@kub66] for the linear response relation in equilibrium. Indeed, say for with perturbation $F_1=-\nabla V$ around equilibrium $\rho \propto \exp[-\beta U]$, we get $$\frac{{\cal L}_1 \rho}{\rho} = -\beta \nabla V\cdot \nabla U + \Delta V = \beta\, LV$$ so that $$\label{kubo}
{{\left< \frac{{{\mathcal L}}_1\rho}{\rho}(0)\,A(\tau) \right>}} = \beta\,\langle LV(0)\,A(\tau)\rangle = \beta\,\frac{{\textrm{d}}}{{\textrm{d}}\tau}\langle V(0)\, A(\tau) \rangle$$ which is the (classical) Kubo formula. The last equality has used that under detailed balance $\langle LV(0)\,A(\tau)\rangle =\langle V(0)\,LA(\tau)\rangle$ (see further details in Sec. \[velo\]). In a way, the Agarwal formula repeats Kubo’s original derivation while stopping short before specifying $\rho$.
Also others have re-found the Agarwal formula, such as in Theorem 2, formul[æ]{} (2.22) and (2.23), of Chapter 2 in [@maj05]. Weidlich gives a quantum version: his equation (2.17) replaces ${{\mathcal L}}_1 \rho \rightarrow \frac{i}{\hbar}[\rho,H_1]$ with the commutator of the perturbing Hamiltonian $H_1$ [@wei71]. Hänggi and Thomas in [@han82] find the Agarwal formula in their equation (3.12) for time-dependent processes. In the review [@mar08] formula (2.70)–(2.73) is the Agarwal formula. We also find these formul[æ]{} such as involving $\log \rho$ presented in the book by Risken, formula (7.10)–(7.13) in [@ris89], and as formula (7) in [@spe06]. There also re-started the emphasis on $\log \rho$ as “generalized” potential.
### Information potential
The formula that Falcioni, Isola and Vulpiani [@fal90] derived for tiny displacements of the initial condition along a unit vector $\hat{e}$, $$\label{vul}
\fbox {$ \displaystyle R(\tau) = {{\left< \hat{e}\cdot\nabla\log \rho (0)\,A(\tau) \right>}} $}$$ (see it also as formula (3.13) in the review [@mar08]) corresponds to the case of diffusion with constant perturbation $F_1=-\hat{e}$ (impulsive constant force in the direction $-\hat{e}$) in . Again, as there is no explicit dependence on the noise level the formula can be readily tried for chaotic dynamics with smooth stationary density [@fal95]. The function ${{\mathcal I}}\equiv -\log \rho$ is sometimes called the information potential. In fact, that potential gets a prominent place in various works on nonequilibrium linear response that follow the Hatano-Sasa formalism [@hat01], such as in the more recent [@ver11] or [@pro09].
Other formul[æ]{} focusing on ${{\mathcal I}}$ can be found in the works by Prost, Joanny and Parrondo [@pro09] and by Speck and Seifert [@sei10]. Consider the stationary density $\rho^h$ for the perturbed dynamics with generator $L^h$ and the linear response $\rho^h(x) - \rho(x) = h\,\rho_1(x) + {{\mathcal O}}(h^2)$. From stationarity ${{\mathcal L}}^h\rho^h =0$ we get ${{\mathcal L}}_1\rho + {{\mathcal L}}\rho_1=0$. For the Agarwal formula we need $$B = \frac{{{\mathcal L}}_1\rho}{\rho} = -\frac{{{\mathcal L}}\rho_1}{\rho}$$ On the other hand, $$-\partial_h {{\mathcal I}}^h = \partial_h \log \rho^h = \lim_{h\to 0}\frac{\rho^h - \rho}{h\,\rho^h} = \frac{\rho_1}{\rho}$$ ($\partial_h$ is understood in $h=0$) so that for inserting in , $B= {{\mathcal L}}(\rho\,\partial_h {{\mathcal I}}^h ) / \rho$. We conclude $$\begin{aligned}
R(\tau) &=& \int {\textrm{d}}x {{\mathcal L}}(\rho\,\partial_h {{\mathcal I}}^h )(x) e^{\tau L}A (x)\nonumber\\
&=& \int {\textrm{d}}x \,\rho(x)\,\partial_h {{\mathcal I}}^h(x)\,Le^{\tau L}A (x) \nonumber\\
&=& \frac{{\textrm{d}}}{{\textrm{d}}\tau}\int {\textrm{d}}x \,\rho(x)\,\partial_h {{\mathcal I}}^h(x)\,e^{\tau L}A (x)
\end{aligned}$$ or $$\label{sei}
\fbox {$ \displaystyle
R(\tau) = \frac{{\textrm{d}}}{{\textrm{d}}\tau}\langle \partial_h {{{\mathcal I}}}^h(0)\,A(\tau)\rangle
$}$$ This formula with the special emphasis on the presence of the information potential appears in [@sei10] (but with a different time-derivative), where ${{\mathcal I}}$ is called stochastic entropy. The result of [@pro09] starts from expanding the Hatano-Sasa identity, which effectively makes a special choice for the observable $A$. There one imagines the process and hence also the $\cal I$ to depend on a family of parameters $\lambda$ and one asks how the expected value of $A= \partial_\lambda \cal I$ at $\lambda=\lambda^*$ changes under a small change in these parameters around a given $\lambda^*$. The answer is provided by , which is equation (5) in [@pro09].
As already noticed for (\[agar\]) a disadvantage of incorporating the information potential into the correlation function of and is that it is generally unknown and not directly measurable. Not only do we not know the stationary density, but also no clear physically relevant interpretation in terms of heat, work or thermodynamic potential has been found for ${{\mathcal I}}$. On the plus side, a clear advantage of (\[vul\]) and is that no detailed information on the dynamics is needed and that one can use parametrized forms of the stationary density $\rho$ to put in $\cal I$. Typically, (quasi-)Gaussian approximations are tried for $\rho$ to make formul[æ]{} , , more explicit and practically useful; see e.g. [@lei75; @maj05; @abr07; @lac09]. Note that there is no reference in formula to the noise strength except through the stationary $\rho$. If one replaces the $\rho$ in the $B$ of by a Gaussian, its variance will effectively reflect the noise level but needs to be fitted.
Finally, recent works on time-dependent processes with feedback rewrite the positivity of the entropy production in terms of the expected information potential (relative entropies) which appears useful for understanding work relations, see e.g. [@sag09; @toy10]. At any rate, is the climax of the analytic approach for smooth probability densities, preserving the Kubo form most faithfully.
### State velocity {#velo}
There is a relation between the information potential ${{\mathcal I}}$ and the state-space velocity $u$. For diffusion processes with generator this state velocity is $$\label{pv}
u \equiv \frac{j_\rho}{\rho} = F - D\nabla \log \rho = F + D\nabla {{\mathcal I}}$$ as can be readily checked from the expression for the stationary probability current $j_\rho$ appearing in the Smoluchowski equation $\partial_t \mu_t + \nabla j_{\mu_t} =0$ expressing conservation of probability $\mu_t$. Of course this probability velocity needs not be related to physical currents, but an interesting observation writes a nonequilibrium response formula as a co-moving (equilibrium) fluctuation-dissipation theorem.
In a stationary process two quantities $A$ and $B$ have time-translationally invariant correlations ${{\left< B(0)A(t) \right>}}={{\left< B(-t)A(0) \right>}}$. As we mentioned above, the Kubo formula (or, the fluctuation-dissipation theorem) holds under detailed balance, i.e., for an unperturbed evolution which gives rise not only to a stationary but also to a time-symmetric distribution of trajectories. Nonequilibrium means breaking time-reversal invariance. We introduce the operator $L^*$ that generates the time-reversed motion to describe for example $${{\left< B(-t)A(0) \right>}} = \int \rho({\textrm{d}}x)\, A(x) \left(e^{t L^*}B\right)(x),\quad t\geq 0$$ One can alternatively write $\int \rho({\textrm{d}}x)\,(Lf)(x)\,g(x) = \int \rho({\textrm{d}}x)\, f(x)\,(L^*g)(x)$, from which we see that $L^*g = {{\mathcal L}}(\rho g)/\rho$. For the overdamped diffusion processes that we have considered so far, from and , $L^*= L -2u\cdot \nabla = -F\nabla + D \Delta + 2 D \nabla \log \rho \nabla $.
Note now that the Kubo formula would follow from the Agarwal formula when in we take $B = -\beta L^*V$. Indeed, $$\langle L^*V(0) A(\tau)\rangle =
\langle V(0) L A(\tau)\rangle = \frac{{\textrm{d}}}{{\textrm{d}}\tau}\langle V(0) A(\tau)\rangle$$ Detailed balance corresponds to $L=L^*$ (i.e., ${{\left< B(-t)A(0) \right>}} = {{\left< B(t)A(0) \right>}}$) and corrections to the Kubo formula will thus arise from choosing in $$\label{loc}
B(x) = -\frac{L+L^*}{2}V\,(x) = -L^*V(x) - \frac{L-L^*}{2}V\,(x)$$ for some function $V$. In other words the antisymmetric part $\frac{L-L^*}{2}V$ in the formula for $B$ will be responsible for violating the Kubo formula and will show the nonequilibrium aspect. Moreover, $L- L^* = 2 u\cdot\nabla$ relates to the state velocity . Substituting these relations via in , we find that for diffusions (with $D=1/\beta$) $$\begin{aligned}
\label{fra1}
R(t,s) &=& \beta \frac{{\textrm{d}}}{{\textrm{d}}s}\int {\textrm{d}}x\, \rho(x)\,
V(x)\, e^{(t-s)L}A(x)\nonumber\\
&& -\beta\int {\textrm{d}}x\, \rho(x)\, u(x) \cdot \nabla V(x)\, e^{(t-s)L}A(x)\end{aligned}$$ abbreviated as $$\label{fra}
\fbox {$ \displaystyle
R(t,s) = \beta\frac{{\textrm{d}}}{{\textrm{d}}s}{{\left< V(s)A(t) \right>}} -
\beta {{\left< [u(s) \cdot\nabla V(s)] A(t) \right>}}
$}$$ The equation shows that the equilibrium Kubo form gets “restored” when describing the system in the Lagrangian frame moving with drift velocity $u$. The passage to the Lagrangian frame of local velocity ${\textrm{d}}/{\textrm{d}}s \rightarrow {\textrm{d}}/{\textrm{d}}s - u\cdot \nabla$ “removes” the non-conservative forcing from the formula, as explained in [@spe06; @che08; @che09]. Still, if we do not know the stationary $\rho$, the formula implies a statistical average over the unknown vector $u$.
Acting on the observable {#sec:det}
------------------------
We go back to the original and we move to an algorithm focusing on the evolution of the observable $A$ rather than that of the density $\rho$. Various contributions to the theory of dynamical systems start exactly from that formula for the formulation of linear response as there still no assumption is needed on the smoothness of the probability $\rho$.
The main player now is $L_1 e^{tL} A$ in (\[dys\]), and $L$ need not to commute with $L_1$. Typically the generator of the perturbation is $L_1 = F_1 \cdot \nabla$ for states $x\in {{\mathbb R}}^d$. We are thus interested in obtaining a useful relation for $\nabla e^{tL} A$. In other words, we need to start from $$\label{rue}
\fbox {$ \displaystyle
R(\tau) = \int\,\rho({\textrm{d}}x) F_1(x)\cdot \nabla e^{\tau L}A\,(x)
$}$$ Ruelle treated a formula with this form for deterministic dynamical systems [@rue98; @rue09], recently applied especially in studies of simple models for climate response [@abr07; @abr08; @luc11]. Indeed (\[rue\]) is especially suited in case the stationary probability law is [*strange*]{} in the sense that its lack of smoothness forbids further partial integration to go back to the Agarwal formula . In that same context however, it is useful to further split into two parts [@rue09], one describing fluctuations along the stable directions of motion and one parallel to unstable directions (those with positive Lyapunov exponents). Informally you would expect that the vector $\nabla e^{\tau L}A\,(x)$ can be given, at least for large $\tau$, in its natural components $\exp[\tau \lambda_{\hat{e}}(x)]\,\hat{e}\cdot\nabla A(x)$ for local Lyapunov exponents $\lambda_{\hat{e}}$ corresponding to the various (stable and unstable) directions $\hat{e}$. Such a decomposition is natural for stationary distributions $\rho$ which belong to the class of Sinai-Ruelle-Bowen (SRB) measures; these have a density in the unstable (expanding) direction (positive Lyapunov exponents) so that there a further [*partial*]{} integration remains possible towards an Agarwal formula . That is exactly what is suggested in the hybrid form of Section 2.3 of [@abr07]. See [@lsy02; @tas98] for an introduction on SRB-measures with their physical interpretation.
### Algorithm for deterministic and stochastic systems {#sec:det-num}
The numerical evaluation of linear response for chaotic dynamics has recently been studied by various groups. In particular, Abramov and Majda have developed new computational approaches based on the theory of SRB-measures [@abr08]. We simplify here the presentation to introduce an algorithm that works also for stochastic systems. To exploit , no knowledge of $\rho$ is required, but we need to get a hold on $\nabla_{\!x}e^{\tau L}A\,(x) = \nabla_{\!x}
\langle A(x_\tau)\rangle_{x_0=x}$ where we have emphasized that the differentiation is on an evolved quantity with respect to the initial condition $x$. To solve the problem that $\nabla$ (state derivative) and $L$ (generating the time-evolution $e^{tL}A(x)$) need not to commute, we unfold the formula a little further. A practical numerical tool for estimating $\nabla_{\!x}A(x_\tau)$ in deterministic dynamics (where $x_\tau$ is uniquely determined from $x_0=x$) was already presented in [@eyi04], but the following numerical method is more efficient for steady states.
Consider first an evolution in discrete time $n=0,1,\ldots$ (but with a parameter $\epsilon$ that will allow a continuous time limit $\epsilon \to 0$), $$\begin{aligned}
\label{dyns}
x_{n+1}= g_n(x_n) = x_n + &v_n(x_n)& ,\qquad x_0=x\\
\mbox{with} &v_n(x)& \!\!\!= \epsilon[F(x) + \xi_n]\nonumber\end{aligned}$$ for $\xi_n$ anything stationary in time, including possible “noise” depending on the time $n$, but that does not depend on the state $x$. This $(\xi_n)$ is considered frozen so that for its given realization we take as a deterministic dynamics. The main point is that $\nabla_x\langle A(x_n)\rangle_x = \langle \nabla_x A(x_n)\rangle_x$ where the $\langle\cdot\rangle$ averages over the “noise” $(\xi_n)$. We then need to deal with $\nabla_{\!x} A(x_n)$ where $x_n$ depends on the initial state $x$ through . By the chain rule, applied recursively, $$\begin{aligned}
\nabla_x A(x_n) &=& (\nabla A)(x_n)\cdot\nabla_x \big(g_{n-1}\circ\ldots\circ g_0\big)\,(x)\nonumber\\
&=& (\nabla A)(x_n)\cdot {{\mathbb G}}_{n-1}\,(x)\nonumber\\
&=& (\nabla A)(x_n)\,(\nabla g_{n-1})(x_{n-1}) \cdot \nabla_x\big(g_{n-2}\circ\ldots\circ g_0\big)\,(x)\nonumber\\
&=&(\nabla A)(x_n)\,(\nabla g_{n-1})(x_{n-1})\cdot {{\mathbb G}}_{n-2}(x)=\ldots \label{Gn}\end{aligned}$$ where $\nabla A$ is a $1\times d$ row vector (easily computed at all times) and ${{\mathbb G}}_k(x) \equiv \nabla \big(g_k\circ g_{k-1}\circ\ldots\circ g_0\big)\,(x)$ is a $d\times d$ matrix obeying the recursive relations \[eq:dro\] \[recG\] $$\begin{aligned}
{{\mathbb G}}_n(x) &=& \nabla g_{n}(x_n) \cdot {{\mathbb G}}_{n-1}(x) \label{recGa}\\
{{\mathbb G}}_0(x) &=& \nabla g_{0}(x) \label{recGb}\end{aligned}$$ Note now that for each time $k$, the analytical form of the matrix $\nabla g_k = {{\mathbb I}}+ \epsilon \nabla F$ does not depend on $k$ or on the “noise” $(\xi_n)$. Hence, $$\begin{aligned}
\label{nabla}
\nabla_x A(x_n) &=& (\nabla A)(x_n)\cdot
\left[{{\mathbb I}}+ \epsilon(\nabla F)(x_{n-1})\right]\nonumber\\
&&\cdot\left[{{\mathbb I}}+ \epsilon(\nabla F)(x_{n-2})\right]\cdots
\left[{{\mathbb I}}+ \epsilon(\nabla F)(x)\right]\end{aligned}$$ The continuous time version ${{\mathbb G}}_t(x)$ can be obtained in a suitable limit $n\to \infty$ for ${{\mathbb G}}_n(x)$ with time $t=n\,\epsilon$ and time step $\epsilon= t/n$. In this limit (\[dyns\]) returns the evolution $\dot{x}_t = F(x_t)$ for the deterministic case, while for stochastic equations an additional suitable rescaling to reproduce e.g. Gaussian noise with $\xi_t$ is needed. At any rate, from ${{\mathbb I}}+ \epsilon (\nabla F)(x) \simeq e^{\epsilon(\nabla F)(x)}$ we get formally $$\label{nablat}
\nabla_x A(x_\tau) = (\nabla A)(x_\tau)\cdot{\textbf{T}}\exp\left[\int_0^\tau ds (\nabla F)(x_s) \right]$$ where each $x_s$ depends deterministically on $x$ for frozen $(\xi_r, 0\leq r < s)$, and **T** indicates a time-ordered integral. The formula or its discretization is ready to be inserted into where the $\langle\cdot\rangle$ will first average there over the possible “noise” and then, with $\rho$, over the initial condition $x$. That provides the main algorithm for the analytic approach working on the observable instead of on the probability distribution.
A similar expression can be found in the Appendix B of Ref. [@eyi04] for the case of deterministic dynamics. In that work, however, the estimate of $ (\nabla A)(x_n){{\mathbb G}}_t(x)$ was performed with an adjoint scheme, by integrating numerically backward in time the final value $(\nabla A)(x_n)$ with the equation $$\partial_t (\nabla A)(x_t) + (\nabla A)(x_t)\cdot (\nabla g)(x_t) = 0$$ (In [@eyi04] they wrote the equation for the column vector rather than for the row vector $\nabla A$, and a transpose of $(\nabla g)$ was used). Although that scheme is equivalent to our (\[nabla\]), it requires a CPU time ${{\mathcal O}}(n^2)$ for a perturbation active during all $n$ iterations, as opposed to the ${{\mathcal O}}(n)$ matrix multiplications (\[recG\]) for estimating matrices ${{\mathbb G}}_k$ for $k\le n$. Our scheme is exploiting stationarity: the propagator from time $n-k$ to time $n$ is the same as ${{\mathbb G}}_k$ from time $0$ to $k$. In transient regimes we would lose this property and we would also need ${{\mathcal O}}(n^2)$ operations.
A further study of such a numerical algorithm is contained in [@abr07; @abr08]; in particular Appendix A of [@abr08] explains the derivation above (without the generalization to stochastic evolutions). For questions of dealing in a similar context with the impact of stochastic perturbations, we refer to [@luc12].
### Numerical illustration
To illustrate the numerical scheme (\[Gn\]) for estimating Ruelle’s linear response formula (\[rue\]) in a simple context, we consider a set of equations introduced in biology to describe circadian cycles, that is the periodicity of biorhythms, for Drosophila [@gol95; @oga06]. The state space has $d=5$ dimensions, with states $x=(P_0$, $P_1$, $P_2$, $P_N$, $M)$. The dynamics couples the concentration $M$ of mRNA with those of four types of proteins, written as $P_0$, $P_1$, $P_2$, and $P_N$ in Ref. [@oga06], where one can find the details of these equations. Denoting $${\left[\frac{P}{K}\right]} \equiv \frac{P}{P+K}$$ we have equations of motion $\dot x=F(x)$ of the form \[eq:dro\] $$\begin{aligned}
\frac{{\textrm{d}}P_0}{{\textrm{d}}t} &=& k_S M - \nu_1 {\left[\frac{P_0}{K_1}\right]}+ \nu_2 {\left[\frac{P_1}{K_2}\right]}\label{eqP0}\\
\frac{{\textrm{d}}P_1}{{\textrm{d}}t} &=& \nu_1 {\left[\frac{P_0}{K_1}\right]}- \nu_2 {\left[\frac{P_1}{K_2}\right]}- \nu_3 {\left[\frac{P_1}{K_3}\right]} + \nu_4 {\left[\frac{P_2}{K_4}\right]}\\
\frac{{\textrm{d}}P_2}{{\textrm{d}}t} &=& \nu_3 {\left[\frac{P_1}{K_3}\right]} - \nu_4 {\left[\frac{P_2}{K_4}\right]} + \nu_d {\left[\frac{P_2}{K_d}\right]} -k_1P_2+k_2P_N\\
\frac{{\textrm{d}}P_N}{{\textrm{d}}t} &=& k_1P_2-k_2P_N\\
\frac{{\textrm{d}}M}{{\textrm{d}}t} &=& \nu_S {\left[\frac{K_I^n}{P_N^n}\right]} - \nu_m {\left[\frac{M}{K_m}\right]}\end{aligned}$$ with parameters as in previous papers, $$\begin{aligned}
&&\left(\nu_S = 0.5,\, \nu_m = 0.3,\, \nu_1 = 6,\, \nu_2=3,\, \nu_3=6,\, \nu_4=3\right)\,
\textrm{nM\,h}^{-1}\\
&&\left(K_m=0.2,\, K_I=2,\, K_1=1.5,\, K_2=2,\, K_3=1.5,\, K_4=2\right)
\,\textrm{nM}\\
&&\left(k_S=2,\, k_1=2,\, k_2=1\right)\,\textrm{h}^{-1},\qquad n=4\end{aligned}$$ Integration of (\[eq:dro\]) was performed with a simple Verlet scheme with a discrete time step $\epsilon=0.0025$. Starting from random concentrations, the model reaches quickly a cyclic regime, as shown in Fig. \[fig:cycle\].
![Evolution of the concentration of mRNA and of the four proteins of the model from random initial conditions, according to (\[eq:dro\]). \[fig:cycle\]](fig_fdt_01.eps){width="9cm"}
We check the response in the mRNA concentration $M$ to a change of the rate $k_S\to k_S(1+h)$, starting from the steady state, i.e., a random phase of the cycle. The dynamical perturbation introduced by that change is $h\,F_1 = (h\, M\, k_S,0,0,0,0)$, since a change in $k_S$ affects only the evolution equation of variable $P_0$, see (\[eqP0\]). The observable $A(x)=M$ has a gradient $(\nabla A)(x) = (0,0,0,0,1)$, which is coupled to the perturbation via the “propagator” matrix ${{\mathbb G}}_n$. With the setup of Sec. \[sec:det-num\], we can estimate ${{\mathbb G}}_n$ by a sequence of matrix multiplications. Given an evolution in discrete time $g(x) = x + F(x)\epsilon$, we are interested in matrix $\nabla g(x) = {{\mathbb I}}+ \nabla F(x) \epsilon$, or in coordinates $(\nabla g)_{ij}(x) = \delta_{ij}+\,dF_i/dx_j\, \epsilon$. The rows of the matrix $(\nabla F)$ are
\[nablaF\] $$\begin{aligned}
&&\left(
-\nu_1{\left[\frac{P_0}{K_1}\right]}_{P_0} ,
\nu_2 {\left[\frac{P_1}{K_2}\right]}_{P_1},
0,
0,
k_S
\right)\\
&&\left(
\nu_1{\left[\frac{P_0}{K_1}\right]}_{P_0} ,
-\nu_2 {\left[\frac{P_1}{K_2}\right]}_{P_1}\!\!\!-\nu_3 {\left[\frac{P_1}{K_3}\right]}_{P_1},
\nu_4 {\left[\frac{P_2}{K_4}\right]}_{P_2} ,
0,
0
\right)\\
&&\left(
0,
\nu_3 {\left[\frac{P_1}{K_3}\right]}_{P_1},
-\nu_4 {\left[\frac{P_2}{K_4}\right]}_{P_2}\!\!\!-\nu_d {\left[\frac{P_2}{K_d}\right]}_{P_2}\!\!\!-k_1,
k_2,
0
\right)\\
&&\left(
0,
0,
k_1,
-k_2,
0
\right)\\
&&\left(
0,
0,
0,
\nu_S{\left[\frac{K_I^n}{P_N^n}\right]}_{P_N},
-\nu_m{\left[\frac{M}{K_m}\right]}_{M}
\right)
\end{aligned}$$ where derivatives with respect to $P$ are denoted as $
{\left[\frac{P}{K}\right]}_P = \frac{1}{P+K} - \frac{P}{(P+K)^2}
$ and similarly for ${\left[\frac{K_I^n}{P_N^n}\right]}_{P_N}$. This matrix with $x_0=(P_0,P_1,P_2,P_N,M)_0$ yields $(\nabla g)(x_0) = {{\mathbb I}}+ (\nabla F)(x_0)\epsilon $, coinciding with ${{\mathbb G}}_0$. Iteratively, ${{\mathbb G}}_1 = (\nabla g)(x_1) \cdot {{\mathbb G}}_0$, and so on.
![Linear response function $R(\tau)$ of the mRNA density $M$ to an impulsive change in the parameter $k_S\to k_S(1+h_0)$ as a function of time and response $\delta{{\left< M(\tau) \right>}}/\delta h_0$ computed with $h_0=10^{-3}$. Units are nM vs. hours. \[fig:resp1\]](fig_fdt_02.eps){width="9cm"}
By sampling many trajectories, in parallel for perturbed and unperturbed dynamics starting from an initial condition from the steady state, the estimate of $\delta M(\tau)/\delta h_0$ has been obtained by setting a small constant $h$ for $\tau>0$ and by evaluating the time derivative of $({{\left< M(\tau) \right>}}^h -{{\left< M(\tau) \right>}})/h $ (we used $h=10^{-3}$). The response function instead has been calculated with (\[Gn\]) and (\[rue\]), and well overlaps with the response, as shown in Fig. \[fig:resp1\]. Asymptotic stability in this model [@oga06] probably favors a good performance of formula (\[rue\]). We computed both the perturbed dynamics and the unperturbed fluctuation formula just to show graphically that the results are the same, but of course there is no additional understanding of the process. Obviously, computing the response only with unperturbed simulations would be convenient as long as the convergence of the method is good. We postpone a more detailed study of the efficiency of this and other numerical schemes to a future work.
Probabilistic approach {#pro}
======================
Linear response makes sense in general only within a statistical theory. That is to say, sensitive dependence on initial and boundary conditions can create strong and relevant effects beyond linear order on the microscopic scale while macroscopic linearity remains valid. For estimating the mobility we do not investigate the microscopic particle’s individual motion at specific times and how it changes under an external field, but we ask for a spatio-temporal averaged current and that includes noise. In fact, for stronger microscopic chaoticity we expect the statistical approach to be more relevant. While various physical variables can show chaotic behavior in their time-evolution, their spatial or temporal averages will typically have a much smoother behavior, see also [@col12] and chapter 6.2 in [@dor99] answering the so called van Kampen objection.
As noise gets important {#asn}
-----------------------
In coarsening dynamics of low temperature spin systems, or in spin glasses, a very long transient regime may exist towards equilibrium, i.e., the nonequilibrium is not imposed by external gradients and the dynamic equations are actually equilibrium ones. It is in this context that an extensive response-literature has been produced during the last decades. We mention briefly some results from 2003, whose focus was mostly to develop zero-field algorithms, in other words, exploiting fluctuation–response relations to estimate numerically responses from fluctuations in unperturbed dynamics. Works of Chatelain [@cha03], Ricci-Tersenghi [@ric03], and Crisanti and Ritort [@cri03] introduced new schemes for computing the response of the system from correlations in the spins. They were followed by Diezeman [@die05] and by Lippiello, Corberi, and Zannetti [@lip05; @and06; @lip07]. The derivations of these results cannot be reduced in our brief scheme and various forms of response relations have been obtained. However, the one by Lippiello et al., a specific case of (\[st1\]) below, emerged as having clear physical significance beyond numerical usefulness, because it matched the form of an earlier study by Cugliandolo, Kurchan and Parisi [@cug94], who proposed a response relation for autocorrelations in Langevin equations. With the present notation (\[lan\]), perturbation $h_s x$ and observable $A(x)=x$, that would read $$\label{ckp}
\fbox {$ \displaystyle
R(t,s) = \frac 1 {2 D} \left\{
\left(\frac {\textrm{d}}{{\textrm{d}}s} - \frac {\textrm{d}}{{\textrm{d}}t}\right){{\left< x_s x_t \right>}}
-\left( {{\left< F(x_t) x_s \right>}} - {{\left< F(x_s) x_t \right>}}\right)
\right\}
$}$$ showing an “asymmetry” term in addition to the form of the Kubo formula. The derivation of (\[ckp\]) is based on the equality ${{\left< x_t\xi_t \right>}}=2 D \beta R(t,s)$, hence it is centered on the presence of noise $\xi_t$. This was new, certainly with respect to derivations presented in previous sections where eventually noise was just an aspect of the dynamics, not fundamental for the derivations. An equation similar to (\[ckp\]) appears in a more recent study for nonequilibrium steady states [@har06], based on a path-space formulation by Harada and Sasa [@har05], see their Appendix B. The Harada-Sasa approach has really pioneered the path-space approach of the next subsection. The consistency between the approaches of Refs. [@cug94; @lip05; @har05] indeed shows that these belong to a general framework with new significant physical content. In the following section we discuss the path-space formulation embracing these results and we mention the physical interpretation that goes beyond merely dissipative aspects. Noise becomes visible then as dynamical activity, ruling the time-symmetric fluctuations.
Path space approach {#path}
-------------------
The origin of dynamical ensembles is the projection of a microscopic Hamiltonian dynamics on the dynamics of reduced variables. That Mori-Zwanzig projection [@zwa61; @mor65] originates from making a physical partition of the phase space, depending on the physical situation at hand, in which each microstate $X$ is mapped (many–to–one) to a reduced state $x(X)$. That induces noise in the reduced dynamics, for example on the mesoscopic level of description. It is then natural to consider dynamical ensembles, i.e., probability distributions on path space where a path refers to a trajectory on the mesolevel. Such was already the approach of Onsager and Machlup starting dynamical fluctuation theory in [@ons53]. These trajectories, under certain limiting conditions (e.g. via a weak coupling limit or via adiabatic elimination), can be described via first-order equations, in which case we meet the Markov processes of the previous Section \[per\]. Dynamical ensembles can however also describe non-Markovian processes describing important memory effects, see e.g. [@boh12] for the application of a response relation in a visco-elastic medium.
Paths are trajectories $\omega = (x_s, 0\leq s\leq t)$ in state space, say looking at the states $x_s$ in the time-interval $[0,t]$. For the sake of simplicity we characterize the unperturbed (perturbed) process by the probability weight $P(\omega)$ \[$P^h(\omega)$\] with respect to some reference ${\textrm{d}}\omega$. The mathematical idea is to turn perturbed expectations ${{\left< A(t) \right>}}^h$ into unperturbed ones ${{\left< A(t) P^h(\omega)/P(\omega) \right>}}$. Defining the relative action ${{\mathcal U}}(\omega) = \log P(\omega)/P^h(\omega)$ and splitting ${{\mathcal U}}(\omega) = [{{\mathcal T}}(\omega)-{{\mathcal S}}(\omega)]/2$ into a time-symmetric ${{\mathcal T}}(\omega)$ and a time-antisymmetric ${{\mathcal S}}(\omega)$, we get \[dynen\] $$\begin{aligned}
\langle A(t)\rangle^h - \langle A(t) \rangle &=&
\int {\textrm{d}}\omega \,P(\omega) A(x_t) \big( e^{-{{\mathcal U}}(\omega)} - 1\big) \\
&=& {{\left< A(t) \,\big( e^{-{{\mathcal U}}(\omega)} - 1\big) \right>}}\\
&=& \frac 1 2 {{\left< A(t){{\mathcal S}}_1(\omega) \right>}} - \frac 1 2 {{\left< A(t){{\mathcal T}}_1(\omega) \right>}}+{{\mathcal O}}(h^2)\label{ST}\end{aligned}$$ where ${{\mathcal T}}_1$ and ${{\mathcal S}}_1$ are the linear contributions in $h$ of ${{\mathcal T}}$ and ${{\mathcal S}}$, respectively, around $h=0$ (where ${{\mathcal U}}=0$). This formula is general but the point is that in physical systems with local detailed balance [@har05] we have a good physical understanding of the path functions ${{\mathcal T}}$ and ${{\mathcal S}}$ [@bai10; @bai09; @bai09b; @mae10]. Let us first look under global detailed balance. We already see here that since the observable $A(t)-A(0)$ is time-antisymmetric (ignoring momenta), under detailed balance $$\begin{aligned}
\label{adynen}
\langle A(t)\rangle^h - \langle A(t) \rangle &=& \langle A(t) - A(0)\rangle^h \nonumber\\
&=& \frac 1 2 {{\left< [A(t)-A(0)]{{\mathcal S}}_1(\omega) \right>}} \nonumber\\
&=& {{\left< A(t)\,{{\mathcal S}}_1(\omega) \right>}} \end{aligned}$$ because ${{\left< [A(t)-A(0)]){{\mathcal T}}_1(\omega) \right>}} = 0$ and ${{\left< A(t)\,{{\mathcal S}}_1(\omega) \right>}} = -{{\left< A(0)\,{{\mathcal S}}_1(\omega) \right>}}$ by time-reversal symmetry of the reference equilibrium process. The result should equal the Kubo formula , and indeed it does as will become clear. When the perturbation is from a potential $-h_s V(x_s)$, it is well established that the time-antisymmetric ${{\mathcal S}}(\omega)={{\mathcal S}}_1(\omega)$ must be the path-dependent entropy flux into the environment as caused by the perturbation potential $V$ [@mae03; @maes03]. If the environment is at uniform temperature, for small constant $h$ for times $t>0$ the change in entropy is ${{\mathcal S}}(\omega)=\beta h [V(t)-V(0)]$, namely dissipated energy $h[V(t)-V(0)]$ divided by temperature. The first term on the right-hand side of has thus the same form one finds in equilibrium , but in general the unperturbed process is in steady nonequilibrium and ${{\mathcal S}}(\omega)=\beta h [V(t)-V(0)]$ is an excess of entropy production with respect to that already generated by the nonequilibrium dynamics. Hence, the derivative $\delta{{\left< A(t){{\mathcal S}}(\omega) \right>}}/\delta h_s=\beta\frac{{\textrm{d}}}{{\textrm{d}}s}{{\left< V(s) A(t) \right>}}$ also mimics the equilibrium Kubo formula.
We now turn to the second term on the right-hand side of . Eq. (\[ST\]) must be true also for a constant $A$, for which the response is zero. We thus get $$\beta \frac {\textrm{d}}{{\textrm{d}}s}{{\left< V(s) \right>}} = \frac \delta {\delta{h_s}} {{\left< {{\mathcal T}}_1(\omega) \right>}} = {{\left< \frac \delta {\delta{h_s}}{{\mathcal T}}_1(\omega) \right>}}$$ If $\frac \delta {\delta{h_s}}{{\mathcal T}}_1(\omega)$ equals a state function $B(s)$, then, since $\frac {\textrm{d}}{{\textrm{d}}s}{{\left< V(s) \right>}} = {{\left< LV(s) \right>}}$, one deduces that $B(s)=\beta LV(s)$, arriving at $$\label{st1}
\fbox {$ \displaystyle
R(t,s) = \frac \beta 2 \frac {\textrm{d}}{{\textrm{d}}s} {{\left< V(s)A(t) \right>}}
- \frac \beta 2 {{\left< L V(s)A(t) \right>}}
$}$$ For diffusion processes with generator and temperature $D=1/\beta$ we see that indeed $LV(x) = (F\cdot \nabla V)(x) + D \Delta V(x)$ is a state function. The same is true for Markov jump processes (for specific derivations of using stochastic calculus, see [@bai09b]). The function $LV$ quantifies the time-symmetric volatility of $V$ under the unperturbed dynamics, also named “frenesy”. It is related to the “dynamical activity” in discrete systems, where it quantifies the change in escape rates; $\cal T(\omega)$ is the excess in dynamical activity over $[0,t]$. We refer to [@bai09; @bai09b; @mae10] for more details and examples. Thus, there is a new contribution next to the first term in (\[st1\]) (taking one half of the Kubo formula and referring as usual to dissipation). In particular, its form now clearly deviates from response formul[æ]{} such as . One then wonders in what sense the decomposition in formul[æ]{} like is intrinsically natural. We think it is, as that splitting makes the first term time-antisymmetric and the second term symmetric in time $s$, which corresponds to the dissipative and the reactive part of the susceptibility, respectively.
Note that in (\[st1\]) one computes averages without needing an explicit knowledge on the stationary probability. On the other hand some knowledge is required on the dynamics sitting in the generator $L$. It is still not clear how much kinetic information is truly needed and how practical that gets. We summarize the general physical idea in the formula $$\label{dec2}
\langle A(t)\rangle^h - \langle A(t)\rangle =
\frac 1{2}\langle \mbox{Entr}^{[0,t]}(\omega)\,A(t)\rangle
- \frac 1{2}\langle \mbox{Esc}^{[0,t]}(\omega)\,A(t)\rangle$$ where Entr$^{[0,t]}(\omega)$ is the excess in entropy flux over the time period $[0,t]$ and Esc$^{[0,t]}(\omega)$ is the excess in dynamical activity due to the perturbation. The physical challenge is to learn to guess or to find that Esc$^{[0,\tau]}(\omega)$ from partial information on the dynamics. For the overdamped dynamics of e.g. it means to consider the expected rate of change of the perturbing potential $V$ under the original dynamics. The response relation out of equilibrium is no longer a fluctuation-dissipation relation but a fluctuation–dissipation-activation relation. In fact, the formula can now be turned around and from measuring violations of the fluctuation–dissipation relation one obtains information about the active forces [@boh12].
Inertial case
-------------
We open a separate subsection on the inertial case of Langevin dynamics because its linear response is much less discussed in the literature despite the obvious interest for example for models of heat conduction. The main point however is that the ideas summarized under and remain unchanged.
For states $(q,p) =
(q^1,q^2,\ldots,q^n;$ $p^1,p^2,\ldots,p^n)\in {{\mathbb R}}^{2n}$ of positions and momenta we attach standard white noise $\xi^i_t$ to each $1\leq i\leq n$, with constant strength $D^i$ and a friction coefficient $\gamma^i$ to model heat baths at temperature $D^i/\gamma^i = T^i$: $$\begin{aligned}
\dot{q}^i &=& p^i \nonumber\\
\dot{p}^i &=&
F^i(q) -\gamma^i p^i +
h_t \frac{\partial V}{\partial q^i} + \sqrt{2D^i}\,\xi^i_t
\label{ud}\end{aligned}$$ The forces $F^i$ can contain a nonconservative part but are confining when we want a stationary regime where the particles typically reside in a bounded region. We already inserted the perturbation $V(q)$ with small time-dependent amplitude $h_s$ for $ s\geq 0$. The linear response is given by formula (17) in [@bai10]: $$\begin{aligned}
\label{genfor}
R(t,s) =& \sum_{i}\frac 1{2T^i}&\!\!\!\!\!\!\!\!\!
{{\left< \frac{\partial V}{\partial q^i}(q_s) \,p^i_s\,A(t) \right>}} \nonumber\\
&-\sum_i\frac 1{2D^i} \Bigg\{&
{{\left< \frac{\partial V}{\partial q^i}(q_s)\,F^i(q_s)\,A(t) \right>}}\nonumber\\
&&
- \frac{{\textrm{d}}}{{\textrm{d}}s} {{\left< \frac{\partial V}{\partial q^i}(q_s)\, p^i_s\,A(t) \right>}}\nonumber\\
&&
+ \sum_j {{\left< \frac{\partial^2 V}{\partial q^j\partial q^i}(q_s)\,p^j_s\,p^i_s\,A(t) \right>}}
\Bigg\}
\end{aligned}$$ The first sum again corresponds to the dissipative part from the entropy fluxes in the reservoirs at temperatures $T^i$. The remaining sums give the frenetic contribution. As not recognized yet, formula can still be rewritten in a similar way as done in . Supposing $D^i=\gamma/\beta$ we must replace there $V\rightarrow \dot{V}/\gamma =p\cdot \nabla_q V/\gamma$, $$\label{st2}
\fbox {$ \displaystyle
R(t,s) = \frac {\beta}{2\gamma} \frac {\textrm{d}}{{\textrm{d}}s} {{\left< p_s\cdot \nabla_q V(s)A(t) \right>}}
- \frac {\beta}{2\gamma} {{\left< L(p_s\cdot \nabla_q V)(s)\,A(t) \right>}}
$}$$ in which the generator $L$ for now reads $$Lf(q,p) =
p\cdot\nabla_q f + (F-\gamma\,p)\cdot\nabla_p f +
D \Delta_p f$$ and is the underdamped version of . Note however that the dissipative last term $D \Delta_p f$ does not contribute in and that these formul[æ]{} do not work in case $D^i=0$ for an $i$ with $\partial V/\partial q^i \neq 0$. In that case the best alternative is probably to apply the algorithm of Section \[sec:det\].
An important application of is to the modification of the Sutherland-Einstein formula relating transport coefficients such as mobility with fluctuation quantities such as the diffusion constant [@bai11].
Further remarks and conclusions {#rema}
===============================
Nonlinear responses
-------------------
There is also a growing number of works on higher-order terms around equilibrium. In fact part of the book by Evans and Morriss is devoted to that [@eva90]. Other references include [@and07], Section 10 in [@maes03] or the more recent [@mar08; @mal11; @lip08; @vil09]. One typical start is the fluctuation symmetry in the distribution of the entropy flux, transient from the reference equilibrium system as also explained in [@eva08] for thermostated dynamics.
As was emphasized in [@col11] the main point is probably not to be able to write formal expansions and formul[æ]{}, but to find useful structures and unifying interpretations. We will not deal with that here, except for mentioning one particular general relation between second order and first order terms that has been largely unnoticed [@sai12], and an instance of which has appeared as identity (25) in [@col11]. $$\label{2r}
\left.\frac{\partial^2}{\partial h^2}\langle A(t)\rangle^h\right|_{h=0}
= \beta\left.\frac{\partial}{\partial h} \langle [A(t)-\bar{A}(0)][V(t)-V(0)]\rangle^h\right|_{h=0}$$ The $\bar{A}$ equals $A$ up to flipping the sign of the momenta. All terms are explicitly expressed as correlation functions in the equilibrium reference process (indicated with the superscript $\langle\cdot\rangle^0$). The derivation of uses linear response around nonequilibrium for non-state functions (i.e., the correlation function in the right-hand side).
Non-state functions
-------------------
In case of observables that depend explicitly on time or are functions of the trajectory over several times (such as products $A_1(t_1)A_2(t_2)\ldots A_n(t_n)$), some of the formul[æ]{} must change. The basic techniques remain however in place. A typical application is how the heat depends on a change in parameters, e.g., for estimating nonequilibrium heat capacities [@bok11]. Heat is not a state function but varies with the trajectory and the applied protocol.
Note on effective temperature
-----------------------------
A traditional approach to violation of the equilibrium fluctuation-dissipation theorem is to imagine an effective temperature in the otherwise unchanged Kubo formula [@cug11]. That idea has had most success with mean field type systems, but it has remained more unclear how the effective temperature can provide a consistent and general tool for realistic systems. Still today it serves as a paradigm for interpreting experimental results, see e.g. [@ben11]. One possible approach for the future would be to associate an effective kinetic temperature to the ratio between the frenetic and the entropic contribution in , see Appendix A in [@bai09b].
Transient case
--------------
Lots of attention have been devoted to the linear response behavior for relaxational processes. We mentioned some of that in Section \[asn\], but there is no way to be complete. For example, the interest in ageing and glassy dynamics has much stimulated the search of modified fluctuation-dissipation relations [@cal05]. Here we emphasize that the analytic approach and in particular the methods of Section \[apr\] loose their simple structure when the unperturbed reference is time-dependent (and not stationary as was assumed). On the other hand, formul[æ]{} like are unchanged when one is not starting at time zero from the stationary distribution, but one needs to take the average in the unperturbed transient regime. Also causality is automatically verified there as the second term in equals the first term when $s > t$. This unification in expressions for the transient and the steady regime is only natural as even the stationary regime is physically and ultimately but a very long transient.
General properties of linear response
-------------------------------------
Some formal properties of linear response go basically unchanged from equilibrium to nonequilibrium contexts. For example, sum rules and Kramers-Kronig relations only depend on the presence of an underlying Hamiltonian dynamics for the total system plus environment or on causality. Of course, specific expressions can differ but there is physically nothing new, see also [@shi10; @shi10b]. Nevertheless, there do exist essential differences. We have already alluded to the fact that the name “fluctuation–dissipation theorem” is no longer so appropriate because of the importance and complimentary character of changes in the dynamical activity (and not only in the dissipation). In fact, the word “fluctuation” also becomes less correct as the time-correlations in the expressions for $R(t,s)$ no longer express a symmetrized time-correlation. To make that point, let us evaluate the equilibrium Kubo formula ) when the perturbing potential $V$ equals the observable $A$: $$\langle V(t)\rangle^h - \langle V\rangle = - \langle V(0)[V(t)- V(0)]\rangle =
\frac 1{2}\langle [V(t)- V(0)]^2\rangle \geq 0$$ which is the variance (or the “diffusion”) of the displacement $V(x_t)-V(x_0)$. That is sometimes called a generalized Einstein relation. In fact all linear transport coefficients for equilibrium can be expressed like that, as was understood already in 1960 by Helfand [@hel60]. For nonequilibrium linear response, that relation including its positivity gets violated, see also [@bai11; @mae11]. In the words of the previous subsection, negative effective temperatures become possible in nonequilibrium. The origin lies in the frenetic contribution, e.g. the second term on the right-hand side of , which can overrun the first dissipative term.
Outlook
-------
Response theory is primarily about predicting the reaction of a system in terms of its unperturbed behavior. As we mostly have in mind response in time, that involves temporal correlations. Therefore, [*dynamical*]{} fluctuation theory can be expected to be most prominent. (See [@mae07] for the distinction between static and dynamic fluctuation theory: statics looks at deviations around the law of large numbers at single times, for example for the average over many copies of the system, while dynamical fluctuations are around the law of large times, for deviations around time-averages.) Around equilibrium, dynamical fluctuations are governed, just as static fluctuations, by the entropy and dissipation functions. That further enables connecting different response coefficients, such as in Onsager reciprocity or via Maxwell relations. Nonequilibrium makes a more drastic difference between static and dynamical fluctuations. In particular, no useful connections between different types of responses have been discovered for nonequilibrium processes. We think of the analogue of relations between compressibilities, heat capacities and conductivities. In our opinion, that challenge will require finding experimental access to quantities like the dynamical activity which is complementing entropic characterizations in the description of nonequilibrium processes.
Conclusions
-----------
We have presented a concise guide (with plenty of references) to the multi-faceted world of linear response for systems out of equilibrium. From there we have discovered similarities but also some missing pieces that otherwise would not be evident. One such piece was the extension to stochastic systems of Ruelle’s formulation, which we have introduced, together with an efficient algorithm. That also allows for numerical calculations where an explicit knowledge of the density of states is not required, as opposed to other (analytic) formulations that we have described. However, the resulting linear response formula contains a correlation function whose physical meaning is not very clear.
In contrast, a rich physical picture emerges from a probabilistic approach based on path space weights, where the stationary distribution is also not needed. Besides the quite different mathematical apparatus compared to the more standard analytical approach, the probabilistic way indeed emerges as the one that currently offers more relations with dynamical fluctuation theory: one has to study also how the system correlates with the activity of the perturbing potential, a time-symmetric quantity complementary to the time-antisymmetric fluxes of entropy. The combination fluctuation-dissipation therefore does not suiffice to characterize the linear response of nonequilibrium systems.
#### Acknowledgments
M.B. thanks M. Colangeli, A. Vulpiani and J. Wouters for useful discussions.
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[^1]: We can of course also use (\[dys2a\]) for discrete jump processes. For each pair $x,y$ of states, the perturbation enters as a modification of jump rates $w(x,y)\to w(x,y)+h\,w_1(x,y)$. The matrix $L_1$ to be put in (\[dys2a\]) has elements $w_1(x,y)$ if $x\ne y$, and $w_1(x,x)=-\sum_y w_1(x,y)$.
|
---
abstract: 'We report spin-selective tunneling of electrons along natural and artificial double-stranded DNA (dsDNA) sandwiched by nonmagnetic leads. The results reveal that the spin polarization strongly depends on the dsDNA sequence and is dominated by its end segment. Both genomic and artificial dsDNA could be efficient spin filters. The spin-filtering effects are sensitive to point mutation which occurs in the end segment. These results are in good agreement with recent experiments and are robust against various types of disorder, and could help for designing DNA-based spintronic devices.'
author:
- 'Ai-Min Guo'
- 'Qing-feng Sun'
title: 'Sequence-dependent spin-selective tunneling along double-stranded DNA'
---
The charge transport along DNA molecule has received significant attention from scientific researchers over the past two decades.[@dc; @erg; @gjc1] In addition to electric charges, the DNA molecule could be also used to manipulate the electron spin. It was reported that self-assembled monolayers of double-stranded DNA (dsDNA) can discriminate the spin of photoelectrons.[@rsg; @gb] These electrons transmitted through the dsDNA monolayers are highly polarized at room temperature and the spin-filtering effects are enhanced with increasing the DNA length.[@gb] Moreover, it was demonstrated that even single dsDNA could be efficient spin filter.[@xz] The underlying physical mechanism arises from the combination of the dephasing, the SO coupling, and the chirality of the DNA molecule.[@gam1] However, the spin polarization vanishes if the dsDNA was changed into single-stranded DNA or damaged by ultraviolet light.[@gb; @xz; @gam1]
The nitrogenous bases guanine (G), adenine (A), cytosine (C), and thymine (T), which are four basic ingredients of the DNA molecule, can constitute thousands of various sequences. While natural DNA molecule can be extracted from the cells of all living organisms, the artificial one could be synthesized in any desired sequence. It was shown that the DNA molecule with different sequences could present any transport behavior of conducting, semiconducting, and insulating.[@zyp; @se; @gx; @sjd] One may thus expect that different dsDNA would display diverse spin-filtering effects. Indeed, the study of spin transport along various dsDNA will provide valuable information to the physical mechanism and the biological processes, and opens up its potential applications in molecular spintronics. In this Letter, we explore spin-selective tunneling of electrons through the dsDNA connected by normal-metal leads. Based on a model Hamiltonian which includes the SO coupling and the dephasing, the conductance and the spin polarization are calculated for a variety of dsDNA. Here, the DNA molecules involve genomic and artificial ones as well as those employed in the experiments.[@gb; @xz] The sequences of several typical DNA samples are listed in Table \[tab:table1\]. The genomic dsDNA is extracted from the sequence of human chromosome 22 (chr22),[@note1] while the artificial dsDNA is taken as random sequence and substitutional one, e.g., Nickel mean (nm), Copper mean (cm), and Triadic Cantor (tc).[@me] All of the substitutional DNA sequences are constructed by initiating from one seed and following a substitution rule. For instance, the nm1 sequence is formed by adopting base A as the seed and the substitution rule A$\rightarrow$AGGG, G$\rightarrow$A.
Name DNA sequence
------- ----------------------------------------------------
sq-26 TTTGTTTGTTTGTTTGTTTTTTTTTT
sq-40 TCTCAAGAATCGGCATTAGCTCAACTGTCAACTCCTCTTT
sq-50 TACTCTACCTTCTCAAGAATCGGCATTAGCTCAACTGTCAACTCCTCTTT
rd1 CAATGCAGTCTATCCACCTGACGGACCCCGACCCGCCTTT
rd2 CAATGCAGTCTATCCACCTGACGGACCCCGACCCGGCTTT
rd3 CAATGCAGTCTATCCACCTGACGGACCCCGACCCGCCATT
hc1 TAAATAAATAAATAAATAAATAAAATAAATAAAAGCCTTT
hc2 GGGCCCTGAGGCATGGGCCCAGAAGCATTCCTGTCCCCTT
hc3 AGCTGGGGAGCAGGGCTCCACTCTGGGAGGGGGGCAGCCT
nm1 AGGGAAAAGGGAGGGAGGGAGGGAAAAGGGAAAAGGGAAA
nm2 ATTTAAAATTTATTTATTTATTTAAAATTTAAAATTTAAA
cm GAAGGGAAGAAGAAGGGAAGGGAAGGGAAGAAGAAGGGAA
tc GAGAAAGAGAAAAAAAAAGAGAAAGAGAAAAAAAAAAAAA
From the study of numerous dsDNA, we find that the spin filtration efficiency presents strong dependence on the DNA sequence and is mainly determined by the end segment with several base-pairs. Both chr22-based and random dsDNA could be very efficient spin filters, while the substitutional one exhibits large spin polarization and conductance. Besides, the spin-filtering effects are sensitive to point mutation which takes place in the end segment of the dsDNA. The high spin polarization still holds even under the environment-induced on-site energy disorder and twist angle disorder. These results could be beneficial for building up DNA-based spintronic devices.
The spin transport along the dsDNA can be described by the Hamiltonian: ${\cal H}={\cal H}_{\rm DNA} + {\cal H}_{\rm so} +{\cal H}_d +{\cal H}_{\rm lead}+{\cal H}_{c},$[@gam1; @gam2] where ${\cal H} _{\rm DNA}= \sum_{j=1}^2(\sum_{n=1}^N \varepsilon_ {jn} c_{jn}^\dag c_{jn}+ \sum_{n=1}^{N-1} t_{jn}c_{jn}^\dag c_{jn+1}) + \sum_{n=1}^N \lambda_n c_{1n}^\dag c_{2n} +\mathrm{H.c.}$ is the Hamiltonian of two-leg ladder model, with $n$ the base-pair index, $j$ the strand label, and $N$ the DNA length. $c_{jn} ^\dag= (c_{jn\uparrow}^\dag, c_{ jn \downarrow } ^\dag)$ is the creation operator, $\varepsilon_ {jn} $ is the on-site energy, $t_{jn}$ is the intrachain hopping integral, and $\lambda_n $ is the interchain hybridization interaction. The second term ${\cal H}_{\rm so}=\sum_{j n}\{ i t_{\rm so} c_{jn}^\dag \sigma _n ^ {(j)} c_{jn+1}+ \mathrm{H.c.} \}$ is the SO Hamiltonian, which stems from the double helix distribution of the electrostatic potential of the dsDNA.[@gam1] $t_{\rm so}$ is the SO coupling strength and $\sigma_n ^{(j ) } =[\sigma_ x ( \sin \varphi_{jn} +\sin \varphi_{jn+1})- \sigma_ y (\cos \varphi_{jn}+\cos \varphi_ {jn+1})] \sin \theta_{jn} +2\sigma_z \cos\theta_{jn}$, with $\sigma_ {x,y,z} $ the Pauli matrices, $\varphi _{jn}$ the cylindrical coordinate of the base, and $\theta _{jn}$ the helix angle between base $n$ and $n+1$ in the $j$th strand. In equilibrium position of the dsDNA, $\varphi _{jn}= (n-1) \Delta\varphi$ and $\theta_ {jn} = \theta$ with $\Delta\varphi$ the twist angle. The third one ${\cal H}_d= \sum_{jnk} ( \varepsilon_ {jnk} b_{jnk}^\dag b_{jnk} + t_d b_{jnk}^\dag c_{jn} +\mathrm {H.c.} )$ is the Hamiltonian of the Büttiker’s virtual leads and their coupling with each base of the dsDNA, simulating the phase-breaking processes due to the inelastic scattering with phonons and counterions.[@bya2; @bya3] The last two terms ${\cal H}_{\rm lead} +{\cal H}_c= \sum_{k,\beta=L,R} \varepsilon_{\beta k} a_{\beta k}^\dag a_{\beta k} + \sum_{jk} ( t_L a_{L k}^\dag c_{j 1} +t_R a_{R k} ^ \dag c_{j N} + \mathrm {H.c.} )$ represent the real leads, and the coupling between these leads and the dsDNA, respectively. Finally, the conductances for spin-up $(G_ \uparrow)$ and spin-down $(G_ \downarrow)$ electrons can be calculated by using the Landauer-Büttiker formula.[@gam1] The spin polarization is $P_s=(G_ \uparrow- G_ \downarrow )/ (G_\uparrow+ G_\downarrow)$. Since the current is flowing from the left real lead to the right one, the terminal of the dsDNA attached to the former is named the beginning, while the other terminal is called the end.
For the dsDNA, $\varepsilon_{jn}$ is chosen as the ionization potential with $\varepsilon_ {\rm G}= 8.3$, $\varepsilon_ {\rm A}= 8.5$, $\varepsilon_ {\rm C}= 8.9$, and $\varepsilon_ {\rm T}= 9.0$, $t_ {jn}$ between identical neighboring bases is taken as $t_{\rm GG}=0.11$, $t_{\rm AA}=0.22$, $t_{\rm CC}=-0.05$, and $t_{\rm TT}=-0.14$, and $\lambda_n =-0.3$. These parameters are extracted from the experimental results[@hns; @dd; @lj1; @tab] and first-principles calculations[@cem; @gfc3; @vaa; @zh; @sk; @hlgd; @av] with the unit eV. $t_ {jn}$ between different neighboring bases X and Y is set to $t_{\rm XY}=(t_{\rm XX}+ t_{\rm YY})/ 2$, in accordance with first-principles results.[@vaa; @zh; @sk; @hlgd] The helix angle and the twist one are $\theta=0.66$ rad and $\Delta \varphi ={\frac \pi 5}$. The SO coupling is estimated to $t_{\rm so}=0.01$. For the real leads, the parameters $\Gamma_L = \Gamma_R =1$ are fixed, while for the virtual ones, the dephasing strength is $\Gamma_ d=0.006$.
![\[fig:one\] Energy-dependent $P_s$ for poly(A)-poly(T) under the on-site energy disorder with degree $W$ (a) and of the twist angle disorder with degree $D$ (b). $\langle G_\uparrow \rangle$ and $\langle P_s\rangle$ vs $W$ (c) and vs $D$ (d). The inset of (c) shows $\langle G_\uparrow \rangle$ in a wider range of $W$ and the dependence can be fitted by the function $\langle G_\uparrow \rangle \propto 10^{-\alpha W}$ with $\alpha =5.80\pm0.09$ (cyan line). $\langle G_\uparrow \rangle$ and $\langle P_s\rangle$ are averaged in the energy region $[9.04, 9.32]$. All of the results are performed for single disorder configuration with $N=40$. Here, $G_0 =e^2/h$ is the quantum conductance.](1.EPS){width="41.00000%"}
It was reported that the ionization potential of the base is affected significantly by both counterions[@brn; @zy] and hydration.[@ksk; @yx; @bl] Consequently, the environmental effects can be properly considered by varying the on-site energies. A random variable $w_{jn}$ is added in each $\varepsilon_{jn}$ to simulate the stochastic population of these counterions and water molecules around the dsDNA, with $w_{jn}$ uniformly distributed within the range $[-\frac W 2, \frac W 2]$ and $W$ the disorder degree. Fig. \[fig:one\](a) shows the spin polarization $P_s$ of poly(A)-poly(T) under the on-site energy disorder, as a function of the energy $E$. It clearly appears that $P_s$ is large for homogeneous poly(A)-poly(T) and is sufficiently robust against the on-site energy disorder. This can be further demonstrated in Fig. \[fig:one\](c), where the averaged spin polarization $\langle P_s\rangle$ is shown. One notices that $\langle P_s\rangle$ fluctuates around its equilibrium value of $5.0\%$ at $W=0$ and the oscillation amplitude is enhanced by $W$. Furthermore, a new energy region of high $P_s$ becomes more distinct in the case of larger $W$ \[see the curves of $W=0.16$ and $0.3$ in Fig. \[fig:one\](a)\]. On the other hand, the averaged conductance $\langle G_ \uparrow\rangle$ is decreased by increasing $W$ as expected, due to the disorder-induced Anderson localization effect. The curve of $\langle G_ \uparrow \rangle $-$W$ can be fitted well by a simple function $\langle G_\uparrow \rangle \propto 10^{-\alpha W}$ \[see inset of Fig. \[fig:one\](c)\].
Besides the on-site energy disorder, each base will waver around its equilibrium position at finite temperature. In this situation, it is reasonable to plus a random variable $d_{jn}$ in each $\varphi_{ jn}$, with $d _{jn}$ distributed in the region $[-\frac D 2, \frac D 2 ]$ and $D$ the disorder degree. By considering constant radius $R$ of the dsDNA and arc length $l_a$ between successive bases to account for the rigid sugar-phosphate backbone,[@gam1; @gj] the helix angle $\theta _{jn}$ will be modulated according to $l_a\cos \theta_{jn} =R (\varphi_{jn+1}- \varphi_{jn})$ and the fluctuations are disregarded in the intrachain hopping integral as a first approximation.[@sk; @gr; @rs1] It can be seen from Fig. \[fig:one\](b) that the curves of $P_s$-$E$ are superposed with each other in the context of the twist angle disorder only. Accordingly, no fluctuations could be observed in the curve of $\langle P_s\rangle$-$D$ \[Fig. \[fig:one\](d)\]. Besides, $\langle G_ \uparrow \rangle$ will not be changed with $D$, because the SO coupling is much smaller than the hopping integral. Therefore, poly(A)-poly(T) remains an efficient spin filter even under the on-site energy disorder and the twist angle disorder.
![\[fig:two\] Energy-dependence of $G_\uparrow $, $G_ \downarrow $, and $P_s$ for the dsDNA used in the experiments. (a) $G_{\uparrow}$ and $G_{\downarrow}$ for the sq-26 sequence. (b) $G_{\uparrow}$ for the sq-40 and sq-50 sequences. (c) $P_s$ for all three dsDNA. The inset of (c) displays $P_s$ with $\Gamma_d =0.01$.](2.EPS){width="36.00000%"}
Then we investigate the spin transport through aperiodic dsDNA in the absence of external environment-induced disorder. Our results still hold if this disorder is included. Let us first discuss the spin transport properties of the dsDNA used in the experiments.[@gb; @xz] Fig. \[fig:two\](a) displays the conductances of spin-up ($G_ \uparrow$) and spin-down ($G_ \downarrow$) electrons for sq-26 sequence, while Fig. \[fig:two\](b) plots $G_ \uparrow$ for sq-40 and sq-50 sequences. As compared with homogeneous dsDNA,[@gam1] the energy spectrum of aperiodic dsDNA is also separated into the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). The conductance is declined by increasing the DNA length, because the electrons experience stronger scattering in longer dsDNA.
In addition, one can see from Fig. \[fig:two\](a) that the discrepancy between $G_ \uparrow$ and $G_ \downarrow$ is more distinct for the LUMO band than the HOMO one. Thus, $P_s$ is larger in the former band than the latter one \[Fig. \[fig:two\](c)\]. Moreover, $P_s$ at $E=9.11$ is, respectively, $9.6\%$, $38\%$, and $38\%$ for the sq-26, sq-40, and sq-50 sequences, in good agreement with the experiment.[@gb] In fact, the obtained $P_s$ is also consistent with the experimental results by adopting different $\Gamma_ d$ from the region $[0.003, 0.01]$, e.g., see $P_s$ of $\Gamma_ d=0.01$ in the inset of Fig. \[fig:two\](c). Besides, although the conductances between the sq-40 and sq-50 sequences are very different, their spin polarizations are almost identical and the difference between the two $P_s$ is within $10^{-6}$ range, due to the fact that the sq-40 sequence is the end part of the sq-50 sequence. This suggests that the spin filtration efficiency of the dsDNA is mainly controlled by its end segment, which will be substantiated below.
![\[fig:three\] Energy-dependent $P_s$ for the random dsDNA (a) and for the chr22-based one (b). (c) Distribution of $P_s$ at $E=9.11$ for various random dsDNA with large $P_s$ (right part) and with small $P_s$ (left part) as a comparison. The results are extracted from $10^5$ DNA samples. Here, only the end segment in the first strand is shown (two sides) and can be obtained for the second one according to the base-pairing rules. (d) $\langle G_ \uparrow \rangle$ and $\langle P_s \rangle$ vs mutation position $L$ for the rd1 sequence.](3.EPS){width="39.00000%"}
Next we turn to study the spin polarization of the random and chr22-based dsDNA. Figs. \[fig:three\](a) and \[fig:three\](b) plot $P_s$ vs $E$ for several typical random and chr22-based sequences, respectively. It is clear from the curves of rd1 and hc1 that both random and chr22-based sequences could be very efficient spin filters with $P_s$ achieving $40\%$. From a statistical study of numerous dsDNA with extremely high $P_s$, it reveals that these sequences are terminated by the segment “CCTTT/GGAAA” in their ends \[Fig. \[fig:three\](c)\]. We emphasize that all of the investigated dsDNA with $N=40$ will exhibit very high $P_s$ around $40 \%$ if their end segments are replaced by “CCTTT/GGAAA”. Besides, the dsDNA could be also very efficient spin filter if it has other end segments, as shown in Fig. \[fig:three\](c), where a distribution of $P_s$ at $E=9.11$ is displayed for different random dsDNA with 16 end segments. It clearly appears that $P_s$ is always large for these dsDNA \[see the right part in Fig. \[fig:three\](c)\], although $P_s$ will vary in a finite range. The dsDNA remains efficient spin filter if it is ended by the moiety “(C)$_{m}$TT/(G)$_{m}$AA” with $m$ the integer (see the curve of hc2). However, $P_s$ can be dramatically reduced by altering the end segment, even if its last base-pair is changed \[see the left part in Fig. \[fig:three\](c)\]. These are due to the fact that the charge will gradually lose its phase and spin memory while transmitting along the dsDNA. The longer distance the charge propagates, the larger the loss of its memory. Accordingly, the spin filtration efficiency of the dsDNA is dominated by its end segment containing several base-pairs.
![\[fig:four\] Distribution functions of $P_s$ for the random and chr22-based dsDNA at $E=9.11$. The inset shows the corresponding statistics of $P_s$ at $E=9.03$. Here, $N=40$.](4.EPS){width="28.00000%"}
To further verify the aforementioned point, we introduce point mutation in the dsDNA, where only one base-pair is modified and replaced by another.[@sct] Here, the point mutation is defined by switching the complementary bases within single base-pair.[@note2] We focus on $P_s$ of the random dsDNA in Fig. \[fig:three\](a). The rd2 and rd3 sequences are derived by introducing the point mutation in the rd1 one.[@note2] One notes that $P_s$ is reduced more significantly if the mutation position is closer to the last base-pair of the rd1 sequence. The largest $P_s$ is decreased from $42\%$ for the rd1 sequence to $25\%$ and $4.7\%$ for the rd2 and rd3 sequences, respectively. $\langle P_s\rangle$ and $\langle G_\uparrow \rangle$ are shown as a function of the mutation position $L$ in Fig. \[fig:three\](d), where the average is obtained within the energy region $[8.98,9.18]$. It is clear that $\langle P_s\rangle$ does not change if the mutation occurs in the very beginning of the rd1 sequence, and fluctuates more strongly if the mutation position becomes closer to its end. $P_s$ is very small if the point mutation takes place within the last three base-pairs, due to the identical sign between $t_{1n}$ and $t_{2n}$.[@gam1] In contrast, $\langle G_\uparrow \rangle$ fluctuates more obviously if the mutation occurs in the beginning of the sequence. And the fluctuation amplitude is more severe in the curve of $\langle P_s\rangle$-$L$ than $\langle G_\uparrow \rangle$-$L$, indicating that the spin polarization is much more sensitive to the modification of the base-pair in the dsDNA than the conductance. In this perspective, the spin transport along the dsDNA may be related to mutation detection in the biological processes and could be beneficial for DNA sequencing.[@zm]
Figure \[fig:four\] shows the statistical properties of $P_s$ at fixed energy for the random and chr22-based dsDNA with $10^5$ samples. It clearly appears that $P_s$ can vary from $42\%$ to negative, implying that the spin polarization direction of the charges transmitted through the dsDNA could be reversed by modifying its sequence. And one can see that many DNA molecules exhibit high $P_s$. From a statistical perspective, the chr22-based dsDNA has more efficient spin filters than the random one. For instance, the number of the dsDNA, of which $P_s$ is larger than $30\%$ (20%), is 458 (1436) and 667 (2020) for the random dsDNA and the chr22-based one, respectively. This can be further demonstrated in the inset of Fig. \[fig:four\], where one notices that the curve of the chr22-based dsDNA is always higher than that of the random one for $P_s>1.3\%$. However, there are also many dsDNA with small $P_s$ at fixed energy. This is attributed to the fact that: (1) $P_s$ depends on $E$ that the energy region of large $P_s$ may differ from one sample to another \[Figs. \[fig:three\](a) and \[fig:three\] (b)\]; (2) the electrons may be not polarized exactly along the helix axis for each dsDNA and the actual spin polarization could be larger.
![\[fig:five\] Energy-dependence of $G_ \uparrow$ and $P_s$ for several substitutional sequences of DNA molecules.](5.EPS){width="39.00000%"}
Finally, we study the spin polarization of the substitutional sequences of DNA molecules, of which the electronic properties have been investigated previously.[@gam3; @rs2] Fig. \[fig:five\] shows $P_s$ and $G_ \uparrow$ for several substitutional dsDNA. It is clear that both $P_s$ and $G_ \uparrow$ are very large for these dsDNA. Therefore, besides homogeneous DNA molecules, other aperiodic DNA sequences can be also efficient spin filters with high spin polarization and conductance.
In summary, we investigate the quantum spin transport through different dsDNA contacted by nonmagnetic leads. We find that the spin polarization strongly depends on the dsDNA sequence and is mainly determined by the end segment. Both natural and artificial dsDNA could be very efficient spin filters. Our results could motivate further experimental studies on DNA spintronics.
This work was financially supported by NBRP of China (2012CB921303 and 2009CB929100) and NSF-China under Grants Nos. 10974236 and 11074174.
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---
abstract: 'Optical modes with different orbital angular momentums (OAMs) per photon may be sorted by Mach-Zehnder interferometers incorporated with beam rotators, without resorting to OAM mode converters.'
author:
- Haiqing Wei
- Xin Xue
title: 'Comment on “Measuring the Orbital Angular Momentum of a Single Photon”'
---
Presented in a recent letter [@Leach02] is an ingenious method of sorting spatial modes of photons with different orbital angular momentum (OAM), which closely resembles, and in a sense complements a scheme of analyzing optical beams with rectangular symmetry [@Xue01]. The method in [@Leach02] employs special Mach-Zehnder (MZ) interferometers as binary branching devices, each of which divides a set of input modes with $l\equiv
k~({\rm mod}~2^n)$ into two groups with $l\equiv k~({\rm
mod}~2^{n+1})$ and $l\equiv 2^n+k~({\rm mod}~2^{n+1})$ respectively, where the integer $l$ denotes the single-photon OAM in units of $\hbar$, $k$ and $n$ are fixed, non-negative integers specific to the individual interferometer. In the Letter, the same condition is assumed implicitly for all the MZ interferometers, that one arm maintains zero phase (or integral multiples of $2\pi$) for all the modes, while the other arm rotates the beam by an angle $\alpha$ so to induce an OAM-dependent phase shift $l\alpha$. The assumption proves rather restrictive and responsible for the necessity of OAM mode converters, which result in optical loss and increase significantly the complexity of the optical setup. The loss of light could impose a serious limitation to quantum optical experiments involving single photons.
The use of OAM mode converters may be avoided by relaxing the aforementioned restriction on the MZ interferometers, namely, by incorporating an adjustable phase shifter in a modified MZ interferometer as depicted in Fig.\[ourMZ\]. In practice, there is always a mechanism of phase adjustment in setting up an optical interferometer. An example implementation is a thin glass film making an adjustable angle to the beam axis. When OAM modes with $l\equiv k~({\rm mod}~2^n)$ are input into the modified MZ interferometer, the beam is rotated by $\alpha=\pi/2^n$ in the upper arm such that a mode with OAM $l$ acquires a phase $l\alpha=l\pi/2^n$, while the phase shifter in the lower arm is tuned to induce a fixed phase shift $k\alpha=k\pi/2^n$ to all the OAM modes. It is easily seen that the modified MZ interferometer segregates the OAM modes into two groups with $l\equiv k~({\rm
mod}~2^{n+1})$ and $l\equiv 2^n+k~({\rm mod}~2^{n+1})$ respectively. Such modified MZ interferometers may be used as binary branching devices to construct an OAM mode sorter, without resorting to OAM mode converters.
The OAM mode sorter is in striking similarity to the Hermite-Gaussian (HG) mode analyzer using MZ interferometers incorporated with fractional Fourier transformers (FRFTs) [@Xue01]. FRFT-incorporated MZ interferometers can even be used after an OAM mode sorter to lift the mode degeneracy due to the radial degree of freedom, so that orthogonal Laguerre-Gaussian (LG) modes can be sorted completely. Moreover, a complete HG mode sorter may be utilized just as a complete LG mode analyzer with the help of HG $\rightleftharpoons$ LG mode converters [@Beijersbergen93]. Finally, it may be noted that the increased information capacity using higher order OAM [@Leach02; @Molina-Terriza02] or HG modes is just the result of an enlarged spatial channel [@Miller98]. To achieve the higher capacity, the spatial channel needs to support optical beams with larger sizes than the fundamental mode.
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|
---
author:
- 'Antoni Szczurek [^1]'
- 'Anna Cisek [^2]'
- 'Marta [Ł]{}uszczak [^3]'
- 'Wolfgang Schäfer [^4]'
title: 'Recent progress in some exclusive and semi-exclusive processes in proton-proton collisions'
---
Introduction
============
Here we briefly summarize our recent results for exclusive production of $J/\psi$ mesons obtained in [@SS2007; @CSS2015] and for photon-photon production of charged dileptons obtained in [@SFPSS2015; @LSS2015].
Exclusive production of vector charmonia
========================================
The Born diagrams for the exclusive production of $J/\psi$ mesons are shown in Fig.\[fig:diagrams\_exclusive\_Jpsi\]. In actual calculations in [@SS2007; @CSS2015] we include also absorption effects due to soft proton-proton interactions.
![Born diagrams for exclusive production of $J/\psi$ mesons.[]{data-label="fig:diagrams_exclusive_Jpsi"}](jpsi_pp_gam_pom.eps "fig:"){height="4.0cm"} ![Born diagrams for exclusive production of $J/\psi$ mesons.[]{data-label="fig:diagrams_exclusive_Jpsi"}](jpsi_pp_pom_gam.eps "fig:"){height="4.0cm"}
All details of the formalism can be found in Refs.[@SS2007] and [@CSS2015]. Here we only sketch the main points.
Imaginary part of the forward $\gamma p \to J/\psi p$ amplitude
$$\begin{aligned}
\Im m \, {\cal M}_{T}(W,\Delta^2 = 0,Q^{2}=0) =
W^2 \frac{c_v \sqrt{4 \pi \alpha_{em}}}{4 \pi^2} \, 2 \,
\int_0^1 \frac{dz}{z(1-z)}
\int_0^\infty \pi dk^2 \psi_V(z,k^2)
\nonumber \\
\int_0^\infty
{\pi d\kappa^2 \over \kappa^4} \alpha_S(q^2)
{\cal{F}}(x_{\rm eff},\kappa^2)
\Big( A_0(z,k^2) \; W_0(k^2,\kappa^2)
+ A_1(z,k^2) \; W_1(k^2,\kappa^2)
\Big) \, .\end{aligned}$$
The full amplitude, at finite momentum transfer is parametrized as: $$\begin{aligned}
{\cal M}(W,\Delta^2) = (i + \rho) \, \Im m {\cal M}(W,\Delta^2=0,Q^{2}=0)
\, \exp(-B(W) \Delta^2/2) \, ,
\label{full_amp}\end{aligned}$$ Then the amplitude for the $p p \to p p J/\psi$ can be written somewhat formally as: $$\begin{aligned}
{\cal M}_{h_1 h_2 \to h_1 h_2 V}^
{\lambda_1 \lambda_2 \to \lambda'_1 \lambda'_2
\lambda_V}(s,s_1,s_2,t_1,t_2) =
{\cal M}_{\gamma \Pom} + {\cal M}_{\Pom \gamma} \nonumber \\
= {\langle {p_1', \lambda_1'} |} J_\mu {| {p_1, \lambda_1} \rangle}
\epsilon_{\mu}^*(q_1,\lambda_V)
{\sqrt{ 4 \pi \alpha_{em}} \over t_1}
{\cal M}_{\gamma^* h_2 \to V h_2}^{\lambda_{\gamma^*} \lambda_2 \to \lambda_V \lambda_2}
(s_2,t_2,Q_1^2) \nonumber \\
+ {\langle {p_2', \lambda_2'} |} J_\mu {| {p_2, \lambda_2} \rangle}
\epsilon_{\mu}^*(q_2,\lambda_V)
{\sqrt{ 4 \pi \alpha_{em}} \over t_2}
{\cal M}_{\gamma^* h_1 \to V h_1}^{\lambda_{\gamma^*} \lambda_1 \to \lambda_V \lambda_1}
(s_1,t_1,Q_2^2) \, .
\label{Two_to_Three}\end{aligned}$$
The auxiliary amplitude in Eq. (\[Two\_to\_Three\]) for the emission of a photon of transverse polarization $\lambda_V$, and transverse momentum ${\mbox{\boldmath $q$}}_1 = - {\mbox{\boldmath $p$}}_1$ can be written as: $$\begin{aligned}
{\langle {p_1', \lambda_1'} |} J_\mu {| {p_1, \lambda_1} \rangle}
\epsilon_{\mu}^*(q_1,\lambda_V)
={ ({\mbox{\boldmath $e$}}^{*(\lambda_V)} {\mbox{\boldmath $q$}}_1) \over \sqrt{1-z_1}}
\, {2 \over z_1} \, \chi^\dagger_{\lambda'}
\Big\{ F_1(Q_1^2)
- { i \kappa_p F_2(Q_1^2) \over 2 m_p }
( {\mbox{\boldmath $\sigma$}}_1 \cdot [{\mbox{\boldmath $q$}}_1,{\mbox{\boldmath $n$}}]) \Big\} \chi_\lambda \, .\end{aligned}$$ Above $F_1$ and $F_2$ are Dirac and Pauli electromagnetic form factors, respectively. In Ref.[@CSS2015] we have included the Pauli form factors for a first time.
Selected results for vector quarkonia
=====================================
Rapidity distributions of $J/\psi$ mesons are show in Fig.\[fig:dsig\_dy\]. In this calculation Gaussian wave functions were used. Only results with UGDFs that include nonlinear effects describe the LHCb experimental distributions [@LHCb]. In our opinion it is too preliminary to conclude that we observe nonlinear effects or onset of gluon saturation. In Ref.[@CSS2015] we obtained similar distributions for $\psi'$ [@CSS2015].
![Rapidity distributions for three different unintegrated gluon distributions. The upper curves were obtained in the Born approximation and the lower band includes absorption effects.[]{data-label="fig:dsig_dy"}](dsig_dy_IN_1S.eps "fig:"){width="4cm"} ![Rapidity distributions for three different unintegrated gluon distributions. The upper curves were obtained in the Born approximation and the lower band includes absorption effects.[]{data-label="fig:dsig_dy"}](dsig_dy_KS_lin_1S.eps "fig:"){width="4cm"} ![Rapidity distributions for three different unintegrated gluon distributions. The upper curves were obtained in the Born approximation and the lower band includes absorption effects.[]{data-label="fig:dsig_dy"}](dsig_dy_KS_nonlin_1S.eps "fig:"){width="4cm"}
Very interesting quantity is the ratio of the cross sections for $\psi'$ and $J/\psi$ production. As shown in Fig.\[fig:ratios\] such a ratio is very sensitive to the functional form of the $c \bar c$ light-cone wave function. We conclude that the Gauss $c \bar c$ wave function much better describes the LHCb data [@LHCb] than Coulomb WF. In some calculation in the literature the wave functions are not included explicitly and only point-like coupling is used [@Ryskin-Martin].
![The ratio of the cross section for $\psi'$ and $J/\psi$.[]{data-label="fig:ratios"}](dsig_dy_ratio_IN.eps "fig:"){width="4cm"} ![The ratio of the cross section for $\psi'$ and $J/\psi$.[]{data-label="fig:ratios"}](dsig_dy_ratio_KS_lin.eps "fig:"){width="4cm"} ![The ratio of the cross section for $\psi'$ and $J/\psi$.[]{data-label="fig:ratios"}](dsig_dy_ratio_KS_nonlin.eps "fig:"){width="4cm"}
$\gamma \gamma$ production of dileptons
=======================================
The $\gamma \gamma$ processes can be categorize according to the final state (see Fig.\[fig:diagrams\_gammagamma\]). All the processes were discussed in [@SFPSS2015; @LSS2015].
![The different categories of $\gamma \gamma$ processes.[]{data-label="fig:diagrams_gammagamma"}](diagram_elaela.eps "fig:"){width="3.5cm"} ![The different categories of $\gamma \gamma$ processes.[]{data-label="fig:diagrams_gammagamma"}](diagram_ineine.eps "fig:"){width="3.5cm"}\
![The different categories of $\gamma \gamma$ processes.[]{data-label="fig:diagrams_gammagamma"}](diagram_elaine.eps "fig:"){width="3.5cm"} ![The different categories of $\gamma \gamma$ processes.[]{data-label="fig:diagrams_gammagamma"}](diagram_ineela.eps "fig:"){width="3.5cm"}
Two different approaches were discussed in the context of $\gamma
\gamma$ production of dileptons.
In the collinear approach the corresponding cross sections are calculated respectively as: $$\begin{aligned}
\frac{d \sigma^{(k,l)}}{d y_1 d y_2 d^2p_t} &=& \frac{1}{16 \pi^2 {\hat s}^2}
x_1 \gamma_{k}(x_1,\mu^2) \; x_2 \gamma_{l}(x_2,\mu^2) \; \overline{|{\cal M}_{\gamma \gamma \to l^+l^-}|^2} \, ,\end{aligned}$$ where $k,l \in \{\rm{el},\rm{in} \}$ stand for the processes with intact ($\rm{el}$) or dissociated $\rm{in}$ proton at the photon vertex. The $\gamma_k(x,\mu^2)$ are the corresponding photon distributions in a proton, with or without the condition of breakup. The elastic photon distributions are calculated with the help of the nucleon electromagnetic form factors, while the inelastic distributions can be obtained by using a combined QCD/QED evolution [@QED-PDF].
In the $k_t$-factorization, proposed recently in [@SFPSS2015; @LSS2015], the fully unintegrated photon flux can be written as (below incoming particles are denoted as $A,B$): $$\begin{aligned}
{d {\cal{F}}_{\gamma^* \leftarrow A} (z,{\mbox{\boldmath $q$}},M_X^2) \over dM_X^2} = {\alpha_{\rm{em}}\over \pi} \, (1-z) \,
\Big( {{\mbox{\boldmath $q$}}^2 \over {\mbox{\boldmath $q$}}^2 + z (M_X^2 - m_A^2) + z^2 m_A^2 }\Big)^2 \, \cdot {p_B^\mu p_B^\nu \over s^2} \, W_{\mu \nu}(M_X^2,Q^2) \, .\end{aligned}$$ Information on the virtual photon-proton interaction is contained in the hadronic tensor, which is obtained in terms of the electromagnetic currents as: $$\begin{aligned}
W_{\mu \nu}(M_X^2,Q^2) = \overline{\sum_X} (2 \pi)^3 \, \delta^{(4)} (p_X - p_A - q) \, {\langle {p} |} J_\mu {| {X} \rangle}{\langle {X} |} J_\nu^\dagger {| {p} \rangle} \, d\Phi_X \, ,
\label{eq:Wmunu}\end{aligned}$$ The virtual photoabsorption cross sections are related to hadronic tensors as: $$\begin{aligned}
\sigma_T(\gamma^* p) = {4 \pi \alpha_{em} \over 4 \sqrt{X}} \, \Big(- {\delta^\perp_{\mu\nu} \over 2} \Big) 2\pi W^{\mu \nu}(M_X^2,Q^2) \, , \,
\sigma_L(\gamma^* p) = {4 \pi \alpha_{em} \over 4 \sqrt{X}} \, e^{0}_\mu e^{0}_\nu \, 2 \pi W^{\mu \nu}(M_X^2,Q^2) \, .\end{aligned}$$ It is customary to introduce dimensionless structure function $F_i(x_{\rm Bj},Q^2), i = T,L$ as $$\begin{aligned}
\sigma_{T,L}(\gamma^* p) = {4 \pi^2 \alpha_{em} \over Q^2} \, {1 \over \sqrt{1 + {4 x^2_{\rm Bj} m_A^2 \over Q^2}} } \, F_{T,L}(x_{\rm Bj},Q^2) \, ,\end{aligned}$$ At high energies, in the calculation of photon fluxes, the contribution of the structure function $$\begin{aligned}
F_2(x_{\rm Bj},Q^2) &=& { F_T(x_{\rm Bj},Q^2) +F_L(x_{\rm Bj},Q^2) \over 1 + {4 x^2_{\rm Bj} m_A^2 \over Q^2}} \end{aligned}$$ dominates.
The unintegrated fluxes enter the cross section for dilepton production as $$\begin{aligned}
{d \sigma^{(i,j)} \over dy_1 dy_2 d^2{\mbox{\boldmath $p$}}_1 d^2{\mbox{\boldmath $p$}}_2} &&=
\int {d^2 {\mbox{\boldmath $q$}}_1 \over \pi {\mbox{\boldmath $q$}}_1^2} {d^2 {\mbox{\boldmath $q$}}_2 \over \pi {\mbox{\boldmath $q$}}_2^2}
{\cal{F}}^{(i)}_{\gamma^*/A}(x_1,{\mbox{\boldmath $q$}}_1) \,
{\cal{F}}^{(j)}_{\gamma^*/B}(x_2,{\mbox{\boldmath $q$}}_2)
{d \sigma^*(p_1,p_2;{\mbox{\boldmath $q$}}_1,{\mbox{\boldmath $q$}}_2) \over dy_1 dy_2 d^2{\mbox{\boldmath $p$}}_1 d^2{\mbox{\boldmath $p$}}_2} \, ,
\label{eq:kt-fact}\end{aligned}$$ where $i,j \in \{ \rm{el},\rm{in} \}$ again refer to elastic and inelastic processes on the proton sides. For brevity we integrated over invariant masses of the possible dissociated system.
The longitudinal momentum fractions carried by photons can be obtained from rapidities and transverse momenta of the charged leptons as: $$\begin{aligned}
x_1 = \sqrt{ {{\mbox{\boldmath $p$}}_1^2 + m_l^2 \over s}} e^{y_1}
+ \sqrt{ {{\mbox{\boldmath $p$}}_2^2 + m_l^2 \over s}} e^{y_2}
\; ,
x_2 = \sqrt{ {{\mbox{\boldmath $p$}}_1^2 + m_l^2 \over s}} e^{-y_1}
+ \sqrt{ {{\mbox{\boldmath $p$}}_2^2 + m_l^2 \over s}} e^{-y_2} \, .\end{aligned}$$
Selected results for $l^+ l^-$ production
=========================================
As an example in Fig.\[fig:dsig\_dMll\_inin\] we show invariant mass distributions for double dissociative processes (both protons undergo electromagnetic dissociation). The calculations have been performed for different experimental conditions specified in the figure legend. We show results for different proton structure functions from the literature. The different structure functions lead to quite different results. The presented results strongly depend on the kinematical regions of longitudinal momentum fraction and photon virtuality ($x$,$Q^2$) which were not sufficiently well studied experimentally and theoretically in which an interplay of perturbative and nonperturbative effects takes place. We observe that the $\gamma \gamma$ contribution constitutes only a small fraction of the cross section compared to the experimental data and Drell-Yan contribution (not shown explicitly here). Similar results for elastic-inelastic channel were shown in [@LSS2015].
![Dilepton invariant mass distributions for different experimental conditions (see the four panels) and for different structure functions (see different lines). []{data-label="fig:dsig_dMll_inin"}](LHCb_in_in.eps "fig:"){width="5cm"} ![Dilepton invariant mass distributions for different experimental conditions (see the four panels) and for different structure functions (see different lines). []{data-label="fig:dsig_dMll_inin"}](ATLAS_in_in.eps "fig:"){width="5cm"}\
![Dilepton invariant mass distributions for different experimental conditions (see the four panels) and for different structure functions (see different lines). []{data-label="fig:dsig_dMll_inin"}](ISR_in_in.eps "fig:"){width="5cm"} ![Dilepton invariant mass distributions for different experimental conditions (see the four panels) and for different structure functions (see different lines). []{data-label="fig:dsig_dMll_inin"}](PHENIX_in_in.eps "fig:"){width="5cm"}
Now we wish to show results which can be directly compared to experimental data [@CMS]. The data were obtained by imposing a kinematical constraint on lepton isolation. The results are shown in Figs.\[fig:CMS\_dsig\_dMll\],\[fig:CMS\_dsig\_dphi\], \[fig:CMS\_dsig\_dptpair\]. A relatively good agreement has been achieved. A better quality results are expected from Run II at the LHC.
![Dilepton invariant mass distributions for two (best) structure functions. The CMS data points are shown for comparison.[]{data-label="fig:CMS_dsig_dMll"}](CMS_Mll_ALLM.eps "fig:"){width="5cm"} ![Dilepton invariant mass distributions for two (best) structure functions. The CMS data points are shown for comparison.[]{data-label="fig:CMS_dsig_dMll"}](CMS_Mll_SY.eps "fig:"){width="5cm"}
![Distribution in relative azimuthal angle between leptons for two (best) structure functions. The CMS data points are shown for comparison.[]{data-label="fig:CMS_dsig_dphi"}](CMS_phi_ALLM.eps "fig:"){width="5cm"} ![Distribution in relative azimuthal angle between leptons for two (best) structure functions. The CMS data points are shown for comparison.[]{data-label="fig:CMS_dsig_dphi"}](CMS_phi_SY.eps "fig:"){width="5cm"}
![Distribution in transverse momentum of the pair of leptons for two (best) structure functions. The CMS data points are shown for comparison.[]{data-label="fig:CMS_dsig_dptpair"}](CMS_ptpair_ALLM.eps "fig:"){width="5cm"} ![Distribution in transverse momentum of the pair of leptons for two (best) structure functions. The CMS data points are shown for comparison.[]{data-label="fig:CMS_dsig_dptpair"}](CMS_ptpair_SY.eps "fig:"){width="5cm"}
Conclusions
===========
Our results and open problems for charmonia production can be summarized as follows. We have found some model dependent indication of presence of nonlinear effects in the small-$x$ gluon distribution of a proton. It is important to remember, that the present experiments are not fully exclusive and rather semi-inclusive processes have to be studied. Experimentally so far proton dissociation has been “extracted” in a model dependent way assuming some functional form in $p_t$. From HERA we have some limited knowledge only about diffractive dissociation. Compared to exclusive production of $J/\psi$ at HERA, in $pp$ collisions there is also electromagnetic dissociation. Interference effects due to the two diagrams in Fig. \[fig:diagrams\_exclusive\_Jpsi\] were predicted. It would be nice to see a modulation in $\phi_{pp}$ due to interference effects between the two diagrams. In the future, the CMS+TOTEM and ATLAS+ALFA experiments could measure purely exclusive reaction and study dependence on many more variables.
Our results for photon-photon induced processes can be summarized as: We discussed two different approaches for $\gamma \gamma$ processes – collinear and $k_t$-factorization, they are not completely equivalent. We have found strong dependence on the structure function input to the photon fluxes in the $k_t$-factorization approach. Semi-exclusive contributions with dissociation are large which is an interesting lesson for $p p \to p pJ/\psi$. The photon-photon contributions are, however, rather small compared to the Drell-Yan contribution. A reasonable description of the CMS data with isolated electrons was obtained (recently also the ATLAS collaboration obtained similar results). So far only the collinear approach has been applied e.g. to the $p p \to (\gamma \gamma) \to W^+ W^- X Y$ processes. Such a reaction is important in searches for Beyond Standard Model effects. A large cross section was found [@LSR2015]. A calculation in $k_t$-factorization approach would be very valuable.
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---
abstract: 'We formalize the intuitive idea of a labelled discrete surface which evolves in time, subject to two natural constraints: the evolution does not propagate information too fast; and it acts everywhere the same.'
author:
- 'Pablo Arrighi [^1]'
- 'Simon Martiel [^2] [^3]'
- Zizhu Wang
bibliography:
- 'biblio\_doi.bib'
title: Causal Dynamics of Discrete Surfaces
---
Introduction
============
Various generalizations of cellular automata, such as stochastics [@ArrighiSCA], asynchronous [@manzoni2012asynchronous] or non-uniform cellular automata [@FormentiNONUNI], have already been studied. In [@ArrighiCGD; @ArrighiIC; @ArrighiCayley; @ArrighiCayleyNesme] the authors, together with Dowek and Nesme, generalize Cellular Automata theory to arbitrary, time-varying graphs. I.e. they formalize the intuitive idea of a labelled graph which evolves in time, subject to two natural constraints: the evolution does not propagate information too fast; and it acts everywhere the same. Some fundamental facts of Cellular Automata theory carry through, for instance these “causal graph dynamics” admit a characterization as continuous functions.
The motivation for developing these Causal Graph Dynamics (CGD) was to “free Cellular Automata off the grid”, so as to be able to model any situation where agents interact with their neighbours synchronously, leading to a global dynamics in which the states of the agents can change, but also their topology, i.e. the notion of who is next to whom. In [@ArrighiCGD; @ArrighiIC; @ArrighiCayley] two examples of such situations are mentioned. The first example is that of a mobile phone network: mobile phones are modelled as vertices of the graph, in which they appear connected if one of them has the other as a contact. The second example is that of particles lying on a smooth surface and interacting with one another, but whose distribution influences the topology the smooth surface (cf. Heat diffusion in a dilating material, or even discretized General Relativity [@Sorkin]). CGD seems quite appropriate for modelling the first situation (or at least a stochastic version of it).
Modelling the second situation, however, is not a short-term perspective. One of the several difficulties we face is that having freed Cellular Automata off the grid, we can no longer interpret our graphs as surfaces, in general. There are, however, a number of formalisms for describing discretized surfaces which are very close to graphs (Abstract simplicial complexes, CW-complexes…[@Hatcher]). These work by gluing triangles alongside so as to approximate any smooth surface. Relying upon these formalisms, can we formalize the idea of a labelled discrete surface which evolves in time — again subject to the constraints that evolution does not propagate information too fast and acts everywhere the same? Can we achieve this by just modelling each triangle as a vertex, and each gluing of two triangles as an edge, and then evolve the graph according to a CGD?
Notice that one could argue that simplicial complexes are not the simplest objects one could use to represent surfaces: planar graphs may seem more natural to some. Our choice is motivated by two reasons. First, the notion of planar graphs can only be used to represent two-dimensional surfaces, and would be limiting when generalizing to higher dimensions (see further work). Second, in a planar graph, the degree of each vertex is not bounded, and thus such graphs would not fit in our model. In order to change this we would have needed to artificially bound this degree by some constant $d$ and lose the generality of planar graphs.
This paper tackles the question of how to give a rigorous definition of “Causal Dynamics of Simplicial Complexes”, focussing on the 2D case for now. It investigates whether CGD can be readily adapted for this purpose, i.e. whether CGD can be “tied up again to discrete 2D surfaces”. It will turn out that this can be done at the cost of two additional restrictions, i.e. a CGD must be rotation-commuting and bounded-star preserving in order to be a valid Causal Dynamics of Simplicial Complexes. The first restriction allows us to freely rotate triangles. The second requirement allows us to map geometrical distances into graph distances. Both restrictions are decidable. This way of modelling simplicial complexes is similar to combinatorial maps defined in [@lienhardt].
Complexes as graphs {#sec:complexesasgraphs}
===================
[**Correspondence.**]{} Our aim is to define a Cellular Automata-like model of computation over [*$2D$ simplicial complexes*]{}. For this purpose, it helps to have a more combinatorial representation of these complexes, as graphs. The straightforward way is to map each triangle to a vertex, and each facet of the triangle to an edge. The problem, then, is that we can no longer tell one facet from another, which leads to ambiguities (see Fig. \[fig:correspondance\] [*Top row.*]{}).
A first solution is to consider [*$2D$ coloured simplicial complexes*]{} instead. In these complexes, each of the three facets of a triangle has a different colour amongst $\{a,b,c\}$. Now each triangle is again mapped to a vertex, and each facet of the triangle to an edge, but this edge holds the colours of the facets it connects at its ends (see Fig. \[fig:correspondance\] [*Bottom row.*]{}). We recover [@ArrighiCGD; @ArrighiIC; @ArrighiCayley] the following definition.
![Complexes as graphs. \[fig:correspondance\] [*Top row*]{}. The straightforward way to encode complexes as graphs is ambiguous. [*Bottom row*]{}. Encoding coloured complexes instead lifts the ambiguity. However, the fact that the extreme triangles share one point or not, is less obvious in the graph representation.](Pictures/corres2.pdf)
\[def:graphs\] A labeled [*graph*]{} $G$ is given by
- A (at most countable) subset $V(G)$ of $V$, whose elements are called [*vertices*]{} and where $V$ is the uncountable set of all possible vertex names.
- A finite set $\pi=\{a,b,c\}$, whose elements are called [*ports*]{}.
- A set $E(G)$ of non-intersecting two element subsets of $V(G):\pi$, whose elements are called edges. Symbol $:$ stands for the cartesian product. An edge $\{u:p,v:q\}$ is to be read “There is an edge linking port $p$ of vertex $u$ and port $q$ of vertex $v$”.
- A function $\sigma: V(G) \rightarrow \Sigma$ associating to each vertex $v$ some label $\sigma(v)$ in a finite set $\Sigma$.
The set of labeled graphs with labels in $\Sigma$ is denoted $\mathcal{G}_{\pi,\Sigma}$ and the set of disks of radius $r$ is denoted $\mathcal{D}^r_{\pi,\Sigma}$. We similarly define the set of (unlabelled) graphs and denote it $\mathcal{G}_{\pi}$.
The following provides a formal interpretation of those graphs into $CW$-complexes.
\[def:interpretation\] Given a graph $G$, its interpretation as a $CW$-complex $K(G)$ is such that:
- its set of triangles $K_2$ is $V(G)$.
- its set of segments $K_1$ is the quotient of $V(G):\pi$ with respect to the equivalence: $u:p\equiv_1 v:q$ if and only if $\{u:p,u:q\}\in E(G)$. Elements of $K_1$ are denoted $u:\overline{p}$, to distinguish them from the following:
- its set of points $K_0$ is the quotient of $V(G):\pi$ with respect to the equivalence: $u:p\equiv_0 v:q$ if and only if $\{u:(p+1),v:(q-1)\}\in E(G)$.
A segment $u:\overline{p}$ has points $\{ u:(p+o) \textrm{ modulo }\equiv_0 \,|\; o\in\{1,2\}\,\}$.\
A triangle $u$ has segments $\{ u:\overline{p} \textrm{ modulo }\equiv_1\,|\; p\in\pi\,\}$. Notice that segments $u:\overline{p}$ and $u:\overline{q}$ have common point $u:\overline{p}\cap\overline{q}$.
![Complexes, Coloured complexes, Oriented Complexes \[fig:complexes\]](Pictures/complex.pdf "fig:")![Complexes, Coloured complexes, Oriented Complexes \[fig:complexes\]](Pictures/complex5.pdf "fig:")![Complexes, Coloured complexes, Oriented Complexes \[fig:complexes\]](Pictures/complex2.pdf "fig:")
This notion of coloured simplicial complex is not so common, however. It is more common to consider a version of coloured complexes where triangles can rotate freely, i.e. where we can permute the colours: $a$ for $b$, $b$ for $c$, $c$ for $a$, so that each triangle has a cyclic ordering of its facets but no privileged facet $a$. The cyclic ordering is then interpreted an orientation: when two facets are glued together in the complex, their orientation must be opposed, so that the two adjacent triangles have the same orientation. This leads to [*oriented $2D$ simplicial complexes*]{}. Fig. \[fig:complexes\] summarizes the three kinds of $2D$ simplicial complexes we have mentioned. Definition \[def:graphs\] captured $2D$ coloured complexes as graphs. How can we capture oriented $2D$ simplicial complexes as graphs?
[First, we]{} define rotations of the vertices of the graphs in a way that corresponds to rotating the triangles of coloured complexes. Namely, vertex rotations simply permute the ports of the vertex, whilst preserving the rest of the graph:
Let $p_{\operatorname{ports}}$ be some cyclic permutation over $\{a,b,c\}$, and $p_{\operatorname{labels}}$ be some bijection from $\Sigma$ to itself such that $p_{\operatorname{labels}}^3=id$. Let $G$ be a graph and $u\in V(G)$ one of its vertices. Then $r_uG=G'$ is such that $V(G')=V(G)$ and:
- $\{v:i,w:j\}\in E(G) \wedge v\neq u \wedge w\neq u \Leftrightarrow \{v:i,w:j\}\in E(G')$.
- $\{u:i,v:j\}\in E(G) \Leftrightarrow \{u:p_{\operatorname{ports}}(i),v:j\}\in E(G')$.
- $\sigma'(u)=p_{\operatorname{labels}}(\sigma(u))$, whereas $\sigma'(v)=\sigma(v)$ for $v\neq u$.
(From now on in order to simplify notations we will drop all labels $\sigma(.)\in \Sigma$, though all the results of this paper carry through to labelled graphs.) [A rotation sequence $\overline{r}$ is a finite composition of rotations $r_{u_1}, r_{u_2}, \ldots$. Since rotations commute with each other, a rotation sequence can be seen as a multiset, i.e. a set whose elements can appear several times. Hence, the union of two rotation sequences $\overline{r_1}\sqcup\overline{r_2}$ refers to multiset union. Moreover, since $r_u^3=Id$, we can consider that each rotation appears at most two times in a rotation sequence.]{}
Second, we define the equivalence relation induced by the rotations. Using this equivalence relation, we can define graphs in which vertices have cyclic ordering of their edges, but no privileged edge $a$.
Two graphs $G$ and $H$ are rotation equivalent if there exists a sequence of rotations $\overline{r}$ such that $\overline{r} G=H$. This equivalence relation is denoted $G\equiv H$.
[**Who is next to whom?**]{} On the one hand in the world of $2D$ simplicial complexes, two simplices are adjacent if they share a point. On the other hand in the world of graphs, two vertices are adjacent if they share an edge. These two notions do not coincide, as shown in Fig. \[fig:correspondance\]. The figure also shows that two triangles share a point if and only if their corresponding vertices are related by a monotonous path:
Let $\Pi=\{a,b,c\}^2$. We say that $u\in\Pi^*$ is a [*path*]{} of the graph $G$ if and only if there is a sequence $u$ of pairs of ports $q_ip_i$ such that it is possible to travel in the graph according to this sequence, i.e. there exists $v_0,\ldots , v_{|u|}\in V(G)$ such that for all $i\in\{0\ldots |u|-1\}$, one has $\{v_i: q_i,v_{i+1}: p_i\}\in E(G)$, with $u_i=q_ip_i$. We say that a path $u=q_0p_0\ldots q_{|u|}p_{|u|}$ alternates at $i= 0\ldots |u|-2$ if either $p_i=q_{i+1}+1$ and $p_{i+1}=q_{i+2}-1$, or $p_i=q_{i+1}-1$ and $p_{i+1}=q_{i+2}+1$. A path is [*$k$-alternating*]{} if and only if it has exactly $k$ alternations. A path is [*monotonous*]{} if and only if it does not alternate.
Thus distance one in complexes is characterized by the existence of a $0$-alternating path. More generally, distance $k+1$ in complexes is characterized by the existence of a $k$-alternating path. Recall that our aim is to define a CA-like model of computation over these complexes. In CA models, each cell must have a bounded number of neighbours (or a bounded “star” in the vocabulary of complexes). This bounded-density of information hypothesis [@Gandy] is the first justification for the following restriction upon the graphs we will consider:
\[def:bsgraphs\] A graph $G$ is [*bounded-star*]{} of bound $s$ if and only if is monotonous paths are of length less or equal to $s$.
Notice that the property is stable under rotation. A further justification for this restriction will be given later.
Causal Graph Dynamics
=====================
We now provide the essential definitions of CGD, through their constructive presentation, namely as localizable dynamics. We will not detail nor explain nor motivate these definitions in order to avoid repetitions with [@ArrighiCGD; @ArrighiIC; @ArrighiCayley]. Still, notice that in [@ArrighiCGD; @ArrighiIC; @ArrighiCayley] this constructive presentation is shown equivalent to an axiomatic presentation of CGD, which establishes the full generality of this formalism. The bottom line is that these definitions capture all the graph evolutions which are such that information does not propagate too fast and which act everywhere the same.
An isomorphism is specified by a bijection $R$ from $V$ to $V$ and acts on a graph $G$ as follow:
- $V(R(G))=R(V(G))$
- $\{u:k,v:l\}\in E(G) \Leftrightarrow \{R(u):k,R(v):l\}\in E(R(G))$
We similarly define the isomorphism $R^*$ specified by the isomorphism $R$ as the function acting on graphs $G$ such that $V(G)\subseteq \mathcal{P}(V.\{\varepsilon,1,...,b\})$ for any bound $b$ as follow:
- $R^*(\{u.i,v.j,...\})=\{R(u).i,R(v).j,...\}$
- $V(R^*(G))=R^*(V(G))$
- $\{u:k,v:l\}\in E(G) \Leftrightarrow \{R^*(u):k,R^*(v):l\}\in E(R^*(G))$
\[Consistent\] Consider two graphs $G$ and $H$ in ${\cal G}_{\pi}$, they are [*consistent*]{} if and only if for all vertices $u,v,w$ and ports $k,l,p$ we have: $$\{u:k,v:l\} \in E(G) \wedge \{u:k,w:p\}\in E(H) \Rightarrow w=v \wedge l=p$$
A function $f:\mathcal{D}^r_\pi \rightarrow \mathcal{G}_\pi$ is called a local rule if there exists some bound $b$ such that:
- For all disk $D$ and $v'\in D$, $v'\in V(f(D)) \Rightarrow v' \subseteq V(D).\{\varepsilon,1,...,b\}$.
- For all graph $G$ and all disks $D_1,D_2 \subset G$, $f(D_1)$ and $f(D_2)$ are consistent.
- For all disk $D$ and all isomorphism $R$, $f(R(D))=R^*(f(D))$[.]{}
The union $G\cup H$ of two consistent graphs $G$ and $H$ is defined as follow:
- $V(G\cup H)=V(G)\cup V(H)$
- $E(G\cup H)=E(G)\cup E(H)$
[@ArrighiCGD; @ArrighiIC; @ArrighiCayley] A function $F$ from ${\cal G}_{ \pi}$ to ${\cal G}_{ \pi}$ is a [*localizable dynamics*]{}, or CGD, if and only if there exists $r$ a radius and $f$ a local rule from ${\cal D}^r_{ \pi}$ to ${\cal G}_{ \pi}$ such that for every graph $G$ in ${\cal G}_{\Sigma,r}$, $$F(G)=\bigcup_{v\in G} f(G^r_v).$$
CGD act on arbitrary graphs. To compute the image graph, they can make use of the information carried out by the ports of the input graph. Thus, they can readily be interpreted as “Causal Dynamics of Coloured Simplicial Complexes”. But what we are really interested in “Causal Dynamics of Bounded-star Oriented Simplicial Complexes”, which we will call “Causal Complexes Dynamics” for short.
Causal Complexes Dynamics
=========================
This section formalizes Causal Complexes Dynamics (CCD).
[**Rotation-commutating.**]{} First, we will restrict CGD so that they may use the information carried out by ports, but only as far as it defines an orientation. Formally, this means restricting to dynamics which commute with graphs rotations.
A function $F$ from $\mathcal{G}_\pi$ to $\mathcal{G}_\pi$ is rotation-commuting if and only if for all graph $G$ and all sequence of rotations $\overline{r}$ there exists a sequence of rotations $\overline{r}^*$ such that $ F(\overline{r} G)=\overline{r}^* F(G) $. Such an $\overline{r}^*$ is called a conjugate of $\overline{r}$. The definition extends naturally to functions from $\mathcal{D}_\pi$ to $\mathcal{G}_\pi$.
For all finite set of graphs $G_1,...,G_n$ and for all set of rotation sequences $\overline{r_1},...,\overline{r_n}$, if $G_1,...,G_n$ are consistent with each other, and $\overline{r_1} G_1,...,\overline{r_n} G_n$ are consistent with each other, then $$\bigcup_{i\in\{1,...,n\}} \overline{r_i}G_i =\left(\displaystyle{\bigsqcup_{i\in\{1,...,n\}}} \overline{r_i}\right) \bigcup_{i\in\{1,...,n\}} G_i$$
#### Proof:
Notice that we only need to prove this result for the union of two graphs. Let us consider two graphs $G_1,G_2$ and two rotation sequences $\overline{r_1},\overline{r_2}$ such that $G_1$ and $G_2$ are consistent and $\overline{r_1}G_1,\overline{r_2}G_2$ are consistent. Let us consider some rotation $r_u$ appearing only once in $\overline{r_1}$. There are two possible cases:
- $r_u \notin \overline{r_2}$: In this case, $r_u$ acts on $G_1\setminus(G_1\cap G_2)$. Indeed, if $u\in V(G_2)$ then $r_u G_1$ and $G_2$ can not be consistent as $u$ has been rotated in the first graph and not in the second. As $u$ only appears in a part of the graph that is left unchanged by the union, we have that $(r_u (\overline{r_1}\setminus r_u) G_1 )\cup (\overline{r_2} G_2)= r_u[(\overline{r_1}\setminus r_u) G_1 \cup \overline{r_2} G_2] $.
- $r_u \in \overline{r_2}$: In this case, the two graphs $(\overline{r_1}\setminus r_u)G_1$ and $(\overline{r_2}\setminus r_u)G_2$ are consistent and the rotations sequences $(\overline{r_1}\setminus r_u)$ and $(\overline{r_2}\setminus r_u)$ leave the vertex $u$ unchanged. It is easy to check that the graphs $r_u\left[ (\overline{r_1}\setminus r_u)G_1\cup(\overline{r_2}\setminus r_u)G_2 \right]$ and $\overline{r_1}G_1\cup\overline{r_2}G_2$ are the same.
The case where $r_u$ appears more than once in $\overline{r_1}$ can be proven similarly. By commuting all the rotations with the $\cup$ operator, we have that: $$(\overline{r_1}\sqcup \overline{r_2})(G_1\cup G_2)= (\overline{r_1}G_1)\cup(\overline{r_2}G_2)$$ $\square$
The next question is “When is a CGD rotation-commuting?”. More precisely, can we decide, given the local rule $f$ of a CGD $F$, whether $F$ is rotation-commuting? The difficulty is that being rotation-commuting is a property of the global function $F$. Indeed, a first guess would be that $F$ is rotation-commuting if and only if $f$ is rotation-commuting, but this turns out to be false.
(Identity function) . Consider the local rule of radius $1$ over graphs of degree $2$ which acts as the identity in every cases but those given in Fig. \[fig:noncom\]. Because of these two cases, the local rule makes use the information carried out by the ports around the center of the neighbourhood. It is not rotation-commuting. Yet, the CGD it induces is just the identity, which is trivially rotation-commuting.
![A non-rotation commuting local rule induces a rotation commuting CGD.[]{data-label="fig:noncom"}](Pictures/noncom.pdf)
Thus, unfortunately, some rotation-commuting $F$ can be induced by a non-rotation-commuting $f$. Yet, fortunately, any rotation-commuting $F$ can be induced by a rotation-commuting $f$.
Let $F$ be a localizable dynamics. $F$ is rotation-commuting if and only if there exists a rotation-commuting local rule $f$ which induces $F$.
#### Proof:
$[\Leftarrow]$ Let us consider a rotation-commuting local rule $f$ of radius $r$ inducing a localizable dynamics $F$. Let $G$ be a graph and $u$ a vertex of $G$. The following sequence of equalities proves that $F$ is rotation-commuting:
$$\begin{array}{lclr}
F(\overline{r} G) & = &\displaystyle{\bigcup_{v\in G}} f(\overline{r} G^r_v) &\\
& = & \displaystyle{\bigcup_{v\in G}} \overline{r}_v^* f(G^r_v) &\textrm{ (using $f$ rotation-commuting) }
\end{array}$$ Using lemma $1$, we can commute the union operator and the sequences of rotations $\overline{r}_v$ as follow: $$F(\overline{r} G)= \left(\displaystyle{\bigsqcup_{v\in G}} \overline{r_v}^*\right) \bigcup_{v\in G} f(G^r_v)=\left(\displaystyle{\bigsqcup_{v\in G}} \overline{r_v}^*\right) F(G)$$ Less formally, we can commute rotations and unions by looking at the highest power with which the rotations appear at the right of the union operator.
$[\Rightarrow]$ Let $F$ be a rotation-commuting localizable function, and $f$ a local rule inducing $F$. Informally, since $f(G_u^r)$ is included in $F(G)$ we know that as far as orientation is concerned $f$ will indeed be rotation-commuting. However it may still happen that $f$, depending upon the orientation of $G_u^r$, will produce a smaller, or a larger, subgraph of $F(G)$. Therefore, we must define some $\tilde{f}$ which does not do that. Let us consider the following function $\tilde{f}$ from $\mathcal{D}_\pi$ to $\mathcal{G}_\pi$:
$$\forall G_u^r, \tilde{f}(G_u^r)= \bigcup_{ \overline{r} } {\overline{r}^*}^{-1} f(\overline{r} G_u^r)$$ with $\overline{r}^*$ a conjugate of $\overline{r}$ (given by $F$ rotation-commuting).
- $\tilde{f}$ is well defined: By definition of $\overline{r}^*$ we have that: $$\begin{array}{llllr}
&\forall \overline{r} ,& f(\overline{r} G_u^r) \subset \overline{r}^* F(G) &\\
\Rightarrow & \forall \overline{r}, & {\overline{r}^*}^{-1} f( \overline{r} G_u^r)\subset F(G)& (*)\\
\Rightarrow & \forall \overline{r_1}, \overline{r_2} , & {\overline{r_1}^*}^{-1} f(\overline{r_1}G_u^r)\ \textrm{and}\ {\overline{r_2}^*}^{-1} f(\overline{r_2}G_u^r)\ \textrm{consistent}&
\end{array}$$
- $\tilde{f}$ is a local rule: we can check that it inherits of the local rule properties of $f$.
- $\tilde{f}$ induces $F$: $$\begin{array}{llll}
\displaystyle{\bigcup_{v\in G}} \tilde{f}(G_u^r)& = & \bigcup_{v\in G} \left[ \displaystyle{\bigcup_{\overline{r}}} {\overline{r}^*}^{-1} f(\overline{r}G_u^r) \right]&\\
&=& \displaystyle{\bigcup_{v\in G}} \left[ f(G_u^r) \cup \left(\displaystyle{\bigcup_{\overline{r} \neq id}} {\overline{r}^*}^{-1} f(\overline{r}G_u^r)\right)\right]&\\
&=& F(G) \cup \displaystyle{\bigcup_{v\in G}}\left(\displaystyle{\bigcup_{\overline{r} \neq id}} {\overline{r}^*}^{-1} f(\overline{r}G_u^r)\right)& \\
&=F(G)& &\textrm{since $(*)$}
\end{array}$$
- $\tilde{f}$ is rotation-commuting: let us consider a sequence of rotations $\overline{s}$. We have: $$\tilde{f}(\overline{s} G_u^r )= \bigcup_{\overline{r} } {\overline{r}^*}^{-1} f(\overline{r}\overline{s}G_u^r)$$ Let us define $\overline{t}=\overline{r}\overline{s}$. As $\overline{r}$ spans all rotations sequences, $\overline{t}$ spans all rotations sequences. We can write:
$$\begin{array}{lllr}
\tilde{f}(\overline{s}G_u^r)& =& \displaystyle{\bigcup_{\overline{t}}} {\overline{r}^*}^{-1}f(\overline{t}G_u^r) &\\
&=& \displaystyle{\bigcup_{\overline{t}}} {\overline{r}^*}^{-1} \overline{t}^* {\overline{t}^*}^{-1} f(\overline{t}G_u^r)&\\
&=&\left(\displaystyle{\bigsqcup_{\overline{t}}} {\overline{r}^*}^{-1} \overline{t}^* \right)\displaystyle{\bigcup_{\overline{t}}} {\overline{t}^*}^{-1} f(\overline{t}G_u^r)&\textrm{using lemma 1}\\
&=& \left(\displaystyle{\bigsqcup_{\overline{t}}} {\overline{r}^*}^{-1} \overline{t}^* \right)\tilde{f}(G_u^r)&
\end{array}$$
$\square$
Given a local rule $f$, it is decidable whether $f$ is rotation-commuting.
#### Proof:
There exists a simple algorithm to verify that $f$ is rotation-commuting. Let $r$ be the radius of $f$. We can check that for all disk $D\in\mathcal{D}^r_\pi$ and for all vertex rotation $r_u$, $u\in V(D)$, we have the existence of a sequence $\overline{r}$ such that $f(r_u D)=\overline{r}f(D)$.\
As the graph $f(D)$ is finite, there is finite number of sequences $\overline{r}$ to test. Indeed, if $|V(f(D))|=k$, we only have $3^k$ different sequences we can apply on $f(D)$ (for each vertex $u$, we can apply $r_u$ 0,1 or 2 times). Notice that as $f$ is a local rule, changing the names of the vertices in $D$ will not change the structure of $f(D)$ and thus we only have to test the commutation property on a finite set of disks. $\square$
[**Bounded-star preserving.**]{} Second, we will restrict CGD so that they preserve the property of a graph being bounded-star. Indeed, we have explained in Section \[sec:complexesasgraphs\] that the graph distance between two vertices does not correspond to the geometrical distance between the two triangles that they represent. By modelling CCD via CGD, we are guaranteeing that information does not propagate too fast with respect to the graph distance, but not with respect to the geometrical distance. The fact that the geometrical distance is less or equal to the graph distance is falsely reassuring: the discrepancy can still lead to an unwanted phenomenon as depicted in Fig. \[fig:boundedstar\].\
![An unwanted evolution: sudden collapse in geometrical distance. \[fig:boundedstar\] [*Left column:*]{} in terms of complexes. [*Right column.*]{} In terms of graph representation.](Pictures/boundedstar.pdf)
Of course we may choose not to care about geometrical distance. But if we do care, then we must make the assumption that graphs are bounded-star. This assumption will not only serve to enforce the bounded-density of information hypothesis. It will also relate the geometrical distance and the graph distance by a factor $s$. As a consequence, the guarantee that information does not propagate too fast with respect to the geometrical distance will be inherited from its counterpart in graph distance. In particular, it will forbid the sudden collapse phenomenon of Fig. \[fig:boundedstar\]. All we need to do, then, is to impose that CCD take bounded-star graphs into bounded-star graphs. This can be decided from its local rule.
A CGD $F$ is *bounded-star preserving* if and only if for all bounded-star graph $G$, $F(G)$ is also bounded-star. A local rule $f$ is bounded-star preserving if and only if it induces bounded-star preserving a global dynamics $F$.
Given a local rule $f$ and a bound $s$, it is decidable whether $f$ is bounded-star preserving with bound $s$.
#### Proof:
We can, for each disk $D\in\mathcal{D}^{r}$ centered on a vertex $u$, consider a disk $H$ of radius $2rs$ centered on $u$ and containing $D$. Considering any $0$-alternating path $p$ of $f(D)$, the two following cases can appear:
- $p$ is strictly contained in $f(D)$ and it can be checked whether its length is greater than $s$,
- $p$ can be extended as a $0$-alternating path in $f(H)$ by a length between $1$ and $s$. In that case, we can also check if its length is strictly less than $s+1$.
By checking this property for each disk $D$ of radius $r$ and each $H$ containing $D$, we can decide whether the image of a graph will contain $0$-alternating path of length greater than $s$. $\square$
Conclusion {#conclusion .unnumbered}
==========
[**Summary.**]{} We have obtained that the following definition captures Causal Complexes Dynamics, i.e. evolutions of discrete surfaces such that information does not propagate too fast and that act everywhere the same:
A function $F$ from ${\cal G}_{ \pi}$ to ${\cal G}_{ \pi}$ is a [*Causal Complexes Dynamics*]{}, or CCD, if and only if there exists $r$ a radius and $f$ a rotation-commuting, bounded-star preserving local rule from ${\cal D}^r_{ \pi}$ to ${\cal G}_{ \pi}$ such that for every graph $G$ in ${\cal G}_{\Sigma,r}$, $F(G)=\bigcup_{v\in G} f(G^r_v)$.
We have also obtained that given a candidate local rule $f$, it is decidable whether it has the required properties. Since CCD are a specialization of CGD, several results follow as corollaries from [@ArrighiCGD; @ArrighiIC; @ArrighiCayley]. For instance, it follows that CCD of radius $1$ are universal, that CCD are composable, that CCD can be characterized as the set of continuous functions from discrete surfaces to discrete surfaces with respect to the Gromov-Hausdorff-Cantor metric upon isomorphism classes. These results deserve to be made more explicit, but they are already indicators of the generality of the model.
[**Further work.**]{} We went constantly back and forth from graph to simplicial complexes, but we have not formalized this relationship. First: Can every such graph be mapped into a $2D$ oriented simplicial complex? On the one hand, it is intuitive that each vertex represents an oriented triangle, and each edge specifies a unique oriented gluing. On the other hand, we are able to represent a sphere, a cylinder, or a torus with just two vertices, whereas these need many triangles in the simplicial complex formalism. Hence the correspondence is to be found with more economical formalisms such as $\Delta$-complexes [@Hatcher].
![Complexes, pseudomanifolds, combinatorial manifolds. \[fig:pseudo\]](Pictures/pseudo.pdf)
Second: Can any $2D$ oriented simplicial complex be represented by such a graph? We are willingly limiting ourselves to those complexes that arise as discretizations of $2D$ manifolds, i.e. combinatorial manifolds with borders [@Lickorish]. In the $2D$ case these are just the complexes obtained by only gluing triangles pairwise and along their sides (See Fig. \[fig:pseudo\]). In $n$-dimensions combinatorial manifolds are harder to characterize, however: the star of every point must be an $n$-ball. Our bounded-star restriction will then play a crucial role.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank Christian Mercat for helping them enter the world of simplicial complexes.
[^1]: This work was supported by the French National Research Agency, ANR-10-JCJC-0208 CausaQ grant
[^2]: This work was supported by the John Templeton Foundation, grant ID 15619
[^3]: This work was supported by the French National Research Agency, ANR-10-JCJC-0208 CausaQ grant
|
---
author:
- Krzysztof Bolejko
bibliography:
- 'backszek-rev.bib'
title: Cosmological backreaction within the Szekeres model and emergence of spatial curvature
---
Introduction {#intro}
============
Backreaction is a process that describes feedback of structure formation on the mean global evolution of the Universe. As long as the evolution of cosmic structures is well within the linear regime, the Universe should successfully be approximated by perturbations around the FLRW model and its global evolution should follow the Friedmann solution. This part of our Universe’s evolution seems to be well understood. What is less understood and still debatable among cosmologists is the current epoch of the Universe’s evolution with its non-linear growth of cosmic structures. Some cosmologists argue that in the non-linear regime the average evolution of the Universe may deviate from the Friedmannian evolution.
The Friedmannian evolution is at the core of the standard cosmological model. Observational data are mostly interpreted within this framework. Also this framework is embedded within $N$-body simulations. The $N$-body simulations are employed to trace the evolution of our Universe in the non-linear regime. Mostly this is done by employing the Newtonian physics on small scales, and assuming that the global evolution follows the Friedmann solution and is unaffected by local interactions. In principle it is possible to construct a Newtonian cosmology based on a system that expands homothetically and obey the Friedmannian evolution [@2014CQGra..31b5003E]. Also the linear perturbations of such a system are known to be consistent with linear perturbations of the FLRW model [@2015CQGra..32e5001E]. However, in the non-linear regime the situation may be different [@Eingorn:2012jm; @Eingorn:2012dg], for example one of the recent studies shows that the relativistic effects lead to Yukawa-type interactions between matter particles [@2016ApJ...825...84E]. Studies based on the weak-filed limit (i.e. applying post-Newtonian corrections) suggest that the mean evolution is well approximated by the Friedmannian evolution [@2013PhRvD..88j3527A; @2014CQGra..31w4006A; @2016JCAP...07..053A]. On the other hand, studies that try to implement backreaction-type effects into the Newtonian Cosmology and $N$-body codes find large effects [@2016arXiv160100110K; @2016arXiv160708797R] — such an approach has sparked recently a debate, see Ref. [@2017arXiv170308809K] and the rebuttal response in Ref. [@2017arXiv170400703B].
The presence of backreaction is a mathematical consequence of the non-linear structure of equations that govern the evolution of the Universe [@2005PhLA..347...38E]. Cosmological backreaction has been debated for the last 30 years [@Buchert:2015iva; @2016arXiv161208222B]. The nature of this debate focuses on the magnitude of backreaction. On one hand, predictions derived from models based on the FLRW framework and $N$-body simulations are consistent with observational data; this is inductive evidence suggesting that the backreaction could be negligibly small. On the other hand, presence of tensions between various data and experiments could be related to backreaction effects. Finally, some people argue that the presence of dark energy, which dominates the energy budget of the standard cosmological model, is just an artefact and nothing else as a manifestation of strong backreaction effects.
This paper aims to be another voice in the debate on the backreaction and presents the study of the backreaction within the Szekeres model, which is an exact inhomogeneous cosmological solution of the Einstein equations. The structure of the paper is as follows: Sec. \[evolution\] continues with the introduction to the phenomenon of backreaction; Sec. \[szekeres\] introduces the Szekeres model; Secs. \[local\] and \[global\] present the results of the analysis of the backreaction phenomenon within the Szekeres model; Sec. \[conclusions\] concludes the analysis.
Evolution of matter in the Universe and backreaction {#evolution}
====================================================
The energy momentum tensor of a viscous fluid with no energy transfer can be written as
$$T_{ab}= \rho u_au_b+ ph_{ab}+ \pi_{ab},$$
where $\rho$ is energy density, $p$ is pressure, $\pi_{ab}$ is the anisotropic stress tensor, and $h_{ab}$ is the spatial part of the metric in 3+1 split
$$h_{ab}= g_{ab} - u_a u_b,$$
and $u_a$ is the matter velocity flow, whose gradient can be decomposed as $$u_{a;b} = \omega_{a b} + \sigma_{a b}
+ \frac{1}{3} h_{a b} \Theta - A_a u_b,$$ where $\omega_{a b} = u_{[a;|\sigma|} h^\sigma{}_{ b]}$ is rotation, $\sigma_{a b} = u_{(a;|\sigma|} h^\sigma{}_{ b)} - \frac{1}{3} h_{a b} \Theta$ is shear, $\Theta = u^a{}_{;a}$ is expansion, and $ A^a = u^a{}_{;b} u^b$ is acceleration.
Evolution of expansion, shear, and rotation are given by [@2009GReGr..41..581E; @2008PhR...465...61T] $$\begin{aligned}
&& \dot{\Theta} = -{1\over3}\,\Theta^2- {1\over2}\,(\rho+3p)-
2(\sigma^2-\omega^2) + {\rm D}^aA_a+ A_aA^a+ \Lambda, \label{ffe1} \\
&& \dot{\sigma}_{\langle ab\rangle} = -{2\over3}\,\Theta\sigma_{ab}-
\sigma_{c\langle a}\sigma^c{}_{b\rangle}- \omega_{\langle
a}\omega_{b\rangle} + {\rm D}_{\langle a}A_{b\rangle}+ A_{\langle
a}A_{b\rangle} - E_{ab}+ {1\over2}\,\pi_{ab},\\
&& \dot{\omega}_{\langle a\rangle} = -{2\over3}\,\Theta\omega_a-
{1\over2}\,{\rm curl} A_a+ \sigma_{ab}\omega^b.\end{aligned}$$ The equations for density, pressure, and anisotropic stress are [@2009GReGr..41..581E; @2008PhR...465...61T] $$\begin{aligned}
&& \dot{\rho} = -\Theta(\rho+p)-
\sigma^{ab}\pi_{ab}, \\
&& (\rho+p)A_a = -{\rm D}_ap- {\rm D}^b\pi_{ab}- \pi_{ab}A^b.\end{aligned}$$ Finally, the evolution of the electric ($E_{ab}$) and magnetic ($H_{ab}$) parts of the Weyl curvature are given by [@2009GReGr..41..581E; @2008PhR...465...61T]
$$\begin{aligned}
&& \dot{E}_{\langle ab\rangle} = -\Theta E_{ab}-
{1\over2}\,(\rho+p)\sigma_{ab}+ {\rm curl} H_{ab}-
{1\over2}\,\dot{\pi}_{ab} -{1\over6}\,\Theta\pi_{ab} \nonumber
\\ && ~~~~~~ +3\sigma_{\langle
a}{}^c\left(E_{b\rangle c}-{1\over6}\,\pi_{b\rangle c}\right) +
\varepsilon_{cd\langle
a}\left[2A^cH_{b\rangle}{}^d-\omega^c\left(E_{b\rangle}{}^d+
{1\over2}\,\pi_{b\rangle}{}^d\right)\right], \\
&& \dot{H}_{\langle ab\rangle} = -\Theta H_{ab}- {\rm curl} E_{ab}+
{1\over2}\,{\rm curl} \pi_{ab} + 3\sigma_{\langle a}{}^cH_{b\rangle c}
-\varepsilon_{cd\langle a}\left(2A^cE_{b\rangle}{}^d+\omega^cH_{b\rangle}{}^d\right). \label{ffe10} \end{aligned}$$
The homogeneous and isotropic FLRW models form a special subset of all possible solutions of the above equations. The FLRW solution is characterised by vanishing Weyl curvature, vanishing rotation and shear, zero anisotropic stress and pressure gradients [@1997icm..book.....K]
$$\begin{aligned}
{\rm FLRW~universe~~} \left\{ \begin{array}{llllll}
E_{ab} \equiv 0 \nonumber \\
H_{ab} \equiv 0 \nonumber \\
\omega_{ab} \equiv 0 \nonumber \\
\sigma_{ab} \equiv 0 \nonumber \\
\pi_{ab} \equiv 0 \nonumber \\
D_a p \equiv 0 \nonumber
\end{array} \right.\end{aligned}$$
In such a case all the above given evolution equations reduce only to 2 equations that fully describe the evolution of a spatially homogeneous and isotropic system $$\begin{aligned}
\dot{\Theta} = -{1\over3}\,\Theta^2- {1\over2}\,(\rho+3p) + \Lambda, \label{hfe1} \\
\dot{\rho} = -\Theta(\rho+p). \label{hfe2}\end{aligned}$$
After some algebra, it can be shown that the above equations are equivalent to the Friedmann equations
$$\begin{aligned}
&& 3 \frac{\ddot{a}}{a} = - 4 \pi G (\rho + 3p) + \Lambda, \label{fes1} \\
&& 3 \frac{\dot{a}^2}{a^2} = 8 \pi G \rho - 3 \frac{k}{a^2} + \Lambda \label{fes2},\end{aligned}$$
where the relation between the scale factor $a(t)$ and the expansion rate is $\Theta = 3 \dot{a}/a$, and the spatial curvature is ${\cal R} = 6 k /a^2$.
If the universe is homogeneous, then the average evolution is exactly the same as the evolution of an individual worldline, and is given by the Friedmann equations. If the universe is inhomogeneous, then the average over all individual worldlines, may deviate from the solution of a uniform universe, and therefore deviate from the Friedmannian evolution. This is what backreaction describes. When studying backreaction, one focuses on all neglected (in the FLRW case) terms \[cf. eq. (\[ffe1\])–(\[ffe10\])\] and investigates if it is possible that all these terms can affect the mean global evolution of the inhomogeneous system. In other words, if global (mean) evolution of the volume of the Universe (i.e. $\Theta$) and/or matter (i.e. $\rho$) is the same as prescribed by the Friedmann solution \[i.e. eqs. (\[hfe1\]) and (\[hfe2\]) or equivalently by eqs. (\[fes1\]) and (\[fes2\])\] or if the contribution from the shear, rotation, and Weyl curvature can affect the expansion rate $\Theta$, and subsequently change its global (mean) evolution compared to the FLRW case.
Because the evolution equations (\[ffe1\])–(\[ffe10\]) are complicated, we still lack a satisfactory description of backreaction for a real Universe. If we limit the analysis only to irrotational and pressureless fluids (the Szekeres model discussed below can only describe irrotational dust) then the scalar parts of the above equations can be averaged and reduced to [@Buchert:1999er]
$$\begin{aligned}
&& 3 \frac{\ddot{a}_{\cal D}}{a_{\cal D}} = - 4 \pi G {\langle{\rho}\rangle}_{\cal D} + \Lambda + \mathcal{Q}_{\cal D}, \label{bucherteq1} \\
&& 3 \frac{\dot{a}^2_{\cal D}}{a^2_{\cal D}} = 8 \pi G {\langle{ \rho}\rangle}_{\cal D} - \frac{1}{2} {\langle{ \mathcal{R} }\rangle}_{\cal D} + \Lambda - \frac{1}{2} \mathcal{Q}_{\cal D}, \label{bucherteq2} \\
&& \big( {\cal Q}_{\cal D} a^6_{\cal D} \dot{\big)} +
a^4_{\cal D} \big( {\langle{ \mathcal{R} }\rangle}_{\cal D} a^2_{\cal D} \dot{\big)} = 0, \label{bucherteq3}\end{aligned}$$
where the dot $\dot{}\,$ denotes partial time derivative $\dot{} \equiv \partial_t$, ${\langle{ \mathcal{R} }\rangle}_{\cal D}$ is an average of the spatial Ricci scalar $\mathcal{R}$, ${\langle{\ }\rangle}_{\cal D}$ is the volume average over the hypersurface of constant time
$${\langle{A}\rangle}_{\cal D} = \frac{\int_{\cal D} d^3x \sqrt{|h|} A }{\int_{\cal D} d^3x \sqrt{|h|}},$$ the scale factor $a_{\cal D}$ is defined as
$$a_{\cal D} = \left( \frac{ V_{\cal D} }{V_{{\cal D},i} } \right)^{1/3},
\label{aave}$$
where V$_{\cal D}$ is the volume of the domain ${\cal D}$, and $V_{ {\cal D}, i}$ is its initial value. Finally the function ${\cal Q}_{\cal D}$ is
$${\cal Q}_{\cal D} = \frac{2}{3} \left( {\langle{ \Theta^2}\rangle}_{\cal D} - {\langle{ \Theta}\rangle}_{\cal D}^2 \right)
- 2 {\langle{ \sigma^2}\rangle}_{\cal D}.$$
Equations (\[bucherteq2\]) can be used to define the Hubble parameter $H_{\cal D}$
$$H_{\cal D}^2= \frac{8 \pi G}{3} {\langle{ \rho}\rangle}_{\cal D} - \frac{1}{6} {\langle{ \mathcal{R} }\rangle}_{\cal D} + \frac{1}{3} \Lambda - \frac{1}{6} \mathcal{Q}_{\cal D},$$ which then can be used to introduce the [*cosmic quartet*]{} [@2008GReGr..40..467B]
$$\begin{aligned}
\Omega_m^{\cal D} &=& \frac{8 \pi G}{3 H_{\cal D}^2 } {\langle{\rho}\rangle}_{\cal D}, \nonumber \\
\Omega_\Lambda^{\cal D} &=& \frac{\Lambda}{3 H_{\cal D}^2}, \nonumber \\
\Omega_\mathcal{R}^{\cal D} &=& -\frac{ {\langle{\mathcal{R}}\rangle}_{\cal D} } { 6 H_{\cal D}^2 }, \nonumber \\
\Omega_\mathcal{Q}^{\cal D} &=& -\frac{ {\mathcal{Q}}_{\cal D} } { 6 H_{\cal D}^2 }.
\label{cqar}\end{aligned}$$
It follows from eq. (\[bucherteq3\]) that iff ${\cal Q}_{\cal D} = 0$ at all times then the average spatial curvature reduces to the Friedmannian curvature, i.e. ${\langle{ \mathcal{R} }\rangle}_{\cal D} \sim a_{\cal D}^{-2}$. This implies that iff ${\cal Q}_{\cal D}$ vanishes at all times then the Buchert equations reduce to the Friedmann equations, meaning that the non-linear effects associated with structure formation cannot affect the average evolution of the Universe. The question and issue of the ongoing debate on backreaction is related to the amplitude of backreaction. Some studies suggest large backreaction , while some suggest otherwise [@Green:2010qy; @Green:2014aga; @Green:2015bma]. The debate on whether the backreaction in realistic models of the Universe is strictly zero has already been settled [@Buchert:2015iva]. However, it is still debated whether a small deviation from zero also means almost Friedmannian evolution, or not. This question is addressed in this paper using the Szekeres solution of the Einstein equations. The investigation is based on a realistic model of the local cosmological environment [@2015arXiv151207364B] and the ensemble of Szekeres models that is based on this type of structures.
Szekeres model {#szekeres}
==============
The Szekeres model [@1975CMaPh..41...55S] is one of the most general inhomogeneous, exact, cosmological solutions of the Einstein equations [@1997icm..book.....K; @2009suem.book.....B]. The metric of the Szekeres model [@1975CMaPh..41...55S; @1975PhRvD..12.2941S] in the spherical coordinates is [@spheszek]
$$\begin{aligned}
&& {\rm d} s^2 = { {\rm d} t^2} - \frac{1}{\varepsilon-K} \left[ R' + \frac{R}{S} \left( S' \cos \theta + N \sin \theta \right) \right]^2 { {\rm d} r^2 }
- R^2 \left( { {\rm d}{\theta}^2} + \sin^2\theta {{\rm d} {\phi}^2} \right) \nonumber \\
&& + \left( \frac{R}{S} \right)^2 \left\{
\left[ S' \sin \theta + N (1 - \cos \theta ) \right]^2 +
\left[ ( \partial_\phi N )(1-\cos \theta) \right]^2 \right\} { {\rm d} r^2} \nonumber \\
&& +2 \left( \frac{R}{S} \right)^2 \left[ S S' \sin \theta + S N (1 -\cos \theta) \right] { {\rm d} r {\rm d} \theta } - 2 \left( \frac{R}{S} \right)^2 S ( \partial_\phi N ) \sin \theta (1-\cos \theta) { {\rm d} r {\rm d} \phi},\label{ds2}\end{aligned}$$
where ${}' \equiv \partial/\partial r$, $ N(r,\phi) \equiv P' \cos\phi + Q'\sin\phi$, $\varepsilon = \pm1,0$ and $K = K(r) \leq \varepsilon$, $S = S(r)$, $P = P(r)$, and $Q = Q(r)$ are arbitrary functions of $r$.
Systems that can be described using the Szekeres solution have no vorticity $\omega_{ab} = 0$, no viscosity $\pi_{ab} =0$, no pressure $p=0$, and no gravitational radiation $H_{ab} =0$. The Szekeres models are of Petrov type D, the shear and electric Weyl tensor can be written as
$$\sigma_{ab} = \Sigma \, {\rm e}_{ab}, \quad E_{ab} = {\cal W} \, {\rm e}_{ab},$$ where ${\rm e}_{ab} = h_{ab} - 3 z_a z_b$ where $z^a$ is a space-like unit vector aligned with the Weyl principal tetrad. As a result the fluid equations (\[ffe1\])–(\[ffe10\]) reduce only to 4 scalars [@2002PhRvD..66h4011H; @2009suem.book.....B; @2012CQGra..29f5018S]
$$\begin{aligned}
& \rho = \frac {2 \left(M' - 3 M E' / E\right)} {R^2 \left(R' - R E' / E\right)},\label{rho} \\
& \Theta = \frac{ \dot{R}' + 2 \dot{R}R'/R - 3 \dot{R} E' / E}{R' - R E' / E}, \\
& \Sigma = -\frac{1}{3} \frac{ \dot{R}' - \dot{R}R'/R}{R' - R E' / E}, \\
& {\cal W} = \frac{M}{3R^3} \frac{3R' - R M'/M} { R' - R E' / E},\end{aligned}$$
where $E'/E = - (S' \cos \theta + N \sin \theta ) /S$. The spatial curvature follows from the “Hamiltonian” constraint $$\frac{1}{6} {\cal R}= \frac{1}{3} \rho + 6 \Sigma^2 - \frac{1}{9} \Theta^2 + \frac{1}{3} \Lambda, \label{hamcon}$$ and is given by $${\cal R} = 2 \frac{K}{R^2} \left( 1 + \frac{ R K'/K - 2 R E' / E}{ R' - R E' / E} \right).$$ Thus, the whole evolution of the system, is reduced only to a single equation for the function $R$ $$\dot{R}^2 = -K +\frac{2 M(r)}{R} +\frac 1 3\Lambda R^2. \label{evo}$$
To define the Szekeres model 5 arbitrary functions of radial coordinate $r$ need to be specified. In this paper, the radial coordinate has been chosen as
$$r= R_{i},$$ where $R_i$ is the value of $R$ at the last scattering instant, and the arbitrary functions were chosen to be
$$\begin{aligned}
M(r) &=& \frac{1}{6} 8 \pi G \rho_{i} \left[ 1 + \frac{1}{2} m_0 \left( 1 - \tanh \frac{r-r_0}{2 \Delta r} \right) \right] r^3, \label{szf1} \\
K(r) &=& \frac{7}{9} 4 \pi G m_0 \rho_i \left( 1 - \tanh \frac{r-r_0}{2 \Delta r} \right) r^2,\label{szf2} \\
S(r) &=& r^\eta, \label{szf3} \\
P(r) &=& 0, \label{szf4} \\
Q(r) &=& 0. \label{szf5}
\label{szekeresfunctions}\end{aligned}$$
Thus, the model is prescribed by following parameters: $r_0$ (size of perturbation), $\Delta r$ (transition zone), $m_0$ (central density contrast of the initial perturbation), $\eta$ (dipole’s size), and $\rho_{i}$ (background density at the initial instant). The initial instant is set to be the last scattering instant, so $\rho_i = (1+z_{CMB})^3 \Omega_m 3 H_0^2/(8 \pi G)$, where $z_{CMB}$ is the CMB’s redshift $z_{CMB} = 1090$.
Local non-linear evolution and emergence of local spatial curvature within a single cosmological environment {#local}
============================================================================================================
The model of a local cosmological environment is specified by fixing the free parameters to
$$\begin{aligned}
& r_0 = 20 {\rm ~Kpc}, \nonumber \\
&\Delta r= \frac{1}{3} r_0, \nonumber \\
& m_0 = -0.00215, \nonumber \\
& \eta = 0.52, \nonumber \\
& \rho_i = (1+z_{CMB})^3 \Omega_m 3 H_0^2/(8 \pi G), \nonumber \\
& \Omega_m = 0.315, \nonumber \\
& \Omega_\Lambda = \frac{\Lambda}{3 H_0^2} = 0.685, \nonumber \\
& H_0 = 67.3 {\rm~km} {\rm~s}^{-1} {\rm~Mpc}^{-1}.\end{aligned}$$
The evolution of the system is calculated by solving eq. (\[evo\]) from the last scattering instant till the present-day instant. The present-day density contrast presented in the right panel of Fig. \[fig1\] is defined as
$$\delta = \frac{\rho-\rho_0}{\rho_0},$$ where $\rho_0$ is the present-day background density, $\rho_0 = 3 H_0^2 \Omega_m/(8 \pi G)$. The evolution of the density contrast at 4 different locations is also presented in Fig. \[fig1\]. The evolution of the density contrast evaluated within this Szekeres model is compared to the evolution of the density contrast $\delta_{lin}$ which follows from the linear approximation [@1980lssu.book.....P]
$$\ddot{\delta}_{lin} + 2 \frac{ \dot{a}}{a} \dot{\delta}_{lin} - 4 \pi G \rho \delta_{lin} =0,
\label{lineara}$$
where the initial conditions for $\delta_i$ and $\dot{\delta}_i$ have been chosen to be the same as in the Szekeres model at the last scattering instant.
Two upper left panels in Fig. \[fig1\] show places where the models is quite homogeneous, and where the evolution is well described by the linear approximation. The other two panels show places where the inhomogeneity is large and the growth is non-linear. A more detailed map on local inhomogeneity and averages is presented in Fig. \[fig2\]. Figure \[fig2\] presents the volume averages evaluated within spherical domains. Each point in Fig. \[fig2\] presents the value of the averaging within a domain centred at that point and of radius 5 Mpc (cf. the matter horizon in Ref. [@2009MNRAS.398.1527E]). The volume averages were evaluated using the code [`SzReD`]{}[^1]. The code uses the Healpix grid to evaluate integrals around any given point in space and time. The average expansion rate and shear at the present instant normalised by $H_0$ are presented in the upper panels. The kinematic backreaction $\Omega_{\cal Q}^{\cal D}$ and spatial curvature $\Omega_\mathcal{R}^{\cal D}$ are presented in the lower panels of Fig. \[fig2\].
![Backreaction and evolution of the cosmological system. From top to bottom, these four panels correspond to the places and panels presented in Fig. \[fig1\]. Each panel presents the evolution of $\Omega$ as defined by eqs. (\[cqar\]). The kinematic backreaction $\Omega_\mathcal{Q}^{\cal D}$ is depicted with a blue curve. In all cases the kinematic backreaction is rather small: in the linear regime $\Omega_\mathcal{Q}^{\cal D} \sim 10^{-5}$, in the departing linear regime $\Omega_\mathcal{Q}^{\cal D} \sim10^{-3}$, inside the void $\Omega_\mathcal{Q}^{\cal D} \sim 10^{-7}$, and inside the overdensity $\Omega_\mathcal{Q}^{\cal D} \sim 0.04-0.07$. The spatial curvature $\Omega_\mathcal{R}^{\cal D}$ is depicted with a green line. Apart from the linear regime, it is non-negligible, and affects the evolution of the cosmological system. The departure from the linear regime (cf. Fig. \[fig1\]) is characterised by $|\Omega_\mathcal{R}^{\cal D}| > 0.05$.[]{data-label="fig3"}](figure3.eps)
By comparing Figs. \[fig1\] and \[fig2\] it is apparent that the non-linear evolution is associated with places where the present-day spatial curvature $\Omega_{\cal R}$ is not negligible. It should be pointed out, that at the initial instant the spatial curvature is negligibly small, which means that the non-linear growth is related to the emergence of the spatial curvature. The emergence of the spatial curvature is better depicted in Fig. \[fig3\]. Figure \[fig3\] shows the evolution of backreaction and emergence of the spatial curvature at 4 regions presented in Fig. \[fig1\]. The evolution of the [*cosmic quartet*]{} (\[cqar\]) at these 4 different locations (presented in Fig. \[fig3\]) shows that regions which evolve into the non-linear regime have also a substantial build-up of the spatial curvature, and the contribution of the spatial curvature to the evolution of the system is comparable with the contribution from matter.
There is nothing new or unexpected about this result. The initial perturbations of the spatial curvature, as seen from the Hamiltonian constraint (\[hamcon\]), are of the same order as density perturbations. These perturbations are then enhanced in the course of evolution. This also happens in the FLRW regime, where the Friedmannian evolution also allows for the change of the spatial curvature
$$\Omega_{\cal R}(t) \rightarrow \Omega_k(t) = -\frac{k}{H^2 a^2}.$$ Thus initial perturbations of curvature are enhanced also by the Friedmannian evolution and its evolution is comparable (but not exactly the same) to the one presented in Fig. \[fig3\]. This shows that a naive expectation that within a spatially flat (globally) universe, the spatial curvature is everywhere the same, is simply not accurate. A more realistic expectation is that within the inhomogeneous regions (i.e. locally) $\Omega_{\cal R}$ (or equivalently $\Omega_k$) is of a comparable amplitude as $\Omega_m$.
Another significant result, presented in Fig. \[fig3\] is that in all cases the contribution from the kinematic backreaction is much smaller than the contribution from the spatial curvature $\Omega_{\cal R}$. This result is consistent with findings reported in Refs. [@2006CQGra..23.6379B; @2009PhRvD..80l3512W; @2013CQGra..30q5006D; @2013PhRvD..87l3503B; @2013JCAP...10..043R], which show that even small perturbations of the kinematic backreaction $\Omega_{\cal Q}$ can lead to non-Friedmannian evolution, or as in the case of this Section to a highly non-linear evolution. This shows that the kinematic backreaction ${\cal Q}$ is merely a part of the backreaction phenomenon and non-linear evolution. Thus, care should be exercised when debating the relevance of backreaction using only arguments based on the amplitude of the kinematic backreaction.
The results reported in this Section were obtained based on a model of a single cosmological environment: a pair of a cosmic void and overdensity, with the size of inhomogeneity below 100 Mpc. Beyond that scale the system is homogeneous. The reason for this is that the size of inhomogeneous structures within the Szekeres model increases with radius $r$, so in order to model realistic structures one has to limit the number of structures to 2 or 3 [@2007PhRvD..75d3508B] (but see [@2015PhRvD..92h3533S; @2016JCAP...03..012S] for an alternative approach). So beyond 100 Mpc in order to eliminate extremely elongated inhomogeneities the model is set to be homogeneous. As a consequence on the scale of 100 Mpc (and beyond) this model is not suitable to study backreaction.
Global evolution and emergence of global spatial curvature within the ensemble of Szekeres models {#global}
=================================================================================================
The model considered in Sec. \[local\] can only be applied to study inhomogeneities on small scales, i.e. much smaller than the scale of homogeneity (i.e. $\sim 100$ Mpc). In order to study backreaction on a much larger, global scale one needs to implement a different approach to the Szekeres model, such as for example the ensemble approach. In this approach we consider an ensemble of Szekeres worldlines, where each worldline is a separate Szekeres model specified by functions (\[szf1\])–(\[szf5\]) with the free parameters set to
$$\begin{aligned}
& r_0 = 5 + {\cal U}_{[0-1]} \times 20 {\rm ~Kpc}, \nonumber \\
& \Delta r = \frac{1}{3} r_0, \nonumber \\
& m_0 = -0.0015 + 0.003 \times {\cal U}_{[0-1]} , \nonumber \\
& \eta = 0.5 + 0.3 \times {\cal U}_{[0-1]}, \nonumber \\
& \rho_i = (1+z_{CMB})^3 \Omega_m 3 H_0^2/(8 \pi G), \nonumber \\
& \Omega_m = 0.315, \nonumber \\
& \Omega_\Lambda = \frac{\Lambda}{3 H_0^2} = 0.685, \nonumber \\
& H_0 = 67.3 {\rm~km} {\rm~s}^{-1} {\rm~Mpc}^{-1}.\label{szes}\end{aligned}$$
where ${\cal U}_{[0-1]} $ is a random number between 0 and 1 (uniform distribution).
For a large number of realisations, the above prescription provides a wide range of inhomogeneous structures from void-like structures (as in Sec. \[local\]) to systems with central overdensity and an adjacent void (as for example in Ref. [@2007PhRvD..75d3508B]).
Two configurations are investigated:
1. Swiss-Cheese-type configuration
The Swiss-Cheese configuration has a lattice of a homogeneous FLRW regions and inhomogeneities smoothly approach the FLRW background (as shown in Fig. \[fig1\]). Through the homogeneous lattice inhomogeneities are joined to other inhomogeneities. The position of a worldline (with respect to the inhomogeneity defined by (\[szes\])) is randomly selected by randomly generating the pseudo-Cartesian coordinates $x,y,z$
$$x = 2 \, {\cal U}_{[0-1]} r_0, \quad y = 2\, {\cal U}_{[0-1]} r_0, \quad z = 2 \, {\cal U}_{[0-1]} r_0,$$ which are then used to evaluate the Szekeres coordinates $$r = \sqrt{x^2 + y^2 +z^2}, \quad \theta = {\rm ~arccos} \frac{z}{r}, \quad \phi = {\rm ~arctan} \frac{y}{x}.$$
With a sufficiently large number of worldlines, this approach reproduces a Swiss-Cheese-type configuration, i.e. each type of inhomogeneity with its asymptotically-approaching FLRW region is well mapped.
2. Styrofoam-type configuration
The Styrofoam-type configuration consists of densely packed closed-cell structures that do not exhibit any fixed FLRW lattice. The inhomogeneity is still defined in the same way as above, i.e. by (\[szes\]), but the asymptotically-approaching FLRW region is excluded from the Monte Carlo simulation by selecting only worldlines that are close to the central inhomogeneity, with coordinates selected as $$r = \sqrt{x^2 + y^2 +z^2}, \quad \theta = {\rm ~arccos} \frac{z}{r}, \quad \phi = {\rm ~arctan} \frac{y}{x},$$ where $$x = \frac{1}{2}\, {\cal U}_{[0-1]} r_0, \quad y = \frac{1}{2} \, {\cal U}_{[0-1]} r_0, \quad z =\frac{1}{2}\, {\cal U}_{[0-1]} r_0.$$
The difference between this type of configuration and the Swiss-Cheese-type configuration is that the inhomogeneous regions are only mapped in their central parts and there is no asymptotically-approaching FLRW region. On one hand this is an advantage – no FLRW lattice. On the other hand, this means that inhomogeneities are just stuck together without any attempt to smoothly join them together, which in the Swiss-Cheese configuration is obtained via the FLRW lattice.
The ensemble consists of $10^7$ different and independent worldlines – since the model is silent ($H_{ab} = 0$, $\nabla_a p =0$, $\pi_{ab} =0$, and $q_a =0$) there is no commutation between the worldlines and therefore each worldline evolves independently. The volume average of a function $A$ within the ensemble of the Szekeres models is
$${\langle{A}\rangle}_{\cal D} = \frac{ \sum_n A_n v_n} {\sum_n v_n},$$
where $A_n$ is the value of a function for a specific worldline, and $v_n$ is the volume around each worldline $$v_n = \sqrt{ | {\rm ~det} \,g| },
\label{volumn}$$ where ${\rm \det} \, g$ is a determinant of the metric (\[ds2\]). Finally, the size of the domain is
$${\cal D} = V^{1/3} = \left( \sum_n v_n \right)^{1/3}.$$ For $10^7$ worldlines, the comoving size of the domain of the ensemble defined by (\[szes\]) is approximately $1050$ Mpc.
As in Sec. \[local\], the initial instant is set to be the last scattering instant. The state of the ensemble at the initial instant presented in Fig. \[fig4\], shows that initially the model is quite homogeneous, with average density ${\langle{\rho}\rangle}$ and domain expansion rate $H_{\cal D}$ quickly approaching the $\Lambda$CDM values, and the spatial curvature and kinematic backreaction approaching $\, 0 \,$ when averaged over a sufficiently large domain. As seen from Fig. \[fig5\], after 13.8 Gyr of evolution the statistical homogeneity is still present at scales beyond 100 Mpc — the averaging over random domains of size ${\cal D} > 100 {\rm~Mpc}$ produces the same results (i.e. the cosmic variance is negligible small at these scales). However, the mean values depend on the global model, and the results differ between the Swiss-Cheese-type and Styrofoam-type models.
The mean evolution within the Swiss-Cheese-type model follows the background $\Lambda$CDM model: average density ${\langle{\rho}\rangle} = \rho_0$, domain expansion parameter $H_{\cal D} = H_0$, the model is practically spatially flat with $\Omega_{\cal R}= 2.4 \times 10^{-4}$, and the kinematic backreaction is negligibly small $\Omega_{\cal Q}= -1.6 \times 10^{-5}$. However, within the Styrofoam-type model the average density falls below the $\Lambda$CDM, i.e. ${\langle{\rho}\rangle} = 0.88 \, \rho_0$ and the domain expansion parameter is faster than in $\Lambda$CDM background $H_{\cal D} = 1.03 \, H_0$. In addition, the mean spatial curvature is $\Omega_{\cal R}= 0.11$, and the kinematic backreaction is $\Omega_{\cal Q}= -1.1 \times 10^{-2}$, which shows significant deviation from the Friedmannian evolution of the background model.
The explanation for the difference between the global behaviour of the Swiss-Cheese-type model and the Styrofoam-type model is presented in Fig. \[fig6\]. Figure \[fig6\] shows the volume fraction of various types of structures within the Styrofoam-type and Swiss-Cheese-type models. The underdense fraction is defined as $$f_u = \frac{ V_{underdense} } {V_{total} },$$ where $V_{underdense}$ is volume occupied by regions with $\rho<0.9 \, \rho_{\Lambda CDM}$. The overdense fraction is defined as $$f_o = \frac{ V_{overdense} } {V_{total} },$$ where $V_{overdense}$ is volume occupied by regions with $\rho>1.1 \, \rho_{\Lambda CDM}$. Finally, the lattice fraction is defined as $$f_l = \frac{ V_{lattice} } {V_{total} },$$ where $V_{lattice}$ is volume occupied by regions with $0.99 \, \rho_{\Lambda CDM} < \rho < 1.01 \, \rho_{\Lambda CDM}$.
As seen the Swiss-Cheese model is dominated by the asymptotically-homogeneous regions that only deviate by less than $1\%$ from the $\Lambda$CDM model. Such regions occupy more than $50\%$ of the total volume of the Swiss-Cheese-type model, while the contribution from the underdense ($\rho<0.9 \, \rho_{\Lambda CDM}$) and overdense ($\rho>1.1 \,\rho_{\Lambda CDM}$) is merely at the percent level, and the rest of the volume is occupied by almost $\Lambda$CDM-like regions. Consequently, the averages and the mean global evolution follows closely the $\Lambda$CDM model. Contrary to the Swiss-Cheese model, within the ensemble of the Szekeres models of the Styrofoam type, the Monte Carlo simulation only probes the central parts of inhomogeneity and avoids the asymptotic FLRW regions. As a result, the volume is dominated by underdense regions with $f_u = 0.75$ at the present-day instant, and the overdense regions occupy $20\%$ of the total volume. The reason why underdense regions dominate the volume is simply because voids expand faster than the overdense regions, and so they quickly start to occupy larger factions of the volume. Interestingly, by comparing the time scales of Fig. \[fig6\] to Fig. \[fig1\] one can see that the instant when voids start to dominate the total volume is when they enter the non-linear regime.
The evolution of the [*cosmic quartet*]{} (\[cqar\]) is presented in Fig. \[fig7\]. Not surprisingly, since the volume of the Swiss-Cheese-type model is dominated by the $\Lambda$CDM-like regions, the evolution of the [*cosmic quartet*]{} follows the $\Lambda$CDM behaviour. As for the Styrofoam-type model, which volume is dominated by voids, the evolution of the [*cosmic quartet*]{}, when compared with Fig. \[fig3\], resembles the void-like behaviour. It is worth pointing out that within the Styrofoam-type model there is no fixed background. The mean density evolves slightly differently than in the $\Lambda$CDM model, which was used to specify the model at the initial instant. In particular, the evolution of the mean spatial curvature is unlike in the $\Lambda$CDM which is spatially flat.
The emergence of the mean global spatial curvature (within the Styrofoam-type model) has an intuitive explanation. Within this model underdense regions dominate from the start, but since density contrast of overdense regions increase at a slightly higher rate – this is because, as seen from eq. (\[ffe1\]) both density and shear negatively contribute to $\dot{\Theta}$ – thus overdense regions more quickly pass the $10\%$-threshold used in the definition of $f_o$ and $f_u$, and thus in the left panel of Fig. \[fig6\] it looks like overdense regions slightly dominate in the first 500 My of evolution. At that instant the spatial curvature of the Styrofoam-type model is $\Omega_{\cal R} \approx 5\times 10^{-3}$. After that instant $f_u > f_o$ and soon afterwards the amplitude of $\Omega_{\cal R}$ becomes non-negligible. This phenomenon can also be understood in terms of the analysis presented in Ref. [@2011CQGra..28p5004R], which focused on the dynamical system of eqs. (\[bucherteq1\])–(\[bucherteq3\]). The analysis of the dynamical system (\[bucherteq1\])–(\[bucherteq3\]) showed that even a tiny perturbation in ${\cal Q}_{\cal D}$ can drive the system into another basin of attraction that is dominated by the averaged curvature [@2011CQGra..28p5004R]. This property of a global gravitational instability of the Friedmannian model has been identified as the reason for large curvature deviations. The Styrofoam-type model based on the ensemble of the Szekeres models provides an explicit realisation of this scenario.
Conclusions
===========
The models considered in this paper (local environment and global ensemble) addressed the issue of the amplitude of backreaction, and its impact on the evolution of a cosmological system. The analysis was based on the exact, cosmological, and non-symmetrical solutions of the Einstein equations, i.e. the Szekeres model, and it provided examples of relativistic models with non-vanishing backreaction (cf. [@Buchert:2015iva]). The obtained results show that the non-linear evolution and backreaction are closely associated with the spatial curvature. The growth of inhomogeneities cannot be separated from the growth of the spatial curvature, which variation across the Universe is comparable with the variation in the matter field, and in addition the global mean spatial curvature does not necessarily average out to zero. This implies that the emergence of the Cosmic Web in the real Universe should also be associated with the emergence of the mean spatial curvature.
There is nothing “exotic” about the obtained results, and the second-order effects do not “magically” appear with an amplitude a few orders of magnitude larger than the first order effects. These findings can easily be understood in a simple and logical manner. Initially, at the last scattering instant, the Universe is fairly homogeneous with only small perturbations present. As long as the growth of structures is linear and perturbations are small, the differences in the expansion rates are negligible and both types of regions: underdense and overdense expand at a similar, background rate. Once the growth of cosmic structure is non-linear (e.g. matter shear $\sigma^2$ is no longer negligible) the expansion rate of overdense regions efficiently slows down (cf. eq. (\[ffe1\])). When that happens the volume of the Universe becomes dominated by voids (cf. Fig. \[fig6\]). Once the volume of the Universe starts to be dominated by voids then the total volume (eg. within the cosmic horizon) increases faster than in the FLRW model, and since matter is conserved, the mean density is lower than in the $\Lambda$CDM model. This explains why, as seen in Fig. \[fig5\], the mean density is below the $\Lambda$CDM matter density, and the expansion rate is slightly higher than $H_0$. As for the spatial curvature, the density perturbations are coupled with expansion rate perturbations and curvature perturbations (cf. eq. \[hamcon\])). Thus, initially at the last scattering instant, not only density but also tiny curvature perturbations are present. In the course of evolution, these perturbations also grow (in the FLRW regime, $\Omega_{\cal R} \to \Omega_k = - k/\dot{a}^2$), which leads to large variations of spatial curvature across cosmic structures (cf. Fig. \[fig2\]). In addition, since the volume of the Universe in the non-linear regime is dominated by voids, the mean global spatial curvature evolves from $\Omega_{\cal R} = 0$ to $\Omega_{\cal R} \approx 0.1$ (cf. Fig. \[fig7\]).
These findings were obtained based on the Styrofoam-type model. In the Swiss-Cheese model, at the local scales the picture is similar (cf. Figs. \[fig1\]–\[fig3\]) — i.e. non-linear evolution leads to large differences between expansion rates of underdense and overdense regions, as well as, large variations of the spatial curvature — however on global scales the results are different, which can be linked to a fact that in the Swiss-Cheese model the volume is dominated by the $\Lambda$CDM-like regions. Since the volume of the real Universe is in fact dominated by cosmic voids [@2012MNRAS.421..926P], it is reasonable to conclude that the Styrofoam-type model is more realistic than the Swiss-Cheese model. As a results, one can expect that the global mean spatial curvature of our Universe should also be dominated by voids, and subsequently it should deviate from zero in the low-redshift Universe, leading to its evolution from spatial flatness ($\Omega_{\cal R} = \Omega_k = 0$) in the early Universe to a negative spatial curvature ($\Omega_{\cal R} \sim \Omega_k > 0$) at the present day epoch, in a similar manner as presented in Fig. \[fig7\].
It is interesting to point out that, in fact, the analysis of low-redshift data, such as a supernova data alone (without combining it with the CMB) implies large spatial curvature, i.e. $\Omega_k \approx 0.2$ and only after inclusion of the CMB reduces to $\Omega_k = 0.005 \pm 0.009$ . Also, supernova data alone point towards slightly lower values of $\Omega_m$ compared to the CMB constraints, which could be understood, not just in terms of various systematics, but also partly in terms of findings presented in Fig. \[fig7\]. In addition, there is a known tension between the values of $H_0$ derived from the CMB and the local measurements [@2016ApJ...826...56R] which again, apart from various systematics [@2016arXiv160802487P], could also be partly explained in terms of findings presented in Fig. \[fig5\].
The models like the one presented in this paper, i.e. Styrofoam-type model, are not perfect realisations of our Universe. There are a number of limitations, for example the lack of rotation $w_{ab}$ excludes presence of virialised regions; the lack of magnetic Weyl tensor $H_{ab}$ excludes multiple eigenvalues of shear; and lack of heat flow $q_a$ excludes energy transfers from one cosmic cell to another. This all means that one should exercise caution when drawing conclusion as to the properties of our Universe. Still, the obtained results should encourage further studies of the cosmological backreaction and development of numerical cosmology towards realistic models of the Universe [@Bentivegna:2015flc; @Mertens:2015ttp; @2017PhRvD..95f4028M]. Also, these results should encourage cosmologists to think outside the box, especially when dealing with low-redshift data, and allow for example for $\Omega_k \ne 0$, even if at the CMB the Universe was spatially flat[^2]. Allowing for $\Omega_k$ to be a free parameter at low-$z$ and independent from high-$z$ constraints, could in principle reduce some of the tensions and inconsistencies in the observed data [@2016IJMPD..2530007B; @2016arXiv161201529C], but the actual analysis remains to be done. In addition, cosmologists should also aim at directly measuring the curvature of the low-redshift Universe to check if it indeed deviates from the CMB constraints [@2015PhRvL.115j1301R].
I would like to thank Thomas Buchert, Jan Ostrowski, Boudewijn Roukema, David Wiltshire, and anonymous referee for their comments and suggestions. This work was supported by the Australian Research Council through the Future Fellowship FT140101270.
[^1]: The code [`SzReD`]{} is not yet publicly available under a free licence. It is accessible on collaboration basis and anyone interested in using the code is advised to contact the author.
[^2]: Separation of the low-$z$ from high-$z$ data has already been implemented by some cosmologists, for example in Ref. [@2007PThPh.117.1067K] the supernova analysis was conducted using different sets of cosmological parameters for low-$z$ and high-$z$ data; in Ref. [@2010JCAP...08..023V] the CMB data was analysed independently from the low-$z$ cosmology that enters via the distance to the last scattering surface.
|
---
abstract: 'The regular polyhedra have the highest order of 3D symmetries and are exceptionally attractive templates for (self)-assembly using minimal types of building blocks, from nano-cages and virus capsids to large scale constructions like glass domes. However, they only represent a small number of possible spherical layouts which can serve as templates for symmetric assembly. In this paper, we formalize the necessary and sufficient conditions for symmetric assembly using exactly one type of building block. All such assemblies correspond to spherical polyhedra which are edge-transitive and face-transitive, but not necessarily vertex-transitive. This describes a new class of polyhedra outside of the well-studied Platonic, Archimedean, Catalan and and Johnson solids. We show that this new family, dubbed *almost-regular* polyhedra, can be parameterized using only two variables and provide an efficient algorithm to generate an infinite series of such polyhedra. Additionally, considering the almost-regular polyhedra as templates for the assembly of 3D spherical shell structures, we developed an efficient polynomial time shell assembly approximation algorithm for an otherwise NP-hard geometric optimization problem.'
author:
- |
Muhibur Rasheed and Chandrajit Bajaj\
\
Center for Computational Visualization\
Institute of Computational Engineering and Sciences\
The University of Texas\
Austin, Texas 78712
title: '**Characterization and Construction of a Family of Highly Symmetric Spherical Polyhedra with Application in Modeling Self-Assembling Structures**'
---
Introduction {#sec:intro}
============
Regular polyhedra are combinatorial marvels. They are simultaneously isogonal (vertex-transitive), isotoxal (edge-transitive), and isohedral (face-transitive). In this context, transitivity means that the vertices (or edges or faces) are congruent to each other; and for any pair of vertices (or edges or faces), there exists a symmetry preserving transformation of the entire polyhedron which isometrically maps one to the other. Transitivity plays a vital role is assembly, especially self-assembly. For instance, if we consider each face as a building block, then face transitivity indicates that a single type of block is sufficient to form a shell-like structure. Similarly, edge-transitivity indicates that there is exactly one way to put any two blocks together.
Viruses, natures smallest organisms, utilize this simplicity by encoding the blueprint (RNA/DNA) for a single type of protein. These proteins can attach within copies of the same proteins in specific ways. The attaching or binding rules is such that when sufficient concentrations of building blocks are available, they combine to assemble into a spherical shell, called a capsid. These spherical shells have icosahedral symmetry. However, unlike a icosahedron, which only has 20 faces, more than half of the viruses have capsids which are formed by a much larger number of proteins. Caspar and Klug [@caspar62] first addressed the layout of virus capsids using triangular tiles. A similar class of assembly is seen in fullerene like particles, with 12 pentagonal and many hexagonal faces, which was first characterized by Goldberg [@Goldberg_1937].
Recently, several researchers have developed efficient constructions and parameterizations of Goldberg-like particles [@Hu_Qiu_2008; @Schwerdtfeger_Wirz_Avery_2013; @Fowler_Rogers_2001]. For instance, Deng et al. [@Deng_Yu_2012] studied extensions of Goldberg’s construction to other platonic solids, but their study is not exhaustive in characterization and enumeration of all the possible cases. In another recent work, Schein and Gayed [@Schein_Gayed_2014] developed a numerical optimization scheme that takes a Goldberg-like polyhedra, which by construction does not have planar faces and is not always convex, and produces strictly convex polyhedra while preserving the edge-lengths. However, the resulting polyhedra no longer have any face-transitivity. Also, even though the edges have the same length, they are not strictly congruent as their neighboring faces are different- hence making such polyhedra unsuitable for using as layouts for assembly.
In this paper, we introduce the *almost-regular* polyhedra, which preserve the global polyhedral symmetry while offer a denser packing with local symmetries, thus ensuring that only one type of building block still suffices. Topologically, these polyhedra are face-transitive and edge-transitive; and has at most two distinct types of vertices (each vertex is transitive to the other vertices of the same type). In some cases, which we have characterized, the polyhedra may become non-convex and numerical optimization is required to keep edge-lengths equal and make the faces as congruent as possible. Note that even the non-convex polyhedra of this family remain spherical (i.e., any ray emanating from the centroid of the polyhedra will intersect it exactly once).
We believe that our results will greatly impact research in several areas including the field of nano-materials which can self-assemble to create nano-structures with desirable properties. For example, gold nanorods for cancer imaging and therapy [@Chen_2005; @XiaGold2014], virus capsids and protein-cages for targeted drug delivery [@Shi_2010; @Smith_2013; @Steinmetz_2009; @Shang_2012]. Advanced scientific computation techniques, to explore and automatically predict possible nano-structures that can be formed symmetrically by one type of building block (eg. an engineered protein) would accelerate the development of new nano-shell structures. Our theoretical groundwork would greatly support such extensive computational techniques. For instance, we show that using our symmetry characterization, a symmetric assembly of $n$ particles can be predicted using a algorithm whose running time is only polynomial in $n$, even though assembly prediction is an NP-hard optimization problem.
Background
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The boundary of a convex polyhedron is homeomorphic to a spherical tiling. While the space of all convex polyhedra is not enumerable (uncountably infinite), the sub-classes which exhibit one or more types of symmetry and congruency conditions are enumerable. Typical congruency conditions considered in this context are face transitivity, edge transitivity and vertex transitivity. A polyhedron is face-transitive (isohedral) if all faces of the polyhedron are congruent and are transitive. In other words, there exists a symmetry transformation of the entire polyhedron which would map any specific face, A, onto another specific face, B. Similarly, it is edge-transitive (isotoxal) if all edges of the polyhedron are congruent and transitive in the same sense as above; and vertex-transitive (isogonal) if all vertices of the polyhedron are congruent and transitive in the same sense as above.
Depending on which of the above properties are satisfied, we have the following classes of polyhedra-
- Johnson solid: Each face is a regular polygon. But the polyhedron does not satisfy any of the transitivity properties. There are exactly 92 such solids- all of them convex. The subclass of Johnson solids which have only equilateral triangular faces are called Deltahedra.
- Catalan solid: Duals of the Archimedean solids. Catalan solids are convex and face-transitive; but not edge-transitive of vertex transitive.
- Semiregular (also called uniform) polyhedra: These are vertex-transitive and each face is a regular polygon (of 2 or more different types). Examples of such polyhedra are the 13 Archimedean solids, and infinite series of convex prisms and anti-prisms with regular polygons as the two parallel faces.
- Quasiregular polyhedra: This class have vertex and edge transitivity. It is easy to show that they can have at most two types of regular faces. There are only two such polyhedra- cuboctahedron and icosidodecahedron.
- Regular polyhedra: A regular polyhedron is face, edge and vertex transitive; and has only one type of regular face. There are only 5 regular polyhedra in 3 dimensions. These are the Platonic solids: Tetrahedron, Cube, Octahedron, Icosahedron and Dodecahedron.
These special classes are all enumerated and there are only a finite number of them, except for the infinite family of prisms and anti-prisms (which are also enumerable as they map directly to the set of natural numbers).
#### A note on symmetry
A Symmetry group consist of a set of symmetry operations (i.e. transformations which map an object to itself) such that the set is closed under the composition (one transformation followed by another) operation. The actions of the polyhedral symmetry groups can be expressed as pure rotations around different axes through the center of the polyhedron, which we assume to be at the origin without loss of generality. We generally refer to the points of intersection of these axes with the polyhedron as the locations of symmetry, and refer to the axes as the symmetry axes. For instance, the octahedron have 6 axes of 4-fold rotational symmetry[^1] going through the four vertices, 8 axes of 3-fold rotational symmetry going through the centers of the faces, and 12 axes of 2-fold rotational symmetry going through the centers of the edges.
The *Almost-Regular* Polyhedra and Their Duals
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We define the family of *almost-regular* polyhedra as all polyhedra which have global polyhedral symmetry, have congruent regular faces, and is face and edge transitive. The first condition means that any such polyhedron must have exactly the same number of rotational symmetry axes with the same symmetry orders as any specific regular polyhedron. We refer to these as the *global* symmetry axes and locations. We shall refer to these global symmetry axes as gv-symmetry axes, ge-symmetry axes and gf-symmetry axes respectively for axes of symmetry going through, respectively, the vertices, edge-centers and face-centers of the regular polyhedron. Additionally, due to the congruency conditions, all vertices, faces and edges of an *almost-regular* polyhedron must have locations of *local* symmetry. Local symmetry operations map the vertices, edges, faces immediately neighboring the location of local symmetry to themselves, but may or may not map the remaining parts of the polyhedron to itself. These axes will be referred to as lv-, le-, and lf-symmetry axes.
![ **Illustration of the criterion for almost-regular polyhedra**. The polyhedra shown in **(a)** and **(d)** are not almost-regular. In **(a)**, the global symmetries are preserved, but the local symmetries are not (for example, it is not locally 2-fold symmetric around the center of DE). In **(d)**, local symmetries are intact (note that DEFG and other creased faces are considered a single face), but the global 3-fold symmetries are not (for example, around the center of ABD).[]{data-label="fig:assemblytheory:quasi"}](quasiregular.pdf){width="0.85\linewidth"}
While this class bear some similarity with the Catalan solids, an important distinction is that the Catalan solids allow non-symmetric faces (hence, are not isotoxal), as long as all the faces are congruent (for example, the solid in Figure \[fig:assemblytheory:quasi\](a) is a Catalan solid, but is not almost-regular.
Note that, the transitivity properties that make regular polyhedra suitable for assembly, is preserved in almost-regular polyhedra, but now the class is richer and more importantly can model structures with more than 20 building blocks.
The dual of an almost-regular polyhedron has regular faces, is isotoxal and isogonal; but may or may not be isohedral (they can hae at most two types of faces). The closest known family is the semi-regular polyhedra (duals of Catalan solids) which are also isotoxal and isogonal. But the semi-regular polyhedra, which includes the 13 Archimedean solids and the family of prisms with regular faces, have at least 2 types of faces.
Related Prior Work
------------------
The construction scheme, that we propose in the next section, closely follows the one proposed by Goldberg [@Goldberg_1937]. The family of polyhedra generated by the Goldberg construction rule [@Goldberg_1937] are fullerene like structures. Fullerene like structures have icosahedral symmetry (symmetry group of the icosahedron), and consists of many hexagonal faces and exactly 12 pentagonal faces. The soccer ball is the smallest example of such structure. See Figure \[fig:assemblytheory:construction\] for an illustrative description of the construction.
![ **Illustration of the Goldberg construction**. The Goldberg construction involves unfolding an icosahedron (see **(a)** and **(b)**), and then mapping the unfolded icosahedron onto a 2D hexagonal lattice scaled and oriented such that all corners of the unfolded icosahedron (its original vertices) falls on the centers of some hexagon of the grid (some example scale and orientations (for one triangle only) are shown **(c)**). Finally, the icosahedron is folded back, along with the hexagonal grid etched onto its faces. For example, for the scaling and orientation of the red triangle in **(c)**, would result in the tiled icosahedron shown in **(d)**. Notice that the new polyhedron has exactly 12 regular pentagonal faces where the icosahedral vertices originally were, and many regular hexagonal faces.[]{data-label="fig:assemblytheory:construction"}](construction.pdf){width="0.85\linewidth"}
Caspar and Klug [@caspar62] proposed a similar approach, but using a triangular lattice, instead of a hexagonal one, and required that the corners of an edge of the unfolded icosahedron falls on the vertices of the lattice (Figure \[fig:assemblytheory:CKdemo\]). Since, the triangular lattice is simply the dual of the hexagonal lattice, the mapping is essentially the same. But the refolded polyhedron now has only regular triangular faces. It has 12 vertices where 5 such faces are incident, and many vertices where 6 faces are incident- the first set are exactly the original vertices of the icosahedron. Notice that this polyhedron is exactly the dual of the one constructed using Goldberg’s method (shown in Figure \[fig:assemblytheory:construction\](d), and also overlayed in Figure \[fig:assemblytheory:CKdemo\](b)).
![ **Illustration of the Caspar-Klug construction**. The Caspar-Klug construction involves unfolding an icosahedron onto a triangular lattice scaled and oriented such that all corners of the unfolded icosahedron (its original vertices) falls on the vertices of the grid (some example scale and orientations (for one triangle only) are shown **(a)**). Then, the icosahedron is folded back, along with the grid etched onto its faces. For example, for the scaling and orientation of the yellow triangle in **(a)**, would result in the tiled icosahedron shown in **(b)**.[]{data-label="fig:assemblytheory:CKdemo"}](CKdemo.pdf){width="0.85\linewidth"}
Polyhedra produced by Caspar and Klug’s construction method are *almost-regular*, and the ones produced by Goldberg’s are duals of *almost-regular*. But notice that both Goldberg, and Caspar and Klug focused only on the icosahedral case and considered a specific unfolding onto specific 2D lattices, and hence only covers a fraction of the possible almost-regular polyhedra and their duals. Separately, Pawley [@Pawley_1961] studied other ways of wrapping different polyhedra using different lattices from the wallpaper group. However, Pawley did not provide any theoretical characterization of the factors related to the possibility or impossibility of such wrappings. We address this issue in the following section.
Characterizing All Possible *Almost-Regular* Polyhedra {#sec:assemblytheory:enumeration}
======================================================
Both Goldberg and Caspar-Klug constructions can be expressed as unfolding a regular polyhedron onto a 2D lattice and then refolding it with the lattice etched onto its faces. Pawley’s wrapping idea is equivalent. We call these procedures the *unfold-etch-refold* method. Here, we prove the conditions that must be satisfied to produce almost-regular polyhedra using the *unfold-etch-refold* idea for any regular solid, unfolded in any way, onto any 2D lattice.
Shepherd’s conjecture [@Shephard_1975] states that all convex polyhedra have a non self-overlapping planar unfolding with only edge-cuts. This conjecture is not proved or disproved yet for all possible convex polyhedra. However, it is true for the set of special classes we are interested in. Hence, in principle it is possible to unfold one such polyhedron and lay it down on a 2D grid, use the grid to draw tiles of the unfolded polyhedron, and then fold it back up to get a tiled polyhedron. However, every polyhedron actually have many unfolding. For example, icosahedron have 43380 unique unfoldings. Caspar and Klug’s construction produced almost-regular polyhedra using 1 such unfolding, but it is not clear whether other unfoldings would also produce similar *almost-regular* polyhedra, or different types of *almost-regular* polyhedra, or not be *almost-regular*. To address this question, we characterize the relationship of the local and global symmetries of the almost-regular polyhedra, and the etched polyhedra (henceforth called *tiling*) produced using unfold-etch-refold construction.
First of all, we prove that the lattice onto which the polyhedron is unfolded must be regular.
\[lemma:assemblytheory:regulargrid\] The polyhedra generated by an unfold-etch-refold using any regular polyhedra and unfolded in any way, cannot be *almost-regular* if a non-regular grid/lattice is used.
The unfold-etch-refold essentially wraps the lattice/grid over a regular polyhedron. Hence, vertices, edges and faces of the regular polyhedron are suppressed, and a new set of vertices, edges and faces appear (all of which belong to the lattice). Now, the *local* symmetry condition of the almost-regular polyhedron requires that every face be symmetric around its center and be congruent to each other. This cannot be satisfied if the lattice itself was not regular, or not symmetric around some points.
There are exactly 3 regular lattices in 2D- the square lattice, the triangular lattice and the hexagonal lattice. The square lattice has 4-fold rotational symmetries at each vertex and face-center, the triangular lattice has 6-fold and 3-fold symmetries at each vertex and face-center; and the hexagonal lattice has 3-fold and 6-fold symmetries at each vertex and face-center. All of them have 2-fold symmetry on the center of each edge.
Among regular polyhedra, the tetrahedron, the octahedron and the icosahedron have 3-fold symmetries at face-centers and respectively 3, 4 and 5-fold symmetries at vertices. The cube has 4-fold symmetries at vertices and face-centers. The dodecahedron has 3-fold and 5-fold symmetries at vertices and face-centers.
Now we prove another lemma addressing global symmetry conditions at gf-, ge-, gv-symmetry axes.
\[lemma:assemblytheory:localglobal\] To satisfy gf-symmetry conditions, all gf-axes must go through a point of the lattice that have $cn$-fold rotational symmetry where $n$ is the order of rotational symmetry around the gf-symmetry axes of the regular polyhedron and $c$ is a positive integer.
Recall that an almost-regular must have exactly the same number of rotational symmetry axes with the same symmetry orders as any specific regular polyhedron, such that there exists a rigid body transformation that perfectly aligns these axes to those of the regular polyhedron. Since, the unfold-etch-refold is a wrapping, the expected locations and axes of global symmetry of the new tiled/etched polyhedra, and the underlying regular polyhedra are already aligned. For example, in Figure \[fig:assemblytheory:CKdemo\](b), A, B and C are locations of 5-fold global symmetry and D is a location of 3-fold local symmetry. Let, $D$ be the location of one axis of symmetry going through the face of the regular polyhedron and $n$ be its symmetry order. Depending on the chosen unfolding and mapping, a particular gf-symmetry axis can have the following three cases-
- If a gf-symmetry axis goes through a vertex of the tiled polyhedra, then the tiled polyhedra can be $n$-fold symmetric around that axis only if the lattice have $cn$-fold rotational symmetry around its vertices.
- If a gf-symmetry axis goes through the center of a face of the tiled polyhedra, then the tiled polyhedra can be $n$-fold symmetric around that axis only if the lattice face is $cn$-regular.
- If a gf-symmetry does not go through a vertex or a face-center, then the tiled polyhedra can not be $n$-fold rotational symmetric around the axis irrespective of the symmetry of the regular grid.
Hence, the lemma is proved for any unfolding/mapping.
A regular 2D grid/lattice can be parameterized using two vectors $e_1, e_2$, and a fixed origin $O$ as follows. Let, $O$ be a vertex of the lattice and that $O$ has at least 2 neighbors $U$ and $V$ such that the vectors $O\rightarrow U$ and $O\rightarrow V$ are not collinear. Then, defining $e_1 = O\rightarrow U$ and $e_2 = O\rightarrow V$, every point of the lattice can be expressed as linear combinations $he_1 + ke_2$ where $h$ and $k$ are integers. This defines a coordinate system $\mathcal{L}$ with $O$ as the origin and $e_1, e_2$ as the primary axes and a co-ordinate $(h,k)$ representing points on the 2D plane, such that if both $h$ and $k$ are integers then, the point lies on the lattice (is a lattice vertex).
![ **Coordinate systems for the 2D regular lattices**. In the figures, $O$ is the origin and $e_1$ and $e_2$ are two coordinate axes. Any point $(x,y)$ can be reached by vector $xe_1+y_e2$ from the origin. Coordinates of some of the lattice points are shown.[]{data-label="fig:assemblytheory:coords"}](coords.pdf){width="0.85\linewidth"}
\[lemma:assemblytheory:sufficient0\] A lattice described in the $\mathcal{L}$ coordinate system is 2-fold rotationally symmetric around a point $(x,y)$ for the following cases-
- Triangular lattice: if and only if both $2x$ and $2y$ are integers
- Square lattice: if and only if both $2x$ and $2y$ are integers
- Hexagonal lattice: if and only if both $2x$ and $2y$ are odd integers, or $x=2(y+1)-3k$ where $x$, $y$ and $d$ are integers.
Triangular lattices have 2-fold (or multiples of 2-fold) symmetries around their vertices and edge-centers, both of which satisfy the condition that $2x$ and $2y$ are integers. Also, no other point satisfy these conditions. Square lattices have 2-fold (or multiples of 2-fold) symmetries around their vertices, face-centers and edge-centers, all of which satisfy the condition that $2x$ and $2y$ are integers. Also, no other point satisfy these conditions. Hexagonal lattices have 2-fold (or multiples of 2-fold) symmetries around their face-centers and edge-centers. The edge-centers satisfy the condition that both $2x$ and $2y$ are odd integers. The face centers are integer solutions of $x$, $y$ for the family of lines $x=2(y+1)-3d$ where $d$ is an integer. No other point satisfy either of these conditions.
\[lemma:assemblytheory:sufficient1\] If a face $\mathcal{T}$ of a regular polyhedron is mapped to a 2D regular lattice in the following ways, then the part of lattice inside $\mathcal{T}$ is $n$-fold rotational symmetric around the center of $\mathcal{T}$, where $n$ is the rotational symmetry around the gf-axes of the regular polyhedron, and the set of faces intersected by each edge of $\mathcal{T}$ is $2$-fold rotationally symmetric around the center of the edge (location of the ge-axes of the regular polyhedron).
1. If $\mathcal{T}$ belongs to either a tetrahedron, an octahedron or a icosahedron and is mapped to triangular lattice, such that all corners have integer coordinates.
2. If $\mathcal{T}$ belongs to either a tetrahedron, a octahedron or a icosahedron and is mapped to a hexagonal lattice such that all corners fall on face centers of the lattice (have integer coordinates $(x,y)$ such that $x=2(y+1)-3d$ where $x$, $y$ and $d$ are integers).
3. If $\mathcal{T}$ belongs to a cube and is mapped to a square lattice, such that either all the corners fall on vertices of the lattice, or all the corners fall on face-centers of the lattice.
First, we show that such mapping is possible.
1. Consider the case when $\mathcal{T}$ is an equilateral triangular face being placed on a triangular lattice. Without loss of generality, we assume that one corner is placed at the origin and another at $(h,k)$ where $h$ and $k$ are integers. Then, it is trivial to show that the third point can be at $(h+k,-h)$.
2. Consider the case when $\mathcal{T}$ is an equilateral triangular face being placed on a hexagonal lattice. Note that the hexagonal lattice is simply the dual of the triangular lattice, hence a mapping that puts corners of $\mathcal{T}$ on vertices of the triangular lattice would put corners of $\mathcal{T}$ on face-centers of the hexagonal lattice.
3. Consider the case when $\mathcal{T}$ is a square being placed on a square lattice. Without loss of generality, we assume that one corner is placed at the origin and another at $(h,k)$ where $h$ and $k$ are integers. Then, it is trivial to show that the other points can be placed at $(h-k, h+k)$ and $(-k, h)$.
When an equilateral triangular face $\mathcal{T}=ABC$ is placed on a triangular lattice such that the corners are at $(0,0), (h,k)$ and $(h+k,-h)$, the the center $O$ of $\mathcal{T}$ is at $(\frac{2h+k}{3},\frac{-h+k}{3})$ which is at a face center, or at a vertex (if $h=k$, or $k=-2h$). Hence $O$ falls on a location of 3-fold or 6-fold symmetry. Hence, gf-symmetry is satisfied. The center $(x,y)$ of any edge would satisfy the condition that $2x$ and $2y$ are integers, and hence falls on a location that have 2-fold (or 6-fold) symmetry and satisfies the ge-condition.
Similar arithmetic can be applied to prove the theorem for the remaining two cases.
We shall refer to the mapping described in Lemma \[lemma:assemblytheory:sufficient1\] a *compatible mapping*.
\[lemma:assemblytheory:sufficient2\] For any unfolding of a regular polyhedron onto a regular lattice, if any face $\mathcal{T}$ is compatibly mapped, then all faces are also compatibly mapped. And, the etching inside each face are congruent.
For any unfolding, there must be at least another face adjacent to $\mathcal{T}$ and shares an edge with it. Let that edge be $AB$. According to Lemma \[lemma:assemblytheory:sufficient0\], for each other point $P$ belonging to $\mathcal{T}$, there exists a point $P'$ produced by rotating $P$ around the center of $AB$ by 180 degrees. Also, if $P$ was on a vertex (or a face-center), then $P'$ will also be the same. Since only a rigid body motion is applied, the new face $\mathcal{T}' = ABP'\ldots$ will also be regular and be congruent to $\mathcal{T}$ (i.e. it is exactly the unfolded face which was neighboring $\mathcal{T}$ along $AB$). Also, for any point $X$ in $\mathcal{T}$, the same transformation would map it to a point $X'$ inside $\mathcal{T}'$ such that the location of $X$ with respect to the corners of $\mathcal{T}$ is the same as the location of $X'$ with respect to the corners of $\mathcal{T}'$. Hence the etching inside $\mathcal{T}$ and $\mathcal{T}'$ are congruent. By propagation of the same argument, all unfolded faces have corners at integer coordinates and are congruent, irrespective of the unfolding.
\[lemma:assemblytheory:sufficient3\] For any unfolding of a regular polyhedron onto a regular lattice, if any face $\mathcal{T}$ is compatibly mapped, then the resulting polyhedron will be *almost-regular* or a dual.
The polyhedron generated by unfold-etch-refold method have local symmetry, due to the lattice being regular.
Lemma \[lemma:assemblytheory:sufficient1\] showed that within the face $\mathcal{T}$, it satisfies global symmetry around the gf-axes. Even after the faces are folded back into the complete polyhedron, the congruency of the all the face (Lemma \[lemma:assemblytheory:sufficient2\]) guarantees that the remaining faces would also map to each other (including the etching inside them).
Lemma \[lemma:assemblytheory:sufficient2\] showed that all faces (and edges) are congruent, and also that the etching is 2-fold symmetric around the edges of the face $\mathcal{T}$. Hence when folded back, the etchings from a neighboring face $\mathcal{T}'$ will match up perfectly (intersect the shared edge of $\mathcal{T}$ and $\mathcal{T}'$ at exactly the same points), and in case there are fractions of a lattice-face inside $\mathcal{T}$, exactly its complement will show up on the other side (inside $\mathcal{T}'$) of the shared edge, thereby all etched-faces are complete. Topologically, the etched-faces that cross the face boundaries and the ones that do not, are identical.
After folding back, the corners of the faces $\mathcal{T}, \mathcal{T}', \ldots$ meet at a point. Note that since all faces are congruent (Lemma \[lemma:assemblytheory:sufficient2\]) and all corners of each face are also symmetric to each other (Lemma \[lemma:assemblytheory:sufficient1\]), gv-symmetry is satisfied. Moreover, when $\mathcal{T}$ was mapped such that the corners fell on vertices of the lattice, then the gv-axis is surrounded by exactly $n$ congruent and regular etched-faces, where $n$ is the rotational symmetry of the gv-axis, and not the rotational symmetry around the lattice-vertices. For instance, mapping a tetrahedron onto a triangular lattice will produce 4 vertices where 3 edges are incident, and many more (depending on the scale of the mapping) where 6 edges are incident. Hence, two types of vertices appear on the polyhedron, the ones coinciding with the gv-axis and the ones that do not. On the other hand, if $\mathcal{T}$ was mapped such that the corners fell on face-centers of the lattice, then the gv-axis is surrounded by exactly a regular polyhedron with $nc$-fold symmetry, where $n$ is the rotational symmetry of the gv-axis and $c$ is an integer. Other etched faces will simply depend on the regularity of the lattice. For example, if an icosahedron is mapped to a hexagonal grid, then there would be 12 pentagonal faces and many hexagonal faces.
Hence, the polyhedron is *almost-regular* if the corners of the faces fell on lattice vertices, or dual if the corners fell on face-centers. Tetrahedron mapped to triangular grid’s face-centers is an exception, where the dual construction also produce *almost-regular* polyhedron.
Figure \[fig:assemblytheory:primaldualconst\] shows a few examples of constructing *almost-regular* polyhedron and their duals by mapping a face of regular polyhedron on regular lattices in a compatible way. Note that the etched-faces that cross an edge of $\mathcal{T}$ are geometrically not identical to the ones that do not. The ones crossing the boundary have a crease inside them, or if they are flattened, they are no longer regular. This is addressed in the Section \[sec:assemblytheory:optimization\].
![ **Illustration of the constructing *almost-regular* polyhedron and their duals**. Top row shows how placing corners of a polyhedral face on vertices of a compatible lattice produces an *almost-regular* polyhedron. The black lines show the original polyhedron, and the red lines show the etching/tiling induced by the lattice. The second row shows and example of placing the corners at face centers and producing duals of *almost-regular* polyhedron. Finally, the bottom row shows examples of both primal and dual construction using square lattice.[]{data-label="fig:assemblytheory:primaldualconst"}](primaldualconst.pdf){width="0.85\linewidth"}
Now we show that any deviation from a compatible mapping results in a violation of some global symmetry condition.
\[lemma:assemblytheory:necessary1\] If a regular polyhedron is mapped to a regular lattice without fulfilling the compatible mapping conditions, then the resulting polyhedron cannot be *almost-regular*.
If some corners of a face $\mathcal{T}$ fall on lattice vertices (or face-centers) and other don’t, then the etching inside the face cannot be symmetric around the center of the face. Hence the polyhedron is not *almost-regular*.
If none of the corners of $\mathcal{T}$ fall on lattice vertices (or face-centers), and the center of the face is not on a lattice vertex or center of a lattice face, then again the polyhedron is not *almost-regular* (by Lemma \[lemma:assemblytheory:localglobal\]).
Finally, we consider the case where none of the corners of $\mathcal{T}$ fall on lattice vertices (or face-centers), but the center $O$ of $\mathcal{T}$ is on a lattice vertex or a face-center. This would not violate the symmetry around the gf-symmetry axes. But we shall show that it would violate the symmetry around the ge-symmetry. Since, we are mapping a regular polyhedron onto a compatible lattice, there exists a mapping that would place a face $\mathcal{T}'$ such that the center falls at $O$, and the corners lie at integer coordinates. We can generate $\mathcal{T}$ by scaling and/or rotating $\mathcal{T}'$ around $O$. Now, we shall prove that any scaling or rotation of $\mathcal{T}$ that moves the vertices off lattice-vertices violate ge-symmetry. The proof depends on the geometry of coordinate system and here we shall only show if for the cube mapped to the square lattice case. Other cases follow the same pattern of proof.
Without loss of generality, we assume that the center $O$ is the origin, and we focus on one edge $AB$ of the face $\mathcal{T}$ with integer coordinates $(x_1,y_1)$ and $(x_2,y_2)$. Hence, the midpoint $C(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$ falls on a face-center, vertex, or edge-center. After rotating around $O$ by $\theta$ degrees, the new location of the center will be $C'(\frac{(x_1+x_2)cos\theta - (y_1+y_2)sin\theta}{2},\frac{(x_1+x_2)sin\theta + (y_1+y_2)cos\theta}{2})$. Hence, $C'$ can be on a 2-fold location if and only if the numerators are integers, which can only happen if $\theta$ is a multiple of $\pi/2$ and in that case, $\mathcal{T}'$ coincides with $\mathcal{T}$.
Now, if $\mathcal{T}$ was scaled by $s$ then the new position of the edge-center $C$ would be $C(\frac{(x_1+x_2)s}{2},\frac{(y_1+y_2)s}{2})$, and again the numerators are integers iff $s$ is an integer. And if $s$ is an integer, then the corners of $\mathcal{T}'$ would also be on lattice vertices or face-centers.
Hence, scaling or rotating $\mathcal{T}$ around its center $O$ will not keep the $C$ on a face-center, vertex, or edge-center unless the corners of the transformed face $\mathcal{T}'$ fall on lattice-vertices. Finally, the fact that a regular lattice is not 2-fold symmetric around any point other than the face-centers, vertices, or edge-centers completes the proof.
Finally, we conclude our characterization with the following theorem.
\[thm:assemblytheory:characterization\] The polyhedron generated by an unfold-etch-refold is *almost-regular* if and only if a compatible mapping of a regular polyhedron onto an unfold-etch-refold compatible lattice is performed.
The proof follows from Lemmas \[lemma:assemblytheory:regulargrid\], \[lemma:assemblytheory:localglobal\], \[lemma:assemblytheory:sufficient3\] and \[lemma:assemblytheory:necessary1\].
Parametrization {#sec:assemblytheory:combinatorics}
---------------
For the sake of simplicity of presentation, the following discussion is focused solely on mapping the icosahedron onto triangular lattices. Other compatible mappings can be discussed in the same manner with almost no difference in the theorems/lemmas presented here except for minor changes in counting. The choice to focus on the icosahedral case is primarily due to two reasons: first, it has the highest level of symmetry among the regular polyhedra which have a compatible mapping, and second, it has applications in modeling viruses, fullerenes etc.
Let $\mathcal{L}$ be a lattice with origin $O$ and axes $H$ and $K$. Any point in the lattice is expressed using coordinates $(h,k)$ where both $h$ and $k$ are integers.
Assuming that one corner $A$ of the face $\mathcal{T}$ of the polyhedron is mapped to the origin $O$ of the lattice (or the nearest face-center for dual constructions). Then specifying the position of another compatibly placed point $B(h,k)$ is sufficient to parametrize the entire mapping.
Since $\mathcal{T}$ is regular, there are exactly 2 possible ways to define the other points of $\mathcal{T}$. Lemma \[lemma:assemblytheory:sufficient1\] showed that both of these choices will result in a compatible mapping as long as $A$ and $B$ are also compatibly mapped. Also by Lemma \[lemma:assemblytheory:sufficient2\], these two are congruent. So, any one of them can be chosen arbitrarily.
We mention the following lemma whose proof is immediate.
Topology of any *almost-regular* polyhedron or its dual can be parametrized using a tuple $<\mathcal{P},\mathcal{L},h,k>$, where $\mathcal{P}$ is a regular polyhedron, $\mathcal{L}$ is a lattice represented using two axes, and $h$ and $k$ are integers.
### Combinatorial Results
We consider the case where $\mathcal{P}$ is an icosahedron whose symmetry group will is $I$, and $\mathcal{L}$ is the triangular lattice which will be denoted as $\mathcal{L}^3$. Now, we discuss some properties of the lattice.
We define each triangle of the lattice $\mathcal{L}^3$ as a small triangle and use $t$ to denote such a triangle. Let us define a triple $<i,j,k>$ where $i$ and $j$ are integers and $k \in \{+,-\}$. Let the triangle produced by the intersections of the lines $h=i$, $k=j$ and $h+k=i+j+1$ (having the vertices $(i,j), (i+1,j)$ and $(i, j+1)$) be denoted $t_{ij+}$. Similarly, the triangle denoted $t_{ij-}$ has vertices $(i,j), (i+1,j-1)$ and $(i+1, j)$, and is produced by the intersections of the lines $h=i+1$, $k=j$ and $h+k=i+j$. \[def:smallt\]
The proof of the following lemma is immediate from this definition.
$t_{i_1j_1k_1}$ coincide with $t_{i_2j_2k_2}$ if and only if $i_1 = i_2$, $j_1 = j_2$ and $k_1 = k_2$. For any small triangle in $\mathcal{L}^3$, there exists a triple $<i,j,k>$ such that $t_{ijk}$ represents that small triangle.
Through etching, the triangular lattice $\mathcal{L}^3$ produces a tiling of a face $\mathcal{T}$ (which will be called a large triangle in this section) of $\mathcal{P}$ where each tile is a small triangle. Now we consider some properties of this tiling.
Assuming $A$ is at $(0,0)$, $B$ is $(h,k)$ such that $h$ and $k$ are integers, the tiling produced by $\mathcal{L}^3$ on $\mathcal{T}$ satisfies:
- The area of $\mathcal{T}$ is $\frac{\sqrt{3}}{4} (h^2+hk+k^2)$, which is equal to the area of $h^2+hk+k^2$ small triangles.
- In addition to $A, B$ and $C$, $\mathcal{T}$ includes exactly $\frac{h^2+hk+k^2-1}{2}$ more vertices of $\mathcal{L}^3$. Note that any vertex that lie on an edge of $\mathcal{T}$ is counted as half a vertex.
- Each edge of $\mathcal{T}$ is intersected by at most $2(h + k) - 3$ lines of the form $h=c$, $k=d$ and $h+k=e$, where $c,d$ and $e$ are integers.
- The number of small triangles intersected by any edge of $\mathcal{T}$ is at most $2(h+k-1)$.
The following are some combinatorial properties of the overall tiled polyhedron:
- There are exactly $20(h^2+hk+k^2)$ small triangles, and the same number of local 3-fold axes.
- The 12 gf-symmetry axes are surrounded by 5 small triangles.
- There are exactly $10(h^2+hk+k^2-1)$ vertices (not lying on the gf-axes) with 6-fold local symmetry.
Similar properties can easily be derived for other mappings as well. The important point to note is that not only the topology, but also the symmetry and combinatorics are also parameterized by only $h$ and $k$.
Constructing all *Almost-regular* Polyhedra
===========================================
In the previous sections we characterized the conditions that must be satisfied by a *unfold-etch-refold* protocol to produce an *almost-regular* polyhedron. The characterization immediately lends itself to efficient generation of families of such polyhedra whose topology can be parameterized using just two variables (discussed below). Furthermore, the symmetry at global and local levels lets us represent the geometry using a minimal set of points. Finally, we show how these properties lead to efficient optimization algorithms for constructing 3D shapes with spherical symmetries.
Note that given a point $P$ with coordinate $(i,j)$ inside $\mathcal{T}$, there exists two other points $Q$ and $R$ such that $P$, $Q$ and $R$ are 3-fold symmetric around the center $D$ of $\mathcal{T}$. The two points $Q$ and $R$ have coordinates $(h-i-j, k+i)$ and $(h+k+j,-h-i)$ respectively. This can be seen by noticing that stepping along the $H$ and $K$ axis by $i$ and $j$ units from $A(0,0)$ is $C^3$ symmetric (around $D$) to stepping in $-H+K$ and $-H$ directions by the same units from $B(h,k)$, and stepping in $-K$ and $H-K$ directions by the same units from $C$. We can further extend it to triangles and deduce the following.
If $A(h_1, k_1)$, $B(h_2, k_2)$ and $C(h_3, k_3)$ are three points in the $HK$ coordinate system such that $h_1, h_2, h_3, k_1, k_2, k_3$ are integers and $ABC$ is an equilateral triangle whose centroid is $O$, then the small triangles $t_{h_1+i,k_1+j,\pm}$, $t_{h_2-i-j-1,k_2+i,\pm}$ and $t_{h_3+j,k_3-i-j-1,\pm}$ are $C^3$ symmetric around $O$. \[lem:trisym2\]
Now, we define the minimal set of points or non-redundant set of points $\mathbb{S}$ such that no two points $s_i,s_j\in S$ are $C^3$ symmetric to each other around $D$, and all points in $\mathbb{S}$ lie inside or on $\mathcal{T}$. Clearly, $|\mathbb{S}| = \lceil\frac{h^2+hk+k^2}{3}\rceil$. Note that applying $C^3$ operations on $\mathbb{S}$ produces all points inside and on $\mathcal{T}$.
Now we are ready to specify a concrete algorithm for computing *almost-regular* polyhedra (see Figure \[fig:ConstructAlmostRegularPolyhedron\]).
The algorithm constructs a minimal geometric representation of the *almost-regular* polyhedron in terms a set of points $\mathbb{S}$ embedded onto the XY plane and a set of 3D transformations $\mathbb{T}_{all}$.
Follows from the definition of $\mathbb{S}$ and Lemma \[lemma:assemblytheory:sufficient3\].
Some polyhedron generated by applying are shown in Figure \[fig:TilingGenResults\].
Note that if the tiles that cross the boundaries of a face $\mathcal{T}$ of $\mathcal{P}$ are not regular, they would look like they have a crease along the edge of the $\mathcal{T}$ by definition of the unfold-etch-refold technique. Tiles generated by this algorithm will also have the same problem and such tiles will be non-regular, and in some cases even non-planar. In the next section we address this issue.
![ **Some polyhderon generated by applying** .[]{data-label="fig:TilingGenResults"}](TilingGenResults.png){width="0.85\linewidth"}
Some polyhedron generated by applying are shown in Figure \[fig:TilingGenResults\].
Note that the tiles that cross the boundaries of a face $\mathcal{T}$ of $\mathcal{P}$ may have a crease along the edge of the $\mathcal{T}$. This is a result of the *unfold-etch-refold* technique. Tiles generated by this algorithm will also have the same problem and such tiles will be non-regular, and in some cases even non-planar.
Curation of Tiles {#sec:assemblytheory:optimization}
-----------------
As mentioned before, sometimes the lattice faces, which corresponds to tiles/faces of the generated *almost-regular* polyhedron, crosses the boundary of the polyhedral face $\mathcal{T}$ embedded on the lattice. During folding, these faces get warped. There can be exactly three types of scenarios (for mapping to a square or triangular lattice).
1. If one corner of $\mathcal{T}$ is at $(0,0)$, and the other corner is at $(h,k)$ such that either $h=0$ or $k=0$, then no lattice face crosses the edges of $\mathcal{T}$, and no curation is required. (see Figure \[fig:assemblytheory:curation\] top row).
2. If one corner of $\mathcal{T}$ is at $(0,0)$, and the other corner is at $(h,k)$ such that $h=k$ then some lattice faces are exactly bisected by the edges of $\mathcal{T}$. The curation is quite trivial in this case. If $h=k$, the center of $\mathcal{T}$ would lie on a lattice vertex. Let the center be $D$, and the face $\mathcal{T}$ be $ABC$. Then, folding along $AD$, $BD$ and $CD$ will not warp any lattice face. Additionally, connecting $D$ to the centers of the neighboring polyhedral faces $\mathcal{T}$ would satisfy all global symmetry conditions as well. This folding will produce a base polytope which actually looks like the *almost-regular* polyhedron with $h=k=1$. In fact, for any integer $i$, to polyhedron generated for $h=k=i$ is nothing but subdivisions of the faces of the $h=k=1$ polyhedron. Interestingly, in some cases, the new folding produces a polyhedron with base geometry like some Catalan solids, but will unlike Catalan solids, these will have regular faces and may be non-convex. For example, the $h=k=1$ polyhedron have the same topology as the pentakis dodecahedron (see Figure \[fig:assemblytheory:curation\] middle row).
3. If one corner of $\mathcal{T}$ is at $(0,0)$, and the other corner is at $(h,k)$ such that $h\neq k, \& h,k>0$; then, for all mappings on the hexagonal lattice, some lattice faces shall cross the edges of $\mathcal{T}$ in variable ways. There does not exist any folding which can avoid the crossing while maintaining global symmetry (as the lines will not meet at the center of the face $\mathcal{T}$). See Figure \[fig:assemblytheory:curation\] bottom row. In this case, no exact solution exists, and we provide a numerical approximation which maximizes the regularity while ensuring that global symmetries are not violated (see below).
![ **Different cases of warping of tiles, and their curations**.[]{data-label="fig:assemblytheory:curation"}](curation.pdf){width="0.85\linewidth"}
We should mention that recently Mannige and Brooks [@Mannige_Brooks_2010] explored such pseudo-irregularities in icosahedral tilings and suggested the existence of three classes depending on the values of $h$ and $k$. Here, we have generalized that to all *almost-regular* polyhedra, and quantified the exact number of faces that gets warped.
### Curation as a Numerical Optimization Problem
The goal of curation would be to make all the faces as regular as possible without violating global symmetry. A similar problem specifically for the family of polyhedra generated by mapping an icosahedron onto a hexagonal lattice was addressed in [@Schein_Gayed_2014]. In that work Schein and Gayed aimed to make all the hexagonal faces that cross the face boundary, and become creased/non-planar, into planar ones while keeping the edge lengths equal. They also showed that it is possible to provide an efficient numerical solution to the problem which ensures that no two hexagonal/pentagonal tiles lie on the same plane and the overall polyhedron is convex. However, the shapes of the hexagons are allowed to get distorted such that the angles are no longer equal, and may vary a lot within the same hexagon. Hence, the faces are no longer congruent (or even nearly congruent) to each other. This makes such a polyhedron non-amenable for modeling structures formed using a single type of building block, for instance viral capsids. In contrast, we want to maintain the congruence of the tiles as much as possible.
When mapping a polyhedron onto a triangular lattice, the generated polyhedra falls under the class called deltahedra, polyhedra whose faces are all equilateral triangles. Even though there are an infinite family of deltahedra [@Trigg_1978] (our families are also infinite), it has been known since Freudenthal and van der Waerden’s work [@Freudenthal_1947], that there are exactly eight convex deltahedra, having 4, 6, 8, 10, 12, 14, 16 and 20 faces; among them only three are regular or have symmetries like the regular ones. So, our family of *almost-regular* polyhedra cannot be convex and regular at the same time. We prioritized regularity.
![ **Numerical curation of warped faces**. Top row: (a) shows a polyhedron with icosahedral symmetry and 260 tiles. The color of the triangles are determined by the ratio of the longest and the shortest edge of the tile. Ratios 1 to 1.3+ are colored using white to red gradient. The triangles at the corners and near the edges of the icosahedron have higher distortion. (b) shows the result of numerical curation. All the triangles are now regular, the worst ratio being 1.034, and the symmetry is preserved. Bottom row: (c-d) show a similar before-after figure for a smaller polyhedra, but one where the warping is more apparent. The numerical optimization brought down the ratio of the worst triangle from 2.13 to 1.004. (e) shows a superposition of the two states to highlight that points are updated symmetrically.[]{data-label="fig:assemblytheory:demo3"}](opt.pdf){width="0.8\linewidth"}
- Let the set of all points on the generated polyhedron be $\mathbb{S}_{all} = \mathbb{T}_{all}(\mathbb{S})$.
- Let $\mathbb{E}_1$ be the set of lattice/tile edges on the generated polyhedron
- Let $\mathbb{E}_2$ be the set of diagonals of all the tiles on the generated polyhedron
- Let, for any point $p \in \mathbb{S}_{all}$, the functions $s(p)$ and $t(p)$ returns respectively a point $q \in \mathbb{S}$ and a transformation $T \in \mathbb{T}_{all}$ such that $p = T(q)$.
- Let $dist(u,v)$ be the square of the Euclidean distance between two points.
In our calculations we shall only update the positions of the points in $\mathbb{S}$, and all other points $p \in \mathbb{S}_{all}$ on the polyhedron will be generated as $t(u)(s(u))$. This ensures that the points are always moved in a symmetric way with respect to the global symmetry axes. Hence global symmetry is never violated.
Let $\mathbb{S}^0$ be the initial positions of the points in $\mathbb{S}$. As we update the locations of the points in $\mathbb{S}$ in our algorithm, the squared displacement of each point $p \in \mathbb{S}$ will be defined as $\delta(p) = (dist(p,p^0))^2$, where $p^0 \in \mathbb{S}^0$ is the initial position of $p$. Also, the squared length of a line segment $e(u,v) \in \mathbb{E}_1 \cap \mathbb{E}_2$ will be computed as $dist(t(u)(s(u)),t(v)(s(v)))^2$ and be denoted $l(e)$. Let us also define $\mu_1 = \frac{1}{|\mathbb{E}_1|} \sum_{e \in \mathbb{E}_1} (l(e))$ and $\mu_2 = \frac{1}{|\mathbb{E}_2|} \sum_{e \in \mathbb{E}_2} (l(e))$.
Finally, we define an energy function $\mathcal{F}(\mathbb{S})$ as follows:
$\mathcal{F}(\mathbb{S}) = \frac{1}{|\mathbb{E}_1|} (\sum_{e \in \mathbb{E}_1} (l(e) - \mu_1)) + \frac{1}{|\mathbb{E}_2|} (\sum_{e \in \mathbb{E}_2} (l(e) - \mu_2)) + \frac{1}{|\mathbb{S}|} (\sum_{p\in \mathbb{S}} \delta(p)).$
Now, we minimize the function $\mathcal{F}(\mathbb{S})$ over the positions of the points in $\mathbb{S}$.
This is clearly a quadratic optimization problem over $h^2+hk+k^2$ variables, and for most practical values of $h$ and $k$ it can be solved efficiently using any numerical solution techniques. We chose to use the limited memory variant of Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm [@Byrd_Lu_Nocedal_Zhu_1995] due to its faster convergence rates. Also since first and second derivatives (hessian) of the energy function are straight-forward to compute, the numerical solution does not require finite-differences and heuristic based hessians, making the solution under BFGS more efficient and stable. Figure \[fig:assemblytheory:demo3\] shows some examples of numerically curating warped faces.
Constructing Shell Structures
=============================
Viruses, as discussed before, have icosahedral symmetric shells formed by multiple copies of the same protein. While several existing works (e.g. [@berger94; @zlotnick05; @zandi05; @Rapaport_2004; @Bona_Sitharam_Vince_2011; @Bahadur_Rodier_Janin_2007; @Carrillo-Tripp_Brooks_Reddy_2008; @Cheng_Brooks_2013]) leverage symmetry to analyze a given shell structure, we are the first to propose a generative algorithm to predict all such structures.
We propose that each virus shell is templated on a particular *almost-regular* polyhedron. We consider the assmebly prediction problem where the number of proteins and the structure of a individual protein is known, and the structure of the whole shell is unknown and must be predicted. We propose the following generation algorithm (Figure \[fig:ShellGen\]) to solve this problem. Note that, while the algorithm specifies icosahedral symmetry as that is the relevant one for viruses, it can easily be extended to handle other cases.
Analysis of the algorithm follows.
Datails of Decoration Rules for Constructing Thick Shell Structures
-------------------------------------------------------------------
We assume that the structures of tiles with the same internal symmetry (e.g. small triangle) remain the same across the shell. And all tiles are decorated by identical building blocks in a cyclic symmetric configuration. In other words, we assume a direct mapping between the internal symmetry of the tile and the set of building blocks used to decorate it. So for example, we decide to use a complex of three identical blocks in a $C^3$ configuration (henceforth called a c-tile) to decorate a small triangle of the layout, as opposed to using 3 independent blocks to decorate each corner of the c-tile.
#### Representation of a tile
Each tile is represented using five parameters $(\mathbf{u_1}, \mathbf{u_2}, f)$.
- $\mathbf{u_1}$ is a unit vector pointing from the origin to the center of symmetry of the tile.
- $\mathbf{u_2}$ is a unit vector pointing from the center of symmetry of the tile to one representative corner of the tile.
- $f$ is the order of symmetry of the decoration to be placed inside the tile.
#### Representation of a c-tile
Each $c$-tile is represented using five parameters $(\mathbf{v_1}, \mathbf{v_2}, \mathbf{c}, o)$.
- $\mathbf{v_1}$ is a unit vector representing the symmetry axis.
- $\mathbf{v_2}$ is a unit vector orthogonal to the symmetry axis pointing to the centroid of one copy from the center of symmetry.
- $\mathbf{c}$ is the center of symmetry.
- $o$ is the order of symmetry.
#### Decorating rules
To decorate a tile $(\mathbf{u_1}, \mathbf{u_2}, f)$ using a $c$-tile $(\mathbf{v_1}, \mathbf{v_2}, \mathbf{c}, o)$, the following must be satisfied-
- $o = f$.
- $c$ is translated such that it lies on the line $m{u_1}=0$ where $m$ is a scalar.
- Rotated such that $v_1$ aligns with $u_1$ and $v_2 \times u_2$ is parallel to $u_1$.
Further, we ensure the following-
- If $C_i$ and $C_j$ are any two $c$-tiles of the same order,
- The $c$-tiles are translated in a symmetric way. i.e. $m_i = m_j$.
- The $c$-tiles are rotated in a symmetric way. i.e. $v_{2_i} \cdot u_{2_i} = v_{2_j} \cdot u_{2_j}$.
The following result is immediate.
The decoration rules are necessary and sufficient for preserving local symmetry (internal to the tile) and global symmetry of the tiling.
Since the algorithm satisfy the decoration rules, it correctly generates symmetric shell structures.
Reults
------
{width="0.9\linewidth"}
\[fig:assemblytheory:demo\]
### Reproducing Known Shell Structures
To verify the effectiveness of the tiling and decoration algorithms, we applied it to predict shell structures for some viruses for which the structure of the individual building blocks (proteins), as well as the entire shell is known. We show some examples here.
![Shells of different sizes using the same protein[]{data-label="fig:virus:sizes"}](multi-size.pdf){width="0.85\linewidth"}
In Figure \[fig:assemblytheory:demo\](a), we present the results of modeling the Tobacco Necrosis Virus which has 60 proteins on its shell. We templated it based on a polyhedra with h=1, k=0 and the resulting computationally predicted shell had less that 5A RMSD error with respect to the known shell. This error is considered acceptable in molecular biology community. There is no topological errors. We had similar success in predicting the structure of Nudaurelia Capensis Virus which requires 240 proteins on its shell and the template polyhedra was constructed with h=2, k=0 (Figure \[fig:assemblytheory:demo\](b)). Finally, Figure \[fig:assemblytheory:demo\](c-d) show the outcome of predicting the shell of Rice Dwarf Virus which has 780 proteins. The layout for this can be either a polyhedra with h=3, k=1; or h=1, k=3; the latter is topologically incorrect if compared to the shell found in nature. Unfortunately, our geometric optimization algorithm and scoring model cannot discriminate between the two.
### Assembling Multiple Sized Shells Using the Same Building Block
In Figure \[fig:virus:sizes\] we show that our algorithm can easily produce shells of different sizes using the same building blocks. Here we used the same protein, but decorated tilings of different complexities and reported the highest scoring models for each size.
Conclusion {#sec:assemblytheory:conclusion}
==========
We have characterized a new family of polyhedra with regular faces such that it is isotoxal, isohedral, and have exactly 2 types of vertices; as well as a dual family which is isogonal, isotoxal and have exactly 2 types of regular faces. We have shown that both of polyhedrons of these families generated by unfolding a regular polyhedron onto a lattice in a compatible way, thereby allowing the lattice vertices, edges and faces to etch out a tiling on the unfolded polyhedron, and finally folding it back again. Further, the compatible ways are specified using only a couple of integer parameters. We also provided a deterministic and efficient algorithm for generating such polyhedra of any size (determined by the two parameters). We have proved that our construction covers all possible polyhedron which satisfies the stated properties. When considering the geometric aspects of the generated polyhedra, we characterized the cases where the faces may become non-regular, and provided solutions for each case.
Finally, we point out that our class of polyhedron is not similar to the known families like Catalan solids, Johnson solids, or Archimedean solids. Some Catalan solids like the tetrakis hexahedron, triakis octahedron, triakis icosahedron, rhombic dodecahedron, rhombic triacontahedron, or pentakis dodecahedron may seem like they satisfy the properties of *almost-regular* polyhedron, but actually all of them violate either the global of the local symmetry conditions. Also Archimedean solids like the truncated cube can be generated by placing the triangles of a tetrahedron on a hexagonal lattice such that the corners of each triangle fall on the centers of 3 faces surrounding a single face. Many other Archimedian solids are isogonal and isotoxal, but are none of them (not even the truncated cube) are duals of of any *almost-regular* polyhedron.
The characterization and construction would greatly aid computational analysis and modeling of dome or spherical shaped objects with symmetry and regular tiles, which is relevant for anti-viral drug design, designing nano-cages for drug delivery and cancer therapy, designing easy to assemble masonry structures etc. Additionally, the regular tilings we produce promises to be an interesting template for arbitrarily refined meshing of manifolds, by diffeomorphic mappings between the manifold and a sphere.
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[^1]: $n$-fold rotational symmetry, also referred to as the symmetry order $n$, means that a rotation by $2\pi/n$ maps the polyhedron to itself
|
---
abstract: 'In this paper, the Newton-Anderson method, which results from applying an extrapolation technique known as Anderson acceleration to Newton’s method, is shown both analytically and numerically to provide superlinear convergence to non-simple roots of scalar equations. The method requires neither a priori knowledge of the multiplicities of the roots, nor computation of any additional function evaluations or derivatives.'
address: 'Department of Mathematics, University of Florida, Gainesville, FL, 32611'
author:
- Sara Pollock
bibliography:
- 'ander.bib'
title: Fast convergence to higher multiplicity zeros
---
Rootfinding ,nonlinear acceleration ,non-simple roots ,Newton’s method , Anderson acceleration
Introduction.
=============
Solving nonlinear equations is a problem of fundamental importance in numerical analysis, and across many areas of science, engineering, finance and mathematics. In general, solving nonlinear equations is an iterative process, accomplished by generating a sequence of approximations to the solution. One of the most common methods of obtaining a solution to the nonlinear problem $f(x) = 0$ is Newton’s method, in which the sequence of approximations to a zero of $f$ is generated, given some initial $x_0$, by $$\begin{aligned}
\label{eqn:newton}
x_{k+1} = x_k - [f'(x_k)]^{-1}f(x_k).\end{aligned}$$
The purpose of this manuscript is to introduce for functions $f: \mathbb R \rightarrow \mathbb R$, the sequence where given $x_0$, $x_1$ is found by , then for $k \ge 1$, $x_{k+1}$ is generated by $$\begin{aligned}
\label{eqn:na-iter}
x_{k+1} = x_k - \frac{x_k - x_{k-1}}
{f(x_k)/f'(x_k)-f(x_{k-1})/f'(x_{k-1})}
(f(x_k)/f'(x_k)).\end{aligned}$$ It will be shown that the iterative scheme gives fast (superlinear) convergence to roots of multiplicity $p > 1$, where the Newton method gives only slower linear convergence. It will also be shown how this sequence is the result of applying an extrapolation method known as Anderson acceleration [@anderson65] to the Newton iteration .
Newton’s method is well-known for its quadratic convergence to simple zeros, supposing the iteration is started close enough to some root of a function $f$. However, some problems may have non-simple (higher multiplicity) roots. For a root $c$ of multiplicity $p > 1$, Newton’s method converges only linearly, and $ \lim_{k \rightarrow \infty }|x_{k+1} -c|/|x_k - c| = 1-1/p$, [@Quarteroni06 Section 6.3]. A modified Newton method $$\begin{aligned}
\label{eqn:modnewton}
x_{k+1} = x_k - p\frac{f(x_k)}{f'(x_k)},\end{aligned}$$ can be seen to restore quadratic convergence. However, this requires knowledge of the multiplicity $p$ of the root which is generally [*a priori*]{} unknown.
Even if $p$ is unknown, it may be approximated in the course of the iterative process. One method that does just that is introduced in [@Quarteroni06 Section 6.6.2] whereby $x_{k+1} = x_k - p_k f(x_k)/f'(x_k)$ gives an approximate or adaptive modified Newton method with $p_0 = 1$ and for $k \ge 1$, $$p_k = \frac{x_{k-1} - x_{k-2}}{2x_{k-1}- x_k - x_{k-2}},$$ where $p_k$ is recomputed on each iteration where the convergence rate is sufficiently stable (see [@Quarteroni06 Program 56]). The method introduced here uses a different approximation to $p_k$, and will be shown to compare favorably to the adaptive method of [@Quarteroni06] in the numerical tests of Section \[sec:numerics\].
Finally it is remarked that another approach to quadratic convergence for non-simple roots discussed in for instance [@mathews89] is a modified Newton-Raphson method $$x_{k+1} = x_k - \frac{f(x_k)f'(x_k)}{(f'(x_k))^2 - f(x_k)f''(x_k)},$$ which bears close resemblence to Halley’s method [@ScTh95]. However, the computation of the second derivative may be considered unnecessarily laborious as it will be seen in Section \[sec:numerics\] that the superlinear convergence of the method introduced here converges very nearly as fast as the modified Newton method but without the [*a priori*]{} knowledge of the multiplicity of the zero.
Anderson accelerating Newton’s method.
======================================
To understand the derivation of the method , the Anderson acceleration algorithm for fixed-point iterations is next introduced. This method, which uses a history of the $m+1$ most recent iterates and update steps to define the next iterate, was introduced by D. G. Anderson in 1965 [@anderson65] in the context of integral equations. It has since increased in popularity and become known as an effective method for improving the convergence rate of fixed-point iterations $x_{k+1} = \phi(x_k)$, and is used in many applications in scientific computing [@EPRX19; @K18; @WaNi11]. The basic Anderson acceleration algorithm with depth $m$ (without damping) applied to the fixed-point problem $\phi(x) = x$ for $\phi: \mathbb R^n \rightarrow \mathbb R^n$, is shown below. To clarify how it is applied to a Newton iteration, the following notation is introduced. The fixed-point iteration may be written as $x_{k+1} = \phi(x_k) = x_k + (\phi(x_k)-x_k) = x_k + w_{k+1}$, where $w_{k+1} = \phi(x_k) - x_k$. Thus, if $\phi(x_k) = x_k - [f'(x_k)]^{-1}f(x_k)$ as in the Newton iteration , the update step is $w_{k+1} = -[f'(x_k)]^{-1}f(x_k)$, (or, $w_{k+1} = -f(x_k)/f'(x_k)$, in the special case of $f: {\mathbb{R}}{\rightarrow}{\mathbb{R}}$).
\[alg:anderson\] [**(Anderson iteration with depth $m$)**]{} Set depth $m \ge 0$. Choose $x_0$. Compute $w_1$. Set $x_1 = x_0 + w_1$.\
For $k = 1, 2, \ldots$, set $m_k = \min\{k,m\}$\
Compute $w_{k+1}$\
Set $F_k= \begin{pmatrix}(w_{k+1}-w_k) & \ldots & (w_{k-m+2} - w_{k-m+1})\end{pmatrix}$, and $E_k= \begin{pmatrix}(x_{k}-x_{k-1}) & \ldots & (x_{k-m+1} - x_{k-m})\end{pmatrix}$\
Compute $\gamma_{k} = \operatorname{argmin}_{\gamma \in {\mathbb{R}}^m} {\left\|{w_{k+1} - F_k \gamma}\right\|}$\
Set $x_{k+1} = x_k + w_{k+1} - \left(E_k + F_k \right)\gamma_{k}$
A discussion on what norm might be used for the optimization in Algorithm \[alg:anderson\], and how the minimization problem is solved, can be found in [@WaNi11]. In the present context of finding a zero of $f: \mathbb R \rightarrow \mathbb R$, one only needs consider depth $m=1$ ($m=0$ is the original fixed-point iteration), and the optimization step reduces to solving a linear equation.
Scalar Newton-Anderson {#subsec:an1}
----------------------
In the scalar case, $f: \mathbb R \rightarrow \mathbb R$, the optimization problem in Algorithm \[alg:anderson\] reduces to a linear equation for a single coefficient, $w_{k+1} - \gamma_k(w_{k+1}- w_k) = 0$, solved by $ \gamma_k = w_{k+1}/(w_{k+1}-w_k).$ Then the accelerated iterate $x_{k+1}$, for $k \ge 1$ is given by $$\begin{aligned}
\label{eqn:seclike}
x_{k+1}
= x_k + w_{k+1} - \frac{w_{k+1}}{w_{k+1}-w_k} ((x_k + w_{k+1} - (x_{k-1} + w_k))
= x_k -\frac{x_k - x_{k-1}}{w_{k+1}-w_k} w_{k+1}.\end{aligned}$$ If the fixed-point scheme $x_{k+1} = x_k + f(x_k)$ is used, then $w_{k+1} = f(x_k)$, which results in the secant method. If the Newton method is accelerated, then plugging in $w_{k+1} = -f(x_k)/f'(x_k)$ yields .
It is remarked here that for $f: \mathbb R^n \rightarrow \mathbb R^n$, (or a more general normed vector space) the Anderson Algorithm \[alg:anderson\] applied to $w_{k+1} = f(x_k)$ results in a method shown to be a type of multi-secant method [@eyert96; @FaSa09], as compared to the standard secant method in the scalar case. One of the motivations for looking at the scalar version of Algorithm \[alg:anderson\] applied to the Newton method was to understand the method in this simpler setting to gain insight into its use in a more general setting. As a result of this investigation, and as demonstrated in [@PoSw19], it was found for $f:{\mathbb{R}}^n \rightarrow {\mathbb{R}}^n$, the Newton-Anderson method can provide superlinear convergence to solutions of degenerate problems, those whose Jacobians are singular at a solution (and for which Newton converges only linearly), as well as nondegenerate problems (where Newton converges quadratically). This paper focuses on $f: {\mathbb{R}}\rightarrow {\mathbb{R}}$, and provides analytical and numerical results to characterize the scalar case. Next, the Newton-Anderson rootfinding method is summarized, then its convergence properties are analyzed in the section that follows.
\[alg:na1\] [**(Newton-Anderson rootfinding method)**]{}\
Choose $x_0$. Compute $w_1 = -f(x_0)/f'(x_0)$. Set $x_1 = x_0 + w_1$\
For $k = 1, 2, \ldots$\
Compute $w_{k+1}= -f(x_k)/f'(x_k)$ $$\text{Set} \quad x_{k+1} = x_k -\frac{x_k - x_{k-1}}{w_{k+1}-w_k}w_{k+1} \hspace{4.25in}$$
Rootfinding.
============
To give further insight into the main result, a trivial case is first considered. The Newton method finds the zero of $f(x) = ax-b$, or in monic form $f(x) = x-c$, exactly in one step. Similarly, it is easily seen that Algorithm \[alg:na1\] locates the zero of $f(x) = (x-c)^p$ with $p> 0$ and $p \ne 1$ after the first optimization step: that is, $x_2 = c$, if the operations are performed in exact arithmetic. The modified Newton method has a similar property: assuming $p$ is known, then given $x_0$, the first iterate $x_1 = x_0 - p(x_0-c)/p = c$. However, one might also show that for $f: \mathbb R^n \rightarrow \mathbb R^n$ defined by an $n \times n$ invertible matrix $A$, and $b \in \mathbb R^n$, the system of $n$ equations given by $f_i(x) = ((Ax - b)_i)^{p_i}$ for exponents $p_i > 0,$ is solved after the first full optimization step of Algorithm with depth $m$ applied to iteration if there are exactly $m$ distinct exponents $p_i$, $i = 1, \ldots, n$ (a numerical demonstration of this is shown in [@PoSw19]). Next, a more general scalar problem is considered. Suppose $c$ is a non-simple root of a function $f(x)$ expressed in the form $f(x) = (x-c)^p g(x)$, $ p > 1$, for some function $g$ which is assumed not to have a zero (or pole) in some neighborhood of $c$. The following lemma shows the Newton-Anderson rootfinding method approximates the modified Newton method ; and, it makes a precise statement regarding how $(x_{k+1}-x_k)/w_{k+1}= (x_k-x_{k-1})/(w_{k+1}-w_k),$ provides an approximation to the multiplicity of the zero of $f$ at $x=c$. The theorem that follows provides a local convergence analysis of Algorithm \[alg:na1\].
An alternative approach to the analysis might be to exploit the interpretation of as a secant method used to find the (simple) zero of $-f(x)/f'(x)$, yielding the usual order of convergence for the secant method, $(1 + \sqrt 5)/2$. The results that follow, however, give a direct proof that the method has an order of convergence of at least $(1+ \sqrt 5)/2$; and, show that it gives an accurate approximation to the multiplicity of the root (also demonstrated numerically in Section \[sec:numerics\]), which can be of use if deflation is used to find additional roots. To fix some notation for the remainder of this section, let $e(x) = c-x$, and let $\mathcal I_k = (\min\{x_k, x_{k-1}\}, \max\{x_k,x_{k-1}\})$.
\[lem:papprox\] Let $f(x) = (x-c)^p g(x)$ for $p > 1$ where $g: \mathbb R \rightarrow \mathbb R$ is a $C^2$ function for which both $g'(x)/g(x)$ and $g''(x)/g(x)$ are bounded in an open interval $\mathcal N$ containing $c$.
Define the constants $$\begin{aligned}
\label{eqn:defMg}
M_0
= \max_{x \in \mathcal N} \frac 1 p \left| \left(\frac{g'(x)}{g(x)}\right)^2
- \frac{g''(x)}{g(x)}\right|,
~\text{ and }~
M_1 = \max_{x \in \mathcal N} \frac 1 p \left| \frac{g'(x)}{g(x)} \right|.\end{aligned}$$
Then, if $x_{k-1},x_k \in \mathcal N_0 \coloneqq
\{ x \in \mathcal N: e(x)^2 < M_0^{-1} ~\text{ and }~ |e(x)| < M_1^{-1}\}$, the iterate $x_{k+1}$ given by Algorithm \[alg:na1\] satisfies $x_{k+1} = x_k + p_k w_{k+1}$ with $$\begin{aligned}
\label{eqn:papprox}
p_k = \left(p - 2 e(\eta_k) \frac{g'(\eta_k)}{g(\eta_k)}
+ \mathcal O(e(\eta_k)^2)\right),
~\text{ for some }~ \eta_k \in \mathcal I_k.\end{aligned}$$
The hypotheses on $g$ maintain that $g$ is reasonably smooth and does not have a zero in the vicinity of $c$.
The Newton update step is $w_k = -f(x_{k-1})/f'(x_{k-1})$ so writing $w_k = w(x_{k-1})$, the update step from Algorithm \[alg:na1\] reads as $$\begin{aligned}
\label{eqn:na1g-001}
x_{k+1} = x_k - \left( \frac{x_k - x_{k-1}}{ w(x_k) - w(x_{k-1})}\right) w_{k+1}
= x_k - p_k w_{k+1},\end{aligned}$$ with $p_k = (x_k - x_{k-1})/(w(x_k) - w(x_{k-1})).$ The aim is now to show $p_k \rightarrow p$ as $e(x_k) \rightarrow 0$.
For $f(x)$ given by $f(x) = (x-c)^p g(x)$, the first two derivatives are given by $$\begin{aligned}
\label{eqn:fderiv}
f'(x) &= (x-c)^{p-1}\big(p g(x) + (x-c) g'(x)\big)
\nonumber \\
f''(x) &= (x-c)^{p-2}\big(p(p-1) g(x) + 2 p (x-c)g'(x) + (x-c)^2 g''(x)\big).\end{aligned}$$ Writing $w(x_k)$ in terms of $f(x_k) = (x-c)^p g(x_k)$, and $f'(x_k)$ given by gives $w(x_{k}) = e(x_k)g(x_k)/(pg(x_k) - e(x_k)g'(x_k)),$ whose denominator is bounded away from zero for $x_k \in \mathcal N_0$. Then by the mean value theorem, there is an $\eta_k \in \mathcal \mathcal I_k$ for which $w(x_k) - w(x_{k-1}) = w'(\eta_k)(x_k - x_{k-1})$, by which reduces to $x_{k+1} = x_k - w_{k+1}/w'(\eta_k)$. Temporarily dropping the subscript on $\eta_k$ for clarity of notation, taking the derivative of $w(\eta) = -f(\eta)/f'(\eta)$ yields $$\frac{-1}{w'(\eta)} = \frac{f'(\eta)^2}{ (f'(\eta))^2 - f(\eta)f''(\eta)}.$$ Applying the expansions of $f'$ and $f''$ from , cancelling common factors of $e(x)^{p-2}$ and simplifying allows $$\begin{aligned}
\label{eqn:na1g-002}
\frac{-1}{w'(\eta)}
&= \frac{\big(p g(\eta) - e(\eta)g'(\eta)\big)^2}
{ \big(p g(\eta) - e(\eta)g'(\eta)\big)^2 -
g(\eta)\big(p(p-1)g(\eta) - 2pe(\eta)g'(\eta) + e(\eta)^2 g''(\eta)\big) }
\nonumber \\
& = \frac{p - 2 e(\eta) \frac{g'(\eta)}{g(\eta)}
+ \frac 1 p e(\eta)^2 \left(\frac{g'(\eta)}{g(\eta)} \right)^2}
{1 + \frac 1 p e(\eta)^2 \left(\left(\frac{g'(\eta)}{g(\eta)}\right)^2 -
\frac{g''(\eta)}{g(\eta)} \right)}.\end{aligned}$$ By hypothesis, $x_{k}$ and $x_{k-1}$ are in $\mathcal N_0$ which implies $\eta_k \in \mathcal I_k \subset \mathcal N_0$, so the denominator of the right hand side of is of the form $1 + \alpha$ with $|\alpha| < 1$. Expanding the denominator in a geometric series shows that $$\begin{aligned}
-\frac{1}{w'(\eta_k)} & = p - 2e(\eta_k)\frac{g'(\eta_k)}{g(\eta_k)}
+ \mathcal O(e(\eta_k)^2).\end{aligned}$$ This shows there is an $\eta_k \in \mathcal I_k$ for which the update of Algorithm \[alg:na1\] satisfies .
The adaptive method of [@Quarteroni06 (6.39)-(6.40)] and the current method both take the form $x_{k+1} = x_k + p_k w_{k+1}$, so it makes sense to compare the two expressions for $p_k$. Letting $\{y_k\}$ represent the sequence generated by [@Quarteroni06 (6.39)-(6.40)], and setting $w_k = -f(y_{k-1})/f'(y_{k-1}),$ the resulting iteration may be written $$\begin{aligned}
y_{k+1} = y_k + \frac{y_{k-1}-y_{k-2}}{ (y_k - y_{k-1}) - (y_{k-1} - y_{k-2})}w_{k+1}
=y_k + \frac{y_{k-1}-y_{k-2}}{ p_{k-1}w_k - p_{k-2}w_{k-1}} w_{k+1}, \end{aligned}$$ which differs from update of Algorithm \[alg:na1\] both in terms of the set of iterates used in the numerator of $p_k$: $\{y_{k-1},y_{k-2}\}$ compared to $\{x_k,x_{k-1}\}$; and, in the form of the denominator $p_{k-1}w_k - p_{k-2}w_{k-1}$ as opposed to $w_{k+1}-w_k$. As such, $p_k$ of the adaptive scheme appears more complicated to analyze as an approximation to $p$, and the two methods will only be compared numerically, in the two examples of Section \[sec:numerics\].
The previous Lemma \[lem:papprox\] shows the update step of Algorithm \[alg:na1\] is of the form $x_{k+1} = x_k + p_k w_k$ where $p_k \rightarrow p$ so long as $x_k \rightarrow c$. The next theorem shows that $x_k \rightarrow c$, and that the order of convergence is greater than one (and, in fact, no worse than $(1 + \sqrt 5)/2$).
\[thm:na1con\] Let $f(x) = (x-c)^p g(x)$, for $p > 1$ where $g: \mathbb R \rightarrow \mathbb R$ is a $C^2$ function for which both $g'(x)/g(x)$ and $g''(x)/g(x)$ are bounded in an open interval $\mathcal N$ containing $c$. Define the interval $\mathcal N_1 = \{ x \in \mathcal N_0 : |e(x)| < 1/(2M_1)\}$, where $\mathcal N_0$ and $M_1$ are given in the statement of Lemma \[lem:papprox\]. Then there exists an interval $\mathcal N_\ast \subseteq \mathcal N_1$ such that if $x_{k-1},x_k \in \mathcal N_\ast$, all subsequent iterates remain in $\mathcal N_\ast$ and the iterates defined by Algorithm \[alg:na1\] converge superlinearly to the root $c$.
Suppose $x_k, x_{k-1} \in \mathcal N_1$. Let $p_k = (x_k - x_{k-1})/(w_{k+1}- w_k)$. Then the error in iterate $x_{k+1}$ satisfies $$\begin{aligned}
\label{eqn:tc-001}
e(x_{k+1}) = c -(x_k - p_k w_{k+1}) = e(x_k) + p_kw_{k+1}.\end{aligned}$$ Similarly to the computations of the previous lemma $$w_{k+1} = -\frac{f(x_k)}{f'(x_k)} = \frac{e(x_k) g(x_k)}{p g(x_k) - e(x_k)g'(x_k)},$$ which together with shows $$\begin{aligned}
\label{eqn:tc-002}
e(x_{k+1}) = e(x_k)\left(1 - \frac{p_k g(x_k)}{pg(x_k) - e(x_k) g'(x_k)} \right)
= e(x_k)\left(1 - \frac{p_k/p }{1 - \frac{1}{p}e(x_k) \frac{g'(x_k)}{g(x_k)}} \right).\end{aligned}$$ For $x_k \in \mathcal N_0$ the denominator of can be expanded as a geometric series to obtain $$\begin{aligned}
\label{eqn:tc-003}
e(x_{k+1})
& = e(x_k)\left(1 - \frac{p_k}{p}
\sum_{j = 0}^{\infty }\left( \frac{1}{p}e(x_k) \frac{g'(x_k)}{g(x_k)} \right)^j
\right).\end{aligned}$$ For $x_k$ and $x_{k-1}$ in $\mathcal N_1 \subset \mathcal N_0$, the results of Lemma \[lem:papprox\] hold, and applying the resulting expansion of $p_k$ to shows $$\begin{aligned}
\label{eqn:tc-004}
e(x_{k+1}) & =
e(x_k) \left\{ 1- \left(1 - \frac{2}{p} e(\eta_k) \frac{g'(\eta_k)}{g(\eta_k)}
+ \mathcal O(e(\eta_k)^2)\right)
\sum_{j = 0}^{\infty }\left( \frac{1}{p}e(x_k) \frac{g'(x_k)}{g(x_k)} \right)^j
\right\}
\nonumber \\& =
e(x_k) \left\{ 1- \left(1 - \frac{2}{p} e(\eta_k) \frac{g'(\eta_k)}{g(\eta_k)}
+ \mathcal O(e(\eta_k)^2)\right)
\left( 1 + \frac{1}{p}e(x_k) \frac{g'(x_k)}{g(x_k)} +
\left( \frac{1}{p}e(x_k) \frac{g'(x_k)}{g(x_k)} \right)^2
\sum_{j = 0}^{\infty }\left( \frac{1}{p}e(x_k) \frac{g'(x_k)}{g(x_k)} \right)^j
\right) \right\}
\nonumber \\& =
e(x_k) \left\{ 1- \left(1 - \frac{2}{p} e(\eta_k) \frac{g'(\eta_k)}{g(\eta_k)}
+ \mathcal O(e(\eta_k)^2)\right)
\left( 1 + \frac{1}{p}e(x_k) \frac{g'(x_k)}{g(x_k)} + \mathcal{O}(e(x_k)^2)
\right) \right\},\end{aligned}$$ for some $\eta_k \in \mathcal I_k$. Multiplying out terms in shows the error satisfies $$\begin{aligned}
\label{eqn:tc-005}
e(x_{k+1})
= e(x_k)e(\eta_k) \frac 2 p\frac{g'(\eta_k)}{g(\eta_k)}
- e(x_k)^2 \frac1 p \frac{g'(x_k)}{g(x_k)} + \mathcal O (e(x_k) e(\eta_k)^2),\end{aligned}$$ which, for $x_k, x_{k-1}$ in an interval $\mathcal N_\ast \subseteq \mathcal N_1$, shows the iterates stay in $\mathcal N_\ast$, and converge superlinearly to $c$.
The standard secant method, when used to approximate a simple root, has an order of convergence of $(1 + \sqrt 5)/2$, and the lowest order term in its error expansion is multiple of $e(x_k) e(x_{k-1})$. From , the lowest order term in the error expansion of Newton-Anderson, when approaching a higher-multiplicity root, is a multiple of $e(x_k) e(\eta_k)$, where $\eta_k$ (from a mean value theorem) is between $x_k$ and $x_{k-1}$. This implies the order of convergence for the method is at least $(1+\sqrt 5)/2$, and generally less than 2 unless $g'/g {\rightarrow}0$.
Numerical examples {#sec:numerics}
==================
In this section, some numerical examples are given to illustrate the efficiency of the Newton-Anderson rootfinding method. In these examples, the proposed method, Algorithm \[alg:na1\], is compared with the Newton method , the modified Newton method (assuming [*a priori*]{} knowledge of the multiplicity $p$ of the zero), and the adaptive method of [@Quarteroni06 Section 6.6.2], implemented as described therein. Additionally, results are shown for the secant method (using $x_0$ as stated, and $x_{-1} = x_0 - 10^{-3}$), and the predictor-corrector (PC) Newton method of [@McWo14]. The secant method is included because, as shown in , scalar Newton-Anderson can be interpreted as a secant method applied to the Newton update step, or a secant method to find the zero of $w(x) = -f(x)/f'(x)$. The predictor-corrector method (which was designed to accelerate Newton’s method for simple roots only) comparatively demonstrates the robustness of Newton-Anderson, which performs comparably in each case tested. In contrast, the predictor-corrector method outperforms the Newton and secant methods in the first example, and is outperformed by both of them in the second.
### Example 1 {#subsubsec:exQ}
The first example is taken from [@Quarteroni06 Example 6.11]. The problem tested is finding the zero of $f(x) = (x^2-1)^q \log x$, which has a zero of multiplicity $p = q+1$ at $x=1$. The condition to exit the iterations are those from [@Quarteroni06 Example 6.11], namely $|x_{k+1}- x_k| < 10^{-10}$. The iteration counts starting from $x_0 = 0.8$ (for standard, adaptive and modifed Newton methods) agree with those stated in [@Quarteroni06]. Tables \[tab:Q6.11-2\]-\[tab:Q6.11-6\] show the respective iteration counts for $q = \{2, 6\}$ for each method starting from initial iterates $x_0 = \{0.8, 2, 10\}$. The final value of $p_k$ is shown in parentheses after the iteration count for the Newton-Anderson and adaptive Newton methods.
Consistent with the analysis from Lemma \[lem:papprox\] and Theorem \[thm:na1con\], the performance of the Newton-Anderson method is linked to its accurate approximation of the root’s multiplicity. For the result below in Tables \[tab:Q6.11-2\]-\[tab:Q6.11-6\], the final value of $p_k$ in Newton-Anderson was accurate to ${\mathcal{O}}(10^{-8})$, except for the last experiment in Table \[tab:Q6.11-2\], where it was ${\mathcal{O}}(10^{-7})$.
### Example 2 {#subsubsec:exexp}
The second example concerns finding the zero of $f(x) = (x-2)^6\exp(-(x-2)^2/2)$, which has a zero of multiplicity 6 at $x=2$. The first 30 differences between consecutive iterates $|x_{k+1}-x_k|$ are shown below in Figure \[fig:exptest6\] starting each iteration from the initial $x_0 = \{0, 1\}$.
For this problem the modified Newton method converges in two fewer iterations than Newton-Anderson (12 less than the adaptive method) starting from $x_0 = 1$, however it fails to converge starting from $x_0 = 0$ (it is attracted to the asymptotic zero as $x \rightarrow \infty$). The Newton-Anderson method has comparable performance and converges to the same zero in both cases, as do the remaining methods, although their convergence is substantially slower.
### Numerical order of convergence
To demonstrate the order of convergence of Newton-Anderson for some instances of each problem, the sequence of approximate convergence orders $q_k = \log|x_k -c|/\log|x_{k-1}-c|$ is shown below in Table \[tab:qk\].
The convergence orders of the first example behave essentially as predicted, generally staying in the range $((1 + \sqrt 5)/2,2)$, whereas the approximate convergence orders from the second example are generally above 2. This can be easily understood however as the constant in front of the lowest order term of is a multiple of $g'(\eta_k)/g(\eta_k)$, which for $g(x) =\exp(-(x-2)^2/2)$, goes to zero as $x$ approaches $c=2$.
Conclusion {#sec:conclusion}
==========
The purpose of this discussion is to understand the iteration derived from applying Anderson acceleration to a scalar Newton iteration. The resulting Newton-Anderson rootfinding method is shown to approximate a modified Newton method that yields quadratic convergence to non-simple roots. The presented method does not require [*a priori*]{} knowledge of the multiplicity of the root, nor does it require additional function evaluations or the computation of additional derivatives. The convergence analysis of the Newton-Anderson method demonstrates a local order of convergence of at least $(1 + \sqrt 5)/2$. The numerical examples demonstrate this and show iteration counts close to that of the modified Newton method, and with less sensitivity to the initial guess. In comparison with the adaptive Newton method of [@Quarteroni06] designed to accomplish the same task, the implementation is simpler as additional heuristics are not involved, and on the examples tested, convergence is faster as a more accurate approximation of the root’s multiplicity is attained. Altogether this makes the Newton-Anderson rootfinding method worthy of consideration in situations involving non-simple roots.
Acknowledgements {#acknowledgements .unnumbered}
================
The author was partially supported by NSF DMS 1852876.
|
---
abstract: 'We show that the local magnetization in the massive boundary Ising model on the half-plane with boundary magnetic field satisfies second order linear differential equation whose coefficients are expressed through Painleve function of the III kind.'
author:
- |
Oleg Miroshnichenko[^1]\
\
[ ]{}*Bogolyubov Institute for Theoretical Physics, Kiev 03143, Ukraine*
title: Differential equation for Local Magnetization in the Boundary Ising Model
---
Introduction
============
In the work [@FZ1] a very simple and elegant derivation of the famous Painleve equations for the spin-spin correlation function in the scaling Ising model with zero magnetic field was given. The approach used in that work was also applied in [@FZ2] to derive finite volume form factors of spin field in the Ising theory and in [@DF] to derive the differential equation for spin-spin correlation functions in the Ising theory on a pseudosphere. Here we apply this approach to derive differential equation for local magnetization (i. e. one-point correlation function of spin field) in the boundary Ising model on the half-plane with boundary magnetic field. It turns out to be a second order linear differential equation whose coefficients are expressed through Painleve function of the III kind. Supplied with appropriate boundary condition it uniquely defines the local magnetization as a function of the distance to the boundary. Besides being interesting in itself, such a representation for local magnetization may be more convenient for numerical calculation in comparison with conventional form factor expansion, especially in the short distance region.
As is well known [@C], there are two essentially different types of conformal boundary conditions (b. c.) in conformal Ising field theory. The so called “free” b. c. corresponds to the universality class represented by the lattice Ising model with unrestricted spins on the boundary. The so called “fixed” b. c. corresponds to the universality class represented by the lattice Ising model with boundary spins all fixed in the same direction (“$+$” or “$-$”, so there is more precisely two different “fixed” b. c.). All this b. c. correspond to the fixed points of the boundary renormalization group flow. The most general local b. c. in the Ising field theory is the “free” b. c. perturbed by the boundary spin operator (which is the only non-trivial relevant boundary operator in the case of “free” b. c.). This b. c. corresponds to the renormalization group flow from “free” b. c. towards one of the “fixed” b. c. [@AL]. More generally, one may consider conformal Ising field theory with “free” b. c., perturbed by both boundary spin operator and bulk thermal operator [@GhZ]. This theory describes the continuum limit in the vicinity of the critical point of the lattice Ising model with zero magnetic field in the bulk and with boundary magnetic field being suitably rescaled. Let us briefly list known results about the local magnetisation in this theory defined on the half-plane.
The form factor expansion for local magnetization $\bar{\sigma}\left(
t\right) $ was written down in [@KLCM] using exact expression for boundary state obtained in [@GhZ]:$$\bar{\sigma}\left( t\right) =\sigma _{0}\exp \left(
\dsum\limits_{k=1}^{\infty }\frac{1}{k}f_{k}\right) \label{1a}$$$$f_{k}=-\frac{1}{\pi ^{2}}\int\limits_{0}^{\infty }du_{1}\ldots
\int\limits_{0}^{\infty }du_{k}\dprod\limits_{l=1}^{k}\frac{\func{ch}u_{l}-1}{\func{ch}u_{l}+\func{ch}u_{l+1}}\left( \frac{\func{ch}u_{l}+1-\lambda }{\func{ch}u_{l}-1+\lambda }\right) e^{-t\func{ch}u_{l}} \label{1b}$$$$t=2m\text{y,\qquad }\lambda =\frac{4\pi h^{2}}{m}\text{,\qquad }\sigma
_{0}=2^{\frac{1}{12}}e^{-\frac{1}{8}}A^{\frac{3}{2}}m^{\frac{1}{8}}
\label{1c}$$where $m\sim T-T_{c}$ is the mass of a particle, $h$ - scaling boundary magnetic field, y - distance from the boundary, $\sigma _{0}$ - magnetization on the infinite plane, $A=1.28243\ldots $ is Glaisher’s constant. Here and later on we always consider the low temperature phase $T<T_{c}$, unless it is specially pointed out. It is also implied conformal normalization of spin field:$$\left\vert x-x^{\prime }\right\vert ^{\frac{1}{4}}\left\langle \sigma \left(
x\right) \sigma \left( x^{\prime }\right) \right\rangle \rightarrow 1\text{,\quad as }x\rightarrow x^{\prime }$$With this normalization $\bar{\sigma}\left( t,\lambda \right) \rightarrow
\sigma _{0}$ as $t\rightarrow \infty $. The expansion (\[1a\])-(\[1c\]) was first obtained in [@B2] from lattice model calculations. It was also shown in [@B1],[@B2] that in the cases of “free” ($h=0$) and “fixed” ($h\rightarrow \pm \infty $) b. c. local magnetization can be expressed through Painleve function of the III kind:$$\bar{\sigma}_{free}\left( t\right) =\sigma _{0}\exp \left\{ \frac{1}{4}\varphi \left( t\right) +\frac{1}{4}\int_{t}^{\infty }\left[ e^{-\varphi
\left( r\right) }-1+\frac{r}{2}\left( \func{sh}^{2}\varphi \left( r\right)
-\left( \varphi ^{\prime }\left( r\right) \right) ^{2}\right) \right]
dr\right\} \label{3}$$$$\bar{\sigma}_{fixed}\left( t\right) =\sigma _{0}\exp \left\{ -\frac{1}{4}\varphi \left( t\right) +\frac{1}{4}\int_{t}^{\infty }\left[ 1-e^{\varphi
\left( r\right) }+\frac{r}{2}\left( \func{sh}^{2}\varphi \left( r\right)
-\left( \varphi ^{\prime }\left( r\right) \right) ^{2}\right) \right]
dr\right\} \label{4}$$where $\varphi \left( r\right) $ is the solution of radial sinh-Gordon equation:$$\varphi ^{\prime \prime }+\frac{1}{r}\varphi ^{\prime }=\frac{1}{2}\func{sh}2\varphi \label{5}$$satisfying asymptotic conditions:$$\varphi \left( r\right) =-\ln \left( -\frac{1}{2}r\Omega \right) +\emph{O}\left( r^{4}\Omega ^{2}\right) \text{, \quad as }r\rightarrow 0\text{,\quad }\Omega =\ln \left( \frac{e^{\gamma }}{8}r\right) \label{6}$$
$$\varphi \left( r\right) =\frac{2}{\pi }K_{0}\left( r\right) +\emph{O}\left(
e^{-3r}\right) \text{,\quad as }r\rightarrow \infty \label{7}$$
where $\gamma $ is the Euler’s constant, $K_{0}\left( x\right) $ is the modified Bessel function of zeroth order. As is known, $\varphi \left(
x\right) $ is related to Painleve function of the III kind $\eta \left(
x\right) $ as $\eta \left( x\right) =e^{-\varphi \left( 2x\right) }$. More about this function see [@MCWTB],[@MCTW].
In the case when bulk is critical ($m=0$) it was shown in [@ChZ] that:$$\bar{\sigma}\left( \text{y}\right) =h2^{\frac{5}{4}}\pi ^{\frac{1}{2}}\left(
2\text{y}\right) ^{\frac{3}{8}}\Psi \left( 1/2,1,8\pi h^{2}\text{y}\right)
\label{8}$$where $$\Psi \left( a,c,x\right) =\frac{1}{\Gamma \left( a\right) }\int_{0}^{\infty
}e^{-xt}t^{a-1}\left( 1+t\right) ^{c-a-1}dt \label{9}$$is a solution of degenerate hypergeometric equation.
The qualitative behavior of $\bar{\sigma}\left( t,\lambda \right) $ is well understood [@KLCM]. On the whole interval $\left( 0,\infty \right) $ $\bar{\sigma}_{free}\left( t\right) $ monotonically increases and $\bar{\sigma}_{fixed}\left( t\right) $ monotonically decreases, both approaching $\sigma
_{0}$ as $t\rightarrow \infty $. For small $t$:[^2]$$\bar{\sigma}_{free}\left( t\right) \sim t^{\frac{3}{8}} \label{10}$$$$\bar{\sigma}_{fixed}\left( t\right) \sim t^{-\frac{1}{8}} \label{11}$$For $0<\lambda <2$ $\bar{\sigma}\left( t,\lambda \right) $ remains monotonically increasing. Its values near the boundary are somewhat enhanced by the presence of boundary magnetic field, the leading term of its short distance asymptotic become dressed by logarithm:$$\bar{\sigma}\left( t\right) \sim t^{\frac{3}{8}}\ln t \label{12}$$For $\lambda >2$ it possesses a maximum in some point. As $\lambda
\rightarrow \infty $ this maximum turns into a very sharp peak located in the region $t\sim \lambda ^{-1}$ near the boundary, its shape being described by (\[8\]), (\[9\]). For $t\ll \lambda ^{-1}$ $\bar{\sigma}\left( t,\lambda \right) $ behaves as (\[12\]), while for $t\gg \lambda
^{-1}$ its behavior coincides with one under “fixed” b. c. (\[4\]). This dependence reflects the renormalization group cross-over between “free” and “fixed” b. c.
The main result of this paper is that for arbitrary $\lambda $:$$\bar{\sigma}\left( t,\lambda \right) =u\left( t,\lambda \right) \bar{\sigma}_{free}\left( t\right) \label{15}$$where $\bar{\sigma}_{free}\left( t\right) $ is given by (\[3\]), and $u\left( t,\lambda \right) $ is the solution of differential equation:$$u^{\prime \prime }-\left( \varphi ^{\prime }-\func{ch}\varphi +\lambda
\right) u^{\prime }+\frac{1}{2}\lambda \left( \varphi ^{\prime }-\func{ch}\varphi +1\right) u=0 \label{16}$$satisfying asymptotic condition:$$u\left( t\right) =1+\emph{O}\left( t^{-\frac{1}{2}}e^{-t}\right) \text{,\quad as }t\rightarrow \infty \label{17}$$(Here $\varphi \left( t\right) $ is the same function as in (\[3\]), ([4]{}) and the strokes stand for detivatives with respect to $t$.)
Let us make some remarks on the equation (\[16\]). One can see that when $\lambda =0$, the only solution of (\[16\]) satisfying (\[17\]) is $u\left( t\right) =1$. When $\lambda \rightarrow \infty $ (\[16\]) turns into a first order differential equation which upon integrating and fixing integration constant with the help of (\[17\]) yields (\[4\]). In the massless limit ($t\rightarrow 0$, $\lambda \rightarrow \infty $, $t\lambda $ kept fixed) (\[16\]) turns into a degenerate hypergeometric equation. Its solution can be fixed by “sewing” its asymptotic as $t\lambda \rightarrow
\infty $ with asymptotic of (\[4\]) as $t\rightarrow 0$, and this yields (\[8\]), (\[9\]). The fact that in massless limit we reproduce the result of [@ChZ] is not very surprising because the approach we used to derive (\[15\]), (\[16\]) is a generalization of one used in [@ChZ]. Concerning the relation between the form factor expansion (\[1a\]), ([1b]{}), (\[1c\]) and our result (\[15\]), (\[16\]) we just note that it seems to be very difficult to show directly that (\[1a\]), (\[1b\]) satisfy (\[15\]), (\[16\]). In any case, it is beyond the analytic abilities of the author.
Being second order linear differential equation, (\[16\]) possesses two linearly independent solutions. Their asymptotics as $t\rightarrow \infty $ are $u_{1}\left( t\right) \sim 1$ and $u_{2}\left( t\right) \sim e^{\left(
\lambda -1\right) t}$. Hence, for $\lambda >1$ the condition that $u\left(
t\right) \rightarrow 1$ as $t\rightarrow \infty $ is sufficient to fix the solution uniquely. For $\lambda \leq 1$ more strict condition (\[17\]) is required, which follows from form factor expansion (\[1a\]), (\[1b\]). Another linearly independent solution in this case also has physical meaning. As explained in [@GhZ], for $\lambda <1$ there exists metastable state characterized by asymptotic behavior $\bar{\sigma}\left(
t\right) \rightarrow -\sigma _{0}$ as $t\rightarrow \infty $ and corresponding to the boundary bound state in the hamiltonian picture with “space” being half-line and “time” axis being parallel to the boundary. The local magnetization $\bar{\sigma}_{1}\left( t,\lambda \right) $ in this state can be obtained from $\bar{\sigma}\left( t,\lambda \right) $ by analytic continuation $h\rightarrow -h$. Clearly, it is also a solution of (\[16\]). As it was shown in [@SchE] its asymptotic as $t\rightarrow
\infty $ is:$$\bar{\sigma}_{1}\left( t,\lambda \right) =-\sigma _{0}+\sigma _{0}\left(
\frac{\lambda }{2-\lambda }\right) ^{\frac{1}{2}}e^{-\left( 1-\lambda
\right) t}+\frac{\sigma _{0}}{4\sqrt{2\pi }}\left( \frac{2}{\lambda }-1\right) t^{-\frac{3}{2}}e^{-t}+\emph{o}\left( t^{-\frac{3}{2}}e^{-t}\right)
\label{18}$$The presence of exponential term $\sim e^{-\left( 1-\lambda \right) t}$ in (\[18\]) agrees with (\[16\]).
In the rest of the paper we present the details of our derivation of ([15]{}), (\[16\]).
Ising field theory in the bulk
==============================
In this section we briefly recall some well known facts [@FZ1] about the structure of the Ising field theory in the bulk needed for further computations. As is known the Ising field theory in zero magnetic field is equivalent to the free Majorana fermion theory with euclidean action:$$S=\frac{1}{2\pi }\int \left( \psi \bar{\partial}\psi +\bar{\psi}\partial
\bar{\psi}-im\bar{\psi}\psi \right) d^{2}x \label{1-1}$$Here we have assumed that the theory is defined on an infinite plane $\mathbb{R}
^{2}$, whose points $x$ are labelled by cartesian coordinates $\left( \text{x},\,\text{y}\right) =\left( \text{x}\left( x\right) ,\,\text{y}\left(
x\right) \right) $, and $d^{2}x\equiv d$x$\,d$y. Complex coordinates are defined as $z\left( x\right) =\,$x$\,+\,i$y, $\bar{z}\left( x\right) =\,$x $+\,i$y, and the derivatives $\partial ,\bar{\partial}$ in (\[1-1\]) stand for $\partial _{z}=\frac{1}{2}\left( \partial _{\text{x}}-i\partial _{\text{y}}\right) $ and $\partial _{\bar{z}}=\frac{1}{2}\left( \partial _{\text{x}}+i\partial _{\text{y}}\right) $ respectively. The ciral components $\psi ,$ $\bar{\psi}$ of fermi field satisfy Dirac’s equations:$$\bar{\partial}\psi =-\frac{im}{2}\bar{\psi}\text{,\qquad }\partial \bar{\psi}=\frac{im}{2}\psi \label{1-2}$$Their normalization in the action (\[1-1\]) corresponds to the following short-distance limit of the operator products$$z\psi \left( x\right) \psi \left( 0\right) \rightarrow 1,\qquad \bar{z}\bar{\psi}\left( x\right) \bar{\psi}\left( 0\right) \rightarrow 1,\qquad \text{as
}x\rightarrow 0 \label{1-2a}$$
The order $\sigma \left( x\right) $ and disorder $\mu \left( x\right) $ fields are semi-local with respect to the fermi fields; the products$$\psi \left( x\right) \sigma \left( 0\right) ,\qquad \psi \left( x\right) \mu
\left( 0\right) ,\qquad \bar{\psi}\left( x\right) \sigma \left( 0\right)
,\qquad \bar{\psi}\left( x\right) \mu \left( 0\right) \label{1-3}$$acquire a minus sign when the point $x$ is taken around zero point. The fields $\psi \left( x\right) $ and $\bar{\psi}\left( x\right) $ in the products (\[1-3\]) can be expanded in the complete set of solutions of Dirac’s equations (\[1-2\]) having this monodromy property:$$\left(
\begin{array}{c}
\psi \left( x\right) \\
\bar{\psi}\left( x\right)\end{array}\right) =\sum_{n\in
\mathbb{Z}
}a_{n}\left(
\begin{array}{c}
u_{-n}\left( x\right) \\
\bar{u}_{-n}\left( x\right)\end{array}\right) +\bar{a}_{n}\left(
\begin{array}{c}
v_{-n}\left( x\right) \\
\bar{v}_{-n}\left( x\right)\end{array}\right) \label{1-4}$$where$$\left(
\begin{array}{c}
u_{n}\left( x\right) \\
\bar{u}_{n}\left( x\right)\end{array}\right) =\left( \frac{m}{2}\right) ^{\frac{1}{2}-n}\Gamma \left( n+\frac{1}{2}\right) \left(
\begin{array}{c}
e^{i\left( n-\frac{1}{2}\right) \theta }I_{n-\frac{1}{2}}\left( mr\right) \\
-ie^{i\left( n+\frac{1}{2}\right) \theta }I_{n+\frac{1}{2}}\left( mr\right)\end{array}\right) \label{1-5}$$$$\left(
\begin{array}{c}
v_{n}\left( x\right) \\
\bar{v}_{n}\left( x\right)\end{array}\right) =\left( \frac{m}{2}\right) ^{\frac{1}{2}-n}\Gamma \left( n+\frac{1}{2}\right) \left(
\begin{array}{c}
ie^{-i\left( n+\frac{1}{2}\right) \theta }I_{n+\frac{1}{2}}\left( mr\right)
\\
e^{-i\left( n-\frac{1}{2}\right) \theta }I_{n-\frac{1}{2}}\left( mr\right)\end{array}\right) \label{1-6}$$(here $r$, $\theta $ are polar coordinates, i. e. $z=re^{i\theta }$, $\bar{z}=re^{-i\theta }$ and $I_{\nu }$ are modified Bessel functions). The coefficients $a_{n}$, $\bar{a}_{n}$ in (\[1-4\]) are understood as operators acting on the space of fields.
It can be easily shown that for any two solutions $\Psi _{1}=\left( \psi
_{1}\left( x\right) ,\bar{\psi}_{1}\left( x\right) \right) $, $\Psi
_{2}=\left( \psi _{2}\left( x\right) ,\bar{\psi}_{2}\left( x\right) \right) $ of Dirac’s equations which change sign after the point $x$ is taken around zero the integral$$\left( \Psi _{1},\Psi _{2}\right) =\frac{1}{2\pi i}\oint\limits_{C_{0}}\psi
_{1}\left( x\right) \psi _{2}\left( x\right) dz-\bar{\psi}_{1}\left(
x\right) \bar{\psi}_{2}\left( x\right) d\bar{z} \label{1-6-1}$$over a contour $C_{0}$ encircling zero (in counter-clockwise direction) does not change under continuous deformation of $C_{0}$ and therefore defines a bilinear form on the space of such solutions. The solutions $U_{n}=\left(
u_{n},\bar{u}_{n}\right) $ and $V_{n}=\left( v_{n},\bar{v}_{n}\right) $ satisfy the following orthogonality properties with respect to this bilinear form:$$\left( U_{n},U_{m}\right) =\delta _{n+m,0}\text{ ,\qquad }\left(
V_{n},V_{m}\right) =\delta _{n+m,0}\text{ ,}\qquad \left( U_{n},V_{m}\right)
=0 \label{1-6-2}$$Let us also write down the following differentiation formulas which are useful in computations with $U_{n}$ and $V_{n}$:$$\begin{aligned}
\partial U_{n} &=&\left( n-\frac{1}{2}\right) U_{n-1}\text{ ,\qquad }\bar{\partial}U_{n}=\frac{m^{2}}{2\left( 2n+1\right) }U_{n+1} \notag \\[0.05in]
\partial V_{n} &=&\frac{m^{2}}{2\left( 2n+1\right) }V_{n+1}\text{ ,\qquad }\bar{\partial}V_{n}=\left( n-\frac{1}{2}\right) V_{n-1} \label{1-6-3}\end{aligned}$$(here we denote $\partial U_{n}\equiv \left( \partial u_{n}\left( x\right)
,\partial \bar{u}_{n}\left( x\right) \right) $, etc.).
Using relations (\[1-6-2\]) one can express operators $a_{n}$, $\bar{a}_{n} $ in terms of contour integrals:$$a_{n}=\frac{1}{2\pi i}\oint\limits_{C_{0}}u_{n}\left( x\right) \psi \left(
x\right) dz-\bar{u}_{n}\left( x\right) \bar{\psi}\left( x\right) d\bar{z}
\label{1-6-4}$$$$\bar{a}_{n}=\frac{1}{2\pi i}\oint\limits_{C_{0}}v_{n}\left( x\right) \psi
\left( x\right) dz-\bar{v}_{n}\left( x\right) \bar{\psi}\left( x\right) d\bar{z} \label{1-6-5}$$This representation can be used to show that they satisfy canonical commutation relations:$$\left\{ a_{n},a_{m}\right\} =\delta _{n+m,0}\text{ ,}\qquad \left\{ \bar{a}_{n},\bar{a}_{m}\right\} =\delta _{n+m,0}\text{ ,}\qquad \left\{ a_{n},\bar{a}_{m}\right\} =0 \label{1-7}$$The fields $\sigma $ and $\mu $ are “primary” with respect to the algebra (\[1-7\]), i. e. they satisfy relations:$$a_{n}\sigma =0\text{ ,}\qquad \bar{a}_{n}\sigma =0\text{ ,\qquad }a_{n}\mu =0\text{ ,\qquad }\bar{a}_{n}\mu =0 \label{1-8}$$for $n>0$, as well as$$\begin{aligned}
a_{0}\sigma &=&\frac{\omega }{\sqrt{2}}\mu \text{ ,}\qquad a_{0}\mu =\frac{\bar{\omega}}{\sqrt{2}}\sigma \notag \\[0.05in]
\bar{a}_{0}\sigma &=&\frac{\bar{\omega}}{\sqrt{2}}\mu \text{ ,}\qquad \bar{a}_{0}\mu =\frac{\omega }{\sqrt{2}}\sigma \label{1-9}\end{aligned}$$where $\omega =e^{i\pi /4}$ and $\bar{\omega}=e^{-i\pi /4}$. These equations define the fields $\sigma $ and $\mu $ up to normalization. In what follows we will assume conformal normalization of fields $\sigma $ and $\mu $:$$\left\vert x\right\vert ^{\frac{1}{4}}\sigma \left( x\right) \sigma \left(
0\right) \rightarrow 1\text{ ,\qquad }\left\vert x\right\vert ^{\frac{1}{4}}\mu \left( x\right) \mu \left( 0\right) \rightarrow 1\text{ ,\qquad as }x\rightarrow 0 \label{1-11}$$As it is shown in [@FZ1], first and second order descendants of $\sigma $ and $\mu $ with respect to the algebra $a_{n}$, $\bar{a}_{n}$ are expressed in terms of coordinate derivatives of $\sigma $ and $\mu $:$$\begin{aligned}
a_{-1}\sigma &=&\frac{\omega }{\sqrt{2}}4\partial \mu \text{ ,}\qquad
a_{-1}\mu =\frac{\bar{\omega}}{\sqrt{2}}4\partial \sigma \notag \\[0.05in]
\bar{a}_{-1}\sigma &=&\frac{\bar{\omega}}{\sqrt{2}}4\bar{\partial}\mu \text{
,}\qquad \bar{a}_{-1}\mu =\frac{\omega }{\sqrt{2}}4\bar{\partial}\sigma
\label{1-12}\end{aligned}$$$$\begin{aligned}
a_{-2}\sigma &=&\frac{\omega }{\sqrt{2}}\frac{8}{3}\partial ^{2}\mu \text{ ,}\qquad a_{-2}\mu =\frac{\bar{\omega}}{\sqrt{2}}\frac{8}{3}\partial ^{2}\sigma
\notag \\[0.05in]
\bar{a}_{-2}\sigma &=&\frac{\bar{\omega}}{\sqrt{2}}\frac{8}{3}\bar{\partial}^{2}\mu \text{ ,}\qquad \bar{a}_{-2}\mu =\frac{\omega }{\sqrt{2}}\frac{8}{3}\bar{\partial}^{2}\sigma \label{1-13}\end{aligned}$$This observation is very important for the method of [@FZ1] to work.
The Majorana theory (\[1-1\]) corresponds to both high and low temperature phases of the Ising model in the vicinity of its critical point $T_{c}$ depending of the choice of the sign of the mass parameter $m$ in (\[1-1\]). Our definition in (\[1-9\]) corresponds to the identification of the case $m>0$ with the ordered phase $T<T_{c}$, while the case $m<0$ is identified with the disordered phase $T>T_{c}$. From now on we will consider the ordered phase $m>0$.
“Free” and “fixed” boundary conditions
======================================
In this section we consider the Ising field theory defined on the half-plane y $>0$. As a warm-up exercise let us first rederive formulas (\[3\]), ([4]{}) for local magnetization in the cases of “free” and “fixed” b. c. Explicit expressions for boundary states for “free” and “fixed” b. c. was obtained in [@C]. It follows from this expressions that the fields $\psi
$ and $\bar{\psi}$ satisfy the following b. c.:$$\begin{aligned}
\left. \left( \psi -\bar{\psi}\right) \right\vert _{\text{y}=0} &=&0\qquad
\text{(for "free" b.~c.)} \label{2-1} \\[0.05in]
\left. \left( \psi +\bar{\psi}\right) \right\vert _{\text{y}=0} &=&0\qquad
\text{(for "fixed" b.~c.)} \label{2-2}\end{aligned}$$Suppose that $\left( \chi \left( x\right) ,\bar{\chi}\left( x\right) \right)
$ is a double-valued solution of Dirac’s equations defined on the half-plane y $>0$ (with punctured point $x_{0}$) such that it changes sign when $x$ is taken around $x_{0}$, decays sufficiently fast as $\left\vert x\right\vert
\rightarrow \infty $, and satisfies b. c.:$$\begin{aligned}
\left. \left( \chi -\bar{\chi}\right) \right\vert _{\text{y}=0} &=&0\qquad
\text{(for "free" b.~c.)} \label{2-3} \\[0.05in]
\left. \left( \chi +\bar{\chi}\right) \right\vert _{\text{y}=0} &=&0\qquad
\text{(for "fixed" b.~c.)} \label{2-4}\end{aligned}$$Then the following identity holds:$$\left\langle {}\right. \left( {}\right. \oint\limits_{C_{x_{0}}}\chi \left(
x\right) \psi \left( x\right) dz-\bar{\chi}\left( x\right) \bar{\psi}\left(
x\right) d\bar{z}\left. {}\right) \mu \left( x_{0}\right) \left.
{}\right\rangle =0 \label{2-5}$$where $C_{x_{0}}$ is a contour encircling the point $x_{0}$. This is because the integral on the left-hand side of (\[2-5\]) does not changes under continuous deformations of the contour $C_{x_{0}}$ and therefore one can deform it in such a way that it constitutes of two parts: $C_{\infty }$ which tends to infinity and $C_{b}$ which passes along the boundary. Then the integral along $C_{\infty }$ is zero because $\chi \left( x\right) $ and $\bar{\chi}\left( x\right) $ decay at infinity and the integral along $C_{b}$ is zero due to (\[2-3\]) or (\[2-4\]). On the other hand one can shrink $C_{x_{0}}$ to a small circle around the point $x_{0}$ and express the left-hand side of (\[2-5\]) in terms of descendents of $\mu $ using the operator product expansions of $\psi \left( x\right) $, $\bar{\psi}\left(
x\right) $ with $\mu \left( x_{0}\right) $. If the singularity of $\left(
\chi \left( x\right) ,\bar{\chi}\left( x\right) \right) $ at the point $x_{0} $ is not too strong only descendents of not higher than the second order appear in this expression. Since they are expressed in terms of coordinate derivatives of $\sigma $ this will lead to a differential equation for local magnetization $\left\langle \sigma \left( x\right)
\right\rangle $. The only problem is to find a solution of Dirac’s equations $(\chi \left( x\right) ,\bar{\chi}\left( x\right) )$ satisfying the above conditions. For this purpose we will use the following trick. We will search for this solution in the form of the linear combination:$$\begin{gathered}
\left(
\begin{array}{c}
\chi \left( x\right) \\
\bar{\chi}\left( x\right)\end{array}\right) =c_{1}\left(
\begin{array}{c}
\left\langle \psi \left( x\right) \sigma \left( x_{0}\right) \mu \left(
Px_{0}\right) \right\rangle _{0} \\
\left\langle \bar{\psi}\left( x\right) \sigma \left( x_{0}\right) \mu \left(
Px_{0}\right) \right\rangle _{0}\end{array}\right) +c_{2}\partial \left(
\begin{array}{c}
\left\langle \psi \left( x\right) \sigma \left( x_{0}\right) \mu \left(
Px_{0}\right) \right\rangle _{0} \\
\left\langle \bar{\psi}\left( x\right) \sigma \left( x_{0}\right) \mu \left(
Px_{0}\right) \right\rangle _{0}\end{array}\right) + \label{2-6} \\[0.05in]
+c_{3}\bar{\partial}\left(
\begin{array}{c}
\left\langle \psi \left( x\right) \sigma \left( x_{0}\right) \mu \left(
Px_{0}\right) \right\rangle _{0} \\
\left\langle \bar{\psi}\left( x\right) \sigma \left( x_{0}\right) \mu \left(
Px_{0}\right) \right\rangle _{0}\end{array}\right) +c_{4}\left(
\begin{array}{c}
\left\langle \psi \left( x\right) \mu \left( x_{0}\right) \sigma \left(
Px_{0}\right) \right\rangle _{0} \\
\left\langle \bar{\psi}\left( x\right) \mu \left( x_{0}\right) \sigma \left(
Px_{0}\right) \right\rangle _{0}\end{array}\right) + \\[0.05in]
+c_{5}\partial \left(
\begin{array}{c}
\left\langle \psi \left( x\right) \mu \left( x_{0}\right) \sigma \left(
Px_{0}\right) \right\rangle _{0} \\
\left\langle \bar{\psi}\left( x\right) \mu \left( x_{0}\right) \sigma \left(
Px_{0}\right) \right\rangle _{0}\end{array}\right) +c_{6}\bar{\partial}\left(
\begin{array}{c}
\left\langle \psi \left( x\right) \mu \left( x_{0}\right) \sigma \left(
Px_{0}\right) \right\rangle _{0} \\
\left\langle \bar{\psi}\left( x\right) \mu \left( x_{0}\right) \sigma \left(
Px_{0}\right) \right\rangle _{0}\end{array}\right)\end{gathered}$$where $\left\langle \ldots \right\rangle _{0}$ denotes a correlation function in the Ising field theory defined on the plane and $P$ denotes the reflection in the line y $=0$ (i. e. $P\left( \text{x, y}\right) =\left(
\text{x, }-\text{y}\right) $). Obviously each term in (\[2-6\]) is non-zero and as a function of $x$ is a solution of Dirac’s equations, change sign when $x$ is taken around $x_{0}$ and decay at infinity. The coefficients in this linear combination can be determined from the requirement that it satisfies (\[2-3\]) or (\[2-4\]). Note that we do not need to know the functions $\chi \left( x\right) $ and $\bar{\chi}\left(
x\right) $ in explicit form. What we really need is several terms of their short-distance asymptotics as $x\rightarrow x_{0}$, but the latter can be expressed in terms of the two-point functions $\left\langle \sigma \left(
x_{0}\right) \sigma \left( Px_{0}\right) \right\rangle _{0}\equiv G\left( 2m\text{y}_{0}\right) $ and $\left\langle \mu \left( x_{0}\right) \mu \left(
Px_{0}\right) \right\rangle _{0}\equiv \tilde{G}\left( 2m\text{y}_{0}\right)
$. As is well known [@MCWTB] (see also [@FZ1]) this functions can be expressed in terms of Painleve function of the III kind as follows:$$G\left( t\right) =\sigma _{0}\func{ch}\left( \frac{1}{2}\varphi \left(
t\right) \right) \exp \left[ \frac{1}{4}\int_{t}^{\infty }r\left( \func{sh}^{2}\varphi \left( r\right) -\left( \varphi ^{\prime }\left( r\right)
\right) ^{2}\right) dr\right] \label{2-6-1}$$$$\tilde{G}\left( t\right) =\sigma _{0}\func{sh}\left( \frac{1}{2}\varphi
\left( t\right) \right) \exp \left[ \frac{1}{4}\int_{t}^{\infty }r\left(
\func{sh}^{2}\varphi \left( r\right) -\left( \varphi ^{\prime }\left(
r\right) \right) ^{2}\right) dr\right] \label{2-6-2}$$where $\varphi \left( t\right) $ is the same function as in (\[3\]), ([4]{}).
Under parity transformation $P$ fermi fields transform as:$$P\left(
\begin{array}{c}
\psi \\
\bar{\psi}\end{array}\right) =\left(
\begin{array}{c}
-i\bar{\psi} \\
i\psi\end{array}\right) \label{2-7}$$We have therefore the following identities which follows from the invariance of correlation functions under parity transformation:$$\begin{aligned}
\left. \left\langle \bar{\psi}\left( x\right) \sigma \left( x_{0}\right) \mu
\left( Px_{0}\right) \right\rangle _{0}\right\vert _{\text{y}=0} &=&\left.
i\left\langle \psi \left( x\right) \mu \left( x_{0}\right) \sigma \left(
Px_{0}\right) \right\rangle _{0}\right\vert _{\text{y}=0} \label{2-8} \\[0.05in]
\left. \left\langle \bar{\psi}\left( x\right) \mu \left( x_{0}\right) \sigma
\left( Px_{0}\right) \right\rangle _{0}\right\vert _{\text{y}=0} &=&\left.
i\left\langle \psi \left( x\right) \sigma \left( x_{0}\right) \mu \left(
Px_{0}\right) \right\rangle _{0}\right\vert _{\text{y}=0} \label{2-9} \\[0.05in]
\left. \left\langle \bar{\partial}\bar{\psi}\left( x\right) \sigma \left(
x_{0}\right) \mu \left( Px_{0}\right) \right\rangle _{0}\right\vert _{\text{y}=0} &=&\left. i\left\langle \partial \psi \left( x\right) \mu \left(
x_{0}\right) \sigma \left( Px_{0}\right) \right\rangle _{0}\right\vert _{\text{y}=0} \label{2-10} \\[0.05in]
\left. \left\langle \bar{\partial}\bar{\psi}\left( x\right) \mu \left(
x_{0}\right) \sigma \left( Px_{0}\right) \right\rangle _{0}\right\vert _{\text{y}=0} &=&\left. i\left\langle \partial \psi \left( x\right) \sigma
\left( x_{0}\right) \mu \left( Px_{0}\right) \right\rangle _{0}\right\vert _{\text{y}=0} \label{2-11}\end{aligned}$$It follows from this identities and Dirac’s equations that from all correlation functions that present in the expression (\[2-6\]) only four are linearly independent functions of $x$ on the line y $=0$ (for example $\left\langle \psi \left( x\right) \sigma \left( x_{0}\right) \mu \left(
Px_{0}\right) \right\rangle $, $\left\langle \psi \left( x\right) \mu \left(
x_{0}\right) \sigma \left( Px_{0}\right) \right\rangle $, $\left\langle
\partial \psi \left( x\right) \sigma \left( x_{0}\right) \mu \left(
Px_{0}\right) \right\rangle $, and $\left\langle \partial \psi \left(
x\right) \mu \left( x_{0}\right) \sigma \left( Px_{0}\right) \right\rangle $). Hence requiring that (\[2-6\]) satisfy (\[2-3\]) or (\[2-4\]) one obtains four linear constraints for six coefficients $c_{1},\ldots c_{6}$. It turns out that they have non-zero solutions. One of the solutions corresponds to the following linear combination (it does not matter what of the solutions to choose):$$\begin{gathered}
\left(
\begin{array}{c}
\chi \left( x\right) \\
\bar{\chi}\left( x\right)\end{array}\right) =\frac{m}{2}\left(
\begin{array}{c}
\left\langle \psi \left( x\right) \sigma \left( x_{0}\right) \mu \left(
Px_{0}\right) \right\rangle _{0} \\
\left\langle \bar{\psi}\left( x\right) \sigma \left( x_{0}\right) \mu \left(
Px_{0}\right) \right\rangle _{0}\end{array}\right) -i\partial \left(
\begin{array}{c}
\left\langle \psi \left( x\right) \sigma \left( x_{0}\right) \mu \left(
Px_{0}\right) \right\rangle _{0} \\
\left\langle \bar{\psi}\left( x\right) \sigma \left( x_{0}\right) \mu \left(
Px_{0}\right) \right\rangle _{0}\end{array}\right) + \label{2-12} \\[0.05in]
+i\frac{m}{2}\left(
\begin{array}{c}
\left\langle \psi \left( x\right) \mu \left( x_{0}\right) \sigma \left(
Px_{0}\right) \right\rangle _{0} \\
\left\langle \bar{\psi}\left( x\right) \mu \left( x_{0}\right) \sigma \left(
Px_{0}\right) \right\rangle _{0}\end{array}\right) -\bar{\partial}\left(
\begin{array}{c}
\left\langle \psi \left( x\right) \mu \left( x_{0}\right) \sigma \left(
Px_{0}\right) \right\rangle _{0} \\
\left\langle \bar{\psi}\left( x\right) \mu \left( x_{0}\right) \sigma \left(
Px_{0}\right) \right\rangle _{0}\end{array}\right) \text{,}\end{gathered}$$for “free” b. c., and$$\begin{gathered}
\left(
\begin{array}{c}
\chi \left( x\right) \\
\bar{\chi}\left( x\right)\end{array}\right) =i\frac{m}{2}\left(
\begin{array}{c}
\left\langle \psi \left( x\right) \sigma \left( x_{0}\right) \mu \left(
Px_{0}\right) \right\rangle _{0} \\
\left\langle \bar{\psi}\left( x\right) \sigma \left( x_{0}\right) \mu \left(
Px_{0}\right) \right\rangle _{0}\end{array}\right) -\partial \left(
\begin{array}{c}
\left\langle \psi \left( x\right) \sigma \left( x_{0}\right) \mu \left(
Px_{0}\right) \right\rangle _{0} \\
\left\langle \bar{\psi}\left( x\right) \sigma \left( x_{0}\right) \mu \left(
Px_{0}\right) \right\rangle _{0}\end{array}\right) + \label{2-13} \\[0.05in]
+\frac{m}{2}\left(
\begin{array}{c}
\left\langle \psi \left( x\right) \mu \left( x_{0}\right) \sigma \left(
Px_{0}\right) \right\rangle _{0} \\
\left\langle \bar{\psi}\left( x\right) \mu \left( x_{0}\right) \sigma \left(
Px_{0}\right) \right\rangle _{0}\end{array}\right) -i\bar{\partial}\left(
\begin{array}{c}
\left\langle \psi \left( x\right) \mu \left( x_{0}\right) \sigma \left(
Px_{0}\right) \right\rangle _{0} \\
\left\langle \bar{\psi}\left( x\right) \mu \left( x_{0}\right) \sigma \left(
Px_{0}\right) \right\rangle _{0}\end{array}\right) \text{,}\end{gathered}$$for “fixed” b. c.
It is now straightforward but somewhat tedious exercise to substitute ([2-12]{}) and (\[2-13\]) in (\[2-5\]) and evaluate the left-hand side. One has to expand $\psi $ and $\bar{\psi}$ using (\[1-4\]), than to evaluate contour integrals using (\[1-6-2\]), (\[1-6-3\]) and than to act by the operators $a_{n}$, $\bar{a}_{n}$ on $\sigma $ and $\mu $ using (\[1-8\]), (\[1-9\]) and (\[1-12\]). Due to (\[1-8\]) all terms with descendants of higher than the first order vanish. Finally, taking into account that $\left\langle \sigma \left( x_{0}\right) \right\rangle \equiv \bar{\sigma}\left( 2m\text{y}_{0}\right) $ depends only on y$_{0}$ due to translation invariance, one obtains the following differential equations:$$2\left( G-\tilde{G}\right) \bar{\sigma}_{free}^{\prime }-\left( G^{\prime }-\tilde{G}^{\prime }+\tilde{G}\right) \bar{\sigma}_{free}=0 \label{2-14}$$$$2\left( G+\tilde{G}\right) \bar{\sigma}_{fixed}^{\prime }-\left( G^{\prime }+\tilde{G}^{\prime }+\tilde{G}\right) \bar{\sigma}_{fixed}=0 \label{2-15}$$(the stroke denotes derivative with respect to $t=2m$y$_{0}$). Integrating this equations, substituting (\[2-6-1\]), (\[2-6-2\]) and fixing integration constants with the help of asymptotic condition $\bar{\sigma}\left( t\right) \rightarrow \sigma _{0}$ as $t\rightarrow \infty $ one obtains (\[3\]) and (\[4\]).
Let us now consider the high temperature phase $T>T_{c}$. The differential equations in this case can be obtained from (\[2-14\]), (\[2-15\]) by substitution $m\rightarrow -m$, $G\rightleftarrows \tilde{G}$:$$2\left( G-\tilde{G}\right) \bar{\sigma}_{free}^{\prime }-\left( G^{\prime }-\tilde{G}^{\prime }+G\right) \bar{\sigma}_{free}=0 \label{2-16}$$$$2\left( G+\tilde{G}\right) \bar{\sigma}_{fixed}^{\prime }-\left( G^{\prime }+\tilde{G}^{\prime }-G\right) \bar{\sigma}_{fixed}=0 \label{2-17}$$The only solution of (\[2-16\]) that does not grow exponentially as $t\rightarrow \infty $ is $\bar{\sigma}_{free}=0$, while from (\[2-17\]) we obtain:$$\bar{\sigma}_{fixed,\;T>T_{c}}=e^{-\frac{1}{2}t}\bar{\sigma}_{fixed,\;T<T_{c}} \label{2-18}$$in agreement with [@B2]. This confirms our identification of the case $m>0$ with the low temperature phase. Had we chosen the other choice, we would obtain the exponentially growing solution for $\bar{\sigma}_{fixed}$ in the high temperature phase.
Boundary magnetic field
=======================
Let us now consider the general case of “free” b. c. perturbed by boundary spin operator $\sigma _{B}$. The latter is identified with degenerate primary boundary field with dimension $\Delta =1/2$ [@C]. It can be written in terms of fermion fields as follows [@GhZ]:$$\sigma _{B}\left( \text{x}\right) =ia\left( \text{x}\right) \left. \left(
\psi \left( x\right) +\bar{\psi}\left( x\right) \right) \right\vert _{\text{y=0}} \label{3-1}$$where $a\left( \text{x}\right) $ is additional fermionic degree of freedom with two-point function$$\left\langle a\left( \text{x}\right) a\left( \text{x'}\right) \right\rangle
_{free}=\frac{1}{2}\,\text{sign}\left( \text{x}-\text{x'}\right) \label{3-2}$$The action of the theory has therefore the following form:$$\begin{gathered}
S=\frac{1}{2\pi }\int\limits_{-\infty }^{\infty }d\text{x}\int\limits_{0}^{\infty }d\text{y}\left( \psi \bar{\partial}\psi +\bar{\psi}\partial \bar{\psi}-im\bar{\psi}\psi \right) + \label{3-3} \\
+\int\limits_{-\infty }^{\infty }\left( -\frac{i}{4\pi }\left. \left( \psi
\bar{\psi}\right) \right\vert _{\text{y}=0}+\frac{1}{2}a\partial _{\text{x}}a\right) d\text{x}+ih\int\limits_{-\infty }^{\infty }a\left( \text{x}\right) \left. \left( \psi +\bar{\psi}\right) \right\vert _{\text{y}=0}d\text{x}\end{gathered}$$It leads to the following b. c. for fermion fields [@GhZ]:$$\frac{\partial }{\partial \text{x}}\left. \left( \psi -\bar{\psi}\right)
\right\vert _{\text{y}=0}=\left. -im\lambda \left( \psi +\bar{\psi}\right)
\right\vert _{\text{y}=0} \label{3-4}$$where $\lambda =4\pi h^{2}/m$. We can now proceed in the same way as in the previous section but now instead of (\[2-3\]) or (\[2-4\]) we should require the functions $\chi $ and $\bar{\chi}$ to satisfy the condition:
$$\frac{\partial }{\partial \text{x}}\left. \left( \chi -\bar{\chi}\right)
\right\vert _{\text{y}=0}=\left. -im\lambda \left( \chi +\bar{\chi}\right)
\right\vert _{\text{y}=0} \label{3-5}$$
in order to write down the Ward identity (\[2-5\]). It turns out that in this case it is necessary to include also terms with second order derivatives in the linear combination (\[2-6\]) in order to satisfy ([3-5]{}). As a result one obtains the following linear combination:$$\begin{gathered}
\left(
\begin{array}{c}
\chi \left( x\right) \\
\bar{\chi}\left( x\right)\end{array}\right) =i\left( \frac{m}{2}\right) ^{2}\left( 1-2\lambda \right) \left(
\begin{array}{c}
\left\langle \psi \left( x\right) \sigma \left( x_{0}\right) \mu \left(
Px_{0}\right) \right\rangle _{0} \\
\left\langle \bar{\psi}\left( x\right) \sigma \left( x_{0}\right) \mu \left(
Px_{0}\right) \right\rangle _{0}\end{array}\right) - \label{3-6} \\[0.05in]
-\frac{m}{2}\left( 1-2\lambda \right) \partial \left(
\begin{array}{c}
\left\langle \psi \left( x\right) \sigma \left( x_{0}\right) \mu \left(
Px_{0}\right) \right\rangle _{0} \\
\left\langle \bar{\psi}\left( x\right) \sigma \left( x_{0}\right) \mu \left(
Px_{0}\right) \right\rangle _{0}\end{array}\right) -\frac{m}{2}\bar{\partial}\left(
\begin{array}{c}
\left\langle \psi \left( x\right) \sigma \left( x_{0}\right) \mu \left(
Px_{0}\right) \right\rangle _{0} \\
\left\langle \bar{\psi}\left( x\right) \sigma \left( x_{0}\right) \mu \left(
Px_{0}\right) \right\rangle _{0}\end{array}\right) + \\[0.05in]
+i\partial ^{2}\left(
\begin{array}{c}
\left\langle \psi \left( x\right) \sigma \left( x_{0}\right) \mu \left(
Px_{0}\right) \right\rangle _{0} \\
\left\langle \bar{\psi}\left( x\right) \sigma \left( x_{0}\right) \mu \left(
Px_{0}\right) \right\rangle _{0}\end{array}\right) +\left( \frac{m}{2}\right) ^{2}\left( 1-2\lambda \right) \left(
\begin{array}{c}
\left\langle \psi \left( x\right) \mu \left( x_{0}\right) \sigma \left(
Px_{0}\right) \right\rangle _{0} \\
\left\langle \bar{\psi}\left( x\right) \mu \left( x_{0}\right) \sigma \left(
Px_{0}\right) \right\rangle _{0}\end{array}\right) - \\[0.05in]
-i\frac{m}{2}\partial \left(
\begin{array}{c}
\left\langle \psi \left( x\right) \mu \left( x_{0}\right) \sigma \left(
Px_{0}\right) \right\rangle _{0} \\
\left\langle \bar{\psi}\left( x\right) \mu \left( x_{0}\right) \sigma \left(
Px_{0}\right) \right\rangle _{0}\end{array}\right) -i\frac{m}{2}\left( 1-2\lambda \right) \bar{\partial}\left(
\begin{array}{c}
\left\langle \psi \left( x\right) \mu \left( x_{0}\right) \sigma \left(
Px_{0}\right) \right\rangle _{0} \\
\left\langle \bar{\psi}\left( x\right) \mu \left( x_{0}\right) \sigma \left(
Px_{0}\right) \right\rangle _{0}\end{array}\right) + \\[0.05in]
+\bar{\partial}^{2}\left(
\begin{array}{c}
\left\langle \psi \left( x\right) \mu \left( x_{0}\right) \sigma \left(
Px_{0}\right) \right\rangle _{0} \\
\left\langle \bar{\psi}\left( x\right) \mu \left( x_{0}\right) \sigma \left(
Px_{0}\right) \right\rangle _{0}\end{array}\right)\end{gathered}$$Substituting it in (\[2-5\]) and evaluating the left-hand side we obtain the following differential equation for local magnetization $\bar{\sigma}\left( t\right) $:$$\begin{gathered}
\left( G+\tilde{G}\right) \bar{\sigma}^{\prime \prime }-\left[ G^{\prime }+\tilde{G}^{\prime }-G+\lambda \left( G+\tilde{G}\right) \right] \bar{\sigma}^{\prime }+ \\[0.05in]
+\frac{1}{4}\left[ G^{\prime \prime }+\tilde{G}^{\prime \prime }-\frac{1}{t}\left( G^{\prime }+\tilde{G}^{\prime }\right) -2G^{\prime }-\tilde{G}+2\lambda \left( G^{\prime }+\tilde{G}^{\prime }+\tilde{G}\right) \right]
\bar{\sigma}=0\end{gathered}$$It can be brought to a simpler form (\[16\]) by means of substitution ([15]{}).
Acknowledgements
================
I am especially grateful to Y. P. Pugai for his interest to my work and encouragement. I am also grateful to my scientific advisers A. I. Bugrij and V. N. Shadura for giving me creative freedom and to N. Z. Iorgov and V. N. Shadura for pointing out the paper [@SchE].
[99]{} P. Fonseca, A. Zamolodchikov, hep-th/0309228
P. Fonseca, A. Zamolodchikov, hep-th/0112167
B. Doyon, P. Fonseca, hep-th/0404136
J. Cardy, Nucl. Phys. B 324, (1989) 581
I. Affleck, A. Ludwig, Phys. Rev. Lett. 67, (1991) 161
S. Ghoshal, A. Zamolodchikov, hep-th/9306002
R. Konik, A. LeClair, G. Mussardo, hep-th/9508099v2
R. Z. Bariev, Theor. Math. Phys. 40, 623 (1980), translated from Teor. Mat. Fiz. 40, 95 (1979)
R. Z. Bariev, Theor. Math. Phys. 77, 1090 (1989), translated from Teor. Mat. Fiz. 77, 127 (1988)
T. T. Wu, B. M. McCoy, C. A. Tracy, E. Barouch, Phys. Rev. B 13, (1976) 316
B. M. McCoy, C. A. Tracy, T. T. Wu, J. Math. Phys. 18, 1058 (1977)
R. Charterjee, A. Zamolodchikov, hep-th/9311165v1
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D. Schuricht, F. H. L. Essler, cond-mat.str-el/0709.1809v1
[^1]: E-mail: oleg2@inbox.ru
[^2]: Note that comparing the coefficient in the short distance asymptotic of $\sigma _{fixed}\left( t\right) $, that follows from (\[4\]), with the result $\sigma _{fixed}\left( \text{y}\right) =2^{\frac{1}{4}}\left( 2\text{y}\right) ^{-\frac{1}{8}}$ (for $m=0$) of [@CL], one obtains the identity $\int_{0}^{\infty }\left( 1-e^{-\varphi \left( r\right) }\right) dr=\ln 2$
|
---
abstract: 'Radio-frequency energy harvesting constitutes an effective way to prolong the lifetime of wireless networks, wean communication devices off the battery and power line, benefit the energy saving and lower the carbon footprint of wireless communications. In this paper, an interference aided energy harvesting scheme is proposed for cooperative relaying systems, where energy-constrained relays harvest energy from the received information signal and co-channel interference signals, and then use that harvested energy to forward the correctly decoded signal to the destination. The time-switching scheme (TS), in which the receiver switches between decoding information and harvesting energy, as well as the power-splitting scheme (PS), where a portion of the received power is used for energy harvesting and the remaining power is utilized for information processing, are adopted separately. Applying the proposed energy harvesting approach to a decode-and-forward relaying system with the three-terminal model, the analytical expressions of the ergodic capacity and the outage capacity are derived, and the corresponding achievable throughputs are determined. Comparative results are provided and show that PS is superior to TS at high signal-to-noise ratio (SNR) in terms of throughput, while at low SNR, TS outperforms PS. Furthermore, considering different interference power distributions with equal aggregate interference power at the relay, the corresponding system capacity relationship, i.e., the ordering of capacities, is obtained.'
author:
- 'Yanju Gu, and Sonia Aïssa'
title: |
RF-based Energy Harvesting in Decode-and-Forward\
Relaying Systems: Ergodic and Outage Capacities
---
Introduction
============
In modern society, wireless communication devices are omnipresent and have become numerous. They are intensively involved in different applications such as video and audio information transmission [@WSN], monitoring in modern healthcare systems [@Monitor] and safety message exchange in vehicular networks [@Proceedings]. On one hand, the energy consumption is tremendous [@Green], which makes energy saving for communication a critical problem to be solved. On the other hand, recharging by traditional wiring method or battery replacement is not feasible for such huge number of small devices, like sensors, implantable medical devices, etc. Although far-field microwave power transfer is a strong candidate to replace cables in long-distance power transfer, additional power beacons need to be settled and deployed, which is not a ready work for nowadays communication systems [@HuangTWC; @XiaICC; @XiaICC2; @XiaTSP].
Energy harvesting has become an appealing solution to such problems [@EH1; @EH5; @wangpower]. Energy from solar, vibration, thermoelectric effects, and so forth [@EH2], can be harvested and converted to electrical energy to support these energy-constrained communication devices [@Harvest-Use2]. A promising harvesting technology is to use the radio frequency (RF) energy, since ambient RF signals, e.g., from TV broadcast and cellular communications, are widely available in urban areas (day and night, indoors and outdoors) [@SIGCOMMbestpaper]. In this technique, the ambient RF radiation is captured by the receive antennas of wireless devices and converted into direct current voltage through appropriate circuits [@EH3; @EH4]. A safe way has been advanced to wirelessly power chips in human body by using such method [@Ada].
As the signal carries information as well as energy at the same time, simultaneous wireless information and power transfer has been studied recently, where the receiver is assumed to be able to decode the information and harvest energy from the same signal [@SEH1; @SEH2]. However, due to practical circuit limitations, it is difficult to harvest the energy and decode the information at the same time. There are two schemes for harvesting energy and decoding information separately [@Neg; @TSplit; @Zhou_AF; @zhang2], one is the time-switching scheme (TS) in which the receiver switches over time between decoding information and harvesting energy; and the other is the power-splitting scheme (PS) in which a portion of the received power is used for energy harvesting and the remaining power is utilized for the information processing. From the perspective of receiver’s complexity, TS is superior to PS in that commercially available circuits that are separately designed for information decoding and energy harvesting can be used.
Simultaneous information decoding and energy harvesting has applications and advantages in wireless systems in general, whether in point-to-point communication or when nodes cooperate together in delivering the source signal to its final destination, [@dingzhiguo; @zhang2]. Indeed, in cooperative networks, by deploying relays between the source and the destination, the cover range and capacity of the communication system can be enhanced. However, the relays may have limited battery and wired charging may be difficult to be implemented when and where needed. To prolong the lifetime of relaying systems, wireless energy harvesting at the relays becomes a necessity [@GuICC14; @Zhou_AF; @R2; @R1].
Since the radio signal propagates freely over space, a receiver would receive the desired signal with a superposition of unwanted signals, namely interferences, which in turn results in low capacity between the transmitter and the receiver. Interference is the primary bottleneck on the data rate capacity of most wireless networks. How to decrease or avoid interference and increase the signal-to-interference-plus-noise ratio (SINR) has always been a big concern in research and industry. Techniques such as frequency reuse [@FreReUse], multi-cell coordination [@Multicell] and interference alignment [@InterferAlignment; @YangGlobecom] have been proposed for interference cancellation. While interference decreases the communication system capacity, from the energy point-of-view the interference signal provides additional energy for the harvesting system. Therefore, investigating the role that the interference plays in energy-harvesting based communication system is of major importance, though still missing.
In this paper, a decode-and-forward (DF) relaying system where the relays need to replenish energy from the received RF signals, is considered. For the limitation of hardware, harvest-use strategy in which no device equipment is dedicated to store the harvested energy, is adopted [@Harvest-Use2]. As opposed to traditional relaying where co-channel interference (CCI) within the same bandwidth as the transmitted signal deteriorates the system performance [@Xia-Tcom12; @Gu], and have to be eliminated by applying interference alignment approach or by decoding the interfering signals when they are strong, in this work, CCI signals are utilized as a new source of power for relay recharging. Specifically, the relays harvest energy from both the information signal and the CCI signals, and then use that energy to decode the source signal and forward it to the destination node. In this way, the interference acts as useful power in the energy harvesting phase and as noise in the information decoding phase. Initial results for the ergodic capacity of a DF relaying system with TS protocol appear in [@GuICC14].
To provide a thorough study and guidelines for practical applications, both TS and PS energy harvesting schemes are investigated. The analysis of the system performance is challenging due to the random feature of the transmission power at the relay in the proposed energy harvesting system. First, the ergodic capacity, which is a fundamental performance indicator for delay-insensitive services, when the codeword length can be sufficiently long to span over all the fading blocks, is investigated. Moreover, for real-time applications, a more appropriate performance metric, the outage capacity, defined as the maximum constant rate that can be maintained over fading blocks with a given outage probability, is studied. Analytical expressions for both the ergodic capacity and the outage capacity are derived and the corresponding achievable throughputs are obtained. In addition, the impacts of the interference power distribution on the ergodic capacity and the outage capacity as well as the corresponding achievable throughputs of the proposed energy harvesting system are also studied based on the majorization theory.
The rest of this paper is organized as follows. The system model and energy harvesting schemes are described in Section II. Considering time-switching and power-splitting protocols, the ergodic capacity and outage capacity are analyzed in Section III and Section IV, respectively. Simulation results which corroborate the analytical results are provided in Section V. Finally, the paper’s conclusion is presented in Section VI.
Energy-Harvesting Based Relaying {#Model}
================================
System and Channel Models
-------------------------
We consider a cooperative DF relaying system, where the source $S$ communicates with the destination $D$ through the help of an energy-constrained intermediate relaying node $R$, as shown in Fig. 1 (a). Each node is equipped with a single antenna and operates in the half-duplex mode in which the node cannot simultaneously transmit and receive signals in the same frequency band. Both, the first hop (source-to-relay) and the second hop (relay-to-destination), experience independent Rayleigh fading with the complex channel fading gains given by $h \sim CN(0,\Omega_{h})$ and $g \sim CN(0,\Omega_{g})$, respectively. The channels follow the block-fading model in which the channel remains constant during the transmission of a block and varies independently from one block to another. The channel state information is only available at the receiver.
Wireless communication networks are generally subjected to CCI due to the aggressive frequency reuse for a more efficient resource utilization. In this vein, we assume that there are $M$ CCI signals affecting the relay. The CCI signals are assumed independent but not identically distributed. Specifically, the channel fading gain between the $i^{\rm th}$ interferer and the relay node $R$, denoted $\beta_i$, is modeled as $\beta_i\sim CN(0,\Omega_{\beta_i})$. Hereafter, the desired channels and the interference channels are assumed to be independent from each other.
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Wireless Energy Harvesting at the Relay
---------------------------------------
In the network under study, the relay is considered to be constrained in terms of energy. That is, it may have limited battery reserves and needs to rely on some external charging mechanism in order to remain active in the network, and assist the communication process between the source, $S$, and the destination, $D$, as required. In the proposed approaches, the received interference and information signals at the relay are exploited to replenish energy for the relay. Both TS and PS architectures for harvesting energy are studied.
*1) Time-Switching Scheme:*
The time-switching based protocol is adopted at the relay node as illustrated in Fig. 1(b), where $T$ is the block time in which a certain block of information is transmitted from the source node to the destination node and $\alpha$, with $0\leq\alpha\leq1$, is the fraction of the block time in which the relay harvests energy from the received interference signal and information signal. The remaining block time is divided into two equal parts, namely $(1-\alpha)T/2$, for information transmission from the source to the relay and from the relay to the destination, respectively. Since there is no energy buffer to store the harvested energy (Harvest-Use) [@Harvest-Use1; @Harvest-Use2], all the energy collected during the harvesting phase is consumed by the relay.
In the first-hop phase, source $S$ transmits signal $s$ with power $P_{_S}$ to the relay $R$. Accordingly, the received signal at the relay is given by $$\label{SR}
y_{_{SR}} = \sqrt {P_{_S}}hs + \sum_{i = 1}^M \sqrt{P_i} \beta_is_i + n_{_R},$$ where $s_i$ and $P_i$ denote the signal and its corresponding power, from the $i^{\rm th}$ interferer, and $n_{_R}$ is the additive white Gaussian noise (AWGN) at the relay with zero mean and variance $\sigma^2_R$.
Accordingly, the received SINR at the relay is given by $$\begin{aligned}
\label{SINR-R}
\gamma_{_{SR}} &=&
\displaystyle\frac{P_{_S}|h|^2}{\sigma_{_R}^2 + \sum_{i = 1}^{M} P_i|\beta_i|^2} \nonumber \\
&=& \displaystyle\frac{\gamma_{h}}{1 + I_R},\end{aligned}$$ where $\gamma_{h} \triangleq \frac{P_{_S}}{\sigma_{_R}^2}|h|^2$ and $I_R \triangleq \sum_{i=1}^{M}\frac{P_i}{\sigma^2_R}|\beta_i|^2$. The received data is correctly decoded if the instantaneous received SINR $\gamma_{_{SR}}$ at the relay is higher than the pre-defined threshold $\gamma_{_\mathrm{th}}$.
When the relay is active, it harvests energy from the received information signal and the interference signal for a duration of $\alpha T$ at each block, and thus, the harvested energy is obtained as $$\label{Eh}
E_{_{H}} = \eta\bigg(P_{_S} |h|^2 + \sum_{i = 1}^{M}P_i|\beta_i|^2\bigg)\alpha T,$$ where $\eta$ is the energy conversion efficiency coefficient, with value varying from $0$ to $1$ depending upon the harvesting circuitry.
Since the processing power required by the transmit/receive circuitry at the relay is generally negligible compared to the power used for signal transmission [@TSplit; @Neg], here we suppose that all the energy harvested from the received signals (the source’s and the CCI’s) is consumed by the relay for forwarding the information to the destination. Therefore, from (\[Eh\]), the transmission power of the relay is readily given by $$\begin{aligned}
\label{P-TS}
P_{_{R}} &=& \displaystyle\frac{E_{_{H}}}{(1 - \alpha)T/2} \nonumber \\
&=& \displaystyle\frac{2\alpha \eta\sigma_R^2}{1 - \alpha } (\gamma_{h} + I_R).\end{aligned}$$ Then, the received signal at the destination node $D$ is given by $$\label{y-D}
y_{_{RD}} = \sqrt {P_{_R}} g s_{_R} + n_{_D},$$ where $s_{_R}$ is the signal transmitted from the relay and $n_{_D}$ is the AWGN noise at the destination, with zero mean and variance $\sigma_D^2$. From (\[y-D\]), the received signal-to-noise ratio (SNR) at the destination node is obtained as $$\begin{aligned}
\label{SNR-D}
\gamma_{_{RD}} &=& \displaystyle\frac{P_{_{R}}|g|^2}{\sigma_{_D}^2} \nonumber \\
&=& \underbrace{\displaystyle \frac{2\alpha\eta}{1-\alpha} \frac{\sigma_R^2}{\sigma_D^2} |g|^2}_{\triangleq W}(\gamma_{h} + I_R),\end{aligned}$$ where the defined random variable $W$ follows the same distribution as of $|g|^2$.
*2) Power-Splitting Scheme:*
In this case, the protocol adopted at the relay node is as illustrated in Fig. 1(c), where $P$ is the power of the received signal and $\theta$, with $0\leq\theta\leq1$, is the fraction of power that the relay harvests from the received interference and information signal. The remaining power is $(1-\theta)P$, which is used for information detection. In this paper, we consider a pessimistic case in which power splitting only reduces the signal power, but not the noise power, which can provide a lower-bound performance measure for relaying networks in practice.
Accordingly, after power-splitting, the received signal at the relay for information detection is given by $$\label{Signal.P}
y_{_{SR}} = \sqrt {(1-\theta)P_{_S}}hs + \sum_{i = 1}^{M} \sqrt{(1-\theta)P_i} \beta_i s_{i} + n_{_{R}}.$$ Then, the received SINR at the relay is obtained as $$\begin{aligned}
\label{SINR.P}
\gamma_{_{SR}} &=& \displaystyle\frac{(1-\theta)P_{_S}|h|^2}{\sigma_R^2 + \sum_{i = 1}^{M} (1-\theta)P_i|\beta_i|^2} \nonumber \\
&=& \displaystyle\frac{\gamma_{h}}{1 + I_R},\end{aligned}$$ where ${\gamma}_h \triangleq\frac{(1-\theta)P_{_S}}{\sigma_{_R}^2}|h|^2$ and ${I}_R \triangleq \sum_{i=1}^{M}\frac{(1-\theta)P_i}{\sigma^2_R}|\beta_i|^2$. Note that ${\gamma}_h $ and ${I}_R$ here have distinct denotations from those in (\[SINR-R\]) for the TS protocol. We use the same symbols ${\gamma}_h $ and ${I}_R$ in order to unify the analysis in the following sections.
Different from TS, for PS, the relay harvests energy from the received information and interference signal for a duration of $T/2$ at each block, and thus, the harvested energy at the relay is obtained as $$\label{Harvest.P}
E_{_{H}}= \eta{\theta}\left(P_{_S} |h|^2 + \sum_{i = 1}^{M}P_i|\beta_i|^2\right)\frac{T}{2}.$$ Suppose that all the harvested energy is consumed by the relay for forwarding the information to the destination node in the second-hop phase. From (\[Harvest.P\]), the transmission power of the relay node is readily given by $$\begin{aligned}
\label{P-PS}
P_{_{R}} &=& \frac{E_{_{H}}}{T/2} \nonumber \\
&=& \frac{{\eta \theta \sigma _R^2}}{1 - \theta }
\left({\gamma}_h+{I}_R\right).\end{aligned}$$ Then, the received SNR at the destination node is expressed as $$\begin{aligned}
\label{SNRD}
\gamma_{_{RD}} &=& \frac{P_R| g |^2}{\sigma _D^2} \nonumber \\
&=& \underbrace{\frac{\eta \theta }{( 1 - \theta) } \frac{\sigma_R^2}{ \sigma _D^2 }{|g|^2}}_{\triangleq W}({\gamma}_h+{I}_R).\end{aligned}$$
By introducing the random variables ${\gamma}_h $, ${I}_R$ and $W$, we unify the derivations of the distribution of the end-to-end SNR and the capacity metrics for these two schemes (TS and PS), as detailed in the following sections.
Ergodic Capacity {#Ergodic Capacity }
================
In this section, the exact closed-form cumulative distribution function (CDF) of the end-to-end SNR is derived. Then, the ergodic capacity and the corresponding achievable throughput are investigated for the energy harvesting DF relaying system with time-switching or power-splitting. The impact of the interference power distribution on the ergodic capacity and achievable throughput is also analyzed, based on the majorization theory.
End-to-End SNR
--------------
All channels, i.e., $h$, $\{\beta_i\}^M_{i=1}$ and $g$ are supposed to be subject to independent Rayleigh fading. Then, the received SNR at the first hop, $\gamma_h$, is of exponential distribution with the probability density function (PDF) given by $$\label{pdfX}
f_{\gamma_{h}}( x ) = \frac{1}{\bar \gamma_h}\exp \left( { - \frac{x}{{{{\bar \gamma }_h}}}} \right),\;\;x \ge 0,$$ where ${\bar\gamma}_h $, equal to $\frac{P_{_S}}{\sigma_R^2}\Omega_h$ for TS and to $\frac{(1-\theta)P_{_S}}{\sigma_R^2}\Omega_h$ for PS, is the average SNR from the source to the relay in a given time slot.
The quantity $I_R$ is the sum of $M$ statistically independent and not necessarily identically distributed (i.n.i.d.) exponential random variables, each with mean $\mu_i = \frac{P_i}{\sigma_R^2}\Omega_{\beta_i}$ for the TS-based scheme and $\mu_i = \frac{(1-\theta)P_i}{\sigma_R^2}\Omega_{\beta_i}$ for the PS-based one. Thus, the PDF of $I_R$ can be explicitly obtained as $$\label{pdfY}
f_{I_R}(y)
= \sum_{i = 1}^{\upsilon ( \bm {A} )} \sum_{j = 1}^{\tau_i(\bm {A})} \chi_{i,j}(\bm {A})
\frac{\mu_{\langle i \rangle }^{- j}}{( j - 1 )!} y^{j - 1}
\exp \bigg(-\frac{y}{\mu_{\langle i \rangle }}\bigg),\;\;y \ge 0,$$ where matrix $\bm {A} = \mathrm{diag}(\mu_1, \mu_2, \ldots, \mu_{_M})$, $\upsilon ( \bm {A} )$ denotes the number of distinct diagonal elements of $\bm {A}$, $\mu_{\langle 1 \rangle } > \mu_{\langle 2 \rangle } > \ldots > \mu_{\langle \upsilon ( \bm {A} ) \rangle}$ are the distinct diagonal elements in decreasing order, $\tau_i(\bm {A})$ is the multiplicity of $\mu_{\langle i \rangle }$, and $\chi_{i,j}(\bm {A})$ is the $(i, j)$th characteristic coefficient of $\bm {A}$ [@Win2]. Note that when the interfering signals are statistically independent and identically distributed (i.i.d.), i.e., $\mu_i = \mu$ for $i=1, 2, \ldots, M$, then $\upsilon ( \bm {A} )=1$, $\tau_1(\bm {A})=M$ and $I_{_R}$ is a sum of $M$ i.i.d. exponential random variables with the central chi-squared distribution given by $f_{I_{_R}}(\gamma)
=\frac{\mu^{- M}}{( M - 1 )!} \gamma^{M - 1}\exp \left(-\frac{\gamma}{\mu}\right)$.
The CDF of $\gamma _{_{SR}}$ is then obtained as $$\begin{aligned}
\label{F_SR}
F_{\gamma_{_{SR}}}(\gamma)
&=& {\mathbb{E}_{{I_R}}}\left\{ {1 - \exp \left[ { - \frac{{\gamma \left( {1 + {I_R}} \right)}}{{{{\bar \gamma }_h}}}} \right]} \right\} \label{F_SR1}\\
&=& 1- \exp \left(-\frac{\gamma}{{\bar \gamma}_h}\right)\sum_{i = 1}^{\upsilon ( \bm {A} )} \sum_{j = 1}^{\tau_i(\bm {A})} \chi_{i,j}(\bm {A})\nonumber \\
& & \times\bigg( 1 + \frac{\mu_{\langle i\rangle}}{{\bar \gamma }_h}\gamma \bigg)^{ - j}, \label{F_SR2}\end{aligned}$$ where $\mathbb{E}[\cdot]$ denotes the statistical expectation operator. In the case when the interfering signals are i.i.d., the CDF of $\gamma _{_{SR}}$ reduces to $F_{\gamma_{_{SR}}}(\gamma)
= 1- \left( 1 + \frac{\mu}{{\bar \gamma }_h}\gamma \right)^{ - M}
\exp \left(-\frac{\gamma}{{\bar \gamma}_h}\right)$.
Similar to the derivation of (\[F\_SR2\]), the PDF of $\gamma_{_{RD}}$, which involves products of two random variables, is determined as $$\begin{aligned}
\label{Eq.10}
F_{\gamma_{_{RD}}}(\gamma)
&=& 1 - \sum_{i = 1}^{\upsilon (\bm {B})} \sum_{j = 1}^{\tau_i(\bm {B})} \frac{\chi_{i,j}(\bm {B})}{(j - 1)!}
2 \bigg(\frac{\gamma}{{\bar \gamma }_g\mu_{\langle i\rangle }} \bigg)^{\frac{j}{2}} \nonumber \\
& & \times K_j\bigg(2\sqrt {\frac{\gamma}{{\bar \gamma }_g\mu_{\langle i \rangle}}} \bigg),\end{aligned}$$ where ${\bar \gamma }_g = \frac{2\alpha \eta}{1 - \alpha }
\frac{\sigma_R^2}{\sigma_D^2}\Omega_g$ for TS and ${\bar \gamma }_g = \frac{\theta \eta}{1 - \theta }
\frac{\sigma_R^2}{\sigma_D^2}\Omega_g$ for PS, $\bm {B} = \mathrm{diag}(\mu_1, \mu_2, \ldots, \mu_{_{M+1}})$ with $\mu_{_{M+1}}= {\bar \gamma }_h$, $\upsilon ( \bm {B} )$ denotes the number of distinct diagonal elements of $\bm {B}$, $\mu_{\langle 1 \rangle } > \mu_{\langle 2 \rangle } > \ldots > \mu_{\langle \upsilon ( \bm {B} ) \rangle}$ are the distinct diagonal elements in decreasing order, $\tau_i(\bm {B})$ is the multiplicity of $\mu_{\langle i \rangle }$, $\chi_{i,j}(\bm {B})$ is the $(i, j)$th characteristic coefficient of $\bm {B}$, and $K_{j}(\cdot)$ stands for the $j$th-order modified Bessel function of the second kind [@Gradshteyn].
Ergodic Capacity and Achievable Throughput
------------------------------------------
Ergodic capacity, in the unit of bit/s/Hz, quantifies the ultimate reliable communication limit of the fading channel. It is only achievable with infinite coding delay. Ergodic capacity can be obtained by averaging the instantaneous capacity over all fading states. In the DF-cooperative communication system under study, the instantaneous capacity is determined by the minimum one of each individual link, i.e., the first- and second-hop links. Therefore, the ergodic capacity is expressed as $$\begin{aligned}
C_\mathrm{erg}\!\!\!\!
&=&\!\!\!\! \mathbb{E}\bigg[\textrm{min}\bigg \{\frac{1}{2}\log_2(1+\gamma_{_{SR}}), \frac{1}{2} \log_2(1+\gamma_{_{RD}})\bigg\}\bigg]
\label{capacity1a}\\
&=&\!\!\! \! \mathbb{E}\bigg[\frac{1}{2}\log_2\left(1+\min \{\gamma_{_{SR}},\gamma_{_{RD}}\}\right)\bigg] \label{capacity1b}\\
&=& \!\!\!\! \frac{1}{2}\int_0^{\infty}\log_2(1+\gamma)f_{\gamma_\textrm{min}}(\gamma)d\gamma,\label{capacity1c}\end{aligned}$$ where $f_{\gamma_\textrm{min}}(\gamma)$ stands for the PDF of the random variable $\min \{\gamma_{_{SR}},\gamma_{_{RD}}\}$. The factor $1/2$ in (\[capacity1a\]) is introduced by the fact that two transmission phases are involved in the system. Expression (\[capacity1b\]) follows from the strictly monotonically increasing property of the logarithm function for non-negative real numbers.
Using the integration-by-parts method, (\[capacity1c\]) can be rewritten as $$\begin{aligned}
C_\mathrm{erg}\label{capacity2a}
\!\!\!\!&=&\!\!\!\!\! \frac{1}{2}\left\{\log_2(1 + \gamma )[ F_{\gamma_{\min}}( \gamma ) - 1 ]\right\}_0^\infty \nonumber\\
&~& -\frac{1}{2\ln 2}\int_0^\infty \!\! \frac{1}{1 + \gamma }[ F_{\gamma_{\min}}( \gamma ) - 1 ] d\gamma \\
&=&\!\!\!\!\! \frac{1}{2\ln 2}\int_0^\infty \!\!\!\!\frac{1}{1 + \gamma }[1- F_{\gamma_{\min}}( \gamma )] d\gamma,
\label{capacity2b}\end{aligned}$$ where in (\[capacity2a\]) the operator $\{f(x)\}_a^b \triangleq f(b)-f(a)$ and $F_{\gamma_\textrm{min}}(\gamma)$ denotes the CDF of the random variable $\min \{\gamma_{_{SR}},\gamma_{_{RD}}\}$ and is given by $$\begin{aligned}
\label{CDF_DM}
F_{\gamma_\textrm{min}}(\gamma)
\!\!\!\!\! &=& \!\!\!\!\! F_{\gamma_{_{SR}}}(\gamma)\!+\! F_{\gamma_{_{RD}}}(\gamma)\!-\!
\Pr \left\{ {\gamma_{_{SR}}\!\!\leq\!\! \gamma, \gamma_{_{RD}}\!\!\leq\!\! \gamma } \right\} \\
\!\!\!\!\! &=& \!\!\!\!\! \Pr \left\{ {\gamma_{_{SR}} \leq \gamma } \right\}+ \Pr \left\{ {\gamma_{_{SR}} > \gamma, \gamma_{_{RD}} \leq \gamma } \right\}. \label{CDF_DM1}\end{aligned}$$ If $\gamma$ is set to be a pre-defined threshold, (\[CDF\_DM1\]) is the expression of the outage probability at the destination, the detailed derivation of which is illustrated in the following section.
The achievable throughput at the destination relates only to the effective information transmission time and is given by $$\begin{aligned}
\label{throughput_TS}
T_\mathrm{erg}
&=& \frac{(1-\alpha)T}{T}C_\mathrm{erg} \nonumber \\
&=& (1-\alpha)C_\mathrm{erg},\end{aligned}$$ for the system with TS protocol, and by $$\label{throughput_PS}
T_\mathrm{erg} = C_\mathrm{erg},$$ for the PS based scheme.
Different from the conventional relaying system with no rechargeable nodes, from (\[throughput\_TS\]) and (\[throughput\_PS\]) it is clear that in the interference aided energy harvesting system, the achievable throughput depends not only on $P_{_S}$, $\sigma_R^2$ and $\sigma_D^2$, but also on $\alpha$ or $\theta$, $\eta$ and $P_i$.
Impact of Interference Power Distribution
-----------------------------------------
In order to provide an analysis of the impact of the interference power distribution on the energy harvesting system performance when the total received interference power is the same, in this section, the Schur-convex property of ergodic capacity and throughput is investigated.
For two vectors $\bm{x}$ and $\bm{y} (\in \mathbb{R}^{n}$) with descending ordered components $x_{1}\geq x_{2}\geq \cdots \geq x_{n}\geq 0$ and $y_{1}\geq y_{2}\geq \cdots \geq y_{n}\geq 0$, respectively, one can say that the vector $\bm {x}$ majorizes the vector $\bm {y}$ and writes $\bm {x}\succeq \bm {y}$ if $\sum\limits_{k = 1}^m {{x_k}} \ge \sum\limits_{k = 1}^m {{y_k}}$ for $m = 1, \ldots , n - 1$, and $\sum\limits_{k = 1}^n {{x_k}} = \sum\limits_{k = 1}^n {{y_k}}$. A real-valued function $\Phi$ defined on $\mathcal{A}\subset \mathbb{R}^{n}$ is said to be Schur-convex on $\mathcal{A}$ if $\bm {x}\succeq \bm {y}$ on $\mathcal{A} \Rightarrow \Phi(\bm {x})\geq \Phi(\bm {y})$.
Assume that $\omega _{1},\ldots ,\omega _{n}$ are i.i.d. random variables according to a given PDF. Furthermore, assume vector $ \bm{\mu} $ to have non-negative entries that are ordered in non-increasing order $\mu_{1}\geq \mu_{2}\geq \cdots \geq \mu_{n}\geq 0$.
\[ExpectedSchur\] Suppose the function $f : \mathbb{R}_{+} \rightarrow \mathbb{R}_{+}$ is concave. Then, $$\label{WeightedSum}
G\left( \bm{\mu} \right) = \mathbb{E}_{\omega _{1},\ldots ,\omega _{n}}\left[ {f\left( {\sum\limits_{k = 1}^n {{\mu _k}{\omega _k}} } \right)} \right]$$ is Schur-concave. Assume $f$ is convex. Then the function $G$ in (\[WeightedSum\]) is Schur-convex [@Majorization].
\[Concave\_Erg\] The ergodic capacity, $C_\mathrm{erg}$, and the achievable throughput, $T_\mathrm{erg}$, of the interference aided energy-harvesting DF relaying system are Schur-convex with respect to $\bm{\mu}$, where $\bm{\mu}=(\mu _{1},\mu _{2},\ldots ,\mu _{M})$ with $\mu_{1}\geq \mu_{2}\geq \cdots \geq \mu_{_M}\geq 0$.
We can see that ${I}_R =\sum_{i=1}^{M}\mu_{i}\omega_{i}$, where $\omega _{1},\ldots ,\omega _{n}$ are i.i.d. standard exponentially distributed with unit mean and $\sum_{i = 1}^{M} {\mu_{i}} = \mathbb{E}\{ {I}_R \}$. According to (\[F\_SR1\]), $$\begin{aligned}
F_{\gamma_{_{SR}}}(\bm{\mu})
&=& {\mathbb{E}_{{I_R}}}\left\{ {1 - \exp \left[ { - \frac{{\gamma \left( {1 + {I_R}} \right)}}{{{{\bar \gamma }_h}}}} \right]} \right\} \nonumber\\
&=& {\mathbb{E}_{{I_R}}}\left\{ f(I_R) \right\}.\end{aligned}$$ Since the second derivative $f''\left( {{I_R}} \right) = - \frac{{{\gamma ^2}}}{{{{\bar \gamma }_h}^2}}\exp \left[ { - \frac{{\gamma \left( {1 + {I_R}} \right)}}{{{{\bar \gamma }_h}}}} \right]\leq 0$, then $f(I_R)$ is a concave function. Thus, according to Lemma \[ExpectedSchur\], $F_{\gamma_{_{SR}}}(\bm{\mu})$ is Schur-concave with respect to $\bm{\mu}$. Similarly, having $$\begin{aligned}
F_{\gamma_{_{RD}}}(\bm{\mu})
&=& \mathbb{E}_{W, {I_R}}\left\{ {1 - \exp \left[ { - \frac{1}{{{{\bar \gamma }_h}}}\left( {\frac{\gamma }{W} - {I_R}} \right)} \right]} \right\} \nonumber\\
&=& \mathbb{E}_{W, {I_R}}\left\{ g(I_R) \right\},\end{aligned}$$ and since $g''\left( {{I_R}} \right) = - \frac{{1}}{{{{\bar \gamma }_h}^2}}\exp \left[ { - \frac{1}{{{{\bar \gamma }_h}}}\left( {\frac{\gamma }{W} - {I_R}} \right)} \right]\leq 0$, the function $F_{\gamma_{_{RD}}}(\bm{\mu})$ conditioned on $W$ is Schur-concave. That is, for any two vectors $\bm{\mu}_1\succeq \bm{\mu}_2$, $F_{\gamma_{_{RD}}}(\bm{\mu}_1|W)\leq F_{\gamma_{_{RD}}}(\bm{\mu}_2|W)$. Averaging over $W$, we have $F_{\gamma_{_{RD}}}(\bm{\mu}_1)\leq F_{\gamma_{_{RD}}}(\bm{\mu}_2)$. Both $f(I_R)$ and $g(I_R)$ are concave functions and using the same arguments, $$\begin{aligned}
F_{\gamma_\textrm{min}}(\bm{\mu})
&=&\!\!\! {\mathbb{E}_{{I_R}}}\left\{ {1 - \exp \left[ { - \frac{{\gamma \left( {1 + {I_R}} \right)}}{{{{\bar \gamma }_h}}}} \right]} \right\} \nonumber\\
&&\!+ \mathbb{E}_{W,{I_R}}\Big\{\exp \left[ { - \frac{{\gamma \left( {1 + {I_R}} \right)}}{{{{\bar \gamma }_h}}}} \right]\nonumber\\
&&- \exp \left[ { - \frac{1}{{{{\bar \gamma }_h}}}\left( {\frac{\gamma }{W} - {I_R}} \right)} \right] \Big\} \nonumber \\
&=& {\exp \left( { - \frac{{\rm{1}}}{{{{\bar \gamma }_g}}}} \right)\mathbb{E}_{{I_R}}}\left\{ f(I_R) \right\} +
\mathbb{E}_{W,{I_R}}\left\{f(I_R) \right\} \nonumber\\
&&+ \mathbb{E}_{W,{I_R}}\left\{g(I_R) \right\}\end{aligned}$$ is also Schur-concave with respect to $\bm{\mu}$. Since $1 - F_{\gamma_\textrm{min}}(\bm{\mu_1}) \geq 1 - F_{\gamma_\textrm{min}}(\bm{\mu_2})\geq 0$, integration with respect to $\gamma$ gives $C_\mathrm{erg}(\bm{\mu}_1) \geq C_\mathrm{erg}(\bm{\mu}_2)$. Therefore, the ergodic capacity, $C_\mathrm{erg}$, and accordingly the achievable throughput, $T_\mathrm{erg}$, are Schur-convex with respect to $\bm{\mu}$.
According to the Schur-convex property of ergodic capacity, under different interference power distributions, the corresponding system capacity relationship, i.e., the ordering of capacities, can be obtained. For example, our results imply that the worst scenario for the capacity performance occurs when the received interfering signals are of equal strength at the relay, whereas the best case happens when there is only one interferer affecting the relay.
Outage Capacity {#Outage Capacity}
================
In this section, the exact closed-form expressions of the outage probability, outage capacity and the achievable throughput are derived for the dual-hop energy harvesting DF relaying system. The impact of the interference power distribution on the outage capacity and the achievable throughput is also analyzed, based on the majorization theory.
Outage Probability
------------------
As an important performance measure of wireless systems, outage probability is defined as the probability that the instantaneous output SNR falls below a pre-defined threshold $\gamma_{_\mathrm{th}}$. This SNR threshold guarantees the minimum quality-of-service requirement of the destination users. Mathematically speaking, $P_{\rm out}(\gamma_{_\mathrm{th}})={\rm Pr}\left\{\gamma \leq \gamma_{_\mathrm{th}}\right\}$. In the DF relaying system under study, if the received SINR $\gamma_{_{SR}}$ at the relay is below $\gamma_{_\mathrm{th}}$, then the data received over that fading block cannot be decoded correctly with probability approaching $1$, and thus, the receiver at the destination declares an outage since the data will not be transmitted to the destination. Therefore, the outage probability at the destination is composed of two parts, that is,
$$\begin{aligned}
\label{CDF_D}
P_{\mathrm{out}}\left( \gamma_{_\mathrm{th}} \right)
&=& 1 - \sum\limits_{i = 1}^{\upsilon \left( {\bm {A}} \right)} {\sum\limits_{j = 1}^{{\tau _i}\left( {\bm {A}} \right)} {{\chi _{i,j}}\left( {\bm {A}} \right)} } {\left( {1 - \frac{{{\mu _{\left\langle i \right\rangle }}}}{{{{\bar \gamma }_h}}}} \right)^{ - j}}
\bigg\{\Gamma\Big(1,\frac{\gamma_{_\mathrm{th}}}{\bar \gamma_h};\frac{\gamma_{_\mathrm{th}}}{{\bar \gamma }_h{\bar \gamma }_g}\Big)
-\frac{1}{{\bar \gamma }_h}\exp \left( {\frac{{{a_{\left\langle i \right\rangle }}\gamma_{_\mathrm{th}}}}{{1 + \gamma_{_\mathrm{th}}}}} \right)
\nonumber \\
& & \times
\sum\limits_{k = 0}^{j - 1}\frac{1}{k!}
\left(\frac{{ - a_{\left\langle i \right\rangle} \gamma_{_\mathrm{th}}}}{1 + \gamma_{_\mathrm{th}}}\right)^k
\sum\limits_{m = 0}^k
{k \choose m }
\frac{b_{\left\langle i \right\rangle }^{-1}}{(-b_{\left\langle i \right\rangle }\gamma_{_\mathrm{th}})^m}
\Gamma\Big(m+1,b_{\left\langle i \right\rangle }\gamma_{_\mathrm{th}};\frac{b_{\left\langle i \right\rangle }\gamma_{_\mathrm{th}}}{\bar \gamma_g}\Big)\bigg\}.\end{aligned}$$ $$\begin{aligned}
\label{Frd2}
P_{\mathrm{out}}\left( \gamma_{_\mathrm{th}} \right)
&=& 1 - \sum\limits_{i = 1}^{\upsilon \left( {\bm A} \right)} {\sum\limits_{j = 1}^{{\tau _i}\left( {\bm A} \right)} {{\chi _{i,j}}\left( {\bm A} \right)} } {\left( {1 - \frac{{{\mu _{\left\langle i \right\rangle }}}}{{{{\bar \gamma }_h}}}} \right)^{ - j}}
\bigg[
\int_{\gamma_{_\mathrm{th}}}^\infty {\frac{1}{{\bar \gamma }_h}\exp \left( { - \frac{\gamma_{_\mathrm{th}}}{{{{\bar \gamma }_g}z}} - \frac{z}{{{{\bar \gamma }_h}}}} \right)} dz \nonumber \\
& & - \sum\limits_{k = 0}^{j - 1} {\frac{{a_{\left\langle i \right\rangle }^k}}{{k!}}} \int_{\gamma_{_\mathrm{th}}}^\infty {\frac{1}{{\bar \gamma }_h} \exp \left( { - \frac{\gamma_{_\mathrm{th}}}{{{{\bar \gamma }_g}z}} - \frac{z}{{{{\bar \gamma }_h}}}} \right)\exp \left( { - {a_{\left\langle i \right\rangle }}\frac{{z - \gamma_{_\mathrm{th}}}}{{1 + \gamma_{_\mathrm{th}}}}} \right){{\left( {\frac{{z - \gamma_{_\mathrm{th}}}}{{1 + \gamma_{_\mathrm{th}}}}} \right)}^k}} dz \bigg].\end{aligned}$$
$$\begin{aligned}
\label{outage1}
P_{\mathrm{out}}\left( {{\gamma_{_\mathrm{th}}}} \right)
&=& \Pr \left\{ {{\gamma_{_{SR}}} \leq {\gamma_{_\mathrm{th}}}} \right\} + \Pr \left\{ {{\gamma_{_{SR}}} > {\gamma_{_\mathrm{th}}}} \right\}\nonumber\\
&&\times\Pr \left\{ {\left. {{\gamma_{_{RD}}} \leq {\gamma_{_\mathrm{th}}}} \right|{\gamma_{_{SR}}} > {\gamma_{_\mathrm{th}}}} \right\} \nonumber\\
&=& \Pr \left\{ {W\left( {{\gamma _h} + {I_R}} \right) \leq {\gamma_{_\mathrm{th}}},\frac{{{\gamma _h}}}{{1 + {I_R}}} > {\gamma_{_\mathrm{th}}}}\right\} \nonumber \\
&&+
\Pr \left\{ {\frac{{{\gamma _h}}}{{1 + {I_R}}} \leq {\gamma_{_\mathrm{th}}}} \right\}\label{outage1b} \\
&=& \Pr \big \{ W(\gamma_{h} + I_R)\mathbbm{1}_{\mathcal{C}} \leq {\gamma_{_\mathrm{th}}}\big \}\label{outage1c},\end{aligned}$$
where $\mathbbm{1}_{\mathcal{C}}$ is the indicator random variable for the set $\mathcal{C}=\{\gamma_{_{SR}} > \gamma_{_\mathrm{th}}\}$, i.e., $\mathbbm{1}_{\mathcal{C}}=1$ if $\gamma_{_{SR}} > \gamma_{_\mathrm{th}}$, otherwise, $\mathbbm{1}_{\mathcal{C}}=0$.
Note that, in contrast to traditional DF relaying system with no rechargeable nodes, the transmission power $P_{_R}$ at the relay in the energy harvesting system is a random variable, which depends on the replenished energy from the interference and information signal. Therefore, the distribution of the received SNR at the destination is determined not only by the distribution of the relay-to-destination channel power gain $|g|^2$, but also by the distribution of the information and interference signal power, i.e., $\gamma_{h}$ and $I_R$. On the other hand, in the counterpart system of conventional DF relaying, when the relay can decode the information correctly, its transmission power $P_{_R}$ is a constant, and thus, the received SNR at the destination only depends on the relay-to-destination channel power gain $|g|^2$.
\[PrPower\] Define $Z \triangleq (\gamma_{h} + I_R)\mathbbm{1}_{\mathcal{C}}$, then the PDF of $Z$ is given by $$\begin{aligned}
\label{PDFZ}
f_Z(z)
\!\!\!\!&=&\!\!\!\!\mathbbm{1}_{\mathcal{Z}}
\frac{1}{{{{\bar \gamma }_h}}}\exp ( - \frac{z}{{{{\bar \gamma }_h}}})\sum\limits_{i = 1}^{\upsilon \left( {\bm {A}} \right)} {\sum\limits_{j = 1}^{{\tau _i}\left( {\bm {A}} \right)} {{\chi _{i,j}}\left( {\bm {A}} \right)} } {\left( {{\rm{1}} - \frac{{{\mu _{\left\langle i \right\rangle }}}}{{{{\bar \gamma }_h}}}} \right)^{ - j}}\nonumber\\
\!\!\!&\times&\!\!\!\! \!\!\!
\left[{{\rm{1}}\!-\! \exp \left( {\!\! - {a_{\left\langle i \right\rangle }}\frac{{z - \gamma_{_\mathrm{th}}}}{{1 + \gamma_{_\mathrm{th}}}}} \right)\sum\limits_{k = 0}^{j - 1} {\frac{{a_{\left\langle i \right\rangle }^k}}{{k!}}} {{\left( {\frac{{z - \gamma_{_\mathrm{th}}}}{{1 + \gamma_{_\mathrm{th}}}}} \right)}^k}} \right]
\label{fZ3}\end{aligned}$$ where $a_{\left\langle i \right\rangle }\triangleq \frac{{\rm{1}}}{{{\mu _{\left\langle i \right\rangle }}}} - \frac{{\rm{1}}}{{{\bar \gamma }_h}}$ and $\mathbbm{1}_{\mathcal{Z}}$ is the indicator random variable for the set $\mathcal{Z}=\{z> \gamma_{_\mathrm{th}}\}$, i.e., $\mathbbm{1}_{\mathcal{Z}}=1$ if $ z > \gamma_{_\mathrm{th}}$, otherwise, $\mathbbm{1}_{\mathcal{Z}}=0$.
The CDF of the random variable $Z = (\gamma_{h} + I_R) \mathbbm{1}_{\mathcal{C}}$ is given by $$\begin{aligned}
\label{CDFZ}
F_Z(z)
&=& \Pr\{Z\leq z\} \nonumber \\
&=& \int\int_{{x,y}\in \mathcal{S}}f_{\gamma_{h},I_R}(x,y) \mathrm{d}x \mathrm{d}y,\end{aligned}$$ where the set $\mathcal{S}={\left\{ {x + y \leq z,{\kern 1pt} \frac{x}{{1 + y}} > \gamma_{_\mathrm{th}}}, x\geq0, y\geq0 \right\}}$. After some set manipulations, we have $\mathcal{S}\neq \emptyset $ if and only if $ z> \gamma_{_\mathrm{th}}$. Since $\gamma_{h}$ and $I_R$ are independent, we get the joint distribution $f_{\gamma_{h},I_R}(x,y) = f_{\gamma_{h}}(x)f_{I_R}(y)$. Then, after some straightforward algebraic derivations, we obtain $$\label{Fz}
F_Z(z) =
\mathbbm{1}_{\mathcal{Z}}
\int_0^{ \frac{z - \gamma_{_\mathrm{th}}}{1 + \gamma_{_\mathrm{th}}}}
\int_{(1+y)\gamma_{_\mathrm{th}}}^{z - y} {f_{\gamma_{h}}(x)f_{I_R}(y)} \mathrm{d}x \mathrm{d}y.$$ Now, substituting (\[pdfX\]) and (\[pdfY\]) into (\[Fz\]) and integrating with respect to $x$ and $y$ yields the CDF of $Z$. Then the PDF of $Z$ follows directly from differentiating $F_Z(z)$ with respect to $z$. Here [@Gradshteyn Eq.(3.351.1)] was used to reach (\[PDFZ\]).
Next, we evaluate the outage probability at the destination by using the above Theorem \[PrPower\].
\[D-SNR\] The outage probability at the destination node of the interference aided energy harvesting DF relaying system is given by (\[CDF\_D\]) shown at the bottom of the page, where ${b_{\left\langle i \right\rangle }} = \frac{1}{{{{\bar \gamma }_h}}} + \frac{{{a_{\left\langle i \right\rangle }}}}{{1 + \gamma_{_\mathrm{th}}}}$ and $\Gamma(a,x;b)$ is the generalized incomplete Gamma function defined by $\Gamma(a,x;b) \triangleq \int_{x}^{\infty}t^{a-1}\exp(-t-bt^{-1})d t$.
We have $$\begin{aligned}
\label{Frd}
P_{\mathrm{out}}\left( \gamma_{_\mathrm{th}} \right)
&=& \Pr \{WZ \leq \gamma_{_\mathrm{th}}\} \nonumber \\
&=& \mathbb{E}_Z \left\{ {1 - \exp \left( - \frac{\gamma_{_\mathrm{th}}}{{{{\bar \gamma }_g}Z}} \right)} \right\} \nonumber \\
&=& 1 - \int_0^\infty {\exp \left( { - \frac{\gamma_{_\mathrm{th}}}{{{{\bar \gamma }_g}z}}} \right){f_Z}} \left( z \right)dz.
\end{aligned}$$ According to Theorem \[PrPower\], by substituting (\[PDFZ\]) into (\[Frd\]), we obtain (\[Frd2\]) shown at the bottom of the page. Next, we focus on the two integrations in (\[Frd2\]). For the first integration, from the definition of the generalized incomplete Gamma function, we have $$\label{A1}
\int_{\gamma_{_\mathrm{th}}}^\infty {\frac{1}{{\bar \gamma }_h}\exp \left( { - \frac{\gamma_{_\mathrm{th}}}{{{{\bar \gamma }_g}z}} - \frac{z}{{{{\bar \gamma }_h}}}} \right)} dz
=\Gamma \Big(1,\frac{\gamma_{_\mathrm{th}}}{\bar \gamma_h};\frac{\gamma_{_\mathrm{th}}}{{\bar \gamma }_h{\bar \gamma }_g}\Big).$$ For the second integration, exploiting the Taylor series expansion of $(z-\gamma_k)^k$ with respect to $z$ and the identity of the generalized incomplete Gamma function lead to (\[CDF\_D\]).
Simplified expressions for the outage probability as 1) the interferences are i.i.d. or 2) the number of interferers equals one, are derived and given by $$\begin{aligned}
\label{CDF_D1}
P_{\mathrm{out}}^{(1)}\left( \gamma_{_\mathrm{th}} \right)
\!\!\!\!&=&\!\!\!\! 1 - {\left( {1 - \frac{{{\mu}}}{{{{\bar \gamma }_h}}}} \right)^{ - M}}
\bigg\{\Gamma\Big(1,\frac{\gamma_{_\mathrm{th}}}{\bar \gamma_h};\frac{\gamma_{_\mathrm{th}}}{{\bar \gamma }_h{\bar \gamma }_g}\Big)
-\frac{1}{{\bar \gamma }_h}
\nonumber \\
\!\!\!\!& &\!\!\!\! \times\exp \left( {\frac{{{a}\gamma_{_\mathrm{th}}}}{{1 + \gamma_{_\mathrm{th}}}}} \right)
\sum\limits_{k = 0}^{M - 1}\frac{1}{k!}
\left(\frac{{ - a \gamma_{_\mathrm{th}}}}{1 + \gamma_{_\mathrm{th}}}\right)^k
\sum\limits_{m = 0}^k \nonumber \\
\!\!\!\!& &\!\!\!\! {k \choose m }
\frac{b^{-1}}{(-b\gamma_{_\mathrm{th}})^m}
\Gamma\Big(m+1,b\gamma_{_\mathrm{th}};\frac{b\gamma_{_\mathrm{th}}}{\bar \gamma_g}\Big)\bigg\}\end{aligned}$$ and $$\begin{aligned}
\label{CDF_N1}
P_{\mathrm{out}}^{(2)}\left( \gamma_{_\mathrm{th}} \right)
\!\!\!\!&=&\!\!\!\! 1 - \frac{{\bar \gamma }_h}{{\bar \gamma }_h-\mu}
\Gamma\Big(1,\frac{\gamma_{_\mathrm{th}}}{\bar \gamma_h};\frac{\gamma_{_\mathrm{th}}}{{\bar \gamma }_h{\bar \gamma }_g}\Big)
+\frac{b^{-1}}{{\bar \gamma }_h-\mu}
\nonumber \\
\!\!\!\!& &\!\!\!\! \times
\exp \left( {\frac{{{a}\gamma_{_\mathrm{th}}}}{{1 + \gamma_{_\mathrm{th}}}}} \right)
\Gamma\Big(1,b\gamma_{_\mathrm{th}};\frac{b\gamma_{_\mathrm{th}}}{\bar \gamma_g}\Big)\end{aligned}$$ respectively, where $a\triangleq \frac{{\rm{1}}}{{{\mu}}} - \frac{{\rm{1}}}{{{\bar \gamma }_h}}$ and ${b} \triangleq \frac{1}{{{{\bar \gamma }_h}}} + \frac{{{a}}}{{1 + \gamma_{_\mathrm{th}}}}$.
Outage Capacity and Achievable Throughput
-----------------------------------------
Outage capacity, in the unit of bit/s/Hz, is defined as the maximum constant rate that can be maintained over fading blocks with a specified outage probability. It is used for slowly varying channels, where the instantaneous SNR $\gamma$ is assumed to be constant for a large number of symbols. In the DF-cooperative communication system under study, the outage capacity in the unit of bit/s/Hz is expressed as $$\label{Outage Capacity}
C_\mathrm{out}= \frac{1}{2}\left[1-P_{\mathrm{out}}( \gamma_{_\mathrm{th}} )\right]\log_2(1+\gamma_{_\mathrm{th}}).$$ The factor $1/2$ accounts for the fact that two transmission phases are involved in the communication between the source $S$ and the destination $D$. The achievable throughput at the destination relates only to the effective information transmission time and is then given by $$\label{through_TS}
T_\mathrm{out}
= (1-\alpha)C_\mathrm{out},$$ for the system employing time switching, and by $$\label{through_PS}
T_\mathrm{out} = C_\mathrm{out},$$ for the system implementing power splitting.
Impact of Interference Power Distribution
-----------------------------------------
\[Concave\_Out\] The outage capacity, $C_\mathrm{out}$, and the achievable throughput, $T_\mathrm{out}$, of the interference aided energy-harvesting DF relaying system is Schur-convex with respect to $\bm{\mu}$, where $\bm{\mu}=(\mu _{1},\mu _{2},\ldots ,\mu _{M})$ with $\mu_{1}\geq \mu_{2}\geq \cdots \geq \mu_{M}\geq 0$.
According to (\[outage1b\]) and using the same arguments as in the proof of Theorem \[Concave\_Erg\], we can see that the outage capacity $C_\mathrm{out}$ and accordingly the achievable throughput $T_\mathrm{out}$ are also Schur-convex with respect to $\bm{\mu}$.
Note that Theorem 1 and Theorem 4 provide engineering insights for design of energy harvesting relay system. With regard to application, the proposed energy harvesting relay system can be seen as building block of a larger cellular network. For instance, consider full frequency reuse for all base stations, which is studied extensively recently for multi-cell cooperation, and where a base station serves farther users through the help of intermediate relaying nodes. From a system design point-of-view, how to choose the relay location to obtain the largest capacity is a meaningful and challenging problem. Relays that are positioned at different geometric locations may suffer the same total received interference power (at the same contour) from neighboring base stations, but with different power distributions. Based on the analysis provided in this paper, the best relay positioning can be identified. Definitely, the detailed application depends on the specific problem, which is beyond the scope of the paper, and can be considered in future extensions of this work.
Numerical Results and Discussions {#simulation}
=================================
In this section, numerical examples are presented and corroborated by simulation results to examine the throughput, $T_\mathrm{erg}$ and $T_\mathrm{out}$, of the DF cooperative communication system, where the energy-constrained relay harvests energy from the received information signal and the CCI signals. Hereafter, and unless stated otherwise, the number of CCI signals at the relay, $M$, is set to $2$ with normalized $\bm{\hat{\mu}}=\frac{\bm{\mu}}{\mathbb{E}\{ {I}_R \}}=(0.6, 0.4)$. The threshold $\gamma_{_\mathrm{th}}$ is set to $8\mathrm{dB}$ and the energy conversion efficiency $\eta$ is set to $1$. To better evaluate the effects of the interferences on the system’s throughput, we define $\frac{P_{_S}\Omega_h}{\sum_{i = 1}^M P_i\Omega_{\beta_i}}$ as the average signal-to-interference ratio (SIR) at the relaying node and $\frac{P_{_S}\Omega_h}{\sigma_R^2}$ as the first-hop average SNR.
\
\
For the system with TS protocol, Fig. \[Fig2\] shows the throughput $T_\mathrm{erg}$ and $T_\mathrm{out}$ versus the energy harvesting ratio $\alpha$ for different values of average SIR received at the relay, where the first-hop average SNR is $20\mathrm{dB}$. It is observed that the analytical results of (\[throughput\_TS\]) and (\[through\_TS\]) match well the simulation results. As the energy harvesting ratio $\alpha$ increases from $0$ to $1$, the throughput of the system increases at first until $\alpha$ reaches the optimal value where the throughput gets its maximum, and thereafter decreases from the maximum to zero. The concave feature of the curves is due to the fact that the energy harvested for the second-hop transmission increases with increasing $\alpha$, which effectively decreases the outage and enhances the capacity of the second hop and, accordingly, improves the throughput of the system. Meanwhile, as $\alpha$ increases, more data are wasted on energy harvesting and less information is decoded for information transmission which heavily reduces the throughput of the system, therefore, the throughput reaches a maximum and then drops down. As SIR increases, the optimal throughput and the optimal $\alpha$ both increase. This means that when the received average SNR at the relay is fixed, an increase in the power of the CCI signals can deteriorate the system performance, but effectively reduces the optimal $\alpha$ required to achieve the optimal throughput.
For comparison purposes, Fig. \[Fig3\] depicts the throughput $T_\mathrm{erg}$ and $T_\mathrm{out}$ versus the energy harvesting ratio $\theta$ for the system with PS protocol under the same simulation settings. It is observed that the analytical results of (\[throughput\_PS\]) and (\[through\_PS\]) match perfectly the simulation results. The concave feature of the curves is due to the fact that the energy harvested for the second-hop transmission increases with increasing $\theta$, which effectively decreases the outage and enhances the capacity of the second hop and, accordingly, improves the throughput of the system. Meanwhile, as $\theta$ increases, more power is harvested for information transmission and less power is left for information decoding which deteriorates the throughput of the system and, thus, the throughput reaches a maximum and then drops down. In both plots (Fig. 2 and Fig. 3), it is seen that the throughput $T_\mathrm{out}$ is less than the throughput $T_\mathrm{erg}$ due to the requirement of the outage capacity that a fixed date rate is maintained in all non-outage channel states.
Next, we compare the throughput performances of the energy harvesting systems with TS and PS protocols, respectively, to facilitate the choice of these two schemes for designing energy harvesting system. Figures \[Fig4\] and \[Fig5\] illustrate the optimal $T_\mathrm{erg}$ and the optimal $T_\mathrm{out}$ versus the first-hop average SNR, respectively, for these two protocols given different values for the average SIR at the relay. It is observed that the PS protocol is superior to the TS protocol at high SNR, in terms of optimal $T_\mathrm{erg}$ and $T_\mathrm{out}$. At relatively low SNR, on the other hand, the TS-based scheme outperforms the PS one in terms of optimal $T_\mathrm{out}$, but with little difference in optimal $T_\mathrm{erg}$. This can be explained as follows. At high SNR, power-splitting with optimal ratio $\theta$ ($\theta$ is around half) would not decrease the received SNR significantly so that the information could still be correctly decoded at the relaying node, but for the time-switching scheme, there always exists an information loss at the energy harvesting phase. Similarly, at low SNR, power splitting with optimal ratio $\theta$ would lead to more decoding errors at the relaying node.
\
\
\
\
The impact of the interference power distribution on the throughput performance is shown in Fig. \[Fig6\] for the system with TS protocol and in Fig. \[Fig7\] for the system with PS protocol. The energy harvesting ratios $\alpha$ and $\theta$ are set to $0.2$ and $0.6$, respectively. The SIR at the relay is $10\mathrm{dB}$. The total interference power is the same at each SNR but with different normalized power distribution: $\bm{\hat{\mu}}_{1}=(1,0,0,0,0)$, $\bm{\hat{\mu}}_{2}=(0.6,0.4,0,0,0)$ and $\bm{\hat{\mu}}_{3}=(0.2,0.2,0.2,0.2,0.2)$. According to the definition of majorization, we have $\bm{\mu}_{1}\succeq \bm{\mu}_{2}\succeq \bm{\mu}_{3}$ and thus, $T_\mathrm{erg}(\bm{\mu}_{1})\geq T_\mathrm{erg}(\bm{\mu}_{2})\geq T_\mathrm{erg}(\bm{\mu}_{3})$ and $T_\mathrm{out}(\bm{\mu}_{1})\geq T_\mathrm{out}(\bm{\mu}_{2})\geq T_\mathrm{out}(\bm{\mu}_{3})$, since the throughput is Schur-convex with respect to $\bm{\mu}$ as proven by Theorems \[Concave\_Erg\] and \[Concave\_Out\]. This is clearly shown by the simulation results, and it implies that the worst scenario for the throughput performance occurs when the interfering signals are of equal received power at the relay, whereas the best case happens when there is only one interferer affecting the relay.
Conclusion {#conclusions}
==========
In this paper, an RF-based energy harvesting relaying system was proposed, where the energy-constrained relay harvests energy from the superposition of received information signal and co-channel interference (CCI) signals, and then uses that harvested energy to forward the correctly decoded signal to the destination. The time-switching (TS) and the power-splitting (PS) protocols were adopted, and their ensuing performance was compared. Different from traditional decode-and-forward relaying system with no rechargeable nodes, the transmission power of the energy constrained relay is not a constant anymore but a random variable depending on the variation of available energy harvested from the received information and CCI signals at the relay. Analytical expressions for the ergodic capacity as well as for the outage capacity were derived to determine the corresponding system achievable throughputs. The PS scheme was demonstrated to be superior to TS at high SNR in terms of the achievable throughput from ergodic or outage capacity, while at relatively low SNR, TS outperforms PS in terms of the achievable throughput from outage capacity. Furthermore, considering different interference power distributions with equal aggregate interference power at the relay, the corresponding system capacity relationship, i.e., the ordering of capacities, was obtained. The results reveal that the worst scenario for the capacity performance occurs when the received interfering signals are of equal strength at the relay, whereas the best case occurs when there is only one interferer affecting the relay.
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---
author:
- |
Satoshi Nagaoka\
High Energy Accelerator Research Organization (KEK)\
Tsukuba, Ibaraki 305-0801, Japan\
E-mail:
title: 'Non-Extreme Black Holes from D-branes at Angles'
---
Introduction
============
A microscopic understanding of the Bekenstein-Hawking entropy of extreme black holes is given in string theory [@SV]. One of the directions of the extension of this result is to generalize away from extremality. Non-extreme black holes play an important role in the study of the properties of realistic black holes. The non-extremality parameter $\mu$, which is the mass of neutral Schwarzschild black hole by setting all charges to zero, interpolates between extremal and Schwarzschild black holes. Entropies of non-extreme black holes are discussed in [@NEBH] in various dimensions. $D=10$ type IIA supergravity can be obtained from the dimensional reduction on a circle of $D=11$ dimensional supergravity. The type IIA theory has the solitonic objects, D$p$-branes, with $p=0,2,4,6$, which preserve $1/2$ of a supersymmetry. Orthogonal intersecting D-branes preserve part of a supersymmetry and extreme black holes are constructed from these configurations. Non-extreme generalization of orthogonal intersecting D-brane solutions of supergravity, which reduce to non-extreme black holes upon toroidal compactification, is shown in [@KT; @CT]. Non-orthogonal intersecting D-branes, which are widely studied as realistic brane models like Standard Model on intersecting D-branes recently, also preserve part of a supersymmetry [@BDL][^1]. Extremal black holes are constructed from branes at angles [@Angle; @Angle2; @CP]. But it is not yet known the non-extreme generalization of non-orthogonal intersecting D-branes [^2].
In this paper, we will present the non-extreme solutions of supergravity from 2-angled non-orthogonal intersecting D-branes. The relation between two configurations, one of which is non-orthogonal intersecting D-branes (A) and the other one is orthogonal D-branes (B), is discussed from supergravity and string theory perspective. In section 2, we construct the non-extreme solutions from intersecting D2-branes. The correspondence between (A) and (B), which is essential for constructing the non-extreme solution, is discussed from supergravity. In section 3, we compare (A) and (B) from string theory. Section 4 is devoted to the conclusion and discussion. In appendix, we calculate mass and entropies of non-extreme black holes which are obtained upon the toroidal compactification.
Non-extreme solutions from branes at angles
===========================================
Part of type IIA supergravity Lagrangian which we need for the analysis is written as $$\begin{aligned}
L=\sqrt{-g} \left( e^{-2 \Phi }(R+4 (\nabla \Phi)^2) -\frac{1}{48} F_4^2
\right) \ ,\end{aligned}$$ where the four-form field strength $F_4$ couples to D2-branes. We adopt the string frame here. Equations of motion for this action are written as $$\begin{aligned}
\notag
R&=-4 \nabla^2 \Phi +4 (\nabla \Phi)^2 \ , \\ \notag
R_{ij}& =-2 \nabla_i \nabla_j \Phi +\frac{1}{12} e^{2\Phi}
(F_i^{\ klm}F_{jklm}-\frac{1}{8} g_{ij} F_4^2) \ , \\
0&=\frac{1}{\sqrt{-g}} {\partial}_i (\sqrt{-g} F^{ijkl}) \ .
\label{eom}\end{aligned}$$ We consider two stacks of D2-branes and both stacks are intersecting each other at angles $2 \theta$ to preserve 1/4 of a supersymmetry. The location of the D2-branes is shown in Fig. 1. For simplicity, we consider these configurations here, but we can generalize the analysis to arbitrary numbers of stacks of intersecting D-branes.
(-190,45)[$x_1$]{} (-10,45)[$x_2$]{} (-90,80)[$y_4$]{} (-250,80)[$y_3$]{} (-230,40)[$2\theta$]{} (-40,40)[$2\theta$]{} (-175,35)[$\otimes$]{}
Now, we would like to construct the non-extreme solution of this system. The idea is to achieve understanding of the properties of the Schwarzschild black hole by doing perturbation theory in $\mu$. An algorithm which leads to a non-extreme solution from a given extreme solution is developed in [@CT]. This procedure is applied for orthogonal intersecting D-branes [@CT], but not yet for non-orthogonal intersecting D-branes.
Let us focus on $q$ D2-branes intersecting at two angles (Fig. 2 (A)). The relation between $Q$ and $q$ is $$\begin{aligned}
Q\equiv q \cdot (\text{D2 charge per unit area}) \ .\end{aligned}$$ We compactify $x_1,x_2, y_3$ and $y_4$ directions with the periods $a_1,a_2,b_1$ and $b_2$. This system has topological charge $$\begin{aligned}
q(1,1) \otimes (1,1) \oplus q(1,-1) \otimes (1,-1)
=2q(1,0) \otimes (1,0) \oplus 2q(0,1) \otimes (0,1) \ ,\end{aligned}$$ where $(q_1,q_3) \otimes (q_2,q_4)$ denotes that branes wrap $q_i$ times along $x_i(y_i)$ directions.
The system (B) also has topological charge $2q(1,0) \otimes (1,0) \oplus 2q(0,1) \otimes (0,1)$.
(-280,75)[$\otimes$]{} (-75,75)[$\otimes$]{} (-280,-10)[(A)]{} (-80,-10)[(B)]{}
Next, we notice energy(tension) of these configurations. Total energy of configuration (A) is $$\begin{aligned}
E_{\text{A}}=2 c \sqrt{a_1^2+b_1^2} \sqrt{a_2^2+b_2^2} \ ,\end{aligned}$$ where $c\equiv \frac{1}{g (2\pi)^2 (\alpha')^{3/2} }$ is D2-brane tension and $g$ is string coupling constant. On the other hand, total energy of configuration (B) is $$\begin{aligned}
E_{\text{B}}=2 c (a_1a_2+b_1b_2) \ .\end{aligned}$$ Using the relation which corresponds to the supersymmetric condition, $$\begin{aligned}
\frac{b_1}{a_1}=\frac{b_2}{a_2} \ ,\end{aligned}$$ we can easily check $E_{\text{A}}=E_{\text{B}}$.
Therefore, we conclude that there are supersymmetric 2-angled non-orthogonal branes (A) which have the same total charge and tension as the orthogonal intersecting branes (B). An non-extremalization procedure for the orthogonal intersecting branes is already known [@CT], then, we can construct the non-extreme non-orthogonal intersecting D-brane solutions by applying this procedure to the configuration (B). It is interesting to construct supersymmetric 3-angled D-brane solutions which preserve $1/8$ of a supersymmetry.
Now, what is the difference between the configurations (A) and (B)? We can not find the difference between (A) and (B) under the supergravity description because supergravity solution is constructed by the global charge, which (A) and (B) have equally. On the other hand, the mass spacing of the open strings connecting the different D-branes is determined by the intersection angle, therefore, configurations (A) and (B) have different mass spacing of the spectrum [^3]. The relations between (A) and (B) will be discussed further in section 3.
We will construct the non-extreme solutions from non-orthogonal intersecting D-branes. Extremal supergravity solutions of orthogonal intersecting D-branes are written as $$\begin{aligned}
\notag
ds^2&=F^{1/2}\Big[
F^{-1} \left( - dt^2 +(1+X_1) (dx_1^2+dx_2^2) +
(1+X_2) (dy_3^2+dy_4^2)
\right) \\ \notag
&+ \sum_{i=5}^9 dx_i^2
\Big] \ , \\ \notag
A_3&=dt \wedge (-\frac{X_1}{1+X_1}dx^1 \wedge dx^2+\frac{X_2}{1+X_2}
dy^3 \wedge dy^4) \ , \\
e^{2 \Phi}&= F^{1/2} \ , \label{ext}\end{aligned}$$ where $$\begin{aligned}
\notag
F&=\Pi_{i=1,2} F_i, \quad F_i=1+X_i \ , \\
X_1&=\frac{Q_1}{r^3}, \quad X_2=\frac{Q_2}{r^3} \ .\end{aligned}$$ $Q_1$ and $Q_2$ are D2-brane charges along $x_1x_2$ plane and $y_3y_4$ plane respectively. Both configurations (A) and (B) have total charge $2q(1,0) \otimes (1,0)
\oplus 2q(0,1) \otimes (0,1)$, which corresponds to $$\begin{aligned}
\notag
Q_1 &= \frac{2Qb_1b_2}{\sqrt{(a_1^2+b_1^2)(a_2^2+b_2^2)}}
=2Q \sin ^2 \theta \ , \\
Q_2 &= \frac{2Qa_1a_2}{\sqrt{(a_1^2+b_1^2)(a_2^2+b_2^2)}}
=2Q \cos ^2 \theta \ . \label{cha}\end{aligned}$$ The solution (\[ext\]) with charge [^4] (\[cha\]) is already found in [@Angle]. Non-extreme procedure for orthogonal intersecting D-branes consists of the steps as $$\begin{aligned}
\notag
dt^2 &\to f(r) dt^2 \ , \\ \notag
\sum_{i=5}^9 dx_i^2 &\to f^{-1} dr^2 +r^2 d \Omega^2_4 \ , \\ \notag
F_i&\to \tilde{F}_i=1+\tilde{X}_i=1+\frac{\tilde{Q}_i}{r^3} \ , \\
A_3&\to \tilde{A}_3
=dt \wedge (-\frac{X_1}{1+\tilde{X}_1}dx^1 \wedge dx^2
+\frac{X_2}{1+\tilde{X}_2}
dy^3 \wedge dy^4) , \end{aligned}$$ where $$\begin{aligned}
f(r)=1-\frac{\mu}{r^3} \ , \end{aligned}$$ and $$\begin{aligned}
\tilde{Q}_i=\mu \sinh^2 \delta_i, \quad Q_i =\mu \sinh \delta_i
\cosh \delta_i \ , \quad i=1,2 \ .\end{aligned}$$ The extremal limit corresponds to $\mu \to 0$ and $\delta_i \to 0$ with $Q_i$ kept fixed. A non-extreme solution of the non-orthogonal intersecting D-branes is obtained as $$\begin{aligned}
\notag
ds^2&={\tilde{F}}^{1/2}\Big[
{\tilde{F}}^{-1} \left( - f dt^2 +(1+\tilde{X}_1) (dx_1^2+dx_2^2) +
(1+\tilde{X}_2) (dy_3^2+dy_4^2)
\right) \\ \notag
&+f^{-1} dr^2 +r^2 d\Omega_4
\Big] \ , \\ \notag
A_3&=\tilde{F}^{-1}dt \wedge (-X_1(1+\tilde{X}_2)dx^1
\wedge dx^2+X_2(1+\tilde{X}_1) dy^3 \wedge
dy^4) \ , \\
e^{2 \Phi}&=\tilde{F}^{1/2} \ , \end{aligned}$$ where $$\begin{aligned}
\tilde{F}=\tilde{F}_1\tilde{F}_2=(1+ \tilde{X}_1)(1+\tilde{X}_2) \ ,\end{aligned}$$ and the functions $\tilde{X}_1$ and $\tilde{X}_2$ are harmonic functions in the transverse space, $$\begin{aligned}
\notag
\tilde{X}_1= \frac{\tilde{Q}_1}{r^{3}}, \quad
\tilde{Q}_1=-\frac{\mu}{2}+\sqrt{(2 Q \sin^2 \theta)^2+
(\frac{\mu}{2})^2} \ , \\
\tilde{X}_2=\frac{\tilde{Q}_2}{r^{3}} , \quad
\tilde{Q}_2=-\frac{\mu}{2}+\sqrt{(2 Q \cos^2 \theta)^2+
(\frac{\mu}{2})^2} \ .\end{aligned}$$ This solution satisfies the equations of motion (\[eom\]). In the extremal limit $\mu \to 0$, we obtain the extreme solution (\[ext\]).
State counting of D-branes
==========================
We consider the number of states of configurations (A) and (B) here. The massless degrees of freedom which contribute to the entropy formula are associated with open strings stretching the intersecting D-branes [@Pol]. Configuration (B) is constructed from $2q$ D2-branes along $x_1x_2$ plane and $2q$ D2-branes along $y_3y_4$ plane. Therefore, the configuration (B) has $4q^2$ massless excitation modes which contribute to the entropy.
The number of massless modes of non-orthogonal intersecting D-branes (A) which contribute to the entropy formula is obtained in [@CP]. We will briefly review the bosonic part of the analysis. Fundamental strings stretching two intersecting D-branes are described by the following boundary condition, $$\begin{aligned}
\notag
&{\partial}_\sigma Z_a +i \tan \theta {\partial}_\tau Z_i |_{\sigma=0}=0, \\
&{\partial}_\sigma Z_a -i \tan \theta {\partial}_\tau Z_i |_{\sigma=\pi}=0 \ ,\end{aligned}$$ where $Z_a=x_a+i y_{a+2}$ for $a=1,2$. The classical solutions of the equations of motion for the complex bosons with these boundary conditions have mode expansions which are written as $$\begin{aligned}
Z_a=z_a+i(\sum_{n=1}^\infty a_{n-\epsilon}^a \phi_{n-\epsilon}
(\tau,\sigma)-\sum_{n=0}^\infty a_{n-\epsilon}^{a\dagger}
\phi_{-n-\epsilon}(\tau,\sigma) ) \ ,\end{aligned}$$ where $$\begin{aligned}
\phi_{n-\epsilon}=\frac{1}{\sqrt{|n-\epsilon|}}\cos ((n-\epsilon)\sigma
+\theta) \exp (-i (n-\epsilon )\tau) \ , \end{aligned}$$ with $\epsilon\equiv \frac{2\theta}{\pi}$. The commutators of zero modes $x_a$ and $y_{a+2}$ are obtained as $$\begin{aligned}
[x_1,y_3]=[x_2,y_4]=i\frac{\pi}{2\tan \theta} \ .\end{aligned}$$ Dirac quantization condition restricts the values of $x_1$ and $x_2$ as $$\begin{aligned}
\notag
x_1&=\frac{n_1p_1p_3a_1}{N_1} \ , \quad n_1: \text{integer} \ ,
\\ \notag
x_2&=\frac{n_2p_2p_4a_2}{N_2} \ , \quad n_2:
\text{integer} \ , \\
N_1&= |p_1q_3-p_3q_1| \ , \quad
N_2=|p_2q_4-p_4q_2| \ ,\end{aligned}$$ where $p_i$ and $q_i$ ($i=1,2,3,4$) are wrapping numbers on cycles $x_i(y_i)$ of the first and second D2-branes. $N_1$ and $N_2$ denote the intersection numbers of $x_1y_3$ plane and $x_2y_4$ plane respectively. The degeneracy of the massless mode is $N_1N_2$. The wrapping number of configuration (A) is $$\begin{aligned}
\notag
(p_1,p_3) \otimes (p_2,p_4)&=(1,1)\otimes(1,1) \ , \\
(q_1,q_3) \otimes (q_2,q_4)&=(1,-1)\otimes(1,-1) \ ,\end{aligned}$$ on each $T^2 \times T^2$, therefore, the total number of massless mode is obtained as $$\begin{aligned}
q^2N_1N_2=q^2|(-1-1)(-1-1)|=4q^2 \ ,\end{aligned}$$ which is equal to the number of massless modes of (B).
On the other hand, the mass spectrum of open string ending on D-branes is written as [@BDL] $$\begin{aligned}
m^2=\frac{2n\theta}{\pi \alpha'} \ , \quad n=0,1,\cdots \ ,\end{aligned}$$ therefore, configuration (A) and (B) have different mass spectrum.
Conclusion and discussion
=========================
It is well-known that the orthogonal intersecting branes can be generalized to the non-extreme solutions, on the other hand, it was not yet known the non-extremalization procedure of the non-orthogonal intersecting D-branes. In this paper, we have constructed the non-extreme black holes from non-orthogonal intersecting D-branes. The essential point for the construction is that we can transform the basis of the brane charge from non-orthogonal D-branes to orthogonal D-branes. After the transformation, we non-extremalize the solutions for each orthogonal intersecting brane charges.
The configuration (A) and (B) have the same mass and global charge, therefore, (A) and (B) reduce to the same black hole. They also have the same number of massless states of D-branes. But they are different configurations in string theory, that is, they have different mass spectrum. In other words, this is something like ‘hair’ of the black hole.
Intersecting D-branes are the fundamental and important objects in string theory, and it has much possibility to bring information from the string theory to the realistic world. In our research directions, first, it is interesting to construct the 3-angled intersecting D-brane solutions, which is the geometrical aspects to interpolate between the string theory and the realistic world. In these configurations, we can consider $D=4$ black holes, and black holes with more than 3-charges. Another context is to realize Standard Model on intersecting D6-branes, which are constructed as 3-cycles wrapped on $(T^2)^3$ with 3 intersection angles. When we consider M-theory lift of intersecting D6-branes (plus O6-planes), the background is a singular 7-manifold with $G_2$ holonomy. It might be interesting to analyze it from this direction. Second, to clarify the relation between supergravity approach and the effective field theory (gauge theory) approach is also interesting. In [@YM], intersecting branes were analyzed by using Yang-Mills theory in the small intersection angles. We have extended the angle parameters to arbitrary value for the non-extreme black brane solutions. It might become some clue to explain the recombination mechanism, which corresponds to Higgs mechanism on Standard Model, on the intersecting D-branes from the supergravity approach. Third, the black hole degrees of freedom should be studied further. It might be interesting to consider the ‘hair’ conjectured by Mathur [@MSS], which is closely related to the information paradox, in this case.
Acknowledgements {#acknowledgements .unnumbered}
================
I would like to thank H. Fuji, K. Hotta, Y. Hyakutake, Y. Sekino, M. Shigemori, H. Shimada, T. Takayanagi and especially Y. Kitazawa for quite useful discussions and comments. I would also like to thank Y. Kitazawa for carefully reading this manuscript.
Mass and entropies of non-extreme solution
==========================================
The area of the horizon at $ r=\mu^{1/3}$ is $$\begin{aligned}
\notag
A_8&=[\frac{8 \pi^2}{3} a_1a_2b_1b_2 r^4 \tilde{F}^{1/2} ]_{r=\mu^{1/3}}
\\
&=\frac{8 \pi^2}{3} a_1a_2b_1b_2 \mu^{4/3} \cosh \delta_1 \cosh \delta_2
\ ,\end{aligned}$$ where internal coordinates $x^a (y^a)$ have period $a_1,a_2,b_1$ and $b_2$. The corresponding 6-dimensional metric is $$\begin{aligned}
ds_6^2=-\tilde{F}^{-1/2} f dt^2+\tilde{F}^{1/2}
[f^{-1}dr^2+r^2 d\Omega_4^2] \ .\end{aligned}$$ The ADM mass is calculated as $$\begin{aligned}
\notag
M&=\frac{4\pi^2 a_1a_2b_1b_2}{3 \kappa^2}
(4 \mu +3 \tilde{Q}_1 +3 \tilde{Q}_2) \\
&=\frac{4 \pi^2a_1a_2b_1b_2}{\kappa^2}
\left(\sqrt{(2 Q \cos^2 \theta)^2+(\frac{\mu}{2})^2}
+\sqrt{(2 Q \sin^2 \theta)^2+(\frac{\mu}{2})^2}+\frac{\mu}{3}\right) \ ,\end{aligned}$$ where $\kappa $ is 6-dimensional Planck constant. In the extremal limit, we obtain $$\begin{aligned}
M=\frac{8 \pi^2a_1a_2b_1b_2 Q}{\kappa^2} \ .\end{aligned}$$ Bekenstein-Hawking entropy is $$\begin{aligned}
\notag
S_{\text{BH}}&=\frac{2 \pi A_8}{\kappa^2} \\ \notag
&=\frac{16\pi^3 a_1a_2b_1b_2}{3 \kappa^2}\mu^{4/3} \cosh \delta_1
\cosh \delta_2 \\
&=\frac{16 \pi^3 a_1a_2b_1b_2}{3 \kappa^2} \mu^{1/3}\left(
\sqrt{(2 Q \sin^2 \theta)^2+(\frac{\mu}{2})^2}+\frac{\mu}{2}
\right)^{\frac{1}{2}}
\left(
\sqrt{(2 Q \cos^2 \theta)^2+(\frac{\mu}{2})^2}+\frac{\mu}{2}
\right)^{\frac{1}{2}} \ ,\end{aligned}$$ and Hawking temperature is $$\begin{aligned}
T=\frac{3 \mu^{2/3}}{4 \pi}
\left(
\sqrt{(2 Q \sin^2 \theta)^2+(\frac{\mu}{2})^2}+\frac{\mu}{2}
\right)^{-\frac{1}{2}}
\left(
\sqrt{(2 Q \cos^2 \theta)^2+(\frac{\mu}{2})^2}+\frac{\mu}{2}
\right)^{-\frac{1}{2}} \ .\end{aligned}$$ In the near extremal limit, we obtain $$\begin{aligned}
\notag
M&=M_0 +\Delta M +{\cal O} (\mu^2) \ , \\
M_0&=\frac{4\pi^2 a_1a_2b_1b_2 Q}{ \kappa^2}, \quad
\Delta M=\frac{4\pi^2 a_1a_2b_1b_2 \mu}{3 \kappa^2} \ ,\end{aligned}$$ and $$\begin{aligned}
\notag
S_{\text{BH}}&=\frac{16 \pi^3 a_1a_2b_1b_2 }{3 \kappa^2} \mu^{1/3}
Q\sin 2 \theta\left(1+\frac{\mu}{2 Q \sin^2 2 \theta }+
{\cal O} (\mu^2)\right) \\
&\sim (\frac{4\pi^2a_1a_2b_1b_2}{3\kappa^2})^{2/3} 4\pi Q \sin 2\theta \ .\end{aligned}$$ By using the Hawking temperature, $S_{\text{BH}}$ is written as $$\begin{aligned}
S_{\text{BH}}=\frac{4 \pi^2 a_1a_2b_1b_2}{\kappa^2}(\frac{4 \pi}{3})^{3/2}
(Q\sin2 \theta)^{3/2} T^{1/2} \ ,\end{aligned}$$ where $$\begin{aligned}
T=\frac{3 \mu^{2/3}}{4 \pi} \left(
\frac{1}{Q \sin 2 \theta}
\right) \ .\end{aligned}$$
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[^1]: Configurations of supersymmetric intersecting D-branes are constructed in many other papers, for example [@Jab].
[^2]: Recently, brane-antibrane systems at finite temperature are analyzed to account for the entropy of the black branes far from extremality [@ddbar].
[^3]: I would like to thank H. Shimada for bringing us this point.
[^4]: $Q_1$ and $Q_2$ might become irrational number for some value $\theta$, but this is no problem because only the ratio of the area becomes irrational and the number of branes remains to be integer. $$\begin{aligned}
\notag
Q_1&=2q \sin^2 \theta \cdot (\text{D2 charge per unit area}) \ , \\
Q_2&=2q \cos^2 \theta \cdot (\text{D2 charge per unit area}) \ .\end{aligned}$$
|
---
abstract: 'The wide public sees solar energy as the future of mankind, and media channels quite oftenly states that our challenge is to improve efficiencies and reduce cost. However, one may point some unconvenient truth’s about the physical limits we are facing, that are barely discussed by the public or even by scientists and institutions that are strongly biased towards a picture of a sustainable oil free energy in the future. In this work we discuss some of those physical limits of photovoltaics based on the principle of the Hubbert’s theory for the oil peak, evidencing that much of the research is focused on photovoltaic efficiencies and this parameter is widely overestimated: better efficiencies oftenly are the result of complex technologies that are expensive and not scalable. In this context, if fossil fuels proved not replaceable, it is very likely that our socioeconomic ideas based in the past will not withstand an energy transition to high cost and low quality sources.'
author:
- Marcos Paulo Belançon
bibliography:
- '/home/mbelancon/Documentos/library.bib'
title: 'Hubbert’s theory and photovoltaics: the nonsense race of breaking energy-conversion records'
---
Improving photovoltaics
=======================
A quick search in the *Web Of Science* for “photovoltaic efficiencies” provides us an insight of how much research is being carried on this subject, or is at least related to it. This data is shown in figure \[WOS\], where in the inset we can see the cumulative photovoltaic (PV) power in the world.
![Number of articles in the Web Of Science database containing the term “photovoltaic efficiencies”.[]{data-label="WOS"}](WOS.jpg)
As one can see, the same exponential behavior is observed in both curves. However, despite growth records of PV’s in the world, forecasts from the *Intergovernmental Panel on Climate Change (IPCC)*, the *Energy Information Administration (EIA)* or from companies such as the *British Petroleum (BP)* still give a picture of a future based mainly on oil and natural gas. Some will say that the potential of solar energy is being underestimated[@Creutzig2017].
Many different dreams about our sustainable energy future are being discussed[@Chu2012; @Haegel2017]. Thousands of researchers are trying to improve solar energy to electricity conversion today; about 14 articles are being published everyday covering from development of new PV’s[@Lee2016; @Albrecht2017] until spectral converters[@Huang2013a] and solar thermal devices[@Kannan2016; @NathanSLewis2016]. A handful of different PV technologies are already very competitive in terms of efficiency[@Rand2007; @Summary2016], however, as it is indicated in figure \[WOS\] our community is worried in further improvements.
The role of energy-conversion efficiencies
==========================================
The unconvenient truth is that PV efficiencies may not help our energy transition. Looking for example to the *National Renewable Energy Laboratory (NREL)* efficiency chart for PV’s, one may see that the Silicon technology widely dominant in the market today is almost 30 years old. By this way one may interprete that, except by improvements in production techniques, Silicon PV’s are still the same. When scientists have developed a new PV technology in the last few decades, even though some energy conversion records were broken, it has produced no net effect in our hability to produce solar energy. Some technologies, such as CdTe PV‘s are competitive today, however the dependende of scarse minerals makes impossible to scale up the production as much as we need.
Even more interesting is to have a look in the NREL PV System Cost Benchmark[@Fu2017] where they modeled the impact of module energy-conversion efficiency ($\eta$) on total system costs. While authors oftenly interpret that improve $\eta$ is “probably the key”[@Green2016] to reduce costs, the NREL model, even considering the optimistic assumption of reach $\eta=60\%$ by the same price of today’s modules, does not project a strong reduction in the total cost. As the data in table \[tabelatempos\] shows, higher efficiencies may reduce costs by only $\sim 20\%$.
------ ------------------------------------------------- -- --
Type **$\eta$ & Cost (US\$/Wp)& Cost if $\eta=60\%$\
Residential & 16.2%&2.90 &2.18\
Commercial & 17.5%&1.84 &1.31\
Utility-Scale & 17.5%&1.09 &0.78\
**
------ ------------------------------------------------- -- --
: Costs and average efficiencies of different PV’s projects [^1][]{data-label="tabelatempos"}
This data should not be miss interpreted: higher $\eta$ itself is desirable, however, if we aim an approach towards global scale applications of PV’s, one may not neglect the scale.
Two centuries ago, the economist Jean-Baptiste Say[@say1803] assumed that the scale of our consumption was so small that natural resources would last forever. This assumption, that natural resources are not produced or consumed is the basis of his work. It is remarkable that this aproximation made by Say has worked by so much time, however, everyone should consider that since his time we multiplied the population several times and the energy consumption per capita even more.
Many natural resources are now being consumed, such as oil and gas; many others are being diluted, such as Silver, Copper and every single element mined on earth. Mining or recycling less concentrated ores consumes more energy, what is a consequence of the laws of thermodynamics. Modern economies worldwide are not considering the implications that today’s reality is imposing to us, and it does not matter if we peak up a liberalist or communist one; the economic theories behind will always consider that the market will find another resource once the first one is exhausted.
Why scale matters?
==================
Even though PV’s cumulative capacity growth may be interpreted as astoshing or promising, one should take care to make a fair comparison between the facts and our expectations. To give some perspective, in figure \[crude\] we have the variation of US crude oil production in each decade since the 1880’s.
![Percentual growth of US oil production per decade.[]{data-label="crude"}](Crude.jpg)
From the begining of the oil exploration until the peak of conventional oil production in the 1970’s the producion has grown pretty fast. Between 1900 and 1929 the production was multiplied by a factor of 15. And then, after the peak in the 1970’s, oil imports contributed to install a chronic debt in the trade balance of the country with the world. Even though the US had already known non-conventional oil and gas sources, the market has choose to buy the cheaper conventional alternative in Venezuela and Midde-east.
It is important to pay attention to some facts in this history. When the cumulative capacity is very low it is common to have growth rates of two or even three digits. In 1900 the US oil production has grown at rates similar to those predicted for the photovoltaics in the coming decade. The other lesson from the US oil peak is that consequences of such event may be huge and difficult to predict, economically and politically.
Hubbert predicted in 1956[@Hubbert1956] the decline of oil production in US and 15 years after the history proved him to be right. Hubbert’s theory can not be applied to solar energy, however, it may be more or less applied to all the raw material we need to build the PV’s. By this way, we should take a look in the real scale of supply, demand and raw material consumption for solar energy. Figure \[pc\] shows a plot of the added capacity (AC) in the last few years, from which we may found that it has grown more or less linearly.
![PV’s world production capacity.[]{data-label="pc"}](added.jpg)
The challenge that PV’s are facing is not only in reducing cost or improving efficiencies; neither is increase production capacity. The challenge is if AC can increase faster than linearly, or at least what slope is sustainable from the perspective of resources and cost. In the last few year the prices were pressured down mainly by overproduction in China, what put many companies in the west in a difficult situation.
On the perspective of available resources, the *World Silver Survey 2017*, from *The Silver Institute*, shows that in 2011 and in 2016 the PV’s industry has consumed the same amouth of Silver, of about 75 million ounces, while the AC speed up by $\sim3$. This achievement is remarkable, however, as it shown in *International Technology Roadmap For Photovoltaics 2017*, it is not so clear that industry will continue to reduce Silver consumption per cell so fast as in the past. There are not much room for improvements when the technology is already mature, and Silver production is a concern[@Sverdrup2014; @Grandell2014].
The world PV’s installed capacity is around 300 GWp today, which corresponds to an average of 60 GW of electricity. This is in the range of the the energy equivalent to the US oil production in the time of the first world war, which was less than 1 million barrels a day. These data discussed here supports the idea that even running far from the scale of mankind’s needs, PV’s are already facing finite resources constraints[@Tao2011].
After China dominated the PV’s market, many companies in US, Europe and Japan lost value or even declared insolvency. To keep AC increasing while maintaining or reducing the prices the PV’s industry needs to keep sharing more or less the same 75 million ounces of Silver they consumed last year. In other words, it is necessary to reduce the silver content[@Grandell2014]. From the materials perspective Copper may be used to replace Silver, however it does not have the same conductivity and chemical resistance. The assumption that we will use less materials, with lower quality but engineered in such way that will result in more efficient devices is too much optimistic and not scientific.
Back to the scale needed, even a conservative projection[@FraunhoferISE2015] considers an AC of PV’s per year three times higher than today. While Silicon technology needs Silver, others are based on elements even more difficult to obtain, such as Cadmium, Tellurium, Indium, Gallium, Selenium or some combination of them. A few of those elements are byproduct mineral[@Bleiwas2010; @Graedel2011; @Grandell2016] and by this way the production of such CdTe, CIS or CIGS PV’s is very unlikely to reach the Silicon PV’s production scale[@Tao2011]. If our best shot to produce solar electricity will be with the 30 years old single junction Silicon technology, we may reduce Silver consumption per cell further, what will enable AC to be increased by the “conservative” (or realistic) factor of three in the next decades.
Production capacity in the future
=================================
To clarify this picture of how far we are from the “Hubbert’s limits" the figure \[projection\] show some very simple scenarios for PC of PV’s. At left in the figure we can see the same data from figure \[pc\], for the AC, as well the linear fit represented by the blue line. The slope for this curve is about 7 GWp/year. This is how much we are speeding up the addition of PV‘s in the world in the last 10 years, on average. If the slope surpasses the growing demand of PV‘s, we should have and excess suply that in will in turn make some companies to crash: a common market driven adjustment.
The three other lines in the figure \[projection\] project what should be the slope if we aim to reach $PC=500$ GWp in 2030, 2040 or 2050. This 500 GWp of PC should be enough to reach in a few decades about 12.5 TWp of installed capacity; which means something like 2.5 TW on average. This value is still a fraction of how much energy we consume worldwide but it is useful to provide a picture of the point discussed here.
![PV’s world production capacity and slopes needed to reach 500 GWp in the next decades.[]{data-label="projection"}](projection.jpg)
If PC keeps increasing at the present rate, we are not going to reach the 500 GWp before 2060. Even to reach this level in 2050 the slope needs to be increased to about 12 GWp/year. This means for example that, to avoid Silver constraints the industry should reduce its consumption per Wp in about 80% in 33 years. Even though this kind of reduction was achieved in the last ten years, it is not clear that such reduction can be achieved again without undesirable performance effects. And this is only one example.
The point here is that the question that should receive more attention from the community is if are all the pieces in the production chain of photovoltaics “free" to grow? Every time one constraint appear, the prices may not decrease, what can reduce demand and investment, compromising the PC growth. Our biggest challenge is than the scale of our needs and a quite short deadline.
Scaling up production instead of improving efficiencies
=======================================================
In a recent report from the HSBC Global Research[@Fustier2017], Fustier et al made estimates that it is very likely that today’s oil field will decline in production by more than 40 million barrels a day until 2040. To replace it, the non-conventional fossil fuels are already been extracted, mainly in Shale plays in US[@Hoegh-guldberg2007; @Hughes2013] and oil sands in Canada.
Since 2008 the shale rush made the US Oil production increase from $\sim5$ to $\sim9$ million barrels a day, deviating from the Hubbert prediction made 70 years ago. 4 million barrels more per day is energy equivalent to much more than what PV’s of the world can deliver today. However, this non-conventional oil source has its own problems: the decline production of individual Shale wells may be as fast as 50% in the first year. So, the Hubbert’s peak for this non-conventional source is very likely to come faster than for conventional sources. By this way, the oil decline is very likely do impact the cost of our energy at the same time it may introduce a kind of deadline to perform a transition to renewables.
I wish PV’s could be the alternative to replace conventional oil and gas. However, considering the prediction made by Fustier, to accomplish that we should multiply our installed PV’s by a factor of about 40 until 2040, which should require a PC of 500 GWp on average from now on; which is clearly not feasible. Anyway, if such oil decline is going to happen, to replace it scientists working in the field should change their research focus.
Across the globe we are working in “photovoltaic efficiencies” as mentioned earlier, and it is very unlikely that such contributions will ever have some effect in our everyday lives. We should than focus on expanding the limits of the cumulative capacity, by increasing life span of PV’s, reducing and eventually replacing scarse materials such as Silver and developing techniques that could make recycling PV’s easier.
All of that could make possible to achieve a higher slope in PC growth; increase photovoltaic efficiences does not help pretty much towards this direction because we are already near the limits of single junction solar cell efficiencies. And we may add that it could be helpful for scientists and the public in general to think more broadly about the energy transition. If replace fossil fuels prove not to be possible in the coming decades, we may need to look to other options.
Other option?
=============
The mankind’s appetite for energy is huge and our institutions are also addicted to it. The IPCC has more specialists in economics than in ecology, and the prevalence of men and europeans is clear[@VandenBergh2017a]. When scientists try to follow the biased view of this kind of institution to pursue grants and publications, science as a whole is in serious danger. The IPCC does not put all option in the table (and in the reports), because only perpetual growth of gross domestic product (GDP) are allowed to be present in the debate. A possible limit to the ideology of unlimited growth[@Kallis2011; @VandenBergh2017a] is not being considered, even though it could solve many aspects of our situation.
Once we are not near of solving our sustainability problem[@Raftery2017], all the options should be present in the scientific debate promoted by institution such as the IPCC. That should include many aspects of the feasibility in the long term of our civilization as it is today. There are socioeconomic options that aim reduce consumption of resources and increase prosperity[@timjackson], however, the institutions did not consider it seriously.
Conclusion
==========
If we choose to replace conventional by non-conventional fossil fuels, the overall cost of energy will rise. On the other hand, if we choose renewables, such as PV‘s, to reach the scale of our needs it is very likely that the overall cost of energy will rise, also.
While scientists are blindly wasting time following what may be proved as useless goals, they are also wasting intellectual resources that could be used to fix our civilization instead of, at maximum, create a new PV company to produce a few more GW’s that may solve a tiny part of the problem. The tragedy of pursuing continuous growth of the energy supply is that the manutention costs of our civilization are growing together[@Tainter2012]; and after the Hubbert’s peak of the key resource we will face deplection while our manutention cost keeps rising. There are no high tech solution that will avoid this dilemma.
In the 1980’s France has built[@Dussud2016] an impressive fleet of nuclear reactors, and even though Uranium production did not peaked, it did not avoid the problem that now the country needs to replace the nuclear reactors to keep the supply of energy. There are other hundreds of nuclear reactors facing the same critical point around the world.
So, the big question is: can we replace fossil fuels by PV‘s or other renewables without replace our socioeconomic status quo? Some scientists will answer that yes, of course. But these answers are about faith and not supported by scientific evidence.
[^1]: data from [@Fu2017], pg 45.
|
---
abstract: 'The interpretation of parasitic gaps is an ostensible case of non-linearity in natural language composition. Existing categorial analyses, both in the typelogical and in the combinatory traditions, rely on explicit forms of syntactic copying. We identify two types of parasitic gapping where the duplication of semantic content can be confined to the lexicon. Parasitic gaps in *adjuncts* are analysed as forms of generalized coordination with a polymorphic type schema for the head of the adjunct phrase. For parasitic gaps affecting *arguments* of the same predicate, the polymorphism is associated with the lexical item that introduces the primary gap. Our analysis is formulated in terms of Lambek calculus extended with structural control modalities. A compositional translation relates syntactic types and derivations to the interpreting compact closed category of finite dimensional vector spaces and linear maps with Frobenius algebras over it. When interpreted over the necessary semantic spaces, the Frobenius algebras provide the tools to model the proposed instances of lexical polymorphism.'
bibliography:
- 'References/bibfile.bib'
nocite: '[@Steedman87; @partee-rooth83; @Morrill19; @KanovichKS19; @Selinger2011]'
title: |
A Frobenius Algebraic Analysis\
for Parasitic Gaps
---
|
---
title: Response to the reviewers
---
Review 1 {#review-1 .unnumbered}
========
- My concerns are mostly addressed (some are a matter of taste and I do not want to force my taste on the authors’ writing).
- Overall, the work has some theoretical value (not major, but fair), and little practical value (the interesting cases are when S is small, where they see very small gain). Having said that, I see no harm in the publication of this work.
- The revised version is improved compared to the previous version. The presentation can still be improved (especially improving the motivation can help). Also, I like to see a paragraph added which clearly discusses the shortcomings of this work (e.g., little practical benefit and the low rate of the codes) and another one which motivates the work mostly from a theoretical point of view (e.g, would the theory behind this work open new research venues?).
Review 2 {#review-2 .unnumbered}
========
The authors have addressed my comments in a satisfactory manner. I still have a few inconveniences and suggestions that are listed below.
Major comments: {#major-comments .unnumbered}
---------------
- Section II, under “repair-by-transfer". The “broader notion" you mention is unclear. This requirement on the entropy seems to invalidate the very essence of “exact repair". If I understand correctly, this merely implies that $X_\ell^{(r)}$ is an injective function of $Y_{\ell,[d]}^{(r,i)}$. If so, how is that a broader interpretation of exact repair? Also, where do you use this assumption in your analysis?
- I strongly suggest to compose a clear list of open problems in a designated location in the paper.
Minor comments: {#minor-comments .unnumbered}
---------------
- Introduction - It is unclear what is the “data recovery criterion", how does it relate to the MDS property, and what is required relation between the parameters in order to satisfy it.
- The repeated use of the notation mod(n,s) is a bit odd. I assume that you mean n mod s.
- Section II - The sentence “Upon receiving ... a function of Y" is unclear.
- Remark 2 - “servers fail" should be “servers that fail" or “failed servers".
- Section II - Under “repair-by-transfer in r’th" should be “in the r’th".
Review 3 {#review-3 .unnumbered}
========
The authors have made reasonable attempts to address my previous concerns, however, the current state of the paper still appears to be a borderline case. Both the technical results and the connection to practical applications are on the weak side. In terms of the presentation, the paper is actually quite nicely written, and there are a few observations that may be useful. Though I do not feel comfortable recommending for an acceptance of the paper at this point, I will also not strongly recommend against it. Below my concerns are (re)-stated.
- My main concern is the motivation and problem formulation itself. The availability issue is of course an important concern in practical data storage systems, but this should not be viewed as of such importance that it needs to be achieved at significant expense of the storage cost. Of course, a small sacrifice in the storage cost may be justifiable. However, due to the Fixed Cluster formulation in this work, the code constructions for the proposed framework are essentially only of theoretical value from the outset, due to the immediate consequence of only allowing very low rate codes in this framework. This understanding is immediately clear even without the effort reported in the current work, and in this regard, the current work simply confirms it with a more quantitative analysis. To make things worse, the focus of the work is on the MBR point instead of the MSR point, which means an even more elevated storage cost. This is exactly the reason that MBR codes for the original regenerating codes did not receive any essential attention in any meaningful application.
- One of my early suggestions was to include results on MSR code construction, but no such code construction was introduced in the revision. The authors did make some efforts to include more converse results on the proposed MBR code construction, however, there are significant restrictions on this optimality result. In the literature, there are three classes of codes proposed: general exact-repair codes, help-by-transfer codes, and repair-by-transfer codes. The last class is the most restrictive, which is the setting the new results are given. Thus this additional result in the revision does not significantly improve the technical contribution of the work.
- The improvement factor (maximum 0.79) of the proposed code, comparing to the original regenerating codes, is small, and thus the result is not very exciting in this aspect. Moreover, this baseline itself should in fact be updated, perhaps to \[19,29\].
Some other comments: {#some-other-comments .unnumbered}
--------------------
- Many references should be updated to their journal versions, essentially for all conference publications or arxiv pre-prints dated on or before 2015.
- The capitalization in the references is not done correctly: e.g., “mbr" should be “MBR" in several places.
|
---
abstract: 'We generalise Ehrhard and Regnier’s Taylor expansion from *pure* to *probabilistic* $\lambda$-terms through notions of probabilistic resource terms and explicit Taylor expansion. We prove that the Taylor expansion is adequate when seen as a way to give semantics to probabilistic $\lambda$-terms, and that there is a precise correspondence with probabilistic Böhm trees, as introduced by the second author.'
author:
- Ugo Dal Lago
- Thomas Leventis
bibliography:
- 'references.bib'
title: |
On the Taylor Expansion of Probabilistic $\lambda$-terms\
(Long Version)
---
Introduction
============
Linear logic is a proof-theoretical framework which, since its inception [@G87], has been built around an analogy between on the one hand linearity in the sense of linear algebra, and on the other hand the absence of copying and erasing in cut elimination and higher-order rewriting. This analogy has been pushed forward by Ehrhard and Regnier, who introduced a series of logical and computational frameworks accounting, along the same analogy, for concepts like that of a differential, or the very related one of an approximation. We are implicitly referring to differential $\lambda$-calculus [@ER03], to differential linear logic [@ER06], and to the Taylor expansion of ordinary $\lambda$-terms [@ER08]. The latter has given rise to an extremely interesting research line, with many deep contributions in the last ten years. Not only the Taylor expansion of pure $\lambda$-terms has been shown to be endowed with a well-behaved notion of reduction, but the Böhm tree and Taylor expansion operators are now known to commute [@ER06b]. This easily implies that the equational theory (on pure $\lambda$-terms) induced by the Taylor expansion coincides with the one induced by Böhm trees.
The Taylor expansion operator is essentially *quantitative*, in that its codomain is not merely the set of resource $\lambda$-terms [@B93; @ER03], a term syntax for promotion-free differential proofs, but the set of *linear combinations* of those terms, with positive real number coefficients. When enlarging the domain of the operator to account for a more quantitative language, one is naturally lead to consider algebraic $\lambda$-calculi, to which giving a clean computational meaning has been proved hard so far [@V09].
But what about *probabilistic* $\lambda$-calculi [@JP89], which have received quite some attention recently (see, e.g. [@EPT18; @BDLGS16; @VKS19]) due to their applicability to randomised computation and bayesian programming? Can the Taylor expansion naturally be generalised to those calculi? This is an interesting question, to which we give the first definite positive answer in this paper. In particular, we show that the Taylor expansion of probabilistic $\lambda$-terms is a conservative extension of the well-known one on ordinary $\lambda$-terms. In particular, the target can be taken, as usual, as a linear combination of *ordinary* resource $\lambda$-terms, i.e., the same kind of structure which Ehrhard and Regnier considered in their work on the Taylor expansion of *pure* $\lambda$-terms. We moreover show that the Taylor expansion, as extended to probabilistic $\lambda$-terms, continues to enjoy the nice properties it has in the deterministic realm. In particular, it is adequate as a way to give semantics to probabilistic $\lambda$-terms, and the equational theory on probabilistic $\lambda$-terms induced by Taylor expansion coincides with the one induced by a probabilistic variation on Böhm trees [@B84]. The latter, noticeably, has been proved to capture observational equivalence, one quotiented modulo $\eta$-equivalence [@B84].
Are we the first ones to embark on the challenge of generalising Taylor’s expansion to probabilistic $\lambda$-calculi, and in general to effectful calculi? Actually, some steps in this direction have recently been taken. First of all, we need to mention the line of works originated by Tsukada and Ong’s paper on rigid resource terms [@TAO17]. This has been claimed from the very beginning to be a way to model effects in the resource $\lambda$-calculus, but it has also been applied to, among others, probabilistic effects, giving rise to quantitative denotational models [@TAO18]. The obtained models are based on species, and are proved to be adequate. The construction being generic, there is no aim at providing a precise comparison between the discriminating power of the obtained theory and, say, observational equivalence: the choice of the underlying effect can in principle have a huge impact on it.
One should also mention Vaux’s work on the algebraic $\lambda$-calculus [@V09], where one can build arbitrary linear combinations of terms. He showed a correspondence between Taylor expansion and Böhm trees, but only for terms whose Böhm trees approximants at finite depths are computable in a finite number of steps. This includes all ordinary $\lambda$-terms but not all probabilistic ones. More recently Olimpieri and Vaux have studied a Taylor expansion for a non-deterministic $\lambda$-calculus [@OV18] corresponding to our notion of *explicit* Taylor expansion (Section \[sec:taylor\_rigid\]).
In the rest of this section, probabilistic Taylor expansion will be informally introduced by way of an example, so as to make the main concepts comprehensible to the non-specialist. In sections \[sec:rigid\] and \[sec:taylor\_rigid\], we introduce a new form of resource term, and a notion of explicit Taylor expansion from probabilistic $\lambda$-terms. These constructions have an interest in themselves (again, see [@OV18]) but in this paper they are just an intermediate step towards proving our main results. Definitionally, the crux of the paper is Section \[sec:taylor\], in which the Taylor expansion of a probabilistic $\lambda$-term is made to produce *ordinary* resource terms. The relationship between the introduced theory and the one induced by Probabilistic Böhm trees [@L18] is investigated in Section \[sec:trees\] and Section \[sec:taylor\_testing\].
The Probabilistic Taylor Expansion, Informally {#the-probabilistic-taylor-expansion-informally .unnumbered}
----------------------------------------------
In this section, we introduce the main ingredients of the probabilistic Taylor expansion by way of an extremely simple, although instructive, example. Let us consider the probabilistic $\lambda$-term $M=\delta(I\oplus\Omega)$, where $\oplus$ is an operator for binary, fair, probabilistic choice, $\delta=\lambda x.xx$, $I=\lambda.x.x$ and $\Omega=\delta\delta$ is a purely diverging, term. As such, $M$ is a term of a minimal, untyped, probabilistic $\lambda$-calculus. Evaluation of $M$, if performed leftmost-outermost is as in Figure \[fig:examplereductiontree\]. In particular, the probability of convergence for $M$ is $\frac{1}{4}$.
[R]{}[.3]{}
Please observe that two copies of the argument $I\oplus\Omega$ are produced, and that the “rightmost” one is evaluated only when the “leftmost” one converges, i.e. when the probabilistic choice $I\oplus\Omega$ produces $I$ as a result.
The main idea behind building the Taylor expansion of any $\lambda$-term $M$ is to describe the dynamics of $M$ by way of *linear approximations* of $M$. In the realm of the $\lambda$-calculus, a linear approximation has traditionally been taken as a *resource $\lambda$-term*, which can be seen as a pure $\lambda$-term in which applications have the form $\linapp{s}{\bag
t}$, where $s$ is a term and $\bag t$ is a *multiset* of terms, and in which the result of firing the redex $\linapp{\lambda x.s}{\bag
t}$ is the linear combination of all the terms obtained by allocating the resources in $\bag t$ to the occurrences of $x$ in $s$. For instance, one such element in the Taylor expansion of $\Delta$ is $\lambda x.(\linapp{x}{[x]})$, where the occurrence of $x$ in head position is provided with only one copy of its argument. If applied to the multiset $[y,z]$, this term would reduce into $\linapp{y}{[z]} + \linapp{z}{[y]}$. Similarly, an element in the Taylor expansion of $\Delta\ I$ would be $\linapp{\lambda
x.\linapp{x}{[x]}}{[I^2]}$, which reduces into $2.\linapp{I}{[I]}$. Another element of the same Taylor expansion is $\linapp{\lambda
x.\linapp{x}{[x]}}{[I^3]}$, but this one reduces into $0$: there is no way to use its resources linearly, i.e., using them without copying and erasing. The actual Taylor expansion of a term is built by translating any application $M\ N$ into an infinite sum $\tay{(M\ N)} = \sum_{n \in
\Nat} \frac{1}{n!}.\linapp{\tay M}{[(\tay N)^n]}$. For instance, the Taylor expansion of $\Delta\ I$ is $\sum_{m,n \in \Nat}
\frac{1}{m!n!}.\linapp{\lambda x.\linapp{x}{[x^m]}}{[I^n]}$. Remark that any summand properly reduces only when $n=m+1$, in which case it reduces to $n!.\linapp{I}{[I^m]}$. In turn $\linapp{I}{[I^m]}$ reduces properly only when $m=1$, and the result is $I$. All the other terms reduce to $0$. In the end the Taylor expansion of $\Delta\ I$ normalises to $\frac{2!}{1!2!}.I=I$.
Extending the Taylor expansion to probabilistic terms seems straightforward, a natural candidate for the Taylor expansion of $M\oplus N$ being just $\frac{1}{2}.\tay M + \frac{1}{2}.\tay N$. When computing the Taylor expansion of $M$ we will find expressions such as $\linapp{\lambda x.\linapp{x}{[x]}}{[(\frac{1}{2}.I + \frac{1}{2}.\tay
\Omega)^2]}$, i.e. $\frac{1}{4}.\linapp{\lambda
x.\linapp{x}{[x]}}{[I^2]} + \frac{1}{4}.\linapp{\lambda
x.\linapp{x}{[x]}}{[\Omega^2]} +\frac{1}{2}.\linapp{\lambda
x.\linapp{x}{[x]}}{[I,\Omega]}$. For non-trivial reasons, the Taylor expansion of any diverging term normalises to $0$, so just like in our previous example, the only element in $\tay M$ which does not reduce to $0$ is $\linapp{\lambda x.\linapp{x}{[x]}}{[I^2]}$. The difference is that this time it appears with a coefficient $\frac{1}{1!2!}\frac{1}{4}$, so $\tay M$ normalises to $\frac{1}{4}.I$. Please notice how this is precisely the “normal form” of the original term $M$. This is a general phenomenon, whose deep consequences will be investigated in the rest of this paper, and in particular in Section \[sec:trees\].
Notations {#notations .unnumbered}
---------
We write $\Nat$ for the set of natural numbers and $\Rpos$ for the set of nonnegative real numbers. Given a set $A$, we write $\Rpos^A$ for the set of families of positive real numbers indexed by elements in $A$. We write such families as linear combinations: an element $S \in \Rpos^A$ is a sum $S = \sum_{a
\in A} S_a.a$, with $S_a \in \Rpos$. The support of a family $S \in
\Rpos^A$ is $\supp S = \{a \in A \mid S_a > 0\}$. We write $\Rpos^{(A)}$ for those families $S \in \Rpos^A$ such that $\supp S$ is finite. Given $a \in A$ we often write $a$ for $1.a \in \Rpos^A$ unless we want to emphasise the difference between the two expressions. We also define *finite multisets* over $A$ as functions $m : A \rightarrow
\Nat$ such that $m(a) \neq 0$ for finitely many $a \in A$. We use the notation $[a_1,\dots,a_n]$ to describe the multiset $m$ such that $m(a)$ is the number of indices $i \leq n$ such that $a_i = a$.
Probabilistic Resource $\lambda$-Calculus {#sec:rigid}
=========================================
Explicit Probabilistic Taylor Expansion {#sec:taylor_rigid}
=======================================
Generic Taylor Expansion of Probabilistic $\lambda$-terms {#sec:taylor}
=========================================================
On the Taylor Expansion and Böhm Trees {#sec:trees}
======================================
Probabilistic Tree Transition Systems and Testing Equivalence {#sect:tree_transition}
=============================================================
\[sec:testing\]
Implementing Tests as Resource Terms {#sec:taylor_testing}
====================================
Conclusion
==========
|
---
abstract: 'We measure the acoustic scale from the angular power spectra of the Sloan Digital Sky Survey III (SDSS-III) Data Release 8 imaging catalog that includes $872,921$ galaxies over $\sim 10,000 {\rm deg}^2$ between $0.45<z<0.65$. The extensive spectroscopic training set of the Baryon Oscillation Spectroscopic Survey (BOSS) luminous galaxies allows precise estimates of the true redshift distributions of galaxies in our imaging catalog. Utilizing the redshift distribution information, we build templates and [fit to the power spectra of the data, which are measured in our companion paper, @Ho11,]{} to derive the location of Baryon acoustic oscillations (BAO) while marginalizing over many free parameters to exclude nearly all of the non-BAO signal. We derive the ratio of the angular diameter distance to the sound horizon scale ${D_A(z)/r_s}= {9.212^{+0.416}_{-0.404}}$ at $z=0.54$, and therefore, ${D\!_A(z)}= 1411\pm 65 {{\rm\;Mpc}}$ at $z=0.54$; the result is fairly independent of assumptions on the underlying cosmology. Our measurement of angular diameter distance ${D\!_A(z)}$ is $1.4 \sigma$ higher than what is expected for the concordance ${\rm {\Lambda CDM}}$ [@Komatsu11], in accordance to the trend of other spectroscopic BAO measurements for $z\gtrsim 0.35$. We report constraints on cosmological parameters from our measurement in combination with the WMAP7 data and the previous spectroscopic BAO measurements of SDSS [@Percival10] and WiggleZ [@Blake11b]. We refer to our companion papers [@Ho11; @dePutter11] for investigations on information of the full power spectrum.'
author:
- 'Hee-Jong Seo, Shirley Ho, Martin White, Antonio J. Cuesta, Ashley J. Ross, Shun Saito, Beth Reid, Nikhil Padmanabhan, Will J. Percival, Roland de Putter, David J. Schlegel, Daniel J. Eisenstein, Xiaoying Xu, Donald P. Schneider, Ramin Skibba, Licia Verde, Robert C. Nichol, Dmitry Bizyaev, Howard Brewington, J. Brinkmann, Luiz Alberto Nicolaci da Costa, J. Richard Gott III, Elena Malanushenko, Viktor Malanushenko, Dan Oravetz, Nathalie Palanque-Delabrouille, Kaike Pan, Francisco Prada, Nicholas P. Ross, Audrey Simmons, Fernando de Simoni, Alaina Shelden, Stephanie Snedden, Idit Zehavi'
title: 'Acoustic scale from the angular power spectra of SDSS-III DR8 photometric luminous galaxies'
---
Introduction
============
Baryon acoustic oscillations (BAO) imprint a distinct feature in the clustering of photons (i.e., cosmic microwave background), mass, and galaxies. Sound waves that propagated through the hot plasma of photons and baryons in early Universe freeze out as photons and baryons decouple and leave a characteristic oscillatory feature in Fourier space and a single distinct peak in the correlation function approximately[^1] at the distance the sound waves have traveled before the epoch of recombination. The distance is called the “sound horizon scale” and determines the physical location of the BAO feature in clustering statistics [e.g., @Peebles70; @SZ70; @Bond84; @Holtzman89; @HS96; @Hu96; @EH98].
Cosmic microwave background (CMB) data provides an independent and precise determination of the sound horizon scale. Therefore, comparing this sound horizon scale to the observed location of the BAO from galaxy clustering statistics allows one to constrain the angular diameter distance and Hubble parameters, thereby providing information on the nature of dark energy. This approach is known as the ‘standard ruler test’ [e.g., @Hu96; @Eisen03; @Blake03; @Linder03; @Hu03; @SE03]. BAO technique is considered an especially robust dark energy probe [@DETF] for various reasons. First, its physical scale is separately measured from CMB data. Second, the nonlinear effects in the matter density field are still mild at the BAO scale $(\sim 150{{\rm\;Mpc}})$ such that the resulting systematic effects are small and can be modeled with low-order perturbation theories [e.g., @Meiksin99; @SE05; @Jeong06; @Crocce06b; @ESW07; @Nishimichi07; @Crocce08; @Mat08; @Pad09; @SSEW08; @Taruya09; @Seo10]. Third, the observational/astrophysical effects such as galaxy/halo bias and redshift distortions are likely smooth in wavenumber and do not mimic BAO such that they can be marginalized over [e.g., @SE05; @Huff07; @Sanchez08; @Pad09; @Mehta11] \[but see @Dalal11 and @Yoo11 for a possibility of an exotic galaxy bias effect\].
In recent years, BAO have been detected in the galaxy distribution and used to constrain cosmology [@Eisen05; @Cole05; @Hutsi06; @Tegmark06; @Percival07a; @Percival07b; @Pad07; @Blake07; @Okumura08; @Estra08; @Gazt09a; @Gazt09b; @Percival10; @Kazin10; @6dF; @Crocce11; @Blake11a; @Blake11b]. Most of these studies have used a 3D distribution of galaxies from spectroscopic surveys to constrain an isotropic distance scale ${D_V}(z)$ (${D_V}(z)\equiv [(1+z)^2D^2_A(z) cz/H(z)]^{1/3}$ where $D_A$ is the angular diameter distance and $H$ is the Hubble parameter) using spherically averaged clustering statistics, while others have constrained $D_A(z)$ and $H(z)$ separately, using anisotropic clustering information.
Retrieving 3D spatial information requires accurate redshift determination (i.e., spectroscopic surveys), demanding specialized spectrographs and surveys that typically take longer times. Multiband imaging surveys, on the other hand, can more quickly cover a large number of galaxies (low shot noise) and a large area of sky but provide only 2D spatial information, assuming a realistic level of photometric redshift error, and therefore fail to retrieve information on $H(z)$[^2]. Another disadvantage of using the imaging data to make BAO measurements is an additional damping of the BAO due to projection effects and difficulty in applying BAO reconstruction. Nevertheless, imaging surveys can, in principle, provide larger and deeper surveys [@SE03; @Amen05; @Blake05; @Dolney06; @Zhan06] and this prospect has motivated current and future imaging BAO surveys such as the Dark Energy Survey[^3] [DES; @DES], the Panoramic Survey Telescope and Rapid Response System[^4] [PanSTARRS; @Pan], the Physics of the Accelerating Universe survey[^5] [PAU; @PAU], the Large Synoptic Survey Telescope[^6] [LSST; @LSST], and EUCLID[^7] [@EUCLID].
A number of previous works have analyzed and reported the cosmological constraints from the galaxy clustering of the imaging surveys [e.g., @Tegmark02; @Blake07; @Ross08; @Sawang09; @Thomas10; @Thomas11; @Crocce11; @Ross11], but there have been only a few published works on the BAO measurement [@Pad07; @Carnero11]. Our goal in this paper is to design a robust method for measuring the location of BAO in the angular power spectrum of imaging surveys and apply it to the final imaging data set of the Sloan Digital Sky Survey III [SDSS; @York00].
We use the DR8 imaging catalog of SDSS-III that includes photometric redshifts of luminous galaxies (hereafter, ‘LGs’) between $0.45<z<0.65$ over $\sim 10,000 {\rm deg}^2$ [@Ross11]; the spectroscopy from the SDSS-III Baryon Oscillation Spectroscopic Survey [BOSS; @Eisen11] is used to create a training sample and therefore to estimate the true redshift distribution of photometric galaxies. The angular power spectra are generated from the data using an optimal quadratic estimator, as presented in detail in [one of our companion papers]{}, @Ho11. Utilizing the estimated true redshift distribution, we construct a theoretical BAO template for the angular power spectrum, fit to the location of the observed BAO feature in the angular power spectrum, and derive the angular diameter distance to $z=0.54$ in a manner independent of underlying dark energy models.
The angular clustering of galaxies contains more cosmological information than the scale of BAO. Redshift distortions on very large scales and the overall shape of the power spectrum (e.g., the matter-radiation equality feature) can provide additional information. However, in this paper, we take a very conservative approach and use only the most robust probe, the location of BAO, while excluding most of the non-BAO information. @Ho11 [Paper I] presents a more extensive study: it includes information of the full power spectrum and derive cosmological constraints. [In parallel, another companion paper, @dePutter11, measures the mass bound on the sum of neutrino masses using the same power spectra.]{}
This paper is organized as follows. In § \[sec:data\], we briefly summarize the imaging data. In § \[sec:power\], we summarize the method used in Paper I to generate angular power spectra. In § \[sec:method\], we describe details of the BAO fitting method used in this paper. In § \[sec:testassume\], we test our method and assumptions using mock data. In § \[sec:results\], we present an analysis of the DR8 imaging data and report the best fit angular diameter distance to sound horizon ratio at $z=0.54$. We show the robustness of our result and [show the effect of correcting for the observational systematics]{}. In § \[sec:discussions\], we discuss constraints on various cosmological parameters when our BAO measurement is combined with the WMAP7 data and other BAO measurements. Finally, in § \[sec:con\], we summarize the results in this paper.
Data {#sec:data}
====
We use the imaging data of the eighth and final data release [DR8; @Aihara11] of SDSS-III [@York00] that is obtained by wide-field CCD photometry in five passbands ($u$, $g$, $r$, $i$, $z$) [see @Fuku96; @Gunn98; @Gunn06; @Pier03 for more technical and data realease details]. We use the photometric redshift catalog constructed as described in @Ross11[^8]. This catalog is selected from DR8 using the same criteria as the SDSS-III BOSS [@Eisen11] targets selected to have approximately constant stellar mass [CMASS; @White11]. Photometric redshifts and probabilities that an object is a galaxy were obtained using a training sample of 112,778 BOSS CMASS spectra \[to be released with Data Release 9 in July 2012\] [^9]. The final catalog covers $9,913\;{\rm deg}^2$ of sky and consists of $872,921$ galaxies between $0.45<z<0.65$, which is an improvement in the survey area compared to the MegaZ-LRG DR7 catalog [@Thomas11 723,556 objects over $7,746\;{\rm deg}^2$ for $0.45<z<0.65$].
The estimated photometric error, $\sigma_{z_{\rm ph}}$, increases from 0.04 to 0.06 over the redshift range [See Figure 10 of @Ross11]. We define four photometric redshift bins, with widths similar to the photometric error, referred to as [CMASS1]{}, [CMASS2]{}, [CMASS3]{}, [CMASS4]{}. Table \[tab:tdndz\] and Figure \[fig:dndz\] show the distribution of the effective[^10] number of galaxies in each redshift bins. [Due to the extensive training sample, our determination of the redshift distribution is expected to be quite accurate; for example, based on the Jack-knife resampling of the training sample, we estimate the error on the mean/median of the distribution of each redshift bin to be less than 0.5%.]{} The median and mean of the combined galaxy distribution are 0.541 and 0.544, respectively.
Optimal quadratic estimator of angular Power spectra {#sec:power}
====================================================
The auto and cross angular power spectra of the four redshift bins were generated using the optimal quadratic estimator in Paper I, to which we refer the readers for more details of the optimal quadratic estimator [OQE, also see @Seljak98; @Tegmark98; @Pad03; @Pad07]. To summarize, we parametrize the power spectrum with 35 step-function band powers and write the data covariance matrix as $$\begin{aligned}
{\mathbb{C}}_{ij} \equiv {\left\langle}\delta_i \delta_j {\right\rangle}= \sum_\beta p_\beta {\mathbb{C}}_{ij}^{(\beta)} + N_{i}\delta^i_{j},\end{aligned}$$ where $\delta_i$ is the galaxy overdensity at the $i^{\rm th}$ pixel at a given redshift bin, $p_\beta$ is the band power for a wave number bin $\beta$, ${\mathbb{C}}^{(\beta)}$ is the derivative of ${\mathbb{C}}$ with respect to the band power $p_\beta$, and $N$ is the shot noise contribution to the covariance matrix, while $\delta^i_{j}$ is the Kronecker delta function. Assuming $\delta_i$ is Gaussian-distributed, requiring the estimator to be unbiased and to have a minimum variance, we derive a band power estimator $$\begin{aligned}
p_\beta = F_{\beta \gamma}^{-1} \left[\frac{1}{2}{\bf\delta}^t {\mathbb{C}}^{-1}{\mathbb{C}}^{(\gamma)}{\mathbb{C}}^{-1} {\bf\delta} -b_{\gamma} \right],\end{aligned}$$ where $b_{\gamma}$ is the contribution from $N$ (i.e., $\frac{1}{2}tr[{\mathbb{C}}^{-1}{\mathbb{C}}^{(\gamma)}{\mathbb{C}}^{-1}N]$), i.e., the shot noise, and $F$ is the Fisher information matrix: $$\begin{aligned}
F_{\beta \gamma}=\frac{1}{2} tr[{\mathbb{C}}^{-1}{\mathbb{C}}^{(\beta)}{\mathbb{C}}^{-1}{\mathbb{C}}^{(\gamma)}].\end{aligned}$$ The variance of the band power is derived by $$\begin{aligned}
Cov[p_{\beta}, p_{\gamma}]=F^{-1}_{\beta \gamma}.\label{eq:Covab}\end{aligned}$$ The expected value of band power, ${\left\langle}p_{\beta} {\right\rangle}$, is related to the power spectrum $p(\ell')$ at an integer $\ell'$ by the band window function $W_{\beta \ell'}$ [e.g., @Knox99]: $$\begin{aligned}
{\left\langle}p_{\beta} {\right\rangle}=\frac{d{\left\langle}p_{\beta} {\right\rangle}}{d p(\ell')}p(\ell')= W_{\beta \ell'} p(\ell'),\end{aligned}$$ and $$\begin{aligned}
W_{\beta \ell'}=F_{\beta \gamma}^{-1} \frac{1}{2}tr[{\mathbb{C}}^{-1}{\mathbb{C}}^{(\gamma)}{\mathbb{C}}^{-1}{\mathbb{C}}^{(\ell')}], \label{eq:Win}\end{aligned}$$ where ${\mathbb{C}}^{(\ell')}$ is the derivative of ${\mathbb{C}}$ with respect to the power at an integer wavenumber $\ell'$. In this paper, we use the band power $p_\beta$ and the covariance matrix of the band power $F^{-1}$ derived in Paper I. The auto power spectra are shown in Figure \[fig:Pk\]. The cross-power estimates between different redshift bins were generated in a similar manner and used only for determining covariance between different redshift bins.
Methods {#sec:method}
=======
Outlines of the fitting method
------------------------------
Our goal is to robustly measure the location of the BAO scale, and therefore the distance scale ${D\!_A(z)}$, while minimizing possible effects from the assumptions made on the nonlinear or/and observational effects as well as cosmology during the fitting procedure. For example, it is non-trivial to properly model the evolution of the broadband power on the nonlinear scales, whether it is due to structure growth, redshift distortions, or galaxy bias. Therefore, we aim to exclude as much non-BAO signal as possible by marginalizing over the effect of the smooth broad-band power. We measure the location of the BAO feature by fitting the observed auto (band) power spectrum ${C_{\rm obs,z_i}{(\ell)}}$[^11] with the following fitting formula using a template power spectrum ${C_{\rm m,z_i}{(\ell/{\alpha})}}$: $$\begin{aligned}
{C_{\rm obs,z_i}{(\ell)}}=B_{z_i}(\ell){C_{\rm m,z_i}{(\ell/{\alpha})}}+A_{z_i}(\ell), \label{eq:fiteq}\end{aligned}$$ where ${\alpha}$, $B_{z_i}(\ell)$, and $A_{z_i}(\ell)$ are fitting parameters. The functional form for $B_{z_i}$ and $A_{z_i}$ is discussed in § \[subsec:BAs\]. The parameter ${\alpha}$ measures the angular location of the BAO relative to that of the fiducial cosmology. That is, $$\begin{aligned}
&&{\alpha}= {\ell_{\rm obs}}/{\ell_{\rm fid}}= [{D_A(z)/r_s}]_{\rm obs}/[{D_A(z)/r_s}]_{\rm fid},\label{eq:ratioal}\end{aligned}$$ where $[{D_A(z)/r_s}]_{\rm fid}$ is the fiducial angular location of the BAO in the template and $[{D_A(z)/r_s}]_{\rm obs}$ is the measured angular location of the BAO. A value of ${\alpha}> 1$ suggests that the observed angular location of the BAO is smaller than that of the fiducial cosmology. For each redshift bin, $z_i$, free parameters $B_{z_i}(\ell)$ and $A_{z_i}(\ell)$ account for the smooth modification of the power spectrum from the template due to nonlinear structure growth and any scale-dependent bias. Finally, we use power spectra for [CMASS1]{}, [CMASS2]{}, [CMASS3]{}, and [CMASS4]{} simultaneously and fit for a universal ${\alpha}$ while marginalizing over $B_{z_i}$ and $A_{z_i}$ independently at each redshift bin.
The template ${C_{\rm m,z_i}{(\ell/{\alpha})}}$ is constructed from the 2-dimensional projection of 3D power spectrum [@Fisher94; @Pad07]. Including linear redshift distortions, $$\begin{aligned}
&&{C_{\rm m,z_i}{}}(\ell)=\frac{2}{\pi} \int \mathrm{d}k k^2 P_m(k,z_i) \nonumber\\
&&\left( \int \mathrm{d}z \frac{\mathrm{d}N_i}{\mathrm{d}z} b(z) \frac{D(z)}{D(z=0)} \left[ j_\ell\left({r(k,z)}\right) -\beta j_\ell^{''}\left({r(k,z)}\right) \right] \right)^2,\label{eq:Clmode}\end{aligned}$$ where $r(k,z)=k(1+z){D_{A, \rm fid}(z)}$, $dN_{z_i}/dz$ is the normalized, true redshift distribution of galaxies for the corresponding $i^{th}$ photometric redshift bin (Figure \[fig:dndz\]), $j_\ell$ is the spherical Bessel function, $j_\ell^{''}$ is the second derivative of the spherical Bessel function with respect to $r(k,z)$, $b(z)$ is a fiducial galaxy bias, $\beta$ is the fiducial redshift distortion parameter[^12], and $D(z)$ is the linear growth rate. [Due to the projection, redshift distortions significantly affect the broad-band shape of the power spectrum only for $\ell < 30$; see the dashed and dotted lines in Figure \[fig:Pk\] [also see e.g., @Nock10].]{}
Knowing $dN_{z_i}/dz$ precisely is critical for constructing the correct BAO location in the template. Note that, thanks to the extensive spectroscopic training set from the BOSS CMASS galaxies ($112,778$ objects), we have an excellent determination of the redshift probability distribution for our photometric luminous galaxy samples. Once we calculate the template ${C_{\rm m,z_i}{}}(l)$ for the fiducial cosmology, we rescale the wavenumber with ${\alpha}$ to generate ${C_{\rm m,z_i}{}}(\ell/{\alpha})$ that fits the observed power spectrum in Equation \[eq:fiteq\] [while marginalizing over the free parameters $B_{z_i}(\ell)$ and $A_{z_i}(\ell)$]{}.
The term ${P_m}$ is generated by degrading the BAO portion of the fiducial, linear power spectrum with a nonlinear damping parameter ${\Sigma_{m}}= 7.527 [D(z)/D(0)]{h^{-1}{\rm\;Mpc}}$ to mimic the nonlinear evolution of the BAO due to the structure growth [@ESW07]: $$\begin{aligned}
\label{eq:Pmodel}
{P_m}(k,z_i)&=&\left[ {P_{\rm lin}}(k)-{P_{\rm nw}}(k)\right] \exp{\left[ -k^2 {\Sigma_{m}}(z)^2/2 \right]} \nonumber \\
&&+{P_{\rm nw}}(k),\end{aligned}$$ where ${P_{\rm lin}}$ is the linear power spectrum at $z=0$ derived from CAMB [@CAMB] and ${P_{\rm nw}}$ is the nowiggle form, i.e., the power without BAO, calculated using the equations in @EH98. The smoothing of the BAO is dominated by the width of the underlying redshift distribution, and the exact choice of ${\Sigma_{m}}(z)$ does not have a significant impact. For clarity we refer to ${C_{\rm m, z_i}(\ell)}$, instead of ${P_m}$, as a ‘template’ (angular) power spectrum and ${P_m}$ as a ‘base’ power spectrum in this paper. We use a fiducial cosmology similar to the WMAP7 results [@Komatsu11] to define ${P_m}$, ${D_{A, \rm fid}(z)}$, and etc: ${\Omega_m}=0.274$, ${\Omega_\Lambda}=0.726$, $h=0.7$, ${\Omega_b}=0.046$, $n_s=0.95$, and ${\sigma}_8=0.8$. These values produce the fiducial sound horizon scale, $r_s=153.14{{\rm\;Mpc}}$, based on @EH98, and the fiducial angular location of the BAO, $[{D_A(z)/r_s}]_{\rm fid}=8.585$ at $z=0.54$.[^13]
Note that, in equation \[eq:Clmode\], we are assuming that rescaling the sound horizon and angular diameter distance is equivalent to rescaling $\ell$ (i.e., the second equality of Eq. \[eq:ratioal\]), which we call ‘$\alpha$ model’. This will be a reasonable approximation when the thickness of the redshift distribution that is projected on the 2D celestial surface is much larger than the scale of the clustering, i.e. in the limit of the Limber approximation. Based on the mock test in § \[sec:testassume\] and the robustness test in § \[sec:results\], the $\alpha$ model appears to be a good enough approximation for our choice of fitting range and parameterization.
The measured band power ${C_{\rm obs,z_i}{(\ell)}}$ has a contribution from a range of wave numbers, which is described by a window function (Eq. \[eq:Win\]). Due to the large and contiguous survey area, the window function is sharply peaked with little correlation with neighboring bands (Figure [\[fig:windowf\]]{}). We account for this window function effect in the fitting. That is, for each wavenumber band $l_A$ for the redshift bin $z_i$, $$\begin{aligned}
&&{C_{\rm obs,z_i}{(\ell_A)}}= \nonumber \\
&& \times \int^{600}_1 [B_{z_i}(\ell'){C_{\rm m,z_i}{(\ell'/{\alpha})}}+A_{z_i}(\ell')]W_i(\ell'|\ell_A)d\ell'. \label{eq:fiteqw}\end{aligned}$$
A choice of $B(\ell)$ and $A(\ell)$ {#subsec:BAs}
-----------------------------------
There are a few considerations to make when choosing the optimal parametrization for $B(\ell)$ and $A(\ell)$. We want $B(\ell)$ and $A(\ell)$ to be flexible enough to model and remove the broad-band shape of the power spectrum. With the flexibility in $B(\ell)$ and $A(\ell)$, we also gain some tolerance on a possible, small difference between the BAO feature in the fiducial template and the observed feature by trading power between $B(\ell){C_{\rm m, z_i}(\ell)}$ and $A(\ell)$. On the other hand, an arbitrarily flexible $B(\ell)$ and $A(\ell)$ will undesirably mimic BAO even with the no-BAO template. An extensive test of the parametrization for $B(\ell)$ and $A(\ell)$ for the template fitting method is discussed in [@SSEW08], where they allow a large number of free parameters for $B$ and $A$ based on the spherically-averaged power spectra from [[*N*]{}-body]{} realizations which correspond to a spectroscopic survey. With our photometric redshift uncertainty, a projection of the BAO from different distances introduces an additional damping in the feature, leaving higher harmonics other than the first much less distinct relative to the noise level. Therefore we are forced to limit the flexibility in $B(\ell)$ and $A(\ell)$ more strictly than the case assumed in [@SSEW08] while still making sure that the BAO scale is correctly recovered. We choose a revised fitting range so that the broadband is well modeled despite the smaller number of $B(\ell)$ and $A(\ell)$.
Based on tests with mock catalogs (§ \[sec:testassume\]), we use a fitting range $30<l<300$ and a linear function in $\ell$ for $B_{z_i}$ and a constant $A_{z_i}$ (abbreviated with ‘A0B1’ hereafter): $$\begin{aligned}
B_{z_i}(\ell)&=&B_{z_i0}+B_{z_i1}\ell \\
A_{z_i}(\ell)&=&A_{z_i0}.\end{aligned}$$ Therefore, for four redshift bins, we fit for a total of 13 parameters including ${\alpha}$.
Parametrizing ${D\!_A(z)}$
--------------------------
The observed location of the BAO peaks in a power spectrum is determined by the angular diameter distance at each redshift. In a sample selected using photometric redshift, the location of the BAO feature for a given redshift slice depends on an integration over the broad range of the true redshift distribution, rather than a distance at a single redshift, as evident in equation \[eq:Clmode\]. In an ideal case, we may attempt to constrain the redshift dependence of ${D\!_A(z)}$ over the entire spectroscopic redshift range in a non-parametric way [e.g., @Percival10] in the fitting process. In reality, the Fisher matrix analysis predicts an error of $\sim 4\%$ on a single measurement of ${D\!_A(z)}$ for all our four redshift bins combined. Also while the expected true redshift distribution has long tails ($0<z_s<1$), the distribution tends to peak sharply for each photometric bin so that most of the information for a given bin is well concentrated within $\pm \sigma_{z_{\rm ph}}$. Third, for a reasonable range of cosmology, the shape of ${D\!_A(z)}$ does not evolve significantly between $z=0.45$ and $z=0.65$. We therefore expect little gain for designing our analysis to measure multiple ${D\!_A(z)}$s, i.e., the evolution of ${D\!_A(z)}$ as a function of redshift. We instead design our fitting method to measure a single, more precise distance measurement at the redshift that contributes the most information. Based on the median and mean redshift of the photometric sample, we assign our measurement of $D_A$ to $z=0.54$.
In detail, we assume ${D_{A, \rm fid}(z)}$, given by the fiducial cosmology, and set $$\begin{aligned}
{D\!_A(z)}= {\alpha}{D_{A, \rm fid}(z)}.\label{eq:daeq}\end{aligned}$$ That is, we fix the shape of the ${D\!_A(z)}$ to be the same as ${D_{A, \rm fid}(z)}$ and measure the amplitude of ${D\!_A(z)}$.
Clustering evolution of lumious galaxies
----------------------------------------
In generating the template ${C_{\rm m, z_i}(\ell)}$, we need to make a prior assumption on the evolution of the galaxy bias of LGs and the linear growth rate (Eq. \[eq:Clmode\]). We consider two extreme cases of the galaxy clustering evolution: first, we assume that the overall clustering, $b^2D^2$, does not change with redshift, which we call as ‘[*con-cluster*]{}’. Second, we assume that the bias does not change with redshift, which we call as ‘[*con-bias*]{}’. The two cases make little difference in the final best fit of ${\alpha}$, mainly because the expected true redshift distribution sharply peaks within $\pm \sigma_{z_{\rm ph}}$, compared to the galaxy clustering evolution. Note that, by marginalizing over $B_{z_i}$ at each photometric redshift bin $z_{i}$, we take into account the evolution of galaxy clustering across different redshift bins whether we use ‘con-cluster’ and ‘con-bias’. As a default, we fix $b=2$ inside ${C_{\rm m, z_i}(\ell)}$ (i.e., ‘[*con-bias*]{}’, and therefore the best fit $B_{z_i}$ can be [*approximately*]{} interpreted as $b^2(z_i)$.
Testing the method {#sec:testassume}
==================
Before applying our fitting method to the real data, we want to validate, using mock catalogs, that our fitting method returns an accurate estimate of the BAO scale. In other words, we want to check that neither our process of deriving optimal quadratic estimators of band powers nor our fitting method biases the measured BAO scale.
[[*N*]{}-body]{} mocks {#subsec:Mocks}
----------------------
As explained in Paper I in detail, we generate mock catalogs of our imaging data making use of the 20 CMASS mocks constructed by [@White11]. We call these ‘[[*N*]{}-body]{} photoz-mocks’. The cosmology used for generating these mocks is the same as our fiducial cosmology. The comoving volume of the original CMASS mock is $[1.5{h^{-1}{\rm\;Gpc}}]^3$ and, to build [[*N*]{}-body]{} photoz-mocks, we extract an octant of a spherical shell between $r=1.33{h^{-1}{\rm\;Gpc}}(z=0.5)$ and $1.45{h^{-1}{\rm\;Gpc}}(z=0.55)$ from the origin (one corner of a simulation box) and project galaxies along the radial direction without introducing photometric redshift errors. [For simplicity,]{} we do not include redshift distortions, which will be visible only on very large scales (Figure \[fig:Pk\]), nor the effect of the mask in generating these mocks. The resulting power spectrum of the projected field has a BAO feature that is quite similar to that expected for [CMASS2]{}(i.e., $0.5<z<0.55$). Each of the resulting [[*N*]{}-body]{} photoz-mocks spans a $\pi/2\;{\rm rad}^2\;(=5157\;{\rm deg}^2)$ and contains $\sim 125,000$ galaxies. While the number of galaxies is smaller than [CMASS2]{}, the amplitude of power spectrum is boosted overall by almost the same factor due to the thinner redshift slice: the signal-to-noise ratio per mode of each mock is therefore similar to [CMASS2]{}. (see Figure \[fig:MockPk\]). Therefore the photoz-mocks serve as reasonable mocks for the observed power spectrum.
We repeat the procedure by placing an origin at eight different corners of each simulation box and generate eight sets (i.e., eight lines of sight) of 20 imaging mocks, i.e., a total of 160 mocks. Note that the eight lines of sight from each simulation box share a portion of volume and therefore are not independent of each other. We generate the template power spectrum based on the galaxy distribution of the mock catalogs using Equation \[eq:Clmode\].
To better detect a possible bias on ${\alpha}$ when using our fitting method, we increase the signal-to-noise ratio by averaging many power spectra. We average power spectra of the 20 mocks for each configuration (i.e., each line of sight) and fit for ${\alpha}$. For the eight configurations, using ‘A0B1’ (i.e., with $B_0+B_1\ell$ and $A_0$), we show the derived best fits and the 68.3% range of the likelihood in Figure \[fig:Nbodyfit\]. The distribution of eight ${\alpha}$ values appears slightly wider than what one would expect from a Gaussian distribution. However, it is not appropriate to compare this result with a Gaussian case, as the number of sets (i.e., eight) is too small to estimate the underlying distribution and the eight configurations are not independent from each other but share some portion of their volumes. Overall, we do not see an obvious indication [either]{} that the best fit ${\alpha}$ from our OQE estimator is biased [or that our fiducial fitting method introduces a bias]{}.
Gaussian CMASS mocks {#subsec:Gmocks}
--------------------
We next test the accuracy of our fitting method using sets of mocks that have exactly the same noise properties as the real data. We generate many Gaussian mock power spectra using the covariance matrix for the real data described in § \[subsec:cov\]. In detail, we find the best fit band powers for the power spectra measurements of [CMASS1]{}, [CMASS2]{}, [CMASS3]{}, and [CMASS4]{} over the entire wavenumber range (i.e., $1<l<600$), while deliberately using a slightly different choice of $B(l)$ and $A(l)$ (A2B0) than our fiducial method; ${\alpha}$ is fixed to be unity during this process. This approach allows us to construct theoretical power spectra for the fiducial cosmology while taking into account the realistic broad-band shape of LGs. Using the four best fit band power spectra and using the covariance matrix for the real data, we generate 500 sets of Gaussian CMASS mocks. For each set, the four CMASS mock power spectra therefore mimic [CMASS1]{}, [CMASS2]{}, [CMASS3]{}, and [CMASS4]{} not only in terms of clustering but also in terms of the covariance among them. One of the mock [CMASS2]{} is shown in the top left of Figure \[fig:Gfour\] in comparison to the real data.
We then apply our fiducial fitting method to the 500 CMASS mocks. Since the mock data and the template use the same fiducial cosmology, we expect the average value of ${\alpha}$, if unbiased, to be unity. The top right panel of Figure \[fig:Gfour\] shows the pdf distribution of the 500 best fit ${\alpha}$ values when we fit to the mock data over $30<l<300$ using A0B1. The mean and the standard deviation of ${\alpha}-1$ is $(0.10 \pm 6.60)\%$. After rescaling the standard deviation by the square root of the number of samples, the mean and the error associated with the mean value of ${\alpha}$ is $(0.10\pm 0.30)\%$, i.e., unbiased within $1-\sigma$. Decreasing the number of free parameters to A0B0 causes a biased estimation of ${\alpha}$ as large as $2\%$ in ${\alpha}$. [We find A1B0 also gives an unbiased result, while we find that parametrizations with more free parameters result in a non-negligible degree of bias on the BAO location.]{}
The distribution has significant ‘tails’ as shown in Figure \[fig:Gfour\]. As a result, the standard derivation (6.60%) is much larger than the range that contains 68.3% of the distribution: ${\alpha}-1(\%)={0.1^{+4.8}_{-4.3}}$. That is, the distribution of ${\alpha}$ appears wider than the Gaussian one. We have excluded a few catastrophic outliers ($\sim 0.8\%$ of the samples for the fiducial case) that show ${\alpha}-1(\%) > 40$ in deriving these statistics. Obviously, these cases are not fitting to the BAO feature. Some of these catastrophic outliers show a factor of a few larger errors on ${\alpha}$ than the rest of the samples, while others show a reasonable error associated with the best fit. We find that the reduced $\chi^2$ of these catastrophic outliers are not necessarily large. While we exclude these extreme outliers so that the statistics of the distribution are not dominated by these occasions, we note that, in analyzing a real data, we expect to derive a very wrong value of BAO scale more likely than would be expected were this distribution perfectly Gaussian.
For the real data, we have only one realization and therefore need to check that the error we quote for the data will be closely approximating the $68.3\%$ range of the sample distribution if we had more than one sample. For each CMASS mock set, we derive errors associated with the best fit ${\alpha}$ by applying $\Delta \chi^2 = \pm 1$ or by deriving the width that contains 68.3% of the likelihood. On average, the resulting error on ${\alpha}$ for an individual CMASS mock is ${\alpha}-1(\%)={0.1^{+4.7}_{-4.4}}$ and ${0.1^{+5.2}_{-4.7}}$ for $\Delta \chi^2$ of $\pm 1$ (dotted red lines in Figure \[fig:Gfour\]) and the 68.3% width of the likelihood (dotted magenta lines), respectively; the latter is slightly larger than the former. These are reasonably similar to the $68.3\%$ range of the best fit distribution of the mocks, which was ${\alpha}-1(\%)={0.1^{+4.8}_{-4.3}}$ (dotted blue lines). We therefore will quote the 68.3% range of the likelihood surface as our formal error for the real data.
Variations in the template {#subsec:vartem}
--------------------------
In Section \[subsec:Gmocks\] we assumed that the fiducial cosmology used for constructing the template matched the true cosmology, and we could recover an unbiased result in this situation. In fact we also need to confirm that our method is unbiased when the cosmology used in the analysis does not match the true cosmology. The shape of the BAO feature is determined by the matter density and the baryon density, which is well measured by the current CMB observations. [@SSEW08] investigated how the results of the template fitting depends on small deviations in ${P_m}$ and found that the effect is negligible (less than 0.02% bias on ${\alpha}$). As explained in § \[subsec:BAs\], our method in this paper uses a smaller set of free parameters due to the lower signal-to-noise level of the real data [^14] and, as a result, it is possible that the fitting result is more sensitive to the deviation in ${P_m}$. We therefore revisit this issue. Since ${\Omega_b h^2}$ is better measured than ${\Omega_m h^2}$ by the CMB data, we only vary ${\Omega_m h^2}$ from the fiducial value, while the current CMB constraint on ${\Omega_m h^2}$ is $0.1326\pm0.0063$ [@Komatsu11]. In order to leave the shape of the angular diameter distance to redshift relation unchanged, we hold ${\Omega_m}$ fixed and vary $h$ accordingly.
The bottom left panel of Figure \[fig:Gfour\] shows the distribution of the best fit ${\alpha}$ of 500 Gaussian CMASS mocks when the template is built using ${\Omega_m h^2}=0.148$ (i.e., $10\%$ away from ${\Omega_m h^2}=0.134$ used for the mocks). We again use the fiducial, A0B1 parameter set and a range of $30<l<300$. We find that using a smaller number of parameters, i.e., A0B0, makes the result more vulnerable to the variations in the template. Based on the sound horizon scale and $h$ in this cosmology, we expect ${\alpha}-1 (\%) = 1.6$ (the dashed vertical line). The average of the best fit is ${\alpha}-1 (\%) = 1.96 \pm 0.31$, and therefore we recover the correct BAO scale. The bottom right panel of Figure \[fig:Gfour\] shows the distribution of ${\alpha}$ using ${\Omega_m h^2}=0.127$, i.e., $-5\%$ away from the true value used for generating the mocks. We expect ${\alpha}-1 (\%)= -2.10$ while we measure $-1.84\pm 0.27$; we recover the expected value. We also have tested A1B0: we find a moderate bias for ${\Omega_m h^2}=0.148$ due to asymmetric tails, while the bias is overall less than 0.6%. This parametrization therefore would be a reasonable choice as well given the level of signal to noise of our data.
In the next section, we will apply the same test to the real data and show that the measured BAO scale does not change as a function of the fiducial cosmology assumed for the template. In addition to the assumption we made for ${P_m}$, we have also assumed a fiducial relation of angular diameter distance to redshift. We will show that we recover the same result over a range of angular diameter distance to redshift relationship.
Summarizing our mock tests, we find that the angular power spectra produced by our OQE code [@Ho11] show no strong sign of a bias in its BAO feature and that ‘A0B1’ is a good choice of parametrization over $30<l<300$ for our data quality and therefore returns the correct BAO scale within $0.3\%$ over a reasonable range of variations in our assumption.
Results: DR8 imaging data {#sec:results}
=========================
Building the covariance matrix {#subsec:cov}
------------------------------
We use Gaussian covariance matrices calculated for the auto power spectra of the four DR8 redshift bins using equation \[eq:Covab\]. As shown in Paper I, we do not find an obvious indication that the Gaussian covariance matrix for the OQE estimator underestimates the true error of the 2-dimensional projection of the nonlinear galaxy field. Also @Takahashi11 and @Ngan11 have shown a negligible effect of non-Gaussian errors on the BAO measurement in multi-parameter fitting. We therefore retain the Gaussian field assumption when deriving the covariance between different redshift bins as well: we use the cross-power spectra between different redshift bins, while taking the window function due to the survey mask into account (see Paper I for more details).
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Best fit angular location of BAO
--------------------------------
We apply our fitting method to the DR8 imaging data and constrain the angular location of the BAO. Figure \[fig:flrgall\] shows our best fit result using the combinations of [CMASS1]{}, [CMASS2]{}, [CMASS3]{}, and [CMASS4]{}. The red lines/points show the best fit $C_l$ with the BAO template in comparison to the measured data (black squares with error bars). The range of red points show the range of the fitting, i.e., $30<l<300$. In the figure, we denote the best universal fit ${\alpha}$ with the associated errors that correspond to the $68.3\%$ range of the likelihood distribution: we derive ${\alpha}-1 (\%) = {6.61^{+4.68}_{-4.82}}$. The reduced $\chi^2$ at the best fit is $1.20$ for 87 degrees of freedom, and the probability of having a reduced $\chi$ value that exceeds this value is 10%. The left panel of Figure \[fig:flrgallchi\] shows the resulting $\chi^2$ surface along ${\alpha}$ when marginalized over other parameters (red line). Note that, due to the oscillatory feature of the BAO both in the data and the template, there are local minima around the global minimum of $\chi^2$. As implied in the figure by the extent of the red line, when we derive the $68.3\%$ range of the likelihood, we only include $\chi^2$ over $0.08<{\alpha}<1.4$, avoiding the local minima beyond this range. The right panel of Figure \[fig:flrgallchi\] shows a stacked $C_l/C_{\rm l,sm}$ of the four panels of Figure \[fig:flrgall\]. We perform this stacking procedure as follows. To better visualize the BAO feature we measured, we shift the wavenumbers of the four power spectra by $D_A(z_{\rm median})/D_A(z=0.54)$, where $D_A(z_{\rm median})$ is the median redshift for each redshift bin. We combine the four band powers, re-bin the combined data while inversely weighting each band power by its error. The solid red line is the best fit for [CMASS2]{} after its wavenumber is rescaled to mimic a result at $z=0.54$.
Interpreting ${\alpha}-1(\%)= {6.61}$ requires our determination of the redshift to which this measurement corresponds. Strictly speaking, the best fit value of ${\alpha}$ represents a constant ratio of the observed ${D_A(z)/r_s}$ to the fiducial ${[D_A(z)/r_{s}]_{\rm fid}}$ assumed in the template. The black solid and dashed lines (with a shade) in Figure \[fig:variousda\] show what the best fit and the $1-\sigma$ error on ${\alpha}$ imply in this strict interpretation. However, although the redshift dependence of ${D\!_A(z)}$ we assume spans $z\sim 0-1$, most of the galaxies are within $z= 0.45$ and $0.65$ with a peak of the distribution near $0.5<z<0.55$. Therefore, it is reasonable to consider that the best fit ${\alpha}{[D_A(z)/r_{s}]_{\rm fid}}$ represents ${D_A(z)/r_s}$ near $z = 0.5-0.55$. The median and the mean of the weighted galaxy distribution are 0.541 and 0.544, respectively. We therefore adopt $z=0.54$ as the characteristic redshift that our BAO measured scale represents.
To show that the best fit BAO scale indeed does not depend on the cosmology we assume for ${[D_A(z)/r_{s}]_{\rm fid}}$, we repeat our fitting with templates constructed using different cosmologies. In detail, we vary the equation of state of dark energy, $w$, by $\pm 0.2$, such that ${[D_A(z)/r_{s}]_{\rm fid}}$ at $z=0.54$ varies by $\sim 3.4-3.7\%$, i.e., slightly less than the $1-\sigma$ range associated with the best fit ${\alpha}$. In other words, we are testing the consistency of our answer by varying the template by $\sim 1-\sigma$ from the fiducial case of ${\alpha}$. Using a template with $w=-1.2$ , the best fit gives ${\alpha}-1= {2.98^{+4.81}_{-4.48}}\%$ and, with $w=-0.8$, ${\alpha}-1= {10.21^{+5.02}_{-4.85}}\%$.
The top panel of Figure \[fig:variousda\] shows the best fit ${D\!_A(z)}/r_s$ ($={\alpha}{D\!_A(z)}/r_{s,\rm fid}$) using the three different templates. The solid lines show the best fit and the dotted lines show a $1\sigma$ range of ${D\!_A(z)}/r_s$. The bottom panel displays ratios of the best fit ${D\!_A(z)}/r_s$ using no-${\rm {\Lambda CDM}}$ templates with respect to the best fit ${D\!_A(z)}/r_s$ using our fiducial ${\rm {\Lambda CDM}}$ template. From the top and bottom panel, one sees that the three different templates have a very similar shape in ${D\!_A(z)}$ over $z=0.45-0.65$, once the absolute difference is absorbed into ${\alpha}$. The three templates return virtually the same ${D_A(z)/r_s}$ at $z=0.54$: they are consistent within $0.3\%$. We therefore quote the best fit using our fiducial ${\rm {\Lambda CDM}}$ template as our official measurement: ${D_A(z)/r_s}= {9.212^{+0.416}_{-0.404}}$ at $z=0.54$. Using the current WMAP7 constraint on the sound horizon at drag epoch, $153.2 \pm 1.7 {{\rm\;Mpc}}$, we derive angular diameter distance ${D\!_A(z)}= 1411\pm 65 {{\rm\;Mpc}}$ at $z=0.54$ [^15]. Table \[tab:fitda\] summarizes our best fit BAO location and the derived distance scale.
We further test the robustness of our result by constructing templates using various cosmologies. Figure \[fig:robustda\] shows the best fit ${D_A(z)/r_s}$ at $z=0.54$ assuming $w$CDM, $o$CDM, ${\rm {\Lambda CDM}}$, and assuming different values of ${\Omega_m h^2}$. For the range of cosmologies we have investigated in this paper, the best fit varies less than $1\%$ in the acoustic scale while the $1\sigma$ error is $\sim 4.7\%$. The errors vary slightly more than the variations in the best fit, especially when the template cosmology deviates substantially from the concordance cosmology. [We also test a different parametrization than the fiducial choice, A1B0, which has marginally passed the mock test (i.e., a likely bias of $\sim + 0.6\%$ on ${\alpha}$ based on the result in § \[subsec:vartem\]). A1B0 gives ${\alpha}-1= 7.5(\%)$, which is consistent with the fiducial result within $1\%$.]{}
If we remove the BAO in the template (the blue line in Figure \[fig:flrgallchi\]), we essentially fail to constrain ${\alpha}$. This means that the flexibility in our fitting is sufficient that the broadband shape information cannot constrain ${\alpha}$. Therefore, we conclude that our measurement of ${D_A(z)/r_s}$ is mainly from the BAO information.
The precision of our measurement is much better than that of @Carnero11, where they measure the BAO location within 9.7%. The discrepancy arises partly because of the larger survey-area coverage in this work, but the main difference is due to the difference in the modeling. @Carnero11 assume a quite general model without utilizing the true redshift distribution of their photometric galaxies, and account for the various scenarios of the deviation between their model and the real data in their error budget. Our method is, on the other hand, a template-based approach utilizing the true redshift distribution of our photometric galaxies that is quite well determined by the extensive training set and generate a template with a precise BAO location given the redshift distribution and cosmology. Our method therefore is quite immune to the most of the systematic errors they account for. We observe approximately 1% of variations depending on the choice of fiducial cosmology and parametrization, which has very small effect when added quadratically to our current error of 4.7%.
### Significance of detection
A reasonable concern regarding our measurement is whether or not we have fitted to only the BAO feature, or whether the result is offset due to noise spikes, which is obviously related to the significance of detection. The conventional method of determining significance of BAO detection is to use a template with (i.e., BAO template) and without BAO information (i.e., no-BAO template) and observe the difference between $\chi^2$ values of the two best fits. Unfortunately, such $\Delta \chi^2$ is very model-dependent. Using A0B1, we derive $\Delta \chi^2=4$, which can be conventionally interpreted as a $2\sigma$ detection of BAO. However, such detection level depends on the choice of parametrization.
We reconsider this issue of the detection level. Various observations including WMAP7 [@Komatsu11] and galaxy surveys [e.g., @Blake11b], have already shown that BAO feature exists. Given the signal-to-noise ratio level of our data, we are interested in how likely we have fitted to a BAO feature not a noise feature, rather than detecting the existence of BAO. Therefore, rather than fitting the power spectra with a no-BAO template, we shall fit [many realizations of]{} the power spectra with a BAO template and see how often we derive the correct BAO scale. For a large sample variance, the BAO feature in the power spectrum may be wiped out by noise [e.g., @Cabre11]. Our mock test, Figure \[fig:Gfour\], shows that, in the presence of the sample variance that is the same as our data, we recover the true acoustic scale within $\sim 4.6\%$ in 68.3% of the time. Obviously, we are not fitting to a BAO feature in the tails of the distribution. We therefore rephrase our detection level: our measurement is likely to recover the true BAO scale within $4.6\%$ in 68.3% of cases, assuming that BAO exists.
Effect of systematics
---------------------
A number of observational systematics can potentially contaminate the observed galaxy clustering: stellar contamination, seeing variations, sky brightness variations, extinction, and color offsets [@Schlafly10]. However, as long as the systematics do not introduce a preferred scale similar to the BAO scale, i.e., if the systematics only introduce a smooth component in the power spectrum [up to a sample variance]{}, our results would not depend on the contamination from systematics. Paper I moreover has shown that the effect of the survey systematics are small. We therefore have not included the systematic corrections for our main result. In this section, however, we use power spectra that were corrected for the systematics using the method introduced in Paper I [See @Ross11 for a similar method for the correlation function] and observe the effect of the systematics on the result.
The method in Paper I assumes that the effect of systematics is small and linear. In Fourier space, therefore, we assume that the following equation holds for each wave band $\ell$: $$\hat{\delta}_{z_i}(\ell) = \hat{\delta}_{g,z_i}(\ell)+\sum_{s_a} \epsilon_{z_i,a}(\ell)\hat{\delta}_{s_a}(\ell),$$ where $\hat{\delta}_{z_i}(\ell)$ is the observed galaxy density field at the $z_i^{\rm th}$ redshift bin, $\hat{\delta}_{g,i}$ is the true galaxy density field, and $\delta_{s_a}$ are the variation of systematics across the sky. We only include the dominant three systematics identified in Paper I: stellar contamination, seeing variations, and sky brightness variations. If we assume that there is no intrinsic correlation between the systematics and the underlying large scale structure, i.e., $<\hat{\delta}_{g,z_i} \hat{\delta}_{s,a}(\ell)> = 0$, we can solve for $\epsilon_{z_i,a}$ using the measurements of galaxy power spectra (i.e., $<\hat{\delta}_{z_i} \hat{\delta}_{z_j}>$) and the cross-power spectra between galaxies and the systematics (i.e., $<\hat{\delta}_{z_i} \hat{\delta}_{s_a}>$), as presented in Paper I. The error on the band power is minimally propagated: the error is quadratically increased by the amount of the final correction, after taking into account the number of wave modes.
Figure \[fig:flrgallSys\] and \[fig:flrgallchiSys\] show the best fit results when we use the power spectra after systematics correction. We derive ${\alpha}-1= {7.012^{+4.71}_{-4.51}}\%$. The reduced $\chi^2$ has slightly improved to be 1.09. The difference in $\chi^2$ between using the BAO template and the no-BAO template has increased to 6.2 after systematics correction, from the previous 4.0 without systematics correction. [The right panel of Figure \[fig:flrgallchiSys\], in comparison to Figure \[fig:flrgallchi\], shows that the systematics correction, while the effect is small, improves the fit on large scales $(l < 100)$. ]{}
Figure \[fig:robustda\] shows that the best fit value with the systematics correction is consistent with the fit before the systematics correction within 1% of ${\alpha}$, demonstrating that the BAO fitting is fairly robust against the systematics effects. The overall improvement in the statistics after the systematics correction, such as on the reduced $\chi^2$, motivates the usage of the method in Paper I for future surveys, which can be further improved with a more careful error propagation during the correction.
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Discussions: Cosmological implications {#sec:discussions}
======================================
We combine our measurements of $D_A(z=0.54)$ with recent spectroscopic BAO measurements. The spectroscopic surveys report ${D_V}(z)$ that contains both the information along the line of sight, $H(z)$, and the information on the transverse direction, $D_A$. In Figure \[fig:baoall\], we present the distance-to-redshift relations of different BAO measurements in a 2-dimensional space of $D_A(z)$ and $H(z)$. Our measurement of $D_A(z=0.54)$ appears as the black horizontal line with the shaded region representing the associated error. We also show the measurements of ${D_V}(z=0.2)/r_s$ and ${D_V}(z=0.35)/r_s$ from @Percival10 for SDSS DR7 [@DR7] as red lines with magenta shades and ${D_V}(z=0.6)/r_s$ from @Blake11b for the WiggleZ data over $0.2 < z < 1$ as a green line with a light green shade. The black square points (along the dotted line) show the expected ${D\!_A(z)}$ and $H$ at $z=0.2$, 0.35, 0.54, and 0.6 for our fiducial ${\rm {\Lambda CDM}}$. Note that the measurements beyond $z=0.35$ have a tendency to imply the location of the BAO at a smaller scale than the concordance ${\rm {\Lambda CDM}}$ (i.e., a larger ${D\!_A(z)}$ than the fiducial cosmology), including our $D_A$ measurement ($\sim 1.4\sigma$ away). Due to nonlinear structure formation and galaxy bias, we expect about a $\sim 0.5\%$ of bias towards a smaller value on the measured BAO scale [@Crocce08; @Pad09; @Seo10; @Mehta11], which has not been accounted in these measurements. Such correction will slightly improve the consistency between the BAO measurements and the concordance ${\rm {\Lambda CDM}}$, but it is overall a very small effect for the current level of errors. The circles along the dashed line and the crosses along the dot-dashed line in Figure \[fig:baoall\] show the expected ${D\!_A(z)}$ and $H(z)$ based on our best fit $w$CDM and $o$CDM cosmologies from COSMOMC [@cosmomc02] that will be explained below.
We use COSMOMC [@cosmomc02] to combine BAO measurements from the various galaxy surveys with the WMAP7 data [@Komatsu11] to derive constraints on cosmological parameters. For BAO measurements, we use ${D_V}(z=0.2)/r_s$ and ${D_V}(z=0.35)/r_s$ from SDSS DR7 [@Percival10], ${D_V}(z=0.44)/r_s$, ${D_V}(z=0.60)/r_s$, and ${D_V}(z=0.73)/r_s$ from WiggleZ[^16], and ${D\!_A(z)}(0.54)/r_s$ from this work. The WMAP7 data provides the sound horizon scale and the distance to the last scattering surface and therefore, in combination of the BAO measurements from the galaxy surveys, we can break the degeneracies and constrain $w$ and $\Omega_m$ (for $w$CDM) or $\Omega_\Lambda$ and $\Omega_m$ (for $o$CDM). [The cosmological parameters that the COSMOMC chain vary are ${\Omega_b h^2}$, $\Omega_{c}h^2$, $\theta$, $\tau$, $n_s$, $\ln A_s$, and $A_{SZ}$, in addition to $w$ (for $w$CDM) or ${\Omega_K}$ (for $o$CDM); here, $\Omega_{c}h^2$ is the dark matter density, $\theta$ is the approximate ratio of the sound horizon scale to the angular diameter distance to recombination, $\tau$ is the optical depth to reionization, $A_s$ is the primordial superhorizon power in the curvature perturbation on $0.05{{\rm\;Mpc}}^{-1}$ scales, and $A_{SZ}$ is the amplitude of the SZ power spectrum.]{}
The left panels of Figure \[fig:womh\] show marginalized 2-D likelihood contour surfaces that enclose 68.3% and 95.5% of the likelihood (reddish shaded contours) on $\Omega_m$ and $w$ (top) and $\Omega_m$ and $h$ (bottom) assuming a flat $w$CDM, in comparison to the case without our measurement (dashed green lined contours for the spectroscopic BAO measurements). The reddish contour lines in the top left show the constraint from our measurement alone using the current CMB prior on ${\Omega_m h^2}$ (i.e, $0.1326\pm 0.0063$). This contour implies that, given the strong prior on ${\Omega_m h^2}$, adding our measurement of the distance scale at $z=0.54$, which is larger than what is expected in the concordance ${\rm {\Lambda CDM}}$, weighs toward a slightly larger ${\Omega_m}$ and therefore a slightly smaller $h$ with respect to the other data sets. We present the marginalized and the best fits of selective cosmological parameters in Table \[tab:cosmomc\]: $\Omega_m = 0.2912 \pm 0.0292$, $w =-1.0185 \pm 0.186$, $h=0.6884 \pm 0.0392$ for a flat $w$CDM. The right panels show the 2-D contour on $\Omega_m$ and $\Omega_\Lambda$ for $o$CDM while holding $w=-1$. The best fit parameters are $\Omega_m=0.2939 \pm 0.0170 $, $\Omega_K=-0.0057\pm 0.0058$, and $h=0.6748\pm 0.0175$ in this case. Overall, an addition of our measurement slightly increases $\Omega_m$ and decreases $\Omega_K$ toward a more negative value. In terms of errors, including our data point provides only a slight improvement on ${\Omega_m}$ and $h$ for $o$CDM.
Conclusion {#sec:con}
==========
We have measured the acoustic scale from the SDSS-III DR8 imaging catalog using $872,921$ galaxies over $\sim 10,000 {\rm deg}^2$ between $0.45<z<0.65$. Galaxies are binned into four different redshift slices where the width of each slice is 0.05, which is approximately the error associated with photometric redshift determination. Angular power spectra are generated using an optimal quadratic estimator, as presented in Paper I. We use $\sim 110,000$ SDSS III BOSS galaxies as a training sample to derive the true redshift distribution of the galaxies in the imaging catalog and therefore build reasonable template power spectra. We fit the templates to the measured angular power spectra and derive the best fit acoustic scale while marginalizing over sufficient free parameters to exclude any non-BAO signal.
We derive ${D_A(z)/r_s}= {9.212^{+0.404}_{-0.416}}$ at $z=0.54$. Using the current WMAP7 constraint on the sound horizon at drag epoch, $153.2\pm1.7 {{\rm\;Mpc}}$, we derive angular diameter distance ${D\!_A(z)}= 1411\pm 65 {{\rm\;Mpc}}$ at $z=0.54$. Without a BAO feature in the template power spectrum, we cannot constrain a distance scale; the distance information we derive is therefore dominated by the BAO feature for our choice of parametrization.
Our measurement of the distance scale is quite insensitive to the fiducial cosmology we assume for building the template. For a wide range of cosmologies we have investigated in this paper, the best fit varies less than $1\%$ in the acoustic scale while the $1\sigma$ error is $\sim 4.7\%$.
The angular distance scale we derive is $1.4 \sigma$ higher than the concordance ${\rm {\Lambda CDM}}$ model. When combined with three other BAO measurements from SDSS DR7 spectroscopic surveys at $z=0.2$ and 0.35 [@Percival10] and WiggleZ [@Blake11b] at $z \sim 0.6$, we find a tendency of cosmic distances measured using BAO to be larger than the concordance ${\rm {\Lambda CDM}}$ for $z \gtrsim 0.35$. Adding our measurement with these BAO measurements in the presence of WMAP7 prior therefore shifts the best fit $\Omega_m$ slightly larger than the concordance cosmology.
[In this paper, we have aimed at deriving a robust and conservative BAO information from the angular clustering of galaxies. We find that an accurate determination of the true redshift distribution of galaxies is crucial for a good photometric BAO measurement. Although the details of the method would and should vary for the conditions of different surveys, we hope that the approach described in this paper serves as a valuable reference for the analyses of future photometric BAO surveys. ]{}
We thank Chris Blake for providing the best fits and covariance matrix of ${D_V}/r_s$ measured using the WiggleZ data. We thank Patrick Mcdonald for helpful discussions.
Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/.
SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, University of Cambridge, University of Florida, the French Participation Group, the German Participation Group, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.
[^1]: The scale observed in the mass is not exactly the distance travelled when recombination occurs as the momentum of the baryonic material means that the motion continues for a short time after recombination, until an epoch known as the baryon-drag epoch.
[^2]: A fractional redshift error of 0.25% in $1+z$ is at least required to recover $H(z)$ [@SE03].
[^3]: www.darkenergysurvey.com
[^4]: www.pan-starrs.ifa.hawaii.edu
[^5]: www.ice.cat/pau
[^6]: www.lsst.org
[^7]: www.sci.esa.int/euclid
[^8]: Available at <http://portal.nersc.gov/project/boss/galaxy/photoz>.
[^9]: We use the redshifts available through MJD 55510
[^10]: We weight each object by the probability that an object is a galaxy.
[^11]: Note that we switched the notation from $p_{\beta}$ to $C(\ell)$.
[^12]: $\beta ={\Omega_m}^{0.56}(z)/b(z)$
[^13]: [Using the exact calculation of drag epoch rather than the fitting formula in @EH98, we find $r_s=149.18{{\rm\;Mpc}}$ for our fiducial cosmology, therefore $[{D_A(z)/r_s}]_{\rm fid}=8.813$ at $z=0.54$.]{}
[^14]: [@SSEW08] uses an [[*N*]{}-body]{} volume of $320{h^{-3}{\rm\;Gpc^3}}$, as a comparison.
[^15]: [Using the exact integration rather than the fitting formula in @EH98, $r_s=149.18{{\rm\;Mpc}}$ and ${D\!_A(z)}={9.456^{+0.427}_{-0.415}}$ at $z=0.54$.]{}
[^16]: For COSMOMC, we use the three-redshift slice representation of the WiggleZ data from @Blake11b, i.e., $0.2<z<0.6$, $0.4<z<0.8$, and $0.6<z<1.0$, accounting for the covariance among them, while in Figure \[fig:baoall\] we show the result for the whole redshift range ($0.2<z<1.0$). [Note that the distance measurements from the WiggleZ data include non-BAO information.]{}
|
---
abstract: 'A natural analogue of the Krein–Milman theorem is shown to fail for [[CAT($0$)]{}]{}spaces.'
address: 'EPFL, 1015 Lausanne, Switzerland'
author:
- Nicolas Monod
bibliography:
- '../BIB/ma\_bib.bib'
title: 'Extreme points in non-positive curvature'
---
Introduction
============
Functional analysis sometimes offers inspiring analogies for the study of complete metric spaces of non-positive curvature such as [[CAT($0$)]{}]{}spaces. Certain classical results from functional analysis translate into fundamental facts, others into open problems, and others yet lead to counter-examples.
Let us illustrate the first outcome on a bounded closed convex subset $C$ in a complete [[CAT($0$)]{}]{}space $X$. Typically $X$ is not locally compact and $C$ not compact. In a reflexive Banach space, such a set would still be *weakly compact*. Here, $C$ is (quasi-) compact for the weakened “convex topology” [@MonodCAT0 Thm. 14], defined as the weakest (not necessarily Hausdorff) topology in which metrically closed *convex* sets are closed.
Furthermore, in a rudimentary analogue of the Ryll-Nardzewski theorem [@Ryll-Nardzewski_s], every isometry of $C$ has a fixed point [@Bridson-Haefliger II.2.8].
The second outcome occurs for instance when trying to generalize Mazur’s compactness theorem [@Mazur30], because it is unknown if the closed convex hull of a compact set in $X$ remains compact [@Gromov91 6.B$_1$(f)].
The purpose of this note is to establish the third outcome for the Krein–Milman theorem about extreme points.
A point in a uniquely geodesic metric space is called *extreme* if it does not lie in the interior of any geodesic segment. This definition is generalized beyond unique geodesics in the presence of a *bicombing*, encompassing notably all linear spaces (see e.g. [@Descombes-Lang] for bicombings).
The Krein–Milman theorem, originating in [@Krein-Milman], states that a non-empty convex compact subset of a locally convex space has extreme points, and indeed is the closed convex hull of the set of extreme points. For more specific classes of Banach spaces, much stronger conclusions hold, even for bounded closed convex sets $C$ that are not compact in any weakened topology. For instance, in spaces with the Radon–Nikodym property, $C$ has extreme points and is even the closed convex hull of its strongly exposed points [@Phelps74].
Turning back to the metric situation, it follows immediately from the [[CAT($0$)]{}]{}condition that any point *maximizing the distance* to some given point must be extreme. For this reason, metrically compact [[CAT($0$)]{}]{}spaces have extreme points. In fact, in the compact case the existence of a convex bicombing is enough to retain the full conclusion of the Krein–Milman theorem [@Buehler_combing].
However, there is generally no point maximizing any distance in a bounded closed convex set. An elementary example is obtained by gluing together at a single endpoint segments of length $1-1/n$ for all $n\in {\mathbf{N}}$. This example has, of course, plenty of extreme points: it still is the convex hull of its extreme points as in Krein–Milman. The purpose of this note is to show that it ain’t necessarily so:
There exists a bounded complete [[CAT($0$)]{}]{}space $X$ without extreme points.
Moreover, one can arrange that $X$ is compact Hausdorff for the convex topology and that every finite collection of points in $X$ is contained in a finite Euclidean simplicial complex of dimension two. Alternatively, one can construct a [[CAT($-1$)]{}]{}example with hyperbolic simplicial complexes of dimension two.
Our proof uses the Pythagorean identity to play off the square-summability of the harmonic series against its non-summability. This will ensure that we remain in a finite radius while the search for extreme points can go on forever. A crucial step is to establish completeness so that there is no extreme point hiding sub rosa in the completion.
(i) Our example is complementary to Roberts’ famous non-locally convex linear counter-example [@Roberts77] to Krein–Milman: since [[CAT($0$)]{}]{}spaces satisfy a strong convexity condition, it is the non-linearity that is to blame here.
(ii) Although there exist powerful barycentric methods for measures on [[CAT($0$)]{}]{}spaces [@Korevaar-Schoen; @Sturm_s], the theorem shows that they cannot afford a Choquet theory [@Choquet56].
(iii) Of course a space as in the theorem cannot be isometrically realized in a space where Krein–Milman holds; therefore, our “rose” $X$ will be folded in an appropriate way. We shall nonetheless press it flat to measure angles, obtaining an infinitely winding spiral of petals.
The rose
========
*C’est le temps que j’ai perdu pour ma rose…*
Antoine de Saint-Exupéry, *Le Petit Prince*, 1943
Define for each $n\in{\mathbf{N}}$ the radius $r_n= \sqrt{\sum_{p=1}^n 1/p^2 }$. The double *petal* $P_n{\subseteq}{\mathbf{R}}^2$ shall be the closed set $$P_n = \Big\{ (x,y) : (n+1) r_n |x| \leq |y|\leq r_n \Big\}$$ with the path-metric induced from ${\mathbf{R}}^2$. In particular, $P_n$ is a compact [[CAT($0$)]{}]{}space; it can also be viewed as two Euclidean triangles glued at a tip.
We refer to the locus $x=0$ as the *central segment* of $P_n$; it has length $2 r_n$. We call the loci $|y|=(n+1) r_n x$ and $|y|= -(n+1) r_n x$ the *right border* and *left border* of $P_n$; they are segments of length $2r_{n+1}$.
The *rose* is the [[CAT($0$)]{}]{}space $X$ obtained by inductively gluing $2^{n-1}$ copies of $P_n$ as follows, starting from a single copy of $P_1$. For each $n\in {\mathbf{N}}$ and for each copy of $P_n$ we assign two new copies of $P_{n+1}$. The first is glued by identifying its central segment with the right border of the given copy of $P_n$. The second is glued by identifying its central segment with the left border. The gluings make sense since these segments have all length $2 r_{n+1}$, and $X$ is defined to be the increasing union of the successively glued spaces (Figure \[fig:3D\]).
This rose $X$ is indeed a [[CAT($0$)]{}]{}space, see [@Bridson-Haefliger II.11.1]. Notice that each verification of the [[CAT($0$)]{}]{}condition takes place in a finite gluing since the convex hull of a set of point does not visit petals of higher index than these points. The rose is bounded because every point of $P_n$ is at distance at most $r_{n+1}$ of the *center*, which is the common intersection of all copies of all petals. Thus $X$ has radius $\pi/\sqrt 6$.
Furthermore, by construction $X$ has no extreme point. Indeed, if a point in some $P_n$ were extreme in $X$, it would a fortiori be extreme in $P_n$ and hence it would be one of the four outer corners of $P_n$. These four points are, however, all midpoints of segments in appropriate copies of $P_{n+1}$.
We now turn to the critical point and prove that $X$ is complete.
Let $(x_j)_{j=1}^\infty$ be a Cauchy sequence in $X$. Suppose for a contradiction that $(x_j)$ does not converge. Upon discarding finitely many terms, we can assume that $(x_j)$ remains at distance at least ${\epsilon}$ of the center for some ${\epsilon}>0$. Define the *index* $n(x)$ of a point $x\in X$ to be the smallest $n$ such that some copy of $P_n$ contains $x$; if $x$ is not the center, then it is contained in at most one other petal, namely a copy of $P_{n(x)+1}$. The indices $n(x_j)$ are unbounded as $j$ varies, because otherwise $(x_j)$ would be confined to the gluing of finitely many petals, which is a compact metric space; this would imply that $(x_j)$ converges.
In order to contradict the Cauchy assumption, it suffices to prove that for any $j$ there is $i>j$ with $d(x_j, x_i) \geq {\epsilon}$. Fix thus $j$ and consider only those $i>j$ with $n(x_i) \geq n(x_j) +2$.
We can assume that the segment $[x_j, x_i]$ avoids the center of the rose since otherwise $d(x_j, x_i) \geq 2 {\epsilon}$. It follows that $[x_j, x_i]$ traverses notably a non-trivial portion of successive copies of $P_{n(x_j)+1}, P_{n(x_j)+2}, \ldots, P_{n(x_i)-1}$, each time entering on the central segment and leaving through a border. Thus, the Alexandrov angle formed at the center of $X$ by $x_j$ and $x_i$ is at least $\sum_{n=n(x_j)+1}^{n(x_i)-1} {\vartheta}_n$, where ${\vartheta}_n$ is the angle between the central segment and the borders of $P_n$ (Figure \[fig:flat\]). That angle satisfies ${\vartheta}_n > \sin {\vartheta}_n > \sqrt 6/\pi(n+1)$. It now follows from the divergence of the harmonic series that if we choose $i$ so that $n(x_i)$ is large enough compared to $n(x_j)$, then the angle at the center between $x_j$ and $x_i$ will be at least $\pi/3$ and hence $d(x_j, x_i) \geq {\epsilon}$.
We observe in passing that the above argument with $\pi/3$ replaced by $\pi$ shows in fact that $[x,y]$ contains the center as soon as $n(y)$ is large enough compared to $n(x)$. Therefore, any sequence $(y_j)$ with $n(y_j)$ going to infinity will converge to the center in the convex topology.
To prove that the convex topology of $X$ is Hausdorff amounts to the following. Given distinct points $x,y\in X$, we need to cover $X$ with finitely many closed convex subsets $U_i, V_i{\subseteq}X$ such that $x\notin U_i$ and $y\notin V_i$ for all $i$. We shall apply the following sufficient condition: it is enough to find a (metrically) compact convex subset $K$ containing $x,y$ such that $X\setminus K$ can be covered by finitely many closed convex subsets $W_i{\subseteq}X$ with $y\notin W_i$ for all $i$. This is indeed sufficient because we can first cover $K$ with finitely many closed convex subsets $U_i, V_i{\subseteq}K$ as above since the convex topology of $K$ coincides with the metric topology [@MonodCAT0 Lem. 17], hence is Hausdorff. These sets $U_i, V_i$ are still closed and convex as subsets of $X$ and it now suffices to add all $W_i$ to the collection of $V_i$ to cover $X$.
We now verify the condition. Upon possibly exchanging $x$ and $y$ we can assume that $y$ is not the center. Choose $n>n(x), n(y)$ and define $K$ to be the union of all copies of $P_m$ over all $m\leq n$ with the gluings of the rose. Then define $W_1, \ldots, W_{2^n}$ to be the $2^n$ connected components of the union of all copies of $P_m$ over $m>n$, with the gluings specified by the construction for $m>n$ only. These sets are all closed convex and satisfy the criteria of the sufficient condition.
This concludes the proof of the theorem for the [[CAT($0$)]{}]{}statement. The [[CAT($-1$)]{}]{}construction is virtually identical with hyperbolic triangles, the key being that the trigonometric estimates remain the same at the first order as the angles ${\vartheta}_n$ converge to zero.
We observe that in either case the full isometry group of the rose is an infinite iterated wreath product of Klein four-groups (see e.g. [@BOERT §IV.4] for such iterated products).
|
---
bibliography:
- 'mnemonic.bib'
- 'ref\_user.bib'
---
[{width="\textwidth"}]{}
[**[Probing the near-IR flux excess in young star clusters ]{}\
Adamo Angela$^{1}$, [Ö]{}stlin G[öran]{}$^{1}$, and Zackrisson Erik$^{1}$** ]{}\
$^{1}$ Department of Astronomy, Stockholm University, Oscar Klein Center, AlbaNova, Stockholm SE-106 91, Sweden\
Abstract {#abstract .unnumbered}
========
We report the results of a recent study of young star clusters (YSCs) in luminous blue compact galaxies (BCGs). The age distributions of the YSCs suggest that the starburst episode in Haro 11, ESO 185-IG13, and Mrk 930 started not more than 30-40 Myr ago. A peak of cluster formation only 3 - 4 Myr old is observed, unveiling a unique sample of clusters still partially embedded. A considerable fraction of clusters (30 - 50 %), mainly younger than 10 Myr, shows an observed flux excess between 0.8 and 2.2 $\mu$m. This so-called near-infrared (NIR) excess is impossible to reproduce even with the most recent spectral synthesis models (that include a self-consistent treatment of the photoionized gas). We have used these YSCs to probe the very early evolution phase of star clusters. In all the three host galaxies, the analysis is limited to the optically brightest objects, i.e., systems that are only partially embedded by their natal cocoons (since deeply embedded clusters are probably too faint to be detected). We discuss possible explanations for this NIR excess, in the context of IR studies of both extragalactic young star clusters and resolved massive star forming regions in the Milky Way and in the nearby Magellanic Clouds.
Introduction \[intro\]
======================
Young star clusters (YSCs) are considered a powerful tool to study the star formation history of their hosts. However, many assumptions and constraints on the evolutionary properties are needed in order to compare YSCs properties and the bulk of the host stellar population. In nearby galaxies, it is possible to study both the resolved stellar population and cluster population . In farther galaxies, however, the study of YSCs is conducted with an analysis of their integrated light at different wavelengths under the assumption of an instantaneous burst, i.e. single stellar population.
The luminous (M$_B< -17.0$, corresponding to total stellar masses of $\sim 10^{9-10}$ M$_{\odot}$) blue compact galaxies (BCGs) show clear signatures of interactions and/or mergers, and the numerous observed massive YSCs are likely the result of these encounters . The very bright ultraviolet and optical luminosities of these systems suggest rather low dust and metallicity content. Spectra dominated by emission lines clearly demonstrate that BCGs are undergoing a starburst episode. The youth of the burst episode is also observed in the recovered age distribution of the star clusters, which shows a peak of cluster formation younger than 5 Myr.
The analysis of the young cluster populations in BCGs is quite challenging due to the rapid evolution a cluster experiences during the first 10 Myr (still partly in an embedded phase). Moreover, this analysis is based on the integrated luminosities of the clusters, which appear unresolved at the distance of the targets. Observations of resolved newly born star clusters in the Milky Way and in the Magellanic Clouds reveal that these are quite complex systems. A cluster forms in a hierarchical medium, from the fragmentation and collapse of giant molecular cloud complexes [@2010IAUS..266....3E]. This implies that stars form in a fractal distribution and only in some cloud cores are their number densities is high enough to eventually form a bound cluster [@2010MNRAS.409L..54B]. This picture is confirmed also by dynamical studies. @2011MNRAS.410L...6G observed a continuous distribution between associations (unbound) and clusters (bound) at very young ages, and a clear distinction at older stages, when the crossing time of unbound systems exceeds their stellar age.
In a newly born cluster, the massive and short-lived stars, rapidly reach the main sequence and produce strong winds and UV radiation, which ionizes the intracluster gas and create bubbles and shells. These H[ii]{} regions surround the optically bright core of stars and significantly contribute to the integrated fluxes. However, a large fraction of the stars is still accreting material from their dusty disks (young stellar objects, YSOs) or contracting (in the pre-main sequence phase, PMS). Moreover, the edges of the clusters are places for triggered [@2011arXiv1101.3112E] and progressive star formation [e.g. @2007ApJ...665L.109C], which can explain the observed age spread in the star forming regions [e.g. @2010ApJ...713..883B].
Evidence for the NIR excess
===========================
A significant fraction of young star clusters in Haro11 [@2010MNRAS.407..870A], ESO185-IG13 [@2011MNRAS.414.1793A], and Mrk930 [@2011MNRAS.tmp..739A] shows a clear signature of a flux excess at near-IR wavebands. The models used include a self-consistent treatment of the the photoionized gas, important during the first few Myrs of the cluster evolution. However, these models are not able to reproduce the NIR observed fluxes of the clusters. In Fig. \[fig1\], we show two representative cases of cluster spectral energy distributions (SEDs). A large fraction of analysed clusters in the 3 galaxies have regular SEDs, easily fitted by our models (see left panel,Fig. \[fig1\]). However, a considerable number of clusters have SEDs similar to cluster \# 43 in the right panel of Fig. \[fig1\]. Three different sets of fits have been performed for each cluster [see @2010MNRAS.407..870A; @2011MNRAS.414.1793A for details] and depending on the outcomes, clusters have been divided into 3 samples: 1) regular SEDs have been fitted from UV to IR; 2) if the source presented an excess at $\lambda > 1.0$ $\mu$m, the fit was performed excluding the IR data (so called IR excess); 3) if the excess affected also $I$ band ($\sim 8000$ Å), then only the data from UV to optical ($\lambda < 8000$ Å) were included in the fit (referred to as NIR excess). The exclusion of the data points with an excess was necessary in order to reproduce the observed UV and optical SEDs of the clusters. The two observed fluxes at 0.5 and 0.6 $\mu$m of cluster \# 43 are produced, respectively, by a narrow filter (F550M) centred on the continuum and a filter (F606W) which transmits H$\alpha$. A jump between these two filters can be used as an age indicator (only clusters younger than 10 Myr have H$\alpha$ in emission). Clearly, \# 43 is a very young cluster. However, if the NIR data points (including $I$ band) are included in the fit, the age obtained for this cluster is 45 Myr [see @2010MNRAS.407..870A for details]. In Table \[tab1\], we give the fractions of clusters with a regular SED or NIR excess.
![\[fig1\] An example of SED analysis of 2 clusters (the name of the host in indicated in the inset). The two objects are representative of different cases: on the left a young cluster with a good fit, i.e. regular SED; on the right, a cluster affected by an excess in flux at wavelengths longer than 8000 Å. The filled black points indicated the observed fluxes for each cluster. The integrated model fluxes are labelled with open squares. They sit above the best fitting spectrum overplotted with a solid line. Red vertical lines indicate which data points have been excluded from the fit. The age and mass of the cluster are shown.](Fig1a_Adamo_A.pdf "fig:"){width="7.6cm"} ![\[fig1\] An example of SED analysis of 2 clusters (the name of the host in indicated in the inset). The two objects are representative of different cases: on the left a young cluster with a good fit, i.e. regular SED; on the right, a cluster affected by an excess in flux at wavelengths longer than 8000 Å. The filled black points indicated the observed fluxes for each cluster. The integrated model fluxes are labelled with open squares. They sit above the best fitting spectrum overplotted with a solid line. Red vertical lines indicate which data points have been excluded from the fit. The age and mass of the cluster are shown.](Fig1b_Adamo_A.pdf "fig:"){width="7.6cm"}
--------- -------------- ----------- ------------
targets regular SEDs IR excess NIR excess
% % %
Haro11 44 14 42
ESO185 68 10 22
Mrk930 62 12 26
--------- -------------- ----------- ------------
: Fraction of clusters in each SED sample, of the three galaxies.
\[tab1\]
Color-color diagrams of the whole Mrk930 cluster population [see @2011MNRAS.tmp..739A for s complete analysis] are shown in Fig. \[fig2\]. The $R-I$ color is used as reference. Generally, a negative $R-I < 0$ color indicates ages younger than 10 Myr and is produced by a strong nebular contribution to the $R$ filter, which includes the bright H$\alpha$ line. However, this assumption is not always valid due to the flux excess in the $I$ band. The clusters, in the diagrams, have different symbols depending of their observed SEDs: black dots indicate normal SEDs, red triangles clusters with an excess at $\lambda \geq 1.0$ $\mu$m (hereafter, IR excess), and blue diamonds SEDs which deviate at $\lambda \geq 0.8$ $\mu$m (hereafter, NIR excess). We notice that clusters with a NIR excess have a $R-I$ color between 0.2 and 1.2 mag redder than the prediction made by the best-fitting SED model. The $R-I$ color of the “blue” cluster-diamonds is such that, if overlooked, causes age (and mass) overestimates, affecting the results of the optical based cluster analysis. The $UV-R$ (left panel) color shows that the clusters detected in the $FUV$ are young, at least younger than 30-40 Myr. Many of the clusters with an NIR excess are located in an area where $R-I > 0$, e.g., corresponding to ages older than 10 Myr. However, their UV color is compatible with being a few Myrs old. The $FUV$ band is sensitive to the reddening. Looking at the color-color diagram, one can see that clusters detected in the $FUV$ have extinctions A$_V \leq 1.0$ ( e.g., the arrow in the plot). Therefore, we consider these very young $FUV$ bright clusters as systems which have already gone through the deeply embedded phase. The inclusion of the IR color (right panel), clearly shows that clusters with a flux excess (clearly young) at the redder wavelengths are mainly located in an area of the diagram where the $R-I > 0$ and $H-R > 1.0$ mag, e.g. apparently older than 1 Gyr. The IR color of the clusters with a red excess suggests that the extinction in these objects should be much higher than the one predicted by the optical and UV colors. In other words, at the NIR wavelengths it is possible that we are probing a different, more deeply embedded stellar populations, indicating that these are very young clusters.
Possible origin of the NIR excess
=================================
The $I$ band excess
-------------------
The $I$ band excess has been found only in very young clusters (usually $<6$ Myr). A viable explanation for this feature is the extended red emission (ERE, see for a review [@2004ASPC..309..115W]). The ERE is observed as a soft rising continuum between $0.6-0.9$ $\mu$m. It is observed around galactic and extragalactic H[ii]{} regions and caused by a photoluminescence reaction on dust grains heated by hard UV radiation. Such energetic photons are mainly produced in short-lived massive stars. This could explain why the $I$ band excess in our clusters is over after 6 Myr.
The IR excess
-------------
Several mechanisms can concur to make the flux at $\lambda > 1.0$ $\mu$m higher than predicted by models. The distance of the galaxy and the resolution achieved - the best with the current accessible facilities - limit our studies to the integrated properties of these YSCs. However, observations of close-by resolved clusters and numerical predictions of stellar populations in clusters can give us an hint of the mechanisms which are likely producing the observed excess.
Among the youngest and massive resolved star clusters, 30 Doradus (hereafter 30 Dor) in the Large Magellanic Cloud (LMC), represents the best reference to understand what a recently born star cluster looks like. 30 Dor is the central region of the extended Tarantula nebula. Multiwavelength studies of this regions have dissected the different components of the complex 30 Dor environment. @1997ApJS..112..457W identified five different stellar populations in the region: the bright core early O-type stars which are part of the compact star cluster R136; in the north and west region embedded massive YSOs; 3 more evolved stellar population groups in the southern and 1.0’ away in the western region. The R136 cluster has a mass of $\sim 10^5$ M$_{\odot}$ and is 3 Myr old. This nuclear region ($\leq 3$ pc) is dust and gas-free. However, the aperture radius we are using in our analysis is much larger (radius of $\approx 36$ pc) than the size of R136. Our apertures are comparable to the size of the image of 30 Dor showed in Figure 1 of @2010MNRAS.405..421C. This suggests that while the optical range is dominated by the stellar and gas emission, the IR transmits also flux from the diffuse dust, heated by the hard UV radiation field, the embedded YSOs formed in triggered star formation events at the edge of the nucleus, where most of the dense gas and dust filaments are located, and the low mass stars still in the PMS phase.
![\[fig2\] Color-color diagrams of the cluster population in Mrk930 [@2011MNRAS.tmp..739A]. Different filter combinations are compared to the $R-I$ color (F606W-F814W). Left: $UV-R$ (F140LP-F606W); Right: $R-H$ (F606W-F160W). In each panel, the Z01 evolutionary tracks are plotted as a solid black line. Where predictions for the used filters were available we included the M08 tracks (dashed lines) as well. Ages are labelled along the tracks. The black dots are clusters with regular SEDs (from UV to IR). The red triangles are cluster with an excess at $\lambda >1.0$ $\mu$m. The blue diamonds shows objects with an excess starting longword $\lambda \geq 0.8$ $\mu$m. The black arrows indicate an visual extinction of A$_V=1$. Mean errors are included. A solid red line divides the plots in two regions: $R-I>0$ (older than 10 Myr) and $R-I<0$ (younger than 10 Myr). ](Fig2a_Adamo_A.pdf "fig:"){width="7.6cm"} ![\[fig2\] Color-color diagrams of the cluster population in Mrk930 [@2011MNRAS.tmp..739A]. Different filter combinations are compared to the $R-I$ color (F606W-F814W). Left: $UV-R$ (F140LP-F606W); Right: $R-H$ (F606W-F160W). In each panel, the Z01 evolutionary tracks are plotted as a solid black line. Where predictions for the used filters were available we included the M08 tracks (dashed lines) as well. Ages are labelled along the tracks. The black dots are clusters with regular SEDs (from UV to IR). The red triangles are cluster with an excess at $\lambda >1.0$ $\mu$m. The blue diamonds shows objects with an excess starting longword $\lambda \geq 0.8$ $\mu$m. The black arrows indicate an visual extinction of A$_V=1$. Mean errors are included. A solid red line divides the plots in two regions: $R-I>0$ (older than 10 Myr) and $R-I<0$ (younger than 10 Myr). ](Fig2b_Adamo_A.pdf "fig:"){width="7.6cm"}
In the literature, studied cases of IR excess in young embedded or partially embedded unresolved extragalactic clusters have explained the red excess as due to an important contribution by YSOs [@2009MNRAS.392L..16F], or hot dust (). Likely, the same mechanism is causing the excess in young star clusters in BCGs.
After several Myr this complex phase is over, so it cannot explain why we still observe objects with an IR excess at older ages. For these evolved clusters a possible source of excess can be an important contribution from red super giants (RSGs). Models usually predict the number of RSGs, assuming that the stars in a cluster fully populate the underlying initial mass function (IMF). However, this assumption is not valid, if the cluster is less massive than $10^4$ M$_{\odot}$, and causes important variations for cluster masses below $5\times10^3$ M$_{\odot}$ . Moreover, it has been observed that in metal-poor environments the number of observed RSGs tends to be higher than the predicted one from the ratio of blue versus RSGs. Therefore, both effects would be observed as a rise in the NIR integrated flux of an unresolved cluster, which our current models cannot fully account for. It is not clear, however, why we do not see any mass dependence between the excess in $H$ band and the mass of the cluster, which could support the stochasticity scenario, or why our models cannot predict the correct number of RSGs only for some of the clusters. NIR spectroscopy is needed to test these scenarios.
Another possible explanation is that a second stellar population is forming in the clusters. If a second population is forming now, then we expect to detect it mainly in the NIR. Multiple stellar populations have been detected in globular clusters and, recently, even in young star clusters [@2011arXiv1106.4560L]. Are we observing the formation of a new stellar population in these massive YSCs?
Finally, a bottom heavy IMF, i.e. a much steeper slope at low mass ranges (higher number of low mass stars), could also explain the excess. However, studies of IMF variations in massive star clusters (see ) have not found any confirmed case of such type of cluster. Moreover, it is not clear how, in the same galaxy, a fraction of clusters could form with a different IMF.
|
---
author:
- 'Ž. Chrobáková, M. López-Corredoira, F. Sylos Labini, H.-F. Wang, R. Nagy'
bibliography:
- 'Refer.bib'
date: 'Received xxxx; accepted xxxx'
subtitle: 'Part III: Rotation curves analysis, dark matter, and Modified Newtonian dynamics tests'
title:
-
- 'Gaia-DR2 extended kinematical maps'
---
[Recent statistical deconvolution methods have produced extended kinematical maps in a range of heliocentric distances that are a factor of two to three larger than those analysed in the Gaia Collaboration based on the same data.]{} [In this paper, we use such maps to derive the rotation curve both in the Galactic plane and in off-plane regions and to analyse the density distribution.]{} [By assuming stationary equilibrium and axisymmetry, we used the Jeans equation to derive the rotation curve. Then we fit it with density models that include both dark matter and predictions of the MOND (Modified Newtonian dynamics) theory. Since the Milky Way exhibits deviations from axisymmetry and equilibrium, we also considered corrections to the Jeans equation. To compute such corrections, we ran N-body experiments of mock disk galaxies where the departure from equilibrium becomes larger as a function of the distance from the centre.]{} [The rotation curve in the outer disk of the Milky Way that is constructed with the Jeans equation exhibits very low dependence on $R$ and $z$ and it is well-fitted both by dark matter halo and MOND models. The application of the Jeans equation for deriving the rotation curve, in the case of the systems that deviate from equilibrium and axisymmetry, introduces systematic errors that grow as a function of the amplitude of the average radial velocity. In the case of the Milky Way, we can observe that the amplitude of the radial velocity reaches $\sim 10\%$ that of the azimuthal one at $R\approx 20$ kpc. Based on this condition, using the rotation curve obtained from the Jeans equation to calculate the mass may overestimate its measurement.]{}
Introduction {#ch1}
============
Substantial progress has been made in the study of the Milky Way rotation curve thanks to the application of a novel range of methods. Inside the solar circle, the tangent-point method has been applied by measuring spectral profiles of the HI and CO line emissions [@burton]. Another approach considers the radial velocity of an object, which requires that its distance be measured independently, for example, by trigonometric or spectroscopic determinations. For this purpose, there is a variety of objects can be adopted, such as OB stars and their associated molecular clouds [@blitz], the thickness of the HI layer [@merrifield], the red giant branch and red clump [@bovy_red_clump; @huang_red], classical Cepheids [@pont_cef; @mroz], and a number of others. Rotation velocities can also be determined by measuring proper motions: when these are provided by Very Long Baseline Interferometry (VLBI) techniques, the rotation curve can be determined with high accuracy [@honma]. The combination of proper motions from USNO-B1 observations with the Two Micron All Sky Survey (2MASS) photometric data has also been used to determine the rotation curve [@martin_rot_curve]. A powerful tool for measuring the rotation curve of the Milky Way is the VLBI Experiment for Radio Astrometry (VERA), which uses trigonometric determinations of three-dimensional positions and velocities of individual maser sources [@reid_vlbi; @honma_vera].
An significant study was carried by [@bhattacharjee] to construct the rotation curve of the Milky Way from $\sim 0.2$ kpc to $\sim 200$ kpc by using a variety of disk and non-disk tracers. In analysing the velocity anisotropy parameter, they also estimated a lower limit for the Milky Way mass. Their work was continued by [@bajkova], who combined circular velocities of masers at low distances with the rotation curve of [@bhattacharjee] and fit the result using a number of models, varying, in particular, the dark matter halo, where they refine parameters for six different models. A comparison of some of our fit parameters with the results of [@bajkova] is given in \[ch5\]. An excellent review of the current status of the study of the rotation curve of the Milky Way is given in [@sofue_review].\
Today, the Gaia mission of the European Space Agency [@gaia2] provides a new possibility for studying the Milky Way with unprecedented accuracy thanks to data that offers the most accurate information about our Galaxy to date. Indeed, the Gaia data offer very precise determinations of position, proper motions, radial velocity measurements, and distance for millions of stars, although the errors of distance measurements increase with the distance from the observer.\
In this paper, we present a systematical analysis of the Milky Way rotation curves derived by means of different methods and by using the Second Data Release (DR2) of the Gaia mission [@gaia]. To calculate the rotation curve, we use the Jeans equation that relates the circular velocity to observational quantities, such as the Galactocentric radial and tangential velocities, along with their respective dispersions. To do so, we must assume that the gravitational potential of Milky Way is axisymmetric and that the Galaxy is in a steady state configuration. In addition, by using numerical N-body experiments of simple disk models, we try to quantify the effect of the deviations from the equilibrium configuration on the determination of the rotation curve through the Jeans equation.
This paper is organized as follows: in \[ch2\], we describe the selection of the data used in this paper and in \[ch3\], we illustrate the method used to measure the Milky Way’s rotation curve and present our determinations. In \[ch4\], we explain the method for calculating the density distribution from the Poisson equation by using the measured rotation curve. In \[ch5\], we fit different density models to our determination of the rotation curve using standard dark matter approaches, that is, by assuming that the Galaxy is embedded in a quasi-spherical halo whose mass can be then derived on the basis of such an hypothesis. In §6, we present our density models based on the Modified Newtonian Dynamics (MOND) theory. We study in \[ch6\] the deviations from the Jeans equation in out-of-equilibrium systems. Finally, in §8, we present our conclusions.
Data selection {#ch2}
==============
@martin [hereafter LS19] have produced extended kinematic maps of the Milky Way by using data from the second Gaia data release DR2 [@gaia] and considering stars with measured radial heliocentric velocities and with parallax error less than $100\%$. Their total sample contains 7 103 123 sources. Such objects were observed by the Radial Velocity Spectrometer [RVS, @cropper], which collects medium-resolution spectra (spectral resolution $\frac{\lambda}{\Delta \lambda}\approx 11 700$) over the wavelength range of 845-872 nm, centred on the Calcium triplet region. Radial velocities are averaged over a 22-month observational time span. Most sources have a magnitude brighter than 13 in the $G$ filter.\
As the parallax error grows with the distance from the observer, LS19 applied a statistical deconvolution of the parallax errors based on the Lucy’s inversion method [@lucy] to statistically estimate the distance. In this way, they derived the extended kinematical maps in the range of Galactocentric distances up to 20 kpc. We chose this method due to its advantage over other Bayesian methods [e.g. @bayes1; @bayes2] as it does not assume any priors about the Milky Way density distribution. Any other method, such as the Lutz-Kelker method [@lutz], is not appropriate here since it would assume a uniform stellar volume density and a constant ratio $\sigma_\pi/\pi$, where $\pi$ is the observed parallax and $\sigma_\pi$ its standard deviation. For more details on this topic, see [@luri_paralaxy], which gives an extensive analysis of different methods if inferring distance from the parallax, along with their respective advantages and disadvantages.
In further detail, the effective temperatures for the sources with radial velocities that LS19 considered are in the range of 3550 to 6900 K. The uncertainties of the radial velocities are: 0.3 km/s at $G_{RVS} < 8$, 0.6 km/s at $G_{RVS}=10$, and 1.8 km/s at $G_{RVS}= 11.75$; along with systematic radial velocity errors of $< 0.1$ km/s at $G_{RVS} < 9$ and 0.5 km/s at $G_{RVS}= 11.75$. The uncertainties of the parallax are: 0.02 –0.04 mas at $G<15$, 0.1 mas at $G=17$, 0.7 mas at $G=20$ and 2 mas at $G=21$. The uncertainties of the proper motion are: 0.07 mas $\mathrm{yr}^{-1}$ at $G<15$, 0.2 mas $\mathrm{yr}^{-1}$ at $G=17$, 1.2 mas $\mathrm{yr}^{-1}$ at $G=20$ and 3 mas $\mathrm{yr}^{-1}$ at $G=21$. For details on radial velocity data processing and the properties and validation of the resulting radial velocity catalogue, see [@sartoretti] and [@katz]. The set of standard stars that was used to define the zero-point of the RVS radial velocities is described in [@soubiran]. LS19 consider the zero-point bias in the parallaxes of Gaia DR2, as found by [@lindegren; @arenou; @stassun; @zinn]; however, they find that the effect of the systematic error in the parallaxes is negligible, so the maps that we use from their study (LS19, Figs. 8-12) do not consider the zero-point correction. We describe the way we use these maps in Sect. \[ch3\] to construct the rotation curves, however, in Fig. 2, we include the zero-point correction to demonstrate that the difference is negligible.
Rotation curves {#ch3}
===============
From the Gaia DR2 catalogue, we estimate, for each object, the parallax $\pi$, the Galactic coordinates ($l,b$), the radial velocity $v_r$ , and two proper motions in equatorial coordinates $\mu_a\mathrm{cos}\delta$ and $\mu_\delta$. For our analysis, we need to know the Galactocentric position of stars in cylindrical coordinates ($R,z,\Phi$), and the Galactocentric velocity in cylindrical coordinates ($v_R,v_\Phi,v_z$). The transformation from these two coordinates systems can be found in LS19.\
We limit the range of vertical distance to $\lvert z \rvert<2.2~$kpc as we find that far off-plane data are affected by larger errors in their parallax determinations. We investigate the disk beyond the solar Galactocentric radius, that is, for $8.4~$kpc$<R< 21.2~$kpc.\
To determine the rotation curve, we consider the one component of Jeans equations in cylindrical coordinates [@binneyb Ch. 4.2, 4-29a]:
$$\begin{aligned}
\label{1}
\frac{\partial(\nu\overline{v_{R}} )}{\partial t}+\nu\left(\frac{\overline{v_{R}^{2}}-\overline{v_{\Phi}^{2}}}{R}+\frac{\partial\Phi}{\partial R}\right)+\frac{\partial(\nu\overline{v_{R}^{2}} )}{\partial R}+\frac{\partial(\nu \overline{v_{R}v_{Z}})}{\partial z} = 0~,\end{aligned}$$
[where $R$ is the Galactocentric radius, $v_R$ is the radial velocity, $v_Z$ is the vertical velocity, $v_\Phi$ is the azimuthal velocity, and $\nu$ is the volume density. The quantity $\overline{v^2}$ is the average square velocity for each component that can be written as $\overline{v^2}=\sigma^2+\overline{v}^2$, where $\sigma$ is the velocity dispersion. For a detailed calculation of the velocities and their respective dispersion, see LS19.]{}
The rotational velocity is defined as [@binneyb] $$\begin{aligned}
\label{5a}
v_{c}^{2}(R,z)=R\frac{\partial\Phi}{\partial R}~.\end{aligned}$$ We use the standard assumption that the volume density can be written as $$\begin{aligned}
\label{3}
\nu(R,z)=\rho_0e^{-\frac{R}{h_{R}}}e^{-\frac{\lvert z \rvert}{h_{z}}}~,\end{aligned}$$ where $h_R$ is the scale length and $h_z$ is the scale height. From Eqs. (\[1\])-(\[3\]) we obtain the rotational velocity as function of $R, z$, that is, the rotation curves,
$$\begin{aligned}
\label{2}
v_{c}^{2}&=&\overline{v_{\Phi}}^{2}+\sigma_{\Phi}^{2}+\left(\overline{v_{R}}^{2}+\sigma_{R}^{2}\right)\frac{R-h_{R}}{h_{R}}-2R\overline{v_{R}}\frac{\partial\overline{v_{R}}}{\partial R} \nonumber \\
&-&R\frac{\partial \sigma_{R}^{2}}{\partial R}+\frac{R}{h_z}\frac{z}{\lvert z \rvert}\overline{v_{R}v_{z}}-R\frac{\partial (\overline{v_{R}v_{z}})}{\partial z}~.\end{aligned}$$
We determined the rotation curves for different values of $z$, in the direction of the anti-center, in bins of size $\Delta R=0.5$ kpc and $\Delta z=0.2$ kpc. For what concerns the scale parameters in Eq.\[3\], we chose values of $h_{R}=2.5$ kpc and $h_{z}=0.3$ kpc [@juric]. Figure \[o1\] shows the results of our fit.
Figure \[o1b\] shows the rotation curves, including the zero-point correction in parallax. The difference from the rotation curves in Fig. \[o1\] is negligible and we do not consider this correction for the rest of the analysis. The results for different scale parameters are almost identical, as we show in Fig. \[o2\]. We observe a flat rotation curve, although it does exhibit some fluctuations. Our rotation curve in the plane of the Galaxy has a small positive gradient of $0.54\pm0.7 (stat.) \pm 0.5 (syst.)~ \mathrm{km~s}^{-1}~\mathrm{kpc}^{-1}$. Recent results have shown an opposite trend: [@eilers] measured rotation curve for Galactocentric distances $5~$kpc$~\leq R \leq 25~$kpc by combining spectral data from the Apache Point Observatory Galactic Evolution Experiment [APOGEE, @apogee] and photometric information from Wide-field Infrared Survey Explorer [WISE, @wise], 2MASS [@2mass], and Gaia DR2, finding a rotation curve with a declining slope of $-1.7\pm0.1~\mathrm{km~s}^{-1}~\mathrm{kpc}^{-1}$, with a systematic uncertainty of $0.46 ~\mathrm{km~s}^{-1}~\mathrm{kpc}^{-1}$. A similar result was obtained by [@mroz], who used classical Cepheids to obtain the rotation curve of the Milky Way for Galactocentric distances $4~$kpc$~\lesssim R \lesssim 20~$kpc, finding a rotation curve with a small negative slope of $-1.34\pm0.21~\mathrm{km~s}^{-1}~\mathrm{kpc}^{-1}$. [@bhattacharjee] have also used the Jeans equation, but only for large distances (R>20 kpc), which we do not consider in our analysis. For the disk tracers, they use the tangent point method for small distances and for higher distances they assume that the tracers follow nearly circular orbit. The advantage is that their method is independent from any density model, although it strongly depends on values of Galactic constants (Sun’s distance from, and circular rotation speed around, the Galactic centre). Nevertheless, their results for rotation curve for various values of Galactic constants are consistent with our findings. We discuss these results more in detail in what follows.\
Density distribution from the Poisson equation {#ch4}
==============================================
Based on the results obtained for the rotation curve, we proceed to determine the density distribution in the Milky Way by considering different approaches. The first one is based on the Poisson equation in cylindrical coordinates and it assumes the dependence of the rotation speed with the azimuth to be negligible:
$$\label{4}
\frac{1}{R}\frac{\partial}{\partial R}\left(R\frac{\partial\Phi}{\partial R}\right)+\frac{\partial^{2}\Phi}{\partial z^{2}}=4\pi G\rho(R,z)~.$$
The first term on left side can be easily obtained by using Eq. (\[5a\]). The second term, on the left side, can be obtained with the same relation and switching derivatives
$$\begin{aligned}
\label{6}
\frac{\partial^2}{\partial z^2}\left(\frac{\partial\Phi}{\partial R}\right)&=& \nonumber \frac{1}{R}\frac{\partial^{2}v_{c}^{2}}{\partial z^{2}}~ \nonumber~, \\
\frac{\partial}{\partial R}\left(\frac{\partial^{2}\Phi}{\partial \nonumber z^{2}}\right)&=&\frac{1}{R}\frac{\partial^{2}v_{c}^{2}}{\partial z^{2}}~. \end{aligned}$$
By integrating the latter relation we find $$\begin{aligned}
\frac{\partial^{2}\Phi}{\partial z^{2}}=-\int_{R}^{R_{max}} \frac{1}{R}\frac{\partial^{2}v_{c}^{2}}{\partial z^{2}} \mathrm{d}R + \Phi(R_{max},z=0)~. \end{aligned}$$ To determine derivatives of $v_c$ with respect to $R,z$, we assume that $v_{c}^2$ has a linear behaviour of the type $$\label{vc2}
v_{c}^2 = a(z)(R-14)+b(z) \,,$$ where clearly $a(z)$ and $b(z)$ must be determined from the data. We find that $a(z)$ and $b(z)$ can be nicely fitted by parabolas and therefore, we can write $$\label{fsl1}
v_{c}^{2}(z) =\left[(\alpha+\beta z^{2})+(\gamma+\delta z^{2})(R-14)\right]\;,$$ where the numerical values of $\alpha,\beta,\gamma,\delta$ are estimated from the data and the values are given below. We use Eq. (\[5a\]) and (\[vc2\]) to express the first term of Eq. (\[4\]) as $$\begin{aligned}
\label{7}
\frac{1}{R}\frac{\partial}{\partial R}\left(R\frac{\partial\Phi}{\partial R}\right)&=&{\frac{1}{R}\frac{\partial}{\partial R}\left(a(z)(R-14)+b(z)\right)} \nonumber \\
&=&\frac{a(z)}{R}\end{aligned}$$ By making the derivative of the fit of the rotational velocity (Eq.\[fsl1\]) with respect to $z,$ we express Eq. (\[6\]) as
$$\begin{aligned}
\label{8}
\frac{\partial^{2}\Phi}{\partial z^{2}}&=&2\beta\mathrm{ln}\left(\frac{R}{R_{max}}\right)+2\delta(R-R_{max}) \nonumber \\
&-&28\delta\mathrm{ln}\left(\frac{R}{R_{max}}\right)+\Phi(R_{max},z=0)~.\end{aligned}$$
We find that the best fit values for $a(z)$ and $b(z)$ are (see Fig. \[o4\]):
$$\label{9}
\begin{split}
a(z)&=(-2200 \pm 400 )z^{2}+(1000 \pm 1000) \\
b(z)&=(11400 \pm 1000 )z^{2}+(53000 \pm 1500) \;.
\nonumber
\end{split}$$
In the Galactic plane, the value of $a(z)$ is positive, which means that in the plane the velocity gradient is positive too: this must be compensated by density increase. That is clearly non-physical as we know that in our Galaxy, the density decreases exponentially in the outwards direction. Therefore, we conclude that we cannot use the Poisson equation to determine the density analytically. This problem may be related to large fluctuations present in the data, as well as by the fact that the system is not in equilibrium, so it does not satisfy the assumptions of the Jeans equation. We analyse the effect of the deviations from equilibrium in greater detail in Section \[ch6\].
[{width="50.00000%"}]{} [{width="50.00000%"}]{}
Density fit with the dark matter model {#ch5}
======================================
Another method to fit the rotation curve data can be done by making use of known density models. By assuming that the system is in equilibrium and made by different mass components, both the density and the rotational velocity can be expressed as $$\begin{aligned}
\label{gal_exp}
\rho&=&\rho_{\mathrm{bulge}}+\rho_{\mathrm{disk}}+\rho_{\mathrm{halo}}~, \\
v_{c}^2 &=& v_{c,\mathrm{bulge}}^2+v_{c,\mathrm{disk}}^2+v_{c,\mathrm{halo}}^2~,\end{aligned}$$ that is, we have decomposed the density and the circular velocity as the sum of three terms: the bulge, the disk, and the halo. Here, we examine each of these terms in more detail.
We do not fit the bulge, as we are interested mainly in outer parts of disk, where contribution of the bulge is negligible: we use $$\begin{aligned}
\label{17}
v_{c,bulge}^2=\frac{GM_{\mathrm{bulge}}}{R}~,\end{aligned}$$ where $M_{\mathrm{bulge}}=2\cdot10^{10} M_{\odot}$ [@valenti].
For the disk, by assuming the balance between the gravitational and centrifugal forces at a generic point $(r,\phi,h)$ (for a detailed derivation see Appendix \[odvodenie\]) we derive: $$\begin{aligned}
\label{16} &&\mkern-20mu\int_{-H/2}^{H/2}\int_{R_{min}}^{R_{max}}\frac{2}{r}\left[\frac{(\hat{r}+r)(\hat{r}-r)+\Delta h^2}{[(\hat{r}-r)^2+\Delta h^2]\sqrt{(\hat{r}+r)^2+\Delta h^2}}E(k) \right. \nonumber \\
&&\mkern-20mu\left.-\frac{1}{\sqrt{(\hat{r}+r)^2+\Delta h^2}}K(k)\right] \nu(\hat{r},\hat{h})
\mathrm{d}\hat{r}\mathrm{d}\hat{h} \nonumber \\
&&\mkern-20mu+A\frac{v_{c,\mathrm{disk}}(r,h)^2}{r}=0~,\end{aligned}$$ where $K(k),E(k)$ are complete elliptic integrals of the first and second kind respectively, and $$\begin{aligned}
k^2=\frac{4\hat{r}r}{(\hat{r}+r)^2+\Delta h^2}~,\end{aligned}$$ where $\Delta h^2=(\hat{h}-h)^2$. For the sake of simplicity, we consider only a thin disk and we approximate $\Delta h \approx h$. For the density in Eq.\[16\] we used the relation: $$\begin{aligned}
\nu(\hat{r},\hat{z}) = \rho_0e^{-\hat{r}/h_{R}}e^{-\lvert \hat{h} \rvert/h_{z}}\;.\end{aligned}$$ In Eq.\[16\] the constant $A$ is the Galactic rotation number defined as $$\begin{aligned}
A=\frac{R_{g,max}V_0^2}{G M_{d,max}}~,\end{aligned}$$ where $M_{d,max}$ is mass of the disk, for which we use the value $M_{d,max}=6.5\cdot10^{10} M_\odot$ [@sofue_mass]. $R_{g,max}$ is the radius of the disk, which we fix at $25$ kpc, $V_0$ is the maximum velocity corresponding to the flat part of the rotation curve in the data-set: $257$ km/s in our case and $G$ is the gravitational constant: $4.302\cdot10^{-6}$ kpc $M_\odot^{-1}$ (km/s)$^2$. We calculate the fit in the Galactic plane, where $\Delta h \rightarrow 0$ and Eq. (\[16\]) becomes $$\begin{aligned}
\label{14}
&&
\int_{R_{min}}^{R_{max}} \left[\frac{E(k)}{\hat{r}-r}-\frac{K(k)}{\hat{r}+r}\right]\rho_0e^{-\hat{r}/h_{R}}\hat{r} \mathrm{d}\hat{r}
\\ \nonumber &&
+ A\frac{v_{c,\mathrm{disk}}(r)^2}{2\lvert h \rvert}=0~,\end{aligned}$$ where $$\begin{aligned}
k^2=\frac{4r\hat{r}}{(\hat{r}+r)^2}~.\end{aligned}$$
To fit the dark matter halo, we assume this is well approximated by the so-called Navarro, Frenk, and White density profile [@nfw] $$\begin{aligned}
\label{nfw}
\rho_{\mathrm{halo}}&=&\frac{\rho_{0h}}{\frac{R}{R_s}\left(1+\frac{R}{R_s}\right)^2}~, \\
v_{c,halo}^2(R)&=&\frac{4\pi G \rho_{0h} R_s^3}{R}\left[\log \left(\frac{R_s+R}{R_s}\right)-\frac{R}{R_s+R}\right]~.\end{aligned}$$ We use the least-squares method to find the best values of the free parameters. As this method requires a long computational time, we fix some well-known parameters and only fit those that are not so well determined. First, we fit only data in the Galactic plane, where we fix $h_R=2.5 $ kpc and $R_s=14.8 $ kpc, which are the values found by [@eilers]. For the free parameters, we obtain the values $\rho_{0h}=2\cdot10^7 M_{\odot}/$kpc$^3$ and $\rho_0=3.83\cdot10^8 M_{\odot}/$kpc$^3$, with the value of the minimal $\chi^2=15.424$ for 107 points. We plot this result in Fig. \[o20\] (a). We see that our rotation curves are well explained by a dominant dark matter halo, with a minimal contribution from the disc. From these values, we calculate the mass of the dark matter halo up to 25 kpc to be $M_h=3.52\cdot10^{11} M_{\odot}$, which is smaller than $7.25\cdot10^{11} M_{\odot}$ found by [@eilers], but higher than $2.9\cdot 10^{11} M_\odot$ found by [@bajkova]. For the disk, we find $M_d=1.41\cdot10^{10} M_\odot$, which is lower than values found in the literature, for example, $6.5\cdot 10^{10} M_\odot$ as found by [@sofue_disk], $0.95 \cdot 10^{11} M_\odot$ as found by [@kafle], or $6.51\cdot 10^{10} M_\odot$ as found by [@bajkova].\
For the off-plane data, we fit rotation curves for different values of $z$ at the same time, using relation (\[16\]), which adds one more free parameter $h_z$ to the fit. Again, to save computational time, we restricted the number of free parameters and fixed $h_R=2.5$ kpc and $R_s=14.8$ kpc. We find $h_z=0.3 $ kpc, $\rho_0=4.1\cdot10^9 M_{\odot}/$kpc$^3$ and $\rho_{0h}=2.389\cdot10^7 M_{\odot}/$kpc$^3$, with the value of minimal $\chi^2=2510.37$ for 4653 points. In Fig. \[o11\], we plot the fit for various values of $z$. We see that in all cases, the dark matter halo is strongly dominant and the contribution from the disk is less important, which is as expected from rotational velocity that does not change with vertical distance.\
This result is in agreement with result of [@eilers], who also fitted their rotation curve with a similar model. They also find a dominant dark matter halo, with free parameter $\rho_{0h}=1.06\cdot10^7 M_{\odot}/$kpc$^3$. However, we disagree with result from [@jalocha], who found that the gross mass distribution in our Galaxy is disk-like, without the need for an halo. [@jalocha] obtained their result based on modelling vertical gradient of azimuthal velocity, assuming the quasi-circular orbit approximation, and relating $v_c$ to $v_\phi$ directly from the balance condition of the radial component of gravitational and inertial force. We guess that the difference between our results comes from the fact that [@jalocha] did not take the Jeans equations into account when deriving rotational velocity. Indeed, in this latter work, assuming quasi-circular orbits $v_\phi$ was directly related to $v_c$ to obtain $v_\phi=\frac{r}{R} v_c$.
Density fit with the MOND model {#ch_mond}
===============================
We tried to fit our results using the MOND theory, without invoking the presence of a heavy dark matter halo. To this purpose we have recalculated the expressions for the disk and the bulge, using relations from MOND [@milgrom_mond]:
$$\begin{aligned}
\label{mond}
a_{M}=\frac{a_{N}}{\mu(\frac{a_{M}}{{a_0}})}~,\end{aligned}$$
where $$\begin{aligned}
\label{15}
\mu\left(\frac{a_{{M}}}{a_0}\right)=
\sqrt{\frac{1}{1+\left(\frac{a_0}{a_{{M}}}\right)^2}}~,\end{aligned}$$ with the value of $a_0=1.2\cdot 10^{-10}~\mathrm{m}\mathrm{s}^{-2}$ [@scarpa]. Solving Eq. (\[mond\]) analytically yields
$$\begin{aligned}
\label{18}
a_{M}=\sqrt{\frac{1}{2}a_{N}^2+\sqrt{\frac{1}{4}a_{N}^4+a_{N}^2a_0^2}}~.\end{aligned}$$
Eq. (\[mond\]) is indeed an approximation, which does not exactly stray from a spherical symmetric mass distribution. The exact solution may be analysed in the context for Bekenstein-Milgrom MOND theory derived from the modification of classical Newtonian dynamics [@brada]. However, the difference between the approximation of Eq. (\[mond\]) and the exact solution is small, so we neglect it here.\
For the fit, we only used the disk and the bulge components. In Fig. \[o20\] (b), we plot the result of the fit for the Galactic plane, nicely matching the observed value. For the free parameters, we found $\rho_0=7.49\cdot 10^8 M_{\odot}/$kpc$^3$ and $h_R=4.8$ kpc. The values of minimal $\chi^2$ is $\chi^2=15.776$ for 107 points, which is similar to the value for Newtonian fit. The mass of the disk up to 25 kpc found with these parameters is $M_d=2.77\cdot10^{10} M_\odot$ which is almost two times higher than that obtained with the dark matter model.\
We tried to fit the off-plane rotation curve with the same approach. Again, we fit data for all $z$ with models for all $z$ at the same time. Thus, we find: $\rho_0=9.15\cdot10^9 M_{\odot}/$kpc$^3$ and $h_R=5.0$ kpc. We fixed the value of scale-height to $h_z=0.3$ kpc. The obtained value of minimal $\chi^2=2677.58$ for 4653 points, which is comparable with the Newtonian case. In Fig. \[o19\], we plot the results of the fit with MOND for different values of $z$. We see that off-plane, the fit is satisfying and there is no preference for the dark matter model over the MOND model. However, our result contradicts that of [@liasnti], who also used Milky Way observables to compare the differences between dark matter and MOND theories. They performed a Bayesian likelihood analysis to compare the predictions of the model with the observed quantities. They find that the dark matter model is preferred, as MOND-like theories struggle to simultaneously explain both the rotational velocity and vertical motion of nearby stars in the Milky Way.
[{width="50.00000%"}]{} [{width="50.00000%"}]{}
Corrections to the Jeans equation {#ch6}
=================================
So far, we used the Jeans equation to determine the rotation curve of the Milky Way and its density profile. We recall that the basic assumptions of the Jeans equation are that the system is collisionless, axisymmetric, and in equilibrium. While the first condition represents a reasonable working hypothesis since collisional effects take place on much longer time scales than those that astrophysically relevant, the recent Gaia data have shown that the Milky Way is not in a stationary situation as there are large-scale gradients in all components of the velocity field and there are clear deviations from axisymmetry [@gaia_kin_maps; @haifeng2; @martin; @haifeng1]. The dynamical origin of such features represents an open problem that has been explored by several authors [@antoja; @binney_equil]. For instance, it has been concluded that the Galactic disk is still dynamically young and was last perturbed less than 1 Gyr ago, therefore modelling it as axisymmetric and in equilibrium is incorrect [@antoja]. The problem of reliability with regard to the Jeans equation was also studied by [@haines], who analysed an N-body simulation of a stellar disk which had been perturbed by the recent passage of a dwarf galaxy and studied the surface density of the system based on the Jeans equation. They found that the Jeans equation gives reasonable results in over-dense regions, but fails in under-dense regions. Thus, the development of non-equilibrium methods for estimating the dynamical matter density locally and in the outer disk is necessary.\
In order to test the effects of the deviations from a stationary configuration and axisymmetry on the Jeans equation, we consider N-body simulations of mock galactic systems that are not completely in an equilibrium configuration. The evolution of these systems was discussed in details in [@Benhaiem+Joyce+SylosLabini_2017; @Benhaiem+SylosLabini+Joyce_2019; @SylosLabini_RCD_DLP_2020].We consider, hereafter, one of these systems, consisting of a thin, rotating, self-gravitating disk embedded in an ellipsoidal dark matter halo with an isotropic velocity dispersion. The inner regions of this system are very close to a stationary configuration, while the outer regions are progressively out-of-equilibrium. The signature of such a situation is represented by the behaviour of the radial velocity averaged in shells: at small distances from the centre this is close to zero, while at large enough distances, it becomes positive: the amplitude grows with the distance from the centre.\
The circular velocity from the Jeans equation is
$$\begin{aligned}
\label{jeans_sim}
v_{c,J}^{2}=\overline{v_{\Phi}}^{2}-\overline{v_{R}}^{2}\left(1+\frac{\partial \mathrm{ln}\nu}{\partial \mathrm{ln}R}+\frac{\partial\mathrm{ln}\overline{v_{R}}^2}{\partial \mathrm{ln}R}\right)~,\end{aligned}$$
where we neglect the cross-term $\overline{v_Rv_\phi}$, as it’s contribution to the final result is negligible $(\sim1\%)$ [@eilers]. By definition, the circular velocity can be computed from the gravitational force:
$$\begin{aligned}
v_{c,F}^{2}=R F_R= \left| \overline{\vec{F}} \cdot \vec{R} \right|
,\end{aligned}$$
where $\overline{\vec{F}}$ is the gravitational force acting of the particles contained in the two-dimensional corona at a distance, $R,$ and thickness, $\Delta R$ (where $R$ is the cylindrical coordinate). Thus, we compute the gravitational force acting of the $i^{th}$ particle as $$\begin{aligned}
\vec{F}_i = G \sum_{j=1}^N m_j m_i
\frac{(\vec{r}_i - \vec{r}_j)}{|\vec{r}_i - \vec{r}_j|^3} \;,
\end{aligned}$$ where $m_i$ is the mass of the $i^{th}$ particle and we compute its average in a corona. If axisymmetry and stationary equilibrium are established, then $v_{c,F} =v_{c,J}$: the difference between these two quantities thus depends on the deviations from the assumptions underlying the Jeans equation. In the Fig. \[sila\], we plot the ratio:
$$\label{theta}
\Theta = \frac{v_{c,J}}{v_{c,F}}$$
as a function of $$\label{zeta}
\zeta = \frac{|v_R|}{|v_\phi|}~.$$ When the radial velocity is small, that is, $\zeta \ll 1,$ then $ \Theta \approx 1$, whereas when the radial velocity becomes larger than $10 \%$ of the azimuthal one, then $\Theta$ becomes larger than one. In the Milky Way, $\zeta \approx 0.1$ at $R \approx 20$ kpc, as found by LS19. We note that in Fig. \[sila\], we have reported the behaviour for two different times, that is, 3 and 9 Gyr; indeed, as the external regions are out-of-equilibrium they continue to evolve over time, while the inner regions are quasi stationary.
![Ratio of the circular velocity from the Jeans equation and from the force (i.e. Eq.\[theta\]) as a function of the ratio between the average radial velocity and the azimuthal velocity (see Eq.\[zeta\]). Black and red circles correspond to the system evolved up to 3/9 Gyr, respectively.[]{data-label="sila"}](Fig7.eps){width="40.00000%"}
We conclude that the Jeans equation is reliable when the radial velocity is smaller than 10% of the azimuthal one, otherwise corrections to the Jeans equation become necessary. In particular, we find that the estimation of the circular velocity though the Jeans equation gives an overestimation with respect to the estimation of the circular velocity through the force. This implies that by using $v_{c,J}$ to compute the mass through the relation, $$M_J(r) = \frac{v_{c,J}^2\times r}{G}
,$$ the real mass is overestimated by a factor that is proportional to $\Theta^2$.
Conclusions
===========
In this paper, we study the rotation curve of the Milky Way from the extended kinematic maps of Gaia-DR2. We calculated the rotation curve in plane and in off-plane regions, using the Jeans equation. Our results show that the rotation curve in the outer disk has very little dependence on $R$ and $z$.\
We fitted the rotation curve using models with dark matter halo or MOND, using the least-squares method. We find that a model based on dark matter fits the data very well, and the results are in good agreement with other works. For the dark matter model, we obtain the minimal $\chi^2= 15.424$ for 107 points in the plane and $\chi^2=2510.37$ for 4653 points off plane. The MOND model in the plane gives $\chi^2= 15.776$ for 107 points, which is comparable with the dark matter model. Off-plane the results are similar as well, with $\chi^2=2677.58$ for 4653 points, which fits the data similarly to the dark matter model.\
We also considered the corrections to the Jeans equation in non-equilibrium and non-axisymmetric systems. Indeed, the Jeans equation assumes that the system is axisymmetric and in equilibrium, which is not the case of the Milky Way. For this reason, we consider N-body simulations of galaxies and calculated the rotational velocity by using the Jeans equation $v_{c,J}$ and by computing the gradient of the gravitational potential $v_{c,F}$. We find that the two ways of calculating the rotational velocity are in good agreement as long as the ratio, $\zeta,$ between the modulus of the radial velocity and of the azimuthal velocity is smaller than $\sim 10\%$. When $\zeta$ becomes larger than this value, then $v_{c,J} > v_{c,F}$ and, thus, we overestimate the Galactic mass if we use the rotational velocity computed through the Jeans equation. For the case of the Milky Way, it was found in LS19 that $\zeta \approx 0.1$ at $R \approx 20$ kpc: this implies that at a larger galactocentric distance, using the Jeans equation leads to an overestimation the mass of the Milky Way. The Gaia DR3 will clarify whether in the range of distances $20 < R < 30$ kpc, such corrections may become large enough to change our view of the Galaxy as a quasi-equilibrium system, thus altering its estimated mass.
We thank the anonymous referee for helpful comments, which improved this paper and Agnes Monod-Gayraud (language editor of A&A) for proof-reading of the text. ZC and MLC were supported by the grant PGC-2018-102249-B-100 of the Spanish Ministry of Economy and Competitiveness (MINECO). HFW is supported by the LAMOST Fellow project, National Key Basic R&D Program of China via 2019YFA0405500 and funded by China Postdoctoral Science Foundation via grant 2019M653504, Yunnan province postdoctoral Directed culture Foundation and the Cultivation Project for LAMOST Scientific Payoff and Research Achievement of CAMS-CAS. RN was supported by the Scientific Grant Agency VEGA No. 1/0911/17. This work has made use of data from the European Space Agency (ESA) mission [*Gaia*]{} (<https://www.cosmos.esa.int/gaia>), processed by the [*Gaia*]{} Data Processing and Analysis Consortium (DPAC, <https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the [*Gaia*]{} Multilateral Agreement.
Derivation of the integral of the thin disk {#odvodenie}
============================================
To derive the rotation curve of a disk in 3D, we derive the equation for balance between the gravitational and the centrifugal force. We consider two points with the coordinates $P(r,\theta,z)$ and $Q(\hat{r},\hat{\theta},\hat{z})$. The distance between these two points can be expressed as $(\hat{r}^2-r^2-2r\hat{r}cos\hat{\theta}+\Delta h^2)^{1/2}$ and the vector projection as $(\hat{r}cos\hat{\theta}-r)$, where $\Delta h$ is the difference in heights $\Delta h=(\hat{h}-h)$. The Newtonian gravitational force on the point $P$ from a body consisting of points $Q$ distributed with a mass density $\hat{\rho}(\hat{r},\hat{h})$ can be expressed as an integral over these points:
$$\begin{aligned}
F_X\mkern-10mu &=&\mkern-10mu \dfrac{G M_g}{R_g^2} \int_{-H/2}^{H/2} \int_{0}^{2\pi} \int_{0}^{1} \dfrac{\hat{r}\cos \hat{\theta} - r}{ ( \hat{r}^2 + r^2 - 2\hat{r}r \cos \hat{\theta} + \Delta h^2)^{3/2}} \nonumber \\
&\cdot&\mkern-15mu\hat{\rho}(\hat{r},\hat{h}) \hat{r} {\hspace*{0.25cm}}d\hat{r} d\hat{\theta} d \hat{h}~.\end{aligned}$$
The centrifugal force can be written simply as
$$\label{CentrifugalForce}
F_c = \dfrac{V^2}{ R }= \dfrac{V_0^2}{R_g} \dfrac{v(r,h)^2}{r}~.$$
Here, we made all the variables dimensionless by measuring the distances in units of the outermost galactic radius $R_g$, mass density $\hat{\rho}$ in units of $M_g/R_g^3$, where $M_g$ is the total galactic mass and velocities in units of the characteristic velocity $V_0$. So the balance between the gravitational and centrifugal force yields
$$\begin{aligned}
\label{a1}
&&\mkern-40mu \int_{-H/2}^{H/2} \int_{0}^{2\pi} \int_{0}^{1} \frac{\hat{r}cos\hat{\theta}-r}{(\hat{r}^2-r^2-2r\hat{r}cos\hat{\theta}+\Delta h^2)^{3/2}}\hat{\rho}(\hat{r},\hat{h})\hat{r}d\hat{r}d\hat{\theta}d\hat{h} \nonumber \\
&+&A\frac{v(r,h)^2}{r}=0~,\end{aligned}$$
where $A$ is the galactic rotation number
$$\begin{aligned}
A=\frac{R_{g}V_0^2}{G M_{g}}~.\end{aligned}$$
We get rid of $\hat{\theta}$ dependency by simplifying the integral
$$\begin{aligned}
\label{a2}
I(r,\hat{r},\Delta h)=\int_{0}^{2\pi} \frac{\hat{r}cos\hat{\theta}-r}{(\hat{r}^2-r^2-2r\hat{r}cos\hat{\theta}+\Delta h^2)^{3/2}}d\hat{\theta}\end{aligned}$$
using complete elliptic integrals of first and second kind. @integral [pages 179 & 182] give the solution to these integrals
$$\begin{aligned}
I_1 = \int \dfrac{ d x }{ ( a - b\cos x )^{1/2} } & = & \dfrac{2}{\sqrt{ a+b }} F(\delta,k)~; \\
I_3 = \int \dfrac{ d x }{ ( a - b\cos x )^{3/2} } & = & \dfrac{2}{ (a-b) \sqrt{a+b} } E(\delta,k)~ ,\end{aligned}$$
where
$$\begin{aligned}
&&\mkern-28mu x\in [0,\pi]; {\hspace*{0.5cm}}\sin \delta = \sqrt{ \dfrac{(a+b)(1-\cos \Phi)}{ 2( a - b \cos \Phi ) } }; {\hspace*{0.5cm}}\\ &&\mkern-28mu k=\sqrt{\dfrac{2b}{a+b}}; {\hspace*{0.5cm}}a>b>0; {\hspace*{0.5cm}}\Phi \in [0,\pi]~.\end{aligned}$$
$F(\delta,k)$ and $E(\delta,k)$ are the incomplete elliptic integrals of the first and second kind
$$\begin{aligned}
\label{EllipticIntegrals}
F( \delta, k) &=& \int_0^{\delta} \dfrac{d \phi}{\sqrt{ 1 - k^2 \sin^2 \phi }}; {\hspace*{0.5cm}}\nonumber\\
E( \delta, k) &=& \int_0^{\delta} \sqrt{ 1 - k^2 \sin^2 \phi } {\hspace*{0.25cm}}d\phi~.\end{aligned}$$
For the angle $\delta=\pi/2,$ we obtain complete elliptic integrals that we can rewrite by substituting $t=\sin \phi$ as
$$\begin{aligned}
\label{EllipticIntegralsComplete}
K(k) &\equiv& F \left( \frac{\pi}{2} , k \right) = \int_0^1 \dfrac{dt}{\sqrt{ (1-t^2)(1 - k^2 t^2) }}; \nonumber \\
E(k) &\equiv& E \left( \frac{\pi}{2},k \right) = \int_0^1 \sqrt{ \dfrac{1 - k^2 t^2}{1-t^2} } {\hspace*{0.25cm}}dt {\hspace*{0.25cm}}.\end{aligned}$$
When we plug our values:
$$\begin{aligned}
&\mkern-10mu a=r^2+\hat{r}^2+\Delta h^2; {\hspace*{0.5cm}}b=2r\hat{r}\end{aligned}$$
to the Eq. \[a1\], we get Eq.(\[16\]):
$$\begin{aligned}
&&\mkern-20mu\int_{-H/2}^{H/2}\int_{R_{min}}^{R_{max}}\frac{2}{r}\left[\frac{(\hat{r}+r)(\hat{r}-r)+\Delta h^2}{[(\hat{r}-r)^2+\Delta h^2]\sqrt{(\hat{r}+r)^2+\Delta h^2}}E(k) \right. \nonumber \\
&&\mkern-20mu\left.-\frac{1}{\sqrt{(\hat{r}+r)^2+\Delta h^2}}K(k)\right]\rho_0e^{-\hat{r}/h_{R}}e^{-\lvert h \rvert/h_{z}}\hat{r}~ \mathrm{d}\hat{r}\mathrm{d}h \nonumber \\
&&\mkern-20mu+A\frac{v_{c,\mathrm{disk}}(r,h)^2}{r}=0~. \nonumber\end{aligned}$$
|
---
abstract: '[ The $\Omega$ baryons with $J^P=3/2^\pm, 1/2^\pm$ are studied on the lattice in the quenched approximation. Their mass levels are ordered as $M_{3/2^+}<M_{3/2^-}\approx M_{1/2^-}<M_{1/2^+}$, as is expected from the constituent quark model. The mass values are also close to those of the four $\Omega$ states observed in experiments, respectively. We calculate the Bethe-Salpeter amplitudes of $\Omega(3/2^+)$ and $\Omega(1/2^+)$ and find there is a radial node for the $\Omega(1/2^+)$ Bethe-Salpeter amplitude, which may imply that $\Omega(1/2^+)$ is an orbital excitation of $\Omega$ baryons as a member of the $(D,L_N^P)=(70,0_2^+)$ supermultiplet in the $SU(6)\bigotimes O(3)$ quark model description. Our results are helpful for identifying the quantum numbers of experimentally observed $\Omega$ states. ]{}'
author:
- |
Jian Liang[^1],${}^{1}$ Wei Sun,${}^1$ Ying Chen[^2],${}^{1,2}$ Wei-Feng Chiu,${}^1$, Ming Gong,${}^{1,2}$ Chuan Liu,${}^3$\
Yu-Bin Liu,${}^4$ Zhaofeng Liu,${}^{1,2}$ Jian-Ping Ma,${}^5$ and Jian-Bo Zhang ${}^6$
title: 'Spectrum and Bethe-Salpeter amplitudes of $\Omega$ baryons from lattice QCD'
---
Introduction
============
There are four $\Omega$ baryon states (the strange number $S$=-3) observed from experiments [@Agashe:2014kda]. Except for the lowest-lying one, $\Omega(1672)$, which is well known as a member of the $J^P=3/2^+$ baryon decuplet, the $J^P$ quantum numbers of the other states, namely, $\Omega(2250)$, $\Omega(2380)$, and $\Omega(2470)$, have not been completely determined from experiments. If they are dominated by the three-quark components, the conventional $SU(6)\bigotimes O(3)$ quark model with a harmonic oscillator confining potential can be used to give them a qualitatively description. In this picture, the baryons made up of $u,d,s$ quarks can be classified into energy bands that have the same number $N$ of the excitation quanta in the harmonic oscillator potential [@Klempt:2009pi]. Each band consists of a number of supermultiplets, specified by $(D,L_N^P)$, where $D$ stands for the irreducible representation of the flavor-spin $SU(6)$ group, $L$ is the total orbital angular momentum, and $P$ is the parity of the supermultiplet. For $\Omega$ baryons whose flavor wave functions are totally symmetric, the ground state of $\Omega$ baryons should be in the $(56,0_0^+)$ supermultiplet with the quantum number $J^P=3/2^+$, namely the $\Omega(1672)$ state. The states in $(70,1_1^-)$ supermultiplet should have a total spin $S=1/2$ and a unit of the orbital excitation, such that their $J^P$ quantum number can be either $3/2^-$ or $1/2^-$. Therefore $\Omega_{3/2^-}$ and $\Omega_{1/2^-}$ are expected to approximately degenerate in mass up to a small splitting due to the different spin wave functions. The $J^P=\frac{1}{2}^+$ $\Omega$ baryons should be in either the $(56,2_2^+)$ or $(70,0_2^+)$ multiplets. Therefore the lowest several $\Omega$ states should have the energy levels ordered as $M_{3/2^+}<M_{3/2^-}\approx M_{1/2^-}<M_{1/2^+}$. On the other hand, for the $(56,2_2^+)$ and $(70,0_2^+)$ multiplets, since they belong to the different $SU(6)$ representations, their spatial wave functions can be different and can serve as a criterion to distinguish them from each other.
However, the quark model is not an ab-initio method and can only give qualitative results, so studies from first principles are desired, such as lattice QCD method. Early lattice QCD works can be found in [@Chiu:2005zc; @Alexandrou:2008tn]. The most recent systematic study with unquenched configurations was carried out in work [@Bulava:2010yg] where the authors find 11 strangeness -3 states with energies near or below 2.5 GeV using sophisticated smearing schemes for operators and variational method for the extraction of energy levels, but have difficulties to distinguish the single $\Omega$ states from possible scattering states. In this work, we explore the excited states of $\Omega$ baryons in the quenched approximation, whose advantage in this topic is that the excited states are free from the contamination of scattering states. We focus on the several lowest-lying $\Omega$ states with $J^P=\frac{1}{2}^\pm, \frac{3}{2}^\pm$. In addition to their spectrum, we also investigate the Bethe-Salpeter amplitudes of these states through spatially extended operators, which may shed lights on the internal structure of these $\Omega$ states.
This paper is organized as follows: Sec. \[operators and correlation functions\] contains our calculation method including the operator constructions, fermion contractions and wave function definitions. The numerical results of the spectrum and the wave functions are presented in Sec. \[numerical details and simulation results\]. The conclusions and a summary can be found in Sec. \[summary\].
operators and correlation functions {#operators and correlation functions}
===================================
Interpolating Operators for $\Omega$ Baryons
--------------------------------------------
The interpolating operator for $\Omega$ baryons can be expressed as $$\mathcal{O}^\mu=\epsilon^{abc}(s^T_a\mathcal{C}\gamma^\mu s_b)s_c,$$ where $\mathcal{C}=\gamma_2\gamma_4$ is the $C$-parity operator, $a,b,c$ are color indices, and $s^T$ means the transpose of the Dirac spinor of the strange quark field $s$. However, $\mathcal{O}^\mu$ has no definite spin and can couple to the $J=3/2$ and $J=1/2$ states [@Weinberg:1995mt]. The $J=3/2$ and $J=1/2$ components of $\mathcal{O}_{\Omega}^\mu$ can be disentangled by introducing the following projectors [@Alexandrou:2008tn] $$\begin{aligned}
\mathcal{P}_{3/2}^{\mu\nu}&=&\delta^{\mu\nu}-\frac{1}{3}\gamma^\mu\gamma^\nu-\frac{1}{3p^2}(p\!\!\!/\gamma^\mu
p^\nu+p^\mu\gamma^\nu p\!\!\!/),\nonumber\\
\mathcal{P}_{1/2}^{\mu\nu}&=&\delta^{\mu\nu}-\mathcal{P}_{3/2}^{\mu\nu}.\end{aligned}$$ In the lattice studies, only the spatial components of $\mathcal{O}^\mu$ are implemented. If we consider the $\Omega$ baryons in their rest frames, the projectors above can be simplified as $$\begin{aligned}
\mathcal{P}_{3/2}^{ij}&=& \delta^{ij}-\frac{1}{3}\gamma^i\gamma^j,\nonumber\\
\mathcal{P}_{1/2}^{ij}&=& \frac{1}{3}\gamma^i\gamma^j.\end{aligned}$$ Thus the spin projected operators with the definite spin quantum number can be obtained as $$\begin{aligned}
\label{irreps}
\mathcal{O}^i_{3/2}&=&\sum_j \mathcal{P}_{3/2}^{ij}\mathcal{O}_{\Omega}^j \nonumber\\
\mathcal{O}^i_{1/2}&=&\sum_j \mathcal{P}_{1/2}^{ij}\mathcal{O}_{\Omega}^j.\end{aligned}$$ Furthermore, one can also use the parity projectors $P^{\pm}=\frac{1}{2}(1\pm\gamma_4)$ to ensure the definite parities of baryons states.
It should be noted that for now all the operators are considered in the continuum case. On a finite lattice, the spatial symmetry group $SO(3)$ breaks down to the octahedral point group $O$, whose irreducible representations corresponding to $J=1/2$ and $J=3/2$ are the two-dimensional $G_1$ representation and the four-dimensional $H$ representation, respectively. Generally, there exist subduction matrices to project the continuum operators to octahedral point group operators [@Edwards:2011jj], say, $$\mathcal{O}(J,\Lambda)_{r}=\sum_{m}S(J,\Lambda)^{m}_{r}\mathcal{O}(J)_{m},$$ where $\mathcal{O}(J)_{m}$ is the continuum operator with total spin $J$ and the third component of spin $m$, $\mathcal{O}(J,\Lambda)_{r}$ is the $r$-th component of the octahedral point group operator under irreducible representation $\Lambda$, $S(J,\Lambda)^{m}_{r}$ is the subduction matrices. In our case, $S(\frac{1}{2},G_1)$ and $S(\frac{3}{2},H)$ are both unit matrices, so that the operators in Eq. \[irreps\], which are actully used in this study, are already the irreducible representations of the lattice symmetry group $O$.
We also consider the spatially extended interpolation operators by splitting $\mathcal{O}^\mu$ into two parts with spatial separations. The explicitly expressions are written as $$\begin{aligned}
\label{operator}
\mathcal{O}_{1}^\mu (r)&=&\sum_{|\vec{r}|}\epsilon^{abc}[s^T_a(x+\vec{r})\mathcal{C}\gamma^\mu s_b(x)]s_c(x),\nonumber\\
\mathcal{O}_{2}^\mu (r)&=&\sum_{|\vec{r}|}\epsilon^{abc}[s^T_a(x)\mathcal{C}\gamma^\mu s_b(x+\vec{r})]s_c(x),\nonumber\\
\mathcal{O}_{3}^\mu (r)&=&\sum_{|\vec{r}|}\epsilon^{abc}[s^T_a(x)\mathcal{C}\gamma^\mu
s_b(x)]s_c(x+\vec{r}).\end{aligned}$$ where the summations are over $\vec{r}$’s with the same ${r=|\vec{r}|}$ in order to guarantee the same quantum number as case of $r=0$. These three splitting procedures have been verified to be numerically equivalent, so we make use of the third type, $\mathcal{O}_3(r)$, in the practical study. These operators are obviously gauge variant, so we carry out the lattice calculation by fixing all the gauge configurations to the Coulomb gauge first. The general form of the two-point function of a baryon of quantum number $J^P$ with $P=\pm$ is $$\label{two-point}
C_J^{\pm,i}(r,t)=\mathrm{Tr}\left[(1\pm\gamma_4)\sum_{\vec{x},j}\langle\mathcal{P}_J^{ij}\mathcal{O}_3^{j}(r,x)
\bar{\mathcal{O}}_3^{j}(0)P_J^{ji}\ra\right]$$ The summation on $\vec{x}$ ensures a zero momentum. For $\Omega$ baryons, there exist six different Wick contractions as shown in the following figure Fig. (\[contract\]).
![Six ways of contraction, we use color indices to label the three $s$ quarks. Solid line means contraction.[]{data-label="contract"}](fig1.eps){height="3.5cm"}
![The effective mass plateaus of $\Omega$ baryons using point source with/without projections. Except for $J=\frac{3}{2}$ states, the point-source two point functions have not good plateaus at the short time range.[]{data-label="point_test"}](fig2.eps){height="5.0cm"}
Source technique
----------------
In principle, all states with the same quantum number $J^P$ contribute to the two-point functions $C_J^{P,i}(r,t)$. For baryons, it is known that the the signal-to-noise ratio of the two-points damps very quickly since the noise decreases as $\sim e^{-3/2m_\pi t}$ in $t$, which is much slower than the decay of the signal $e^{-M_B t}$, where $M_B$ is the baryon mass. Therefore, in order to obtain clear and reliable signals of the ground state from two-point functions in the available early time range, some source techniques are implemented by replacing the local operator $O_3^j(\mathbf{0},0)$ by some versions of spatially extended source operators $O_3^{j,(s)}(0)$ which enhance the contribution of the ground state and suppress that from excited states. The extended source operator $O_3^{j,(s)}$ is usually realized by calculating the quark propagators through a source vector with a spatial distribution $\phi(\mathbf{x})$, $$M(x;y)S_F^{(s)}(y;t_0)=\sum\limits_{\mathbf{z}}\delta(\mathbf{x}-\mathbf{z})\delta(t-t_0)\phi(\mathbf{z}),$$ thus the effective propagator $S_F^{(s)}(y;t=0)$ relates to the normal point source propagator $S_F(y;\mathbf{z},t_0)$ as $$S_F^{(s)}(y;t_0)=\sum\limits_{\mathbf{z}}\phi(\mathbf{z})S_F(y;\mathbf{z},t_0).$$ When one calculates a baryon two-point function using the same Wick contraction by replacing the point-source propagators with the effective propagators, it is equivalent to using the spatially extended source operator $$O^{(s)}(t_0)=\sum\limits_{\mathbf{z,w,v}}\phi(\mathbf{z})\phi(\mathbf{w})\phi(\mathbf{v})
\psi(\mathbf{z},t_0)\psi(\mathbf{w},t_0)\psi(\mathbf{v},t_0),$$ where $\psi\psi\psi$ stands for the original baryon operator (the color indices and corresponding $\gamma$ matrices are omitted for simplicity. Note that gauge links should be considered if one requires the gauge invariance of spatially extended operators). The matrix element of $O^{(s)}$ between the vacuum and the baryon state $|B\rangle$, which manifests the coupling of this operator to the state, can be expressed as, $$\label{overlap}
\langle 0 |O^{(s)}|B\rangle =
\sum\limits_{\mathbf{z,w,v}}\phi(\mathbf{z})\phi(\mathbf{w})\phi(\mathbf{v})\Phi_B(\mathbf{z},
\mathbf{w},\mathbf{v})\zeta_B,$$ where $\zeta_B$ is the spinor reflecting the spin of $|B\rangle$, and $\Phi_B(\mathbf{z},\mathbf{w},\mathbf{v})$ is its Bethe-Salpeter amplitude, which is defined as the corresponding matrix element of the original operator, $$\langle 0|\psi(\mathbf{z})\psi(\mathbf{w})\psi(\mathbf{v})|B\rangle \equiv
\Phi_B(\mathbf{z},\mathbf{w},\mathbf{v})\zeta_B,$$ In order to enhance the coupling $\langle 0 |O^{(s)}|B\rangle$ and suppress the related coupling of excited states, the essence is to tune the parameters in $\phi(\mathbf{x})$ such that $\phi(\mathbf{z})\phi(\mathbf{w})\phi(\mathbf{v})$ resembles $\Phi_B(\mathbf{z},\mathbf{w},\mathbf{v})$ as closely as possible and the overlap integration in Eq. (\[overlap\]) (actually summations over the spatial lattice sites) can be maximized. If the BS amplitudes can be approximately interpreted to be the spatial wave function of a state, the coupling of this operator to excited states can be minimized subsequently according to the orthogonality of the wave functions. The commonly used source techniques include the Gaussian smeared source [@Gusken:1989qx; @Marinari:1988tw]and the wall source in a fixed gauge. The Gaussian smeared source corresponds to the function $\phi(\mathbf{x})\sim
e^{-{\sigma^2|\mathbf{x}|^2}}$ with $\sigma^2$ a tunable parameter, while the wall source in a fixed gauge is the extreme situation of the Gaussian smeared source when $\sigma\rightarrow
\infty$. The Gaussian smeared source usually works well for states whose BS amplitude has no radial nodes. This is similar to the case in the quantum mechanics where a Gaussian-like function serves as a good trial wave function of the ground state in solving a bound state problem using the variational method with $\sigma$ the variational parameter.
For the case of this work, we try first the Gaussian smeared source for $\Omega$ baryons and find it work surely good for $\Omega_{\frac{3}{2}^+}$. It is not surprising since the $\Omega_{\frac{3}{2}^+}$ is the ground state whose spatial wave functions is $(1s)(1s)(1s)$ in the standard quark model with a harmonic oscillator potential. However for other states, especially for $\Omega_{\frac{1}{2}^+}$, we cannot get a good effective mass plateau before the signals are undermined by noise. Similar phenomena are also observed by previous works (see Ref. [@Alexandrou:2008tn] for example). Inspired by the quark model description that the $J^P=\frac{1}{2}^+$ decuplet baryons belong to the higher excitation energy bands, we conjecture that the BS amplitude of $\Omega_{\frac{1}{2}^+}$ has radial node(s), and thereby propose a new type of source which reflects some node structure, say, $$\phi(\mathbf{x})=(1-A|\mathbf{x}|^2)e^{-\sigma^2 |\mathbf{x}|^2},
$$ where $\sigma$ and $A$ are parameters to be tuned to give a good effective mass plateau in the early time range. The effects of the extended source operator on the effective masses of different states are illustrated in Fig. (\[smear\_test\]). For $J^P=\frac{3}{2}^\pm, \frac{1}{2}^-$ states, we use the Gaussian smeared sources which improve the qualities of the effective mass plateaus as expected. For the $J^P=\frac{1}{2}^+$ state, the new type of the source operators with the nodal structure makes the effective mass plateaus fairly satisfactory in contrast to the case of point source. We advocate that this new type of source operators can be potentially applied to other studies on radial excited states of hadrons.
![Smeared source $\Omega$ spectrum. For $J^p=\frac{3}{2}^{\pm}$ and $J^p=\frac{1}{2}^-$, we use common Gaussian smeared source, for $J^p=\frac{1}{2}^+$, we use a novel “smeared source" with a radial node.[]{data-label="smear_test"}](fig3.eps){height="5.0cm"}
numerical details and simulation results {#numerical details and simulation results}
========================================
The gauge configurations used in this work are generated on two anisotropic ensembles with the tadpole-improved gauge action [@Morningstar:1997ff]. The anisotropy $\xi\equiv a_s/a_t=5$ and the lattice sizes are $L^3\times T=16^3\times 96$ and $24^3\times 144$, respectively. The relevant input parameters are listed in Tab. \[latticesetup\], where the $a_s$ values are determined through the static potential with the scale parameter $r_0^{-1} = 410(20) $MeV. The spatial extensions of the two lattices are larger than 3 fm, which are expected to be large enough for $\Omega$ baryons such that the finite volume effects can be neglected. We use the tadpole improved Wilson clover action [@Liu:2001ss] to calculate the quark propagators with the bare strange quark mass parameter being tuned to reproduce the physical $\phi$ meson mass value. We use a modified version of a GPU inverter [@Clark:2009wm] to calculate all the inversions in this work.
$\beta$ $\xi$ $a_s/r_0$ $a_s$(fm) $La_s$(fm) $L^3\times T$ $N_{\rm conf}$
--------- ------- ----------- -------------- -------------- ------------------ ----------------
2.4 5 0.461(4) 0.222(2)(11) $\sim 3.55$ $16^3\times 96$ 1000
2.8 5 0.288(2) 0.138(1)(7) $\sim 3.31 $ $24^3\times 144$ 1000
: The input parameters for the calculation. Values of the coupling $\beta$, anisotropy $\xi$, the lattice size, and the number of measurements are listed. $a_s/r_0$ is determined by the static potential, the first error of $a_s$ is the statistical error and the second one comes from the uncertainty of the scale parameter $r_0^{-1}=410(20)$ MeV.
\[latticesetup\]
As mentioned before, the spatially extended operators we use for $\Omega$ baryons are not gauge invariant, so we calculate the corresponding two-point functions in the Coulomb gauge by first carrying out the gauge fixing to the gauge configurations. By the use of the source vectors with properly tuned operators, we generate the quark propagators in this gauge, from which the two-point functions in different channels are obtained. Since we focus on the ground states in each channel, the related two-point functions are analyzed with the single-exponential function form in properly chosen time windows, $$C_2^J(r,t)\overset{t\to\infty}{\sim}N^J\Phi^J(r)e^{-m^Jt},$$ where $J$ denotes different quantum numbers, $N^J$ stands for a irrelevant normalization constant, $\Phi^J(r)$ is the BS amplitude and $m^J$ is the mass. In order to take care of the possible correlation, we fit $C_2^J(r,t)$ with different $r$ simultaneously through a correlated miminal-$\chi^2$ fit procedure, where the covariance matrix are calculated by the bootstrap method. As such, in addition to the masses $m_J$, we can also obtain the $r$-dependence of the the BS amplitudes $\Phi^J(r)$. Figure (\[mass\]) shows the effective mass plateaus for $C^J(r=0,t)$ and the fit range. We quote the bootstrap errors as the statistical ones for masses and BS amplitudes.
![Effective mass plots of the $\Omega$ system. The two ensembles are both included. The points with errorbars are lattice data, while the colored bands are fit results indicating both the fit range and the fit error.[]{data-label="mass"}](fig4a.eps "fig:"){height="5.0cm"} ![Effective mass plots of the $\Omega$ system. The two ensembles are both included. The points with errorbars are lattice data, while the colored bands are fit results indicating both the fit range and the fit error.[]{data-label="mass"}](fig4b.eps "fig:"){height="5.0cm"}
The masses for different $\Omega$ states on the two lattice are listed in Tab. (\[masstable\]), where the mass values are expressed in the physical units using the lattice spacings in Tab. \[latticesetup\]. The masses of these states are insensitive to the lattice spacings which implies that the discretization uncertainty is small for these states. It is seen that the mass of the $J^P=\frac{3}{2}^+$ $\Omega$ we obtain is consistent with the physical mass of $\Omega(1672)$, the masses of $J^P=\frac{3}{2}^-$ and $\frac{1}{2}^-$ are almost degenerate, as expected from the quark model, but lower than the experimental states $\Omega(2250)$ and $\Omega(2380)$. For the $J^P=\frac{1}{2}^+$ state, we get a mass of 2.464(26) GeV on the coarse lattice while 2.492(14) GeV on the fine lattice, which is in agreement with the mass of $\Omega(2470)$.
--------- --------------------- --------------------- --------------------- ---------------------
$\beta$ $m_{\Omega_{3/2+}}$ $m_{\Omega_{3/2-}}$ $m_{\Omega_{3/2-}}$ $m_{\Omega_{1/2+}}$
(GeV) (GeV) (GeV) (GeV)
2.4 1.668(9) 2.176(26) 2.189(13) 2.464(26)
2.8 1.695(4) 2.153(5) 2.125(14) 2.492(14)
--------- --------------------- --------------------- --------------------- ---------------------
: The spectrum of the $\Omega$ baryons on the two lattices. The errors of masses are all statistical. We do not include the error owing to the uncertainty of $r_0^{-1}=410(2)$ MeV here.
\[masstable\]
The BS amplitudes for the $\frac{3}{2}^+$ and $\frac{1}{2}^+$ states are plotted in Fig. (\[wavef\]) (normalized as $\Phi_J(r=0)=1$). In order to compare the results from different lattices, we plot the $x-$axis in physical units. From the figure one can see that the discretization artifacts are also small for BS amplitudes. We do observe a radial node in the BS amplitude of $\frac{1}{2}^+$ state. We use the following functions $$\begin{aligned}
\label{fitting function}
\Phi_{\frac{3}{2}^+}(r)&=&e^{-(r/r_0)^\kappa}, \nonumber\\
\Phi_{\frac{1}{2}^+}(r)&=&(1-b~r^\kappa)e^{-(r/r_0)^\kappa},\end{aligned}$$ to fit the data points, which are also plotted in curves in the figure. The fit results are summarized in Tab. (\[wave\_fit\]).
$J^p$ $\beta$ $r_0$ (fm) $\kappa$ $b$
------------------- --------- ------------ ---------- --------
$\frac{3}{2}^{+}$ 2.4 0.504(3) 1.49(2)
$\frac{3}{2}^{+}$ 2.8 0.494(4) 1.55(2)
$\frac{1}{2}^{+}$ 2.4 0.568(5) 1.74(3) 3.7(1)
$\frac{1}{2}^{+}$ 2.8 0.529(8) 1.73(4) 4.5(2)
: Fit results of the BS amplitudes for $\Omega_{\frac{3}{2}^+}$ and $\Omega_{\frac{1}{2}^+}$.
\[wave\_fit\]
![The BS amplitudes of $\Omega\frac{3}{2}^+$ and $\Omega\frac{1}{2}^+$. The dots are lattice results while the lines are the fitting functions in Eq. (\[fitting function\]). A radial node of the BS amplitude of the $\Omega_{\frac{1}{2}^+}$ is observed.[]{data-label="wavef"}](fig5.eps){height="5.0cm"}
Now we resort to the non-relativistic quark model to understand the radial behavior of the BS amplitude of $J^P=1/2^+$ $\Omega$. In the non-relativistic approximation, the relativistic quark field $\psi$ can be expressed in terms of its non-relativistic components through the Foldi-Wouthuysen-Tani transformation $$\psi=\exp \left(\frac{\mathbf{\gamma}\cdot \mathbf{D}}{2m_s}\right) \left(
\begin{array}{c}\chi \\ \eta
\end{array}
\right),$$ where the Pauli spinor $\chi$ annihilates a quark and $\eta$ creates an anti-quark, and $\mathbf{D}$ is the covariant derivative operator. $\eta$ and $\chi$ satisfy the conditions $$\chi|0\rangle=0,~\langle 0|\chi^{\dagger}=0,~\eta^{\dagger}|0\rangle=0,~\langle 0|\eta=0.$$ With this expansion, the operator $\mathcal{O}_{\Omega}^i$ can be expressed as $$\begin{aligned}
\mathcal{O}_{\Omega}^i &\sim&
\epsilon^{abc}\left[\chi^{aT}\left(1+\frac{(\mathbf{\sigma}\cdot\overleftarrow{{\mathbf
D}})^2}{4m_s^2}\right)\sigma_2\sigma_i\left(1+\frac{(\mathbf{\sigma}\cdot\overrightarrow{{\mathbf
D}})^2}{4m_s^2}\right)\chi^b\right.
\nonumber\\
&&\left. -\chi^{aT}\frac{\mathbf{\sigma}\cdot\overleftarrow{{\mathbf
D}}}{2m_s}\sigma_2\sigma_i\frac{\mathbf{\sigma}\cdot\overrightarrow{{\mathbf
D}}}{2m_s}\chi^b\right] \left(
\begin{array}{c}
\left(1+\frac{(\mathbf{\sigma}\cdot\overrightarrow{{\mathbf
D}})^2}{4m_s^2}\right)\chi^c\\
\frac{\mathbf{\sigma}\cdot\overrightarrow{\mathbf{D}}}{2m_s}\chi^c
\end{array}
\right)\nonumber\\
&+&\ldots.\end{aligned}$$ We would like to caution that this expansion is not justified rigorously for the strange quark since its relativistic effect in the hadron might be important. However, the non-relativistic quark model are usually used to given reasonable descriptions of hadron spectrum, so we tentatively follow this direction to make the following discussion. The non-relativistic wave function for a baryon state in its rest frame is defined in principle as $$\Psi_J(\mathbf{x_1},\mathbf{x_2},\mathbf{x_3})\zeta\sim \langle
0|\epsilon^{abc}\chi^{aT}(\mathbf{x_1})\chi^b(\mathbf{x_2})\chi^c(\mathbf{x_3})|\Omega_J\rangle$$ where $\zeta$ stands for the spin wave function for $\Omega_J$. If we introduce the Jacobi’s coordinates, $$\begin{aligned}
\mathbf{R}&=&\frac{1}{3}(\mathbf{x}_1+\mathbf{x}_2+\mathbf{x}_3)\nonumber\\
\mathbf{\rho}&=&\frac{1}{\sqrt{2}}(\mathbf{x}_1-\mathbf{x}_2)\nonumber\\
\mathbf{\lambda}&=&\frac{1}{\sqrt{6}}(\mathbf{x}_1+\mathbf{x}_2-2\mathbf{x}_3),\end{aligned}$$ as is usually done in the non-relativistic quark model study of baryons, in the rest frame of $\Omega_{1/2^+}$ ($\mathbf{R}=0$), the matrix element of $\mathcal{O}_J^i(\mathbf{x}_1,\mathbf{x}_2,\mathbf{x_3})$ between the vacuum and the $\Omega$ state can be written qualitatively as $$\begin{aligned}
\label{BS-amplitude}
&&\langle 0|\mathcal{O}_J^i(\mathbf{x}_1,\mathbf{x}_2,\mathbf{x_3})|\Omega_J\rangle\nonumber\\
&\sim&\left(D^i+ A^i\frac{\partial^2}{\partial
\rho^2}+B^i\frac{\partial^2}{\partial\rho\partial\lambda}+C^i\frac{\partial^2}{\partial\lambda^2}\right)
\Psi_J(\mathbf{\rho},\mathbf{\lambda})\zeta\nonumber\\\end{aligned}$$ where we approximate the covariant derivative $\mathbf{D}$ by the spatial derivative $\mathbf{\nabla}$.
In the standard non-relativistic quark model with harmonic oscillator potentials for baryons, baryons can be sorted into energy bands of the the two independent oscillators, the so-called $\rho$-oscillator and $\lambda$-oscillator, which are depicted by the the radial and orbital quantum numbers $(n_\lambda, l_\lambda)$ and $(n_\rho, l_\rho)$ [@Klempt:2009pi]. For baryons made up of $u,d,s$ quarks, these energy bands are labelled as $(D,L_N^P)$, where $D$ is the irreducible representation of the flavor-spin $SU(6)$ group, $L=|l_\rho-l_\lambda|,
|l_\rho-l_\lambda|+1,\ldots, l_\rho+l_\lambda$ is the total orbital angular momentum, $N=2(n_\rho+n_\lambda)+(l_\rho+l_\lambda)$ is the total number of the excited quanta of the harmonic oscillators, and $P$ is the parity of baryons. For the flavor symmetric $\Omega$ baryons, the lowest $J^P=\frac{1}{2}^+$ states can be found in the supermultiplets $(56, 2_2^+)$, and $(70,0_2^+)$. $(56,2_2^+)$ has the excitation mode $(n_\lambda,n_\rho)=(0,0)$ and $(l_\lambda,l_\rho)=(2,0)$ or $(0,2)$ with the total spin $S=\frac{3}{2}$, and gives the quantum number $J^P=\frac{1}{2}^+, \frac{3}{2}^+, \frac{5}{2}^+, \frac{7}{2}^+$. $(70,0_2^+)$ has the excitation mode $(n_\lambda,n_\rho)=(0,0)$ and $(l_\lambda,l_\rho)=(1,1)$ with $S=\frac{1}{2}$, which corresponds to the quantum number $J^P=\frac{1}{2}^+$. In this picture, the spatial wave function of the $(56,2_2^+)$ multiplet can be written qualitatively (here we ignore the angular part) [@Karl:1969iz; @Faiman:1968js] $$\label{wavef1}
\Psi(\rho,\lambda)\sim (\rho^2+\lambda^2)e^{-\alpha (\rho^2+\lambda^2)},$$ while the spatial wave function of the $(70,0_2^+)$ multiplet is either $$\label{wavef2}
\Psi(\rho,\lambda)\sim (\rho^2-\lambda^2)e^{-\alpha(\rho^2+\lambda^2)},$$ or $$\label{wavef3}
\Psi(\rho,\lambda)\sim
\rho\lambda e^{-\alpha(\rho^2+\lambda^2)},$$ where the parameter $\alpha$ depends on the constituent quark mass and the parameters in the potential. Obviously, the local operators correspond to $\lambda=\rho=0$, such that their coupling to the $J^P=\frac{1}{2}^+$ state can be largely suppressed. Recalling that the interpolation operator we use for $\Omega$ baryons is $\mathcal{O}_3(r)$, which corresponds to $\rho=0$ and $\lambda\propto r$. As such, we have the qualitatively radial behavoirs of the Bethe-Salpeter amplitudes $$\langle 0|\mathcal{O}_{3,\frac{1}{2}^+}^i(r)|\Omega_{\frac{1}{2}^+}\rangle \sim(
A'+B'r^2+C'r^4)e^{-\alpha r^2}\zeta^i,$$ if we use the wave functions in Eq. (\[wavef1\]) and Eq. (\[wavef2\]), and $$0|\mathcal{O}_{3,\frac{1}{2}^+}^i(r)|\Omega_{\frac{1}{2}^+};(70,0_2^+)\rangle \sim(
A''+B''r^2)e^{-\alpha r^2}\zeta^i$$ for the wave function form in Eq. (\[wavef3\]). Obviously, the former may has two nodes in the $r$ direction, while the later has only one. In this sense, the radial behaviors of the BS amplitudes in Fig. \[wavef\] may imply that the $J^P=\frac{1}{2}^+$ $\Omega$ baryon we have observed is possibly mainly the $(70,0_2^+)$ state, whose spatial wave function may have the qualitative form in Eq. (\[wavef3\]). It should be noted that these discussions are very tentatively and the the reality can be much more complicated. This can be seen in Table \[wave\_fit\] where the parameters $\kappa$ deviate substantially from $\kappa=2$ which corresponds to the harmonic oscillator potential.
summary
=======
We carry out a lattice study of the spectrum and the Bethe-Salpeter amplitudes of $\Omega$ baryons in the quenched approximation. In the Coulomb gauge, we propose a new type of source vectors for the calculation of quark propagators, which is similar in spirit to the conventionally used Gaussian smearing source technique, but is oriented to increase the coupling to the states whose Bethe-Salpeter amplitude may have more complicated nodal behavior than that of the ground state. As for a excited states, either an orbital excitation or a radial excitations, it is expected that their BS amplititude may have radial nodes, so we use source vectors with nodal structures, which resemble the node structure of its BS amplitude. This technique works in practice, since we can obtain fairly good effective mass plateaus for $J^P=\frac{1}{2}^+$ at the early time slices.
With the quark mass parameter tuned to be at the strange quark mass using the physical mass of the $\phi$ meson, we calculate the spectrum of $\Omega$ baryons with the quantum number $J^P=\frac{3}{2}^\pm, \frac{1}{2}^\pm$ on two anisotropic lattices with the spatial lattice spacing set at $a_s=0.222(2)$ fm and $a_s=0.138(1)$ fm, respectively. On both lattices, the $J^P=\frac{3}{2}^-$ and $\frac{1}{2}^-$ $\Omega$ baryons have almost degenerate mass in the range from 2100 MeV to 2200 MeV. This is compatible with the expectation of the non-relativistic quark model that they are in the same supermultiplet $(70,1_1^-)$ with the same excitation mode, say, $(n_\lambda,n_\rho)=(0,0)$ and $(l_\rho,l_\lambda)=(1,0)$ or (0,1), and the same total quark spin $S=\frac{1}{2}$. For the $\frac{1}{2}^+$ $\Omega$ baryon, we obtain its mass at roughly 2400-2500 MeV. Furthermore, we also calculate the BS amplitude of the $\frac{1}{2}^+$ $\Omega$ baryon in the Coulomb gauge and observe a radial node, which can be qualitatively understood as the reflection of the second order differential of the non-relativistic wave function of $(70,0_2^+)$ baryons. Therefore it is preferable to assign the $\frac{1}{2}^+$ $\Omega$ state we observe to be a member of $(70,0_2^+)$ supermultiplet instead of that of $(56,2_2^+)$.
We notice that the latest $N_f=2+1$ full-QCD lattice calculation has obtained 11 energy levels of the $\Omega$ spectrum around and below 2500 MeV, but has difficulties in the assignment of their status for the sake of no reliable criterion to distinguish single particle states from the would-be scattering states. Fortunately we are free of this kind of trouble with the quenched approximation, such that the masses we obtain can be taken as those of the [*bare*]{} $\Omega$ baryon states before their hadronic decays are switch on. In comparison with the experiments, our predicted masses of $J^P=\frac{3}{2}^-$ and $\frac{1}{2}^-$ $\Omega$ baryons are close to that of $\Omega(2250)$, and the mass of $J^P=\frac{1}{2}^+$ is consistent with $\Omega(2470)$. This observation may be helpful in determining their $J^P$ quantum numbers.
ACKNOWLEDGEMENTS {#acknowledgements .unnumbered}
================
The numerical calculations are carried out on Tianhe-1A at the National Supercomputer Center (NSCC) in Tianjin. This work is supported in part by the National Science Foundation of China (NSFC) under Grants No. 11105153, No. 11335001, and 11405053. Z.L. is partially supported by the Youth Innovation Promotion Association of CAS. Y.C. and Z.L. also acknowledge the support of NSFC under No. 11261130311 (CRC 110 by DFG and NSFC).
[10]{} K. A. Olive [*et al.*]{} \[Particle Data Group Collaboration\], Chin. Phys. C [**38**]{}, 090001 (2014). E. Klempt and J. M. Richard, Rev. Mod. Phys. [**82**]{}, 1095 (2010) \[arXiv:0901.2055 \[hep-ph\]\]. T. W. Chiu and T. H. Hsieh, Nucl. Phys. A [**755**]{}, 471 (2005) \[hep-lat/0501021\]. C. Alexandrou [*et al.*]{} \[European Twisted Mass Collaboration\], Phys. Rev. D [**78**]{}, 014509 (2008) \[arXiv:0803.3190 \[hep-lat\]\]. J. Bulava, R. G. Edwards, E. Engelson, B. Joo, H. W. Lin, C. Morningstar, D. G. Richards and S. J. Wallace, Phys. Rev. D [**82**]{}, 014507 (2010) \[arXiv:1004.5072 \[hep-lat\]\]. S. Weinberg, “The Quantum theory of fields. Vol. 1: Foundations”. R. G. Edwards, J. J. Dudek, D. G. Richards and S. J. Wallace, Phys. Rev. D [**84**]{}, 074508 (2011) \[arXiv:1104.5152 \[hep-ph\]\]. S. Gusken, Nucl. Phys. Proc. Suppl. [**17**]{}, 361 (1990). E. Marinari, Nucl. Phys. Proc. Suppl. [**9**]{}, 209 (1989). C. J. Morningstar and M. J. Peardon, Phys. Rev. D [**56**]{}, 4043 (1997) \[hep-lat/9704011\]. C. Liu, J. h. Zhang, Y. Chen and J. P. Ma, Nucl. Phys. B [**624**]{}, 360 (2002) \[hep-lat/0109020\]. M. A. Clark, R. Babich, K. Barros, R. C. Brower and C. Rebbi, Comput. Phys. Commun. [**181**]{}, 1517 (2010) \[arXiv:0911.3191 \[hep-lat\]\]. G. Karl and E. Obryk, Nucl. Phys. B [**8**]{}, 609 (1968). D. Faiman and A. W. Hendry, Phys. Rev. [**173**]{}, 1720 (1968).
[^1]: liangjian@ihep.ac.cn
[^2]: cheny@ihep.ac.cn
|
---
author:
- 'Mohamad Shalaby (Mohamad@aims.ac.za)'
bibliography:
- 'trial.bib'
nocite: '[@*]'
title: |
Dynamics and Light Propagation\
in a Universe with Discrete Matter Content
---
Abstract {#abstract .unnumbered}
========
We discuss a model for a universe with discrete matter content instead of the continuous perfect fluid taken in FRW models. We show how the redshift in such a universe deviates from the corresponding one in an FRW cosmology. This illustrates the fact that averaging the matter content in a universe and then evolving it in time, is not the same as evolving a universe with discrete matter content. The main reason for such deviation is the fact that the photons in such a universe mainly travel in an empty space rather than the continuous perfect fluid in FRW geometry.
|
---
bibliography:
- 'partRatio.bib'
---
|
---
abstract: 'Impact of single particle onto a rigid substrate leads to its deformation and fragmentation. The flow associated with the particle spreading on a solid substrate after impact is extremely complicated. In this theoretical study a simplified model for the plastic flow with the rate-dependent yield strength is developed. The flow in the particle is approximated by an incompressible inviscid flow past a thin rigid disk. The expression for the pressure field distribution is obtained in the vicinity of the impact axis. The total momentum balance of the particle is used to derive the equations of the particle deformation by impact. The theoretical predictions of the typical geometrical parameters of the particle, the peak force and the evolution of the force in time are compared with the existing experimental data. The agreement is rather good.'
author:
- |
Ilia V. Roisman\
Institute for Fluid Mechanics and Aerodynamics, Technische Universität Darmstadt, Darmstadt, Germany\
`roisman@sla.tu-darmstadt.de`\
title: Dynamics of collision of a spherical ice crystal particle with a perfectly rigid substrate
---
Introduction
============
Ice particle impact onto a solid substrate is influenced by the impact parameters and material properties of the target. At relatively low velocities the impact leads to the elastic rebound. Starting from some limiting velocity the collision leads to a significant particle plastic deformation and even break-up. If the target or the particle is wet the impact could lead to the particle deposition and further agglomeration.
Important industrial processes are associated with single ice particle impact. One of them is the ice crystal ice accretion in the low pressure compression system of an aircraft [@currie2014experimental; @mason2011understanding; @currie2012fundamental; @bucknell2018experimental; @saunders1998laboratory; @veres2013modeling; @ayan2015prediction; @veres2012model] where the walls are hot enough to partially melt the particles during collision. Ice crystal accretion on internal components of aircraft engines and further ice layer shedding [@mason2011understanding; @mazzawy2007modeling] may lead to flameout, mechanical damage, rollback, etc. Sensors on the aircraft can also be affected by ice crystal accretion.
The physics of ice crystal accretion is not yet completely understood. It is known that the inception of accretion can be associated with the cooling of an initially hot substrate. As soon as the wall exposed to the flow of cold ice crystals reaches the melting temperature the ice layer starts to grow by accumulating of the fragments of the impacted particles [@lowe2016inception; @hauk2016investigation]. Another important reason leading to the particle deposition is the capillary forces associated with the liquid bridges formed between wet contacting particles. In the case of impacting particle the liquid bridge may decelerate the rebounding particle significantly [@antonyuk2009influence; @cruger2016coefficient; @gollwitzer2012coefficient; @mullins2003particle; @fu2005experimental]. The particle will agglomerate on the substrate if the typical particle deposition time, determined by the viscous, inertial and capillary forces in the liquid bridge, is shorter than the breakup time of the liquid bridge [@brulin2020pinch; @weickgenannt2015pinch; @qian2011motion; @curran2005liquid].
Ice crystal impact studies are also motivated by the atmospheric and meteorological sciences, since ice collision phenomena influence the physical processes in clouds, thunderstorm electrification processes,[@saunders1993review; @brooks1997effect; @saunders1998laboratory] can lead to the generation of fine crystal particles in clouds [@vardiman1978generation; @murphy2004particle].
In general particle collision with solid surfaces and between each other is a phenomenon relevant to many other technological processes, mainly governed by the dynamics of granular media. Among the examples of such applications are mineral processing, agricultural products, detergents, pharmaceuticals, food products and chemicals. The breakage of the particles leads to the evolution of the size distribution of the particles [@reynolds2005breakage], while coalescence (promoted by a presence of a liquid phase), or sintering in some cases, can cause a catastrophic defluidisation [@geldart1972effect; @antony2004granular]. Collision phenomena, including coagulation, rebound, and fragmentation, are also important physical processes controlling the dynamics of Saturn’s rings [@higa1998size; @supulver1995coefficient] or formation processes of satellites and planets in the outer Solar System [@kato1995ice].
Many existing theoretical approaches to the modeling of the impact, deformation and penetration of solid bodies are based on the hydrodynamic theories [@rosenberg1990hydrodynamic; @tate1986long; @frankel1990hydrodynamic; @lou2014long; @seguin2011dense; @anderson2017analytical; @yarin2017collision]. Such models are often based on a kinematically admissible flow, which satisfies the continuity equations, the boundary conditions and the momentum balance conditions in a simplified form, for example in integral form of in certain regions of the deforming media. Such models are able to successfully predict integral parameters of the problem, like penetration depth, projectile deformation, its residual velocity after penetration of a finite target.
Approximate model of deformation of a long plastic rod has been developed in [@taylor1948use]. The model is based on the assumption that the rod is deformed by the propagation of a plastic wave, generated by impact onto a perfectly rigid target. Similar ideas have been applied in [@Hauk.2015; @roisman2015impact] for the description of deformation and breakup of an ice particle. However, these simplified approaches are not sufficient to model the processes of particle fragmentation or heat transfer. For such advanced modeling some details of the velocity field in the deforming particle are required [@yarin2000model].
The main objective of this study is the theoretical model for deformation of a plastic particle during its collision with a perfectly rigid target. The flow in the particle is approximated by an instantaneous potential flow past a disc. The disc in the model is associated with the circular spot of the deforming particle and the rigid substrate. In the plastic flow in the particle the yields strength $Y$ is a function of the effective strain rate. Such material behaviour is typical to quasi-brittle ice crystal particles [@tippmann2013experimentally]. The momentum balance equation is solve in the proximity to the particle impact axis. The stress field in the deforming particle allows to estimate the total contact force and to predict evolution of the particle deformation. The predicted evolution of the force applied to the target by an impacting ice particle agrees well with the experimental data existing in the literature.
Flow field in the deforming particle
====================================
The flow field in a particle deforming by a collision with a rigid target is determined by the material properties of the ice, like yield strength $Y$, elastic shear modulus $G$, density. The flow can be also influenced by the developments of cracks leading to the particle fragmentation. Nevertheless, in many cases a simplified kinematically admissible flow can be a rather good approximation for the velocity field in the particle, especially if the inertial effects in the flow are dominant. Such approach is useful for the description of the velocity fields in the penetration mechanics [@yarin2017collision]. In this study a flow in the deforming particle is approximated by an incompressible potential flow, associated with the axisymmetric flow around a thin disc. The assumption of the particle incompressibility is valid for the impact velocities much smaller that the speed of sound in ice. The speed of sound for pressure waves at $-10 ^\circ$C is 3865 m/s, which is much higher than the typical impact velocities of 100 m/s relevant to the aircraft applications.
Consider a cylindrical coordinate system $\{r,\theta,z\}$ with the corresponding base unit vectors $\{\bm e_r, \bm e_\theta, \bm e_z\}$. The flow field in the deforming particle is approximated by the flow around a flat disc of the radius $a$, where $a$ is the instantaneous radius of the impression. The velocity potential for this flow relative to the wall is obtained from the well-known solution [@batchelor1967introduction] in the form $$\label{eq_phi}
\phi = \frac{2 a U}{\pi} \cos\eta \left[\sinh\xi \cot^{-1}(\sinh\xi) -1\right] - U z,$$ where $\xi, \eta$ are dimensionless elliptic coordinates defined through $$\label{xieta}
\xi + i \eta= \sinh^{-1}\left(\frac{z+i r}{a}\right).$$
This velocity satisfies the impenetrability conditions at the solid substrate $z=0, r < a$. The velocity field at the infinity approaches to the uniform flow $u_z=-U$.
Equation (\[xieta\]) allows to explicitly derive the expressions for the coordinates in the cylindrical system
$$\begin{aligned}
\xi &=& \frac{1}{2} \ln \left[\frac{(z+S \cos \beta)^2+(r+S \sin \beta)^2}{a^2} \right]\\
\eta &=& \tan^{-1}\left[\frac{r+S \sin \beta}{z+S \cos \beta}\right]\\
S&=& \left[4 r^2 z^2 +(a^2-r^2+z^2)^2\right]^{1/4},\\
\beta &=& \left\{\begin{matrix}
\frac{1}{2} \tan^{-1}\left(\frac{2 r z}{z^2+a^2-r^2}\right) & \mathrm{if}\, r^2< z^2+a^2 \\
\frac{1}{2} \tan^{-1}\left(\frac{2 r z}{z^2+a^2-r^2}\right)+\frac{\pi}{2} & \mathrm{if}\, r^2 >z^2+a^2\\
\frac{\pi}{2} & \mathrm{if}\, r^2 =z^2+a^2
\end{matrix}\right.\label{eq_beta}\end{aligned}$$
The components of the velocity field in the cylindrical coordinate system can be now obtained from the potential function $$\label{eq_u}
\bm u = \bm\nabla \phi= (\phi_{,\xi} \xi{,_r} + \phi_{,\eta} \eta_{,r})\bm e_r + (\phi_{,\xi} \xi{,_z} + \phi_{,\eta} \eta_{,z})\bm e_z.$$
Velocity field in the vicinity of the impact axis
-------------------------------------------------
The velocity components can be obtained from (\[eq\_u\]) with the help of (\[eq\_phi\])-(\[eq\_beta\]). Near the axis the expressions for the velocity components can be reduced to
$$\begin{aligned}
u_r & =& U\left[\frac{2 a^3 r U}{\pi \left(a^2+z^2\right)^2} + \mathcal{O}\left(\frac{r^3}{a^3}\right)\right]\\
u_z &= &U\left[\frac{2 \cot ^{-1}\left(\frac{z}{a}\right) }{\pi }-\frac{2 a z }{\pi
\left(a^2+z^2\right)}-1 -\frac{4 a^3 r^2 z}{\pi \left(a^2+z^2\right)^3} + \mathcal{O}\left(\frac{r^3}{a^3}\right)\right] \label{eq:uz}\end{aligned}$$
The corresponding rate-of-strain tensor $\bm E$ is the symmetric part of the velocity gradient at the impact axis $$\begin{aligned}
\bm E &=& \frac{\dot\gamma}{2} \left(\bm e_r \otimes \bm e_r-\frac{4r z}{a^2+z^2}(\bm e_r \otimes \bm e_z+\bm e_z \otimes \bm e_r) + \bm e_\theta \otimes \bm e_\theta - 2 \bm e_z \otimes \bm e_z\right),\label{eq:strainr}\\
\dot\gamma&=&\frac{4 a^3 U}{\pi \left(a^2+z^2\right)^2},\label{eq:strainrateeq}\end{aligned}$$ where the symbol $\otimes$ denotes the usual tensor product, and $\dot\gamma \equiv \sqrt{2/3}\sqrt{\bm E:\bm E}$ is the equivalent rate of strain.
Dimensionless particle dislodging during impact
-----------------------------------------------
Let us introduce a dimensionless particle dislodging $\zeta(t)$ scaled by the particle initial radius $R$ $$\zeta = \frac{1}{R} \int_0^t U(t) \mathrm{d}t.$$
The evolution of the particle height can be estimated by the numerical integration of the equation $\mathrm{d}h/\mathrm{d}t = u_z(z=h)$ leading to
\[eq:difh\] $$\begin{aligned}
\frac{1}{R} \frac{\mathrm{d} h}{\mathrm{d} \zeta}& =& \frac{2 \cot ^{-1}\left(\frac{h}{a}\right)}{\pi }-\frac{2 a h }{\pi\left(a^2+h^2\right)}-1,\\
h &=& 2 R, \quad \mathrm{at}\quad \zeta = 0.\end{aligned}$$
![Theoretically predicted dimensionless shapes of the impacting particle at different values of the dimensionless displacement $\zeta$.[]{data-label="fig:shapes"}](shape.jpg){width="60.00000%"}
It is known that in the case of impact of an inviscid drop the flow near the wall does not influence the outer flow. The value of $U$ is thus constant and the dislodging is reduced to the dimensionless time $\zeta = U_0/R$, where $U_0$ is the impact velocity. Theoretically predicted shapes of the impacting particle, computed using the velocity field (\[eq\_u\]), is shown in Fig. \[fig:shapes\]. These predictions do not account for the ejection of the lamella along the substrate. Nevertheless, the shapes are very similar to the observed forms of the deforming liquid drop during its high-velocity impact onto a dry substrate.
In Fig. \[fig:Height\] the predicted particle dimensionless height $h/R$ is shown as a function of $\zeta$ in comparison with the computations of a liquid drop impacts with very high Reynolds and Weber numbers [@roisman2009inertia]. These computations are validated by comparison with the numerous experimental data. The agreement between the present theory and the CFD computations is rather good, although no adjustable parameters have been introduced in the theory.
![Theoretically predicted dimensionless particle height $h/R$ as a function of $\zeta$, in comparison with the computations for high-speed impact of a liquid drop [@roisman2009inertia].[]{data-label="fig:Height"}](Height.jpg){width="60.00000%"}
Stress fields in the deforming plastic particle
===============================================
Plastic stresses near the particle axis
---------------------------------------
The stress tensor in the particle is determined by the velocity field and by the yield strength $Y$ $$\label{eq:sigPL}
\bm \sigma = - p \bm I +\bm \sigma',\quad \bm\sigma'= \left(\frac{2}{3}\right)^{1/2} \frac{Y}{\sqrt{\bm E:\bm E}} \bm E$$ where $p$ is the pressure and $\bm I$ is the unit tensor.
In this study the effect of the strain rate on the hardening of the particle material is taken into account. The material hardening is an important property of the ice crystals [@tippmann2013experimentally]. This means the uniaxial yield strength is modeled in the form: $$\label{formY}
Y = Y_0 y(\dot\gamma),$$ where $y$ is a dimensionless function of the the equivalent rate of strain $\dot\gamma$, determined in (\[eq:strainrateeq\]) and $Y_0$ is the static yield strength at $\dot \gamma\rightarrow 0$.
Near the particle axis, $r\rightarrow 0$, the deviatoric part of the stress is obtained using (\[eq:strainr\]) and (\[eq:sigPL\]) and further linearization it for small $r$ $$\label{sigmaStr}
\bm\sigma' = \frac{Y}{3} \left(\bm e_r \otimes \bm e_r-\frac{4r z}{a^2+z^2}(\bm e_r \otimes \bm e_z+\bm e_z \otimes \bm e_r) + \bm e_\theta \otimes \bm e_\theta - 2 \bm e_z \otimes \bm e_z\right).$$
The momentum balance equation in the particle flow can be written in the form $$\label{Mom}
\bm\nabla\left(\rho \frac{\partial \phi}{\partial t}+\frac{\rho}{2} \bm\nabla\phi\cdot\bm\nabla\phi + p \right) = \bm\nabla\cdot\bm\sigma'.$$ where the divergence of the deviatoric stress at $r\rightarrow 0$ is obtained from (\[sigmaStr\]) $$\label{divMom}
\bm\nabla\cdot\bm\sigma' =\left[\frac{32 Y_0 a^3 U z}{3\pi \left(a^2+z^2\right)^3} \frac{\mathrm{d} y(\dot\gamma)}{\mathrm{d}\dot\gamma} - \frac{8 z Y_0 }{3(a^2+z^2)}y(\dot\gamma)\right] \bm e_z.$$
Integration of (\[Mom\]) with the help of (\[divMom\]) yields $$p = - \frac{2 Y}{3}+\frac{2 Y_0}{3} \int_{\dot\gamma(0)}^{\dot\gamma}\frac{y(\dot\gamma)}{\dot\gamma}\mathrm{d}\dot\gamma -\frac{\rho}{2} \bm\nabla\phi\cdot\bm\nabla\phi -\rho \frac{\partial \phi}{\partial t} + f(t),$$ where $f(t)$ is a function of time which has to be determined from the boundary conditions. The expression for the pressure at the axis of the particle for the velocity potential defined in (\[eq\_phi\]) is therefore $$\begin{aligned}
p &=& f(t) - \frac{2 Y(z)}{3}+\frac{2 Y_0}{3} \int_{\dot\gamma(0)}^{\dot\gamma}\frac{y(\dot\gamma)}{\dot\gamma}\mathrm{d}\dot\gamma\nonumber \\
&-& \frac{\rho z}{\pi }\left[\pi -2 \cot ^{-1}\left(\frac{z}{a}\right)\right] \frac{\mathrm{d} U}{\mathrm{d} t} -\frac{2\rho z^2 U}{\pi \left(a^2+z^2\right)} \frac{\mathrm{d} a}{\mathrm{d} t}\nonumber\\
&-& \frac{\rho}{2} U^2 \left\{\frac{2 \cot ^{-1}\left(\frac{z}{a}\right)}{\pi }-\frac{2 a z }{\pi\left(a^2+z^2\right)}-1\right\}^2 \label{eq:p0t}\end{aligned}$$
The function $f(t)$ is determined from the condition at the rear tip of the particle $z=h(t)$ where the normal stress $\sigma_{zz}= -p - 2 Y(h)/3$ vanishes: $$\begin{aligned}
f(t) &=& -\frac{2 Y_0}{3} \int_{\dot\gamma(0)}^{\dot\gamma(h)}\frac{y(\dot\gamma)}{\dot\gamma}\mathrm{d}\dot\gamma\\
&+& \frac{\rho h}{\pi }\left[\pi -2 \cot ^{-1}\left(\frac{h}{a}\right)\right] \frac{\mathrm{d} U}{\mathrm{d} t} +\frac{2\rho h^2 U}{\pi \left(a^2+h^2\right)} \frac{\mathrm{d} a}{\mathrm{d} t} \nonumber\\
&+& \frac{\rho}{2} U^2 \left\{\frac{2 \cot ^{-1}\left(\frac{h}{a}\right)}{\pi }-\frac{2 a h }{\pi\left(a^2+h^2\right)}-1\right\}^2 \label{eq:ft}\end{aligned}$$
Finally, the normal stress at the wall surface, $z=0, \,r=0$, is obtained from (\[eq:p0t\]) and (\[eq:ft\]) in the form
$$\begin{aligned}
\label{eq:szzG}
-\sigma_{zz} &=& A(\zeta) \rho U\frac{\mathrm{d} U}{\mathrm{d} \zeta} + B(\zeta) \rho U^2 + C Y_0, \label{eq:szz}\\
A &=& \frac{h}{R}\left[1 - \frac{2}{\pi} \cot ^{-1}\left(\frac{h}{a}\right)\right] , \\
B&=& \frac{2 h^2}{\pi R \left(a^2+h^2\right)} \frac{\mathrm{d} a}{\mathrm{d} \zeta}+ \frac{1}{2} \left\{\frac{2 \cot ^{-1}\left(\frac{h}{a}\right)}{\pi }-\frac{2 a h }{\pi\left(a^2+h^2\right)}-1\right\}^2, \label{eq:B}\\
C &=& \frac{2}{3} \int_{\dot\gamma(h)}^{\dot\gamma(0)}\frac{y(\dot\gamma)}{\dot\gamma}\mathrm{d}\dot\gamma.\label{funC}\end{aligned}$$
where $A(\zeta)$ and $B(\zeta)$ are dimensionless functions of the dimensionless particle dislodging $\zeta$, and $C$ is a dimensionless function based on the distribution of the of the strain rates.
Momentum balance of an entire particle
--------------------------------------
Let us assume the total axial momentum of the deforming particle in integral form $$\label{eq:defKY}
P(\zeta) = \pi \rho R^3 U K(\zeta),$$ where $K(\zeta)$ is a dimensionless function of $\zeta$. The momentum balance equation $$\frac{\mathrm{d}P}{\mathrm{d}t} \approx \pi a^2 \sigma_{zz},$$ can be rewritten with the help of (\[eq:szzG\]) and (\[eq:defKY\]) in the form $$\begin{aligned}
\left(K + \frac{a^2 A}{R^2} \right) U\frac{\mathrm{d}U}{\mathrm{d}\zeta} &+&\left(\frac{\mathrm{d}K}{\mathrm{d}\zeta} + \frac{a^2 B}{R^2}\right)U^2 + \frac{ a^2 C Y_0}{ R^2\rho} =0. \label{eqMomC}\end{aligned}$$
The momentum balance equation can be solved numerically if the dependence of the dimensionless particle axial momentum $K(\zeta)$ is known. This function is determined from the consideration of a high-speed particle impact governed exclusively by the inertial terms.
The effect of the elastic region in the deforming particle is neglected in this study since the strains in the particle grow rather fast as soon the particle is deforming. The analysis of the elastic region is described in \[appa\], where the minor effect of the elastic stresses is demonstrated.
High impact velocity approximation in the limit $Y\ll \rho U^2$
===============================================================
In the case of very high impact velocity the effect of the yield stress can be much smaller than the inertia. Therefore the terms associated with the yield strength in (\[eqMomC\]) can be neglected.
Moreover, since the inviscid is flow is disturbed only in a finite region near the wall of the size comparable with $a$, the value of $U$ far from the wall is assumed to be constant. The following expression for the axial momentum balance is obtained from (\[eqMomC\]) for $U(\zeta)=U_0$ $$\begin{aligned}
\label{eqforK}
\frac{\mathrm{d}K}{\mathrm{d}\zeta} + \frac{a^2 B}{R^2}&=&0.\\
\zeta &=& \frac{t U_0}{R}.\end{aligned}$$
The impression radius at early times can be roughly approximated by the radius of the truncated sphere $a\approx R\sqrt{2 \zeta}$. This assumption is confirmed by the numerous experimental data [@rioboo2002time]. In Fig. \[fig:pinviscid\] the theoretical predictions for $B(\zeta)$ are compared with the CFD computations [@roisman2009inertia; @eggers2010drop] for the initial stage of a high Reynolds number impact of a liquid drop. The agreement between the theory and the computations is rather good which means that the model accounts for the main physical players. Note that no adjustable parameters have been used in the present theory.
Integration of the ordinary differential equation (\[eqforK\]) subject the initial conditions $K=4/3$ at $\zeta =0$, corresponding to the axial momentum of a rigid sphere, yield the following expression for $K(\zeta)$ which can be reduced to $$\label{eq:defK}
K=\frac{4 }{3} - 2 \int_0^\zeta \zeta B(\zeta)\mathrm{d} \zeta.$$
The computed values of the dimensionless axial momentum of the drop $K(\zeta)$ is shown in Fig. \[fig:Kinviscid\].
Note that the present solution is valid only for relatively small values of $\zeta$. For long times $\zeta \gg 1$ the spreading of the drop is governed by a radially expanding flow in a thin lamella. Such flow has been intensively studied in the literature [@yarin1995impact; @yarin2017collision; @roisman2009inertia] and is out of the scope of this study.
Equations of motion of the deforming plastic particle
=====================================================
The governing equations of particle deformation can be now written in the dimensionless form using $$\begin{aligned}
a &=& \overline{a}(\zeta) R, \quad h = \overline h(\zeta) R, \quad t = \overline t R/U_0, \label{scales}\\
U &=& \overline U U_0, \quad Y_0 = \rho U_0^2 \overline Y.\label{scalesb}\end{aligned}$$
Expression (\[eqMomC\]) in the dimensionless form is written assuming $\overline a \approx \sqrt{2\zeta}$ with the help of (\[scales\]), (\[scalesb\]) and (\[eq:difh\])
\[eq:momY\] $$\begin{aligned}
\overline U\frac{\mathrm{d} \overline U}{\mathrm{d}\zeta} &=&- \frac{2 \zeta C \overline Y}{K + 2\zeta A} .\label{eqMomCdl}\\
\frac{\mathrm{d} \overline h}{\mathrm{d} \zeta}& =& -\frac{\overline h \sqrt{2} g }{\pi\sqrt{\zeta}}-\frac{A}{\overline h},\\
\frac{\mathrm{d}\overline t}{\mathrm{d} \zeta} &=& \frac{1}{\overline U},\end{aligned}$$ where $$\begin{aligned}
A &\equiv& \overline h \left[1 - \frac{2}{\pi} \cot ^{-1}\left(\frac{\overline h}{\sqrt{2 \zeta}}\right)\right],\\
B&\equiv& \frac{\overline h^2 g}{\sqrt{2}\pi \zeta^{3/2} } + \frac{1}{2} \left[\frac{A}{\overline h}+\frac{2\overline h g }{\pi\sqrt{2\zeta}}\right]^2, \label{eq:Bdl}\\
% G &\equiv& \frac{4\zeta^{1-b/2}\eta}{3 b} \left(\frac{4 U_0 \overline U}{\sqrt{2} \pi R }\right)^b \left[1-g^{2b}\right]- \frac{8\zeta}{3} \ln g,\\
g &\equiv& \frac{2\zeta}{2\zeta+\overline h^2}.\end{aligned}$$
The system of the ordinary differential equations (\[eq:momY\]) can be integrated numerically with the help of the solution of the differential equation (\[eq:difh\]) for $\overline h$ and of the expression for the dimensionless axial momentum of the particle (\[eq:defK\]) subject the initial conditions $$\zeta =0, \quad \overline U=1,\quad \overline h = 2, \quad \mathrm{at}\quad \overline t = 0.$$
The total force produced by the particle impact on the target can be now evaluated in the form $$\label{eq:forceZ}
F_{z}(\zeta)\approx -\pi a^2 \sigma_{zz}=2\zeta\pi \rho R^2 U_0^2\left( B\overline U^2+\frac{C K \overline Y}{K+2 A \zeta}\right)$$
Results and discussion
======================
The relation of the dependence of the yield strength on the strain rate can be estimated by examining the existing values of the maximum impact force. For a perfectly plastic material the function $y(\dot \gamma)$ defined in (\[formY\]) is constant, $y =1$. But this simplification is not sufficient for a reliable description of ice particle impact.
In this study we use the value for the static yield strength of the ice particle $Y_0=5.6$ MPa determined in [@tippmann2013experimentally]. The dimensionless evolution of the yield stress, defined in (\[formY\]), is assumed in the form $$\label{eq:formy}
y(\dot\gamma)= 1+\chi-\chi\exp(- \tau \dot\gamma)$$ which corresponds to the conditions $$y(0) = 1,\quad y(\infty)\rightarrow 1+\chi,$$ which ensures finite stresses even at very high strain rates, typical to the very early stages of particle impact $\zeta\ll 1$. The dimensionless constant $\chi$ and the characteristic time $\tau$ have to be determined from by the fitting with the experimental data.
Integration of the expression (\[funC\]) for the function $C$ with the help of (\[eq:strainrateeq\]) and (\[eq:formy\]) yields $$\begin{aligned}
C &=& \frac{4}{3}(\chi+1) \ln \frac{2\zeta+\overline h^2}{2\zeta} +\frac{2\chi}{3} \mathrm{Ei}\left[-\frac{8\sqrt{2}\tau U_0\zeta^{3/2} \overline U}{\pi R\left(2\zeta + \overline h^2\right)^2}\right]\nonumber\\
&-& \frac{2\chi}{3} \mathrm{Ei}\left[-\frac{2\sqrt{2}\tau U_0 \overline U}{\pi R\zeta^{1/2}}\right].\end{aligned}$$
In Fig. \[fig:maxforce\] the theoretical predictions of the peak force computed using (\[eq:forceZ\]) is plotted as a function of the impact velocity $U_0$ in comparison with the experimental data from [@tippmann2013experimentally]. In all the computations the fitting values $\chi=5.0$ and $\tau=0.0002$ s are used. The agreement is rather good for various ice particle initial diameters $D_0$. Moreover, in Fig. \[fig:maxforce1\] the assumed dependence of the yield strength $Y(\dot\gamma)$ on the strain rate is compared with the estimations from the full CFD computations [@rhymer2012force]. The values of the yield strength are of the same order despite the fact that a very simplified model is used in this study. It should be noted that the form (\[eq:formy\]) is chosen accounting for the possibility of the derivation of the solutions for the stresses in an explicit form. We have also tried to minimize the number of the fitting parameters. Generally any form of the function $y(\dot\gamma)$ can be used in the solution for different particle materials.
![Theoretically predicted force evolution $F(t)$ in comparison with the experimental data from [@kim2000modeling]. []{data-label="fig:forcekim"}](FE.jpg "fig:"){width="49.00000%"}![Theoretically predicted force evolution $F(t)$ in comparison with the experimental data from [@kim2000modeling]. []{data-label="fig:forcekim"}](FE1.jpg "fig:"){width="49.00000%"}
![Theoretically predicted force evolution $F(t)$ in comparison with the experimental data from [@rhymer2012force]. []{data-label="fig:forcerhymer"}](FE2.jpg "fig:"){width="49.00000%"}![Theoretically predicted force evolution $F(t)$ in comparison with the experimental data from [@rhymer2012force]. []{data-label="fig:forcerhymer"}](FE3.jpg "fig:"){width="49.00000%"}
The theoretical predictions of the force $F(t)$ computed using expression (\[eq:forceZ\]) are shown in Figs. \[fig:forcekim\] and \[fig:forcerhymer\] in comparison with the experimental data from [@kim2000modeling] and [@rhymer2012force], respectively. In all the cases the theory allows to predict rather well the peak value and the order of magnitude of the contact duration. In Fig. \[fig:forcekim\] the predicted time corresponding to the force peak is slightly underpredicted. As already mentioned in [@kim2000modeling] this delay can be explained by the not perfectly spherical shape of the ice particle which leads to the difficulties in the precise determination of the instant of contact. Moreover, the target used in the experiments, which is a carbon/epoxy composite panel, can be deflected by particle impact if the impact velocity is high enough. The possible target deformation and damage can be evaluated using the forces predicted by this theory, however their effect on the particle deformation is not yet accounted for.
The predictions in Fig. \[fig:forcerhymer\] agree with the experimental data much better than in Fig. \[fig:forcekim\]. The initial instant in the experiments shown in the graph in Fig. \[fig:forcerhymer\] is defined by the inception of the rise of the measured force.
In some cases the theory slightly overpredicts the value of the contact force at the later stages of drop deformation after the force reaches the maximum value. This overprediction can be explained by the particle fragmentation. The effect of the particle fragmentation on the value of the effective yield strength is not considered in this theoretical model.
The value of the maximum dimensionless particle dislodging $\zeta_\mathrm{max}$ at which the particle velocity vanishes can serve as a measure of the particle total deformation. The theoretically predicted values of $\zeta_\mathrm{max}$ are shown in Fig. \[fig:zMaxTheory\] as a function of the impact velocity for several particle initial radii. The value of $\zeta_\mathrm{max}$ is mainly determined by the impact velocity $U_0$. However it reduces also for smaller particles. This effect is caused by the particle hardening for higher strain rates, which are of order $\dot\gamma \sim U_0/R$. For smallest particle sizes the dependence on the radius is only minor. However, accurate experimental data with such relatively small particles are necessary for the confirmation of these predictions.
Conclusions
===========
A theoretical model has been developed for the deformation of a spherical particle due to its normal collision with a perfectly rigid target. The stress field in the deforming particle takes into account the hardening effects, namely the increase of the yield strength $Y$ on the local effective strain rate. This model is applied to the description of an impact of ice crystal particle.
The model is based on the potential flow field around a thin disc. It satisfies the continuity and momentum balance equations for initial stages of impact of an spherical particle. The momentum balance equations is solved in the vicinity of the impact axis. The obtained stress at the target interface is used for the estimation of the total force.
In the limit $Y=0$ the flow is reduced to the flow in spreading inviscid drop. The theoretical predictions for the evolution of the drop height in time and for the evolution of the pressure at the substrate agree very well with the existing computational results for drop impact with very high value of the Reynolds and Weber numbers.
The dependence of the yield strength $Y$ on the local strain rate is estimated by the fitting of the theoretical predictions for the maximum force with the experiments. Finally, the model is applied for the prediction of the time evolution of the force. The theoretical predictions agree rather well with the experiments data for various particle impact velocity and initial diameter. This indicates that the theory is able not only to predict the total force but also to estimate the characteristic time of collision process and the characteristic strain rate. These results indicate that the approach in this study can be potentially applied to the prediction of the heat transfer during collision. The flow field can be used for the description of the particle fragmentation.
Acknowledgement {#acknowledgement .unnumbered}
===============
The funding received within the European Union’s Horizon 2020 research and innovation program under grant agreement MUSIC-haic-767560 is gratefully acknowledged.
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Elastic region of the particle {#appa}
==============================
The strain tensor $\bm \varepsilon$ in the elastic region can be estimated through $$\label{strainvec}
\frac{\mathrm{d}\bm \varepsilon}{\mathrm{d} t} = \bm E.$$
The strain tensor can be expressed using (\[eq:uz\])-(\[strainvec\])
$$\begin{aligned}
\bm \varepsilon &=&\varepsilon (\bm e_r \otimes \bm e_r + \bm e_\theta \otimes \bm e_\theta - 2 \bm e_z \otimes \bm e_z),\\
\frac{\mathrm{d} \varepsilon}{\mathrm{d} \zeta} &=& \frac{2 R a^3 }{\pi \left(a^2+z^2\right)^2},\\ \frac{1}{R}\frac{\mathrm{d} z}{\mathrm{d} \zeta} &=& \frac{2 \cot ^{-1}\left(\frac{z}{a}\right) }{\pi }-\frac{2 a z }{\pi
\left(a^2+z^2\right)}-1,\label{dzdzeta}\end{aligned}$$
The impression radius $a=a(\zeta)$ is determined only by $\zeta$ and the initial shape of the particle.
The elastic-plastic boundary can be now determined from the yield condition $$G \sqrt{6\bm \varepsilon \cdot \bm \varepsilon} = Y,$$ where $G$ is the shear modulus of ice particle. This condition at the axis yields $$\varepsilon = \frac{Y}{6 G}.$$
![Evolution of the strain $\varepsilon$ along the particle axis.[]{data-label="fig:epsilon"}](epsilons.jpg){width="60.00000%"}
For a spherical particle of radius $R$, the coordinate $z=z^\star$ of the elastic-plastic boundary is obtained using the relation $a=R\sqrt{1-(1-\zeta)^2}$. In Fig. \[fig:epsilon\] the computed values of the strain $\varepsilon$ at the particle axis are shown for various $s$. Since for ice $Y/G\sim 10^{-3}$ already at relatively small particle displacements the particle is completely plastic.
In the further analysis for very high impact velocities leading to significant particle deformation the effect of the elasticity at the initial stage is neglected.
|
---
abstract: 'Event sequence, asynchronously generated with random timestamp, is ubiquitous among applications. The precise and arbitrary timestamp can carry important clues about the underlying dynamics, and has lent the event data fundamentally different from the time-series whereby series is indexed with fixed and equal time interval. One expressive mathematical tool for modeling event is point process. The intensity functions of many point processes involve two components: the background and the effect by the history. Due to its inherent spontaneousness, the background can be treated as a time series while the other need to handle the history events. In this paper, we model the background by a Recurrent Neural Network (RNN) with its units aligned with time series indexes while the history effect is modeled by another RNN whose units are aligned with asynchronous events to capture the long-range dynamics. The whole model with event type and timestamp prediction output layers can be trained end-to-end. Our approach takes an RNN perspective to point process, and models its background and history effect. For utility, our method allows a black-box treatment for modeling the intensity which is often a pre-defined parametric form in point processes. Meanwhile end-to-end training opens the venue for reusing existing rich techniques in deep network for point process modeling. We apply our model to the predictive maintenance problem using a log dataset by more than 1000 ATMs from a global bank headquartered in North America.'
author:
- |
Shuai Xiao$^{1}$[^1], Junchi Yan$^{23}$, **Stephen M. Chu**$^3$, **Xiaokang Yang**$^1$, **Hongyuan Zha**$^4$\
$^1$ Shanghai Jiao Tong University $^2$ East China Normal University $^3$ IBM Research – China $^4$ Georgia Tech\
{benjaminforever,xkyang}@sjtu.edu.cn, jcyan@sei.ecnu.edu.cn, schu@us.ibm.com, zha@cc.gatech.edu
bibliography:
- 'xiao-yan.bib'
title: Modeling The Intensity Function Of Point Process Via Recurrent Neural Networks
---
Introduction
============
Event sequence is becoming increasingly available in a variety of applications such as e-commerce transactions, social network activities, conflicts, and equipment failures etc. The event data can carry rich information not only about the event attribute (e.g. type, participator) but also the timestamp $\{z_i,t_i\}_{i=1}^N$ indicating when the event occurs. Being treated as a random variable when the event is stochastically generated in an asynchronous manner, the timestamp makes the event sequence fundamentally different from the time series [@MontgomeryTSBook15] with equal and fixed time interval, whereby the time point only serves as a role for index $\{y_t\}_{t=1}^T$. A major line of research [@AalenPP2008] has been devoted to study event sequence, especially exploring the timestamp information to model the underlying dynamics of the system, whereby point process [@SnyderPPBook12] has been a powerful and compact framework in this direction.
Recently there are many machine learning based models for scalable point process modeling. We attribute the progressions in this direction in part to the smart mathematical reformulations and optimization techniques e.g. [@LewisJNS2011; @ZhouICML13; @ZhouAISTATS13], as well as novel parametric forms for the conditional intensity function [@shen2014modeling; @ErtekinRPP2015; @xu2016icu] as carefully designed by researchers’ prior knowledge to capture the character of the dataset in study. However, one major limitation of the parametric forms of point process is due to their specialized and restricted expression capability for arbitrary distributed event data which trends to be oversimplified or even infeasible for capturing the problem complexity in real applications. Moreover, it runs at the risk of model underfitting due to the misjudgement on model choice. Recent works e.g. [@ZhouICML13] start to turn to non-parametric form to fit the structure of a point process, but their method is under the Hawkes process formulation,while this formulation runs at the risk of model mis-choice.
In this paper, we view the conditional intensity of a point process as a nonlinear mapping between the predicted transient occurrence intensity of events with different types, and the model input information of event participators, event profile and the system history. Such a nonlinear mapping is expected to be complex and flexible enough to model various characters of real event data for its application utility.
In fact, deep learning models, such as Convolutional Neural Networks (CNNs) [@LeCunIEEE98], Recurrent Neural Networks (RNNs) [@PascanuICML13] have attracted wide attention in recent vision, speech and language communities, and many of them has dominated the competing results on perceptual benchmark tasks e.g. [@ILSVRC15]. In particular, we turn to RNNs as a natural way to encode such nonlinear and dynamic mapping, in an effort for modeling an end-to-end nonlinear intensity mapping without any prior knowledge.
**Key idea and highlights** Our model interprets the conditional intensity function of a point process as a nonlinear mapping, which is synergetically established by a composite neural network with two RNNs as its building blocks. As illustrated in Fig.\[fig:ppts\], time series (top row) and event sequence (bottom row) are distinct to each other in that time series is more suitable to carry the synchronously (i.e. in a fixed pace) and regularly updated or constant profile features, while the event sequence can compactly catch event driven, more abrupt information, which can affect the condition intensity function over longer time period. More specifically, the highlights of this paper are:
1\) We first make an observation that many conditional intensity functions can be viewed as an integration of two effects: i) spontaneous background component inherently affected by the internal (time-varying) attributes of the individual and the event type; ii) effects from history events. Meanwhile, most information in real world can also be covered by continuously updated features like age, temperature, and asynchronous event data such as clinical records, failures. This motivates us to devise a general approach.
2\) Then we use an RNN whose units are aligned with the time points of a time series as its units, and an RNN whose units are aligned with events. The time series RNN can timely track the spontaneous background while the event sequence RNN is used to efficiently capture the long-range dependency over history with arbitrary time intervals. This allows to fit arbitrary dynamics of point process which otherwise will be very difficult or often impossible to be specified by a parameterized model under certain assumptions.
3\) To our best knowledge, this is the first work to *fully* interpret and instantiate the conditional intensity function with fused time series and event sequence RNNs. This opens up the room for connecting the neural network techniques to traditional point process that emphasizes more on specific model driven by domain knowledge. More importantly the introduction of a full RNN treatment lessen the efforts for the design of (semi-)parametric point process model and its complex learning algorithms which often call for special tricks that prohibiting the wide use for practitioners. In contrast, neural networks and specifically RNN, is becoming off-the-shelf tools and getting widely used recently.
4\) Our model is simple, general and can be end-to-end trained. We target to a predictive maintenance problem. The data is from a global bank headquartered in North America, consisting decades of thousands of event logs for a large number of Automated Teller Machines (ATMs). The state-of-the-art performance on failure type and timestamp prediction corroborates its suitability to real-world applications.
Related Work and Motivation
===========================
We view the related concepts and work in this section, which is mainly focused on Recurrent Neural Networks (RNNs) and their applications in time series and sequences data, respectively. Then we give our point of view on existing point process methods and their connection to RNNs. All these observations indeed motivate the work of this paper.
**Recurrent neural network** The building blocks of our model are the Recurrent Neural Networks (RNNs) [@ElmanCS90; @PascanuICML13] and its modern variant Long Short-Term Memory (LSTM) units [@HochreiterNC97; @GravesArxiv13]. RNNs are dynamical systems whose next state and output depend on the present network state and input, which are more general models than the feed-forward networks. RNNs have long been explored in perceptual applications for many decades, however it can be very difficult to train RNNs to learn long-range dynamics perhaps in part due to the vanishing and exploding gradients problem. LSTMs provide a solution by incorporating memory units that allow the network to learn when to forget previous hidden states and when to update hidden states given new information. Recently, RNNs and LSTMs have been successfully applied in large-scale vision [@GregorDraw15], speech [@GravesICML14] and language [@SutskeverNIPS14] problems.
**RNNs for series data** From application perspective, we view RNNs works by two scenarios as particularly considered in this paper: i) RNNs for synchronized series with evenly spaced interval e.g. time series or indexed sequence with pure order information e.g. language; ii) asynchronous sequence with random timestamp e.g. event data.
*i) Synchronized series*: RNNs have been a long time a natural tool for standard time series modeling and prediction [@ConnorTNN94; @HanTSP04; @ChandraNC12; @ChenJH13], whereby the indexed series data point is fed as input to an (unfold) RNN. In a broader sense, video frames can also be treated as time series and RNN are widely used in recent visual analytics works [@JainICRA16; @TripathiBMVC16] and so for speech [@GravesICML14]. RNNs are also intensively adopted for sequence modeling tasks [@ChungArxiv14; @BengioNIPS15] when only order information is considered.
*ii) Asynchronous event*: In contrast, event sequence with timestamp about their occurrence, which are asynchronously and randomly distributed over the continuous time space, is another typical input type for RNNs [@DuKDD16; @ChoiArxiv16; @EstebanArxiv16] and [@CheArxiv16] (despite its title for ’time series’). One key differentiation against the first scenario is that the timestamp or time duration between events (together with other features) is taken as input to the RNNs. By doing so, (long-range) event dependency can be effectively encoded.
**Point process** Point process has been a principled framework for modeling event data [@AalenPP2008]. The dynamics of the point process can be well captured by its conditional intensity function whose definition is briefly reviewed here: for a short time window $[t,t+dt)$, $\lambda(t)$ represents the rate for the occurrence of a new event conditioned on the history $\mathcal{H}_t = \{z_i,t_i|t_i < t\}$: $$\notag
\lambda(t)=\lim_{\Delta t\rightarrow 0}\frac{\mathbb{E}(N(t+\Delta t)-N(t)|\mathcal{H}_t)}{\Delta t}=\frac{\mathbb{E}(dN(t)|\mathcal{H}_t)}{dt}$$ where $\mathbb{E}(dN(t)|\mathcal{H}_t)$ is the expectation of the number of events happened in the interval $(t, t + dt]$ given the historical observations $\mathcal{H}_t$. The conditional intensity function has played a central role in point processes and many popular processes vary on how it is parameterized.
1\) *Poisson process* [@KingmanPP92]: the homogeneous Poisson process has a very simple form for its intensity function: $\lambda(t)=\lambda_{0}$. Poisson process and its time-varying generalization are both assumed to be independent of the history.
2\) *Reinforced poisson processes* [@PemantlePS07; @shen2014modeling]: the model captures the ‘rich-get-richer’ mechanism characterized by a compact intensity function, which is recently used for popularity prediction [@shen2014modeling].
3\) *Hawkes process* [@HawkesBiometrika71]: Hawkes process has received wide attention recently in social network analysis [@ZhouAISTATS13], viral diffusion[@yang2013mixture] and criminology [@lewis2010self] etc. It explicitly uses a triggering term to model the excitation effect from history events and is originally motivated to analyze the earthquake and its aftershocks[@OgataJASA88].
4\) *Reactive point process* [@ErtekinRPP2015]: it can be regarded as a generalization for the Hawkes process by adding a self-inhibiting term to account for the inhibiting effects from history events.
5\) *Self-correcting process* [@IshamSPP79]: its background part increases steadily, while it is decreased by a constant $e^{-\alpha}<1$ every time a new event appears.
We reformulate these intensity functions in their general form in Table \[tab:intensity\]. It tries to separate the spontaneous background component and history event effect explicitly.
**Predictive maintenance** Predictive maintenance [@MobleyBH02] is a sound testbed for our model which refers to a practice that involves equipment risk prediction to allow for proactive scheduling of corrective maintenance. Such an early identification of potential concerns helps deploy limited resources more cost effectively, reduce operations costs and maximize equipment uptime [@GrallTR02]. Predictive maintenance is adopted in a wide variety of applications such as fire inspection [@MadaioKDD16], data center [@SirbuPPW15] and electrical grid [@ErtekinRPP2015] management. For its practical importance in different scenarios and relative rich event data for modeling, we target our model to a real-world dataset of more than 1,000 automated teller machines (ATMs) from a global bank headquartered in North America.
\[tab:intensity\]
Network Structure and End-to-End Learning
=========================================
Taking a sequence $\{\textbf{x}\}_{t=1}^T$ as input, the RNN generates the hidden states $\{\textbf{h}\}_{t=1}^T$ and outputs a sequence [@ElmanCS90; @PascanuICML13]. Specifically, we implement our RNN with Long Short Term Memory (LSTM) [@HochreiterNC97; @GravesArxiv13] for its popularity and well-known capability for efficient long-range dependency learning. In fact other RNN variant e.g. Gated Recurrent Units (GRU) [@ChungArxiv14] can also be alternative choice. We reiterate the formulation of LSTM: $$\begin{aligned}
\notag
\textbf{i}_t&= \sigma(\textbf{W}_i\textbf{x}_t+\textbf{U}_i\textbf{h}_{t-1}+\textbf{V}_i\textbf{c}_{t-1}+\textbf{b}_{i}),\\\notag
\textbf{f}_t&= \sigma(\textbf{W}_f\textbf{x}_t+\textbf{U}_f\textbf{h}_{t-1}+\textbf{V}_f\textbf{c}_{t-1}+\textbf{b}_{f}),\\\notag
\textbf{c}_t&= \textbf{f}_t\textbf{c}_{t-1}+\textbf{i}_t\odot\text{tanh}(\textbf{W}_c\textbf{x}_t+\textbf{U}_c\textbf{h}_{t-1}+\textbf{b}_c),\\\notag
\textbf{o}_t&= \sigma(\textbf{W}_o\textbf{x}_t+\textbf{U}_o\textbf{h}_{t-1}+\textbf{V}_o\textbf{c}_{t}+\textbf{b}_{o}),\\\notag
\textbf{h}_t&=\textbf{o}_t\odot\text{tanh}(\textbf{c}_t)\end{aligned}$$where $\odot$ denotes element-wise multiplication and the recurrent activation $\sigma$ is the Logistic Sigmod function. The above system can be reduced into an LSTM equation: $$\notag
(\textbf{h}_t,\textbf{c}_t)= \text{LSTM}(\textbf{x}_t,\textbf{h}_{t-1}+\textbf{c}_{t-1})$$We consider two types of input: i) continuously and evenly distributed time-series data e.g. temperature; ii) event data whose occurrence time interval is random. The network is comprised by two RNNs using evenly spaced time series $\{y_t\}_{t=1}^T$ to model the background intensity of events occurrence and event sequence $\{z_i,t_i\}_{i=1}^N$ to capture long-range event dependency. As a result, we have: $$\begin{aligned}
(\textbf{h}^y_t,\textbf{c}^y_t)&= \text{LSTM}_{y}(\textbf{y}_t,\textbf{h}^y_{t-1}+\textbf{c}^y_{t-1}),\\
(\textbf{h}^z_t,\textbf{c}^z_t)&= \text{LSTM}_{z}(\textbf{z}_t,\textbf{h}^z_{t-1}+\textbf{c}^z_{t-1}),\\
\textbf{e}_t &= \text{tanh}(\textbf{W}_f[\textbf{h}^y_t,\textbf{h}^z_t]+\textbf{b}_f),\\
\textbf{U}_t &=\text{softMax}(\textbf{W}_U\textbf{e}_t+\textbf{b}_U),\\
\textbf{u}_t &=\text{softMax}(\textbf{W}_u[\textbf{e}_t,\textbf{U}_t]+\textbf{b}_u)\\
{s}_t &= \textbf{W}_s\textbf{e}_t+{b}_s,\end{aligned}$$where $U$ and $u$ denotes the main type and subtype of events respectively. $s$ is the timestamp associated with each event. The total loss is the sum of the time prediction loss and the cross-entropy loss for event type:
$$\sum_{j=1}^N\left(-W_U^j\log(U^j_{t})-w_u^j\log(u^j_{t})-\log\left(f(s^j_t|h^j_{t-1})\right)\right)
\label{eq:loss}$$
where $N$ is the number of training samples indexed by $j$, and $s^j_t$ is the timestamp for the coming event, while $h^j_{t-1}$ is the history information. The underlying rationale for the third term is that we not only encourage correct classification of the coming event type, but also reinforce the corresponding timestamp of the event shall be close to the ground truth. We adopt a Gaussian penalty function with a fixed $\sigma^2=10$: $$f(s^j_t|h^j_{t-1})=\frac{1}{\sqrt{2\pi\sigma}}\exp\left(\frac{-(s^j_{t}-\tilde{s}^j_{t})^2}{2\sigma^2}\right)$$ The output $\tilde{s}^j_{t}$ from the timestamp prediction layer is fed to the classification loss layer to compute the above penalty given the actual timestamp $s^j_{t}$ for sample $i$.
Following the importance weighting methodology for skewed data of model training [@rosenberg2012classifying], the weight parameters $W, w$ for both main-type and subtype are set as the inverse of the sample number ratio in that type against the total size of samples, in order to weight more on those classes with fewer training samples. For the loss of independent main-type or subtype prediction as shown in Fig.\[fig:flat\_pred\], we set the weight parameter $w$ and $W$ to zero respectively.
We adopt RMSprop gradients [@RmspropArxiv15] which have been shown to work well on training deep networks to learn these parameters.
Experiments on Real-world Data
==============================
We use failure prediction for predictive ATMs maintenance as a typical example of event based point process modeling. We have no prior knowledge on the dynamics of the complex system and the task can involve arbitrarily working schedules and heterogeneous mix of conditions. It takes much cost or even impractical to devise specialized models.
\[tab:type\_statistics\]
Problem and real data description
---------------------------------
In maintenance support services, when a device fails, the equipment owner raises a maintenance service ticket and technician will be assigned to repair the failure. In fact, the history log and relevant profile information about the equipment can be indicative signals for the coming failures.
The studied dataset is comprised of the event logs involving error reporting and failure tickets, which is originally collected from a large number of ATMs owned by an anonymous global bank headquartered in North America. The bank is also a customer of the technical support service department of a Fortune 500 IT company.
**ATM models** The training data consists of 1085 ATMs and testing data has 469 ATMs, in total 1557 Wincor ATMs that cover 5 ATM machine models: ProCash 2100 RL (980, 430), 1500 RL (19, 5), 2100 FL (53, 21), 1500 FL (26, 10), and 2250XE RL (7, 3). The numbers in the bracket indicate the number of machines for training and testing.
**Event type** There are two main types ‘ticket’ and ‘error’ from Sep. 2014 to Mar. 2015. Statistics is presented in Table \[tab:type\_statistics\]. Moreover ‘error’ is divided into 6 subtypes regarding in which component the error occurs: 1) printer (PRT), 2) cash dispenser module (CNG), 3) internet data center (IDC), 4) communication part (COMM), 5) printer monitor (LMTP), 6) miscellaneous e.g. hip card module, usb (MISC).
**Features** The input features for the two RNNs are: **1)** *Time series RNN*: For each sub-window of length 7 days, for the time series RNN, we extract features including: i) the inventory information: ATM models, age, location, etc; ii) Event statistics, including tickets events from maintenance records, and errors from system log. Their occurrence frequencies are used as features. The concatenation of the above two categories of features serves as the features for each sub-window i.e. time series point. **2)** *Event sequence RNN*: event type and the time interval between two events.
**Model setting** We use a single layer LSTM of size 32 with Sigmoid gate activations and tanh activation for hidden representation. The embedding layer is fully connected and it uses tanh activation and outputs a 16 dimensional vector. One-hot or embedding can be used for event type representation. For a large number of types, embedding representation is compact and efficient. For time series RNN, we set the length of each sub-window (i.e. the evenly spaced time interval) to be 7 days and the number of sub-window to be 5. In this way, our observation length is 35 days for time series. For event-dependency, the length of event sequence can be arbitrarily long. Here we take it by 7.
We also test degraded versions of our model as follows:
**1)** **Time series RNN**: the input is event sequence (the right half in the yellow part of Fig.\[fig:overview\]) is removed. Note this design is in spirit similar to many LSTM models [@JainICRA16; @TripathiBMVC16] used for video analytics, whereby the frame sequence can be treated as time series as the input to LSTMs. **2)** **Event (sequence) RNN**: the RNN whose input is time series (the left half in the yellow part of Fig.\[fig:overview\]) is removed; **3)** **Intensity RNN**: two RNN are fused as shown in Fig.\[fig:overview\]. For the above three methods, the output layer is directly the fine-grained subtype of events with no hierarchical structure as shown in the top left part of Fig.\[fig:overview\]) in dark green. We also term three ‘hierarchical’ versions whose two hierarchical prediction layers in Fig.\[fig:overview\] are used: **4)** **Time series hRNN**, **5)** **Event (sequence) hRNN**, **6)** **Intensity hRNN**.
In addition, we compare three major peer methods. For Logistic model, the input are the concatenation of feature vectors for all active time series RNN sub-windows (set to 5 in this paper). For RMTPP and Hawkes process, we train the model on the event sequences with associated information. In fact, RMTPP will further process the event data into the similar input information to our event RNN.
1\) **Logistic model**: we use Logistic regression for event timestamp prediction and use another independent Logistic classification model for event type prediction.
2\) **Recurrent Marked Temporal Point Processes (RMTPP)**: [@DuKDD16] uses neural network to model the event dependency flexibly. The method can only sample transient time series features when an event happens and use partially parametric form for the base intensity.
3\) **Hawkes Process**: To enable multi-type event prediction, we use a Multi-dimensional Hawkes process. Similar to [@ZhouAISTATS13], we also add a sparsity regularization term on the mutual infection matrix but the low-rank assumption is removed as we only have 6 subtypes.
**Evaluation metrics** We use several popular prediction metrics for performance evaluation. For the coming event type prediction, we adopt *Precision*, *Recall*, *F1 Score* and *Confusion matrix* over 2 main types (‘error’, ‘ticket’) as well as Confusion matrix over 6 subtypes under ‘error’. Note all these metrics are computed for each type, and then are averaged over all types. For event time prediction, we use the *Mean Absolute Error (MAE)* which measures the absolute difference between the predicted time point and the actual one. These settings are similar to [@DuKDD16].
To evaluate the type and timestamp prediction jointly, we devise two more strict metrics. For type prediction, we narrow down to test the samples whose timestamp prediction error *MAE*$<3$ days and we compute the new *F1 score+*. For timestamp, we recompute the new *MAE+* only for the samples whose coming event is correctly predicted.
**Platform** The code is based on Theano running on a Linux server with 32G memory, 2 CPUs with 6 cores for each: Intel(R) Xeon(R) CPU E5-2603 v3 @ 1.60GHz. We also use 4 GPU:GeForce GTX TITAN X for acculturation.
\[tab:performance\]
\
\
\
\
Results and discussion
----------------------
All evaluations are performed on the testing dataset distinctive to training set whose statistics are shown in Table \[tab:type\_statistics\].
**Averaged performance** Table \[tab:performance\] shows the averaged performance among various types of events. As shown in Fig.\[fig:hierarchical\], we test two architectures of the event type prediction layer, i.e. hierarchical predictor (Fig.\[fig:hier\_pred\]) and flat independent predictors (Fig.\[fig:flat\_pred\]). The main type includes ‘ticket’ and ‘error’ and the subtype include ‘ticket’ and the other six subtypes under ‘error’ as we describe earlier in the paper.
**Confusion matrix** The confusion matrix for the six subtypes under ‘error’ event, as well as for the two main types ‘ticket’ and ‘error’ are shown in Fig.\[fig:confusion\] by various methods.
We make observations and analysis based on the results:
1\) As shown in \[tab:performance\], for main-type, the flat architecture that directly predicts the main types outperforms the hierarchical one in different settings of the input RNN as well as varying evaluation metrics. This can be explained that the loss function focuses on the main-type misclassification only. While for the subtype prediction, the hierarchical layer performs better since it fuses the output from the main-type prediction layer and the embedding layer as shown in Fig.\[fig:hier\_pred\].
2\) No surprisingly, for both event type and timestamp prediction, our main approach, i.e. intensity RNN that fuses two RNNs outperforms its counterparts time series RNN and event sequence RNN by a notable margin. While the event RNN also often performs better than the time series counterpart. This suggests at least in the studied dataset, history event effects are important for the future event occurrence.
3\) Our main method intensity RNN is almost always superior against other methods except for the main-type prediction task, whereby the Logistic classification model performs better. However for more challenging tasks i.e. subtype prediction and event timestamp prediction, our method significantly outperforms especially for subtype prediction task. Interestingly, all point process based models obtain better results on this task which suggests the point process models are more promising compared with classical classification models. Indeed, our methodology provides an end-to-end learning mechanism without any pre-assumption in modeling point process. All these empirical results on real-world tasks suggest the efficacy of our approach.
![The evolving of point process modeling.[]{data-label="fig:evolve"}](pic/evolve){width="48.00000%"}
Conclusion
==========
We use Fig.\[fig:evolve\] to conclude and further position our model in the development of (implicit and explicite) modeling the intensity function of point process. In fact, Hawkes process uses a full explicit parametric model and RMTPP misses the dense time series features to model time-varying base intensity and assumes a partially parametric form for it. We make a further step by a full implicit mapping model. Our model (see Fig.\[fig:overview\]) is simple, general and can be learned end-to-end with standard backpropagation and opens up new possibilities for borrowing the advances in neural network learning to the area of point process modeling and applications. The representative study in this paper has clearly suggests its high potential to real-world problems, even we have no domain knowledge on the problem at hand. This is in contrast to existing point process models where an assumption about the dynamics is often need to be specified beforehand.
[^1]: Correspondence author is Junchi Yan. This research was partially supported by The National Key Research and Development Program of China (2016YFB1001003), NSFC (61602176, 61672231, 61527804, 61521062), STCSM (15JC1401700, 14XD1402100), China Postdoctoral Science Foundation Funded Project (2016M590337), the 111 Program (B07022) and NSF (IIS-1639792, DMS-1620345).
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---
abstract: 'We consider the asymptotic behavior of a family of gradient methods, which include the steepest descent and minimal gradient methods as special instances. It is proved that each method in the family will asymptotically zigzag between two directions. Asymptotic convergence results of the objective value, gradient norm, and stepsize are presented as well. To accelerate the family of gradient methods, we further exploit spectral properties of stepsizes to break the zigzagging pattern. In particular, a new stepsize is derived by imposing finite termination on minimizing two-dimensional strictly convex quadratic function. It is shown that, for the general quadratic function, the proposed stepsize asymptotically converges to the reciprocal of the largest eigenvalue of the Hessian. Furthermore, based on this spectral property, we propose a periodic gradient method by incorporating the Barzilai-Borwein method. Numerical comparisons with some recent successful gradient methods show that our new method is very promising.'
author:
- 'Yakui Huang[^1]'
- 'Yu-Hong Dai[^2]'
- 'Xin-Wei Liu[^3]'
- 'Hongchao Zhang[^4]'
title: 'On the asymptotic convergence and acceleration of gradient methods[^5]'
---
gradient methods, asymptotic convergence, spectral property, acceleration of gradient methods, Barzilai-Borwein method, unconstrained optimization, quadratic optimization
90C20, 90C25, 90C30
Introduction {#intro}
============
The gradient method is well-known for solving the following unconstrained optimization $$\label{eqpro}
\min_{x\in\mathbb{R}^n}~f(x),$$ where $f: \mathbb{R}^n \to \mathbb{R}$ is continuously differentiable, especially when the dimension $n$ is large. In particular, at $k$-th iteration gradient methods update the iterates by $$\label{eqitr}
x_{k+1}=x_k-\alpha_kg_k,$$ where $g_k=\nabla f(x_k)$ and $\alpha_k>0$ is the stepsize determined by the method.
One simplest nontrivial nonlinear instance of is the quadratic optimization $$\label{qudpro}
\min_{x\in\mathbb{R}^n}~f(x)=\frac{1}{2}x {^{\sf T}}Ax-b {^{\sf T}}x,$$ where $b\in\mathbb{R}^n$ and $A\in\mathbb{R}^{n\times n}$ is symmetric and positive definite. Solving efficiently is usually a pre-requisite for a method to be generalized to solve more general optimization. In addition, by Taylor’s expansion, a general smooth function can be approximated by a quadratic function near the minimizer. So, the local convergence behaviors of gradient methods are often reflected by solving . Hence, in this paper, we focus on studying the convergence behaviors and propose efficient gradient methods for solving efficiently.
In [@cauchy1847methode], Cauchy proposed the steepest descent (SD) method that solves by using the exact stepsize $$\label{eqsd}
\alpha_k^{SD}=\arg\min_{\alpha}~f(x_k-\alpha g_k)=\frac{g_{k} {^{\sf T}}g_{k}}{g_{k} {^{\sf T}}Ag_{k}}.$$ Although $\alpha_k^{SD}$ minimizes $f$ along the steepest descent direction, the SD method often performs poorly in practice and has linear converge rate [@akaike1959successive; @forsythe1968asymptotic] as $$\label{sdrate}
\frac{f(x_{k+1})-f^*}{f(x_{k})-f^*}\leq\left(\frac{\kappa-1}{\kappa+1}\right)^2,$$ where $f^*$ is the optimal function value of and $\kappa=\lambda_n/\lambda_1$ is the condition number of $A$ with $\lambda_1$ and $\lambda_n$ being the smallest and largest eigenvalues of $A$, respectively. Thus, if $\kappa$ is large, the SD method may converge very slowly. In addition, Akaike [@akaike1959successive] proved that the gradients will asymptotically alternate between two directions in the subspace spanned by the two eigenvectors corresponding to $\lambda_1$ and $\lambda_n$. So, the SD method often has zigzag phenomenon near the solution. In [@forsythe1968asymptotic], Forsythe generalized Akaike’s results to the so-called optimum $s$-gradient method and Pronzato et al. [@pronzato2006asymptotic] further generalized the results to the so-called $P$-gradient methods in the Hilbert space. Recently, by employing Akaike’s results, Nocedal et al. [@nocedal2002behavior] presented some insights for asymptotic behaviors of the SD method on function values, stepsizes and gradient norms.
Contrary to the SD method, the minimal gradient (MG) method [@dai2003altermin] computes its stepsize by minimizing the gradient norm, $$\label{mg}
\alpha_k^{MG}=\arg\min_{\alpha}~\|g(x_k-\alpha g_k)\|=\frac{g_{k} {^{\sf T}}Ag_{k}}{g_{k} {^{\sf T}}A^2g_{k}}.
$$ It is widely accepted that the MG method can also perform poorly and has similar asymptotic behavior as the SD method, i.e., it will asymptotically zigzag in a two-dimensional subspace. In [@zhou2006gradient], the authors provide some interesting analyses on $\alpha_k^{MG}$ for minimizing two-dimensional quadratics. However, rigorous asymptotic convergence results of the MG method for minimizing general quadratic function are very limit in literature.
In order to avoid the zigzagging pattern, it is useful to determine the stepsize without using the exact stepsize because it would yield a gradient perpendicular to the current one. Barzilai and Borwein [@barzilai1988two] proposed the following two novel stepsizes: $$\label{sBB}
\alpha_k^{BB1}=\frac{s_{k-1} {^{\sf T}}s_{k-1}}{s_{k-1} {^{\sf T}}y_{k-1}}~~\textrm{and}~~
\alpha_k^{BB2}=\frac{s_{k-1} {^{\sf T}}y_{k-1}}{y_{k-1} {^{\sf T}}y_{k-1}},$$ where $s_{k-1}=x_k-x_{k-1}$ and $y_{k-1}=g_k-g_{k-1}$. The BB method performs quite well in practice, though it generates a nonmonotone sequence of objective values. Due to its simplicity and efficiency, the BB method has been widely studied [@dai2005asymptotic; @dhl2018; @dai2002r; @fletcher2005barzilai; @raydan1993barzilai] and extended to general problems and various applications, see [@birgin2000nonmonotone; @huang2016smoothing; @huang2015quadratic; @jiang2013feasible; @liu2011coordinated; @raydan1997barzilai]. Another line of research to break the zigzagging pattern and accelerate the convergence is occasionally applying short stepsizes that approximate $1/\lambda_n$ to eliminate the corresponding component of the gradient. One seminal work is due to Yuan [@yuan2006new; @yuan2008step], who derived the following stepsize: $$\label{syv}
\alpha_k^{Y}=\frac{2}{\frac{1}{\alpha_{k-1}^{SD}}+\frac{1}{\alpha_{k}^{SD}}+
\sqrt{\left(\frac{1}{\alpha_{k-1}^{SD}}-\frac{1}{\alpha_{k}^{SD}}\right)^2+
\frac{4\|g_k\|^2}{(\alpha_{k-1}^{SD}\|g_{k-1}\|)^2}}}.$$ Dai and Yuan [@dai2005analysis] further suggested a new gradient method with $$\label{sdy}
\alpha_k^{DY}=\left\{
\begin{array}{ll}
\alpha_k^{SD}, & \hbox{if mod($k$,4)$<2$;} \\
\alpha_k^{Y}, & \hbox{otherwise.}
\end{array}
\right.$$ The DY method is a monotone method and appears very competitive with the nonmonotone BB method. Recently, by employing the results in [@akaike1959successive; @nocedal2002behavior], De Asmundis et al. [@de2014efficient] show that the stepsize $\alpha_k^{Y}$ converges to $1/\lambda_n$ if the SD method is applied to problem . This spectral property is the key to break the zigzagging pattern.
In [@dai2006new2], Dai and Yang developed the asymptotic optimal gradient (AOPT) method whose stepsize is given by $$\label{saopt}
\alpha_k^{AOPT}=\frac{\|g_k\|}{\|Ag_k\|}.$$ Unlike the DY method, the AOPT method only has one stepsize. In addition, they show that $\alpha_k^{AOPT}$ asymptotically converges to $\frac{2}{\lambda_1+\lambda_n}$, which is in some sense an optimal stepsize since it minimizes $\|I-\alpha A\|$ over $\alpha$ [@dai2006new2; @elman1994inexact]. However, the AOPT method also asymptotically alternates between two directions. To accelerate the AOPT method, Huang et al. [@huang2019gradient] derived a new stepsize that converges to $1/\lambda_n$ during the AOPT iterates and further suggested a gradient method to exploit spectral properties of the stepsizes. For the latest developments of exploiting spectral properties to accelerate gradient methods, see [@de2014efficient; @de2013spectral; @di2018steplength; @gonzaga2016steepest; @huang2019gradient].
In this paper, we present the analysis on the asymptotic behaviors of gradient methods and the techniques for breaking the zigzagging pattern. For a uniform analysis, we consider the following stepsize $$\label{glstep1}
\alpha_k=\frac{g_k {^{\sf T}}\Psi(A) g_k}{g_k {^{\sf T}}\Psi(A)A g_k},$$ where $\Psi$ is a real analytic function on $[\lambda_1,\lambda_n]$ and can be expressed by Laurent series $$\Psi(z)=\sum_{k=-\infty}^\infty c_kz^k,~~c_k\in\mathbb{R},$$ such that $0<\sum_{k=-\infty}^\infty c_kz^k<+\infty$ for all $z\in[\lambda_1,\lambda_n]$. Apparently, $\alpha_k$ is a family of stepsizes that would give a family of gradient methods. When $\Psi(A)=A^u$ for some nonnegative integer $u$, we get the following stepsize $$\label{glstep2}
\alpha_k=\frac{g_k {^{\sf T}}A^{u} g_k}{g_k {^{\sf T}}A^{u+1} g_k}.$$ The $\alpha_k^{SD}$ and $\alpha_k^{MG}$ simply correspond to the cases $u=0$ and $u=1$, respectively.
We will present theoretical analysis on the asymptotic convergence on the family of gradient methods whose stepsize can be written in the form , which provides justifications for the zigzag behaviors of all these gradient methods including the SD and MG methods. In particular, we show that each method in the family will asymptotically alternate between two directions associated with the two eigenvectors corresponding to $\lambda_1$ and $\lambda_n$. Moreover, we analyze the asymptotic behaviors of the objective value, gradient norm, and stepsize. It is shown that, when $\Psi(A)\neq I$, the two sequences $\Big\{\frac{\Delta_{2k+1}}{\Delta_{2k}} \Big\}$ and $\Big\{ \frac{\Delta_{2k+2}}{\Delta_{2k+1}} \Big\}$ may converge at different speeds, while the odd and even subsequences $\Big\{\frac{\Delta_{2k+3}}{\Delta_{2k+1}}\Big\}$ and $\Big\{ \frac{\Delta_{2k+2}}{\Delta_{2k}}\Big\}$ converge at the same rate, where $\Delta_k = f(x_k) - f^* $. Similar property is also possessed by the gradient norm sequence. In addition, we show each method in has the same worst asymptotic rate.
In order to accelerate the gradient methods , we investigate techniques for breaking the zigzagging pattern. We derive a new stepsize $\tilde{\alpha}_k$ based on finite termination for minimizing two-dimensional strictly convex quadratic function. For the $n$-dimensional case, we prove that $\tilde{\alpha}_k$ converges to $1/\lambda_n$ when gradient methods are applied to problem . Furthermore, based on this spectral property, we propose a periodic gradient method, which, in a periodic mode, alternately uses the BB stepsize, stepsize and our new stepsize $\tilde{\alpha}_k$. Numerical comparisons of the proposed method with the BB [@barzilai1988two], DY [@dai2005analysis], ABBmin2 [@frassoldati2008new], and SDC [@de2014efficient] methods show that the new gradient method is very efficient. Our theoretical results also significantly improve and generalize those in [@akaike1959successive; @nocedal2002behavior], where only the SD method (i.e., $\Psi(A)=I$) is considered. We point out that [@pronzato2006asymptotic] does not analyze the asymptotic behaviors of the objective value, gradient norm, and stepsize, though is similar to the $P$-gradient methods in [@pronzato2006asymptotic]. Moreover, we develop techniques for accelerating these zigzag methods with simpler analysis. Notice that $\alpha_k^{AOPT}$ can not be written in the form . Thus, our results are not applicable to the AOPT method. On the other hand, the analysis of the AOPT method presented in [@dai2006new2] can not be applied directly to the family of methods .
The paper is organized as follows. In Section \[secasmg\], we analyze the asymptotic behaviors of the family of gradient methods . In Section \[secnewstep\], we accelerate the gradient methods by developing techniques to break its zigzagging pattern and propose a new periodic gradient method. Numerical experiments are presented in Section \[secnum\]. Finally, some conclusions and discussions are made in Section \[seccls\].
Asymptotic behavior of the family {#secasmg}
==================================
In this section, we present a uniform analysis on the asymptotic behavior of the family of gradient methods for general $n$-dimensional strictly convex quadratics.
Let $\{\lambda_1,\lambda_2,\cdots,\lambda_n\}$ be the eigenvalues of $A$, and $\{\xi_1,\xi_2,\ldots,\xi_n\}$ be the associated orthonormal eigenvectors. Noting that the gradient method is invariant under translations and rotations when applying to a quadratic function. For theoretical analysis, we can assume without loss of generality that $$\label{formA}
A=\textrm{diag}\{\lambda_1,\lambda_2,\cdots,\lambda_n\},~~0<\lambda_1<\lambda_2<\cdots<\lambda_n.$$
Denoting the components of $g_k$ along the eigenvectors $\xi_i$ by $\mu_k^{(i)}$, $i=1,\ldots,n$, i.e., $$\label{gmu}
g_k=\sum_{i=1}^n\mu_k^{(i)}\xi_i.$$ The above decomposition of gradient $g_k$ together with the update rule gives that $$\label{gupeq}
g_{k+1}=g_k-\alpha_kAg_k=\prod_{j=1}^k(I-\alpha_jA)g_0=\sum_{i=1}^n\mu_{k+1}^{(i)}\xi_i,$$ where $$\label{upmuk}
\mu_{k+1}^{(i)}=(1-\alpha_k\lambda_i)\mu_k^{(i)}=\mu_0^{(i)}\prod_{j=1}^k(1-\alpha_j\lambda_i).$$ Defining the vector $q_k=\left(q_k^{(i)}\right)$ with $$\label{defpk}
q_k^{(i)}=\frac{(\mu_k^{(i)})^2}{\|\mu_k\|^2}$$ and $$\label{defgama}
\gamma_k=\frac{1}{\alpha_k}=\frac{g_k {^{\sf T}}\Psi(A)A g_k}{g_k {^{\sf T}}\Psi(A) g_k}
=\frac{\sum_{i=1}^n\Psi(\lambda_i)\lambda_iq_k^{(i)}}{\sum_{i=1}^n\Psi(\lambda_i)q_k^{(i)}},$$ we can have from , and that $$\label{pk1}
q_{k+1}^{(i)}=\frac{(\lambda_i-\gamma_k)^2q_k^{(i)}}
{\sum_{i=1}^n(\lambda_i-\gamma_k)^2q_k^{(i)}}.$$ In addition, by the definition of $q_k$, we know that $q_k^{(i)}\geq0$ for all $i$ and $$\sum_{i=1}^nq_k^{(i)}=1,~~\forall~~k\geq1.$$
Before establishing the asymptotic convergence of the family of gradient methods , we first give some lemmas on the properties of the sequence $\{q_k\}$.
\[lm1\] Suppose $p\in\mathbb{R}^n$ satisfies (i) $p^{(i)}\geq0$ for all $i=1,2,\ldots,n$; (ii) there exist at least two $i's$ with $p^{(i)}>0$; and (iii) $\sum_{i=1}^np^{(i)}=1$. Define $T: \mathbb{R}^n\rightarrow\mathbb{R}$ be the following transformation: $$\label{tranT}
(Tp)^{(i)}=\frac{(\lambda_i-\gamma(p))^2p^{(i)}}
{\sum_{i=1}^n(\lambda_i-\gamma(p))^2p^{(i)}},$$ where $$\label{defgama2}
\gamma(p)=\frac{\sum_{i=1}^n\Psi(\lambda_i)\lambda_ip^{(i)}}{\sum_{i=1}^n\Psi(\lambda_i)p^{(i)}}.$$ Then we have $$\label{thtinq}
\Theta(Tp)\geq\Theta(p),$$ where $$\label{deftht}
\Theta(p)=\frac{\sum_{i=1}^n\Psi(\lambda_i)(\lambda_i-\gamma(p))^2p^{(i)}}
{\sum_{i=1}^n\Psi(\lambda_i)p^{(i)}}.$$ In addition, holds with equality if and only if there are two indices, say $i_1$ and $i_2$, such that $p^{(i)}=0$ for all $i\notin\{i_1,i_2\}$ and $$\label{sum2eig}
\gamma(Tp)+\gamma(p)=\lambda_{i_1}+\lambda_{i_2}.$$
It follows from the definition of $Tp$ that $$\begin{aligned}
\label{thtp}
\Theta(Tp)&=\frac{\sum_{i=1}^n\Psi(\lambda_i)(\lambda_i-\gamma(Tp))^2(Tp)^{(i)}}
{\sum_{i=1}^n\Psi(\lambda_i)(Tp)^{(i)}}
\nonumber\\
&=\frac{\sum_{i=1}^n\Psi(\lambda_i)(\lambda_i-\gamma(Tp))^2(\lambda_i-\gamma(p))^2p^{(i)}}
{\sum_{i=1}^n\Psi(\lambda_i)(\lambda_i-\gamma(p))^2p^{(i)}}.\end{aligned}$$ Let us define two vectors $w = (w_i) \in \mathbb{R}^n$ and $z = (z_i) \in \mathbb{R}^n$ by $$w_i=\sqrt{\Psi(\lambda_i)}(\lambda_i-\gamma(Tp))(\lambda_i-\gamma(p))\sqrt{p^{(i)}}$$ and $$z_i=\sqrt{\Psi(\lambda_i)}\sqrt{p^{(i)}}.$$ Then, we have from the Cauchy-Schwarz inequality that $$\begin{aligned}
\label{thtpinq}
\|w\|^2\|z\|^2&=\left(\sum_{i=1}^n\Psi(\lambda_i)(\lambda_i-\gamma(Tp))^2(\lambda_i-\gamma(p))^2p^{(i)}\right)
\left(\sum_{i=1}^n\Psi(\lambda_i)p^{(i)}\right)
\nonumber\\
&\geq (w {^{\sf T}}z)^2=\left(\sum_{i=1}^n\Psi(\lambda_i)(\lambda_i-\gamma(Tp))(\lambda_i-\gamma(p))p^{(i)}\right)^2.\end{aligned}$$ Using the definition of $\gamma(p)$, we can obtain that $$\begin{aligned}
\label{keyeq}
&& \sum_{i=1}^n\Psi(\lambda_i)(\lambda_i-\gamma(Tp))(\lambda_i-\gamma(p))p^{(i)}
-\sum_{i=1}^n\Psi(\lambda_i)(\lambda_i-\gamma(p))^2p^{(i)} \nonumber \\
& = &(\gamma(p)-\gamma(Tp))\sum_{i=1}^n\Psi(\lambda_i)(\lambda_i-\gamma(p))p^{(i)}
=0,\end{aligned}$$ which together with gives $$\begin{aligned}
\label{thtpinq2}
&& \left(\sum_{i=1}^n\Psi(\lambda_i)(\lambda_i-\gamma(Tp))^2(\lambda_i-\gamma(p))^2p^{(i)}\right)
\left(\sum_{i=1}^n\Psi(\lambda_i)p^{(i)}\right)
\nonumber\\
& \geq & \left(\sum_{i=1}^n\Psi(\lambda_i)(\lambda_i-\gamma(p))^2p^{(i)}\right)^2.\end{aligned}$$ Then, the inequality follows immediately.
The equality in holds if and only if $$\label{equv2}
\sqrt{\Psi(\lambda_i)}(\lambda_i-\gamma(Tp))(\lambda_i-\gamma(p))\sqrt{p^{(i)}}
=C\sqrt{\Psi(\lambda_i)}\sqrt{p^{(i)}},~~i=1,\ldots,n$$ for some nonzero scalar $C$. Clearly, holds when $p^{(i)}=0$. Suppose that there exist two indices $i_1$ and $i_2$ such that $p^{(i_1)},p^{(i_2)}>0$. It follows from that $$(\lambda_{i_1}-\gamma(Tp))(\lambda_{i_1}-\gamma(p))
=(\lambda_{i_2}-\gamma(Tp))(\lambda_{i_2}-\gamma(p)).$$ So, by the assumption , we have $$\lambda_{i_1}+\lambda_{i_2}=\gamma(Tp)+\gamma(p),$$ which again with assumption imply that holds if and only if $p$ has only two nonzero components and holds.
\[tpeqp\] Let $p_*\in\mathbb{R}^n$ satisfy the conditions of Lemma \[lm1\] and $T$ be the transformation . If $p_*$ has only two nonzero components $p_*^{(i_1)}$ and $p_*^{(i_2)}$, we have $$\label{tps1}
(Tp_*)^{(i_1)}=\frac{\Psi^2(\lambda_{i_2})p_*^{(i_2)}}
{\Psi^2(\lambda_{i_1})p_*^{(i_1)}+\Psi^2(\lambda_{i_2})p_*^{(i_2)}},$$ $$\label{tps2}
(Tp_*)^{(i_2)}=\frac{\Psi^2(\lambda_{i_1})p_*^{(i_1)}}
{\Psi^2(\lambda_{i_1})p_*^{(i_1)}+\Psi^2(\lambda_{i_2})p_*^{(i_2)}},$$ $$\label{tsqstar}
(T^2p_*)^{(i_1)}=p_*^{(i_1)},~~(T^2p_*)^{(i_2)}=p_*^{(i_2)},$$ and $$\label{gmaptp}
\gamma(p_*)+\gamma(Tp_*)=\lambda_{i_1}+\lambda_{i_2},$$ where the function $\gamma$ is defined in . Moreover, $p_*=Tp_*$ if and only if $$\label{ps1}
p_*^{(i_1)}=\frac{\Psi(\lambda_{i_2})}{\Psi(\lambda_{i_1})+\Psi(\lambda_{i_2})} \quad \mbox{and} \quad
p_*^{(i_2)}=\frac{\Psi(\lambda_{i_1})}{\Psi(\lambda_{i_1})+\Psi(\lambda_{i_2})}.$$
By the definition of $\gamma(p)$, we have $$\label{gmsr1}
\gamma(p_*)=\frac{\Psi(\lambda_{i_1})\lambda_{i_1}p_*^{(i_1)}+\Psi(\lambda_{i_2})\lambda_{i_2}p_*^{(i_2)}}
{\Psi(\lambda_{i_1})p_*^{(i_1)}+\Psi(\lambda_{i_2})p_*^{(i_2)}},$$ which indicates that $$\lambda_{i_1}-\gamma(p_*)=\frac{\Psi(\lambda_{i_2})p_*^{(i_2)}(\lambda_{i_1}-\lambda_{i_2})}
{\Psi(\lambda_{i_1})p_*^{(i_1)}+\Psi(\lambda_{i_2})p_*^{(i_2)}}, \quad
\lambda_{i_2}-\gamma(p_*)=\frac{\Psi(\lambda_{i_1})p_*^{(i_1)}(\lambda_{i_2}-\lambda_{i_1})}
{\Psi(\lambda_{i_1})p_*^{(i_1)}+\Psi(\lambda_{i_2})p_*^{(i_2)}}.$$ Then, it follows from the definition of transformation $T$ that $$\begin{aligned}
(Tp_*)^{(i_1)}&=\frac{(\Psi(\lambda_{i_2})p_*^{(i_2)})^2p_*^{(i_1)}}
{(\Psi(\lambda_{i_2})p_*^{(i_2)})^2p_*^{(i_1)}+(\Psi(\lambda_{i_1})p_*^{(i_1)})^2p_*^{(i_2)}}
\\
&=\frac{\Psi^2(\lambda_{i_2})p_*^{(i_2)}}
{\Psi^2(\lambda_{i_1})p_*^{(i_1)}+\Psi^2(\lambda_{i_2})p_*^{(i_2)}}.
\end{aligned}$$ This gives . can be proved similarly. By and , we have $$\begin{aligned}
(T^2p_*)^{(i_1)}
&=\frac{\Psi^2(\lambda_{i_2})(Tp_*)^{(i_2)}}
{\Psi^2(\lambda_{i_1})(Tp_*)^{(i_1)}+\Psi^2(\lambda_{i_2})(Tp_*)^{(i_2)}}\\
&=\frac{\Psi^2(\lambda_{i_1})\Psi^2(\lambda_{i_2})p_*^{(i_1)}}
{\Psi^2(\lambda_{i_1})\Psi^2(\lambda_{i_2})p_*^{(i_2)}+
\Psi^2(\lambda_{i_1})\Psi^2(\lambda_{i_2})p_*^{(i_1)}}\\
& =\frac{p_*^{(i_1)}}
{p_*^{(i_1)}+p_*^{(i_2)}}=p_*^{(i_1)}.
\end{aligned}$$ $(T^2p_*)^{(i_2)}$ follows similarly. This proves .
Again by , and the definition of function $\gamma$ in , we have $$\label{gmsrtp}
\gamma(Tp_*)=\frac{\lambda_{i_1}\Psi(\lambda_{i_2})p_*^{(i_2)}+
\lambda_{i_2}\Psi(\lambda_{i_1})p_*^{(i_1)}}
{\Psi(\lambda_{i_1})p_*^{(i_1)}+\Psi(\lambda_{i_2})p_*^{(i_2)}}.$$ Then, the equality follows from and . For , let $$p_*^{(i_1)}=\frac{\Psi^2(\lambda_{i_2})p_*^{(i_2)}}
{\Psi^2(\lambda_{i_1})p_*^{(i_1)}+\Psi^2(\lambda_{i_2})p_*^{(i_2)}}.$$ Rearranging terms and using $p_*^{(i_1)}+p_*^{(i_2)}=1$, we have $$\Psi^2(\lambda_{i_1})(p_*^{(i_1)})^2=\Psi^2(\lambda_{i_2})(p_*^{(i_2)})^2,$$ which implies that $$\Psi(\lambda_{i_1})p_*^{(i_1)}=\Psi(\lambda_{i_2})p_*^{(i_2)}.$$ This together with the fact $p_*^{(i_1)}+p_*^{(i_2)}=1$ yields .
\[lm3\] Let $p\in\mathbb{R}^n$ satisfy the conditions of Lemma \[lm1\] and $T$ be the transformation . Then, there exists a $p_*$ satisfying $$\label{t2k}
\lim_{k\rightarrow\infty}T^{2k}p=p_*~~and~~
\lim_{k\rightarrow\infty}T^{2k+1}p=Tp_*,$$ where $p_*$ and $Tp_*$ have only two nonzero components satisfying $$\label{limp3}
p_*^{(i_1)}+p_*^{(i_2)}=1, ~~p_*^{(i)}=0,~~i\neq i_1,i_2,$$ $$\label{limp4}
(Tp_*)^{(i_1)}+(Tp_*)^{(i_2)}=1,~~(Tp_*)^{(i)}=0,~~i\neq i_1,i_2,$$ for some $i_1,i_2\in\{1,\ldots,n\}$. Hence, , , and hold.
Let $p_0=T^0p=p$ and $p_{k}=T{p_{k-1}}=T^{k}{p_0}$. Obviously, for all $k\geq0$, $p_{k}$ satisfies (i) and (iii) of Lemma \[lm1\]. Let $i_{\min} = \min\{i \in \mathcal{N}: p_0^{(i)}>0\}$ and $i_{\max} = \max \{i \in \mathcal{N}: p_0^{(i)}>0\}$, where $\mathcal{N} = \{1, \ldots, n\}$. From the definition of $\gamma$, we know $\lambda_{i_{\min}}< \gamma(p)<\lambda_{i_{\max}}$. Thus, by the definition of $T$, we have $p_1^{({i_{\min}})}>0$ and $p_1^{({i_{\max}})}>0$. Then, by induction, for all $k\geq0$, $p_{k}$ satisfies (ii) of Lemma \[lm1\]. So, by Lemma \[lm1\], $\{\Theta(p_k)\}$ is a monotonically increasing sequence. Since $\lambda_1\leq \gamma(p)\leq\lambda_n$, we have $(\lambda_i-\gamma(p))^2\leq(\lambda_n-\lambda_1)^2$. Hence, we have from the definition of $\Theta$ that $\Theta(p_k)\leq (\lambda_n-\lambda_1)^2 $. Thus, $\{\Theta(p_k)\}$ is convergent. Let $\Theta_*=\lim_{k\rightarrow\infty}\Theta(p_k) >0$.
Denote the set of all limit points of $\{p_k\}$ by $P_*$ with cardinality $|P_*|$. Since $\{p_k\}$ is bounded, $|P_*|\geq1$. For any subsequence $\{p_{k_j}\}$ converging to some $p_*\in P_*$, we have $$\lim_{j\rightarrow\infty}\Theta(p_{k_j})=\Theta(p_*) \quad \mbox{and} \quad \lim_{j\rightarrow\infty}\Theta(Tp_{k_j})=\Theta(Tp_*),$$ by the continuity of $\Theta$ and $T$. Notice $p_{k_j+1}=Tp_{k_j}$, we have $ \Theta_*=\Theta(p_*)=\Theta(Tp_*)$.
Since $p_{k}$ satisfies (i)-(iii) of Lemma \[lm1\] for all $k\geq0$, $p_*$ must satisfy (i) and (iii). If $p_*$ has only one positive component, we have $\Theta(p_*)=0$ which contradicts $\Theta(p_*) = \Theta_* > 0 $. Hence, by Lemma \[lm1\], Lemma \[tpeqp\] and $\Theta(p_*)=\Theta(Tp_*)$, $p_*$ has only two nonzero components, say $p_*^{(i_1)}$ and $p_*^{(i_2)}$, and their values are uniquely determined by the indices $i_1$, $i_2$ and the eigenvalues $\lambda_{i_1}$ and $\lambda_{i_2}$. This implies $|P_*| < \infty$. Furthermore, by Lemma \[tpeqp\], for any $p_*\in P_*$, $Tp_*$ is given by and , and $Tp_*\in P_*$.
We now show that $|P_*|\leq2$ by way of contradiction. Suppose $|P_*| \ge 3 $. For any $p_*\in P_*$ and $Tp_*\in P_*$, denote $\delta_1$ and $\delta_2$ to be the distance from $p_*$ to $P_*\setminus \{p_*\}$ and from $Tp_*$ to $P_*\setminus \{Tp_*\}$, respectively. Since $ 3 \le |P_*| < \infty$, we have $\delta_1>0$, $\delta_2>0$ and there exists an infinite subsequence $\{p_{k_j}\}$ such that $$p_{k_j}\rightarrow p_*, \quad \mbox{and} \quad
p_{k_j+1}=Tp_{k_j}\rightarrow Tp_*,$$ but $p_{k_j+2}\notin\mathcal{B}\left(p_*,\frac{1}{2}\delta\right)\cup
\mathcal{B}\left(Tp_*,\frac{1}{2}\delta\right)$, where $\delta=\min\{\delta_1,\delta_2\}$ and $\mathcal{B}(p_*,r)=\{p: \|p-p_*\|\leq r\}$. However, by we have $T^2p_*=p_*$. Hence, by continuity of $T$, $$\lim_{j\rightarrow\infty}p_{k_j+2}=\lim_{j\rightarrow\infty}Tp_{k_j+1}=\lim_{j\rightarrow\infty}T^2p_{k_j}= p_*,$$ which contradicts the choice of $p_{k_j+2}\notin\mathcal{B}\left(p_*,\frac{1}{2}\delta\right)$. Thus, $\{p_k\}$ has at most two limit points $p_*$ and $Tp_*$, and both have only two nonzero components.
Now, we assume that $p_*$ is a limit point of $\{p_{2k}\}$. Since $T^2p_*=p_*$, all subsequences of $\{p_{2k}\}$ have the same limit point, i.e., $p_{2k}=T^{2k}p\rightarrow p_*$. Similarly, we have $T^{2k+1}p\rightarrow Tp_*$. Then, and follow directly from the analysis.
Based on the above analysis, we can show that each gradient method in will asymptotically reduces its search in a two-dimensional subspace spanned by the two eigenvectors $\xi_1$ and $\xi_n$.
\[th1\] Assume that the starting point $x_0$ has the property that $$\label{assp1}
g_0 {^{\sf T}}\xi_1\neq0~~and~~g_0 {^{\sf T}}\xi_n\neq0.$$ Let $\{x_k\}$ be the iterations generated by applying a method in to solve problem . Then $$\label{mu2k}
\lim_{k\rightarrow\infty}\frac{(\mu_{2k}^{(i)})^2}{\sum_{j=1}^n(\mu_{2k}^{(j)})^2}=
\left\{
\begin{array}{ll}
\displaystyle\frac{1}{1+c^2}, & \hbox{if $i=1$,} \\
0, & \hbox{if $i=2,\ldots,n-1$,} \\
\displaystyle\frac{c^2}{1+c^2}, & \hbox{if $i=n$,}
\end{array}
\right.$$ and $$\label{mu2k1}
\lim_{k\rightarrow\infty}\frac{(\mu_{2k+1}^{(i)})^2}{\sum_{j=1}^n(\mu_{2k+1}^{(j)})^2}=
\left\{
\begin{array}{ll}
\displaystyle\frac{c^2\Psi^2(\lambda_{n})}{\Psi^2(\lambda_{1})+c^2\Psi^2(\lambda_{n})}, & \hbox{if $i=1$,} \\
0, & \hbox{if $i=2,\ldots,n-1$,} \\
\displaystyle\frac{\Psi^2(\lambda_{1})}{\Psi^2(\lambda_{1})+c^2\Psi^2(\lambda_{n})}, & \hbox{if $i=n$,}
\end{array}
\right.$$ where $c$ is a nonzero constant.
By the assumption , we know that $q_0$ satisfies (i)-(iii) of Lemma \[lm1\]. Notice that $q_k = T^k q_0$. Then, by Lemma \[lm3\], there exists a $p_*$ such that the sequences $\{q_{2k}\}$ and $\{q_{2k+1}\}$ converge to $p_*$ and $Tp_*$, respectively, which have only two nonzero components satisfying , for some $i_1,i_2\in\{1,\ldots,n\}$, and holds. Hence, if $1 \le i_1<i_2<n$, we have $$\label{p2klim}
\lim_{k\rightarrow\infty}q_{2k}^{(n)}=0, \qquad \lim_{k\rightarrow\infty}\frac{q_{2k}^{(i_2)}}{q_{2k+2}^{(i_2)}}=1,$$ and $$\lim_{k\rightarrow\infty}(\gamma(q_{2k})+\gamma(q_{2k+1}))= \gamma(p_*)+\gamma(Tp_*) = \lambda_{i_1}+\lambda_{i_2}.$$ In addition, since $q_0^{(1)} > 0 $ and $q_0^{(n)} > 0$ by , we can see from the proof of Lemma \[lm3\] that $q_k^{(1)}>0$, $q_k^{(n)}>0$ for all $k\geq0$. Thus, we have $$\begin{aligned}
\lim_{k\rightarrow\infty}\frac{q_{2k+2}^{(n)}}{q_{2k}^{(n)}}
& =\lim_{k\rightarrow\infty}\frac{q_{2k+2}^{(n)}}{q_{2k}^{(n)}}\frac{q_{2k}^{(i_2)}}{q_{2k+2}^{(i_2)}}
=\lim_{k\rightarrow\infty}\frac{(\lambda_n-\gamma(q_{2k+1}))^2(\lambda_n-\gamma(q_{2k}))^2}
{(\lambda_{i_2}-\gamma(q_{2k+1}))^2(\lambda_{i_2}-\gamma(q_{2k}))^2} \nonumber \\
&=\lim_{k\rightarrow\infty} \left(\frac{\lambda_n^2-(\gamma(q_{2k})+\gamma(q_{2k+1}))\lambda_n+\gamma(q_{2k})\gamma(q_{2k+1})}
{\lambda_{i_2}^2-(\gamma(q_{2k})+\gamma(q_{2k+1}))\lambda_{i_2}+\gamma(q_{2k})\gamma(q_{2k+1})}\right)^2 \nonumber \\
&=\left(\frac{\lambda_n^2-(\lambda_{i_1}+\lambda_{i_2})\lambda_n+\tilde{\gamma}}
{\lambda_{i_2}^2-(\lambda_{i_1}+\lambda_{i_2})\lambda_{i_2}+\gamma(p_*)\gamma(Tp_*)}\right)^2 \nonumber\\
&=\left(1+\frac{(\lambda_n-\lambda_{i_1})(\lambda_n-\lambda_{i_2})}
{\lambda_{i_2}^2-(\lambda_{i_1}+\lambda_{i_2})\lambda_{i_2}+\gamma(p_*)\gamma(Tp_*)}\right)^2 = : \rho. \label{huang111}\end{aligned}$$ Since $\lambda_{i_1} < \gamma(p_*) < \lambda_{i_2}$ and $\lambda_{i_1} < \gamma(Tp_*) < \lambda_{i_2}$, we have $$\begin{aligned}
\lambda_{i_2}^2-(\lambda_{i_1}+\lambda_{i_2})\lambda_{i_2}+\gamma(p_*)\gamma(Tp_*)
&= &\lambda_{i_2}^2-( \gamma(p_*)+\gamma(Tp_*) )\lambda_{i_2}+\gamma(p_*)\gamma(Tp_*)\\
&= & (\lambda_{i_2} - \gamma(p_*)) (\lambda_{i_2} - \gamma(Tp_*)) > 0.\end{aligned}$$ Hence, it follows from that $\rho > 1$. So, $q_{2k}^{(n)}\rightarrow+\infty$, which contradicts . Then, we must have $i_2=n$. In a similar way, we can show that $i_1=1$. Finally, the equalities in and follow directly from Lemma \[tpeqp\].
In the following, we refer $c$ as the same constant in Theorem \[th1\]. By Theorem \[th1\] we can directly obtain the asymptotic behavior of the stepsize.
\[cor1\] Under the conditions of Theorem \[th1\], we have $$\label{salp1}
\lim_{k\rightarrow\infty}\alpha_{2k}=
\frac{\Psi(\lambda_{1})+c^2\Psi(\lambda_{n})}{\lambda_{1}(\Psi(\lambda_{1})+c^2\kappa\Psi(\lambda_{n}))}$$ and $$\label{salp2}
\lim_{k\rightarrow\infty}\alpha_{2k+1}=
\frac{\Psi(\lambda_{1})+c^2\Psi(\lambda_{n})}{\lambda_{1}(\kappa\Psi(\lambda_{1})+c^2\Psi(\lambda_{n}))},$$ where $\alpha_k$ is defined in and $\kappa = \lambda_n/\lambda_1$ is the condition number of $A$. Moreover, $$\label{sum2reps}
\lim_{k\rightarrow\infty}\left(\frac{1}{\alpha_{2k}}+\frac{1}{\alpha_{2k+1}}\right)
=\lambda_{1}+\lambda_{n}.$$
The next corollary interprets the constant $c$. A special result for the case $\Psi(A)=I$ (i.e., the SD method) can be found in Lemma 3.4 of [@nocedal2002behavior].
\[cvalue\] Under the conditions of Theorem \[th1\], we have $$\label{eqcvalue}
c=\lim_{k\rightarrow\infty}\frac{\mu_{2k}^{(n)}}{\mu_{2k}^{(1)}}
=-\frac{\Psi(\lambda_{1})}{\Psi(\lambda_{n})}\lim_{k\rightarrow\infty}\frac{\mu_{2k+1}^{(1)}}{\mu_{2k+1}^{(n)}}.$$
It follows from Theorem \[th1\] that $$\label{csqr}
\lim_{k\rightarrow\infty}\frac{(\mu_{2k}^{(n)})^2}{(\mu_{2k}^{(1)})^2}
=\frac{\Psi^2(\lambda_{1})}{\Psi^2(\lambda_{n})}
\lim_{k\rightarrow\infty}\frac{(\mu_{2k+1}^{(1)})^2}{(\mu_{2k+1}^{(n)})^2}=c^2.$$ Note that $1/\lambda_n<\alpha_k<1/\lambda_1$ by the assumption . And we have by that $$\mu_{2k+2}^{(1)}= \prod_{\ell=1}^2 (1-\alpha_{2k+\ell}\lambda_1)\mu_{2k}^{(1)} \quad \mbox{and} \quad
\mu_{2k+2}^{(n)}= \prod_{\ell=1}^2 (1-\alpha_{2k+\ell}\lambda_n)\mu_{2k}^{(n)}.$$ Thus, the sequence $\Big\{\frac{\mu_{2k}^{(n)}}{\mu_{2k}^{(1)}}\Big\}$, and similarly for $\Big\{\frac{\mu_{2k+1}^{(1)}}{\mu_{2k+1}^{(n)}}\Big\}$, do not change its sign. Hence, without loss of generality, we can assume by that $$\label{eqcvalue2}
c=\lim_{k\rightarrow\infty} \mu_{2k}^{(n)} / \mu_{2k}^{(1)}.$$ Then, by , and , we have $$\lim_{k\rightarrow\infty}\frac{\mu_{2k+1}^{(1)}}{\mu_{2k+1}^{(n)}}
=\lim_{k\rightarrow\infty}\frac{\mu_{2k}^{(1)}(1-\alpha_{2k}\lambda_1)}{\mu_{2k}^{(n)}(1-\alpha_{2k}\lambda_n)}
=-c\frac{\Psi(\lambda_{n})}{\Psi(\lambda_{1})},$$ which gives .
We have the following results on the asymptotic convergence of the function value.
\[thconfk\] Under the conditions of Theorem \[th1\], we have $$\label{f2k2f1k}
\lim_{k\rightarrow\infty}\frac{f(x_{2k+1})-f^*}{f(x_{2k})-f^*}
=R_f^1
\quad \mbox{and} \quad \lim_{k\rightarrow\infty}\frac{f(x_{2k+2})-f^*}{f(x_{2k+1})-f^*}
=R_f^2,$$ where $$\label{ratefk1}
R_f^1=\frac{c^2(\kappa-1)^2(\Psi^2(\lambda_{1})+c^2\kappa\Psi^2(\lambda_{n}))}
{(\Psi(\lambda_{1})+c^2\kappa\Psi(\lambda_{n}))^2(c^2+\kappa)},$$ $$\label{ratefk2}
R_f^2=\frac{c^2(\kappa-1)^2(c^2+\kappa)\Psi^2(\lambda_{1})\Psi^2(\lambda_{n})}
{(c^2\Psi(\lambda_{n})+\kappa\Psi(\lambda_{1}))^2(\Psi^2(\lambda_{1})+c^2\kappa\Psi^2(\lambda_{n}))}.$$ In addition, if $\Psi(\lambda_{n})=\Psi(\lambda_{1})$ or $c^2=\Psi(\lambda_{1})/\Psi(\lambda_{n})$, then $R_f^1=R_f^2$.
Let $\epsilon_k=x_k-x^*$. Since $g_k=A\epsilon_k$, by , we have $$\epsilon_k=\sum_{i=1}^n\lambda_i^{-1}\mu_k^{(i)}\xi_i.$$ By Theorem \[th1\], we only need to consider the case $\mu_k^{(i)}=0$, $i=2,\ldots,n-1$, that is, $$\epsilon_k=\lambda_1^{-1}\mu_k^{(1)}\xi_1+\lambda_n^{-1}\mu_k^{(n)}\xi_n.$$ Thus, $$\begin{aligned}
\label{fk}
f(x_k)-f^*&=\frac{1}{2}\epsilon_k {^{\sf T}}A\epsilon_k
=\frac{1}{2}\frac{\lambda_n(\mu_k^{(1)})^2+\lambda_1(\mu_k^{(n)})^2}{\lambda_1\lambda_n}.
\end{aligned}$$ Since $$g_k=\mu_k^{(1)}\xi_1+\mu_k^{(n)}\xi_n \quad \mbox{and} \quad \alpha_k=\frac{\Psi(\lambda_{1})(\mu_k^{(1)})^2+\Psi(\lambda_{n})(\mu_k^{(n)})^2}
{\lambda_1\Psi(\lambda_{1})(\mu_k^{(1)})^2+\lambda_n\Psi(\lambda_{n})(\mu_k^{(n)})^2},$$ by the definition of $\epsilon_k$ and the update rule , we further have that $$\begin{aligned}
\epsilon_{k+1}&=\epsilon_k-\alpha_kg_k
=(\lambda_1^{-1}-\alpha_k)\mu_k^{(1)}\xi_1+(\lambda_n^{-1}-\alpha_k)\mu_k^{(n)}\xi_n\\
&=\frac{\Psi(\lambda_{n})(\lambda_n-\lambda_1)(\mu_k^{(n)})^2\mu_k^{(1)}}
{\lambda_1\left(\lambda_1\Psi(\lambda_{1})(\mu_k^{(1)})^2+\lambda_n\Psi(\lambda_{n})(\mu_k^{(n)})^2\right)}\xi_1\\
&+\frac{\Psi(\lambda_{1})(\lambda_1-\lambda_n)(\mu_k^{(1)})^2\mu_k^{(n)}}
{\lambda_n\left(\lambda_1\Psi(\lambda_{1})(\mu_k^{(1)})^2+\lambda_n\Psi(\lambda_{n})(\mu_k^{(n)})^2\right)}\xi_n\\
&=
\frac{(\lambda_n-\lambda_1)\left(\lambda_n\Psi(\lambda_{n})(\mu_k^{(n)})^2\mu_k^{(1)}\xi_1
-\lambda_1\Psi(\lambda_{1})(\mu_k^{(1)})^2\mu_k^{(n)}\xi_n\right)}
{\lambda_1\lambda_n\left(\lambda_1\Psi(\lambda_{1})(\mu_k^{(1)})^2+\lambda_n\Psi(\lambda_{n})(\mu_k^{(n)})^2\right)}.
\end{aligned}$$ Hence, we obtain $$\begin{aligned}
\label{fk1}
&f(x_{k+1})-f^*
=\frac{1}{2}\epsilon_{k+1} {^{\sf T}}A\epsilon_{k+1}\nonumber\\
=&
\frac{1}{2}\frac{(\lambda_n-\lambda_1)^2(\mu_k^{(1)})^2(\mu_k^{(n)})^2
\left(\lambda_n\Psi^2(\lambda_{n})(\mu_k^{(n)})^2+\lambda_1\Psi^2(\lambda_{1})(\mu_k^{(1)})^2\right)}
{\lambda_1\lambda_n\left(\lambda_1\Psi(\lambda_{1})(\mu_k^{(1)})^2+\lambda_n\Psi(\lambda_{n})(\mu_k^{(n)})^2\right)^2}.\end{aligned}$$ Combining with yields that $$\begin{aligned}
& \frac{f(x_{k+1})-f^*}{f(x_{k})-f^*} =\frac{\epsilon_{k+1} {^{\sf T}}A\epsilon_{k+1}}{\epsilon_k {^{\sf T}}A\epsilon_k}\\
= &\frac{(\mu_k^{(1)})^2(\mu_k^{(n)})^2(\kappa-1)^2
\left(\kappa\Psi^2(\lambda_{n})(\mu_k^{(n)})^2+\Psi^2(\lambda_{1})(\mu_k^{(1)})^2\right)}
{\left(\Psi(\lambda_{1})(\mu_k^{(1)})^2+\kappa\Psi(\lambda_{n})(\mu_k^{(n)})^2\right)^2
\left(\kappa(\mu_k^{(1)})^2+(\mu_k^{(n)})^2\right)},\end{aligned}$$ which gives by substituting the limits of $(\mu_k^{(1)})^2$ and $(\mu_k^{(n)})^2$ in Theorem \[th1\].
Notice $\kappa>1$ by our assumption. So, $R_f^1=R_f^2$ is equivalent to $$\frac{\Psi^2(\lambda_{1})+c^2\kappa\Psi^2(\lambda_{n})}
{(\Psi(\lambda_{1})+c^2\kappa\Psi(\lambda_{n}))^2(c^2+\kappa)}
=\frac{(c^2+\kappa)\Psi^2(\lambda_{1})\Psi^2(\lambda_{n})}
{(c^2\Psi(\lambda_{n})+\kappa\Psi(\lambda_{1}))^2(\Psi^2(\lambda_{1})+c^2\kappa\Psi^2(\lambda_{n}))},$$ which by rearranging terms gives $$c^4\Psi^2(\lambda_{n})(\Psi(\lambda_{n})-\Psi(\lambda_{1}))=\Psi^2(\lambda_{1})(\Psi(\lambda_{n})-\Psi(\lambda_{1})).$$ Hence, $R_f^1=R_f^2$ holds if $\Psi(\lambda_{n})=\Psi(\lambda_{1})$ or $c^2=\Psi(\lambda_{1})/\Psi(\lambda_{n})$.
Theorem \[thconfk\] indicates that, when $\Psi(A)=I$ (i.e., the SD method), the two sequences $\Big\{\frac{\Delta_{2k+1}}{\Delta_{2k}} \Big\}$ and $\Big\{ \frac{\Delta_{2k+2}}{\Delta_{2k+1}} \Big\}$ converge at the same speed, where $\Delta_k = f(x_k) - f^*$. Otherwise, the two sequences may converge at different rates.
To illustrate the results in Theorem \[thconfk\], we apply gradient method with $\Psi(A)=A$ (i.e., the MG method) to an instance of , where the vector of all ones was used as the initial point, the matrix $A$ is diagonal with $$\label{exfkrate}
A_{ii}=i\sqrt{i},~~i=1,\ldots,n,$$ and $b=0$. Figure \[confk\] clearly shows the difference between $R_f^1$ and $R_f^2$.
![Problem with $n=10$: convergence history of the sequences $\big\{1- \frac{\Delta_{2k+1}}{\Delta_{2k}}\big\}$ and $\big\{1- \frac{\Delta_{2k+2}}{\Delta_{2k+1}}\big\}$ generated by gradient method with $\Psi(A)=A$ (i.e., the MG method).[]{data-label="confk"}](frate.eps "fig:"){width="75.00000%" height="52.00000%"}\
The next theorem shows the asymptotic convergence of the gradient norm.
\[thcongk\] Under the conditions of Theorem \[th1\], the following limits hold, $$\label{gk2g1}
\lim_{k\rightarrow\infty}\frac{\|g_{2k+1}\|^2}{\|g_{2k}\|^2}
=R_g^1
\quad \mbox{ and } \quad \lim_{k\rightarrow\infty}\frac{\|g_{2k+2}\|^2}{\|g_{2k+1}\|^2}
=R_g^2,$$ where $$\label{rg}
R_g^1=\frac{c^2(\kappa-1)^2(\Psi^2(\lambda_{1})+c^2\Psi^2(\lambda_{n}))}{(1+c^2)(\Psi(\lambda_{1})+c^2\kappa\Psi(\lambda_{n}))^2},$$ $$\label{rg2}
R_g^2=\frac{c^2(1+c^2)(\kappa-1)^2\Psi^2(\lambda_{1})\Psi^2(\lambda_{n})}
{(c^2\Psi(\lambda_{n})+\kappa\Psi(\lambda_{1}))^2(\Psi^2(\lambda_{1})+c^2\Psi^2(\lambda_{n}))}.$$ In addition, if $\Psi(\lambda_{n})=\kappa\Psi(\lambda_{1})$ or $c^2=\Psi(\lambda_{1})/\Psi(\lambda_{n})$, then $R_g^1=R_g^2$.
Using the same arguments as in Theorem \[thconfk\], we have $$\|g_{k}\|^2=(\mu_k^{(1)})^2+(\mu_k^{(n)})^2$$ and $$\|g_{k+1}\|^2=\epsilon_{k+1} {^{\sf T}}A^2\epsilon_{k+1}
=
\frac{(\lambda_n-\lambda_1)^2(\mu_k^{(1)})^2(\mu_k^{(n)})^2
\left(\Psi^2(\lambda_{n})(\mu_k^{(n)})^2+\Psi^2(\lambda_{1})(\mu_k^{(1)})^2\right)}
{\left(\lambda_1\Psi(\lambda_{1})(\mu_k^{(1)})^2+\lambda_n\Psi(\lambda_{n})(\mu_k^{(n)})^2\right)^2},$$ which give that $$\frac{\|g_{k+1}\|^2}{\|g_{k}\|^2}
=
\frac{(\kappa-1)^2(\mu_k^{(1)})^2(\mu_k^{(n)})^2
\left(\Psi^2(\lambda_{n})(\mu_k^{(n)})^2+\Psi^2(\lambda_{1})(\mu_k^{(1)})^2\right)}
{\left(\Psi(\lambda_{1})(\mu_k^{(1)})^2+\kappa\Psi(\lambda_{n})(\mu_k^{(n)})^2\right)^2
\left((\mu_k^{(1)})^2+(\mu_k^{(n)})^2\right)}.$$ Thus, follows by substituting the limits of $(\mu_k^{(1)})^2$ and $(\mu_k^{(n)})^2$ in Theorem \[th1\].
Notice $\kappa>1$ by our assumption. So, $R_g^1=R_g^2$ is equivalent to $$\frac{\Psi^2(\lambda_{1})+c^2\Psi^2(\lambda_{n})}{(1+c^2)(\Psi(\lambda_{1})+c^2\kappa\Psi(\lambda_{n}))^2}
=\frac{(1+c^2)\Psi^2(\lambda_{1})\Psi^2(\lambda_{n})}
{(c^2\Psi(\lambda_{n})+\kappa\Psi(\lambda_{1}))^2
(\Psi^2(\lambda_{1})+c^2\Psi^2(\lambda_{n}))},$$ which by rearranging terms gives $$c^4\Psi^2(\lambda_{n})(\kappa\Psi(\lambda_{1})-\Psi(\lambda_{n}))=
\Psi^2(\lambda_{1})(\kappa\Psi(\lambda_{1})-\Psi(\lambda_{n})).$$ Hence, $R_g^1=R_g^2$ holds if $\Psi(\lambda_{n})=\kappa\Psi(\lambda_{1})$ or $c^2=\Psi(\lambda_{1})/\Psi(\lambda_{n})$.
\[gratemk\] Theorem \[thcongk\] indicates that the two sequences $\Big\{\frac{\|g_{2k+1}\|^2}{\|g_{2k}\|^2}\Big\}$ and $\Big\{\frac{\|g_{2k+2}\|^2}{\|g_{2k+1}\|^2}\Big\}$ generated by the MG method (i.e., $\Psi(A)=A$) converge at the same rate. Otherwise, the two sequences may converge at different rates.
By Theorems \[thconfk\] and \[thcongk\], we can obtain the following corollary.
\[ratefg\] Under the conditions of Theorem \[th1\], we have $$\label{ratefeq}
\lim_{k\rightarrow\infty}\frac{f(x_{2k+3})-f^*}{f(x_{2k+1})-f^*}
=\lim_{k\rightarrow\infty}\frac{f(x_{2k+2})-f^*}{f(x_{2k})-f^*}
=R_f^1R_f^2,
$$ $$\label{rategeq}
\lim_{k\rightarrow\infty}\frac{\|g_{2k+3}\|^2}{\|g_{2k+1}\|^2}
=\lim_{k\rightarrow\infty}\frac{\|g_{2k+2}\|^2}{\|g_{2k}\|^2}
=R_g^1R_g^2.
$$ In addition, $$\label{ratefgeq}
R_f^1R_f^2=R_g^1R_g^2=\frac{c^4(\kappa-1)^4\Psi^2(\lambda_{1})\Psi^2(\lambda_{n})}
{(\Psi(\lambda_{1})+c^2\kappa\Psi(\lambda_{n}))^2(c^2\Psi(\lambda_{n})+\kappa\Psi(\lambda_{1}))^2}.$$
\[fgrate\] Corollary \[ratefg\] shows that the odd and even subsequences of objective values and gradient norms converge at the same rate. Moreover, we have $$\label{ratefgupbd}
R_f^1R_f^2=R_g^1R_g^2
=\frac{(\kappa-1)^4}
{(1+\kappa/t+t\kappa+\kappa^2)^2}
\leq\left(\frac{\kappa-1}{\kappa+1}\right)^4,$$ where $t= c^2 \Psi(\lambda_{n})/\Psi(\lambda_{1})$. Notice that the right side of only depends on $\kappa$, which implies these odd and even subsequences generated by all the gradient methods will have the same worst asymptotic rate independent of $\Psi$.
Now, as in [@nocedal2002behavior], we define the *minimum deviation* $$\label{defmsig}
\sigma=\min_{i\in\mathcal{I}}~\left| \frac{2 \lambda_i- (\lambda_1+\lambda_n)}{ \lambda_n-\lambda_1 }\right|,$$ where $$\mathcal{I}=\{i: \lambda_1<\lambda_i<\lambda_n,~g_0 {^{\sf T}}\xi_i\neq0,~\textrm{and}~\lambda_i\neq\alpha_k~\textrm{for all}~k\}.$$ Clearly, $\sigma\in(0,1)$. We now close this section by deriving a bound on the constant $c$ defined in Theorem \[th1\]. The following theorem generalizes the results in [@akaike1959successive; @nocedal2002behavior], where only the case $\Psi(A)=I$ (i.e., the SD method) is considered.
\[bdc2\] Under the conditions of Theorem \[th1\], and assuming that $\mathcal{I}$ is nonempty, we have $$\label{eqbdc2}
\frac{\Psi(\lambda_{1})}{\Psi(\lambda_{n})}\frac{1}{\phi_\sigma}\leq c^2\leq\frac{\Psi(\lambda_{1})}{\Psi(\lambda_{n})}\phi_\sigma,$$ where $$\label{defphi}
\phi_\sigma=\frac{2+\eta_\sigma+\sqrt{\eta_\sigma^2+4\eta_\sigma}}{2} \quad \mbox{and} \quad \eta_\sigma=4\left(\frac{1+\sigma^2}{1-\sigma^2}\right).$$
Let $p=q_0$. By the definition of $T$, we have that $$\label{tpk2i21}
\frac{(T^{k+2}p)^{(i)}}{(T^{k+2}p)^{(1)}}=
\frac{(T^{k}p)^{(i)}}{(T^{k}p)^{(1)}}
\frac{(\lambda_i-\gamma(T^{k}p))^2(\lambda_i-\gamma(T^{k+1}p))^2}
{(\lambda_1-\gamma(T^{k}p))^2(\lambda_1-\gamma(T^{k+1}p))^2}.$$ It follows from Theorem \[th1\] and Lemma \[lm3\] that $$\label{tpki}
\frac{(T^{k}p)^{(i)}}{(T^{k}p)^{(1)}}\rightarrow0,~i=2,\ldots,n-1.$$ By the continuity of $T$ and in Lemma \[lm3\], we always have that $$\label{eq67}
\frac{(\lambda_i-\gamma(T^{k}p))^2(\lambda_i-\gamma(T^{k+1}p))^2}
{(\lambda_1-\gamma(T^{k}p))^2(\lambda_1-\gamma(T^{k+1}p))^2}\rightarrow
\frac{(\lambda_i-\gamma(p_*))^2(\lambda_i-\gamma(Tp_*))^2}
{(\lambda_1-\gamma(p_*))^2(\lambda_1-\gamma(Tp_*))^2},$$ which together with and implies that $$\label{lbdgama1}
\frac{(\lambda_i-\gamma(p_*))^2(\lambda_i-\gamma(Tp_*))^2}
{(\lambda_1-\gamma(p_*))^2(\lambda_1-\gamma(Tp_*))^2}
\leq1,~i=2,\ldots,n-1,$$ where $p_*$ is the same vector as in Lemma \[lm3\]. Clearly, also holds for $i=1$. As for $i=n$, it follows from in Lemma \[tpeqp\] and Theorem \[th1\] that $$\label{gmaptp7}
\gamma(p_*)+\gamma(Tp_*)=\lambda_1+\lambda_n,$$ which yields that $$\frac{(\lambda_n-\gamma(p_*))^2(\lambda_n-\gamma(Tp_*))^2}
{(\lambda_1-\gamma(p_*))^2(\lambda_1-\gamma(Tp_*))^2}=1.$$ Thus, holds for $i=1,\ldots,n$. Hence, we have $$\begin{aligned}
\label{ineqsig2}
&& \left(\lambda_i-\delta
-\left(\gamma(p_*)-\delta\right)\right)^2
\left(\lambda_i-\delta
-\left(\gamma(Tp_*)-\delta\right)\right)^2
\\
& \leq & \left(\lambda_1-\delta
-\left(\gamma(p_*)-\delta\right)\right)^2
\left(\lambda_1-\delta
-\left(\gamma(Tp_*)-\delta\right)\right)^2, \nonumber
\end{aligned}$$ where $\delta=\frac{\lambda_1+\lambda_n}{2}$. By and , we obtain $$\begin{aligned}
&& \left(\lambda_i-\delta
-\left(\gamma(p_*)-\delta\right)\right)^2
\left(\lambda_i-\delta
+\left(\gamma(p_*)-\delta\right)\right)^2 \\
&\leq & \left(\frac{\lambda_1-\lambda_n}{2}
-\left(\gamma(p_*)-\delta\right)\right)^2
\left(\frac{\lambda_1-\lambda_n}{2}
+\left(\gamma(p_*)-\delta\right)\right)^2,
\end{aligned}$$ which implies that $$\label{ineqsig1}
\left(\frac{\lambda_1-\lambda_n}{2}\right)^2+
\left(\lambda_i-\delta\right)^2
\geq
2\left(\gamma(p_*)-\delta\right)^2.$$ By Lemma \[tpeqp\] and Theorem \[th1\], we have that $$\gamma(p_*)=\frac{\lambda_1\Psi(\lambda_{1})p_*^{(1)}+\lambda_n\Psi(\lambda_{n})p_*^{(n)}}
{\Psi(\lambda_{1})p_*^{(1)}+\Psi(\lambda_{n})p_*^{(n)}}.$$ Substituting $\gamma(p_*)$ into , we obtain $$\label{ineqsig3}
\left(\frac{\lambda_1-\lambda_n}{2}\right)^2+
\left(\lambda_i-\delta\right)^2
\geq
\frac{(\lambda_n-\lambda_1)^2(\Psi(\lambda_{n})c^2-\Psi(\lambda_{1}))^2}
{2(\Psi(\lambda_{n})c^2+\Psi(\lambda_{1}))^2},$$ which gives $$\label{inqsig}
4\left(\frac{1+\sigma_i^2}{1-\sigma_i^2}\right)\geq\frac{(c^2\Psi(\lambda_{n})-\Psi(\lambda_{1}))^2}{c^2\Psi(\lambda_{1})\Psi(\lambda_{n})},
\quad \mbox{where} \quad \sigma_i=\frac{2 \lambda_i-(\lambda_1+\lambda_n)}
{\lambda_n-\lambda_1}.$$ Noting that holds for all $i\in\mathcal{I}$. Thus, we have $$\label{inqsig2}
\frac{(c^2\Psi(\lambda_{n})-\Psi(\lambda_{1}))^2}{c^2\Psi(\lambda_{1})\Psi(\lambda_{n})}\leq\eta_\sigma,$$ which implies . This completes the proof.
Techniques for breaking the zigzagging pattern {#secnewstep}
==============================================
As shown in the previous section, all the gradient methods asymptotically conduct its searches in the two-dimensional subspace spanned by $\xi_1$ and $\xi_n$. By , if either $\mu_k^{(1)}$ or $\mu_k^{(n)}$ equals to zero, the corresponding component will vanish at all subsequent iterations. Hence, in order to break the undesired zigzagging pattern, a good strategy is to employ some stepsize approximating $1/\lambda_1$ or $1/\lambda_n$. In this section, we will derive a new stepsize converging to $1/\lambda_n$ and propose a periodic gradient method using this new stepsize.
A new stepsize {#subsecns}
--------------
Our new stepsize will be derived by imposing finite termination on minimizing two-dimensional strictly convex quadratic function, see [@yuan2006new] for the case of $\Psi(A)=I$ (i.e., the SD method). We mention that the key property used by Yuan [@yuan2006new] is that two consecutive gradients generated by the SD method are perpendicular to each other, which may not be true for all the gradient methods . However, we have by the stepsize definition that $$\label{orthg}
g_k {^{\sf T}}\Psi(A)g_{k+1}=g_k {^{\sf T}}\Psi(A)g_k-\alpha_kg_k {^{\sf T}}\Psi(A)Ag_k=0.$$
Consider the two-dimensional case. Suppose we want to find the minimizer of with $n=2$ after the following $3$ iterations: $$x_1 = x_0 - \alpha_0g_0, \quad
x_2 = x_1 - \alpha_1g_1, \quad
x_3 = x_2 - \alpha_2g_2,$$ where $g_i \ne 0$, $i=0,1,2$, $\alpha_0$ and $\alpha_2$ are stepsizes given by , and $\alpha_1$ is to be derived by ensuring $x_3$ is the solution.
By , we have $g_0 {^{\sf T}}\Psi(A)g_1=0$. Hence, all vectors $x_k$ can be expressed by the linear combination of $\frac{\Psi^r(A)g_0}{\|\Psi^r(A)g_0\|}$ and $\frac{\Psi^{1-r}(A)g_1}{\|\Psi^{1-r}(A)g_1\|}$ for any given $r \in \mathbb{R}$. Now, consider $$\begin{aligned}
\label{ftu1}
\varphi(t,l) &:= &
f\left(x_1+t\frac{\Psi^r(A)g_0}{\|\Psi^r(A)g_0\|}+l\frac{\Psi^{1-r}(A)g_1}{\|\Psi^{1-r}(A)g_1\|}\right) \\
& = &
f(x_1) +G {^{\sf T}}\begin{pmatrix}
t \\
l \\
\end{pmatrix}
+ \frac{1}{2}
\begin{pmatrix}
t \\
l \\
\end{pmatrix} {^{\sf T}}H
\begin{pmatrix}
t \\
l \\
\end{pmatrix}, \nonumber\end{aligned}$$ where $$\label{eqgrad}
G= B g_1 = \begin{pmatrix}
\frac{g_1 {^{\sf T}}\Psi^r(A)g_0}{\|\Psi^r(A)g_0\|}\\
\frac{g_1 {^{\sf T}}\Psi^{1-r}(A)g_1}{\|\Psi^{1-r}(A)g_1\|}
\end{pmatrix}
\mbox{ with } B=\begin{pmatrix}
\frac{\Psi^r(A)g_0}{\|\Psi^r(A)g_0\|},
\frac{\Psi^{1-r}(A)g_1}{\|\Psi^{1-r}(A)g_1\|} \\
\end{pmatrix} {^{\sf T}}$$ and $$\label{eqhes}
H= B A B {^{\sf T}}= \begin{pmatrix}
\frac{g_0 {^{\sf T}}\Psi^{2r}(A)Ag_0}{\|\Psi^r(A)g_0\|^2} & \frac{g_0 {^{\sf T}}\Psi(A) Ag_1}{\|\Psi^r(A)g_0\|\|\Psi^{1-r}(A)g_1\|} \\
\frac{g_0 {^{\sf T}}\Psi(A)Ag_1}{\|\Psi^r(A)g_0\|\|\Psi^{1-r}(A)g_1\|} & \frac{g_1 {^{\sf T}}\Psi^{2(1-r)}(A)Ag_1}{\|\Psi^{1-r}(A)g_1\|^2}\\
\end{pmatrix}.$$ Note that $B {^{\sf T}}B = B B {^{\sf T}}= I$ since $n=2$. The minimizer $(t^*, l^*)$ of $\varphi$ satisfy $$G
+
H
\begin{pmatrix}
t^* \\
l^* \\
\end{pmatrix}=0,
\quad \Longrightarrow \quad
\begin{pmatrix}
t^* \\
l^* \\
\end{pmatrix}
=-H^{-1}G.
$$ Suppose $x_3$ is the solution, that is $$x_3=x_1+t^*\frac{\Psi^r(A)g_0}{\|\Psi^r(A)g_0\|}+l^*\frac{\Psi^{1-r}(A)g_1}{\|\Psi^{1-r}(A)g_1\|}.$$ Then, since $x_3 = x_2 - \alpha_2 g_2$, we have $x_3-x_2$ is parallel to $g_2$, i.e., $$\label{parallel-g2}
B {^{\sf T}}\begin{pmatrix}
t^* \\
l^* \\
\end{pmatrix}
+ \alpha_1 g_1
\quad \mbox{ is parallel to } \quad g_2,$$ which is equivalent to $$\label{g2pal1}
\begin{pmatrix}
t^* \\
l^* \\
\end{pmatrix}
-(-\alpha_1G)
= -(H^{-1}G-\alpha_1 G)
\quad \mbox{and} \quad G+H(-\alpha_1G)$$ are parallel. Denote the components of $G$ by $G_i$, and the components of $H$ by $H_{ij}$, $i,j=1,2$. By , we would have $$\begin{pmatrix}
H_{22}G_1-H_{12}G_2-\alpha_1\Delta G_1 \\
H_{11}G_2-H_{12}G_1-\alpha_1\Delta G_2\\
\end{pmatrix}
\quad \mbox{and} \quad
\begin{pmatrix}
G_1-\alpha_1(H_{11}G_1+H_{12}G_2) \\
G_2-\alpha_1(H_{12}G_1+H_{22}G_2)\\
\end{pmatrix}.$$ are parallel, where $\Delta=\textrm{det}(H) = \textrm{det}(A) >0$. It follows that $$\begin{aligned}
&& (H_{22}G_1-H_{12}G_2-\alpha_1\Delta G_1)
[G_2-\alpha_1(H_{12}G_1+H_{22}G_2)]\\
& =& (H_{11}G_2-H_{12}G_1-\alpha_1\Delta G_2)
[G_1-\alpha_1(H_{11}G_1+H_{12}G_2)],\end{aligned}$$ which gives $$\label{eqalp1}
\alpha_1^2\Delta \Gamma-\alpha_1(H_{11}+H_{22})\Gamma+\Gamma=0,$$ where $$\Gamma=(H_{12}G_1+H_{22}G_2)G_1-(H_{11}G_1+H_{12}G_2)G_2.$$ On the other hand, if holds, we have holds, which by , $H^{-1} = B A^{-1} B {^{\sf T}}$ and $B {^{\sf T}}B=I$ implies that $$- B {^{\sf T}}H^{-1} G + \alpha_1 g_1 = - A^{-1} g_1 + \alpha_1 g_1 = - A^{-1}(g_1- \alpha_1 A g_1)
= - A^{-1} g_2$$ is parallel to $g_2$. Hence, $g_2$ is an eigenvector of $A$, i.e. $Ag_2 = \lambda g_2$ for some $\lambda > 0$, since $g_2 \ne 0$. So, by , $\alpha_2 = \Psi(\lambda) g_2 {^{\sf T}}g_2 /(\lambda \Psi(\lambda) g_2 {^{\sf T}}g_2 ) = 1/\lambda$. Therefore, $g_3 = g_2 - \alpha_2 A g_2 = g_2 - \alpha_2 \lambda g_2 = 0$, which implies $x_3$ is the solution. So, guarantees $x_3$ is the minimizer.
Hence, to ensure $x_3$ is the minimizer, by , we only need to choose $\alpha_1$ satisfying $$\label{eqalp2}
\alpha_1^2\Delta -\alpha_1(H_{11}+H_{22})+1=0,$$ whose two positive roots are $$\frac{(H_{11}+H_{22})\pm\sqrt{(H_{11}+H_{22})^2-4\Delta}}{2\Delta}.$$ These two roots are $1/\lambda_1$ and $1/\lambda_2$, where $0 < \lambda_1 < \lambda_2$ are two eigenvalues of $A$ (Note that $A$ and $H$ have same eigenvalues). For numerical reasons (see next subsection), we would like to choose $\alpha_1$ to be the smaller one $1/\lambda_2$, which can be calculated as $$\begin{aligned}
\label{new2}
\alpha_1&= \frac{2}{(H_{11}+H_{22})+\sqrt{(H_{11}+H_{22})^2-4\Delta}}
\nonumber\\
&= \frac{2}{(H_{11}+H_{22})+\sqrt{(H_{11}-H_{22})^2+4H_{12}^2}}.\end{aligned}$$
To check this finite termination property, we applied the above described method with $\alpha_1$ given by , and $\Psi(A)=A$ in , (i.e., $\alpha_0$ and $\alpha_2$ use the MG stepsize) to minimize two-dimensional quadratic function with $$\label{twoquad}
A=\textrm{diag}\{1,\lambda\} \quad \mbox{and} \quad b=0.$$ We run the algorithm for 3 iterations using ten random starting points and the averaged values of $\|g_3\|$ and $f(x_3)$ are presented in Table \[tb2ft\]. We can observe that for different values of $\lambda$, the $\|g_3\|$ and $f(x_3)$ obtained by the method in three iterations are numerically very close to zero. This coincides with our analysis.
Spectral property of the new stepsize {#astilalp}
-------------------------------------
Notice that $g_1=g_0-\alpha_0Ag_0$ and $g_0 {^{\sf T}}\Psi(A)g_1=0$. So, we have $$g_0 {^{\sf T}}\Psi(A)Ag_1=-(g_1 {^{\sf T}}\Psi(A)g_1)/\alpha_0.$$ Hence, the matrix $H$ given in can be also written as $$\label{eqhes2}
H= \begin{pmatrix}
\frac{g_0 {^{\sf T}}\Psi^{2r}(A)Ag_0}{\|\Psi^r(A)g_0\|^2} & -\frac{g_1 {^{\sf T}}\Psi(A) g_1}{\alpha_0\|\Psi^r(A)g_0\|\|\Psi^{1-r}(A)g_1\|} \\
-\frac{g_1 {^{\sf T}}\Psi(A) g_1}{\alpha_0\|\Psi^r(A)g_0\|\|\Psi^{1-r}(A)g_1\|} & \frac{g_1 {^{\sf T}}\Psi^{2(1-r)}(A)Ag_1}{\|\Psi^{1-r}(A)g_1\|^2} \\
\end{pmatrix}.$$ So, for general case, we could propose our new stepsize at the $k$-th iteration as $$\label{newn}
\tilde{\alpha}_{k}
=\frac{2}{(H_{11}^k+H_{22}^k)+\sqrt{(H_{11}^k-H_{22}^k)^2+4(H_{12}^k)^2}},$$ where $H_{ij}^k$ is the component of $H^k$: $$\label{eqhesk}
H^k= \begin{pmatrix}
\frac{g_{k-1} {^{\sf T}}\Psi^{2r}(A)Ag_{k-1}}{\|\Psi^r(A)g_{k-1}\|^2} & -\frac{g_k {^{\sf T}}\Psi(A) g_k}{\alpha_{k-1}\|\Psi^r(A)g_{k-1}\|\|\Psi^{1-r}(A)g_k\|} \\
-\frac{g_k {^{\sf T}}\Psi(A) g_k}{\alpha_{k-1}\|\Psi^r(A)g_{k-1}\|\|\Psi^{1-r}(A)g_k\|} & \frac{g_k {^{\sf T}}\Psi^{2(1-r)}(A)Ag_k}{\|\Psi^{1-r}(A)g_k\|^2} \\
\end{pmatrix}$$ and $\alpha_{k-1}$ is given by . Clearly, $\alpha_k^Y$ in can be obtained by by setting $\Psi(A)=I$ in . In addition, by we have that $$\label{newbd}
\frac{1}{H_{11}^k+H_{22}^k}
\leq\tilde{\alpha}_k
\leq\frac{1}{\max\{H_{11}^k,H_{22}^k\}}.$$
The next theorem shows that the stepsize $\tilde{\alpha}_k$ enjoys desirable spectral property.
\[thtilalp\] Suppose that the conditions of Theorem \[th1\] hold. Let $\{x_k\}$ be the iterations generated by any gradient method in to solve problem . Then $$\label{tidlbdn}
\lim_{k\rightarrow\infty}\tilde{\alpha}_k=\frac{1}{\lambda_n}.$$
It follows from and of Theorem \[th1\] that $$\begin{aligned}
\lim_{k\rightarrow\infty}H_{11}^k
&=\lim_{k\rightarrow\infty}\frac{g_{k-1} {^{\sf T}}\Psi^{2r}(A)Ag_{k-1}}{\|g_{k-1}\|^2}
\frac{\|g_{k-1}\|^2}{\|\Psi^r(A)g_{k-1}\|^2}\\
&=\frac{\lambda_1(c^2\Psi^{2r}(\lambda_{1})\Psi^2(\lambda_{n})+\kappa\Psi^{2r}(\lambda_{n})\Psi^2(\lambda_{1}))}
{c^2\Psi^{2r}(\lambda_{1})\Psi^2(\lambda_{n})+\Psi^{2r}(\lambda_{n})\Psi^2(\lambda_{1})}\end{aligned}$$ and $$\begin{aligned}
\lim_{k\rightarrow\infty}H_{22}^k&=
\frac{g_k {^{\sf T}}\Psi^{2(1-r)}(A)Ag_k}{\|g_k\|^2}
\frac{\|g_k\|^2}{\|\Psi^{1-r}(A)g_k\|^2} \\
&=\frac{\lambda_1(\Psi^{2(1-r)}(\lambda_{1})+\kappa c^2\Psi^{2(1-r)}(\lambda_{n}))}{\Psi^{2(1-r)}(\lambda_{1})+ c^2\Psi^{2(1-r)}(\lambda_{n})}\\
&=\frac{\lambda_1(\Psi^2(\lambda_{1})\Psi^{2r}(\lambda_{n})+\kappa c^2\Psi^2(\lambda_{n})\Psi^{2r}(\lambda_{1}))}
{\Psi^2(\lambda_{1})\Psi^{2r}(\lambda_{n})+ c^2\Psi^2(\lambda_{n})\Psi^{2r}(\lambda_{1})},\end{aligned}$$ which give $$\label{mgsd}
\lim_{k\rightarrow\infty}(H_{11}^k+H_{22}^k)
=\lambda_1(\kappa+1)$$ and $$\label{mgsd2}
\lim_{k\rightarrow\infty}(H_{11}^k-H_{22}^k)
=\frac{\lambda_1(\kappa-1)(\Psi^2(\lambda_{1})\Psi^{2r}(\lambda_{n})- c^2\Psi^2(\lambda_{n})\Psi^{2r}(\lambda_{1}))}
{\Psi^2(\lambda_{1})\Psi^{2r}(\lambda_{n})+ c^2\Psi^2(\lambda_{n})\Psi^{2r}(\lambda_{1})}.$$ Then, by the definition of $\alpha_k$, we have $$g_{k} {^{\sf T}}\Psi(A)g_{k}=-\alpha_{k-1}g_{k-1} {^{\sf T}}\Psi(A)Ag_{k-1}
+\alpha_{k-1}^2g_{k-1} {^{\sf T}}\Psi(A)A^2g_{k-1},$$ which together with in Theorem \[th1\] and in Corollary \[cor1\] yields that $$\begin{aligned}
&\lim_{k\rightarrow\infty}(H_{12}^k)^2\\
=&\lim_{k\rightarrow\infty}\frac{g_{k} {^{\sf T}}\Psi(A)g_{k}}{\alpha_{k-1}^2\|\Psi^r(A)g_{k-1}\|^2}
\frac{g_k {^{\sf T}}\Psi(A) g_k}{\|\Psi^{1-r}(A)g_k\|^2}\\
= &\lim_{k\rightarrow\infty}
\left(-\frac{1}{\alpha_{k-1}}\frac{g_{k-1} {^{\sf T}}\Psi(A)Ag_{k-1}}{\|\Psi^r(A)g_{k-1}\|^2}
+\frac{g_{k-1} {^{\sf T}}\Psi(A)A^2g_{k-1}}{\|\Psi^r(A)g_{k-1}\|^2}\right)
\frac{g_k {^{\sf T}}\Psi(A) g_k}{\|\Psi^{1-r}(A)g_k\|^2}\\
=&
\Bigg[-\frac{\lambda_{1}(\kappa\Psi(\lambda_{1})+c^2\Psi(\lambda_{n}))}{\Psi(\lambda_{1})+c^2\Psi(\lambda_{n})}
\frac{\lambda_1(c^2\Psi(\lambda_{1})\Psi^2(\lambda_{n})+\kappa\Psi(\lambda_{n})\Psi^2(\lambda_{1}))}
{c^2\Psi^{2r}(\lambda_{1})\Psi^2(\lambda_{n})+\Psi^{2r}(\lambda_{n})\Psi^2(\lambda_{1})}
+\\
&
\frac{\lambda_1^2(c^2\Psi(\lambda_{1})\Psi^2(\lambda_{n})+\kappa^2\Psi(\lambda_{n})\Psi^2(\lambda_{1}))}
{c^2\Psi^{2r}(\lambda_{1})\Psi^2(\lambda_{n})+\Psi^{2r}(\lambda_{n})\Psi^2(\lambda_{1})}
\Bigg]
\frac{(\Psi(\lambda_{1})+ c^2\Psi(\lambda_{n}))\Psi^{2r}(\lambda_{1})\Psi^{2r}(\lambda_{n})}{\Psi^2(\lambda_{1})\Psi^{2r}(\lambda_{n})+ c^2\Psi^2(\lambda_{n})\Psi^{2r}(\lambda_{1})}
\\
=& \frac{\lambda_1^2c^2(\kappa-1)^2\Psi^{2+2v}(\lambda_{1})\Psi^{2+2v}(\lambda_{n})}
{(\Psi^2(\lambda_{1})\Psi^{2r}(\lambda_{n})+ c^2\Psi^2(\lambda_{n})\Psi^{2r}(\lambda_{1}))^2}.\end{aligned}$$ Then, from the above equality and , we obtain that $$\label{mgsd3}
\lim_{k\rightarrow\infty}\sqrt{(H_{11}^k-H_{22}^k)^2+4(H_{12}^k)^2}
=\lambda_1(\kappa-1).$$ Combining and , we have that $$\lim_{k\rightarrow\infty}\tilde{\alpha}_k=
\frac{2}{\lambda_1(\kappa+1)+\lambda_1(\kappa-1)}=\frac{1}{\lambda_n}.$$ This completes the proof.
When $r=1$, we have from that $\tilde{\alpha}_k \le 1/H_{22}^k = \alpha_k^{SD}$. Hence, using this stepsize $\tilde{\alpha}_k$ will give a monotone gradient method. Theorem \[thtilalp\] indicates that the general $\tilde{\alpha}_k$ will have the asymptotic spectral property , and hence will be asymptotically be smaller than $\alpha_k^{SD}$ independent of $r$. But a proper choice $r$ will facilitate the calculation of $\tilde{\alpha}_k$. This will be more clear in the next section.
Using the similar arguments, we can also show the larger stepsize derived in subsection \[subsecns\] converges to $1/\lambda_1$.
Let $$\bar{\alpha}_k=
\frac{2}{(H_{11}^k+H_{22}^k)-\sqrt{(H_{11}^k-H_{22}^k)^2+4(H_{12}^k)^2}}.$$ Under the conditions of Theorem \[thtilalp\], we have $$\lim_{k\rightarrow\infty}\bar{\alpha}_k=\frac{1}{\lambda_1}.$$
To present an intuitive illustration of the asymptotic behaviors of $\tilde{\alpha}_k$ and $\bar{\alpha}_k$, we applied the gradient method with $\Psi(A)=A$ (i.e., the MG method) to minimize the quadratic function with $$\label{tp1}
A=\textrm{diag}\{a_1,a_2,\ldots,a_n\} \quad \mbox{and} \quad b=0,$$ where $a_1=1$, $a_n=n$ and $a_i$ is randomly generated between 1 and $n$ for $i=2,\ldots,n-1$. From Figure \[appstep1\], we can see that $\tilde{\alpha}_k$ approximates $1/\lambda_n$ with satisfactory accuracy in a few iterations. However, $\bar{\alpha}_{k}$ converges to $1/\lambda_1$ even slower than the decreasing of gradient norm. This, to some extent, explains the reason why we prefer $\tilde{\alpha}_k$ to the short stepsize.
![Problem with $n=1,000$: convergence history of the sequences $\{\tilde{\alpha}_k\}$ and $\{\bar{\alpha}_k\}$ for the first 5,000 iterations of the gradient method with $\Psi(A)=A$ (i.e., the MG method).[]{data-label="appstep1"}](hsn-lbd.eps "fig:"){width="65.00000%" height="42.00000%"}\
A periodic gradient method {#secmethod}
--------------------------
A method alternately using $\alpha_k$ in and $\tilde{\alpha}_k$ to minimize a $2$-dimensional quadratic function will monotonically decrease the objective value, and terminates in $3$ iterations. However, for minimizing a general $n$-dimensional quadratic function, this alternating scheme may not be efficient for the purpose of vanishing the component $\mu_k^{(n)}$. One possible reason is that, as shown in Figure \[appstep1\], it needs tens of iterations before $\tilde{\alpha}_k$ being a good approximation of $1/\lambda_n$ with satisfactory accuracy. In what follows, by incorporating the BB method, we develop an efficient periodic gradient method using $\tilde{\alpha}_k$.
Figure \[BBMG\] illustrates a comparison of the gradient method using $\Psi(A)=A$ (i.e., the MG method) with a method using 20 BB2 steps first and then MG steps on solving problem . We can see that by using some BB2 steps, the modified MG method is accelerated and the stepsize $\tilde{\alpha}_k$ will approximate $1/\lambda_n$ with a better accuracy. Thus, our method will run some BB steps first. Now, we investigate the affect of reusing a short stepsize on the performance of the gradient method . Suppose that we have a good approximation of $1/\lambda_n$, say $\alpha=\frac{1}{\lambda_n+10^{-6}}$. We compare MG method with its two variants by applying (i) $\alpha_0=\alpha$ or (ii) $\alpha_0=\ldots=\alpha_9=\alpha$ before using the MG stepsize. Figure \[BBMGstep\] shows that reusing $\alpha$ will accelerate the MG method. Hence, we prefer to reuse $\tilde{\alpha}_k$ for some consecutive steps when $\tilde{\alpha}_k$ is a good approximation of $1/\lambda_n$. Finally, our new method is summarized in Algorithm \[al1\], which periodically applies the BB stepsize, $\alpha_k$ in and $\tilde{\alpha}_k$. The $R$-linear global convergence of Algorithm \[al1\] for solving can be established by showing that it satisfies the property in [@dai2003alternate], see Theorem 3 of [@dhl2018] for example.
![Problem with $n=10$: the MG method (i.e., $\Psi(A)=A$) with different stepsizes.[]{data-label="BBMGstep"}](repeat_first_steps_MG_largeg.eps "fig:"){width="60.00000%" height="42.00000%"}\
The BB steps in Algorithm \[al1\] can either employ the BB1 or BB2 stepsize in . The idea of using short stepsizes to eliminate the component $\mu_k^{(n)}$ has been investigated in [@de2014efficient; @de2013spectral; @gonzaga2016steepest]. However, these methods are based on the SD method, that is, occasionally applying short steps during the iterates of the SD method. One exception is given by [@huang2019gradient], where a method is developed by employing new stepsizes during the iterates of the AOPT method. But our method periodically uses three different stepsizes: the nonmonotone BB method, the gradient method and the new stepsize $\tilde{\alpha}_k$.
Numerical experiments {#secnum}
=====================
In this section, we present numerical comparisons of Algorithm \[al1\] and the following methods: BB with $\alpha_k^{BB1}$ [@barzilai1988two], Dai-Yuan (DY) [@dai2005analysis], ABBmin2 [@frassoldati2008new], and SDC [@de2014efficient].
Notice that the stepsize rule for Algorithm \[al1\] can be written as $$\label{sal1}
\alpha_{k}=
\begin{cases}
\alpha_{k}^{BB},& \text{if $\textrm{mod}(k,K_b+K_m+K_s)<K_b$}; \\
\alpha_{k}(\Psi(A)),& \text{if $K_b\leq\textrm{mod}(k,K_b+K_m+K_s)<K_b+K_m$}; \\
\tilde{\alpha}_{k}(\Psi(A)),& \text{if $\textrm{mod}(k,K_b+K_m+K_s)=K_b+K_m$}; \\
\alpha_{k-1},& \text{otherwise},
\end{cases}$$ where $\alpha_k^{BB}$ can either be $\alpha_k^{BB1}$ or $\alpha_k^{BB2}$, $\alpha_{k}(\Psi(A))$ and $\tilde{\alpha}_{k}(\Psi(A))$ are the stepsizes given by and , respectively. We tested the following four variants of Algorithm \[al1\] using combinations of the two BB stepsizes and $\Psi(A)=I$ or $A$:
- BB1SD: $\alpha_k^{BB1}$ and $\Psi(A)=I$ in
- BB1MG: $\alpha_k^{BB1}$ and $\Psi(A)=A$ in
- BB2SD: $\alpha_k^{BB2}$ and $\Psi(A)=I$ in
- BB2MG: $\alpha_k^{BB2}$ and $\Psi(A)=A$ in
Now we derive a formula for the case $\Psi(A)=A$, i.e., $\alpha_{k}(\Psi(A))=\alpha_{k}^{MG}$. If we set $r=0$, by , we have $$\label{newmg1}
\tilde{\alpha}_k=\frac{2}{\left(\frac{1}{\alpha_{k-1}^{SD}}+ \frac{g_k {^{\sf T}}A^3g_k}{g_k {^{\sf T}}A^2g_k}\right)+
\sqrt{\left(\frac{1}{\alpha_{k-1}^{SD}}- \frac{g_k {^{\sf T}}A^3g_k}{g_k {^{\sf T}}A^2g_k}\right)^2+
\frac{4(g_k {^{\sf T}}Ag_k)^2}{(\alpha_{k-1}^{MG})^2\|g_{k-1}\|^2g_k {^{\sf T}}A^2g_k}}},$$ which is expensive to compute directly. However, if we set $r=1/2$, we get $$\label{newsmg}
\tilde{\alpha}_{k}=
\frac{2}{\frac{1}{\alpha_{k-1}^{MG}}+\frac{1}{\alpha_k^{MG}}+
\sqrt{\left(\frac{1}{\alpha_{k-1}^{MG}}-\frac{1}{\alpha_k^{MG}}\right)^2
+\frac{4g_k {^{\sf T}}A g_k}{(\alpha_{k-1}^{MG})^2g_{k-1} {^{\sf T}}A g_{k-1}}}}.$$ This formula can be computed without additional cost because $g_{k-1} {^{\sf T}}A g_{k-1}$ and $g_k {^{\sf T}}A g_k$ have been obtained when computing the stepsizes $\alpha_{k-1}^{MG}$ and $\alpha_k^{MG}$.
All the methods in consideration were implemented in Matlab (v.9.0-R2016a) and carried out on a PC with an Intel Core i7, 2.9 GHz processor and 8 GB of RAM running Windows 10 system. We stopped the algorithm if the number of iteration exceeds 20,000 or the gradient norm reduces by a factor of $\epsilon$.
We randomly generated quadratic problems proposed in [@dhl2018], where $A=QVQ {^{\sf T}}$ with $$Q=(I-2w_3w_3 {^{\sf T}})(I-2w_2w_2 {^{\sf T}})(I-2w_1w_1 {^{\sf T}}),$$ $w_1$, $w_2$, and $w_3$ are unitary random vectors, and $V=diag(v_1,\ldots,v_n)$ is a diagonal matrix where $v_1=1$, $v_n=\kappa$, and $v_j$, $j=2,\ldots,n-1$, are randomly generated between 1 and $\kappa$ by the *rand* function in Matlab. We tested seven sets of different distributions of $v_j$ as shown in Table \[tbspe\] with different values of the condition number $\kappa$ and tolerance $\epsilon$. In particular, $\kappa$ were set to $10^4, 10^5, 10^6$ and $\epsilon$ were set to $10^{-6}, 10^{-9}, 10^{-12}$. For each value of $\kappa$ or $\epsilon$, 10 instances were generated and there are totally 630 instances. For each instance, the entries of $b$ were randomly generated in $[-10,10]$ and $e=(1,\ldots,1) {^{\sf T}}$ was used as the starting point.
-- -----------------------------------------------------------------
$\{v_2,\ldots,v_{n-1}\}\subset(1,\kappa)$
$\{v_2,\ldots,v_{n/5}\}\subset(1,100)$
$\{v_{n/5+1},\ldots,v_{n-1}\}\subset(\frac{\kappa}{2},\kappa)$
$\{v_2,\ldots,v_{n/2}\}\subset(1,100)$
$\{v_{n/2+1},\ldots,v_{n-1}\}\subset(\frac{\kappa}{2},\kappa)$
$\{v_2,\ldots,v_{4n/5}\}\subset(1,100)$
$\{v_{4n/5+1},\ldots,v_{n-1}\}\subset(\frac{\kappa}{2},\kappa)$
$\{v_2,\ldots,v_{n/5}\}\subset(1,100)$
$\{v_{n/5+1},\ldots,v_{4n/5}\}\subset(100,\frac{\kappa}{2})$
$\{v_{4n/5+1},\ldots,v_{n-1}\}\subset(\frac{\kappa}{2},\kappa)$
$\{v_2,\ldots,v_{10}\}\subset(1,100)$
$\{v_{11},\ldots,v_{n-1}\}\subset(\frac{\kappa}{2},\kappa)$
$\{v_2,\ldots,v_{n-10}\}\subset(1,100)$
$\{v_{n-9},\ldots,v_{n-1}\}\subset(\frac{\kappa}{2},\kappa)$
-- -----------------------------------------------------------------
: Distributions of $v_j$.[]{data-label="tbspe"}
The parameter $K_b$ for Algorithm \[al1\] was set to 100 for the first and fifth sets and 30 for other sets. Other two parameters $K_m$ and $K_s$ were selected from $\{9,13,15\}$. As in [@frassoldati2008new], the parameter $\tau$ of the ABBmin2 method was set to 0.9 for all instances. The parameter pair $(h,s)$ used for the SDC method was set to $(8,6)$, which is more efficient than other choices for this test.
Table \[tbrandpBB1SD\] shows the averaged number of iterations of BB1SD and other four compared methods for the seven sets of problems listed in Table \[tbspe\]. We can see that, for the first problem set, our BB1SD method performs much better than the BB, DY and SDC methods, although the ABBmin2 method seems surprisingly efficient among the compared methods. For the second to the last problem sets, our method with different settings performs better than the BB, DY, ABBmin2 and SDC methods. Moreover, for all the settings and different tolerance levels, our method outperforms all the compared four methods in terms of total number of iterations.
Tables \[tbrandpBB1MG\], \[tbrandpBB2SD\] and \[tbrandpBB2MG\] present the averaged number of iterations of BB1MG, BB2SD and BB2MG, respectively. For comparison purposes, the results of the BB, DY, ABBmin 2 and SDC methods are also listed in those tables. As compared with the BB, DY, ABBmin 2 and SDC methods, similar results to those in Table \[tbrandpBB1SD\] can be seen from these three tables. For the comparison of BB1SD and BB1MG, we can see from Tables \[tbrandpBB1SD\] and \[tbrandpBB1MG\] that BB1MG is slightly better than BB1SD for the second to fourth, sixth, and the last problem sets. In addition, BB1MG is comparable to BB1SD for the first and the fifth problem sets. The results in Tables \[tbrandpBB2SD\] and \[tbrandpBB2MG\] do not show much difference between BB2SD and BB2MG. In general, BB1MG performs slightly better than BB1SD, BB2SD and BB2MG for most of the problem sets.
We further compared these methods in Figures \[fig:fBB1\] and \[fig:fBB2\] by using the performance profiles of Dolan and Moré [@dolan2002] on the iteration metric. In these figures, the vertical axis shows the percentage of the problems the method solves within the factor $\rho$ of the metric used by the most effective method in this comparison. We select the results of our four methods corresponding to the column $(15,15)$ in the above tables. It can be seen that all our methods BB1SD, BB1MG, BB2SD and BB2MG clearly outperform the other compared methods. For comparison of BB1SD, BB1MG, BB2SD and BB2MG, Figure \[fBBSDMG\] shows that BB1MG is slightly better than the other three methods, while BB1SD, BB2SD and BB2MG do not show much difference in this test.
![Performance profiles for BB1SD, BB1MG, BB2SD and BB2MG, iteration metric, 630 instances of the problems in Table \[tbspe\].[]{data-label="fBBSDMG"}](BBSDMG1.eps){width="50.00000%" height="37.00000%"}
Conclusions and discussions {#seccls}
===========================
We present theoretical analyses on the asymptotic behaviors of a family of gradient methods whose stepsize is given by , which includes the steepest descent and minimal gradient methods as special cases. It is shown that each method in this family will asymptotically zigzag in a two-dimensional subspace spanned by the two eigenvectors corresponding to the largest and smallest eigenvalues of the Hessian. In order to accelerate the gradient methods, we exploit the spectral property of a new stepsize to break the zigzagging pattern. This new stepsize is derived by imposing finite termination on minimizing two-dimensional strongly convex quadratics and is proved to converge to the reciprocal of the largest eigenvalue of the Hessian for general $n$-dimensional case. Finally, we propose a very efficient periodic gradient method that alternately uses the BB stepsize, $\alpha_k$ in and our new stepsize. Our numerical results indicate that, by exploiting the asymptotic behavior and spectral properties of stepsizes, gradient methods can be greatly accelerated to outperform the BB method and other recently developed state-of-the-art gradient methods.
As a final remark, one may also break the zigzagging pattern by employing the spectral property in . In particular, we could use the following stepsize $$\label{sum2reps2}
\hat{\alpha}_k=\left(\frac{1}{\alpha_{2k}}+\frac{1}{\alpha_{2k+1}}\right)^{-1},$$ to break the zigzagging pattern. By , $ \hat{\alpha}_k$ satisfies $$\lim_{k\rightarrow\infty}\hat{\alpha}_k
= \frac{1}{\lambda_{1}+\lambda_{n}}.$$ Hence, $ \hat{\alpha}_k$ is also a good approximation of $1/\lambda_n$ when the condition number $\kappa=\lambda_n/\lambda_1$ is large. One may see the strategy used in [@de2013spectral] for the case of the SD method.
Tables
======
-- -- --------- ---------- ---------- ---------- ------------ ----------- ---------- ------------ ------------- --------- --------- --------- ---------
$(9,9)$ $(9,13)$ $(9,15)$ $(13,9)$ $(13,13)$ $(13,15)$ $(15,9)$ $(15,13)$ $(15,15)$
367.6 339.7 372.3 346.1 352.0 344.9 336.8 317.6 368.0
1232.3 983.7 1312.7 1149.2 1149.5 1281.4 1011.4 1086.7 1150.6
1849.9 1514.1 1812.8 1760.6 1780.3 1792.6 1518.1 1605.0 1465.4
242.1 244.4 235.8 238.2 229.3 236.7 249.6 233.9 240.9
816.3 790.9 765.5 840.0 729.8 737.0 758.9 750.3 746.8
1255.9 1222.6 1207.4 1305.8 1179.1 1154.8 1211.5 1187.4 1178.1
297.1 288.5 275.9 284.6 283.1 273.3 283.6 279.7 270.0
829.0 816.2 796.5 848.0 758.5 763.2 796.9 791.7 743.7
1330.5 1241.0 1252.8 1345.9 1224.3 1176.1 1275.2 1189.9 1178.9
358.0 331.8 343.5 331.6 331.6 318.4 331.5 326.6 342.6
882.6 823.2 825.3 917.8 814.2 817.4 860.3 808.6 832.4
1422.0 1327.9 1347.9 1324.3 1232.8 1271.9 1318.4 1288.9 1258.0
838.4 829.4 850.8 851.9 836.4 855.5 874.1 856.5 844.5
3147.9 3086.3 2985.3 2932.6 3004.0 3062.6 3093.1 3086.7 3094.2
4942.5 4996.4 4688.7 4542.9 5020.5 4921.7 4900.0 4845.1 4868.1
155.1 140.8 140.3 138.9 139.4 137.8 132.8 137.9 137.3
554.3 557.4 541.4 590.8 513.8 500.1 559.9 539.4 512.9
905.9 883.1 897.7 939.9 801.1 824.3 925.9 895.9 814.9
455.6 437.0 430.8 457.9 432.9 424.0 445.8 411.3 424.8
905.8 876.0 828.2 922.4 870.5 869.8 925.6 851.0 859.6
1349.8 1323.1 1265.4 1374.2 1278.5 1267.2 1319.2 1252.1 1240.3
2713.9 2611.6 2649.4 2649.2 2604.7 2590.6 2654.2 [2563.5]{} 2628.1 4367.2 3613.0 3773.4 2748.3
8368.2 7933.7 8054.9 8200.8 [7840.3]{} 8031.5 8006.1 7914.4 7940.2 18480.9 16549.1 10970.3 10837.4
13056.5 12508.2 12472.7 12593.6 12516.6 12408.6 12468.3 12264.3 [12003.7]{} 30219.9 28427.1 16885.8 17947.3
-- -- --------- ---------- ---------- ---------- ------------ ----------- ---------- ------------ ------------- --------- --------- --------- ---------
: Number of averaged iterations of BB1SD, BB, DY, ABBmin2 and SDC on the problems in Table \[tbspe\].[]{data-label="tbrandpBB1SD"}
-- -- --------- ---------- ---------- ------------ ----------- ----------- ------------ ----------- ------------- --------- --------- --------- ---------
$(9,9)$ $(9,13)$ $(9,15)$ $(13,9)$ $(13,13)$ $(13,15)$ $(15,9)$ $(15,13)$ $(15,15)$
378.0 366.2 344.9 354.3 364.5 341.7 338.1 374.1 362.1
1187.6 1369.2 1192.8 1029.0 1297.6 1040.6 1124.6 1201.2 1095.8
1909.2 1809.4 1666.2 1558.3 1784.7 1577.6 1578.7 1862.7 1485.3
216.5 211.0 227.0 218.2 211.2 228.5 223.3 225.5 230.2
729.7 679.9 703.0 665.9 674.4 686.0 675.6 665.8 680.9
1199.7 1079.8 1130.7 1076.6 1076.3 1067.1 1096.8 1081.7 1059.3
258.3 265.4 273.7 273.3 249.2 254.1 253.1 246.2 252.3
810.6 743.8 756.7 707.6 720.5 694.0 731.2 723.2 701.6
1208.6 1137.4 1182.6 1112.4 1128.5 1102.7 1153.6 1108.9 1099.7
309.8 325.1 305.4 315.2 309.3 312.6 315.1 304.9 315.8
871.1 753.9 764.6 771.7 766.2 748.4 766.8 749.2 768.3
1268.6 1186.6 1203.9 1164.3 1162.0 1140.8 1200.9 1159.1 1181.2
856.8 833.5 847.7 862.7 847.2 848.3 843.7 906.7 865.1
3197.5 3014.6 3216.2 2988.8 3015.1 3088.4 3137.5 3155.4 3042.1
4937.7 4769.0 4986.6 4933.8 4709.7 4861.1 4944.6 5167.5 4869.2
129.1 125.6 126.0 132.5 126.1 135.4 128.6 127.0 137.3
510.8 498.9 510.1 496.3 452.1 471.3 461.6 487.2 447.6
841.4 799.5 789.0 808.8 712.1 780.5 754.2 748.2 699.8
400.6 417.1 382.8 423.1 407.0 405.6 402.0 415.8 402.7
841.3 815.6 788.3 832.9 820.8 794.4 825.4 844.7 814.5
1245.0 1193.1 1161.9 1218.1 1202.7 1190.3 1210.3 1238.0 1167.7
2549.1 2543.9 2507.5 2579.3 2514.5 2526.2 [2503.9]{} 2600.2 2565.5 4367.2 3613.0 3773.4 2748.3
8148.6 7875.9 7931.7 [7492.2]{} 7746.7 7523.1 7722.7 7826.7 7550.8 18480.9 16549.1 10970.3 10837.4
12610.2 11974.8 12120.9 11872.3 11776.0 11720.1 11939.1 12366.1 [11562.2]{} 30219.9 28427.1 16885.8 17947.3
-- -- --------- ---------- ---------- ------------ ----------- ----------- ------------ ----------- ------------- --------- --------- --------- ---------
: Number of averaged iterations of BB1MG, BB, DY, ABBmin2 and SDC on the problems in Table \[tbspe\].
\[tbrandpBB1MG\]
-- -- --------- ---------- ---------- ------------ ----------- ----------- ------------ ------------- ----------- --------- --------- --------- ---------
$(9,9)$ $(9,13)$ $(9,15)$ $(13,9)$ $(13,13)$ $(13,15)$ $(15,9)$ $(15,13)$ $(15,15)$
347.9 357.2 365.1 349.4 344.3 325.0 338.1 349.4 369.2
1132.2 1454.1 1247.4 1192.7 1224.4 1274.7 1237.7 1291.9 1209.6
1985.3 2429.8 1986.8 1838.2 2062.1 2181.2 1958.2 1961.0 1927.2
219.4 223.9 220.5 226.0 229.3 224.4 217.8 220.4 226.4
749.4 723.3 713.9 746.6 720.1 711.2 728.1 729.4 713.3
1235.9 1188.4 1168.4 1167.9 1158.1 1158.3 1165.2 1186.0 1130.9
248.5 259.0 253.8 254.0 246.3 261.6 252.6 262.8 267.4
780.5 757.1 754.2 759.3 738.4 767.2 793.6 774.4 759.3
1229.4 1230.7 1227.8 1216.0 1214.8 1182.3 1215.2 1227.7 1210.6
320.8 315.1 305.5 313.6 315.9 310.9 318.4 307.5 317.1
805.0 823.3 813.4 819.5 813.5 789.0 779.5 836.1 802.5
1348.7 1298.3 1244.4 1242.8 1276.1 1238.6 1250.0 1269.9 1246.3
860.0 847.3 848.7 831.2 799.3 825.5 804.4 809.5 862.0
3066.6 3191.0 2998.8 2918.1 3049.0 3038.7 2995.5 2995.7 3095.7
5272.4 5133.8 5106.8 4962.9 4867.3 4894.1 5083.6 4775.5 5100.4
129.1 138.8 124.8 128.4 135.3 133.7 122.2 130.8 133.4
560.3 549.5 531.5 514.6 520.9 538.9 516.5 530.4 525.1
912.8 892.1 940.0 913.5 928.1 873.5 892.5 873.3 845.2
418.4 393.6 406.6 410.6 409.7 418.4 394.8 429.7 405.9
898.0 835.8 849.0 852.9 847.6 847.8 868.4 873.3 848.4
1324.7 1238.8 1221.1 1290.1 1263.3 1265.1 1302.7 1279.2 1267.4
2544.1 2534.9 2525.0 2513.2 2480.1 2499.5 [2448.3]{} 2510.1 2581.4 4367.2 3613.0 3773.4 2748.3
7992.0 8334.1 7908.2 [7803.7]{} 7913.9 7967.5 7919.3 8031.2 7953.9 18480.9 16549.1 10970.3 10837.4
13309.2 13411.9 12895.3 12631.4 12769.8 12793.1 12867.4 [12572.6]{} 12728.0 30219.9 28427.1 16885.8 17947.3
-- -- --------- ---------- ---------- ------------ ----------- ----------- ------------ ------------- ----------- --------- --------- --------- ---------
: Number of averaged iterations of BB2SD, BB, DY, ABBmin2 and SDC on the problems in Table \[tbspe\].
\[tbrandpBB2SD\]
-- -- --------- ---------- ------------- ---------- ----------- ----------- ---------- ------------ ----------- --------- --------- --------- ---------
$(9,9)$ $(9,13)$ $(9,15)$ $(13,9)$ $(13,13)$ $(13,15)$ $(15,9)$ $(15,13)$ $(15,15)$
355.7 365.1 341.9 322.6 350.7 327.5 337.9 313.6 321.1
1209.5 1327.4 908.0 1064.7 1206.9 1209.7 965.6 1255.1 1351.1
1858.7 1772.7 1477.3 1640.8 1701.6 1877.9 1651.6 1889.2 1751.7
235.1 237.9 238.2 233.0 229.2 239.2 236.4 235.2 238.0
822.7 778.9 752.8 805.0 747.0 762.7 785.7 748.0 737.0
1273.8 1233.0 1212.6 1294.3 1144.2 1193.2 1248.0 1178.3 1167.0
273.8 265.6 287.9 264.6 271.2 274.4 275.2 263.1 281.9
866.7 831.4 793.5 862.2 777.6 789.1 804.3 786.0 786.6
1313.6 1318.9 1244.3 1361.4 1219.6 1234.7 1313.4 1271.2 1251.8
333.7 335.8 341.9 353.0 319.9 317.4 331.7 333.0 329.1
876.9 877.7 853.3 863.8 844.5 836.6 881.4 804.5 800.1
1364.3 1329.9 1307.0 1351.1 1296.9 1259.4 1337.0 1275.7 1286.7
806.4 836.7 837.7 807.1 842.2 862.9 817.8 814.9 819.9
3106.8 3101.1 3008.3 3102.0 3169.6 3058.9 3073.8 2997.6 3097.9
4996.6 5100.9 4749.5 5079.1 5012.9 5004.8 5090.7 5094.0 4708.6
137.1 138.9 135.9 143.4 135.1 139.0 135.1 136.9 138.9
612.6 571.2 535.3 588.6 543.6 523.8 504.2 569.0 523.0
933.9 874.6 870.0 1026.1 864.7 830.9 862.3 910.9 861.2
462.7 430.8 434.4 454.2 428.2 438.8 440.8 437.9 435.1
957.1 932.7 904.4 935.3 868.1 889.4 933.5 917.1 869.6
1383.7 1337.3 1281.5 1344.8 1288.7 1323.3 1373.0 1310.1 1277.8
2604.5 2610.8 2617.9 2577.9 2576.5 2599.2 2574.9 [2534.6]{} 2564.0 4367.2 3613.0 3773.4 2748.3
8452.3 8420.4 [7755.6]{} 8221.6 8157.3 8070.2 7948.5 8077.3 8165.3 18480.9 16549.1 10970.3 10837.4
13124.6 12967.3 [12142.2]{} 13097.6 12528.6 12724.2 12876.0 12929.4 12304.8 30219.9 28427.1 16885.8 17947.3
-- -- --------- ---------- ------------- ---------- ----------- ----------- ---------- ------------ ----------- --------- --------- --------- ---------
: Number of averaged iterations of BB2MG, BB, DY, ABBmin2 and SDC on the problems in Table \[tbspe\].
\[tbrandpBB2MG\]
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[^1]: Institute of Mathematics, Hebei University of Technology, Tianjin 300401, China ().
[^2]: LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China (, [http://lsec.cc.ac.cn/\\string\~dyh/](http://lsec.cc.ac.cn/\string~dyh/)).
[^3]: Institute of Mathematics, Hebei University of Technology, Tianjin 300401, China ().
[^4]: Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918, USA (, [https://www.math.lsu.edu/\\string\~hozhang/](https://www.math.lsu.edu/\string~hozhang/)).
[^5]: August 19, 2019, This research was supported by the National Natural Science Foundation of China (11701137, 11631013, 11671116), by the National 973 Program of China (2015CB856002), by the China Scholarship Council (No. 201806705007), and by the USA National Science Foundation (1522654, 1819161).
|
---
abstract: 'Ca$_{2}$Y$_{2}$Cu$_{5}$O$_{10}$ is build up from edge-shared CuO$_4$ plaquettes forming spin chains. From inelastic neutron scattering data we extract an in-chain nearest neighbor exchange $J_{1} \approx-170\,{\rm K}$ and the frustrating next neighbor $J_{2} \approx 32\,{\rm K}$ interactions, both significantly larger than previous estimates. The ratio $\alpha=|J_{2}/J_{1}|\approx0.19$ places the system very close to the critical point $\alpha_c=0.25$ of the $J_1$-$J_2$ chain, but in the [*ferromagnetic*]{} regime. We establish that the vicinity to criticality only marginally affects the dispersion and coherence of the elementary spin-wave-like magnetic excitations, but instead results in a dramatic $T$-dependence of high-energy Zhang-Rice singlet excitation intensities.'
author:
- 'R.O. Kuzian'
- 'S. Nishimoto'
- 'S.-L. Drechsler'
- 'J. Málek'
- 'S. Johnston'
- 'J. van den Brink'
- 'M. Schmitt'
- 'H. Rosner'
- 'M. Matsuda'
- 'K. Oka'
- 'H. Yamaguchi'
- 'T. Ito'
title: ' Ca$_{2}$Y$_{2}$Cu$_{5}$O$_{10}$: the first frustrated quasi-1D ferromagnet close to criticality '
---
[^1]
Frustrated low-dimensional magnets serve as breeding grounds for novel and exotic quantum many-body effects. Ca$_2$Y$_2$Cu$_5$O$_{10}$ (CYCO) and the closely related Li$_2$CuO$_2$ (LCO) are considered candidates for this type of unconventional and challenging physics [@Matsuda01; @Matsuda05]. These systems are in the family of frustrated edge-shared chain cuprates (ESC) and their magnetic excitation spectra, as probed by inelastic neutron scattering (INS), indicate striking puzzles. It was claimed that the dispersion of the magnetic excitations in CYCO shows an anomalous [*double*]{} branch [@Matsuda01; @Matsuda05] while LCO exhibits a single, but weakly dispersing mode[@Boehm98]. These observations would point at a strong deviation of the dispersion from standard linear spin wave theory (LSWT), in any realistic ESC parameter regime. This motivates scenarios with more sophisticated many-body physics, e.g. due to the presence of antiferromagnetic (AFM) interchain couplings (IC), causing the branch doubling in CYCO [@Matsuda01; @Matsuda05]. However, such a scenario invoking strong quantum effects seems to be at odds with the observed large and almost saturated magnetic moments $ \sim 0.9\mu_{\rm B}$ at $T\ll T_{\mbox{\tiny N}}=29.5$ K [@Matsuda99; @Fong99], which suggests strongly [*suppressed*]{} quantum fluctuations. To resolve the situation it is essential to identify the precise values of the exchange interactions in these ESC’s, both within and between the spin chains. To this end it is key to measure and at the same time calculate the elementary magnetic excitations, ideally for directions of momentum transfer in which the excitations depend most sensitively on the strength of the in-chain couplings. From scattering along the $a$-axis of CYCO, which does [*not*]{} fulfill the latter condition, a moderate value of the ferromagnetic (FM) nearest neighbor (NN) coupling $J_1\approx-93$ K has been extracted [@Matsuda01] with a tiny frustrating AFM next-nearest neighbor (NNN) exchange $J_2\approx 4.7$ K (see Fig. 1).
![(Color) Schematic view of the structure of the CuO$_2$ chain layer and the main exchange paths of CYCO (see text). []{data-label="fig2"}](FIG1.eps){width="0.79\columnwidth"}
From a theoretical point of view this is rather unexpected for the ESC chain geometry due to the presence of sizable O-O 2$p$ hopping along the chains. A recent reassessment of the exchange strengths based on INS on isotopically clean $^7$Li$_2$CuO$_2$ [@Lorenz09] has revealed a relatively large FM coupling $|J_1| > 200$ K, which is more than a factor 2 larger than earlier theoretical values [@Mizuno98]. Given the structural similarities to CYCO, one expects a larger value $J_1$ here too. In fact, high-$T$ $^{89}$Y-NMR data on CYCO appear difficult to reconcile with small $|J_1|$’s [@choi]. Here we show that indeed by measuring with INS the magnetic excitations in CYCO along $\mathbf{Q}=(H,0,1.5)$, a direction where they are little affected by inter-chain couplings, one extracts a substantial in-chain $J_{1} \approx-170\,{\rm K}$ and a frustrating $J_{2} \approx 32\,{\rm K}$, so that $\alpha=|J_{2}/J_{1}|\approx0.19$. This indicates an exceptional position of CYCO within the ESC family: close to the critical point (CP) of the $J_1$-$J_2$ model $\alpha _c=1/4$, but on the FM side of its phase diagram and in contrast with Li$_2$ZrCuO$_4$ ($\alpha$= 0.3 [@Drechsler07]) and LCO ($\alpha=0.33$ [@Lorenz09]) which are on the [*spiral*]{} side of the critical point. We compare the obtained $J$’s to a realistic 5-band extended Hubbard $pd$ model and L(S)DA+$U$ calculations, which are in good agreement. The resulting magnetic excitations calculated with exact diagonalization compare well to the ones obtained with LSWT, implying that the coherence of the elementary, spin-wave like, magnetic excitations are marginally affected by $\alpha$ and quantum fluctuations. However, the relatively large $J_1$ and $\alpha$ values obtained affect the thermodynamics [@Haertel11]. The vicinity to the critical point is probed by $\varepsilon=\alpha-\alpha_c$ strongly affects both the magnitude and de- or increasing $T$-dependence of the Zhang-Rice singlet (ZRS) excitation intensity for $\varepsilon > 0$ and $\varepsilon < 0$, respectively. This is manifest in resonant inelastic x-ray scattering (RIXS), EELS, and optics [@Malek08] measurements as we will show.
CYCO has edge-shared CuO$_{2}$ chains along the $a$-axis. The Cu$^{2+}$ spins are aligned FM along the $a$-axis (see Fig. 1). $ac$-planes with CuO$_2$ chains alternate along the $b$-axis with magnetically inactive cationic planes containing incommensurate and partially disordered CaY-chains which produce a non-ideal geometry in the CuO$_2$ chains. These mutual structural peculiarities might be responsible for the puzzling strong damping at large transferred neutron momenta [@Matsuda01; @Matsuda05] to be addressed elsewhere.
Our INS-study has been performed with a fixed final neutron energy of 14.7 meV on a 3-axis neutron spectrometer TAS-2 installed at the JRR-3 by the Japan Atomic Energy Agency. To analyze the dispersion of the magnetic excitations we adopt the model given in Ref. . Then, CYCO has the following main couplings $J(\mathbf{R})$, $\mathbf{R}\equiv(xa,yb,zc)$: NN and NNN couplings along the chain $J(1,0,0)\equiv J_{1},$ $J(2,0,0)\equiv J_{2}$, and the interchain coupling (IC) $J(0.5,0,0.5)\equiv J'_{ic},$ $J(1.5,0,0.5)\equiv J_{ic}$, $J(0,1,0)\equiv J_{b}$=-0.06 meV, and $J(0.5,0.5,0)\equiv J_{ab}$=-0.03 meV. For the small interplane FM couplings $J_{b}$ and $J_{ab}$ we adopt the values from Ref. . Their contribution to the inchain dispersion is negligible. Within the LSWT the magnetic excitations dispersion is given by Eq. (2) of Ref. : $
\omega ^{2}({\mathbf{q}})
%&
=
%&
A_{\mathbf{q}}^{2}-B_{\mathbf{q}}^{2},\label{eq:wq}
$ $
A_{\mathbf{q}}
%&
\equiv
%&
J_{\mathbf{q}}-J_{\mathbf{0}}+
\tilde{J}_{\mathbf{0}}-D, %\label{eq:eq}
\quad %\\
B_{\mathbf{q}}
%&
\equiv
%&
\tilde{J}_{\mathbf{q}},\label{eq:gq}
%\\\nonumber
%\vspace{-0.3cm}
%\end{equation}
$ where $J_{\mathbf{q}}=
(1/2)\sum_{\mathbf{r}}J_{\mathbf{r}}\exp\left(\imath\mathbf{q}\cdot\mathbf{r}\right)$ is the Fourier transform of intrasublattice interactions and analogously for the intersublattice interactions $\tilde{J}_{\mathbf{q}}$. The dispersion along $(0,0,L)$ shown in Fig. 3 (c) of Ref. depends only on $J_{s}=J'_{ic}+J_{ic}$. Its value, as well as the anisotropy parameter $D$, may be found from the INS data for $\mathbf{q}=(0,0,0),(0,0,1.5)$: $
J_{s}^{2} = \frac{1}{4}\left[\omega ^{2}(0,0,1.5)-\omega ^{2}(0,0,0)\right]
$, $
D = 2J_s-\omega (0,0,1.5).
$ Using $\omega (0,0,1.5)=5.03\pm0.03$, and $\omega (0,0,0)=1.63\pm0.01$ meV we obtain $J_{s}\approx$ 2.35 meV, and $D\approx-0.27$ meV [@rem1] which is very close to $J_{s}=2.24$ meV, and $D=-0.26$ meV found in Ref. .
![(Color online) Dispersion along three lines of the first Brillouin zone parallel to the $a$-axis, obtained from constant $\mathbf{q}$ scans. The LSWT-fit was refined only for the dispersion along (H,0,1.5) and it is shown by thin lines. []{data-label="FIG2"}](fig2.eps){width="0.82\columnwidth"}
The dispersion along the line $(H,0,1.5)$ depends only on the inchain $J$’s (if the tiny interplane $J_{ab}=-0.03$ meV is ignored). It reads $
\omega (\mathbf{q})= A_{\mathbf{q}}=J_{\mathbf{q}}-J_{\mathbf{0}}+
\omega (0,0,1.5).
$ These $J$’s may be accessed from the dispersion along this line with much higher precision than from the previously reported data along $(H,0,0)$ and we have [@rem1]: $$\begin{aligned}
J_{1} & = & -14.69\pm0.5\ (4\%)\,{\rm meV}\approx-170.4\,{\rm K},\\
J_{2} & = & 2.78\pm0.2(7.6\%)
\,{\rm meV}\approx32.2\,{\rm K}.\end{aligned}$$ The dispersions along $(H,0,0)$ (reported in Ref. ) and $(H,0,1.25)$ (given here) are affected by $J'_{ic}$ and $J_{ic}$. As we have determined only their sum, we adopt $J'_{ic}/J_{ic}=\tau \approx 4/9$ for the sake of concreteness (suggested by our band structure calculations). Notice that our $\tau$-value differs from 2 adopted in Ref. . It might be refined empirically, if one measures also along (H=1/6,K,L) for any K value. With the aim to detect quantum effects beyond the LSWT, we calculated the dynamical structure factor $S(\omega,q)$ using exact diagonalizations (see Fig. 3). In Fig. \[FIG2\] the INS data together with the refined new LSWT-fit are shown. $S(\omega,q)$ for our set and that of Ref. 1 are shown in Fig. 3. The peak positions always nicely follow the LSWT-curves, however our set gives a better description of the INS-data than that in Ref. (Fig. 4 therein) where the artificial double branching was ascribed to AFM IC. Indeed, it induces some intensity apart from the LSWT-curve, but these intensities are far too weak to be considered as a branch doubling. Notice the inflection point at $\pi/2$ for finite $\alpha $. The total dispersion width is given solely by 2$| J_1| $.
![(Color) Magnetic dynamical structure factor $S(\omega,q)$ for the $J$-set of Ref. (left) and for our set (right) from exact diagonalizations with $L=14\times 2$ and $15\times 2$ adopting $\tau=D=0$ and $J_{ic}=2.24$ meV, (see Ref. . Red line: dispersion from LSWT for the two parameter sets (see Fig. 2 and text). []{data-label="FIG3"}](FIG3.eps){width="0.76\columnwidth"}
As an application of our data, we consider the 1D-magnetic susceptibility $\chi(T)$ in the isotropic limit (see Fig. 4).
![(Color) Spin susceptibility $\chi(T)$ within the isotropic 1D $J_1$-$J_2$ model using the transfer matrix renormalization group method, $\chi_0 $ - background susceptibility. []{data-label="figchi"}](FIG4.eps){width="0.6\columnwidth"}
Despite the uncertainty caused by the unknown background susceptibility $\chi_0$, the relatively large value of $| J_1 | $ doesn’t allow to extract directly $\Theta_{CW}$ from a $1/\chi(T)$ plot using only data up to 300 K. Instead a much broader $T$-interval up to about 800 K would be required to reach the asymptotic high-$T$ limit necessary for a proper quasi-linear behavior. Alternatively, higher orders in the high-$T$ expansion can be applied [@Schmidt11]. Hence, the reported [*AFM*]{} values $\Theta_{CW}\sim -15$ K [@Yamaguchi99] or weak FM ones $\Theta_{CW}\sim +8$ K [@Kudo05] are rather artificial. Using the $J$’s from our INS-fits we predict instead a markedly larger [*FM*]{} value $$\Theta_{CW}=0.5(| J_1| -J_2) -J'_{ic}-J_{ic}-J_{ab}=+43.4\
\mbox{K}.$$ Since we found $\alpha < \alpha_c$, we readily predict the value of the 2D saturation field $H_s$ which is here determined solely by the total AFM IC like for LCO [@Nishimoto11]: $
gH_s=4(J'_{ic}+J_{ic})=\frac{2}{g}77.4\ \mbox{T}\approx 64.8\ \mbox{T},
$ refining an estimate of 70 T for $H \parallel b$ and $g=2.39$ from low-field magnetization data [@Kudo05]. Next we consider the magnetic moment in the ordered state at low $T$. Within the LSWT the reduction due-to quantum fluctuations is about 6.8% which yields 1.07$\mu_{\rm B}$ to be compared with the experimental value of 0.92$\pm 0.08 \mu_{\rm B}$ [@Fong99] which however is affected by the chemical reduction effect since about 0.22$\pm 0.04$ of the local moment resides on the O 2$p$ orbitals.
The exchange coupling strengths can also be determined by DFT+$U$ calculations. For this we used the full potential scheme FPLO [@koepernik99] (vers. fplo9.01) and performed super cell calculations for different collinear spin arrangements applying LDA and GGA functionals [@pw; @pbe]. The Coulomb repulsion $U_{3d}$ was varied in the physical relevant range from 5 to 8eV for a fixed $J_{3d}=1$eV. In our calculations the incommensurate crystal structure of CYCO can be treated only approximatively. Thus, we neglect (i) the modulation of the Cu-O distances within the CuO$_2$ chains and its buckling and (ii) the incommensurability of the CuO$_2$ and the CaY subsystems. In particular, the CuO$_2$ chains were treated as ideal planar chains reflecting an averaged Cu-O distance of 1.92 Å and a Cu-O-Cu bond angle of 94.5$^\circ$. Furthermore, we modelled the CaY layer by a Na layer to preserve the half filling of the system. The structure of the simplified model systems is given in Ref. . These structural simplifications allow a reliable modelling of $J_1$ yielding an FM value: $\sim$ -150 K [@supp]. In previous studies of closely related ESC [@li2zrcuo4; @linarit] for the effects of (i) chain buckling and (ii) the cation related crystal fields it was shown that $J_1$ is rather robust whereas $J_2$ is strongly reduced by a factor 2 to 3. Thus, our $-J_1
\sim 150~K$ is considered as a rather reliable lower estimate by about 10 to 20% with respect to the buckled chain geometry in CYCO. However, in view of the drastic dependence of $J_2$ on these parameters [@li2zrcuo4; @linarit], a derivation of a reliable value from the applied model structure is difficult. Thus, CYCO fits the general experience of a sizable FM $J_1$-value for ESC in contrast to the assignments of only a few K, proposed for LiVCuO$_4$ [@Enderle] and NaCu$_2$O$_2$ [@Capogna]. Such small $J_1$’s would put them in a region of strong quantum fluctuations, harboring the difficulty that the observed pitch angle cannot be described classically [@NishimotoEPL].
Whereas the vicinity of CYCO to the quantum critical point only weakly affects the dispersion and coherence of the elementary, spin-wave like, magnetic excitations, we will show that the amplitude to excite Zhang-Rice singlets (ZRS) at typical high-energies, probed by spectroscopic means depends strongly on the frustrating $J_2$. and temperature. We have also performed exact diagonalization calculations using an extended 5-band Cu 3$d$ O$2p$ Hubbard model for CYCO with a standard parameters [@remarkparameters]. To fit the INS-derived value of $J_2=$32 K, the $t_{px,px}=0.59$ eV has been slightly reduced as compared with LCO (0.84 eV, $J_2=$ 76 to 66 K) which simulates probably the deviations from the ideal chain geometry. Mapping the spin states of the 5-band Hubbard model onto a frustrated spin model, we obtain $J_1=-177.5$ K and $J_2=32.3$ K in full accord with the LSWT-analysis of the INS data given above. We stress that the value of $J_1$ is mainly determined by the direct FM exchange coupling $K_{pd}=65$ meV and not by the Hund’s rule coupling on the O sites $J_p=0.5\left( U_p-U_{p_xp_y}\right)
=0.6$ eV as often adopted. The significant value of $J_1$ is generic for ESC with a Cu-O-Cu bond angle $\varphi <
96^\circ$ at variance from the case of CuGeO$_3$ with $\varphi \approx 98^\circ$ causing an AFM $J_1$.
![(Color) $T$-dependent O-K RIXS-spectrum for $xx$-polarization from a Cu$_4$O$_{10}$ cluster within exact diagonalization using a Lorentzian broadening (half width $\Gamma=0.13$ eV). []{data-label="figoptzrs"}](Fig5.eps){width="0.85\columnwidth"}
As a modern spectroscopy, RIXS provides valuable insights into the correlated orbital and electronic structure (for a review see Ref. ). Therefore, we also studied the $T$-dependent O-K edge RIXS-spectra for a Cu$_4$O$_{10}$ cluster within exact diagonalization (see Fig. 5). We find a strong [*decrease*]{} of the intensity for the ZRS excitons with decreasing $T$, which is qualitatively in accord with general considerations for EELS and optics [@Malek08]. The not yet assigned feature observed for CYCO at 300 K near 527 eV (Fig. 5 in Ref. , counted from the 530 eV excitation energy) corresponds just to these excitations. It perfectly agrees with 2.95 eV obtained here. Notice that when one sets $t_{px,px}=0$, thereby strongly suppressing $J_2\propto t^2_{px,px}$, no ZRS exciton is observed even at 300 K. Hence, the frustrated FM differs qualitatively from a pure FM in its high-energy response.
To summarize, we have shown that Ca$_2$Y$_2$Cu$_5$O$_{10}$ is a frustrated quasi-1D ferromagnet close to criticality. This edge-sharing spin-chain compound has pronounced FM correlations in the presence of a sizable in-chain frustration. The main intensity of magnetic excitations in $S(\omega,q)$ is reasonably well described within LSWT. The signatures of the sizable in-chain frustration found here cause (i) a characteristic curvature of the in-chain dispersion of magnetic excitations and (ii) Zhang-Rice singlet features at $\sim 3$ eV that are strongly $T$-dependent and are detectable by spectroscopies. The spin alignment between the chains below $T_N$ is supported by a specific AFM inter-chain coupling which determines the saturation field. It causes interesting quantum effects that go beyond a linear spin wave description. These effects, and the interplay of $J'_{ic}$ and $J_{ic}$, provide an interesting problem not yet investigated in full detail. However, such quantum effects are neither strong enough to cause a breakdown of LSWT, nor to induce an additional branch of magnetic excitations, as suggested previously. Our results can further aid in the correct assignment of the frustration [@Enderle; @Capogna] in other ESC, including multiferroics, but especially for the complex chain-ladder system (La,Sr,Ca)$_{14}$Cu$_{24}$O$_{41}$ [@carter96] which also possibly contains frustrated FM CuO$_2$ chains as suggested by the edge-sharing geometry and sizable NNN transfer integrals [@schwing07] giving rise to significant AFM $J_2$’s. Its nonideal chains, as with CYCO, challenge one to look for more sophisticated but yet solvable theoretical models that include incommensurate and disorder effects.
We thank the DFG for financial support. In particular, the Emmy-Noether program is acknowledged for founding. We also acknowledge fruitful discussions with J. Richter, J. Geck, and V. Bisogni.
[99]{} M. Matsuda et al., Phys.Rev.B, **63**, 180403 (2001). M. Matsuda [*et al.*]{} J. Phys. Soc. Jpn. [**74**]{}, 1578 (2005). M. Boehm [*et al.*]{}, EPL [**43**]{}, 77 (1998). M. Matsuda [*et al.*]{} J. Phys. Soc. Jpn. [**68**]{}, 269 (1999). H.F. Fong [*et al.*]{}, Phys. Rev. B [**59**]{} 6873 (1999). W.E.A. Lorenz [*et al.*]{}, EPL [**88**]{} 37002 (2009). Y. Mizuno [*et al.*]{} Phys. Rev. B [**57**]{}, 5326 (1998). H.-J. Choi [*et al.*]{}, Low Temp. Phys., Parts A & B (Ed. Y. Takano) Book Ser.: AIP Conf. Proc. [**850**]{}, 1019 (2006). S.-L. Drechsler [*et al.*]{}, Phys. Rev. Lett. [**98**]{}, 077202 (2007). M. Härtel [*et al.*]{} Phys. Rev. B [**84**]{}, 104411 (2011). J. Málek [*et al.*]{}, Phys. Rev. B [**78**]{}, 060508(R) (2008). These $J$’s belong to a fit including both const. $E$ and $\mathbf q$-scans. A fit with const. $\mathbf q$ scans, only, yields close values: $J_1=-164.9$ K (5.95%), $J_2=31.1$ K for $J_3=0$ fixed and $D=-3.1$ K, i.e. $\alpha \approx 0.19$.
H.-J. Schmidt [*et al.*]{}, Phys. Rev. B [**84**]{}, 104443 (2011). H. Yamaguchi [*et al.*]{}, Physica C [**320**]{}, 167 (1999). K. Kudo [*et al.*]{}, Phys. Rev. B [**71**]{}, 104413 (2005). S. Nishimoto [*et al.*]{}, Phys. Rev. Lett. [**107**]{} 097201 (2011). K. Koepernik [*et al.*]{} Phys. Rev. B [**59**]{}, 1743 (1999). P. Perdew [*et al.*]{} Phys. Rev. B [**45**]{}, 13244 (1992). P. Perdew [*et al.*]{}, Phys. Rev. Lett. [**77**]{}, 3865 (1997).
See EPAPS Document No. \[\]. For more information on EPAPS, see http://www.aip.org/pubservs/epaps.html M. Schmitt [*et al.*]{}, Phys. Rev. B [**80**]{}, 205111 (2009). M. Schmitt [*et al.*]{} to be published.
M. Enderle [*et al.*]{}, EPL [****]{} (2005). L. Capogna [*et al.*]{}, Phys. Rev. B [**71**]{} 140402 (2005) S. Nishimoto [*et al.*]{}, EPL submitted (arXiv:1105.2810v2)
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We used $U_d=8.5$ eV, $U_p=4.1$ eV, the interorbital onsite Coulomb repulsion on O sites $U_{p_xp_y}=2.9$ eV, charge transfer energies $\Delta_{p_xd}=3.5$ eV, and $\Delta_{p_yd}=3.8$ eV (close to Ref. with an isotropic value $\Delta_{pd}=3.8$ eV). We further include the intersite Coulomb interaction $V_{pd}=0.5$ eV neglected in Ref. . This seems to be the main reason for the difference of the ZRS-exciton energies, upshifted by $\sim$ 0.3 eV compared to Ref. .
S.A. Carter [*et al.*]{}, Phys. Rev. Lett. [**77**]{}, 1378 (1996). U. Schwingenschlögl [*et al.*]{} Eur. Phys. J. B [**55**]{}, 43 (2007).
**EPAPS supplementary online material:**
**‘" Ca$_{2}$Y$_{2}$Cu$_{5}$O$_{10}$: the first frustrated quasi-1D ferromagnet close to criticality ‘"**
R.O. Kuzian, S. Nishimoto, S.-L. Drechsler, J. Málek, J. van den Brink\
IFW Dresden, P.O. Box 270116, D-01171 Dresden, Germany\
M. Schmitt, and H. Rosner\
Max-Planck Institute for Chemical Physics of Solids, 01187 Dresden, Dresden, Germany\
M. Matsuda,\
Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA\
K. Oka, H. Yamaguchi and T. Ito\
AIST, Tsukuba, Ibaraki 305-8562, Japan\
[In the present EPAPS supplementary online material we present (i) exact diagonalization studies of the dynamical structure factor for coupled chains with various interchain couplings of the type as realized in CYCO. (ii) We show our prediction and details of the calculated optical conductivity $\sigma(\omega)$ (iii) We demonstrate the influence of interchain coupling and spin anisotropy on the magnetic susceptibility. (iv) we provide the reader with further details of the LSDA+$U$ and GGA+$U$ calculations. (v) we illustrate the influence of inchain frustration on the dispersion of magnetic excitations focussing on its curvature behavior and in particular on the positions of inflection points within linear spin-wave theory.]{}\
The spin-Hamiltonian with uniaxial anisotropy and CYCO geometry {#the-spin-hamiltonian-with-uniaxial-anisotropy-and-cyco-geometry .unnumbered}
---------------------------------------------------------------
In general, spin waves are the quanta of small oscillations of spins around the classical ground state of the spin-Hamiltonian. So, for the same Hamiltonian, the form of the dispersion may be radically different for different values of parameters, since the classical ground state may be different from the quantum one. The spin-Hamiltonian for CYCO reads $$\begin{aligned}
\hat{H} & = & \hat{H}_{A}+\hat{H}_{B}+\hat{H}_{AB}\label{H}\\
\hat{H}_{A(B)} & = & \frac{1}{2}\sum_{\mathbf{m}\in A(B)}
\left\{ \sum_{\mathbf{r}}
\left[J_{\mathbf{r}}^{z}\hat{S}_{\mathbf{m}}^{z}\hat{S}_{\mathbf{m}+\mathbf{r}}^{z}
+J_{\mathbf{r}}\hat{S}_{\mathbf{m}}^{+}\hat{S}_{\mathbf{m}+\mathbf{r}}^{-}
\right]\right. \nonumber \\
& + & \left.\sum_{\mathbf{R}}
\left[J_{\mathbf{R}}^{z}\hat{S}_{\mathbf{m}}^{z}\hat{S}_{\mathbf{m}+\mathbf{R}}^{z}
+J_{\mathbf{R}}\hat{S}_{\mathbf{m}}^{+}\hat{S}_{\mathbf{m}+\mathbf{R}}^{-}
\right]\right\} \label{eq:HA} \\
\hat{H}_{AB} & = & \sum_{\mathbf{m}\in A}\sum_{\mathbf{f}}
\left[\tilde{J}_{\mathbf{f}}^{z}\hat{S}_{\mathbf{m}}^{z}
\hat{S}_{\mathbf{m}+\mathbf{f}}^{z}
\right. \nonumber \\
& + & \left.
\frac{\tilde{J}_{\mathbf{f}}}{2}\left(\hat{S}_{\mathbf{m}}^{+}
\hat{S}_{\mathbf{m}+\mathbf{f}}^{-}+
\hat{S}_{\mathbf{m}}^{-}\hat{S}_{\mathbf{m}+\mathbf{f}}^{+}
\right)\right]\label{eq:HAB}\end{aligned}$$ where $\mathbf{m}$ enumerates the sites in one sublattice, ($\mathbf{r}=\pm n\mathbf{a},\: n=1,2\ldots$) determines the neighboring sites within the chain, $\mathbf{a}$ being the lattice vector along the chain, vector $\mathbf{f}$ connects the sites of different chains in the $\mathbf{ac}$ plane, $\mathbf{R}$ connets the sites in different planes, but in the same sublattice in CYCO. We have allowed for an uniaxial anisotropy of the exchange interactions.
Linear spin-wave theory
-----------------------
For the collinear AFM the classical ground state is the Neél state, the spins on the $A$ sublattice are directed up, and down in the $B$ sublattice. We introduce two different sets of spin-deviation operators $$\begin{aligned}
\hat{S}_{\mathbf{m}\in A}^{+} & \equiv & \sqrt{2S}f_{\mathbf{m}}(S)a,
\ \hat{S}^{-}\equiv\sqrt{2S}a^{\dagger}f_{\mathbf{m}}(S),\label{SA}\\
\hat{S}^{z} & \equiv & S-\hat{n}_{\mathbf{m}},\
\hat{n}_{\mathbf{m}\in A} = a_{\mathbf{m}}^{\dagger}a_{\mathbf{m}}\label{nmA}\\
\hat{S}_{\mathbf{m}\in B}^{-} & \equiv & \sqrt{2S}f_{\mathbf{m}}(S)b,
\ \hat{S}^{+}\equiv\sqrt{2S}b^{\dagger}f_{\mathbf{m}}(S),\label{eq:SB}\\
\hat{S}^{z} &\equiv & -S+\hat{n}_{\mathbf{m}},\
\hat{n}_{\mathbf{m}\in B} = b_{\mathbf{m}}^{\dagger}b_{\mathbf{m}},\label{nmB}\\
f_{\mathbf{m}}(S) & = & \sqrt{1-\hat{n}_{\mathbf{m}}/2S} \\
&=& 1-\left(\hat{n}_{\mathbf{m}}/4S\right)-
\frac{1}{32}\left(\hat{n}_{\mathbf{m}}/S\right)^{2}+\cdots\label{eq:fm}\end{aligned}$$ The Neél state (AFM ordering of FM chains in the $\mathbf{ac}$ plane) is the vacuum state for the operators $b\left|\mathrm{Neel}\right\rangle =0$. So, the operator $b(a)$ annihilates the spin deviation from the Neél order (which means a spin-flip for $s=1/2$) on the sublattice $B(A)$. The Hamiltonian (\[H\]) can be rewritten as $$\begin{aligned}
\hat{H} & = & \hat{H}_{0}+\hat{H}_{int},\label{Hviab}\\
\hat{H}_{0} & = & \sum_{\mathbf{q}}
\left[A_{\mathbf{q}}\left(a_{\mathbf{q}}^{\dagger}a_{\mathbf{q}}
+ b_{\mathbf{q}}^{\dagger}b_{\mathbf{q}}\right) \right. \nonumber \\
&+& \left. B_{\mathbf{q}}\left(a_{\mathbf{q}}b_{-\mathbf{q}}
+a_{\mathbf{q}}^{\dagger}b_{-\mathbf{q}}^{\dagger}\right)\right],
\label{H0}\\
A_{\mathbf{q}} & \equiv &
S\left[\sum_{\mathbf{r},\mathbf{R}}J_{\mathbf{r}}\exp\left(\imath\mathbf{qr}\right)-
\sum_{\mathbf{r},\mathbf{R}}J_{\mathbf{r}}^{z}+
\sum_{\mathbf{f}}\tilde{J}_{\mathbf{f}}^{z}\right],\label{eq:eq}\\
B_{\mathbf{q}} & \equiv &
S\sum_{\mathbf{f}}I_{\mathbf{f}}\exp\left(\imath\mathbf{qf}\right) \quad .
\label{eq:gq}\end{aligned}$$ The Fourier transform of e.g. $b_{\mathbf{q}}$ reads $b_{\mathbf{q}}=\sqrt{2/N}\sum_{\mathbf{m}\in B}\exp(-\imath\mathbf{qm})b_{\mathbf{m}}$; $N$ denotes the total number of sites. The transverse part of $\hat{H}$ (\[H\]) defines the one-particle hoppings in $\hat{H}_{0}$ (\[H0\]), the Ising part contributes on-site energy values. The terms, which contain more than two spin-wave operators enter $\hat{H}_{int}$. The magnons $\alpha_{\mathbf{q}},\beta_{\mathbf{q}}$ in an AFM are introduced as a mean-field solution of Eq. (\[Hviab\]) [@Oguchi60], which *neglects* $\hat{H}_{int}$. $$\begin{aligned}
\left[\alpha_{\mathbf{q}},\hat{H}\right] & \approx &
\omega_{\mathbf{q}}\alpha_{\mathbf{q}},\;
\left[\beta_{\mathbf{q}},\hat{H}\right]\approx
\omega_{\mathbf{q}}\beta_{\mathbf{q}}\label{eq:mf}\\
\omega_{\mathbf{q}} & = & \sqrt{A_{\mathbf{q}}^{2}-
B_{\mathbf{q}}^{2}},\label{eq:wSW}\\
\alpha_{\mathbf{q}} & = & \cosh\theta_{\mathbf{q}}a_{\mathbf{q}}+
\sinh\theta_{\mathbf{q}}b_{-\mathbf{q}}^{\dagger},\label{eq:uv}\\
\beta_{\mathbf{q}} & = & \cosh\theta_{\mathbf{q}}b_{-\mathbf{q}}+
\sinh\theta_{\mathbf{q}}a_{\mathbf{q}}^{\dagger}\nonumber \\
\tanh2\theta_{\mathbf{q}} & = & B_{\mathbf{q}}/A_{\mathbf{q}}.\nonumber \end{aligned}$$ Now we specify the expressions for $A_{\mathbf{q}},B_{\mathbf{q}}$, which result from the geometry of CYCO and the spin value $s=1/2$. In the summations over $\mathbf{r},\mathbf{R},\mathbf{f}$ we retain only the following terms. For the in-chain exchanges we retain $J_{1},\: J_{2}$, which correspond to $\mathbf{r}=\mathbf{a},\:2\mathbf{a},$ respectively. For the interchain interactions we retain $J_{ic}^{\prime},\: J_{ic},\: J_{b},\: J_{ab}$, which correspond to $\mathbf{f}^{\prime}=\left(\mathbf{a}+\mathbf{c}\right)/2$, $\mathbf{f}=\left(3\mathbf{a}+\mathbf{c}\right)/2$, $\mathbf{R}_{b}=\mathbf{b}$, $\mathbf{R}_{ab}=\left(\mathbf{a}+\mathbf{b}\right)/2$. Then we find $$\begin{aligned}
A_{\mathbf{q}} & = &
J_{1}\left(\cos q_{a}a-1\right)+J_{2,}\left(\cos2q_{a}a-1\right) \nonumber \\
&+& J_{b}\left(\cos q_{b}b-1\right)+\nonumber \\
& + & 2J_{ab}\left(\cos\frac{q_{a}a}{2}\cos\frac{q_{b}b}{2}-1\right)+
2
J_s-D,\label{eq:eqli}\\
B_{\mathbf{q}} & = & 2\left[J_{ic}^{\prime}\cos\frac{q_{a}a}{2}
+J_{ic}\cos\frac{3q_{a}a}{2}\right]\cos\left(\frac{q_{c}c}{2}
\right).\label{eq:gqli}\end{aligned}$$ Note that interplane interactions enter $A_{\mathbf{q}}$ as they represent intra-sublattice interactions. The value of the gap at $\mathbf{q}=0$ is $$\Delta = \sqrt{A_{0}^{2}-B_{0}^{2}},$$ the anisotropy enters the dispersion via the integral value $D$, which is defined as $$A_{0}-B_{0}\equiv D=\sum_{\mathbf{f}}\left(\tilde{J}_{\mathbf{f}}^{z}-
\tilde{J}_{\mathbf{f}}\right)-\sum_{\mathbf{r},\mathbf{R}}
\left(J_{\mathbf{r}}^{z}-J_{\mathbf{r}}\right),\label{eq:Ddef}$$ and is related to the gap value by the relations $$\begin{aligned}
\Delta & = & \sqrt{D\left(D+2B_{0}\right)},\nonumber \\
D & = & \left(\sqrt{B_{0}^{2}+\Delta^{2}}-B_{0}\right).\label{eq:Dval}\end{aligned}$$
Dispersion of magnetic excitations and inchain frustration {#dispersion-of-magnetic-excitations-and-inchain-frustration .unnumbered}
----------------------------------------------------------
Ignoring tiny interactions like $J_b$ for $qc=2(m+1)\pi$, $m= 0,1,2, ...$ the dispersion along (H,0,1.5) within the LSWT reads (see Eq. 2 in Ref. ): $$\begin{aligned}
\frac{\omega}{\mid J_1 \mid}&=& 1-\cos x +\alpha \left( \cos 2x -1\right) + \nonumber \\
&& 2\beta +\gamma \left( \cos 3x -1 \right) + ..., \ ., \nonumber \\\end{aligned}$$ where $x=qa$, $\gamma= J_3/\mid J_1 \mid $ denotes the third neighbor inchain coupling. The dimensionless interchain coupling constant slightly “renormalized” by the anisotropy parameter $D$: $\beta_D
=(J_s-0.5D)/\mid J_1 \mid \approx 0.17$ has been omitted in Fig. S1 since it doesn’t affect the curvature
![(Color) The influence of the frustration $\alpha$ on the dispersion of magnetic excitation for a direction not affected by the interchain coupling. e.g. (H,0,1.5) (see Fig. 2 of the main text). Uper: position of the inflection point. Blue curve Eq. (S25). lower: the full dispersion for various $\alpha$-values. []{data-label="frust1"}](frust1.eps){width="0.75\columnwidth"}
![(Color) The influence of the frustration $\alpha$ on the dispersion of magnetic excitation for a direction affected by the interchain coupling (IC) e.g. (H,0,0) (see Figs. 2 and 3 of the main text). Black curves: no inchain frustration, red curves: inchain frustration included. Broken curves curves: $J_{ic}$ included, but we adopt $J'_{ic}$=0 for the sake of simplicity.[]{data-label="frust2"}](frust2.eps){width="0.7\columnwidth"}
of the dispersion curve in the case under consideration. According to our results of the mapping of the 5-band extended Hubbard $pd$-model on a frustrated spin model $\gamma$ is usually FM but very small $\sim 10^{-2}$ to $10^{-3}$. In many cases it can be therefore neglected. The whole width of the dispersion $W$ is given by the odd Fourier coefficients, only: $ W = 2 (\mid J_1 \mid +J_3 + ...)$. In the limit $x \ll 1$ Eq. (S22) can be rewritten as $$\frac{\omega}{\mid J_1 \mid}-2\beta \approx
\frac{1-4\alpha -9\gamma}{2}x^2
-\frac{1-16\alpha -81\gamma}{24}x^4 +O(x^6) \ .$$ Thus, the inchain frustration $\alpha >0$ reduces the quadratic term (which vanishes approaching the critical point, see Fig. S1) and strongly affects the quartic term which is [*negative*]{} for the unfrustrated FM ($\alpha=0$) but changes it sign for a sizable value of $\alpha > (1-81\gamma )/16 \approx $ 1/16 to 1/8, well realized for CYCO with $\alpha=0.19$. Thus, a [*positive*]{} sign of the quartic term gives direct evidence for a pronounced frustrated ferromagnet. Just the opposite behavior is realized near the maximum of the dispersion near the BZ boundary. There the expansion in terms of $\varepsilon =\pi -qa$ reads $$\begin{aligned}
\frac{\omega}{\mid J_1 \mid}-2\beta \approx
2(1-\gamma) -
\frac{1+4\alpha -9\gamma}{2}x^2+\nonumber \\
+\frac{1+16\alpha -81\gamma}{24}x^4
+O(x^6), \end{aligned}$$ the inchain frustration $\alpha$ enhances the negative quadratic term (see also Fig. S1) In the case $J_3 = 0 $ approximately fulfilled for almost all ELC, the inflection point $q_{ip}$ of the dispersion curve (Eq. (S22)) is given by $$q_{ip}a=\cos ^{-1}\left( \frac{1-\sqrt{1+128\alpha^2}}{16 \alpha } \right) .$$ It provides also a clear measure of the inchain frustration $\alpha =J_2/\mid J_1 \mid$ present in the system (see Fig. S1). To summarize, analyzing the full dispersion curve along the line (H,0,1.5) the presence of the inchain frustration $\alpha$ is easily detected.
However, we admit that in many cases where the full dispersion curve is not available, a fit of a significant part of the total curve might provide a more precise value of $\alpha$ or $J_2$. In such cases the extraction of $\alpha$ from the $T$-dependence of the Zhang-Rice exciton intensity shown in the main text might provide a reasonable alternative.
Now we briefly consider the (H,0,0) scattering direction where the interchain coupling (IC) is involved. Compared with the former “1D”-case we show in Fig. S2 that the IC has a sizable influence on the shape and on the curvature of the dispersion. Note that a small quadratic term exists only for finite anisotropy $-D>0$. $$\frac{\omega}{\mid J_1 \mid}=\sqrt{
\Delta^2_0 +\frac{1}{2} [ (1-4\alpha ) \beta_D
+2\beta ( \beta +8\beta_2 ) ] x^2 +O(x^4) },$$ where $\delta=-D/|J_1|$, $\beta=J_s/\mid J_1\mid$, and $\beta_2=J_{ic}/|\mid J_1\mid $. In the isotropic limit $D$ or $\delta=0$, i.e. when the spin gap $\Delta_0=\sqrt{4\delta\beta +\delta^2}$ vanishes, the dispersion becomes linear (compare Fig. 3 (left) in the main text).
Like in the former case one observes that the in-chain frustration $\alpha$ strongly enhances the higher order terms near the $\Gamma$ point. But the interchain coupling reduces this enhancement. Under such circumstances the extraction of the IC and $\alpha$ from a a scattering direction affected by both types of interactions is unconvenient.
Magnetic excitations within the model of coupled chains {#magnetic-excitations-within-the-model-of-coupled-chains .unnumbered}
-------------------------------------------------------
We fix the sum of the two interchain couplings $J_s=J'_{ic}+J_{ic}=2.241$ meV at the value derived from the INS data and change only their ratio $\tau$. First we switch off the frustrating NNN AFM inchain coupling $J_2$ and adopt for $J_1$ the value suggested in Ref. (see Fig. 5) One observes roughly two kinds of dispersing peaks: one with a dominant intensity well discribed by the LSWT and second one which much less intensity. The latter type of curves has been interpreted as “evidence” for an additional anomalous second ME branch in Refs. . A similar behavior is observed for all cases with a dominant NN interchain coupling $J'_
{ic}$. In the opposite case of a dominant $J_{ic}$ that second curve almost invisible. For our parameter set that minority peaks are much less pronounced. Thus we conclude, that this quantum effect depends on the details of the two main AFM interchain couplings and becomes weaker with increasing $\mid J_1 \mid$ in contrast to what has been suggested previously. Anyhow, a systematic study including the examination of finite size scaling is left for a future investigation since it is of less relevance for CYCO.
![The dynamical structure factor from exact diagonalizations of two coupled chains with 14 sites for each chain at various energies $\omega$ and momenta $q$. Red line: projected maximum position of $S(\omega, q)$ very close to the LSWT result. Blue dashed line: intensity from the weak minority peaks and the inchain parameters proposed in Ref. 1. []{data-label="fig1"}](fig_supl_01.eps){width="0.7\columnwidth"}
![(Color) The same as in Fig. S3 for the new parameter set given in the main text. []{data-label="fig1"}](fig_supl_02.eps){width="0.7\columnwidth"}
The reduction of the magnetic moment at $T=0$ {#the-reduction-of-the-magnetic-moment-at-t0 .unnumbered}
---------------------------------------------
In contrast to the case of FM ordering, the vacuum state for the antiferro-magnons $\alpha_{\mathbf{q}},\beta_{\mathbf{q}}$ (\[eq:uv\]) does not coincide with the classical Neel ground state $\left|\mathrm{Neel}\right\rangle \neq\left|0\right\rangle $. Thus, the vacuum $\left|0\right\rangle $ contains an finite number of spin deviations (\[nmA\]),(\[nmB\]) even at $T=0$, $\left\langle 0\right|\hat{n}_{\mathbf{m}}\left|0\right\rangle \neq0$. Expressing the spin deviation operators $a_{\mathbf{q}}$ via $\alpha_{\mathbf{q}},\beta_{\mathbf{q}}$ (\[eq:uv\]), we obtain $$\begin{aligned}
& & \left\langle 0\right|a_{\mathbf{m}}^{\dagger}a_{\mathbf{m}}
\left|0\right\rangle =
\frac{2}{N}\sum_{\mathbf{q}}
\left\langle 0\right|a_{\mathbf{q}}^{\dagger}a_{\mathbf{q}}
\left|0\right\rangle \nonumber \\
&=& \frac{2}{N}\sum_{\mathbf{q}}\sinh^{2}\theta_{\mathbf{q}}
\left\langle 0\right|\beta_{\mathbf{q}}\beta_{\mathbf{q}}^{\dagger}
\left|0\right\rangle =
\frac{1}{N}\sum_{\mathbf{q}}
\left(\frac{A_{\mathbf{q}}}{\omega_{\mathbf{q}}}-1\right). \nonumber \\\end{aligned}$$ Note, that in our geometry, the summation over $\mathbf{q}$ runs over the magnetic Brillouin zone (BZ) which in the present case coincides with the lattice BZ $\frac{2\pi}{a}\times\frac{2\pi}{b}\times\frac{2\pi}{c}$. Finally, for the sublattice magnetization at $T=0$, we have $$\left\langle 0\right|\hat{S}_{\mathbf{m}}^{z}\left|0\right\rangle =
S\left(1-\frac{\left\langle 0\right|\hat{n}_{\mathbf{m}}
\left|0\right\rangle }{S}\right). \label{m}$$
The optical conductivity within the 5-band extended Hubbard model {#the-optical-conductivity-within-the-5-band-extended-hubbard-model .unnumbered}
------------------------------------------------------------------
Focusing on the Zhang-Rice exciton, we note that a similar picture as in the RIXS spectra is shown in the predicted optical conductivity and EELS spectra (not shown here). Some insight in the corresponding transitions can be gained from the zoomed figure with an artificial small broadening to resolve these transitions (see Fig. S5).
site x/a y/b z/c
------ ----- ----- -------
Cu 3/8 0 1/4
Cu 7/8 0 1/4
Cu 1/8 1/2 1/4
Cu 5/8 1/2 1/4
O 3/8 0 .6270
O 7/8 0 .6270
O 1/8 1/2 .6270
O 5/8 1/2 .6270
O 3/8 0 .8730
O 7/8 0 .8730
O 1/8 1/2 .8730
O 5/8 1/2 .8730
Na 1/8 1/4 0
Na 5/8 1/4 0
Na 1/8 1/4 1/2
Na 5/8 1/4 1/2
: \[struc\] Crystal structure of the [ *commensurate* ]{} approximate effective Na$_2$CuO$_2$ compound for DFT+$U$ calculation (enlarged unit cell and reduced symmetry to allow different spin configuration). Space group P2/M (SG 10), $a=5.6306~$Å, $b=6.286~$Å and $c=10.5775~$Å.
Additional information on the L(S)DA+$U$ calculations {#additional-information-on-the-lsdau-calculations .unnumbered}
------------------------------------------------------
To model the mutually incommensurate crystal structure of CYCO we neglect (i) the modulation of the Cu-O distances within the CuO$_2$ chains and (ii) the incommensurability of the CuO$_2$ and CaY subsystems as a good approximate to estimate the NN exchange along the CuO$_2$ chains. The crystal structure data for this simplified model structure are given in Tab. S2. To calculate the NN exchange $J_1$ for the approximated crystal structure we constructed a super cell by the doubling of the unit cell and the reduction of its symmetry to allow for different spin configurations (see Tab. S1). The obtained FM NN exchange $J_1$ depending on the Coulomb repulsion $U_{3d}$ and the specific functional are depicted in Fig. \[S5\]. $J_1$ depends only weakly on these parameters. The given NN exchange $J_1$=-150K is the average between the LDA and GGA result at $U_{3d}$=6.5eV.
site x/a y/b z/c
------ ----- ----- -------
Cu 0 0 0
O 0 0 0.623
Na 1/2 1/4 1/4
: \[struc\] Crystal structure of the [*commensurate*]{} approximate effective Na$_2$CuO$_2$ structure as a starting model for CYCO. Space group FMMM (SG 69), $a=2.8153$ Å, $b=6.286$ Å and $c=10.5775$ Å. []{data-label="struc2"}
Aspects of the magnetic susceptibility {#aspects-of-the-magnetic-susceptibility .unnumbered}
---------------------------------------
In addition to 1D susceptibilities, for the isotropic as well as the easy-axis anisotropic case calculated using the transfer matrix renormaliztion group theory method, also the 3D case has been examined treating the adopted isotropic interchain coupling (IC) within the RPA (random phase approximation) (see Fig. S7 and Eq. (S29)). $$\chi_{\mbox{\tiny 3D}}(T)\approx
\frac{
\chi_{\mbox{\tiny 1D}}(T)}{1+k\chi_{\mbox{\tiny 1D}}(T)} \quad .$$
![ (Color) Left: spin susceptibility $\chi(T)$ fitted within the 1D $J_1$-$J_2$ model supplemented with isotropic AFM IC in 2D treated within the RPA . The latter is measured by the parameter $k=2\left( J'_{ic }+J_{ic} \right)/\mid J_1\mid $. Right: The same as left for an anisotropic easy-axis inchain coupling $J_1$. []{data-label="figchi1"}](chiaca018persite150.eps "fig:"){width="0.47\columnwidth"} ![ (Color) Left: spin susceptibility $\chi(T)$ fitted within the 1D $J_1$-$J_2$ model supplemented with isotropic AFM IC in 2D treated within the RPA . The latter is measured by the parameter $k=2\left( J'_{ic }+J_{ic} \right)/\mid J_1\mid $. Right: The same as left for an anisotropic easy-axis inchain coupling $J_1$. []{data-label="figchi1"}](chiacapersite.eps "fig:"){width="0.47\columnwidth"}
$[1]$ M. Matsuda [*et al.*]{}, Phys. Rev. B, [**63**]{}, 180403 (2001).\
$[2]$ T. Oguchi, Phys. Rev. [**117**]{}, 117 (1960).\
[^1]: Corr. author, E-mail: s.l.drechsler@ifw-dresden.de
|
---
abstract: 'We use 3D hydrodynamical numerical simulations and show that jittering bipolar jets that power core-collapse supernova (CCSN) explosions channel further accretion onto the newly born neutron star (NS) such that consecutive bipolar jets tend to be launched in the same plane as the first two bipolar jet episodes. In the jittering-jets model the explosion of CCSNe is powered by jittering jets launched by an intermittent accretion disk formed by accreted gas having a stochastic angular momentum. The first two bipolar jets episodes eject mass mainly from the plane defined by the two bipolar axes. Accretion then proceeds from the two opposite directions normal to that plane. Such a flow has an angular momentum in the direction of the same plane. If the gas forms an accretion disk, the jets will be launched in more or less the same plane as the one defined by the jets of the first two launching episodes. The outflow from the core of the star might have a higher mass flux in the plane define by the jets. In giant stellar progenitors we don’t expect this planar morphology to survive as the massive hydrogen envelope will tend to make the explosion more spherical. In SNe types Ib and Ic, where there is no massive envelope, the planar morphology might have an imprint on the supernova remnant. We speculate that planar jittering-jets are behind the morphology of the Cassiopeia A supernova remnant.'
author:
- Oded Papish and Noam Soker
title: 'A PLANAR JITTERING-JETS PATTERN IN CORE COLLAPSE SUPERNOVA EXPLOSIONS'
---
INTRODUCTION {#sec:intro}
============
One class of core collapse supernova (CCSN) explosion models is based on neutrino [@Colgate1966], mainly the delayed neutrino mechanism (e.g., @bethe1985 [@Burrows1985; @Burrows1995; @Fryer2002; @Ott2008; @Marek2009; @Nordhaus2010; @Kuroda2012; @Hanke2012; @Janka2012; @Bruenn2013]). However, recent 3D numerical studies have shown that the desired explosions are harder to achieve [@Couch2013; @Jankaetal2013; @Takiwakietal2013] than what 2D numerical simulations had suggested (for a summary of problems of the delayed neutrino mechanism see @Papishetal2014). The problems of the delayed-neutrino mechanism can be overcome if there is a strong wind, either from an accretion disk [@kohri2005] or from the newly born neutron star (NS). Such a wind is not part of the delayed-neutrino mechanism, and most researchers consider this wind to have a limited contribution to the explosion.
Another class of explosion mechanisms is the jittering-jet scenario [@Soker2010; @Papish2011; @Papish2012a; @Papish2012b; @PapishSoker2014; @GilkisSoker2013]. Processes for CCSN explosion by jets were considered before the development of the jittering-jet scenario (e.g. @LeBlanc1970 [@Meier1976; @Bisnovatyi1976; @Khokhlov1999; @MacFadyen2001; @Hoflich2001; @Woosley2005; @Burrows2007; @Couch2009; @Couch2011; @Lazzati2011]). However, most of these MHD models require a rapidly spinning core before collapse starts, and hence are limited to a small fraction of all CCSNe. The jittering-jet scenario posits that [*all CCSNe are exploded by jets*]{}. Recent observations (e.g. @Milisavljevic2013 [@Lopez2013; @Ellerbroek2013]) show indeed that jets might have a much more general role in CCSNe than what is expected in the neutrino-driven mechanisms and mechanisms that require rapidly rotating cores.
In the jittering-jets scenario the sources of the angular momentum for disk formation are the convective regions in the core [@GilkisSoker2013] and instabilities in the shocked region of the collapsing core, e.g., neutrino-driven convection or the standing accretion shock instability (SASI). Recent 3D numerical simulations show indeed that neutrino-driven convection and SASI are well developed in the first second after core bounce [@Hankeetal2013; @Takiwakietal2013] and the unstable spiral modes of the SASI can amplify magnetic fields [@Endeveetal2012]. The spiral modes with the amplification of magnetic fields build the ingredients necessary for jets’ launching.
When the average specific angular momentum of the matter in the pre-collapse core is small relative to the amplitude of the specific angular momentum of these instabilities, intermittent jets-launching episodes with random directions occur. The two launching axes of the first two launching episodes define a plane. Using the FLASH numerical code we set a numerical study of the accretion pattern that is likely to be formed after the first two launching episodes. The code and numerical set-up for the 3D simulations are described in section \[sec:setup\]. The accretion pattern following two jets-launching episodes, followed by a third episode, is described in section \[sec:accretion\]. Our summery is in section \[sec:summary\].
NUMERICAL SETUP {#sec:setup}
===============
We study the accretion structure that result from multiple jets-launching episodes using the [flash]{} gasdynamical numerical code version 4.2 [@Fryxell2000]. The widely used [flash]{} code is a publicly available code for supersonic flow suitable for astrophysical applications.
The simulations are done using the split PPM solver of [flash]{}. We use 3D Cartesian coordinates with an adaptive mesh refinement (AMR) grid. [[[[ Fig. \[fig:res\] shows the resolution in our simulations as a function of distance from the center. In the inner part the cells size is 10 km, increasing with radius.]{} ]{}]{}]{}
[[[[We treat the spherical inner region of up to $100 \km$ from the center as a hole.]{} ]{}]{}]{} This means that we are not simulating the NS itself, nor the assumed accretion disk. The boundary condition at the edge of the hole is inflow only, meaning the velocity cannot be positive in the radial direction unless we inject a jet at that specific zone.
![ [[[ [The dependence of the grid cell size on the distance from the center of the simulation grid. The thin line represent the simulation used to check the convergence of our results.]{} ]{}]{}]{} []{data-label="fig:res"}](resolution){width="80.00000%"}
[[[ [ To check the sensitivity of our simulations to the resolution used we run a test case with higher resolution (see Fig. \[fig:res\]). A compression with the higher resolution run, that will be presented in section \[subsec:res\] (Figs. \[fig:acc\], \[fig:high\]), shows that the low and high resolutions runs give results that are within $\sim 10 \%$ from each other. ]{} ]{}]{}]{}
We start the simulations after the core has collapsed and bounced back. The initial conditions are taken for a $15 M_\odot$ model from the 1D simulations of [@Liebend2005] at a time of $t \simeq 0.2 \s$ after bounce. We map their results into 3D, including the chemical composition. [[[[ We did not included nuclear reactions in our calcualtions. ]{} ]{}]{}]{} We set outflow boundary conditions at the exteriors of the simulation’s domain.
In each episode we inject one pair of two opposite jets (bipolar jets). All jets’ axes are in one plane, that we take to be the $y=0$ plane of the numerical Cartesian grid. We take the $z$ axis of the numerical grid to be at $40^\circ$ to the direction of the symmetry axis of the first jets pair. The $n$ launching episode results in two opposite jets at some angle $\theta_n$ from the direction of the first jets. All jets have an initial conical shape, with a half opening angle of $10^\circ$.
ACCRETION PATTERN {#sec:accretion}
=================
Simulated Cases {#cases}
---------------
We start each simulation by injecting two opposite jets, the first jets’ launching episode. The second bipolar jets pair is injected either at an angle of $40^\circ$ or $70^\circ$ relative to the first episode. Each jet launching episode lasts $0.05 \s$, and the second episode is launched immediately after the end of the first episode. The different simulated cases are summarized in Table \[table\]. The direction of the jets launched in the first two episodes are presented in Fig. \[fig:grid\]. Note that in that presentation one of the jets in each pair is represented by two regions.
------------- ------------ ------------ -------------
Run $\theta_1$ $\theta_2$ $\theta_3$
Active($s$) $0-0.05$ $0.05-0.1$ $0.1-0.15$
A1 $0^\circ$ $40^\circ$ $80^\circ$
A2 $0^\circ$ $40^\circ$ $-40^\circ$
B1 $0^\circ$ $70^\circ$ $55^\circ$
B2 $0^\circ$ $70^\circ$ $110^\circ$
------------- ------------ ------------ -------------
: A1, A2, B1, and B2 are the three different simulated cases. The angles $\theta_{2}$ and $\theta_{3}$ are the angles of the jets in the second and third jets’ launching episodes relative to the direction of the first jets that are injected at $40^\circ$ from the $z$ axis. All jets are injected in the $y=0$ plane. Each episode lasts $0.05 \s$.[]{data-label="table"}
![The simulations grid in the Mollweide projection. Shown are the directions of the first and second episodes in the different Runs. In each episode two opposite jets are launched. One of the jets in each episode is represented by two halves in this diagram.[]{data-label="fig:grid"}](grid_all){width="80.00000%"}
First jet-launching episode {#first}
---------------------------
In Fig. \[fig:dens-temp-05\] we present the density, left panel, and the velocity and temperature, right panel, maps at the end of the first jets launching episode. The flow structure has the typical structure of jet-inflated bubbles (see review by @Sokeretal2013). Here we should note the following. ($i$) The jets shock material to high temperatures and densities, where nucleosynthesis takes place. These regions have high velocities and can have imprint on the distribution of different isotopes in the SN remnant at later times. This will not be studied here. ($ii$) The pre-jets flow is that of a collapsing core onto a newly formed NS. After the first jets launching episode accretion continues from directions near the plane perpendicular to the jets’ axis.
![Flow pattern in the $y=0$ plane at the end of the first jets launching episode ($t=0.05 \s$), common to all simulated cases. Left Panel: density, with a color coding in logarithmic scale and units of $\g \cm^{-3}$. Right Panel: temperature in log scale and in units of $\K$, and a velocity map. Velocity is proportional to the arrow length, with inset showing an arrow for $30,000 \km \s^{-1}$. []{data-label="fig:dens-temp-05"}](X-Y_t05deg40){width="\textwidth"}
Let us elaborate on the accretion pattern. In Fig. \[fig:one\_jets500\] we present the inflow mass flux on spheres of radii $500$ and $1000 \km$. We present the local mass inflow rate $\dot M_{\rm loc}$, defined as if the entire sphere would have the same inflow mass flux as in the given location, $$\dot M_{\rm loc}= 4 \pi r^2 \rho v_{\rm in},
\label{eq:phil}$$ where $\rho$ and $v_{\rm in}$ is the density and the radial inward velocity at the point. The first jets pair is able to penetrate thorough the inflowing matter to beyond $1000 \km$ [@PapishSoker2014]. As the jets gas is in outflow, it appears as white areas in Fig. \[fig:one\_jets500\] and the following similar figures.
![The inflow mass flux at times $t=0, 0.026,0.05 \s$, i.e., at the beginning, middle and end of the first jets-launching episode, given on spheres of radius $r= 500 \km$ (upper row) and $r= 1000 \km$ (lower row). Mass inflow rate at each point is calculated as if the entire sphere would have the same inflow mass flux as in the given point (eq. \[eq:phil\]). White areas are regions with outflow, i.e., the jets.[]{data-label="fig:one_jets500"}](t000-050deg40)
A bipolar accretion pattern {#bipolar}
---------------------------
We run two cases of a second jets-launching episode, in directions of $40^\circ$ (Run A) and $70^\circ$ (Run B) relative to the direction of the jets’ axis in the first episode. All jets’ axes are in the $y=0$ plane. The density maps, left panels, and the temperature maps with velocity arrows, right panels, of the two cases at the end of the second episode, $t=0.1 \s$, are presented in Fig. \[fig:global2s\]. All plots are in the $y=0$ plane. From these panels, two for each Run, we learn that the strongest inflow is from two opposite directions, through which we draw a dashed line on the density maps of the figure at angles of $\alpha_1 = 35^\circ$, and $\alpha_2 = 54^\circ$ respectively relative to the x-axis.
![Flow structure at the end of the second jets episode at $t=0.1\s$ in the $y=0$ plane, for Run A in the upper panels, and for Run B in the lower panels. Left Panels: density, with a color coding in log scale and units of $\g \cm^{-3}$. The dashed line indicate the intersect of the plane shown in Fig. \[fig:global2s\]. Right Panels: temperature in log scale in units of $\K$, and velocity arrows. Velocity is proportional to the arrow length, with inset showing an arrow for $30,000 \km \s^{-1}$. []{data-label="fig:global2s"}](X-Y_t1deg40-70){width="\textwidth"}
To identify the inflow pattern of the gas, most of which will be eventually accreted by the NS, we present in Fig. \[fig:NormalPlane\] the flow structure in a plane perpendicular to the $y=0$ plane, and cutting it along the dashed lines drawn in the density maps of Fig. \[fig:global2s\]. Note that the two planes for Run A and Run B are not identical. The left panels of Fig. \[fig:NormalPlane\] present the density maps of the two Runs, while the right panels present the inflow mass flux and flow velocity maps. The mass flux is $\dot M_{\rm loc}$ as define in equation (\[eq:phil\]). In is evident that the inflow close to the NS at the center is mainly from two general opposite directions, the $+y$ and $-y$ directions, that are perpendicular to the axes of the two jets’ launching episodes. When the angle between the two first jet-pairs is small, $40^\circ$, the high inflow mass flux is from two extended opposite areas. These becomes smaller when the angle between the two episodes are large, $70^\circ$. A bipolar accretion flow pattern has emerged.
![Flow structure at the end of the second jets episode at $t=0.1\s$, as in Fig. \[fig:global2s\], but in a plane perpendicular to the $y=0$ plane and intersecting it at the dashed line drawn on the left panels of Fig. \[fig:global2s\]. The coordinate along the dashed line is define as $W$, and it is along the line $z=-0.7x$ for Run A (upper panels) and $z=-1.38x$ for Run B (lower panels). Note that the panels here show a small inner region of the grid. Partition of panels as in Fig. \[fig:global2s\], but in the right panels the color represent mass influx rate $\dot M_{loc}$.[]{data-label="fig:NormalPlane"}](normal_t1deg40-70){width="\textwidth"}
We can also use the Mollweide-projection (see Fig. \[fig:grid\]) to present the emergence of the bipolar inflow (accretion) pattern. This is shown in Fig. \[fig:second\_jets-40\] for Run A, and in Fig. \[fig:second\_jets-70\] for Run B. Presented are the inflow mass fluxes on spheres of radii $500 \km$ and $1000 \km$, as in Fig. \[fig:one\_jets500\] for the first jets. Again, the emergence of a bipolar inflow (accretion) structure is evident, particularly in Run B presented in Fig. \[fig:second\_jets-70\].
![The inflow mass flux at $t=0.062, 0.084, 0.1 \s$, from left to right, i.e., at the beginning, middle and end of the second jets-launching episode, for Run A, given on spheres of radius $r= 500 \km$ (upper row) and $r= 1000 \km$ (lower row). Mass inflow rate at each point is calculated as if the entire sphere would have the same inflow mass flux as in the given point (eq. \[eq:phil\]. White area are regions no inflow, some with outflow, i.e., the jets. Note the emergence of two opposite high accretion rate regions. []{data-label="fig:second_jets-40"}](t062-100deg40){width="\textwidth"}
![Like Fig. \[fig:second\_jets-40\], but for Run B, where the jets of the second episode are launched at $\theta_1=70^\circ$ to the first jets’ axis, rather than $\theta_1=40^\circ$. The bipolar accretion patter is clearly seen in the dark-red areas. []{data-label="fig:second_jets-70"}](t062-100deg70){width="\textwidth"}
Later evolution {#third}
---------------
As we discussed in the next section, jets launched by accretion disks formed from the bipolar accretion pattern are likely to be lunched perpendicular to the direction of accretion. Namely, they will be launched in, or close to, the $y=0$ plane defined by the jets’ axes of the first two episodes. We are not in the numerical stage to follow the angular momentum of the accreted gas and from that to find the direction to launch the next jets episode. We therefore arbitrarily run 4 cases for the third jets launching episode as listed in Table \[table\]. In all these cases the third jets launching episode is active in the time period $t=0.1-0.15 \s$. We present the flow structure for two cases, A1 and B2, in Fig. \[fig:third1\], at the end of the active phase $t=0.15 \s$. In all cases we find that a third jets-launching episode strengthens the bipolar accretion pattern and the inflow becomes further concentrated to two opposite directions. In Fig. \[fig:NormalPlane-3\] we use the Mollweide-projection (see Fig. \[fig:grid\]) to present the inflow rates through a spheres of radius $r=500 \km$ for the four Runs at the end of the third episode. The jets of the third episode make the bipolar accretion pattern more prominent.
![Flow structure in the $y=0$ plane at the end of the third jets launching episode, $t=0.15 \s$, for Run A1 (upper panels) and B2 (lower panels). Left Panels: density, with a color coding in logarithmic scale and units of $\g \cm^{-3}$. The three arrows depict the direction of jets’ launching in the three episodes as numbered. Right Panels: temperature in log scale in units of $\K$, and velocity map. Velocity is proportional to the arrow length, with inset showing an arrow for $30,000 \km \s^{-1}$. []{data-label="fig:third1"}](X-Y_t15deg40-70){width="\textwidth"}
![Like Fig. \[fig:second\_jets-40\] but at $t=0.15 \s$ and on a sphere of radius $r=500 \km$. The four cases are as indicated on the panels and according to Table \[table\]. The third jets episode strengthens the bipolar accretion pattern. []{data-label="fig:NormalPlane-3"}](4plot){width="\textwidth"}
To further quantify the bipolar accretion pattern we construct two opposite cones whose common vertex is at the center, and calculate the ratio of the average inflow mass flux through the cones to the average inflow mass flux through the entire sphere at the same radius. The cones’ axis is perpendicular to the $y=0$ plane, i.e., perpendicular to the jets’ axes, and through the center. The quantity we use is $$\xi \equiv \frac{ \dot M_{\rm cone} ({\rm inflow})/\Omega }{ M_{\rm total} ({\rm inflow})/ 4 \pi} ,
\label{eq:xi1}$$ where $\Omega$ is the solid angle covered by the two cones. The evolution of $\xi$ with time for two cone-pairs and at two radii are presented in Fig. \[fig:acc\], for the 4 different Runs listed in Table \[table\]. In one cones pair each cone has an opening angle of $30^\circ$, and in the other each cone has an opening angle of $45^\circ$. We note that the flow becomes more concentrate, i.e., $\xi$ increases with time, to the perpendicular direction as more jets are launched in the $y=0$ plane. The mass flux per unit solid angle for the $30^\circ$ cones is substantially larger than that for the $45^\circ$ cones, $\xi(30^\circ) > \xi(45^\circ)$. This implies that the inflow has the pattern of two opposite stream columns. This is what we refer to as a bipolar accretion pattern.
![The normalized mass inflow rate $\xi$, as given in equation \[eq:xi1\], within two opposite cones of opening angle $\alpha$ and at two radii. The four cases marked are according to Table \[table\]. [[[[The thick line in case A1 is a test run with higher resolution (see section \[subsec:res\]).]{} ]{}]{}]{} []{data-label="fig:acc"}](line "fig:"){width="100.00000%"} .
Resolution dependency of the results {#subsec:res}
------------------------------------
[[[ [ To check the dependency of our simulations on resolution we ran a test case of case A1 with a higher resolution (see Fig. \[fig:res\] for the resolution vs radius). We find the accretion rates in the low and high resolution runs to be within $10 \%$ of each other, as presented by the thick dashed line in the upper-left panel of Fig. \[fig:acc\]. In Fig. \[fig:high\] we compare the flow structure in the low and high resolution runs at one time. As expected, the features are sharper in the high-resolution run, but the large-scale flow structure is very similar. ]{} ]{}]{}]{}
![[Shown are results for case A1 at time $t=0.15 \s$ with higher resolution (left) and standard resolution (right). Beside the expected sharper features in the high-resolution run, the large scale flow is very similar.]{}[]{data-label="fig:high"}](X-Y_t15deg40+high){width="80.00000%"}
We next turn to discuss the implication of the bipolar accretion pattern.
IMPLICATIONS AND SUMMARY {#sec:summary}
========================
In this study we assumed that CCSN explosions are driven by jittering-jets [@Papish2011], an examined the pattern by which jets are launched. The angular momentum of the core is not large, such that the jets’ axis is determined, at least in part, by stochastic processes such as instabilities and convection in the pre-collapse core [@GilkisSoker2013]. We conducted 3D numerical simulations with the FLASH code [@Fryxell2000]. In each jets-launching episode we launched one pair of bipolar jets (Fig. \[fig:dens-temp-05\]). Under these assumptions, the directions of the two first jets-launching episode are more or less random. However, the first two episodes, if not along the same direction, define a plane. We set this plane to be $y=0$ in our study. The jets will eject mass outward along their propagation direction, leaving inflow in perpendicular directions. After the first episode the inflow is from a belt region perpendicular to the jets’ axis, as demonstrated in Fig. \[fig:one\_jets500\]. After the second episode the inflow is concentrated in two opposite directions, as clearly seen in Figs. \[fig:NormalPlane\], \[fig:second\_jets-40\] and \[fig:second\_jets-70\]. A bipolar accretion flow with its axis perpendicular to the $y=0$ plane has been formed.
An inflowing gas along a direction normal to the $y=0$ plane has an angular momentum direction within that plane. This implies that bipolar accretion flow makes it more likely that the axis of the jets in the third episode will be in the $y=0$ plane, as jets are launched along the angular momentum axis. Namely, the third episode jets’ axis will be in about the same plane as the axes of the first two episodes.
We simulated two directions for the second jets episode, and for each of these we simulated two directions in the same plane for a third jets episode. These four cases are summarized in Table \[table\]. The asymmetry in the accretion flow is presented for these cases in Fig. \[fig:acc\], quantified by the the parameter $\xi$ defined in equation (\[eq:xi1\]). The bipolar pattern becomes more prominent as the inflow becomes more concentrated along two opposite directions. This implies that the following launching of jets will be in, or near, the $y=0$ plane.
The bubbles inflated by the jets grow and expand to all directions as they move toward lower density core gas. Eventually they will close on directions perpendicular to their axis as well, and expel the outer core and the rest of the star in all directions. In addition, some of the jets will not be exactly in the same plane, but will have some stochastic variations from the $y=0$ plane. This will also help in expelling the rest of the star in all directions. This later evolutionary phase will be studied in the future.
Can this *planar jittering pattern* have any observational consequences? We don’t expect prominent signature in in Type II SNe where the shock will become more spherical as it propagate through the extended massive hydrogen envelope. In type Ic SNe the imprint might exist in the supernova remnant (SNR). We raise here the possibility that the torus morphology of a tilted thick disk with multiple jets in Cassiopeia A SNR (@DeLaneyetal2010 [@MilisavljevicFesen2013]) is a result of a planar jittering pattern. The last jet was more free to expand, as all core has been removed, and it is now observed as the high velocity jet like outflow along the northeast direction of the SNR.
Finally, we point out that a planar jet launching patter might have taken place during galaxy formation, where a feedback between accretion of cold gas and jet activity might have taken place.
[[[ [We thank the referee, Jason Nordhaus, for helpful comments.]{} ]{}]{}]{} This research was supported by the Asher Fund for Space Research at the Technion, and a generous grant from the president of the Technion Prof. Peretz Lavie. OP is supported by the Gutwirth Fellowship. [[[ [The software used in this work was developed in part by the DOE NNSA ASC- and DOE Office of Science ASCR-supported Flash Center for Computational Science at the University of Chicago. This work was supported by the Cy-Tera Project (ΝΕΑ $ \rm Y \Pi O \Delta O M H / \Sigma T P A T H$/0308/31), which is co-funded by the European Regional Development Fund and the Republic of Cyprus through the Research Promotion Foundation.]{}]{}]{}]{}
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abstract: 'We prove that the class of topological knot types that are both Legendrian simple and satisfy the uniform thickness property (UTP) is closed under cabling. An immediate application is that all iterated cabling knot types that begin with negative torus knots are Legendrian simple. We also examine, for arbitrary numbers of iterations, iterated cablings that begin with positive torus knots, and establish the Legendrian simplicity of large classes of these knot types, many of which also satisfy the UTP. In so doing we obtain new necessary conditions for both the failure of the UTP and Legendrian non-simplicity in the class of iterated torus knots, including specific conditions on knot types.'
author:
- 'Douglas J. LaFountain'
title: |
Studying uniform thickness I:\
Legendrian simple iterated torus knots
---
ł Ł ø v [C]{} u *[[SL(2,C)]{}]{} PS. [[PSL(2,C)]{}]{}*
[^1]
Introduction
============
In this paper we begin a general study of the [*uniform thickness property*]{} (UTP) in the context of iterated torus knots that are embedded in $S^3$ with the standard tight contact structure. Our goal in this study will be to determine the extent to which iterated torus knot types fail to satisfy the UTP, and the extent to which this failure leads to cablings that are Legendrian or transversally non-simple. The specific goal of this note is to address both questions by establishing new necessary conditions for the failure of the UTP, as well as new necessary conditions for slopes of cablings that are Legendrian non-simple. In the process we will show that, in some sense, most iterated torus knot types are Legendrian simple, and many satisfy the UTP, including many iterated cablings that begin with knots which fail the UTP.
Specifically, we will begin by showing that the class of knots that are both Legendrian simple and satisfy the UTP is closed under cabling, and hence all iterated cablings that begin with negative torus knots are Legendrian simple. We will then study, for arbitrary numbers of iterations, iterated cablings that begin with positive torus knots, and demonstrate the Legendrian simplicity of many of these knot types, some of which also satisfy the UTP. Our analysis will result in a precise class of iterated torus knot types that may fail the UTP, as well as the identification of many solid tori representatives that may fail to thicken. We will also obtain a precise class of iterated torus knots that may be Legendrian non-simple. A forthcoming note, [*Studying uniform thickness II*]{}, will then more directly address the related problems of determining whether these two classes indeed fail the UTP and are Legendrian non-simple.
To bring the above goals into focus, we recall the definition of the [*uniform thickness property*]{} as given by Etnyre and Honda [@[EH1]]. For a knot type $K$, define the [*contact width*]{} of $K$ to be
$$w(K)=\textrm{sup}\frac{1}{\textrm{slope}(\Gamma_{\partial N})}$$
In this equation the $N$ are solid tori having representatives of $K$ as their cores, and $\textrm{slope}(\Gamma_{\partial N})$ refers to the slope of the [*dividing curves*]{} on the convex torus $\partial N$. Slopes are measured using the preferred framing coming from a Seifert surface for $K$, and slopes are calculated so that the longitude has slope $\infty$; the supremum is taken over all solid tori $N$ representing $K$ where $\partial N$ is convex. Any knot type $K$ satisfies the inequality $\overline{tb}(K) \leq w(K) \leq \overline{tb}(K) + 1$, where $\overline{tb}$ is the maximal Thurston-Bennequin number for $K$.
A knot type $K$ satisfies the UTP if the following hold:
- $\overline{tb}(K)=w(K)$.
- Every solid torus $N$ representing $K$ can be thickened to a standard neighborhood of a maximal $tb$ Legendrian knot.
Using this definition, Etnyre and Honda identified necessary conditions for the existence of Legendrian non-simple iterated torus knot types [@[EH1]]. Specifically, they showed that if all iterated torus knots were to satisfy the UTP, then they would all be Legendrian simple; hence if some iterated torus knot fails to be Legendrian simple, then there must exist an iterated torus knot which fails the UTP. They subsequently established that the $(2,3)$ torus knot fails the UTP and indeed has a cabling which is Legendrian non-simple, namely the $((2,3),(2,3))$ iterated torus knot. They also established, for arbitrary numbers of iterations, iterated torus knots that are Legendrian simple, where at each iteration the knot type satisfies the UTP, and cabling fractions $P/q$ are less than the contact width.
In this note, we extend Etnyre and Honda’s work; we begin by proving the following theorem:
\[UTP theorem\] Let $K$ be a topological knot type. If $K$ is Legendrian simple and satisfies the UTP, then all of its cablings are Legendrian simple and satisfy the UTP.
Most of the content of this theorem was proved by Etnyre and Honda in Theorems 1.1 and 1.3 in [@[EH1]]; we prove the satisfaction of the UTP for cabling fractions $P/q$ that are greater than the contact width. As an immediate consequence, using the fact that negative torus knots are Legendrian simple and satisfy the UTP [@[EH1]; @[EH2]], we have the following result:
\[Cablings of negative torus knots\] All iterated cabling knot types that begin with negative torus knots are Legendrian simple; that is, if $K_r = ((P_1,q_1),...,(P_r,q_r))$ is an iterated torus knot type where $(P_1,q_1)$ is a negative torus knot, then $K_r$ is Legendrian simple.
We then undertake an analysis of iterated cablings that begin with positive torus knots, and identify, for arbitrary numbers of iterations, Legendrian simple classes of such iterated torus knots. In order to obtain a precise statement of these other results, we will first need to recall and introduce some terminology. However, at this point the reader may wish to look ahead to Figure \[fig:SchematicNewestB2\], where in graphical form we combine Etnyre and Honda’s results with ours to provide a summary of what is known concerning the uniform thickness and Legendrian classification of iterated torus knots.
Recall that for Legendrian knots embedded in $S^3$ endowed with the standard tight contact structure, there are two classical invariants of Legendrian isotopy classes, namely the Thurston-Bennequin number, $tb$, and the rotation number, $r$. For a given topological knot type, we can represent Legendrian isotopy classes by points on a grid whose horizontal axis plots values of $r$ and whose vertical axis plots values of $tb$. This plot takes the visual form of a [*Legendrian mountain range*]{}. For a given topological knot type, if the ordered pair $(r,tb)$ completely determines the Legendrian isotopy classes, then that knot type is said to be [*Legendrian simple*]{}. Previous examples of Legendrian simple knot types include the unknot [@[EF]], as well as torus knots and the figure eight knot [@[EH2]].
[*Iterated torus knots*]{}, as topological knot types, can be defined recursively. Let 1-iterated torus knots be simply torus knots $(P_1,q_1)$ with $P_1$ and $q_1$ co-prime nonzero integers, and $|P_1|, q_1 > 1$. Here $P_1$ is the algebraic intersection with a longitude, and $q_1$ is the algebraic intersection with a meridian in the preferred framing for a torus representing the unknot. Then for each $(P_1,q_1)$ torus knot, take a solid torus regular neighborhood $N((P_1,q_1))$; the boundary of this is a torus, and given a framing we can describe simple closed curves on that torus as co-prime pairs $(P_2,q_2)$, with $q_2 > 1$. In this way we obtain all 2-iterated torus knots, which we represent as ordered pairs, $((P_1,q_1),(P_2,q_2))$. Recursively, suppose the $(r-1)$-iterated torus knots are defined; we can then take regular neighborhoods of all of these, choose a framing, and form the $r$-iterated torus knots as ordered $r$-tuples $((P_1,q_1),...,(P_{r-1},q_{r-1}),(P_r,q_r))$, again with $P_r$ and $q_r$ co-prime, and $q_r > 1$.
For ease of notation, if we are looking at a general $r$-iterated torus knot type, we will refer to it as $K_r$; a Legendrian representative will usually be written as $L_r$. Note that we will use the letter $r$ both for the rotation number and as an index for our iterated torus knots; context will distinguish between the two uses.
We will study iterated torus knots using two framings. The first is the standard framing for a torus, where the meridian bounds a disc inside the solid torus, and we use the preferred longitude which bounds a Seifert surface in the complement of the solid torus. We will refer to this framing as $\mathcal{C}$. The second framing is a non-standard framing using a different longitude that comes from the cabling torus. More precisely, to identify this non-standard longitude on $\partial N(K_r)$, we first look at $K_r$ as it is embedded in $\partial N(K_{r-1})$. We take a small neighborhood $N(K_r)$ such that $\partial N(K_r)$ intersects $\partial N(K_{r-1})$ in two parallel simple closed curves. These curves are longitudes on $\partial N(K_r)$ in this second framing, which we will refer to as $\mathcal{C'}$. Note that this corresponds to the $\mathcal{C'}$ framing in [@[EH1]], and is well-defined for any cabled knot type. Moreover, for purpose of calculations there is an easy way to change between the two framings, which is presented in [@[EH1]] and which we will review in the body of this note.
Given a simple closed curve $(\mu, \lambda)$ on a torus, measured in some framing as having $\mu$ meridians and $\lambda$ longitudes, we will say this curve has slope of $\lambda/\mu$; i.e., longitudes over meridians. Therefore we will refer to the longitude in the $\mathcal{C'}$ framing as $\infty^\prime$, and the longitude in the $\mathcal{C}$ framing as $\infty$. The meridian in both framings will have slope $0$. This way of representing slopes corresponds to that in [@[EH1]]; in short, slopes are the reciprocals of cabling fractions $\mu/\lambda$.
A new convention we will be using is that meridians in the standard $\mathcal{C}$ framing, that is, algebraic intersection with $\infty$, will be denoted by upper-case $P$. On the other hand, meridians in the non-standard $\mathcal{C}'$ framing, that is, algebraic intersection with $\infty'$, will be denoted by lower-case $p$.
Given an iterated torus knot type $K_r = ((p_1,q_1),...,(p_r,q_r))$ where the $p_i$’s are measured in the $\mathcal{C}'$ framing, we define two quantities, whose meaning will be revealed in the body of this note. The two quantities are:
$$\label{ArBr}
\displaystyle A_r := \sum_{\alpha=1}^{r}p_\alpha \prod_{\beta=\alpha+1}^{r}q_\beta \prod_{\beta=\alpha}^{r}q_\beta \ \ \ \ \ \ \ \ B_r := \sum_{\alpha=1}^r \left(p_\alpha \prod_{\beta=\alpha+1}^r q_\beta \right) + \prod_{\alpha=1}^r q_\alpha$$
Note here we use a convention that $\displaystyle\prod_{\beta=r+1}^{r}q_\beta := 1$. Also, if we restrict to the first $i$ iterations, that is, to $K_i = ((p_1,q_1),...,(p_i,q_i))$, we have an associated $A_i$ and $B_i$. For example, $\displaystyle A_i := \sum_{\alpha=1}^{i}p_\alpha \prod_{\beta=\alpha+1}^{i}q_\beta \prod_{\beta=\alpha}^{i}q_\beta$.
Finally, for convenience in stating our theorems, we will define a particular class of iterated torus knot types, each member of which we will denote by $\breve{K}_r$:
[*$\breve{K}_r = ((p_1,q_1),..., (p_i,q_i),...,(p_r,q_r))$ is an $r$-iterated torus knot type, where we require that $r \geq 1$, $q_i > 1$ for all $i$, $p_1 > 1$, and for $i \geq 1$ we have $q_{i+1}/p_{i+1} \notin (-1/B_i,0)$; at each iteration we use the $\mathcal{C'}$ framing.*]{}
We will show that the following is an equivalent definition for $\breve{K}_r$ in the $\mathcal{C}$ framing: form an iterated torus knot by beginning with a positive $(P_1,q_1)$ torus knot, and then at each iteration take cabling fractions $P_{i+1}/q_{i+1}$ greater than $w(K_i)$. Note also that for $\breve{K}_r$ we will show that $A_r > B_r > 0$.
We can now state our remaining results; our first is that the $\breve{K}_r$ are Legendrian simple:
\[main theorem\] Each $\breve{K}_r$ is Legendrian simple, and has a Legendrian mountain range with a single peak at $\overline{tb}=A_r-B_r=-\chi(\breve{K}_r)$ and $r=0$.
The Legendrian classification of the $\breve{K}_r$ generalizes that of positive torus knots, as their Legendrian mountain ranges are vertical translates of those for positive torus knots. A result of Etnyre and Honda is that the $(2,3)$ torus knot fails the UTP; hence many of the $\breve{K}_r$ are iterated cablings that begin with knots failing the UTP.
We then determine more cablings of these $\breve{K}_r$ that are also Legendrian simple, and futhermore satisfy the UTP:
\[main theorem 3\] Let $K_{r+1}$ be a $(p_{r+1},q_{r+1})$ cabling of $\breve{K}_r$, where $q_{r+1}/p_{r+1} \in (-1/A_r,0)$, as measured in the $\mathcal{C'}$ framing. Then $K_{r+1}$ is Legendrian simple, $\overline{tb}=A_{r+1}$, and the Legendrian mountain range can be determined based on the Legendrian classification of $\breve{K}_r$. Moreover, $K_{r+1}$ satisfies the UTP.
Note that by Theorem \[UTP theorem\], all iterated cablings beginning with these $K_{r+1}$ are Legendrian simple.
Taken together, these two theorems show that all cablings of $\breve{K}_r$ with slopes in the complement of the interval $[-1/B_r,-1/A_r]$ are Legendrian simple. This is not by accident; it will be shown that the slopes of dividing curves on the boundary of solid tori representing $\breve{K}_r$ that may fail to thicken will be contained within the interval $[-1/B_r,-1/A_r)$.
We will prove these two theorems using the $\mathcal{C}'$ framing, as they are stated. However, after changing from $\mathcal{C}'$ to $\mathcal{C}$ via a change of coordinates, Theorems \[main theorem\] and \[main theorem 3\] will immediately imply the following Corollaries \[fractionslargerthanwidth\] and \[negativecablings\], respectively, both of which are stated in the $\mathcal{C}$ framing:
\[fractionslargerthanwidth\] Let $K_r = ((P_1,q_1),...,(P_i,q_i),...,(P_r,q_r))$ be an iterated torus knot where $(P_1,q_1)$ is a positive torus knot, and such that $P_{i+1}/q_{i+1} > w(K_i) = \overline{tb}(K_i)$ for $1 \leq i < r$. Then $K_r$ is Legendrian simple and has a Legendrian mountain range with a single peak at $\overline{tb}(K_r) = -\chi(K_r)$ and $r=0$.
\[negativecablings\] Let $K_{r+1}= ((P_1,q_1),...,(P_i,q_i),...,(P_r,q_r), (P_{r+1},q_{r+1}))$ be an iterated torus knot such that $P_{r+1} < 0$, $(P_1,q_1)$ is a positive torus knot and $P_{i+1}/q_{i+1} > w(K_i)$ for $1 \leq i < r$. Then $K_{r+1}$ is Legendrian simple and satisfies the UTP.
Also note that the above classes of knots are transversally simple, since Legendrian simplicity implies transversal simplicity (see Theorem 2.10 in [@[EH2]]).
Figure \[fig:SchematicNewestB2\] is a schematic indicating what is known and what is unknown about the uniform thickness and the Legendrian simplicity of iterated torus knots. What is known is boxed; what is unknown is in bold with question marks.
![[]{data-label="fig:SchematicNewestB2"}](SchematicNewestRevisedv2B){width=".80\textwidth"}
Combining Theorem \[UTP theorem\], Corollaries \[fractionslargerthanwidth\] and \[negativecablings\], and the fact that negative torus knots are simple and satisfy the UTP, yields the following necessary conditions for failure of the UTP for iterated torus knots.
\[necessarycondforUTP\] Suppose $K_r$ is an iterated torus knot type that fails the UTP. Then either:
- $K_r = \breve{K}_r$.
- $K_r = ((P_1,q_1),..., (P_i,q_i),...,(P_r,q_r))$, where for some $1 \leq i < r$ we have $K_i = \breve{K}_i$ and $P_{i+1}/q_{i+1} \in (0,w(K_i))$.
Finally, again combining Theorem \[UTP theorem\], Corollaries \[fractionslargerthanwidth\] and \[negativecablings\], and the fact that negative torus knots are simple and satisfy the UTP, we obtain the following necessary conditions for Legendrian non-simplicity of iterated torus knots:
\[necessarycondfornonsimple\] Suppose $K_r$ is an iterated torus knot type that is Legendrian non-simple. Then $K_r = ((P_1,q_1),..., (P_i,q_i),...,(P_r,q_r))$, where for some $1 \leq i < r$ we have $K_i = \breve{K}_i$ and $P_{i+1}/q_{i+1} \in (0,w(K_i))$.
We will be using tools developed by Giroux, Kanda, and Honda, and used by Etnyre and Honda in their work, namely convex tori and annuli, the classification of tight contact structures on solid tori and thickened tori, and the Legendrian classificaton of torus knots. Most of the results we use can be found in [@[H]], [@[EH1]], or [@[EH2]], and if we use a lemma, proposition, or theorem from one of these works, it will be specifically referenced. We will also briefly make use of facts involving the classical invariant for transversal isotopy classes, namely the self-linking number, $sl$.
With this in mind, this note will proceed as follows. In §2 we prove Theorem \[UTP theorem\]. In §3 we perform preliminary calculations that allow us to outline a strategy for proving Theorem \[main theorem\]. This leads us to §4, where we examine solid tori representing $\breve{K}_r$, obtaining necessary conditions for those that fail to thicken, as well as calculating $w(\breve{K}_r)$. In §5 we prove Theorem \[main theorem\], and in §6 we prove Theorem \[main theorem 3\].
Acknowledgements {#acknowledgements .unnumbered}
----------------
This work composes part of my PhD thesis at the University at Buffalo under the advisement of William Menasco, whom I wish to thank for many helpful discussions and suggestions.
Cabling preserves simplicity and the UTP
========================================
We first review some facts about Legendrian knots on convex tori. Recall that the characteristic foliation induced by the contact structure on a convex torus can be assumed to have a standard form, where there are $2n$ parallel [*Legendrian divides*]{} and a one-parameter family of [*Legendrian rulings*]{}. Parallel push-offs of the Legendrian divides gives a family of $2n$ [*dividing curves*]{}, referred to as $\Gamma$. For a particular convex torus, the slope of components of $\Gamma$ is fixed and is called the [*boundary slope*]{} of any solid torus which it bounds; however, the Legendrian rulings can take on any slope other than that of the dividing curves by Giroux’s Flexibility Theorem [@[G]]. A [*standard neighborhood*]{} of a Legendrian knot $L$ will have two dividing curves and a boundary slope of $1/tb(L)$.
For a topological knot type $K$, if $N$ is a solid torus having a representative of $K$ as its core and convex boundary, then $N$ [*fails to thicken*]{} if for all $N' \supset N$, we have $\textrm{slope}(\Gamma_{\partial N'}) = \textrm{slope}(\Gamma_{\partial N})$.
Given a ruling curve $L = (P,q)$ on a convex torus $\partial N(K)$, then recall that section 2.1 in [@[EH1]] provides a relationship between the framings $\mathcal{C}'$ and $\mathcal{C}$ on $\partial N(L)$. In terms of a change of basis, we can represent slopes $\lambda/\mu$ as column vectors and then get from a slope $\lambda/\mu'$, measured in $\mathcal{C}'$ on $\partial N(L)$, to a slope $\lambda/\mu$, measured in $\mathcal{C}$, by:
$$\displaystyle \left(\begin{array}{cc}
1 & Pq\\
0 & 1\end{array}\right)\left(\begin{array}{c}
\mu '\\
\lambda\end{array}\right) = \left(\begin{array}{c}
\mu\\
\lambda\end{array}\right)\nonumber$$
In other words, $\mu = \mu ' + Pq\lambda$. If we then define $t$ to be the twisting of the contact planes along $L$ with respect to the $\mathcal{C}'$ framing on $\partial N(L)$, equation 2.1 in [@[EH1]] gives us:
$$\label{5}
tb(L) = Pq + t(L)$$
Observe that $t(L)$ is also the twisting of the contact planes with respect to the framing given by $\partial N$, and so is equal to $-1/2$ times the geometric intersection number of $L$ with $\Gamma_{\partial N}$. The maximal twisting number with respect to this framing will be denoted by $\overline{t}$.
Finally, recall that if $\mathcal{A}$ is a convex annulus with Legendrian boundary components, then dividing curves are arcs with endpoints on either one or both of the boundary components; an annulus with no boundary-parallel dividing curves is said to be [*standard convex*]{}.
We can now prove Theorem \[UTP theorem\]:
Recall that we have a knot $K$ that is Legendrian simple and satisfies the UTP. By Theorem 1.3 in [@[EH1]], we know that $(P,q)$ cables are simple and satisfy the UTP, provided $P/q < w(K)$. Thus we only need to look at the case where $P/q > w(K)$. We will refer to the $(P,q)$ cable as $K_{(P,q)}$. From Theorem 3.2 in [@[EH1]], we know that $K_{(P,q)}$ is Legendrian simple and that $\overline{t}(K_{(P,q)}) < 0$. Moreover, we know from the same theorem that $K_{(P,q)}$ achieves $\overline{tb}(K_{(P,q)})$ as a Legendrian ruling curve on a convex torus with boundary slope $1/w(K)$ and two dividing curves.
To prove that $K_{(P,q)}$ satisfies the UTP, it suffices to show that any solid torus $N_{(P,q)}$ representing $K_{(P,q)}$ thickens to a standard neighborhood of a Legendrian knot at $\overline{tb}(K_{(P,q)})$. So given a solid torus $N_{(P,q)}$, let $\mathcal{A}$ be a convex annulus connecting $\partial N_{(P,q)}$ to itself, with $\partial \mathcal{A}$ being two $\infty^\prime$ rulings so that $\partial N_{(P,q)} \backslash \partial \mathcal{A}$ consists of two annuli, one of which, along with $\mathcal{A}$, bounds a solid torus $\widehat{N}$ representing $K$ with $\widehat{N} \supset N_{(P,q)}$. Now since $K$ satisfies the UTP, $\widehat{N}$ can be thickened to a standard neighborhood of a Legendrian knot at $\overline{tb}(K)$, which we call $N$. See part (a) in Figure \[fig:UTP1BB\].
![[]{data-label="fig:UTP1BB"}](UTP1BB){width="80.00000%"}
We now let $L$ be a Legendrian core curve representing $K$ in $\widehat{N} \backslash N_{(P,q)}$, and let $\widetilde{\mathcal{A}}$ be a convex annulus joining $\partial N$ to $\partial N(L)$ inside $N \backslash N_{(P,q)}$, with boundary components $(P,q)$ Legendrian rulings. See part (b) in Figure \[fig:UTP1BB\]. We may assume that we have topologically isotoped $L$ so that the Thurston-Bennequin number is maximized over all such topological isotopies for the space $N \backslash N_{(P,q)}$. $N(L)$ will have dividing curves of slope $1/m$ in $\mathcal{C}$, where $m\in \mathbb{Z}$. We claim that in fact $m=\overline{tb}(K)$. For if $m < \overline{tb}(K)$, then by the Imbalance Principle, there must exist bypasses on the $\partial N(L)$-edge of $\widetilde{\mathcal{A}}$, since the $\partial N$-edge of $\widetilde{\mathcal{A}}$ is at maximal twisting (see Prop 3.17 in [@[H]]). But such a bypass would induce a destabilization of $L$, thus increasing its $tb$ by one – see Lemma 4.4 in [@[H]]. To satisfy the conditions of this lemma, we are using the fact that $P/q > w(K)$. Thus $m=\overline{tb}(K)$ and $\widetilde{\mathcal{A}}$ is standard convex.
Finally, note that now $N_{(P,q)}$ thickens to $\widetilde{N}_{(P,q)}=N \backslash (N(\widetilde{\mathcal{A}}) \cup N(L))$. We can calculate the boundary slope of $\widetilde{N}_{(P,q)}$. We choose $(P',q')$ to be a curve on $N$ and $N(L)$ such that $Pq'-P'q=1$, and we change coordinates to a basis $\mathcal{C}''$ via the map $((P,q),(P',q')) \mapsto ((0,1),(-1,0))$. Under this map we obtain $$\textrm{slope}(\Gamma_{\partial N}) = \textrm{slope}(\Gamma_{\partial N(L)})= \frac{q'w(K)-P'}{qw(K)-P}$$
We then obtain in the $\mathcal{C}'$ framing, after edge-rounding, that
$$\begin{aligned}
\textrm{slope}(\Gamma_{\partial \widetilde{N}_{(P,q)}})&=&\frac{q'w(K)-P'}{qw(K)-P} -\frac{q'w(K)-P'}{qw(K)-P}+\frac{1}{qw(K)-P}\nonumber\\
&=&\frac{1}{qw(K)-P} = \frac{1}{\overline{t}(K_{(P,q)})} \end{aligned}$$
Hence the boundary slope of $\widetilde{N}_{(P,q)}$ must be $1/\overline{tb}(K_{(P,q)})$ with two dividing curves in the standard $\mathcal{C}$ framing. Thus $K_{(P,q)}$ satisfies the UTP.
Preliminary calculations
========================
In this section we collect some identities and lemmas that will be useful in our analysis of iterated cablings that begin with positive torus knots.
First suppose $K_r = ((p_1,q_1),...,(p_r,q_r))$ is a general $r$-iterated torus knot type, with $p_i$’s measured in the $\mathcal{C}'$ framing. We first obtain a formula for the $P_i$’s as measured in the standard $\mathcal{C}$ framing. To this end, from equation \[ArBr\] we obtain two useful identities:
$$\label{ArBrrecursive}
A_r = q_r^2 A_{r-1} + p_rq_r \qquad \qquad B_r = q_r B_{r-1} + p_r$$
Now suppose we have a $((p_1,q_1),...,(p_r,q_r))$ iterated torus knot as described above, and let $P_i$ be the meridians for the $i$-th iteration, but as measured in the standard $\mathcal{C}$ framing. To determine $P_{i+1}$, the algebraic intersection with the preferred longitude, we use the change of basis mentioned in §2 to obtain $P_{i+1} = q_{i+1}P_iq_i + p_{i+1}$. We then can prove the following lemma:
\[Pr\] $P_r=q_rA_{r-1} + p_r$ for $r \geq 2$ and $A_r = P_r q_r$ for $r \geq 1$.
First observe that $P_1=p_1$ and so equation \[ArBr\] immediately gives us $A_1=P_1q_1$. We then use induction, beginning with a base case of $r=2$. From the comments above we have $P_2=q_2A_1 + p_2$, and thus $A_2=P_2q_2$. But then inductively we can assume that $A_{r-1}=P_{r-1}q_{r-1}$, and so again by the above comments $P_r=q_rA_{r-1}+p_r$, and hence $A_r = P_rq_r$.
Note that as a consequence of this lemma, the change of coordinates from the $\mathcal{C}'$ framing to the $\mathcal{C}$ framing on $\partial N(K_r)$ becomes left multiplication by $\displaystyle \left(\begin{array}{cc}
1 & A_r\\
0 & 1\end{array}\right)$.
We now focus in on those particular iterated torus knot types $\breve{K}_r$ with $r \geq 1$, $q_i > 1$ for all $i$, $p_1 > 1$, and where for $i \geq 1$ we have $q_{i+1}/p_{i+1} \notin (-1/B_i,0)$. We first prove a preliminary lemma concerning $A_r$, $B_r$, and $P_r$.
$A_r > B_r > 0$ and $P_r > 0$ for any iterated torus knot type $\breve{K}_r$.
Observe that since $A_1 > B_1 > 0$ and $P_1=p_1 > 0$ for positive torus knots, we can assume inductively that $A_{r-1} > B_{r-1} > 0$ and that $P_{r-1} > 0$. Then if $p_r > 0$, we certainly have $P_r > 0$ by Lemma \[Pr\]; moreover, $A_r = q_r^2 A_{r-1} + p_rq_r > q_r A_{r-1} + p_r > q_rB_{r-1} + p_r = B_r > 0$. In the other case, if $q_r/p_r < -1/B_{r-1}$, that means that $q_rB_{r-1}+p_r =B_r > 0 $. Moreover, $P_r = q_rA_{r-1}+p_r > q_rB_{r-1}+p_r > 0$. Finally, note that the previous proof that $A_r > B_r$ works for this case too.
Recall that Lemma 2.2 in [@[EH1]] provides us with a way of calculating $r(L_r)$ from $r(\partial D)$ and $r(\partial \Sigma)$, where $D$ is a convex meridional disc for $N_{r-1}$ and $\Sigma$ is a convex Seifert surface for the preferred longitude on $N_{r-1}$. Specifically, we have the equation:
$$\label{6}
r(L_r) = P_r r(\partial D) + q_r r(\partial \Sigma)$$
We now can prove the following lemma:
\[tblemma\] $\overline{sl}(\breve{K}_r)=\overline{tb}(\breve{K}_r)=A_r - B_r = -\chi(\breve{K}_r)$
We first show that $\chi(\breve{K}_r) = B_r - A_r$; as a consequence, from the Bennequin inequality we obtain $\overline{sl}(\breve{K}_r) \leq A_r - B_r$ and $\overline{tb}(\breve{K}_r) \leq A_r - B_r$.
To this end, we use a formula for $\chi(K_r)$ given at the end of the proof of Corollary 3 in [@[BW]]. The notation used in that paper is that an iterated torus knot $K_r$ is given by a sequence $(e_1(p_1,q_1),e_2(p_2,q_2),\cdots,e_r(p_r,q_r))$ where $p_i,q_i > 0$, $e_i = \pm 1$ indicates the parity of the cabling (either positive or negative), $e_1(p_1,q_1)$ is a torus knot, and for $i > 1$ the $p_i$ represent (efficient) geometric intersection with a meridian, while the $q_i$ represent (efficient) geometric intersection with a preferred longitude. Using this notation from [@[BW]], therefore, the formula of interest is:
$$\displaystyle\chi(K_r) = \prod_{i=1}^r p_i - \sum_{i=1}^r e_iq_i(p_i-1)\prod_{j=i+1}^r p_j - \sum_{i=1}^r (1-e_i)q_i(p_i-1)\prod_{j=i+1}^r p_j\nonumber$$
We need to translate this formula into our notation. For our $\breve{K}_r$ we have $e_i = 1$, since we are cabling positively at each iteration; also, our $(P_i,q_i)$ corresponds to $(q_i,p_i)$ in [@[BW]] for $i > 1$. Thus our formula for $\chi(\breve{K}_r)$ is:
$$\displaystyle \chi(\breve{K}_r) = P_1 \prod_{i=2}^r q_i - q_1(P_1 - 1)\prod_{i=2}^r q_i - \sum_{i=2}^r P_i(q_i-1)\prod_{j=i+1}^r q_j \nonumber$$
To show that this is equal to $B_r - A_r$, we need to rewrite it in terms of our $p_i$’s. To do this, we note that $P_1 = p_1$, and from Lemma 3.1 we have for $i \geq 2$ that $\displaystyle P_i = p_i + q_i \sum_{\alpha = 1}^{i-1} p_\alpha \prod_{\beta=\alpha+1}^{i-1}q_\beta\prod_{\beta=\alpha}^{i-1}q_\beta$. Our equation then becomes:
$$\begin{aligned}
\displaystyle \chi(\breve{K}_r) &=& p_1 \prod_{i=2}^r q_i - q_1(p_1 - 1)\prod_{i=2}^r q_i\\
& &-\sum_{i=2}^r \left(p_i + q_i \sum_{\alpha = 1}^{i-1} p_\alpha \prod_{\beta=\alpha+1}^{i-1}q_\beta\prod_{\beta=\alpha}^{i-1}q_\beta\right)(q_i-1)\prod_{j=i+1}^r q_j \nonumber\end{aligned}$$
If we distribute a few times and collect terms with plus signs, we obtain
$$\begin{aligned}
\displaystyle \chi(\breve{K}_r) &=& p_1 \prod_{i=2}^r q_i + \prod_{i=1}^r q_i + \sum_{i=2}^r p_i \prod_{j=i+1}^r q_j -p_1 \prod_{i=1}^r q_i - \sum_{i=2}^r p_i \prod_{j=i}^r q_j \\
& & -\sum_{i=2}^r \left(q_i \sum_{\alpha = 1}^{i-1} p_\alpha \prod_{\beta=\alpha+1}^{i-1}q_\beta\prod_{\beta=\alpha}^{i-1}q_\beta\right)(q_i-1)\prod_{j=i+1}^r q_j\nonumber\end{aligned}$$
The top line of the equation with plus signs is $B_r$; for the terms with minus signs, for each $i$ we can collect terms of like $p_i$ and obtain:
$$\begin{aligned}
\displaystyle\chi(\breve{K}_r) &=& B_r - \sum_{i=1}^r p_i \left(\prod_{j=i}^r q_j +\sum_{j=i+1}^r q_j \prod_{\beta=i+1}^{j-1}q_\beta \prod_{\beta=i}^{j-1}q_\beta (q_j-1)\prod_{\beta=j+1}^r q_\beta\right)\\
&=& B_r - \sum_{i=1}^r p_i \left(\prod_{j=i}^r q_j \left[1+\sum_{j=i+1}^r (q_j-1)\prod_{\beta=i+1}^{j-1}q_\beta\right]\right) = B_r - A_r\nonumber\end{aligned}$$
where in the last line we have used that $\displaystyle \left[1+\sum_{j=i+1}^r (q_j-1)\prod_{\beta=i+1}^{j-1}q_\beta\right] = \prod_{j=i+1}^r q_j$ (along with a notational convention that $\displaystyle \sum_{r+1}^r = 0$).
Then inductively we can assume $\overline{tb}(\breve{K}_{r-1})=A_{r-1}-B_{r-1}$ and there is a representative at that $tb$ value having $r=0$, since this is true for positive torus knots [@[EH2]]. Then look at the $(p_r,q_r)$ cabling on a standard neighborhood of that representative of $\breve{K}_{r-1}$ at $tb=A_{r-1}-B_{r-1}$ and $r=0$. Then the longitude and meridian both have $r=0$, and the twisting of the cabling equals $-B_r$. Thus there is a representative of $\breve{K}_r$ at $tb=A_r-B_r$ and $r=0$, and hence $\overline{tb}(\breve{K}_r)= A_r-B_r$. Moreover, by taking a positive transverse push-off, this proves $\overline{sl}(\breve{K}_r)= A_r-B_r$.
Now in the Legendrian mountain range for $\breve{K}_r$, the outer left slope contains all Legendrian isotopy classes whose positive transverse push-offs are at $\overline{sl}$. By the proof above, this slope must intersect the $r=0$ axis at $\overline{tb}=A_r-B_r$. Since the mountain range is symmetric about the $r=0$ axis, we thus have the following corollary:
The Legendrian mountain range for $\breve{K}_r$ consists of isotopy classes contained in a single peak centered around the line $r=0$ and with height at $\overline{tb}=A_r-B_r$.
The following will thus suffice to prove that $\breve{K}_r$ is Legendrian simple:
- Show that there is a unique Legendrian isotopy class at $\overline{tb}=A_r-B_r$.
- Show that if $tb(L_r) < A_r-B_r$, then $L_r$ Legendrian destabilizes.
Recall from the work of Etnyre and Honda that a convenient way to find destabilizations of Legendrian knots embedded in tori is to find bypasses attached to these tori. These bypasses can be found on either the interior or exterior of the solid tori, but with possible restrictions due to the failure of the UTP. Thus, before we can prove Theorem \[main theorem\], we must turn our attention to the thickening of solid tori.
Necessary conditions for solid tori $\breve{N}_r$ that do not thicken
=====================================================================
We begin with two new definitions that will be useful in this section.
[*Let $N$ be a solid torus with convex boundary in standard form, and with $\textrm{slope}(\Gamma_{\partial N})=a/b$ in some framing. If $|2b|$ is the geometric intersection of the dividing set $\Gamma$ with a longitude ruling in that framing, then we will call $a/b$ the [*intersection boundary slope*]{}*]{}.
Note that when we have an intersection boundary slope $a/b$, then $2\textrm{gcd}(a,|b|)$ is the number of dividing curves.
[*For $r \geq 1$ and nonnegative integer $k$, define $N_r^k$ to be any solid torus representing $\breve{K}_r$ with intersection boundary slope of $-(k+1)/(A_rk+B_r)$, as measured in the $\mathcal{C}'$ framing. Also define the integer $n_r^k :=\textrm{gcd}((k+1),(A_rk+B_r))$.*]{}
Note that $N_r^k$ has $2n_r^k$ dividing curves.
We will show that any solid torus $N_r$ representing $\breve{K}_r$ can be thickened to an $N_r^k$ for some nonnegative integer $k$, and that any solid torus with the same boundary slope as $N_r^k$ which fails to thicken must have at least $2n_r^k$ dividing curves. Another way of saying this is that every solid torus $N_r$ is contained in some $N_r^k$, and that if $N_r$ fails to thicken, then boundary slopes do not change in passing to the $N_r^k \supset N_r$, although the number of dividing curves may decrease.
Our analysis proceeds by induction, where the base case is positive torus knots. The following lemma is proved for the $(2,3)$ torus knot in [@[EH1]], and there it is noted that there is a corresponding lemma for a positive $(p,q)$ torus knot. However, the calculation is not explicitly provided, so for completeness we prove the general lemma here.
\[basecasethickening\] Let $N$ be a solid torus with core $\breve{K}_1 = (p,q)$ where $p,q > 1$ and co-prime. Then $N$ can be thickened to an $N_1^k$ for some nonnegative integer $k$. Moreover, if $N$ fails to thicken, then it has the same boundary slope as some $N_1^k$, as well as at least $2n_1^k$ dividing curves.
We first construct the setting. Let $T$ be a torus which bounds solid tori $V_1$ and $V_2$ on both sides in $S^3$, and which contains a $(p,q)$ torus knot $\breve{K}_1$. We will think of $T=\partial V_1$ and $T=-\partial V_2$. Let $F_i$ be the core unknots for $V_i$. We know $\overline{tb}(\breve{K}_1)=pq-p-q$ (see [@[EH2]]); measured with respect to the coordinate system $\mathcal{C}'$, for either $i$, $\overline{t}(\breve{K}_1)=-p-q$.
Now let $L_i$, $i=1,2$, be a Legendrian representative of $F_i$ with $tb=-m_i$, where $m_i > 0$ (recall that $\overline{tb}(\textrm{unknot})=-1$). If $N(L_i)$ is a regular neighborhood of $L_i$, then $\textrm{slope}(\Gamma_{\partial N(L_i)})=-1/m_i$ with respect to $\mathcal{C}_{F_i}$. Consider an oriented basis $((p,q),(p',q'))$ for $T$, where $pq'-qp'=1$; we map this to $((0,1),(-1,0))$ in a new framing $\mathcal{C}''$. This corresponds to the map $\displaystyle \Phi_1 = \left(\begin{array}{cc}
q &-p\\
q' &-p'\end{array}\right)$. Then $\Phi_1$ maps $(-m_1,1) \mapsto (-qm_1-p, -q'm_1-p')$. Since we are only interested in slopes, we write this as $(qm_1+p, q'm_1+p')$.
Similarly, we change from $\mathcal{C}_{F_2}$ to $\mathcal{C}''$. The only thing we need to know here is that $(-m_2,1)$ maps to $(pm_2+q,p'm_2+q')$.
This concludes the construction of the setting; we can now prove the lemma. Let $N$ be a solid torus representing $\breve{K}_1$. Let $L_i$ be Legendrian representatives of $F_i$ which maximize $tb(L_i)$ in the complement of $N$, subject to the condition that $L_1 \sqcup L_2$ is isotopic to $F_1 \sqcup F_2$ in the complement of $N$. Now suppose $qm_1+p \neq pm_2 + q$. This would mean that the twisting of Legendrian ruling representatives of $\breve{K}_1$ on $\partial N(L_1)$ and $\partial N(L_2)$ would be unequal. Then we could apply the Imbalance Principle (see Proposition 3.17 in [@[H]]) to a convex annulus $\mathcal{A}$ in $S^3 \backslash N$ between $\partial N(L_1)$ and $\partial N(L_2)$ to find a bypass along one of the $\partial N(L_i)$. This bypass in turn gives rise to a thickening of $N(L_i)$, allowing the increase of $tb(L_i)$ by one (see Lemma 4.4 in [@[H]]). Hence, eventually we arrive at $qm_1+p = pm_2 + q$ and a standard convex annulus $\mathcal{A}$.
Since $m_i > 0$, the smallest solution to $qm_1+p = pm_2 + q$ is $m_1=m_2=1$. All the other positive integer solutions are therefore obtained by taking $m_1=pk+1$ and $m_2=qk+1$ with $k$ a nonnegative integer. We can then compute the intersection boundary slope of the dividing curves on $\partial(N(L_1) \cup N(L_2) \cup \mathcal{A})$, measured with respect to $\mathcal{C}'$, after edge-rounding. This will be the intersection boundary slope for $\widetilde{N} \supset N$. We have:
$$-\frac{q'(pk+1)+p'}{pqk+p+q} + \frac{p'(qk+1)+q'}{pqk+p+q}-\frac{1}{pqk+p+q} = -\frac{k+1}{pqk+p+q}=-\frac{k+1}{A_1k+B_1}$$
This shows that any $N$ thickens to some $N_1^k$, and if $N$ fails to thicken, then it has the same boundary slope as some $N_1^k$. Suppose, for contradiction, that $N$ fails to thicken and has $2n$ dividing curves, where $n < n_1^k$. Then using the construction above we know that outside of $N$ in $S^3$ are neighborhoods of the two Legendrian unknots $L_i$ with $\breve{K}_1$ rulings that intersect the dividing set on $\partial N(L_i)$ exactly $2(A_1k+B_1)$ number of times. However, since $n < n_1^k$, the $\infty'$ rulings on $N$ intersect the dividing set less than $2(A_1k+B_1)$ number of times. Thus by the Imbalance Principle there exists bypasses off of the $\breve{K}_1$ rulings on the $\partial N(L_i)$, and so the $L_i$ can destabilize in the complement of $N$ to smaller $k$-value, allowing for a slope-changing thickening of $N$. This is a contradiction.
We now can prove the following general result by induction using the above lemma as our base case:
\[nonthickening inductive step\] Let $N_r$ be a solid torus representing $\breve{K}_r$, for $r \geq 1$. Then $N_r$ can be thickened to an $N_r^k$ for some nonnegative integer $k$. Moreover, if $N_r$ fails to thicken, then it has the same boundary slope as some $N_r^k$, as well as at least $2n_r^k$ dividing curves.
Inductively we can assume that the lemma is true for solid tori $N_{r-1}$ representing $\breve{K}_{r-1}$. Let $N_r$ be a solid torus representing $\breve{K}_r$. Let $L_{r-1}$ be a Legendrian representative of $\breve{K}_{r-1}$ in $S^3\backslash N_r$ and such that we can join $\partial N(L_{r-1})$ to $\partial N_r$ by a convex annulus $\mathcal{A}_{(p_r,q_r)}$ whose boundaries are $(p_r,q_r)$ and $\infty'$ rulings on $\partial N(L_{r-1})$ and $\partial N_r$, respectively. Then topologically isotop $L_{r-1}$ in the complement of $N_r$ so that it maximizes $tb$ over all such isotopies; this will induce an ambient topological isotopy of $\mathcal{A}_{(p_r,q_r)}$, where we still can assume $\mathcal{A}_{(p_r,q_r)}$ is convex. In the $\mathcal{C}'$ framing we will have $\textrm{slope}(\Gamma_{\partial N(L_{r-1})})=-1/m$ where $m > 0$, since $\overline{t}(\breve{K}_{r-1})=-B_{r-1} < 0$. Now if $m=B_{r-1}$, then there will be no bypasses on the $\partial N(L_{r-1})$-edge of $\mathcal{A}_{(p_r,q_r)}$, since the $(p_r,q_r)$ ruling would be at maximal twisting. On the other hand, if $m > B_{r-1}$, then there will still be no bypasses on the $\partial N(L_{r-1})$-edge of $\mathcal{A}_{(p_r,q_r)}$, since such a bypass would induce a destabilization of $L_{r-1}$, thus increasing its $tb$ by one – see Lemma 4.4 in [@[H]]. To satisfy the conditions of this lemma, we are using the fact that either $p_r > 0$ or $q_{r}/p_{r} < -1/B_{r-1}$. Furthermore, we can thicken $N_r$ through any bypasses on the $\partial N_r$-edge, and thus assume $\mathcal{A}_{(p_r,q_r)}$ is standard convex. See (a) in Figure \[fig:non-thickening1B\].
\[nonthickening1\] ![[]{data-label="fig:non-thickening1B"}](nonthickening1BB "fig:"){width=".85\textwidth"}
Now let $N_{r-1}:=N_r \cup N(\mathcal{A}_{(p_r,q_r)}) \cup N(L_{r-1})$. By our inductive hypothesis we can thicken $N_{r-1}$ to an $\widetilde{N}_{r-1}$ with intersection boundary slope $-(k_{r-1} + 1)/(A_{r-1}k_{r-1} + B_{r-1})$, and we can assume that $k_{r-1}$ is minimized for all such thickenings. Then consider a convex annulus $\mathcal{\widetilde{A}}$ from $\partial N(L_{r-1})$ to $\partial \widetilde{N}_{r-1}$, such that $\mathcal{\widetilde{A}}$ is in the complement of $N_r$ and $\partial \mathcal{\widetilde{A}}$ consists of $(p_r,q_r)$ rulings. See (b) in Figure \[fig:non-thickening1B\]. We will show that $\mathcal{\widetilde{A}}$ is standard convex. Certainly there are no bypasses on the $\partial N(L_{r-1})$-edge of $\mathcal{\widetilde{A}}$; furthermore, any bypasses on the $\partial \widetilde{N}_{r-1}$-edge must pair up via dividing curves on $\partial \widetilde{N}_{r-1}$ and cancel each other out as in part (a) of Figure \[nonthickening2\], for otherwise a bypass on $\partial N(L_{r-1})$ would be induced via the annulus $\mathcal{\widetilde{A}}$ as in part (b) of Figure \[nonthickening2\]. As a consequence, allowing $\widetilde{N}_{r-1}$ to thin inward through such bypasses does not change the boundary slope, but just reduces the number of dividing curves. But then inductively we can thicken this new $\widetilde{N}_{r-1}$ to a smaller $k_{r-1}$-value, contradicting the minimality of $k_{r-1}$. Thus $\mathcal{\widetilde{A}}$ is standard convex.
\[nonthickening2\] ![[]{data-label="fig:non-thickening2B"}](nonthickening2BB "fig:"){width="80.00000%"}
Now four annuli compose the boundary of a solid torus $\widetilde{N}_r$ containing $N_r$: the two sides of a thickened $\mathcal{\widetilde{A}}$; $\partial \widetilde{N}_{r-1} \backslash \partial \mathcal{\widetilde{A}}$; and $\partial N(L_{r-1}) \backslash \partial \mathcal{\widetilde{A}}$. We can compute the intersection boundary slope of this solid torus. To this end, recall that $\textrm{slope}(\Gamma_{\partial N(L_{r-1})})=-1/m$ where $m > 0$. To determine $m$ we note that the geometric intersection of $(p_r,q_r)$ with $\Gamma$ on $\partial \widetilde{N}_{r-1}$ and $\partial N(L_{r-1})$ must be equal, yielding the equality
$$p_r + mq_r = p_rk_{r-1} + p_r + q_r(A_{r-1}k_{r-1} + B_{r-1})$$
This gives
$$m=p_r\frac{k_{r-1}}{q_r}+A_{r-1}k_{r-1} + B_{r-1}$$
We define the integer $k_r:=k_{r-1}/q_r$. We now choose $(p'_r,q'_r)$ to be a curve on these two tori such that $p_rq'_r-p'_rq_r=1$, and as in Lemma \[basecasethickening\], we change coordinates to $\mathcal{C}''$ via the map $((p_r,q_r),(p'_r,q'_r)) \mapsto ((0,1),(-1,0))$. Under this map we obtain $$\textrm{slope}(\Gamma_{\partial \widetilde{N}_{r-1}}) = \frac{q'_r(A_{r-1}k_{r-1}+B_{r-1})+p'_r(q_rk_r+1)}{A_rk_r+B_r}$$
$$\textrm{slope}(\Gamma_{\partial N(L_{r-1})})=\frac{q'_r(p_rk_r +A_{r-1}k_{r-1}+B_{r-1}) +p'_r}{A_rk_r+B_r}$$
We then obtain in the $\mathcal{C}'$ framing, after edge-rounding, that the intersection boundary slope of $\widetilde{N}_r$ is
$$\begin{aligned}
\textrm{slope}(\Gamma_{\partial \widetilde{N}_r})&=&\frac{q'_r(A_{r-1}k_{r-1}+B_{r-1})+p'_r(q_rk_r+1)}{A_rk_r+B_r}\nonumber\\
&-&\frac{q'_r(p_rk_r +A_{r-1}k_{r-1}+B_{r-1}) +p'_r}{A_rk_r+B_r}\nonumber\\
&-&\frac{1}{A_rk_r+B_r}\nonumber\\
&=&-\frac{k_r+1}{A_rk_r+B_r}\end{aligned}$$
This shows that any $N_r$ representing $\breve{K}_r$ can be thickened to one of the $N_r^k$, and if $N_r$ fails to thicken, then it has the same boundary slope as some $N_r^k$. We now show that if $N_r$ fails to thicken, and if it has the minimum number of dividing curves over all such $N_r$ which fail to thicken and have the same boundary slope as $N_r^k$, then $N_r$ is actually an $N_r^k$.
To see this, as above we can choose a Legendrian $L_{r-1}$ that maximizes $tb$ in the complement of $N_r$ and such that we can join $\partial N(L_{r-1})$ to $\partial N_r$ by a convex annulus $\mathcal{A}_{(p_r,q_r)}$ whose boundaries are $(p_r,q_r)$ and $\infty'$ rulings on $\partial N(L_{r-1})$ and $\partial N_r$, respectively. Again we have no bypasses on the $\partial N(L_{r-1})$-edge, and in this case we have no bypasses on the $\partial N_r$-edge since $N_r$ fails to thicken and is at minimum number of dividing curves.
As above, let $N_{r-1}:=N_r \cup N(\mathcal{A}_{(p_r,q_r)}) \cup N(L_{r-1})$. We claim this $N_{r-1}$ fails to thicken. To see this, take a convex annulus $\mathcal{\widetilde{A}}$ from $\partial N(L_{r-1})$ to $\partial N_{r-1}$, such that $\mathcal{\widetilde{A}}$ is in the complement of $N_r$ and $\partial \mathcal{\widetilde{A}}$ consists of $(p_r,q_r)$ rulings. We know $\mathcal{\widetilde{A}}$ is standard convex since the twisting is the same on both edges and there are no bypasses on the $\partial N(L_{r-1})$-edge. A picture is shown in Figure \[fig:non-thickening3BB\].
![[]{data-label="fig:non-thickening3BB"}](nonthickening3BB){width="40.00000%"}
Now four annuli compose the boundary of a solid torus containing $N_r$: the two sides of the thickened $\mathcal{\widetilde{A}}$, which we will call $\mathcal{\widetilde{A}}_+$ and $\mathcal{\widetilde{A}}_-$; $\partial N_{r-1} \backslash \partial \mathcal{\widetilde{A}}$, which we will call $\mathcal{A}_{r-1}$; and $\partial N(L_{r-1}) \backslash \partial \mathcal{\widetilde{A}}$, which we will call $\mathcal{A}_{L_{r-1}}$. Any thickening of $N_{r-1}$ will induce a thickening of $N_r$ to $\widetilde{N}_r$ via these four annuli.
Suppose, for contradiction, that $N_{r-1}$ thickens outward so that $\textrm{slope}(\Gamma_{\partial N_{r-1}})$ changes. Note that during the thickening, $\mathcal{A}_ {L_{r-1}}$ stays fixed. We examine the rest of the annuli by breaking into two cases.
**Case 1:** After thickening, suppose $\mathcal{\widetilde{A}}$ is still standard convex; that means both $\mathcal{\widetilde{A}}_+$ and $\mathcal{\widetilde{A}}_-$ are standard convex. Since we can assume that after thickening $\mathcal{A}_{r-1}$ is still standard convex, this means that in order for $\textrm{slope}(\Gamma_{\partial N_{r-1}})$ to change, the holonomy of $\Gamma_{\mathcal{A}_{r-1}}$ must have changed. But this will result in a change in $\textrm{slope}(\Gamma_{\partial N_r})$, since $\mathcal{A}_ {L_{r-1}}$ stays fixed and any change in holonomy of $\Gamma_{\mathcal{\widetilde{A}}_+}$ and $\Gamma_{\mathcal{\widetilde{A}}_-}$ cancels each other out and does not affect $\textrm{slope}(\Gamma_{\partial N_r})$. Thus we would have a slope-changing thickening of $N_r$, which by hypothesis cannot occur.
**Case 2:** After thickening, suppose $\mathcal{\widetilde{A}}$ is no longer standard convex. Now note that there are no bypasses on the $\partial N(L_{r-1})$-edge of $\mathcal{\widetilde{A}}$; furthermore, any bypass for $\mathcal{\widetilde{A}}_+$ on the $\partial N_{r-1}$-edge must be cancelled out by a corresponding bypass for $\mathcal{\widetilde{A}}_-$ on the $\partial N_{r-1}$-edge as in part (a) of Figure \[fig:non-thickening2B\], so as not to induce a bypass on the $\partial N(L_{r-1})$-edge as in part (b) of the same figure. But then again, in order for $\textrm{slope}(\Gamma_{\partial N_r})$ to remain constant, the holonomy of $\Gamma_{\mathcal{A}_{r-1}}$ must remain constant, and thus $\textrm{slope}(\Gamma_{\partial N_{r-1}})$ must also have remained constant, with just an increase in the number of dividing curves.
This proves the claim that $N_{r-1}$ does not thicken, and we therefore know that its boundary slope is $-(k_{r-1}+1)/(A_{r-1}k_{r-1}+B_{r-1})$. Furthermore, we know the number of dividing curves is $2n$ where $n \geq n_{r-1}^{k_{r-1}}$. Suppose, for contradiction, that $n > n_{r-1}^{k_{r-1}}$. Then we know we can thicken $N_{r-1}$ to an $N_{r-1}^{k_{r-1}}$, and if we take a convex annulus from $\partial N_{r-1}$ to $\partial N_{r-1}^{k_{r-1}}$ whose boundaries are $(p_r,q_r)$ rulings, by the Imbalance Principle there must be bypasses on the $\partial N_{r-1}$-edge. But these would induce bypasses off of $\infty'$ rulings on $N_r$, which by hypothesis cannot exist. Thus $n = n_{r-1}^{k_{r-1}}$, and by a calculation as above we obtain that the intersection boundary slope of $N_r$ must be $-(k_{r}+1)/(A_{r}k_{r}+B_{r})$ for the integer $k_r=k_{r-1}/q_r$.
Note the following inequality, which, among other things, shows that the boundary slopes of solid tori representing $\breve{K}_r$ that may fail to thicken are contained in the interval $[-1/B_r, -1/A_r)$.
$$\label{simple 1}
-\frac{1}{B_r} < -\frac{2}{A_r + B_r} < -\frac{3}{2A_r + B_r} < \cdots < -\frac{k_r + 1}{A_rk_r + B_r} < \cdots < -\frac{1}{A_r}$$
To conclude this section, we have the following lemma:
\[widthKr\] $w(\breve{K}_r)=\overline{tb}(\breve{K}_r)$
Using the inequality above, it suffices to show that any solid torus $N_r$ representing $\breve{K}_r$ can be thickened to a solid torus with boundary slope $-(k_r + 1)/(A_rk_r + B_r)$ for some nonnegative integer $k_r$, for then to prevent overtwisting it would have to be the case that $\textrm{slope}(\Gamma_{\partial N_r}) \in [-1/B_r,0)$. But by the above lemma this is true.
Legendrian simplicity of $\breve{K}_r$
======================================
We now use the strategy outlined in §3 to prove Theorem \[main theorem\]. Since Theorem \[main theorem\] is true for positive torus knots [@[EH2]], we can inductively assume that it holds for $\breve{K}_{r-1}$. We then prove it true for $\breve{K}_r$. The proof will parallel the proof from [@[EH1]] that $K$ being simple and satisfying the UTP guarantees simplicity of cablings for cabling fractions that are greater than the contact width. However, in our case $\breve{K}_{r-1}$ may not satisfy the UTP, so we will need appropriate modifications for our proof.
We begin by showing that if $L_r$ and $L_r^\prime$ have maximal $\overline{tb}(\breve{K}_r)=A_r-B_r$, then they are Legendrian isotopic. Now $\overline{t}(\breve{K}_r)=-B_r < 0$, so we can assume that both $L_r$ and $L_r^\prime$ exist as Legendrian rulings on convex tori $\partial N_{r-1}$ and $\partial N_{r-1}^\prime$. Let $\textrm{slope}(\Gamma_{\partial N_{r-1}})=-\frac{a}{b}$ be an intersection boundary slope where $a,b>0$. Then $-a/b \geq -1/B_{r-1}$, and we have $b \geq aB_{r-1}$. But since $t(L_r)=-B_r$, we also have $ap_r + bq_r = B_r$. Combining this equality and inequality we obtain $B_r \geq ap_r + aq_rB_{r-1}=aB_r$, which implies $a=1$ and $b=B_{r-1}$. Hence, we can assume that $L_r$ lies on a convex torus with boundary slope $-1/B_{r-1}$, and similarly for $L_{r}^\prime$.
Now by Proposition 4.3 in [@[H]], each solid torus with boundary slope $-1/B_{r-1}$ is contact isotopic to the standard neighborhood of a Legendrian representative of $K_{r-1}$ with $t(L_{r-1})=-B_{r-1}$; both $L_{r}$ and $L_{r}^\prime$ are Legendrian rulings on such a boundary torus. But inductively there is only one such Legendrian $L_{r-1}$ at maximal $\overline{t}(K_{r-1})=-B_{r-1}$. Thus, as in the proof of Lemma 3.4 in [@[EH1]], we may assume that $L_{r}$ and $L_{r}^\prime$ are Legendrian rulings on the same boundary torus, and hence Legendrian isotopic via the rulings.
We now show that if $tb(L_r) < \overline{tb}(\breve{K}_r)$ then $L_r$ destabilizes using a bypass. To this end, we note that since $q_r > 1$, we have
$$\label{simple 2}
-\frac{1}{(A_r/q_r)} < -\frac{2}{A_r + B_r}$$
We first suppose that $t(L_r) = -m$, where $B_r < m \leq (A_r/q_r)$ (note that $B_r < (A_r/q_r)$ for $r>1$). Then $N(L_r)$ has boundary slope $-1/m \leq -1/(A_r/q_r)$, and this, combined with Lemma \[nonthickening inductive step\] and inequalities \[simple 1\] and \[simple 2\], allows us to conclude that $N(L_r)$ can be thickened to a solid torus $N_r$ with intersection boundary slope $-1/B_r$. Then an $\infty^\prime$ ruling on $N(L_r)$ can be destabilized using a bypass on a convex annulus joining the two tori.
Now suppose alternatively that $m > (A_r/q_r)$. In this case, we look at $L_r$ as a $(p_r,q_r)$ Legendrian ruling on the convex boundary of a solid torus $N_{r-1}$ with boundary slope $s$. We may assume that $L_r$ intersects the dividing set efficiently, for otherwise $L_r$ immediately destabilizes. Note first that if $L_r^\prime$ is a $(p_r,q_r)$ ruling on a solid torus with intersection boundary slope $-1/A_{r-1}$, then $t(L_r^\prime)=-(A_r/q_r)$. In light of this, note that by Lemmas \[nonthickening inductive step\] and \[widthKr\] and inequality \[simple 1\], as well as Lemmas 3.15 and 3.16 in [@[EH2]], we must have $N_{r-1}$ either containing a solid torus with intersection boundary slope $-1/A_{r-1}$ (if $s < -1/A_{r-1}$), or $N_{r-1}$ must thicken to a solid torus of intersection boundary slope $-1/A_{r-1}$ (if $s > -1/A_{r-1}$). Either way, we can connect $L_r$ to an $L_r^\prime$ via a convex annulus and destabilize $L_r$ using a bypass.
This proves Theorem \[main theorem\]; a change of coordinates from $\mathcal{C}'$ to $\mathcal{C}$ then yields Corollary \[fractionslargerthanwidth\].
Legendrian simple cablings of $\breve{K}_r$ that satisfy the UTP
================================================================
We now prove Theorem \[main theorem 3\]:
Recall that we are given $q_{r+1}/p_{r+1} \in (-1/A_r,0)$. Note first that in this case $P_{r+1} = p_{r+1}+q_{r+1}A_r < 0$ in the $\mathcal{C}$ framing. Moreover, since $w(\breve{K}_r)=A_r-B_r > 0$, we have that $P_{r+1}/q_{r+1} < w(\breve{K}_r)$. Our proof for this case will thus parallel the proof in [@[EH1]] that $K$ being Legendrian simple and satisfying the UTP, along with $P/q < w(K)$, guarantees that the $(P,q)$ cabling is also Legendrian simple and satisfies the UTP. In our case, $\breve{K}_r$ does not necessarily satisfy the UTP, and thus we will need appropriate modifications for our proof.
The proof will require five steps:
- Show that $\overline{tb}(K_{r+1})=A_{r+1}$.
- Show that $K_{r+1}$ satisfies the UTP.
- Calculate $r(L_{r+1})$ at $\overline{tb}$ and show that Legendrian isotopy classes at $\overline{tb}$ are determined by their rotation numbers.
- Show that if $tb(L_{r+1}) < \overline{tb}$, then $L_{r+1}$ destabilizes.
- Show that if $L_{r+1}$ is in a valley of the Legendrian mountain range (ie, $(r(L_{r+1})\pm 1,tb(L_{r+1})+ 1)$ have images in the mountain range, but $(r(L_{r+1}),tb(L_{r+1})+2)$ does not), then $L_{r+1}$ can destabilize both positively and negatively.
**Step 1:** Our analysis in the first two steps will draw heavily from ideas in the proof of Theorem 1.2 in [@[EH1]] that negative torus knots satisfy the UTP. We first examine representatives of $K_{r+1}$ at $\overline{tb}$. Since there exists a convex torus representing $\breve{K}_r$ with Legendrian divides that are $(p_{r+1},q_{r+1})$ cablings (inside of the solid torus representing $\breve{K}_r$ with $\textrm{slope}(\Gamma)=-1/A_r$) we know that $\overline{tb}(K_{r+1}) \geq P_{r+1}q_{r+1}=A_{r+1}$. To show that $\overline{tb}(K_{r+1}) = A_{r+1}$, we show that $\overline{t}(K_{r+1})=0$ by showing that the contact width $w(K_{r+1},\mathcal{C}')=0$, since this will yield $\overline{tb}(K_{r+1}) \leq w(K_{r+1})=A_{r+1}$. So suppose, for contradiction, that some $N_{r+1}$ has convex boundary with $\textrm{slope}(\Gamma_{\partial N_{r+1}})=s > 0$, as measured in the $\mathcal{C}'$ framing, and two dividing curves. After shrinking $N_{r+1}$ if necessary, we may assume that $s$ is a large positive integer. Then let $\mathcal{A}$ be a convex annulus from $\partial N_{r+1}$ to itself having boundary curves with slope $\infty'$. Taking a neighborhood of $N_{r+1} \cup \mathcal{A}$ yields a thickened torus $R$ with boundary tori $T_1$ and $T_2$, arranged so that $T_1$ is inside the solid torus $N_r$ representing $\breve{K}_r$ bounded by $T_2$.
Now there are no boundary parallel dividing curves on $\mathcal{A}$, for otherwise, we could pass through the bypass and increase $s$ to $\infty'$, yielding excessive twisting inside $N_{r+1}$. Hence $\mathcal{A}$ is in standard form, and consists of two parallel nonseparating arcs. We now choose a new framing $\mathcal{C}''$ for $N_r$ where $(p_{r+1},q_{r+1}) \mapsto (0,1)$; then choose $(p'',q'') \mapsto (1,0)$ so that $p''q_{r+1}-q''p_{r+1}=1$ and such that $\textrm{slope}(\Gamma_{T_1})=-s$ and $\textrm{slope}(\Gamma_{T_2})=1$. As mentioned in the third paragraph of the proof of Theorem 1.2 in [@[EH1]], this is possible since $\Gamma_{T_1}$ is obtained from $\Gamma_{T_2}$ by $s+1$ right-handed Dehn twists. Then note that in the $\mathcal{C'}$ framing, we have that $q_{r+1}/p_{r+1} > \textrm{slope}(\Gamma_{T_2})=(q''+q_{r+1})/(p''+p_{r+1}) > q''/p''$, and $q_{r+1}/p_{r+1}$ and $q''/p''$ are connected by an arc in the Farey tessellation of the hyperbolic disc (see section 3.4.3 in [@[H]]). Thus, since $-1/A_r$ is connected by an arc to $0/1$ in the Farey tessellation, we must have that $(q''+q_{r+1})/(p''+p_{r+1}) > -1/A_r$. Thus we can thicken $N_r$ to a standard neighorhood with $\textrm{slope}(\Gamma)=-1/A_r$. Then, just as in Claim 4.2 in [@[EH1]], we have the following:
- inside $R$ there exists a convex torus parallel to $T_i$ with slope $q_{r+1}/p_{r+1}$;
- $R$ can thus be decomposed into two layered [*basic slices*]{};
- the tight contact structure on $R$ must have [*mixing of sign*]{} in the Poincar$\acute{\textrm{e}}$ duals of the relative half-Euler classes for the layered basic slices;
- this mixing of sign cannot happen inside the universally tight standard neighborhood with $\textrm{slope}(\Gamma)=-1/A_r$.
This contradicts $s > 0$. So $\overline{tb}(K_{r+1})=P_{r+1}q_{r+1}=A_{r+1}$.
**Step 2:** Here we show that any $N_{r+1}$ can be thickened to a standard neighborhood of $L_{r+1}$ with $t(L_{r+1})=0$. So suppose that $N_{r+1}$ has convex boundary with $\textrm{slope}(\Gamma_{\partial N_{r+1}})=s$, as measured in the $\mathcal{C}'$ framing, where $-\infty' < s < 0$. Construct $R$ as in Step 1 above, and look at the convex annulus $\mathcal{A}$, which in this case may not be standard convex. If all dividing curves on $\mathcal{A}$ are boundary parallel arcs, then $N_{r+1}$ can be thickened to have boundary slope $\infty'$. On the other hand, if there are nonseparating dividing curves on $\mathcal{A}$ after going through bypasses, then the resulting $T_2$ will have negative boundary slope in the $\mathcal{C}''$ framing, and we can thicken $N_r$ to obtain a convex torus outside of $R$ on the $T_2$-side with slope $q_{r+1}/p_{r+1}$ in the $\mathcal{C}'$ framing, since $q_{r+1}/p_{r+1} > -1/A_r$ and thickening can occur. Then using the Imbalance Principle we can thicken $N_{r+1}$ to have boundary slope $\infty'$.
It remains to show that we can achieve just two dividing curves for this $N_{r+1}$. Note that $N_{r+1}$ is contained in a thickened torus $R$ representing $\breve{K}_r$ with $\partial R = T_2-T_1$ and where the dividing curves on $T_i$ have slope $q_{r+1}/p_{r+1}$. The key now is that since $q_{r+1}/p_{r+1} \in (-1/A_r,0)$, there is twisting on both sides of $R$. We can thus reduce the number of dividing curves on $N_{r+1}$ by either finding bypasses in $R\backslash N_{r+1}$ or by finding bypasses along $T_1$ or $T_2$ that can be extended into $R$, as in the proofs of Claims 4.1 and 4.3 in [@[EH1]].
**Step 3:** We now show that the $L_{r+1}$ at $\overline{tb}$ are distinguished by their rotation numbers. To do this, we first note that since $q_{r+1}/p_{r+1} > -1/A_r$, there exists an integer $n \geq A_r$ with $-1/A_r \leq -1/n < q_{r+1}/p_{r+1} < -1/(n+1)$. Changing to the standard $\mathcal{C}$ framing yields $-1/(n-A_r) < q_{r+1}/P_{r+1} < -1/((n+1)-A_r)$. This thickened torus bounded by the tori with slopes $-1/(n-A_r)$ and $-1/((n+1)-A_r)$ is a universally tight [*basic slice*]{} in the sense of [@[H]], and thus by an argument identical to that in Lemma 3.8 in [@[EH1]] we have that the set of rotation numbers achieved by $L_{r+1}$ at $\overline{tb}$ is:
$$\label{rlr+1}
r(L_{r+1}) \in \left\{\pm (P_{r+1} + q_{r+1}(n-A_r+r(L_r))) | tb(L_r)=A_r-n\right\}$$
Changing to $p_j$’s and $q_j$’s yields:
$$r(L_{r+1}) \in \left\{\pm (p_{r+1} + nq_{r+1}+q_{r+1}r(L_r)) | tb(L_r)=A_r-n\right\}$$
Now we know from the Legendrian classification of $\breve{K}_r$ that if $tb(L_r)=A_r-n$, then
$$r(L_r) \in \left\{-(n-B_r), -(n-B_r)+2, \cdots, (n-B_r)-2,(n-B_r)\right\}$$
Plugging these values of $r(L_r)$ just into the values $r(L_{r+1})=p_{r+1} + nq_{r+1}+q_{r+1}r(L_r)$ yields $r(L_{r+1})$ that begin on the left at $B_{r+1} < 0$, and then increase by $2q_{r+1}$, ending at $p_{r+1} + nq_{r+1}+q_{r+1}(n-B_r)$. Reflecting these values across the $r=0$ axis yields the $r(L_{r+1})=-(p_{r+1} + nq_{r+1}+q_{r+1}r(L_r))$; these two distributions interleave to form one total distribution of $r$-values. Thus, if we define $s=-p_{r+1}-nq_{r+1}$ we have that the distribution of $r(L_{r+1})$ when $\overline{tb}(L_{r+1})=A_{r+1}$ is as follows:
$$B_{r+1} < B_{r+1}+2s < B_{r+1}+2q_{r+1} < \cdots < -(B_{r+1}+2q_{r+1}) < -(B_{r+1}+2s) < -B_{r+1}\nonumber$$
Note that $q_{r+1} > s > 0$. Algorithmically, the distribution of values for $r(L_{r+1})$ is achieved as follows: begin on the left at $B_{r+1}$, and then move right to the next $r$-value by alternating lengths of $2s$ and $2(q_{r+1}-s)$, until one reaches $-B_{r+1}$. As mentioned in [@[EH1]], a way to see where these rotation numbers come from is noting that to each $L_r$ with $tb(L_r)=A_r-n$, there corresponds two $L_{r+1}^{\pm}$ at $\overline{tb}$, where $r(L_{r+1}^{\pm})=q_{r+1}r(L_r)\pm s$. $L_{r+1}^{\pm}$ is obtained by removing a standard neighborhood of $N(S_{\pm}(L_r))$ from $N(L_r)$ and taking a Legendrian divide on a torus with slope $q_{r+1}/p_{r+1}$ inside $N(L_r) \backslash N(S_{\pm}(L_r))$. Here $S_+$ indicates positive stabilization and $S_-$ means negative stabilization. As a consequence, if $L_{r+1}$ and $L'_{r+1}$ are both at $\overline{tb}$ and have the same rotation number, then they must exist in basic slices that are associated to $L_r$ and $L'_r$ at $tb=A_r-n$ and having the same rotation number, as well as the same parity of stabilization for $L_r$ and $L'_r$. These basic slices are thus contact isotopic since $\breve{K}_r$ is Legendrian simple, yielding a Legendrian isotopy from $L_{r+1}$ to $L'_{r+1}$ using a linearly foliated torus – see Lemma 3.17 in [@[EH2]].
**Step 4:** We now show that if $tb(L_{r+1}) < \overline{tb}$, then $L_{r+1}$ destabilizes. To see this, note that since $\overline{t}(K_{r+1})=0$, if $L_{r+1}$ has $tb(L_{r+1}) < \overline{tb}$, we know that $L_{r+1}$ is a Legendrian ruling on the boundary of a solid torus $N_r$ and that $N_r$ either contains a solid torus with $\textrm{slope}(\Gamma)=q_{r+1}/p_{r+1}$ or can be thickened to a solid torus with such a boundary slope, since $q_{r+1}/p_{r+1} > -1/A_r$. Thus $L_{r+1}$ will destabilize by the Imbalance Principle.
**Step 5:** We now show that if $L^v_{r+1}$ is in a valley of the Legendrian mountain range, that is $(r(L^v_{r+1})\pm 1,tb(L^v_{r+1}) + 1)$ have images in the mountain range, but $(r(L^v_{r+1}),tb(L^v_{r+1})+2)$ does not, then there are two Legendrian representatives of $K_{r+1}$ at $\overline{tb}$, namely the two closest peaks $L^+_{r+1}$ and $L^-_{r+1}$, such that $L^v_{r+1}=S^m_+(L^-_{r+1})=S^m_-(L^+_{r+1})$ for some $m>0$.
To see this, first note that from the distribution of rotation numbers at $\overline{tb}$, there are two types of valleys, those with depth $s$, and those with depth $q_{r+1}-s$. We first consider valleys of depth $s$. Such a valley falls between two peaks represented by Legendrian knots at $\overline{tb}$, where $r(L^+_{r+1})=q_{r+1}r(L_r)+s$ and $r(L^-_{r+1})=q_{r+1}r(L_r)-s$. So $r(L^v_{r+1})=q_{r+1}r(L_r)$ and $t(L^v_{r+1})=p_{r+1}+nq_{r+1}$; hence $L^v_{r+1}$ is a $(p_{r+1},q_{r+1})$ ruling on a standard neighborhood of $L_r$ where $t(L_r)=-n$. Then we can stabilize $L_r$ both positively and negatively to obtain two different basic slices having boundary slopes $-1/n$ and $-1/(n+1)$. In the one, there will be a boundary parallel torus with $t(L_{r+1})=0$ and a convex annulus that results in $s$ positive destabilizations of $L^v_{r+1}$; in the other there will be a convex annulus to a similar torus that results in $s$ negative destabilizations of $L^v_{r+1}$.
Now consider a valley of depth $q_{r+1}-s$. Then such a valley falls between two peaks represented by $r(L^+_{r+1})$ and $r(L^-_{r+1})$ where $r(L^+_{r+1})=q_{r+1}r(L_r) - s$. Thus $r(L^v_{r+1})=q_{r+1}(r(L_r)-1)$ and $t(L^v_{r+1})=-p_{r+1}-(n+1)q_{r+1}$; hence $L^v_{r+1}$ is a $(p_{r+1},q_{r+1})$ ruling on a standard neighborhood of $S_-(L_r)$. Now note that if $r(L_r)=-(n-B_r)$, that would imply that $r(L^+_{r+1})=B_{r+1}$, which is not true. Thus a consideration of the Legendrian mountain range for $\breve{K}_r$ allows us to conclude that $S_-(L_r)$ destabilizes both positively and negatively to obtain two different basic slices having boundary slopes $-1/n$ and $-1/(n+1)$. In the one, there will be a boundary parallel torus with $t(L_{r+1})=0$ and a convex annulus that results in $q_{r+1} - s$ positive destabilizations of $L^v_{r+1}$; in the other there will be a convex annulus to a similar torus that results in $q_{r+1} - s$ negative destabilizations of $L^v_{r+1}$.
This proves Theorem \[main theorem 3\]; a change of coordinates from $\mathcal{C}'$ to $\mathcal{C}$ then yields Corollary \[negativecablings\].
\
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[^1]: 2000 *Mathematics Subject Classification*. Primary 57M25, 57R17; Secondary 57M50
|
**Antimonotonous quadratic forms\
and partially ordered sets**
**L.A. Nazarova $^*$, A.V. Roiter $^{*}$, M.N. Smirnova $^{*}$**
$^{*}$ Institute of Mathematics of National Academy of Sciences of Ukraine,\
Tereshchenkovska str., 3, Kiev, Ukraine, ind. 01601\
E-mail: roiter@imath.kiev.ua
Let $P$ be a bounded set in $n$-dimensional space ${\mathbb R}_n$, $f(x_1,\ldots, x_n)=f(x)$ ($x\in \mathbb{R}_n$) be a continuous function. By the second Weierstrass theorem ([@13], s. 163) $\inf
\{f(\overline{P})\}$ $(=\inf\limits_{\overline P}f(x))$ is reached. We will call a function $f$ [*$P$-faithful*]{}, if $\inf
\{f(\overline{P})\}$ is not reached on $\overline{P}\setminus P$ and $\inf\{f(\overline{P})\}>0$ (i. e. $f$ is positive on $\overline P$). Remark that if $n=1$, $P=(a,b)$, then it is necessary for $P$-faithfulness that a function is not monotonous.
Further we will suppose that $P=P_n=\{(x_1,\ldots,x_n)\,|\,0<
x_i\leq 1, \,i=\overline{1,n},\, x_1+\cdots+x_n=1\}$. If $n>1$ then $x_i<1$, $i=\overline{1,n}$. Then $\overline{P}=\{(x_1,\ldots,x_n) | 0\leq
x_i\leq 1,\; i=\overline{1,n},\; x_1+\ldots +x_n=1\}$. In this case the $P$-faithfullness $f$ is essentially connected with behavior of function on the hyperplane $H_n=\{(x_1,\ldots,x_n)\,|\,
x_1+\cdots+x_n=0\}$.
For differentiable function $f$ put $C^-(f)=\{h\in H_n\setminus \{0\}\,|\, \frac{\partial f}{\partial
x_i}(h)\leq 0,\, i=\overline{1,n}\}$, $C^+(f)=\{h\in H_n\setminus
\{0\}\,|\, \frac{\partial f}{\partial x_i}(h)\geq 0,\, i=\overline{1,n}\}$, $C(f)=C^+(f)\cup C^-(f)$. A function $f$ is [*antimonotonous*]{} if $C(f)=\varnothing$. If $n=1$ then $P_1=(1)$, $H_1\setminus
\{0\}=\varnothing$ and any function is antimonotonous.
In s. 3 (Proposition 1) we prove that any $P$-faithful quadratic form is antimonotonous and, therefore, in this case antimonotonousness is a generalization of $P$-faithfulness.
[**Example 1.**]{} A linear function $f=\sum\limits_{i=1}^n a_i x_i$ is antimonotonous only if all $a_i=0$, i. e. $f=0$. The quadratic forms $x_1^2+x_2^2$, $x_1^2+x_2^2+x_1x_2$ are antimonotonous, but the forms $x_1^2-x_2^2$, $x_1^2+x_2^2+x_3^3+x_1x_2+x_1x_3$ are not.
The problem of effective criterion of antimonotonousness even for quadratic forms is probably difficult.
In this article we solve this problem for quadratic form $f_S$, attached to (finite) partially ordered set (poset) $S=\{s_1,\ldots,s_n\}$: $f_S(x_1,\ldots,x_n)=\sum\limits_{s_i\leq s_j}x_i x_j$ [@1] under the additional condition of positive semidefiniteness form $f_S$ (i. e. $f_S(x)\geq 0$). Posets with antimonotonous form generalize $P$-faithful posets, defined in [@2] and studied in [@2; @3; @4; @5; @6] and (it follows from this work) coincide with them not only for positive definite forms, but for positive semidefinite ones.
A direct construction of a vector contained in $C(f_S)$ permits essentially to simplify the proof of criterion of $P$-faithfulness [@2; @4; @5; @6] avoiding consideration of many different cases.
We also take out evident formula for calculating of $\inf\{f_S(\overline{P})\}$ for $P$-faithful $S$, on base of this formula we give simple proofs of the criterions of finite representativity [@7] (see also [@8] and tameness [@9], (see also [@10]) of partially ordered sets.
[**1.**]{} In s. 1 $f$ is a differentiable function, defined on ${\mathbb R}_n$. We call vectors the elements of ${\mathbb R}_n$.
Let ${\mathbb
R}^+_n=\{x\in{\mathbb R}_n\,|\,
x_i>0,\,i=\overline{1,n}\}$, ${\overline{\mathbb
R}}^+_n=\{x=(x_1,\ldots,x_n)\in {\mathbb R}_n\,|\, 0\leq x_i,
i=\overline{1,n}; x\not=0\}$, $P_n=\overline{P}_n\cap {\mathbb R}^+_n$ ($\mathbb{R}^+=\mathbb{R}_1^+$).
If $f_1$, $f_2$ are defined respectively on ${\mathbb R}_m$ and on ${\mathbb R}_n$ we put $(f_1\oplus f_2)(x_1,\ldots,x_m$, $x_{m+1},\ldots,x_{n+m})=f_1(x_1,\ldots,x_{m})+
f_2(x_{m+1},\ldots,x_{m+n})$.
We call a twice differentiable function $f$ [*concave*]{}, if the following conditions hold:
a\) $\frac{\partial f}{\partial x_i}(0)=0,\; i=\overline{1,n}$,
b\) ${\partial^2f\over \partial x_i \partial x_j} \geq 0$, $i,j=\overline{1,n}$,
[*$q$-conceve*]{}, $q\in\mathbb{R}^+$, if a), b) and
c\) ${\partial^2f\over \partial x_i^2} \geq q$, $i=\overline{1,n}$ hold.
A quadratic form $f_S$ attached to poset $S$ is, in particular, 2-concave.
[**Remark 1.**]{} By Lagrange theorem b) implies $I$, c) implies II$_{q}$.
I\) $\frac{\partial f}{\partial x_i}(x_1,\ldots,x_{j-1},x_j+d,x_{j+1},\ldots,x_n)\geq
\frac{\partial f}{\partial x_i}(x_1,\ldots,x_n)$, $(i, j\in 1,\ldots,n)$.
II$_q$) $\frac{\partial f}{\partial x_i}(x_1,\ldots,x_{i-1}, x_{i}+d,x_{i+1},\ldots,x_n)\geq
\frac{\partial f}{\partial x_i}(x_1,\ldots,x_n)+qd$, $(i=\overline{1,n})$;
Put $\widehat C^-(f)=\Big\{x\in {\mathbb R}_n\setminus
\{0\}\,|\,\sum\limits_{i=1}^n x_i\geq 0,\ {\partial f\over\partial
x_i}(x)\leq 0,\ i=\overline{1,n}\Big\}$; $\widehat C^+(f)=\Big\{x\in
{\mathbb R}_n\setminus \{0\}\,|\,\sum\limits_{i=1}^n x_i\leq 0,\ {\partial
f\over\partial x_i}(x)\geq 0,\ i=\overline{1,n}\Big\}$.
[**Lemma 1.**]{} [*If $f$ is a concave function then it is antimonotonous if and only if $\widehat C^+(f)\cup \widehat
C^-(f)=\varnothing$.*]{}
Let $x\in \widehat C^-(f)$ (the case $x\in \widehat C^+(f)$ is analogous), $\sum\limits_{i=1}^d x_i=d\in {\mathbb R}^+$. Then $\{x_1-d,x_2,\dots,x_n\}\in C(f)$ (according to I) if only $x\neq
(d,0,\dots,0)$. But at the last case $(d,-d,0,\dots,0)\in C(f)$. If $y\in C(f)$ then it is clearly that $y\in \widehat C^+(f)\cup \widehat
C^-(f)$.
------------------------------------------------------------------------
[**Lemma 2.**]{} [*If $f_1$ and $f_2$ are concave then the function $f_1\oplus f_2$ is antimonotonous if and only if $f_1$ and $f_2$ are antimonotonous.*]{}
We shall prove that $C(f_1\oplus f_2)\neq\varnothing$ if and only if either $C(f_1)\neq\varnothing$ or $C(f_2)\neq\varnothing$. If $(x_1,\dots,x_{n_1},y_1\dots,y_{n_2})\in C(f_1\oplus f_2)$ then either $(x_1,\dots,x_{n_1})\in \widehat C^+(f_1)\cup \widehat C^-(f_1)$ or $(y_1,\dots,y_{n_2})\in \widehat C^+(f_2)\cup \widehat C^-(f_2)$ and by Lemma 1 in the first case $C(f_1)\neq\varnothing$ and in the second case $C(f_2)\neq \varnothing$.
If $x=(x_1,\dots,x_{n_1})\in C(f_1)$ then $(x_1,\dots,x_{n_1},\underbrace{0,\dots,0}\limits_{n_2})\in
C(f_1\oplus f_2)$ using $a)$; if $(y_1,\dots,y_{n_2})\in
C(f_2)$ then $(\underbrace{0,\dots,0}\limits_{n_1},y_1,\dots,y_{n_2})\in
C(f_1\oplus f_2)$.
------------------------------------------------------------------------
We say that a nonzero vector $d\in \mathbb{Z}_n$ is [*$m$-Dynkin*]{}) $(1\leq m\leq n)$ for $q$-concave function $f$, if 1) $0\leq \frac{\partial f}{\partial x_m}(d)\leq q$; 2) $\frac{\partial f}{\partial x_j}(d)=0$ if $j\not= m$, $j=\overline{1,n}$.
We call a function $f$ [*$m$-isolated*]{}, if ${\partial f\over\partial x_k}(s_m)=0$ for $1\leq
k\leq n$, $k\neq m$, $s_m=(\underbrace{0,\dots,0}\limits_{m-1},1,0,\dots,0).$
[**Lemma 3.**]{} [*Let $f$ be $q$-concave not $m$-isolated function. If there exists $m$-Dynkin vector then $C(f)\not=\varnothing$.*]{}
Let $\sum\limits_{i=1}^n d_i=\overline d$. If $\overline d\leq 0$ then $d\in \widehat C^+(f)$ and $C(f)\not=\varnothing$ by Lemma 1. Let $\overline d>0$. Put $u_j=d_j$ if $j\not =m$ and $u_m=d_m-\overline d$, and we prove that $u=(u_1,\ldots,u_n)\in C(f)$. It is clear that $u\in H_n$. $\frac{\partial f}{\partial x_j}(u)\leq 0$ if $j\not=m$ according to I and 2, and $\frac{\partial f}{\partial x_m}(u)\leq 0$ according to II$_q$ and 1).
Now we shall prove that $u\neq 0$. If $u=0$ then $d=\lambda s_m$, $\lambda\neq 0$ (since $d\neq 0$). But then non $m$-isolateness $f$ implies that ${\partial f\over\partial
x_k}(d)\neq 0$, $k\neq m$.
------------------------------------------------------------------------
[**Example 2.**]{} Let $S=\{s_1,s_2,s_3,s_4,s_5\,|\,
s_1<s_i,\,i=\overline{2,5}\}$, $f_S=\sum\limits_{i=1}^5 x_i^2+x_1\sum\limits_{j=2}^5 x_j$, $d=(-2,1,1,1,1)$ — $i$-Dynkin vector for $f_S$ $(i=\overline{1,5})$. Vectors $(-2,1,1,1,-1)$, $(-2,1,1,-1,1)$, $(-2,1,-1,1,1)$, $(-2,-1,1,1,1)$ and $(-4,1,1,1,1)$ belong to $C(f_S)$.
Consider $P$-faithfulness. Put ${\rm St}(f)=\{a\in{\mathbb R}^+_n\,|\,
\frac{\partial f}{\partial x_i}(a)=\frac{\partial f}{\partial x_j}(a)$, $i,j=\overline{1,n}\}$; ${\rm St}^+(f)=\{a\in{\rm St}\,|\,
\frac{\partial f}{\partial x_i}(a)>0\}$.
Vector $u\in P_n$ is [*$P$-faithful*]{} for function $f$, if $f(u)>0$ and $w\in \overline{P}_n$ implies $f(u)\leq f(w)$, moreover if $w\not\in P_n$, then $f(u)<f(w)$.
Denote $\widetilde{{\rm St}}(f)$ the set of $P$-fathful vectors for $f$. $P$-faithfulness of $f$ is equivalent to $\widetilde{{\rm St}}(f)\not=0$.
[**Lemma 4.**]{} [*$\widetilde{{\rm St}}(f)\subseteq
{\rm St}(f)$ for any $f$.*]{}
Let $n>1$. We express $x_n=1-\sum\limits_{i=1}^{n-1}x_i$, and obtain the function $\widehat{f}(x_1,\ldots,x_{n-1}) = f(x_1,\ldots, x_{n-1},
1-\sum\limits_{i=1}^{n-1}x_i)$. If $u=(u_1,\ldots,u_n)$ is $P$-faithful vector for $f$ then $\widehat{f}(\widehat{u})$ is minimum when $\widehat{u}=(u_1,\ldots,u_{n-1})$. $\frac{\partial \hat f}{\partial
x_i}=\frac{\partial f}{\partial x_i}+ \frac{\partial f}{\partial x_n}\cdot
\frac{\partial x_n}{\partial x_i}$; $x_n=1-\sum\limits_{i=1}^{n-1}x_i$; $\frac{\partial x_n}{\partial x_i}=-1$ $(i=\overline{1,n-1})$. Therefore $\frac{\partial \hat f}{\partial x_i}=\frac{\partial f}{\partial x_i}-
\frac{\partial f}{\partial x_n}=0$.
------------------------------------------------------------------------
Let $f$ be a homogeneous function of degree $k$ (i. e. $f(\lambda x_1,\ldots,\lambda x_n)=\lambda^k f(x_1,\ldots,x_n)$). For homogeneous $f$, if $k\not=1$ and $\inf\{f(P)\}>0$ put $P(f)=\inf
\{f(P)\}^{\frac{1}{1-k}}$. In particular, for $k=2$, $P(f)=\inf
\{f(\overline{P})\}^{-1}$.
[**Lemma 5.**]{} [*Let $f_1(x_1,\ldots,x_{n_1})$, $f_2(x_{n_1+1},\ldots, x_{n_2})$ be two homogeneous functions of degree $k$, $n_1+n_2=n$, $\inf\{f_j(P_{n_j})\}>0$, $j=1,2$. Then $P(f_1\oplus f_2)=P(f_1)+P(f_2)$.*]{}
Values of homogeneous function $f$ on ${\overline{\mathbb
R}}^+_n$ are defined by its values on $\overline{P}_n$, namely for $y\in
{\overline{\mathbb R}}^+_n$, $f(y)=\lambda^kf(u)$ where $u\in
\overline{P}_n$, $\lambda=\sum\limits_{i=1}^ny_i$, $u=\lambda^{-1}y$. Therefore $\inf\{(f_1\oplus
f_2)(\overline{P}_n)\}=\inf\limits_{0\leq\lambda\leq 1}\bigg[\lambda^k
\inf\{f_1(\overline{P}_{n_1})\}+(1-\lambda)^k\inf\{f_2(\overline{P}_{n_2})\}
\ b i g g ] $ . Put $\inf\{f_1(\overline{P}_{n_1})\}=a$, $\inf\{f_2(\overline{P}_{n_2})\}=b$.
Consider function $\Phi_{ab}(\lambda)=a\lambda^k+b(1-\lambda)^k$, $a>0$, $b>0$ and find $\inf\limits_{0\leq\lambda\leq
1}\Phi_{ab}(\lambda)$. The derivative of $\Phi_{ab}(\lambda)$ with respect to $\lambda$ (consider $u$ and $v$ to be constants) is $(\Phi_{ab}(\lambda))'_\lambda=ka\lambda^{k-1}-kb(1-\lambda)^{k-1} $ . Let $\overline\lambda$ be a positive root of equation $(\Phi_{ab}(\lambda))'_\lambda =0$. Substitute $\overline\lambda$ and obtain $a\overline\lambda^{k-1}=b(1-\overline\lambda)^{k-1}$. Raise the both parts of this equality to ${1\over{k-1}}$ power and obtain $a^{{1\over{k-1}}}\overline\lambda= b^{1\over{k-1}}(1-\overline\lambda)$, $\overline\lambda = {b^{1\over{k-1}}\over
a^{1\over{k-1}}+b^{1\over{k-1}}}$. Thus, $\inf\limits_{0\leq\lambda\leq 1}
\Phi_{ab}(\lambda)=\min\{\Phi_{ab}(0),\Phi_{ab}(1),
\Phi_{ab}(\overline\lambda)\}=\min\{a,b,\Phi_{ab}(\overline\lambda)\}$.
We will prove that $\Phi_{ab}(\overline\lambda)<\Phi_{ab}(0)=b$. Really, $a\overline\lambda^{k-1}=b(1-\overline\lambda)^{k-1}$. $\Phi_{ab}(\overline\lambda)=a\overline\lambda^k+b(1-\overline\lambda)^k=
b(1-\overline\lambda)^{k-1}\overline\lambda+b(1-\overline\lambda)^k=
b(1-\overline\lambda)^{k-1}$. $\Phi_{ab}(\overline\lambda)<b$, since $a>0$, $b>0$, $0<\overline\lambda<1$ Analogously $\Phi_{ab}(\overline\lambda)=a\overline{\lambda}^{k-1}
<\Phi_{ab}(1)=a$. Therefore $\inf\{(f_1\oplus f_2)(\overline{P}_n)\}=\Phi_{ab}(\overline\lambda)=
{ab\over\left(a^{1\over k-1}+b^{1\over k-1}\right)^{k-1}}$.
Let us return to $P(f_1\oplus f_2)$, $P(f_1)$, $P(f_2)$. We have $P(f_1)=a^{1\over{1-k}}$, $P(f_2)=b^{1\over 1-k}$, $P(f_1\oplus f_2)=\Bigg({ab\over\left(a^{1\over k-1}+b^{1\over
k-1}\right)^{k-1}}\Bigg)^{1\over 1-k}=b^{1\over 1-k}+a^{1\over
1-k}$.
------------------------------------------------------------------------
[**Corollary 1.**]{} [*Under the conditions of the lemma $f_1\oplus f_2$ is $P$-faithful if and only if $f_1$ and $f_2$ are $P$-faithful.*]{}
[**2.**]{} Further $f=\sum\limits_{i,j=1}^n a_{ij} x_i x_j$ $(a_{ij}=a_{ji})$ is a quadratic form over field ${\mathbb R}$; $A=(a_{ij})$ is a symmetric matrix of quadratic form $f$. $\frac{\partial^2 f}{\partial x_i \partial x_j} = 2a_{ij}$. $f$ is a $2$-concave if $a_{ii}\in{\mathbb N}$, $a_{ij}+a_{ji}\in{\mathbb N}_0$ $(i,j=\overline{1,n})$. For $x=(x_1,\ldots,x_n)\in{\mathbb R}_n$ (for fixed $f$) put $$x'_i=\frac{\partial f}{\partial
x_i}(x_1,\ldots,x_n)=2\sum_{j=1}^na_{ji}x_j, \quad
x'=(x'_1,\ldots,x'_n)=2xA.$$
We will use the following identity, which is easy to check: $$\begin{gathered}
f(u+v)=f(u)+f(v)+\sum_{i=1}^n u'_i v_i, \quad
u,v\in {\mathbb R}_n,\quad\mbox{that implies}\label{0*}\\ \sum_{i=1}^n u'_i
v_i=\sum_{i=1}^n v'_i u_i, \quad f(u+\varepsilon
v)=f(u)+\varepsilon^2f(v)+\varepsilon \sum_{i=1}^n u_iv'_i, \quad
\varepsilon\in{\mathbb R}.\label{*} \end{gathered}$$
Put in $u=v$, and obtain (see. [@13], s. 178):
$$\label{*1}
f(u)=\frac 12 \sum_{i=1}^n u_i u'_i.$$
By means of (3) we can reformulate Lemma 4 by the following way.
[**Lemma 4$'$.**]{} [*$\widetilde{{\rm
St}}(f)\subset {\rm St}^+(f)$ for quadratic form $f$.*]{}
Put $\widetilde{C}(f)=\{(v_1,\ldots,v_n)\in C(f)\,|\,
(v'_1,\ldots,v'_n)\not=0\}$. Since $\frac{\partial f}{\partial
x_i}(-x)=-\frac{\partial f}{\partial x_i} (x)$ then if $C(f)\neq
\varnothing$, then $C^-(f)\neq \varnothing$ and $C^+(f)\neq \varnothing$. Therefore, choosing a vector $v\in C(f)\neq\varnothing$, further we will suppose that $v\in C^-(f)$.
[**Proposition 1.**]{} [*For any quadratic form $f$ $1)$ at least one of the sets ${\rm St}(f)$ and $\widetilde{C}(f)$ is empty $2)$ at least one of the sets $C(f)$ and $\widetilde{\rm St}(f)$ is empty.*]{}
1\) Let $u\in {\rm St}(f)$, $v\in \widetilde{C}(f)$. Then $\sum\limits_{i=1}^nu'_iv_i=u'_1\sum\limits_{i=1}^nv_i=0$. In the other hand, if $v'_j<0$ then $\sum\limits_{i=1}^nu_iv'_i<0$ (since $v'_i\leq 0$, $u_i>0$, $i=\overline{1,n}$), that contradicts .
2\) Let $u\in\widetilde{\rm St}(f)$, $v\in C(f)$. If $v\in\widetilde{C}(f)$ then 1) implies the statement. Let $v\in
C(f)\setminus\widetilde{C}(f)$, i. e. $v'_i=0$ for $i=\overline{1,n}$. Then $f(v)=0$ () and so $f(u+\varepsilon v)=f(u)$ for any $\varepsilon$. Put $|\varepsilon|=\min\limits_i\frac{u_i}{|v_i|}$. The sign of $\varepsilon$ is opposite sign of one of those $v_i$, for which the minimum is reached. Then $u+\varepsilon v\in\overline{P}_n\setminus P_n$, that contradicts $P$-faithfulness of $u$.
------------------------------------------------------------------------
[**Corollary 2.**]{} [*$P$-faithful quadratic form is antimonotonous.*]{}
[**Example 3.**]{} Let $f=\sum\limits_{i=1}^4x_i^2+(x_1+x_2)(x_3+x_4)$. Then $$A= \left(\begin{array}{cccc}
1&0&{1\over 2}&{1\over 2}\\
0&1&{1\over 2}&{1\over 2}\\
{1\over 2}&{1\over 2}&1&0\\
{1\over 2}&{1\over 2}&0&1
\end{array}\right),\quad St(f)\ni (1,1,1,1),\quad
C(f)\ni (1,1,-1,-1).$$ Proposition 1 implies $\widetilde{{\rm St}}(f)=\varnothing$, $\widetilde{C}(f)=\varnothing$.
In this example $|A|=0$.
[**Proposition 2.**]{} [*If $|A|\neq 0$ then one of the sets $C(f)$, ${\rm St}(f)$ is not empty, but the other one is empty.*]{}
At first suppose that $\varnothing\neq C(f)\ni v$ and $\varnothing\neq {\rm St}(f)\ni u$. If $v\in\widetilde{C}(f)$ then ${\rm St}(f)=\varnothing$ by Proposition 1. If $v\in
C(f)\setminus\widetilde{C}(f)$ then $v'=0$, and $vA={1\over 2}v'=0$. Hereof $v=0$ that contradicts $v\in H_n\setminus \{0\}$ (the definition of $C(f)$).
Now we prove that either ${\rm St}(f)\neq\varnothing$, or $C(f)\neq\varnothing$. Let $e_n=(1,\dots,1)\in \mathbb{R}_n$, $y=e_nA^{-1}$, $yA=e_n$. If $y\in \mathbb{R}_n^+$ or $-y\in \mathbb{R}_n^+$, then $y\in {\rm St}(f)$. If $\{y,-y\}\cap \mathbb{R}_n^+=\varnothing$ then either for certain $k$ $y_k=0$, or for certain $s$ and $t$ $y_s<0$, $y_t>0$. It is easy to see that in both cases there exists $w\in\overline{\mathbb{R}}_n^+$ such that $wy^T
\Big(=\sum\limits_{i=1}^nw_iy_i\Big)=0$ (at the first case we can put $w_k>0$, $w_i=0$ for $i\neq k$, at the second case $w_s=y_t$, $w_t=-y_s$, $w_i=0$ for $i\not\in \{s,t\}$, $i=\overline{1,n}$). We prove that $v=-wA^{-1}\in C(f)$. $-v'=wA^{-1}A=w\in \overline{\mathbb R}^+_n$, and so $v_i'\leq 0$. $v\neq 0$, since $w\neq 0$ and $|A|\neq 0$. It remains to prove that $v\in H_n$, that is equivalent to $v e_n^T=0$. $v
e_n^T=-wA^{-1}e_n^T$; $y^T=(A^{-1})^Te_n^T=A^{-1}e_n^T$ (since $A^T=A$). Therefore $-wA^{-1}e_n^T=-wy^T=0$.
------------------------------------------------------------------------
[**Proposition 3**]{} (see [@6], p. II, remark to theorem 1). [*$1)$ If $\widetilde{{\rm St}}(f)\not=\varnothing$, then $f$ is positive definite. $2)$ If $f$ is positive definite then $\widetilde{{\rm St}}(f)={\rm St}(f) \cap
P_n$ (and therefore $\widetilde{{\rm St}}(f)=\varnothing$ if and only if ${\rm St}(f)=\varnothing$).*]{}
1\) We suppose contrary: $f(v)\leq 0$ $(v\not=0)$, $u\in \widetilde{{\rm St}}(f)$. a) At first we suppose that $v\in H_n$, i. e. $\sum\limits_{i=1}^n v_i=0$ and $f(v)<0$. Then $f(u+\varepsilon v)=f(u)+\varepsilon^2 f(v) +\varepsilon
\sum\limits_{i=1}^n u'_i v_i$. By Lemma 4 $u\in{\rm St}(f)$ and, therefore, $\varepsilon\sum\limits_{i=1}^n u'_i v_i=0$, i. e. $f(u+\varepsilon
v)=f(u)+\varepsilon^2 f(v)$. Since $f(v)<0$ then $f(u+\varepsilon
v)<f(u)$, that contradicts $P$-faithfulness of $u$.
b\) Now let $v\in H_n$, $f(v)=0$. Then $f(u+\varepsilon v)=f(u)$ for any $\varepsilon$. Put $\varepsilon=\min\limits_i\frac{u_i}{|v_i|}$. The sign of $\varepsilon$ is opposite to the sign of one of those $v_i$, for which this minimum is reached. Then $u+\varepsilon v\in \overline{P}_n\setminus P_n$, again we have the contradiction with $P$-faithfulness of $u$.
c\) Let finally $\sum\limits_{i=1}^n v_i\not=0$. We can admit $\sum\limits_{i=1}^n v_i=1$. Put $w=u-v$. for $\varepsilon=-1$ and Lemma 4 imply $f(w)=f(u)+f(v)-u'$, $u'=u'_i$, $i=\overline{1,n}$. implies $f(u)=\frac{u'_1}{2}$ and, therefore, $f(w)=f(v)-\frac{u'_1}{2}$. By Lemma 4$'$ $u'>0$, $f(w)<0$; $w\in H_n$. Hereat $w\not=0$, since if $w=0$ then $u=v$, according to , but $f(v) \leq 0$. Thus, we reduced c) to a).
2\) Let $u\in{\rm St}(f)\cap P_n$, $v\in \overline{P}_n$, $v\not=u$. $u\neq 0$ since $u\in {\rm St}(f)$ so $f(u)>0$. We will prove that $f(u)<f(v)$. $0<f(u-v)\,\overset{\mbox{\footnotesize formula (2)}}{=}\,
f(u)+f(v)-\sum\limits_{i=1}^n u'_i v_i \,\overset{\mbox{\footnotesize
formula (3)}}{=}\, \frac{u'}{2}+f(v) -u'= f(v)-\frac{u'}{2}=f(v)-f(u)$, i. e. $f(v)>f(u)$.
------------------------------------------------------------------------
[**3.**]{} Further we will consider 2-concave form $f_S$ for poset $S=\{s_1,\ldots,s_n\}$, $f_S=\sum\limits_{s_i\leq s_j} x_ix_j$. Put $C(S)=C(f_S)$. $S$ is [*antimonotonous*]{} if $f_S$ is antimonotonous.
A poset $S$ is [*$P$-faithful*]{}, if $\widetilde{{\rm
St}}(f_S)\not=\varnothing$. (It is equivalent to the definition of $P$-faithfulness of the poset from [@2]) In this case Proposition 1 implies $C(S)=\varnothing$. Remark that $\inf\{f_S(\overline{P})\}>0$ since $a_{ij}\geq 0$, $i,j=\overline{1,n}$, $A\not= (0)$.
Hasse quiver (orgraph) $Q(S)$ of poset $S$ is a quiver, whose vertices are elements of $S$ and two vertices are connected by an arrow $s_i\to s_j$ if $s_i<s_j$ and there is no $s_k\in S$ such that $s_i<s_k<s_j$. Drawing lines (edges) instead of arrows, we obtain (nonoriented) graph Hasse $\Gamma(S)$ of partially ordered set $S$. Finite poset $S$ is usually depicted by a diagram, i. e. by graph $\Gamma(S)$ assuming that lesser element is drawn below than greater.
We denote elements of poset $S$ and corresponding elements of $Q(S)$ and $\Gamma(S)$ by the same symbol.
Put ${\rm St}(S)={\rm St}(f_S)$, $\widetilde{\rm St}(S)=\widetilde{\rm
St}(f_S)$. Let $\mathcal{T}_S(x_1,\ldots,x_n)=\sum\limits_{i=1}^n
x_i^2-\sum\limits_{s_i-s_j} x_ix_j$ be a quadratic Tits form of graph $\Gamma(S)$ (the second sum is taken by all edges of graph $\Gamma(S)$). We denote matrix of the form $\mathcal{T}_S$ either $\mathcal{A}$ or $\mathcal{A}(S)$.
Let $Q$ be a quiver without loops and parallel (i. e. having the same origin and the same terminus) pathes with vertices $s_1,\ldots,s_n$. $\widetilde{Q}$ is a matrix, in which $\widetilde Q_{ij}=1$, if there is an arrow from $s_i$ to $s_j$, $\widetilde Q_{ij}=0$ in the opposite case $(i,j=\overline{1,n})$. Then $({\widetilde{Q}}^t)_{ij}$ is equal to the number of pathes of length $t$ from $s_i$ to $s_j$, ${\widetilde{Q}^n}=0$. If, moreover, $Q=Q(S)$ then $A=\frac{1}{2}[(E+\widetilde{Q}+\cdots+\widetilde{Q}^{n-1})+(E+\widetilde{Q}
+ \cdots+ \widetilde{Q}^{n-1})^T]$ ($A$ is the matrix of $f_S$). It is easy to see that $(E+\widetilde{Q}+\cdots+\widetilde{Q}^{n-1})=(E-\widetilde{Q})^{-1}$. Put $E-\widetilde{Q}=\hat Q$, $|\hat Q|=1$. $\mathcal{A}=\frac 12 (\hat Q+\hat
Q^T)$. $A=\frac{1}{2}(\widehat{Q}^{-1}+(\widehat{Q}^{-1})^T)$.
[**Proposition 4 ([@3])[^1]**]{} [*If there are no parallel pathes in $Q(S)$ then the forms $\mathcal{T}_S$ and $f_S$ are equivalent over $\mathbb{Z}$.*]{}
Really, $\hat Q^{-1}\mathcal{A}_T(\hat Q^{-1})^T=\frac 12 \hat Q^{-1}(\hat
Q+\hat Q^T) (\hat Q^{-1})^T=\frac 12 [(\hat Q^{-1})^T +\hat
Q^{-1}]=A$.
------------------------------------------------------------------------
Propositions 1, 2, 3, 4 imply
[**Corollary 3.**]{}
*Let $\Gamma(S)$ be acyclical and at least one of the forms $f_S$, ${\mathcal T}_S$ be positive definite (it holds if $\Gamma(S)$ is a Dynkin graph, see s. 4). Then the other form is positive definite as well, and the following statements are equivalent:*
a\) $S$ is antimonotonous;
b\) $S$ is faithful;
c\) ${\rm St}(S)\neq \varnothing$.
We denote $I(s_i)$ for $s_i\in S$ the number of edges of graph $\Gamma(S)$, having $s_i$ its terminus.
$s_i$ is [*a terminal point*]{} if $I(s_i)\leq 1$, $s_i$ is [*a branch point*]{} if $I(s_i)\geq 3$. $s_i$ is [*a junction point*]{}, if it is either a terminus of at least two arrows, or an origin of at least two arrows of quiver $Q(S)$. We denote $S^\times$ the set of junction points.
[**Example 4.**]{}
$$S=\mbox{\raisebox{-10mm}[0pt][0pt]{\includegraphics{rys7.eps}}}$$
Here $S^\times =S$, $C(S)\cup \, \widetilde{\rm St}(S)=\varnothing$. Really, values of $f_S$ can be negative ($f_S(1,1,1,-1$, $-1,-1)=-2$), consequently, by Proposition 3, $\widetilde{\rm St}(S)=\varnothing$, but ${\rm St}(S)\ni (1,2,1,1,2,1)$, $|A|=-48$. Proposition 2 implies $C(S)=\varnothing$.
We will assume further (except Appendix) graph $\Gamma(S)$ to be connected.
We remind that a cycle in graph $\Gamma$ is a sequence $\{s_1,\dots s_m\}$ of different vertices $s_i$ of graph $\Gamma$ where $m\geq 3$ such that a vertex $s_i$ is connected with $s_{i+1}$ for $i=\overline{1,m}$, and a vertex $s_m$ is connected with $s_1$. We call a cycle $\{s_1,\dots,s_m\}$ [*simple*]{}, if there are no other vertices between $s_1,\dots,s_m$. We call graph $\Gamma$ and poset $S$ such that $\Gamma(S)=\Gamma$ [*cyclical*]{}, if $\Gamma$ contains a cycle, and [*acyclical*]{} in opposite case. it is easy to see that a cyclical graph contains a simple cycle, and, correspondingly, a cyclical poset $S$ contains a subset $S'$ such that $\Gamma(S')=\widetilde{A}_m$. It is clear that if $S$ is acyclical then $Q(S)$ has no parallel pathes.
If $\Gamma(S)=\Gamma(\overrightarrow S)$ then $Q(\overrightarrow S)$ can be obtained from $Q(S)$ by “reorientation” (i. e. by alternation of direction) of several arrows. If $\Gamma(S)$ is acyclic and quiver $\overrightarrow Q$ is obtained by reorientation of arrows from $Q(S)$, then there exists $\overrightarrow S$ such that $\overrightarrow Q=Q(\overrightarrow S)$.
We call quiver $Q=Q(S)$ and poset $S$ [*standard*]{}, if $I(s_i)=2$ implies that $s_i$ is origin of one arrow and terminus also of one arrow, and $I(s_i)\neq 2$ implies that $s_i$ is either origin of $I(s_i)$ arrows or terminus of $I(s_i)$ arrows ($i=\overline{1,n}$). It is easy to see that exactly one standard poset is attached to each acyclical graph up to antiisomorphism. If $S^*$ is antiisomorphic to $S$ then $\Gamma(S)=\Gamma(S^*)$, and $Q(S^*)$ is obtained from $Q(S)$ by reorientation of all arrows.
If $\varphi$ is an arrow of $Q(S)$, then we denote $S(\varphi)$ poset, obtaining from $S$ by overturn of an arrow $\varphi$. $A_\varphi$ is matrix of $f_{S(\varphi)}$. It is clear that $\mathcal{A}(S(\varphi))=\mathcal{A}(S)$.
We call a point $s_m\in S$ [*Dynkin*]{}, if there exists $m$-Dynkin vector for form $f_S$.
[**Remark 2.**]{} Function $f_S$ is $m-$isolated in sense of s.1, if $s_m$ is not comparable with other points of $S$. Therefore for connected $S$ the condition of non-isolateness of $f_S$ in Lemma 3 holds automatically.
[**Lemma 6.**]{} [*Let $\Gamma(S)$ be acyclical, $s_i\overset{\varphi}\rightarrow
s_j\in Q(S)$; $d\neq 0$ is such a vector that $d'_i=2(dA)_i=0$, $d'_j=2(dA)_j=0$. Then there exists a vector $\widehat{d}\neq 0$ such that $dA=\widehat{d}A_\varphi$.*]{}
It follows from the proof of Proposition 4 that $\widetilde{Q}^{-1}\mathcal{A}(\widetilde{Q}^{-1})^T=A$, $\widetilde{Q}^{-1}_{\varphi} \mathcal{A}_{\varphi}
(\widetilde{Q}^{-1}_{\varphi})^T = A_{\varphi}$, $\widetilde{Q}_\varphi^{-1}\widetilde{Q}A
\widetilde{Q}^T(\widetilde{Q}_\varphi^T)^{-1}
=A_\varphi$ ($\mathcal{A}_{\varphi}=\mathcal{A}$). Put $\widehat{d}=d\widetilde{Q}^{-1}\widetilde{Q}_\varphi$, $(\widehat{d}\neq 0)$, $\widehat{d}A_\varphi=d\widetilde{Q}^{-1}\widetilde{Q}_\varphi
\widetilde{Q}_\varphi^{-1}\widetilde{Q}A\widetilde{Q}^T(Q_\varphi^T)^{-1}
= dA\widetilde{Q}^T(\widetilde{Q}_\varphi^T)^{-1}$.
Put (in view of $d'_i=d'_j=0$), $dA=\sum\limits_{k\not\in\{i,j\}}\alpha_ks_k=b$. We shall prove that $b\widetilde{Q}^T(\widetilde{Q}_\varphi^T)^{-1}=b$, i. e. that $s_k\widetilde{Q}^T(\widetilde{Q}_\varphi^T)^{-1}=s_k$, i. e. that $s_k\widetilde{Q}^T=s_k\widetilde{Q}_\varphi^T$, that follows from the definition of $\widetilde{Q}$ and from $k\not\in\{i,j\}$.
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[**Lemma 7.**]{} [*If $\Gamma(S)$ is acyclical, $s_t$ is a Dynkin terminal point of $S$ then it is Dynkin point for poset $\overrightarrow S$ if $\Gamma(\overrightarrow S)=\Gamma(S)$.*]{}
If $\overrightarrow{S}=S(\varphi)$ then the statement follows from Lemma 6. Considering the general case ($\overrightarrow S$ is not $S(\varphi)$), we remark firstly that if $S^*$ and $S$ are antiisomorphic then $f_S=f_{S^*}$ and $s_t$ is a Dynkin point also for $S^*$.
We denote $\psi$ (resp. $\hat\psi$) unique arrow $Q(S)$ (resp. $Q(\overrightarrow{S}$)) for which $s_t$ is either terminus or origin. Then the condition $s_t\not\in\{s_i,s_j\}$ is equal to $\varphi\not=\psi$.
Without loss of generality suppose that $\psi$ in $Q(S)$ and $\hat\psi$ in $Q(\overrightarrow S)$ have the same orientation, otherwise pass to $\overrightarrow S^*$. In this case we can pass from $S$ to $\overrightarrow S$, returning several arrows, different from $\psi$, and, therefore, partial cases $\overrightarrow S=S(\varphi)$ (Lemma 6) and $\overrightarrow S=S^*$ considered by us imply the statement of lemma.
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[**4.**]{} Let $\Gamma$ be a connected acyclical graph with one branch point and three terminal points. $\Gamma$ is a union of three chains $A_{n_1}$, $A_{n_2}$, $A_{n_3}$, intersecting in a branch point $s_1$. $\Gamma=A_{n_1}\cup A_{n_2}\cup A_{n_3}$, $A_{n_1}\cap A_{n_2}=A_{n_1}\cap A_{n_3}=A_{n_2}\cap A_{n_3}=\{s_1\}$, $|A_{n_j}|=n_j$, $j=\overline{1,3}$, $|\Gamma|=n_1+n_2+n_3-2$. We will denote $\Gamma$ by $\Gamma( n_1,n_2,n_3)$ (graph does not change if we permutate $n_j$).
All Dynkin graphs besides $A_n$ (i. e. $D_n$, $E_6$, $E_7$, $E_8$) and extended Dynkin graphs $\widetilde{E}_6$, $\widetilde{E}_7$, $\widetilde{E}_8$ have the form $\Gamma(n_1,n_2,n_3)$. It is well-known that $\Gamma(n_1,n_2,n_3)$ is a Dynkin graph if and only if $n_1^{-1}+n_2^{-1}+n_3^{-1}>1$, and is an extended Dynkin graph when $n_1^{-1}+n_2^{-1}+n_3^{-1}=1$.
Namely, $\Gamma(n_1,n_2,n_3)$ is $E_6$, $E_7$, $E_8$, $D_n$, if respectively $(n_1,n_2,n_3)=(3,3,2)$; $(2,4,3)$; $(2,3,5)$; $(1,1,n-2)$. $\Gamma( m_1,m_2,m_3)$ is respectively $\widetilde{E}_6$, $\widetilde{E}_7$, $\widetilde{E}_8$. for $(m_1,m_2,m_3)=(3,3,3)$; $(2,4,4)$; $(2,3,6)$.
We fixed here the numeration of $m_j$ and $n_j$ so that $m_1\leq m_2\leq m_3$ and for $E_n$ $(n=6,7,8)$, $n_1=m_1$, $n_2=m_2$, $n_3=m_3-1$.
Remark, that (in all cases) $m_3$ is divided by $m_1$ and $m_2$.
[**Proposition 5.**]{} [*If $\Gamma(S)=\Gamma(n_1,n_2,n_3)$ is Dynkin graph or extended Dynkin graph (i. e. $D_n$, $E_6$, $E_7$, $E_8$, $\widetilde{E}_6$, $\widetilde{E}_7$ or $\widetilde{E}_8$) then $S$ contains terminal Dynkin point.*]{}
By Lemma 7 we can, without loss of generality suppose $S$ to be standard.
For any $\Gamma(m_1,m_2,m_3)$ $(I(s_1)=3)$ we construct a vector $\widetilde d$, putting $\widetilde{d}_1=-m_3$; $\widetilde{d}_i=\frac{m_3}{m_j}$, for $s_i\in A_{n_j}$, $i\not=1$. It is easy to see that $\widetilde{d}'_i=0$, for $i\not=1$, $\widetilde{d}'_1=m_3(1-m_1^{-1}-m_2^{-1}-m_3^{-1})$. If, moreover, $\Gamma(S)$ is extended Dynkin graph then $\widetilde{d}\in
\mathbb{Z}_n$ and $\widetilde{d}'_1=0$, i. e. $\widetilde d$ is in this case a $i$-Dynkin vector for any $i$.
Let $\Gamma(S)$ be $E_n$, $|S|=n$. $\widetilde S$ is such standard poset that $\Gamma(\widetilde
S)=\widetilde{E}_n$, $|\widetilde S|=n+1$, $S\subset \widetilde S$, $\widetilde S\setminus S=\{s_{n+1}\}\subset A_{n_3}$. We construct for $S$ a Dynkin vector $d$, modifying Dynkin vector $\widetilde d$ for $\Gamma(\widetilde S)$. Put $d_i=\widetilde{d}_i$ for $i<n$, and $d_n=2$ $(=\widetilde{d}_n+\widetilde{d}_{n+1})$, $d'_n=1$; ($d'_i=0,\;
i=\overline{1,n-1}$).
Let $$\Gamma(S)=D_n=\raisebox{-9mm}[+10mm][0pt]{\mbox{\includegraphics{rys1.eps}}},$$
then $w=(w_1,\ldots,w_n)$ where $w_1=-2$, $w_2=w_3=1$, $w_n=2$, $w_i=0$, $i\not\in \{1,2,3,n\}$ is $n$-Dynkin vector $s_n$ $(w'_n=2)$.
------------------------------------------------------------------------
We write out evidently Dynkin vector for standard $S$ if $\Gamma(S)=E_6,E_7,E_8$ $$\mbox{\includegraphics{rys2.eps}}\qquad\mbox{\includegraphics{rys3.eps}}\qquad
\mbox{\includegraphics{rys4.eps}}$$
[**Example 5.**]{} Dynkin vector for standard poset $S$ such that $\Gamma(S)=\widetilde{D}_n$, $n>4$ (for $n=4$ see Example 2) has the following form (all points are Dynkin points)
$$\mbox{\includegraphics{rys6.eps}}
\qquad %\raisebox{9mm}[0mm][0pt]$$
[**5.**]{} Let poset $V=\raisebox{-9mm}[+10mm][10mm]{\mbox{\includegraphics{rys8.eps}}}$, poset $W^{2k}=\{s_1^-,\ldots,s_k^-,s_1^+,\ldots,s_k^+\,|\, s_i^-<s_i^+, s_i^-<s_{i+1}^-,
s_k^-<s_1^+, i=\overline{1,k}\}$, $k>1$, in particular, $W^4=\raisebox{-9mm}[+10mm][10mm]{\mbox{\includegraphics{rys9.eps}}}$ (see Example 3).
[**Lemma 8.**]{} [*If poset $S$ is cyclical and each $S'\subset S$ is acyclical then $S$ is either $V$, or $W^{2k}$ $(k\geq 2)$.*]{}
Without loss of generality suppose that $\Gamma(S)$ is a simple cycle $\widetilde{A}_n=
\raisebox{-3mm}[7mm][4mm]{\mbox{\includegraphics{rys12.eps}}}$. Let $S\setminus S^{\times}\ni s$, then $s^-<s<s^+$ and in $\Gamma(S)$ $s^-$ and $s^+$ are connected with $s$ by edges. Then consider $S'=S\setminus\{s\}$. If there is an edge $s^-$—$s^+$ in graph $\Gamma(S')$ then $\Gamma(S')$ is a cycle (that contradicts the condition of lemma).
If $s^-$ and $s^+$ are not connected by edge in $\Gamma(S')$ then there is a point $\bar s\not=s$ in $S$ such that $s^-<\bar s<s^+$, $\bar s>\!\!\!\!< s$ (i. e. $\bar s$ and $s$ are not comparable) because in the opposite case $s^-$ and $s^+$ should be not connected with $s$ in $\Gamma(S)$. And so $\{s^-,s^+,s,\bar s\}=V$ and $S=V$.
So, if $S\neq V$ then $S^{\times}=S$ and then (since $\Gamma(S)=\widetilde{A}_n$) it is easy to see that $S=W^{2k}$ for some $k>1$.
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[**Lemma 9.**]{} [*If $S\supseteq V$ and $S\not\supset W^4$ then $C(S)\not=\varnothing$.*]{}
An arbitrary vector $v\in {\mathbb R}_n$ can be considered as a function on $S$ with values in ${\mathbb R}$.
Let $v: S\to {\mathbb R}$. $v(h^{-})=v(h^+)=-1$, $v(h_1)=v(h_2)=1$, $v(t)=0$ for $t\in S\setminus V$; $v\in H_n$. We prove that $v'(s)\leq 0$ for $s\in S$. $v'(h^-)=v'(h^+)=-1$, $v'(h_1)=v'(h_2)=0$. If $t$ is not comparable niether with $h_1$ nor with $h_2$ then it is clear that $v'(t)\leq 0$. If $t$ is comparable only with one of $h_1$, $h_2$, then it is comparable either with $h^-$ or with $h^+$, and also $v'(t)\leq 0$. Let $t$ be comparable with $h_1$ and with $h_2$. Suppose $t<h_1$ (a case $t>h_1$ is analogous) then $t<h_2$. ($h_1<t<h_2$ is impossible, so $t<h^+$). Then if $t$ is comparable also with $h^-$ then $v'(t)=0$, and in the opposite case $S\supset
W^4=\{t,h_2,h^-,h_1\}$.
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[**Lemma 10.**]{} [*If $S\supseteq W^{2k}$ $(k\geq 2)$ and form $f_S$ is positive semidefinite then $C(S)\not=\varnothing$.*]{}
Let $t\in T=S\setminus W^{2k}$. Put $S^-(t)=|\{s_i^-\,|\, t<s_i^-\}|\cup \{s_i^-\,|\, t>s_i^-\}|$, $S^+(t)=|\{s_i^+\,|\, t<s_i^+\}\cup \{s_i^+\,|\, t>s_i^+\}|$. We prove that positive semidefiniteness of $f_S$ implies $S^-(t)=S^+(t)$.
Really, let $S^-(t_0)>S^+(t_0)$ for fixed $t_0\in T$ (a case $S^-(t_0)<S^+(t_0)$ is analogous). Consider $x: S\to \mathbb{R}_n$ where $x(s_i^-)=-1$, $x(s_i^+)=1$, ($i=\overline{1,k}$), $x(t_0)=\varepsilon$, $0<\varepsilon<1$ and $x(t)=0$ for $t\in T\setminus \{t_0\}$. It is easy to see that $f_S(x)<0$.
Now consider vector $v: S\to {\mathbb R}_n$ where $v(s_i^-)=-1$, $v(s_i^+)=1$, $v(t)=0$ for $t\in T$. $S^{-1}(t)=S^+(t)$ $(t\in T)$ implies $v'(s)=0$ for any $s\in S$. It is clear, that $v\in H_n$ and therefore $v\in C(S)$
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[**Proposition 6.**]{} [*If $S$ is an antimonotonous poset and form $f_S$ is positive semidefinite then $\Gamma(S)=A_n$.*]{}
If $S$ is cyclical, then Lemmas 8,9,10 imply the statement. If $S$ is acyclical, then by Proposition 4 the Tits form $\mathcal{T}_S$ is positive semidefinite, so $\Gamma(S)$ is one of $A_n$, $D_n$, $E_6$, $E_7$, $E_8$, $\widetilde{D}_n$, $\widetilde{E}_6$, $\widetilde{E}_7$, $\widetilde{E}_8$ ($\Gamma(\widetilde{A}_n)$ is cyclical). If $\Gamma(S)\neq A_n$ then Proposition 5, Examples 2 and 5 and Lemma 7 imply the existence of the Dynkin point and, by Lemma 3 (and Remark 2), $C(S)\neq
0$.
[**6.**]{} We consider now $\Gamma(S)=A_n$. In this case poset $S$ up to antiisomorphism is defined by its order and by subset $S^\times$ consisting of junction points (see s .3). It is clear that $S^\times=\emptyset$ if and only if $S$ is a chain.
Let poset $W^{k,k+1}=\{s_1^-,\ldots,s_k^-,s_1^+,\ldots,s_{k+1}^+\,
|\, s_i^-<s_i^+, s_i^-<s_{i+1}^+,i=\overline{1,k}\}$, $W^{k+1,k}=\{s_1^-,\ldots,s_{k+1}^-,s_1^+,\ldots,s_{k}^+\,
|\, s_i^+>s_i^-, s_i^+>s_{i+1}^-,i=\overline{1,k}\}$.
[**Lemma 11.**]{} [*If $\Gamma(S)=A_n$ and $\Gamma(S)\supseteq W$ of the form $W^{k,k+1}$ (resp. $W^{k+1,k}$), moreover $s_1^+,s_{k+1}^+\not\in S^\times$ (resp. $s_1^-,s_{k+1}^-\not\in S^\times$) then $C(S)\not=\varnothing$.*]{}
Let, for the sake of definiteness, $S\supset W^{k,k+1}$. Consider vector $v$ such that $v(s_i^-)=-2$, $i=\overline{1,k}$; $v(s_i^+)=+2$, $i=\overline{2,k}$; $v(s_1^+)=v(s_{k+1}^+)=1$; $v(t)=0$ for $t\in S\setminus W^{k,k+1}$. We prove that $v\in C(f)$.
Indeed, $v\in H_n$; $v'(s_1^-)=v'(s_k^-)=-1$; $v'(s_i^-)=0$ for $i=\overline{2,k-1}$; $v'(s_i^+)=0$ for $i=\overline{1,k+1}$. Absence of branch points implies that if $t\not\in W$ is comparable with $w\in W$ then $w\in \{s_1^+,s_{k+1}^+\}$. If $t$ is comparable with both $s_1^+$, $s_{k+1}^+$ then $S$ would be cyclical. $s_1^+$, $s_{k+1}^+$ $\not\in S^{\times}$ implies that it can be only $t > w$. Therefore every $t$ either is comparable exactly with one $s_i^-$ and one $s_i^+$, or is not comparable with any $w\in W$, hereof $v'(t)\leq 0$.
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We call a poset $\zeta$ [*a wattle*]{} [@2] if it is the union of not intersecting chains $Z_i$, $|Z_i|\geq 2$, $i=\overline{1,t}$, $t>1$, in which the minimal element of $Z_i$ is less than the maximal element of $Z_{i+1}$ and there are no other comparisons between elements of different $Z_i$. $\Gamma(\zeta)=A_n$. According to [@2] we denote $\zeta=\langle
n_1,\ldots,n_t\rangle$ where $n_i=|Z_i|$.
For a poset $S$ we consider a disconnected subgraph $\Gamma(S^{\times})$ of $\Gamma(S)$. Denote $S_i^{\times}$ its connected components.
[**Lemma 12.**]{} [*A poset $S$ where $\Gamma(S)=A_n$ is either a chain or a wattle if (and only if) the orders of all $S_i^\times$ are even.*]{}
If $S$ is a wattle then the statement is evident (and we will not use it). We will prove the converse statement induction by $|S|$. The base is evident. Let $|S|=n+1$. $\Gamma(S)=\cdots s_{n-1}$—$s_n$—$s_{n+1}$ where $s_{n+1}$ is a terminal point (therefore $s_{n+1}\not\in
S^\times$). For the sake of definiteness we will suppose that $s_n>s_{n+1}$. So $s_{n+1}$ is minimal. Put $S'=S\setminus\{s_{n+1}\}$ and $S''=S\setminus\{s_{n+1},s_{n}\}$. We have two possibilities: 1)$s_{n-1}>s_n$; 2)$s_{n-1}<s_n$.
1\) $s_n\not\in S^\times$, $(S')^\times=S^\times$. By assumption of the induction $S'$ is a wattle in which $s_n$ is a minimal terminal point. It is clear that $S$ is either a wattle or a chain. (If $S'=\langle n_1,\dots,n_t\rangle $ then $S'=\langle
n_1,\dots,n_t+1\rangle$).
2\) $s_n\in S^\times$ ($S'$ does not satisfy the induction’s assumption!). $s_n\in S^\times_p$, $|S_p^\times|\equiv 0 (\mod 2)$. So $s_{n-1}\in S_p^\times\subset S^\times$. $s_{n-1}$ is a terminal point of $S''$ and so $s_{n-1}\not\in(S'')^\times$. If $S^\times=\bigcup\limits_{i=1}^{p}S_i^\times$ then $(S'')^\times=\bigcup\limits_{i=1}^{p-1}S_i^\times
\cup(S^\times_p\setminus\{s_n,s_{n+1}\})$.
Consequently $S''$ satisfies the induction’s assumption and so it is either a chain or a wattle, in which $s_{n-1}$ is a minimal point. If $S''=\langle n_1,\dots n_t\rangle $ then $S=\langle n_1,\dots,n_t,2\rangle$.
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[**Proposition 7.**]{} [*If a form $f_S$ is positive semidefinite and $C(S)=\varnothing$, then $S$ is either a chain or a wattle.*]{}
Proposition 6 implies $\Gamma(S)=A_n$.
If $S$ is neither a chain nor a wattle then by Lemma 12 there exists $S^\times_p$ such that $|S_p^\times|\equiv 1(\mod 2)$. It is easy to see that $S_p^\times$ is $W^{k,k+1}$ or $W^{k+1,k}$. Let, for the sake of definiteness $S_p^\times=W^{k,k+1}=
\raisebox{-8mm}[7mm][8mm]{\mbox{\includegraphics{rys15.eps}}}$. The facts that $s_1^+\in S^\times$ and $S_p^\times$ is a connected component in $S^\times$, imply that there exists $s_0^-\in S\setminus S^\times$ such that $s_0^-\rightarrow s_1^+$. Analogously there exists $s_{k+1}^-\in
S\setminus S^\times$, $s_{k+1}^-\rightarrow s_{k+1}^+$. It is easy to see that $S_p^\times\cup\{s_0^-,s_{k+1}^-\}=W^{k+2,k+1}$ and $C(S)\neq\emptyset$ by Lemma 11.
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Example 4 (s. 3) implies that the condition of positive semidefiniteness of $f_S$ can not, generally speaking, be excluded.
[**Hypothesis.**]{} [*If $S$ is acyclical and $\Gamma(S)\neq A_n$ then $C(S)\not=\varnothing$.*]{}
In some cases the existence of $v\in C(f)$ for acyclical $S$ is evident. We give however the example of an acyclic poset $S$ and $v\in C(S)$ which we constructed only by means of computer.
[**Example 6.**]{} $$\raisebox{-65mm}[0mm][0mm]{\mbox{\includegraphics{rys10.eps}}}$$
[**7.**]{} Let $\zeta=\langle n_1,\ldots,n_t\rangle$ $(t>1)$ be a wattle, where $n_i=|Z_i|$, $\sum\limits_{i=1}^t n_i=n$, $n_i>1$, $i=\overline{1,t}$.
In [@2] the minimal points of chains $Z_i$, $i=\overline{1,t-1}$ are denoted by $z_i^-$, and maximal points of chains $Z_i$, $i=\overline{2,t}$ are denoted by $z_i^+$, $z_i^-<z_{i+1}^+$. The rest (i. e. not junction) points are called [*common*]{} (including the maximal point of chain $Z_1$ and a minimal point of chain $Z_t$). They are compared only with points from its chain.
A [*width*]{} $\omega(S)$ of a partially ordered set $S$ is the maximal number of its pairwise uncomparable elements. We attach to any poset $S$ a rational number $r(S)=\frac{n+1}{t}-1$ where $n=|S|$, $t=\omega(S)$. If $S$ is a chain then $w(s)=1$, $r(S)=n$.
It is clear that there exist many wattles with the same $r$. We prove below, however, that to any noninteger $r>1$ it is corresponded exactly one (uniform in sense [@2]) $P$-faithful ($=$ antimonotonous, see Corollary 3) wattle, which we will call here $r$-wattle and denote $\zeta(r)$.
For a positive rational $a$ we put $\{a\}=a-[a]$. Let $r$ be a positive noninteger rational number greater than 1. $q/t$ is the representation of $\{r\}$ in the form of irreducible fraction. We write out the sequence of integers $n_1,\ldots,n_t$, which will be the orders of sets $Z_i$ in $\zeta(r)$. Put $n_1=n_t=[r]+1$, $n_i=[ri]-[r(i-1)]+1$ for $i=\overline{2,t-1}$. It is clear, that $[\{r\}i]-[\{r\}(i-1)]$ is either 1 or 0. Therefore $n_i$ is either $1+[r]$ or $2+[r]$. The number of those $i$ for which $n_i=2+[r]$ is $q-1$; $n=t([r]+1)+q-1$, $r(\zeta(r))=r$
Remark, that $r$-wattles are uniform in sense of [@2] and conversely.
Thus, we corresponded to each noninteger rational number $r>1$ a wattle $\zeta(r)$. We can consider also integer numbers, putting for natural $r$ that $\zeta(r)$ is a chain of length $r$. We will call $r$-sets all posets of form $\zeta(r)$ $r\geq 1$ (i. e. uniform wattles and chains).
[**Theorem.**]{} [*Let form $f_S$ be positive semidefinite ($\Gamma(S)$ be connected). Then $C(S)=\varnothing$, if and only if $S$ is $r$-set.*]{}
If $r$ is integer then the statement is evident (see [@2]). Therefore as a matter of fact we need to prove, counting Proposition 7, that $C(\zeta)=\varnothing$ if and only if $\zeta$ is $r$-wattle.
We attach to $r$-wattle $\zeta(r)$ a vector $x: \zeta \to {\mathbb R}^+$, $x(s)=1$ for $s\in\zeta\setminus\zeta^\times$, $x(z_i^-)=\{ir\}$, $x(z_i^+)=1-x(z_{i-1}^-)$. $(q,t)=1$ implies $x(s)>0$ for any $s\in \zeta(r)$.
It is posiible to check that $x\in {\rm St}(\zeta)$ either spontaneously, using the definition of ${\rm St}(\zeta)$, or using the following lemma.
[**Lemma 13 (see [@2], Lemma 5).**]{}
*Vector $x: \zeta \to {\mathbb
R}$ is contained in ${\rm St}(\zeta)$ if and only if there exist such positive $\alpha$, $\beta$ that*
$1)$ $x(s)=\alpha$ for $s\in \zeta\setminus \zeta^\times$ (we can suppose $\alpha=1$ multiplying $x$ by $\lambda\in{\mathbb R}^+$);
$2)$ $x(z_i^-)+x(z_{i+1}^+)=\alpha$ for $i=\overline{1,t-1}$;
$3)$ $\sum\limits_{s\in Z_i} x(s)=\beta$, $i=\overline{1,t}$.
Proof of the lemma is almost evident. Remark merely that at first 2) should be proved (it follows from $\frac{\partial f_S}{\partial
z_i^-}(x) = \frac{\partial f_S}{\partial z_i^+}(x)$, $i=\overline{2,t-1}$), and then 1).
Vector $x$ constructed above evidently satisfies the conditions 1), 2) of Lemma 13.It is easy to check (for $\alpha=1$) that
$$\begin{gathered}
\sum\limits_{s\in Z_i} x(s)=r (i=\overline{1,t});\label{4*}\\
x'(s)=1+r;\label{5*}\\
\sum\limits_{s\in \zeta(r)} x(s)=tr,\label{6*}\end{gathered}$$
Thus, $x\in{\rm St}(\zeta)$, so $\widetilde{\rm St}(\zeta)\neq \varnothing$, and $C(S)=\varnothing$ by Corollary 3.
It remains to prove that any $P$–faithful wattle $\zeta$ is $r$-wattle (where $[z]=|Z_1|-1$, $\{r\}=x(z_1^-)$). This follows from the next statement.
[**Lemma 14.**]{} [*Let $\zeta=\langle z_1,\ldots,z_t\rangle$ and $\hat \zeta=\langle \hat z_1,\ldots,\hat z_t\rangle$ are two $P$-faithful wattles, $x\in{\rm St}(\zeta)$, $\hat x\in {\rm St}(\hat
\zeta)$ $(\hat{\alpha}=\alpha=1)$. Then, if $Z_1=\hat Z_1$ and $x(s)=\hat
x(s)$ for $s\in Z_1=\hat Z_1$ then $\zeta=\hat \zeta$ and $x(s)=\hat x(s)$ for $s\in\zeta$.*]{}
It is sufficient to prove that if $m\leq\max\{t,\hat t\}$, then $z_i=\hat z_i$ for $i\leq m$ and $x(s)=\hat x(s)$ for $s\in \cup_{i=1}^m Z_i$. Lemma 13 implies this by induction with respect to $m$ (see [@2]).
------------------------------------------------------------------------
------------------------------------------------------------------------
There was introduced the numerical function $\rho(r)=1+\frac{r-1}{r+1}$ where $r\in
{\mathbb N}$ [@2]. We spread this definition on the case $r\geq
1$ is rational. Put $\rho(r_1,\ldots,r_t)=\sum\limits_{i=1}^t\rho (r_i)$. If $Z_n$ is a chain of order $n$ then $P(Z_n)=\rho(n)$ [@2]. Let $\zeta(r)$ be a wattle. Vector $\overline{x}=(tr)^{-1}x\overset{(6)}{\in}
P_n \cap {\rm St} (\zeta(r))$ (where $x$ is a vector constructed in the proof of the theorem).
$$\begin{gathered}
P(\zeta(r))=f_{\zeta(r)}^{-1}(\overline{x})=
(tr)^2f_{\zeta(r)}^{-1}(x)\overset{(3)}{=}\\
=2(tr)^2 \left(\sum\limits_{s\in\zeta(r)}x'(s)x(s)\right)^{-1}
\overset{\eqref{5*},\; \eqref{6*}}{=}
\frac{2t^2r^2}{(1+r)tr}=\frac{2tr}{1+r}=t\rho(r).\end{gathered}$$
This formula is true if $t=1$ (i. e. in the case of chain). For any positive rational $r=\frac{l}{t}$ $((l,t)=1)$ $t\rho(r)=\frac{2lt}{l+t}$. Introduce the function $P(r)=\frac{2lt}{l+t}$ (for $n\in {\mathbb N}$ $P(n)=\rho(n)$). Thus, for any $r\geq 1$
$$P(\zeta(r))=t\rho(r)=P(r)\label{z4}.$$
[**Appendix.**]{} We call a poset $S$ [*connected*]{} if graph $\Gamma(S)$ is connected. The theorem and Corollary 3 imply that a connected poset $S$ is $P$-faithful if and only if it is $r$-set [@2; @3; @4; @5; @6]. On the other hand, [@2; @3; @4; @5; @6] imply our theorem only if $f_S$ is positive definite (but not positive semidefinite).
We remind about role of $P$-faithful posets in representation theory. We will write $S=S_1\bigsqcup S_2$, if $S=S_1 \cup S_2$, $S_1\cap
S_2=\varnothing$ and elements $S_1$ are not comparable with elements $S_2$. $S= Z_1 \bigsqcup \cdots\bigsqcup
Z_p$ is [*primitive*]{}, if $Z_i$ are chains, $i=\overline{1,p}$. We denote it $(n_1,\ldots,n_p)$ if $n_i=|Z_i|$. The characterization of antimonotonous disconnected posets follows from the theorem and Lemma 2.
Any poset $S=\bigsqcup\limits_{i=1}^p S_i$ where $S_i$ are connected components. According to Lemma 5, $P(S)=\sum\limits_{i=1}^p P(S_i)$, and if $S$ is primitive then $P(S)=\sum\limits_{i=1}^p \rho(n_i)= \rho(n_1,\ldots,n_t)$.
A role of quadratic forms in the theory of representations of quivers and posets is well-known [@11].
The norm of a relation $\|S,\leq\|=\inf\limits_{u\in \overline{P}_n}
f_S(u)$ was introduced in [@1] on base of form $f_S$. In view of Lemma 5 it is naturally to consider instead of $\|S,\leq\|$ the function $P(S)=\|S,\leq\|^{-1}$ [@1].
[**Proposition 8.**]{} [*$S$ has finite (respectively tame) type if and only if $P(S)<4$ (respectively $P(S)=4$).*]{}
With this point of view the Kleiner’s list of the critical posets [@7] is the list of [*$P$-faithful*]{} posets $S_i$, for which $P(S)=4$.
4 posets of Kleiner’s list are primitive: $$\mbox{(I)}.\quad (1,1,1,1), \quad (2,2,2), \quad (1,3,3),\quad (1,2,5),
\quad \mbox{and the fifth is} \quad (4)\bigsqcup K,$$ where $K=\raisebox{-4mm}[7mm][5mm]{\mbox{\includegraphics{rys11.eps}}}=
\langle 2;2\rangle=\zeta(1\frac 12)$. It is easy to see that any chain is $P$-faithful, and s. 7 implies that $K$ is also $P$-faithful $(P(K)=2,4)$. By Lemma 5 a disconnected poset is $P$-faithful if and only if all its components are $P$-faithful.
The list of critical sets [@9]: $$\mbox{(II)}.\quad (1,1,1,1,1),\quad (1,1,1,2),\quad (2,2,3),\quad
(1,3,4),\quad (1,2,6),\quad (6)\bigsqcup K$$ can be characterized as the list of $S$, having the following properties:
1\) $P(S)>4$,
2\) if $S'\subset S$ then $P(S')\leq 4$.
The following statement play central role in the theory of representations of posets [@7], [@9].
[*A poset $S$ is finitely represented (respectively tame), if and only if $S$ does not contain subsets I (respectively II).*]{}
It has been naturally to suppose that all $P$-faithful poset are either chains or some sets, for which $K$ is the least representative. This was a reason to introduce $P-$faithful posets [@2].
Now we show how the lists (I), (II) can be obtained using characteristic of (connected) $P$-faithful sets and formula . It is easy to check that $P(S)=4$ for $S\in \mbox{I}$ and $P(S)>4$ for $S\in \mbox{II}$ (in view of Lemma 5 and formula ).
We call a $P$-faithful poset $S$ [*utmost*]{} if $P(S)\geq
4$ and $P(S')\leq 4$ for any $S'\subset S$ (hereat $S'$ can be supposed $P$-faithful).
[**Lemma 15.**]{} [*Not primitive utmost $S$ is of the form $K\bigsqcup Z_m$ where $m$ is 4 or 5.*]{}
Let $S$ contains a connected component $\zeta(r)$ where $\{r\}=\frac{q}{t}$, $t>1$, $q<t$, $(q,t)=1$. The characterization of $P$-faithful posets implies $\omega(S)<4$ since in the opposite case $S\supset S'=(2,1,1,1)$, $\rho(2,1,1,1)=4\frac{1}{3}>4$. Consequently $t\leq 3$, moreover if $t=3$ then $\zeta(r)=S$.
Let $t=3$, $1\leq q\leq 2$. If $[r]\geq 2$ then $S\supset
S'=S\setminus\{z_1^-,z_3^+\}$. $S'$ is a primitive poset containing $(2,3,2)$, $\rho(2,2,3)=4\frac{1}{6}$, $\rho(S')>4$. If $[r]=1$ then either $r=1\frac{1}{3}$ or $r=1\frac{2}{3}$. We obtain $\rho(r)\leq 1\frac{1}{4}$, then $P(S)=3\rho(r)<4$ (see ).
Let $t=2$ and $S\neq
\zeta(1\frac{1}{2})\bigsqcup\hat{S}$. If $S=\zeta(r)$ then $P(S)=2\rho(r)<4$ since $\rho(r)<2$ for any $r$. So, $S=\zeta(r)\bigsqcup\hat{S}$, $r>1\frac{1}{2}$, i. e. $r\geq 2\frac{1}{2}$, $\zeta(r)\supset\{\zeta(r)\setminus
z_1^-\}\supseteq(2,3)$. For $|\hat{S}|>1$, $S\supset S'$, contains $(2,2,3)$ or $(1,1,2,3)$ and $P(S')>4$. Hence $|\hat{S}|=1$. Then $[r]<3$, since otherwise $\zeta(r)\supset S'=\zeta(r)\setminus
z_1^-\supseteq(3,4)$ and $P(S'\bigsqcup(1))>4$ since $\rho(3,4,1)>4$. For $[r]=2$ we obtain $P(S)<4$ since $P(\zeta(2\frac{1}{2}))=2\cdot 1\frac{3}{7}$, $P(\zeta(2\frac{1}{2})\bigsqcup(1))=2\frac{6}{7}+1$ (Lemma 5).
Let, finally, $S=\zeta(1\frac{1}{2})+\widehat{S}$ ($S\neq\zeta(1\frac{1}{2})$) since $P(\zeta(1\frac{1}{2}))=2,4$), then if $w(\widehat S)>1$ then $S\supset
S'=\{\zeta(1,\frac{1}{2})\setminus
z_1^-\}\bigsqcup(1,1)=(2,1,1,1)$, $\rho(S')>4$. If $\widehat S=Z_m$ then for $m<4$ $\rho(m)<1,6$ and $P(S)<4$, and for $m>5$, $S\supset S'=(\zeta(1\frac{1}{2})\bigsqcup Z_5)$, $\rho(S')>4$, $P(K\bigsqcup Z_4)=4$, $P(K\bigsqcup Z_5)=4\frac{1}{15}$.
------------------------------------------------------------------------
[**Proposition 9.**]{} [*$P$-faithful $S$ is utmost if and only if $S\in
\mbox{\rm I}\cup \mbox{\rm II}$.*]{}
If $S$ is not primitive then Lemma 15 implies the statement. Let $S$ is primitive. Then $w(S)>2$ and $S\not\in\{(1,1,n),(1,2,2),(1,2,3),(1,2,4)\}$ (otherwise $P(n_1,\dots,n_t)=\rho(n_1,\dots,n_t)<4$). In the rest cases we can see that if $S\not\in \mbox{I}\cup \mbox{II}$ then $S\supset S'\in \mbox{II}$, and if $S\in \mbox{I}\cup \mbox{II}$ then $S\not\supset S'\in \mbox{II}$.
------------------------------------------------------------------------
Propositions 8, 9 (in view of $P(S)=4$ for $S\in \mbox{I}$ and $P(S)>4$ for $S\in \mbox{II}$) imply the main theorems [@7] and [@9].
$P-$faithful posets, for which $P=4$, play important role in representation theory. We don’t know whether $P-$faithful posets with $P=n>4$ play some analogous role. In [@14] primitive posets with $P=5$ are written out. We give the example (probably unique) of a not primitive poset $S=\zeta(3\frac{1}{2})\bigsqcup(17)$ for which $P(S)=5$ (see and Lemma 3).
It is Example 4 (s. 3) where $C(S)=\varnothing$ but $S$ is not $P-$faithful. We hope that studying of $C(S)$ can be interesting for the representation theory. Remark, that in [@1] the norm $\|P\|$ of an arbitrary binary relation $P$ (on finite set) and corresponding notion of $P$-faithfulness are defined (these notions can be used for locally scalar representations (see [@8]) in Hilbert spaces). However in this case it is more complicate to review $P$-faithful sets. Such investigation would be seemingly rather difficult and interesting problem.
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[^1]: A.I. Sapelkin in [@5] called this statement as Zeldich lemma. M.V. Zeldich in [@6] called it “important and surprising”, with what authors quite agree, in spite of brevity of the proof
|
---
abstract: |
We extend the Colombeau algebra of generalized functions to arbitrary (paracompact) $C^\infty$ $n$-manifolds $M$. Embedding of continuous functions and distributions is achieved with the help of a family of $n$-forms defined on the tangent bundle $TM$, which form a partition of unity upon integration over the fibres.
PACS numbers: 9760L, 0250
---
[Herbert BALASIN ]{}\
[ *Institute for Theoretical Physics, University of Alberta\
Edmonton, T6G 2J1, CANADA* ]{}\
Introduction {#introduction .unnumbered}
============
Colombeau theory [@Col1; @ArBi] set out to give a mathematically consistent way of multiplying distributions. From a physical point of view the Colombeau algebra provides a sound framework to accommodate calculations involving regularization methods employed to deal with singular quantities that actually arise as products of distributions. The most prominent example in this regard is provided by the renormalization procedure of perturbative quantum field theory. Recently, there have been some attempts [@Par; @BaNa1; @BaNa2; @Ba1; @LoSo; @ClVi] to apply a similar formalism to spacetime singularities that arise in general relativity. Unless one is willing to restrict to manifolds that are topologically ${\mathbb R}^n$ the application of the Colombeau formalism requires its generalization to arbitrary manifolds.
Although it is a pretty recent development the definition of the Colombeau algebra has undergone various changes, starting from a functional analytic motivation, involving the delicate subject of differential calculus in locally convex vector spaces [@Col0].
The definition [@Col2] we are going to generalize in this work considers the elements of the Colombeau algebra ${\mathcal G}({\mathbb R}^n)$ as moderate one-parameter families $(f_\epsilon)$ of $C^\infty$ functions denoted by $C_M^\infty({\mathbb R}^n)$ up to negligible families denoted by $C_N^\infty({\mathbb R}^n)$, where the adjectives refer to certain growth conditions in the parameter $\epsilon$ of the family. The embedding of continuous functions and distributions into algebra is achieved with the help of a smoothing-kernel, which has to obey certain properties. The above quotient is then such that $C^\infty({\mathbb R}^n)$ becomes a subalgebra of ${\mathcal G}({\mathbb R}^n)$ thereby reconciling the two different embeddings as constant sequences or with the aid of the smoothing kernel.
Although the definition of the Colombeau algebra used above is fairly straightforwardly generalized to an arbitrary manifold [@Ba1], the notion of the smoothing kernel presents some difficulties since it draws heavily upon concepts specific to ${\mathbb R}^n$. On the other hand the smoothing kernels play an important role for the embedding of distributions as linear subspaces into the Colombeau algebra. Although there have been proposals [@CoMe] which weaken the condition on the moments, we will stick to the original condition by taking the smoothing kernel to be an $n$-form on the tangent-bundle $TM$ of the manifold $M$ (more precisely a family of $n$-forms of this type, which becomes a partition of unity upon integration along the fibre). Since diffeomorphisms of $M$ act linearly on the fibres of $TM$ this approach gives (invariant) meaning to the conditions on the moments familiar from ${\mathbb R}^n$. The embedding of smooth and continuous functions with the aid of the above smoothing kernel produces locally defined moderate functions, whose sum defines the corresponding Colombeau-object.
Our work will be organized as follows. In chapter one we recall the basic notions of the Colombeau algebra. Section two will be devoted to a brief survey of distribution theory on arbitrary manifolds, thereby giving a manifestly covariant formulation of the latter. Finally in chapter three we present our generally covariant formulation of the Colombeau algebra motivated by the extension of smoothing kernels which we will consider to be a suitable family of $n$-forms on the $2n$-dimensional tangent-bundle.
1) Moderate and negligible functions, association and all that {#moderate-and-negligible-functions-association-and-all-that .unnumbered}
==============================================================
The basic idea of Colombeau’s approach for the multiplication of distributions is to find a (differential) algebra large enough to contain all the usual $C^\infty$ functions as a subalgebra and the distributions as a linear subspace. The construction starts by considering one-parameter families $(f_\epsilon)$ of $C^\infty$ functions subject to the condition
$$\begin{aligned}
\label{Mod}
&& C^\infty_M = \{ (f_\epsilon)\vert f_\epsilon \in
C^\infty({\mathbb R}^n)\quad \forall K \subset {\mathbb R}^n compact,
\forall \alpha\in {\mathbb N}^n\quad \\
&&\hspace*{1cm}\exists\, N\in {\mathbb N},\exists\> \eta > 0,\exists\> c>0
\quad s.t. \sup_{x\in K}\vert D^\alpha f_\epsilon(x)\vert \leq
\frac{c}{\epsilon^N}\quad\forall 0<\epsilon< \eta\},\nonumber\\
&&\mbox{where}\quad D^\alpha =
\frac{\partial^{|\alpha|}}{(\partial x^1)^{\alpha_1}
\cdots(\partial x^n)^{\alpha_n}}.\nonumber\end{aligned}$$
One way of getting some intuition for these objects is by considering them as regularizations of functions that (possibly) become singular in the limit $\epsilon\to 0$. All operations like addition and multiplication are defined pointwise, and one easily proves that (\[Mod\]) is an algebra, which will be denoted $C^\infty_M$.
$C^\infty$-functions are canonically embedded into $C_M^\infty$ as constant sequences, whereas continuous functions (and distributions) of at most polynomial growth require a smoothing kernel $\varphi \in {\mathcal S}$ [^1]. Since it represents an approximate $\delta$-function one requires that $$\label{smooth}
\int d^nx \varphi(x)=1 \qquad \int d^nx\> x^\alpha \varphi(x)=0 \quad
|\alpha|\geq 1$$ The embedding is done by convoluting the (rescaled and shifted) smoothing kernel $\varphi$ with $f$, i. e. $$f_\epsilon(x) = \int d^ny \frac{1}{\epsilon^n} \varphi(\frac{y-x}{\epsilon})
f(y).$$
In order to reconcile the different embeddings of $C^\infty$ functions one identifies them by employing a suitable ideal $C^\infty_N({\mathbb R}^n)$. Its members will be addressed as negligible functions in the following. $$\begin{aligned}
\label{Neg}
&& C^\infty_N({\mathbb R}^n) = \{ (f_\epsilon)\vert (f_\epsilon) \in
C^\infty_M({\mathbb R}^n)\quad
\forall K \subset {\mathbb R}^n compact, \forall \alpha\in {\mathbb N}^n,
\forall N\in {\mathbb N}\nonumber\\
&&\hspace*{1cm}\exists\> \eta > 0,\exists\> c>0,\quad
s.t. \sup_{x\in K}\vert D^\alpha f_\epsilon(x)\vert \leq
c\epsilon^N\quad\forall\> 0<\epsilon< \eta\}\end{aligned}$$ Considering the difference $f(x) - \int d^ny (1/\epsilon^n)
\varphi(\frac{y-x}{\epsilon})f(y)$, where $f$ denotes an arbitrary $C^\infty$ function of at most polynomial growth, one easily checks that it is a negligible function. The Colombeau algebra ${\mathcal G}({\mathbb R}^n)$ is therefore defined to be the quotient of $C^\infty_M({\mathbb R}^n)$ with respect to $C^\infty_N({\mathbb R}^n)$. A Colombeau object is thus a moderate family $(f_\epsilon(x))$ of $C^\infty$ functions modulo negligible families.
The usual distribution theory arises from coarse graining the Colombeau algebra employing an equivalence relation called association. Two Colombeau objects $(f_\epsilon)$ and $(g_\epsilon)$ will be considered associated if $$\label{assoc}
\lim_{\epsilon\to 0} \int d^nx (f_\epsilon(x) -g_\epsilon(x))\varphi(x) =0
\qquad \forall \varphi \in {\mathcal D}.$$ This equivalence relation is much coarser than the one used for the definition of $\mathcal G$. It is compatible with addition, differentiation and multiplication by $C^\infty$ functions. It is, however not compatible with multiplication. Intuitively speaking different Colombeau objects are packaged together into one association-class. One might think of such a class as containing different regularizations of the same (possibly singular) non-smooth function.
Let us give a simple (by now classical [@Col2]) example showing the power of the association calculus. Consider $\theta^n$ which as piecewise continuous function gives rise to the same (regular) distribution as $\theta$. Upon naive differentiation one obtains $$\theta^n(x) = \theta(x) \Rightarrow n \theta(x) \theta'(x) = \theta'(x)
\Rightarrow \theta(0) = \frac{1}{n},$$ which is a contradiction since $\theta(0)$ is independent of $n$. With regard to the Colombeau algebra $\theta^n$ is no longer equal to $\theta$, they are however associated. Since association respects differentiation we have $$\theta^n(x) \approx \theta(x) \Rightarrow n \theta^{n-1}(x) \theta'(x)
\approx \theta'(x) \Rightarrow \theta^n(x)\theta'(x) \approx
\frac{1}{n+1}\delta(x),$$ which only tells us that we are allowed to replace $\theta^n\theta'$ by $\delta/(n+1)$. Since multiplication breaks association we do not encounter any ambiguities.
2) Distributions on arbitrary manifolds {#distributions-on-arbitrary-manifolds .unnumbered}
=======================================
The main goal of this section is to give a manifestly covariant formulation of distribution theory, which shows that the distribution concept relies only on the differentiable structure of the underlying manifold $M$ and does not require any additional notions such as the existence of a metric or a volume-form [@ChBr; @BaNa1]. Usually distributions on ${\mathbb R}^n$ are defined as elements of the (topological) dual of test-function space $\mathcal D$, which in the simplest case is considered to consist of all $C^\infty$ functions with compact support. Locally integrable functions $f(x)$ are embedded into distribution space as so-called regular functionals $$\label{regular}
(f,\varphi) = \int d^n x f(x)\varphi(x).$$ Operations like differentiation and multiplication by arbitrary $C^\infty$ functions are defined via the corresponding operations on $\mathcal D$, namely $$\begin{aligned}
(D^\alpha f,\varphi) &:=& (-)^{|\alpha|}(f,D^\alpha \varphi)\quad \alpha \in
{\mathbb N}^n \\
( g\> f,\varphi) &:=& (f,g \> \varphi),\end{aligned}$$ which reduce to standard integral relations if one restricts to regular distributions generated by differentiable functions. Both of the above operations are well-defined since they map $\mathcal D$ onto itself.
In order to generalize the above concepts to arbitrary manifolds $M$, we first have to decide what to do about test-function space. Although $C^\infty$ functions with compact support are a concept that makes sense on arbitrary manifolds this would not allow us to embed locally integrable functions in the same way as we did in ${\mathbb R}^n$, where we made use of the natural volume form $d^n x$, unless we are willing to single out a volume form. Thinking, however, of $\varphi$ and $d^nx$ as parts of [*one*]{} object, we are immediately lead to consider $C^\infty$ $n$-forms $\tilde{\varphi}$ with compact support as the natural generalization of test-functions. Taking into account that the latter are sections of a vector-bundle and by employing a partition of unity (which requires the underlying manifold $M$ to be paracompact) we basically may construct the locally convex vector-space topology in very much the same way as in ${\mathbb R}^n$. Distributions are then once again defined as the elements of the topological dual of this space.
The concepts of multiplication by $C^\infty(M)$ functions is without problems. In order to generalize differentiation we make use of the Lie-derivative along an arbitrary $C^\infty $ vector-field and the classical identity
$$(L_X f,\tilde{\varphi}) = \int_M L_X f \tilde{\varphi} =
\int_M d (i_X (f \tilde{\varphi})) - \int_M f L_X\tilde{\varphi} =
-(f,L_X\tilde{\varphi})$$ where the last equality is taken to be true for arbitrary distributions. Let us once again emphasize that the embedding of locally integrable functions led us to generalizing test-function space to $C^\infty$ $n$-forms with compact support.
3) Generally covariant formulation of the\
Colombeau algebra {#generally-covariant-formulation-of-the-colombeau-algebra .unnumbered}
==========================================
This chapter is devoted to a formulation of the Colombeau algebra suitable for arbitrary manifolds $M$. Considering the definition of moderate and negligible functions (\[Mod\],\[Neg\]) one sees that they immediately generalize to $M$, since the conditions required for the definition remain invariant under coordinate transformations. A manifestly covariant definition may be given with the aid of the Lie-derivative $$\begin{aligned}
C^\infty_M(M) &=& \{ (f_\epsilon)| f_\epsilon \in C^\infty(M) \forall\>
K \subset M\> compact,
\forall\>\{X_1,\dots,X_p\}\>\\
&&\hspace*{0.5cm}X_i\in\Gamma(TM),[X_i,X_j ]=0,\exists N\in{\mathbb N},\exists
\eta> 0,
\exists c>0\> \\
&&\hspace*{0.5cm} s.t. \sup_{x\in K}| L_{X_1}\dots L_{X_p}
f_\epsilon(x)|
\leq \frac{c}{\epsilon^N}\quad 0<\epsilon<\eta\},\end{aligned}$$
$$\begin{aligned}
C^\infty_N(M) &=& \{ (f_\epsilon) \in C^\infty_M(M) | \forall\>
K \subset M\> compact,\forall\>\{X_1,\dots,X_p\}\> \\
&&\hspace*{0.5cm}X_i\in\Gamma(TM),
[X_i,X_j]=0,\forall q\in{\mathbb N},\exists \eta> 0, \exists c>0\>\\
&&\hspace*{0.5cm} s.t. \sup_{x\in K}| L_{X_1}\dots
L_{X_p}f_\epsilon(x)|
\leq c\epsilon^q\quad 0<\epsilon<\eta\}.\end{aligned}$$
However, looking at the smoothing kernel $\varphi$ required for the embedding of continuous functions (and distributions) one realizes that the condition on the moments does not remain invariant under coordinate transformations. One way of remedying this situation was presented in [@CoMe], where instead of a fixed smoothing kernel $\varphi$ a whole family $\varphi_\epsilon$ was considered, which allowed a weakening of the conditions on the moments.
We will try to follow a different path, which allows us to keep the conditions on the moments in a coordinate invariant manner. This seemingly paradoxical statement is easily understood in terms of the tangent bundle $TM$ of $M$. Let us consider a bundle-atlas of $TM$ induced by an atlas $M$ and let $(x,\xi)$ denote the respective coordinates of an arbitrary chart. Now fix a differential $n$-form $$\label{smoothfrm}
\tilde{\varphi} = \varphi(x,\xi)(d\xi^1+ N^1{}_idx^i)\wedge\cdots\wedge
(d\xi^n+ N^n{}_idx^i)$$ which we require to obey $$\begin{aligned}
\label{smoothprp}
&&\int\limits_{T_xM} i_x^*\tilde{\varphi} = \int \varphi(x,\xi) d^n\xi
=1\qquad\int\limits_{T_xM} \xi^\alpha i_x^* \tilde{\varphi} =
\int \xi^\alpha\varphi(x,\xi) d^n\xi = 0\\
&&\mbox{where }\> i_x:T_xM\to TM\qquad
i_x^*\tilde{\varphi}=\varphi(x,\xi)d^n\xi\nonumber\\
&&\hspace*{2cm}\xi\mapsto (x,\xi)\nonumber\end{aligned}$$ The advantage of the tangent-bundle formulation comes from the fact that diffeomorphisms in $M$ induce a specific type of diffeomorphism on $TM$, namely fibre-preserving bundle morphisms, which act [*linearly*]{} on the fibres thereby leaving (\[smoothprp\]) invariant. Moreover, the rescaling and shift operations are now naturally interpreted as action of the structure group on $TM$, given by $$\begin{aligned}
\label{structgr}
&&\phi_\epsilon:TM\to TM\qquad (x,\xi)\mapsto (x,\frac{1}{\epsilon}\xi),
\nonumber\\
&&\phi_a:TM\to TM\qquad (x,\xi)\mapsto (x,\xi+a)\end{aligned}$$ which are specific $IGL(n,{\mathbb R})$[^2] transformations. Let us now use the smoothing-form $\tilde\varphi$ to embed a continuous function $f$ $$f_\epsilon(x):= \int\limits_{T_xM}\phi_\epsilon^*\phi_{-x}^*i_x^*
\tilde{\varphi}f
= \int d^n\xi\frac{1}{\epsilon^n} \varphi(x,\frac{\xi-x}{\epsilon})f(\xi)=
\int d^n\xi \varphi(x,\xi)f(x+\epsilon\xi).$$ The last relation makes explicit use of the coordinate representation of the function $f$ with respect to a given coordinate chart (which we assume to be ${\mathbb R}^n$) in order to lift $f$ to a function on $T_xM$. This entails that $f_\epsilon$ is also only defined locally. However, using a $C^\infty$ partition of unity $(\rho_i)$ subordinate to the cover $(U_i)$ allows us to patch the local objects together to a global one $$\label{glemb}
f_\epsilon(x) := \sum\limits_i \rho_i(x)f_{i,\epsilon}(x),$$ where the sum is always finite due to the local finiteness of the cover $(U_i)$. Note that actually every term in the sum is a well-defined $C^\infty$–object on $M$. The above construction may be completely absorbed into the $n$-form $\tilde{\varphi}$ by taking a family of $n$-forms $\tilde{\varphi}_i$ subordinate to the cover $(U_i)$ such that $$\sum\limits_i \int\limits_{T_x M}i_x^*\tilde{\varphi}_i = 1$$ That is to say $\rho_i(x) := \int\limits_{T_x M}i_x^*\tilde{\varphi}_i$ defines a partition of unity subordinate to the (locally finite) cover $(U_i)$. It is now easy to show that the two different embeddings of $C^\infty$ functions differ only by elements belonging to $C^\infty_N(M)$. $$f_\epsilon(x)-f(x)=\sum\limits_i \int d^n\xi\varphi_i(x,\xi)
(f(x+\epsilon \xi) - f(x)) = {\mathcal O}(\epsilon^q),$$ where the last equality is derived from Taylor-expanding $f(x+\epsilon \xi)$. Although the embedding (\[glemb\]) depends on the atlas employed, since we made specific use of the local representation of the function in order to lift it to the tangent-bundle, this dependence disappears for $C^\infty$ functions $$\begin{aligned}
\bar{f}_\epsilon(\bar{x})&=&\int d^n\bar{\xi}
\bar{\varphi}(\bar{x},\bar{\xi})\bar{f}(\bar{x}+\epsilon\bar{\xi})
=\int d^n\xi\varphi(x,\xi)f(\mu^{-1}(\mu(x)+\epsilon
\frac{\partial\mu}{\partial x}\xi))\\
&=&\int d^n\xi\varphi(x,\xi)f(x+\epsilon \xi) + {\mathcal O}(\epsilon^q)
=f_\epsilon(x) + {\mathcal O}(\epsilon^q),\end{aligned}$$ where once again the result was obtained by Taylor-expanding up to order $q$.
[**Acknowledgement:**]{} The author wants to thank Michael Oberguggenberger and Michael Kunzinger for introducing him to the subject of Colombeau theory, during his stay in Innsbruck. Furthermore the author wants to thank the relativity and cosmology group at the University of Alberta and especially Werner Israel, for their kind hospitality, during the final stage of this work.
Conclusion {#conclusion .unnumbered}
==========
In this paper we generalized the Colombeau algebra to arbitrary manifolds. The main problem that had to be overcome was the embedding of continuous functions and distributions with the help of a smoothing kernel. The definition of the latter used in the standard ${\mathbb R}^n$-approach required all its moments to vanish. Unfortunately, this concept does not remain invariant under coordinate transformations and does therefore not generalize to arbitrary manifolds. Taking advantage of the tangent bundle we were, however, able to maintain the condition on the moments by taking the smoothing kernel to be a differential $n$-form defined on the $2n$-dimensional tangent bundle. The coordinate invariance of this construction is guaranteed by linear action of $M$-diffeomorphisms on the fibres of $TM$.
[99]{} Colombeau J, [*New Generalized Functions and Multiplication of Distributions* ]{} Mathematics Studies [**84**]{}, North Holland (1984). Aragona J and Biagioni H, [*Analysis Mathematica*]{} [**17**]{}, 75 (1991). Parker P, [*J. Math. Phys. *]{} [**20**]{}, 1423 (1979). Balasin H and Nachbagauer H, [*Class. Quantum Grav. *]{} [**10**]{}, 2271 (1993). Balasin H and Nachbagauer H, [*Class. Quantum Grav. *]{} [**11**]{}, 1453 (1994). Balasin H, [*Geodesics of impulsive pp-waves and the multiplication of distributions*]{} gr-qc/9607076 . Loustó C and Sánchez N [*Nucl. Phys. *]{}[**B383**]{} 377 (1992). Clark C, Vickers J and Wilson J, [*Class. Quantum Grav. *]{} [**13**]{}, 2485 (1996). Colombeau J, [*Differential calculus and holomorphy*]{} Mathematics Studies [**64**]{}, North Holland (1982). Colombeau J, [*Multiplication of Distributions* ]{} [**LNM 1532**]{}, Springer (1992). Colombeau J and Meril A, [*Journal of Mathematical Analysis and Applications*]{} [**186**]{}, 357-364 (1994). Choquet-Bruhat Y, Morette-DeWitt C and Dillard-Bleick M, [*Analysis Manifolds and Physics*]{}, North-Holland (1982).
[^1]: $\mathcal S$ denotes the space of rapidly decreasing $C^\infty$ functions
[^2]: we actually consider the fibres $T_xM$ to be affine spaces
|
---
abstract: 'We suggest a new mechanism for generation of large-scale magnetic field in the hot plasma of early universe which is based on the parity violation in weak interactions and depends neither on helicity of matter turbulence resulting in the standard $\alpha$-effect nor on general rotation. The mechanism can result in a self-excitation of an (almost) uniform cosmological magnetic field.'
author:
- 'V. B. Semikoz$^1$ and D. D. Sokoloff$^2$'
title: 'Large-scale magnetic field generation by $\alpha$-effect driven by collective neutrino-plasma interaction'
---
$^1$ The Institute of the Terrestrial Magnetism, the Ionosphere and Radio Wave Propagation of the Russian Academy of Sciences,\
IZMIRAN, Troitsk, Moscow region, 142190, Russia\
Email: semikoz@izmiran.rssi.ru
$^2$ Department of Physics, Moscow State University, 119899, Moscow, Russia\
Email: sokoloff@dds.srcc.msu.su
The large-scale magnetic field self-excitation in astrophysical bodies like Sun, stars, galaxies etc is usually connected with so-called $\alpha$-effect, i.e. a specific term in the Faraday electromotive force ${\cal E =
\alpha {\bf B}}$ that is proportional to the large-scale magnetic field $%
{\bf B}$. This term is connected with a violation of mirror symmetry of a rotating stratified turbulence or convection: the number of right-handed vortices systematically differs from the number of left-handed vortices due to the Coriolis force action. In this sense, $\alpha$ is determined by helicity of turbulent motions.
The differential rotation $\Omega$ usually participates with the $\alpha$-effect in dynamo action (so-called $\alpha \Omega$-dynamo), however $\alpha$-effect induced in a rigidly rotating turbulent body could lead to a dynamo action alone (so-called $\alpha^2$-dynamo) while the differential rotation alone is unable to result in a dynamo action [@ZRS]. The $\alpha$-effect is induced by Coriolis force which destroys the mirror symmetry of turbulent motions.
On the one hand, the $\alpha$-effect is impossible in electrodynamics of classical nonmoving media because of its mirror-symmetry. On the other hand, the mirror asymmetry of the matter happens at the level of particle physics and we can expect that an $\alpha$-effect could be based on this asymmetry. The aim of our paper is to present such mechanism based on parity violation in weak interactions.
We remind that the main problem of most particle physics mechanisms of the origin of seed fields is how to produce them coherently on cosmological (large) scales. There are many ways allowing to generate seed [*small-scale random magnetic fields*]{} in early universe, e.g. at phase transitions [@Vachaspati], however, the following growth of the correlation length, e.g. in the inverse cascade with the merging of such small-scale fields [@BEO], hardly could produce a substantial large-scale fields at present time [@Becketal]. We do not consider an evolution of correlated domains and the corresponding growth of correlation length considered e.g. in the review [@Enqvist] and concentrate here on the generation of a mean magnetic field (amplification of its strength) via $%
\alpha$-effect if such mean field has been already seeded somehow from small-scale magnetic fields.
Let us consider hot plasma of early universe after electroweak phase transition, $T\ll T_{EW}\simeq 10^5~{\rm {MeV}}$, when we may use point-like (Fermi) approximation for weak interactions and where at the beginning a weak random magnetic field has a small macroscopic scale comparing with the horizon, $\Lambda\ll l_H$, while within a domain of the volume $\sim
\Lambda^3$ such magnetic field can be uniform and directed along an arbitrary z-axis, ${\bf B}= (0,0, B)$. Obviously, this does not violate the isotropy of universe as a whole with many randomly oriented domains.
Within a domain with an uniform magnetic field obeying the WKB limit $\mid
e\mid B\ll T^{2}$ the single quantum (spin) effect remains for electrons and positrons which populate the main Landau level only and contribute to the lepton gas magnetization, $M_{j}^{(\sigma )}=\mu _{B}<\bar{\psi}_{\sigma
}\gamma _{j}\gamma _{5}\psi _{\sigma }>=\delta _{jz}\mu _{B}({\rm sgn}\,$ $%
\sigma )n_{0\sigma }\sim ({\rm sgn}\,\sigma )B$ [@OS], where $\mu _{B}$ is the Bohr magneton, $n_{0\sigma }$ is the number density at the main Landau level for the electrons and positrons ($\sigma =e^{-}$, $\bar{\sigma}%
=e^{+}$),
$$n_{0\sigma }\approx n_{0\bar{\sigma}}=\frac{\mid e\mid B}{2\pi ^{2}}%
\int_{0}^{\infty }f_{{\rm eq}}^{(\sigma )}(\varepsilon _{p})dp\simeq \frac{%
\mid e\mid B~T\ln 2}{2\pi ^{2}}~. \label{mainlevel}$$
The magnetization $M_{j}^{(\sigma )}$ changes sign for electrons and positrons , $({\rm sgn}$ $\sigma )=\pm 1,$ effectively due to the opposite spin projections on the magnetic field at the main Landau levels.
For a small magnitude of magnetic fields we neglect small polarization of other components: muons, tau-leptons, quarks or nucleons. Obviously, densities (\[mainlevel\]) are very small comparing the total lepton densities$n_{\sigma } =\int (d^{3}p/(2\pi )^{3})f_{{\rm eq}}^{(\sigma )}(\varepsilon
_{p})\approx 0.183~T^{3}$, $n_{0\sigma }\ll n_{\sigma }$. Here $\sigma
=e^{\mp }$; $f_{{\rm eq}}^{(\sigma )}(\varepsilon _{p})$ is the Fermi distribution; $e$, $\varepsilon _{p}=\sqrt{p^{2}+m_{e}^{2}}$, $T$ are the lepton electric charge, the energy and the temperature of lepton gas correspondingly.
In magnetized plasma the pseudovector $M_{j}^{(\sigma )}$ enters weak interaction of the charged $\sigma $-fluid component with neutrinos (antineutrinos) through the axial part of the point-like ${\rm current}%
\times {\rm current}$ interaction Hamiltonian,$$V_{\sigma }^{(A)}=G_{F}M_{j}^{(\sigma )}\cdot \delta j_{j}^{(\nu )}/\mu
_{B},$$where $G_{F}=10^{-5}/m_{p}^{2}$ is the Fermi constant, $\delta {\bf j}^{(\nu
)}={\bf j}_{\nu }-{\bf j}_{\bar{\nu}}$ is the neutrino current density asymmetry.
Such interaction provides a force ${\bf F}_{\sigma }^{{\rm weak}}$ (see below Eq. (\[Euler\])) that is additive to the Lorentz force $q_{\sigma }(%
{\bf E}+{\bf V}_{\sigma }\times {\bf B})$ acting in MHD plasma on charged particles of the kind $\sigma $ and obviously depends on gradients of the interaction potential, ($F_{\sigma }^{{\rm weak}})_{i}\sim -\partial
_{i}V_{\sigma }^{(A)}$, or for an uniform magnetization within a domain ($%
M_{j}={\rm constant}$) on derivatives of the neutrino current density asymmetry, $\partial _{i}\delta j_{j}^{(\nu )}$.
The electric field ${\bf E}$ being common for all charged particles is obtained multiplying the motion equations for each charged components by the corresponding electric charge $q_{\sigma }$ with the following summing over components that leads to $\sum_{\sigma }q_{\sigma }^{2}{\bf E}$ in the Lorentz force and the remarkable addition of $n_{0\sigma }$ in the weak electromotive force term for electron-positron components, $E_{j}^{{\rm weak}%
}=\alpha B_{j}\sim (n_{0+}+n_{0-}),$ due to the independence of the product $q_{\sigma}\partial_iV_{\sigma}^{(A)}\sim
q_{\sigma}G_FM_j^{(\sigma)}\cdot\partial_i\delta j_j^{(\nu)}/\mu_B$ on the sign of the electric charge since $M_j^{(\sigma)}\sim ({\rm sgn}\, \sigma)$. Such term violates parity and provides a new particle physics origin of $%
\alpha $-effect for magnetic field generation, $\partial _{t}{\bf B}=-\nabla
\times {\bf E}^{{\rm weak}}$.
The pair motion equation in the one-component MHD is derived after the summation of Euler equations for comoving electrons and positrons for which the standard (polar vector) electric field cancels since $q_{\sigma}=\pm
\mid e\mid$, the standard Lorentz force $\mid e\mid ({\bf V}_+ - {\bf V}%
_-)\times {\bf B}= {\rm {rot}{\bf B}\times {\bf B}/4\pi n_e}$ arises while the weak force term depends in hot lepton plasma on the negligible difference of densities, $(n_{0+} - n_{0-})$ as well as the neutrino axial vector potential $V^{(A)}$ describing a probe neutrino in the electron-positron plasma [@Nunokawa] when $\delta j_j^{(\nu)}\to k_j$ and the sum over $\sigma$ leads to $V^{(A)}=G_F\sqrt{2}(n_{0+}- n_{0-}){\bf B%
}\cdot {\bf k}/Bk$.
Now we estimate the $\alpha$-effect originated in early universe by particle physics effects. Let us note that in an external large-scale magnetic field $%
{\bf B}$ a polarized equilibrium lepton plasma is characterized by the density matrix, $$\label{matrix}
\hat{f}^{(\sigma)}(\varepsilon_p)=\frac{\delta_{\lambda^{\prime}\lambda}}{2}
f_{{\rm eq}}^{(\sigma)}(\varepsilon_p) + \frac{(\vec{\sigma}\hat {\vec{b}}%
)_{\lambda^{\prime}\lambda}}{2}S_{{\rm eq}}^{(\sigma)}(\varepsilon_p)~,$$ where $\vec{\sigma}$ is the Pauli matrix; $\hat{{\bf b}}= {\bf B}/B$ is the ort directed along the magnetic field; $f_{{\rm eq}}^{(\sigma)}(%
\varepsilon_p)$ is the Fermi distribution; $S_{{\rm eq}}^{(\sigma)}(%
\varepsilon_p)= -(\mid e\mid B/2\varepsilon_p){\rm {d}{\it f}%
_{eq}^{(\sigma)}(\varepsilon_p)/ {d}\varepsilon_p}$ is the spin equilibrium distribution that defines the number density at the main Landau level ([mainlevel]{}), $\int d^3pS^{(\sigma)}_{{\rm eq}}(\varepsilon_p)/(2\pi)^3=n_{0%
\sigma}$; $\lambda=\pm 1$ is the spin projection on magnetic field.
Then we start from the linearized relativistic kinetic equations (RKE) derived in Vlasov approximation for a magnetized lepton plasma in the standard model (SM) of electroweak interactions after the summing over spin variables as given in Eq. (30) of Ref. [@OS], $$\begin{aligned}
\label{RKE}
&&\frac{\partial \delta f^{(\sigma)}({\bf p}, {\bf x},t)}{\partial t} + {\bf %
v} \frac{\partial \delta f^{(\sigma)}({\bf p}, {\bf x},t)}{\partial {\bf x}}+
\nonumber \\
&& + q_{\sigma}{\bf E} \frac{\partial f^{(\sigma)}_{{\rm eq}}(\varepsilon_p)%
}{\partial {\bf p}} + [{\bf v}\times {\bf B}]\frac{\partial \delta
f^{(\sigma)}({\bf p}, {\bf x},t)}{\partial {\bf p}} + \nonumber \\
&& +{\bf F}^{(V)}_{{\rm weak}}\frac{\partial f^{(\sigma)}_{{\rm eq}%
}(\varepsilon_p)}{\partial {\bf p}} + {\bf F}^{(A)}_{{\rm weak}}\frac{%
\partial S^{(\sigma)}_{{\rm eq}}(\varepsilon_p)}{\partial {\bf p}}=0.\end{aligned}$$ Here weak forces ${\bf F}^{(V)}_{{\rm weak}}$, ${\bf F}^{(A)}_{{\rm weak}}$ that appear due to the generalization in SM of the standard Boltzman equation have the form: $$\begin{aligned}
\label{vector}
{\bf F}^{(V)}_{{\rm weak}}=&&({\rm {sgn}~\sigma)G_F\sqrt{2}\sum_ac^{(V)}_a%
\left[- \nabla \delta n_{\nu_a} - \frac{\partial \delta {\bf j}_{\nu_a}}{%
\partial t} +\right.} \nonumber \\
&&\left. + {\bf v}\times \nabla\times \delta {\bf j}_{\nu_a} \right],\end{aligned}$$ and $$\begin{aligned}
\label{axialvector}
{\bf F}^{(A)}_{{\rm weak}}= &&-({\rm {sgn}~\sigma)G_F\sqrt{2}\sum_ac^{(A)}_a%
\left[- \frac{\partial \delta n_{\nu_a}\hat{{\bf b}}}{\partial t} -\right.}
\nonumber \\
&&\left. -{\bf v}\times \nabla\times \delta n_{\nu_a}\hat{{\bf b}} + \frac{%
m_e}{\varepsilon_p}\nabla ({\bf a}({\bf p})\cdot\delta {\bf j}_{\nu_a})%
\right];\end{aligned}$$ $c^{(V)}_a=2\xi \pm 0.5$, $c^{(A)}_a=\mp 0.5$ are the vector and axial couplings correspondingly (upper sign for electron neutrino) where subindex $%
a=e, \mu, \tau$ characterizes the kind of neutrino, $\xi=\sin^2\theta_W%
\simeq 0.23$ is the Weinberg parameter; $\delta j_{\nu_a}^{\mu}=
j^{\mu}_{\nu_a} - j^{\mu}_{\bar{\nu}_a}$ is the neutrino four-current density asymmetry, $$j^{\mu}_{\nu_a,\bar{\nu}_a}({\bf x },t)\equiv (n_{\nu_a,\bar{\nu}_a}, {\bf j}%
_{\nu_a,\bar{\nu}_a})=\newline
\int \frac{d^3k}{(2\pi)^3}\frac{k^{\mu}}{\varepsilon_k}f^{(\nu_a, \bar{\nu}%
_a)}({\bf k}, {\bf x},t)$$ is the neutrino (antineutrino) four-current density; $\delta n_{\nu_a}=
n_{\nu_a} - n_{\bar{\nu}_a}$ is the neutrino density asymmetry that plays an important role in the generation of magnetic field (see below (\[helicity\]) and in [@Dolgov]).
Finally ${\bf a}({\bf p})$ in the last term in (\[axialvector\]) is the three-vector component of the four-vector $a_{\mu}$ that is the analogue of the Pauli-Lubański four-vector $a_{\mu}(p)= {\rm Tr}~(\rho
\gamma_5\gamma_{\mu})/4m_e=\left(\vec
{p}\vec{\zeta}/m_e;~ {\bf \zeta} +
\vec{p}(\vec{p}\cdot \vec{\zeta})/m_e(\varepsilon_p + m_e)\right)$ with the change of the spin $\vec{\zeta}$ to $\hat{{\bf b}}$.
Notice that we can substitute the total number density distribution function $f^{(\sigma)}({\bf p},{\bf x},t)=f^{(\sigma)}_{{\rm eq}}(\varepsilon_p) +
\delta f^{(\sigma)}({\bf p}, {\bf x},t)$ normalized on the total density $n_{\sigma}=\int (d^3p/(2\pi)^3)f^{(\sigma)}({\bf p},{\bf x},t)\approx \int
(d^3p/(2\pi)^3)f_{{\rm eq}}^{(\sigma)}$ into all the terms of RKE in the first and second lines (\[RKE\]) restoring its standard Boltzman form.
In addition, RKE (\[RKE\]) obeys the continuity equation, $\partial
j_{\mu}^{(\sigma)}/\partial x_{\mu}=0$, or the lepton number is conserved, where $j_{\mu}^{(\sigma)}({\bf x},t)=\int
d^3p(p_{\mu}/\varepsilon_p)f^{(\sigma)}({\bf p},{\bf x},t)/(2\pi)^3$ is the lepton four-current density.
Then we can use the standard method [@Pitaevsky] for transition from kinetic equations to the hydrodynamical ones. Multiplying RKE (\[RKE\]) by the momentum ${\bf p}$ and integrating it over $d^3p$ with the use of the standard definitions of the fluid velocity ${\bf V}_{\sigma}=
n_{\sigma}^{-1}\int d^3p{\bf v}f^{(\sigma)}({\bf p},{\bf x},t)/(2\pi)^3$ and the generalized momentum ${\bf P}_{\sigma}= n_{\sigma}^{-1}\int d^3p{\bf p}%
f^{(\sigma)}({\bf p},{\bf x},t)/(2\pi)^3$ one obtains the Euler equation for the fluid species $\sigma$ in plasma with the additive collision terms taken in the $\tau$-approximation, $$\begin{aligned}
\label{Euler}
\frac{{\rm {d}{\bf P_{\sigma}}}}{{\rm {d}t}}=&& - \nu_{\sigma}^{{\rm em}%
}\delta {\bf P}_{\sigma} - (\nu_{\sigma \nu} + \nu_{\sigma \bar{\nu}}){\bf P}%
_{\sigma} - \frac{\nabla p_{\sigma}}{n_{\sigma}} + \nonumber \\
&& + q_{\sigma}({\bf E} + [{\bf V}_{\sigma}\times {\bf B}]) + {\bf F}%
_{\sigma}^{{\rm weak}}~.\end{aligned}$$ Here $\sigma=e^-, \mu^-, \tau^-, q_u, q_d,...$ ($\bar{\sigma}= e^+, \mu^+,
\tau^+, \bar{q}_u, \bar{q}_d,...$ for antiparticles) enumerates the plasma components; $\nu^{{\rm em}}_{\sigma}$ is the electromagnetic collision frequency which leads to the fast equilibrium in plasma and defines plasma conductivity; $\nu_{\sigma \nu}$, $\nu_{\sigma \bar{\nu}}$ are the weak collision frequencies providing the generation mechanism suggested in [Dolgov]{} for neutrino scattering off electrons and positrons before neutrino decoupling; $p_{\sigma}$ is the fractional pressure.
We isolate in the weak ponderomotive force vector and axial parts, i.e. $%
{\bf F}_{\sigma }^{{\rm weak}}={\bf F}_{\sigma }^{(V)}+{\bf F}_{\sigma
}^{(A)}$. The first term ${\bf F}_{\sigma }^{(V)}$ coming from (\[vector\]) was found by the independent (Lagrangian) method in [@Brizard] and is irrelevant to the magnetic field generation mechanism considered here. The axial vector force ${\bf F}_{\sigma }^{(A)}$ appearing from ([axialvector]{}) due to the polarization of lepton gas in a weak magnetic field ${\bf B}$, is
$$\begin{aligned}
\label{axial}
{\bf F}^{(A)}_{\sigma}=&& \frac{G_F\sqrt{2}\delta_{\sigma e}({\rm {sgn}%
~\sigma)}}{n_{\sigma}} \sum_{a=e,\mu,\tau}c_{\sigma \nu_a}^{(A)}\left[%
n_{0\sigma}\hat{{\bf b}} \frac{\partial \delta n_{\nu_a}}{\partial t} +
\right. \nonumber \\
&&\left.+ N_{0\sigma}\nabla (\hat{{\bf b}}\cdot \delta {\bf j}_{\nu_a})%
\right].\end{aligned}$$
Finally the relativistic polarization term $N_{0\sigma}$, $$\label{relativ}
N_{0\sigma}= \frac{n_{0\sigma}}{3} + \frac{4\pi \mid e\mid B m_e}{9(2\pi)^3}%
\int_0^{\infty}f_{eq}^{(\sigma)}(\varepsilon_p )dp\frac{\partial}{\partial p}%
[v(3-v^2)]~,$$ in the non-relativistic case tends to the lepton density at the main Landau level given by Eq. (\[mainlevel\]), $N_{0\sigma}\to n_{0\sigma}$. Obviously, the weak force (\[axial\]) changes sign for positrons, $%
\sigma\to \bar{\sigma}$, due to the signature function.
For the hot plasma multiplying the Euler equation (\[Euler\]) by the electric charge $q_{\sigma}$, summing over $\sigma$ and dividing by $%
\sum_{\sigma}q_{\sigma}^2=Q^2$ we find the electric field ${\bf E}$ including all known polar vector terms plus the new axial vector ${\bf E}%
\sim \alpha {\bf B}$ originated by electron-positron polarizations which violates parity. This is similar to the derivation of ${\bf E}$ in [Brizard]{} for unpolarized plasma, and , in particular, from the Lorentz force one obtains the term $-\sum_{\sigma}(q_{\sigma}^2/Q^2){\bf V}%
_{\sigma}\times {\bf B}$ that obviously leads from the Maxwell equation $%
\partial_t {\bf B}= - \nabla\times {\bf E}$ to the dynamo effect in Faradey equation. Thus, we arrive to a governing equation for magnetic field evolution $$\label{Faradey}
\frac{\partial {\bf B}}{\partial t} = \nabla\times \alpha {\bf B} + \eta
\nabla^2 {\bf B}~,$$ where we omitted dynamo term neglecting any macroscopic rotation in plasma of early universe [^1].
Here we approximate the tensor $\alpha_{ij}$ coming in ${\bf E}$ from the axial vector force (\[axial\]) by the first diagonal ($\sim
\alpha\delta_{ij}$) term: $$\begin{aligned}
\label{helicity}
&&\alpha = \frac{G_F}{2\sqrt{2}\mid e\mid B}\sum_ac^{(A)}_{e\nu_a}\left[%
\left(\frac{n_{0-} + n_{0+}}{n_{e}}\right)\frac{\partial \delta n_{\nu_a}}{%
\partial t}\right]\simeq \nonumber \\
&&\simeq \frac{\ln 2}{4\sqrt{2}\pi^2}\left(\frac{10^{-5}T} {%
m_p^2\lambda^{(\nu)}_{{\rm fluid}}}\right)\left(\frac{\delta n_{\nu}} {%
n_{\nu}}\right)~,\end{aligned}$$ where densities $n_{0\pm}$ are given by Eq. (\[mainlevel\]), [*equilibrium*]{} densities obey $n_{\nu}/n_e =0.5$, and we assume a scale of neutrino fluid inhomogeneity $t\sim \lambda_{{\rm fluid}}^{(\nu)}$, that is small comparing with a large $\Lambda$-scale of the mean magnetic field $%
{\bf B}$, $\lambda^{(\nu)}_{{\rm fluid}}\ll \Lambda$. Let us stress that the addition of positron and electron contributions in $\alpha$ stems from the change of the sign in the weak force (\[axial\]).
The origin of the [*scalar*]{} $\alpha$-coefficient (\[helicity\]) from weak interactions leads to its important difference from the [*pseudoscalar*]{} coefficient $\alpha=<{\bf v}\cdot(\nabla\times {\bf v})>$ in the standard MHD based on fluid electrodynamics, where $C,P,T$ symmetries are conserved separately.
Really, while in the last case (standard MHD) all terms in induction equation are pseudovectors obeying $P{\bf B}P^{-1}= {\bf B}$, etc., in our case the first term in the r.h.s of Eq. (\[Faradey\]) turns out to be a pure vector, violating parity, $P(\nabla \times \alpha {\bf B})P^{-1}= -
\nabla \times \alpha {\bf B}$.
Nevertheless, all terms in the generalized induction equation (\[Faradey\]) obey $CP$-invariance as it should be for electroweak interactions in SM since the new coefficient $\alpha$ (\[helicity\]) is CP-odd, $%
CP\alpha(CP)^{-1}= - \alpha$, as well as curl-operator $(\nabla\times...)$. This is due to the changes $n_0^{(-)}\leftrightarrow n_0^{(+)}$ and $\delta
n_{\nu_a}\to - \delta n_{\nu_a}$ in (\[helicity\]), provided by the well-known properties: particle helicities are P-odd and particles become antiparticles under charge conjugation operation C, in particular, the active left-handed neutrinos convert into the active right-handed antineutrinos under CP- operation, $\nu_a\to \bar{\nu}_a$.
Finally the diffusion coefficient $\eta \approx (4\pi \times 137~T)^{-1}$ is given by the relativistic plasma conductivity. We do not present in Eq. [Faradey]{} standard terms like differential rotation etc which seems to be not very important in the problem under consideration.
This is our main result. We stress that the Eq. \[Faradey\] is the usual equation for mean magnetic field evolution (see e.g. [@KR]) with $\alpha$-effect based on particle effects rather on the averaging of turbulent pulsations. It is well-known (see e.g. [@ZRS]) that Eq. \[Faradey\] describes a self-excitation of a magnetic field with the spatial scale $%
\Lambda\approx \eta/\alpha $ and the growth rate $\alpha^2/4 \eta$.
Let us estimate these values for the early universe. For a small neutrino chemical potential $\mu_{\nu}$, $\xi_{\nu_a}(T)=\mu_{\nu_a}(T)/T\ll 1$, the neutrino asymmetry in the r.h.s. of Eq. (\[helicity\]) is the algebraic sum following the sign of the axial coupling, $c^{(A)}_{e\nu_a}= \pm 0.5$,
$$\label{asymmetry}
\frac{\delta n_{\nu}}{n_{\nu}}\equiv \sum_ac^{(A)}_{e\nu_a}\frac{\delta
n_{\nu_a}}{n_{\nu_a}}= \frac{2\pi^2}{9\zeta(3)}[\xi_{\nu_{\mu}}(T) +
\xi_{\nu_{\tau}}(T) - \xi_{\nu_e}(T)]~.$$
We take for crude estimations below $\xi_{\nu_{\mu}}(T) +
\xi_{\nu_{\tau}}(T)- \xi_{\nu_e}(T)\approx - 2\xi_{\nu_e}(T)$ because different chemical potentials almost compensate each other for high temperatures [@Dolgov1], i.e. $\xi_{\nu_e}(T) + \xi_{\nu_\mu}(T) +
\xi_{\nu_{\tau}}(T)\approx 0$.
As a result, we arrive to the following estimate of the $\alpha$-coefficient (\[helicity\]), $$\alpha= 2.8\times 10^{-34}(T/{\rm {MeV})^6(l_{\nu}(T)/\lambda_{fluid}^{(%
\nu)})\mid \xi_e\mid,}$$ where a free parameter for our collisionless mechanism - scale $%
\lambda^{(\nu)}_{{\rm fluid}}$ is normalized on the neutrino mean free path $%
l_{\nu}(T)=\Gamma_W^{-1}$ given by the weak rate $\Gamma_W=5.54\times
10^{-22}(T/{\rm MeV})^5~{\rm MeV}$.
Substituting $\alpha$ into $\Lambda=\eta/\alpha$ we arrive now to the estimate $$\label{scale}
\frac{\Lambda}{l_H} = 1.6\times 10^9\left(\frac{T}{{\rm {MeV}}}\right)^{-5}
\left(\frac{\lambda^{(\nu)}_{{\rm fluid}}}{l_{\nu}(T)}\right) (\mid
\xi_{\nu_e}(T)\mid)^{-1}~,$$ where $l_H(T)=(2H)^{-1}$ and $H=4.46\times 10^{-22}(T/{\rm {MeV})^2~{MeV}}$ is the Hubble parameter.
If the neutrino fluid inhomogeneity scale $\lambda _{{\rm fluid}}^{(\nu )}$ is of the order $l_{\nu }(T_{0})\sim 4~{\rm cm}\ll l_{H}(T_{0})\sim 10^{6}~%
{\rm cm}$, we have $\Lambda /l_{H}\geq 1$ at the beginning of the lepton era ($T=T_{0}\sim 10^{2}~{\rm {MeV,}}$ redshift $z\sim 3\times 10^{11}$), or more correctly, accounting for the BBN limit $\xi _{\nu _{e}}(T)\lesssim 0.07
$ [@Dolgov1] obtained for $T_{BBN}=0.1$ ${\rm MeV}$ , the mean magnetic field will be uniform in whole universe, $\Lambda /l_{H}\geq 1$, at $T\sim
118$ ${\rm MeV}$. If this neutrino parameter would be much smaller at high temperatures $T_{0},$ $\xi _{\nu _{e}}(T_{0})\ll 0.07,$ one can choose another free neutrino parameter $\lambda _{{\rm fluid}}^{(\nu )}\ll l_{\nu
}(T_{0})$ in such a way that the ratio $\lambda _{{\rm fluid}}^{(\nu
)}/(l_{\nu }(T_{0})\mid $ $\xi _{\nu _{e}}(T_{0})\mid )$ remains invariant and our conclusion about the tendency to a global uniform field is still valid. Note that for the neutrino gas the macroscopic parameter $\lambda _{%
{\rm fluid}}^{(\nu )}$ varies in a wide region $T^{-1}\ll \lambda _{{\rm %
fluid}}^{(\nu )}\leq l_{\nu }(T).$
The magnetic field time evolution is given by
$$B(t)=B_{\max }\exp \left( \int_{t_{\max }}^{t}\frac{\alpha ^{2}(t^{\prime })%
}{4\eta (t^{\prime })}dt^{\prime }\right) ~, \label{dynamo}$$
where $B_{\max }$ is some seed value at the instant $T_{\max }\ll T_{EW}\sim
100{\rm GeV}$ (here we imbed the standard estimates of $\alpha ^{2}$-dynamo into the context of expanding Universe).
For $\lambda _{{\rm fluid}}^{(\nu )}(T)\sim l_{\nu }(T)$ we can estimate the index in the exponent (\[dynamo\]) substituting in the integrand the expansion time$t(T)$$=3.84\times 10^{21}(T/{\rm {MeV})^{-2}{MeV}^{-1}/\sqrt{g^{\ast }}}$ with the effective number of degrees of freedom $g^{\ast }\sim 100$ at the temperatures $T\;\raise0.3ex\hbox{$>$\kern-0.75em\raise-1.1ex\hbox{$\sim$}}%
\;1~{\rm {GeV}}$. Then from our estimates of $\alpha (T)$ and $\eta (T)$ with the change of the variable $(T/2\cdot 10^{4}{\rm {MeV})\rightarrow {\it %
x}}$ one finds the fast growth of the mean field (\[dynamo\]) in hot plasma, ${\it x}\leq 1$ with a conservative estimate, $$B(x)=B_{\max }\exp \left( 25\int_{x}^{1}\left( \frac{\xi _{\nu _{e}}({\it %
x^{\prime }})}{0.07}\right) ^{2}{\it x^{\prime }}^{10}d{\it x^{\prime }}%
\right) \label{last}$$given by the upper limit $x_{\max }=1,$ $T_{\max }=20{\rm GeV}$. Such upper limit defines entirely the magnetic field amplification due to the steep dependence on the temperature and still obeys the point-like Fermi interaction we rely on. As in the case of magnetic field scales ([scale]{}) the second free parameter $\lambda _{{\rm fluid}}^{(\nu )}$ can be chosen much smaller, $\lambda _{{\rm fluid}}^{(\nu )}\ll l_{\nu }(T)$ , providing the invariant ratio $l_{\nu }(T)\mid \xi _{\nu _{e}}\mid /\lambda
_{{\rm fluid}}^{(\nu )}$ for very small neutrino chemical potential $\xi
_{\nu _{e}}(T)\ll 0.07$ and resulting in an enhancement of a small mean magnetic field $B_{\max }\ll T_{\max }^{2}/\mid e\mid \ll T_{EW}^{2}/\mid
e\mid $ by collective neutrino-plasma interactions considered here in Eq. (\[last\]).
Note that the inflation mechanism (with a charged scalar field fluctuations at super-horizon scales) explains the origin of mean field at cosmological scales, however, the value of this field is too small for seeding the galactic magnetic fields [@Giovan].
The amplification mechanism suggested in our paper can improve this very low estimate by a substantial factor from Eq. (\[last\]).
Thus, while in the temperature region $T_{EW}\gg T\gg T_{0}=10^{2}~{\rm {MeV}%
}$ there are many small random magnetic field domains, a mean magnetic field turns out to be developed into the uniform [*global*]{} magnetic field. The global magnetic field can be small enough to preserve the observed isotropy of cosmological model [@Zeld] while strong enough to be interesting as a seed for galactic magnetic fields. This scenario was intensively discussed by experts in galactic magnetism [@Kulsrud], however until now no viable origin for the global magnetic field has been suggested. We believe that the $\alpha ^{2}$-dynamo based on the $\alpha $-effect induced by particle physics solves this fundamental problem and opens a new and important option in galactic magnetism.
[99]{} Ya.B. Zeldovich, A.A. Ruzmaikin, and D.D. Sokoloff, [*Magnetic fields in Astrophysics*]{} (Gordon & Breach, New York, 1983); E. Parker, [*Cosmological Magnetic Fields*]{} (Oxford Univ. Press, Oxford, 1979);A.A. Ruzmaikin, A.A. Shukurov and D.D. Sokoloff, [*Magnetic Fields of Galaxies*]{} (Kluwer, Dordrecht,1988).
T. Vachaspati, Phys. Lett. B [**265**]{}, 258 (1991);K. Enqvist and P. Olesen, [*ibid.,*]{} [**319**]{}, 178 (1993);K. Enqvist and P. Olesen, [*ibid.,*]{} [**329**]{}, 195 (1994).
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R.Beck, A.Brandenburg, D.Moss, A.Shukurov and D.Sokoloff, Ann. Rev. Astron. Astrophys. [**34**]{}, 155 (1996).
K. Enqvist, Int. J. Mod. Phys., [**D7**]{}, 331 (1998).
V.N. Oraevsky, V.B. Semikoz, Astropart. Phys. [**18**]{}, 261 (2002);A.I. Rez, V.B. Semikoz, hep-ph/0201259.
H.Nunokawa et al., Nucl. Phys. B[**501**]{}, 17 (1997).
A.D. Dolgov, D. Grasso, Phys. Rev. Lett. [**88**]{}, 011301 (2002).
E.M. Lifshits and L.P. Pitaevsky, [*Physical Kinetics*]{} (Pergamon Press: Oxford, New York, 1981).
A.J. Brizard, H. Murayama and J.S. Wurtele, Phys. Rev. E [**61**]{}, 4410 (2000).
V.B. Semikoz, in talk given at AHEP-03 International Workshop, Valencia, October 2003, published in JHEP Proceedings, [http://jhep.sissa.it/archive/prhep/preproceeding/010/059/semikoz.pdf]{}
F.Krause and K.-H.Rädler, [*Mean-Field Electrodynamics and Dynamo Theory*]{} (Pergamon Press, Oxford, 1980).
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[^1]: The detailed derivation of full Faradey equation including dynamo, Bierman battery terms, both vector weak interaction cotribution [@Brizard] and new axial vector interaction terms is given also in [@Semikoz] where lepton MHD equations in SM for unpolarized and polarized media are derived from RKE’s [@OS].
|
---
abstract: 'This paper examines the uplink user capacity in a two-tier code division multiple access (CDMA) system with hotspot microcells when user terminal power is limited and the wireless channel is [*finitely-dispersive*]{}. A finitely-dispersive channel causes variable fading of the signal power at the output of the RAKE receiver. First, a two-cell system composed of one macrocell and one embedded microcell is studied and analytical methods are developed to estimate the user capacity as a function of a dimensionless parameter that depends on the transmit power constraint and cell radius. Next, novel analytical methods are developed to study the effect of variable fading, both with and without transmit power constraints. Finally, the analytical methods are extended to estimate uplink user capacity for [*multicell*]{} CDMA systems, composed of multiple macrocells and multiple embedded microcells. In all cases, the analysis-based estimates are compared with and confirmed by simulation results.'
author:
- 'Shalinee Kishore, Larry J. Greenstein, H. Vincent Poor, Stuart C. Schwartz [^1]'
title: 'Uplink User Capacity in a CDMA System with Hotspot Microcells: Effects of Finite Transmit Power and Dispersion'
---
Cellular systems, code division multiple access, microcells.
Introduction
============
Wireless operators often install low-power [*hotspot microcell*]{} base stations to provide coverage to small high-traffic regions within a larger coverage area. These microcells enhance the user capacity and coverage area supported by the existing high-power [*macrocell*]{} base stations. In this paper, we study these gains for a [*two-tier code division multiple access (CDMA) network*]{} in which both the macrocells and microcells use the same set of frequencies. Specifically, we study the effects, on the uplink capacity and coverage area, of transmit power constraints and variable power fading due to multipath.\
\
Earlier studies have examined the uplink performance of these single-frequency, two-tier CDMA systems, e.g., [@Shapira]-[@skishore3]. In [@skishore1], we showed how to compute the uplink user capacity for a single-macrocell/single-microcell (two-cell) system using exact and approximate analytical methods that accounted for random user locations, propagation effects, signal-to-interference-plus-noise ratio (SINR)-based power control, and various methods of base station selection. In [@skishore2], we calculated capacity gains under similar assumptions for a system composed of multiple macrocells and/or multiple microcells; the results pointed to a roughly linear growth in capacity as the number of these base stations increase, where the constants of the linear curve were derivable solely from the analysis for the two-cell case. This linear approximation was conjectured based on the trends observed from several simulations. More recently, we have developed an [*analytical*]{} approximation to the user capacity of a multicell system which is much tighter than the empirical result of [@skishore2]; and is valid over a larger number of embedded microcells. Interestingly, this new analytical approach also depends on constants obtained from a far-simpler two-cell analysis. These studies therefore highlight the importance of understanding two-cell performance in computing the performance of larger multicell systems.\
\
The capacity gains demonstrated in these earlier works were based on assumptions that 1) user terminals have unlimited power; and 2) that the wireless channel is so wideband that user signals have constant output power after RAKE receiver processing. This latter condition is equivalent to assuming that user signals go through an infinitely-dispersive channel before reception.[^2] In this paper, we remove these conditions. Portions of this work were presented in [@skishore4] and a companion paper on [*downlink*]{} capacity [@skishore5] showed that this type of system is uplink-limited. Here, we improve the presentation in [@skishore4] and present several new results. Specifically, we show how the capacity/coverage tradeoff under finite power constraints can vary significantly with shadow fading. Further, we bridge our study of finite power constraints and finite channel dispersion by presenting a new analysis of user capacity in two-cell systems under [*both*]{} maximum transmit power constraints and variable power fading. Finally and most significantly, we develop and verify analytical expressions for the total user capacity of [*multicell*]{} two-tier CDMA systems under finite dispersion.\
\
Section \[sysdes\] describes the basic two-cell system and the model used to capture the power gain between a user and a base; and it summarizes our previous results for total uplink capacity. Section \[pmax\_sec\] assumes a limit on terminal transmit power and approximates total capacity as a function of this power and cell size. Section \[varfading\_sec\] relaxes the condition on infinite dispersion and approximates the total capacity, for both limited and unlimited terminal power, as a function of the multipath delay profile. Section \[multimulti\_sec\] extends the results for finite dispersion and unlimited terminal power to the case of multiple macrocells and multiple microcells. Simulations confirm the accuracy of the approximation methods over a wide range of practical conditions and assumptions.
System and Channel Description {#sysdes}
==============================
[**The System:**]{} We first consider a system with a macrocell and an embedded microcell which together contain $N$ total users, comprising $N_M$ in the macrocell and $N_\mu=N-N_\mu$ in the microcell. We assume users have random codes of length $W/R$, where $W$ is the system bandwidth and $R$ is the user rate. Each user is power-controlled by its base so as to achieve a required uplink power level. It was shown in [@skishore1] that the required received power levels to meet a minimum SINR requirement of $\Gamma$ are $$S_{M}= \eta W \frac{K-N_\mu+I_M}{(K-N_\mu)(K-N_M)-I_MI_\mu}
\label{pwr1}$$ and $$S_{\mu}= \eta W \frac{K-N_M+I_\mu}{(K-N_\mu)(K-N_M)-I_MI_\mu}
\label{pwr2}$$ at the macrocell and microcell base stations, respectively. Here, $K =
1+(W/R)/\Gamma$ denotes the single-cell pole capacity [@gilhousen], $\eta$ is the power spectral density of the additive white Gaussian noise, and $I_M$ and $I_\mu$ are [*normalized cross-tier interference*]{} terms. Specifically, $S_\mu I_M$ is the total interference power at the macrocell due to microcell users and $S_M I_\mu$ is the total interference power at the microcell due to macrocell users. Thus, $$I_M = \sum_{j \in U_\mu}\frac{T_{Mj}}{T_{\mu j}}; \mbox{~~~~~} I_\mu = \sum_{j \in U_M}\frac{T_{\mu j}}{T_{Mj}},
\label{IM_Imu}$$ where $U_\mu$ represents the set of microcell users, $U_M$ represents the set of macrocell users, and $T_{Mk}$ and $T_{\mu k}$ are the path gains from user $k$ to the macrocell and microcell base stations, respectively. We say that a given arrangement of user locations and path gains is [*feasible*]{} if and only if the common denominator in (\[pwr1\]) and (\[pwr2\]) is positive. The fraction of all possible cases (i.e., all combinations of $I_M$ and $I_\mu$) for which this occurs is called the [*probability of feasibility*]{}. The cross-tier interference terms depend on the path gains for all users to both base stations and on the method by which users select base stations, leading to the sets $U_\mu$ and $U_M$.\
\
[**Path Gain Model:**]{} We model the instantaneous power gain (i.e., the sum of the power gains of the resolvable multipaths) between user $k$ and base station $l$ as $$\begin{aligned}
T_{lk}=\left\{ \begin{array}{cc} H_l \left( \frac{b_l}{d_{lk}} \right)^2 10^{\zeta/10}\rho & d_{lk}
< b_l \\ H_l \left( \frac{b_l}{d_{lk}} \right)^4 10^{\zeta/10}\rho & d_{lk} \geq b_l \end{array}
\right.,\end{aligned}$$ where $d_{lk}$ is the distance between user $k$ and base station $l$; $b_l$ is the breakpoint distance of base station $l$ at which the slope of the decibel path gain versus distance changes; $H_l$ is a gain factor that captures the effects of antenna height and gain at base station $l$; $10^{\zeta/10}$ represents [*shadow fading*]{}; and $\rho$ is the variable fading due to multipath [@Erceg]-[@rap]. Note that $\zeta$ and $\rho$ are different for each user-base pair ($l,k$) but, for convenience, we suppress the subscripts for these terms. Since the antenna height and gain are greater at the macrocell, we can assume $H_M > H_\mu$. We model $\zeta$ as a zero mean Gaussian random variable with variance $\sigma^2_l$ for base station $l$. The fading due to multipath, $\rho$, is scaled to be a unit-mean random variable (${\mbox{E}}\{ \rho \} =1$). In an infinitely dispersive channel, $\rho
= 1$. Let $\tilde{T}_{lk}$ denote the path gain between user $k$ and base $l$ in an infinitely dispersive channel. Thus, $\tilde{T}_{lk}$ represents the local spatial average of $T_{lk}$, and $T_{lk}=\tilde{T}_{lk} \rho$.\
\
[**Base-Selection Scheme:**]{} We assume a user $k$ chooses to communicate with the macrocell base station if $\tilde{T}_{Mk}$ exceeds $\tilde{T}_{\mu k}$ by a fraction $\delta$, called the [*desensitivity*]{} parameter. Thus, in both finitely-dispersive and infinitely dispersive channels, the user selects base stations according to the local mean path gains.[^3] With no loss in generality, we assume $\delta =1$ in this study.\
\
[**Capacity:**]{} The path gain model and the selection scheme described above imply that the cross-tier interference terms are random variables; they depend on the random variations due to shadow and multipath fading and on the random user locations. In [@skishore1], we demonstrated how to compute the cumulative distribution functions (CDF’s) of the two cross-tier interference terms in (\[IM\_Imu\]) under the condition of infinite dispersion. Let $\tilde{I}_M$ and $\tilde{I}_\mu$ denote these cross-tier terms under that condition. The CDF’s of $\tilde{I}_M$ and $\tilde{I}_\mu$ were used to compute the exact uplink user capacity in a two-cell system with unlimited terminal power. In addition, the mean values of $\tilde{I}_M$ and $\tilde{I}_\mu$ were used to reliably approximate attainable user capacity for the two-cell system as $$\tilde{N}=\frac{2K}{1+\sqrt{\tilde{v}_M \tilde{v_\mu}}},
\label{Nunif}$$ where $\tilde{v}_M$ and $\tilde{v}_\mu$ are the mean values (over all user locations and shadowing fadings) of single terms in the sums $\tilde{I}_M$ and $\tilde{I}_\mu$, respectively.[^4] The total user capacity of the two-cell system thus depends on $K$ (a system parameter) and the product $\tilde{v}_M\tilde{v}_\mu$. We computed this product under various conditions and observed that it (and therefore $\tilde{N}$) are robust to variations in propagation parameters, separation between the two base stations, and user distribution. Its value under all conditions examined was about 0.09, leading to the robust result $\tilde{N} \approx 1.5K$.
User Capacity with Limited Terminal Power {#pmax_sec}
=========================================
Analysis {#pmax_sec1}
--------
In this section, we assume an infinitely dispersive channel and study a system where each user terminal has a maximum transmit power constraint of $P_{\max}$. The two-cell system is unable to support a given set of $N$ users if the users are [*infeasible*]{} (i.e., unsupportable even with unlimited terminal power) or if any one of the $N$ users requires a transmit power higher than $P_{\max}$. We have already computed the probability of infeasibility of $N$ total users in [@skishore1]; we now determine the probability that any one of $N$ feasible users exceeds the terminal power limit of $P_{\max}$. The combined event, when the system is either infeasible or feasible but a user terminal exceeds the transmit power requirements, is referred to as [*outage*]{}. The probability of outage for $N$ users can then be written as $$P_{out}(N) = P_{inf}(N) + (1-P_{inf}(N)) \cdot \mbox{Pr~}[P>P_{\max}|N],
\label{totaloutage}$$ where $P_{inf}(N)$ is the probability of infeasibility of $N$ users and $\mbox{Pr~}[P>P_{\max}|N]$ is the probability that the transmit power of any one of the $N$ feasible users exceeds $P_{\max}$. We seek the values of $N$ that a two-cell system can support for a specified probability of outage, i.e., we desire the largest $N$ such that $P_{out}(N) \leq
\alpha_{out}$.\
\
We assume $b_M=b_\mu=b$. For analytical purposes, it is convenient to write the path gain between user $k$ and base station $l$ as: $$\tilde{T}_{lk}=H_l \left( \frac{b}{d_{\max}} \right)^4 \tilde{T}'_{lk},$$ where $d_{\max}$ is the desired maximum distance from the macrocell base station to a system user (it can be regarded as the macrocell radius), and $$\begin{aligned}
\tilde{T}'_{lk}=\left\{ \begin{array}{cc} \left( \frac{d_{\max}}{d_{lk}} \right)^2 \left( \frac{d_{\max}}{b} \right)^2 10^{\zeta/10}, & d_{lk} < b \\
\left( \frac{d_{\max}}{d_{lk}} \right)^4 10^{\zeta/10}, & d_{lk} \geq b
\end{array} \right. \mbox{~.}
\label{mod_tgain}\end{aligned}$$ Microcell user $j$ transmits at power $P=S_{\mu j}/\tilde{T}_{\mu j}$; thus $P$ exceeds $P_{\max}$ if $$\frac{\tilde{S}'_\mu}{\tilde{T}'_{\mu j}} > \frac{P_{\max} H_\mu}{\eta W}
\left( \frac{b}{d_{\max}} \right)^4 \label{peruser_outage_mu} \equiv F$$ where $\tilde{S}'_\mu = S_\mu/\eta W$. Similarly, the required transmit power of macrocell user $i$ exceeds $P_{\max}$ if $$\frac{\tilde{S}_M'}{\tilde{T}_{Mi}} > \frac{P_{\max} H_M}{\eta W}
\left( \frac{b}{d_{\max}} \right)^4 = FH,$$ where $\tilde{S}_M'=S_M/\eta W$ and $H=H_M/H_\mu$.\
\
Note that $\tilde{S}_M$ and $\tilde{S}_\mu$ and all path gains are random variables, related to the randomness of user locations and shadow fadings. Following the steps outlined in [@skishore1] to obtain the probability distributions of $\tilde{I}_M$ and $\tilde{I}_\mu$, we can extend the analysis to compute the distributions of $\tilde{S}'_M$ and $\tilde{S}'_\mu$. Next, we can determine the distributions of $\tilde{T}'_M$ and $\tilde{T}'_\mu$ for a random macrocell user and a random microcell user, respectively. With the distributions of $\tilde{S}'_M$, $\tilde{S}'_\mu$, $\tilde{T}'_{M}$ and $\tilde{T}'_{\mu}$, we can then determine $\mbox{Pr~}[P>P_{\max}|N]$ as a function of $F$: $$\begin{aligned}
\mbox{Pr~} [P>P_{\max}|N]=\sum_{n=0}^N \left( \begin{array}{c} N \\ n \end{array} \right) p^nq^{N-n} (1-p_M^n p_\mu^{N-n}),
\label{outage_eq}\end{aligned}$$ where $p$ is the probability of a user assignment to the macrocell;[^5] $q=1-p$; $$p_M = {\mbox{P}}[\tilde{S}_M'/\tilde{T}_M' \leq FH]; \mbox{~~~~~}p_\mu = {\mbox{P}}[\tilde{S}_\mu'/\tilde{T}_\mu' \leq F].$$ Finally, we can determine the largest $N$ for which $P_{out}(N) \leq \alpha_{out}$ for a given value of $F$. However, instead of computing the distributions of $\tilde{S}'_M$ and $\tilde{S}'_\mu$, we propose an approximate method that works quite well, namely, using the means of $\tilde{S}'_M$ and $\tilde{S}'_\mu$. This approach retains (\[totaloutage\]) and (\[outage\_eq\]) but uses the following modified values for $p_M$ and $p_\mu$: $$\begin{aligned}
\tilde{p}'_M &=& \mbox{Pr~} [\mbox{E~} \{ \tilde{S}'_M \} /\tilde{T}'_{M}\leq FH]; \\
\tilde{p}'_\mu &=& \mbox{Pr~} [\mbox{E~} \{ \tilde{S}'_\mu \} /\tilde{T}'_{\mu} \leq F].\end{aligned}$$ This greatly simplifies calculating the probability of outage, with minor loss in accuracy.
Numerical Results
-----------------
We use simulation to study the two-cell system in Figure \[fxy\]. Specifically, we assume a square geographic region, with sides of length $S$. At a distance $D$ from the center of this square is a smaller square region, with sides of length $s$. A portion of the total user population $N$ is uniformly distributed over the larger square region and the remaining users are uniformly distributed over the smaller square region. The smaller square thus represents a traffic hotspot within the larger coverage region. A macrocell base with antenna height $h_M$ is at the center of the larger square, while a microcell base with antenna height $h_\mu$ is at the center of the smaller square. We assume each user terminal has antenna height $h_m$. These antenna heights are used to compute the distance between transmit antenna at each user terminal and the receive antenna at the two base stations. The breakpoint of the path loss to the microcell base is engineered to be $s/2$, to help ensure that the microcells provide strong signals to those users contained within the high density region, and sharply diminishing signals to users outside it. Finally, we assume that on average half of the $N$ users are uniformly distributed over the hotspot region surrounding the microcell base. This is done to obtain a roughly equal number of users for each base, i.e., to ensure that the maximal user capacity occurs with high probability.
\
\
Using the parameter values listed in Table 1, we determine the total user capacity that can be sustained with 5% outage probability for various values of $P_{\max}$. Figure \[pwr\_out1\] shows the resulting user capacity as a function of $F$ as defined in (\[peruser\_outage\_mu\]). We observe the close correspondence between the approximate and simulation results. We also note that the user capacity goes to an asymptotic value as $F$ gets large and that, below a critical value of $F$ (denoted by $F^*$), the user capacity decreases sharply as $F$ decreases. (From the figure, $F^* \approx 1$.) In other words, there are critical combinations of $P_{\max}$ and $d_{\max}$ such that, if we either reduce the transmit power constraint or increase the desired coverage area, the overall user capacity noticeably drops; and, for values of $F>F^*$, the system operates as if there is unlimited terminal power.
\
\
In an earlier study [@thesis], we found that total user capacity is nearly invariant to $\sigma_M$ and $\sigma_\mu$, increasing [*very*]{} slightly as $\sigma_M$ and $\sigma_\mu$ increase. This finding was for unlimited terminal power. We now consider the uplink user capacity as a function of $F$ for various pairs of $(\sigma_M,\sigma_\mu)$. The results, obtained via simulation, are in Figure \[sigma\_curves\]. Even though the capacity for unlimited transmit power ($F$ infinite) is slightly larger for $\sigma_M=12$,$\sigma_\mu=6$ [@thesis], the curves increase much faster for smaller values of $\sigma_M$ and $\sigma_\mu$. Thus, although for unlimited transmit power the total user capacity is roughly invariant to $\sigma_M$ and $\sigma_\mu$, it can vary significantly with these parameters under transmit power constraints.
User Capacity with Variable Fading: Two-Cell System {#varfading_sec}
===================================================
Analysis {#analysis}
--------
We now consider [*finite dispersion*]{} (i.e., a finite number of significant multipaths), which causes the sum of the multipath powers to fade with time as a user moves in the environment. We assume that users still select bases according to the [*slowly*]{} varying path gains, $\tilde{T}_{lk}$, but the fluctuations of signals and interferences due to multipath lead to instantaneous occurrences of outage.
### Channel Delay Profile
The instantaneous path gain from base $l$ to user $k$ is $T_{lk}=\rho \tilde{T}_{lk}$, where $\rho$ is independent and identically distributed (i.i.d.) over all $(l,k)$. Let $r_n$ represent the instantaneous power gain of the $n$-th multipath component of a particular uplink channel. Then, for that channel, $$\rho=\sum_{n=1}^{L_p} r_n \label{rho_def},$$ where $L_p$ is the total number of multipaths and we assume a scaling such that ${\mbox{E}}\{ \rho \} =1$. We consider four possible multipath delay profiles, for each of which $\rho$ has a different statistical nature. One is for the so-called [*uniform channel*]{}, in which the $L_p$ multipaths are i.i.d. and Rayleigh-fading, i.e., each has a power gain that is exponentially distributed with a mean of $1/L_p$.\
\
The other three delay profiles studied here are based on cellular channel models for the typical urban (TU), rural area (RA), and hilly terrain (HT) environments provided in third-generation standards. See, for example, [@3gpp], which tabulates ${\mbox{E}}\{ r_n \}$ in dB for these environments.
### Methodology
Base selections (and thus, the sets $U_\mu$ and $U_M$) are made according to the local average path gains, $\tilde{T}_{lk}$. The instantaneous cross-tier interference powers are $$\begin{aligned}
I_M&=&\sum_{i \in U_\mu} \frac{T_{Mi}}{T_{\mu i}} = \sum_{i \in U_\mu} \frac{\rho_{Mi}}{\rho_{\mu i}} \cdot \frac{\tilde{T}_{Mi}}{\tilde{T}_{\mu i}}; \label{IM_new}\\
I_\mu&=&\sum_{i \in U_M} \frac{T_{\mu i}}{T_{Mi}} = \sum_{i \in U_M} \frac{\rho_{\mu i}}{\rho_{M i}} \cdot \frac{\tilde{T}_{\mu i}}{\tilde{T}_{Mi}} \label{Imu_new}\end{aligned}$$ where $\rho_{Mi}$ and $\rho_{\mu i}$ are i.i.d. random variables, each distributed as $\rho$ in (\[rho\_def\]). We see from the definitions of $I_M$ and $I_\mu$ that, at any one instant, the system could become infeasible or, more generally, experience outage, depending on the values of $\rho_{Mi}$ and $\rho_{\mu i}$. Here again, we seek the largest number of users, $N$, that can be supported such that $P_{out}(N) \leq \alpha_{out}$.\
\
The exact method for calculating $N$ requires that the distributions of $I_M$ and $I_\mu$ be computed. The procedure is even more complex than before since we must incorporate the effects of the additional random quantities, $\rho_{Mi}$ and $\rho_{\mu i}$. Thus, we resort to two analytical approximations. The first estimates total user capacity under finite dispersion but for large $F$ (i.e., $F > F^*$); and the second estimates capacity under both finite dispersion and finite transmit power requirements.
### Total User Capacity for $F$ Large {#fading_f_large}
#### Estimated User Capacity in a Uniform Channel
The total user capacity in a uniform channel, $N_u$, can be approximated by modifying the mean method presented in (\[Nunif\]). Specifically, we postulate that $$N_u = \frac{2K}{1+\sqrt{\overline{v}_M \overline{v}_\mu}},
\label{Nfading_approx}$$ where $\overline{v}_M={\mbox{E}}\{ \kappa \} \tilde{v}_M$ and $\overline{v}_\mu={\mbox{E}}\{ \kappa \} \tilde{v}_\mu.$ The mean values $\tilde{v}_M$ and $\tilde{v}_\mu$ were defined earlier; and $\kappa=\rho_1/\rho_2$, where $\rho_1$ and $\rho_2$ are i.i.d. Gamma random variables with unit mean and $L_p$ degrees of freedom. In other words, $\rho_1$ represents either $\rho_{Mi}$ or $\rho_{\mu i}$, and $\rho_2$ represents the other. For a uniform channel with $L_p$ paths, the probability density of $\rho$ (and therefore of $\rho_1$ or $\rho_2$) is $$f_\rho (x) = \frac{L_p}{(L_p-1)!}(x L_p)^{L_p-1}e^{-x L_p}, \mbox{~~~}x>0.
\label{rho_density}$$ We can compute the mean of $\kappa$ as ${\mbox{E}}\{ \kappa \} = {\mbox{E}}\{ \rho_1 \} \cdot {\mbox{E}}\left\{ \rho_2^{-1} \right\}$, where ${\mbox{E}}\{\rho_1 \}=1$ and $$\begin{aligned}
{\mbox{E}}\left\{ \frac{1}{\rho_2} \right\} &=& \frac{L_p}{L_p-1} \int_0^{\infty} \frac{1}{(L_p-2)!} y^{L_p-2}e^{-y} \,dy \\
&=& \frac{L_p}{L_p-1} \mbox{,~~~~~}L_p>1.\end{aligned}$$ We can therefore relate $N_u$ to $L_p$, as follows: $$N_u(L_p) = \frac{2K}{1+\frac{L_p}{L_p-1}\sqrt{\tilde{v}_M \tilde{v}_\mu}}.
\label{Nu_approx}$$ Clearly, this approximation breaks down as $L_p$ approaches 1, as it predicts zero capacity in that case.[^6] For $L_p>1$, we obtain a simple relationship between user capacity and the degree of multipath dispersion. As $L_p$ becomes infinite, user capacity converges to the estimate given by (\[Nunif\]).
#### Estimated User Capacity in a Non-Uniform Channel
The result in (\[Nu\_approx\]) can be used to approximate user capacity for [*any*]{} delay profile. Consider a channel delay profile having some arbitrary variation of ${\mbox{E}}\{ r_n \}$ with $n$ over the $L_p$ paths. We can compute a [*diversity factor*]{}, $DF$, defined as the ratio of the square of the mean to the variance of $\rho$ in (\[rho\_def\]). For independent Rayleigh paths, $DF$ can be shown to be [@lola] $$DF = \frac{\left( \sum_{n=1}^{L_p} {\mbox{E}}\{r_n\} \right)^2}{\sum_{n=1}^{L_p} ({\mbox{E}}\{r_n \})^2}.
\label{DF_definition}$$ From this definition, we see that the diversity factor for a uniform channel is precisely $L_p$.[^7] The proposed approximation makes use of this fact by first calculating the diversity factor for a given (non-uniform) channel. The approximate user capacity is then the value of $N_u$ corresponding to $L_p=DF$, i.e., $N_u(DF)$ in (\[Nu\_approx\]).
### Total User Capacity with Finite Terminal Power {#fading_f_small}
To approximate user capacity under both finite dispersion and finite terminal power, we incorporate the approximations of Sections \[pmax\_sec\] and \[fading\_f\_large\]. We begin by first studying the uniform delay profile and then develop an approximation for a general non-uniform environment.
#### Estimated User Capacity in a Uniform Channel
As in Section \[pmax\_sec\], we seek the largest $N$ such that the probability of outage is $\alpha_{out}$ or lower. In calculating outage, we use the expressions in (\[totaloutage\]) and (\[outage\_eq\]) but replace $p_M$ and $p_\mu$ with $p'_M$ and $p'_\mu$, respectively, so that $$\begin{aligned}
p'_M &=& \mbox{Pr~}[\mbox{E~}\{ S'_M \} /\rho \tilde{T}'_M \leq FH];\\
p'_\mu &=& \mbox{Pr~}[\mbox{E~} \{ S'_\mu \} / \rho \tilde{T}'_\mu \leq F], \label{pMmu_new}\end{aligned}$$ where $\mbox{E~} \{ S'_M \} = \mbox{E~} \{ S_M \} / \eta W$, $\mbox{E~} \{ S'_\mu \} = \mbox{E~} \{S_\mu \} / \eta W$, and $\rho$ is a random variable with density given in (\[rho\_density\]). The steps used to compute ${\mbox{E}}\{S'_M \}$ and ${\mbox{E}}\{ S'_\mu \}$ are identical to the steps outlined in Section \[pmax\_sec\] for the calculations of ${\mbox{E}}\{ \tilde{S}'_M \}$ and ${\mbox{E}}\{ \tilde{S}'_\mu \}$. The difference here is that the densities of $S'_M$ and $S'_\mu$ depend on the densities of $I_M$ and $I_\mu$, (\[IM\_Imu\]), which account for variable fading, whereas $\tilde{S}'_M$ and $\tilde{S}'_\mu$ were obtained using the cross-tier interference terms in infinitely-dispersive channels. The densities of $I_M$ and $I_\mu$ depend on the densities of the individual terms in their sums. The terms in each sum are i.i.d. random variates, denoted here by $v_M$ and $v_\mu$.\
\
For a uniform channel with $L_p$ paths, we note that $v_M = \kappa \tilde{v}_M$, where $\kappa = \rho_1 / \rho_2$, and $\rho_l$ ($l \in {1,2}$) has probability density given in (\[rho\_density\]). Thus, the density of $v_j$ ($j \in \{ M, \mu \}$) can be computed as $$f_{v_j}(v) = \int_{0}^{\infty} f_{\tilde{v}_j}(v/x)\cdot f_{\kappa}(x) \,dx, \mbox{~~~}v_j \in \{ M, \mu \},$$ where the probability density of $\tilde{v}_j$ is given in [@skishore1] and $$f_{\kappa}(x) = \frac{1}{(L_p-1)!} \sum_{i=0}^{L_p-1} \frac{(L_p-1+i)!}{i!} \cdot \frac{x^{i-1}(L_px-i)}{(x+1)^{L_p+1+i}}.$$ We obtain $f_{\kappa}(x)$ from its CDF, which can be computed as[^8]$$\begin{aligned}
F_{\kappa}(x) &=& {\mbox{P}}[\kappa \leq x] = {\mbox{P}}[\rho_1 \leq \rho_2 x] \\
&=& 1-\frac{1}{(L_p-1)!} \sum_{i=0}^{L_p-1} \frac{x^i}{i!} \cdot \frac{(L_p-1+i)!}{(x+1)^{L_p+i}}.\end{aligned}$$ With the density of $v_M$ and $v_\mu$ at hand, we use the analysis in [@skishore1] to compute the densities of $I_M$ and $I_\mu$ for a given value of $N$. We then use the densities of these cross-tier interferences to compute the densities of $S'_M$ and $S'_\mu$, which in turn give us the mean values needed in (\[pMmu\_new\]). The outage probability for this capacity $N$ is then determined using (\[totaloutage\]) and (\[outage\_eq\]) with the substitutions for $p_M$ and $p_\mu$ indicated in (\[pMmu\_new\]). Finally, we can approximate the maximum number of users supported for a desired outage level ($\alpha_{out}$) for the uniform multipath channel.
#### Estimated User Capacity in a Non-Uniform Channel
Given a non-uniform channel, we propose an approximation to user capacity for a given value of $F$ that uses the above technique for the uniform channel. Specifically, we use the results of Section \[fading\_f\_large\] to claim that a system in a non-uniform channel with diversity factor $DF$ supports roughly the same number of users as the same system in a uniform channel with $DF$ paths. Thus, for a given non-uniform delay profile, we first compute $DF$ from (\[DF\_definition\]). We then set $L_p$ as the integer closest to $DF$ and use the analysis above to estimate the total user capacity with outage $\alpha_{out}$ in a uniform channel with $L_p$ paths. This value approximates user capacity in the given non-uniform channel.
Numerical Results
-----------------
We use the two-cell system described in Figure \[fxy\] and Table 1 to obtain simulation results for finitely-dispersive channels. For a given configuration of $N$ users, i.e., a fixed set of user locations and shadow fadings, we generate 1000 values of $\rho_M$ and $\rho_\mu$ for each user (based on the given channel delay profile) and record the instances of outage. We then calculate this outage measurement over 999 other random user locations and shadow fadings (assuming a fixed value of $N$). The total instances of outage over all these trials determine the probability of outage for $N$.\
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Using this simulation method, we first calculate the user capacity with 5% outage probability when there are no transmit power constraints ($F=\infty$).[^9] This is done for the uniform channel and the RA, HT, and TU environments. For the uniform channel, we determine user capacity as a function of $L_p$; Figure \[NvsL\_fading\] shows results for both the simulation and approximation methods. The approximation curve follows the simulation curve closely, especially for smaller values of $L_p$. Figure \[NvsL\_fading\] also shows the infinite-$F$ capacity for the RA, HT, and TU environments (from simulation). We show each of these capacities at a value of $L_p$ equal to its diversity factor. The diversity factors for the RA, HT, and TU environments are 1.6, 3.3, and 4.0, respectively. We see that, using the diversity factor, the capacity of a non-uniform channel can be mapped to the computed curve for the uniform channel, yielding a simple and reliable analytical approximation to user capacity. For channels with $DF > 5$, the total user capacity appears to be within 5% of that for an infinitely dispersive channel.
\
\
We also obtained user capacity (with 5% outage probability) as a function of $F$ for the 2-path and 4-path uniform channels. This was done via both simulation and analytical approximation, and the results are presented in Figure \[NvsF\_fading\]. We observe, first, that the approximation matches the simulation results for the 2-path and 4-path channels in the region of interest, i.e., for $N \geq K$. Overall, the approximations improve as $L_p$ increases. Furthermore, the user capacities noticeably decrease for $F < F^*$, where $F^* \approx 1$ for all three channels studied here, a result which is consistent with the infinitely-dispersive case (Figure \[pwr\_out1\]).
\
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In Figure \[NvsF\_fading\_2\], we plot user capacity versus $F$ using the approximation method of Section \[fading\_f\_small\] for $L_p = 2,3,$ and 4 and simulation results for the RA, HT, and TU environments. The RA results closely match those for $L_p=2$; the HT results closely match those for $L_p=3$; and the TU results closely match those for $L_p=4$. Thus, the diversity factor method of Section \[fading\_f\_small\] allows us to reliably estimate user capacity versus $F$ for any realistic delay profile.
\
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Finally, we note that as in Figure \[NvsF\_fading\], the critical value $F^*$ is around 1 for all cases in Figure \[NvsF\_fading\_2\]. Thus, we find that, regardless of the delay profile, $F^* \approx 1$. For the remainder of the paper, we assume practical values of $P_{\max}$ and $d_{\max}$ such that $F > 1$.
User Capacity with Variable Fading: Multicell Systems {#multimulti_sec}
=====================================================
Estimated User Capacity in Infinitely-Dispersive Channels {#multi_multi_inf_subsec}
---------------------------------------------------------
In computing the approximate user capacity in an infinitely dispersive channel, we use a result in [@skishore2] which states that maximal user capacity results when each cell in a multicell system supports an equal number of users. Let us denote $N_p$ as the (equal) number of users supported in each cell and $N_{\infty}(L,M)$ as the total number of users supported in a system with $L$ microcells, $M$ macrocells, and an infinitely dispersive channel.[^10] We assume initially that all $L$ microcells are embedded within one of the $M$ macrocells (macrocell 1) and that the remaining $M-1$ macrocells surround this macrocell. We first write $L+M$ inequalities representing the minimum SINR requirements at the $L+M$ bases. The received SINR’s are determined using average interference terms. For example, the SINR requirement at macrocell base station 1 (which contains embedded microcells) is $$\frac{\frac{W}{R} S_{M 1}}{N'_p S_{M1} + \sum_{l=1}^L S_{\mu l} N_p
\tilde{v}_M+\sum_{k =2}^M S_{M k} N_p \tilde{v}_{MM} + \eta W} \geq \Gamma ,
\label{multi_sinr_1}$$ where $N'_P=N_p-1$, $S_{M j}$ is the required received power at macrocell $j$, $N_p=N(L,M)/(L+M)$ (as per the optimal equal distribution of users), and the same-tier interference between macrocells is represented by its average value, i.e., $N_p \tilde{v}_{MM}$, which we assume to be the same for all macrocells. Here $\tilde{v}_{MM}$ is the interference into a macrocell caused by a user communicating with a neighboring macrocell, averaged over all possible user locations in $\mathcal R$ and over shadow fading. In our calculations, we simplify the SINR requirement in (\[multi\_sinr\_1\]) by assuming that $\tilde{v}_{MM} \approx \tilde{v}_M$, i.e., interference due to two users in neighboring macrocells is roughly equal to the interference at a macrocell due to an embedded microcell user. Although macrocells transmit at higher powers than microcells, they are further apart from each other than is a macrocell base from an embedded microcell user. This approximation assumes that the larger distance between macrocells balances the impact of higher transmit powers.\
\
Next, we determine the SINR requirement at macrocell $j$ ($j \geq 2$). To do so, we assume (as above) that $\tilde{v}_{MM} \approx \tilde{v}_M$. We further assume that the interference from neighboring macrocells is much larger than the interference caused by transmissions in a few isolated microcells in a nearby macrocell, i.e., $\sum_{k \neq k}S_{Mk} \tilde{v}_{MM} N_p \gg \sum_{l=1}^L S_{\mu l}\tilde{v}_{M \mu} N_p$, where $\tilde{v}_{M \mu}$ is the interference at any macrocell base due to a microcell user embedded in a neighboring macrocell averaged over all possible user locations and shadow fadings. Under these two assumptions, the SINR requirement at macrocell $j$ ($j \geq 2$) is approximately $$\frac{\frac{W}{R}S_{Mj}}{N'_pS_{Mj}+\sum_{k \neq j}S_{Mk}N_P \tilde{v}_{M} +\eta W} \geq \Gamma.$$ Finally, we calculate the SINR requirements at the $L$ microcell bases. Here, we assume first that $\tilde{v}_{\mu M} \approx \tilde{v}_{\mu}$, where $\tilde{v}_{\mu M}$ represents the interference into any microcell base due to a user in a neighboring macrocell averaged over user location and shadow fading. In other words, we assume the interference at a microcell from neighboring macrocells is comparable to interference caused by transmission to the umbrella macrocell. Note that this is a somewhat pessimistic assumption as $\tilde{v}_{\mu M}$ will typically be less than $\tilde{v}_\mu$. This pessimistic approach is balanced by assuming further that the microcell-to-microcell interference $\tilde{v}_{\mu \mu}$ is negligible. Here $\tilde{v}_{\mu \mu}$ is the average interference into any microcell base due to a user communicating with any other microcell. Thus, the required SINR equation at microcell $l$ is $$\frac{\frac{W}{R}S_{\mu l}}{N'_pS_{\mu l} + S_{M1} N_p \tilde{v}_\mu +
\sum_{k =2}^M S_{Mk}N_p \tilde{v}_{\mu} + \eta W} \geq
\Gamma.$$ These $L+M$ inequalities can be used to show [@skishore3] that $S_{Mj} \geq 0$ and $S_{\mu l} \geq 0$ for all $j \in \{ 1,2,\ldots M \}$ and $l \in \{1,2, \ldots L \}$ if and only if $$N_{\infty}(L,M) = (L+M)N_p \leq \frac{K (L+M)}{1+\sqrt{(L+M-1) \tilde{v}_M \tilde{v}_\mu}}.
\label{eq:cap_multi}$$ A benefit of this simplified capacity condition is that it allows for capacity calculation using $\tilde{v}_M$ and $\tilde{v}_\mu$ alone. As a result, it is no more complex to determine than is $N_T$, the user capacity when $M=1$ and $L=1$, Section 2. Further, as the product $\tilde{v}_M \tilde{v}_\mu$ is robust to variations in propagation parameters and user distributions, so is the approximation in (\[eq:cap\_multi\]). Despite its simplified form, this approximation yields a fairly accurate estimate to the attainable user capacity.\
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A simple extension of the capacity expression in (\[eq:cap\_multi\]) can now be used to find the user capacity in a [*general*]{} multicell system, i.e., a system in which the $L$ microcells can be embedded anywhere within the coverage areas of the $M$ macrocells. We first note that, in the general system, each macrocell contains [*on average*]{} $L/M$ microcells. Next, we use (\[eq:cap\_multi\]) to determine the user capacity for a multicell system in which there are $M$ macrocells but only the center macrocell contains $L/M$ microcells. In this case, each cell contains, $N_p$ users, where $$N_p=\frac{K}{1+\sqrt{(\frac{L}{M}+M-1) \tilde{v}_M
\tilde{v}_\mu}}.
\label{eq:cap_multi_2}$$ Next, we assume that this value of $N_p$ hardly changes when some other macrocell contains (on average) $L/M$ embedded microcells. Thus, we can finally approximate the total attainable capacity for the general multicell system as $(L+M)$ times the result in (\[eq:cap\_multi\_2\]), i.e., $$N_{\infty}(L,M) \approx \frac{K (L+M)}{1+\sqrt{(\frac{L}{M}+M-1) \tilde{v}_M
\tilde{v}_\mu}}.
\label{eq:cap_multi_3}$$
Estimated User Capacity with Variable Fading
--------------------------------------------
The goal here is to develop a calculation for the multicell user capacity in [*finitely dispersive channels*]{} which is as straightforward to calculate as the capacity expression in (\[eq:cap\_multi\_3\]). Although there are several approaches to this problem, we propose an extremely simple and highly accurate approximation method. First, we compute the diversity factor for the given multipath channel. Next, we make the assumption that systems with the same diversity factor support equivalent numbers of users and we seek to find $N_{DF}(L,M)$, the user capacity in a system with $L$ microcells, $M$ macrocells, and a multipath channel with diversity factor $DF$. We then use the results of Section \[varfading\_sec\] to compute the total user capacity for a two-cell system, i.e, $$N_{DF}(1,1) = \frac{2K}{1+\frac{DF}{DF-1}\sqrt{\tilde{v}_M \tilde{v}_\mu}}.$$ We also compute the total user capacity for a two-cell system with infinite dispersion, i.e., $$N_{\infty}(1,1) = \frac{2K}{1+\sqrt{\tilde{v}_M \tilde{v}_\mu}}.$$ We use these two capacities to determine the capacity percentage loss due to dispersion, $p_{loss}(DF)$, for the two-cell system. This loss is given as $$p_{loss} = \frac{N_{DF}(1,1)}{N_{\infty}(1,1)} = \frac{1+\sqrt{\tilde{v}_M \tilde{v}_\mu }}{1 + \frac{DF}{DF-1} \sqrt{\tilde{v}_M \tilde{v}_\mu }}.$$ Finally, we claim and demonstrate through extensive numerical results that the total attainable user capacity in a multicell system with $L$ microcells, $M$ macrocells, and finite dispersion $DF$ is simply $p_{loss}(DF)$ times the total attainable user capacity of a system with $L$ microcells, $M$ macrocells, and infinite dispersion, i.e., $$\begin{aligned}
N_{DF}(L,M) \approx p_{loss}(DF) \cdot N_{\infty}(L,M) \qquad \qquad \\
= \frac{1+\sqrt{\tilde{v}_M \tilde{v}_\mu }}{1 + \frac{DF}{DF-1} \sqrt{\tilde{v}_M \tilde{v}_\mu }} \cdot \frac{K(L+M)}{1+\sqrt{\left( \frac{L}{M} + M - 1 \right)\tilde{v}_M \tilde{v}_\mu }}.
\label{multi_fading_capacity}\end{aligned}$$
Numerical Results
-----------------
### Single-Macrocell/Multiple-Microcell System
We begin with a large square region having side $S$ and a macrocell base at its center. A fraction of the total system users are uniformly distributed over this region. This larger square is divided into $n^2$ squares with side $s$, as shown in Figure \[singlesquares\] for $n=5$. The smaller squares represent potential high-density regions. In each simulation trial, we randomly select $L$ high density regions (excluding the center square which contains the macrocell base) and place a microcell base at the center. Finally, we assume that the $L$ microcells have identical values for $b_\mu$, $H_\mu$, and $\sigma_\mu$.
\
\
Our simulations assume $n=5$, meaning 24 possible candidate high density regions and the parameters in Table 1. For a given value of $L$, the simulation randomly selects $L$ of the 24 high density regions. A portion of the total user population is uniformly-distributed over the large square region and the remaining users are uniformly distributed over the selected high density regions. The average fraction of the total user population placed in each region is $\frac{1}{L+1}$. As before, this was done to ensure that maximal user capacity occurs with high probability [@skishore2]-[@skishore3]. The simulation then determines the total number of users supported with 5% outage. This is done for 23 other random selections of $L$ high density regions, and the average over these selections is computed as a function of $L$.\
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We performed the above simulations for uniform channels with $L_p=2$ and $L_p = 4$, and for the infinitely dispersive channel. The results are in Figure \[ul\_multicell\_1a\]. In addition, we show the attainable user capacities predicted by (\[eq:cap\_multi\_3\]) and (\[multi\_fading\_capacity\]). The simulation points match the corresponding approximations very tightly up to about $L=12$, corresponding to a microcell [*fill factor*]{} (the fraction of macrocellular area served by microcells) of about one half. In other words, our approximation technique works reliably in the domain [*hotspot*]{} microcells.
### Multiple-Macrocell/Multiple-Microcell System
We simulated a multiple-macrocell/multiple-microcell system by extending the system studied above. Specifically (Figure \[multimulti\_fig\]), there are $m^2$ macrocells and $m^2(n^2-1)$ candidate high density regions (locations for microcell bases). We randomly select $L$ high density regions from this pool. We again accrue the average total capacity and its one-standard-deviation spread for the two extreme cases of dispersion, i.e., infinite dispersion and $L_p=2$. The results for $m=3$, $n=5$ are in Figure \[multi\_multi\_curve\]. Also shown are the capacities predicted by (\[eq:cap\_multi\_3\]) and (\[multi\_fading\_capacity\]) for $M=9$ and $DF=2$. The approximations closely match simulation points up to at least $L=72$, which translates to roughly eight microcells per macrocell (fill factor of about one third). Thus, the approximation method is reliable in the domain of hotspot microcells.
Conclusion
==========
We first examined the effect of transmit power constraints and cell size on the uplink performance of a two-cell two-tier CDMA system. We then investigated the effect of finite channel dispersion (which causes variable fading of the output signal power) on this capacity. Analysis was presented to approximate the uplink capacity of [*any*]{} channel delay profile using the uniform multipath channel, both with and without terminal power constraints. Finally, we presented an approximation to the uplink user capacity in larger multicell systems with finite dispersion and demonstrated its accuracy using extensive simulation.\
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While a particular set of system parameters was used in obtaining our numerical results, the analytical approximation methods, and their confirmation via simulations, are quite general. What we have shown here, first, is that two-tier uplink user capacity can be reliably estimated in a two-cell system for arbitrary pole capacity, channel delay profile, shadow fading variances, transmit power limit, and cell size. Moreover, under realistic conditions on transmit power and cell size ($F>1$), the method can be extended to the case of multiple macrocells with multiple microcells. Thus, we have developed simple, general and accurate approximation methods for treating a long-standing problem in two-tier CDMA systems.
Acknowledgment {#acknowledgment .unnumbered}
==============
We thank the editor, Dr. Halim Yanikomeroglu, for his helpful suggestions on expanding the scope of our study.
[1]{} J. Shapira, “Microcell engineering in cdma cellular networks,” , vol. 43, no. 4, pp. 817–825; Nov. 1994. J.S. Wu, et al., “Performance study for a microcell hot spot embedded in cdma macrocell systems,” , vol. 48, no. 1, pp. 47–59; Jan. 1999. J. J. Gaytán and D. Mu [n]{}oz Rodríguez, “Analysis of capacity gain and ber performance for cdma systems with desensitized embedded microcells,” in [*Proc. of International Conf. on Universal Personal Commun. ’98*]{}, 1998, vol. 2, pp. 887–891; Oct. 1998. S. Kishore, et al., “Uplink capacity in a cdma macrocell with a hotspot microcell: exact and approximate analyses,” , vol. 2, no. 2, pp. 364–374; Mar. 2003. S. Kishore, et al., “Uplink user capacity of a multi-cell cdma system with hotspot microcells,” in [*Proc. of Vehic. Techn. Conf. S-02*]{}, vol. 2, pp. 992–996; May 2002. S. Kishore, et al., “Uplink user capacity of multi-cell cdma system with hotspot microcells,” submitted to [*IEEE Trans. on Wireless Comm.*]{}, Sept. 2003, http://www.eecs.lehigh.edu/126 skishore/research/inprogress.htm. S. Kishore, et al., “Uplink user capacity in a cdma macrocell with a hotspot microcell: Effects of finite power constraints and channel dispersion,” in [*Proc. of IEEE Globecom*]{}, vol. 3, pp. 1558–1562, Dec. 2003. S. Kishore, et al., “Downlink user capacity in a cdma macrocell with a hotspot microcell,” in [*Proc. of IEEE Globecom*]{}, vol. 3, pp. 1573–1577, Dec. 2003. K. S. Gilhousen, et al., “On the capacity of a cellular cdma system,” , vol. 40, no. 2, pp. 303–312; May 1991. V. Erceg, et al., “An empirically based path loss model for wireless channels in suburban environments,” , vol. 17, no. 7, pp. 1205–1211; July 1999. T. S. Rappaport, , Prentice Hall, 1996; Chapter 3. S. Kishore, , Ph.D. Thesis, Department of Electrical Engineering, Princeton University; Jan. 2003. “Deployment Aspects,” 3GPP Specifications, TR 25.943 V4.0.0, (2001-2006), 2001. A. Awoniyi, et al., “Characterizing the orthogonality factor in wcdma downlink,” , vol. 2, no. 4, pp. 611–615; July 2003.
------------ ------------ -------------- -------
$W/R$ 128 $h_m$ 1.5 m
$\Gamma_M$ 7 dB $\Gamma_\mu$ 7 dB
$h_M$ 60 m $h_\mu$ 9 m
$b_M$ 100 m $b_\mu$ 100 m
$H_M$ $10 H_\mu$ $D$ 300 m
$\sigma_M$ 8 dB $\sigma_\mu$ 4 dB
$s$ 200 m $S$ 1 km
------------ ------------ -------------- -------
: System parameters used in simulation.
\
\[sys\_param\]
[Shalinee Kishore]{} received the B.S. and M.S. degrees in Electrical Engineering from Rutgers University in 1996 and 1999, respectively, and the M.A. and Ph.D. degrees in Electrical Engineering from Princeton University in 2000 and 2003, respectively.
Dr. Kishore is an Assistant Professor in the Department of Electrical and Computer Engineering at Lehigh University. From 1994 to 2002, she has held numerous internships and consulting positions at AT&T, Bell Labs, and AT&T Labs-Research. She is the recipient of the National Science Foundation CAREER Award, the P.C. Rossin Assistant Professorship, and the AT&T Labs Fellowship Award. Her research interests are in the areas of communication theory, networks, and signal processing, with emphasis on wireless systems.
[Larry Greenstein]{} (M’59-SM’80-F’87-LF’02) received the B.S., M.S., and Ph.D. degrees in Electrical Engineering from Illinois Institute of Technology, Chicago, in 1958, 1961, and 1967, respectively.
From 1958 to 1970, he was with IIT-Research Institute, Chicago, IL, working on radio-frequency interference and anti-clutter airborne radar. He joined Bell Laboratories, Holmdel, NJ, in 1970. Over a 32-year AT&T career, he conducted research in digital satellites, point-to-point digital radio, lightwave transmission techniques, and wireless communications. For 21 years during that period (1979-2000), he led a research department renowned for its contributions in these fields. His research interests in wireless communications have included measurement-based channel modeling, microcell system design and analysis, diversity and equalization techniques, and system performance analysis and optimization. Since April 2002 he has been a research professor at Rutgers WINLAB, Piscataway, NJ, working in the areas of ultra-wideband systems, sensor networks, relay networks and channel modeling. Dr. Greenstein is an AT&T Fellow and a member-at-large of the IEEE Communications Society Board of Governors. He has been a guest editor, senior editor and editorial board member for numerous publications.
[H. Vincent Poor]{} (S’72,M’77,SM’82,F’97) received the Ph.D. degree in EECS from Princeton University in 1977. From 1977 until 1990, he was on the faculty of the University of Illinois at Urbana-Champaign. Since 1990 he has been on the faculty at Princeton, where is the George Van Ness Lothrop Professor in Engineering. Dr. Poor’s research interests are in the area of statistical signal processing and its applications in wireless networks and related fields. Among his publications in these areas is the recent book [*Wireless Networks: Multiuser Detection in Cross-Layer Design*]{} (Springer:New York, NY, 2005).
Dr. Poor is a member of the National Academy of Engineering, and is a Fellow of the Institute of Mathematical Statistics, the Optical Society of America, and other organizations. In 1990, he served as President of the IEEE Information Theory Society, and in 1991-92 he was a member of the IEEE Board of Directors. He is currently serving as the Editor-in-Chief of the [*IEEE Transactions on Information Theory*]{}. Recent recognition of his work includes the Joint Paper Award of the IEEE Communications and Information Theory Societies (2001), the NSF Director’s Award for Distinguished Teaching Scholars (2002), a Guggenheim Fellowship (2002-2003), and the IEEE Education Medal (2005).
[Stuart Schwartz]{} received the B.S. and M.S. degrees from M.I.T. in 1961 and the Ph.D. from the University of Michigan in 1966. While at M.I.T. he was associated with the Naval Supersonic Laboratory and the Instrumentation Laboratory (now the Draper Laboratories). During the year 1961-62 he was at the Jet Propulsion Laboratory in Pasadena, California, working on problems in orbit estimation and telemetry. During the academic year 1980-81, he was a member of the technical staff at the Radio Research Laboratory, Bell Telephone Laboratories, Crawford Hill, NJ, working in the area of mobile telephony.
He is currently a Professor of Electrical Engineering at Princeton University. He was chair of the department during the period 1985-1994, and served as Associate Dean for the School of Engineering during the period July 1977-June 1980. During the academic year 1972-73, he was a John S. Guggenheim Fellow and Visiting Associate Professor at the department of Electrical Engineering, Technion, Haifa, Israel. He has also held visiting academic appointments at Dartmouth, University of California, Berkeley, and the Image Sciences Laboratory, ETH, Zurich. His principal research interests are in statistical communication theory, signal and image processing
[^1]: S. Kishore is with the Department of Electrical and Computer Engineering, Lehigh University, Bethlehem, PA, USA. L.J. Greenstein is with WINLAB, Rutgers University, Piscataway, NJ, USA. H.V. Poor and S.C. Schwartz are with the Department of Electrical Engineering, Princeton University, Princeton, NJ, USA. This research was jointly supported by the New Jersey Commission on Science and Technology, the National Science Foundation under CAREER Grant CCF-03-46945, Grant ANI-03-38807, and the AT&T Labs Fellowship Program. Contact: skishore@eecs.lehigh.edu.
[^2]: By “infinitely dispersive,” we mean that the channel has an infinitude of significant paths so that, after RAKE processing, the receiver signal output has constant power.
[^3]: In [@skishore1], we studied this path gain-based selection method as well as the selection scheme in which users are assigned to base stations such that each terminal requires the least transmit power. The result show that the latter scheme out-performs the path-gain-based method by roughly 15% in capacity.
[^4]: In computing $\tilde{N}$, we used the fact, proved in [@skishore2], that maximal total capacity results when $N_M=N_\mu$.
[^5]: $p$ can be computed using the methods in [@skishore1].
[^6]: The true capacity for a single-path Rayleigh fading channel is in fact quite poor, though not zero.
[^7]: Furthermore, via a simple implementation of the Schwarz inequality, it can be shown that the diversity factor for a non-uniform channel with $L_p$ paths is less than $L_p$.
[^8]: We use the following to compute the CDF of $\kappa$: $\int_0^c \frac{x^m}{m!}e^{-x}\,dx = 1 - e^{-c} \sum_{i=0}^m \frac{c^i}{i!}$ and $\int_0^\infty x^n e^{- \alpha x} \,dx = n!/\alpha^{n+1}$.
[^9]: For $F$ large, 5% outage is the same as 5% infeasibility.
[^10]: Note that $N_{\infty}(L,M)=(L+M)N_p$.
|
---
abstract: 'We succeeded in the synthesis of high-*J$_{c}$* MgB$_{2}$ bulks via high energy ball-milling of elemental Mg and B powder at ambient temperatures. Alternatively to long-time mechanical alloying technique, the mixed powder was ball-milled for only 1h and the completed reaction is achieved by subsequent annealing. The correlations among synthesis parameters, microstructures and superconducting properties in MgB$_{2}$ bulks were investigated. Samples were characterized by XRD and SEM, and the magnetization properties were examined in a SQUID magnetometer. The high-*J$_{c}$*, approximately 1.7$\times\,$10$^{6}$A/cm$^{2}$(15K, 0.59T), and improved flux pinning were attribute to a large number of grain boundaries provided by small grain size.'
address:
- 'Northwest Institute for Nonferrous Metal Research, P. O. Box 51, Xian, Shaanxi 710016, P. R. China'
- 'Northwestern Polytechnical University, Xi’an 710012, P.R.China'
- 'Southwest Jiaotong University, Si Chuan 610031, P.R.China'
author:
- 'Y. F. Wu'
- 'Y. Feng'
- 'G. Yan'
- 'J. S. Li'
- 'H. P. Tang'
- 'S. K. Chen'
- 'Y. Zhao'
- 'M. H. Pu'
- 'H. L. Xu'
- 'C. S. Li'
- 'Y. F. Lu'
title: 'The Influence of Synthesis Parameters on Microstructures and Superconducting Properties of MgB$_{2}$ Bulks'
---
bulk MgB$_{2}$, high energy ball-milling, high-*J$_{c}$*, flux pinning
Introduction {#section:intro}
============
The improvement of the intrinsic properties of MgB$_{2}$ was recognized as a decisive goal to enable potential applications [@Larbalestier:01]. Now, MgB$_{2}$ is showing higher *H$_{c2}$* than that of conventional NbTi and Nb$_{3}$Sn and becoming the first promising metallic superconductor applicable at 20K. In particular, very high upper critical field, *H$_{c2}$*(0), exceeding 30T has been reported for MgB$_{2}$ wires, tapes and bulks. However, it was recognized that the irreversibility field *H$_{irr}$*(T) of the samples prepared by standard solid state reaction is apparently lower than *H$_{c2}$* due to weak flux pinning. A typical relationship, *H$_{irr}$*(T)$\sim$0.5*H$_{c2}$*(T), has been observed for the undoped MgB$_{2}$ bulks [@Yamamoto:05; @Gumbel:02]. It was also shown that for MgB$_{2}$, contrast to high *T$_{c}$* superconductors, grain boundaries in MgB$_{2}$ are not acting as impediments for superconducting currents [@Larbalestier:01; @Bugoslavsky:01]. Unfortunately, a rapid drop of *J$_{c}$* at high fields, probably related to weak-link-like behavior, can be seen in most studies [@Giunchi:03; @Civale:02; @Suo:01; @Dou:02; @Fujii:02; @Flukiger:03]. The predominant pinning mechanism in MgB$_{2}$ is still controversy. However, a microstructure with very small-sized defects (e.g. grain boundary) should be favorable for the optimum pinning of magnetic flux lines. The high energy ball-milling technique facilitates the formation of an optimal microstructure with small particle sizes. Hence, a high grain boundary density is obtained, which is expected to enhance the magnetic flux pinning ability and to improve the critical currents in external magnetic fields.
Experimental {#section:Experimental}
============
In this study Mg(99.8%) and amorphous B(95+%) powders with 5wt% Mg surplus were filled under purified Ar-atmosphere into an agate milling container and milling media. The milling was performed on a SPEX 8000M mill for 1h using a ball-to-powder mass ratio of 3. The milled powders were then cold pressed to form pellets with a diameter of 20mm and a height of 3mm. The pellets were placed in an alumina crucible inside a tube furnace under ultra-high purity Ar-atmosphere. The samples were heated at different temperatures for different times, then cooled down to the ambient temperature.
A Quantum Design SQUID magnetometer was used to measure the AC magnetic susceptibility of the samples over a temperature range of 5 to 50 K under an applied field of 1Oe. Magnetization versus magnetic field (M-H) curves were also measured on rectangular-shaped samples at temperatures of 10 and 15K under magnetic fields up to 90000Oe to determine the critical current density *J$_{c}$*(*H*).
The phase compositions of the samples were characterized by the APD1700 X-ray diffraction instrument. The surface morphology and microstructures of the samples were characterized by the JSM-6460 and the JSM-6700F scanning electron microscope.
Results and discussion {#section3}
======================
The influence of sintering temperature to the phase composition, microscopy and superconducting properties of bulk MgB$_{2}$ {#section3.1}
----------------------------------------------------------------------------------------------------------------------------
The X-ray diffraction patterns of MgB$_{2}$ bulks sintered at different temperatures are shown in figure 1. For the samples sintered at $650\,^{\circ}\mathrm{C}$, $700\,^{\circ}\mathrm{C}$ and $750\,^{\circ}\mathrm{C}$, almost single phase MgB$_{2}$ appears in the patterns with minor fraction of MgO. The percentage of MgO is apparently increased for the sample sintered at $800\,^{\circ}\mathrm{C}$, indicating easy oxidation of Mg at higher temperature.
Shown in Fig.2(a) is the surface morphology of the sample sintered at $650\,^{\circ}\mathrm{C}$. Well-developed coarse columnar grains can be seen in the figure. Evidently, it could not connect very well in the surface. There are also some black impurity phases for the sample sintered at $650\,^{\circ}\mathrm{C}$,see Fig.2(b). The energy spectrum analysis showed that the black one is the B-rich phase, which is due to the stability of the higher borides at lower temperature, see Fig.3. In contrast, the grains of the samples sintered at $700\,^{\circ}\mathrm{C}$ and $750\,^{\circ}\mathrm{C}$ are equiaxial, fine and well-connected,shown in Fig.2(c)and Fig.2(e). Additionly, there are only a few impurity phases for these samples, see Fig.2(d) and Fig.2(f). It exhibits the sintering temperatures are appropriate right here. A broad grain size distribution can be seen for the sample sintered at $800\,^{\circ}\mathrm{C}$, as shown in Fig.2(g). The large number of existing impurity phases,see Fig.2(h) would obviously take bad effect to the grain connectivity and lower the superconducting properties of the sample. It is clearly not the suitable temperature for preparation of bulk MgB$_{2}$.
Fig.4 shows the AC magnetic susceptibility as a function of the temperature for the samples sintered at different temperatures. A constant magnetic field of 1Oe was applied. As we can see, all samples have sharp transitions. But it is interesting that the superconducting transition of the samples sintered at $700\,^{\circ}\mathrm{C}$, $750\,^{\circ}\mathrm{C}$ and $800\,^{\circ}\mathrm{C}$ ( *T$_{c}$*$\sim$1.5 K) is about 2.5 times sharper than that of the samples sintered at $650\,^{\circ}\mathrm{C}$ ( *T$_{c}$*$\sim$4 K). It is not clear what caused the transition broaden. However, the broadening could be related to microstructural changes induced by the heat treatment temperature. It seems that the columnar grains lead to worse grain connectivity than the impurity phases.
The magnetization curves (M-H) for the samples sintered at different temperatures are shown in Fig.5. *J$_{c}$*(H) was determined by Bean critical state model, as shown in Fig.6. The *J$_{c}$* at 0K is nearly equal for all the samples. However, the sample sintered at $750\,^{\circ}\mathrm{C}$ has a significantly higher *J$_{c}$* than the other samples in magnetic field, indicating improved magnetic flux pinning of it. The differences in *J$_{c}$* among them increase with field. The sample sintered at $650\,^{\circ}\mathrm{C}$ shows a steep drop in *J$_{c}$* at higher fields (H$>$4T and T = 10K). It is most probably because of the columnar grains existing in the sample lead to bad grain connectivity, which severely limits its *J$_{c}$* performance. No such steep drop in *J$_{c}$* is observed for the other samples.
The influence of the holding time to the phase composition, microscopy and superconducting properties of bulk MgB$_{2}$
-----------------------------------------------------------------------------------------------------------------------
The X-ray diffraction patterns of bulk MgB$_{2}$ sintered for different times are shown in figure 7. Mainly peaks of MgB$_{2}$ are visible but some peaks of MgO are present. The differences in the percentage of MgO phase are difficult to perceive in the patterns.
Shown in Fig.8(a) is the surface morphology of the samples sintered for 0.5h. As we can see, the reaction between Mg and B is incomplete. The big grains could not disappear completely. Meanwhile, a large number of small grains have become visible in the grain boundaries. It reveals that the grains are not well connected in the surface for the sample sintered for short time. Some dark grey and light grey impurity phases can be seen for the sample sintered for 0.5h,see Fig.8(b). The energy spectrum analysis showed that the dark one is the B-rich phase (Fig.9) and the light one is the magnetism oxidation compound (Fig.10). However, the grains of the sample sintered for 1h are equiaxial, fine and well-connected,see Fig.8(c). It only has a few small-sized impurity phases,see Fig.8(d). Although the grains are fine as well for the sample sintered for 3h, see Fig.8(e), a great many existing impurities would apparently deteriorate its *J$_{c}$* performance as shown in Fig.8(f). It indicates undue shorter or longer holding time lead to much more impurity phases and inferior microstructures.
Shown in Fig.11 are magnetization-field loops for samples sintered for different times. *J$_{c}$*(H) was deduced from the hysteresis loops using the Bean model, see Fig. 12. The sample sintered for 1h has a significantly higher *J$_{c}$* and than other samples in the magnetic field due to its optimum microstructure. The *H$_{irr}$* value is about 6.3T at 15K, as determined from the closure of hyseresis loops with a criterion of *J$_{c}$*=10$^{2}$A/cm$^{2}$. The differences in *J$_{c}$* among them increase with field. The sample sintered for 0.5 and 3h shows a steep drop in *J$_{c}$* at higher fields (H$>$4T and T = 15K). It indicates both columnar grains and excessive impurity phases lead to the bad grain connectivity and weak flux pinning, especially in high magnetic field.
Summary {#section:summary}
=======
In conclusion, the high-*J$_{c}$* MgB$_{2}$ bulks were prepared by short-time high energy ball milling. The introduction of impurities (especially oxygen) during the ball-milling process was minimized. A close relation among microstructure, impurities and superconducting properties with the synthesis parameters was detected. The irreversibility field of the optimum sample reaches 6.3T at 15K and *J$_{c}$* is high around 1.7$\times\,$10$^{6}$A/cm$^{2}$(15K, 0.59T) at 15K in 0.59T. The improved pinning of this material seems to be caused by enhanced grain boundary pinning provided by the large number of grain boundaries in the sample. Another crucial point during preparation of MgB$_{2}$ is to avoid weak-link-like behavior at grain boundaries due to a non-superconducting surface layer of magnetism oxidation or B-rich compounds.
ACKNOWLEDGMENT {#section:Acknowledgment}
==============
This work was partially supported by National Natural Science Foundation project(grant 50472099)and National 973 project(grant 2006CB601004).
[00]{}
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**Figure captions**
Fig.1 XRD patterns of MgB$_{2}$ bulks sintered at (a)$650
\,^{\circ}\mathrm{C}$, (b) $700\,^{\circ}\mathrm{C}$, (c)$750\,^{\circ}\mathrm{C}$ and (d) $800\,^{\circ}\mathrm{C}$. Peaks of MgB$_{2}$ and MgO are marked by solid circles and squares, respectively.
Fig.2 The SEM photograph of the MgB$_{2}$ bulks sintered at different temperatures: (a)$650\,^{\circ}\mathrm{C}$$\times\,$15,000 (b)$650\,^{\circ}\mathrm{C}$$\times\,$4,000 (c)$700\,^{\circ}\mathrm{C}$$\times\,$15,000 (d)$700\,^{\circ}\mathrm{C}$$\times\,$4,000 (e)$750\,^{\circ}\mathrm{C}$$\times\,$15,000 (f)$750\,^{\circ}\mathrm{C}$$\times\,$4,000 (g)$800\,^{\circ}\mathrm{C}$$\times\,$15,000 and (h)$800\,^{\circ}\mathrm{C}$$\times\,$4,000.
Fig.3 The EDX analysis of the second phase in the sample sintered at $650\,^{\circ}\mathrm{C}$.
Fig.4 The AC Magnetic susceptibility as a function of temperature of MgB$_{2}$ bulks. A constant magnetic field of 1Oe was applied.
Fig.5 Magnetization *M* as a function of magnetic field *H* at 10K for the samples sintered at different temperatures.
Fig.6 Magnetization critical current density *J$_{c}$* as a function of magnetic field *H* at 10K for the samples sintered at different temperatures.
Fig.7 The XRD patterns of MgB$_{2}$ sintered at $750\,^{\circ}\mathrm{C}$ for different holding times.Peaks of MgB$_{2}$ and MgO are marked by solid circles and squares, respectively.
Fig.8 The SEM photograph of the MgB$_{2}$ bulks sintered for different holding times:(a)0.5h$\times\,$15,000(b)0.5h$\times\,$4,000 (c)1h$\times\,$15,000(d)1h$\times\,$4,000 (e)3h$\times\,$15,000 and (f)3h$\times\,$4,000.
Fig. 9 The EDX analysis of dark gray second phase in the 0.5h sintered sample.
Fig.10 The EDX analysis of light gray second phase in the 0.5h sintered sample.
Fig.11 Magnetization *M* as a function of magnetic field *H* at 15K for samples sintered for different holding times.
Fig.12 Magnetization critical current density *J$_{c}$* as a function of magnetic field H at 15 K for the samples sintered for different holding times.
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---
abstract: 'We explore the correlations between primordial non-Gaussianity and isocurvature perturbation. We sketch the generic relation between the bispectrum of the curvature perturbation and the cross-correlation power spectrum in the presence of explicit couplings between the inflaton and another light field which gives rise to isocurvature perturbation. Using a concrete model of a Peccei-Quinn type field with generic gravitational couplings, we illustrate explicitly how the primordial bispectrum correlates with the cross-correlation power spectrum. Assuming the resulting ${f_{\rm NL}}\sim {{\cal O}}(1)$, we find that the form of the correlation depends mostly upon the inflation model but only weakly on the axion parameters, even though ${f_{\rm NL}}$ itself does depend heavily on the axion parameters.'
---
1.0 cm
1.0cm
[Jinn-Ouk Gong$^{a,b}$ and Godfrey Leung$^{a}$ ]{}
0.5cm
1.2cm
Introduction
============
Inflation [@inflation; @Linde:1981mu] is widely considered as the best candidate to explain the origin of the primordial fluctuations in our universe, and has been well tested by cosmic microwave background (CMB) observations, most recently by the Planck mission [@Ade:2015lrj]. The simplest scenario involves a single scalar field slowly rolling down a flat potential [@Linde:1981mu]. However, when embedded in ultraviolet complete theories such as string theory [@infreview], it is not likely that such a simple description of single field inflation remains legitimate throughout the whole inflationary epoch, as in general additional degrees of freedom become dynamically relevant during inflation.
It is illustrative to consider the field trajectory in the field space to see how these additional degrees of freedom affect the inflationary predictions. While in single field inflation we have only a single direction along which the inflaton moves straightly, in multi-field space we have more than one direction and thus the trajectory is in general curved. As the component of field fluctuations along the trajectory is associated with the curvature perturbation, those orthogonal to the trajectory are responsible for the isocurvature perturbations [@Gordon:2000hv]. What is important here is that since the field trajectory is in general curved, the fluctuation along the trajectory at a moment receives contributions from those orthogonal to the trajectory before. In other words, the curvature and isocurvature perturbations are correlated [@Gordon:2000hv; @correlation].
Thus, on general ground the existence of isocurvature perturbations can be interpreted as a proof of additional degrees of freedom in the early universe. Even if the isocurvature perturbations themselves are too small to be detected, their correlation to the curvature perturbation can lead to distinctive observational signatures. It is thus useful and important to study how observables are correlated with and sourced by isocurvature perturbations on completely general ground. In this article, we concentrate on the correlations between primordial non-Gaussianity and isocurvature perturbations due to interactions at horizon-crossing, whereas previous studies focus on correlations due to non-linear evolution on super-horizon scales [@other_ng-iso]. The primordial non-Gaussianity we are considering might not necessarily be that associated with the CMB scales, but also on smaller scales which are relevant for large scale structure for example. By studying the correlation structure, we can put further constraints on the model parameters compared to that coming from studying the observables individually.
This article is organised as follows: in Section \[sec:general\_structure\], we discuss the general structure of three-point interaction terms between the inflaton and an isocurvature field, and how the resulting bispectrum $B_{{\cal R}}$ correlates with the cross-correlation power spectrum ${{\cal P}}_{{\cal C}}$. We then illustrate this general feature using a concrete example involving a Peccei-Quinn (PQ) type field, which interacts with the inflaton via a gravitationally induced coupling in Section \[sec:example\]. We subsequently conclude in Section \[sec:conc\]. Detailed derivations of the results are given in the Appendix.
General structure of correlations {#sec:general_structure}
=================================
For simplicity, let us consider a generic two-field system, one being the inflaton $\phi$ and the other being a spectator field $\theta$ that survives after thermalisation and becomes responsible for the isocurvature perturbation ${{\cal I}}$ at late times. We also assume the curvature perturbation ${{\cal R}}$ is purely sourced by the inflaton fluctuations. In such a system, the three-point correlation function of the curvature perturbation ${{\cal R}}$ may receive additional contributions from $\theta$ [@Bartolo:2001cw] via the quadratic interaction $H_2^I \sim \theta\phi$ which is responsible for the cross-correlation power spectrum ${{\cal P}}_{{\cal C}}$. The resulting bispectrum of the curvature perturbation $B_{{\cal R}}$ is therefore correlated with ${{\cal P}}_{{\cal C}}$, giving additional constraints to the spectator field $\theta$.
To see this explicitly, we decompose the cubic interaction Hamiltonian as $$\label{H3}
H_3 = H_3^\text{(self1)} + H_3^\text{(cross1)} + H_3^\text{(cross2)} + H_3^\text{(self2)} \, ,$$ where $H_3^\text{(self1)} \sim \phi^3$, $H_3^\text{(cross1)} \sim g_1\theta\phi^2$, $H_3^\text{(cross2)} \sim g_2\theta^2\phi$ and $H_3^\text{(self2)} \sim g_3\theta^3$ with $g_i$’s being the couplings. Note that these are schematic forms only, which could also include derivative interactions such as $(\nabla\theta)^2\phi$ and $\dot\theta^3$. In standard slow-roll inflation $H_3^\text{(self1)}$ usually contributes to a slow-roll suppressed, negligible non-Gaussianity of ${{\cal O}}(\epsilon)$ [@Maldacena:2002vr]. We therefore neglect $H_3^\text{(self1)}$ here. Through the transfer vertex $H_2^I$, each cubic interaction Hamiltonian term contributes to the primordial bispectrum $B_{{\cal R}}$ as $$\label{generalcorr}
\begin{split}
H_3^\text{(cross1)} & \mapsto B_{{\cal R}}\sim g_1{{\cal P}}_{{\cal R}}{{\cal P}}_{{\cal C}}\, ,
\\
H_3^\text{(cross2)} & \mapsto B_{{\cal R}}\sim g_2\sqrt{{{\cal P}}_{{\cal R}}}{{\cal P}}_{{\cal C}}^2 \, ,
\\
H_3^\text{(self2)} & \mapsto B_{{\cal R}}\sim g_3{{\cal P}}_{{\cal C}}^3 \, .
\end{split}$$ Diagrammatic representation of each term is shown in Figure \[fig:interaction\]. However, the detail of each contribution, such as the momentum-dependent shape and amplitude, depends on the coupling and derivative structure of the corresponding cubic interaction Hamiltonian. In the following section using a concrete example we demonstrate that indeed the generic relation is valid.
(440,100)(0,0) (40,60)[20]{}[0.7]{} (40,80)(40,100) (10,30)(26,46) (70,30)(54,46) (85,60)\[\][$=$]{} (130,60)(130,100) (100,30)(130,60) (160,30)(130,60) (130,15)\[\][$H_3^\text{(self1)} \sim \phi^3$]{} (175,60)\[\][$+$]{} (220,60)(190,30) (220,60)(250,30) (220,60)(220,80)[2]{} (220,80)(220,100) (220,80)[3]{} (207,15)\[\][1 $H_2^I$ and]{} (222,0)\[\][$H_3^\text{(cross1)} \sim g_1\theta\phi^2$]{} (265,60)\[\][$+$]{} (310,60)(340,30) (310,60)(310,80)[2]{} (310,80)(310,100) (310,80)[3]{} (310,60)(295,45)[2]{} (280,30)(295,45) (295,45)[3]{} (300,15)\[\][2 $H_2^I$’s and]{} (313,0)\[\][$H_3^\text{(cross2)} \sim g_2\theta^2\phi$]{} (355,60)\[\][$+$]{} (400,60)(400,80)[2]{} (400,80)(400,100) (400,80)[3]{} (400,60)(385,45)[2]{} (385,45)(370,30) (385,45)[3]{} (400,60)(415,45)[2]{} (415,45)(430,30) (415,45)[3]{} (395,15)\[\][3 $H_2^I$’s and]{} (400,0)\[\][$H_3^\text{(self2)} \sim g_3\theta^3$]{}
0.3cm
Before moving on to discuss a concrete example, let us comment on the relative size of the various bispectrum contributions above. Naively, one may think this somewhat suggests the above bispectrum contributions are small compared to that from the intrinsic inflaton self-interaction $H_3^\text{(self1)}$ since $H_3^\text{(self1)}\mapsto B_{{\cal R}}\sim {{\cal P}}_{{\cal R}}^2$, if ${{\cal P}}_{{\cal R}}>{{\cal P}}_{{\cal C}}$. Besides, one might think there exists a hierarchy between the various contributions, with $B_{{\cal R}}^\text{(cross1)}>B_{{\cal R}}^\text{(cross2)}>B_{{\cal R}}^\text{(self2)}$. However, this is not necessarily the case. In general, the resulting contributions are momentum and shape dependent. Depending on the forms of $H_3^\text{(cross1)}$, $H_3^\text{(cross2)}$ and $H_3^\text{(self2)}$, the various contributions might all have different momentum dependences and peak at different shapes. Furthermore, ${{\cal P}}_{{\cal C}}$ and ${{\cal I}}$ also depend implicitly on the ratio of matter density of isocurvature modes to that of the total matter at late time, i.e. $\Omega_\theta/\Omega_m$, whereas the various bispectrum contributions above do not. If $\Omega_\theta/\Omega_m \ll 1$, it is possible to realise scenario where ${{\cal P}}_{{\cal C}}\ll {{\cal P}}_{{\cal R}}$ yet having $B_{{\cal R}}^\text{(cross1)}$ $B_{{\cal R}}^\text{(cross2)}$, $B_{{\cal R}}^\text{(self2)}$ unsuppressed.
Axion with gravitationally induced interactions {#sec:example}
===============================================
Having discussed concisely the generic correlation structure between the bispectrum of the curvature perturbation $B_{{\cal R}}$ and the cross-correlation power spectrum ${{\cal P}}_{{\cal C}}$ in a generic two-field system, we now explicitly illustrate this generic behaviour using a concrete model. The model Lagrangian in the matter sector is [@Kadota:2014hpa] $$\label{eq:the_model}
{{\cal L}}= \sqrt{-g} \left[ -\frac{1}{2}(\partial_\mu\phi)^2 - |\partial_\mu\Phi|^2 - V_\text{inf}(\phi) - V_\text{axion}(\Phi) - V_\text{int}(\phi,\Phi) \right] \, ,$$ where $\phi$ is the inflaton field and $V_\text{inf}(\phi)$ is its potential. We do not envisage any specific form of $V_\text{inf}(\phi)$, as long as it can successfully support an inflationary epoch. The complex field $\Phi$ is a PQ type field [@PQ_field], which can be decomposed into real radial and angular components, $r$ and $\theta$ respectively, as $$\Phi = \frac{re^{i\theta}}{\sqrt{2}} \, ,$$ The corresponding potential is the standard symmetry-breaking type, $V_\text{axion}(\Phi) = \lambda\left( |\Phi|^2 - f_a^2/2 \right)^2$, so that the radial field is minimised at $r_0 = f_a$ with $f_a$ being the symmetry breaking scale, which is usually taken to be smaller than ${m_{\rm Pl}}$. Assuming the radial field completely settles down at the minimum, we may identify the axion as the angular field $a \equiv f_a\theta$.
For concreteness, we consider a toy but generic dimension-5 interaction that can be gravitationally induced [@Kamionkowski:1992mf], $$\label{eq:V_interaction}
V_\text{int}(\phi,\Phi) = g\frac{\phi\Phi^4}{{m_{\rm Pl}}} + {\rm h.c.} \, .$$ Additionally, overall we may add a cosmological constant to make the entire potential nearly vanishing at the global minimum. We shall associate the inflaton and axion fluctuations with the curvature and isocurvature perturbations respectively in the standard manner as [@defn] $$\label{eq:curv_iso_defn}
{{\cal R}}\equiv -\frac{H}{\dot{\phi_0}}\phi \quad \text{and} \quad {{\cal I}}\equiv \frac{2\theta}{\theta_0} \, .$$ Here the subscript $0$ denotes the homogeneous background field values. We have implicitly assumed $\Omega_\theta/\Omega_m \sim 1$ as we are interested in the case where the isocurvature perturbations are large enough to be potentially observed in future experiments.
Given the Lagrangian, we can straightly compute the correlation functions of the curvature and isocurvature perturbations using the in-in formalism [@Maldacena:2002vr; @in-in; @Weinberg:2005vy]. The leading order contributions to the interaction Hamiltonian come from the gravitationally induced interaction term in the matter sector, which at quadratic and cubic orders are $$\begin{aligned}
\label{eq:H2}
H_2^I = -a^3\alpha \int d^3x \phi\theta \quad \text{with} \quad \alpha & \equiv 2g\frac{r_0^4}{{m_{\rm Pl}}}\sin(4\theta_0) \, ,
\\
\label{eq:selfH3}
H_3^\text{(self2)} = a^3\beta \int d^3x \theta^3 \quad \text{with} \quad \beta & \equiv \frac{16}{3}g\frac{\phi_0}{{m_{\rm Pl}}}r_0^4\sin(4\theta_0) \, ,
\\
\label{eq:crossH3}
H_3^\text{(cross2)} = -a^3\gamma \int d^3x \phi\theta^2 \quad \text{with} \quad \gamma & \equiv 4g\frac{r_0^4}{{m_{\rm Pl}}}\cos(4\theta_0) \, .\end{aligned}$$ The power spectra of the curvature and isocurvature perturbations, and their cross-correlation are $$\begin{aligned}
\label{eq:power_spectra_PR}
{{\cal P}}_{{\cal R}}(k) & = \left(\frac{H}{2\pi}\right)^2 \left(\frac{H}{\dot{\phi}_0}\right)^2 \left[ 2^{\nu_\phi-3/2} \frac{\Gamma(\nu_\phi)}{\Gamma(3/2)} \left(-k\tau\right)^{3/2-\nu_\phi} \right]^2 \, ,
\\
\label{eq:power_spectra_PI}
{{\cal P}}_{{\cal I}}(k) & = \left(\frac{H}{2\pi}\right)^2 \left(\frac{2}{\theta_0 r_0}\right)^2 \left[ 2^{\nu_\theta-3/2} \frac{\Gamma(\nu_\theta)}{\Gamma(3/2)} \left(-k\tau\right)^{3/2-\nu_\theta} \right]^2 \, ,
\\
\label{eq:power_spectra_PC}
{{\cal P}}_{{\cal C}}(k) & = -\frac{\pi\alpha}{2r_0H^2} {{\cal A}}\sqrt{ {{\cal P}}_{{\cal R}}(k) {{\cal P}}_{{\cal I}}(k) } \, ,\end{aligned}$$ where the indices of Hankel functions are defined as $\nu_\phi \equiv \sqrt{9/4 - m_\phi^2/H^2}$ with $m^2_\phi$ being the mass of $\phi$ and similarly for $\nu_\theta$. Here $\tau$ is the conformal time defined as ${\rm d}t=a{\rm d}\tau$, whereas $H$ and $\dot{\phi}_0$ are evaluated at horizon-crossing. Current observations suggest that the power spectra of isocurvature perturbations and the cross-correlation are much smaller than that of curvature perturbation on CMB scales, i.e. ${{\cal P}}_{{\cal R}}\gg {{\cal P}}_{{\cal C}}, {{\cal P}}_{{\cal I}}$, see the following. Here ${{\cal A}}$ is an integral involving the Hankel functions, defined as $
{{\cal A}}\equiv \Re \left[ i\int_0^{\infty} dx/x H^{(2)}_{\nu_\phi}(x)H^{(2)}_{\nu_\theta}(x) \right]
$. If we naively perform this integral to leading order in $\nu_\phi \approx 3/2$ and $\nu_\theta \approx 3/2$ with a lower cutoff $x_c$, ${{\cal A}}\sim \log{x_c}$ so it is logarithmically divergent. But taking into account the asymptotic behaviour of the curvature perturbation at later times, we may evaluate ${{\cal A}}$ at an arbitrary value of $x_c$ as long as $\exp\left[-1/{{\cal O}}(\epsilon)\right] < x_c < 1$ [@Gong:2001he], giving ${{\cal A}}\sim -0.45$. Details of the calculation are given in Appendix A.
Next we consider the primordial bispectrum of the curvature perturbation $B_{{\cal R}}$ that arises from the cubic interaction Hamiltonian. There are two contributions: one from the axion cubic self-interaction $H_3^\text{(self2)} \sim \theta^3 $ and another from the cubic cross-interaction $H_3^\text{(cross2)} \sim \theta^2\phi$. We therefore expect the resulting bispectra scale as ${{\cal P}}_{{\cal C}}^3$ for the former interaction and ${{\cal P}}_{{\cal C}}^2\sqrt{{{\cal P}}_{{\cal R}}}$ for the latter one. After some calculations similar as in [@calculation], we find the resulting bispectra of the curvature perturbation are of the form, to leading order in the limit $\nu_\phi\approx3/2$ and $\nu_\theta\approx3/2$, $$\begin{aligned}
\label{eq:BcalR_self}
B_{{\cal R}}^\text{(self2)}(k_1,k_2,k_3) & = \frac{k_1^3+k_2^3+k_3^3}{(k_1k_2k_3)^3} {{\cal P}}_{{\cal C}}^3 \frac{64\pi^{3}}{3{{\cal A}}^3} \frac{\theta_0^3\beta}{H^4} \, ,
\\
\label{eq:BcalR_cross}
B_{{\cal R}}^\text{(cross2)}(k_1,k_2,k_3) & = -\frac{k_1^3+k_2^3+k_3^3}{(k_1k_2k_3)^3} {{\cal P}}_{{\cal C}}^2\sqrt{{{\cal P}}_{{\cal R}}} \frac{16\pi^3}{3{{\cal A}}^2} \frac{\theta_0^2\gamma}{H^3} \, .\end{aligned}$$ Details of the calculation can be found in Appendix C. The above results of the two bispectrum contributions, ${{\cal P}}_{{\cal C}}$ and ${{\cal P}}_{{\cal R}}$ are evaluated on relevant scales when the gravitationally induced interaction becomes important, which do not necessarily correspond to CMB scales, but some smaller scales.
The amplitude of the bispectrum is usually measured in terms of the non-linear parameter ${f_{\rm NL}}$ [@Komatsu:2001rj]. The current constraints on ${f_{\rm NL}}$ by Planck are ${f_{\rm NL}}^{\text{(local)}}=0.8\pm 5.0$, ${f_{\rm NL}}^{\text{(eq)}}=-4\pm43$ and ${f_{\rm NL}}^{\text{(ortho)}}=-26\pm21$ at 68% confidence level [@Ade:2015ava] for $k=0.05{\rm Mpc}$. Having in mind ${f_{\rm NL}}$, we define a dimensionless shape function as $$\label{eq:fnl_defn}
{f_{\rm NL}}(k_1,k_2,k_3) \equiv \frac{10}{3} \frac{k_1k_2k_3}{k_1^3+k_2^3+k_3^3} \frac{(k_1k_2k_3)^2B_{{\cal R}}}{(2\pi)^4{{\cal P}}_{{\cal R}}^2} \, .$$ The corresponding shape functions for the bispectra $B_{{\cal R}}^\text{(self2)}$ and $B_{{\cal R}}^\text{(cross2)}$ are then $$\label{fnls_1}
{f_{\rm NL}}^\text{(self2)}(k_1,k_2,k_3) = \frac{40}{9\pi{{\cal A}}^3} \frac{{{\cal P}}_{{\cal C}}^3}{{{\cal P}}_{{\cal R}}^2} \frac{\theta_0^3\beta}{H^4}
\quad \text{and} \quad
{f_{\rm NL}}^\text{(cross2)}(k_1,k_2,k_3) = -\frac{10}{9\pi{{\cal A}}^2} \frac{{{\cal P}}_{{\cal C}}^2}{{{\cal P}}_{{\cal R}}^{3/2}} \frac{\theta_0^2\gamma}{H^3} \, ,$$ or purely in terms of the model parameters as $$\label{fnls}
{f_{\rm NL}}^\text{(self2)} \sim \left(\frac{\dot{\phi_0}}{H}\right) \left(\frac{\alpha}{r_0 H^2} \right)^3 \left(\frac{\beta}{r_0^3H^2} \right)
\quad \text{and} \quad
{f_{\rm NL}}^\text{(cross2)}\sim \left(\frac{\dot{\phi_0}}{H}\right) \left(\frac{\alpha}{r_0 H^2} \right)^2 \left(\frac{\gamma}{r_0^2H^2} \right)\, ,$$ Again might suggest $|{f_{\rm NL}}^\text{(self2)}|<|{f_{\rm NL}}^\text{(cross2)}|$ since ${{\cal P}}_{{\cal C}}<{{\cal P}}_{{\cal R}}$. This is not necessarily the case though as both of the non-linear parameters depend upon the coefficients $\beta$ and $\gamma$, which implicitly depend on the axion model parameters such as the misalignment angle $\theta_0$, as seen in . It is possible to tune the axion model parameters such that $|f_{\rm NL}^{\rm (self2)}|>|f_{\rm NL}^{\rm (cross2)}|$ while having ${{\cal P}}_{{\cal C}}$ small.
In the following we give an estimate about the size of contributions to the non-linear parameter ${f_{\rm NL}}$ on CMB scales in simple inflationary models. Assuming simple chaotic inflation with a quadratic potential, we have $\phi_0=15{m_{\rm Pl}}$ and the Hubble parameter at horizon-exit $H\approx 10^{-4}{m_{\rm Pl}}$. From one then deduce $$\begin{aligned}
\label{fnls_example}
{f_{\rm NL}}^\text{(self2)} \sim & - \frac{1280}{27} g^4 \sin^4(4\theta_0) (\sqrt{2\epsilon}) \left(\frac{\phi_0}{{m_{\rm Pl}}} \right) \left(\frac{r_0}{{m_{\rm Pl}}} \right)^2\left(\frac{r_0}{H} \right)^8 \, , \nonumber \\
{f_{\rm NL}}^\text{(cross2)}\sim & \,16 g^4 \sin^2(4\theta_0) \cos(4\theta_0) (\sqrt{2\epsilon}) \left(\frac{r_0}{{m_{\rm Pl}}} \right)^2\left(\frac{r_0}{H} \right)^6 \, ,\end{aligned}$$ and $|{f_{\rm NL}}^\text{(self2)}/{f_{\rm NL}}^\text{(cross2)}| \sim g \tan(4\theta_0)\sin(4\theta_0) (\phi_0/{m_{\rm Pl}})(r_0/H)^2$. Note that $r_0>H$ here since we are considering scenario where spontaneous symmetry breaking happens at higher energy scales than $H$. Therefore we expect $|{f_{\rm NL}}^\text{(self2)}|>|{f_{\rm NL}}^\text{(cross2)}|$ in this simple example unless the coupling is very small, i.e. $g\lesssim 10^{-3}$. In theory the value of ${f_{\rm NL}}^\text{(self2)}$ could span a wide range depending on the combination of $r_0$ and $g$. For instance, assuming the symmetry breaking scale is around the GUT scale such that $r_0\sim 10^{-2}{m_{\rm Pl}}$, then we find ${f_{\rm NL}}^\text{(self2)}\sim 1$ for $g\sim {{\cal O}}(10^{-3})$.
From , we can also see that the resulting non-linear parameters are shape independent as opposed to non-canonical single field models such as K-inflation, where the non-Gaussianity induced due to non-linear interactions at horizon-crossing (or equivalently reduced sound speed $c_s$) typically peaks in the equilateral shape [@nonG_cs]. This is because the forms of non-linear interactions are completely different between our case and the single field case. An another example where non-Gaussianity induced by non-linear interactions at horizon-crossing does not peak exactly in the equilateral shape is quasi-single field inflation [@Chen:2009we]. In general models, however, the resulting non-linear parameters are not necessarily shape independent as we discussed in the last section. For our model, since ${f_{\rm NL}}$ is shape independent, the tightest observational constraint comes from ${f_{\rm NL}}^{\text{(local)}}$ above if the gravitationally induced interaction becomes relevant on CMB scales.
It is also convenient to express ${{\cal P}}_{{\cal C}}$ in terms of fractions of the isocurvature perturbation and cross-correlation, defined as[^1] $$\label{eq:def_fractions}
\beta_{{\cal I}}\equiv \frac{{{\cal P}}_{{\cal I}}}{{{\cal P}}_{{\cal R}}}
\quad \text{and} \quad
\beta_{{\cal C}}\equiv \frac{{{\cal P}}_{{\cal C}}}{\sqrt{{{\cal P}}_{{\cal R}}{{\cal P}}_{{\cal I}}}} \, .$$ For generic cold dark matter isocurvature perturbation, at $k = 0.002 \text{Mpc}^{-1}$ the upper bound from Planck TT, TE, EE + lowP + WP is $\beta_{{\cal I}}\lesssim 0.021$ [@Ade:2015lrj]. On the other hand, the constraint on $\beta_{{\cal C}}$ from Planck TT, TE, EE + lowP + WP is $-0.07\lesssim\beta_{{\cal C}}\lesssim 0.21$ [@Ade:2015lrj]. Recent study has shown $|\beta_{{\cal C}}|$ needs to be of ${{\cal O}}(0.1)$ for forthcoming CMB experiments to be sensitive to the isocurvature cross-correlation [@Kadota:2014hpa]. With the definitions , we can see how the observables are correlated in general $${f_{\rm NL}}^{\text{(self2)}} \sim \beta_{{\cal C}}^3\beta_{{\cal I}}^{3/2}{{\cal P}}_{{\cal R}}^{5/2}
\quad \text{and} \quad
{f_{\rm NL}}^{\text{(cross2)}} \sim \beta_{{\cal C}}^2\beta_{{\cal I}}{{\cal P}}_{{\cal R}}^{1/2} \, .$$
Consistency checks {#sec:consistency}
------------------
There are implicit non-trivial constraints for the perturbation theory to remain valid, which also constrain the conditions for realising a large ${f_{\rm NL}}$ given by . In order to treat the transfer vertex $H_2^I$ as a small perturbation, the correction to the curvature perturbation power spectrum ${{\cal P}}_{{\cal R}}$ due to $H_2^I$ must be subleading compared to the leading piece . Explicitly, the correction is given by $$\label{eq:deltaPR}
\Delta{{\cal P}}_{{\cal R}}(k) = \left( -\frac{\pi\alpha}{2r_0H^2} \right)^2 {{\cal D}}{{\cal P}}_{{\cal R}}(k) \, ,$$ where ${{\cal D}}$ is an integral involving the Hankel functions defined as $$\label{eq:calD}
{{\cal D}}\equiv \Re\left\{\int_0^\infty\frac{{\rm d}x_1}{x_1}\left[H_{\nu_\phi}^{(1)}(x_1) + H_{\nu_\phi}^{(2)}(x_1) \right] H_{\nu_\theta}^{(1)}(x_1) \int_{x_1}^{\infty} \frac{{\rm d}x_2}{x_2} H_{\nu_\phi}^{(2)}(x_2) H_{\nu_\theta}^{(2)}(x_2)\right\} \sim \frac{(\log x_c)^2}{2!} \, ,$$ with $x_c$ being the lower cutoff, so that approximately ${{\cal D}}\sim {{\cal A}}^2$. Thus for perturbative calculations to be valid, from we require $$\label{eq:condition_2point}
\left| \frac{\pi\alpha}{2r_0H^2} \right| \lesssim {{\cal O}}(1) \, .$$ Notice that this also means ${{\cal P}}_{{\cal C}}\lesssim \sqrt{{{\cal P}}_{{\cal R}}{{\cal P}}_{{\cal I}}}$. For detailed derivations, see Appendix B.
From , the non-linear parameter ${f_{\rm NL}}^{\text{(self2)}}$ given in terms of the model parameter scales as $\alpha^3\beta/(r_0^6H^8)$. Since the coefficients $\alpha$ and $\beta$ are in fact related by $\beta = -(4\phi_0/3)\alpha$ in our model, by using and taking $r_0=f_a$, we can see ${f_{\rm NL}}^{\text{(self2)}}$ is as large as ${{\cal O}}(1)$ only if $$\label{eq:fnl_self_large_general}
\frac{\phi_0{m_{\rm Pl}}}{f_a^2} \gg 1 \, .$$ This can be achieved by setting $\phi_0 \sim {{\cal O}}({m_{\rm Pl}})$ and $f_a\ll {{\cal O}}({m_{\rm Pl}})$. Similarly, the coupling $\gamma$ is related to the mass of the PQ field $m_\theta$. The massless axion limit, i.e. $m_\theta^2/H^2 \rightarrow 0$, thus corresponds to $$\label{eq:condition_m_theta}
\frac{\phi_0\gamma}{r_0^2H^2} \ll 1 \, .$$ From , the non-linear parameter ${f_{\rm NL}}^{\text{(cross2)}}$ given in terms of the model parameter scales as $\alpha^2\gamma/(r_0^4H^6)$. As a result, we can see ${f_{\rm NL}}^{\text{(cross2)}}\sim {{\cal O}}(1)$ only if $$\label{eq:fnl_cross_large_general}
\frac{{m_{\rm Pl}}}{\phi_0} \gg 1 \, ,$$ which is possible only if $\phi_0$ is sub-Planckian. We can see the two conditions and are mutually exclusive in general. We therefore conclude only one of them can be made large and the resulting non-linear parameter ${f_{\rm NL}}$ either scales as ${{\cal P}}_{{\cal C}}^3$ or ${{\cal P}}_{{\cal C}}^2\sqrt{{{\cal P}}_{{\cal R}}}$. These results appear mainly due to the fact that the quadratic and cubic interaction Hamiltonians all arise from the same gravitationally induced interaction term .
Our results also apply to more general cases where the gravitationally induced interaction between the inflaton $\phi$ and the PQ field $\Phi$ is of the form $$g \frac{\phi^m\Phi^n}{{m_{\rm Pl}}^{m+n-4}} \, .$$ For $m\geq 2$, the cross-interaction $\phi^2\theta$ is also present, and contributes to ${f_{\rm NL}}$ which is directly correlated with $\beta_{{\cal C}}$, $${f_{\rm NL}}^{\text{(cross1)}} \sim \beta_{{\cal C}}\, .$$ Whether or not this contribution can be comparable to ${f_{\rm NL}}^{\text{(self2)}}$ and/or ${f_{\rm NL}}^{\text{(cross2)}}$ depends upon $\phi_0$, which is model-dependent. The relative size between ${f_{\rm NL}}^{\text{(cross1)}}$ and ${f_{\rm NL}}^{\text{(self2)}}$ depends on the model of inflation with ${f_{\rm NL}}^{\text{(cross1)}}/{f_{\rm NL}}^{\text{(self2)}}\sim {m_{\rm Pl}}/\phi_0$, whereas for ${f_{\rm NL}}^{\text{(cross2)}}$ the ratio go as $\sim\tan(n\theta_0)$.
Conclusions {#sec:conc}
===========
In this article, we have studied the correlations between primordial non-Gaussianity and isocurvature perturbation due to horizon-crossing interactions on general ground. In the presence of explicit couplings between the inflaton and another light field that later produces isocurvature perturbation, the bispectrum of the curvature perturbation receives contributions from the conversion of isocurvature perturbation. These contributions can be written in terms of the cross-correlation power spectrum. Taking the gravitationally induced coupling of an axion-like field as an explicit example, we have shown that the primordial bispectrum correlates with the cross-correlation power spectrum as $B_{{\cal R}}\sim {{\cal P}}_{{\cal C}}^3$ or $B_{{\cal R}}\sim {{\cal P}}_{{\cal C}}^2\sqrt{{{\cal P}}_{{\cal R}}}$, depending on the inflation model. If we do observe primordial non-Gaussianity on smaller than CMB scales in future experiments, we might be able to say something about or even put constraints on inflation models where the inflaton interacts with axion-like fields in the early universe.
Besides the standard cold inflation scenario we have considered here, isocurvature perturbations can also arise naturally in warm inflation models via warm baryogenesis [@warm_iso]. In such scenario, one should consider an inflaton-fluid system instead. We hope to investigate the possible correlations between the resulting isocurvature perturbation and primordial non-Gaussianity in such scenario in the future.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We would like to thanks the referee for his/her valuable comments. We acknowledge the Max-Planck-Gesellschaft, the Korea Ministry of Education, Science and Technology, Gyeongsangbuk-Do and Pohang City for the support of the Independent Junior Research Group at the Asia Pacific Center for Theoretical Physics. This work is also supported in part by a Starting Grant through the Basic Science Research Program of the National Research Foundation of Korea (2013R1A1A1006701).
Appendix {#appendix .unnumbered}
========
A. Computation of the two-point statistics {#a.-computation-of-the-two-point-statistics .unnumbered}
------------------------------------------
Given the model Lagrangian , we can work out the corresponding Hamiltonian at appropriate order. At quadratic order, assuming slow-roll and the off-diagonal term of the mass matrix is small, we can take the free field Hamiltonian density, ${{\cal H}}_{\rm free}$ , and the interaction Hamiltonian density, ${{\cal H}}^I_2$ ,as $$\begin{aligned}
\label{eq:free_int_Hamiltonian}
{{\cal H}}_{\rm free} = & \frac{a^3}{2}\dot{\phi}^2 + \frac{a}{2} (\nabla\phi)^2 + \frac{a^3r_0^2}{2}\dot{\theta}^2 + \frac{a r_0^2}{2}(\nabla\theta)^2 + a^3\delta V_{\rm inf} - 8a^3 g \left(\frac{\phi_0r_0^4}{{m_{\rm Pl}}}\right)\cos(4\theta_0)\theta^2 \,, \nonumber \\
{{\cal H}}^I_2 = & -2a^3 g\frac{r_0^4}{{m_{\rm Pl}}} \sin(4\theta_0) \phi\theta = -a^3 \alpha \phi\theta \,,\end{aligned}$$ where $\alpha$ is defined as in . The equations of motion of the free fields $\phi$ and $\theta$ follow from ${{\cal H}}_{\rm free}$.
To compute the two-point correlation functions, we first promote the fields to operators and decompose them into annihilation and creation operators in Fourier space $$\begin{aligned}
\label{eq:field_perturbation_fourier_operator}
\hat{\phi}_{\textbf{k}}= & \hat{a}_{\textbf{k}}u_{\textbf{k}}+ \hat{a}_{-{\textbf{k}}}^{\dagger} u^*_{\textbf{k}}\,, \nonumber \\
\hat{\theta}_{\textbf{k}}= & \hat{b}_{\textbf{k}}v_{\textbf{k}}+ \hat{b}_{-{\textbf{k}}}^{\dagger} v^*_{\textbf{k}}\,,\end{aligned}$$ where $ \hat{a}_{\textbf{k}}$ and $ \hat{b}_{\textbf{k}}$ satisfying the commutation relations $$\begin{aligned}
\label{eq:operator_commutation}
\left[\hat{a}_{\textbf{k}}, \hat{a}_{{\textbf{k}}'}^{\dagger} \right] = \left[\hat{b}_{\textbf{k}}, \hat{b}_{{\textbf{k}}'}^{\dagger} \right] = (2\pi)^3\delta^{(3)}({\textbf{k}}-{\textbf{k}}') \,,\end{aligned}$$ and zero otherwise. The mode functions $u_{\textbf{k}}$ and $v_{\textbf{k}}$ satisfy the following equations $$\begin{aligned}
\label{eq:EOM_field_perturbation}
u_{\textbf{k}}'' - \frac{2}{\tau} u_{{\textbf{k}}}' + \left(k^2 + \frac{m_\phi^2}{H^2\tau^2}\right) u_{\textbf{k}}=& 0 \,, \nonumber \\
v_{\textbf{k}}'' - \frac{2}{\tau} v_{{\textbf{k}}}' + \left(k^2 + \frac{m_\theta^2}{H^2\tau^2}\right) v_{\textbf{k}}=& 0 \,,\end{aligned}$$ where $$\label{eq:mass_inflaton_axion}
m_\phi^2 \equiv \frac{\partial^2 V_{\rm inf}}{\partial\phi^2} \,\,\, {\rm and} \,\,\, m_\theta^2 \equiv -8g\frac{\phi_0}{{m_{\rm Pl}}}r_0^2 \cos(4\theta_0) \,.$$ Assuming the last terms in are small such that both fields behave as massless fields up to slow-roll approximation, we find the solutions of the mode functions $$\begin{aligned}
\label{eq:mode_func_soln}
u_{\textbf{k}}(\tau) =& -i \exp\left[i\left(\nu_\phi + \frac{1}{2} \right)\frac{\pi}{2}\right] \frac{\sqrt{\pi}}{2}H(-\tau)^{3/2}H^{(1)}_{\nu_\phi} (-k\tau) \\
v_{\textbf{k}}(\tau) =& -\frac{i}{r_0} \exp\left[i\left(\nu_\theta + \frac{1}{2} \right)\frac{\pi}{2}\right]\frac{\sqrt{\pi}}{2}H(-\tau)^{3/2}H^{(1)}_{\nu_\theta} (-k\tau) \end{aligned}$$ with $\nu_\phi \equiv \sqrt{9/4 - m_\phi^2/H^2}$ and similarly for $\nu_\theta$, for $0\leq \nu_j\leq 3/2$. Here $H^{(1)}_\nu$ is the Hankel function of the first kind.
The correlation functions can then be computed using the in-in formalism [@in-in]. Using the commutator form, the expectation values of some operators $\hat{O}(t)$ at the time $t$ is given by [@Weinberg:2005vy] $$\begin{aligned}
\label{eq:in_in_correlation_commutator}
\left\langle \hat{O} (t) \right\rangle = & \left\langle 0\bigg| \hat{O} (t) \bigg| 0 \right\rangle + i \int_{t*}^t {\rm d}t_1 \left\langle 0\bigg| \left[H_I(t_1), \hat{O} (t)\right] \bigg| 0 \right\rangle \nonumber \\
& + i^2 \int_{t*}^t {\rm d}t_1 \int_{t*}^{t_1}{\rm d}t_2 \left\langle 0\bigg| \left[ H_{I}(t_1), \left[H_I(t_2), \hat{O} (t)\right] \right] \bigg| 0 \right\rangle + O(H_{I}^{3})\end{aligned}$$ where $\left.|0\right\rangle$ is the interaction vacuum at some initial time $t_*$. The resulting power spectra - can be found by inserting the interaction into for $\hat{O}=\left\{\hat{\phi}_{{\textbf{k}}_1}\hat{\phi}_{{\textbf{k}}_2}, \hat{\theta}_{{\textbf{k}}_1}\hat{\theta}_{{\textbf{k}}_2}, \hat{\phi}_{{\textbf{k}}_1}\hat{\theta}_{{\textbf{k}}_2}\right\}$. For instance, derivations of ${{\cal P}}_{{\cal C}}$ and ${{\cal A}}$ can be found in [@Kadota:2014hpa].
B. Computation of the correction to the leading order ${{\cal P}}_{{\cal R}}$ {#b.-computation-of-the-correction-to-the-leading-order-cal-p_cal-r .unnumbered}
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In section \[sec:consistency\], we argue that the condition have to be satisfied for consistency in order to treat ${{\cal H}}^I_2$ as a small perturbation compared to ${{\cal H}}_{\rm free}$. As stated in the section, this comes from the fact that the non-zero contribution to the inflaton power spectrum, or in other words, ${{\cal P}}_{{\cal R}}$, coming from ${{\cal H}}^I_{2}$ should be subleading compared to the free-field contribution .
To see, we compute the correction to ${{\cal P}}_{{\cal R}}$ due to ${{\cal H}}^I_{(2)}$ using the commutator form of the in-in formalism in a similar fashion as in [@calculation]. Labeling this correction by $\Delta{{\cal P}}_{{\cal R}}$, this is of order ${{\cal O}}[({{\cal H}}^{I}_{2})^2]$ and is given by
$$\begin{aligned}
\label{eq:inflaton_power_spectrum_correction}
\Delta{{\cal P}}_{{\cal R}}(k) = &\int_{t_*}^t {\rm d}t_1\int_{t_*}^t {\rm d}t_2\left\langle 0 \left | H^I_{2}(t_1)\hat{\phi}_{{\textbf{k}}_1}(t)\hat{\phi}_{{\textbf{k}}_2}(t)H^I_{2}(t_2)\right |0\right\rangle \nonumber \\
& -2\Re\left[ \int_{t_*}^t {\rm d}t_1\int_{t_*}^{t_1} {\rm d}t_2\left\langle 0 \left |\hat{\phi}_{{\textbf{k}}_1}(t)\hat{\phi}_{{\textbf{k}}_2}(t)H^I_{2}(t_1
)H^I_{2}(t_2)\right |0\right\rangle \right] \, .\end{aligned}$$
Assuming inflationary background such that $a \sim (-1/H\tau)$ and $H\approx {\rm const.}$ and taking the large wavelength limit after horizon-exit, we finally arrive $$\label{eq:inflaton_power_spectrum_correction_final}
\Delta{{\cal P}}_{{\cal R}}(k) = \left|u_{\textbf{k}}(0)\right|^2\frac{k^3}{2\pi^2} \left(\frac{\pi\alpha}{2r_0H^2} \right)^2 {{\cal D}}= \left( \frac{\pi\alpha}{2r_0H^2} \right)^2 {{\cal D}}{{\cal P}}_{{\cal R}}(k) \, , \\$$ where ${{\cal D}}$ is an integral involving the Hankel functions defined as $$\label{eq:calD}
{{\cal D}}\equiv \Re\left\{\int_0^\infty\frac{{\rm d}x_1}{x_1}\left[H_{\nu_\phi}^{(1)}(x_1) + H_{\nu_\phi}^{(2)}(x_1) \right] H_{\nu_\theta}^{(1)}(x_1) \int_{x_1}^{\infty} \frac{{\rm d}x_2}{x_2} H_{\nu_\phi}^{(2)}(x_2) H_{\nu_\theta}^{(2)}(x_2)\right\} \, ,$$ In the massless limit where $\nu_\phi, \nu_\theta \approx 3/2$, ${{\cal D}}$ reduces to $$\label{eq:inflaton_power_spectrum_correction_integral_massless}
{{\cal D}}= \left(\frac{4}{\pi^2}\right) \Re\left\{\int_0^\infty \frac{{\rm d}x_1}{x_1^{4}} \left[ (1-i x_1)e^{i x_1} - {\rm c.c.}\right] (1-ix_1)e^{ix_1} \int_{x_1}^{\infty} \frac{{\rm d}x_2}{x_2^{4}} (1+ix_2)^2e^{-2ix_2} \right\} \,,$$ and is logarithmically divergent in the IR limit. Here ${\rm c.c.}$ stands for complex conjugate. To regularsie this, we introduce a lower cut-off $x_c$ in the above integral. Then we can see ${{\cal D}}$ approximately scale as $$\label{eq:inflaton_power_spectrum_correction_integral_massless_IR}
{{\cal D}}\sim \frac{(\log x_c)^2}{2!} \sim {{\cal A}}^2 \,.$$ For the correction to be small such that $\Delta{{\cal P}}_{{\cal R}}(k) / {{\cal P}}_{{\cal R}}(k) \ll {{\cal O}}(1)$, we therefore generically require $|\pi\alpha/2r_0 H^2| \ll {{\cal O}}(1)$, i.e. the condition .
C. Computation of the bispectrum {#c.-computation-of-the-bispectrum .unnumbered}
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The resulting primordial bispectrum can be computed in a similar fashion as the two-point functions. Assuming slow-roll, at cubic order, the corresponding interaction Hamiltonian $H_3$ dominated by the contribution from the gravitationally induced interaction and is given by $$\begin{aligned}
\label{eq:action_cubic_interaction}
H_3 = & \int d^3x {{\cal H}}_3 = a^3\frac{g r_0^4}{{m_{\rm Pl}}} (2\pi)^{(3)} \delta^{(3)}({\textbf{k}}_1 + {\textbf{k}}_2 + {\textbf{k}}_3)
\int \frac{d^3k_1}{(2\pi)^3}\int \frac{d^3k_2}{(2\pi)^3} \int \frac{d^3k_3}{(2\pi)^3} \nonumber \\
\times & \left[ \frac{16}{3} \phi_0 \cos(4\theta_0) \hat{\theta}_{{\textbf{k}}_1}\hat{\theta}_{{\textbf{k}}_2}\hat{\theta}_{{\textbf{k}}_3} - \frac43 \sin(4\theta_0) \hat{\phi}_{{\textbf{k}}_1}\hat{\theta}_{{\textbf{k}}_2}\hat{\theta}_{{\textbf{k}}_3} + {\rm perm.} \right] \,, \nonumber \\\end{aligned}$$ where ${\rm perm.}$ denotes permutation over momenta ${\textbf{k}}_j$ of the second term in the square bracket. Splitting $H_3$ in terms of the form of the interaction, we therefore arrive at and .
### C1. Contribution from $H_3^\text{(self2)}$ {#c1.-contribution-from-h_3textself2 .unnumbered}
Expressed in the nested commutator form of the in-in formalism, the leading contribution to the fully connected inflaton three-point function $\left\langle\phi^3 \right\rangle$ from the axion self-interaction Hamiltonian $H_3^\text{(self2)}$ is of cubic order in $H_2^I$ and is given by $$\label{eq:3point_fun_inflaton_cross}
\left\langle \phi^3 \right\rangle^\text{(self2)} = \int_{t_0}^{t} {\rm d}t_1 \int_{t_0}^{t_1} {\rm d}t_2 \int_{t_0}^{t_2} {\rm d}t_3\int_{t_0}^{t_3} {\rm d}t_4 \left\langle \left[H^I(t_4),\left[H^I(t_3),\left[H^I(t_2),\left[H^I(t_1),\hat{\phi}_I(t)^3 \right]\right]\right]\right]\right\rangle \,,$$ where $H^I$ corresponds to an interaction Hamiltonian, with one being $H_3^\text{(self2)}$ and the others being $H_2^I$. The overall contribution is therefore [@calculation] $$\begin{aligned}
\label{eq:3point_fun_inflaton}
\left\langle \phi^3 \right\rangle^\text{(self2)} = & 12\alpha^3\beta \left[u_{{\textbf{k}}_1}(0)u_{{\textbf{k}}_2}(0)u_{{\textbf{k}}_3}(0)\right] \Re \left[ \int_{-\infty}^0 d\tau_1 \int_{-\infty}^{\tau_1} d\tau_2 \int_{-\infty}^{\tau_2} d\tau_3\int_{-\infty}^{\tau_3} d\tau_4 \prod_{i=1}^4 a^{4}(\tau_i)\right. \nonumber \\
&\times \left.\left(A + B + C\right)\right] (2\pi)^3\delta^{(3)}(\sum_i {\textbf{k}}_i) + 5 \, {\rm perm.} \, ,\end{aligned}$$ where the terms $A$, $B$ and $C$ are $$\begin{aligned}
\label{eq:3point_fun_inflaton_A_term}
A =&-[u_{{\textbf{k}}_1}(\tau_1) - {\rm c.c.}][v_{{\textbf{k}}_1}(\tau_1)v_{{\textbf{k}}_1}^*(\tau_2) - {\rm c.c.}][v_{{\textbf{k}}_3}(\tau_2)v_{{\textbf{k}}_3}^*(\tau_4)u_{{\textbf{k}}_3}(\tau_4) - {\rm c.c.}]
v_{{\textbf{k}}_2}(\tau_2)v_{{\textbf{k}}_2}^*(\tau_3)u_{{\textbf{k}}_2}^*(\tau_3) \, , \\
\label{eq:3point_fun_inflaton_B_term}
B =&-[u_{{\textbf{k}}_1}(\tau_1) - {\rm c.c.}][u_{{\textbf{k}}_2}(\tau_2) - {\rm c.c.}][v_{{\textbf{k}}_1}^*(\tau_1)v_{{\textbf{k}}_2}^*(\tau_2)v_{{\textbf{k}}_1}(\tau_3)v_{{\textbf{k}}_2}(\tau_3) - {\rm c.c.}]
v_{{\textbf{k}}_3}(\tau_3)v_{{\textbf{k}}_3}^*(\tau_4)u_{{\textbf{k}}_3}^*(\tau_4) \, , \\
\label{eq:3point_fun_inflaton_C_term}
C =&[u_{{\textbf{k}}_1}(\tau_1) - {\rm c.c.}][u_{{\textbf{k}}_2}(\tau_2) - {\rm c.c.}][u_{{\textbf{k}}_3}(\tau_3) - {\rm c.c.}]
v_{{\textbf{k}}_1}^*(\tau_1)v_{{\textbf{k}}_2}^*(\tau_2)v_{{\textbf{k}}_3}^*(\tau_3)v_{{\textbf{k}}_1}(\tau_4)v_{{\textbf{k}}_2}(\tau_4)v_{{\textbf{k}}_3}(\tau_4) \, , \end{aligned}$$ corresponding to replacing $H^I(t_2)$, $H^I(t_3)$ or $H^I(t_4)$ with the axion self-interaction Hamiltonian $H_3^\text{(self2)}$ respectively. Note that replacing $H^I(t_1)$ with $H_3^\text{(self2)}$ gives zero contribution since $H_3^\text{(self2)}\propto \hat{\theta}_I^3$, which does not contract with any of the external legs $\hat{\phi}_I(t)$. Here we also have used the fact that $u_{\textbf{k}}(0)$ is real. Plugging in the solutions for the mode functions $u$ and $v$, - written in terms of the Hankel functions are $$\begin{aligned}
\label{eq:3point_fun_inflaton_3_terms_Hankel}
A =& - {{\cal W}}\left(-\tau_1\right)^3\left(-\tau_2\right)^{9/2}\left(-\tau_3\right)^3\left(-\tau_4\right)^3 \Re\left[H_{\nu_\phi}^{(1)}(-k_1\tau_1)\right] \Im \left[H_{\nu_\theta}^{(1)}(-k_1\tau_1)H_{\nu_\theta}^{(2)}(-k_1\tau_2)\right] \nonumber \\
\times &
\Re\left[H_{\nu_\theta}^{(1)}(-k_3\tau_2)H_{\nu_\theta}^{(2)}(-k_3\tau_4)H_{\nu_\phi}^{(2)}(-k_3\tau_4)\right]
H_{\nu_\theta}^{(1)}(-k_2\tau_2)H_{\nu_\theta}^{(2)}(-k_2\tau_3)H_{\nu_\phi}^{(2)}(-k_2\tau_3) \, , \\
B =& {{\cal W}}\left(-\tau_1\right)^3\left(-\tau_2\right)^{3}\left(-\tau_3\right)^{9/2}\left(-\tau_4\right)^3\Re\left[H_{\nu_\phi}^{(1)}(-k_1\tau_1)\right] \Re\left[H_{\nu_\phi}^{(1)}(-k_2\tau_2)\right] \nonumber \\
\times &
\Im \left[H_{\nu_\theta}^{(2)}(-k_1\tau_1)H_{\nu_\theta}^{(2)}(-k_2\tau_2)H_{\nu_\theta}^{(1)}(-k_1\tau_3)H_{\nu_\theta}^{(1)}(-k_2\tau_3)\right]
H_{\nu_\theta}^{(1)}(-k_3\tau_3)H_{\nu_\theta}^{(2)}(-k_3\tau_4)H_{\nu_\phi}^{(2)}(-k_3\tau_4) \, ,
\\
C =& -i{{\cal W}}\left(-\tau_1\right)^3\left(-\tau_2\right)^{3}\left(-\tau_3\right)^{3}\left(-\tau_4\right)^{9/2} \Re\left[H_{\nu_\phi}^{(1)}(-k_1\tau_1)\right] \Re\left[H_{\nu_\phi}^{(1)}(-k_2\tau_2)\right]\Re\left[H_{\nu_\phi}^{(1)}(-k_3\tau_3)\right] \nonumber \\
\times &
H_{\nu_\theta}^{(2)}(-k_1\tau_1)H_{\nu_\theta}^{(2)}(-k_2\tau_2)H_{\nu_\theta}^{(2)}(-k_3\tau_3)
H_{\nu_\theta}^{(1)}(-k_1\tau_4)H_{\nu_\theta}^{(1)}(-k_2\tau_4)H_{\nu_\theta}^{(1)}(-k_3\tau_4) \, ,\end{aligned}$$ where we have defined ${{\cal W}}\equiv 8/r_0^6(H\sqrt{\pi}/2)^9$. Assuming ${{\cal R}}$ is sourced purely by the inflaton fluctuations, the resulting curvature perturbation bispectrum is simply $$\label{eq:3point_fun_R}
\left\langle {{\cal R}}({\textbf{k}}_1){{\cal R}}({\textbf{k}}_2){{\cal R}}({\textbf{k}}_3) \right\rangle = -\left(\frac{H}{\dot{\phi}_0}\right)^3 \left\langle\phi({\textbf{k}}_1)\phi({\textbf{k}}_2)\phi({\textbf{k}}_3) \right\rangle \, ,$$ where the corresponding bispectrum $B_{{\cal R}}^\text{(self2)}(k_1,k_2,k_3)$ and the dimensionless shape function ${f_{\rm NL}}^\text{(self2)}(k_1,k_2,k_3)$ as defined in are given by $$\begin{aligned}
\label{eq:3point_fun_R_fnl_gen}
\left\langle {{\cal R}}({\textbf{k}}_1){{\cal R}}({\textbf{k}}_2){{\cal R}}({\textbf{k}}_3) \right\rangle = & (2\pi)^3 \delta^{(3)}\left(\sum_i {\textbf{k}}_i\right) B_{{\cal R}}^\text{(self2)}(k_1,k_2,k_3) \nonumber \\
= & (2\pi)^3 \delta^{(3)}\left(\sum_i {\textbf{k}}_i\right) \left(\frac{3}{10} \right) {f_{\rm NL}}^\text{(self2)}(k_1,k_2,k_3) \frac{k_1^3+k_2^3+k_3^3}{(k_1k_2k_3)^3} (2\pi)^4{{\cal P}}_{{\cal R}}^2 \, .\end{aligned}$$ Here $H$ and $\dot{\phi}_0$ are evaluated at horizon exit. Factorising ${f_{\rm NL}}^\text{(self2)}(k_1,k_2,k_3)$ in terms of the shape dependent and independent parts, we have $${f_{\rm NL}}(k_1,k_2,k_3)^\text{(self2)} = \mathcal{F}^\text{(self2)} s^\text{(self2)}(k_1,k_2,k_3) \frac{(XYZ)^{3/2}}{X^3+Y^3+Z^3} \, .$$ For convenience, we have introduced a normalisation wavenumber $K$ and have defined $x_i \equiv -K\tau_i$, $X\equiv k_1/K$, $X\equiv k_2/K$ and $Z\equiv k_3/K$ in the above expression. In the massless limit $\nu_\theta$, $\nu_\phi\approx 3/2$, $\mathcal{F}^\text{(self2)}$ is $$\mathcal{F}^\text{(self2)} = \left(\frac{\dot{\phi}_0}{H}\right) \left(\frac{5\alpha^3\beta}{H^8r_0^6} \right) 2^{-3/2}\pi^{9/2} \, ,$$ and the dimensionless shape function $s^\text{(self2)}(k_1,k_2,k_3)$ is $$\begin{aligned}
\label{eq:3point_fun_R_shape_fun_self}
& s^\text{(self2)}(k_1,k_2,k_3) \nonumber \\
\equiv & \Re\left\{ \int_{0}^{\infty} \frac{{\rm d}x_1}{x_1} \int_{x_1}^{\infty} \frac{{\rm d}x_2}{x_2} \int_{x_2}^{\infty} \frac{{\rm d}x_3}{x_3} \int_{x_3}^{\infty} \frac{{\rm d}x_4}{x_4} \,\, \Re\left[H_{\nu_\phi}^{(1)}(X x_1)\right] \nonumber \right. \\
\times &\left[ -\left(x_2\right)^{3/2} \Im \left[H_{\nu_\theta}^{(1)}(X x_1)H_{\nu_\theta}^{(2)}(X x_2)\right]
\Re \left[H_{\nu_\theta}^{(1)}\left(Z x_2\right)H_{\nu_\theta}^{(2)}\left(Z x_4\right)H_{\nu_\phi}^{(2)}\left(Z x_4\right)\right] \right. \nonumber \\
\times & H_{\nu_\theta}^{(1)}\left(Y x_2\right)H_{\nu_\theta}^{(2)}\left(Y x_3\right)H_{\nu_\phi}^{(2)}\left(Y x_3\right) \nonumber \\
+ & \left(x_3\right)^{3/2} \Re \left[H_{\nu_\phi}^{(1)}\left(Y x_2\right)\right]
\Im \left[H_{\nu_\theta}^{(2)}(X x_1)H_{\nu_\theta}^{(2)}\left(Y x_2\right) H_{\nu_\theta}^{(1)}(X x_3)H_{\nu_\theta}^{(1)}\left(Y x_3\right)\right] \nonumber \\
\times & H_{\nu_\theta}^{(1)}\left(Z x_3\right)H_{\nu_\theta}^{(2)}\left(Z x_4\right)H_{\nu_\phi}^{(2)}\left(Z x_4\right) \nonumber \\
- & i \left(x_4\right)^{3/2}\Re \left[H_{\nu_\phi}^{(1)}\left(Y x_2\right)\right]\Re\left[H_{\nu_\phi}^{(1)}\left(Z x_3\right)\right] \nonumber \\
\times & \left.\left. H_{\nu_\theta}^{(2)}(X x_1)H_{\nu_\theta}^{(2)}\left(Y x_2\right)H_{\nu_\theta}^{(2)}\left(Z x_3\right)
H_{\nu_\theta}^{(1)}(X x_4)H_{\nu_\theta}^{(1)}\left(Y x_4\right)H_{\nu_\theta}^{(1)}\left(Z x_4\right) \right] \right\} + 5 \,\, {\rm perm.} \end{aligned}$$ Here $s^\text{(self2)}(k_1,k_2,k_3)$ diverges in the IR in the massless limit. Introducing a lower cut-off $x_c$ as before, one finds $$s^\text{(self2)}(k_1,k_2,k_3) \approx \left(\frac{44}{243}\right) \pi^{-9/2} \left(\log x_c\right)^4 \frac{X^3+Y^3+Z^3}{(XYZ)^{3/2}} \, ,$$ to leading order. In the following, we briefly summarise how to compute the above result.
To compute the leading order term in the IR limit, we take the following asymptotic form of the Hankel function in the $x\ll 1$ limit $$\begin{aligned}
\label{eq:asymptotic_form_Hankel}
H_{3/2}^{(1)}(x) \rightarrow & -i \frac{2^{3/2}\Gamma(3/2)}{\pi} x^{-3/2} - i \frac{2^{-1/2}\Gamma(3/2)}{\pi(1/2)} x^{1/2} \nonumber \\
&+ \left(- i \frac{\cos(3\pi/2)\Gamma(-3/2)}{2^{3/2}\pi} + \frac{1}{2^{3/2}\Gamma(5/2)}\right) x^{3/2} + ... \end{aligned}$$ whereas that of $H_{3/2}^{(2)}(x)$ is the complex conjugate of . Here we can see the real and the imaginary parts scale as ${{\cal O}}(x^{3/2})$ and ${{\cal O}}(x^{-3/2})$ respectively at leading order. Using this, we can see how the integrand in scales with $x_c$. For the second line in , at leading order in the IR limit, it scales as $$\label{eq:integrand_asympt_form_first_line}
\prod_{i=1}^{4} (x_i)^{-1} \Re \left[H_{\nu_\phi}^{(1)}(X x_1)\right] \rightarrow \prod_{i=1}^{4} (x_i)^{-1} \left[\frac{1}{2^{3/2}\Gamma(5/2)}\right](X x_1)^{3/2} + ...$$ whereas for the third line, the leading order piece is $$\begin{aligned}
\label{eq:integrand_asympt_form_second_line}
& -\left(x_2\right)^{3/2} \Im \left[H_{\nu_\theta}^{(1)}(X x_1)H_{\nu_\theta}^{(2)}(X x_2)\right]
\Re \left[H_{\nu_\theta}^{(1)}\left(Z x_2\right)H_{\nu_\theta}^{(2)}\left(Z x_4\right)H_{\nu_\phi}^{(2)}\left(Z x_4\right)\right] \nonumber \\
\,\,\, &\rightarrow \left[\frac{\Gamma(3/2)}{\Gamma(5/2)\pi}\right]^2 \frac{2^{5/2}\Gamma(3/2)}{\pi} \left(\frac{x_2}{x_1}\right)^{3/2} Z^{-3/2} + ...\end{aligned}$$ Here we have taken $H_{\nu_\theta}^{(1)}(X x_1)$, $H_{\nu_\theta}^{(1)}\left(Z x_2\right)$ and either $H_{\nu_\theta}^{(2)}\left(Z x_4\right)$ or $H_{\nu_\phi}^{(2)}\left(Z x_4\right)$ to be imaginary. For the fourth line in , it must be real in order to keep the whole integrand real. The leading order term scales as $$\label{eq:integrand_asympt_form_third_line}
H_{\nu_\theta}^{(1)}\left(Y x_2\right)H_{\nu_\theta}^{(2)}\left(Y x_3\right)H_{\nu_\phi}^{(2)}\left(Y x_3\right) \rightarrow 2\left[\frac{\Gamma(3/2)}{\Gamma(5/2)\pi}\right] \left[\frac{2^{3/2}\Gamma(3/2)}{\pi}\right](Y x_2)^{-3/2} + ...$$ Here we have taken $H_{\nu_\theta}^{(1)}\left(Y x_2\right)$ and either $H_{\nu_\theta}^{(2)}\left(Y x_3\right)$ or $H_{\nu_\phi}^{(2)}\left(Y x_3\right)$ to be imaginary. Together the above leading order pieces in the three lines therefore give the following integrand $$\label{eq:integrand_asympt_form_A}
\prod_{i=1}^{4} (x_i)^{-1} \left[\frac{\Gamma(3/2)}{\Gamma(5/2)\pi}\right]^4\left[\frac{\Gamma(3/2)2^{7/2}}{\pi}\right] \left(\frac{X}{YZ}\right)^{3/2} \,,$$ and the corresponding contribution to $s^\text{(self2)}(k_1,k_2,k_3)$ scales as $(\log x_c)^4$ in the IR limit. Taking other choices of the leading order pieces in the Hankel functions give subleading terms. For instance, taking $H_{\nu_\theta}^{(1)}(X x_1)$ to be real and $H_{\nu_\theta}^{(2)}(X x_2)$ to be imaginary instead in the second line, together with the above leading order terms in the first and third lines, the resulting integral scales at most as $(-\log x_c)^3$.
Similarly, for the fifth line in , the leading order piece is $$\begin{aligned}
\label{eq:integrand_asympt_form_fourth_line}
& \left(x_3\right)^{3/2} \Re \left[H_{\nu_\phi}^{(1)}\left(Y x_2\right)\right]
\Im \left[H_{\nu_\theta}^{(2)}(X x_1)H_{\nu_\theta}^{(2)}\left(Y x_2\right) H_{\nu_\theta}^{(1)}(X x_3)H_{\nu_\theta}^{(1)}\left(Y x_3\right)\right] \nonumber \\
\,\,\, &\rightarrow (x_3)^{3/2} \left[\frac{\Gamma(3/2)}{\Gamma(5/2)\pi}\right]^2 \frac{2^{3/2}\Gamma(3/2)}{\pi} (X x_1)^{-3/2} \left[\left( \frac{X}{Y}\right)^{3/2} + \left(\frac{Y}{X}\right)^{3/2} \right] + ...\end{aligned}$$ Here we have taken $H_{\nu_\theta}^{(2)}(X x_1)$, $ H_{\nu_\theta}^{(2)}\left(Y x_2\right)$ and either $H_{\nu_\theta}^{(1)}(X x_3)$ or $H_{\nu_\theta}^{(1)}\left(Y x_3\right)$ to be imaginary. In order for the overall contribution together with the second and fifth lines to the integral to be real, the leading order piece in the sixth line must be real and is given by $$\label{eq:integrand_asympt_form_fifth_line}
H_{\nu_\theta}^{(1)}\left(Z x_3\right)H_{\nu_\theta}^{(2)}\left(Z x_4\right)H_{\nu_\phi}^{(2)}\left(Z x_4\right) \rightarrow \left[\frac{\Gamma(3/2)}{\Gamma(5/2)\pi}\right]\left[\frac{2^{5/2}\Gamma(3/2)}{\pi}\right](Z x_3)^{-3/2} + ...$$ Here we have taken $H_{\nu_\theta}^{(1)}\left(Z x_3\right)$ and either $H_{\nu_\theta}^{(2)}\left(Z x_4\right)$ or $H_{\nu_\phi}^{(2)}\left(Z x_4\right)$ to be imaginary. Together the second, fifth and sixth lines in give the following leading order contribution to $s^\text{(self2)}(k_1,k_2,k_3)$ in the IR limit $$\label{eq:integrand_asympt_form_B}
\prod_{i=1}^{4} (x_i)^{-1} \left[\frac{\Gamma(3/2)}{\Gamma(5/2)\pi}\right]^4\left[\frac{2^{5/2}\Gamma(3/2)}{\pi}\right] \left[\left(\frac{X}{YZ}\right)^{3/2} + \left(\frac{Y}{XZ}\right)^{3/2} \right] \, .$$ Again similarly, the leading order term in the seventh line in is $$\label{eq:integrand_asympt_form_sixth_line}
- i\left(x_4\right)^{3/2} \Re \left[H_{\nu_\phi}^{(1)}\left(Y x_2\right)\right]\Re \left[H_{\nu_\phi}^{(1)}\left(Z x_3\right)\right] \rightarrow - i(x_4)^{3/2} \left[\frac{1}{2^{3/2}\Gamma(5/2)}\right]^2 (Y x_2)^{3/2} (Z x_3)^{3/2} + ...$$ whereas for the eighth line, the leading order term must be imaginary in order to keep the overall contribution real and thus is given by $$\begin{aligned}
\label{eq:integrand_asympt_form_seventh_line}
& H_{\nu_\theta}^{(2)}(X x_1)H_{\nu_\theta}^{(2)}\left(Y x_2\right)H_{\nu_\theta}^{(2)}\left(Z x_3\right)
H_{\nu_\theta}^{(1)}(X x_4)H_{\nu_\theta}^{(1)}\left(Y x_4\right)H_{\nu_\theta}^{(1)}\left(Z x_4\right) \nonumber \\
& \rightarrow i \left[\frac{2^{3/2}\Gamma(3/2)}{\pi}\right]^4 \left[\frac{\Gamma(3/2)}{\Gamma(5/2)\pi}\right] (XYZ)^{-3/2} ( x_1x_2x_3)^{-3/2} \left[\left( \frac{X}{YZ}\right)^{3/2} + \left( \frac{Y}{XZ}\right)^{3/2} + \left( \frac{Z}{XY}\right)^{3/2} \right] + ...\end{aligned}$$ Here we have taken either $H_{\nu_\theta}^{(1)}(X x_4)$, $H_{\nu_\theta}^{(1)}\left(Y x_4\right)$ or $H_{\nu_\theta}^{(1)}\left(Z x_4\right)$ to be real. Together the second, seventh and eighth lines give the following leading order contribution to $s^\text{(self2)}(k_1,k_2,k_3)$ in the IR limit $$\label{eq:integrand_asympt_form_C}
\prod_{i=1}^{4} (x_i)^{-1} \left[\frac{\Gamma(3/2)}{\Gamma(5/2)\pi}\right]^4 \frac{2^{3/2}\Gamma(3/2)}{\pi} \left[\left( \frac{X}{YZ}\right)^{3/2} + \left( \frac{Y}{XZ}\right)^{3/2} + \left( \frac{Z}{XY}\right)^{3/2} \right] \, .$$ After factorising out the common factor $\prod_{i=1}^{4} (x_i)^{-1}$ in , and , we can see the integral goes as $$\label{eq:integral_IR_limit_scaling}
\int_{x_c} {\rm d}x_1 \int_{x_1} {\rm d}x_2 \int_{x_2} {\rm d}x_3 \int_{x_3} {\rm d}x_4 \prod_{j=1}^{4} (x_i)^{-1} \sim \frac{(\log x_c)^4}{4!} \,,$$ assuming the integral vanishes in the UV limit $x_i\rightarrow \infty$. Thus finally, to leading order in the IR limit, $s^\text{(self2)}(k_1,k_2,k_3)$ is given by $$\begin{aligned}
\label{eq:s_self_IR_limit}
s^\text{(self2)}(k_1,k_2,k_3) \underset{\rm IR}{\rightarrow} & (XYZ)^{-3/2}\left[\frac{\Gamma(3/2)}{\Gamma(5/2)\pi}\right]^4\left[\frac{2^{3/2}\Gamma(3/2)}{\pi}\right] \frac{(\log x_c)^4}{4!}\nonumber \\
& \times \left[4X^3 + 2(X^3+Y^3) + (X^3 + Y^3 + Z^3) \right] + \,\,5 \,\, {\rm perm.} \, .\end{aligned}$$ Summing over all permutations, it is not difficult to see $s^\text{(self2)}(k_1,k_2,k_3)$ scales as $(X^3+Y^3+Z^3)/(XYZ)^{3/2}$ and thus ${f_{\rm NL}}(k_1,k_2,k_3)^\text{(self2)}$ is approximately shape independent. Using the results of the cross-correlation spectrum ${{\cal P}}_{{\cal C}}$ and expressing ${f_{\rm NL}}(k_1,k_2,k_3)^\text{(self2)}$ in terms of ${{\cal P}}_{{\cal C}}$, we therefore arrive at .
### C2. Contribution from $H_3^\text{(cross2)}$ {#c2.-contribution-from-h_3textcross2 .unnumbered}
The corresponding leading order contribution to the inflaton bispectrum $\left\langle \phi^3 \right\rangle$ from the axion cross-interaction $H_3^\text{(cross2)}$ is of quadratic order in $H^I_2$. Expressed in commutator form, this contribution is given by $$\label{eq:3point_fun_inflaton_cross}
\left\langle \phi^3 \right\rangle^\text{(cross2)} = \int_{t_0}^{t} {\rm d}t_1 \int_{t_0}^{t_1} {\rm d}t_2 \int_{t_0}^{t_2} {\rm d}t_3 \left\langle\left[H^I(t_3),\left[H^I(t_2),\left[H^I(t_1),\phi_I(t)^3 \right]\right]\right]\right\rangle \,,$$ where one of the $H^I$ is given by $H_3^\text{(cross2)}$. Expanding this, we get $$\begin{aligned}
\label{eq:3point_fun_inflaton_B_expand}
\left\langle \phi^3 \right\rangle^\text{(cross2)} =& -4\alpha^2\gamma \left(\prod_{i=1}^3 u_{k_i}(0)\right)(2\pi)^3\delta^{(3)}\left({\textbf{k}}_1+{\textbf{k}}_2+{\textbf{k}}_3\right) \nonumber \\
\times & \Re \left[-i\int_{-\infty}^{0}{\rm d}\tau_1\int_{-\infty}^{\tau_1}{\rm d}\tau_2\int_{-\infty}^{\tau_2}{\rm d}\tau_3 \prod_{i=1}^3 a^4(\tau_i)\left(D+E+F\right)\right] + 5\,\, {\rm perm.} \,,\end{aligned}$$ where $D$, $E$ and $F$ correspond to contributions coming from replacing $H_I(t_1)$, $H_I(t_2)$ and $H_I(t_3)$ respectively with $H_I^{(3),{\rm cross}}$: $$\begin{aligned}
\label{eq:3point_fun_inflaton_B_terms}
& D = [u_{k_1}(\tau_1) - {\rm {\rm c.c.}}][v_{k_2}^*(\tau_1)v_{k_2}(\tau_2)u_{k_2}(\tau_2) - {\rm c.c.}] v_{k_3}(\tau_1)v_{k_3}^*(\tau_3)u_{k_3}^*(\tau_3) \, , \\
& E = [u_{k_1}(\tau_1) - {\rm c.c.}][v_{k_1}^*(\tau_1)v_{k_1}(\tau_2)u_{k_2}(\tau_2) - {\rm c.c.}] v_{k_3}(\tau_2)v_{k_3}^*(\tau_3)u_{k_3}^*(\tau_3) \, ,\\
& F = [u_{k_1}(\tau_1) - {\rm c.c.}][u_{k_2}(\tau_2) - {\rm c.c.}]v_{k_1}(\tau_1)v_{k_2}(\tau_2)v_{k_1}^*(\tau_3)v_{k_2}^*(\tau_3)u_{k_3}^*(\tau_3) \, ,\end{aligned}$$ Here we have again used the fact that $u_{k}(0)$ is real. Written in terms of the Hankel functions, they are given by $$\begin{aligned}
\label{eq:3point_fun_inflaton_B_terms}
D = & i {{\cal N}}(-\tau_1)^{9/2}(-\tau_2)^{3}(-\tau_3)^{3} \Re \left[H_{\nu_\phi}^{(1)}(-k_1\tau_1)\right] \Re\left[H_{\nu_\theta}^{(2)}(-k_2\tau_1)H_{\nu_\theta}^{(1)}(-k_2\tau_2)H_{\nu_\phi}^{(1)}(-k_2\tau_2)\right] \nonumber \\
\times & H_{\nu_\theta}^{(1)}(-k_3\tau_1)H_{\nu_\theta}^{(2)}(-k_3\tau_3)H_{\nu_\phi}^{(2)}(-k_3\tau_3) \, ,\\
E = & - i {{\cal N}}(-\tau_1)^{3}(-\tau_2)^{9/2}(-\tau_3)^{3} \Re \left[H_{\nu_\phi}^{(1)}(-k_1\tau_1)\right]\Re \left[H_{\nu_\theta}^{(2)}(-k_1\tau_1)H_{\nu_\theta}^{(1)}(-k_1\tau_2)H_{\nu_\phi}^{(1)}(-k_2\tau_2)\right] \nonumber \\
\times & H_{\nu_\theta}^{(2)}(-k_3\tau_2) H_{\nu_\theta}^{(1)}(-k_3\tau_3)H_{\nu_\phi}^{(1)}(-k_3\tau_3) \, ,\\
F = & -i {{\cal N}}(-\tau_1)^{3}(-\tau_2)^{3}(-\tau_3)^{9/2} \Re\left[H_{\nu_\phi}^{(1)}(-k_1\tau_1)\right] \Re\left[H_{\nu_\phi}^{(1)}(-k_2\tau_2)\right] \nonumber \\
\times & H_{\nu_\theta}^{(2)}(-k_1\tau_1)H_{\nu_\theta}^{(2)}(-k_2\tau_2)
H_{\nu_\theta}^{(1)}(-k_1\tau_3)H_{\nu_\theta}^{(1)}(-k_2\tau_3)H_{\nu_\phi}^{(1)}(-k_3\tau_3) \, ,\end{aligned}$$ where ${{\cal N}}\equiv 4/r_0^4(H\sqrt{\pi}/2)^7$. The resulting the curvature bispectrum $B_{{\cal R}}^\text{(cross2)}(k_1,k_2,k_3)$ is simply given by multiplying with $-(H/\dot{\phi}_0)^3$. Similar to the $H_3^\text{(self2)}$ case, we factorise the corresponding shape function ${f_{\rm NL}}^\text{(cross2)}(k_1,k_2,k_3)$ into shape independent and dependent parts as follows $${f_{\rm NL}}(k_1,k_2,k_3)^\text{(cross2)} = \mathcal{F}^\text{(cross2)} s^\text{(cross2)}(k_1,k_2,k_3) \frac{(XYZ)^{3/2}}{X^3+Y^3+Z^3} \, .$$ In the massless limit $\nu_\theta$, $\nu_\phi\approx 3/2$, $\mathcal{F}^\text{(cross2)}$ is given by $$\mathcal{F}^\text{(cross2)} = \left(\frac{\dot{\phi}_0}{H}\right) \left(\frac{\alpha^2\gamma}{H^6r_0^4} \right) \left(\frac{\pi}{2}\right)^{7/2} \, ,$$ whereas $s^\text{(cross2)}(k_1,k_2,k_3)$ is $$\begin{aligned}
\label{eq:3point_fun_R_shape_fun_cross}
& s^\text{(cross2)}(k_1,k_2,k_3) \nonumber \\
= & \Re\left\{\int_{0}^{\infty} \frac{{\rm d}x_1}{x_1} \int_{x_1}^{\infty} \frac{{\rm d}x_2}{x_2} \int_{x_2}^{\infty} \frac{{\rm d}x_3}{x_3} \,\, \Re \left[H_{\nu_\phi}^{(1)}(X x_1)\right] \left\{\left(x_1\right)^{3/2} \Re \left[H_{\nu_\theta}^{(2)}\left(Y x_1\right) H_{\nu_\theta}^{(1)}\left(Y x_2\right)H_{\nu_\phi}^{(1)}\left(Y x_2\right)
\right] \right.\right. \nonumber \\
\times & H_{\nu_\theta}^{(1)}\left(Z x_1\right)H_{\nu_\theta}^{(2)}\left(Z x_3\right) H_{\nu_\phi}^{(2)}\left(Z x_3\right) - \left(x_2\right)^{3/2} \Re\left[H_{\nu_\theta}^{(2)}(X x_1) H_{\nu_\theta}^{(1)}(X x_2)H_{\nu_\phi}^{(1)}\left(Y x_2\right)
\right] \nonumber \\
\times & H_{\nu_\theta}^{(2)}\left(Z x_2\right) H_{\nu_\theta}^{(1)}\left(Z x_3\right)H_{\nu_\phi}^{(1)}\left(Z x_3\right) - \left(x_3\right)^{3/2} \Re\left[H_{\nu_\phi}^{(1)}\left(Y x_2\right)\right] H_{\nu_\theta}^{(2)}(X x_1)H_{\nu_\theta}^{(2)}\left(Y x_2\right) \nonumber \\
\times & \left. \left. H_{\nu_\theta}^{(1)}(X x_3)
H_{\nu_\theta}^{(1)}\left(Y x_3\right)H_{\nu_\phi}^{(1)}\left(Z x_3\right)\right\} \right\}
+ 5 \,\, {\rm perm.} \end{aligned}$$ Here we have again defined $x_i \equiv -K\tau_i$, $X\equiv k_1/K$, $Y\equiv k_2/K$ and $Z\equiv k_3/K$. Similar to the $H_3^\text{(self2)}$ case, $s^\text{(cross2)}(k_1,k_2,k_3)$ diverges in the IR in the massless limit. Introducing a lower cut-off $x_c$ and following similar approach as in the $H_3^\text{(self2)}$ case, one finds $$\begin{aligned}
\label{eq:s_cross_IR_limit}
s^\text{(cross2)}(k_1,k_2,k_3) \underset{\rm IR}{\rightarrow} & -(XYZ)^{-3/2}\left[\frac{\Gamma(3/2)}{\Gamma(5/2)\pi}\right]^3\left[\frac{2^{3/2}\Gamma(3/2)}{\pi}\right] \left[\frac{(\log x_c)^{3}}{3!}\right]\nonumber \\
& \left[4X^3 - 2(X^3+Y^3) - (X^3 + Y^3+Z^3) \right] + \,\,5 \,\, {\rm perm.} \, .\end{aligned}$$ Summing over all permutations, it is not difficult to see $s^\text{(cross2)}(k_1,k_2,k_3)$ also scales as $(X^3+Y^3+Z^3)/(XYZ)^{3/2}$ and thus ${f_{\rm NL}}(k_1,k_2,k_3)^\text{(cross2)}$ is approximately shape independent. Rewriting ${f_{\rm NL}}(k_1,k_2,k_3)^\text{(cross2)}$ in terms of ${{\cal P}}_{{\cal C}}$, we therefore arrive at .
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[^1]: Note this definition of $\beta_{{\cal I}}$ is slightly different from $\beta_\text{iso}$ used in [@Ade:2015lrj]. They are related by $\beta_{{\cal I}}= \beta_\text{iso}/\left(1-\beta_\text{iso}\right)$.
|
---
abstract: 'E-commerce Web-servers often face overload conditions during which revenue-generating requests may be dropped or abandoned due to an increase in the browsing requests. In this paper we present a simple, yet effective, mechanism for overload control of E-commerce Web-servers. We develop an E-commerce workload model that separates the browsing requests from revenue-generating transaction requests. During overload, we apply LIFO discipline in the browsing queues and use a dynamic priority model to service them. The transaction queues are given absolute priority over the browsing queues. This is called the *LIFO-Pri* scheduling discipline. Experimental results show that LIFO-Pri dramatically improves the overall Web-server throughput while also increasing the completion rate of revenue-generating requests. The Web-server was able to operate at nearly 60% of its maximum capacity even when offered load was 1.5 times its capacity. Further, when compared to a single queue FIFO system, there was a seven-fold increase in the number of completed revenue-generating requests during overload.'
author:
- |
Naresh Singhmar Vipul Mathur Varsha Apte D. Manjunath\
Indian Institute of Technology - Bombay\
Powai, Mumbai, 400 076, India\
`{naresh,vipul,varsha}@cse.iitb.ac.in, dmanju@ee.iitb.ac.in `
bibliography:
- 'main.bib'
title: 'A Combined LIFO-Priority Scheme for Overload Control of E-commerce Web Servers'
---
***Keywords:*** E-commerce, overload control, Web-servers, LIFO, priority.
Introduction {#sec:Intro}
============
The capacity of a Web-server is measured in terms of the rate of requests/second that it can fulfill. When the request rate to a Web-server exceeds its capacity, the server is overloaded, its response time increases to an unacceptable level, and requests start timing out, i.e., they are abandoned, typically after some service has been received. Abandonments lead to retries, and the effective load on the server increases further. In this situation, in the absence of an overload control mechanism, the server ends up being busy doing unproductive work and the throughput degrades. E-commerce Web-servers, e.g., retail Web sites, often experience such overload situations, triggered by events such as closing time of a sale or intense shopping days [@Christmas].
Occurrence of overload situations can be minimized by appropriately sizing the server centers and by using techniques such as load balancing. However, overloads are not completely avoidable—unexpected consumer demand, partial server failures, or other such events can trigger unexpected overloads. We therefore need mechanisms to protect the Web-server from being pushed to an unproductive state during overloads. In this paper we propose and experimentally analyze one such mechanism. Specifically, we focus on E-commerce Web-servers, e.g., the server for an on-line store. For such Web-servers, the requirement is not only to be productive during overload, but to be able to differentiate between direct *revenue-generating requests* and *browsing requests* that generate revenue only indirectly. On typical shopping Web sites, the load due to the browsing requests far exceeds that of the revenue-generating requests and it is imperative that the browsing requests do not prevent revenue-generating requests from getting completed.
Overload control of telecommunication switches has been studied extensively, e.g., [@Doshi86], and some of the principles developed there can be applied to Web-servers. However, there is an important difference between a Web-server and a telecommunication switch. The former is typically modeled as a single queue (with single or possibly multiple servers) while the latter is a multi-queue system. Furthermore, since the servers take the form of processor threads, the service rate of the servers is a decreasing function of the number of active servers. Thus it is not clear if the overload control methods developed for telecommunication switches will be directly applicable. Therefore experimental evaluation like the one that we do in this paper is necessary.
Overload control of Web-servers has gained much attention in the recent past. Approaches include admission control [@Cherkasova02] or sophisticated scheduling policies [@Chen02], or both [@Elnikety04]. The fact that Web usage is session-oriented has been recognized, and several overload control mechanisms are based on that. A mechanism that does not admit new sessions at overload was proposed in [@Cherkasova02]. The mechanism proposed by [@Chen02] employs a dynamic weighted fair sharing policy to process requests from those sessions that are more likely to complete. This is done by dynamically adjusting the weights of the queues, as calculated by maximizing a productivity function. Elnikey et al [@Elnikety04] propose and implement an admission control and request scheduling policy, in which the resource requirement of a request is estimated by an external entity, and admission control is done based on that. Furthermore, a shortest job first scheduling approach is utilized for improving response times. A control theory based approach to overload control is described in [@Abdelzaher02]. The authors use a feedback control loop based mechanism to prevent overload by monitoring the utilization of server resources and switching to a degraded QoS level in overload conditions. However, their solution is meant primarily for static content as it relies on the availability of an alternate ‘degraded’ set of objects to be served. Thus, it does not take into account the variable execution time of scripts that are involved in serving dynamic content. Hence the approach of [@Abdelzaher02] is not directly applicable to an E-commerce scenario such as the one we have considered here.
Our survey suggests that although a number of mechanisms have been proposed, none of the work focuses on the essential difference between revenue-generating and browsing requests, that are a characteristic of an E-commerce Web site. In our work, we specifically recognize this difference, and work from there. We assume that the ultimate goal of an E-commerce Web site is to complete as many revenue-generating requests as possible—any work that an E-commerce Web-server does should be in support of this final goal. We propose a simple combination of priority queuing and last-in-first-out (LIFO) scheduling during overload, to achieve this goal. We have implemented and analyzed our mechanism experimentally, by emulating a typical E-commerce Web site. We use a session-based workload model that emulates realistic user behavior—variable abandonments, variable retries, and session abandonments as a result of request abandonments. We show that our mechanism performs well under all such realistic conditions.
Note that the use of LIFO for overload control when dealing with impatient customers is not new and has been proposed for telecommunication systems. Doshi and Heffes [@Doshi86] provide an excellent analysis of this family of service disciplines for overload control. They have analytically shown that LIFO based schemes are more attractive at overload from both throughput and delay points of view. Note though, that in the absence of overload, the response time of LIFO will have a higher variance than that of FIFO and can hence cause more abandonments than FIFO. In fact, we have experimental results to show that this does happen in the case of Web-servers.
The rest of the paper is organized as follows. In Section \[sec:Proposal\] we propose an E-commerce workload model and describe our LIFO-priority based overload control mechanism. In Section \[sec:Experiment\] we describe the experimental setup and discuss the results. We conclude in Section \[sec:Conclude\] with discussions and suggestions for future work.
Proposed Overload Control Scheme for E-Commerce Web-Servers {#sec:Proposal}
===========================================================
The goal of an E-commerce Web site, is revenue generation, which it achieves by allowing visitors to browse through its merchandise (if it is a retailing Web site), and then buy. Since a large fraction of the browsing visitors do not intend to buy, it is important that those that have shown the intent to buy by beginning the buying process must be helped to complete the transaction without timing out and abandoning the transaction. This is especially important during overload conditions when the server strains under an increased overhead. Before describing the overload control scheme we present our model about a typical E-commerce workload.
An E-Commerce Workload Model {#E-Com-Workload}
----------------------------
![FSM diagram of a session representing a retail Web site. We also show example transition probabilities for the case of the FSM representing a Markov Chain. The self-loops and the exit probabilities from each stage are not shown. In our experiments the transitions shown as dashed lines are assumed to have zero probability.[]{data-label="bro_buy_stages"}](figure1.eps){width="3.2in"}
We assume that in an E-commerce Web site most of the users browse the site for some time and leave, while a few of these browsing users proceed towards a revenue-generating transaction that is a multi-step (multiple Web page) process. For example, in an online retail site, the user first visits the home page and then possibly browses or searches through the catalog. If the preferred product is available then more details about that product may be sought. We term these requests as the *browsing requests*. Most of the users leave the site at this point; few who have the intention of buying some product proceed to the first step in a sequence of transactions, e.g., the ‘login’ page. From this point onward, the user is led through a multi-stage sequenced transaction (involving, e.g., entering payment and shipping details), usually culminating in a ‘confirm’ request, that finalizes the transaction. We term these the *transaction requests*. This movement of the user between the different types of pages can be represented by a finite state machine as shown in Fig. \[bro\_buy\_stages\]. To construct a tractable model that can simplify simulation and analysis, we assume that the transitions between the states are memoryless and that the probabilities are stationary. Thus the user behavior can be modeled as a stationary finite state Markov chain with states corresponding to the pages.
LIFO-Pri Overload Control Algorithm {#sec:LIFO-Pri-describe}
-----------------------------------
Recall that our objectives are twofold—(1) maximize the throughput of revenue-generating requests while (2) improving overall throughput of the Web-server during overload. The mechanism that we propose in the sequel will be called *LIFO-Pri*.
To achieve the first goal of maximizing the throughput of revenue-generating requests, we employ a priority mechanism. Separate queues are maintained for each type of request. The transaction request queues are given a simple non-preemptive priority over the browsing request queues. We make a simplifying assumption that we would never want to serve any browsing request if a transaction request is waiting to be served. Between the transaction queues, the queue for the last request, e.g. ‘confirm’, in the multi-step transactions has the highest priority. The queue for the request, e.g. ‘payment’, just before ‘confirm’ has the second highest priority, and so on.
To achieve the second goal of maximizing the overall throughput, we propose a load-based LIFO mechanism—a FIFO policy during normal load and a LIFO policy during overload. As noted earlier LIFO based policies provide better throughput and delay performance at overload as compared to FIFO. This can be explained as follows. Since the mean delay at overload is high, the high variance of the delay works in our favor by having more requests that do not time out than would happen with FIFO.
We make the reasonable assumption that overload is primarily due to browsing requests. Hence we employ LIFO during overload only for the browsing queues while serving the transaction queues according to FIFO.
We also propose a dynamic priority mechanism for selecting requests from the browsing queues to allow those that may have a higher chance of leading to a transaction request to complete with a higher probability. We use dynamic priorities because static absolute priorities can lead to starvation of low priority queues. The proposed scheme is as follows. For the browsing queues, two different attributes are maintained for each queue:
- Number of pending requests in that queue ($N_{i}$).
- Utility of that queue ($U_{i}$).
The queue priority at any time is then given by $U_i \times N_i$. The *utility* is an indicator of the relative importance of the queues. This [utility]{} could, for example, be based on the ‘revenue generation potential’, i.e., if the ‘details’ page request is more likely to lead to a buy request than a ‘search’ page request, then the ‘details’ page can be given a higher [utility]{}. The values for the utility may be obtained from the Markov chain describing the user behavior. By including the queue length in obtaining the priority, we prevent the lower priority queues from getting starved.
SET\_DISCIPLINE:
CPU\_Util $\Leftarrow$ Utilization measured over an interval Browsing\_Policy $\Leftarrow$ LIFO Browsing\_Policy $\Leftarrow$ FIFO
DYNAMIC\_PRIORITY:
$DP_i \Leftarrow N_i \times U_i$ $Q \Leftarrow \argmax_i (DP_i)$ Read a request from queue $Q$ according to current service discipline. Assign worker thread to request.
The service discipline used by [LIFO-Pri]{} for the browsing requests depends on the CPU utilization. If the CPU utilization crosses a predefined *upper threshold*, then it starts serving the browsing requests according to LIFO, and it continues with this discipline while the CPU utilization is above *lower threshold*. Recall that the transaction requests are always served in FIFO order.
The above discussion is summarized in Algorithm \[algo:LIFO-Pri\]. Note that the two parts of the algorithm—SET\_DISCIPLINE and DYNAMIC\_PRIORITY have to be executed in parallel, typically by separate threads.
Experimental Results and Discussions {#sec:Experiment}
====================================
The overload control policy as described in the above section was implemented in a Web-server. The Web-server architecture is as depicted in Fig. \[Web\_server\]. Experiments were carried out to verify the performance of our overload control mechanism, by varying load on the Web-server that we have built. The experiments done can be divided into two parts:
- Experiments to compare FIFO and LIFO service order.
- Experiments with an E-commerce setup to test the LIFO-Pri policy.
![Web-server architecture[]{data-label="Web_server"}](figure2.eps){width="3.2in"}
The first set of experiments separately characterize performance of LIFO and FIFO under non-overload and overload conditions on the Web-servers. These experiments offer several insights that will be discussed later in this section. The second set of experiments test the effectiveness of the LIFO-Pri overload control mechanism.
The test-bed contains a server and a client machine. The server machine is based on an Intel P-IV 1.6 GHz CPU with 256 MB RAM, running Debian GNU/Linux Sid. The Web-server runs on this machine with a maximum limit of 30 worker threads. It must be noted that the priority assignment is only for the assignment of a worker thread and the transaction queues are not given priority in execution by the operating system. The client machine is based on an Intel P-IV 2.4 GHz CPU with 256 MB RAM running Debian GNU/Linux Sid. The client is used to generate load on the Web-server using `httperf` [@httperf98].
Comparison of FIFO and LIFO {#sec:comparison}
----------------------------
Since we are specifically comparing the performance of LIFO and FIFO service disciplines, we carried out a set of experiments on a basic Web-server with a single queue and not with the E-commerce setup model of Fig. \[bro\_buy\_stages\]. The load is generated by repeatedly making a request for a *CPU-intensive* CGI file. The distribution of the inter-arrival time between requests is exponential.
### Experimental Setup {#sec:setup1}
The Web-server is configured with a single queue with a buffer capacity of fifty. To compare the FIFO and LIFO approaches, we repeat the experiments with the following three different service policies. In the first case, called *Always-FIFO*, the Web-server always serves the requests in FIFO order. In the second case, termed [Always-LIFO]{}, the Web-server always serves the requests in LIFO order. In the third case, that we call *LIFO-at-overload*, the service discipline alternates between LIFO and FIFO as is done in LIFO-Pri.
### Results {#sec:results}
The experiments were performed to study the server response as a function of increasing load. In this set of experiments, we use a fixed timeout value for all the requests.
![Response time distribution at $\rho=0.941$ with a timeout of 20 seconds.[]{data-label="Log_LF_rate2-8_TO18_dis"}](figure3.ps){width="2.2in"}
[|l|c|c|c|]{}\
Percentage (%) & Always-FIFO & Always-LIFO & LIFO-at-overload\
Requests Completed & 86.7 & 84.4 & 84.6\
Requests Timeout & 0.0 & 2.3 & 2.0\
Requests Dropped & 13.3 & 13.4 & 13.4\
\
Requests Completed & 21.9 & 81.0 & 76.8\
Requests Timeout & 64.9 & 5.4 & 9.7\
Requests Dropped & 13.3 & 13.6 & 13.4\
Denote the server intensity (ratio of arrival rate to service rate) by $\rho$. When the offered load is below the capacity of the server, i.e., $\rho < 1.0$, the number of requests that are either dropped or timed out is almost zero for all the three cases. Fig. \[Log\_LF\_rate2-8\_TO18\_dis\] shows the unconditional complementary distribution of the response time[^1] for $\rho=0.941$. Observe the longer tail for the case of [Always-LIFO]{} implying that a significant fraction of requests have a long response time. This effect is not seen when [LIFO-at-overload]{} is used. Thus, using LIFO is not appropriate when the load is less then the capacity of the Web-server.
When the offered load is higher than the capacity of the server, requests are dropped or are abandoned due to timeouts. We consider two timeout values—40 seconds and 20 seconds to model less patient customers. It can be seen in Table \[LOG\_lifo\_fifo\] that the percentage of requests dropped is almost identical for all the three service schemes but the abandonment rate depends significantly on the timeout value. First, consider the case when the timeout value is 40 seconds. Here, as is to be expected, [Always-FIFO]{} has the lowest percentage of abandoned requests. A large timeout value favors FIFO, because the FIFO response time does not have the “long tail” of LIFO. Note that even for the same average queue length, LIFO may result in much larger response time values than FIFO (a request that gets “pushed” to the end of the queue may never get served, and will eventually timeout). With a 20 second timeout, the [Always-FIFO]{} policy now shows a much larger abandonment rate than the LIFO policies. Further, the FIFO policy is able to achieve only 21.9% success rate as opposed to about 80% for the LIFO policies.
[cc]{} &\
\
Histogram & Distribution
[cc]{} &\
\
Histogram & Distribution
Fig. \[plot:LOG\_LF\_rate5\_TO40\] shows the response time histogram and distribution for the case of $\rho=1.47$ and a timeout of 40 seconds. Observe that for [Always-FIFO]{} the mode is at 20 seconds. Also see that for [Always-FIFO]{} all the requests have a response time of less than 24 seconds (which explains no abandonments, since the timeout is 40 seconds). For the two LIFO-based policies we observe two interesting phenomena—the mode occurs at about 7 seconds but a significant number of the requests have a very large response time, even as large as 40 seconds. This is also reflected in the long tail of the LIFO response time distribution.
Fig. \[plot:LOG\_LF\_rate5\_TO20\] shows the histogram and the distribution of the response time with a timeout of 20 seconds. Comparing with the 40 second timeout case, we observe that the difference in histograms for [Always-LIFO]{} and for [LIFO-at-overload]{} does not change significantly with the timeout value except that the tail is shortened. However, for the case of [Always-FIFO]{} the mode of the distribution is at about 18 seconds. Also, for [Always-LIFO]{} and for [LIFO-at-overload]{}, nearly 80% of the requests have a response time of less then 10 seconds, whereas for [Always-FIFO]{}, less than 5% of requests experience this response time.
Thus by using [LIFO-at-overload]{} approach we have achieved not only higher throughput, but also significantly better response time distribution at higher load.
Experimental Analysis of LIFO-Pri {#sec:LIFO-Pri-Analysis}
---------------------------------
In the experiments described in the previous section the workload consisted of a random sequence of requests for URLs and did not correspond to a transaction. We verified the claim that using [LIFO]{} service discipline improves the performance of a Web-server during overload in the presence of impatient users. We now present results of experiments that were performed to test the proposed LIFO-Pri mechanism.
### Experimental Setup {#experimental-setup}
For validating our mechanism, we set up a Web site that emulates the characteristics of a typical E-commerce Web site as per our model of Fig. \[bro\_buy\_stages\]. Some of the possible transitions in the model were assigned a probability of zero so as to minimize the effect of ‘unknown’ factors in the controlled experiments.
The eight types of pages shown in Fig. \[bro\_buy\_stages\] are generated using Perl CGI scripts that have interleaved random busy and waiting periods. The busy periods represent local processing and the waiting periods represent time spent in the back-end server calls such as database lookups. Table \[cgi-exe-time\] shows the mean execution times (including the delay in servicing back-end requests) of these CGI scripts.
We use `httperf` with the `–wsesslog` option to generate the E-commerce workload. i.e., `httperf` reads session descriptions from a file of 1000 randomly generated session descriptions according to the Markov chain shown in Fig. \[bro\_buy\_stages\] and keeps cycling through them until a specified total number of sessions have been completed. This is necessary because we did not have a load generator that could generate such a randomly distributed workload. Each session consists of a sequential set of requests which must be completed for the session to succeed. The session arrival process is modeled to be Poisson.
As in the previous section, we model the ‘impatience’ of the users by using timeouts for the requests. `httperf` supports two kinds of timeouts. The basic timeout is called `–timeout` and it is the amount of time that the load generator waits for a server reaction, i.e., forward progress must be made within this timeout value while creating a TCP connection, sending a request, waiting for or receiving a reply. An additional `–think-timeout` is added to the basic timeout while waiting for a reply after issuing a request. This is used to allow for the additional response time that the server might need to *initiate* sending a reply for a request, since we are running time-consuming CGI scripts and not merely fetching a static file. The `think-timeout` is particularly important in our case because it directly corresponds to the ‘impatience’ of customers. `httperf` (up to version 0.8) supported only fixed values for these timeouts. We modified the code to implement exponentially distributed `–think-timeout` values. This allowed us to use variable and random timeouts in our experiments to enable us to more reasonably model user impatience. Note that most other experimental works assume fixed timeout values.
Since we are modeling abandonments by timeouts, we must also model the user behavior of retrying an abandoned request. The retry model that we use is as follows. Whenever a request times out, it retries with a probability of $p$ and abandons with probability of $1-p$. The number of retries per request is upper bounded by $M$. We added this new functionality, which is accessed with the `–retry-model` option, to `httperf`.
If any request in a session fails, even after the retries, the entire session is considered to have failed. The remaining requests in that session are not issued in such a case. This is the realistic model for a Web-server because users would most likely ‘give up’ and leave the Web site, after failing to load a desired page. Thus, for a transaction request to be generated, all the preceding browsing requests of that session must have been completed successfully. This clearly implies that to have a higher amount of revenue generation under overload conditions, we must also increase the number of browsing requests that are completed. This would increase the chances of success for a session that would result in a revenue-generating transaction. Our proposal of giving a strictly higher priority to a transaction request over browsing requests would then ensure that if a transaction request is generated, it has a very high chance of completion.
Request Mean execution time (mS)
------------------ --------------------------
Main Page (Br-1) 200
Browsing (Br-2) 300
Searching (Br-3) 300
Details (Br-4) 222
Login (Tr-1) 280
Shipping (Tr-2) 420
Payment (Tr-3) 500
Confirm (Tr-4) 300
: Mean execution time of CGI scripts.[]{data-label="cgi-exe-time"}
The server is configured with eight queues: four queues for browsing requests and four for transaction requests. Thus each queue, and each type of request has its own parameters and handling mechanisms. We perform three sets of experiments as follows.
1. *SQ*: Single queue to store all the requests and served according to FIFO. The queue is assumed to have capacity to queue 100 requests.
2. *8Q-AF*: Eight queues, one for each type of request with all of them always serving in FIFO order. The browsing queues are assumed to have capacity to store 50 requests while the transaction queues have capacity for 25 requests.
3. *8Q-LIFO-Pri*: Eight queues, one for each type of request with LIFO at overload for browsing queues and FIFO for transaction queues and dynamic priority. Buffer capacities are as in 8Q-AF.
The utility of the browsing queues and the transaction queues is assigned in proportion to the probability that a request of that type eventually results in a ‘confirm’ (Tr-4) transaction. For instance, from Fig. \[bro\_buy\_stages\], the probability that a ‘browse’ page (Br-2) will lead to a final ‘confirm’ transaction Tr-4 is 0.022, whereas the probability that a ‘details’ (Br-4) page will lead to a final ‘confirm’ transaction is 0.073. The utility for each type of request for the parameters shown in Fig. \[bro\_buy\_stages\] is shown in Table \[table:utility\_values\]. Also, note that we assign the utility to the queues in such a way that the transaction requests always have higher priority than the browsing ones while their relative priority changes dynamically.
Request queue Utility
------------------ ---------
Main Page (Br-1) 27
Browsing (Br-2) 22
Searching (Br-3) 36
Details (Br-4) 73
Login (Tr-1) 3650
Shipping (Tr-2) 4050
Payment (Tr-3) 4500
Confirm (Tr-4) 5000
: Utility values for queues.[]{data-label="table:utility_values"}
The `–timeout` value for these experiments is 8 seconds and the `–think-timeout` is exponentially distributed with a mean of 12 seconds. Thus the mean total timeout value for receiving a reply is 20 seconds. If a request is timed out, the retry mechanism described earlier comes into operation. The request is retried with a probability of 0.4 ($p=0.4$) whenever a timeout occurs, up to a maximum of 5 ($M=5$) retries per request. Also, the upper threshold of the CPU utilization for the switch-over from FIFO to LIFO in the browsing queues is 0.99 while the lower threshold value for the change from LIFO to FIFO is 0.95.
### Experimental Results
![Overall throughput vs. load[]{data-label="LOG_s_load_th"}](figure6.ps){width="2.2in"}
Fig. \[LOG\_s\_load\_th\] shows the overall throughput as a function of the offered load. We can see that when the load is below the capacity of the server, i.e., $\rho=1$, (corresponds to 5.6 requests/second for this workload model), all the three schemes have similar throughput. When $\rho > 1$, the throughput of the [SQ]{} system drops significantly and is the minimum of the three cases for $\rho > 1.3$. In the 8Q-AF system a larger number of transaction requests complete and we can see a marginal improvement in the throughput. The best performance is clearly in the 8Q-LIFO-Pri system with a throughput of almost 3.5 requests/second (about 63% of the server capacity) even for $\rho=2.0$ .
**$\rho$** **Case** **Requests** **Browsing** **Tr-1** **Tr-2** **Tr-3** **Tr-4**
------------ ------------- --------------- -------------- ---------- ---------- ---------- ----------
54480 888 792 720 648
**Completed** 54480 888 792 720 648
SQ **Timed out** 0 0 0 0 0
**Dropped**
**Completed** 54480 888 792 720 648
0.85 8Q-AF **Timed out** 0 0 0 0 0
**Dropped** 0 0 0 0 0
**Completed** 54480 888 792 720 648
8Q-LIFO-Pri **Timed out** 0 0 0 0 0
**Dropped** 0 0 0 0 0
**Generated**
**Completed** 16170 20 15 9 8
SQ **Timed out** 20029 18 5 1 1
**Dropped**
**Generated** 43324 24 20 19 15
**Completed** 19852 23 19 19 15
1.4 8Q-AF **Timed out** 16305 1 1 0 0
**Dropped** 7167 0 0 0 0
**Generated** 44826 195 137 99 53
**Completed** 30851 187 127 87 50
8Q-LIFO-Pri **Timed out** 4075 8 10 12 3
**Dropped** 9900 0 0 0 0
$\rho$
--------------- ----- ------- ------------- ------ ------- -------------
Case SQ 8Q-AF 8Q-LIFO-Pri SQ 8Q-AF 8Q-LIFO-Pri
Completed 100 100 100 29.9 36.6 57.5
Timed out 0 0 0 36.8 29.9 7.5
Dropped 0 0 0 10.6 13.1 18.2
Not Generated 0 0 0 22.8 20.4 16.8
We now discuss the results in more detail and analyze it at the requests level. Table \[LOG\_s\_results\_all\] shows the composition of requests for each value $\rho$, along with the number of requests completed, requests timed out, and requests dropped for each scheme ([SQ]{}, [8Q-AF]{} and [8Q-LIFO-Pri]{}) from each of the queues. For 8Q-AF and 8Q-LIFO-Pri, the data for the browsing queues is combined. Table \[LOG\_s\_rate\] shows the overall percentage of requests completed, requests dropped, requests timed out and requests that were not generated because the session aborted before completion.
We can see that when the offered load is less then the capacity of the server, ($\rho=0.85$ case) the percentages are the same in all the three schemes with 100% of the sessions getting completed.
When the offered load exceeds server capacity, requests timeout and generate retries which further increases the offered load to the server. However, since some sessions are aborted, the requests after the session abortion are not offered and this can cause some reduction in the offered load. This effect is seen in the reduced number of browsing and transaction requests *generated* under each policy—42,029 requests are generated in SQ as compared to 43,402 in 8Q-AF and 45,310 requests in 8Q-LIFO-Pri. The end result is that the number of the Tr-4 requests (the *direct* revenue-generating request) completed[^2] increases from 8 in SQ to 15 in 8Q-AF and to 50 in 8Q-LIFO-Pri. Recall that this number should be the primary measure of performance of an E-commerce Web-server.
Our experimental setup represents the fact that browsing requests are important in the sense that they are the *source* of transaction arrivals. Increasing the browsing request completion rate, coupled with priority to transaction service, results in an overall increase in the transaction completion rate.
Table \[LOG\_s\_results\_all\] shows that with $\rho=1.4$, the [LIFO-Pri]{} scheme increases the number of ‘login’ requests generated to 195, out of which 187 are actually completed, only 8 time out and there are zero drops. This is due to the fact that a larger number of browsing requests are completed, which in turn leads to the generation of transaction requests.
![Response time distribution for ‘main’ page (Br-1) for $\rho=1.4$[]{data-label="resp-dist-br1-high"}](figure7.ps){width="2.2in"}
Some more observations from Table \[LOG\_s\_results\_all\]:
- The number of timed-out transactions is higher for LIFO-Pri than for 8Q-AF. Although this may seem surprising, observe that that a significantly larger number were generated, e.g., 195 Tr-1 requests for LIFO-Pri as compared to 24 for 8Q-AF.
- The effect of LIFO on reducing abandonments is clearer from the browsing requests where the difference in the number generated is not very significant (44,826 *vs.* 43,324). However, only 19,852 completed in 8Q-AF *vs.* 30,851 in LIFO-Pri—a result of the reduction of the number of request abandonments from 16,305 to 4,075.
- For 8Q-AF no transaction requests are dropped even at high loads because these queues have a high priority and also because very few are offered.
- Using LIFO in the browsing queues along with priority for transaction queues as in 8Q-LIFO-Pri retains the benefits of giving high priority to the transaction requests. This can be seen in Table \[LOG\_s\_results\_all\] where the number of transaction requests dropped in 8Q-LIFO-Pri is zero even in overload conditions.
![Average response time vs. load[]{data-label="LOG_s_load_resp"}](figure8.ps){width="2.2in"}
Fig. \[resp-dist-br1-high\] shows the response time distribution of Br-1 requests for $\rho=1.4$; we see that for 8Q-LIFO-Pri, nearly 80% of the requests have a response time less then 5 seconds, whereas in SQ and in 8Q-AF only about 10% of the requests achieve this.
Fig. \[LOG\_s\_load\_resp\] shows the graph between the average response time (of completed transactions) as a function of $\rho$ for the three policies. The response time with the LIFO-Pri policy is significantly better during overload. Given the improved throughput performance of LIFO-Pri, as was observed from Table \[LOG\_s\_results\_all\], this is not surprising because in the presence of abandonments it is necessary to improve response time performance to be able to increase throughput. Improving response time reduces request abandonments, which in turn causes fewer session abandonments and an increased overall throughput.
Summary and Discussion {#sec:Conclude}
======================
In this paper, we proposed and experimentally evaluated an overload control scheme for Web-servers under a reasonably realistic model of for E-commerce workload. The LIFO-Pri scheme proposed in this paper is an extremely simple, yet effective, mechanism for overload control. The experimental results are highly encouraging—the server could operate at nearly 60% of its maximum capacity even when offered a load 1.5 times its capacity and has a factor of 7 increase in the number of direct revenue-generating requests completed as compared to a single queue model during overload.
The benefits of LIFO were observed by Dalal and Jordan [@Dalal01], however, the results were not for an E-commerce environment, and no implementation and experiments were done (validation was by simulation). We believe our work confirms experimentally the truly remarkable effect on performance during overload of the LIFO policy along with a priority for revenue-generating requests. Although the LIFO service policy seems to always imply high variability and unfairness, the abandonment and retry behavior of users during overload, turns LIFO into a compelling choice.
Future work includes having better indicators for overload—this work assumed that the CPU was the bottleneck resource and used CPU utilization as the indicator. We would like to extend this to cases where we do not know the bottleneck resource. Work is also needed to model user behavior even more appropriately (e.g. longer response times should discourage ‘repeat’ visits). Lastly, analytical models are necessary to gain further insight into overload control mechanisms for Web-servers.
[^1]: All response time distribution graphs in this paper are the unconditional complementary distributions. This allows us to treat the response time of the timed out or dropped requests to be infinity.
[^2]: The seemingly disproportionate decrease in these numbers as compared to the non-overload case can be attributed to the lack of a load generator that could generate our randomly distributed workload. However, the numbers are sufficient for highlighting the performance improvement in LIFO-Pri as compared to other schemes in overload conditions.
|
---
abstract: 'By connecting light to magnetism, cavity-magnon-polaritons (CMPs) can build links from quantum computation to spintronics. As a consequence, CMP-based information processing devices have thrived over the last five years, but almost exclusively been investigated with single-tone spectroscopy. However, universal computing applications will require a dynamic control of the CMP on demand and within nanoseconds. In this work, we perform fast manipulations of the different CMP modes with independent but coherent pulses to the cavity and magnon system. We change the state of the CMP from the energy exchanging beat mode to its normal modes and further demonstrate two fundamental examples of coherent manipulation: First, a dynamic control over the appearance of magnon-Rabi oscillations, i.e., energy exchange, and second, a complete energy extraction by applying an anti-phase drive to the magnon. Our results show a promising approach to control different building blocks for a quantum internet and pave the way for further magnon-based quantum computing research.'
author:
- Tim Wolz
- Alexander Stehli
- Andre Schneider
- Isabella Boventer
- Rair Macêdo
- 'Alexey V. Ustinov'
- Mathias Kläui
- Martin Weides
bibliography:
- 'FMR\_TD2.bib'
title: 'Introducing coherent time control to cavity-magnon-polariton modes'
---
\[sec:introduction\][Introduction]{} {#secintroductionintroduction .unnumbered}
====================================
The cavity-magnon-polariton (CMP) [@huebl_high_2013; @zhang_strongly_2014; @tabuchi_hybridizing_2014; @goryachev_high-cooperativity_2014] is a hybrid particle arising from strong coupling between photon and magnon excitations. It interconnects light with magnetism being an excellent candidate to combine quantum information with spintronics [@lachance-quirion_hybrid_2019; @karenowska_magnon_2016]. The first CMP-based devices, such as a gradient memory [@zhang_magnon_2015] and radio-frequency-to-optical transducers [@hisatomi_bidirectional_2016] have already been developed. Especially the latter ones are crucial devices for a quantum internet, for instance, because they bridge microwave-frequency based quantum processors to long range optical quantum networks. Since the recent emergence of this hybrid particle, three different models, in particular, have helped to unravel the physics of CMPs over the last years: first, the picture of two coupled oscillators, which is the most intuitive one; the underlying physics, however, is only revealed from an electromagnetic viewpoint, which is the second model and shows a phase correlation between cavity and magnon excitation [@bai_spin_2015]; and finally, the quantum description of the CMP, which has, for instance, given the theoretical framework for a coupling of magnons to a superconducting qubit [@tabuchi_coherent_2015; @lachance-quirion_resolving_2017]. Many spectroscopic experiments have led to new insights about loss channels [@tabuchi_hybridizing_2014; @kosen_microwave_2019; @pfirrmann_magnons_2019], their temperature dependence [@boventer_complex_2018; @zhang_cavity_2015; @golovchanskiy_interplay_2019], and to the observation of level attraction [@grigoryan_synchronized_2018; @harder_level_2018; @boventer_control_2019]. These spectroscopic measurements, however, are performed under continuous driving, and while they have yielded great physical insight into these hybrid systems, flexible and universal information processing requires the manipulation of such physical states on demand and on nanosecond timescales. Despite this necessity for fast manipulation, the literature about time resolved experiments with either an yttrium iron garnet (YIG) waveguide [@van_loo_time-resolved_2018] or CMPs [@zhang_strongly_2014; @zhang_magnon_2015; @morris_strong_2017; @match_transient_2019-1] is scarce and confined to cavity-pulsing. A simultaneous and coherent control over both subsystems has yet to be demonstrated, which is the subject of this work. We establish the control over the cavity *and* magnon system by using coherent manipulation pulses on the timescale of nanoseconds. We observe the transition from maximum energy exchange to no energy exchange between the two quasi-particles depending on the applied pulses. Furthermore, we employ these results for a dynamic control of the different modes and for the extraction of the total energy from the system by destructive interference within the sample.
In our experiments the electromagnetic resonance of a copper cavity interacts with the Kittel mode - the uniform ferromagnetic resonance (FMR) [@kittel_theory_1948] - of a YIG-sphere mounted inside the cavity. The Landau-Lifshitz-Gilbert (LLG) equation [@landau_theory_1935] describes the Kittel-mode as a macrospin with dynamic magnetization $m(t) = m \rm{e}^{-\rm{i} \mathit{\omega t}}$ in an external magnetic field $\bm{H}$. The cavity resonance can be modeled as an RLC circuit. Following Ref. [@bai_spin_2015], a linear coupling between both systems arises from their mutual back actions, leading to a phase correlation. The changing magnetization of the FMR induces an electric field in the cavity according to Faraday’s law. Following Ampère’s law, the cavity field gives rise to a cavity current, which produces a magnetic AC-field $h(t) = h \rm{e}^{-\rm{i} \mathit{\omega t}}$ driving the FMR. Combining the LLG, the RLC equation and Maxwell’s laws yields a system of coupled equations for $h$ and $m$ (Supplementary). If both subsystems are close to resonance, these equations can be simplified to the eigenvalue equations of two coupled harmonic oscillators with constant coupling strength $g$: $$\left(\begin{array}{cc}
\omega-\tilde{\omega}_{{\rm c}} & g\\
g & \omega-\tilde{\omega}_{\rm r}
\end{array}\right)\left(\begin{array}{c}
h\\
m
\end{array}\right)=0.
\label{eq:coupled_pendulums}$$ Here, $\tilde{\omega}_{{\rm c}}$ and $\tilde{\omega}_{\rm r}$ denote the complex eigenfrequencies of the cavity and magnon system, respectively. They are defined as $\tilde{\omega}_{{\rm c}} = \omega_{\rm c} - \rm{i}\beta \omega_{\rm c}$ and $\tilde{\omega}_{\rm r} = \omega_{\rm r} - \rm{i}\alpha \omega_{\rm c}$ with bare cavity frequency $\omega_{{\rm c}}=1/\sqrt{LC}$, bare magnon frequency $\omega_{{\rm r}}=\gamma\sqrt{|H|(|H|+M_0)}$, where $\alpha$, $\beta$ are damping factors, $\gamma$ the gyromagnetic ratio of the Kittel-mode and $M_0$ its saturation magnetization. If both subsystems are exactly on resonance, i.e, at their crossing point, $\omega_{{\rm c}}=\omega_{\rm r}=\omega_0$, the eigenfrequencies of Eq. (\[eq:coupled\_pendulums\]) are given by $\omega_{\pm}=\omega_0\pm g$ with the eigenvectors $\xi_{\pm}=\left(\begin{array}{cc}
1, & \pm1\end{array}\right)$, the so-called normal modes. A single, short pulse to the cavity prepares the system in the non-eigenstate $\xi_{0, \rm{c}} = \xi_+ + \xi_- = \left(\begin{array}{cc}
1, & 0 \end{array}\right)$, known as beat mode. The excitation, and therefore the energy, periodically oscillates between cavity and magnon. Hence, the system displays classical magnon-Rabi oscillations [@zhang_strongly_2014; @match_transient_2019-1]. Figures. \[fig:setup\] illustrate these different modes in the intuitive picture of two coupled pendula. To observe the normal modes $\xi_{\pm}$, where no energy is exchanged, one has to coherently and simultaneously excite the cavity and magnon system while recording the cavity response (Fig. \[fig:setup\]a).
Results {#sec:sample .unnumbered}
=======
![**Experimental setup and mode visualization.** **a** Typical pulse sequence used to prepare the system in its normal mode. **b** The time domain setup comprises the following three parts: the magnon manipulation line, the cavity manipulation line, and the recording line for the cavity response. A continuous signal of the microwave source is up-converted with pulses from the AWG, which then excites cavity and magnon system. The reflected and down-converted signal from the cavity is recorded by an ADC-card. **c-e** Pendula representation of the different CMP modes: in-phase mode $\xi_{+}$, anti-phase mode $\xi_{-}$, and beat-mode $\xi_{0}$. []{data-label="fig:setup"}](setup_sample_pulses_easy_with_pendula.png){width="\columnwidth"}
\[sec:res\] Our sample is a copper reentrant cavity [@goryachev_high-cooperativity_2014] resonating at $\omega_{r}/2\pi=\SI{6.58}{\giga \hertz}$. An additional stripline with a second microwave port is fixed to the bottom of the cavity. This port allows for the direct manipulation of the magnon mode in a YIG sphere with a diameter of . The sphere is placed close to the magnetic antinode of the cavity. The cavity’s magnetic AC-field, the stripline AC-field and the external bias field stand all perpendicular to each other (Supplementary), which minimizes unwanted crosstalk between the two AC-fields. Measurements are performed with a time-domain setup (Fig. \[fig:setup\]b) comprising three parts: magnon manipulation, cavity manipulation and recording. It enables us to independently but coherently pulse the two subsystems and record the reflected signal, i.e., the outgoing photons, of the cavity. An arbitrary phase offset between all pulses can be chosen. Additionally, the applied power to the magnon can be adjusted allowing for an amplitude matching and thus equal excitation of cavity and magnon system.
The avoided level crossing data (Fig. \[fig:chevrons\]c), measured spectroscopically, shows a coupling strength of $g/2\pi=\SI{24.6}{\mega \hertz}$, as theoretically expected for this cavity-magnon-system [@boventer_complex_2018], and identical decay rates (Supplementary) at the crossing point of $\kappa_{\rm{crp}}/2\pi = \SI{2.1}{\mega \hertz}$ due to equal hybridization. We hence conclude that our system is strongly coupled. This result is also validated in the time domain (Fig. \[fig:chevrons\]a). The external field, and therefore $\omega_{{\rm r}}$, is swept and the cavity is excited with a single short pulse in between the sweep steps. The reflected signal shows clear Rabi-oscillations confirming the coupling strength of $g/2\pi=\SI{24.6}{\mega \hertz}$ and thus exhibiting an oscillation period of $t_{\rm R}=2\pi/g=\SI{40.6}{\nano \second}$ for $\omega_{{\rm c}}=\omega_{{\rm r}}$. The measured decay time of $\tau = \SI{77.6}{\nano \second}$ is also in good agreement with $1/\kappa_{\rm{crp}} = \SI{75.8}{\nano \second}$. Both, time resolved and spectroscopic data exhibit another weakly coupled magnon-mode at around , which slightly distorts the signal of the pure Kittel mode but is not of interest for our experiments.
![**Time-resolved and spectroscopic cavity response for the different CMP modes.** **a** Time evolution of the reflected cavity signal revealing magnon-Rabi oscillations after a single pulse to the cavity. Between each recorded time trace the external field is swept. Close to another spurious mode is visible, particularly between and . **b** Cavity time evolution after phase and amplitude matched pulses to both cavity and magnon. Rabi oscillations on resonance (green dashed line) are suppressed since the system is prepared in one normal mode. **c** Avoided level crossing of the CMP, probed spectroscopically. **d** Comparison of the cavity’s time evolution in one normal mode $\xi_{\pm}$ (green line) and beat mode $\xi_{0}$ (gray line) at the crossing point. Time traces are line cuts along the dashed lines in **a** and **b**. []{data-label="fig:chevrons"}](chevrons.jpg){width="\columnwidth"}
![**Mode composition of the CMP** depending on applied power ratio and phase offset $\varphi$ between the two pulses. Experimental data of the cavity response (**a**) are fitted to Eq. (\[eq:timetrace\]) and can then be compared to analytic data (**b**) . The chosen attenuation in the magnon line (y-axis of **a**) corresponds to the power ratio of the drive pulses used in **b** with an experimentally inaccessible offset. The parameter $\lambda$ translates to the mode composition of the CMP. Within the red ellipses, the CMP is predominantly excited in its normal modes. Experimentally found normal mode ellipses are slightly shifted to lower phase values and differ in power ratio compared to the simulated data due to a minimal timing mismatch of the applied pulses, direct crosstalk or small frequency drifts in the system (Supplementary). []{data-label="fig:trans_modes"}](sine_amp_phase_power_mpl2.jpg){width="\columnwidth"}
After the characterization of our system, we apply an additional pulse directly to the magnon system. The cavity response of such a two-pulse experiment is shown in Fig. \[fig:chevrons\]b. The two pulses are phase and amplitude-matched for the on-resonance-case in order to prepare the system in its normal mode. Since no energy is exchanged in the normal modes, a pure exponential decay is expected and observed at the crossing point. However, if $\omega_{{\rm c}} \neq \omega_{{\rm r}}$, the amplitude and phase-matching does not hold and the Rabi oscillations are visible. But when the system approaches its crossing point (green dashed line in Fig. \[fig:chevrons\]b), the dips of the oscillations become more shallow than in Fig. \[fig:chevrons\]a, until they are almost completely suppressed. The slight remaining oscillations left are due to experimental imperfections. Figure \[fig:chevrons\]d shows a line cut at the crossing point of the single and two-pulse experiment emphasizing the different responses of the normal mode and beat mode. The two-pulse response reveals the expected exponential decay of either one of the normal modes $\xi_{\pm}$. Following the external drive pulses, the cavity and magnon field have the same amplitude and oscillate in-phase (anti-phase) for $\xi_{+}$ ($\xi_{-}$), which is characteristic for the normal modes [@bai_spin_2015].
We also monitor the transition from $\xi_0$ to $\xi_{\pm}$ by sweeping the phase offset and amplitude ratio between the two pulses at the crossing point. The cavity response during free evolution is recorded for every set of phase-offset $\varphi$ and applied power ratio, and fitted to the following formula (See supplementary for details on the derivation): $$P_{{\rm c}}(t)=p_0\left[(1-\lambda)+(1+\lambda)\cos^{2}(g\,t+\phi_0)\right]{\rm e}^{-t/\tau},
\label{eq:timetrace}$$ which describes the cavity response during free evolution. The parameter $\lambda$ has inherent bounds of $-1$ to $1$ and determines the behavior of the system. $\lambda=1$ gives a damped sine function and thus represents the beat mode $\xi_{0}$, whereas $\lambda=-1$ yields the pure exponential decay of the normal modes $\xi_{\pm}$. A proportionality constant $p_0$ normalizes the different input powers, $\phi_0$ describes the initial phase of the beating and $\tau$ the decay time. Figure \[fig:trans\_modes\] displays the extracted values for $\lambda$ corresponding to the different modes for the measured data and can be compared to the analytic solution. As expected from the oscillator and the electromagnetic model [@bai_spin_2015], where the phase between magnon field and cavity field at the crossing point is locked to either in phase or complete anti-phase for the two eigenmodes, the system is prepared in the normal modes (red regions) for matching powers and phase offsets of 180 and 360. In the experiment, the red regions are shifted to lower phase offset values by roughly 30. This phase shift translates to a timing mismatch between the two applied pulses below , which is beyond the precision of our setup. The slope between the normal mode regions is either due to crosstalk or little drifts of the external magnetic field. We have verified the reasons for these deviations from the exact analytic solution by numerical simulations of two coupled oscillators with short drive pulses (Supplementary). Apart from these little discrepancies, which are purely limitations of the experimental setup, our collected data agrees well with theory. We can change the parameter $\lambda$ and therefore the CMP mode composition continuously. With a second pulse, the phase relation between cavity current and magnon magnetization can be set to an arbitrary value. Thus, these results extend the work of Bai *et. al*, where the phase relation between the two systems is fixed by the external field [@bai_spin_2015]. This pulsed mode control of the CMP may hence benefit future spin rectification [@harder_electrical_2016] experiments and applications.
![ **Coherent and dynamic control over the CMP.** **a** Time trace of the cavity response showing dynamic control over the time span of energy exchange. The time trace is divided into three parts: (I) The system is in its normal mode with almost no energy exchange; (II) An additional pulse to the magnon system introduces an energy difference leading to Rabi oscillations; (III) A third pulse to the magnon extracts energy out of the system by destructive interference, bringing it back to the normal mode. **b** Cavity time trace showing energy extraction by an anti-phase drive of the magnon, which counters the energy coming from the cavity. Photons interfere destructively and the CMP is completely deexcited by destructive interference after the magnon pulse. Solid brown and blue lines represent the applied pulse sequences for the cavity and magnon system, respectively (pulse-height not scaled). The ring up of cavity and magnon system, $t<\SI{0}{\nano \second}$, is omitted for clarity. Insets show the corresponding coupled pendula visualizations. []{data-label="fig:coherence_control"}](coherent_control_final_hf2.pdf){width="\columnwidth"}
The external control of the CMP mode composition, which we showed and described, is directly linked to the amplitude control of Rabi oscillation and thus to the amount of energy transferred between the two subsystems. Seizing this opportunity, we now demonstrate a coherent and also dynamic control over the CMP during one single decay by choosing an arbitrary period in which the magnon-Rabi oscillations are allowed to occur (Fig. \[fig:coherence\_control\]a). Having prepared the system on resonance, we excite both magnon and cavity with phase and amplitude-matched pulses, to bring the CMP into its normal mode $\xi_{\pm}$ and observe a pure exponential decay. We then increase the energy of the magnon subsystem by pulsing it again with a short pulse. The whole system is now in a superposition of $\xi_{+}$ and $\xi_{-}$, i.e., in its beat mode. Rabi oscillations are visible and energy is exchanged. After a few oscillations, a third pulse in anti-phase to the incoming photons from the cavity and with lower amplitude, due to energy loss in the system, extracts the additional energy, previously introduced to the magnon subsystem, by destructive interference and brings the whole system back to its normal mode. The Rabi oscillations stop and a simple exponential decay is visible, again. In the picture of two coupled pendula the three different segments of the decay corresponds to (I) both pendula oscillating in phase, (II) a strong drive of one pendulum introduces energy leading to the beat mode, and (III) and a careful short deacceleration brings the system back to the normal mode.
In a second experiment (Fig. \[fig:coherence\_control\]b), we apply the technique used for active noise control [@elliott_active_1990; @kuo_active_1999] in acoustics to the CMP: The cavity is excited by a short pulse and the energy is transferred to the magnon system and back to the cavity. During the second energy transfer to the magnon, we drive the magnon in an anti-phase manner to the oscillation of the incoming photons. A destructive interference extracts all the stored energy from the system and thus the reflected power of the cavity drops within a few nanoseconds by roughly , before the signal reaches the baseline of the measurement setup. This behavior can also be understood intuitively in the picture of the coupled pendula: A first pulse tilts only one pendulum, the energy is transferred with time and the second pendulum starts oscillating. Exactly in the moment when all energy is transferred, i.e., the first pendulum is at rest, the other pendulum is stopped abruptly by the external second pulse and the whole system is deexcited. These two experiments presented here demonstrate the fundamentals of dynamic and coherent control over the CMP.
Discussion {#discussion .unnumbered}
==========
We presented coherent time-domain control of both cavity and magnon while recording the cavity response, as well as real-time manipulation of the CMP. We also showed the transition from the beat mode to the normal modes of the CMP and explained it with the theory models provided in Ref. [@bai_spin_2015]. The CMP can be set in an arbitrary superposition of its eigenmodes depending on the phase offset and amplitude of the applied pulses. This pulse influence agrees well with theory considering the finite time resolution of our setup. Furthermore, we demonstrated a coherent control over the CMP, with which the amount of transferred energy as well as the total amount of energy in the system can be manipulated at any given time.
Spectroscopic two-tone experiments predicted and observed the regime of level attraction [@zhang_observation_2017; @grigoryan_synchronized_2018; @boventer_complex_2018], which has also been linked to an entanglement of photon and magnon [@yuan_steady_2019]. Our technique would allow to prepare the CMP in this regime and then observe its time-evolution. Although our demonstration was purely classical, the presented control can readily be applied at cold temperatures, i.e., in the single magnon regime and the predicted entanglement in the level attraction regime may be verified. Moreover, with the demonstration of dynamic and coherent control, we have added another instrument to the toolbox for the construction of a quantum internet [@kimble_quantum_2008]. Together with magnon based storage [@zhang_magnon_2015] and qubit magnon coupling [@tabuchi_coherent_2015; @lachance-quirion_hybrid_2019], we believe that our work will advance the encoding of qubit / Fock states in magnons [@rezende_coherent_1969] similar to superconducting resonators [@hofheinz_generation_2008; @leghtas_deterministic_2013] and subsequently the implementation of bosonic gates [@vlastakis_deterministically_2013; @heeres_implementing_2017]. The CMP’s significant potential as an interface from radio frequency to optics [@hisatomi_bidirectional_2016] balances the short lifetime of magnons compared to superconducting resonators. Thus, our results promise a link between the different building blocks for a quantum network and open new ways for magnon based quantum computation research.
Finally, our work demonstrates the fundamental principle of time-control of the individual components in hybrid systems. Applied to other compound devices featuring polaritons from the strong coupling of electromagnetic waves with electric or magnet excitations, such as optomechanics [@aspelmeyer_cavity_2014] or electromechanics [@regal_cavity_2011], it provides a flexible platform that intrigues fundamental coherent control of the strong light-matter interaction dynamics.
Methods {#methods .unnumbered}
=======
Experimental setup {#experimental-setup .unnumbered}
------------------
The experimental setup is adapted from quantum simulation experiments with superconducting qubits [@braumuller_analog_2017]. Its core components are a microwave source, an arbitrary waveform generator (AWG) with two sets of DACs, and a two-channel ADC-card. For our experiments it is vital that the phase between magnon and cavity control pulses is independently controllable but also stable over the entire experiment. We ensure this by using a single microwave source and two DAC sets in combination with the internal clock of the AWG for both DAC sets. The continuous signal generated by the microwave source is up-converted to $\omega_0$ via separate but identical IQ mixers and short IQ-pulses with a carrier frequency of from the AWG. The up-conversion preserves the phase offset and the envelope of the IQ-pulses emitted by the AWG. A voltage controllable attenuator in combination with a amplifier inserted in the magnon line enables us to adjust the excitation amplitude and hence vary the power ratio between magnon and cavity excitation pulses.
Data acquisition {#data-acquisition .unnumbered}
----------------
The cavity response is recorded by measuring the IQ components of the down converted, filtered and amplified signal with the ADC-card. A subsequent digital down conversion removes the carrier frequency and yields amplitude and phase data. All data acquisition and analysis are performed with the open source measurement suite qkit <https://github.com/qkitgroup/qkit>.
Experimental technique {#experimental-technique .unnumbered}
----------------------
The initial two pulses for the cavity and magnon system have to reach the sample simultaneously in order to ensure a good phase and amplitude matching (Supplementary). We therefore calibrate the cable delay between the two input lines by emitting two Gaussian shaped pulses simultaneously at the AWG, which are sent to both subsystems. Due to undeterrable crosstalk, a part of the pulse applied to magnon system is transferred to the recording line of the cavity. Recording the reflected cavity pulse and the transmitted magnon pulse, we find a cable delay of $\SI{7}{ns}$ by fitting the pulses and extracting their mean values. However due to simplicity, square pulses are used for most experiments. Because of inaccessible and fluctuating parameters, such as uncorrectable cable delays below , drifts in the external fields and unknown reflected parts of the emitted pulses, the correct phase offsets and power ratio for the specific experiments are found experimentally by a sweep of these two parameters. Although the eigenfrequencies are shifted by $g$ compared to the bare resonator frequency, the system is always pulsed at $\omega_0$. This gives the best experimental compromise for equally exciting the different modes of the system. All experiments are performed in the linear regime (Supplementary).
Sample details {#sample-details .unnumbered}
--------------
The employed YIG-sphere is commercially available from Ferrisphere Inc. The stripline is matched and open-ended. It is made from a Rogers TMM10i copper cladded () substrate with a thickness of .
We acknowledge valuable discussions with Konrad Dapper, Bimu Yao and Can-Ming Hu. This work was supported by the European Research Council (ERC) under the Grant Agreement 648011, Deutsche Forschungsgemeinschaft (DFG) within Project No. WE4359/7-1 and INST 121384/138-1, through SFB TRR 173/Spin+X, and the Initiative and Networking Fund of the Helmholtz Association. T.W. acknowledges financial support by Helmholtz International Research School for Teratronics (HIRST), A.St. by the Landesgraduiertenförderung (LGF) of the federal state Baden-Württemberg, A.Sch. by the Carl-Zeiss-Foundation and R.M. by the Leverhulme Trust. A.V.U. acknowledges partial support from the Ministry of Education and Science of the Russian Federation in the framework of the contract No. K2-2017-081.
Author contributions {#author-contributions .unnumbered}
====================
T.W. and M.W. conceived the experiment. T.W. performed the measurements with support by A.St., A.Sch., and I.B. T.W. carried out data analysis with contributions from A.St. and R.M. T.W. wrote the manuscript with input from and discussions with all co-authors. A.V.U., M.K, and M.W. supervised the project.
Supplementary Material {#supplementary-material .unnumbered}
======================
Visualization of the CMP coupling mechanism
-------------------------------------------
According to Ref. [@bai_spin_2015] four equations govern the behavior of the CMP: (i) the LLG equation describes the dynamics of the macrospin representing the FMR; (ii) the RLC equation gives the current dynamics of the cavity; (iii) Ampère’s law shows the coupling from cavity to magnon because the cavity current drives the magnetization of the FMR; (iv) a time dependent change in the magnetization produces a voltage acting on the cavity, according to Faraday’s law. Fig. \[fig:pendulums\] illustrates this model. Combining all these equations, one obtains the following coupled system of equations: $$\left(\begin{array}{cc}
\omega^{2}-\omega_{\rm c}^{2}+2{\rm i}\beta\omega_{\rm c}\omega & {\rm i}\omega^{2}K_{\rm c}\\
-{\rm i}\omega_{\rm m}K_{\rm m} & \omega-\omega_{\rm{r}}+{\rm i}\alpha\omega
\end{array}\right)\left(\begin{array}{c}
j\\
m
\end{array}\right)=0.
\label{eq:coupled_cmp_electro}$$ Here, $K_{\rm m}$ and $K_{\rm c}$ are coupling constants, $\omega_m = \gamma M_0$ and all other variables as defined in the main text. This model describes the discussed phase correlation. On resonance, the equations can be simplified to the coupled oscillator model, given by Eq. (\[eq:coupled\_pendulums\]) in the main text.
![**Illustration of the electrodynamic CMP model** and its coupling mechanisms, according to Bai *et al.*[@bai_spin_2015].[]{data-label="fig:pendulums"}](coupling_only.pdf)
Field arrangement inside the cavity
-----------------------------------
The magnetic field arrangement is depicted in Fig. \[fig:fields\]. The external field penetrating the cavity is aligned in parallel to the two posts inside the cavity. The magnetic field of the cavity mode circulates around the post and interferes constructively in the middle between the posts. Here, the magnetic AC-field of the cavity is aligned parallel to the stripline. According to Ampère’s law, the stripline’s AC-field circulates around the stripline, giving a perpendicular orientation towards the other fields.
![**Field arrangement inside the reentrant cavity.** The external magnetic field $H_{\rm{ext}}$ and the two AC-fields, $h_{\rm{strp}}$ and $h_{\rm{cav}}$, are aligned perpendicular towards each other. Signals are coupled into the cavity via an inductive loop and into the magnon system via a stripline.[]{data-label="fig:fields"}](fields_in_cavity.pdf)
Line widths and decay times
---------------------------
\[fig:dec\_times\]
We use a Lorentzian fit to extract the off-resonant linewidth (HWHM), i.e., decay rate of the cavity from the avoided level crossing data with the magnon detuned (Fig. \[fig:dec\_times\]a) and find $\kappa_{\rm{c}}/2\pi = \SI{3.0}{\mega \hertz}$ corresponding to $\beta=\kappa_{\rm{c}}/\omega_{\rm{c}}=\SI{0.046}{\percent}$ ($Q_L=1090$). Tuning in the magnon, the linewidth of the cavity decreases as expected [@bai_spin_2015] until, at the crossing point, both dips hybridize with equal amounts (Fig. \[fig:dec\_times\]b) leading to decay rates of $\kappa_{\rm{crp}}/2\pi = \SI{2.1}{\mega \hertz}$ corresponding to $(\alpha + \beta)/2 = \SI{0.032}{\percent}$. The frequency independent Gilbert damping factor $\alpha$ of the magnon system can then be calculated as $\alpha = \SI{0.014}{\percent}$ ($Q=3570$).
Estimation of the local magnetic field strength
-----------------------------------------------
We explained our results from the viewpoint of two coupled linear oscillator. Here, we present a rough estimate verifying that the magnon system is indeed only driven linearly. We use Ref. [@gurevich_magnetization_1996] for the saturation field of a ferromagnetic resonance in a sphere, which is given by $$\mu_0 H_{\rm{sat}} = 0.5\, \mu_0 \, \Delta H,$$ where $\Delta H$ denotes the FWHM of the spin wave resonance. Detuned from the cavity resonance we find for our setup $\Delta \omega_r = \SI{1.5}{\mega \hertz}$. With the dispersion relation of the Kittel-mode, this value corresponds to $\mu_0 \Delta H \approx \SI{54}{\micro \tesla}$. Thus, the magnetic AC-field of our drive must not exceed half ot this value. The rf-power reaching the sample, i.e., the power emitted by the microwave source but then attenuated through cables and microwave components ranges from $\SI{-5}{\decibel m}$ to $\SI{+5}{\decibel m}$. The maximum power corresponds to a current of $I=\SI{8}{\milli \ampere}$. We now approximate the magnon transmission line as a cylindrical conductor and use Ampère’s law $$\mu_0 H=\frac{\mu_0 I}{2 \pi r},$$ to calculate the AC-field strength with a distance $r=\SI{2}{\milli \meter}$. This gives a field strength of $\mu_0 H_{\rm{ac, max}}=\SI{5}{\micro \tesla}$ as an absolute maximum value and thus $$\mu_0 H_{\rm{ac}} \leq \SI{5}{\micro \tesla} \ll \mu_0 H_{\rm{sat}} \approx \SI{27}{\micro \tesla}.$$ Hence, we can conclude that all experiments were conducted in the linear regime.
Time evolution of the cavity energy
-----------------------------------
Eq. (\[eq:timetrace\]) in the main text describes the reflected power of the cavity, which is proportional to the stored energy inside the cavity. It is written in a form that the energy of the normal modes and beat mode occur in two separate terms with the parameter $\lambda$ giving the ratio of the two different modes. For our experiment, $\lambda$ should be a function of phase offset $\varphi$ and applied power ratio. At the crossing point, the two systems hybridize leading to same decay rates. Assuming similar coupling to the input ports, i.e., transmission lines, the applied power ratio equals the ratio of stored energy $\Delta^{2}=A_{\mathrm{m}}^{2}/A_{\mathrm{c}}^{2}$, where $A_{i}$ describes the maximum amplitude of the oscillation with only potential energy present. To derive Eq. (\[eq:timetrace\]) and find an analytic expression for $\lambda$, we start with the general solution for two coupled oscillators: $$\begin{aligned}
x_{{\rm c}}= & A\cos((\omega_{0}+g)\,t+\phi_0)+B\cos((\omega_{0}-g)\,t-\phi_0)\label{eq:cp_equ}\\
x_{{\rm r}}= & A\cos((\omega_{0}+g)\,t+\phi_0)-B\cos((\omega_{0}-g)\,t-\phi_0).\label{eq:cp_equ2}\end{aligned}$$ $A$, $B$ and $\phi_0$ are free parameters depending on the initial condition, i.e., the state of the oscillator when the pulses have stopped. Damping is neglected because the exponential decay can be factored out. The energy of the cavity oscillator is given by $$E_{{\rm c}}(t)=\frac{1}{2}\omega_{0}^{2}x_{{\rm c}}^{2}+\frac{1}{2}\dot{x}_{{\rm c}}^{2}. \label{eq:energy}$$ Substituting Eq. (\[eq:cp\_equ\]) and its derivative into Eq. (\[eq:energy\]) while neglecting terms proportional to $g$, since $g\ll\omega_{0}$, yields $$E_{c}(t)=\frac{1}{2}\omega^{2}(\left|A\right|^{2}+\left|B\right|^{2})+\omega^{2}\left|AB\right|-2\omega^{2}\left|AB\right|\sin(g\,t+\phi_0)^{2}. \label{eq:energy_AB}$$ Here, we have already denoted $A$ and $B$ as complex parameters with their absolute values. In a next step, we have to transform the initial conditions of the non-diagonal system, i.e., $\varphi$ and $\Delta$ into $A$ and $B$: $$\left(\begin{array}{c}
A\\
B
\end{array}\right)=\left(\begin{array}{cc}
1 & 1\\
1 & -1
\end{array}\right)\left(\begin{array}{c}
1\\
\Delta{\rm e^{{\rm i\varphi}}}
\end{array}\right)=\left(\begin{array}{c}
1+\Delta{\rm e^{{\rm i\varphi}}}\\
1-\Delta{\rm e^{{\rm i\varphi}}}
\end{array}\right).\label{eq:AB_vectors}$$ Inserting these vectors into Eq. (\[eq:energy\_AB\]) gives $$\begin{aligned}
E = & \frac{\omega_{0}^{2}}{2}\,\left(2\left(1+\Delta^{2}\right)+2\sqrt{\left(1-\Delta^{2}\right)^{2}+4\Delta\sin^2\varphi}-4\sqrt{\left(1-\Delta^{2}\right)^{2}+4\Delta\sin^2\varphi}\sin(g\,t+\phi_0)^{2}\right)\\
E = & \frac{\omega_{0}^{2}}{2}\,\left(\underbrace{2\left(1+\Delta^{2}\right)-2\sqrt{\left(1-\Delta^{2}\right)^{2}+4\Delta\sin^2\varphi}}_{c(1-\lambda)}+\underbrace{4\sqrt{\left(1-\Delta^{2}\right)^{2}+4\Delta\sin^2\varphi}}_{c(1+\lambda)}\cos(g\,t+\phi_0)^{2}\right). \label{eq:E_complete}\end{aligned}$$
Eq. (\[eq:E\_complete\]) is now in the same form as Eq. (\[eq:timetrace\]). We can identify $(1-\lambda)$ and $(1+\lambda)$ and hence solve for $\lambda$ and $c$: $$\begin{aligned}
\label{eq:a_rabi}
\lambda & = \frac{-\Delta^2+3\sqrt{\left(\Delta^2+1\right)^2 - 4\Delta^2\cos^2\varphi}-1}{\Delta^2+\sqrt{\left(\Delta^2+1\right)^2 - 4\Delta^2\cos^2\varphi}+1}\\
c & = \Delta^2+\sqrt{\left(\Delta^2+1\right)^2 - 4\Delta^2\cos^2\varphi}+1.\end{aligned}$$ The results of Eq. (\[eq:a\_rabi\]) for different $\Delta^2$ and $\varphi$ are plotted in Fig. \[fig:trans\_modes\]b in the main text.
Influence of experimental imperfections and numerical simulations
-----------------------------------------------------------------
Investigating the shifted ellipses found in the measurement values of Fig. \[fig:trans\_modes\] in the main text, we perform numerical simulations of two coupled oscillator with a short driving pulse. This gives us the possibility to test the influence of crosstalk, timing mismatches and a detuning between magnon and cavity. The equations of motions for two coupled oscillators, which lead to Eq. (\[eq:coupled\_pendulums\]), and with drives included, are given by $$\begin{aligned}
\ddot{x}_{\rm c}+2\beta \omega_{\rm c}\dot{x}_{\rm c}-g^{2}\omega_{\rm c}x_{\rm r}& = (1-\zeta) F_{\rm c} \, + \, \zeta F_{\rm r} \label{eq:coupled_systems1} \\
\ddot{x}_{\rm r}+2\alpha \omega_{\rm r}\dot{x}_{2}-g^{2}\omega_{\rm r}x_{\rm c}& = (1-\zeta)F_{\rm r} \, + \, \zeta F_{\rm c},
\label{eq:coupled_systems2}\end{aligned}$$ with $$\begin{aligned}
F_{\rm c}& = \cos(\omega_{0}t)\,\theta(t-t_{1,1})\,\theta(t_{1,2}-t), \\
F_{\rm r}& = \Delta \cos(\omega_{0}(t-\delta t) - \varphi)\,\theta(t-t_{2,1})\,\theta(t_{2,2}-t).\end{aligned}$$ $x_{\rm r}$ and $x_{\rm c}$ denote the amplitudes of the oscillators, $\alpha$ and $\beta$ the damping factors. The drives are represented by $F_1$ and $F_2$ with theta functions allowing for short square pulses, as in our experiment. A possible timing mismatch can be introduced by $\delta t = t_{2,1} - t_{1,1} = t_{2,2} - t_{1,2}$. The parameter $\zeta$ defines the amount of crosstalk. The theoretical case without any crosstalk in the system is thus given by $\zeta=0$. A possible detuning of $\Delta f = (\omega_{\rm r} - \omega_{\rm c})/2\pi$ can also be investigated. The cavity frequency and the drive frequency $\omega_0$ are kept constant. Eq. (\[eq:coupled\_systems1\]) and Eq. (\[eq:coupled\_systems2\]) are numerically solved with the scipy integrate package. We then use the resulting simulated time traces and their derivative to calculate $E(t)$ and fit it to Eq. (\[eq:timetrace\]) in the main text. The parameter $\lambda$ is plotted for different configurations of $\delta t$, $\Delta f$ and $\zeta$ in Fig. \[fig:delay\_plots\]. It is evident that even a slight timing mismatch leads to a deviation from the ideal case, where the normal modes occur at exactly the same applied power to both systems and phase offset values of and . Instead, with increasing time delay $\delta t = t_{\rm{mag}} - t_{\rm{cav}}$, the phase interval between in-phase and anti-phase mode (starting with the in-phase mode) becomes greater than and increases with $\delta t$. For negative $\delta t$, the opposite behavior is the case. One can observe a slope between the two elliptical normal mode regions if crosstalk is allowed. This is due to the phase depending constructive or destructive interference of the two pulses at the sample. A similar effect occurs if the magnetic field drifts and hence $\omega_{{\rm c}} \neq \omega_{{\rm r}}$. Then one system is closer to $\omega_{0}$ and receives therefore more energy than the other system.
\[fig:delay\_plots\]
|
---
abstract: 'The band gap of SrTiO$_{3}$ retards its photocatalytic application. Regardless narrowing the band gap of the anion doped SrTiO$_{3}$, the anion doping structures have low photoconversion efficiency. The co-cation dopings are used to modify the band gap and band edges positions of SrTiO$_{3}$ to enhance the photocatalyitic properties by extending the absorption to longer visible-light. Using density functional calculations with the Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional for exchange-correlation, the crystal and electronic structures as well as the optical properties of the mono and codped SrTiO$_{3}$ are investigated. The (Mo, Zn/Cd) and (Ta, Ga/In) codoped SrTiO$_{3}$ are explored in the way to retain the semiconductor characteristics of the latter. It is found that (Ta, Ga/In) codoping does not enhance the photocatalytic activity of SrTiO$_3$ due to its large band gap. Moreover, the position of the conduction band edge of the (Mo, Cd) monodoping impedes the photocatalytic efficiency. The obtained results indicate that (Mo, Zn) codoped SrTiO$_3$ can potentially improve the photocatalyitic activity.'
author:
- 'M.M. Fadlallah$^{1,2}$'
- 'T.J.H. Vlugt$^{3}$'
- 'M.F. Shibl$^{4,5}$'
title: 'Towards an efficient perovskite visible-light active photocatalyst'
---
Introduction
============
Designing efficient materials to minimize or exclude utilizing hazardous substances is the heart of sustainable (green) life. Hydrogen production from solar water splitting as renewable green energy resources and minimizing environmental pollution by photoreduction of carbon dioxide via photocatalysis are few examples. Although photocatalysis efficiency has been extensively studied both experimentally and theoretically for decades, finding efficient photocatalyst is still the focus of many researchers. New course of metal-oxide materials has contributed to photocatalysis. However, most of the metal-oxide photocatalysts comprise large band gap causing lack of photoactivity under visible light. For instance, the ability of perovskites to absorb the visible spectrum is limited by their optical band gaps. Consequently, many experimental efforts to reduce perovskites optical gaps by chemical doping have been made [@A1; @A2; @A3; @A4].
SrTiO$_{3}$ as a perovskite candidate is an oxide semicoductor with a band gap of $3.2$ eV at room temperature. SrTiO$_{3}$ can be considered as a promising photocatalyst because of its ability to split water into H$_{2}$ and O$_{2}$ without the application of an external electric field [@A5; @A6; @A7; @A8]. The large band gap (3.2 eV) activates the photocatalytic properties of SrTiO$_{3}$ by absorbing only UV radiation, which represents $3$% of solar spectrum. In order to enhance the photocatalytic efficiency of SrTiO$_{3}$ in the visible light region, introducing doping states into the band gap and/or narrowing the latter are the common methods. Furthermore, the doping into a semiconductor can create a new optical absorption edge which is very important in the photocatalysis process.
Many experimental and theoretical publications have been performed to develop the photocatalytic efficiency of SrTiO$_{3}$ via doping in either cationic sites (mostly Ti) or the oxygen anionic site. In the band structure of SrTiO$_{3}$, oxygen 2p-orbitals dominate in the valence band (VB) whereas Ti 3d-orbitals prevail in the conduction band (CB). Generally, anion doping serves mostly to modify the VB of SrTiO$_{3}$ due to different p-orbital energy of the dopant material, while cation doping usually produces gap states in the forbidden region or resonates with the bottom of the CB. For anionic dopant material, the N-doping has been tackled theoretically and experimentally. N-doping increases the photocatalytic activity by introducing mid-gap states which can prevent the charge carrier mobility resulting in decreasing the photoconversion efficiency [@A9; @A10; @A11; @A12]. Moreover, C, F, P, S and B dopings have been studied to improve the photocatalytic properties, it is found that C doping at the Ti site decreases the band gap and thus improving their photocatalytic properties [@A12; @A13; @A14]. On the other hand, cation doping using transition metals can increase the visible light activity but cannot activate water splitting such as Cr [@A15; @A16; @A8], Mn, Ru, Rh and Pd [@A2; @A17; @A18; @A19]. Furthermore, anion-cation, anion-anion or cation-cation codoping pairs have been used to enhance the photoresponse [@A10; @A17; @A18; @A19; @A20]. However, to our knowledge, all the co-cation doping studies have been performed without giving attention to SrTiO$_{3}$ semiconductor characteristics.
In this contribution, designing a novel and efficient visible-light active photocatalyst (SrTiO$_{3}$) employing cation codoping is inspected taking into account the following criteria: the co-cation i) should not change the SrTiO$_{3}$ semiconductor characteristics, ii) does not make any significant distortion in the clean SrTiO$_{3}$ crystal, iii) reduces the band gap to absorb the visible light and iv) changes the band gap edge of the CB to fulfill the redox potential requirements for water splitting. Since the electronic configuration of Ti$^{4+}$ is \[Ar\] 3d$^{0}$4s$^{0}$, the most convenient cation dopant should have empty or completely filled valence d-orbital, i.e. d$^0$ or d$^{10}$ electronic configuration. Four co-cation dopings have been assumed to maximize the visible light activity of the photocatalyst that are (Mo$^{6+}$, Zn$^{2+}$/Cd$^{2+}$) and (Ta$^{5+}$, Ga$^{3+}$/In$^{3+}$).
The manuscript is organized as follows: in section \[calc\], the methods utilized in the calculations are represented. The pristine SrTiO$_3$ is discussed as a reference system for the rest of the calculations in section \[res\]. Moreover, section \[res\] outlines various cation monodoping collection which is the entry of different cation co-doping combinations emphasizing the band structure and its effects on the optical properties and photocatalytic activity. Finally, section \[conc\] concludes.
Calculation Methods {#calc}
===================
The spin polarized density functional theory (DFT) calculations are performed using the projector augmented wave (PAW) pseudopotentials in the Vienna *ab initio* Simulations Package (VASP) code [@B1; @B2]. For the exchange and correlation energy density functional, the generalized gradient approximation (GGA) [@B3; @B4] in the scheme of Perdew-Bueke-Ernzerhof (PBE) [@B5] is utilized to get the optimzed structures. The PAW potentials with the valance electron Sr (4s$^{2}$ 4p$^{6}$ 5s$^{2}$), Ti (4s$^{2}$ 3d$^{2}$), O (2s$^{2}$ 2p$^{4}$), Mo (5s$^{2}$ 4d$^{4}$), Zn (4s$^{2}$ 3d$^{10}$), Cd (5s$^{2}$ 4d$^{10}$), Ta (6s$^{2}$ 5d$^{3}$), Ga (4s$^{2}$ 4p$^{1}$) and In (5s$^{2}$ 5p$^{1}$) are employed. The wave functions are expanded in plane waves up to cutoff energy of 600 eV. A Monkhorst-Pack $k$ point mesh [@B6] of 8 $\times$ 8 $\times$ 8 is used for geometry optimization until the largest force on the atoms becomes smaller than 0.01 eV/Å and the tolerance of total energy reaches 10$^{-6}$ eV. A 2 $\times$ 2 $\times$ 2 supercell containing 40 atoms is structured to simulate SrTiO$_{3}$. The stability, the electronic structures and the optical properties are carried out using the hybrid functional Heyd-Scuseria-Ernzerhof (HSE06) [@B7; @B8; @B9]. The exchange-correlation energy in the hybrid functional (HSE06) is formulated as: $$E^{HSE}_{xc} = 0.28E^{SR}_{x}(\mu)+(1-0.28)E^{PBE,SR}_{x}(\mu)+E^{PBE,LR}_{x}(\mu)+E^{PBE}_{c} ,$$ The mixing coefficient (0.28) and the screening ($\mu=0.2 \AA^{-1}$) parameters are considered to get the closer to the experimental band gap value [@B10; @B11]. A 3 $\times$ 3 $\times$ 3 kpoint in the Brioullioun zone is utilized in the HSE06 calculations. Using the frequency dependent dielectric function as implemented in VASP, the optical properties are investigated. Figure 1 illustrates the crystal structure of pristine, mono-doped, and codoped SrTiO$_{3}$ which are used in this study
![(Color online) 2 x 2 x 2 supercell for pristine, mono-doped, and codoped SrTiO$_{3}$ crystal structures. Blue, red, green, pink, and yellow spheres represent Sr, Ti, O, first mono (M1) doped, and second doped (M2) atoms, respectively.[]{data-label="fig1"}](C.eps "fig:"){width="30.00000%"} ![(Color online) 2 x 2 x 2 supercell for pristine, mono-doped, and codoped SrTiO$_{3}$ crystal structures. Blue, red, green, pink, and yellow spheres represent Sr, Ti, O, first mono (M1) doped, and second doped (M2) atoms, respectively.[]{data-label="fig1"}](d1.eps "fig:"){width="30.00000%"} ![(Color online) 2 x 2 x 2 supercell for pristine, mono-doped, and codoped SrTiO$_{3}$ crystal structures. Blue, red, green, pink, and yellow spheres represent Sr, Ti, O, first mono (M1) doped, and second doped (M2) atoms, respectively.[]{data-label="fig1"}](d2.eps "fig:"){width="30.00000%"}
Results and discussion {#res}
======================
Pristine SrTiO$_{3}$
--------------------
In order to study the effect of dopants on the electronic structure of SrTiO$_{3}$, first the pristine SrTiO$_{3}$ is addressed. The relaxed Sr$-$O and Ti$-$O bond lengths are 2.755 Å and 1.945 Å respectively, which are in good agreement with previous experimental results and theoretical calculations [@A1; @A4]. The density of states (DOS) and projected density of states (PDOS) of pristine structure are depicted in Figure 2. The top of valance band is dominated by O states and the contribution of Ti states can be considered below $-3$ eV which refers to the covalent bond Ti$-$O. However Ti states dominate in the bottom of conduction band and O states have good contributions in the whole range of conduction band. A small density of Sr states is observed in the whole energy range which reveals ionic interaction between Sr and TiO$_{6}$ octahedron. The band gap is found to be $3.2$ eV which is in line with the experimental values [@A6].
![(Color online) Density of states of pristine SrTiO$_{3}$.[]{data-label="fig2"}](pristin.eps "fig:"){width="50.00000%"}
mono-doped SrTiO$_{3}$
----------------------
[**Mo/Ta doping**]{}\
It is useful to discuss the influence of mono-doping on the SrTiO$_{3}$ electronic structure to understand the co-doping effect. First a Ti atom is replaced by a Mo one in the pristine SrTiO$_{3}$. The optimized structure shows that the Mo$-$O bond length ($1.944$ Å) does not change significantly as compared to Ti$-$O in pristine crystal due to comparable ionic sizes of Mo$^{6+}$ ($R=0.60$ Å) and Ti$^{4+}$ ($R=0.61$ Å) [@C0; @A3; @C01]. To determine the stability of the mono-doped structures, the defect formation energy ($E_{f}$) has been calculated using the following expression:
$$E_{f} = E_{M-SrTiO_{3}}+\mu_{Ti}-(E_{SrTiO_{3}}+\mu_{M}),$$
where $E_{M-SrTiO_{3}}$ and $E_{SrTiO_{3}}$ are the total energies of metal-doped SrTiO$_{3}$ and pristine SrTiO$_{3}$, respectively. The $\mu_{Ti}$ and $\mu_{M}$ stand for the chemical potential for Ti and dopant atom, respectively, which are assumed as the energy of one metal atom in their corresponding metal bulk structure [@C011]. The defect formation energy for Mo doping is found to be $3.36$ eV. Figure 3 (left) displays the calculated density of states (DOS) and the projected one of the Mo-doped SrTiO$_{3}$, the band gap becomes $2.0$ eV which is less than that of the pristine structure ($3.2$ eV). The Fermi energy is shifted towards the edge of the CB suggesting an n-type conducting character. This is due to the excess two electrons compiled by Mo atom (Mo is in its 6+ oxidation state). The magnetic moment of the Mo-doped SrTiO$_{3}$ becomes $2$ $\mu_{B}$. The mid gap states are created at $-0.8$ eV below the conduction band. The band structure is shifted towards the low energy compared to the pristine structure and the created states arise from the contribution of the localized Mo states.
![(Color online) Density of states of (left) Mo- and (right) Ta-doped SrTiO$_{3}$.[]{data-label="fig3"}](Mo.eps "fig:"){width="50.00000%"} ![(Color online) Density of states of (left) Mo- and (right) Ta-doped SrTiO$_{3}$.[]{data-label="fig3"}](Ta.eps "fig:"){width="50.00000%"}
For Ta-doped structure, the bond length of Ta$-$O becomes ($1.95$ Å) which is slightly larger than that of Ti$-$O due to the comparable ionic sizes of Ta$^{5+}$ ($R=0.64$ Å) and Ti$^{4+}$. The calculated defect formation energy using eq. (2) for Ta-doping gives $-3.21$ eV suggesting a more stable and lower energy cost than Mo-doped structures. The effect of Ta-doping on the electronic structure is illustrated in Figure 3 (right). The Fermi level located in the CB indicates an n-type conducting behavior (similar to Mo doping) due to the 5+ oxidation state of Ta. The band gap does not change as compared to the pristine structure. The low contribution of Ta-localized states appears in the CB but the O and Ti ones still prevail as seen in the pristine SrTiO$_{3}$. Moreover, like Mo-doping, the band structure is moved towards the low energy.
[**Zn/Cd doping**]{}\
Since, the ionic size of Zn$^{2+}$ ($R=0.7$ Å) is slightly larger than that of Ti, there is a little local distortion of the crystal structure where the bond length of Zn$-$O ($R=1.982$ Å) becomes larger than that of Ti$-$O. The local distortion and the magnetic moment (2 $\mu_{B}$) of the structure is related to the oxidation state of Zn$^{2+}$. The defect formation energy in this case is $12.9$ eV indicating less stablility of Zn-doped structure than Ta and Mo-doped ones. The localized Zn states contribute at the top of the valance band and the band gap becomes less than the pristine structure by $0.8$ eV as exhibited in Figure 4 (left). The Zn doping can experimentally improve the photocatalytic properties [@C1; @C2]. In the case of Cd-doped SrTiO$_{3}$, the ionic size of Cd$^{2+}$ ($R=0.95$ Å) is larger than that of Ti$^{4+}$, which reflects the increasing in the bond length of Cd$-$O ($2.034$ Å) compared to the bond length of Ti$-$O in the pristine structure. Cd$^{2+}$ causes similar effect on the magnetic moment and distortion like Zn$^{2+}$ does. The defect formation energy becomes $15.48$ eV. The contribution of Cd-states appears at the top of valance band with O states. This contribution of Cd-dopant material reduces the band gap to $1.8$ eV, see Figure 4 (right).
![(Color online) Density of states of (left) Zn- and (right) Cd-doped SrTiO$_{3}$.[]{data-label="fig4"}](Zn.eps "fig:"){width="50.00000%"} ![(Color online) Density of states of (left) Zn- and (right) Cd-doped SrTiO$_{3}$.[]{data-label="fig4"}](Cd.eps "fig:"){width="50.00000%"}
[**In/Ga doping**]{}\
The last two single dopant atoms are In and Ga. The ionic size of In$^{3+}$ ($R=0.82$ Å) is larger than that of Ti$^{4+}$. This size effect is manifested in the defect formation energy value ($10.58$ eV) which is larger than Ta-doped SrTiO$_{3}$. Furthermore, the bond length of In$-$O becomes $2.055$ Å. The magnetic moment is 1 $\mu_{B}$ due to 3+ oxidation state of In. The influence of In-dopant on the electronic structure of SrTiO$_{3}$ is shown in Figure 5 (right). The trivalent oxidation state for In creates a vacancy in the system and analysing the PDOS indicates that the In-states have insignificant contributions. The O states at the top of the valance band is disturbed along the spin down component and the band gap does not change compared to the pristine SrTiO$_{3}$. Regarding Ga-doped SrTiO$_{3}$, Ga$^{3+}$ has ionic size ($0.62$ Å) which is slightly larger than Ti$^{4+}$. Hence, the Ga$-$O bond length ($R=1.971$ Å) is slightly larger than Ti$-$O and the defect formation energy becomes $7.48$ eV. The magnetic moment and the band gap of Ga-doped SrTiO$_{3}$ is very similar to In-doped one, see Figure 5 (left).
![(Color online) Density of states of (left) In- and (right) Ga-doped SrTiO$_{3}$.[]{data-label="fig5"}](In.eps "fig:"){width="50.00000%"} ![(Color online) Density of states of (left) In- and (right) Ga-doped SrTiO$_{3}$.[]{data-label="fig5"}](Ga.eps "fig:"){width="50.00000%"}
Codoped SrTiO$_{3}$
-------------------
[**(Mo, Zn/Cd) codoping**]{}\
So far, the mono-doped SrTiO$_{3}$ has been discussed in a bit details but the question now is whether the co-doping improves the properties of the pristine material. First, one can substantially look at the geometry and stability of the codoped material. In the case of (Mo/Zn) co-doping, both Mo and Zn dopants change the bond lengths of Mo$-$O and Zn$-$O as compared to the corresponding ones in Mo and Zn mono-doped SrTiO$_{3}$. The Mo$-$O becomes shorter ($1.903$ Å) while Zn$-$O turns out to be longer ($2.012$ Å). The corresponding defect formation energy of the codoped SrTiO$_{3}$ is given by:
$$E_{f} = E_{M1,M2-SrTiO_{3}}+2\mu_{Ti}-(E_{SrTiO_{3}}+\mu_{M1}+\mu_{M2}),$$
where $E_{M1,M2-SrTiO_{3}}$ is the energy of the codoped SrTiO$_{3}$ structure, $\mu_{M1}$ and $\mu_{M1}$ represent the chemical potential for first and second metals, respectively. The defect formation energy of the (Mo, Zn) codoped, $12.4$ eV, is very close to that of Zn-doped SrTiO$_{3}$. Moreover, the stability of the codoped structure can be determined using the defect pair binding energy which is calculated by [@C03; @C031]:
$$E_{b} = E_{M1-SrTiO_{3}}+E_{M2-SrTiO_{3}}+2\mu_{Ti}-(E_{M1,M2-SrTiO_{3}}+E_{SrTiO_{3}}),$$
where $E_{M1-SrTiO_{3}}$ and $E_{M2-SrTiO_{3}}$ are the energies of the mono-doped first and second metals, respectively. The defect pair binding energy is calculated as $E_{b}=4.45$ eV where the positive value indicates that the codoped structure is sufficiently stable [@A1].
Second, it is necessary to explore how the electronic structure of SrTiO$_{3}$ changes upon (Mo, Zn) co-doping. The Fermi energy is located right above the valance band, similar to the pristine structure. The band gap decreases significantly to $1.8$ eV as compared to the pristine SrTiO$_{3}$, which allows the structure to absorb visible light. Furthermore, the band gap is similar to that of Mo-mono-doped SrTiO$_{3}$. The top of the valance band consists of O and Zn states with a little contribution of Ti and Mo states. However, Mo dominates in the bottom of the conduction band with the O states. (Mo, Zn) codoped material reduces the charge carrier loss and forms a charge compensated (n-p compensated) without emerging gap states or splitting the valance band.
![(Color online) Density of states of (left) (Mo, Zn)- and (right) (Mo, Cd)-codoped SrTiO$_{3}$.[]{data-label="fig6"}](MoZn.eps "fig:"){width="50.00000%"} ![(Color online) Density of states of (left) (Mo, Zn)- and (right) (Mo, Cd)-codoped SrTiO$_{3}$.[]{data-label="fig6"}](MoCd.eps "fig:"){width="50.00000%"}
For (Mo, Cd) codoped SrTiO$_{3}$, the bond length Mo$-$O in the codoped structure ($1.923$ Å) is shorter than the corresponding one in the Mo-mono-doped structure. Whereas, the Cd$-$O bond length in the codoped structure ($2.134$ Å) is longer than that in the Cd-mono-doped structure. The formation energy for (Mo, Cd) codoped, $14.94$ eV, is higher than that of (Mo, Zn) codoped SrTiO$_{3}$ due to the large value of the formation energy of Cd. However the defect binding energy ($4.18$ eV) is lower than the corresponding one of the (Mo, Zn) co-doping. Figure 6 (right) reveals the density of states and the projected ones of the (Mo, Cd) codoped SrTiO$_{3}$. The band gap becomes $1.6$ eV. Cd and Ti states contribute at the top of the valance band with dominated O states. Mo and O states have good contributions at the bottom of the conduction band with a lower contribution of the Ti states on contrast to the pristine junction where the latter contribute at the top of conduction band.\
[**(Ta, In/Ga) codoping**]{}\
Last but not least, (Ta, In/Ga) co-doping is explored in this subsection. The bond lengths of Ta$-$O and In$-$O are $1.972$ Åand $2.085$ Å, respectively, which are slightly longer than the corresponding bond length in Ta and In mono-doped SrTiO$_{3}$. The defect formation and pair binding energies are $3.84$ eV which are less than of (Mo, Zn/Cd) codoped SrTiO$_{3}$. The band gap ($3.5$ eV) becomes larger than the pristine structure. The In states contribute at the top of the valance band similar to Ti states but O states still dominate. The bottom of the conduction band consists of analogous contribution of the Ti and O states but lower contribution of the Ta states.
![(Color online) Density of states of (left) (Ta, In)- and (right) (Ta, Ga)-codoped SrTiO$_{3}$.[]{data-label="fig2"}](InTa.eps "fig:"){width="50.00000%"} ![(Color online) Density of states of (left) (Ta, In)- and (right) (Ta, Ga)-codoped SrTiO$_{3}$.[]{data-label="fig2"}](GaTa.eps "fig:"){width="50.00000%"}
The bond lengths of Ta$-$O and Ga$-$O are $ 1.97$ Å which are a bit longer than the corresponding bond lengths in Ta and In mono-doped SrTiO$_{3}$. The defect formation energy in this structure ($7.65$ eV) is higher than (Ta, In) codoped analogue. However the defect pair binding energy ($3.82$ eV) is lower than the (Ta, In) codoped system. The DOS of (Ta, Ga) codoped SrTiO$_{3}$ shows that the band gap becomes ($3.5$ eV) in line with that of (Ta, In) codoped system. The contribution of the Ga states at the top of valance band is small compared to Ti and O states. On the other hand, Ta contributes little to the bottom of the conduction band compared to Ti and O states. Ti states dominate in contrast to (Ta, In) codoped structure. Further, the band gap is larger than the pristine structure. Compared to the pristine structure, (Ta, In) and (Ta, Ga) co-doping cannot be used to extend the absorption edge to the longer-wavelength visible light. Fortunately, the gaps of the (Mo, Zn) and (Mo, Cd) codoped are smaller than the pristine SrTiO$_{3}$. Zn 3d and Cd 4d are located at the top of the valance band and the Mo states prevail at the bottom of the former. Due to the movement of the conduction band bottom to the lower energy, the band gap tightens without any gap states. The mid-gap or localized states are not created due to the n-p compensated co-doping in all studied structures. Therefore, Mo together with Zn or Cd represent perfect co-doping candidates to enhance the photocatalytic properties. Since the ionic size plays an important role from the experimental point of view, the Mo$^{6+}$ and Zn$^{2+}$ dopants which are of comparable ionic radii to that of Ti$^{4+}$ suggest appropriate co-doping materials. Notice that, the effect of dopant concentration on the electronic structure is done using 2 $\times$ 2 $\times$ 1 and 2 $\times$ 2 $\times$ 3 supercells. It is found that the behavior of DOS does not change as compared to that of 2 $\times$ 2 $\times$ 2 supercell in line with the effect of different dopant concentrations on SrTiO$_{3}$ [@A1; @A3].
Optical properties
------------------
The optical absorption properties of a semiconductor photocatalyst is to a large extent linked with its electronic structure which, in turn, affect the photocatalytic activity [@C3]. Absorption in the visible light range is crucial for improving the photocatalytic activity of the doped SrTiO$_{3}$. The optical properties are determined by the angular frequency ($\omega$) dependent complex dielectric function $\varepsilon_{1}(\omega)=\varepsilon_{1}(\omega)+i\varepsilon_{2}(\omega)$, which depends solely on the electronic structure. The imaginary part of the dielectric function, $\varepsilon_{2}(\omega)$, can be calculated from the momentum matrix elements between the occupied and unoccupied states. In addition, the real part can be calculated from the imaginary part by the Kramer-Kronig relationship. The corresponding absorption spectrum, was evaluated as implemented in VASP [@A4; @C4]:
$$\alpha(\omega) = \sqrt{2}\omega\left(\sqrt{\varepsilon_{1}^{2}(\omega)+\varepsilon_{2}^{2}(\omega)}-\varepsilon_{1}^{2}(\omega)\right)^{1/2},$$
![(Color online) The absortion coefficient of the codoped SrTiO$_{3}$.[]{data-label="fig2"}](ab1.ps "fig:"){width="40.00000%"}
where $\alpha$ is the optical absorption coefficient. It can be seen that the pristine SrTiO$_{3}$ can only absorb the narrow UV light (360 nm) and shows no absorption activity in the visible light region, Figure 8. The calculated optical absorption spectra for (Mo, Cd) and (Mo, Zn) codoped SrTiO$_{3}$ show absorption activity at the visible-light region in the range (450-800 nm). The optical absorption of compensated (Mo, Cd) codoped SrTiO$_{3}$ exhibits much more favorable visible-light absorption than compensated (Mo,Zn) codoped system, these results agrees with the earlier discussion
Photocatalytic activity
-----------------------
In this section, the desired thermodynamic conditions for the water splitting photocatalyst is discussed. The most influential criterion for the water splitting photocatalysis is the position of the relative band edges, i.e. the VB edge should lie below the H$_2$O/O$_2$ oxidation level and the CB edge is located above the H$^+$/H$_2$ reduction level. Assuming fixed VB edge position, the closer the CB edge to the H$^+$/H$_2$ reduction level is, the better the photocatalytic efficiency becomes as this reduces the band gap. Figure \[ws\] shows that the calculated band edges positions of the pristine SrTiO$_{3}$ reveal water splitting photocatalytic activity. However, the large band gap does not promote the visible light absorption. In the case of (Mo, Cd) and (Mo, Zn) codoped SrTiO$_{3}$, The valance band edge for both codoped SrTiO$_{3}$ are located below the H$_2$O/O$_2$ oxidation level indicating their ability to release O$_2$ in water splitting process. For conduction band edge, (Mo, Cd) codoped SrTiO$_{3}$ lies below the H$^+$/H$_2$ reduction level but, in the case of (Mo, Zn) dopant, it is very close to the reduction level suggesting that the former may not be capable of releasing hydrogen in the water splitting. Hence, the codoping of (Mo, Zn) can be considered a good photocatalytic material.
![(Color online) The calculated band edges positions of the pristine and codoped SrTiO$_{3}$.[]{data-label="ws"}](ws.eps "fig:"){width="60.00000%"}
Conclusion {#conc}
==========
Density functional theory is used to examine the structure, stability, electronic structure and optical properties of the co-catoinic doping of SrTiO$_{3}$ as well as those of monodopants. Based on the HSE hybrid exchange correlation functional, the calculated band gap of pristine SrTiO$_{3}$ is in good agreement with the experimental finding. Although the Mo and Ta monodopants decrease the band gap of the pristine SrTiO$_{3}$, the semiconductor properties are changed which change the characteristics of the structure. The (Ta, In/Ga) codoping is energetically more favorable than (Mo, Zn/Cd) system. However, codoping with (Ta, In/Ga) does not improve the photocatalyitic properties due to the larger band gap as compared to the pristine system. (Mo, Zn/Cd) codoping reduces the band gap of SrTiO$_{3}$. Since Mo and Zn comprise comparable inoic sizes with Ti, the (Mo, Zn/Cd) co-cationic doping diminishes the band gap of pristine SrTiO$_{3}$ without emerging any undesirable mid-gap states. (Mo, Cd) co-cationic doping has less band gap than (Mo, Zn) one but the CB edge position prevents (Mo, Cd) SrTiO$_{3}$ of being a photocatalyst for water splitting. In this case, the bottom of the conduction band is shifted downwards while the valance band remains unaffected. Eventually, the calculations of the absorption coefficient verified the improvement of the (Mo, Zn) codoped SrTiO$_{3}$ water splitting photocatalytic properties.
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---
author:
- |
Mario Nicodemi$^{a,b}$(\*), Barbara Panning$^c$,\
and Antonella Prisco$^d$\
$^a$ Department of Physics and Complexity Science, University\
of Warwick, UK, and $^b$ INFN Napoli, Italy\
$^c$ Department of Biochemistry and Biophysics,\
University of California San Francisco, California, USA\
$^d$ CNR Inst. Genet. and Biophys. ‘Buzzati Traverso’,\
Via P. Castellino 111, Napoli, Italy
title: A thermodynamic switch for chromosome colocalization
---
A general model for the early recognition and colocalization of homologous DNA sequences is proposed. We show, on a thermodynamic ground, how the distance between two homologous DNA sequences is spontaneously regulated by the concentration and affinity of diffusible mediators binding them, which act as a switch between two phases corresponding to independence or colocalization of pairing regions.
Chromosome recognition and pairing is a general feature of nuclear organization. In particular, these phenomena have a prominent role (and are comparatively better studied) in meiosis, the specialized cell division necessary for the production of haploid gametes from diploid nuclei. During the prophase of the first meiotic division, homologous chromosomes identify each other and pair via a still mysterious long-distance reciprocal recognition process [@Kleckner98; @ScottHawley05; @Zickler06].
Many hypotheses exist on the mechanisms underlying the early stages of coalignment of homologs along their length (see ref.s in [@Kleckner98; @ScottHawley05; @Zickler06]). A longstanding idea is that pairing may occur via unstable interactions, such as a direct physical contact between DNA duplexes (the “kissing model", see, e.g., [@KlecknerWeiner93]). Pairing initially based on non permanent interactions has the important advantage of preventing ectopic association between non-homologous chromosomes, and avoid topologically unacceptable entanglements, leaving space to adjustments [@KlecknerWeiner93]. Several mechanisms could contribute to the outcome of the process, e.g., costrained motion of chromosome in territories, bouquet formation at telomeres, tethering to the nuclear envelope. While chromosome full alignment includes several stages, the early physical contact and colocalization could be driven by specific chromosomal regions bridged by molecular mediators. In this complex scenario, though, the crucial question on the mechanical origin of early recognition and pairing remains unexplained.
Here we explore the thermodynamic properties of a recognition/pairing mechanism based on weak, biochemically unstable interactions between specific DNA sequences and molecular mediators binding them. We show that randomly diffusing molecules can produce a long-distance interaction mechanism whereby homologous sequences spontaneously recognize and become tethered to each other. This colocalization mechanism is tunable by two “thermodynamic switches", namely the concentration of molecular mediator and their affinity for their binding sites. When threshold values in the concentration, or affinity, of mediators are exceeded, homologous sequences are joined together, else they move independently.
[**Model:** ]{} Our model includes (see Fig.\[transitionf\]) two homologue segments involved in mutual recognition and pairing, described as a self-avoiding bead chains, a well established model of polymer physics [@EdwardsDoi], and a concentration, $c$, of Brownian molecular factors having a chemical affinity, $E_X$, for them. We investigate the thermodynamics properties of the system by Monte Carlo (MC) computer simulations [@Binder]. For computational purposes, chromosomal segments and molecules are placed in a volume consisting of a cubic lattice with spacing $d_0$ (our space unit, of the order of the molecular factors length) and linear sizes $L_x=2L$, $L_y=L$ and $L_z=L$ (see Fig.\[transitionf\]). In each simulation, the ‘beads’ of the chromosomal segments start from a straight, vertical line configuration, at a distance L from each other, and molecular mediators from a random initial distribution. Diffusing molecules randomly move from one to a nearest neighbor vertex on the lattice. On each vertex no more than one particle can be present at a given time. The chromosomal segments diffuse as well on such a lattice performing a Brownian motion under the constraint that two proximal ‘beads’ on the string must be within a distance $\sqrt{3}d_0$ from each other (i.e., on next or nearest next neighboring sites on the lattice). For the sake of simplicity, we disregard here the rest of the chromosomes and DNA segment ends are costrained to move tethered to the bottom and top plane of the system volume (Fig.\[transitionf\]). When neighboring a chromosomal chain, molecules interact with it via a binding energy $E_X$. Below, we mainly discuss the case where $E_X$ is of the order of a “weak” hydrogen bond-like energy, say 3 kJ/mole, which at room temperature corresponds to $E_X=1.2kT$ [@Watson]. In our simulations, at each time unit (corresponding to a MC lattice sweep) the probability of a particle to move to a neighboring empty site is proportional to the Arrhenius factor $r_0\exp(-\Delta E/kT)$, where $\Delta E$ is the energy barrier in the move, $k$ the Boltzmann constant and $T$ the temperature [@Watson; @Stanley]. The factor $r_0$ is the reaction kinetic rate, depending on the nature of the molecular factors and of the surrounding viscous fluid, and sets the time scale. We employ $r_0=30 sec^{-1}$, a typical value in biochemical kinetics. Averages are over up to 2048 runs from different initial configurations.
[**Results:** ]{} First we show how the interaction of chromosomes with molecular mediators drives colocalization. To this aim, we calculated the thermodynamic equilibrium value of the average square distance (relative to the system linear size $L$) between the two chromosomal segments: $$d^2=\frac{1}{N}\sum_{z=1}^N {\langle r^2(z)\rangle \over L^2}$$ where $N$ is the number of beads in each string (here $N=L$) and $\langle r^2(z)\rangle$ is the average (over MC simulations) of the square distance of the beads at ‘height’ $z$. The average value of $d^2$ is maximal when the two ‘chromosomes’ float independently and decreases if parts of the polymers become colocalized, approaching zero when a perfect alignment is attained.
The equilibrium distance, $d^2$, depends on the concentration, $c$, of mediators. At low concentration (see Fig.\[transition\], e.g., $c<c_1$) $d^2$ has a value of the order of the system size (around $40\%$ of $L^2$), corresponding to the expected average distance of two independent strings undergoing Brownian motion in a box of size $L$; a typical configuration for $c=0.3\%$ being shown in Fig.\[transitionf\] panel A). Indeed, the physical basis for the independence of chromosomes exposed to a low concentration of mediating molecules is intuitive: pairing can occur when bridges are formed by molecules attached to couples of binding sites. A single bridging event, however, can be statistically quite unlikely since ‘weak’ bonds are biochemically unstable and to form a bridge a diffusing molecule must first find (and bind) a site on one chromosome and then together they have to successfully encounter the second one.
Fig.\[transition\] shows, however, that when $c$ is higher than a threshold value, $c_{tr}$ (for $E_X=1.2kT$, $c_{tr}\simeq 0.7\%$), $d^2$ collapses to zero: this is the sign that the two ‘chromosomes’ have colocalized; a typical picture of the system state, for $c=2.5\%$, is shown in panel C) of Fig.\[transitionf\]. Actually, when $c$ is high enough chances increase to form multiple bridges and, as they reinforce each other, configurations where molecules hold together the two polymers become stabilized. The threshold concentration value, $c_{tr}$, corresponds to the point where such a positive mechanisms becomes winning, and can be approximately defined by the inflection point of the curve $d^2(c)$. Alike phase transitions in finite-size systems [@Binder; @Stanley] (see below), around $c_{tr}$ there is a crossover region which can be located, for instance, between the concentrations $c_1$ and $c_2$ (see Fig.\[transition\]) defined by the criterion that $d^2$ is close within $5\%$ to the random or zero plateau value (for $E_X=1.2kT$, $c_1\simeq 0.3\%$ and $c_2\simeq 2\%$).
In Fig.\[transition\], along with the distance between chromosomes, $d^2$, we plot the squared fluctuations of the distance (i.e., its statistical variance), $\Delta d^2(c)$, as a function of the concentration of mediators. For $c<c_1$, both $d^2(c)$ and $\Delta d^2(c)$ have the non zero value found for non interacting Brownian strings in the independent diffusion regime ($\Delta d^2\sim 30\%$); instead, $\Delta d^2(c)=0$ for $c>c_2$ in the tight colocalization regime. Interestingly, in the crossover region, $d^2(c)$ is smaller than in the purely random regime, although it has marked fluctuations ($\Delta d^2(c)$ can be even larger than $d^2(c)$). This situation is illustrated by a picture of a typical configuration, for $c=0.9\%$, shown in panel B) of Fig.\[transitionf\]. In such an intermediate regime chromosome couples are continuously formed and disrupted.
Summarizing, our results show that colocalization is spontaneously induced by the ‘collective’ binding of molecular mediators and occurs only when $c$ is above a critical value, $c_{tr}$, i.e., in the ‘colocalization phase’. Conversely, when $c$ is below $c_{tr}$, $d^2(c)$ has the same value found for two non interacting Brownian strings. This is the ‘random phase’, where chromosomes are independent. The concentration of mediators acts as a switch between the two phases, while around the critical threshold chromosomes undergo transient interactions.
A similar effect is found when, for a given (high enough) concentration, $c$, the chemical affinity, $E_X$, of binding sites is changed (see Fig.\[transition\] lower panel): when $E_X$ is smaller than a threshold value, $E_{tr}$, the two polymers float independently one from the other. Around $E_{tr}$ a crossover region is found, and as soon as $E_X$ gets larger than $E_{tr}$, an effective attraction between polymers is established and they are spontaneously colocalized. Another potential layer of regulation of the system is the number of binding sites for molecular mediators. In fact, a reduction in the number of binding sites produces the same effect of a reduction in the affinity of mediators, that is, chromosomes become unable to find and bind each other.
The pairing mechanisms illustrated above has a thermodynamics origin. It is a ‘phase transition’ [@Stanley] occurring when entropy loss due to polymer colocalization is compensated by particle energy gain as they bind both polymers, the lower $E_X$ the higher the concentration, $c$, required. Actually, the transition is found in a broad region of the $(E_X,c)$ plane, as shown in Fig.\[ph\_diag\] where the system phase diagram is plotted in a range of typical biochemical values of “weak” binding energies $E_X$. For very low values of $E_X$ the colocalization can be, instead, impossible. The overall properties of such a phase diagram (independent v.s. colocalized chromosomes) are robust to changes in the model details, though the precise location of the different phases can be affected [@Stanley]. Summarizing, when soluble mediators bind a specific recognition sequence on homologous chromosomes, recognition and colocalization of homologs can occur, as a result of a robust and general thermodynamic phenomenon, namely a phase transition occurring in the system. The higher the affinity of mediators for chromosomal binding sites, the lower is the threshold concentration of mediators that promotes colocalization (see Fig.\[ph\_diag\]).
[**Discussion:** ]{} We described a general colocalization mechanism, grounded on thermodynamics, whereby specific regions of a pair of chromosomes can spontaneously recognize each other and align. Physical juxtaposition is mediated by sequence-specific molecular factors that bind DNA via weak, non permanent, biochemical interactions. When the concentration/affinity of molecular mediators is above a critical threshold an effective attraction between their binding regions is generated, leading to a close alignment; else chromosomes float away from each other by Brownian motion. In the threshold crossover region, pairing sites undergo transient interactions: the average distance is shorter than in the purely random regime, but marked fluctuations are observed.
In our simulations, the two homologous pairing regions are described as polymers diffusing with their ends tethered to the upper and lower planes of the system box. This recalls telomeres tethering to the nuclear envelope observed at meiosis. While it is not a prerequisite for the switch mechanism, on the other hand, it can enhance the switch effects [@Kleckner98; @ScottHawley05; @Zickler06]. Releasing such a constraint doesn’t change the general results, but pairing regions would collapse in a more disordered geometry. The overall properties of the phase diagram (independent vs. colocalized chromosomes) are robust to changes in the model details [@noinpreparation]. A model including many a pair of chromosomes has longer equilibration times, as expected in a crowded environment, yet, its phase diagram is unchanged. The scenario is also unaltered in the case of mediators that interact with each other and aggregate.
An implication of this model is that a cell can regulate the initiation of homologous chromosome interaction by up-regulating the concentration of mediators or their affinity for DNA sites (e.g., through changes in the chromatin or by a chemical modification of the mediator). This switch has general and robust roots in a thermodynamics phase transition [@Stanley], irrespective of ultimate molecular and biochemical basis. In real cells, specific short chromosomal regions (“pairing centers") could mediate the early steps of homolog recognition, and act as a seed and reference point to a subsequent stable long scale chromosomal pairing, which could involve additional mechanisms. A speculation is that the threshold effect can be exploited to ensure a precise control of pairing formation/release, while the presence of a crossover region in concentration to reduce undesired entanglements. The initial binding molecules could, in turn, help the sequences in recruiting complexes later used to other purposes (e.g., in pairing stabilization, synapsis, recombination).
In the present model individual mediators do not need to be strongly binding to glue homologous chromosomes together, and any molecules with above threshold affinity can induce attraction. Specificity of colocalization among many chromosome pairs could be, indeed, obtained by sets of molecules binding, with higher affinities, specific homologous sequences. While the molecular mediators considered here are supposed to have more than one “DNA binding domain", proteins that can bind a single DNA site, but are able to make protein-protein interactions, could also mediate co-localization. As a pair of linked proteins is, in fact, a single molecular mediator the thermodynamics picture is unchanged. Finally, direct DNA duplex interactions [@KlecknerWeiner93] could replace, or help, binding molecules. A duplex kissing site would correspond in our model to a binding site with a molecular mediator already attached, so the overall behavior should be similar.
Experimental discoveries on meiotic pairing have accomplished huge progresses, but the mechanisms for homologue early coalignment are still unclear [@Kleckner98; @ScottHawley05; @Zickler06]. In [*C. elegans*]{}, for instance, homologs proper pairing is primarily regulated by special telomeric regions, known as “pairing centers" (PCs) [@McKim88; @Villeneuve94; @MacQueen05]. Homologous PCs interact, during early prophase, with HIM/ZIM Zn-finger proteins which are necessary to mediate pairing [@Phillips05; @Phillips06]. Specific sites and proteins are also involved in meiotic pairing of [*Drosophila*]{}. In male, on the X and Y chromosomes, a 240bp repeated sequence in the intergenic spacer of rDNA acts as a pairing center, and autosomes pair, as well, by the interaction of a number of sites (see ref.s in [@ScottHawley05; @Zickler06]). A similar behavior is observed in [*Drosophila*]{} female [@Hawley92; @Dernburg96]. In [*Drosophila*]{} males, special proteins, SNM and MNM, have been also discovered which bind X-Y and autosomal pairing sites at prophase I, and are required for pairing [@Thomas05]. The question is open whether the present model applies to such an experimental scenario. In a picture where pairing is mediated by unstable interactions, thermodynamics dictates, anyway, a precise framework showing that minimal “ingredients", such as soluble DNA binding molecules and homologous arrays of binding sites, can in fact be sufficient for pairing if the balance of mediator concentration and DNA affinity is appropriate.
Our thermodynamic switch theory is prone to experimental tests (e.g., the existence of threshold effects in mediator concentration, $c$). It can be exploited, as well, for a quantitative understanding of the effects on pairing, e.g., of deletions (which can be modeled here by reducing the binding site number within $L$), or of chemical modifications of binding sequences (modeled by changes in $E_X$), and to guide the search for candidates for chromosomal sites and interaction mediators. Finally, the general message of the model may be applicable to various cellular processes that involve the spatial reorganization of DNA in nuclear space (e.g., organization of chromosomal loci and territories, justapposition of DNA sequences in transcriptional regulation, somatic pairing, pairing of X chromosomes at the onset of X inactivation [@Kleckner98; @ScottHawley05; @Zickler06; @Lee06; @ap14; @mario; @nicodemi07b; @Lanctot; @Misteli07]).
We thank N. Kleckner and A. Storlazzi for very helpful discussions and critical reading of the manuscript.
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---
abstract: 'We present the results from [[*Suzaku*]{} ]{}observations of the Perseus galaxy cluster, which is relatively close, the brightest in the X-ray sky and a relaxed object with a cool core. A number of exposures of central regions and offset pointing with the X-ray Imaging Spectrometer cover a region within radii of $20''-30''$. The central data are used to evaluate the instrumental energy-scale calibration with accuracy confirmed to within around 300 [km s$^{-1}$]{} by the spatial and temporal variation of the instruments. These deep and well-calibrated data are used to measure X-ray redshifts of the intracluster medium. A hint of gas bulk motion, with radial velocity of about $-(150-300)$ [km s$^{-1}$]{} relative to the main system was found at $2-4$ arcmin (45–90 kpc) west of the cluster center, where an X-ray excess and a cold front were found previously. No other velocity structure was discovered. Over spatial scales of 50–100 kpc and within 200 kpc radii of the center, the gas-radial-velocity variation is below 300 [km s$^{-1}$]{}, while over scales of 400 kpc within 600 kpc radii, the variation is below 600 [km s$^{-1}$]{}. These X-ray redshift distributions are compared spatially with those of optical member galaxies for the first time in galaxy clusters. Based on X-ray line widths gas turbulent velocities within these regions are also constrained within 1000–3000 [km s$^{-1}$]{}. These results of gas dynamics in the core and larger scales in association with cluster merger activities are discussed and future potential of high-energy resolution spectroscopy with [ASTRO-H ]{}is considered.'
author:
- 'T. Tamura, N. Yamasaki, R. Iizuka'
- 'Y. Fukazawa'
- 'K. Hayashida, S. Ueda'
- 'K. Matsushita, K. Sato'
- 'K. Nakazawa'
- 'N. Ota'
- 'M. Takizawa'
- 'Department of Physics, Yamagata University, 1-4-12 Kojirakawa-machi, Yamagata, 990-8560, Japan'
title: 'GAS BULK MOTION IN THE PERSEUS CLUSTER MEASURED WITH [[*Suzaku*]{} ]{}'
---
INTRODUCTION {#intro}
============
Galaxy clusters are the largest and youngest gravitationally-bound cosmic structure and form hierarchically through collisions and mergers of smaller systems. Spatial and radial velocity distributions of member galaxies and X-ray observations of the intracluster medium (ICM) have revealed that some systems are still forming and unrelaxed. Along with gravitational lensing observations, these studies have provided dynamical measurements of dark matter in clusters.
As the structure forms, gravitational potential governed by dark matter pulls and thus heats the ICM through shocks. Sharp X-ray images obtained by [[*Chandra*]{} ]{}revealed shocks and density discontinuities ( so-called “cold fronts”) suggesting supersonic or transonic gas motions, respectively [see @mv07 for a review]. These heating processes would then develop gas turbulence, and accelerate particles, which in turn generate diffuse radio halos and relics. The activities of central massive black holes also disturb and warm up the gas around cluster cores. The motion of gas should be measured to facilitate direct understanding of these energy flows. Constraining total energy distribution in a system allows the total gravitational mass to be measured, as required for precise cosmology. Energies in gas random or ordered motions could be the key uncertainty to calibrate total mass distribution, as suggested by numerical simulations [e.g. @evrard96; @Nagai07; @Takizawa10; @Suto13].
Gas bulk or turbulent motions can be measured directly using the Doppler shift or by broadening of X-ray line emissions. However, the limited energy resolutions of current X-ray instruments continues to hinder such measurements. For example typical CCD energy resolution is about 130 eV (in full width half maximum; FWHM), compared with a possible energy shift of $\sim 20$ eV corresponding to a radial velocity of 1000 [km s$^{-1}$]{}, all at the Fe-K line energy.
The [[*Suzaku*]{} ]{}XIS [@koyama07] is currently the optimal X-ray spectrometer for gas motion search with Fe-K lines, as was demonstrated when gas bulk motion was detected in a merging cluster, Abell 2256, by [@tamura2011]. In addition, XIS data imposed tight constraints on gas motion in a number of clusters, as summarized in Table \[tbl:suzaku-res\].
[lll]{} Centaurus & $\Delta v < 1400$ km s$^{-1}$ & 1\
Oph. & $\Delta v < 3000 $ km s$^{-1}$ & 2\
AMW 7 & $\Delta v < 2000 $ km s$^{-1}$ & 3\
A 2319 & $\Delta v < 2000$ km s$^{-1}$ & 4\
A 2256 & $\Delta v = 1500 \pm 300 ({\rm stat.}) \pm 300 ({\rm sys.}) $km s$^{-1}$ & 5\
Coma & $\Delta v < 2000$ km s$^{-1}$ & 6\
A 3627 & $\Delta v < 800$ km s$^{-1}$ & 7\
Numerical studies have striven to clarify cluster formation histories and the associated origins, development, and dissipation of various types of gas motions. However, by nature cluster evolution is governed by random processes, which means systematic measurements of systems in a wide range of evolution stages are also required as well as these theoretical efforts. The geometrically-complex nature of gas dynamics should be spatially resolved. For example, gas dynamics in apparently relaxed clusters should be studied as demonstrated by [@ota07 hereafter OTA07] using early [[*Suzaku*]{} ]{}observations of the Centaurus cluster. To improve the limit of gas bulk motion in a relaxed system, we analyzed a large set of [[*Suzaku*]{} ]{}data of the Perseus cluster.
The Perseus cluster, the brightest in X-ray, is a prototype of a nearby and relaxed object. At the same time, it exhibits a number of past and current activities at core and larger scales, while at the cluster center, the very peculiar central galaxy NGC 1275 displays a complex network of emission-line nebulosity. Based on the thread-like structure of filaments resolved by the Hubble Space Telescope, [@Fabian2008] suggested the magnetic support of the structure and ordered gas motions not highly turbulent over scales of $<10$ kpc around the galaxy. The cluster X-ray emission peaks sharply toward the galaxy and its central active galactic nucleus (AGN), 3C 84. The AGN also develops jets and radio lobes where high-energy particles and X-ray-emitting-hot gas interact, as discovered by [@Boehringer1993]. Associated with these current and possibly past AGN activities, fine gas structures such as bubbles, ripples, and weak shock fronts were revealed by deep [[*Chandra*]{} ]{}imaging [@Fabian2011 and references therein]. These processes dissipate energies into various types of gas motions. Independently, based on the lack of resonant scattering of Fe-K line emissions, @chura04 suggested gas motions in the core of a velocity of at least half the speed of sound.
West of NGC 1275, a chain of galaxies is distributed toward another radio galaxy IC 310, while other member galaxies and clusters virtually aligned with this chain form the Pisces-Perseus supercluster. On the other hand, the ICM X-ray emission has been traced up to at least 2 Mpc with an elongation in the east-west direction. Based on the X-ray morphology of early observations, [@Hirayama1978] hypothesized that the gas and galaxies are rotating with a velocity of close to 3000 [km s$^{-1}$]{}. [@Furusho2001] found asymmetrical ICM temperature distribution and suggested a past merger in the direction parallel to the line of sight. Using [[*XMM-Newton*]{} ]{}@chura03 further studied the asymmetric structures around the core ($R<20'$) and revealed a hot “horseshoe” surrounding the cool core. They also assumed a minor merger along the east-west direction in the sky plane. [@Simionescu2012] studied the gas distribution over wider spatial scale ranges and suggested mergers and associated swirling gas motions.
The cluster center has been observed twice-yearly as a calibration target of the XIS and we used all available data. The integrated deep exposure of this unique object along with the high sensitivity of the XIS provides one of the best quality X-ray spectra from clusters. We also used some offset region ($R<30'$) data and derived larger scale ICM emission properties.
Parts of the [[*Suzaku*]{} ]{}Perseus data set were used for a number of measurements. @t09 reported the first detection of Cr and Mn X-ray lines from clusters and ICM elemental abundance measurements, while [@Matsushita2013] and [@Werner13] reported abundance measurements out to the cluster outskirt. Using offset XIS data along with the HXD data, @nishino10 reported an upper temperature limit in the outer region. @Simionescu2011 [@Simionescu2012] measured large-scale gas thermal distribution over the viral radius.
In the next section, we describe observations and data reduction, while in section \[cal\] we evaluate the accuracy of the XIS energy response, which is crucial to calibrate our measurement. In sections \[sect:ana-center\] and \[ana-large\] we measure gas bulk velocities within the central pointing and over a larger scale respectively, for the first time in this system, which enables one of the most accurate measurements of gas bulk motion in relaxed clusters. Finally these results are summarized and discussed in the last section.
Throughout this paper, we assume the flowing cosmological parameters: $H_0 = 70$ km s$^{-1}$Mpc$^{-1}$, $\Omega_\mathrm{m} = 0.3$, and $\Omega_\mathrm{\Lambda} = 0.7$. At the cluster redshift of 0.0183, one arcmin corresponds to 22.2 kpc. We use the 68% ($1\sigma$) confidence level for errors, unless otherwise stated.
OBSERVATIONS AND DATA REDUCTION {#sect:obs}
===============================
The Perseus cluster center is a calibration target and the central galaxy, NGC 1275, was always located at the CCD center. We call these observations [*central*]{} pointings. Here we use the data obtained from 2006 to 2013 taken in the normal window mode and Table \[obs:obs-center\] shows an observation log. The total exposure time is 576 ks. The XIS 2 sensor was only available until November 2006. These data were used in the instrument calibration reported by @koyama07, @ozawa09, and @uchiyama09.
The off-center regions were also observed many times. We call these [*offset*]{} pointings and use the data given in Table. \[obs-offset\]. The total exposure time is 327 ks. Fig. \[obs:rosat2\] shows the field of views of the observations.
Detailed descriptions of the [[*Suzaku*]{} ]{}observatory, XIS instrument, and X-ray telescope are found in @mitsuda07, @koyama07, and @serlemitsos07, respectively.
The XIS field of view is a square of $17'.8 \times 17'.8$ with $1024 \times 1024$ pixels. Accordingly, 1 pixel $\simeq$ 1.04 arcsec. The energy resolution is about 50 eV at 1 keV and 130 eV at 6 keV in FWHM. Initially the XIS are operated normally in the spaced-row charge injection (SCI) off mode. Since 2006 October, the SCI on has been used in normal mode.
We started the analysis from archived cleaned event files, which were filtered with standard selection criteria, and used the latest calibration file as of 20012 November 6.
We examined light curves in the 0.3–2.0 keV band excluding the central bright region events ($R<6'$), for stable-background periods. None of the data showed any flaring event.
The XIS energy data have an original energy channel width of 3.65 eV. We grouped these by common bin sizes of about one-third of the FWHM of the energy resolution. Around the Fe-K line, the width is 8 channels or about 30 eV. To establish the XIS energy response function, we used [xisrmfgen]{} software alongside [makepi]{} (version-20121009) to assign PI values. The energy bin size is 1 eV ($\sim 0.15\times 10^{-3}$ of the Fe-K line energy), which means we can resolve the change of energy at less than the systematic limit of about $10^{-3}$. For spectra fitting, we use XSPEC [version-12.8; @Arnaud1996] and maximum likelihood ($\chi^2$) statistics implemented in the same. Note that the above binning gives about 15 counts per bin in the lowest-count spectrum and more than 20 counts per bin in other spectra analyzed below.
[llllcc]{} CEN-0602 & 2006 Feb & 800010010 & off & 43.7 & 260.2\
CEN-0608 & 2006 Aug & 101012010 & off/on & 92.0 & 66.0\
CEN-0702 & 2007 Feb & 101012020 & on & 40.0 & 258.7\
CEN-0708 & 2007 Aug & 102011010 & on & 35.1& 83.4\
CEN-0802 & 2008 Feb & 102012010 & on & 34.9 & 255.2\
CEN-0808 & 2008 Aug & 103004010 & on & 34.1 & 86.8\
CEN-0902 & 2009 Feb & 103004020 & on & 46.3 & 256.1\
CEN-0908 & 2009 Aug & 104018010 & on & 34.2 & 67.0\
CEN-1002 & 2010 Feb & 104019010 & on & 33.6 & 277.3\
CEN-1008 & 2010 Aug & 105009010 & on & 29.6 & 66.6\
CEN-1102 & 2011 Feb & 105009020 & on & 32.9 & 259.7\
CEN-1108 & 2011 Aug & 106005010 & on & 34.1 & 83.8\
CEN-1202 & 2012 Feb & 106005020 & on & 41.1& 262.0\
CEN-1208 & 2012 Aug & 107005010 & on & 33.2& 72.6\
CEN-1302 & 2013 Feb & 107005020 & on & 41.2 & 256.3\
[lllllrrr]{} A & 2006 Sep & 801049010 & off& 25 & 36 & 345\
B & 2006 Sep & 801049020 & off& 27 & 31 & 260\
C & 2006 Sep & 801049030 & off& 31 & 28 & 140\
D & 2006 Sep & 801049040 & off& 8 & 33 & 56\
E & 2011 Feb & 805045010 & on & 27 & 18 & 180\
F & 2011 Feb & 805046010 & on & 18 & 16 & 0\
G & 2011 Feb & 805047010 & on & 17 & 14 & 110\
H & 2011 Feb & 805048010 & on & 15 & 17 & 58\
E1 & 2009 Jul & 804056010 & on & 7 & 16 & 90\
N1 & 2009 Aug & 804063010 & on & 14 & 15 & 330\
S1 & 2010 Aug & 805096010 & on & 8 & 16 & 200\
W1 & 2010 Aug & 805103010 & on & 6 & 16 & 280\
SE1 & 2011 Aug & 806099010 & on & 11 & 15 & 140\
SW1 & 2011 Aug & 806106010 & on & 12 & 14 & 225\
NNE1 & 2011 Aug & 806113010 & on & 9 & 16 & 10\
F\_2 & 2013 Feb & 807019010 & on & 14 & 16 & 0\
S1\_ 2 & 2013 Feb & 807020010 & on & 23 & 16 & 200\
H\_2 & 2013 Feb & 807021010 & on & 18 & 19 & 58\
W1\_2 & 2013 Feb & 807022010 & on & 23 & 16 & 280\
G\_2 & 2013 Feb & 807023010 & on & 14 & 19 & 110\
ENERGY RESPONSE CALIBRATION {#cal}
===========================
Motivations {#cal-mot}
-----------
The main goal of this paper is to measure Doppler shifts of cluster K-shell iron lines in X-rays. These astronomical shifts should be separated from instrumental effects on the energy scale (i.e. recorded pulse-height vs. energy). The instrumental effects include (i) CCD-to-CCD variation, (ii) intra-chip [*spatial*]{} variations attributable to charge-transfer inefficiency (CTI) and by CCD segment-to-segment qualities, (iii) observation-to-observation [*temporal*]{} variation, (iv) absolute energy determination, and unknown systematics. These instrumental characteristics have been calibrated and extensively evaluated by the instrument team, which is an advantage of the [[*Suzaku*]{} ]{}XIS over other CCD-type spectrometers. These results have also been demonstrated; not only in the cluster redshift analysis stated in section 1 but also in X-ray spectroscopy in the Galactic center and ridge [e.g. @koyama07] and in supernova remnants [e.g. @Hayato2010]. We summarize the latest calibration status in subsection \[sect:cal-status\]. Using the latest calibration and the Perseus data, we further evaluate accuracy in subsection \[sect:cal-cal\].
Among the above instrumental variations, factor (i) can be checked by comparing observed data from different CCDs in any observations. The built-in calibration source lines are used to evaluate factors (iv) for each observation.
When analyzing the Perseus central data (section \[sect:ana-center\]), factor (ii) is crucial and (iii) can be internally checked against the data. For the offset data analysis (section \[ana-large\]), (iii) is important. Moreover, (ii) also affects the result because each pointing observation includes different distributions of events over the detector position.
Reported Status {#sect:cal-status}
---------------
@koyama07 estimated the systematic uncertainty of absolute energy in the Fe-K band to be within $+0.1$ and $-0.05$%, based on Fe lines observed from the Galactic center alongside Mn K$_\alpha$ and K$_\beta$ lines (at 5895 and 6490 eV respectively) from the built-in calibration source ($^{55}$Fe). Independently, OTA07 investigated the XIS data and evaluated the energy-scale calibration in detail. Using early calibration and early Perseus cluster data (sequence 800010010), they estimated the systematic error of the [*spatial*]{} gain non-uniformity (ii) to be $\pm 0.13$% (68% confidence level). A similar analysis was performed by @tamura2011 using two Perseus cluster datasets with a new calibration and the accuracy was confirmed. Furthermore, @ozawa09 systematically examined the XIS data obtained from the start of the operation in July 2005 until December 2006 (SCI off mode data). They reported that the [*spatial*]{} dependence of the energy scale (ii) was well corrected for charge-transfer inefficiency and cited a time-averaged uncertainty of absolute energy of $\pm$ 0.1%. @uchiyama09 reported the energy scale calibration for SCI-on mode. @Sawada2012 reported an improved CTI correction. They archived energy accuracies of $<0.7$% and $<0.1$% at 1 keV and 7 keV, respectively, over the entire CCD position in the SCI-on mode data. We based on these calibration methods in the latest calibration files.
The energy resolution changed over time and was measured at 140 eV (July 2005; SCI off), 200 eV (April 2007 ; SCI off), and 160 eV (March 2008; SCI on) for the FI CCD at Mn K$\alpha$ in FWHM. This gradual change in energy resolution was also calibrated; the typical uncertainty of resolutions is 10–20 eV in FWHM.
Calibration Source Data {#sect:cal-cal}
-----------------------
To evaluate the absolute energy scale in different CCD segments (A and D in XIS-0,1,3) we use the Perseus central pointing data (Table \[obs:obs-center\]) and extracted spectra of calibration sources which illuminate two corners of each CCD. These spectra within the energy range 5.3–7.0 keV are fitted with two Gaussian lines (zero intrinsic width) for the Mn K${\alpha}$ and K${\beta}$ along with a bremsstrahlung continuum component. We parametrize the energy scale using a common “redshift” deviated from the expected value of 5895 eV (Mn K${\alpha}$) and 6490 eV (Mn K${\alpha}$). Spectra from different observations, CCDs, and segments were fitted separately.
As shown in Table \[cal:cal-var\], the average redshifts are all below $10^{-3}$, which indicates that the absolute energy scale is calibrated accurately if averaged over observations. The variations within each segment are comparable with statistical errors, which suggests that systematic errors in instrumental variations are below statistical ones.
Following OTA07 and based on the obtained distribution of redshifts we measure residual variation by calculating $\sigma_{\rm sys}$ such that $\chi^2 = \Sigma (z_i - <z>)^2/(\sigma_i^2 + \sigma_{\rm sys}^2)$ equals the degree of freedom. Here $z_i, <z>$, and $\sigma_i$ are the obtained redshift, its average, and statistical errors, respectively. We found $\sigma_{\rm sys}$ to be $0.8\times 10^{-3}$. This variation includes CCD-to-CCD (i), segment-to-segment (ii), and temporal (iii) factors. The calibration source is attached to the opposite side of read-out nodes of each CCD; hence the above calibration source accuracy energy indicates the precision of the CTI calibration.
Averaging the four segments of the FI-CCDs (Table \[cal:cal-var\]), we obtain a redshift close to $-0.1\times 10^{-3}$, while BI redshift is close to $0.8 \times10^{-3}$. This relatively significant BI spectral deviation, which is still within statistical errors, could introduce systematic error. Because of better sensitivity of the FI at higher energy bands, FI is calibrated better than BI at these energy bands. Moreover the FI spectra also provide better energy resolution. In view of these properties, we use only the FI data in the following analysis.
To evaluate energy resolution calibration, we fitted the Mn lines by adding an artificial line width ([$\sigma_{\rm add}$]{}) and found that the best-fit [$\sigma_{\rm add}$]{} tended to be zero. The 1 $\sigma$ upper limit of [$\sigma_{\rm add}$]{} ranged from 5 to 80 eV depending on the line counts with an average of 25 eV. These confirm the reported energy resolution calibration. We assume this 25 eV $\sim 1250$ km s$^{-1}$ to be a 1$\sigma$ systematic error in line width at around the Fe-K energy.
[lcccc]{} XIS0/A & 0.23 & 1.24 & 2.7 & 1.0\
XIS0/D & 0.31 & 1.37 & 2.6 & 0.7\
XIS3/A & -0.11 & 0.63 & 1.6 & 1.1\
XIS3/D & -0.72 & 1.46 & 4.2 & 1.0\
XIS1/A & -0.65 & 1.28 & 3.5 & 0.9\
XIS1/D & -0.95 & 1.43 & 2.8 & 1.4\
The Perseus Central Observations {#cal-per}
--------------------------------
### Motivations {#motivations}
The spatial variation in the XIS energy scale \[(ii) in subsection \[cal-mot\]\] was calibrated using Perseus cluster data as one of the primal sources. Here we use the same source but all available data (Table \[obs:obs-center\]) alongside the latest calibration information and evaluate the accuracy.
We should note that the Perseus data themselves are used to correct the CTI, assuming the cluster has uniform Fe-line energy over the central region ($17'.8\times 17'.8$). Possible deviations from this assumption could add systematic errors. We assume these systematic errors to be insignificant on the followings grounds. Firstly, the CTI parameters are determined using Perseus data observed in different roll-angles , meaning the CTI correction is not solely dependent on cluster-intrinsic distribution of Fe-line energy. Secondly, the CTI correction is not tuned for each observation, but based on instrumental characteristics as a function of the number of CCD read-out transfer (ACT-Y coordinates), CCD segments, observation time, and X-ray energy. Thirdly, built-in calibration source data along with other astronomical sources such as the Galactic center and supernova remnants (at lower energies) were used to correct and evaluate the calibration. We check the assumption more quantitatively in this subsection.
Here we group observations by their roll-angles. One group (“winter”) data were obtained in February each year with roll-angles of 255–277. The other (“summer”) data were obtained in August with roll-angles of 66–86. Note that pointing centers are close to each other within about 30 arcsec in all data. Within each group, different observations share roll angles close to each other, meaning a certain detector position allows a particular sky region to be observed. Using these data sets with two different extraction methods, we checked both [*instrumental*]{} and [*cluster-*]{}intrinsic variations in the energy scale.
In the first extraction (subsection \[cel-per:det\]) we integrated spectra from different sky regions sorted by detector positions and found no significant variation in the line centers measured among different detector positions nor between the two-roll groups. In the second extraction, given in section \[sect:ana-center\], we integrated spectra from different detector positions sorted by sky regions. We found no significant variation among different sky regions nor between the two-roll groups. These analyses are not completely independent and we cannot completely separate [*instrumental*]{} and [*cluster*]{} variations. Nevertheless, both indicate that both [*instrumental*]{} and [*cluster*]{} variations are small and within certain error level.
### Detector-sorted spectra {#cel-per:det}
We divided the XIS field of view into $4 \times 4$ cells of size $4'.5\times4'.5$ in detector coordinates (DETX and DETY) as illustrated in Fig. \[cal:detector\]. We call these cells DET00, DET01 and so on up to DET33.
Based on each cell and observation, a spectrum is extracted, whereupon each spectrum in the 5.7–7.3 keV band is fitted with two Gaussian lines for He-like K${\alpha}$ ($\sim 6700$ eV) and H-like K${\alpha}$ ($\sim 6966$ eV) and a bremsstrahlung continuum model.
As stated in subsection \[ana-center-lw\], since we found no need for additional line width for Gaussian lines, we fixed the width at zero henceforth. We used the common redshift of the two lines as a fitting parameter. Examples of the spectral fitting are given in Appendix (Fig. \[cal-app:sp\]).
We found that the instrumental non-X-ray background is well-below the source count and subtracting the estimated background does not affect the obtained redshift, so we decided not to subtract the background from the central pointing data henceforth. We confirmed that the two FI CCD (XIS-0 and -3) spectra always give consistent line energies and this validates that CCD-to-CCD variation (i) is well calibrated, at least within the FI CCDs. We therefore combine these two CCD data henceforth. We fitted a set of spectra from individual cells (DET$ij$) from different observations within each roll-angle group using a common redshift. We also fitted all observations using a common redshift. The obtained redshift variations are given in Fig. \[cal-per:reg-z\] and Table \[cal-per:det-var\].
There are two special cells, DET03 and DET33. Because these include calibration source areas, they are much smaller in area (See Fig. \[cal:detector\]) and hence lower in counts than others. These areas are not used here and in the Perseus spectral extraction given below. In particular, the count of DET33 are lower than 10% of those in others. DET33 also shows the largest deviation from average, which could be attributable to instrumental systematic effects but also statistically. DET03 shows larger redshifts in both the two groups. In this case, the deviation is more likely caused by an instrumental effect. These deviations in DET03 and DET33 are as small as $(1-3)\times 10^{-3}$. In addition, the sum of these two positions occupies only a small percentage of the whole CCD area, which means any systematic errors have little effect on our measurements.
We found that standard deviations, [$\sigma_{\rm sd}$]{}, among the detector positions for each roll-group and for all observations are $(0.8-1.6)\times 10^{-3}$ in redshift (Table \[cal-per:det-var\]). If we exclude the DET33 result, these are reduced to $\sim 0.6 \times 10^{-3}$. These variations are attributable to the combination of [*instrumental*]{} and [*cluster*]{} variations.
In the DET23 cell, two-roll-group spectra reveal the largest difference, $2.0\times 10^{-3}$, in redshifts, but elsewhere, they give consistent redshifts. Noting that different sky regions are observed by both groups in the same detector position, this indicates that [*cluster-*]{}intrinsic variation is below these differences.
Further analysis of smaller spatial scales is limited by statistics, but we assume that the instrumental energy scale does not change on such scales.
Based on these analysis, we conclude that the CCD-to-CCD variation \[(i) in subsection \[cal-mot\]\] and intra-chip [*spatial*]{} variation (ii) within spatial scales of a few arcminutes is below 0.1% ($10^{-3}$ in redshift or 300 [km s$^{-1}$]{} around the Fe-K energy) over the whole CCD position.
[llccc]{}\
winter & 18.2 & 1.6 & 5.9 & 0.39\
summer & 18.1 & 0.78 & 1.7 & 0.33\
all & 18.2 & 1.0 & 3.4 & 0.23\
\
winter & 18.5 & 0.66 & 0.91 & 0.28\
summer & 18.2 & 0.59 & 0.41 & 0.25\
all & 18.4 & 0.56 & 0.60 & 0.18\
CENTRAL POINTING SPECTRA {#sect:ana-center}
========================
Here we use a set of central pointing data of the Perseus cluster and investigate the spatial distribution of spectral properties within the central region ($\sim 17'.6\times 17'.6$).
Spectral Extraction
-------------------
We divided the central sky region into cells of sizes $2'.2\times2'.2$ or $4'.4\times4'.4$, as illustrated in Fig. \[ana-center:regions\]. We call inner and smaller cells CS$i$ and larger cells CL$i$, with $i$ ranging from 0 to 15, starting from the south-west corner toward the north-east. Note that the sum of CL5,CL6,CL9,and CL10 overlap with CS0–CS15. From each cell and each observation one spectrum is extracted.
Before measuring redshifts, we evaluate spectra by fitting them in the 2.0–9.0 keV energy band with a single temperature model and the Galactic absorption (hydrogen column density of $1.5\times 10^{-21}$cm$^{-2}$) and determine the temperature and metal abundance of each region. We found a two-dimensional temperature variations of 4.5–6.5 keV. Conversely, the metallicity increased radially from 0.3 to 0.45 solar toward the center. These results match previous measurements and confirm that these spectra can be used to measure local properties from different sky regions.
Redshift Measurements {#ana-center-z}
---------------------
We measure redshifts based on the Fe-K line center and the same model (two Gaussians and a bremsstrahlung continuum) as in section \[cal-per\]. Fig. \[ana-center-z:ex-sp\] shows fitting examples. The redshift variation as a function of observation and hence observation date is shown in Fig. \[ana-center-z:seq\_z\]. Spectra from the first observation (sequence 800010010) show systematically smaller redshift. Other than this, however, there is no systematic change over the observation period, confirming that the [*temporal*]{} variation in the energy scale is effectively corrected.
To compensate possible observation-dependence of the energy scale, we measure the redshift relative to that of a bright cell (CS5 or CL5) for each observation and each sky cell, $\Delta z$. Fig.\[ana-center-z:reg\_z\] shows $\Delta z$ averaged over observations. Here we measure the error ([$\sigma_{\rm sd,all}$]{}) for each cell based on standard deviation among all observations. These observations differ in terms of the detector orientation (i.e. roll-angle) and observation-date. Therefore the scope of error includes not only statistical errors but also at least partially systematic errors due to instrumental energy-scale variations. Along with average values from all observations, those from summer and winter roll-angle observations are given separately. Differences between the two-roll-angle observations enable systematic errors from instrumental effects to be estimated. In all cases, however, the differences are below $2\times 10^{-3}$, confirming that the [*instrumental*]{} variation is insignificant as described in section \[cal-per\].
The largest $|\Delta z|$ is $1.5\times 10^{-3}$ at CL0, where [$\sigma_{\rm sd,all}$]{} is about $1.9\times 10^{-3}$. All other regions have $|\Delta z|$ below $1.0\times 10^{-3}$. Averages and standard deviations of the best-fit $\Delta z$ along with averages of [$\sigma_{\rm sd,all}$]{} are given in Table \[ana-center:rms\]. These results show uniform $\Delta z$ distribution and hence no gas motion exceeding these variations and errors. The errors for each sky region range from $\pm 0.5\times 10^{-3}$ to $\pm 2\times 10^{-3}$, corresponding to radial velocities of 150 and 600 [km s$^{-1}$]{} respectively.
Within the inner cells, CS1–CS3 show lower $\Delta z$ in the best-fit value. To further search for possible spatial variation in redshifts, we use data differently from the above case. Here we fit all spectra for each cell simultaneously with a common redshift, assuming no instrumental and systematic variation in the energy scale. We remove one observation (sequence 800010010) to avoid systematic error as mentioned above. We sorted the result as a function of the position angle of the cell with respect to the cluster center, as shown in Fig. \[ana-center-z:pa\_z\]. The obtained redshift are basically consistent with the above result in Fig. \[ana-center-z:reg\_z\]., although the given error declines because these include only statistical errors. From the inner cells (left panel Fig. \[ana-center-z:pa\_z\]) we found lower redshifts in easterly and westerly directions and higher redshifts toward the south. From the larger cells (left panel Fig. \[ana-center-z:pa\_z\]) we found some variations but more random and comparable to the possible systematic error (section \[cal-per\]; $\sim 10^{-3}$ in redshift or 300 [km s$^{-1}$]{}). We cannot find all variations if we use the data in summer or winter observations separately (See Fig. \[ana-center-z:reg\_z\]). Based on these careful analyses, we suggest a hint of lower redshift regions at $2-4'$ (45–90 kpc; CS1–CS3) west of the cluster center. Other variations are insignificant and possibly attributable to instrumental effects.
In a search for redshift variation on a finer scale, we divided the central $3'\times 3'$ region into 9 cells of sized $1'\times 1'$. We found a uniform redshift distribution within about $\pm 1\times 10^{-3}$ in redshift.
[lccc]{} CS0–CS15 & -0.1 & 0.35 & 0.92\
CL0–CL15 & 0.12 & 0.52 & 1.3\
Line Broadening and Turbulent Velocity {#ana-center-lw}
--------------------------------------
The observed spectra are used to constrain the turbulent Doppler broadening of the emission line. Following OTA07 and similar to our analysis stated in subsection \[sect:cal-cal\], we model the Fe-line spectra by including an additional line width ([$\sigma_{\rm add}$]{}; a Gaussian width).
Here we note that the cluster-intrinsic lines are not a single Gaussian line, but integrate many lines. For example the emission around 6.7 keV includes not only a dominant He-like resonance line at 6700 eV but also intercombination lines at 6667-6682 eV and a forbidden line at 6636 eV along with other weak lines between the resonance and forbidden lines. Effective widths for this He-like triplet are about 30 eV for the temperature of 2–4 keV gas (OTA07). In addition, gas bulk motions unresolved spatially within the spectral extraction region and projected in the line of sight contribute to [$\sigma_{\rm add}$]{}.
For each observation and region, we obtained [$\sigma_{\rm add}$]{} and its upper limit. We found that in most cases [$\sigma_{\rm add}$]{} was consistent with zero and that even using a non-zero [$\sigma_{\rm add}$]{} did not change the best-fit redshift. These are averaged over different observations and presented in Fig. \[ana-center-z:lw\]. The best-fit [$\sigma_{\rm add}$]{} are all below 20 eV, except for that from CL15 where the statistics are significantly lower than others. Given the systematic error and intrinsic cluster line width, these [$\sigma_{\rm add}$]{} are all consistent with no additional broadening due to turbulent motion. Statistical upper limits of [$\sigma_{\rm add}$]{} are 20–50 eV, which equates to the upper limits for turbulent motion of 900–2200 [km s$^{-1}$]{}. We may add systematic uncertainty of 1250 [km s$^{-1}$]{}(subsection \[sect:cal-cal\]).
LARGE-SCALE SPECTRA {#ana-large}
===================
Here we use a set of offset pointing data with offset angles of $14'-36'$ from the cluster center (Table \[obs-offset\] and Fig. \[obs:rosat2\]) and study the spatial redshift distribution on a large scale.
We extract a spectrum from the entire CCD field of view for each offset observation. Here we use the same Gaussian model as in subsection \[ana-center-z\] to fit the FI 5.7–7.0 keV band spectra and measure the redshift distribution.
There are some sets of two observations ( X and X\_2) sharing a pointing direction but with different roll-angles. We found that these data provide consistent results within the set, hence in these pointing sets, spectra are fitted simultaneously with a common model. Examples fittings are shown in Fig. \[ana-large-z:ex\].
The obtained redshifts are shown in Fig. \[ana-large-z:z1\]. We notice that G and SE regions show larger and smaller redshifts, respectively, compared with the average. These two regions, however, are in proximity each other and overlap in the sky regions by about half the area of the entire field of view. Therefore, these deviations are likely statistical or systematic errors, and do not originate entirely from the cluster gas redshift variation. When we combine these two regions, a redshift of $0.020\pm 0.001$ consistent with other regions, is obtained, as shown in Fig. \[ana-large-z:z1\] (a star mark). Region A also shows a relatively large deviation alongside significant statistical errors. Including these deviated regions, the error-weighted average, standard deviation, and average statistical error are found to be $19.2 \times 10^{-3}$, $1.8 \times 10^{-3}$, and $1.2 \times 10^{-3}$, respectively.
Based on the above results, we conclude that gas redshifts are uniform within $\pm (1-2) \times 10^{-3}$ over the cluster core region ($R<20-30'$) in a spatial scale of about $15'$ (300 kpc). In other words, we found no systematic velocity structure exceeding $\pm (300-600)$ km s$^{-1}$.
The average and standard deviation of redshifts in the central region given in subsection \[ana-center-z\]) are $18.4 \times 10^{-3}$ and $0.6 \times 10^{-3}$, respectively. The redshift averaged over the offset regions is consistent with that of the central region within these variations and errors.
![ Example of spectral fitting from offset pointing data, which are used to measure redshifts. From top left to bottom right, the spectra from regions A, B, C, F, G, and H are shown. For each plot, in the upper panels the data, best-fit model, and model components in counting units of s$^{-1}$ keV$^{-1}$ are shown. In the lower panels fit residuals in terms of the data to model ratio are shown. []{data-label="ana-large-z:ex"}](fig10a.ps "fig:"){height="0.35\textheight"} ![ Example of spectral fitting from offset pointing data, which are used to measure redshifts. From top left to bottom right, the spectra from regions A, B, C, F, G, and H are shown. For each plot, in the upper panels the data, best-fit model, and model components in counting units of s$^{-1}$ keV$^{-1}$ are shown. In the lower panels fit residuals in terms of the data to model ratio are shown. []{data-label="ana-large-z:ex"}](fig10b.ps "fig:"){height="0.35\textheight"} ![ Example of spectral fitting from offset pointing data, which are used to measure redshifts. From top left to bottom right, the spectra from regions A, B, C, F, G, and H are shown. For each plot, in the upper panels the data, best-fit model, and model components in counting units of s$^{-1}$ keV$^{-1}$ are shown. In the lower panels fit residuals in terms of the data to model ratio are shown. []{data-label="ana-large-z:ex"}](fig10c.ps "fig:"){height="0.35\textheight"} ![ Example of spectral fitting from offset pointing data, which are used to measure redshifts. From top left to bottom right, the spectra from regions A, B, C, F, G, and H are shown. For each plot, in the upper panels the data, best-fit model, and model components in counting units of s$^{-1}$ keV$^{-1}$ are shown. In the lower panels fit residuals in terms of the data to model ratio are shown. []{data-label="ana-large-z:ex"}](fig10d.ps "fig:"){height="0.35\textheight"} ![ Example of spectral fitting from offset pointing data, which are used to measure redshifts. From top left to bottom right, the spectra from regions A, B, C, F, G, and H are shown. For each plot, in the upper panels the data, best-fit model, and model components in counting units of s$^{-1}$ keV$^{-1}$ are shown. In the lower panels fit residuals in terms of the data to model ratio are shown. []{data-label="ana-large-z:ex"}](fig10e.ps "fig:"){height="0.35\textheight"} ![ Example of spectral fitting from offset pointing data, which are used to measure redshifts. From top left to bottom right, the spectra from regions A, B, C, F, G, and H are shown. For each plot, in the upper panels the data, best-fit model, and model components in counting units of s$^{-1}$ keV$^{-1}$ are shown. In the lower panels fit residuals in terms of the data to model ratio are shown. []{data-label="ana-large-z:ex"}](fig10f.ps "fig:"){height="0.35\textheight"}
![ Redshift distribution from offset observations as a function of position angles of the pointing. The star mark shows the redshift obtained by a simultaneous fit of G and SE regions. The horizontal solid and dashed lines indicate the average and standard variation from the central pointing (subsection \[ana-center-z\]). []{data-label="ana-large-z:z1"}](fig11.ps)
To constrain line broadening and turbulent velocity, we performed a similar spectral fitting as in subsection \[ana-center-lw\]. However, lower statistics mean the limits obtained are typically weaker than those in the central regions above. We found the obtained [$\sigma_{\rm add}$]{} to be consistent with zero turbulent velocity and statistical upper limits of [$\sigma_{\rm add}$]{} of 20–70 eV, corresponding to velocities of 900–3000 km s$^{-1}$.
SUMMARY AND DISCUSSION
======================
Spatial Mixing {#dis-limit}
--------------
The photon mixing caused by the [[*Suzaku*]{} ]{}telescopes and projection effects meant the measured velocity variation was diluted compared with the intrinsic three dimensional variation.
The point spread function of the telescopes include half-power diameters of $1'.8-2'.3$ and 90% fraction encircled-energy radii of $2'.5-3'.0$, meaning variations below these scales would be smoothed. It is also difficult to separate the gas velocity of a faint emission component. This effect was examined by a ray-tracing simulation in OTA07 for the [[*Suzaku*]{} ]{}measurements in the Centaurus cluster. They found that when $2'\times 2'$ cells are used at the core, about 50-60% of the events collected by each cell originate from the surrounding eight cells. Because the Perseus cluster has a central surface brightness distribution similar to that of Centaurus, our measurement in the central region with $2'.2$ width cells (CS0 to CS15) is similarly affected. For the outer cells with a $4'.4$ width (CL0 to CL15), mixing effects should be much smaller ($<10$ %).
In the large-scale measurements (section \[ana-large\]), photon mixing due to stray X-rays from the bright center could be an additional contamination. We estimate, however, that photons scattered from the central region to offset regions with a typical offset angle of $15-20'$ represent a minute percentage of all photons within each offset pointing and we could hence ignore this mixing.
Summary of Results and Comparison with Previous Studies {#dis-com}
-------------------------------------------------------
[@dupke01-per] claimed a radial velocity difference in the Perseus cluster using the ASCA GIS. They compared spectra collected at the center and offsets separated by more than $40'$. These spatial scales exceed our study within radii of $30'$, which means these two measurements cannot be compared.
We summarize the limits we determined for gas motion along with related parameters in Table \[dis-com:summary\]. For comparison, results of OTA07 are also presented. Both results of the [[*Suzaku*]{} ]{}XIS are limited by similar levels of systematic errors from the energy scale calibration. The levels of statistical errors between the two are also comparable.
As described in section 1, gas velocity measurements are important to calibrate the hydrostatic equilibrium (H.E.) mass estimation. OTA07 converted velocity variation obtained into rotational motion and approximated the uncertainty of the H.E. mass by gas kinetic to thermal energy ratio. The ratio is proportional to $f M^2 = f (\frac{v}{c_s})^2$, where $f$, $v$ and $M$ are the fraction, velocity, and Mach number of the moving gas and $c_s$ is the gas sound velocity (given in Table \[dis-com:summary\]). The uncertainty of the rotation angle introduces another error.
Our limits on gas bulk velocities and hence on $M$ are comparable to those in OTA07. The wider physical size covered by our observations (row R in Table \[dis-com:summary\]) allowed us to constrain the gas motion and hence the total mass within a volume significantly exceeding that in OTA07. This is an important step to measure total masses in cluster scales. Provided the Fe-K line emission of the Perseus central region is exclusively brighter than other nearby clusters, this system could represent a unique measurement of the gas motion at the cluster scale via current CCD-type instruments. The decreasing X-ray surface brightness hinders efforts to constrain the motion beyond our achieved scale. It will also remain challenging to reach out beyond cluster core regions even with near-future high energy resolution instruments such as SXS [@mitsuda10] onboard [ASTRO-H ]{}[@takahashi10] without a large collecting area.
[llccc]{} redshift & & 0.0183 & – & 0.0104\
$1'$ & (kpc) & 22.2 & – & 15.3\
$kT$ & (keV) & 4.5 & 6 & 3\
$c_s$ & ([km s$^{-1}$]{}) & 1100 & 1300 & 900\
region size & & $2'.2/4.4'$ & $18'$ & $2'.1$\
& (kpc) & 50/100 & 400 & 30\
R & (kpc) & 200 & 600 & 150\
$\Delta v_{\rm bulk}$ & ([km s$^{-1}$]{}) & 300–400 & 550 & 700\
& & $\pm 300$ (sys.) & $\pm$ 300 (sys.) & $\pm$500 (sys.)\
[$v_{\rm turb}$]{} & ([km s$^{-1}$]{}) & 900–2200 & 900–3000 & 900\
& ([km s$^{-1}$]{}) & $\pm$ 1250 (sys.) & $\pm$ 1250 (sys.) & $\pm$ 1250 (sys.)\
Dynamics in the Core {#dis-core}
--------------------
In and around NGC 1275, complex interactions occur among various gaseous components, high-energy particles from the AGN and radiation from stars and AGN, and probably magnetic fields. Nearby galaxies as well as the central galaxy may have moved around within the growing gravitational potential. In this active environment, the ICM should have various types of motions. Observationally, the lack of resonant scattering of the Fe-K line at the Perseus core indicates gas motions of $M>0.5$ [@chura04]. It is crucial to measure the energies in these gas motions in our study.
We found a hint of gas bulk motions within the cluster core (Fig. \[ana-center-z:pa\_z\]; $R<4'.4$), while the key detection was lower velocities in regions $2-4'$ west of center. West of center, clear X-ray enhancement was found as seen in Fig. \[ana-center:regions\]. [@chura03] interpreted this structure as a result of a past minor merger in an east-west direction. The obtained gas motion is consistent with this minor merger model and additionally suggests a velocity component in the line of sight. Because we observe a mix of photons from the sub component and main cluster due to the sky projection and the telescope point spread function, the real gas velocity of the sub exceeds the observed figure ($v_{\rm obs}=150-300$ [km s$^{-1}$]{}). If we assume $f$ to be the emission fraction from the sub, $v_{\rm obs}$ can be approximated with sub and main velocities ($v_{\rm sub}$ and $v_{\rm main}$) as $v_{\rm obs} = f v_{\rm sub} + (1-f) v_{\rm main}$. In this case, the relative velocity with respect to $v_{\rm main}$ can be estimated as $\Delta v_{\rm sub} \sim f^{-1} \Delta v_{\rm obs}$. From the relative deviation map of the surface brightness (Fig. \[ana-center:regions\]), we estimate $f$ to be 0.2–0.4. In this case, $\Delta v_{\rm sub}$ could be 500–1000 [km s$^{-1}$]{}. Interestingly, the second optically-bright member galaxy, NGC 1272, sitting on a rim of the west sub structure ($\sim 5'$ WWS of NGC 1275), also has a lower radial velocity than that of NGC 1275 by $\sim 1100$ [km s$^{-1}$]{}. Furthermore, the chain of galaxies further west has an average radial velocity lower than the cluster center by about 1500 [km s$^{-1}$]{}. We suggest the association of the obtained gas velocity structure with these galaxies. We discuss these gas and galaxy relations on a larger scale in the next subsection.
Using the obtained measurements, we estimate kinetic energy ($E_{\rm kin}$) by gas bulk motion. As stated in section \[dis-com\], $E_{\rm kin}$ relative to the thermal energy ($E_{\rm th}$) in the gas can be presented by $f M_{\rm bulk}^2 = f (\frac{\Delta v_{\rm bulk}}{c_s})^2$. Using the relation between $\Delta v_{\rm sub} \rightarrow \Delta v_{\rm bulk}$ and $\Delta v_{\rm obs}$ given above, $E_{\rm kin}/E_{\rm th}$ becomes $f^{-1} (\frac{v_{\rm obs}}{c_s})^2$. Assuming $c_s$ of 1100 [km s$^{-1}$]{} and $f$ of $0.2-0.4$, the observed $\Delta v_{\rm obs}$ (150–300 [km s$^{-1}$]{}) gives $E_{\rm kin}/E_{\rm th} = 0.1-0.3$. [@Fabian2011] estimated the energy associated with non-radial structure ($E_{\rm d}$) based on variation in the gas pressure map. This energy should be closely related to $E_{\rm kin}$ estimated here. They estimated $E_{\rm d}/ E_{\rm th}$ to be within a few percent at radii of $\sim 110$ kpc corresponding to the west sub component region. Because both estimations are order of magnitude, we cannot compare these quantitatively. These estimations are among the first attempts to measure all the energy components in the cluster gas and understand the energy distribution of clusters.
The west system is the largest substructure within the core. Our measurement also indicates that it has the largest bulk velocity, at least in a radial direction, which means this structure could have the largest kinetic energy due to bulk motion. We found no other systematic motions around the core with scales $>20$ kpc and bulk velocities exceeding half the sound velocity ($\sim 550$ km s$^{-1}$). This is consistent with the observation that any other features revealed around the core are below 10 kpc ($30''$) in size. These structures are also likely to have velocities not exceeding the virial velocity of the central galaxy, or about $200$ km s$^{-1}$. The lack of heated gas or strong shock [@Fabian2003] also pointed to no sonic or super sonic motion. We conclude that within much of the cluster core volume, the gas remains in hydrostatic equilibrium.
In Table \[dis-cd:z\] we present radial velocities of NGC 1275 and the cluster from optical data and this X-ray results. The optical radial velocity of NGC 1275 differs from that of a mean of cluster member by only about 200 km s$^{-1}$, demonstrating that the galaxy remains at the bottom of the cluster potential. The absolute radial velocity of the gas at the core is consistent with that of the galaxy in optical within the errors from X-ray data. This indicates that large parts of stars and the gas in and around the galaxy stay together at the bottom of the cluster potential center. The cluster gas on a large scale also has a mean velocity consistent with that in optical, confirming that these two baryonic components beyond galaxy scale remain together at the same potential center. These comparisons in absolute redshifts are among the first attempts involving clusters.
Gas dynamics in cluster cool cores has been studied by numerical simulations. For example, [@Asca06] simulated off-axis minor mergers and gas sloshing and reproduce cold fronts. They presented evolutions of gas velocity structures. Compared with large velocity variations in their representative runs, our measured variations (Figs. \[cal-per:reg-z\] and \[ana-center-z:pa\_z\]) are more uniform spatially and close to their results for late stages [e.g. 6/4.8 Gyr in Fig.5/10 of @Asca06]. Note that our Doppler measurements are not sensitive to motions in the plane of the sky and diluted by projection. In fact in the Perseus cluster gas spiral flows in the plane of the sky are suggested from X-ray images [e.g. @Fabian2011; @Simionescu2012]. Focusing on AGN jet-driven bubbles, [@Heinz10] simulated high-resolution X-ray spectroscopy of nearby clusters including the Perseus cluster. They found the velocity structure of the bubbles in this core with energy shifts of about 350 [km s$^{-1}$]{}, which are close to our limit. However, these structures are too small in size ($<10$ arcsec) and too faint in X-ray brightness contrast to resolve by the current data.
[lccl]{} NGC 1275 & 5250 & 20 & 1\
member galaxy mean & 5470 & 100 & 2\
X-ray gas at center & 5520 & 300 & section \[sect:ana-center\] and \[ana-large\]\
X-ray gas at $R<15'$ & 5760 & 600 & section \[ana-large\]\
Large-Scale Dynamics
--------------------
We found an absence of gas bulk motion on a large scale. Our upper limit for the radial gas velocity is about 600 km s$^{-1}$, which implies either that gas motion in the cluster is predominantly in the plane of the sky or subsonic at most.
In subsection \[dis-core\] we discussed the potential of the gas motion around the western core being associated with galaxy motions. The associated chain of galaxies is further distributed toward west. To study the gas and galaxies relation in the west and larger areas, we compare our derived gas dynamics with that in galaxies in the cluster.
We use the following two catalogs as the deepest collections of galaxies in the Perseus cluster. The CfA redshift catalog [@Huchra1995] includes 94 (120) galaxies with radial velocities within $30'$ ($60'$) of the cluster center, while the 2MASS redshift survey [@Huchra2012] includes 66 (103) galaxies in the same regions but with complete photometric data. Fig. \[dis-large:pa-vh\] shows the radial velocities of galaxies within a radius of $30'$ of the center as a function of azimuthal angles relative to the center. Most galaxies are distributed around the cluster dynamical center at redshift of $\sim 0.018$, corresponding to that of NGC 1275. This is consistent with early studies of galaxy distributions [e.g @kent1983]. Some galaxies have higher or lower velocities ($\Delta v > 2000$ km s$^{-1}$), which may be contamination dynamically unrelated to the main system.
We include gas velocities from Fig. \[ana-large-z:z1\] into Fig.\[dis-large:pa-vh\], which is the first such comparison. In galaxy velocities, there are some local structures, the clearest of which is a group at an angle of around 260 with lower velocities ($-2000$ km s$^{-1} < \Delta v < 1000 $ km s$^{-1}$). This is part of the chain of galaxies mentioned in subsection \[dis-core\], which should be before or after a collision into the main cluster centered on NGC 1275. Based on the X-ray distribution around the core, [@chura03] suggested that the substructure is after a collision about 0.25 Gyr ago. Based on comparison between the galaxy and gas velocities around this angle at the cluster west, we notice that the gas is closer to the cluster center than galaxies at this azimuth. Note that these galaxies in total exceed the cD galaxy in terms of luminosity and hence mass. If these two baryonic components are associated, what explains this difference? We note a difference in effective radii between the two wavelength bands. The X-ray emission is weighted toward the inner region, due not only to the positions of spectral extractions (Fig. \[obs:rosat2\]) but also the density-squared X-ray emission intensity. Considering these effects, the effective radius of the X-ray data can be approximated around $10'$. On the other hand, galaxies in the west are uniformly distributed over the $30'$ radius region. These galaxies are point sources in X-ray without any bright extended excess emission [@Fabian2011]. In addition, this difference suggests a segregation between the two components during a violent collision. Such detachments in the sky position have been observed in some merging systems such as the Bullet [@Clowe06]. Our measurement is a new approach to reveal such actions not only in spatial terms but also in the radial velocity space. Note that in the major-merger system Abell 2256 @tamura2011 revealed that gas and galaxies move in pairs within radial velocity and spatial spaces.
Based on ASCA observations [@Furusho2001] observed an extended cool region $10-20'$ east of the cluster center, alongside a ring-like region surrounding the east cool region and the core. They suggested a past collision of a poor cluster in a direction nearly parallel to the line of sight. Our observation covers a substantial fraction of the east cool region as shown in Fig. \[obs:rosat2\]. To limit the current velocity of the east sub system we must assume the X-ray emission fraction of the moving system within the extracted spectra. We estimate this at 0.3–0.5 according to the relative deviation map of surface brightness in [@chura03]. Combining this with our measurement in Fig. \[ana-large-z:z1\] ($\Delta v <600$ [km s$^{-1}$]{}), we constrain the radial velocity of the sub system $<(1200-2000)$ [km s$^{-1}$]{}.
In future we would cover more galaxies by conducting deeper surveys and deep X-ray spectra covering larger volumes in a number of systems. Gravitational lensing would also measure dark matter distributions. Direct comparisons between these three components in dynamical and spatial distributions should reveal a number of local events and measure the dynamical cluster age, thus giving a systematic picture of structure formation of gas and galaxies but invisibly controlled by dark matter.
Considering that the virial radius ($r_{\rm vir}$) of the cluster is 2.2 Mpc (or $\simeq$ 100’), we measured the gas motion typically around $R \simeq 0.1-0.3 r_{\rm vir}$. The obtained upper limits for the bulk motions are typically $(\Delta v_{\rm bulk}/c_s) \lesssim 0.5$ taking account of the systematic errors. The turbulent motions are more loosely constrained ($v_{\rm turb} \gtrsim c_s$). Cosmological cluster simulation results suggest that the kinematic energy of the gas is expected to be left only in the level of $\sim 10$ % of the thermal energy in the corresponding radius range [e.g. @Lau09; @Vazza09]. Furthermore, this value tends to be even lower for a sub sample of relaxed clusters. Therefore, our results of the absence of both the bulk and turbulent motions except for the some hints at west of the core are consistent with these simulations.
![ Radial velocity distributions of galaxies and gas as a function of the position angle. The small (black) and large (red) circles indicate galaxies from the CfA redshift catalog (94 galaxies) and bright galaxies (K-band magnitude $<10.7$ ) from the 2MASS redshift catalog. There are some overlaps between the two samples. The velocity of NGC 1275 is shown by a solid line. Star marks indicate the gas velocity taken from Fig. \[ana-large-z:z1\]. The central velocity of the gas components is shown by a dashed line. []{data-label="dis-large:pa-vh"}](fig12.ps)
Turbulent Motion
----------------
In addition to the gas bulk motion, we searched for the turbulent motion using the line width. We obtained the limits generally larger than the sound speed as given in Table \[dis-com:summary\]. OTA07 also reported a limit of the turbulence using the XIS data of the Centaurus integrated over a large region of $18' \times 18'$ (Table \[dis-com:summary\]). These two limits are similar to each other, because that both limits depend largely on the energy resolution of the XIS and its calibration. Our result is the first attempt to constrain the turbulent motion based on the spatially-resolved line X-ray spectra.
The gas turbulent (random) motion could introduce an error on the H.E. mass estimate. The additional mass or kinetic energy due to the turbulence can be approximated by the factor given in subsection \[dis-core\], $M^2$ with the turbulent velocity, [$v_{\rm turb}$]{} replaced as $v$. We obtained the limit on [$v_{\rm turb}$]{} a few times larger than that on the bulk velocity (Table \[dis-com:summary\]). Therefore the uncertainty from the turbulent motion dominates that from the ordered one.
The [[*XMM-Newton*]{} ]{}RGS was used to constrain the gas turbulent motions in hot gas in clusters and galaxies [e.g. @Xu2002]. @Sanders2013 analyzed RGS spectra of dozens of sources and found evidence for $>400$ [km s$^{-1}$]{} velocity line broadening and limits down to 300 [km s$^{-1}$]{} for some systems. The RGS spectra can be used only for centrally-peaked X-ray line emission below 1.5 keV (dominated by Fe-L lines) and results are coupled with the brightness distribution of the line emission which is not straightforward to model.
As described in subsection \[ana-center-lw\], the limit on [$v_{\rm turb}$]{} is mostly limited by the instrumental energy resolution. Therefore to improve the limit significantly high energy resolution instruments such as X-ray calorimeters are necessary. For example, direct and robust study will be obtained by SXS onboard [ASTRO-H ]{}, providing 5–7 eV (FWHM) energy resolution, corresponding to 300 [km s$^{-1}$]{} at the Fe-K line. A 100 ksec SXS observation of the Perseus center provides more than 10,000 counts in the He-like triplet emission and measurements of the turbulent along with those of bulk velocity structure.
We thank the referee for useful comments and suggestions. [[*Chandra*]{} ]{}image of the Perseus cluster was provided kindly by J.Sanders. We thank all the [*Suzaku*]{} team member for their supports. We acknowledge the support by a Grant-in-Aid for Scientific Research from the MEXT, No.24540243 (TT), No.25400231 (NO), and No.A2411900 (SU).
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Energy scale calibratios
========================
![Example of the fitting of spectra sorted by detector coordinates. The axis units are the same as those in previous spectral plots. These are used in section \[cal\]. []{data-label="cal-app:sp"}](fig13a.ps "fig:") ![Example of the fitting of spectra sorted by detector coordinates. The axis units are the same as those in previous spectral plots. These are used in section \[cal\]. []{data-label="cal-app:sp"}](fig13b.ps "fig:")
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abstract: 'We describe how the notion of optical beam shifts (including the spatial and angular Goos-Hänchen shift and Imbert-Federov shift) can be understood as a classical analogue of a quantum measurement of the polarization state of a paraxial beam by its transverse amplitude distribution. Under this scheme, complex quantum weak values are interpreted as spatial and angular shifts of polarized scalar components of the reflected beam. This connection leads us to predict an extra spatial shift for beams with a radially-varying phase dependence.'
address: |
$^1$H H Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK\
$^2$Max-Planck-Institute for the Physics of Complex Systems, Noethnitzer Str. 38, 01187 Dresden, Germany
author:
- 'Mark R Dennis$^1$ and Jörg B Götte$^{1,2}$'
title: The analogy between optical beam shifts and quantum weak measurements
---
Introduction
============
There are many analogies between phenomena in quantum theory and in classical wave optics, including the connection between Heisenberg’s uncertainty principle and the bandwidth theorem [@gabor], transverse optical polarization and spin half as 2-state systems [@Aspect], and the Schödinger equation and paraxial equation of light [@kogelnik], connecting the time propagation of quantum wavepackets and the propagation of narrow, coherent light beams. Although the underlying physics differs in these cases, the similarity in the underlying mathematics gives rise to analogous phenomena.
Our aim in this paper is to describe and explore the strong analogy between the spatial and angular shifts that light beams experience on reflection from a planar interface, and the notion of weak quantum measurement, incorporating quantum weak values of operators [@Aharonov+:PRL60:1988; @AharonovRohrlich:WVCH:2005; @Mitchinson+]. Such a connection was identified by Hosten and Kwiat [@HostenKwiat:SCI:2008] in the experimental measurement of the spin Hall effect of refracted light beams via quantum weak values. Here, we generalize this approach, and show that the shift of a paraxial light beam, and its polarized components, is determined by the appropriate average value of an ‘Artmann operator’ related to the reflection matrix for the beam.
Spatial beam shifts are usually small: they are typically comparable to an optical wavelength, which is much smaller than the physical width of the paraxial beam, even when focused on the reflecting interface. Angular shifts of the direction of the beam, when they exist, are also small, of the order of the (narrow) spectral width of the beam in Fourier space. The effect originates physically from the Fresnel coefficients’ dependence on incidence angle – as the plane wave components making up the beam deviate from their mean, the complex amplitude and possibly polarization of the reflected plane wave components varies. The magnitude and direction of the beam shift thus depends on the incidence angle of the beam and its incident polarization, and while we concentrate in this paper on reflection, the case of transmission is very similar. We will not derive all beam shift formulas here, as they involve some technical details unimportant to the connection to quantum measurements. For these details (in an explicitly optical approach), we refer the reader to a companion paper [@GoetteDennis:NJP:2012].
Historically, the first beam shift to be discovered was for light beams polarized linearly in, or perpendicular to, the plane of incidence ($p$- and $s$-polarizations respectively), by Goos and Hänchen in 1943 [@GoosHaenchen:AndP436:1947]. In this case, the shift is in the plane of incidence (referred to as ‘longitudinal’), given by the famous Artmann formula [@Artmann:AndP437:1948]. If the incident beam has circular polarization (or, more generally, has any polarization not linear in or perpendicular to the plane of incidence), there is an ‘Imbert-Federov shift’ transverse to the plane of incidence, which has been explained in terms of spin-orbit coupling [@BliokhBliokh:PRL96:2006]. Goos-Hänchen and Imbert-Federov shifts are largest in the regime of total reflection, close to the critical angle; when reflection is partial (reflection by a denser medium, or inside the critical angle), Fourier filtering of the spectrum leads to an angular shift of the propagation direction of the light beam [@RaBertoniFelsen; @AntarBoerner; @ChanTamir:OL10:1985], which has been the subject of much recent experimental attention [@Merano+:NatPhot3:2009; @Merano+:PRA82:2010].
![\[fig:shift\] Schematic illustration of a spatial shift $D_x$ and angular shift $\Delta_x$ of a reflected beam in the plane of incidence. If reflection of the incident beam (green) was purely specular, one would observe the ‘virtual beam’ (red). However, due to angle-dependence of the reflection coefficients, the real beam is displaced both in position and propagation direction (blue). ](Fig1.pdf){width="9cm"}
A schematic showing the geometry of a longitudinal shift, both spatial and angular, is shown in Fig. \[fig:shift\]. It is natural to introduce the concept of a ‘virtual beam’, represented in red in the figure. Inspired by the method of images, this is an ideal, unphysical reflected beam whose position and direction is as if the reflection were purely specular. Rather than refer to spatial and angular shifts with respect to the incident beam, we instead denote the shifts with respect to the expected position and direction of the virtual beam. It therefore suffices to work in coordinates based on the virtual beam, which propagates in the $z$-direction, and the plane of incidence is the $xz$-plane, as shown in the figure. This avoids introducing complicated 3D geometry for the reflection with respect to the incident beam.
Of course, the usual Goos-Hänchen and Imbert-Federov shifts apply to the centroid of the intensity of the light. Since the reflected beam also acquires a weakly inhomogenous polarization, analyzing the reflected beam with a polarizer reveals further shifts of polarized components, such as the transverse spin Hall effect of light [@BliokhBliokh:PRL96:2006; @HostenKwiat:SCI:2008] of circular components when linear polarization is incident; in this case, each circular component undergoes a shift even when there is no net transverse shift to the overall beam.
A particular interest of beam shifts is that the nature and magnitude of the resulting shifts are independent of the spatial profile of the light beam provided the incident beam is axisymmetric and focused on the interface. When this condition is relaxed, for instance with a beam with an azimuth-dependent phase vortex $\exp(\rmi \ell \phi)$ [@Allen+:PRA45:1992; @Dennis+:PO53:2009], there is an additional ‘vortex-induced shift,’ where the spatial shift involves a combination of the Artmann expressions for the spatial and angular shifts [@Bliokh+:OL34:2009; @Merano+:PRA82:2010].
Each of the notions above corresponds to a similar notion in quantum measurement via a weak von Neumann interaction, and our purpose in this paper is to make these explicit. In the optical reflection case, the interaction is between the polarization and spatial structure of the beam, and the weakness follows from paraxiality (i.e. localization in Fourier space). Before describing the details of the analogy, we review the relevant notions from quantum measurements, largely following the description and notation of the very clear exposition of the quantum case by Jozsa [@Josza:PRA76:2007].
Weak quantum measurement {#sec:weak}
========================
The weak quantum measurement of an observable $A$ of a quantum system in initial state $|\psi_{\mathrm{i}} \rangle$ is achieved by considering the entire system in a product state of the system to be measured, together with the state of the measurement pointer $|\varphi\rangle,$ i.e. $|\psi_{\mathrm{i}}\rangle|\varphi\rangle.$ The interaction between the measured system and the pointer is weak, in the sense that the interaction hamiltonian $H_{\mathrm{int}} \equiv g A p,$ acting at a single instant in time, where $g \ll 1$ is a small coupling constant, and $p$ is the momentum operator generating translations of the pointer state $|\varphi\rangle.$ The unitary evolution operator corresponding to the interaction, in units with $\hbar = 1,$ is $$\exp(-\rmi H_{\mathrm{int}}) \approx 1 - \rmi g A p,
\label{eq:unitaryapprox}$$ following from the fact that the coupling constant is small, i.e. the interaction is weak.
Now, the measurement may be made in the usual von Neumann sense, where the measured system’s freedoms are ‘traced over’, by taking the (sub-)inner product with $|\psi_{\mathrm{i}}\rangle$, giving rise to the expectation value $$\langle A \rangle \equiv \frac{\langle \psi_{\mathrm{i}} | A | \psi_{\mathrm{i}}\rangle}{\langle \psi_{\mathrm{i}} | \psi_{\mathrm{i}}\rangle}.
\label{eq:expa}$$ Alternatively, the projection of the measured system in a certain postselected final state $|\psi_{\mathrm{f}}\rangle$ may be considered, giving rise to the so-called ‘weak value’ of the operator [@Aharonov+:PRL60:1988; @AharonovRohrlich:WVCH:2005] $$A_w \equiv \frac{\langle \psi_{\mathrm{f}} | A | \psi_{\mathrm{i}}\rangle}{\langle \psi_{\mathrm{f}} | \psi_{\mathrm{i}}\rangle}.
\label{eq:wea}$$ When $A$ is hermitian, $\langle A \rangle$ is necessarily real, although it is not for more general $A.$ However, for any $A,$ the weak value $A_w$ is usually complex-valued, and depends on both the pre- and postselected states.
Furthermore, the weak value may take on very large values (‘superweak’ [@BerryShukla:JPA45:2012]), particularly when $|\psi_{\mathrm{i}}\rangle$ and $|\psi_{\mathrm{f}}\rangle$ are almost orthogonal (so the denominator of Eq. (\[eq:wea\]) becomes vanishingly small). We will denote the average value of $A,$ whether the expectation or weak value, by the generally complex $$a \equiv \frac{\langle \psi | A|\psi_{\mathrm{i}}\rangle}{\langle \psi | \psi_{\mathrm{i}}\rangle},$$ where $\langle \psi |$ is $\langle \psi_{\mathrm{i}} |$ or $\langle \psi_{\mathrm{f}} |.$
The final pointer state, after the weak interaction and possibly postselection, is therefore $$\fl \langle \psi | \exp(-i H_{\mathrm{int}}) |\psi_{\mathrm{i}}\rangle|\varphi \rangle
\approx \langle \psi |\psi_{\mathrm{i}} \rangle \left( 1 - \rmi g a p\right)|\varphi\rangle
\approx \langle \psi |\psi_{\mathrm{i}} \rangle \exp(-\rmi g a p)|\varphi\rangle.$$ The result of the measurement is a shift in the mean position of the pointer wavefunction $\langle q | \varphi \rangle = \varphi(q) \equiv |\varphi(q)| \rme^{\rmi \chi(q)},$ in the position representation, where $\chi$ is the wavefunction’s phase. When $a$ is real, this simply corresponds to a translation of $\varphi(q)$ by the small amount $g a;$ when $a$ is complex, the expectation value of $q$ is shifted by $$D = g \mathrm{Re}(a) - g \mathrm{Im}(a) \int \rmd q\, q^2 J'(q)
= g \mathrm{Re}(a) + 2 g \mathrm{Im}(a) \langle q \chi'(q) \rangle,
\label{eq:jtheorem1}$$ where the second term is nonzero when the pointer wavefunction $\varphi(q)$ at the moment of interaction has a varying phase (so the probability current $J(q) = |\varphi|^2 \chi'$ is nonzero), and we have assumed the unshifted expectation of position $\langle q \rangle = 0.$ The second line here follows from integration by parts, assuming $\chi$ is sufficiently well behaved; in [@Josza:PRA76:2007], the term proportional to $g {\mathrm{Im}}(b)$ was found to be $m \partial_t \langle q^2 \rangle$ by the continuity equation for probability and Schrödinger’s equation. In Fourier space, the pointer’s conjugate wavefunction $\widetilde{\varphi}(p)$ also undergoes a shift to its mean position [@Steinberg:PRA52:1995; @Mitchinson+], by $$\Delta = g \langle p^2 \rangle \mathrm{Im} (a),$$ proportional to the imaginary part of $a$ and the width (variance) of the Fourier transform $\widetilde{\varphi}(p).$ We note that it is common in the quantum mechanical literature to refer to the weak interaction between measured and pointer systems as a ‘weak measurement’, referring to the small magnitude of the coupling constant $g;$ the measurement is weak regardless of whether the weak or expectation value of the operator is being measured.
The analogy between beam shifts and quantum measurement {#sec:analogy}
=======================================================
We are now in a position to draw the analogy between beam shifts and quantum weak measurements. As previously in classical optics analogies with weak values [@Ritchie+:PRL66:1990; @HostenKwiat:SCI:2008], the polarization of the beam, represented by the constant transverse vector $\boldsymbol{E},$ is identified with the measured system, and the complex amplitude of the beam corresponds to the pointer. The measured system is therefore effectively a 2-state system (i.e. a 2-dimensional Jones vector), and the pointer wavefunction (a normalized, position-dependent complex amplitude) $\varphi(\boldsymbol{r})$ depends on $\boldsymbol{r} = (x,y)$ in the plane perpendicular transverse to the propagation in $z.$ A homogeneously polarized light beam, in the quantum language, therefore corresponds to the product state. Free paraxial propagation is analogous to free quantum time evolution according to the Schrödinger equation. The paraxial approximation applied to the beam follows from its strong localization around a mean propagation direction in direction (Fourier) space. A regular or weak measurement of the beam is made depending on whether the centroid of the total intensity of the final reflected beam, or a polarized component of it, are considered.
When a *plane wave* encounters a planar dielectric interface, it is reflected [@Jackson:JohnWiley:1998]: it changes direction according to the law of specular reflection, and its polarization $\boldsymbol{E}$ undergoes reflection according to the appropriate Fresnel coefficients $r_s$ and $r_p$ (evaluated at the incidence angle): the $p$-direction perpendicular to the plane of incidence, and the $s$-direction parallel to it, are distinguished as eigenpolarizations of the incident polarization $\boldsymbol{E}.$ The resulting polarization is therefore $\mathbf{R}\cdot\boldsymbol{E},$ where $\mathbf{R}$ is a reflection matrix, diagonal in the $s,p$ basis with entries given by the appropriate reflection coefficients. It is therefore natural to choose the beam coordinates to correspond to these eigenpolarization directions: $x$ corresponding to $p,$ in the plane of reflection, and $y$ perpendicular to it ($s$ polarization), as in Fig. \[fig:shift\].
When the homogeneously polarized *beam* – rather than simply a plane wave – encounters a planar dielectric interface, it approximately undergoes the same change: it is specularly reflected, and its polarization changes according to its central wavevector component. However, the small variation of wavevector directions about the propagation direction gives rise to a small variation in the incidence angle and plane of incidence of each Fourier component, and it is this small variation, ignored in the approximation of beam as plane wave, that corresponds to weak quantum interaction.
This is made explicit by the introduction of the virtual beam, that is, the imaginary beam propagating in the $z$-direction (i.e. after specular reflection) as discussed above. The virtual beam is simply specularly reflected, and its polarization is constant; the ‘weak interaction’ is then the residual small correction (to first order) from the angle dependence. It is convenient to perform the analysis in the coordinates of the virtual beam.
Spherical angles for beam coordinates will be denoted by azimuth $\alpha$ and colatitude $\delta,$ i.e. the transverse wavevector (whose length $k$ is fixed) $$\boldsymbol{K} = (K_x,K_y) = k \sin\delta \, (\cos\alpha,\sin\alpha) \approx k \delta \, (\cos\alpha,\sin\alpha),$$ since we assume the spread of the beam in Fourier space is small: paraxiality implies that the variance $\langle \delta^2 \rangle \equiv \frac{1}{k^2}\int\rmd^2\boldsymbol{K}\, |\boldsymbol{K}|^2 |\widetilde{\varphi}(\boldsymbol{K})|^2 \ll 1.$ The polarization of the virtual beam is assumed to be uniform, and determined by $\overline{\mathbf{R}},$ the reflection matrix of the mean wavevector. $\overline{\mathbf{R}}$ is the reflection matrix for a plane wave incident at $\theta_0,$ the incidence angle of the centre of the beam.
The virtual beam is therefore in a product state, whose amplitude distribution is the same as the initial beam, and whose homogeneous polarization is given by $\boldsymbol{E}_{\mathrm{i}} \equiv \overline{\mathbf{R}}\cdot\boldsymbol{E}.$ It is this virtual beam which plays the role of the preselected quantum product state: the main change to the beam on reflection is its specular change in direction and plane wave reflection-like change to its polarization: the smaller shifts in mean position and direction follow from the weak $\boldsymbol{K}$-dependent variation in $\mathbf{R}.$
The physically reflected beam, in its Fourier representation, is found by multiplication of the virtual beam Fourier transform with the appropriate $\boldsymbol{K}$-dependent reflection matrix $\mathbf{R}.$ Each $\boldsymbol{K}$ has its own particular plane of incidence, and $\mathbf{R}$ applies the appropriate reflection coefficients in the local $s,p$ basis. Since the spread of $\boldsymbol{K}$-directions around the propagation direction is small, the beam is sensitive only to a low-order Taylor expansion of this matrix $\mathbf{R}$ with respect to $\delta,$ that is $$\mathbf{R} \approx \overline{\mathbf{R}} + \delta \, (\cos\alpha,\sin\alpha)\cdot(\overline{\mathbf{R}_x},\overline{\mathbf{R}_y}),$$ where $\overline{\mathbf{R}_x}$ and $\overline{\mathbf{R}_y}$ represent the mean derivatives of $\mathbf{R}$ in the longitudinal and transverse directions respectively, evaluated at the mean wavevector. In this paper, we will not perform the extra geometrical calculations required to find explicit forms of $\overline{\mathbf{R}_x}$ and $\overline{\mathbf{R}_y};$ we derive these in the companion paper [@GoetteDennis:NJP:2012] in detail.
It should be clear by analogy with the exposition in Section \[sec:weak\] above that in this approximation, the reflection operator may be rewritten directly as the action of the mean $\overline{\mathbf{R}}$ followed by an interaction operator entangling the position and polarization. This can be represented by the mean reflection $\overline{\mathbf{R}}$ left-multiplied by a weak interaction operator (as described above in Section \[sec:weak\]), $$\mathbf{R} \approx \left(\boldsymbol{1} - \rmi \boldsymbol{K}\cdot(\mathbf{A}_x,\mathbf{A}_y) \,\right)\overline{\mathbf{R}}
\approx \exp\left(-\rmi \boldsymbol{K}\cdot(\mathbf{A}_x,\mathbf{A}_y)\,\right) \overline{\mathbf{R}},$$ where $(\mathbf{A}_x,\mathbf{A}_y)$ is a vector of $2\times 2$ matrices we call ‘Artmann operators’, $$\mathbf{A}_j \equiv \frac{\rmi}{k}\overline{\mathbf{R}_j} \, \overline{\mathbf{R}}^{-1}, \quad j = x, y.$$ If the reflection is total (i.e. the reflection coefficients are unimodular), it is straightforward to see (but omitted here) that each $\mathbf{A}_j$ matrix is hermitian. However, if the reflection is partial, some of the incident light is lost through transmission, resulting in the $\mathbf{A}_j$ matrices being nonhermitian. The corresponding evolution operator is nonunitary as it does not preserve normalization, as some light is refracted.
In Fourier space, the reflection operator acts as an impulsive evolution of the virtual beam under the ‘interaction hamiltonian’ $\boldsymbol{K}\cdot(\mathbf{A}_x,\mathbf{A}_y),$ which is weak since the transverse momentum $\boldsymbol{K}$ is necessarily small for the paraxial beam. Apart from the generalization to two dimensions (which is straightforward unless $\varphi(\boldsymbol{r})$ is complex, for which see Section \[sec:complex\]), the connection with the quantum measurement described above is immediate: the centre of the physical beam in real space, and its direction in Fourier space, are effectively a weak measurement-like shift to the pointer $\varphi(\boldsymbol{r}),$ and its Fourier transform $\widetilde{\varphi}(\boldsymbol{K})$ by the real and imaginary parts of the (possibly complex-valued) average of the Artmann operators $\mathbf{A}_x, \mathbf{A}_y.$
In the case of optical polarization, ‘tracing out’ the polarization degrees of freedom is simply considering the overall intensity of the beam ignoring the polarization; in this case, the spatial shift in $j = x,y$ is given by the general shift (for a real $\varphi(\boldsymbol{r})$) $$\begin{aligned}
D_j & = & {\mathrm{Re}}\frac{\boldsymbol{E}_{\mathrm{i}}^* \cdot \mathbf{A}_j \cdot\boldsymbol{E}_{\mathrm{i}}}{\boldsymbol{E}_{\mathrm{i}} \cdot \boldsymbol{E}_{\mathrm{i}}}
= {\mathrm{Re}}\left( \frac{\rmi}{k}\frac{\boldsymbol{E}^*\cdot \overline{\mathbf{R}}^{\dagger} \overline{\mathbf{R}_j} \overline{\mathbf{R}}^{-1}\overline{\mathbf{R}}\cdot\boldsymbol{E}}{\boldsymbol{E}_{\mathrm{i}} \cdot \boldsymbol{E}_{\mathrm{i}}}\right) \nonumber \\
& = & -{\mathrm{Im}}\left( \frac{1}{k}\frac{\boldsymbol{E}\cdot \overline{\mathbf{R}}^{\dagger} \overline{\mathbf{R}_j} \cdot\boldsymbol{E}}{\boldsymbol{E}\cdot \overline{\mathbf{R}}^{\dagger}\overline{\mathbf{R}} \cdot \boldsymbol{E}}\right).
\label{eq:spatial}\end{aligned}$$ When $\boldsymbol{E}$ is linearly polarized in the $x$- or $y$-directions ($p$ and $s$ polarized respectively), the corresponding longitudinal Goos-Hänchen shift $D_x$ is given by the famous Artmann formulas $-\frac{1}{k}{\mathrm{Im}}r'_p/r_p, -\frac{1}{k}{\mathrm{Im}}r'_s/r_s$ [@Artmann:AndP437:1948], which follow directly from the spatial shift formula in Eq. (\[eq:spatial\]), and $D_y = 0$ in this case. When $\boldsymbol{E}$ is circularly polarized (right- $(+)$ or left- $(-)$ handed), there is a transverse Imbert-Federov shift $D_y$ given by $\pm \frac{1}{k}|r_s+r_p|^2 \cot\theta_0$ [@imbert; @fedorov; @BliokhBliokh:PRL96:2006; @Aiello; @GoetteDennis:NJP:2012]. It can be shown that $\overline{\mathbf{R}_y},$ originating from spin-orbit coupling, is a constant times the Pauli matrix $\boldsymbol{\sigma}_1$ in the $s,p$ basis, so in fact $D_y$ is zero if $\boldsymbol{E}$ is linearly polarized.
If reflection is total so the $\mathbf{A}_j$ are hermitian, these spatial shifts are the only shifts which occur. However, if reflection is partial, there is a shift to the Fourier transform $\tilde{\varphi}(\boldsymbol{K}),$ which is $k$ times an angular shift $\Delta_j$ to the direction of the beam in the $x$ and $y$-directions. From above, this is proportional to the variance of the beam’s Fourier distribution $\langle |\boldsymbol{K}|^2 \rangle = k^2 \langle \delta^2 \rangle,$ where $\langle \delta^2 \rangle$ is the variation of the beam in direction space. Thus $$\begin{aligned}
\Delta_j = & = & \frac{\langle |\boldsymbol{K}|^2 \rangle}{k} {\mathrm{Im}}\frac{\boldsymbol{E}_{\mathrm{i}}^* \cdot \mathbf{A}_j \cdot\boldsymbol{E}_{\mathrm{i}}}{\boldsymbol{E}_{\mathrm{i}} \cdot \boldsymbol{E}_{\mathrm{i}}} \nonumber \\
& = & \langle \delta^2 \rangle {\mathrm{Re}}\left( \frac{\boldsymbol{E}\cdot \overline{\mathbf{R}}^{\dagger} \overline{\mathbf{R}_j} \cdot\boldsymbol{E}}{\boldsymbol{E}\cdot \overline{\mathbf{R}}^{\dagger}\overline{\mathbf{R}} \cdot \boldsymbol{E}}\right)
\label{eq:angular}\end{aligned}$$ This equation is derived purely from the optical viewpoint in [@GoetteDennis:NJP:2012]. Despite the recent interest in the angular shift, the simple form here in terms of the variance in direction space has not previously been emphasized. The universality, coming from the relation with weak interactions (although the value of the nonhermitian Artmann operator is not weak here) is one of the main new statements in optical beam shifts in this paper.
What of weak values themselves? As discussed above, postselection is clearly here represented by the presence of a polarizing analyzer $\boldsymbol{F},$ which projects the beam onto a ‘final’ polarization state. In terms of optical physics, the analogue of the entanglement between measured system and pointer is a position-dependent polarization pattern: the mean position of different final polarization states varies (Fig. \[fig:polarized\]), depending on the details of the incident polarization and reflection matrix. The component shift, or ‘weak’ shift, can therefore be written as follows: the component spatial shift is $$D_{wj} = {\mathrm{Re}}\frac{\boldsymbol{F}^* \cdot \mathbf{A}_j \cdot\boldsymbol{E}_{\mathrm{i}}}{\boldsymbol{F} \cdot \boldsymbol{E}_{\mathrm{i}}}
= -\frac{1}{k} {\mathrm{Im}}\left( \frac{\boldsymbol{F}^*\cdot \overline{\mathbf{R}_j} \cdot\boldsymbol{E}}{\boldsymbol{F}^*\cdot \overline{\mathbf{R}} \cdot \boldsymbol{E}}\right),
\label{eq:wspatial}$$ again for $j = x, y,$ and the component angular shift is $$\Delta_{wj} = \frac{\langle |\boldsymbol{K}|^2 \rangle}{k} {\mathrm{Im}}\frac{\boldsymbol{F}^* \cdot \mathbf{A}_j \cdot\boldsymbol{E}_{\mathrm{i}}}{\boldsymbol{F}^* \cdot \boldsymbol{E}_{\mathrm{i}}}
= \langle \delta^2 \rangle {\mathrm{Re}}\left( \frac{\boldsymbol{F}^*\cdot \overline{\mathbf{R}_j} \cdot\boldsymbol{E}}{\boldsymbol{F}^*\cdot \overline{\mathbf{R}} \cdot \boldsymbol{E}}\right) .
\label{eq:wang}$$ Of course, the weak shift typically has an angular contribution even when the $\mathbf{A}_j$ are hermitian, representing the inhomogeneity of the polarization pattern in direction space, and weak shifts can be rather different directions to their strong counterparts (or exist when these are zero, as is the case of the spin Hall effect for light [@HostenKwiat:SCI:2008]).
![\[fig:polarized\] Examples of shifted, reflected beams with weakly varying polarization, in units of $k^{-1}.$ In each case, the incident beam has polarization $\boldsymbol{E} = \frac{1}{\sqrt{2}}(1,1)$, with an incident gaussian profile with width $100k^{-1}$ at incidence angle $\theta_0 = \arcsin 3/4.$ Reflection is between air and glass, with (a) total reflection ($n = 3/2$), (b) partial reflection ($n = 2/3$). The plots, in virtual beam coordinates, show contours of overall intensity (grey), and (a) magnitude of components in $p$- and $s$-polarization (purple and green respectively); (b) magnitude of right-hand and left-hand circular components (cyan and turquoise respectively). Despite the beam shifts being different, the variation in polarization from the virtual beam polarization $\boldsymbol{E}_0$ is imperceptible at this scale (blue ellipses and lines). ](Fig2.pdf){width="12cm"}
An illustration of optical beam shifts, including component shifts, is shown in Fig. \[fig:polarized\]. In each case, on the lengthscale of $k^{-1},$ the inhomogeneous polarization is extremely weak, although both longitudinal and transverse shifts are present. A systematic theoretical study of the component and beam shifts for different choices of $\boldsymbol{E}$ and $\boldsymbol{F}$ is made in [@GoetteDennis:NJP:2012].
For certain choices of $\boldsymbol{F},$ the component shift may be very large, with a rather small overall amplitude. However, it is important to note that such superweak shifts occur when $\boldsymbol{F}^*\cdot \boldsymbol{E}_{\mathrm{i}} = \boldsymbol{F}^*\cdot \overline{\mathbf{R}} \cdot \boldsymbol{E}$ is small, and the inner product between initial and final polarizations $\boldsymbol{F}^*\cdot \boldsymbol{E}$ does not directly play a role. This is of course because the analogue of the prepared quantum state is the virtual beam.
These optical shifts completely mirror quantum mechanical pointer shifts, and the 2-dimensional nature does not change any of the essential description. However, if the complex pointer amplitude $\varphi(\boldsymbol{r})$ has radial- and azimuthal-varying complex phase, the situation is more complicated than the one-dimensional case described in [@Josza:PRA76:2007], and we describe this in the following section.
The case of radial and azimuthally varying pointer phase {#sec:complex}
========================================================
In this section, we will represent the weak or expectation value of the Artmann operator by the complex 2-dimensional vector $\boldsymbol{a} = (a_x,a_y),$ which is defined by one of the following: $$a_j = \frac{\boldsymbol{E}\cdot \overline{\mathbf{R}}^{\dagger} \overline{\mathbf{R}_j} \cdot\boldsymbol{E}}{\boldsymbol{E}\cdot \overline{\mathbf{R}}^{\dagger}\overline{\mathbf{R}} \cdot \boldsymbol{E}}
\quad\hbox{or}\quad \frac{\boldsymbol{F}^*\cdot \overline{\mathbf{R}_j} \cdot\boldsymbol{E}}{\boldsymbol{F}^*\cdot \overline{\mathbf{R}} \cdot \boldsymbol{E}}, \quad j = x,y.$$ Although our main example is optical beams, the discussion applies to any weakly interacting quantum system with two noncommuting operators $A_x, A_y$ multiplying the two components of pointer momentum, and our aim is to generalize the contribution of ${\mathrm{Im}}(a)$ to the spatial pointer shift.
Following [@Josza:PRA76:2007], we consider the modulus squared (intensity or probability distribution) of the reflected wave after tracing out or postselecting the polarization state, $$|(1 - \rmi \boldsymbol{K}\cdot \boldsymbol{a})\varphi(\boldsymbol{r})|^2 \approx |\rme^{-{\mathrm{Re}}(\boldsymbol{a})\cdot\nabla}\varphi|^2 - \rmi \, {\mathrm{Im}}(\boldsymbol{a})\cdot (\varphi^* \nabla \varphi - \varphi \nabla \varphi^*)
\label{eq:jshift}$$ where $\boldsymbol{K},$ as the momentum, corresponds to the generator of translations $-\rmi \nabla,$ and only terms up to the first order in $\boldsymbol{a}$ are kept as usual. The first term represents the intensity pattern shifted by ${\mathrm{Re}}(\boldsymbol{a})$, and the second term represents additional interference caused between the differently-weighted Fourier components with varying incident phases.
This second term generalizes the second term on the RHS of (\[eq:jtheorem1\]) to 2-dimensional pointer wavefunctions; the first term here – a pure translation – is just the usual spatial shift. The shift is the mean of $\boldsymbol{r}$ with respect to this distribution, and we assume that the mean position of the unshifted (virtual) beam is the origin.
We begin our analysis of the second term of Eq. (\[eq:jshift\]) by rewriting it as the (optical or probability) current vector $\boldsymbol{J} \equiv -\frac{\rmi}{2}(\varphi^*\nabla\varphi - \varphi\nabla\varphi^*),$ $$- \rmi \, {\mathrm{Im}}(\boldsymbol{a})\cdot (\varphi^* \nabla \varphi - \varphi \nabla \varphi^*) = 2 \,{\mathrm{Im}}(\boldsymbol{a})\cdot \boldsymbol{J}.$$ Its contribution to the $x$-component of the spatial shift is therefore $$D_x^{\mathrm{extra}} \equiv 2 \int \rmd^2 \boldsymbol{r}\, x \,{\mathrm{Im}}(\boldsymbol{a})\cdot \boldsymbol{J} = -\int \rmd^2 \boldsymbol{r}\, x^2 \partial_x\left[{\mathrm{Im}}(\boldsymbol{a})\cdot \boldsymbol{J}\right],
\label{eq:shiftcorr}$$ with the equality following from Green’s theorem. An equivalent expression holds for the shift in the $y$-direction.
In the one-dimensional case [@Josza:PRA76:2007], this is related through conservation of current and the Schrödinger equation to the rate of change of the variance of the wavefunction as in Eq. (\[eq:jtheorem1\]). This is not possible in two or higher dimensions since the integral does not involve the divergence of $\boldsymbol{J},$ and the general situation appears rather more complicated.
However, in most optical fields of interest, any varying phase factor factorizes into an azimuthal part (say with a vortex with quantum number $\ell$) and a radial part, i.e. for azimuthal angle $\phi$ and radius $r,$ $$\varphi(\boldsymbol{r}) = \sqrt{I(r)} \exp(\rmi \ell \phi + \rmi f(r) ),
\label{eq:phaseansatz}$$ for intensity $I(r) = |\varphi|^2,$ and $f(r)$ some radius-dependent phase function. It should be noted that, for optical reflection, the azimuthal index $\ell$ reverses sign on reflection [@Fedoseyev:OC:2001; @leachetal]. Our convention here is that Eq. (\[eq:phaseansatz\]) refers to the virtual beam, so the azimuthal dependence of the incident beam is $\exp(-\rmi \ell \phi).$
It is easy to see that, for the virtual beam (\[eq:phaseansatz\]) on returning to cartesian cordinates, $$\boldsymbol{J} = I(r) \left[ \frac{\ell}{r^2} (-y,x) + \frac{f'(r)}{r} (x,y) \, \right].$$ Using this form in the formula for the extra shift in Eq. (\[eq:shiftcorr\]) in $x,$ and after integrating out the azimuth $\phi,$ we have $$\begin{aligned}
\fl D_x^{\mathrm{extra}}
= -\frac{1}{8} {\mathrm{Im}}(\boldsymbol{a})\cdot\int_0^{\infty} \rmd r\, r \left(r I(r)[f'(r)+3r f''(r)]+3r^2 f'(r)I'(r), \ell[3r I'(r)-2I(r)]\right) \nonumber \\
= {\mathrm{Im}}(\boldsymbol{a})\cdot(\langle r f'(r)\rangle, \ell), \end{aligned}$$ where the second line follows from normalization and integration by parts, assuming $f(r)$ is reasonably well-behaved. An identical argument for $y$ gives $$D_y^{\mathrm{extra}} = {\mathrm{Im}}(\boldsymbol{a})\cdot(-\ell,\langle r f'(r) \rangle),$$ so the net total shift in two dimensions, analogous to (\[eq:jtheorem1\]) for a pointer with the assumed form, is $$\boldsymbol{D} = {\mathrm{Re}}(\boldsymbol{a}) + (\langle r f'(r) \rangle +\rmi \ell \boldsymbol{\sigma}_2)\cdot{\mathrm{Im}}(\boldsymbol{a}),$$ where $\sigma_2$ is the second Pauli matrix.
This is the form of the the spatial shift for complex $\boldsymbol{a}$ and complex $\varphi(\boldsymbol{r}).$ The $\ell$-dependent parts correspond to the ‘vortex-induced shift’ [@Fedoseyev:OC:2001; @dasguptagupta; @Bliokh+:OL34:2009; @Merano+:PRA82:2010]: the $\ell$-fold twisting of the phase fronts results in ${\mathrm{Im}}(a_y),$ usually determining the angular shift in $y,$ contributing to the real shift in $x$ weighted by $\ell$ and similarly (in a manner preserving the sense of circulation), $-\ell{\mathrm{Im}}(a_x)$ contributes to the shift in $y.$ Apart from the azimuthal quantum number $\ell,$ no other features of the incident beam contribute to this part of the shift.
The other part, given by ${\mathrm{Im}}(\boldsymbol{a})\langle r f'(r)\rangle,$ directly generalizes the corresponding term in (\[eq:jtheorem1\]). In optical beams, a radially-varying phase is interpreted as curved phasefront (rather than a beam at its focus, where the phasefronts are flat and $f(r)=0$), due, for instance, to the Gouy phase [@Gouy:CRAS110:1890; @Siegman:USB:1986]. This radial phase variation leads to a term proportional to the expectation value of phasefront gradient $\langle rf'(r)\rangle$ times the usual angular shift.
A simple example to consider is a gaussian light beam (gaussian wavepacket) of width $w$ propagating according to the paraxial equation, $\varphi(\boldsymbol{r},z) = \exp(-r^2/[w^2(1+ \rmi z/z_{\mathrm{R}})])/(1+\rmi z/z_{\mathrm{R}}),$ where $z_{\mathrm{R}} = k w^2/2,$ the Rayleigh distance. In this case, $\langle r f'(r) \rangle = z/z_{\mathrm{R}},$ implying the radial part of the extra shift is small when the beam is focused close to the interface; this result naturally is similar to the one-dimensional [@Josza:PRA76:2007] constant related to the rate of expansion of the transverse waveform. To our knowledge, this additional ‘defocusing-induced shift’ has not been experimentally measured, but is present as an additional feature of a more complicated amplitude pattern, corresponding to a more complicated quantum mechanical pointer wavefunction [@AharonovBotero:PRA72:2005; @BerryShukla:JPA45:2012]. In particular it is different to a focal shift, that is a shift of the focus along the propagation axis of the beam as discussed in [@McGuirkCarniglia:JOSA67:1977].
Discussion {#sec:disc}
==========
We have shown how the reflection of a narrow optical beam is a classical wave analogue to a quantum weak measurement. Although the experiment could be done for an ensemble of single photons, nothing in our analysis relies strictly on the quantum nature of the light: the Hilbert space is completely classical, describing polarization and (transverse) position of the optical beam, with entanglement of the degrees of freedom interpreted as a position-dependent polarization pattern [@SimonGori2010].
As such, this phenomenon adds to many well-known classical optical analogues of quantum phenomena involving polarization and complex optical amplitudes. Although our paper highlights the analogy between the weak values and optical beam shifts of polarization components, the discussion applies to the whole beam without a polarizing analyzer, which is itself analogous to a usual expectation value. In particular, the effect of the weak measurement on the imaginary part of an operator’s average value is immaterial of the kind of average (weak or expectation); the shift to the momentum wavefunction and possibly extra spatial shift always occurs for average values of nonhermitian operators, realised here in partial reflection.
The main effect of reflection, of course, is to change a beam’s propagation direction and polarization, giving what we call the virtual beam. It is the virtual beam which is analogous to the quantum prepared system, whose position and polarization weakly interact, causing it to be shifted in both position and direction. Rather than coming from a small coupling constant, the weakness comes from paraxiality, that is narrowness of the beam about its mean propagation direction in Fourier space. In this case, as we have seen, the main change to a beam on reflection is the small shift in position and possibly direction; for nonparaxial beams (such as the dipole radiation field considered in [@Berry:PRLSLA467:2011]), the shifts are present, but the reflected field is much more complicated.
The effects described here are only to first order, as are most phenomena studied in the physics of weak measurement. Second order effects become important for superweak values, whenever the analyzer is almost orthogonal to the polarization of the virtual beam. The magnitude of the spatial and angular shifts corresponding to superweak values is explored in an optical setting elsewhere [@GoetteDennis:OL37:2012]. With the exception of the extra shift coming from the pointer wavefunction’s varying phase, all of the shifts are independent of the spatial distribution of the amplitude of the beam (assuming its modulus is radially symmetric). This simplifies previous descriptions of the angular shift in optics, as the shift has a universal form when measured in units of the angular spread $\langle \delta^2 \rangle.$ Higher-order effects involve changes to the shape of the beam, and these will reveal more subtle structures associated with the entanglement between position and polarization.
It should be stressed that the shift of a beam on reflection does not require polarization, only an incidence angle-dependent reflection coefficient $r(\theta_0).$ This might be achieved with acoustic waves at a lossy wall (similar to the partial reflection case), or Robin boundary conditions (similar to total reflection) [@DennisGoette:SPIE7950:2011]. From the discussion above, it is clear that even in this case there is a complex scalar shift $a = (\rmi/k) (r'/r) ,$ whose real and imaginary parts contribute in the ways described above: the real part gives rise to a longitudinal spatial shift, the imaginary part to a longitudinal angular shift, and possibly to an additional longitudinal spatial shift (if the wave has a radius-dependent phase) and a transverse shift (if there is an azimuthal phase). Most of the transverse shifts described above are absent as they require spin (polarization) interacting with azimuthal (orbital) terms.
The fact that weak values generalise so simply to classical wave physics cements their significance as physical quantities. Furthermore, the generality of the beam shift framework as we have outlined suggests generalization to other kinds of waves, not only acoustic waves as suggested above, but also elastic waves and shifts to matter waves [@deHaan+:PRL104:2010; @balasz]. This suggests that further analysis of such simple physical phenomena as the reflection of a beam at a dielectric might reveal new insights into quantum physics.
Acknowledgements {#acknowledgements .unnumbered}
================
We are grateful to Michael Berry and Kostya Bliokh for valuable comments. This work was carried out when JG was a Royal Society Newton Fellow. MRD is a Royal Society University Research Fellow.
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|
---
abstract: 'This paper documents the fiber R and D for the CMS hadron barrel calorimeter (HCAL). The R and D includes measurements of fiber flexibility, splicing, mirror reflectivity, relative light yield, attenuation length, radiation effects, absolute light yield, and transverse tile uniformity. Schematics of the hardware for each measurement are shown. These studies are done for different diameters and kinds of multiclad fiber.'
address: 'Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627'
author:
- 'H. S. Budd, A. Bodek, P. de Barbaro, D. Ruggiero, E. Skup'
title: Fiber R and D for the CMS HCAL
---
epsf
UR-1513 ER-40685-909
To be published in the proceeding of SCIFI97 Conference,\
November 3-6, 1997, Notre Dame, Indiana.
The CMS HCAL optical design is similar to the CDF Plug Upgrade optical design [@cdf_end_plug] [@note]. A wavelength shifting (WLS) fiber, containing K27 waveshifter, embedded in the tile collects the scintillation light. Outside the tile, the WLS fiber is spliced to a clear fiber. The clear fiber takes the light to a connector at the edge of the pan. An optical cable brings the light to the optical readout box. The readout box assembles the light from layers to towers and brings the light to the photodetectors.
The fibers tested for this paper are all multiclad. The diameters of the fibers range from 0.83 mm to 1.0 mm. CMS has chosen fiber of 0.94 mm diameter. We test four types of Kuraray fiber, non-S (S-25), S-35, S-50, and S (S-70) [@kuraray]. The most flexible Kuraray fiber is S type and the least flexible fiber is non-S type. We test two batches of WLS Bicron fiber, BCF91A-MC[@bicron]. Batch 1 is an earlier version than Batch 2. We test one batch of Bicron clear fiber, BCF-98 MultiClad, which was made at the same time at Batch 1 WLS fiber. The waveshifter for all WLS fiber is K27.
We use R580-17 phototubes. This tube is a Hamamatsu 1.5 inch diameter, 10 stage tube with a green extended photocathode. The R580-17 photocathode and the photocathode for the CMS photodetectors, HPDs, are the same.
For most measurements a tile excited by a radioactive source generates the light. This insures that the spectrum of light is the same for these tests as the spectrum from the CMS HCAL calorimeter. We use both a Cs-137 $\gamma$ source and a Ru-106 $\beta$ source. The Cs-137 source is collimated by a lead cone. The widest diameter of the cone is 7.5 cm. A picoammeter reads out the phototube. The data aquisition (DAQ) program averages 20 measurements from the picoammeter and creates a pedestal subtracted data file. The absolute light yield measurement uses the Ru-106 source. Its DAQ consists of a 2249A Lecroy ADC triggered by a coincidence of two scintillation counters.
The optical connectors used for these measurements were developed by DDK [@ddk] and CDF Plug Upgrade Group [@optical_conn] [@cdf_upgrade]. Their part numbers are MCP-10P-1 (0.83 mm fiber), MCP-10P-2 (0.90 mm fiber), MCP-10P-3 (1.00 mm fiber), and MCP-10A (the connector housing).
Fiber flexibility {#fiber-flexibility .unnumbered}
=================
We have studied Kuraray non-S fiber for flexibility by looking at the change in light transmission after the fibers are bent. Fibers with a change in transmission greater than 2% always have cracks or crazing in the bent portions. They have light leaking out from these cracks. Hence, we test for flexibility by looking for light leaking out of the bend portions.
Fiber type Fiber not damaged Fiber Damaged
---------------------- ------------------- ---------------
Kuraray 1 mm non-S 2 1/2 in 2 in
Kuraray 0.94 mm S-35 3/4 in 5/8 in
Kuraray 0.94 mm S-50 5/8 in -
Kuraray 0.83 mm S 5/8 in -
Bicron 1.00 mm 2 in 1 1/2 in
: Flexibility of different fibers. Column 1 gives the kind of fiber. Column 2 gives the smallest bend diameter without the fiber damage. Column 3 gives the largest bend diameter with fiber damage.
\[flexibility\]
We test the flexibility by wrapping the fibers around dowels and looking for cracks where light leaks out. Table \[flexibility\] gives the result. The test lasted 1/2 year. If the fiber develops cracks, the cracks appear 1/2 - 2 days after the fibers are wrapped around the dowel. The smallest fiber bend diameter for the HCAL barrel is 3 inches. Both the Kuraray S-35 fiber and Bicron fiber are flexible enough for CMS HCAL.
Fiber Splicing {#fiber-splicing .unnumbered}
==============
Fiber splicing is done with the semi-automated splicer developed by the CDF Plug Upgrade Group [@splicer]. Splice transmission of WLS fibers is measured by scanning across a splice using the CDF automated UV scanner [@splicer], see Figure \[cdf\_uvscanner\_eva\]. Figure \[splice\_green\]a shows the results of splicing tests. Table \[splice\] lists the results of the tests. Only splicing tests done at the same time should be compared. The Dec 96 splicing test shows that non-S fiber splices have higher transmission than S type fiber. This result confirms CDF’s measurement of the difference in splice transmission between non-S and S type fibers [@splicer]. The Nov 97 splicing test shows the splice transmission for non-S and S-35 fiber is the same. Splice transmission for S-50 is worse than non-S and S-35 fiber. We have chosen S-35 fiber for the HCAL preproduction prototype because of its excellent flexibility and high splice transmission. The Sept 96 splicing test shows that the splice transmission for Kuraray non-S fiber and Bicron fiber is the same.
The fiber ends must be polished well for good splice transmission. We have compared splice transmission using two different polishing techniques. One polishing technique uses the Avtech polisher [@cdf_upgrade]. The Avtech polisher is a single fiber polisher which CDF used for its production. The second technique is ice polishing. Ice polishing was pioneered by Fermilab Charmonium experiment E835 [@E835], and was used by Fermilab experiment HyperCP E871 [@E871], and D0. The technique involves freezing fibers in water and polishing the fiber/ice combination. Many fibers can be polished at once with ice polishing. Figure \[splice\_green\]b shows the results of splicing Kuraray 0.94 mm, S-35 fibers with these two techniques. We conclude that both polishing techniques give the same transmission. CMS HCAL has chosen to ice polish their fibers.
We have tested the splice transmission of clear fibers. Figure \[clear\_splice\_test\_eva\] shows the setup used to measure the transmission of the splice. A connector and fiber assembly with WLS fiber at the nonconnector end is called a “pigtail”. The WLS fibers in a “pigtail” are inserted into a tile and then the tile is excited by a radioactive source to readout the light. The pigtail for this test consists of 20 cm WLS fibers spliced to 99 cm clear fibers in a connector. The WLS fiber inserted into a tile injects a constant amount of light into the clear fiber. After the clear fiber is cut and spliced, the pigtail is remeasured. The ratio of the measurement before and after the splice is defined as the splice transmission. The results are shown in Figure \[splice\_clear\]. The test shows that the splice transmission for clear non-S and clear S-35 fiber is the same. Both clear and WLS S-35 splicing tests are done with the same splicing machine by the same operator. Table \[splice\] lists all the fiber splicing results.
=5.7in\
\[cdf\_uvscanner\_eva\]
=2.7in\
\[splice\_green\]
Fiber type Fluor Polish Date Number Mean RMS
----------------------- ------- -------- -------- -------- ------- -------
Kuraray 0.83 mm non-S K27 Avtech Dec 96 3 0.948 0.006
Kuraray 0.83 mm S K27 Avtech Dec 96 3 0.876 0.009
Kuraray 0.94 mm non-S K27 Avtech Nov 97 2 0.928 –
Kuraray 0.94 mm S-35 K27 Avtech Nov 97 9 0.930 0.009
Kuraray 0.94 mm S-50 K27 Avtech Nov 97 4 0.879 0.019
Kuraray 0.83 mm non-S K27 Avtech Sep 96 5 0.908 0.009
Bicron 0.83 mm K27 Avtech Sep 96 5 0.902 0.003
Kuraray 0.94 mm S-35 K27 Avtech Mar 98 5 0.908 0.018
Kuraray 0.94 mm S-35 K27 Ice Mar 98 5 0.930 0.008
Kuraray 0.94 mm non-s clear Avtech Nov 97 5 0.904 0.018
Kuraray 0.94 mm S-35 clear Avtech Nov 97 5 0.893 0.011
: Splice transmission of different kinds of fibers and different polishing techniques. Column 1 gives the kind of fiber. Column 2 gives the fluor. Column 3 gives the kind of polish. Column 4 gives the date of the test. Column 5 gives the number of fibers tested. Column 6 gives the mean of the splice distribution. Column 7 gives the RMS of the distribution. Only tests done on the same date should be compared.
\[splice\]
=5.5in\
\[clear\_splice\_test\_eva\]
=2.7in \[splice\_clear\]
Mirror Reflectivity {#mirror-reflectivity .unnumbered}
===================
The ends of the fibers are mirrored using vacuum deposition in Lab 7 at Fermilab. A brief description of the mirroring procedure is given in reference [@E871]. We have studied the mirror reflectivity for 2 types of fibers and both polishing techniques.
The reflectivity is measured using the automated UV scanner (see Figure \[cdf\_uvscanner\_eva\]). A mirrored fiber is put in a connector and measured with the UV scanner near the mirror. Next, the mirror is cut off at 45$^\circ$ to the fiber axis. The end of the fiber is painted black to prevent any reflection from the end of the fiber. The fiber is remeasured with the automated UV scanner. The reflectivity is defined as (measurement with mirror)/(measurement without mirror) - 1.
Figure \[mirror\] shows the mirror reflectivity results. We have measured the reflectivity of a 3 1/2 year old CDF pigtail, which was a spare for the CDF End Plug Hadron Calorimeter [@cdf_upgrade]. The pigtail was made of 0.83 mm, non-S fibers polished with the Avtech polisher. The mirror was dipped in Red Spot UV curable coating to protect the mirror. The measurement shows no degeneration in the mirror after 3 1/2 years. A measurement of recently mirrored non-S type fiber gives the same reflectivity. The reflectivity of S-35 fibers is roughly 5% lower. The reflectivity for ice polished fibers seems to be slightly lower than Avtech polished fibers. Figure \[mirror\]b shows the mirror reflectivity for Kuraray and Bicron fiber are very similar.
=2.7in\
\[mirror\]
We have measured the light increase from the mirror. A pigtail with three fibers, shown in Figure \[light\_from\_mirror\_eva\], is made and scanned with the UV scanner. The pigtail is also measured with a tile and phototubes as shown in Figure \[light\_from\_mirror\_eva\]. Next, the mirror is cut off at 45$^\circ$ to the fiber axis, and the end of the fiber is painted black. The pigtail is remeasured with both setups. We get the following: $$\normalsize \frac{\left( \frac
{{\displaystyle}\text{light from tile with mirror}}
{{\displaystyle}\text{light from tile with no mirror}}\right)}
{ \left( \frac
{{\displaystyle}\text{light from UV scanner with mirror}}
{{\displaystyle}\text{light from UV scanner with no mirror}}\right)}
= \text{0.865 ~~~with RMS = 0.004}$$ This is used to transfer reflectivity measurements using the UV scanner into reflectivity measurements done using a scintillator tile. Figure \[mirror\]a states that light increase from the mirror for ice polished S-35 fibers from the UV scanner is 1.77. The corresponding increase for a tile is 1.53.
=5.7in
\
\[light\_from\_mirror\_eva\]
Relative Light Yield For Different Fibers {#relative-light-yield-for-different-fibers .unnumbered}
=========================================
We measured the light response of different kinds of fibers. Figure \[rel\_light\_diff\_fiber\_eva\] shows the setup used for the relative light yield test. Fibers are inserted into a tile made of Kuraray SCSN-81 scintillator. A clear “cable” connects the pigtail from the connector to the phototube. Table \[relative\_light\_yield\] gives the result for different types of WLS fibers using both 103 cm and 251 cm Kuraray 1 mm S type clear cables. Column 2 and 3 are separately normalized to the Kuraray 0.83 mm, 250 ppm, non-S result (labeled by $\equiv$). For Kuraray fiber, the 0.94 mm WLS fiber yields 6% more light than a 0.83 mm WLS fiber. There is no difference in light between a 0.94 mm WLS fiber and a 1.0 mm WLS fiber connected to 1 mm clear fiber. Table \[fiber\_diameter\] gives the core diameter and fiber diameter. One sample of each fiber is measured. The increase in light is smaller than the increase in either the core diameter or the fiber diameter.
=5.7in
\
\[rel\_light\_diff\_fiber\_eva\]
Type of WLS fiber 251 cm Cable 103 cm Cable
---------------------------------- --------------- ---------------
Bicron 1.00 mm, 200 ppm, batch 1 0.93 0.93
Bicron 1.00 mm, 200 ppm, batch 2 1.00 1.02
Kuraray 0.83 mm, 250 ppm, non-S $\equiv$ 1.00 $\equiv$ 1.00
Kuraray 0.94 mm, 250 ppm, S-35 1.07 1.06
Kuraray 0.94 mm, 250 ppm, S-50 1.06 1.06
Kuraray 1.00 mm, 200 ppm, non-S 1.06 1.05
Kuraray 1.00 mm, 300 ppm, S 1.08 1.07
: Relative light yield. The cables are made of 1 mm S type Kuraray fiber. Column 2 gives the light using a 251 cm cable and Column 3 gives the light using a 103 cm cable. The entries marked $\equiv$ are defined to be 1.00.
\[relative\_light\_yield\]
Fiber type Core diameter Inner cladding diameter Outside diameter
------------------------------- --------------- ------------------------- ------------------
Bicron, 0.83 mm, WLS 0.789 $\dagger$ 0.850
Kuraray, 0.83 mm non-S, WLS 0.742 0.786 0.844
Kuraray, 0.83 mm non-S, clear 0.737 $\ddagger$ 0.838
Kuraray, 0.94 mm non-S, WLS 0.841 0.903 0.959
Kuraray, 0.94 mm non-S, clear 0.838 $\ddagger$ 0.946
Kuraray, 1.00 mm non-S, WLS 0.887 0.955 1.008
Kuraray, 1.00 mm non-S, clear 0.902 0.958 1.010
: Core and fiber diameter. All units are mm. One fiber is measured for each type. $\dagger$ means fiber has only one visible cladding. $\ddagger$ means interface between outer cladding and inner cladding was not distinguishable enough to measure it. Kuraray fiber has 2 claddings. Multiclad Bicron fiber has one visible cladding, as the outer cladding is too thin to be visible.
\[fiber\_diameter\]
Attenuation Length of Clear Fibers {#attenuation-length-of-clear-fibers .unnumbered}
==================================
We have looked at the relative light transmission of different clear fibers by using two different cables. Figure \[rel\_light\_diff\_fiber\_eva\] shows the setup. Table \[ratio\_cable\_light\] gives the ratio of light for the two different cables. Column 2 and column 3 give the cables used. The length of the cables for column 2 are all 251 cm. Column 4 gives the ratio of the measurement of column 2 over column 3. The results are consistent with an equal attenuation length for 1 mm S, 0.94 mm S-35, and 0.94 S-50 Kuraray fiber.
Pigtail Cable 1, L=251 cm Cable 2 Cab 1/Cab 2
-------------------- ------------------- ------------------------- -------------
Kur 1.00 mm, non-S Kur 1.00 mm S 103 cm Kur 1.00 mm S 0.79
Kur 0.94 mm, S-35 Kur 0.94 mm S-35 103 cm Kur 0.94 mm S-35 0.80
Kur 0.94 mm, S-50 Kur 0.94 mm S-50 103 cm Kur 0.94 mm S-50 0.80
Bicron 1.00 mm Bicron 1.00 mm 251 cm Kur 1.00 mm S 0.89
: Comparison of light yields using two different clear cables.
\[ratio\_cable\_light\]
=5.7in
\
\[cable\_cdf\_att\_len\_eva\]
=5.7in
\
\[clear\_kur\_bic\_eva\]
=2.7in\
\[clear\_kur\_bic\]
We have measured the attenuation length of the clear fiber in a cable. Figure \[cable\_cdf\_att\_len\_eva\] gives the apparatus for measuring Kuraray 0.9 mm S type fibers, and Figure \[clear\_kur\_bic\]a gives the result. Each point is a measurement of one fiber in a cable. Two separate pigtails are used and they give the same attenuation length of the clear fiber. The combined results of both pigtails for the attenuation length is 732 $\pm$ 13 cm. The RMS of the normalized light about the exponential curve in Figure \[clear\_kur\_bic\]a is 5.6%. The test shown in Figure \[clear\_kur\_bic\_eva\] measures the difference in attenuation between the Kuraray 1 mm fiber and Bicron 1 mm fiber. Figure \[clear\_kur\_bic\]b gives the result. Each point is the average of the three fibers in the pigtail. The attenuation lengths for Kuraray fibers given in Figure \[clear\_kur\_bic\]a and Figure \[clear\_kur\_bic\]b agree.
The data for the clear fiber attenuation measurements are fit to ae$^{-x/b}$. The data and curve are normalized by setting a = 1. A fit to the data used for Figure \[clear\_kur\_bic\]b gives a$_{kur}$/a$_{bic}$ = 1.02. a$_{kur}$/a$_{bic}$ is the amount of light accepted by the Kuraray clear fibers divided by the amount of light accepted by the Bicron clear fibers. Theoretically, this should be the same as ratio of numerical apertures of the two kinds of fibers. Hence, Kuraray fiber and Bicron fiber have the same numerical aperture.
=5.7in
\
\[clear\_att\_len\_eva\]
=2.7in\
\[fib\_att\_clear\]
We have looked at the attenuation length of the clear fiber when one end is spliced to a WLS fiber and the other end is glued in a connector. Figure \[clear\_att\_len\_eva\] shows the apparatus used to measure the clear attenuation length, and Figure \[fib\_att\_clear\] gives the results. For Test A ( Figure \[fib\_att\_clear\]a), one pigtail with 3 fibers is made for each length of clear fiber. A single exponential fit gives $\lambda$ = 6.4 m with the cable and $\lambda$ = 6.55 m without the cable. Hence, the attenuation length of the clear fiber does not depend on whether the cable is present.
For Test B (Figure \[fib\_att\_clear\]b), one pigtail was made with three 4.2 m clear fibers. The pigtail is measured. The pigtail connector is cut off and a new connector is put on with the clear fiber reduced to 3 m. The pigtail is measured and the process continues until the pigtail has 0.1 m of clear fiber left. Test B uses the same splice between the clear and green fibers for all the clear lengths and should give a better measurement of the attenuation length.
The measurement shown in Figure \[clear\_kur\_bic\]a gives the best measurement of the attenuation length of the clear Kuraray fiber. A single exponential with $\lambda$ = 7.3 m gives an adequate description of the attenuation length of all Kuraray clear fiber for lengths $<$ 4 m. The other measurements of the clear Kuraray fiber are consistent with this measurement.
Attenuation Length of WLS Fibers {#attenuation-length-of-wls-fibers .unnumbered}
================================
=2.7in\
\[fib\_att\_all\]
=5.7in
\
\[minos\_app\_eva\]
We have measured the attenuation lengths of various WLS fibers. The attenuation lengths are measured with a setup provided by the MINOS experiment, see Figure \[minos\_app\_eva\] . The MINOS setup provides an easy and quick way of measuring the attenuation length of many WLS fibers using scintillator material as a light source. One end of the fiber is polished, and the other end is cut at 45$^\circ$ and painted black. The fiber is inserted into a long hole in the scintillator and the polished end is pushed up against a light mixer on the phototube. The source is moved across the scintillator and the phototube current is read out with a picoammeter. The data are fit to $ae^{-x/l_1}+be^{-x/l_2}$. Figure \[fib\_att\_all\]a plots the distance from source to the phototube vs the normalized light. Table \[K27\_atten\] gives the numerical results of the fits.
Almost all of our WLS fibers are 3 m long. We have some 4 m Kuraray 0.94 mm, 250 ppm, non-S fiber. To determine if the attenuation lengths using 3 m and 4 m pieces agree, we measure two 4 m pieces in the MINOS setup. We cut the 2 pieces to 3 m and remeasured them. Figure \[fib\_att\_94\_nons\_4m\] plots the results. The measurement shows that an attenuation length measured with a 3 meter piece can be extrapolated to 4m.
=2.7in \[fib\_att\_94\_nons\_4m\]
WLS Fiber type Setup $a$ $l_1$ $b$ $l_2$
--------------------------------- ------------ ------- ------- ------- -------
Bicron 1.00 mm, 200 ppm, bat-1 MINOS 0.488 35 0.512 254
Bicron 1.00 mm, 200 ppm, bat-2 MINOS 0.357 33 0.643 343
Kuraray 0.83 mm, 250 ppm, non-S MINOS 0.323 34 0.677 379
Kuraray 0.94 mm, 250 ppm, non-S MINOS 0.372 39 0.628 366
Kuraray 0.94 mm, 250 ppm, S-35 MINOS 0.345 33 0.655 320
Kuraray 0.94 mm, 250 ppm, S-50 MINOS 0.348 34 0.652 317
Kuraray 1.00 mm, 200 ppm, non-S MINOS 0.317 31 0.683 326
Bicron 1.00 mm, 200 ppm, bat-1 UV 0.304 63 0.696 611
Kuraray 0.83 mm, 250 ppm, non-S UV 0.102 33 0.898 375
Kuraray 1.00 mm, 200 ppm, non-S UV 0.188 79 0.812 431
Kuraray 1.00 mm, 300 ppm, S UV 0.524 131 0.476 1407
Kuraray 0.94 mm, 250 ppm, S-35 SLIDE 0.287 31 0.713 366
Bicron 1.00 mm, 200 ppm, bat-1 SLIDE 0.381 31 0.619 303
Bicron 1.00 mm, 200 ppm, bat-1 MINOS RAD 0.484 30 0.516 259
Bicron 1.00 mm, 200 ppm, bat-1 BEFORE RAD 0.462 39 0.538 332
Kuraray 1.00 mm, 200 ppm, non-S MINOS RAD 0.343 33 0.657 250
Kuraray 1.00 mm, 200 ppm, non-S BEFORE RAD 0.335 42 0.665 361
Kuraray 1.00 mm, 300 ppm, S MINOS RAD 0.362 31 0.638 235
Kuraray 1.00 mm, 300 ppm, S BEFORE RAD 0.331 36 0.669 323
Bicron 1.00 mm, 200 ppm, bat-1 UV RAD 0.305 58 0.695 358
Kuraray 1.00 mm, 200 ppm, non-S UV RAD 0.395 148 0.605 332
Kuraray 1.00 mm, 300 ppm, S UV RAD 0.733 152 0.267 1056
: Results of fit to attenuation data for WLS fibers. The data are fit to $ae^{-x/l_1}+be^{-x/l_2}$. All units for $l_1$ and $l_2$ are cm. The column marked Setup gives the apparatus used to measure the fibers. MINOS uses the MINOS apparatus, UV used the CDF UV scanner, and SLIDE is the sliding fiber apparatus. MINOS RAD are the radiated fibers measured with the MINOS setup. BEFORE RAD are the same fibers measured before irradiation with the MINOS apparatus with the correction function. UV RAD are the same radiated fibers measured with the UV setup. These are the same fibers labeled UV in the Setup column.
\[K27\_atten\]
We compared the above measurement with a measurement using a UV light and pin diodes to read out the fibers, see Figure \[cdf\_uvscanner\_eva\]. Figure \[fib\_att\_all\]b plots the attenuation of the fibers vs the distance to the connector. The MINOS measurement gives a greater difference between the Kuraray fiber and batch 1 Bicron fiber than the UV measurement. The greater sensitivity of pin diodes in the UV setup to long wavelengths light may be the reason. Table \[K27\_atten\] gives the results of the fits in the rows labeled UV.
Fibers can be measured quickly with either the MINOS setup or UV scanner. Both tests are useful for comparing fibers. However, the setups may not give the correct attenuation length of the WLS fibers which are relevant for CMS HCAL design. Figure \[sliding\_fiber\_eva\] shows the setup designed to give a more accurate attenuation length. Figure \[fib\_att\_slide\]a shows the attenuation length of two kinds of fibers measured with the sliding fiber setup. We have shown the measurements of the same fibers with the MINOS measurement. Table \[K27\_atten\] gives the results for the sliding fiber test for those entries marked SLIDE in column 2. The results of the sliding fiber setup for 0.94 mm Kuraray are used in the design of CMS HCAL.
Figure \[fib\_att\_slide\]a shows the difference in attenuation length of a fiber measured in two different ways, but with the same photodetector and light injection. The difference seen is due to the extra clear fiber cable and the clear to WLS splice of the sliding fiber setup. Figure \[fib\_att\_slide\]b compares the sliding tile measurement with the MINOS measurement with x=0 set 18 cm (Kuraray) and 25 cm (Bicron) away from the phototube into the scintillator. The comparison shows 18 cm (Kuraray) of WLS fiber is acting like the clear fiber and cable.
=5.7in\
\[sliding\_fiber\_eva\]
=2.7in\
\[fib\_att\_slide\]
Radiation Study of Fibers {#radiation-study-of-fibers .unnumbered}
=========================
Some of the WLS fibers were irradiated with 127 krad at an electron source at Florida State University.
We encountered a problem in measuring the irradiated fibers. The fibers were first measured with the MINOS setup in Lab 5 at Fermilab. While the fibers were being irradiated, the MINOS setup was moved to the Muon Lab. We measured the irradiated fibers with the MINOS setup in Muon Lab.
We measured the same three Kuraray 0.83 mm fibers in both Lab 5 and the Muon Lab. The normalized light at 100 cm was .57 in Lab 5 and .52 in the Muon Lab. We have no explanation for the difference, since the setups are the same. The normalized light difference between different fiber types is not affected by this problem, but the absolute normalized light is affected by this problem. The measurement of the 0.83 mm fiber in both Lab 5 and Muon Lab is used to get a correction function for the Lab 5 measurement. The normalized light yield of the fibers before irradiation is multiplied by the correction function. Figure \[fib\_att\_rad\] compares the normalized light of the fibers before and after radiation measured with both the MINOS setup and the CDF UV scanner. The results of the fits of the fibers before and after radiation are given in Table \[K27\_atten\].
=2.7in\
\[fib\_att\_rad\]
Light vs Tile Size and Absolute Light {#light-vs-tile-size-and-absolute-light .unnumbered}
=====================================
Figure \[light\_vs\_tile\_size\_lay\_10\_eva\] and Figure \[light\_vs\_tile\_size\_lay\_16\_eva\] show the setup used to measure the light vs tile size for CMS HCAL tower 10 and 16, respectively. Tower 10 is a tower in the middle of the barrel, while tower 16 is at the high eta edge of the barrel. To measure tower 10, we make 6 tiles each for layers 1, 7, and 16, for a total of 18 tiles. We use the same pigtail to measure the light from each of the tiles. The results are plotted as light vs perimeter/area [@lovera1] [@lovera2]. The perimeter measures the length of fiber in the tile, since the distance the fiber groove is from the edge of the scintillator is kept at 0.3 cm for these measurements. Figure \[l\_over\_a\]a gives the result. This result includes the additional length of WLS fiber outside the tile for layers 1 and 7. Figure \[l\_over\_a\]b gives the result with the WLS fiber attenuation removed. Figure \[l\_over\_a\]b gives the light vs tile size. The overall normalization is not measured by this measurement. The mean of the data for tower 10 is normalized to 1 for both Figure \[l\_over\_a\]a and Figure \[l\_over\_a\]b. The data on Figure \[l\_over\_a\]b is fit to a straight line. The 3.8 is a relative number that depends on the how the light measurement is normalized. The measurement measures the coefficent 0.077/cm, which is the intercept divided by the slope of the line.
To measure tower 16, we make 5 tiles each for layers 1, 5, and 8. Again, we use the same pigtail to measure the light from each of the tiles. The pigtails for the tower 10 measurement and tower 16 measurement are different. Hence, the normalization of the tower 10 and tower 16 measurements are independent. The mean of the data for tower 16 in Figure \[l\_over\_a\]a is set to 1. For Figure \[l\_over\_a\]b the normalization for tower 16 is set so that the mean of the perimeter/area and mean of the normalized light lie on the straight line for tower 10. This enables us to see how consistent the two measurements are.
For the CMS design we used two models for the variation of light vs tile size. The first model assumes the light yield is a linear function of perimeter/area. The line is given in Figure \[l\_over\_a\]b. We notice that the points for Tower 16 do not follow a straight line. In the measurement for Figure \[l\_over\_a\]a the same fibers are used to measure all the tiles in a tower. The total variation of the points for Figure \[l\_over\_a\]a for both Tower 10 and 16 is 4%. The second model assumes that a fiber with the same length green inserted into all the CMS tiles gives the same light. It assumes the change in the light by changing the tile size is compensated for by the attenuation in the green fiber.
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\[light\_vs\_tile\_size\_lay\_10\_eva\]
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\[light\_vs\_tile\_size\_lay\_16\_eva\]
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\[l\_over\_a\]
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\[abs\_lite\_eva\]
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\[abs\_light\]
Figure 22 \[abs\_lite\_eva\] shows the apparatus used to measure the absolute light yield of CMS tiles. A Ru-106 $\beta$ source is used instead of the Cs-137 source. Figure \[abs\_light\] gives the result. A CMS HCAL barrel tile gives roughly 2 photoelectrons at the photodetector with a green extended photocathode.
The light for the tiles in HCAL barrel can be predicted using the attenuation length of the green and clear fiber, the model of the light vs tower size, and the absolute light yield. From this we can get the total light of a tower. All layers of a tower, except for the first, go to the same photodetector. The longitudinal variation of light within a tower should be less than 10%. By varying the position of the splice, we make the light uniform longitudinally in a tower.
Transverse Tile Uniformity {#transverse-tile-uniformity .unnumbered}
==========================
We have studied the transverse uniformity of the tiles. We constructed four tiles with the dimensions of tower 14, layer 14, which is the largest tile in the CMS HCAL Barrel. Two tiles have the fibers inserted parallel to the short side, called short side fiber insertion. The tiles with short side fiber insertion are shown in Figure \[tile\_unif\_cms\_short\_eva\]. Two tiles have the fibers inserted parallel to the long side, called long side fiber insertion. The tiles for long side fiber insertion are shown in Figure \[tile\_unif\_cms\_long\_eva\]. CMS HCAL tiles have long side fiber insertion. The edges of the tiles are painted with white TiO$_2$ paint [@white_paint].
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\[tile\_unif\_cms\_short\_eva\]
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\[tile\_unif\_cms\_long\_eva\]
The uniformity is measured with a collimated Cs-137 $\gamma$ source. The collimator constrains the radiation to a 7.5 cm diameter circle. The central transverse size of a hadron shower is approximately 7.5 cm. Hence, the collimated source simulates the transverse size of a hadron shower. Figure \[tile\_plot\_cms\_short\] shows the uniformity across the tile with the short side fiber insertion. The uniformity was measured with both Kuraray fibers and Bicron fibers. The Kuraray measurement uses 0.94 mm S-35 fibers for both the pigtail and the cable. For the Bicron measurement, the WLS 1.0 mm Bicron fiber (Batch 2) is spliced to 1.0 mm non-S Kuraray fiber. The cable for the Bicron measurement was made with S type 1.0 mm Kuraray fiber. For both kinds of fibers the tile is very uniform with a 10% increase at the boundary between the 2 tiles. The increase is due to increased light collection at the fiber. Figure \[tile\_plot\_cms\_long\] shows the uniformity with long side fiber insertion. The transverse uniformity has an RMS $\sim$ 3% regardless of the fiber type or fiber insertion point. The resolution of the CMS calorimeter is $120\%/\sqrt{E}~\oplus~5\%$ [@cms_calorimeter_talk]. The transverse uniformity across the tile should be somewhat less than the constant term, 5%, to prevent transverse uniformity from affecting the constant term in the resolution.
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\[tile\_plot\_cms\_short\]
=2.7in \[tile\_plot\_cms\_long\]
The HCAL CMS scintillator design has individual tiles glued together with TiO$_2$ loaded epoxy resin [@glue], to form a “megatile”. The configuration at the boundary between tiles is shown in Figure \[cms\_tile\_boundary\_eva\]. A 0.9 mm wide “separation groove” is cut to separate 2 tiles, with 1/4 mm of scintillator left uncut on the bottom of the groove. The groove is filled with TiO$_2$ loaded epoxy. The scintillator is marked with a black mark made with the narrow end of a black marker pen [@doubleshot]. The black mark is underneath the separation groove. The black mark is about 1.5 mm wide, slightly wider than the separation groove. The black mark reduces the light cross talk through the 1/4 mm of scintillator left at the tile boundary.
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\[cms\_tile\_boundary\_eva\]
We constructed a glued megatile consisting of 2 tiles inside a piece of scintillator. The tiles for that glued megatile tile are from tower 10, layer 1. Figure \[tile\_unif\_glue\_tile\_eva\] shows the apparatus used to measure the glued megatile. Figure \[tile\_plot\_mark\] shows the result. The transverse RMS is roughly 1.7%. The transverse uniformity does not increase the constant term of the resolution.
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\[tile\_unif\_glue\_tile\_eva\]
=2.7in \[tile\_plot\_mark\]
We have measured the cross talk between the glued tiles. The 1/4 mm of scintillator, left uncut between the two tiles, provides a path for light to pass between 2 tiles. The cross talk is measured by first putting a fiber in one of the tiles. Next, we measure the current with the source on the following three locations: the tile with the fiber, the tile without the fiber, and just off the tile with the fiber. The last location is used to measure the source cross talk. The cross talk is $\sim$ 1%.
The black mark decreases the light output. We measured the light from glued megatile. Next, the black marks are made on 3 sides of the tile, similar to the way the HCAL CMS barrel tiles will be marked. The light goes down roughly 8%.
Conclusion {#conclusion .unnumbered}
==========
The CMS R and D enables us to design the optics of HCAL Barrel Calorimeter and predict its performance. We have chosen Kuraray S-35 fiber for the HCAL preproduction prototype because of its excellent flexibility, excellent mirror reflectivity, and high splice transmission. CMS HCAL has chosen to ice polish the fibers, since it enables us to polish many fibers at once. We predict the light of each tile in the barrel using the attenuation lengths of fibers and the absolute light vs the tile size. By varying the position of the splice for each tile, we can optimize the light distribution in a tower. CMS has chosen to have the same length WLS fiber for all layers in a tower. Measurements of the transverse uniformity shows that it does not effect the resolution of the calorimeter.
P. de Barbaro et al., [*CDF End Plug Upgrade Calorimeter Design*]{}, University of Rochester Preprint UR-1360, Jul. 1994, CDF Note 2545.
P. de Barbaro and A. Bodek, [*Scintillator Tile-Fiber Calorimeters for High Energy Physics: The CDF End Plug Upgrade, Selected Articles*]{}, University of Rochester preprint UR 1389, October 1994. References [@cdf_end_plug], [@cdf_upgrade], [@splicer], [@lovera1], [@lovera2]. and [@glue] are in this document.
Kuraray Co., LTD., 8F, Maruzen Building, 3-10, 2-Chome, Nihonbashi, Chuo-ku, Tokyo, 103-0027, Japan.
Bicron Corporation, 12345 Kinsman Road, Newbury, Ohio 44065-9677.
DDK Electronics Inc., 3001 Oakmead Village Drive, Santa Clara, Calif. 95051, and DDK Co. Ltd, Yoyogi 2-7-12, Shibuya-ku, Tokyo 151.
S. Aota et al., Nucl. Instr. and Meth. A357 (1995) 71.
G. Apollinari, P. de Barbaro, and M. Mishina, [*CDF End Plug Calorimeter Upgrade Project*]{}, Proceedings of the IV International Conference on the Calorimetry in High Energy Physics, La Biodola Elba, Sep 19-25, 1993.
J.P. Mansour, C. Bromberg, J. Huston, S. Joy, B. Miller, R. Richards, B. Tannenbaum (Michigan State U.), [*A Semiautomated Splicer for Plastic Optical Fibers*]{}, Notre Dame 1993, Proceedings, Scintillating fiber detectors, 534-541.
R. Mussa et al., [*Performance Measurements of Histe-V VLPC Photon Detectors for E835 at FNAL*]{}, paper submitted to these proceedings.
C. Durandet et al., [*The Use of WLS Fibers in a Hadronic Calorimeter for the HyperCP Experiment*]{}, paper submitted to these proceedings.
P. de Barbaro et al., [*Recent R&D Results on Tile/Fiber Calorimeter*]{}, University of Rochester preprint UR-1299, SDC-93-407, January 1993.
P. Koehn, [*Tile/Fiber Results for the Upgraded Plug Hadron Calorimeter*]{}, IEEE 1993 Nuclear Science Symposium and Medical Imaging Conference, San Francisco, October 1993.
Bicron BC-620 white paint, Bicron Corp, Newbury, OH. P. de Barbaro et al., [*Performance of a Prototype CMS Hadron Barrel Calorimeter in a Test Beam*]{}, Proceedings of the CALOR97, VII International Conference on Calorimetry in High Energy Physics November 9 - 14, 1997 University of Arizona Tucson, Arizona.
M. Olsson et al., [*Techniques for Optical Isolation and Construction of Megatiles*]{}, University of Rochester preprint UR-1370.
“Doubleshot” ink marker (No. 11120), Pentech International Inc., Edison, NJ.
|
---
abstract: 'Let $\Uereslsl2$ be the restricted integral form of the quantum loop algebra $U_q(L\mathfrak{sl}_2)$ specialised at a root of unity $\varepsilon$. We prove that the Grothendieck ring of a tensor subcategory of representations of $\Uereslsl2$ is a generalised cluster algebra of type $C_{l-1}$, where $l$ is the order of $\varepsilon^2$. Moreover, we show that the classes of simple objects in the Grothendieck ring essentially coincide with the cluster monomials. We also state a conjecture for $\Uesl3$, and we prove it for $l=2$.'
author:
- 'Anne-Sophie Gleitz'
title: Quantum affine algebras at roots of unity and generalised cluster algebras
---
Introduction {#intro}
============
Cluster algebras have been introduced in 2001 by Fomin and Zelevinski [@FZ1]. These rings have special generators, called *cluster variables*. For every cluster $\mathbf{x}$, and every cluster variable $x\in\mathbf{x}$, there is a unique cluster $\left( \mathbf{x}\setminus \{x\}\right)\cup\{x'\}$, and an exchange relation $$\label{clu0}
xx' = m_+ + m_-$$ where $m_\pm$ are exchange monomials in $\mathbf{x}\setminus\{x\}$. Fomin and Zelevinsky [@FZ2] have proved a classification theorem for cluster algebras with finitely many clusters (also called of finite type), in terms of Cartan matrices.
We are interested in *generalised cluster algebras*, introduced by Shapiro and Chekhov in 2011 [@CS]. The difference with standard cluster algebras resides in the exchange relations, whose right-hand side can include polynomials with more than two terms, unlike (\[clu0\]). Otherwise, finite type classification and combinatorial behaviour stay the same [@CS].
We focus on a generalised cluster algebra $\mathcal{A}_n$ of Cartan type $C_n$, with a particular choice of coefficients, and describe its inner combinatorics. In particular, we describe several $\mathbb{Z}$-bases of $\mathcal{A}_n$.
On the other hand, the theory of finite-dimensional representations of the quantum loop algebra $U_q(L{\mathfrak{g}})$ for $q\in\mathbb{C}^*$ not a root of unity is well established. In this paper, we are interested in the case where $q=\varepsilon$ is a root of unity. The algebra $\Uqlg$ is then replaced by the restricted integral form $\Uereslg$, introduced and studied by Chari and Pressley [@CP2], and later by Frenkel and Mukhin [@FM]. In the spirit of Hernandez and Leclerc’s papers [@HL] and [@HL2], we consider a certain tensor category $\CZres$ of finite-dimensional $\Uereslg$-modules, and we show that when $\mathfrak{g}=\mathfrak{sl}_2$, the Grothendieck ring of $\CZres$ is isomorphic to $\mathcal{A}_{l-1}$ (see Theorem \[conjA1\]), where $l$ is the order of $\varepsilon^2$. Moreover, under this isomorphism, the basis of classes of simple objects of $\CZres$ coincides with the basis of (generalised) cluster monomials, multiplied by Tchebychev polynomials in the single generator of the coefficient ring. This is proved by combining tools from the theory of generalised cluster algebras (see Section \[s21\]), and from the representation theory of $\Uereslsl2$ (see Section \[s22\]).
For $\mathfrak{g}=\mathfrak{sl}_3$ and $l=2$, we prove a similar result, where $\mathcal{A}_{l-1}$ is replaced by a generalised cluster algebra of type $G_2$. Extensive computations with Maple allow us to formulate a conjecture for $\mathfrak{g}=\mathfrak{sl}_3$ and $l>2$. However, the generalised cluster algebras occurring in this conjecture are of infinite type, and we still lack the proper tools to prove it.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The author would like to thank B. Leclerc for his invaluable advice and insight throughout this work.
Cluster algebras {#s21}
================
We are interested in a structure that generalises the notion of cluster algebras, defined by Shapiro and Chekhov in [@CS].
Generalised cluster algebras {#s212}
-----------------------------
We recall, following [@CS], the definition and the main structural properties of generalised cluster algebras, see also [@N].
For a fixed integer $n\in\mathbb{N}^*$, let $B=(b_{ij})\in\nolinebreak\mathcal{M}_n(\mathbb{Z})$ be a skew-symmetrisable matrix, i.e. such that there exists an integer diagonal matrix $\tilde{D}=\mathrm{diag}(\tilde{d}_1\dots\tilde{d}_n)$ such that $\tilde{D}B$ is skew-symmetric.
Suppose that for each index $k\in\llbracket 1,n\rrbracket$, there is an integer $d_k\in\mathbb{N}$ that divides all coefficients $b_{ j k}$ in the $k$-th column. Introduce the notation $$\beta_{jk} := \displaystyle \frac{b_{jk}}{d_k} \in\mathbb{Z}.$$
Let $(\mathbb{P},\cdot,\oplus)$ be a commutative semifield, called the *coefficient group*. For example, one can take for $\mathbb P$ the tropical semifield $\mathrm{Trop}(\lambda_1,\dots,\lambda_n)$ generated by some indeterminates $\lambda_1,\dots,\lambda_n$. This is by definition the set of Laurent monomials in the $\lambda_i$’s, with ordinary multiplication and tropical addition $$\left(\displaystyle\prod_i \lambda_i^{a_i}\right) \oplus \left(\displaystyle\prod_i \lambda_i^{b_i}\right) = \left(\displaystyle\prod_i \lambda_i^{\min(a_i,b_i)}\right).$$ Let $\mathcal{F}=\mathbb{ZP}(t_1,\dots,t_n)$ be the ambient field of rational functions in $n$ independent variables, where $\mathbb{ZP}$ is the integer group ring of $\mathbb{P}$.
For a collection of variables $\mathbf{p}_i=(p_{i,0}, p_{i,1},\dots, p_{i,d_i})\in\mathbb{P}^{d_i+1}\quad(i\in\llbracket 1,n\rrbracket)$, define the corresponding homogeneous *exchange polynomial* $$\theta_i\lbrack \mathbf{p}_i\rbrack(u,v) := \displaystyle \sum_{r=0}^{d_i} p_{i,r} u^r v^{d_i-r}\in\mathbb{ZP}\lbrack u,v\rbrack.$$
A *generalised seed* is a triple $(\mathbf{x},\bar{\mathbf{p}},B)$ where
1. the tuple $\mathbf{x}=\{x_1,\dots,x_n\}$, called a *cluster*, is a collection of algebraically independent elements of $\mathcal F$, called *cluster variables*, which generate $\mathcal F$ over $\mathrm{Frac}\:\mathbb{ZP}$;
2. the matrix $B=(b_{ij})\in\mathcal{M}_n(\mathbb{Z})$, called the *exchange matrix*, is skew-symmetrisable;
3. $\bar{\mathbf{p}}=(\mathbf{p}_1,\dots,\mathbf{p}_n)$ is a *coefficient tuple*, where for each $i\in\llbracket 1,n\rrbracket$, the tuples $\mathbf{p}_i=(p_{i,0},p_{i,1},\dots,p_{i,d_i})\in\mathbb{P}^{d_i+1}$ are the coefficients of the $i$-th exchange polynomial $\theta_i$.
The triple $(\mathbf{x},\{\theta_1,\dots,\theta_n\},B)$ is also called a generalised seed.
\[genmut\] The *generalised mutation in direction $k\in\llbracket 1,n\rrbracket$*, is the operation that transforms a generalised seed $(\mathbf{x},\bar{\mathbf{p}},B) $ into another generalised seed $\mu_k(\mathbf{x},\bar{\mathbf{p}},B):=(\mathbf{x}',\bar{\mathbf{p}'},B')$ given by
1. : the matrix $B'=(b'_{ij})$ is defined by $$b'_{ij} =\left\{\begin{array}{ll}
-b_{i j} &\mbox{if } i=k \mbox{ or } j=k\\
b_{ij} + \displaystyle\frac{1}{2}\left( |b_{ik}| b_{k j} + b_{ik} |b_{kj}|\right) &\mbox{otherwise}
\end{array} \right.$$
2. $$\left\{\begin{array}{ll}
x'_i = x_i &\mbox{if } i\neq k\\
x_k x'_k = \theta_k\lbrack \mathbf{p}_k\rbrack(u_k^+,u_k^-)
\end{array}\right.$$ where we define $$u_k^+ := \displaystyle \prod_{j=1}^n x_j^{\lbrack \beta_{jk}\rbrack_+} \quad\mbox{and}\quad u_k^- := \displaystyle \prod_{j=1}^n x_j^{\lbrack -\beta_{jk}\rbrack_+} .$$
3. $$\left\{\begin{array}{ll}
p'_{k,r}=p_{k,d_k-r}\\\\
\displaystyle \frac{p'_{i,r}}{p'_{i,r-1}}&= \left\{ \begin{array}{ll}
(p_{k,d_k})^{\beta_{k i}} \displaystyle \frac{p_{i,r}}{p_{i,r-1}} &\mbox{if } i\neq k \mbox{ and } b_{ki}\geq 0\\\\
(p_{k,0})^{\beta_{k i}} \displaystyle \frac{p_{i,r}}{p_{i,r-1}} &\mbox{if } i\neq k \mbox{ and } b_{ki}\leq 0
\end{array} \right.
\end{array}\right.$$
For $r\in\llbracket 1,n\rrbracket$, write $\mu_r(B):=(b'_{ij})$. It follows easily from the definition of matrix mutation that for each $k\in\llbracket 1,n\rrbracket$, the integer $d_k$ divides all coefficients in the $k$-th column of $\mu_r(B)$. Moreover, note that $\mu_r$ is an involution. We say that two generalised seeds are *mutation-equivalent* if one can be obtained from the other by performing a finite sequence of mutations.
Observe that if $d_i=1$ for all $i$, then the exchange polynomials are of the form $\theta_i(u,v)=p_{i,0}u+p_{i,1}v$. We then recover the ordinary notions of seed and seed mutation from [@FZ1] and [@FZ2] by setting $p_i^+=p_{i,1}$ and $p_i^-=p_{i,0}$.
The *generalised cluster algebra* $\mathcal{A}(\bar{\mathbf{p}},B)=\mathcal{A}(\mathbf{x},\{\theta_1,\dots,\theta_n\},B)$ of *rank* $n$, corresponding to the generalised seed $(\mathbf{x},\{\theta_1,\dots,\theta_n\},B)$, is the $\mathbb{ZP}$-subalgebra of $\mathcal{F}$ generated by all cluster variables from all the seeds that are mutation-equivalent to the initial seed $(\mathbf{x},\{\theta_1,\dots,\theta_n\},B)$.
We say that a generalised cluster algebra is of *finite type* if it has finitely many cluster variables. The Laurent phenomenon from [@FZ1] remains true for generalised cluster algebras.
\[geneLau\] Every generalised cluster variable is a\
Laurent polynomial in the initial cluster variables.
Two generalised cluster algebras $\mathcal{A}(\bar{\mathbf{p}},B)\subset\mathcal{F}$ and $\mathcal{A}(\bar{\mathbf{p}}',B')\subset\mathcal{F}'$ over the same semifield $\mathbb{P}$ are called *strongly isomorphic* if there is a $\mathbb{ZP}$-isomorphism $\mathcal{F}\rightarrow\mathcal{F}'$ that sends any generalised seed of $\mathcal{A}(\bar{\mathbf{p}},B)$ onto a generalised seed $\mathcal{A}(\bar{\mathbf{p}}',B')$. This induces a bijection between the sets of generalised seeds, as well as an algebra isomorphism $\mathcal{A}(\bar{\mathbf{p}},B)\cong\mathcal{A}(\bar{\mathbf{p}}',B').$ Every generalised cluster algebra $\mathcal{A}(\bar{\mathbf{p}},B)$ over a semifield $\mathbb{P}$ belongs to a series $\mathcal{A}(-,B)$, consisting in all the generalised cluster algebras $\mathcal{A}(\bar{\mathbf{p}},B)$ where $B$ is fixed and $\bar{\mathbf{p}}$ may vary. We say that two series $\mathcal{A}(-,B)$ and $\mathcal{A}(-,B')$ are *strongly isomorphic* if $B$ and $B'$ are mutation-equivalent, modulo simultaneous relabeling of rows and columns.
Let $M=(m_{ij})\in\mathcal{M}_n(\mathbb{Z})$. The Cartan counterpart of $M$ is the generalised Cartan matrix $A=A(M)=(a_{ij})\in\mathcal{M}_n(\mathbb{Z})$ defined by $$a_{ij}=\left\{\begin{array}{ll}
2&\mbox{if }i=j\\
-|m_{ij}|&\mbox{if }i\neq j.
\end{array} \right.$$
\[geneCla\] Generalised cluster algebras of finite type follow the same Cartan-Killing classification as standard cluster algebras. Namely, there is a canonical bijection between the Cartan matrices of finite type and the strong isomorphism classes of series of generalised cluster algebras of finite type. Under this bijection, a Cartan matrix $A$ of finite type corresponds to the series $\mathcal{A}(-,B)$, where $B$ is a skew-symmetrisable matrix such that $A(B)=A$.
Finally, we recall that a *cluster monomial* in $\mathcal A(\bar{\mathbf p},B)$ is a monomial in the cluster variables involving only variables belonging to a single cluster.
A generalised cluster algebra of type Cn {#s213}
----------------------------------------
### Combinatorics and exchange relations {#s231}
The *exchange graph* of a cluster algebra is the graph whose vertices are the clusters, and two clusters are linked by an edge if they can be obtained from each other by one mutation.
We know from [@FZ2 Section 12.3] that for a cluster algebra of type $C_{n}$, the exchange graph is isomorphic to the $n$-dimensional cyclohedron. It has a nice description in terms of triangulations of a regular $(2n+2)$-gon $\mathbf{P}_{2n+2}$.
More precisely, each cluster variable can be associated with either a centrally symmetric pair of diagonals, or a diameter. Under this bijection, each vertex of the exchange graph corresponds to a centrally symmetric triangulation, and two such triangulations are linked by an edge if they can be obtained from each other either by a flip involving two diameters, or by a pair of centrally symmetric flips. Note that each centrally symmetric triangulation contains a unique diameter.
For a standard cluster algebra, the exchange relations correspond to Ptolemy relations in the appropriate quadrilaterals. For a generalised cluster algebra, certain formulas are slightly more complicated, see Proposition \[pCn\] below.
Let us identify the set $\Sigma$ of vertices of $\mathbf{P}_{2n+2}$ with the cyclic group $$\mathbb{Z}/(2n+2)\mathbb{Z}\cong 2 \mathbb{Z}/(4n+4)\mathbb{Z},$$ by labelling the vertices clockwise: 0,2,4,$\dots$,$2n$,\
$2n+2$,$2n+4$,$\dots$, $4n+2$, with the natural additive law induced by the cyclic group. We rename half of the vertices in the following way: for each $k\in\llbracket 0,n\rrbracket$, write $$(2n+2)+2k := \overline{2k}.$$ In particular, $2n+2=\bar{0}$ and $\overline{2n}+2=0$. This makes it easier to identify centrally symmetric pairs of diagonals. It might seem odd to use “$2k$” instead of just “$k$”, but this notation will turn out to be the most natural one for Section \[s3\].
Let $\mathscr{C}$ be the circle in which $\mathbf{P}_{2n+2}$ is inscribed, and let $\Theta$ be the central symmetry around the center of $\mathscr{C}$. Consider a pair $\{\lbrack a,b \rbrack, \lbrack \bar{a},\bar{b}\rbrack\}$ of centrally symmetric diagonals.We may choose $\lbrack a,b\rbrack$ to represent the $\Theta$-orbit of this pair. The segment $\lbrack a,b\rbrack$ divides the circle $\mathscr{C}$ into two arcs. The $\Theta$-orbits of the vertices of $\mathbf{P}_{2n+2}$ that lie on the smallest arc form a set denoted by $\mathcal{O}_{ab}$. For example, if $a<b\in\llbracket 0,2n\rrbracket$, the set $\mathcal{O}_{ab}$ consists of the $\Theta$-orbits of $a+2$, $a+4$,$\dots$, $b-2$. In general, we have $\mathcal{O}_{ab}=\mathcal{O}_{ba}=\mathcal{O}_{\bar{a}\bar{b}}=\mathcal{O}_{\bar{b}\bar{a}}.$
In type $C_3$, the vertices of the regular octagon $\mathbf{P}_8$ will be numbered as in Figure \[fig:octa\]. For example, the pair of centrally symmetric diagonals $\{\lbrack 2,\bar{0}\rbrack,\lbrack \bar{2},0\rbrack\}$ corresponds to the set $\mathcal{O}_{2,\bar{0}}$, which consists of the $\Theta$-orbits $\{4,\overline{4}\}$ and $\{6,\overline{6}\}$.
=1.1cm (0:) in [45,90,...,360]{} [ – (:) ]{} – cycle (180:) node\[left\] [0]{} – cycle (270: ) node\[below\] [$\bar{4}$]{} – cycle (315: ) node\[below right\] [$\bar{2}$]{} –cycle (225: ) node\[below left\] [$\bar{6}$]{} – cycle (0:) node\[ right\] [$\bar{0}$]{} – cycle (135:) node\[above left\] [2]{} – cycle (90:) node\[above\] [4]{} – cycle (45:) node\[above right\] [6]{};
We label cluster variables by the corresponding $\Theta$-orbits of diagonals. Namely, if $b\neq \overline{a}$, the variable $x_{ab}$ corresponds to the pair of diagonals $\{\lbrack a,b\rbrack, \lbrack \bar{a},\bar{b}\rbrack\}$. If $b=\overline{a}$, the variable $x_{a\overline{a}}$ corresponds to the diameter $\{\lbrack a,\overline{a}\rbrack\}$. Thus we have $$x_{ab}=x_{ba}=x_{\bar{a}\bar{b}}=x_{\bar{b}\bar{a}}.$$ By convention, if $a$ and $b$ are neighbours in $\mathbf{P}_{2n+2}$, we set $x_{ab}=1$. Note that each cluster variable $x_{ab}$ may also be labelled by the set $\mathcal{O}_{ab}$.
Theorem \[geneCla\] allows us to use the same labeling system for a generalised cluster algebra of type $C_n$. In particular, mutations can be seen as flips between triangulations of $\mathbf{P}_{2n+2}$.
The following example is a particular case of the more general Definition \[dCn\].
In type $C_3$, consider the following initial seed, with coefficient group $\mathbb{P}=\mathrm{Trop}(\lambda)=\mathbb Z \lbrack \lambda^{\pm 1}\rbrack$: $$\Pi_0:=(\mathbf{x}^{(0)},\{\theta_1,\theta_2,\theta_3\},B)$$ where we set $$\mathbf{x}^0 = (x_{2\bar{6}},x_{4\bar{6}},x_{6\bar{6}}),\quad \left\{ \begin{array}{l}
\theta_1(u,v)=u+v\\
\theta_2(u,v)=u+v\\
\theta_3(u,v)=u^2 + \lambda u v + v^2
\end{array}\right.,B=\left( \begin{array}{ccc}
0&1&0\\-1&0&2\\0&-1&0
\end{array}\right).$$ Because of the special choice of coefficients, the exchange polynomials remain unaffected by mutation. Mutating $\Pi_0$, we obtain twelve cluster variables, which can be organised in 20 clusters, as in Figure \[ex:c3\]. This corresponds to the 3-dimensional cyclohedron whose vertices are the non-crossing centrally symmetric triangulations of the octagon (Figure \[fig:cyclo3d\]).
$\xymatrix @R=-0.5pc @C=-0.4pc @M=0.0pc {
&&&\mbox{$\begin{array}{c}x_{2\bar{6}},x_{4\bar{6}},\\x_{4\bar{4}}\end{array}$}\ar@{-}[rr] &&\mbox{$\begin{array}{c}x_{2\bar{6}},x_{2\bar{4}},\\x_{4\bar{4}}\end{array}$}\ar@{-}[rrd] &&&&&\\
&\mbox{$\begin{array}{c}x_{04},x_{4\bar{6}},\\x_{4\bar{4}}\end{array}$}\ar@{-}[rru]\ar@{-}[ld]&&&&&&\mbox{$\begin{array}{c}x_{0\bar{4}},\\x_{2\bar{4}},\\x_{4\bar{4}}\end{array}$}&&&&&&&&&&&&&&&&&&\\
\mbox{$\begin{array}{c}x_{04},x_{4\bar{6}},\\x_{6\bar{6}}\end{array}$}\ar@{-}[rr] & &\mbox{$\begin{array}{c}x_{2\bar{6}},\\x_{4\bar{6}},\\x_{6\bar{6}}\end{array}$}\ar@{-}[uur]\ar@{-}[dr]&&\mbox{$\begin{array}{c}x_{04},\\x_{06},\\x_{4\bar{4}}\end{array}$}\ar@{.}[ulll]\ar@{.}[dd]\ar@{.}[urrr]&&\mbox{$\begin{array}{c}x_{2\bar{6}},\\x_{2\bar{4}},\\x_{2\bar{2}}\end{array}$}\ar@{-}[uul]\ar@{-}[dl]\ar@{-}[rr] &&\mbox{$\begin{array}{c}x_{0\bar{4}},x_{2\bar{4}},\\x_{2\bar{2}}\end{array}$}\ar@{-}[ul]&&&&&&&&&&&&&&&&&&&\\
\mbox{$\begin{array}{c}x_{04},x_{06},\\x_{6\bar{6}}\end{array}$}\ar@{-}[ddr]\ar@{-}[u]\ar@{-}[dr]&&&\mbox{$\begin{array}{c}x_{2\bar{6}},\\x_{26},\\x_{6\bar{6}}\end{array}$}\ar@{-}[rr]\ar@{-}[dll]&&\mbox{$\begin{array}{c}x_{2\bar{6}},\\x_{26},\\x_{2\bar{2}}\end{array}$}\ar@{-}[drr]&&&\mbox{$\begin{array}{c}x_{0\bar{4}},x_{0\bar{2}},\\x_{2\bar{2}}\end{array}$}\ar@{-}[ddl]\ar@{-}[u]\ar@{-}[dl]&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&\\
&\mbox{$\begin{array}{c}x_{06},x_{26},\\x_{6\bar{6}}\end{array}$}\ar@{-}[ddrr]&&&\mbox{$\begin{array}{c}x_{04},\\x_{0\bar{4}},\\x_{0\bar{0}}\end{array}$}\ar@{.}[dlll]\ar@{.}[drrr]&&&\mbox{$\begin{array}{c}x_{26},x_{0\bar{2}},\\x_{2\bar{2}}\end{array}$}\ar@{-}[ddll]&&&&&&&&&&&&&&&&&&&&&&&\\
&\mbox{$\begin{array}{c}x_{04},x_{06},\\x_{0\bar{0}}\end{array}$}\ar@{-}[rrd]&&&&&&\mbox{$\begin{array}{c}x_{0\bar{4}},x_{0\bar{2}},\\x_{0\bar{0}}\end{array}$}&&&&&&&&&&&&&&
\\
&&&\mbox{$\begin{array}{c}x_{26},x_{06},\\x_{0\bar{0}}\end{array}$}\ar@{-}[rr]&&\mbox{$\begin{array}{c}x_{26},x_{0\bar{2}},\\x_{0\bar{0}}\end{array}$}\ar@{-}[rru]&&&&&&&&&&&&&&&&&&&
}$
\[dCn\] Let $\mathbb{P}=\mathrm{Trop}(\lambda)$. For an integer $n\in\mathbb{N},\:n\geq 2$, we denote by $\overline{\mathcal{A}}_n=\mathcal{A}(\mathbf{x},\{\theta_1^0,\dots,\theta_n^0\},B)$ the generalised cluster algebra defined by the initial seed $$\theta_i^0(u,v)=u+v\:\:(i\in\llbracket 1,n-1\rrbracket), \qquad \theta_n^0(u,v)=u^2+\lambda uv+v^2,$$ and $$B:=\left( \begin{array}{ccccccc}
0&1&0&0&0&\dots&0\\
-1&0&1&0&0&\dots&0\\
0&-1&0&1&0&\dots&0\\
0&0&-1&0&\ddots&\ddots&\vdots\\
\vdots&\vdots&\ddots&\ddots&\ddots&1&0\\
0&0&\dots&0&-1&0&2\\
0&0&0&\dots&0&-1&0
\end{array} \right).$$
Thus $\overline{\mathcal{A}}_n$ is the $\mathbb Z\lbrack \lambda^{\pm 1}\rbrack$-subalgebra of $\mathcal F$ generated by the cluster variables. We will rather work with a variant of $\overline{\mathcal{A}}_n$, in which the coefficient $\lambda$ is not assumed to be invertible.
\[dAn\] Let $\mathcal A_n$ be the $\mathbb Z\lbrack\lambda\rbrack$-subalgebra of $\mathcal F$ generated by the cluster variables of $\overline{\mathcal{A}}_n$.
As above, we label the cluster variables of $\overline{\mathcal{A}}_n$ (or $\mathcal{A}_n$) by $\Theta$-orbits of diagonals of $\mathbf{P}_{2n+2}$. The initial cluster variables corresponding to the initial seed are as follows (see Figure \[fig:initclu\]): $$x_k:=x_{\overline{2n},2k}\quad(k\in\llbracket 1,n\rrbracket).$$
1. It follows from Definition \[genmut\], that the exchange polynomials $\theta_i^0, \:i<n$, are not affected by mutation; moreover, they coincide with standard exchange relations.
2. There is exactly one variable of the form $x_{a,\bar{a}}$ in each cluster. Indeed, as noted above, any centrally symmetric triangulation contains exactly one diameter. Moreover, the exchange polynomial $\theta_n^0$ also remains unaffected by mutation. Therefore, mutating $x_{a,\overline{a}}$ will yield a variable of the form $x_{b,\overline{b}}$. This can also be seen in terms of triangulations: flipping a diameter while keeping a non-crossing, centrally-symmetric triangulation of $\mathbf{P}_{2n+2}$, gives another diameter (it is easy to see that the quadrilaterals in which diameters are flipped, are always rectangles).
\[pCn\] In the generalised cluster algebra $\mathcal{A}_n$, the following exchange relations between variables $x_{ab}$ and $x_{cd}$ hold, up to rotation (i.e. index shifting):
1. If $a\neq\bar{b}$, $c\neq \bar{d}$, and the quadrilateron $\lbrack acbd\rbrack$ is contained in one half of the circle $\mathscr{C}$, we have a standard exchange relation of the form $$\label{gexc1}
x_{\overline{2n},2k+2} x_{2d-2,2r+2}= x_{\overline{2n},2r+2} x_{2d-2,2k+2} + x_{\overline{2n},2d-2} x_{2k+2,2r+2},$$ which corresponds to the Ptolemy rule in the first diagram of Figure \[fig:diagex\].
2. if $a=\bar{b}$ and $c=\bar{d}$, we have a generalised exchange relation of the form $$\label{gexc2}
x_{\overline{2n},2n}x_{\overline{2k},2k} = x_{\overline{2n},2k}^2+x_{2k,2n}^2+\lambda x_{\overline{2n},2k}x_{2n,2k}.$$ The monomials with coefficient 1 correspond to the Ptolemy rule in the second diagram of Figure \[fig:diagex\].
The identity (\[gexc1\]), for $d=r=k+1$, is by definition true in the initial seed. Moreover, the exchange polynomials $\theta_1^0,\dots,\theta_{n-1}^0$ are exactly the ones that appear in a standard cluster algebra of type $C$, and they are unaffected by mutation: indeed, only the monomials $u_k^\pm$ change, in accordance with the mutations of the exchange matrix. Therefore, cluster variables that do not correspond to diameters behave the same way as in a standard cluster algebra of type $C$, as described in [@FZ2]. The proof for the first case is thus similar to those found in [@CFZ] and [@FZ2].
The second equation (\[gexc2\]), for $k=n-1$, is also by definition true in the initial seed. Since every cluster contains exactly one variable of the form $x_{a,\overline{a}}$, and any mutation of a variable $x_{a,\overline{a}}$ yields a variable corresponding to another diameter, we can deduce from the initial cluster that all variables $x_{a,\overline{a}}$ are linked by a mutation in direction $n$. In the initial cluster $(x_{\overline{2n},2k},\:k\in\llbracket 1,n\rrbracket )$, we have $$\label{rellambda}x_{\overline{2n},2n}x_{\overline{2n-2},2n-2}=x_{\overline{2n},2n-2}^2 +\lambda x_{\overline{2n},2n-2} +1.$$ The general relation (\[gexc2\]) can be obtained directly in the following cluster (see Figure \[fig:mutclu\]): $$\begin{array}{l}
\mu_{n-1}\mu_{n-2}\dots \mu_{k+1}(x_{\overline{2n},2k},\:k\in\llbracket 1,n\rrbracket)\\\qquad\qquad\quad = (x_{\overline{2n},2},x_{\overline{2n},4},\dots,x_{\overline{2n},2k}, x_{2k,2k+4},x_{2k,2k+6},\dots,x_{2k,2n},x_{\overline{2n},2n}),\end{array}$$ where performing the mutation $\mu_n$ maps $x_{\overline{2n},2n}$ to $x_{\overline{2k},2k}$, and $\theta_n^0$ gives (\[gexc2\]).
Indeed, recall that $\theta_n^0$ is unaffected by mutation, so that in order to understand $\mu_n$, it is enough to know how the matrix $B$ mutates, namely in the standard way (Definition 2). This determines the variables $x_{ab}$ appearing in the monomials $u_n^+$ and $u_n^-$ in the mutated cluster above, thus yielding (\[gexc2\]). $\Box$
Exchange relations do not cover every possibility for multiplication of cluster variables that are not in the same cluster. For example, we also have the following useful identity for multiplying a diagonal by a diameter: if $a\neq \bar{b}$ and $c=\bar{d}$ (Figure \[fig:diagnex\]), relations are of the form $$\label{rel2}
\begin{array}{ll}
x_{\overline{2n},2k}x_{2d,\overline{2d}}=\lambda x_{\overline{2n},2d}x_{2d,2k} + x_{\overline{2n},2d}x_{2k,\overline{2d}}+ x_{2d,2k}x_{\overline{2d},\overline{2n}}.
\end{array}$$
### Bases {#s232}
We call a pair of centrally symmetric diagonals of $\mathbf{P}_{2n+2}$ *small* if the corresponding set $\mathcal{O}_{ab}$ contains only one element. The attached variables $x_{ab}$ are also called *small*. Thus in type $C_3$, there are four small variables: $$x_{04}, x_{26}, x_{4\bar{0}}, x_{6\bar{2}}.$$
\[genbasisA\] The set $\mathcal{S}$ of all monomials in the small variables forms a $\mathbb{Z}$-basis of $\mathcal{A}_n$. Equivalently, $\mathcal A_n$ is the polynomial ring with coefficients in $\mathbb Z$ in the small variables.
We first prove that $\mathcal{S}$ spans $\mathcal{A}_n$ over $\mathbb{Z}$. Since $\mathcal{A}_n$ is generated by the elements $x_{ab}$, it is enough to show that each $x_{ab}$ is a polynomial in the small variables. We will argue by induction on $\mathrm{Card} \mathcal{O}_{ab}$.
Let $k\geq 2$, and suppose that variables $x_{ab}$ such that $\mathrm{Card}\:\mathcal{O}_{ab}\leq k-1$ can be written as $\mathbb{Z}$-linear combinations of elements of $\mathcal{S}$.
Let $x_{ab}=x_{2d,2d+2k+2}$ be a cluster variable wih $\mathrm{Card}\:\mathcal{O}_{ab}=k\leq n$. Applying Proposition \[pCn\] (1) in the quadrilateron $\lbrack 2d,2d+2k-2,2d+2k,2d+2k+2\rbrack$ yields $$\begin{array}{ll} x_{2d,2d+2k+2} &= x_{2d+2k-2,2d+2k+2}\, x_{2d,2d+2k} - x_{2d,2d+2k-2},\end{array}$$ and by induction, the right-hand side is a $\mathbb Z$-linear combination of elements of $\mathcal S$. Moreover, $\lambda$ itself is a polynomial in the $x_{2r,2r+4}$: indeed, by , we have $$\label{lambda1} x_{\overline{2n},2n} x_{2n-2,\overline{0}} = \lambda + x_{\overline{0},\overline{2n}} + x_{\overline{2n},2n-2}.$$ Therefore, $\mathcal{S}$ spans $\mathcal{A}_n$ over $\mathbb{Z}$.
Let $\mathcal M_0$ be the set of cluster monomials of $\mathcal A_n$, and let $\mathcal M$ be the set of cluster monomials multiplied by powers of $\lambda$.
Note that we can specialise $\lambda$ to 0 in $\mathcal A_n$, and this gives a standard cluster algebra $A_n$ of type $C_n$. Moreover, in this specialisation, the set $\mathcal M_0$ becomes the set $M$ of cluster monomials in $A_n$, which is free over $\mathbb Z$. This implies that $\mathcal M_0$ is free over $\mathbb Z\lbrack \lambda\rbrack$. Indeed, if $\mathcal{M}_0$ were not free, there would be a non-trivial $\mathbb{Z}\lbrack \lambda \rbrack$-linear dependence relation between elements of $\mathcal M_0$, of the form $$\label{m0}
\displaystyle \sum_{t=0}^N P_t(\lambda)\cdot m_t=0,\quad P_t\in\mathbb{Z}\lbrack\lambda\rbrack,\: m_t\in\mathcal{M}_0\quad(t\in\llbracket 0,N\rrbracket).$$ Dividing if necessary by a suitable power of $\lambda$, we may assume that at least one $P_t(\lambda)$ is not divisible by $\lambda$, i.e. $P_t(0)=a_t\neq 0$. The relation (\[m0\]) above would then specialise, for $\lambda=0$, into a non-trivial $\mathbb{Z}$-linear dependence relation between the cluster monomials of $M$. Thus $\mathcal{M}_0$ is free over $\mathbb{Z}\lbrack \lambda\rbrack$, and therefore $\mathcal{M}$ is free over $\mathbb{Z}$.
To show that $\mathcal S$ is free over $\mathbb Z$, let us now prove that $\mathcal{M}$ and $\mathcal{S}$ can be linked by an infinite unitriangular matrix $U$. For a monomial $m\in\mathcal{M}$ of the form $m=\lambda^e \cdot\prod x_{ab}^{m_{ab}}$, define its degree $$\deg(m):=(n+1)\cdot e+\sum m_{ab}\cdot\mathrm{Card}\:\mathcal{O}_{ab}.$$ Choose a total order on $\mathcal{M}$ such that for any $m,m'\in\mathcal{M}$, $$\deg(m)<\deg(m')\Rightarrow m< m' .$$
Let $\Phi:\mathcal{M}\rightarrow\mathcal{S}$ be the map that sends a monomial $m=\lambda^e \cdot\prod x_{ab}^{m_{ab}}\in\mathcal{M}$ to the monomial $$\Phi(m):= \displaystyle\left( \prod_{\mathrm{Card}\mathcal{O}_{ab}=1} x_{ab}\right)^e \cdot \prod\left( \prod_{ 2k\in\mathcal{O}_{ab}} x_{2k-2,2k+2} \right)^{m_{ab}}\in\mathcal{S}.$$ We show that $\Phi$ is a bijection by constructing an inverse map $\Psi:\mathcal{S}\rightarrow\mathcal{M}$. To a monomial $s=\displaystyle\prod_{k=0}^n x_{2k,2k+4}^{a_k}\in\mathcal{S}$, we attach the multiset $M(s)$ containing $a_k$ times the integer $2k+2$ for each $k=0,\dots,n$. A subset of $M(s)$ of the form $$\llbracket 2k,2\ell \rrbracket := \{2k,2k+2,\dots,2\ell-2,2\ell\}\quad(1\leq k\leq \ell\leq n+1)$$ is called a *segment* of length $\ell-k+1$. Let $r$ be the number of distinct copies of $\llbracket 2,2n+2\rrbracket$ contained in $M(s)$, and let $M^{(1)}(s)$ be the multiset obtained from $M(s)$ by removing these $r$ maximal segments. Then it is an elementary combinatorial fact that $M^{(1)}(s)$ has a unique decomposition into a union of segments pairwise in *generic position*. Here we say that two segments $\Sigma_1=\llbracket 2k_1,2\ell_1\rrbracket$ and $\Sigma_2=\llbracket 2k_2,2\ell_2\rrbracket$ are in generic position if the corresponding diagonals $(2k_1-2,2\ell_1+2)$ and $(2k_2-2,2\ell_2+2)$ do not intersect or are equal. Let $m_{ab}$ be the number of copies of $\llbracket a+2,b-2\rrbracket$ in this decomposition. Then $$\Phi(s):=\lambda^r\displaystyle\prod x_{ab}^{m_{ab}}$$ is in $\mathcal M$ and $\Psi\circ\Phi(m)=m$, $\Phi\circ\Psi(s)=s$. We then order $\mathcal S$ by $$(s<s')\Leftrightarrow(\Phi(s)<\Psi(s')).$$
Let $U$ be the matrix $U=(u_{ms})_{m\in\mathcal{M},s\in\mathcal{S}}$ where the entries $u_{ms}\in\mathbb{Z}$ are defined by the infinite system of equations $$m= \displaystyle \sum_{s\in\mathcal{S}} u_{ms} s \quad(m\in\mathcal{M}).$$ The entries $u_{ms}$ are computed using the relations (\[gexc1\]) to (\[rel2\]) above. The rows and columns are ordered using the above total orders on $\mathcal{M}$ and $\mathcal{S}$.
By recursion on the degree, we are going to prove, with relations (\[gexc1\])-(\[rel2\]), that $U$ is lower unitriangular, that is, every monomial $m\in\mathcal{M}$ can be written as $$\label{monoMS}
m=\Phi(m) + \displaystyle\sum_{s<\Phi(m),\:s\in\mathcal{S}} u_{ms}\cdot s.$$ First, if $m=x_{ab}$, we can deduce from relation (\[gexc1\]), that $$\label{monoxab}
x_{ab}=\Phi(x_{ab}) + \displaystyle \sum_{\deg(s)<\deg(x_{ab}),\:s\in\mathcal{S}} u_{x_{ab},s}\cdot s.$$ Indeed, we can write $x_{ab}=x_{2k,2k+2d}$ for some $k,d\in\llbracket 0,n\rrbracket$. We have $x_{2k,2k+2}=1$ and $\Phi(x_{2k,2k+4})=x_{2k,2k+4}$. We also deduce from that $$x_{2k,2k+6}= x_{2k,2k+4}x_{2k+2,2k+6}-1=\Phi(x_{2k,2k+6})-1.$$ In general, suppose that the relation (\[monoxab\]) holds, up to a certain degree $d-1<n$ of $x_{ab}.$ Then we use relation (\[gexc1\]), up to shifting of the indices, to deduce that $x_{\overline{2n},2d}$ is equal to $\Phi(x_{\overline{2n},2d})$ plus some terms of degree $<d$. This can be seen for each term displayed above. Therefore, we obtain (\[monoxab\]).
The reasoning is similar for $\lambda$. It suffices to take the variables in a special case of (\[gexc2\]), and replace them with the expressions obtained from (\[monoxab\]), to get an expression of the form $$\label{monolambda}
\lambda = (x_{\overline{2n},2} x_{04}x_{26}\dots x_{2n-4,\overline{2n}}x_{2n-2,\overline{0}}) + \displaystyle \sum_{\deg s \leq n} u_{\lambda,s} \cdot s.$$ The right-hand side above is equal to $\Phi(\lambda)$, which is of degree $n+1$, plus some terms of degree $\leq n$, hence (\[monolambda\]) is true. The identities (\[monoxab\]) and (\[monolambda\]) yield the first lower triangular rows of $U$.
The relation now follows from and because of the compatibility of the orderings with multiplication. More precisely, note that we clearly have, for any two non-trivial monomials $m,m'\in\mathcal{M}$, $$\begin{array}{l}\Phi(m\cdot m')=\Phi(m)\Phi(m'),\quad \:\Phi(m)<\Phi(mm'),\quad\Phi(m')<\Phi(mm'),\\\mathrm{and}\: \:\:\deg(mm')=\deg(m)+\deg(m').\end{array}$$ Thus for any two cluster variables $x_{ab}$ and $x_{cd}$, we have $$\begin{array}{ll}
x_{ab}x_{cd}&=\Phi(x_{ab}x_{cd}) + \displaystyle\sum_{s<\Phi(x_{ab})} u_{x_{ab},s} s\Phi(x_{cd}) + \displaystyle\sum_{s'<\Phi(x_{cd})} u_{x_{cd},s'} \Phi(x_{ab}) s' \\\\&\qquad+ \displaystyle \sum_{s<\Phi(x_{ab}),s'<\Phi(x_{cd})} u_{x_{ab},s}u_{x_{cd},s'} s s',
\end{array}$$ where each term in the three sums is of degree $<\deg x_{ab}+\deg x_{cd}$.
Likewise, for each monomial $m_0\in\mathcal{M}_0$, the element $\lambda m_0$ is equal to $\Phi(\lambda m_0)$ plus some terms of degree $<n+1+\deg m_0$. This product compatibility immediately implies (\[monoMS\]).
Finally, the unitriangularity of $U$ readily implies that, since $\mathcal M$ is free over $\mathbb Z$, then $\mathcal S$ is free over $\mathbb Z$. In conclusion, $\mathcal S$ is a $\mathbb Z$-basis of $\mathcal A_n$. $\Box$
It follows from the proof of Proposition \[genbasisA\] that the set $\mathcal M_0$ os cluster monomials is a $\mathbb Z\lbrack \lambda\rbrack$-basis of $\mathcal A_n$. We now use it to introduce another interesting $\mathbb Z$-basis of $\mathcal{A}_n$, which will be meaningful in representation theory.
For $k\in\mathbb{N}$, denote by $S_k(u)\in\mathbb{Z}\lbrack u\rbrack$ the $k$-th Tchebychev polynomial of the second kind, given by $$S_k(u)^2=S_{k-1}(u)S_{k+1}(u)+1$$ with initial conditions $S_0(u)=1$ and $S_1(u)=u$. Recall from the proof of Proposition \[genbasisA\] that $\mathcal{M}_0$ is the set of cluster monomials of $\mathcal{A}_n$ that do not contain powers of $\lambda$. Then the set $$\label{basisB}
\mathcal{B}:=\{ S_k(\lambda)\cdot m,\: k\in\mathbb{N}, m\in \mathcal{M}_0\}$$ is a $\mathbb Z$-basis of $\mathcal{A}_n$. This new basis will later correspond (Section \[s3\]) to the basis of classes of simple modules in the Grothendieck ring of a category of representations of $\Uereslsl2$.
Representations of quantum affine algebras {#s22}
==========================================
Let $l$ be an integer, $l\geq 2$. Introduce the root of unity $$\label{epsilon}\varepsilon :=\left\{ \begin{array}{ll}
\exp\left(\displaystyle\frac{i\pi}{l}\right) &\mbox{if } l \mbox{ is even}\\\\
\exp\left(\displaystyle\frac{2 i\pi}{l}\right) &\mbox{if } l \mbox{ is odd}
\end{array} \right.$$ Thus $l$ is the order of $\varepsilon^2$, and we have $\varepsilon^{2l}=1$. Following [@FM], let us also write $$\varepsilon^* := \varepsilon^{l^2}=1.$$
Quantum affine algebras and their specialisations {#s221}
-------------------------------------------------
Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra of simply-laced type, with Dynkin diagram $\delta$, vertex set $I=\llbracket 1,n\rrbracket$ and Cartan matrix $C=(a_{ij})_{i,j\in I}$. Denote by $\alpha_i$ the simple roots, by $\varpi_i$ the fundamental weights and by $P$ the weight lattice [@B].
Let $q$ be an indeterminate; then $\mathbb{C}(q)$ is the field of rational functions of $q$ with complex coefficients, and $\mathbb{C}\lbrack q,q^{-1}\rbrack$ is the ring of complex Laurent polynomials in $q$.
Let $U_q(\widehat{\mathfrak{g}})$ be the quantum affine algebra associated with ${\mathfrak g}$ [@FR2]. This is a Hopf algebra over $\mathbb C (q)$. Denote by $\Uqlg$ the quantum loop algebra, which is isomorphic to a quotient of $U_q(\widehat{\mathfrak{g}})$ where the central charge is mapped to 1. Therefore, $\Uqlg$ inherits a Hopf algebra structure. For more information on $L\mathfrak g$, $\widehat{\mathfrak g}$ and their quantum enveloping algebras, we refer the reader to [@CP], [@CH] and [@Lec].
We will be interested in finite-dimensional representations of $U_q(\widehat{\mathfrak{g}})$, on which the central charge acts trivially. It is therefore sufficient to consider finite-dimensional representations of $\Uqlg$, and we will focus on these onwards.
Let $\Uqreslg$ be the restricted integral form corresponding to $\Uqlg$ [@CP2]. This is a Hopf algebra over $\mathbb C \lbrack q,q^{-1}\rbrack$.
Let us now specialise $\Uqreslg$ at the root of unity $\varepsilon$, by setting $$\begin{array}{l}
\Uereslg:= \Uqreslg\otimes_{\mathbb{C}\lbrack q,q^{-1}\rbrack} \mathbb{C}\\
\end{array}$$ via the algebra homomorphism $$\begin{array}{ccc}
\mathbb{C}\lbrack q,q^{-1}\rbrack&\longrightarrow&\mathbb{C}\\
q&\longmapsto&\varepsilon
\end{array}$$ For an element $x$ of $\Uqreslg$ , we denote the corresponding element of $\Uereslg$ also by $x$.
Representations of Uq(Lg) {#s222}
-------------------------
#### The category $\mathcal{C}_q$.
Let $\mathcal{C}_q$ be the category of finite-dimensional type 1 $\Uqlg$-modules (see [@CP Section 11.2]). It is known that $\mathcal{C}_q$ is a monoidal, abelian, non semisimple category.
An object $V$ in $\mathcal{C}_q$ has a $q$-character $\chi_q(V)$ [@FR2], which is a Laurent polynomial with positive integer coefficients in variables $Y_{i,a}$, $i\in I$, $a\in\mathbb C(q)$. Any irreducible object or $\mathcal{C}_q$ is determined, up to isomorphism, by its $q$-character. Such irreducible representations are parametrised [@FR2] by the highest dominant monomial of their $q$-characters, which is a *dominant monomial*, i.e. it contains only positive exponents.
Let $\mathcal{M}_q$ be the set of Laurent monomials in the $Y_{i,a}$, and let $\mathcal{M}_q^+$ be the subset of dominant monomials in $\mathcal{M}_q$. If $S$ is a simple object of $\mathcal{C}_q$ such that the highest monomial of $\chi_q(S)$ is $m\in\mathcal{M}_q^+$, then $S$ will be denoted by $L(m)$ [@Lec]. For $i\in I$ and $a\in\mathbb{C}(q)$, the simple modules $L(Y_{i,a})$ are called fundamental modules. A *standard module* is a tensor product of fundamental modules.
Let $K_0(\mathcal{C}_q)$ be the Grothendieck ring of $\mathcal{C}_q$. It is known ([[@FR2 Corollary 2]]{}) that $$K_0(\mathcal{C}_q) \cong \mathbb{Z}\lbrack \lbrack L(Y_{i,a})\rbrack, i\in I, a\in\mathbb{C}(q)\rbrack.$$
For $i\in I,\:k\in\mathbb{N}^*,\: a\in\mathbb{C}(q)$, the simple object $$W_{k,a}^{(i)} = L(Y_{i,a} Y_{i,aq^2}\dots Y_{i,aq^{2(k-1)}})$$ is called a *Kirillov-Reshetikhin module*. In particular, we have $W_{1,a}^{(i)}=L(Y_{i,a})$ and by convention, $W_{0,a}^{(i)}=\mathbf{1}$ for any $a,i$.
The classes $\lbrack W_{k,a}^{(i)}\rbrack$ in $K_0(\mathcal C_q)$ satisfy a system of equations called $T$-system: $$\lbrack W_{k,a}^{(i)}\rbrack\lbrack W_{k,aq^2}^{(i)}\rbrack = \lbrack W_{k+1,a}^{(i)}\rbrack\lbrack W_{k-1,aq^2}^{(i)}\rbrack+\displaystyle \prod_{j\sim i} \lbrack W_{k,aq}^{(j)}\rbrack\quad (i\in I,k\in\mathbb{N}^*,a\in\mathbb{C}(q)),$$ where $j\sim i$ means that $j$ is a neighbour of $i$ in the Dynkin diagram $\delta$.
If ${\mathfrak{g}}={\mathfrak{sl}_2}$, the $T$-system reads $$\lbrack W_{k,a}^{(1)}\rbrack\lbrack W_{k,aq^2}^{(1)}\rbrack = \lbrack W_{k+1,a}^{(1)}\rbrack\lbrack W_{k-1,aq^2}^{(1)}\rbrack+1\quad (k\in\mathbb{N}^*).$$
#### The category $\CZ$.
Let us now define a subcategory of $\mathcal{C}_q$, following [@HL].
Since the Dynkin diagram $\delta$ is a bipartite graph, there is a partition of the vertices $I=I_0 \sqcup I_1$, where each edge connects a vertex of $I_0$ with a vertex of $I_1$. For $i\in I$, set $$\xi_i:=\left\{ \begin{array}{ll}
0&\mbox{if } i\in I_0\\
1&\mbox{if }i\in I_1
\end{array}\right.$$ The map $i\mapsto \xi_i$ is determined by the choice of $\xi_{i_0}\in\{0,1\}$ for a single vertex $i_0$. Therefore, there are only two possible collections of $\xi_i$.
Let $\CZ$ be the full subcategory of $\mathcal{C}_q$ whose objects $M$ have all their composition factors $L(m)$ such that $m$ contains only variables of the form $Y_{i,q^{2k+\xi_i}}$ $(k\in\mathbb Z,\:i\in I)$. This is a tensor subcategory of $\mathcal{C}_q$.
The ring $R_\mathbb{Z} := K_0(\mathcal{C}_{q^{\mathbb{Z}}})$ is the subring of $K_0(\mathcal{C}_q)$ generated by the classes of the form $\lbrack L(Y_{i,q^{2k+\xi_i}})\rbrack\:(i\in I,k\in\mathbb{Z}).$
Representations of Ueres(Lg) {#s223}
----------------------------
Let $\Cres$ be the category of finite-dimensional type 1 $\Uereslg$-modules. Let $K_0(\Cres)$ be its Grothendieck ring.
An object $V$ in $\Cres$ has an $\varepsilon$-character $\chi_\varepsilon(V)$ [@FM], which is a Laurent polynomial with positive integer coefficients in variables $Y_{i,a}$, $i\in I$, where $a\in\mathbb C^*$.
The parametrisation of the simple objects by their highest monomials also holds on $\mathcal C_\varepsilon$, with $q$ replaced by $\varepsilon$ (see [@CP2; @FM]). Let $\mathcal{M}_\varepsilon$ be the set obtained from $\mathcal{M}_q$ by replacing $q$ by $\varepsilon$, and let $\mathcal M_\varepsilon^+$ be the subset of dominant monomials in $\mathcal M_\varepsilon$. The simple module whose highest weight monomial is $m\in\mathcal{M}_\varepsilon^+$ will be denoted by $L(m)$.
In particular, the *fundamental modules* of $\Uereslg$ are the simple objects\
$L(Y_{i,a})$, where $i\in I,\:a\in\mathbb{C}^*$, and the *standard modules* are the tensor products of fundamental modules. The simple module $L(m)$ is called *prime* if it cannot be written as a tensor product of non-trivial modules.
The ring $K_0(\Cres)$ is the ring of polynomials with integer coefficients in variables $\lbrack L(Y_{i,a})\rbrack$, $i\in I,a\in\mathbb{C}^*$ (see [@FM Section 3.1]).
For a simple object $V$ of $\mathcal{C}_q$, with highest weight vector $v$, it is known [@FM Proposition 2.5] that the $\Uqreslg$-module $V^{\mathrm{res}}:= \Uqreslg\cdot v$ is a free $\mathbb C\lbrack q,q^{-1}\rbrack$-module. Put $V_\varepsilon^{\mathrm{res}}=V^{\mathrm{res}}\otimes_{\mathbb{C}\lbrack q,q^{-1}\rbrack} \mathbb C$, where as above $q$ acts on $\mathbb C$ by multiplication by $\varepsilon$. This is a $\Uereslg$-module called the *specialisation of $V$ at $q=\varepsilon$*.
For $i\in I$ and $a\in\mathbb{C}^*$, introduce the following notation: $$\mathbf{Y}_{i,a}:= \displaystyle \prod_{j=0}^{l-1} Y_{i,a\varepsilon^{2j+\xi_i}}.$$ Note that since $\varepsilon^{2l}=1$, we have $\mathbf{Y}_{i,\varepsilon^{2r}}=\mathbf{Y}_{i,1}$ for any $r\in\mathbb{Z}$. A monomial in the variables $Y_{i,a}$ is called *$l$-acyclic* if it is not divisible by $\mathbf{Y}_{j,b}$ for any $j\in I,\:b\in\mathbb C^*$.
Let $\CZres$ be the full subcategory of $\Cres$ whose objects $M$ have all their composition factors $L(m)$ such that $m\in\mathcal{M}_\varepsilon^+$ contains only variables of the form $Y_{i,\varepsilon^{2k+\xi_i}}$, where $i\in I$ and $k\in\mathbb{Z}$. For example, the modules $z_i:= L(\mathbf{Y}_{i,1})$ are objects of $\CZres$.
Let $R_{\varepsilon^{\mathbb{Z}}}=K_0(\CZres)$ be the Grothendieck ring of $\CZres$. This is the subring of $K_0(\Cres)$ generated by the classes $\lbrack L(Y_{i,\varepsilon^{2k+\xi_i}})\rbrack, i\in I,\:k\in\mathbb{Z}.$
Representations of Ueres(Lg) {#s225}
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Consider the category $\mbox{Rep }\Ueereslg$ of finite-dimensional type 1 representations of $\Ueereslg$. Since we are in the case where $\varepsilon^{*}=\varepsilon^{l^2}=1$, this category is equivalent to the category $\Rep L{\mathfrak{g}}$ of finite-dimensional $ L{\mathfrak{g}}$-modules.
For $a\in\mathbb{C}^*$, consider the evaluation morphism $\phi_a: L{\mathfrak{g}}\cong \mathfrak{g}\lbrack t,t^{-1}\rbrack \rightarrow \mathfrak{g}$ that maps a Laurent polynomial $P(t)$ to its evaluation $P(a)$ at $a$. For an irreducible representation $V_\lambda$ of $\mathfrak{g}$ that has highest weight $\lambda$, the pullback $V_\lambda(a):=\phi_a^*(V_\lambda)$ is an irreducible $L {\mathfrak{g}}$-module. It is known [@CP1] that any simple object $S\in\Rep L{\mathfrak{g}}$ is a tensor product of evaluation modules $V_{\lambda_1}(a_1)\otimes \dots \otimes V_{\lambda_n}(a_n)$, such that $a_i\neq a_j$ for every $i\neq j$.
Let $V$ be a representation of $\mathfrak{g}$, with weight decomposition $V=\bigoplus_\mu V_\mu$. Recall that the character $\chi(V)$ of $V$ is a polynomial in variables $y_i^{\pm 1},i\in I$ defined by $$\chi(V)=\sum_\mu \dim V_\mu \cdot y^\mu,$$ where for a weight $\mu=\sum_{i\in I} \mu_i\varpi_i$, we set $y^\mu = \prod_{i\in I}y_i^{\mu_i}$.
From q-characters to e-characters {#s226}
---------------------------------
Frenkel and Mukhin ([@FM Proposition 2.5]) prove that there is a surjective ring morphism $K_0(\mathcal{C}_q)\rightarrow K_0(\Cres)$ that maps the isomorphism class $\lbrack V\rbrack$ of a simple object $V$ of $\mathcal{C}_q$ to the class $\lbrack V_\varepsilon^{\mathrm{res}}\rbrack$.
Since the map $\chi_q:K_0(\mathcal{C}_q)\rightarrow \mathbb{Z}\lbrack Y_{i,a}^{\pm 1},\: i\in I,\:a\in\mathbb{C}(q)\rbrack$ is an injective ring morphism (see [@FR2 Theorem 3]), Frenkel and Mukhin ([@FM Theorem 3.2]) prove that the $\varepsilon$-character map $\chi_\varepsilon:K_0(\Cres)\rightarrow \mathbb{Z}\lbrack Y_{i,a}^{\pm 1},\:i\in I,\:a\in\mathbb{C}^*\rbrack$ is also an injective ring morphism. Moreover, Theorem 3.2 in [@FM] also states that for a simple module $V\in\mathcal{C}_q$, the $\varepsilon$-character $\chi_\varepsilon(V_\varepsilon^{\mathrm{res}})$ is obtained by substituting $q\mapsto \varepsilon$ in $\chi_q(V)$. The $\varepsilon$-characters $\chi_\varepsilon$ thus satisfy combinatorial properties similar to $q$-characters ([@FM Section 3.2]).
The Frobenius pullback {#s227}
----------------------
Following Lusztig [@Lu], Frenkel and Mukhin [@FM] describe a *quantum Frobenius map* $\mathrm{Fr}:\Uereslg\rightarrow\Ueereslg$ that gives rise to the *Frobenius pullback* $$\mbox{Fr}^* : K_0(\Rep\Ueereslg)\longrightarrow K_0(\Cres).$$
\[frob\] The Frobenius pullback $$\mathrm{Fr}^* :\begin{array}{ccc} K_0(\Rep\Ueereslg) &\longrightarrow& K_0(\Cres)\end{array}$$ is the injective ring homomorphism such that $\mathrm{Fr}^*(\lbrack L(Y_{i,a}) \rbrack)= \lbrack L(\mathbf{Y}_{i,a})\rbrack$.
Decomposition theorem {#s228}
---------------------
Let $m\in\mathcal M_\varepsilon^+$. There is a unique factorisation $m=m^0 m^1$ where $m^1$ is a monomial in the variables $\mathbf{Y}_{i,a}$, and $m^0$ is $l$-acyclic. The monomial $m^0$ is called the $l$-acyclic part of $m$. The following theorem was proved by Chari and Pressley for roots of unity of odd order [@CP2] and generalised by Frenkel and Mukhin [@FM] to roots of unity of arbitrary order.
\[decthm\] Let $L(m)$ be a simple object of $\Cres$. Then $$L(m)\cong L(m^0)\otimes L(m^1).$$
Note that by Proposition \[frob\], $L(m^1)$ is the Frobenius pullback of an irreducible $\Ueereslg$-module.
Characters of Ue\*reslg-modules
-------------------------------
Since $\varepsilon^*=1$, the category $\Rep\Ueereslg$ is equivalent to $\Rep L{\mathfrak{g}}$, and in order to compute $\varepsilon^*$-characters we just need to know $\chi_1(V_\lambda(a))$, which is obtained from $\chi(V_\lambda)$ by replacing each $y_i^{\pm 1}$ by $Y_{i,a}^{\pm 1}$.
For an irreducible $\Ueereslg$-module $L(m)$, the pullback $\mathrm{Fr}^*(L(m))$ is the module $L(M)$, where $M$ is the monomial obtained from $m$ by replacing each $Y_{i,a^l}^{\pm 1}$ with $\mathbf{Y}_{i,a}^{\pm 1}$. The $\varepsilon$-character $\chi_\varepsilon(\mathrm{Fr}^*(L(m)))$ of $L(m)\in \Rep\Ueereslg$ is obtained from $\chi_{\varepsilon^*}(L(m))$ by replacing each $Y_{i,a^l}^{\pm 1}$ by $\mathbf{Y}_{i,a\varepsilon^{\xi_i}}^{\pm 1}$.
Therefore, the computation of $\varepsilon$-characters of simple objects of $\Cres$ is reduced, by Theorem \[decthm\], to understanding the $\varepsilon$-characters of all representations $L(m^0)$ where $m^0$ is $l$-acyclic.
The case g=sl2 {#s310}
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### Prime simple modules
Let us introduce the following notation: $$W_\varepsilon(k,a,i):=\left( W_{k,a}^{(i)}\right)^{\mathrm{res}}_\varepsilon\quad(i\in I,\:k\in\mathbb N,\: a\in\mathbb C^*).$$ Here $I=\{1\}$, so we drop the index $i$ in the above notation. The specialisation of a Kirillov-Reshetikhin module of $\CZ$ to an object of $\CZres$ is not always simple.
To each module $W_\varepsilon(k,\varepsilon^{2d})$ with $k<l$ and $d\in\llbracket 1,l\rrbracket$, attach the diagonal $\lbrack 2d-2,2d+2k\rbrack$ of the $2l$-gon $\mathbf{P}_{2l}$ defined in Section \[s231\]. The following result is equivalent to a special case of a theorem from Chari and Pressley [@CP2].
\[KRA\] For $\mathfrak g = \mathfrak{sl}_2$, the simple objects of $\CZres$ are exactly the tensor products of the form $$\displaystyle \bigotimes_{t=1}^r L(Y_{1,\varepsilon^{2d_t}} \dots Y_{1,\varepsilon^{2(d_t+k_t-1)}})^{\otimes a_t}\otimes L(\mathbf{Y}_{1,1}^{ a}) = \displaystyle \bigotimes_{t=1}^r W_\varepsilon (k_t,\varepsilon^{2d_t})^{\otimes a_t}\otimes L(\mathbf{Y}_{1,1}^{ a})$$ where $r\in\mathbb N^*,\: k_1,\dots,k_r\in\llbracket 0,l-1\rrbracket,\:d_1,\dots,d_r\in\mathbb Z,\: a_1,\dots,a_r,a\in\mathbb N$, under the condition that for every $t\neq s\in\llbracket 1,r\rrbracket$, the diagonals $\lbrack 2d_t-2,2d_t+2k_t\rbrack$ and $\lbrack 2d_s-2,2d_s+2k_s\rbrack$ do not intersect inside $\mathbf{P}_{2l}$.
It follows from Theorem \[KRA\] that the *prime* simple modules of $\Uereslsl2$ are the modules $W_\varepsilon(k,\varepsilon^{2d})$ $(k<l,\:r\in\llbracket 1,l\rrbracket)$ and the Frobenius pullbacks $L(\mathbf{Y}_{1,1}^a)$ $(a\in\mathbb N^*)$. From now on, we drop the index $i=1$ in the variables $Y_{1,\varepsilon^n}$ and introduce the notation $$Y_{n}:= Y_{1,\varepsilon^{n}}\quad\mbox{and}\quad \mathbf{Y}_1=Y_0Y_2\dots Y_{2l-2}$$ for any integer $n$. We then have $Y_{2l+n}=Y_n$ for every $n$.
With this new notation, we have $$\label{eq:KR}\begin{array}{l}W_\varepsilon(k,\varepsilon^{2r})= L(Y_{2r}Y_{2r+2}Y_{2r+4}\dots Y_{2(r+k-1)}) \quad(k,r\in\llbracket 0,l-1\rrbracket),\\ z:=z_1= L(\mathbf{Y}_{1}).\end{array}$$
### e-Characters
In type $A_1$, we have an explicit expression for $q$-characters: $$\label{expA1}\begin{array}{ll}
\chi_q(W_{k,a}^{(1)}) &= Y_{1,a}Y_{1,aq^2}\dots Y_{1,aq^{2(k-2)}}Y_{1,aq^{2(k-1)}} \\&\quad+Y_{1,a}Y_{1,aq^2}\dots Y_{1,aq^{2(k-2)}}Y_{1,aq^{2k}}^{-1}\\&\quad+Y_{1,a}Y_{1,aq^2}\dots Y_{1,aq^{2(k-3)}}Y_{1,aq^{2(k-1)}}^{-1}Y_{1,aq^{2k}}^{-1}\\&\quad+\dots + Y_{1,aq^2}^{-1}Y_{1,aq^4}^{-1}\dots Y_{1,aq^{2(k-1)}}^{-1}Y_{1,aq^{2k}}^{-1} .
\end{array}$$ Each module $W_{k,q^{2d}}^{(1)}$ specialises to $W_\varepsilon(k,\varepsilon^{2d})$, which is irreducible if $k<l$. We can then directly translate the formula above into the $Y_n$ notation for $k<l$. Moreover, since $z_1=L(\mathbf{Y}_{1,1})$ is the pullback of $L(y_1)$, and $\chi(L(y_1))=y_1+y_1^{-1}$, we have $$\label{foz1}\chi_\varepsilon( z_1) = Y_0Y_2\dots Y_{2(l-1)}+Y_0^{-1}Y_2^{-1}\dots Y_{2(l-1)}^{-1}.$$ This implies that the specialised Kirillov-Reshetikhin modules $W_\varepsilon(k,\varepsilon^{2d})$, for $k<l-1$, satisfy the $T$-system, and the $\varepsilon$-characters behave the same way as their $q$-character counterparts. Namely, for $k\leq l-2$, we have $$\label{tsyse}\begin{array}{l}\chi_\varepsilon(L(Y_0Y_2\dots Y_{2k}))\chi_\varepsilon(L(Y_2Y_4\dots Y_{2k+2} ))\\\qquad =\chi_\varepsilon(L(Y_2\dots Y_{2k}))\chi_\varepsilon(L(Y_0\dots Y_{2k+2}))+1.\end{array}$$ The difference of behaviour between $q$-characters and $\varepsilon$-characters resides in the $(l-1)$-th equation of the $T$-system, which will later be seen as a generalised exchange relation:
\[carac\] We have $$\label{caracl}\begin{array}{ll}\chi_\varepsilon(L(Y_0Y_2Y_4\dots Y_{2(l-2)}))\chi_\varepsilon(L(Y_2Y_4\dots Y_{2(l-1)}))\\\quad=
\chi_\varepsilon(L(Y_2Y_4\dots Y_{2(l-2)})) \cdot \chi_\varepsilon( z_1) +1 + \chi_\varepsilon(L(Y_2Y_4\dots Y_{2(l-2)}))^2.\end{array}$$
This follows immediately from an explicit computation using formulas and . $\Box$
The cluster structure on Ko(CZres) in type A1 {#s3}
=============================================
We can now state the main theorem of this paper. Recall the generalised cluster algebra $\mathcal{A}_n$ defined in Section \[s213\], Definition \[dAn\]. Let $R=R_{\varepsilon^{\mathbb Z}}$ be the Grothendieck ring of $\CZres$ for $\mathfrak g = \mathfrak {sl}_2$ and $\varepsilon$ as in .
\[conjA1\] There exists a ring isomorphism $\varphi:\mathcal{A}_{l-1}\rightarrow R$, such that $$\varphi (x_{2r,2d})=\lbrack L(Y_{2r+2}\dots Y_{2d-2})\rbrack\:(r,d\in\llbracket 0,l-1\rrbracket, |r-d|<l),\quad \varphi(\lambda)= \lbrack z_1\rbrack.$$ The $\mathbb{Z}$-basis $\mathcal{S}$ of $\mathcal{A}_{l-1}$ is mapped by $\varphi$ to the basis of classes of standard modules in $R$. The $\mathbb{Z}$-basis $\mathcal{B}$ of $\mathcal{A}_{l-1}$ consisting in generalised cluster monomials (see (\[basisB\])) is mapped to the basis $B$ of classes of simple modules in $R$.
We established an isomorphism $\mathcal{A}_{l-1}\cong \mathbb{Z}\lbrack x_{2r-2,2r+2},\:r\in\llbracket 0,l-1\rrbracket\rbrack$ in Section \[s213\] (Proposition \[genbasisA\]). We also know from Section \[s223\] that there is an isomorphism $R\cong \mathbb{Z}\lbrack \lbrack L(Y_{2k})\rbrack,\:k\in\llbracket 0,l-1\rrbracket \rbrack$. Therefore, we may fix a ring isomorphism $\varphi$, which sends each variable $x_{2r-2,2r+2}$ to the class $\lbrack L(Y_{2r})\rbrack$ in the Grothendieck ring $R$. Clearly, $\varphi$ maps the basis $\mathcal S$ to the basis of classes of standard modules.
It is easy to deduce from Proposition \[pCn\] that the cluster variables in $\mathcal{A}_n$ are built from the $x_{2r-2,2r+2}$ using relation (\[gexc1\]), with $k=r-d$. On the other hand, implies that the $\varepsilon$-characters of the simple Kirillov-Reshetikhin modules in $R$ are built from the $\varepsilon$-characters of fundamental modules $\chi_\varepsilon(L(Y_{2r}))$ using the same relations. Thus $\varphi(x_{2r,2d})=\lbrack L(Y_{2r+2}\dots Y_{2d-2})\rbrack$. Moreover, comparing and , we also get $\varphi(\lambda)=\lbrack z_1\rbrack$.
Let us now move on to the correspondence between the bases $\mathcal B$ and $B$. We know from Theorem \[decthm\] that the class of every simple module can be written as $F\cdot M$, where $M$ is $l$-acyclic and $F$ is the Frobenius pullback of the class of an irreducible $\mathfrak{sl}_2$-module. We know that $\lbrack z_1\rbrack$ corresponds to $\mathrm{Fr}^*(\lbrack V(\varpi)\rbrack)$, the Frobenius pullback of the 2-dimensional representation of $\mathfrak{sl}_2$. It then follows from the classical theory of characters for $\mathfrak{sl}_2$ that $\mathrm{Fr}^*(\lbrack V(k\varpi)\rbrack)$ is the Tchebychev polynomial of the second kind $S_k(\lbrack z_1\rbrack)$. Therefore, the basis of classes of simple modules in $R$ consists of elements of the form $S_k(\lbrack z_1\rbrack)\cdot M$, where $M$ is the class of a tensor product of simple Kirillov-Reshetikhin modules that satisfy the condition from Theorem \[KRA\]. This geometrical condition on diagonals of $\mathbf{P}_{2l}$ corresponds exactly to cluster variable compatibility in $\mathcal{A}_{l-1}$: indeed, recall that two cluster variables of $\mathcal{A}_{l-1}$ are in the same cluster if and only if their attached diagonals do not cross inside $\mathbf{P}_{2l}$. Therefore, Theorem \[decthm\] allows us to conclude that the image of the basis $\mathcal{B}$ under the isomorphism $\varphi$, is the basis $B$ of classes of simple modules in $R$. $\Box$
Type A2 {#s4}
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The case A2,l=2. {#s42}
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We start by studying the Grothendieck ring of $\CZres$ for $\mathfrak g = \mathfrak{sl}_3$ and $l=2$, in terms of $\varepsilon$-characters. For any integer $n\in\mathbb Z$ and any vertex $i=1,2$ of the Dynkin diagram, we set $$Y_{i,n}:=Y_{i,\varepsilon^n},$$ where we consider the second index modulo $4$. We also write $$\mathbf{Y}_1=Y_{1,0}Y_{1,2}\quad\mbox{and}\quad\mathbf{Y}_2=Y_{2,1}Y_{2,3}.$$ In this case, the simple modules of $\CZres$ are of the form $L(m)$, where $$m=Y_{1,0}^{a_{10}}Y_{1,2}^{a_{12}}Y_{2,1}^{a_{21}}Y_{2,3}^{a_{23}}\quad(a_{10},\dots,a_{23}\in\mathbb N).$$
\[caracA2\] The Laurent polynomial $\chi_\varepsilon(L(Y_{1,0}))^k\chi_\varepsilon(L(Y_{1,0}Y_{2,1}))^\ell$ contains a unique dominant monomial. It follows that $$\chi_\varepsilon(L(Y_{1,0}))^k\chi_\varepsilon(L(Y_{1,0}Y_{2,1}))^\ell = \chi_\varepsilon(L(Y_{1,0}^{k+\ell}Y_{2,1}^\ell)).$$
It is easy to compute the following $\varepsilon$-characters: $$\label{A22}\begin{array}{lll}
\chi_\varepsilon(L(Y_{1,0}))&=& Y_{1,0} + Y_{2,1}Y_{1,2}^{-1} + Y_{2,3}^{-1}\\
\chi_\varepsilon(L(Y_{1,0}Y_{2,1}))& =& Y_{1,0}Y_{2,1} + Y_{1,0}Y_{1,2}Y_{2,3}^{-1} + Y_{2,1}^2 Y_{1,2}^{-1} + 2 Y_{2,1}Y_{2,3}^{-1}\\&&\quad + Y_{2,1}Y_{1,0}^{-1}Y_{1,2}^{-1} + Y_{1,2}Y_{2,3}^{-2} + Y_{1,0}^{-1} Y_{2,3}^{-1}.
\end{array}$$ One can directly check that the only way to obtain a dominant monomial by multiplying terms of the sums above, is to involve only $Y_{1,0}$ and $Y_{1,0}Y_{2,1}$. Thus the only dominant monomial of $\chi_\varepsilon(L(Y_{1,0}))^k\chi_\varepsilon(L(Y_{1,0}Y_{2,1}))^\ell$ is $Y_{1,0}^{k+\ell}Y_{2,1}^\ell$. Therefore, the $\varepsilon$-character $\chi_\varepsilon(L(Y_{1,0}))^k\chi_\varepsilon(L(Y_{1,0}Y_{2,1}))^\ell$coincides with $\chi_\varepsilon(L(Y_{1,0}^{k+\ell}Y_{2,1}^\ell))$. $\Box$
Theorem \[decthm\] allows us to deduce the following property.
\[decG\] Any simple finite-dimensional $U_\varepsilon^{\mathrm{res}}(L\mathfrak{sl}_3)$-module $L(m)$ can be written $$L(m)= L(\mathbf{Y}_{1}^k\mathbf{Y}_{2}^\ell)\otimes L(m^0),$$ where $L(m^0)$ is one of the following eight tensor products: $$\begin{array}{rllrl}
(i)&L(Y_{1,0})^{\otimes a}\otimes L(Y_{1,0}Y_{2,1})^{\otimes b} & & (ii)&L(Y_{2,1})^{\otimes a}\otimes L(Y_{1,0}Y_{2,1})^{\otimes b} \\
(iii)&L(Y_{1,0})^{\otimes a}\otimes L(Y_{1,0}Y_{2,3})^{\otimes b} & &(iv)&L(Y_{2,3})^{\otimes a}\otimes L(Y_{1,0}Y_{2,3})^{\otimes b} \\
(v)&L(Y_{1,2})^{\otimes a}\otimes L(Y_{1,2}Y_{2,1})^{\otimes b} & & (vi)&L(Y_{2,1})^{\otimes a}\otimes L(Y_{1,2}Y_{2,1})^{\otimes b} \\
(vii)&L(Y_{1,2})^{\otimes a}\otimes L(Y_{1,2}Y_{2,3})^{\otimes b} & & (viii)& L(Y_{2,3})^{\otimes a}\otimes L(Y_{1,2}Y_{2,3})^{\otimes b} . \\
\end{array}$$
In this case, any $l$-acyclic dominant monomial $m^0$ is of the form $$m^0=Y_{1,0}^{a_{10}}Y_{1,2}^{a_{12}}Y_{2,1}^{a_{21}}Y_{2,3}^{a_{23}},$$ with $a_{10}a_{12}=0$ and $a_{21}a_{23}=0$. The proof is the same for all eight situations, so we only check $(i)$. Case $(i)$ corresponds to $a_{12}=a_{23}=0$ and $a_{10}\geq a_{21}$. Then we have $m^0= (Y_{1,0})^{a_{10}-a_{21}} (Y_{1,0}Y_{2,1})^{a_{21}},$ and we deduce from Lemma \[caracA2\] that $
L(m^0)= L(Y_{1,0})^{\otimes a_{10}-a_{21}}\otimes L(Y_{1,0}Y_{2,1})^{\otimes a_{21}}.
$ $\Box$
The modules $L(m)$ where $m$ is $l$-acyclic satisfy some interesting relations.
\[relGc\] The following identities hold, for $i\in\{0,2\}$ and $j\in\{1,3\}$. $$\begin{array}{ll}
\chi_\varepsilon(L(Y_{1,i}))\chi_\varepsilon(L(Y_{2,j}))&=\chi_\varepsilon(L(Y_{1,i}Y_{2,j})) +1 \\
\chi_\varepsilon(L(Y_{1,i}Y_{2,1}))\chi_\varepsilon(L(Y_{1,i}Y_{2,3}))&= \chi_\varepsilon(L(Y_{1,i}))^3 + \chi_\varepsilon(L(Y_{1,i}))^2 \chi_\varepsilon(L(\mathbf{Y}_{2})) \\&\quad+ \chi_\varepsilon(L(Y_{1,i})) \chi_\varepsilon(L(\mathbf{Y}_{1})) + 1 \\
\chi_\varepsilon(L(Y_{1,0}Y_{2,j}))\chi_\varepsilon(L(Y_{1,2}Y_{2,j}))&= \chi_\varepsilon(L(Y_{2,j}))^3 + \chi_\varepsilon(L(Y_{2,j}))^2 \chi_\varepsilon(L(\mathbf{Y}_{1})) \\&\quad+ \chi_\varepsilon(L(Y_{2,j})) \chi_\varepsilon(L(\mathbf{Y}_{2})) + 1.
\end{array}$$
In addition to the expressions of , we have the following formulas: $$\begin{array}{ll}
\chi_\varepsilon(L(Y_{2,1})) &= Y_{2,1} + Y_{1,2}Y_{2,3}^{-1} + Y_{1,0}^{-1}\\
\chi_\varepsilon(L(\mathbf{Y}_{1}))& = Y_{1,0}Y_{1,2} + Y_{2,1}Y_{2,3}Y_{1,0}^{-1}Y_{1,2}^{-1}+ Y_{2,1}^{-1}Y_{2,3}^{-1}\\
\chi_\varepsilon(L(\mathbf{Y}_{2})) &= Y_{2,1}Y_{2,3}+ Y_{1,0}Y_{1,2}Y_{2,1}^{-1}Y_{2,3}^{-1} + Y_{1,0}^{-1}Y_{1,2}^{-1}.
\end{array}$$ All the relations can then be obtained by straightforward computations. $\Box$
Just like for $\mathfrak{sl}_2$, Section \[s223\] implies that the Grothendieck ring $R:=K_0(\CZres)$ for $\mathfrak g = \mathfrak{sl}_3,\: l=2,$ is isomorphic to the polynomial ring: $$\label{isocarG}
R \cong \mathbb{Z} \lbrack \lbrack L( Y_{1,0}) \rbrack, \lbrack L( Y_{1,2})\rbrack, \lbrack L( Y_{2,1})\rbrack, \lbrack L( Y_{2,3})\rbrack \rbrack.$$
Recall that for $\mathfrak g = \mathfrak{sl}_3$, the Grothendieck ring of the finite-dimensional representations of $\mathfrak g$ is isomorphic to the polynomial ring $\mathbb Z\lbrack \lbrack V(\varpi_1)\rbrack, \lbrack V(\varpi_2)\rbrack $. In this case, the simple modules are of the form $V(a_1\varpi_1+a_2\varpi_2)$ and can be written as polynomials in $\lbrack V(\varpi_1)\rbrack,\lbrack V(\varpi_2)\rbrack$, which creates a 2-parameter family of polynomials in 2 variables, denoted by $S_{a_1,a_2}(\lbrack V(\varpi_1)\rbrack,\lbrack V(\varpi_2)\rbrack)$. These polynomials can be computed inductively using the Littlewood-Richardson rule. The Frobenius pullback maps the class $\lbrack V(a_1\varpi_1+a_2\varpi_2)\rbrack $ to the class $\lbrack L(\mathbf{Y}_{1}^{a_1} \mathbf{Y}_{2}^{a_2})\rbrack$, which can then be written as the polynomial $S_{a_1,a_2}(\lbrack L(\mathbf{Y}_{1})\rbrack ,\lbrack L(\mathbf{Y}_{2})\rbrack)$.
Let $\overline{\mathcal{G}}$ be the generalised cluster algebra of type $G_2$, with $\mathbb{P}=\mathrm{Trop}(\lambda_1,\lambda_2)$, initial cluster variables $x_1,x_2$, exchange matrix $$B=\left( \begin{array}{cc}
0&3\\
-1&0
\end{array} \right),$$ and initial exchange polynomials $$\theta_1^0 (u,v)=u+v, \qquad \theta_{2}^0(u,v)= u^3 + \lambda_1 u^2 v + \lambda_2uv^2+ v^3.$$ There are eight cluster variables $x_1,\dots,x_8$, organised in eight clusters, as in Figure \[ex:g2\]. Note that the exchange polynomials are not affected by mutation.
$$\xymatrix @R=1.2pc @C=1pc @M=0.0pc {& (x_1,x_8)\ar@{-}[r]^{\mu_1} & (x_7,x_8)\ar@{-}[dr]^{\mu_2}&\\
(x_1,x_2)\ar@{-}[ur]^{\mu_2}\ar@{-}[d]^{\mu_1} &&&(x_7,x_6)\ar@{-}[d]^{\mu_1}\\
(x_3,x_2)\ar@{-}[dr]^{\mu_2}&&&(x_5,x_6)\ar@{-}[dl]^{\mu_2}\\
&(x_3,x_4)\ar@{-}[r]^{\mu_1} &(x_5,x_4)
}$$
Let $\mathcal G$ be the $\mathbb Z\lbrack \lambda_1,\lambda_2\rbrack$-subalgebra of the ambient field $\mathcal F$ generated by the cluster variables of $\overline{\mathcal{G}}$. As for $\mathcal{A}_n$, one can also check that $\mathcal G$ is isomorphic to a polynomial ring: $$\label{isoG}
\mathcal G \cong \mathbb Z \lbrack x_1,x_3,x_5,x_7\rbrack.$$ Denote by $\mathcal M_0$ the set of generalised cluster monomials of $\mathcal G$. Then the set $$\mathcal H := \{ S_{a_1,a_2}(\lambda_1,\lambda_2)\cdot m,\:a_1,a_2\in\mathbb N,\:m\in\mathcal{M}_0\}$$ is a $\mathbb Z$-basis of $\mathcal G$. Let us now exhibit a cluster structure on $R$.
\[thmA2l2\] For $l=2$ and $\mathfrak g = \mathfrak{sl}_3$, there exists a ring isomorphism $\eta :\mathcal G \rightarrow R$ such that $$\label{isoGcorr}
\begin{array}{ll}
\eta(x_1)= \lbrack L( Y_{1,0}) \rbrack & \eta(x_2)= \lbrack L( Y_{1,0}Y_{2,3}) \rbrack \\
\eta(x_3)= \lbrack L( Y_{2,3}) \rbrack & \eta(x_4)= \lbrack L( Y_{1,2}Y_{2,3}) \rbrack \\
\eta(x_5)= \lbrack L( Y_{1,2}) \rbrack & \eta(x_6)= \lbrack L( Y_{1,2}Y_{2,1}) \rbrack \\
\eta(x_7)= \lbrack L( Y_{2,1}) \rbrack & \eta(x_8)= \lbrack L( Y_{1,0}Y_{2,1}) \rbrack \\
\eta(\lambda_1)= \lbrack L(\mathbf{Y}_{1})\rbrack & \eta(\lambda_2)= \lbrack L(\mathbf{Y}_{2})\rbrack \\
\end{array}$$ The $\mathbb Z$-basis $\mathcal E\subset\mathcal G$ of monomials in $x_1,x_3,x_5,x_7$ is mapped by $\eta$ to the basis of classes of standard modules in $R$. Moreover, the $\mathbb Z$-basis $\mathcal H$ of generalised cluster monomials of $\mathcal G$ is mapped to the basis $B$ of classes of simple modules in $R$.
More precisely, the clusters can be organised as in Figure \[mod:g2\].
$\xymatrix @R=1pc @C=0.8pc @M=0.0pc { &\mbox{$\begin{array}{c}\lbrack L(Y_{1,0})\rbrack,\\\lbrack L(Y_{1,0}Y_{2,1})\rbrack\end{array} $} \ar@{-}[r]^{\mu_1}\ar@{-}[dl]^{\mu_2} & \mbox{$\begin{array}{c}\lbrack L(Y_{2,1})\rbrack,\\\lbrack L(Y_{1,0}Y_{2,1})\rbrack\end{array}$} \ar@{-}[dr]^{\mu_2} &\\
\mbox{$\begin{array}{c}\lbrack L(Y_{1,0})\rbrack,\\\lbrack L(Y_{1,0}Y_{2,3})\rbrack\end{array}$}\ar@{-}[d]^{\mu_1} &&&\mbox{$\begin{array}{c}\lbrack L(Y_{2,1})\rbrack,\\\lbrack L(Y_{1,2}Y_{2,1})\rbrack\end{array}$}\ar@{-}[d]^{\mu_1} \\
\mbox{$\begin{array}{c}\lbrack L(Y_{2,3})\rbrack,\\\lbrack L(Y_{1,0}Y_{2,3})\rbrack\end{array}$}\ar@{-}[dr]^{\mu_2}&&&\mbox{$\begin{array}{c}\lbrack L(Y_{1,2})\rbrack,\\\lbrack L(Y_{1,2}Y_{2,1})\rbrack\end{array}$}\ar@{-}[dl]^{\mu_2} \\
& \mbox{$\begin{array}{c}\lbrack L(Y_{2,3})\rbrack\\\lbrack L(Y_{1,2}Y_{2,3})\rbrack\end{array}$}\ar@{-}[r]^{\mu_1}& \mbox{$\begin{array}{c}\lbrack L(Y_{1,2})\rbrack\\\lbrack L(Y_{1,2}Y_{2,3})\rbrack\end{array}$}
}$
The proof is analogous to the $A_1$ case. The isomorphisms and allow us to fix a ring isomorphism $\eta$, which sends each cluster variable $x_1,x_3,x_5,x_7$ to the class of the corresponding fundamental module as written above in . Moreover, Proposition \[relGc\] and the definition of $\mathcal G$ imply that the cluster variables and their images under $\eta$ satisfy the same relations, where $\chi_\varepsilon(L(\mathbf{Y}_{i}))$ is replaced by $\lambda_i$. Therefore, we have $\eta(\lambda_i)= \lbrack L(\mathbf{Y}_{i}) \rbrack$ for $i=1,2$, and the correspondence between the basis $\mathcal E$ and the basis of classes of standard module immediately follows.
Moreover, Theorem \[decG\] and the description of the classes $\lbrack L(\mathbf{Y}_{1}^{a_1} \mathbf{Y}_{2}^{a_2})\rbrack$ as the polynomials $S_{a_1,a_2}(\lbrack L(\mathbf{Y}_{1})\rbrack ,\lbrack L(\mathbf{Y}_{2})\rbrack)$, imply that the basis of classes of simple modules in $R$ consists of elements of the form $$S_{a_1,a_2}(\lbrack L(\mathbf{Y}_{1})\rbrack ,\lbrack L(\mathbf{Y}_{2})\rbrack)\cdot M,$$ where $a_1,a_2\in\mathbb N$ and $M$ is the class of one of the eight tensor products described in Theorem \[decG\] (which corresponds to an element of the set $\mathcal M_0$ in $\mathcal G$). This allows us to conclude that the basis $\mathcal H$ of $\mathcal G$ is mapped to the basis $B$ of $K_0(\CZres)$. $\Box$
The case A2,l>2
------------------
Our various computations have led us to the following conjecture.
\[conjA2\] For $l\geq 2$, the Grothendieck ring of $\CZres$, is isomorphic to a generalised cluster algebra $\mathcal G_l$ of rank $2l-2$. An initial seed of $\mathcal G_l$ is given by the exchange matrix $$\left( \begin{array}{cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc}
0&-1&1&0&\dots&0&\dots&0& 0&0&0\\
1&0&-1&1&0&0 &\dots&0& 0&0&0 \\
-1&1&0&-1&1&0 &\ddots& 0&0&0&0 \\
0&-1&1&0&-1&1&0&\vdots&\vdots&\vdots&\vdots \\
0&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots& 0&0&0\\
0&\dots&0&-1&1&0&-1&1& 0&0&0\\
0&\dots&&0&-1&1&0&-1& 1&0&0\\
&&&0&0&-1&1&0& -1&1&0\\
\vdots&\ddots&&0&0&0&-1&1& 0&-2&3\\
&&&0&0&0&0&-1& 2&0&-3\\
0&&\dots&0&0&0&0&0& -1&1&0
\end{array} \right),$$ the cluster variables correspond to $$\begin{aligned}
&&x_{2k+1}= \lbrack L(Y_{1,0}Y_{1,2l-2}\dots Y_{1,2l-2k})\rbrack\quad(k\in\llbracket 0,l-2\rrbracket),\\&&x_{2k}=\lbrack L(Y_{2,2l-1}Y_{2,2l-3}\dots Y_{2,2l-2k+1})\rbrack\quad(k\in\llbracket 1,l-2\rrbracket),\\&&
x_{2l-2}=\lbrack L(Y_{1,0}Y_{1,2l-2}Y_{1,2l-4}\dots Y_{1,4}Y_{2,2l-1}Y_{2,2l-3}Y_{2,2l-5}\dots Y_{2,5}Y_{2,3})\rbrack.\end{aligned}$$ The coefficients are $\lambda_i=\lbrack L(\mathbf{Y}_{i})\rbrack,\:i=1,2,$ where $$\mathbf{Y}_1= Y_{1,0}Y_{1,2}\dots Y_{1,2l-2}\quad\mbox{and}\quad \mathbf{Y}_2=Y_{2,1}Y_{2,3}\dots Y_{2,2l-1}.$$
The initial exchange polynomials are $\theta_r^0 (u,v)=u+v$ for $r\in\llbracket 1,2l-3\rrbracket$, and $\theta_{2l-2}^0(u,v)= u^3 + \lambda_1 u^2 v + \lambda_2uv^2+ v^3$.
Moereover, the generalised cluster monomials are mapped to classes of simple modules.
For $l>2$, the above generalised cluster algebras are of infinite type, so we can only hope for an inclusion of the set of cluster monomials in the set of classes of simple modules $L(m)$ where $m$ is $l$-acyclic.
[7]{}
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L. Chekhov, M. Shapiro, *Teichmüller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables*, [International Mathematics Research Notices]{} **016** (2013) 22 pp., arXiv:1111.3963
E. Frenkel, E. Mukhin, *The $q$-characters at roots of unity*, [Adv. Math.]{} **171**, no. 1 (2002) 139-167
E. Frenkel, N. Reshetikhin, *The q-characters of representations of quantum affine algebras*, Recent developments in quantum affine algebras and related topics, Contemp. Math. **248** (1999) 163-205
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LMNO, CNRS UMR 6139, Université de Caen, 14032 Caen cedex, France\
email: [anne-sophie.gleitz@unicaen.fr](anne-sophie.gleitz@unicaen.fr)
|
Introduction {#sec:intro}
============
We study the Fibonacci substitution $\varphi$ given by $$\varphi:\quad 0\rightarrow\,01,\;1\rightarrow 0.$$ The infinite Fibonacci word ${w_{\rm F}}$ is the unique one-sided sequence (to the right) which is a fixed point of $\varphi$: $${w_{\rm F}}=0100101001\dots.$$ We also consider one of the two two-sided fixed points ${x_{\rm F}}$ of $\varphi^2$: $${x_{\rm F}}=\dots01001001\!\cdot\!0100101001\dots.$$ The dynamical system generated by taking the orbit closure of ${x_{\rm F}}$ under the shift map $\sigma$ is denoted by $(X_\varphi,\sigma)$.
The question we will be concerned with is: what are the substitutions $\eta$ which generate a symbolical dynamical system topologically isomorphic to the Fibonacci dynamical system? Here topologically isomorphic means that there exists a homeomorphism $\psi: X_\varphi\rightarrow X_\eta$, such that $\psi\sigma=\sigma\psi$, where we denote the shift on $X_\eta$ also by $\sigma$. In this case $(X_\eta, \sigma)$ is said to be conjugate to $(X_\varphi,\sigma)$.
This question has been completely answered for the case of constant length substitutions in the paper [@CDK]. It is remarkable that there are only finitely many injective primitive substitutions of length $L$ which generate a system conjugate to a given substitution of length $L$. Here a substitution $\alpha$ is called *injective* if $\alpha(a)\ne \alpha(b)$ for all letters $a$ and $b$ from the alphabet with $a\ne b$. When we extend to the class of all substitutions, replacing $L$ by the Perron-Frobenius eigenvalue of the incidence matrix of the substitution, then the conjugacy class can be infinite in general. See [@Dekking-TCS] for the case of the Thue-Morse substitution. In the present paper we will prove that there are infinitely many injective primitive substitutions with Perron-Frobenius eigenvalue $\Phi=(1+\sqrt{5})/2$ which generate a system conjugate to the Fibonacci system—see Theorem \[th:inf\].
In the non-constant length case some new phenomena appear. If one has an injective substitution $\alpha$ of constant length $L$, then all its powers $\alpha^n$ will also be injective. This is no longer true in the general case. For example, consider the injective substitution $\zeta$ on the alphabet $\{1,2,3,4,5\}$ given by $$\zeta: \qquad 1\rightarrow 12,\;
2\rightarrow 3,\;
3\rightarrow 45,\;
4\rightarrow 1,\;
5\rightarrow 23.$$ An application of Theorem \[th:Nblock\] followed by a partition reshaping (see Section \[sec:reshaping\]) shows that the system $(X_\zeta,\sigma)$ is conjugate to the Fibonacci system. However, the square of $\zeta$ is given by $$\zeta^2: \qquad 1\rightarrow 123,\;
2\rightarrow 45,\;
3\rightarrow 123,\;
4\rightarrow 12,\;
5\rightarrow 345,$$ which is *not* injective. To deal with this undesirable phenomenon we introduce the following notion. A substitution $\alpha$ is called a *full rank* substitution if its incidence matrix has full rank (non-zero determinant). This is a strengthening of injectivity, because obviously a substitution which is not injective can not have full rank. Moreover, if the substitution $\alpha$ has full rank, then all its powers $\alpha^n$ will also have full rank, and thus will be injective.
Another phenomenon, which does not exist in the constant length case, is that non-primitive substitutions $\zeta$ may generate uniquely defined minimal systems conjugate to a given system. For example, consider the injective substitution $\zeta$ on the alphabet $\{1,2,3,4\}$ given by $$\zeta:\qquad 1\rightarrow 12,\quad
2\rightarrow 31,\quad
3\rightarrow 4,\quad
4\rightarrow 3.$$ With the partition reshaping technique from Section \[sec:reshaping\] one can show that the system $(X_\zeta,\sigma)$ is conjugate to the Fibonacci system (ignoring the system on two points generated by $\zeta$). In the remainder of this paper we concentrate on primitive substitutions.
The structure of the paper is as follows. In Section \[sec:Nblock\] we show that all systems in the conjugacy class of the Fibonacci substitution can be obtained by letter-to-letter projections of the systems generated by so-called $N$-block substitutions. In Section \[sec:C3\] we give a very general characterization of symbolical dynamical systems in the Fibonacci conjugacy class, in the spirit of a similar result on the Toeplitz dynamical system in [@CKL08]. In Section \[sec:reshaping\] we introduce a tool which admits to turn non-injective substitutions into injective substitutions. This is used in Section \[sec:C1\] to show that the Fibonacci class has infinitely many primitive injective substitutions as members. In Section \[sec:two\] we quickly analyse the case of a 2-symbol alphabet. Sections \[sec:equi\] and \[sec:mat\] give properties of maximal equicontinuous factors and incidence matrices, which are used to analyse the 3-symbol case in Section \[sec:C2\]. In the final Section \[sec:L2L\] we show that the system obtained by doubling the 0’s in the infinite Fibonacci word is conjugate to the Fibonacci dynamical system, but can not be generated by a substitution.
$N$-block systems and $N$-block substitutions {#sec:Nblock}
=============================================
For any $N$ the $N$-block substitution $\hat{\theta}_N$ of a substitution $\theta$ is defined on an alphabet of $p_\theta(N)$ symbols, where $p_\theta(\cdot)$ is the complexity function of the language ${{\cal L}}_\theta$ of $\theta$ (cf. [@Queff p. 95]). What is *not* in [@Queff], is that this $N$-block substitution generates the $N$-block presentation of the system $(X_\theta,\sigma)$.
We denote the letters of the alphabet of the $N$-block presentation by $[a_1a_2\dots a_N]$, where $a_1a_2\dots a_N$ is an element from ${{\cal L}}_\theta^N$, the set of words of length $N$ in the language of $\theta$. The $N$-block presentation $(X^{[N]}_\theta,\sigma)$ emerges by applying an sliding block code $\Psi$ to the sequences of $X_\theta$, so $\Psi$ is the map\
$$\Psi(a_1a_2\dots a_N)=[a_1a_2\dots a_N].$$ We denote by $\psi$ the induced map from $X_\theta$ to $X^{[N]}_\theta$: $$\psi(x)=\dots\Psi(x_{-N},\dots,x_{-1})\Psi(x_{-N+1},\dots,x_{0})\dots.$$ It is easy to see that $\psi$ is a conjugacy, where the inverse is $\pi_0$ induced by the 1-block map (also denoted $\pi_0$) given by $\pi_0([a_1a_2\dots a_N])=a_1$.
The $N$-block substitution $\hat{\theta}_N$ is defined by requiring that for each word $a_1a_2\dots a_N$ the length of $\hat{\theta}_N([a_1a_2\dots a_N])$ is equal to the length $L_1$ of $\theta(a_1)$, and the letters of $\hat{\theta}_N([a_1a_2\dots a_N])$ are the $\Psi$-codings of the first $L_1$ consecutive $N$-blocks in $\theta(a_1a_2\dots a_N)$.
\[th:Nblock\] Let $\hat{\theta}_N$ be the $N$-block substitution of a primitive substitution $\theta$. Let $(X^{[N]}_\theta,\sigma)$ be the $N$-block presentation of the system $(X_\theta,\sigma)$. Then the set $X^{[N]}_\theta$ equals $X_{\hat{\theta}_N}$.
[*Proof:*]{} Let $x$ be a fixed point of $\theta$, and let $y=\psi(x)$, where $\psi$ is the $N$-block conjugacy, with inverse $\pi_0$. The key equation is $\pi_0\,\hat{\theta}_N=\theta\,\pi_0$. This implies\
$$\pi_0\,\hat{\theta}_N(y)=\theta\,\pi_0(y)=\theta\,\pi_0(\psi(x))=\theta(x)=x.$$
Applying $\psi$ on both sides gives $\hat{\theta}_N(y)=\psi(x)=y$, i.e., $y$ is a fixed point of $\hat{\theta}_N$. But then $X^{[N]}_\theta=X_{\hat{\theta}_N}$, by minimality of $X^{[N]}_\theta$. $\Box$
It is well known (see, e.g., [@Queff p. 105]) that $p_\varphi(N)=N+1$, so for the Fibonacci substitution $\varphi$ the $N$-block substitution $\hat{\varphi}_N$ is a substitution on an alphabet of $N+1$ symbols.
We describe how one obtains $\hat{\varphi}_2$. We have ${{\cal L}}_\varphi^2=\{00, 01, 10\}$. Since 00 and 01 start with 0, and 10 with 1, we obtain $$\hat{\varphi}_2:\quad [00]\mapsto [01][10],\;[01]\mapsto [01][10],\; [10]\mapsto [00],$$ reading off the consecutive 2-blocks from $\varphi(00)=0101,\, \varphi(01)=010$ and $\varphi(10)=001$. It is useful to recode the alphabet $\{[00],[01],[10]\}$ to the standard alphabet $\{1,2,3\}$. We do this in the order in which they appear for the first time in the infinite Fibonacci word ${w_{\rm F}}$— we call this the *canonical coding*, and will use the same principle for all $N$. For $N=2$ this gives $[01]\rightarrow 1,\; [10]\rightarrow 2,\; [00]\rightarrow 3$. Still using the notation $\hat{\varphi}_2$ for the substitution on this new alphabet, we obtain $$\hat{\varphi}_2(1)=12 \quad \hat{\varphi}_2(2)=3, \quad \hat{\varphi}_2(3)=12.$$ In this way the substitution is in standard form (cf. [@CDK] and [@Dekking-2016]).
The Fibonacci conjugacy class {#sec:C3}
=============================
Let $F_n$ for $n=1,2,\dots$ be the Fibonacci numbers $$F_1=1,\, F_2=1,\, F_3=2,\, F_4=3,\, F_5=5, \dots.$$
Let $(Y,\sigma)$ be any subshift. Then $(Y,\sigma)$ is topologically conjugate to the Fibonacci system $(X_\varphi,\sigma)$ if and only if there exist $n\ge 3 $ and two words $B_0$ and $B_1$ of length $F_n$ and $F_{n-1}$, such that any $y$ from $Y$ is a concatenation of $B_0$ and $B_1$, and moreover, if $\cdots B_{x_{-1}} B_{x_0} B_{x_1}\cdots B_{x_k}\cdots$ is such a concatenation, then $x=(x_k)$ is a sequence from the Fibonacci system.
*Proof:* First let us suppose that $(Y,\sigma)$ is topologically isomorphic to the Fibonacci system. By the Curtis-Hedlund-Lyndon theorem, there exists an integer $N$ such that $Y$ is obtained by a letter-to-letter projection $\pi$ from the $N$-block presentation $(X^{[N]}_\varphi, \sigma)$ of the Fibonacci system. Now if $B_0$ and $B_1$ are two decomposition blocks of sequences from $X^{[N]}_\varphi$ of length $F_n$ and $F_{n-1}$, then $\pi(B_0)$ and $\pi(B_1)$ are decomposition blocks of sequences from $Y$ with lengths $F_n$ and $F_{n-1}$, again satisfying the concatenation property. So it suffices to prove the result for $X^{[N]}_\varphi$. Note that we may suppose that the integers $N$ pass through an infinite subsequence; we will use $N=F_n$,where $n=3,4,\dots$. Useful to us are the *singular words* $w_n$ introduced in [@WenWen]. The $w_n$ are the unique words of length $F_{n+1}$ having a different Parikh vector from all the other words of length $F_{n+1}$ from the language of $\varphi$. Here $w_1=1, w_2=00$, $w_3=101$, and for $n\ge4$ $$w_n=w_{n-2}w_{n-3}w_{n-2}.$$ The set of return words of $w_n$ has only two elements which are $u_n=w_nw_{n+1}$ and $v_n=w_nw_{n-1}$ (see page 108 in [@HuangWen]). The lengths of these words are $|u_n|=F_{n+3}$ and $|v_n|=F_{n+2}$. Let $w_n^-$ be $w_n$ with the last letter deleted. Define for $n\ge5$ $$B_0=\Psi(u_{n-3}w_{n-3}^-), \quad B_1=\Psi(v_{n-3}w_{n-3}^-),$$ where $\Psi$ is the $N$-block code from ${{\cal L}}_\varphi^N$ to ${{\cal L}}_{\varphi^{[N]}}$, with $N=F_{n-2}$. Then these blocks have the right lengths, and by Theorem 2.11 in [@HuangWen], the two return words partition the infinite Fibonacci word ${w_{\rm F}}$ according to the infinite Fibonacci word—except for a prefix $r_{n,0}$: $${w_{\rm F}}=r_{n,0}u_nv_nu_nu_nv_nu_n\dots.$$ By minimality this property carries over to all two-sided sequences in the Fibonacci dynamical system.
For the converse, let $Y$ be a Fibonacci concatenation system as above. Let $C_0=\varphi^{n-2}(0)$ and $C_1=\varphi^{n-2}(1)$. We define a map $g$ from $(Y,\sigma)$ to a subshift of $\{0,1\}^{\mathbb{Z}}$ by $$g:\quad \cdots B_{x_{-1}} B_{x_0} B_{x_1}\cdots B_{x_k}\cdots\; \mapsto \; \cdots C_{x_{-1}} C_{x_0} C_{x_1}\cdots C_{x_k}\cdots,$$ respecting the position of the $0^{\rm th}$ coordinate. Since $|C_0|=|B_0|$ and $|C_1|=|B_1|$, $g$ commutes with the shift. Also, $g$ is obviously continuous. Moreover, since for any sequence $x$ in the Fibonacci system $\varphi^{n-2}(x)$ is again a sequence in the Fibonacci system, $g(Y)\subseteq X_\varphi$. So, by minimality, $(X_\varphi,\sigma)$ is a factor of $(Y,\sigma)$. Since $g$ is invertible, with continuous inverse, $(Y,\sigma)$ is in the conjugacy class of the Fibonacci system. $\Box$
[**Example**]{} The case $(F_n,F_{n-1})=(13,8)$. Then $n=7$, so we have to consider the singular word $w_4=00100$ of length 5.
The set of $5$-blocks is $\{01001,\,10010,\,00101,\,01010,\,10100,\,00100\}.$\
These will be coded by the canonical coding $\Psi$ to the standard alphabet $\{1,2,3,4,5,6\}$. Note that $\Psi(w_4)=6$. Further, $w_3=101$ and $w_5=10100101$. So $u_n=0010010100101$ and $v_n=00100101$. Applying $\Psi$ gives the two decomposition blocks $B_0 = 6123451234512$ and $B_1 = 61234512$.
Reshaping substitutions {#sec:reshaping}
=======================
We call a language preserving transformation of a substitution a reshaping. An example is the prefix-suffix change used in [@Dekking-TCS]. Here we consider a variation which we call a *partition reshaping*.
We give an example of this technique. Take the $N$-block representation of the Fibonacci system for $N=4$. All five 4-blocks occur consecutively at the beginning of the Fibonacci word ${w_{\rm F}}$ as $\{0100,\,1001,\,0010,\, 0101,\, 1010\}.$ The canonical coding to $\{1,2,3,4,5\}$ gives the 4-block substitution $\hat{\varphi}_4$: $$\hat{\varphi}_4:\qquad 1\rightarrow 12,\;
2\rightarrow 3,\;
3\rightarrow 45, \;
4\rightarrow 12,\;
5\rightarrow 3.$$
Its square is equal to $$\hat{\varphi}_4^2: \qquad 1\rightarrow 123,\;
2\rightarrow 45,\;
3\rightarrow 123,\;
4\rightarrow 123,\;
5\rightarrow 45.$$ Since the two blocks $B_0=123$ and $B_1=45$ have no letters in common this permits to do a partition reshaping. Symbolically this can be represented by
-------------- -------- -------------- -------------- --- -------------- --- -------------- --
\[.005cm\] 1 2 3 4 5
$\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$
1 2 3 4 5 1 2 3
1 2 $\|$ 3 4 $\|$ 5 1 $\|$ 2 3 $\|$
-------------- -------- -------------- -------------- --- -------------- --- -------------- --
: Partition reshaping.[]{data-label="tab:table1"}
Here the third line gives the images $\hat{\varphi}_4(B_0)=\hat{\varphi}_4(123)=12345$ and $\hat{\varphi}_4(B_1)=\hat{\varphi}_4(45)=123$; the fourth line gives a *another* partition of these two words in three, respectively two subwords from which the new substitution $\eta$ can be read of: $$\eta: \qquad 1\rightarrow 12,\;
2\rightarrow 34,\;
3\rightarrow 5 ,\;
4\rightarrow 1,\;
5\rightarrow 23.$$ What we gain is that the partition reshaped substitution $\eta$ generates the same language as $\hat{\varphi}_4$, but that $\eta$ is injective—it is even of full rank.
The Fibonacci class has infinite cardinality {#sec:C1}
============================================
\[th:inf\] There are infinitely many primitive injective substitutions with Perron-Frobenius eigenvalue the golden mean that generate dynamical systems topologically isomorphic to the Fibonacci system.
We will explicitly construct infinitely many primitive injective substitutions whose systems are topologically conjugate to the Fibonacci system. The topological conjugacy will follow from the fact that the systems are $N$-block codings of the Fibonacci system, where $N$ will run through the numbers $F_n-1$. As an introduction we look at $n=5$, i.e., we consider the blocks of length $N=F_5-1=4$. With the canonical coding of the $N$-blocks we obtain the 4-block substitution $\hat{\varphi}_4$—see Section \[sec:reshaping\]: $$\hat{\varphi}_4:\qquad 1\rightarrow 12,\,
2\rightarrow 3,\,
3\rightarrow 45, \,
4\rightarrow 12,\,\
5\rightarrow 3.$$
An *interval* $I$ starting with $a\in A$ is a word of length $L$ of the form $$I=a,a+1,...,a+L-1.$$
Note that $\hat{\varphi}_4(123)=12345$, and $\hat{\varphi}_4(45)=123$, and these four words are intervals.
This is a property that holds in general. First we need the fact that the first $F_n$ words of length $F_n-1$ in the fixed point of $\varphi$ are all different. This result is given by Theorem 2.8 in [@Chuan-Ho]. We code these $N+1$ words by the canonical coding to the letters $1,2,\dots,F_n$. We then have $$\label{eq:Fib}\hat{\varphi}_N(12...F_{n-1})=12\dots F_{n}, \qquad \hat{\varphi}_N(F_{n-1}\!+1,\dots F_n)=12\dots F_{n-1}.$$ This can be seen by noting that $\pi_0 \hat{\varphi}_N^n=\varphi^n \pi_0,$ for all $n$, and that the fixed point of $\varphi$ starts with $\varphi^{n-2}(0)\varphi^{n-3}(0)$.
We continue for $n \ge 5$ with the construction of a substitution $\eta=\eta_n$ which is a partition reshaping of $\hat{\varphi}_N$. The $F_n$ letters in the alphabet $A^{[N]}$ are divided in three species, S, M and L (for Small, Medium and Large). $${\rm S}:={1,...,F_{n-3}}, \quad {\rm M}:={F_{n-3}+1,...,F_{n-1}},\quad {\rm L}:={F_{n-1}\!+1,...,F_n}.$$ Note that ${\rm Card}\, {\rm M}=F_{n-1}-F_{n-3}=F_{n-2}=F_{n}-F_{n-1}={\rm Card}\, {\rm L}.$
An important role is played by $a_{ \rm M}:=F_{n-3}+1$, the smallest letter in M, and $a_{ \rm L}:=F_{n-1}+1$, the smallest letter in L.
For the letters in M (except for $a_{ \rm M})$ the rules are very simple: $$\eta(a)= a+F_{n-2}$$ (i.e., a single letter obtained by addition of the two integers). The first letter in M has the rule $$\eta(a_{ \rm M})=\eta(F_{n-3}\!+1)= F_{n-1}, F_{n-1}\!+1= F_{n-1},a_{ \rm L} .$$ The images of the letters in L are intervals of length 1 or 2, obtained from a partition of the word $12\dots F_{n-1}$. Their lengths are coming from $\varphi^{n-4}(0)$, rotated once (put the 0 in front at the back). This word is denoted $\rho(\varphi^{n-4 }(0))$. The choice of this word is somewhat arbitrary, other choices would work. The properties of $v:=\rho(\varphi^{n-4}(0))$ which matter to us are
(V1) $\ell:=|v|=F_{n-2}$.
(V2) $v_1=1$, $v_\ell=0$.
(V3) $v$ does not contain any 11.
Now the images of the letters in L are determined by $v$ according to the following rule: $|\eta(a_{ \rm L}+k-1)|=2-v_k$, for all $k=1,\dots,F_{n-2}$. Note that this implies in particular that for all $n\ge 5$ one has by property (V2) $$\eta(a_{\rm L})=\eta(F_{n-1}\!+1)= 1, \;\qquad \eta(F_n)= F_{n-1}-1,F_{n-1}.$$
The images of the letters in S are then obtained by choosing the lengths of the $\eta(a)$ in such a way that the largest common refinement of the induced partitions of the images of S and L is the singleton partition.
[**Example**]{} The case $n=7$, so $ F_n=13$, $ F_{n-1}=8$, and $ F_{n-2}=5$.
Then ${\rm S}=\{1,2,3\},\, {\rm M}=\{4,5,6,7,8\},\, {\rm L}=\{9,10,11,12,13\}.$
Rules for M: $4\rightarrow 89,\;5\rightarrow 10, \;6\rightarrow11, \;7\rightarrow 12, \;8\rightarrow13.$ Now $$\varphi^3(0)=01001\; \Rightarrow\; v= 10010\; \Rightarrow\; {\rm the\, partition\, is}\; 1|23|45|6|78.$$ This partition gives the following rules for L: $$9\rightarrow1,\; 10\rightarrow23,\; 11\rightarrow45,\; 12\rightarrow6,\; 13\rightarrow 78.$$ The induced partition for the images of the letters in S is $|12|34|567|8$, yielding rules $$1\rightarrow12,\; 2\rightarrow34,\; 3\rightarrow567.$$
In summary we obtain the substitution $\eta=\eta_7$ given by : $$\begin{aligned}
{\rm S}:
\begin{cases}
1& \rightarrow 1,2\\
2& \rightarrow 3,4\\
3& \rightarrow 5,6,7
\end{cases}
\qquad {\rm M}:
\begin{cases}
4& \rightarrow 8,9\\
5& \rightarrow 10\\
6& \rightarrow 11\\
7& \rightarrow 12\\
8& \rightarrow 13
\end{cases}
\qquad {\rm L}:
\begin{cases}
\,\,9\!\!& \rightarrow 1\\
10& \rightarrow 2,3\\
11& \rightarrow 4,5\\
12& \rightarrow 6\\
13& \rightarrow 7,8.
\end{cases}\end{aligned}$$ The substitution $\eta$ is primitive because you ‘can go’ from the letter 1 to any letter and from any letter to the letter 1. This gives irreducibility; there is primitivity because periodicity is impossible by the first rule $1\rightarrow 1,2$.
The substitution $\eta$ has full rank because any unit vector $$e_a=(0,\dots,0,1,0,\dots,0)$$ is a linear combination of rows of the incidence matrix $M_\eta$ of $\eta$. For $a\in {\rm L}\setminus\{9\}$ this combination is trivial, and for the other letters this is exactly forced by the choice of lengths in such a way that the largest common refinement of the induced partitions of the images of S and L is the singleton partition. In more detail: denote the $a^{\rm th}$ row of $M_\eta$ by $R_a$. Then $e_1=R_9$, and thus $e_2=R_1-R_9$, $e_3=R_{10}-e_2=R_{10}-R_1+R_9$, etc.
The argument yielding the property of full rank will hold in general for all $n\ge5$. To prove primitivity for all $n$ we need some more details.
The substitution $\eta=\eta_n$ is primitive for all $n \ge 5$.
*Proof:* The proposition will be proved if we show that for all $a\in A$ the letter $a$ will occur in some iteration $\eta^k(1)$, and conversely, that for all $a\in A$ the letter $1$ will occur in some iteration $\eta^k(a)$. The first part is easy to see from the fact that $\eta(1)=1,2$ and that $\eta^2(1,\dots,F_{n-2})=1,\dots,F_n-1$, plus $\eta^2(a_{\rm M})=F_n,1$. For the second part, we show that A) for any $a\in$ M$\cup$L a letter from S will occur in $\eta^k(a)$ in $k\le$ Card M$\cup$L steps (see Lemma \[lem:dec\]) and B), that for any $a\in$ S the letter 1 will occur in $\eta^k(a)$ in $k\le$ 2Card $A$ steps (see Lemma \[lem:occ1\]). $\Box$
\[lem:dec\] Let $f:A\rightarrow A$ be the map that assigns the first letter of $\eta^2(a)$ to $a$. Then $f$ is strictly decreasing on L $\cup$ M$\backslash \{a_{\rm M}\}$.
*Proof:* First we consider $f$ on ${\rm L}$. We have $$\eta^2(a_{\rm L}\dots F_n)=\eta(1,\dots, F_{n-1}-1, F_{n-1})=1\dots F_n.$$ Since $$\eta^2(F_n)=\eta( F_{n-1}-1,F_{n-1})=F_{n-1}-1+F_{n-2},F_{n-1}+F_{n-2}=F_n-1, F_n,$$ we obtain $f(F_n)=F_n-1<F_n$. This implies that also the previous letters in ${\rm L}$ are mapped by $f$ to a smaller letter.
Next we consider $f$ on M$\backslash \{a_{\rm M}\}$. Here $$\eta^2(a_{\rm M}+1,\dots, F_{n-1})=\eta(a_{\rm L}+1,\dots, F_{n})=2,3,\dots, F_{n-1}.$$ Now $$\eta^2(F_{n-1})=\eta( F_{n})=F_{n-1}-1,F_{n-1}.$$ So we obtain $f(F_{n-1})=F_{n-1}-1<F_{n-1}$. This implies that also the previous letters in ${\rm M}$ are mapped by $f$ to a smaller letter.$\Box$
\[lem:occ1\] For all $a\in S$ there exists $k \le 2\,{\rm Card}\, A$ such that the letter 1 occurs in $\eta^{k}(a)$.
*Proof:* The substitution $\eta^2$ maps intervals $I$ to intervals $\eta^2(I)$, provided $I$ does not contain $a_{\rm M}$ or $a_{\rm L}$. By construction, since the $\eta(b)$ for $b\in {\rm L}$ have length 1 or 2, the length of $\eta(a)$ for $a\in {\rm S}$ is 2 or 3, and so $\eta(a)$ contains a word $c, c+1$ for some $c\in A$. Since $\rho\varphi^{(n-4)}(0)$ does not contain two consecutive 1’s (property (V3)), the image $\eta^2(c,c+1)$ has at least length 3. Since[^1] any word of length at least 3 in the language of $\eta$ contains an interval of length 2, the length increases by at least 1 if you apply $\eta^2$. It follows that for all $n\ge 5$ and all $a\in {\mathrm S}$ one has $|\eta^{2n+1}(a)| \ge n+2$. But then after less than ${\rm Card}\, A$ steps a letter $a_{\rm M}$ or a letter $a_{\rm L}$ must occur in $\eta^{2n+1}(a)$. This implies that the letter 1 occurs in $\eta^{2n+3}(a)$, since both $\eta^2(a_{\rm M})$ and $\eta^2(a_{\rm L})$ contain a 1.$\Box$
The 2-symbol case {#sec:two}
=================
The eigenvalue group of the Fibonacci system is the rotation over the small golden mean $\gamma=(\sqrt{5}-1)/2$ on the unit circle, and any system topologically isomorphic to the Fibonacci system must have an incidence matrix with Perron Frobenius eigenvalue the golden mean or a power of the golden mean (cf. [@Pytheas Section 7.3.2]). Thus, modulo a permutation of the symbols, on an alphabet of two symbols the incidence matrix with Perron-Frobenius eigenvalue the golden mean has to be $\left( \begin{smallmatrix} 1 \, 1\\ 1\, 0 \end{smallmatrix} \right).$ There are two substitutions with this incidence matrix: Fibonacci $\varphi$, and reverse Fibonacci ${{\varphi_{\textsc{\tiny R}}}}$, defined by $${{\varphi_{\textsc{\tiny R}}}}: \qquad 0\rightarrow\,10,\;1\rightarrow 0.$$ These two substitutions are essentially different, as they have different standard forms (see [@Dekking-2016] for the definition of standard form).
However, it follows directly from Tan Bo’s criterion in his paper [@Tan] that ${{\varphi_{\textsc{\tiny R}}}}$ and $\varphi$ have the same language[^2], but then they also generate the same system. Conclusion: the conjugacy class of Fibonacci with Perron-Frobenius eigenvalue the golden mean restricted to two symbols consists of Fibonacci and reverse Fibonacci.
Maximal equicontinuous factors {#sec:equi}
==============================
Let $T$ be the mapping from the unit circle $Z$ to itself defined by $Tz=z+\gamma \mod 1$, where $\gamma$ is the small golden mean. This, being an irrational rotation, is indeed an equicontinuous dynamical system – the usual distance metric is an invariant metric under the mapping. The factor map from the Fibonacci dynamical system $(X_\varphi,\sigma)$ to $(Z,T)$ is given by requiring that the cylinder sets $\{x:x_0=0\}$ and $\{x:x_0=1\}$ are mapped to the intervals $[0,\gamma]$ and $[\gamma,1]$ respectively, and requiring equivariance. If we take any point of $Z$ not of the form $n\gamma \mod 1$ ($n$ any integer), then the corresponding sequence is unique. If, however, we use an element in the orbit of $\gamma$, then for this point there will be two codes, a “left" one and a “right" one.
We want to understand more generally why two or more points map to a single point. Suppose $x$ and $y$ are two points of a system $(X,\sigma)$ that map to two points $x'$ and $y'$ in an equicontinuous factor. Then for any power of $T$ (the map of the factor system) the distance between $T^n(x')$ and $T^n(y')$ is just equal to the distance between $x'$ and $y'$. So $x$ and $y$ map to the same point $x'$ if either all $x_n$ and $y_n$ are equal for sufficiently large $n$, or all $x_n$ and $y_n$ are equal for sufficiently large $-n$. We say that $x$ and $y$ are respectively *right asymptotic* or *left asymptotic*
A pair of letters $(b,a)$ is called a *cyclic pair* of a substitution $\alpha$ if $ba$ is an element of the language of $\alpha$, and for some integer $m$ $$\alpha^m(b)=\dots b \quad{\rm and}\quad \alpha^m(a)=a\dots.$$ Such a pair gives an infinite sequence of words $\alpha^{mk}(ba)$ in the language of $\alpha$, which—if properly centered—converge to an infinite word which is a fixed point of $\alpha^m$. With a slight abuse of notation we denote this word by $\alpha^{\infty}(b)\cdot \alpha^{\infty}(a)$.
For the Fibonacci substitution $\varphi$, $(0,0)$ and $(1,0)$ are cyclic pairs, and the two synchronized points $\varphi^\infty(0)\cdot\varphi^\infty(0)$ and $\varphi^\infty(1)\cdot\varphi^\infty(0)$, are right asymptotic so they map to the same point in the equicontinuous factor.
Because of these considerations we now define $Z$-triples. Let $\eta$ be a primitive substitution. Call three points $x$, $y$, and $z$ in $X_\eta$ a $Z$-*triple* if they are generated by three cyclic pairs of the form $(b,a),\, (b,d)$ and $(c,d)$, where $a,b,c,d \in A$. Then $x$, $y$, and $z$ are mapped to the same point in the maximal equicontinuous factor.
\[th:Zth\] Let $(X_\eta,\sigma)$ be any substitution dynamical system topologically isomorphic to the Fibonacci dynamical system. Then there do not exist $Z$-triples in $X_\eta$.
*Proof:* Since $(X_\eta,\sigma)$ is topologically isomorphic to $(X_\varphi,\sigma)$, its maximal equicontinuous factor is $(Z,T$), and the factor map is at most 2-to-1. Suppose $(b,a),\, (b,d)$ and $(c,d)$ gives a $Z$-triple $x,y,z$ in $X_\eta$. Noting that $$x=\eta^\infty(b)\cdot\eta^\infty(a), \quad y=\eta^\infty(b)\cdot \eta^\infty(d)$$ are left asymptotic, and $y=\eta^\infty(b)\cdot \eta^\infty(d)$ and $z=\eta^\infty(c)\cdot\eta^\infty(d)$ are right asymptotic, this would give a contradiction. $\Box$
[**Example**]{} Let $\eta$ be the substitution given by $$\eta:\qquad 1\rightarrow 12,\,
2\rightarrow 34,\,
3\rightarrow 5 ,\,
4\rightarrow 1,\,
5\rightarrow 23.$$ Then $\eta$ generates a system that is topologically isomorphic to the Fibonacci system ($\eta$ is the substitution at the end of Section \[sec:reshaping\]). Quite remarkably, $\eta^6$ admits 5 fixed points generated by the cyclic pairs $(1, 2),\, (2, 3),\, (3, 1),\, (4, 5)$ and $(5, 1)$. Note however, that no three of these form a $Z$-triple.
Fibonacci matrices {#sec:mat}
==================
Let $\mathcal{F}_r$ be the set of all non-negative primitive $r\times r$ integer matrices, with Perron-Frobenius eigenvalue the golden mean $\Phi = (1+\sqrt{5})/2$.\
We have seen already that $\mathcal{F}_2$ consists of the single matrix $\left( \begin{smallmatrix} 1 \, 1\\ 1\, 0 \end{smallmatrix} \right).$
\[th:F3\] The class $\mathcal{F}_3$ essentially consists of the three matrices
$ \left( \begin{smallmatrix} 0\, 1\, 0\\ 1\, 0\, 1\\ 1\, 1\, 0 \end{smallmatrix} \right),\;
\left( \begin{smallmatrix} 0\, 1\, 0\\ 0\, 0 \,1 \\ 1\, 2\,0 \end{smallmatrix} \right),\;
\left( \begin{smallmatrix} 0\, 1\, 0\\ 1 \,0\, 1\\ 1\, 0\, 1 \end{smallmatrix} \right).$
Here essentially means that in each class of 6 matrices corresponding to the permutations of the $r=3$ symbols, one representing member has been chosen (actually corresponding to the smallest standard form of the substitutions having that matrix).
*Proof:* Let $M$ be a non-negative primitive $3\times 3$ integer matrix, with Perron-Frobenius eigenvalue the golden mean $\Phi = (1+\sqrt{5})/2$. We write\
\[-0.7cm\]
$$M= \left( \begin{matrix} a\; b\; c\\ d\; e\; f\\ g\; h\; i \end{matrix} \right).$$ The characteristic polynomial of $M$ is $\chi_M(u)=u^3-Tu^2+Fu-D,$ where $T=a+e+i$ is the trace of $M$, and $$\label{eq:FandD}
F=ae+ai+ei-bd-cg-fh,\quad
D=aei+bfg+cdh-afh-bdi-ceg.\quad$$ Of course $D$ is the determinant of $M$. Since $\Phi$ is an eigenvalue of $M$, and we consider matrices over the integers, $u^2-u-1$ has to be a factor of $\chi_M$. Performing the division we obtain $$\chi_M(u)=\big(u-(T-1)\big)\big(u^2-u-1\big),$$ and requiring that the remainder vanishes, yields $$\label{eq:DF}
F=T-2,\quad D=1-T.$$ Note that the third eigenvalue equals $\lambda_3=T-1$. From the Perron-Frobenius theorem follows that this has to be smaller than $\Phi$ in absolute value, and since it is an integer, only $\lambda_3=-1, 0, 1$ are possible. Thus there are only 3 possible values for the trace of $M$: $T=0,\, T=1$ and $T=2$.
The smallest row sum of $M$ has to be smaller than the PF-eigenvalue $\Phi$ (well known property of primitive non-negative matrices). Therefore $M$ has to have one of the rows $(0,0,1)$, $(0,1,0)$ or $(0,0,1)$. Also, because of primitivity of $M$, the 1 in this row can not be on the diagonal. By performing permutation conjugacies of the matrix we may then assume that $M$ has the form $$M= \left( \begin{matrix} 0\;\; 1\;\; 0\\ d\;\; e\;\; f\\ g\;\; h\;\; i \end{matrix} \right).$$ The equation combined with then simplifies to $$\label{eq:DF2}
T-2=F=ei-d-fh, \quad
1-T=D=fg-di.$$
[**Case $\mathbf{{\emph T}=0}$**]{}
In this case $e=i=0$, so simplifies to $$\label{eq:T0F}
-2=F=-d-fh, \quad 1=D=fg.$$ Then $f=g=1$, and so $d+h=2$. This gives three possibilities leading to the matrices $ \left( \begin{smallmatrix} 0\, 1\, 0\\ 1\, 0\, 1\\ 1\, 1\, 0 \end{smallmatrix} \right),\;
\left( \begin{smallmatrix} 0\, 1\, 0\\ 0\, 0 \,1 \\ 1\, 2\,0 \end{smallmatrix} \right),\;
\left( \begin{smallmatrix} 0\, 1\, 0\\ 2 \,0\, 1\\ 1\, 0\, 0 \end{smallmatrix} \right).$ Here the third matrix is permutation conjugate to the second one.
[**Case $\mathbf{{\emph T}=1}$**]{}
In this case $e=1, i=0$, or $e=0, i=1$.
First case: $e=1, i=0$. Now simplifies to $$\label{eq:T1F}
-1=F=-d-fh, \quad 0=D=fg.$$ Then $g=0$, since $f=0$ is not possible because of primitivity. But $g=0$ also contradicts primitivity, as $d+fh=1$, gives either $d=0$ or $h=0$.
Second case: $e=0, i=1$. Now simplifies to $$\label{eq:T1F2}
-1=F=-d-fh, \quad 0=D=fg-d.$$ Then $d=0$ would imply that $f=h=1$. But, as $g>0$ because of primitivity, we get a contradiction with $fg=d=0$.
On the other hand, if $d>0$, then $d=1$ and $f=0$ or $h=0$. But $fg=d=1$ gives $f=g=1$, so $h=0$, and we obtain the matrix $ \left( \begin{smallmatrix} 0\, 1\, 0\\ 1\, 0\, 1\\ 1\, 0\, 1 \end{smallmatrix} \right).$
[**Case $\mathbf{{\emph T}=2}$**]{}
In this case becomes $$\label{eq:T1F}
0=F=ei-d-fh, \quad -1=D=fg-di.$$ Since $ei=0$ would lead to $f=0$, which is not allowed by primitivity, what remains is $e=1,i=1$. Then, substituting $d=fg+1$ in the first equation gives $0=f(g+h)$. But both $f=0$ and $g=h=0$ contradict primitivity.
Final conclusion: there are three matrices in $\mathcal{F}_3$, modulo permutation conjugacies. $\Box$
[**Remark**]{} It is well-known that the PF-eigenvalue lies between the smallest and the largest row sum of the matrix. One might wonder whether this largest row sum is bounded for the class $\mathcal{F}=\cup_r\mathcal{F}_r$. Actually the class $\mathcal{F}_r$ contains matrices with some row sum equal to $r-1$ for all $r\ge 3$: take the matrix $M$ with $M_{1,j}=1$ for $j=2,\dots,r$, $M_{2,2}=1$ and $M_{i,{i+1}}=1$, for $i=2,\dots,r-1$, $M_{r,1}=1$ and all other entries 0.
Now note that $(1, \Phi,...,\Phi)$ is a left eigenvector of $M$ with eigenvalue $\Phi$ (since $\Phi^2=1+\Phi$). Since the eigenvector has all entries positive, it must be a PF-eigenvector (well known property of primitive, non-negative matrices), and hence $M$ is in $\mathcal{F}_r$.
The 3-symbol case {#sec:C2}
=================
There are two primitive injective substitutions $\eta$ and $\zeta$ on a three letter alphabet $\{a,b,c\}$ that generate dynamical systems topologically isomorphic to the Fibonacci system. These are given[^3] by\
$$\eta(a)=b,\,\eta(b)=ca,\, \eta(c)=ba,\quad \zeta(a)=b,\,\zeta(b)=ac,\, \zeta(c)=ab.$$
*Proof:* The possible matrices for candidate substitutions are given in Theorem \[th:F3\]. Let us consider the first matrix $ \left( \begin{smallmatrix} 0\, 1\, 0\\ 1\, 0\, 1\\ 1\, 1\, 0 \end{smallmatrix} \right)$. There are four substitutions with this matrix as incidence matrix: $$\begin{aligned}
\eta_1: \; a& \rightarrow b,\, b\rightarrow ca,\,c\rightarrow ba,& \eta_2: \; a \rightarrow b,\, b\rightarrow ca,\,c\rightarrow ab,\\
\eta_3: \; a& \rightarrow b,\, b\rightarrow ac,\,c\rightarrow ba,& \eta_4: \; a \rightarrow b,\, b\rightarrow ac,\,c\rightarrow ab,\end{aligned}$$ Here $\eta_1=\eta$. To prove that the system of $\eta$ is conjugate to the Fibonacci system consider the letter-to-letter map $\pi$ given by $$\pi(a)=1,\quad \pi(b)=\pi(c)=0.$$ Then $\pi$ maps $X_\eta$ onto $X_\varphi$, because $\pi\eta=\varphi\pi$. Moreover, $\pi$ is a conjugacy, since if $x\ne y$ and $\pi(x)=\pi(y)$, then there is a $k$ such that $x_k=b$ and $y_k=c$. But the words of length 2 in the language of $\eta$ are $ab, ba, bc$ and $ca$, implying that $x_{k-1}=a$ and $y_{k-1}=b$, contradicting $\pi(x)=\pi(y)$.
Since $\zeta$ is the time reversal of $\eta$, and we know already that the system of ${{\varphi_{\textsc{\tiny R}}}}$ is conjugate to the Fibonacci system, the system generated by $\eta_4=\zeta={{\eta_{\textsc{\tiny R}}}}$ is conjugate to the Fibonacci system.
It remains to prove that $\eta_2$ and $\eta_3$ generate systems that are *not* conjugate to the Fibonacci system. Again, since $\eta_3$ is the time reversal of $\eta_2$, it suffices to do this for $\eta_2$. The language of $\eta_2$ contains the words $ab, bb$ and $bc$. These words generate fixed points of $\eta_2^6$ in the usual way. But these three fixed points form a $Z$-triple, implying that the system of $\eta_2$ can not be topologically isomorphic to the Fibonacci system (see Theorem \[th:Zth\]).
The next matrix we have to consider is $\left( \begin{smallmatrix} 0\, 1\, 0\\ 0\, 0 \,1 \\ 1\, 2\,0 \end{smallmatrix} \right).$ There are three substitutions with this matrix as incidence matrix: $$\begin{aligned}
\eta_1: \; a& \rightarrow b,\, b\rightarrow c,\,c\rightarrow abb,& \eta_2: \; a \rightarrow b,\, b\rightarrow c,\,c\rightarrow bab,\\
\eta_3: \; a& \rightarrow b,\, b\rightarrow c,\,c\rightarrow bba.\end{aligned}$$ Again, the system of $\eta_1$ contains a $Z$-triple generated by $ab, bb$ and $bc$. So this system is not conjugate to the Fibonacci system, and neither is the one generated by $\eta_3$ (time reversal of $\eta_1$). The system generated by $\eta_2$ behaves similarly to the Fibonacci system, *but* is has an eigenvalue $-1$ (it has a two-point factor via the projection $a,c\rightarrow 0, b\rightarrow 1$.)
Finally, we have to consider the matrix $\left( \begin{smallmatrix} 0\, 1\, 0\\ 1 \,0\, 1\\ 1\, 0\, 1 \end{smallmatrix} \right).$ There are four substitutions with this matrix as incidence matrix: $$\begin{aligned}
\eta_1: \; a& \rightarrow b,\, b\rightarrow ac,\,c\rightarrow ac,& \eta_2: \; a \rightarrow b,\, b\rightarrow ac,\,c\rightarrow ca,\\
\eta_3: \; a& \rightarrow b,\, b\rightarrow ca,\,c\rightarrow ac,& \eta_4: \; a \rightarrow b,\, b\rightarrow ca,\,c\rightarrow ca.\end{aligned}$$ Here $\eta_1$ and $\eta_4$ generate systems conjugate to the Fibonacci system, but the substitutions are not injective. The substitution $\eta_2$ has all 9 words of length 2 in its language, and all of these generate fixed points of $\eta_2^6$. So the system of $\eta_2$ is certainly not topologically isomorphic to the Fibonacci system. The proof is finished, since $\eta_3$ is the time reversal of $\eta_2$. $\Box$
Letter-to-letter maps {#sec:L2L}
=====================
By the Curtis-Hedlund-Lyndon theorem all members in the conjugacy class of the Fibonacci system can be obtained by applying letter-to-letter maps $\pi$ to $N$-block presentations $(X^{[N]},\sigma)$. Here we analyse the case $N=2$. The 2-block presentation of the Fibonacci system is generated by (see Section \[sec:Nblock\]) the 2-block substitution $$\hat{\varphi}_2(1)=12 \quad \hat{\varphi}_2(2)=3, \quad \hat{\varphi}_2(3)=12.$$ There are (modulo permutations of the symbols) three letter-to-letter maps from $\{1,2,3\}$ to $\{0,1\}$. Two of these project onto the Fibonacci system, as they are projections on the first respectively the second letter of the 2-blocks. The third is $\pi$ given by $$\pi(1)=0,\quad \pi(2)=0, \quad \pi(3)=1.$$ What is the system $(Y,\sigma)$ with $Y=\pi\big(X^{[2]}\big)$?
First note that $(Y,\sigma)$ is conjugate to the Fibonacci system since $\pi$ is clearly invertible. Secondly, we note that the points in $Y$ can be obtained by doubling the 0’s in the points of the Fibonacci system. This holds because $\pi(12)=00,\, \pi(3)=1$, but also $$\pi(\hat{\varphi}_2(12))=\pi(123)=001,\;\pi(\hat{\varphi}_2(3))=\pi(12)=00.$$ Thirdly, we claim that the system $(Y,\sigma)$ cannot be generated by a substitution. This follows from the fact that the sequences in $Y$ contain the word 0000, but no other fourth powers. This is implied by the $4^{\rm th}$ power free-ness of the Fibonacci word, proved in [@Karhumaki].
A fourth property is that the sequence $y^+$ obtained by doubling the 0’s in ${w_{\rm F}}$, where ${w_{\rm F}}$ is the infinite Fibonacci word is given by $$y^+_n=[(n+2)\Phi]-[n\Phi]-[2\Phi], \qquad {\rm for\;} n\ge 1,$$ according to [@Wolfdieter], and [@OEIS-Fib] (here $[\cdot]$ denotes the floor function).
Finally we remark that Durand shows in the paper [@Durand] that the Fibonacci system is prime *modulo topological isomorphism*, and ignoring finite factors and rotation factors. This implies that all the projections are automatically invertible, if the projected system is not finite.
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[^1]: This follows from the fact that any word in the language of $\eta$ occurs in some concatenation of the two words $12\dots F_{n}$ and $12\dots F_{n-1}$.
[^2]: This follows also directly from the well-known formula ${{\varphi_{\textsc{\tiny R}}}}^{\!2n}(0)\,10=01\,\varphi^{2n}(0)$ for all $n\ge1$ (see [@Berstel p.17]).
[^3]: Standard forms: replace $a,b,c$ by $1,2,3$.
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abstract: 'Zigzag phosphorene nanoribbons are metallic owing to the edge states, whose energies are inside the gap and far from the bulk bands. We show that – through electrical manipulation of edge states – electron propagation can be restricted to one of the ribbon edges or, in case of bilayer phosphorene nanoribbons, to one of the layers. This finding implies that edge and layer can be regarded as tunable equivalents of the spin-one-half degree of freedom i.e., the pseudospin. In both layer- and edge-pseudospin schemes, we propose and characterize a pseudospin field-effect transistor, which can generate pseudospin-polarized current. Also, we propose edge- and layer-pseudospin valves that operate analogously to conventional spin valves. The performance of valves in each pseudospin scheme is benchmarked by the pseudomagnetoresistance (PMR) ratio. The edge-pseudospin valve shows a nearly perfect PMR, with remarkable robustness against device parameters and disorder. These results may initiate new developments in pseudospin electronics.'
author:
- 'S. Soleimanikahnoj'
- 'I. Knezevic'
title: Pseudospin Electronics in Phosphorene Nanoribbons
---
Introduction
============
The internal degrees of freedom of electrons in nanostructures are an important focal point in modern condensed matter physics. In cases where these degrees of freedom are tunable by an external field, they can be employed in electronic devices for digital information processing [@ney2003programmable; @yuasa2004giant]. The most prominent example is electron spin. As spin couples to magnetic fields, it can be harnessed for spin electronics and quantum information applications [@wolf2001spintronics; @vzutic2004spintronics; @awschalom2007challenges; @kane1998silicon; @gershenfeld1997bulk]. Analogous applications can also be realized by exploiting other discrete degrees of freedom, referred to as the pseudospin. In materials such as multilayer graphene [@novoselov2004electric] and transition-metal dichalcogenides (TMDs), [@mak2010atomically] both the layer [@san2009pseudospin; @min2008pseudospin; @pesin2012spintronics; @xu2014spin] and valley [@gunawan2006valley; @rycerz2007valley; @xiao2007valley; @kim2014ultrafast; @gong2013magnetoelectric] pseudospin degrees of freedom are present. In TMDs, the layer pseudospin is controlled by electrical polarization, while the spin and valley degrees of freedom can also be tuned by magnetic and optical means [@gong2013magnetoelectric]. Moreover, in TMDs, the strong coupling of layer and valley degrees of freedom offers a convenient platform for reliable spintronics implemented in two dimensions. Unfortunately, these features tend to disappear upon tailoring the material into a ribbon [@botello2009metallic], which limits the system’s possible applications in on-chip electronics. As a result, the quest for finding a practical, adjustable pseudospin degree of freedom in sub-two-dimensional nanostructures continues.
In this paper, we show that zigzag phosphorene nanoribbons (ZPNRs) [@liu2014phosphorene; @lu2014plasma] provide a platform for pseudospin electronics. In particular, charge transport in bilayer ZPNRs can be limited to one of the layers by applying a perpendicular electric field. This gives rise to the concepts of “up” or “down” pseudospin when charge transport takes place in the top or bottom layer, respectively. Analogously, applying an in-plane electric field across either single layer or bilayer ZPNRs restricts charge transport to the “left” or “right” edge of the ribbon, where electrons are considered to have “left” or “right” pseudospin polarization, respectively. In both cases, the pseudospin can be flipped by changing the sign of the applied electric field. In each case, we introduce nonmagnetic analogues of the spin field-effect transistor and spin valve, in which the role of the magnetic field is replaced by an electric field. For the pseudospin valves, a nonmagnetic counterpart of the magnetoresistance ratio, called the pseudomagnetoresistance ratio (PMR), is introduced and a large value is obtained at room temperature for both edge- and layer-pseudospin valve. Furthermore, we evaluate how the PMR is affected by the geometry of pseudospin-valves and presence of disorder. In particular, the PMR in the layer-pseudospin scheme is sensitive to the length of the valve and strong disorder, while in edge-pseudospin scheme is completely robust. These findings show that phosphorene nanoribbons provide key features necessary for pseudospin electronics that is compatible with current nanotechnology.
Pseudospin Field-Effect Transistor
==================================
![\[fig1\] Schematic of a bilayer ZPNR. (a) Top view: The left and the right edges are zigzag. The gray rectangle denotes a unit cell for this ZPNR. (b) Side view: $a = 3.31$ $\mathrm{\AA}$ is the length of the unit cell.](Figure1.pdf){width=".9\columnwidth"}
![\[fig2\] (a) Band structure of unbiased bilayer ZPNR. (b) The on-site probability density for the states with energies marked by “x” in panel (a). The red circles denote the probability density, with a larger circle diameter corresponding to higher probability density.](Figure2.pdf){width=".7\columnwidth"}
The crystal structure of a bilayer ZPNR is presented in Fig. \[fig1\]. A unit cell is denoted by the solid gray rectangle. The length of the unit cell (a) is $3.31$ [Å]{}. The band structure of a ZPNR, calculated using the fifteen-nearest-neighbors tight-binding Hamiltonian [@rudenko2015toward], is shown in Fig. \[fig2\](a). The width of the ribbon is chosen to be $5$ nm. In this figure, one can see the presence of midgap bands (in red), disconnected from the bulklike bands (in blue). Each midgap band is twofold degenerate; therefore, a total of four midgap bands are present in the case of a bilayer ZPNR. The Fermi level (dashed line) passes through the midgap bands, but is energetically far from the bulklike bands. Consequently, the ribbon is metallic, with low-field charge transport solely conducted through the states associated with the midgap bands. The probability density for the states associated with the four midgap bands at the zone center (wave number $k=0$), marked with $1-4$ in Fig. \[fig2\](a), is plotted across the ribbon in Fig. \[fig2\](b). The probability density peaks near the edges and decays toward the middle [@carvalho2014phosphorene]. The dispersion of the midgap bands and the localization of their corresponding wave function at the edges is not dependent on the width of the ribbon; rather, it is dictated by a topological invariant, fixed by the hopping elements of the Hamiltonian [@ezawa2014topological].
{width="1.9\columnwidth"}
The position of the midgap bands in the energy diagram of ZPNRs can be shifted by applying an in-plane electric field (along the y-direction in Fig. \[fig1\], the ribbon width direction) or out-of-plane electric field (along z direction in Fig. \[fig1\]). In both cases, the applied electric field can alter the energy of the midgap bands associated with the edge states. The effect of the electric field is incorporated in the tight-binding model through the diagonal terms of the Hamiltonian. The ribbons are assumed to be infinitely long. The resulting band structure from applying an in-plane electric field in the width direction (y) is shown in Fig. \[fig3\](a). The right edge is fixed at the potential $+0.3$ V and the left fixed at $-0.3$ V and we assume that the potential varies linearly between the edges [@yuan2016quantum]. The applied bias moves bands 1 and 2 upward, while it pushes bands 3 and 4 downward, and leaves the bulk bands unchanged. The probability densities of the states 1 and 2 are pushed to the left edge and those of states 3 and 4 are confined to the right edge \[see Fig. \[fig3\](e)\]. Under these circumstances, if the ribbon is connected to a source and drain, whose Fermi levels \[one denoted by the dashed and the other by the solid horizontal line in Fig. \[fig3\](a)\] are slightly offset with respect to one another owing to a small applied bias, the electronic transport can be channeled exclusively through the states 3 and 4, without contribution from any other states, so the current is carried only by the states associated with energy bands 3 and 4, which now have wave functions mostly located at the left edge of ZPNR \[Fig. \[fig2\](e)\]. Therefore, applying an in plane electric field along the width of the ZPNR leads to the generation of edge-pseudospin-polarized current in ZPNRs. In a similar manner, one can generate a current with the opposite pseudospin polarization by switching the sign of the applied bias on the two edges. When the right edge potential is fixed at $-0.3$ V instead of $+0.3$ V and the left edge bias is changed from $-0.3$ V to $+0.3$ V, states 3 and 4 move upward in the energy diagram, while 1 and 2 are pushed downward \[see Fig. \[fig3\](b)\]. According to Fig. \[fig3\](b), at this bias, bands 1 and 2 are the only ones within the transport window designated by the source and drain Fermi levels and thus carrying current. The corresponding states are at the right edge \[Fig. \[fig3\](f)\]. Therefore, a current with pseudospin right (carried only by electrons at the right edge) can be produced.
The feasibility of edge-pseudospin current generation is not limited to bilayer ZPNRs. The band-structure analysis given above can be expanded to single-layer ZPNRs, where there are two midgap bands instead of four. Applying an electric field along the width of a single-layer ZPNR leads to separation of these two midgap bands in energy. This separation in energy is associated with the confinement of the corresponding wave function at the opposite edges of the single-layer ZPNR. In the presence of a source and drain, where the transport window between the Fermi levels intersects only one midgap band, we can obtain an edge-pseudospin current, carried through the states localized near only one edge associated with that band.
To conclude, in ZPNRs, edge-pseudospin-polarized current can be obtained by the application of a lateral (in-plane) electric field. This can realized by using a side-gate field-effect transistor (FET) \[see Fig. \[fig4\](a)\]. Here, a lateral field is created by applying voltages $V_{g1}$ and $V_{g2}$ to gates, which separates the states in the active region of the device (solid-line box in Fig. \[fig4\](a)). Also the carrier transport energy window is controlled by the biases on the source ($V_S$) and the drain ($V_D$). Side-gate FETs have been proven to be experimentally feasible as they have been used with graphene as the channel material for other applications [@hahnlein2012side; @molitor2007local].
If an applied electric field is perpendicular to the surface of the ribbon (along the z-direction in Fig. \[fig1\]), similar midgap-band separation occurs, where the bands are pushed apart in pairs \[see Fig. \[fig3\](c)\]. Here, the voltage at the top of the ZPNR is fixed at $0.3$ V and at the bottom is $-0.3$ V and the voltages varies linearly in between. Once more, by attaching the ribbon to a source and drain, with the transport window between their Fermi levels chosen as shown in Fig. \[fig3\](c), current is carried solely by the states associated with bands 3 and 4. Since the corresponding wave function are located in the bottom layer, the current generated can be referred to as “pseudospin down” current. Similar to the edge-pseudospin scheme, the pseudospin polarization of the generated current can be changed by switching the sign of the applied biases. With the top of the ribbon fixed at $-0.3$ V bias and the bottom at $+0.3$ V, bands 1 and 2 are in the carrier-transport energy window \[see Fig. \[fig3\](d)\]; their associated wave functions are located in the top layer \[see Fig. \[fig3\](h)\]. The current produced under these conditions will have pseudospin up (carried only by electrons at the upper layer).
The FET in Fig.\[fig4\](b) generates current with layer-pseudospin. Voltages on the top ($V_{TG}$) and bottom ($V_{BG}$) gates are applied to the active region of the device (solid line box). This causes a potential difference (electric field) across the layers which leads to midgap band separation. The source and drain voltages tune the energy window of carrier transport. Dual-gate structures of this sort have been realized for bulk phosphorene transistors [@tayari2016dual; @kim2015dual]. Layer-pseudospin current generation obviously cannot be achieved in single-layer ZPNRs.
To get a clearer understanding of the pseudospin FETs operation, we calculate electrical current using nonequilibrium Green’s functions (NEGF) coupled with a Poisson solver [@datta1997electronic]. Numerical implementation is described in the Appendix. All the simulations are done at room Temperature ($T = 300$ K). Figures \[fig5\](a) and \[fig5\](b) show the current–voltage relation of the edge-pseudospin FET \[schematic shown in Fig.\[fig4\](a)\], where single-layer and bilayer ZPNR were chosen as the channel material, respectively. The Fermi level of source ($E_{fS}$) is set at $E_{fS} = -275$ meV and drain’s Fermi level ($E_{fD}$) is kept at $E_{fD} = -325$ meV. $E_{fS,D}$ are offset with respect to one another by $50$ meV and have an average of $E = -0.3$ eV \[this contact Fermi level is denoted by the dashed black line in the band structures portrayed in Fig. \[fig5\](a)-\[fig5\](c)\]. $E_{fS,D}$ are kept at these values throughout the paper. In relation to Fig. \[fig4\](a), the ribbon width ($W$) and the gate-oxide widths ($W_o$) are $5$ nm and the ribbons are very long (we will discuss length effects further below). As can be seen, for both single and bilayer ZPNRs the edge-pseudospin field-effect device has three different regimes of operation, depending on the voltages on the side gates. At low bias \[region II in Fig. \[fig5\](a) and (b)\], the average contact Fermi level passes through all the the midgap states, leading to a current with no particular pseudospin polarization. By increasing the magnitude of gate voltages, the midgap bands are pushed apart. When the voltage on the gates is increased to highlighted region I (III), the narrow transport window close to the Fermi level passes through two out of the four midgap bands, whose wave function is confined to the right (left) edge. This results in a pseudospin-polarized current, with edge-pseudospin right (left).
![\[fig4\] Schematic of the proposed field-effect transistor used to generate (a) edge-pseudospin and (b) layer-pseudospin current in ZPNRs. Gate oxides shown in green are $\mathrm{Al_{2}O_{3}}$ slabs. Gate electrodes, source, and drain are made of Au. The materials chosen are used in fabrication of bulk-phosphorene transistors and phosphorene dual-gate structures [@tayari2016dual; @kim2015dual].](Figure4.pdf){width=".8\columnwidth"}
The I–V curve of the layer-pseudospin FET \[Fig. \[fig4\](b)\] is shown in Fig. \[fig5\](c). For this structure, thickness of the gate oxide ($T_O$) is $5$ nm. The width of the bilayer ZPNR is $5$ nm and the active region is assumed very long. Similar to the case of edge-pseudospin FET, depending on the bias of the top gate $V_{TG}$ and bottom gate $V_{BG}$, the ribbon can be in three different regimes of operation. Once again, at low bias \[region II in Fig.\[fig5\](c)\] the current is carried by all the four midgap bands present in bilayer ZPNRs. This current does not have any particular pseudospin polarization. By tuning the bias voltages to region I (III), the midgap bands are separated in pairs, and only the midgap bands with their wave function in the upper (lower) layer overlap with the energy window imposed by source and drain Fermi levels. This results in a current with layer-pseudospin up (down).
In all cases discussed above, increasing the bias on the gates beyond regions I and III leads to further separation of midgap states. Consequently, the average Fermi level of the contacts does not cross any bands, and the ribbon shows insulating behavior [@soleimanikahnoj2016tunable]. These regions of operation are omitted from Figs. \[fig5\](a)-\[fig5\](c), as they occur at high electric field and bear no physical significance in the pseudospin scheme presented here. Simulations on wider ribbons ($W > 15$ nm) show that the regions of operations and the current–gate voltage curve remain the same as in Fig. \[fig5\], which underscores the topological nature of the midgap states that govern transport [@ezawa2014topological].
It is important to note that the response of the midgap bands to applied electric field depends on the length of the ribbon. Since the generation of the pseudospin current is based upon electric-field tuning of ZPNRs, the pseudospin polarization of the current is expected to be length dependent, as well. The polarization of the current can be measured by a population imbalance of electrons with opposite pseudospin in the active region of the pseudospin FET. Figure \[fig6\] shows a population percentage of electrons as a function of length ($L$) in pseudospin FETs. Panels (a) and (b) correspond to edge-pseudospin FET with single layer and bilayer ZPNR as the channel material respectively. Voltages on the gates are tuned so that the FET generates a current with pseudospin-left ($V_{G1} = -V_{G2} = 0.35$ V). $N_{\rightarrow}$ ($N_{\leftarrow}$) is the population percentage of electrons in the right (left) edge. Figure \[fig6\](c) shows the percentage of electrons in layer-pseudospin FET in the top ($N_{\uparrow}$) and bottom ($N_{\downarrow}$) layer, where the FET gates are biased so pseudospin-down current is generated ($V_{TG} = -V_{BG} = 0.65$ V). As it is shown, with increasing the ribbon length, electric field tuning of the midgap bands becomes more analogous to that in infinite-length ribbons and the population imbalance increases. In all cases, length $L \geq 90a \simeq 29.7$ nm guarantees an electron population percentage of over $90$ % ($N_{\leftarrow},N_{\downarrow} > 90\%$) of the expected pseudospin. Hence, the generation of current with high pseudospin polarization happens only when the quasi-one-dimensional nature of the ribbon is pronounced, i.e., when the ribbon is very long.
![\[fig5\] (a) and (b) I–V characteristics of the edge-pseudospin transistor portrayed in Fig.\[fig4\](a), where the channel material is the single-layer and bilayer ZPNR, respectively. (c) I–V characteristic of the layer-pseudospin transistor in Fig.\[fig4\](b).](Figure5.pdf){width=".9\columnwidth"}
![\[fig6\] Population percentage of electrons in the edge- and layer-pseudospin field effect transistors shown in Figs. 4(a) and 4(b), respectively. Panels (a) and (b) correspond to edge-pseudospin FET where the single-layer and bilayer ZPNR were used, respectively. (c) Population percentage of electrons in the layer-pseudospin FET.](Figure6.pdf){width="1.01\columnwidth"}
Pseudospin Valve
================
As in conventional spin devices, one would expect that the flow of the pseudospin-polarized current can be controlled by tuning the bias along the direction of electron transport [@baibich1988giant; @viret1996spin]. This idea can be implemented through a pseudospin-based counterpart of the spin valve. Figure \[fig7\](a) shows the pseudospin valve in the edge scheme. The valve mode of operation changes through a variation of electrical resistance. The change in electrical resistance is controlled by the voltages on the gates at the opposite ends of the valve ($V_{G1-4}$). For instance, if in Fig.\[fig7\](a), the voltages on gates 1 and 2 are opposites of each other ($V_{G1} = -V_{G2}$) and are tuned to region I (II) of Figs. \[fig5\](a),(b), the ZPNR area sandwiched between gates 1 and 2 will carry current with edge-pseudospin left (right) only. If the second row of gates $\mathrm{G_{3,4}}$ are tuned to the same region of operation as the first row of gates, the electron pseudospin polarization will remain intact as they propagate through the valve. In this case the valve is said to be in the *parallel configuration*. On the other hand, if the gates are tuned such that the region of operation changes as the current flows through the valve ($V_{G1,2}$ are tuned to region I and $V_{G3,4}$ biases are at region III or vice versa), the valve is in its *antiparallel configuration*. In this configuration, by going through the valve, electrons are forced to “rotate” their pseudospin (going from the left to the right edge or vice versa). Considering that the overlap between the wave functions of electron at the opposite edges is vanishingly small, the forcible change of the edge-pseudospin in the antiparallel biasing configuration results in a much higher resistance than the resistance in the parallel configuration, where no edge switching of electrons is imposed.
A similar analysis can be given for the layer-pseudospin valve shown in Fig.\[fig7\](b). When all the the gates are tuned according to Fig. \[fig5\](c) such that the regions at the opposite ends of the valve carry the same layer-pseudospin current, the device is said to be in its parallel configuration.
![\[fig7\] Schematics of (a) an edge-pseudospin valve and (b) a layer-pseudospin valve.](Figure7.pdf){width="1\columnwidth"}
In the parallel configuration, electrons stay in the same layer while propagating through the valve intralayer electronic transport is dominant and the conductance is high. However, if the gates at one end of the valve are biased to region I (II) and the the gates at the other end are tuned to region II (I), the valve is in the antiparallel configuration. In this case, electrons are forced to “rotate” their pseudospin, going from the upper (lower) layer to the lower (upper) layer. Considering that the intralayer hopping terms ($t_{\parallel}$) are much larger than their interlayer counterparts ($t_{\perp}$) [@rudenko2015toward], electron interlayer movement in the antiparallel biasing configuration results in a much higher resistance than the intralayer movement characteristic of the parallel configuration. This resistance increase in the antiparallel configuration is analogous to the resistance increase due to spin scattering off domain walls in conventional spin devices in the antiparallel configuration [@viret1996spin].
In spintronic applications, the giant magnetoresistance (GMR) ratio is the standard for benchmarking spin-valve performance. Equivalently, the nonmagnetic version of the GMR ratio called the pseudomagnetoresistance (PMR) ratio characterizes the pseudospin-valve operation: $$\label{eq1}
\mathrm{PMR} = \frac{R_{AP} - R_{p}}{R_{AP}}.$$ Here, $R_{p}$ and $R_{AP}$ are the electrical resistance of the valve in parallel and antiparallel configurations respectively. For a perfect valve, the PMR ratio would be $100\%$.
Here, we calculated the PMR ratio for $5$-nm wide ZPNRs for both edge and layer pseudospin scheme using the self-consistent NEGF method described in the Appendix. The length of the active region ($L$) is $10a = 3.31$ nm. The phosphorene areas sandwiched between the oxides are assumed to be infinitely long. The oxide width ($W_O$) in the edge valve and the oxide thickness ($T_O$) in the layer valves are assumed to be 5 nm. All simulations are done at room temperature ($T = 300$ K).
![\[fig8\] (a),(b) Current vs. gate voltage characteristic of (a) single-layer and (b) bilayer edge-pseudospin valve. (c) Current vs. gate voltage characteristic of the layer-pseudospin valve. In each panel, $I_P$ ($I_{AP}$) is the current in parallel (antiparallel) configuration. In all three panes, the pseudomagnetoresistance \[\[eq1\]\] is shown by the green curve.](Figure8.pdf){width=".9\columnwidth"}
The calculated PMR ratio versus gate bias for the edge valve is shown in Figs. \[fig8\](a) and \[fig8\](b), where a single-layer ZPNR and a bilayer ZPNR were used as the channel material, respectively. The highlighted voltage-magnitude intervals in Fig. \[fig8\](a) is ($ 0.18$ V$ < |V_{G1-4}| < 0.50$ V) and in Fig. \[fig8\](b) ($ 0.2$ V$ < |V_{G1-4}| < 0.60$ V) are the regions of overlap between regions I and III from Figs. \[fig5\](a) and \[fig5\](b), respectively. Depending on the relative sign of applied biases, if the ZPNR at both ends carries the same pseudospin current, the device is in its parallel configuration; the corresponding current ($I_P$) in this case is shown in Fig. \[fig8\]. Likewise, if the region of operation changes along the valve, the device operates in its antiparallel configuration and a change in the pseudospin polarization is imposed; the corresponding current $I_{AP}$ is also shown in Fig. \[fig8\]. As expected, $I_P$ is significantly higher than $I_{AP}$ in the highlighted voltage interval. This major difference is also mirrored in the PMR ratio, as shown in Figs. \[fig8\](a) and \[fig8\](b). As can be seen, for both single-layer and bilayer ZPNR in the highlighted region, the PMR ratio exceeds $99\%$, indicating a nearly perfect valve. Similarly, the PMR ratio versus gate bias for the layer valve is portrayed in Fig. \[fig8\](c), where the highlighted region ($ 0.4$ V$ < |V_{TG1,2}|,|V_{BG1,2}| < 1.0$ V) is the overlap between region I and III in Fig. \[fig5\](c). Inside the overlap region, when both ends of the valve are in the same region of operation the valve is in parallel configuration. Conversely, if the region of operation changes along the charge transport direction the valve is in antiparallel configuration. The imposed change of layer-psudospin in antiparallel configuration leads to a considerable difference between the current in this case compared to parallel configuration \[see Fig. \[fig8\](c)\]. The difference between $I_P$ and $I_{AP}$ leads to a large value of the PMR ratio ($>92\%$) inside the highlighted region shown in Fig.\[fig8\](c).
![\[fig9\] (a) The pseudomagnetoresistance ratio and (b) polarization as a function of valve length in the edge and layer schemes. For the edge-pseudospin valve $|V_{G1-4}| = 0.35$ V for both single-layer and bilayer ZPNR. For the layer-pseudospin valve $|V_{TG1,2}| = |V_{BG1,2}| = 0.65$ V.](Figure9.pdf){width="1\columnwidth"}
In conventional spin valves, the magnetoresistance ratio tends to decrease by increasing the length of the valve [@shim2008large; @barraud2010unravelling]. The PMR ratio, being the nonmagnetic version of magnetoresistance ratio, also decreases as the length of the valve increases. Figure \[fig9\](a) shows the PMR ratio as a function of the valve length ($L$) in the layer and edge schemes. The PMR ratio of the layer valve drops rapidly as a function of length, demonstrating the short relaxation length of the layer-pseudospin degree of freedom. In contrast, the edge-pseudospin valve maintains its high value of the PMR ratio even for large lengths ($L = 70a \simeq 23.2$ nm) for both single-layer and bilayer ZPNRs, showing the long relaxation length of the edge-pseudospin degree of freedom. This originates from the small overlap of the electron wave functions at the opposite edges of the ZPNRs, which makes transport of electrons from one edge to the other highly improbable. As a result, electrons tend to keep their edge-pseudospin polarization over much longer distances than compared to their layer-pseudospin counterpart.
The relaxation of the pseudospin along the valve can also be traced to the population imbalance between electrons with opposite pseudospins, as they propagate through the valve. The population imbalance of electrons with opposite edge-pseudospin can be represented using the pseudospin polarization, $P(x)$, defined as:
$$P(x) = \frac{|n_{\leftarrow}(x) - n_{\rightarrow}(x)|}{n_{\leftarrow}(x) + n_{\rightarrow}(x)}.$$
Here, $n_{\leftarrow}(x)$ ($n_{\rightarrow}(x)$) is the population of electrons with edge-pseudospin left (right) at position $x$ along the valve. The polarization of layer-pseudospin is the same as above with $n_{\substack{\leftarrow \\ \rightarrow}}$ replaced with $n_{\substack{\uparrow \downarrow}}$. Figure \[fig9\](b) shows the pseudospin polarization of electrons along the length for edge- and layer-pseudospin valve. Here, $L = 70a \simeq 23.1$ nm. As can be seen, electrons are injected from the left with a nearly perfect pseudospin polarization ($P \simeq 100 \%$). As they move through the valve, the population imbalance of electrons with opposite pseudospin decreases. As a result, pseudospin polarization of electrons decreases. The drop in the polarization of layer-pseudospin is more significant than for edge-pseudospin. This verifies the fast relaxation of layer-pseudospin in comparison with edge-pseudospin.
As the layer-valve works based on the asymmetry between the interlayer and intralayer hopping, it is also realizable in other bilayer Van der Waals materials. In such valves, the PMR ratio and polarization of pseudospin would be dependent on the value of the interlayer hopping elements ($t_{\perp}$) with respect to intralyer hopping elements ($t_{\parallel}$). Assuming the hopping elements of phosphorene, preliminary calculation on a valve with $L = 70a$ shows that reducing the interlayer hopping elements by 33.3% ($t_{\perp} \rightarrow \frac{2}{3} t_{\perp}$) improves the PMR ratio from 8% to 34% and the polarization at the end of the valve from 16% to 41%. This shows that layered materials with a weaker Van der Waals force between the layers are better candidates for layer-pseudospin electronics.
As a further matter, phosphorene samples are found to be sensitive to the environment, which makes the role of impurities significant [@wood2014effective; @island2015environmental]. In particular, potential fluctuations caused by charged impurities play a crucial role in electronic transport of two-dimensional materials [@ong2014anisotropic; @dean2010boron; @ni2009probing]. Here, the effect of charged impuirities is added to our model in the form of superposition of Gaussian potential fluctuations [@paez2016disorder], $$\label{eq2}
U(\mathbf{r}_i) = \sum_{k=1}^{N_{\mathrm{imp}}}U_ke^{|\mathbf{r}_i - \mathbf{R}_k|^2/2\xi^2}.$$ Here, $\mathbf{r}_i$ denotes the position of the lattice site $i$. $N_{\mathrm{imp}}$ is the number of scatterers that are located at $\{\mathbf{R}_k\}_{k = 1,N_{\mathrm{imp}}}$. These locations are chosen by a random uniform distribution. The scatterers have amplitudes $\{U_k\}_{k = 1,N_{\mathrm{imp}}}$ taken from a uniform distribution $[-\delta U/2,\delta U/2]$. $\xi$ is the correlation length. Density of scatterers is $n_{\mathrm{imp}} = N_{\mathrm{imp}}/N$ where $N$ is the number of phosphorus atoms in the active region of the valve. At each point of the lattice, the potential term from Eq. (\[eq2\]) is calculated and added to the diagonal term of the main Hamiltonian that corresponds to that lattice point. This model has been previously used for modeling of charged impurities in phosphorene [@paez2016disorder] and graphene [@mucciolo2010disorder; @mivskovic2012ionic; @wehling2009impurities; @vierimaa2017scattering], where a good agreement with experiment was obsereved [@tan2007measurement; @adam2008density]. Figure \[fig10\] shows the PMR ratio as a function of impurity density ($n_{\mathrm{imp}}$). $\delta U = 1$ eV and $\xi$ is fixed at 5 Å. The values within this range were shown to capture the role of charged-impurity in carrier transport of nanostrips effectively [@mucciolo2010disorder]. Each data point is an average over 200 configurations. The PMR ratio in the layer-pseudospin scheme drops as the number of scatterers increases. In contrast, the PMR ratio for the edge valve is completely robust against $n_{\mathrm{imp}}$.
![\[fig10\] Pseudomagnetoresistance ratio as a function of impurity abundance. Edge-pseudospin valve biases are $|V_{G1-4}| = 0.35$ V for both single-layer and bilayer ZPNR. For the layer-pseudospin valve, $|V_{TG1,2}| = |V_{BG1,2}| = 0.65$ V. Data points are from numerical simulation with $\delta U = 1$ eV and $\xi = 5$ Å.](Figure10.pdf){width=".8\columnwidth"}
The reason for this difference between the layer and edge valves originates from the mechanisms based on which they operate. The layer valve works based on the imbalance between interlayer and intralayer electron transport in ZPNRs. By increasing the scatterer density, the difference between interlayer and intralayer hopping elements is overshadowed by the scattering in a larger area of the ribbon. As a result, the layer PMR ratio decreases. In contrast, the edge-valve operation is based upon the very existence of edge states, which endure in the presence of scatterers [@ryu2002topological]. In particular, the existence of edge states in ZPNRs is associated with a topological winding number, which is independent of the diagonal elements of the Hamiltonian. As the presence of disorder changes only the diagonal terms of the Hamiltonian, edge states remain intact. Hence, the edge PMR ratio is unchanged in the presence of impurities.
In general, implementation of conventional spintronic applications requires efficient generation and detection of spin-polarized current. The former is typically obtained by generating a spin current in a magnetically tuned ferromagnetic nanofilm, followed by injection into a semiconductor via an ohmic contact; the latter is carried out by the spin valve. The pseudospin schemes described in this paper integrate the generation and detection of pseudospin current into a single material, which makes for easier miniaturization. The applications of spintronics, such as spin-logic gates, are found to be attainable in pseudospin schemes [@schaibley2016valleytronics]. Also, the advantages of spin-based logic devices over conventional metal-oxide semiconductor (MOS) field-effect devices in terms of power consumption and speed are also expected in their pseudospin-based counterparts [@banerjee2009bilayer].
The pseudospin devices discussed in this paper can also be realized using the recently discovered skewed-armchair phosphorene nonoribbons [@soleimanikahnoj2016tunable]. Similar to ZPNRs, midgap states are present in the energy dispersion of skewed-armchair phosphorene nanoribbons, which will facilitate pseudospin electronics.
Conclusions
===========
In summary, electron transport in metallic ZPNRs is governed by the states localized near the edges, whose energies belong to the midgap bands that are energetically far from the bulk bands. These states can be electrically manipulated by gating ZPNRs in two different ways, which bring about two practical versions of the pseudospin. One is the edge pseudospin, where pseudospin “left” (“right”) is associated with the conducting electrons located near the left (right) edge. The other is realized in bilayer ZPNRs, where limiting electron transport to the “top” (“bottom”) layer gives rise to the concept of “up” (“down”) pseudospin. In each scheme, we proposed two devices: an FET for the generation of pseudospin-polarized current and a pseudospin valve. The PMR ratio is calculated for both edge and layer-pseudospin valves, where the edge-pseudospin valve is nearly perfect and robust against variations in device parameters and disorder. Although the results presented here are promising, we acknowledge the experimental challenges in the fabrication of zigzag phosphorene nanoribbons with perfect edges. Nevertheless, nanoribbons with atomically precise edges have been realized in graphene [@cai2010atomically] and TMDs [@chen2017atomically] which belong to the same family of two-dimensional honeycomb-lattice materials as phosphorene. Thus, the results presented here should be viewed as a start, which will hopefully encourage further experimental and theoretical work.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors gratefully acknowledge support by the US Department of Energy under Award No. DE-SC0008712 (Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, Physical Behavior of Materials Program). Related preliminary work was funded as a seed project by the UW–Madison MRSEC (NSF award DMR-1121288). The work was performed using the compute resources and assistance of the UW-Madison Center for High Throughput Computing (CHTC) in the Department of Computer Sciences. The authors thank Farhad Karimi for his valuable comments.
Appendix: IMPLEMENTATION DETAILS
================================
The modeling of electrical devices was performed through a self-consistent solution of two equations. The first is the retarded Green’s $G$ function, which describes the dynamics of electrons inside the active region of the devices [@datta1997electronic] $$G_{r,r'}(E) = [E - H_{r,r'} - U_{r,r} - \Sigma^S_{r,r'}(E) - \Sigma^D_{r,r'}(E)].$$ $E$ is the energy at which the Green’s function is being calculated. $r$ and $r'$ are the lattice-point positions. $H$ is the Hamiltonian of the active region and $U$ is the self-consistent potential being applied to the active region. $\Sigma^{S(D)}$ is the self energy of the source (drain). The effects of metal contacts have been ignored and the contact self-energies were calculated using the Sancho-Rubio iterative scheme [@sancho1985highly]. The number of electrons ($n$) and holes ($p$) at each lattice point is calculated using $ n(r)= 2\int \frac{dE}{2\pi}G^n_{r,r}$ and $h(r) = 2\int \frac{dE}{2\pi}G^p_{r,r}$, respectively. Here, $G^n_{r,r'}$ and $G^p_{r,r'}$ are analytical functions: $\Gamma^{S(D)} = -2Im[\Sigma^{S(D)}]$, the Green’s function, the Fermi level of the source ($E_{fS}$) and the drain ($E_{fD}$), and the Fermi-Dirac distribution function ($f(E)$):
$$G^n = G[\Gamma^Sf(E-E_{fS}) + \Gamma^Df(E-E_{fD})]G^{\dagger},$$
$$G^p = G[\Gamma^S(1 - f(E-E_{fS})) + \Gamma^D(1 - f(E-E_{fD}))]G^{\dagger}.$$
The second equation that was solved self-consistently with the first is Poisson’s equation: $$\nabla(\epsilon(r)\nabla U(r)) = -\rho (r)$$ which determines the self-consistent potential $U$ for a given charge distribution $\rho (r)$. Charge distribution is a function of the electron and hole occupation obtained from the Green’s function. The dielectric function was assumed to be position dependent to take into account the different materials (phosphorene sheet; $\mathrm{Al_2O_3}$). In the simulations, the dielectric function of $\mathrm{Al_2O_3}$ was assumed to be $9.5\epsilon_0$ and those of single-layer and bilayer phosphorene were taken to be $1.12\epsilon_0$ and $1.72\epsilon_0$ respectively [@wang2015native]. The effect of substrate was ignored and gates were taken into account via Dirichlet boundary condition. Neumann boundary conditions were assumed at the remaining boundaries.
The Green’s function and the Poisson equation were solved self-consistently until convergence was obtained. The converged Green’s function was then used to obtain the current ($I$): $$I = \frac{2q}{h}\int_{-\infty}^{+\infty}dE \ T\left(E\right)[f(E-E_{fS}) - f(E-E_{fD})].$$ Here, $T(E)$ is the transmission function, calculates as $$T = Trace(\Gamma^SG\Gamma^DG^{\dagger}).$$
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---
abstract: 'This paper uses robots to assemble pegs into holes on surfaces with different colors and textures. It especially targets at the problem of peg-in-hole assembly with initial position uncertainty. Two in-hand cameras and a force-torque sensor are used to account for the position uncertainty. A program sequence comprising learning-based visual servoing, spiral search, and impedance control is implemented to perform the peg-in-hole task with feedback from the above sensors. Contributions are mainly made in the learning-based visual servoing of the sequence, where a deep neural network is trained with various sets of synthetic data generated using the concept of domain randomization to predict where a hole is. In the experiments and analysis section, the network is analyzed and compared, and a real-world robotic system to insert pegs to holes using the proposed method is implemented. The results show that the implemented peg-in-hole assembly system can perform successful peg-in-hole insertions on surfaces with various colors and textures. It can generally speed up the entire peg-in-hole process.'
author:
- 'Joshua C. Triyonoputro$^{1}$, Weiwei Wan$^{1,2,*}$, Kensuke Harada$^{1,2}$[^1]'
bibliography:
- 'paperJoshua.bib'
title: '**Quickly Inserting Pegs into Uncertain Holes using Multi-view Images and Deep Network Trained on Synthetic Data**'
---
Peg-in-hole, deep learning, domain randomization, multi-view images
Introduction
============
One goal in robotics is to automate product assembly. At present, most robotic assembly systems, such as the ones implemented in the production of cars, still follow the basic principle of teaching and playback. The teaching and playback concept is useful when the target objects are fixed. When variations exist, the usefulness of teaching and playback principle is limited. This leads people to use automatic motion planning to assemble a diverse range of products with variations.
An important issue of automatic motion planning is the accumulated position errors. The goal pose after execution could be very different from the goal pose in the simulation after motion planning. People usually use visual detection or scanning search using force sensors to locate the hole and avoid the position errors. However, both methods have shortages: Visual detection requires mild color, texture, and reflection, etc.; Scanning search using force sensors is slow.
![ The workflow of the proposed peg-in-hole assembly system. The system uses learning-based visual servoing to quickly move the peg closer to the hole, uses spiral search to precisely align the peg and the hole, and uses impedance control to fully insert the peg into the hole. The learning-based visual servoing is our main contribution.[]{data-label="fig:first_image"}](imgs/first_image.jpg){width="0.8\columnwidth"}
This leads us to study how to quickly assemble pegs into uncertain holes on surfaces with different colors and textures. We develop a peg-in-hole assembly system that uses learning-based visual servoing to quickly move the peg closer to the hole, uses spiral search to precisely align the peg and the hole, and uses impedance control to fully insert the peg into the hole. Fig.\[fig:first\_image\] shows the outline of the developed peg-in-hole program. The search phase comprises the learning-based servoing and spiral search. The insertion phase comprises the impedance control.
Specifically, our main contribution is the learning-based visual servoing. We use synthesized data to train a deep neural network to predict the position of a hole, and use iterative visual servoing to iteratively moves a peg towards the hole. Various experiments and analysis using both simulation and real-world experiments are performed to (1) analyze the performance of the learning-based visual servoing against uncertain holes on surfaces with different colors and textures, and (2) compare the efficiency of executions under different initial hole positions. The results show that the proposed method is robust to various surface backgrounds and can generally speed up the entire peg-in-hole process.
Related Work {#related_work}
============
This paper focuses on the problem of peg-in-hole assembly using deep learning. Thus, this section reviews the related work in peg-in-hole assembly and the applications of deep learning in industrial robots.
Peg-in-hole assembly {#peginholeassembly}
--------------------
Peg-in-hole assembly refers to the task of inserting a peg to a hole. The task generally has two phases – the search phase and the insertion phase. The insertion phase refers to the phase when the peg is being inserted, and it has been studied extensively. The search phase is the stage of finding a hole when position uncertainty exceeds the clearance of a hole. It is less studied.
### Insertion phase
One of the earliest studies about the insertion phase is Shirai and Inoue[@shirai1973guiding], where they used visual feedback to perform insertion. About a decade later, researchers shifted from the use of visual feedback to the use of compliance to accommodate the motion of the end-effector during insertion[@whitney1982quasi][@mason1981compliance][@balletti2012towards][@zhang2017peg]. In the 1990s, the quasi-static contact analysis was used to guide the insertion[@nguyen1995fuzzy][@kim1999active][@su2012sensor]. The state-of-the-art method for insertion is impedance control[@broenink1996peg][@cho2012strategy]. It is widely used in many practical systems.
### Search phase
The search phase is before the insertion and is used to align the peg and the hole. Below, related work about the search phase, sometimes followed by insertion, is reviewed. The studies can generally be categorized by the types of sensors used: vision sensors, force sensors, or both.
The first category uses vision sensors. Yoshimi and Allen[@yoshimi1994active] dealt with visual uncertainty for peg-in-hole by attaching a camera to the end-effector and rotating the camera around the last axis of the robot. Morel et al.[@morel1998impedance] employed 2D visual servoing (search phase) followed by force control (insertion phase) to successfully performed peg-in-hole assembly with large initial offsets. Huang et al.[@huang2013fast] used high-speed cameras to align a peg to a hole. More recently, the visual coaxial system was used to perform precise alignment in peg-in-hole assembly[@nagarajan2016vision][@yang2018coaxial].
The second category uses force-torque sensors. Newman et al.[@newman2001interpretation] proposed the use of force/torque maps to guide the robot to the hole. Sharma et al.[@sharma2013intelligent] generalized the work of Newman et al.[@newman2001interpretation] to tilted pegs. Chhatpar and Branicky[@chhatpar2001search] explored various blind search methods such as tilting and covering the search space using paths like spiral path (a.k.a. spiral search). Spiral search and its variants were also discussed in [@jasim2014position][@marvel2018multi]. The problem of spiral search is that it is time consuming, given that the robot just blindly searches for the hole. Tilting, often considered as a method intuitive to human, was also explored in several studies[@marvel2018multi][@park2013intuitive][@jasim2014contact][@abdullah2015approach]. The limitation of tilting is that it assumes that the initial offset is small.
More recently, studies such as [@nguyen2019probabilistic] used a combination of visual sensors and force-torque sensors to track the uncertainties of object poses and sped up the search process. This paper similarly employs both visual sensors and a force-torque sensor. Specifically, we explore the use of a combination of visual servoing using two in-hand RGB cameras, followed by the spiral search using force sensors, to perform a peg-in-hole assembly.
Deep learning in industrial robotics
------------------------------------
Deep learning has in the recent years gained prominence in robotics. Some studies used real-world data to train deep neural networks. For example, Pinto and Gupta[@pinto2016supersizing] collected 700 robot hours of data and used them to train a robot to grasp objects. Levine et al.[@levine2018learning] similarly made robots perform bin-picking randomly for days, before finally using the data obtained to train a deep network for bin-picking. Inoue et al.[@inoue2017deep] proposed deep reinforcement networks for precise assembly tasks. Lee et al.[@lee2018making] trained multimodal representations of contact-rich tasks and trained a robot to perform peg-in-hole. Thomas et al.[@thomas2018learning] used CAD data to help improve the performance of end-to-end learning for robotic assembly. Yang et al.[@yang2017repeatable] and Ochi et al.[@ochi2018deep] took data by performing teleoperations and used them to make the robot learn specific motions. De Magistris et al.[@de2018experimental] took labeled force-torque sensor data to train a robot to perform multi-shape insertion. Although the aforementioned studies showed the possibilities of using real-world data, developing such systems are difficult. Collecting and labeling the real-world data is time-consuming and labor intensive.
For this reason, robotic searchers began to study training deep neural networks using synthesized data. Dwibedi et al.[@dwibedi2017cut] cut and pasted pictures of objects on random backgrounds to train deep neural networks for object recognition. Mahler et al.[@mahler2017dex] used synthetic data of depth images to train robots for bin-picking. Unfortunately, the use of synthetic data is limited due to reality gap[@jakobi1995noise].
To overcome the reality gap, one method is transfer learning [@pan2010survey]. Domain adaptation is an example of transfer learning, where synthetic data and real-world data are both used[@csurka2017domain][@zhang2017adversarial] [@bousmalis2018using][@fang2018multi]. Domain randomization highly promotes the use of synthesized data[@sadeghi2016cad2rl][@tobin2017domain]. It suggests that synthetic data with randomization can be helpful to allow transfer without the need for real-world data. Using domain randomization in data synthesis were widely studied[@tremblay2018training][@andrychowicz2018learning][@prakash2018structured][@sundermeyer2018implicit].
The learning-based visual servoing of this work is based on domain randomization. It is used to synthesize the training data for a deep neural network. It is also used to synthesize various testing data sets to analyze the performance of the neural network.
Methods {#method}
=======
This section gives a general explanation of the proposed peg-in-hole assembly method, with a special focus on the learning-based visual servoing, and the synthetic data generation.
Overview of the proposed peg-in-Hole assembly {#proposed_peg_in_hole_method}
---------------------------------------------
The work flow of the proposed method is shown in Fig.\[fig:peg\_in\_hole\_approach\]. It includes a search phase (Fig.\[fig:peg\_in\_hole\_approach\](a, b)) and an insertion phase (Fig.\[fig:peg\_in\_hole\_approach\](c)). The search phase has two steps: Learning-based visual servoing (Fig.\[fig:peg\_in\_hole\_approach\](a)) and spiral search (Fig.\[fig:peg\_in\_hole\_approach\](b)). The learning-based visual servoing quickly moves a peg closer to the hole, while the spiral search can precisely align the peg and the hole.
![The proposed peg-in-hole assembly includes two phases: A search phase and an insertion phase. The search phase has two steps: (a) Learning-based visual servoing and (b) Spiral search. They move and align the peg to the hole. The insertion phase uses (c) impedance control to insert the peg into the hole.[]{data-label="fig:peg_in_hole_approach"}](imgs/approach.jpg){width="0.95\columnwidth"}
![Definition of the coordinate systems $\Sigma_{eef}$, $\Sigma_{img}$, and $\Sigma_{spiral}$. (The definition is for the image captured by Camera 1. For the other camera discussed later, the coordinate system reverses.)[]{data-label="fig:coordinate_system"}](imgs/coordinate_system.jpg){width="0.95\columnwidth"}
The details of the learning-based visual servoing will be discussed in Section \[visual\_alignment\]. It is the main contribution of the work.
The spiral search is conducted along the $xy$-plane of the end-effector coordinate system $\Sigma_{eef}(x_{eef},y_{eef})$, shown in Fig.\[fig:coordinate\_system\]. The reference coordinate system for spiral search $\Sigma_{spiral}(x_{spiral},y_{spiral})$ is of the same orientation as $\Sigma_{eef}$ with the origin $r$ away from the initial peg position. The path for spiral search is given in Eqn.. $$\label{eq:spiral_search}
x_{spiral} = r\cos{\theta}, y_{spiral} = r\sin{\theta}$$ where $\theta$ and $r$ start from 0. $\theta$ increases by $\delta\theta$ every timestep, while $r$ increases for $\delta$$r$ for every 1 full rotation. The robot will move following the discrete spiral path described above, and continue until the force at the -$z$ direction of $\Sigma_{eef}$ is less than $F_{max}$ or $r$ reaches a predefined threshold.
The assembly switches to the insertion phase when the condition for the -$z$ direction force is fulfilled. Impedance control is used to perform insertion in the insertion phase.
Learning-based visual servoing {#visual_alignment}
------------------------------
The learning-based visual servoing uses (1) a deep neural network and multi-view images to predict the position of the hole and (2) continuous visual servoing to move the peg towards the hole.
### Predicting the position of the hole {#hole_detector}
We train a neural network that can map an image $I$ to an output of ($x$,$y$), where ($x$,$y$) indicates the distance between the center of the peg and the hole as seen in the image in pixels on coordinate system $\Sigma_{img}$ shown in Fig.\[fig:coordinate\_system\].
![VGG-16 network architecture [@simonyan2014very] used in the proposed method.[]{data-label="fig:vgg16"}](imgs/vgg16.jpg){width="\columnwidth"}
VGG-16 network is used following the suggestion of [@tobin2017domain]. The VGG-16 network is adjusted for regression instead of classification. The input of the network is adjusted to a grayscale image of size 160$\times$160. The output is a predicted hole position ($x$,$y$). Fig.\[fig:vgg16\] shows the diagram of the VGG-16 network. Following [@tobin2017domain], the dropout components of the network are removed to avoid local minima. The network is trained using Adam, with settings following [@kingma2014adam]. Mean squared error (MSE) is selected as the loss function.
The input image is a concatenation of two images from two in-hand cameras installed on two sides of a robot hand. Fig.\[fig:real\_img\_preproc\] shows the concatenation. To define the size of the cropped image, a bounding box of size 160$\times$80 around the center of the peg is predefined, such that the concatenated image reaches 160$\times$160.
![Concatenating two multi-view images into the input image to the VGG-16 network. (a) Configurations of the two in-hand cameras. (b) An area around the center of the peg of each image is cropped. (c-d) Concatenate the cropped area into a 160$\times$160 image.[]{data-label="fig:real_img_preproc"}](imgs/real_image_preprocessing.jpg){width="1.\columnwidth"}
### Iterative visual servoing
While the network outputs ($x$,$y$) in pixels, the image is not exactly a 2D image parallel to the surface of where the hole is. Thus, instead of directly moving the peg to the predicted position, we use the $\operatorname{sgn}$ function to classify the outputted values into 4 quadrants, as shown in Fig.\[fig:quadrant\_visual\_servoing\], and iteratively moves the peg towards the quadrants.
![Definition of the quadrants. Top images show how the gripper and the hole are relatively positioned from a front view. The bottom images are the concatenated image from the two cameras. (a) Quadrant “Topleft”. (b) Quadrant “Bottomleft”. (c) Quadrant “Bottomright”. (d) Quadrant “Topright”.[]{data-label="fig:quadrant_visual_servoing"}](imgs/quadrant.jpg){width="0.95\columnwidth"}
Consider the coordinate system $\Sigma_{h}$ which shares the same orientation as $\Sigma_{eef}$ and has an origin at the center of the hole. Assuming at discrete timestep $t$, where $t$ is a non-negative integer that starts from 0, the peg is located at ($x_{h}[t]$,$y_{h}[t]$) and the values outputted by the trained network is ($x[t]$,$y[t]$). We can move the peg closer with only the quadrant information using Eqn..
$$\label{eq:visual_servoing}
\begin{bmatrix}
x_{h}[t+1] \\
y_{h}[t+1]
\end{bmatrix}
=\begin{bmatrix}
x_{h}[t] \\
y_{h}[t]
\end{bmatrix}-\lambda[t]\begin{bmatrix}
-\operatorname{sgn}(x[t]) \\
-\operatorname{sgn}(y[t])
\end{bmatrix}$$
where $\lambda$ (unit=$mm$/px) is a time dependent coefficient with decreasing values and converges to 0 along with time. $\lambda$ is defined as: $$\label{eq:visual_servoing_coef}
\lambda[t]=\frac{A(n-t)}{n}$$ where $A$ is the maximum allowable relative moving distance. $\lambda[t]$ converges to 0 at time $n$. By repeating this for $n_{run}$ times, where $n_{run}<n$, the peg will get closer to the center of the hole as long as the quadrant the hole is at relative to the peg can be correctly predicted by the deep neural network. The method is robust to the prediction errors of the deep neural network since it is not directly using the predicted numbers.
Synthetic Data Generation Method {#synthetic_data_gen}
--------------------------------
Synthetic data generation is used to get a large amount of training. The basic idea is to change the background of the cameras with various images. First, we get a gripper template mask following Fig.\[fig:get\_template\_mask\]. The purpose of having a gripper template mask is to simulate the view of the gripper in the cameras. Then, the gripper template mask is attached to some random images with a circle (the hole) to make a synthesized assembly data. Fig.\[fig:synthetic\_data\_gen\] shows attaching process. There are four kinds of randomization in the attaching. (1) The background of the image is randomized. (2) The size of the circle (the hole) is randomized. (3) The darkness of the circle (the hole) is randomized. (4) Gaussian noises are added to randomize the gripper template mask. By using random background images captured from the Internet and likening the hole to a dark-colored circle, a large number of synthetic images with known labels ($x$,$y$) can be quickly synthesized.
![Obtaining the gripper template mask for generating synthetic data. (a) The images from the in-hand cameras. (b) Rotate the image from camera 2, mark the gripper mask, locate the center of the peg, and define the bounding box on each image. (c) Crop the image according to the bounding box and concatenate.[]{data-label="fig:get_template_mask"}](imgs/get_template_mask.jpg){width="0.87\columnwidth"}
![Attaching the gripper template mask to some random images with a circle (the hole) to make a synthesized assembly data. The hole is added to the background image in (d). The gripper template mask is added in (e). Gaussian noises are added in (f).[]{data-label="fig:synthetic_data_gen"}](imgs/synthetic_data_gen.jpg){width="1.0\columnwidth"}
Experiments
===========
The experiments section is divided into two parts. In the first part, we compare and analyze the performance of the neural network under different training data. In the second part, we analyze the real-world visual servoing and insertions using the best performing network.
Performance of the neural network
---------------------------------
The specification of the computer used for the neural network is Intel(R) Core(TM) I5-6500 @3.20 GHz, 16GB RAM, with an Nvidia Geforce GTX 1080 card.
### Training data
The synthetic training data was generated using the method described in Section \[synthetic\_data\_gen\]. Six categories of random images were prepared, as shown in Fig.\[fig:6categories\]. 776 positions (194 positions per quadrant) are evenly sampled in each image to define the position of a hole. These positions have a maximum of 4 $cm$ uncertainty (the range is \[-66 px, 66 px\]). The darkness of a hole was randomized in range \[10, 70\] (0 is fully black, 255 is fully white). The diameter of a hole (in pixels) was randomized in range \[10 px , 35 px\] (around \[3 $cm$, 1 $cm$\]). In total, we generated 69,840 synthetic images using each category of random images. It took an average time of 22 minutes without using the Graphics Processing Unit (GPU).
![Six categories of randomly collected images are used for synthesizing the training data. Images of category “Plain” is generated manually. The rest are downloaded from the Internet using category names as the search keywords.[]{data-label="fig:6categories"}](imgs/6categories.jpg){width="1.\columnwidth"}
The details of the synthesis are as follows. Using the six categories of images, we synthesized 9 sets of training data. They are “Plain”, “Image”, “Textures+Scenery(18)”, “Textures+Scenery(30)”, “Textures+Scenery(45)”, “Textures+Scenery(90)”, “Textures(45)”, “Metallic(45)”, “Scenery(45)”. For the “Plain” training set, the background images were randomly selected from the “Plain” category, resulting into 69,840 synthetic images with background color ranges \[0,255\]. For the “Image” training set, the background images were randomly selected from the “Image” category where 776 random images searched using the keyword “Image” were downloaded. For the “Textures(45)”, “Scenery(45)”, and “Metallic(45)” training set, 45 images randomly selected from their corresponding categories. For the “Textures+Scenery(45)” training set, 22 images from the “Textures” category and 23 images from the “Scenery” category were randomly selected and combined. The “Textures+Scenery(18)”, “Textures+Scenery(30)”, and “Textures+Scenery(90)” training data sets were prepared similarly to “Textures+Scenery(45)”, except that different number of images (9-9, 15-15, and 45-45 respectively) were selected from the corresponding categories. Images from the “Food” category were not used in the training data. They were prepared for testing.
### Testing data
Generation of testing data was done similarly, except with 584 random positions instead of 776. In total, 6 sets of testing data were prepared. They are “Plain”, “Light plain”, “Textures”, “Metallic”, “Scenery”, and “Food”. The background images of these testing data set were randomly selected from their corresponding categories. Especially, for the “Light plain” testing set, 35 different colors of range \[125,255\] were selected instead of \[0,255\], making it different from the “Plain” testing set.
### Training
Several different VGG-16 networks were trained and compared using the various training data sets. The names of the networks are the same as the training data sets to clearly show the correspondence. The parameter settings of the VGG-16 neural network was shown in Fig.\[fig:vgg16\]. The initial weights were random. The learning rate was set to 1e-5. The training data set used for each network was divided by a ratio of 8:2 for training and validation. The epoch was set to 40. Convergence was faster for less random images (“Plain”, “Textures”, “Metallic surface”). The loss at the end of epoch 40 for these less varied image categories was also smaller, albeit overfitting existed in all trained networks, similar to [@tobin2017domain]. Each training time was on average 11 hours.
### Results
Table \[Tab:performance\_synthetic\_dataset\] shows the results of the trained networks and their performance on the 6 sets of testing data.
The results show that the ${MSE}_{all}$ on most testing data sets are quite large. The reason is because of the existence of outliers (the images which predicted outputs are completely off from the true outputs). Without counting the outliers, the MSE ($MSE_{no\_outlier}$) drops significantly. Thus, to minimize the effect of the outliers, the quadrants and iterative visual servoing method explained in section \[visual\_alignment\] was adopted.
Fig.\[fig:plain\_vs\_image\] shows how the use of images instead of plain backgrounds improves the network’s robustness. There is an increase in average performance ($R_{quadrant}$ of Table \[Tab:performance\_synthetic\_dataset\]) on all testing data sets. The network trained with the “Plain” data set performs better on the “Plain” testing data set due to overfitting, especially in cases where backgrounds have similar darkness to the hole.
![Performance of the networks trained with the “Plain” and “Image” data set. The vertical axis is the $R_{quadrant}$ of Table \[Tab:performance\_synthetic\_dataset\] in %.[]{data-label="fig:plain_vs_image"}](imgs/plain_vs_image.jpg){width=".97\columnwidth"}
Fig.\[fig:45backgrounds\] shows the $R_{quadrant}$ of the “Metallic(45)”, “Textures+Scenery(45)”, “Textures(45)”, and “Scenery(45)” training sets. The comparison indicates that: (1) With exception on the “Light plain”, the networks trained with a certain data set generally perform better on similar test sets. (2) Networks trained on certain data sets cannot perform as well on a background with high randomness like the “Food” test set. Thus, training with images of a certain data set can improve the network’s performance on similar test sets. On the other hand, for categories with less variety like “Metallic” or “Textures” (less than “Food”), improvements on the network performance can be obtained by training on data of higher variety, like “Scenery”. This is from the observation that the network trained with “Scenery” performs similarly to the one trained with “Textures” on the “Textures” test set, and performs better compared to the networks trained with “Metallic” on the “Metallic” test set.
![Performance the networks trained with “Metallic(45)”, “Textures+Scenery(45)”, “Textures(45)”, and “Scenery(45)” training sets. The vertical axis is the $R_{quadrant}$ of Table \[Tab:performance\_synthetic\_dataset\] in %.[]{data-label="fig:45backgrounds"}](imgs/45backgrounds.jpg){width="\columnwidth"}
Fig.\[fig:different\_number\_background\] compares the performance of networks trained data sets synthesized with different numbers of similar background images. The results imply that the number is not the only crucial parameter. Other forms of randomization including hole sizes and hole darkness also play important roles in the synthesis of the training data.
![Performance of the networks trained with “Textures+Scenery” data sets synthesized with different numbers of similar background image. The vertical axis is the $R_{quadrant}$ of Table \[Tab:performance\_synthetic\_dataset\] in %.[]{data-label="fig:different_number_background"}](imgs/different_number_background.jpg){width="\columnwidth"}
Real-world Experiments {#exp_real_world}
----------------------
We performed real-world experiments using the networks trained with the “Image” data set and the “Plain” data set, for they have the best and worst performance. Four different surfaces, as is shown in Fig.\[fig:surface\_real\_world\_exp\], were used in the experiments. A success execution is judged to be when the peg is inserted within 90 sec.
![The four surfaces used in real-world experiments. (a) White. (b) Brown. (c) Pink. (d) Sky.[]{data-label="fig:surface_real_world_exp"}](imgs/surface_real_world_exp.jpg){width=".97\columnwidth"}
The robot we used to do real-world experiments is a UR3 robot with a FT300 force sensor and a Roboti-85 gripper. Two in-hand cameras were used to collect multi-view images. Fig.\[fig:experiment\_setup\] shows the experimental setup. The robot was made to insert a 75$\times$10 $mm$ peg into a hole. The taskboard in the left of Fig.\[fig:experiment\_setup\] is the base with the hole to be inserted. The lenience of the hole for the experiment is 0.4$mm$. The specification of the computer used was the same as the one used to train the deep neural network.
![The experimental setup. (a) The overall view on the experimental setup. (b) A close-up view of the Robotiq-85 gripper with the two in-hand RGB cameras.[]{data-label="fig:experiment_setup"}](imgs/experiment_env.jpg){width=".95\columnwidth"}
The parameters for the spiral search were $r_{init}$=0.3$mm$, $r_{max}$=7.0$mm$, $\delta\theta$=12.5$^\circ$, $\delta$$r$=0.3$mm$, and $F_{max}$=20$N$. The parameters for the iterative visual servoing were $A$=10$mm$, $n$=10, $n_{run}$=5. The parameters of the impedance control were $c$ = \[50,50,50,1,1,1\] for the damper and $k$ = \[100,100,100,100,100,100\] for the spring.
Table \[Tab:performance\_real\_world\] shows the success rates of the real-world experiments. Ten times of trial are performed for each training set and testing surface combination. The network trained with the “Image” data set can successfully ignore the variations of the “White”, “Brown”, and “Pink” surfaces and correctly predict the correct quadrants. Fig.\[fig:real\_exp\_image\](a) shows an example of the success sequence on the “Pink” surface. The network trained with “Plain” can also correctly predict the quadrant and successfully performs insertion on the “White” and “Brown” surfaces. Although a few overshoots were spotted, the quadrant-based visual servoing compensated them after several iterations. An example is shown in Fig.\[fig:real\_exp\_image\](b).
![Some snapshots of the real-world executions from the results shown in Table \[Tab:performance\_real\_world\]. (a) A successful execution using the network trained with “Image” and tested on the “Pink” surface ($\oplus$ category). (b) A successful execution (with overshoots) using the network trained with “Plain” and tested on the “White” surface ($\ominus$ category). (c) A failure case using the network trained with “Image” and tested on the “Sky” surface ($\bigtriangleup$ category). (d) A failure case using the network trained with “Plain” and tested on the “Sky” surface ($\times$ category).[]{data-label="fig:real_exp_image"}](imgs/real_exp_image.png){width="1.\columnwidth"}
The failure appears with the “Sky” surface for both networks and the “Pink” surface for the network trained with the “Plain” data set. For the network trained with the “Image” data set and examined using the “Sky” surface and the network trained with the “Plain” data set and examined using the “Pink” surface, errors sometimes occur, resulting into 3/10 and 4/10 success rates in Table \[Tab:performance\_real\_world\] respectively. Fig.\[fig:real\_exp\_image\](c) shows an example of the failure sequence. For the network trained with the “Plain” data set and examined using the “Sky” surface, the success rate is 0/10. Fig.\[fig:real\_exp\_image\](d) is an example of the failure.
Table \[Tab:real\_world\_time\] compares the time cost of 10 executions on the “White” surface with and without the learning-based visual servoing. The initial positions errors were randomly set (as is shown in the first column of Table \[Tab:real\_world\_time\]). When the initial position error is large, the robot system could finish an insertion in less than 70 sec. with the learning-based visual servoing. In contrast, the execution costs larger than 90 sec. or fails. An exception is $D_{euclidean}$=4.0$mm$. In this case, the initial position error is small. A simple spiral search could quickly find the hole. The result demonstrates that the proposed method improves search efficiency when the position of the hole has large uncertainty.
Conclusions {#conclusion}
===========
In conclusion, we developed a learning-based visual servoing method to quickly insert pegs into uncertain holes. The method used a deep neural network trained on synthetic data to predict the quadrant of a hole, and used iterative visual servoing to move the peg towards the hole step by step. The synthetic data was generated by cutting and pasting gripper template masks on random images, which allowed extremely fast synthetic data generation. Performance of different training data sets was compared. Training with images from categories of higher variety can lead to better performance, even for testing with images from categories of less variety. Real-world experiments showed that the proposed method is robust to various surface backgrounds. The system is generally faster compared to a peg-in-hole assembly using only a spiral search, unless the initial error is very small.
[^1]: $^{1}$[Graduate School of Engineering Science, Osaka University, Japan.]{} $^{2}$[National Inst. of AIST, Japan.]{} \*[Correspondent author: Weiwei Wan, ]{}[wan@sys.es.osaka-u.ac.jp]{}
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---
author:
- |
Pedro F. González-Díaz.\
Instituto de Matemáticas y Física Fundamental\
Consejo Superior de Investigaciones Científicas\
Serrano 121, 28006 Madrid (SPAIN)\
date: 'March 1, 1993'
title: The Quantum Sphaleron
---
A gravitational instanton is found that can tunnel into a new more stable vacuum phase where diffeomorphism invariance is broken and pitchfork bifurcations develop. This tunnelling process involves a double sphaleron-like transition which is associated with an extra level of quantization which is above that is contained in quantum field theory.
1\. Many field theories have a degenerate vacuum structure showing more than one potential minimum. Such a complicate vacuum structure makes it possible the occurrence at zero energy of quantum transitions describable as instantons between states lying in the vecinity of different vacua. Instantons are localized objects which correspond to solutions of the Euclidean equations of motion with finite action \[1,2\]. On the other hand, for nonzero energies, transitions between distinct vacua may also occur classically by means of sphalerons. Whereas instantons tunnel from one potential minimum to another by going below the barrier, sphalerons do the transit over the barrier. Sphalerons correspond to the top of this barrier and are unstable classical solutions to the field equations which are static and localized in space as well \[3,4\].
In this letter, we want to explore the possibility of new sphaleron-like transitions which are classically forbidden though they may still take place in the quantum-mechanical realm. Such transitions would typically pertain to nonlinear systems showing bifurcations phenomena. In order to see how these nonclassical transitions may appear, let us consider a theory with the Euclidean action $$S_{E}(\varphi , x)
=-\frac{1}{2}\int_{\eta_{i}}^{\eta_{f}}d\eta\varphi^{2}((\frac{dx}{d\eta})^{2}
+x^{2}-\frac{1}{2}m^{2}\varphi^{2}x^{4}),$$ where $m$ is the generally nonzero mass of a dimensionless constant scalar field $\varphi$ and $\eta$ denotes a dimensionless Euclidean time $\eta=\int\frac{d\tau}{x}$. The sign for the Euclidean action would be positive when we choose the usual Wick rotation $t\rightarrow -i\tau$ (clockwise) or negative for a Wick rotation $t\rightarrow +i\tau$ (anti-clockwise). If the ’energy’ of the particles is positive, then one should use clockwise rotation, but if it is negative the rotation would be anti-clockwise \[5\]. It will be seen later on that (1) corresponds to the case of a massive field conformally coupled to gravity, with $x$ playing the role of the Robertson-Walker scale factor. Thus, since the gravitational energy associated with the scale factor is negative, one should rotate $t$ not to $-i\tau$, but to $+i\tau$, such as it is done in (1), and therefore the semiclassical path integral involving action $S_{E}$ may be generally interpreted as a probability for quantum tunnelling \[5\]. In the classical case, if the field $\varphi$ is axionic, then it becomes pure imaginary, i.e. $\varphi=i\varphi_{0}$, with $\varphi_{0}$ real. In this case, the potential becomes $$V(x) =
\varphi_{0}^{2}(\frac{1}{2}x^{2}+\frac{1}{4}m^{2}\varphi_{0}^{2}x^{4}),$$ the theory has just one zero-energy vacuum, and the solution to the classical equations of motion is $\bar{x}=0$, with Euclidean action $S_{E}=0$.
Expanding about the classical solution we obtain the usual path integral at the semiclassical limit \[6\] $$<0\mid e^{-HT/\hbar}\mid
0>=N[det(-\partial_{\tau}^{2}+\omega^{2})]^{-\frac{1}{2}}(1+O(\hbar)),$$ where $N$ is a normalization constant and $\omega^{2}=V''(0)$, with $'$ denoting differentiation with respect to $x$. The ground-state solution to the wave equation for the operator $-\partial_{\tau}^{2}+\omega^{2}$ which corresponds here to a harmonic oscillator with substracted zero-point energy can be written as $$\psi_{0}=\omega^{-1}e^{-\omega T}\sinh[\omega(\tau+\frac{T}{2})].$$ Then, the path integral becomes constant and given by $$<0\mid e^{-HT/\hbar}\mid
0>=(\frac{\omega}{\pi\hbar})^{\frac{1}{2}}(1+O(\hbar)).$$ If the zero-point energy had not been substracted, then Eqn. (4) would also contain the well-known time-dependent factor $e^{-\frac{1}{2}\omega T}$ \[6\].
Let us now consider $\varphi^{2}$ as being the control parameter for the nonlinear dynamic problem posed by action (1). All the values of $\varphi^{2}$ corresponding to an axionic classical field will be negative and, in the classical case, can be continuously varied first to zero (a zero potential critical point) and then to positive values (the field $\varphi$ has become real, no longer axionic). The associated variation of the dynamics will represent a typical classical bifurcation process that can finally lead to spontaneous breakdown of a given symmetry \[7\]. In the semiclassical theory this generally is no longer possible however. Not all values of the squared field $\varphi^{2}$ are then equally probable. For most cases, large ranges of $\varphi^{2}$-values along the bifurcation itinerary are strongly suppressed, and hence the bifurcation mechanism would not take place. Nevertheless, there could still be sudden reversible quantum jumps from the most probable negative values to the most probable positive values of $\varphi^{2}$. Such jumps would be expressible as analytic continuations in the field $\varphi$ to and from its real axis, or alternatively, in $x$ to and from its imaginary values, lasting a very short time. The system will first go from the bottom of potential (2) for $\varphi^{2}<0$ to the sphaleron point of potential (see Fig.1) $$V(x)=\varphi
'_{0}^{2}(-\frac{1}{2}x^{2}+\frac{1}{4}m^{2}\varphi_{0}'^{2}x^{4}),$$ without changing position or energy, and then will be perturbed about the sphaleron saddle point to fall into the broken vacua where the broken phase condenses for a short while, to finally redo all the way back to end up at the bottom of potential (2) for $\varphi^{2}<0$ again. The whole process may be denoted as a quantum sphaleron transition and it is assumed to last a very short time and to occur at a very low frequency along the large time $T$. Therefore, one can use a dilute spahaleron approximation which is compatible with our semiclassical approach. Thus, for large $T$, besides individual quantum sphalerons, there would be also approximate solutions consisting of strings of widely separated quantum sphalerons. In analogy with the instanton case \[6\], we shall evaluate the functional integral by summing over all such configurations, with $n$ quantum sphalerons centered at Euclidean times $\tau_{1}$,$\tau_{2}$,...,$\tau_{n}$. If it were not for the small intervals containing the quantum sphalerons, $V''$ would equal $\omega^{2}$ over the entire time axis, and hence we would obtain the same result as in (4). However, the small intervals with the sphalerons correct this expression. In the dilute sphaleron approximation, instead of (4), one has $$(\frac{\omega}{\pi\hbar})^{\frac{1}{2}}(-\frac{1}{2}K_{sph})^{n}(1+O(\hbar)),$$ where $K_{sph}$ is an elementary frequency associated to each quantum sphaleron, and the sign minus accounts for the feature that particles acquire a negative energy below the sphaleron barrier (Fig.1). Note that the action changes sign as one goes from potential (2) to potential (5) below such a barrier. The factor $\frac{1}{2}$ has been introduced to account for the feature that the sphaleron transition must take two particles, both at the same time, from the bottom of the potential (2) for $\varphi^{2}<0$ to make the transition to the two minima of potential (5) and back to the bottom of potential (2) simultaneously. After integrating over the locations of the sphaleron centers, the sum over $n$ sphalerons produces a path integral $$<0\mid e^{-HT/\hbar}\mid
0>_{sph}=(\frac{\omega}{\pi\hbar})^{\frac{1}{2}}
e^{-\frac{1}{2}K_{sph}T}(1+O(\hbar)),$$ which is proportional to the semiclassical probability of quantum tunnelling from $\bar{x}=0$ first to $\bar{x}_{\pm}=\pm(m\varphi'_{0})^{-1}$ and then to $\bar{x}=0$ again.
In Eqn.(7) we have summed over any number of sphalerons, since all the small time intervals start and finish on the axis $x=0$, at the bottom of potential (2). Approximation (7) corresponds \[6\] to a ground-state energy $E_{0}=\frac{1}{2}\hbar K_{sph}$. Thus, the effect of the quantum sphalerons should be the creation of an extra nonvanishing zero-point energy which must correspond to a further level of quantization which is over and above that is associated with the usual second quantization of the harmonic oscillator.
2\. As pointed out before, an Euclidean action with the same form as (1) arises in a theory where a scalar field $\Phi$ with mass $m$ couples conformally to Hilbert-Einstein gravity. Restricting to a Robertson-Walker metric with scale factor $a$ and Wick rotating anti-clockwise, the Euclidean action for this case becomes $$I=-\frac{1}{2}\int d\eta
N(\frac{\dot{\chi}^{2}}{N^{2}}+\chi^{2}
-\frac{\dot{a}^{2}}{N^{2}}-a^{2}+m^{2}a^{2}\chi^{2}),$$ where the overhead dot means differentiation with respect to the conformal time $\eta=\int d\tau/a$, $\chi=(2\pi^{2}\sigma^{2})^{\frac{1}{2}}a\Phi$, $N$ is the lapse function and $\sigma^{2}=2G/3\pi$. The equations of motion derived from (8) are (in the gauge $N=1$) $\ddot{\chi}=\chi+m^{2}a^{2}\chi$ and $\ddot{a}=a-m^{2}\chi^{2}a$. We note that these two equations transform into each other by using the ansatz $\chi=ia$. Invariance under such a symmetry manifests not only in the equations of motion, but also in the Hamiltonian action and four-momentum constraints. The need for Wick rotating anti-clockwise becomes now unambiguous \[8\], since all the variable terms in the action become associated with energy contributions which are negative if symmetry $\chi=ia$ holds. Without loss of generality, the equations of motion can then be written as the two formally independent expressions $\ddot{\chi}=\chi-m^{2}\chi^{3}$ and $\ddot{a}=a+m^{2}a^{3}$. If $\chi=ia$, then $\Phi$ becomes a constant axionic field $\Phi=i(2\pi^{2}\sigma^{2})^{-\frac{1}{2}}$. Therefore, re-expressing action $I$ in terms of the field $\chi$ alone, one can write for the Lagrangian in the gauge $N=1$ $$L(\varphi,a)=-(\frac{1}{2}\varphi^{2}\dot{a}^{2}+\frac{1}{2}\varphi^{2}a^{2}
-\frac{1}{4}m^{2}\varphi^{4}a^{4}+\frac{1}{2}R_{0}^{2}),$$ where $\varphi=\frac{\Phi}{m_{p}}$, $m_{p}$ is the Planck mass, and $R_{0}^{2}$ is an integration constant which has been introduced to account for the axionic character of the constant field implied by $\chi=ia$; such an imaginary field would require a constant additional surface term to contribute the action integral. Besides this constant term $\frac{1}{2}R_{0}^{2}$, (9) exactly coincides with the Lagrangian in (1). For the axionic field case where the symmetry $\chi=ia$ holds, $\varphi^{2}=-\varphi_{0}^{2}$, with $\varphi_{0}$ real. Then the solution to the equations of motion and Hamiltonian constraint is $$a(\tau)=(m\varphi_{0})^{-1}[(1+2m^{2}R_{0}^{2})^{\frac{1}{2}}
\cosh(2^{\frac{1}{2}}m\varphi_{0}\tau)-1]^{\frac{1}{2}},
\chi=ia(\tau),$$ which represents an axionic nonsingular wormhole spacetime.
If we rewrite (9) as a Lagrangian density $L(\Phi,a)=m_{p}^{2}L(\varphi,a)/a^{4}$, it turns out that, although the Lagrangian density as written in the form $L(\Phi,a)$ looks formally similar (except the last term in the potential) to that for an isotropic and homogeneous Higgs model in Euclidean time, it however preserves all the symmetries of the theory intact. Indeed, the field $\Phi$ is not but a simple imaginary constant $\Phi=i(2\pi^{2}\sigma^{2})^{-\frac{1}{2}}$. None the less, if $\Phi$ is shifted by some variable $\rho$, such that $\Phi\rightarrow\phi=i\xi+\rho$, where $\xi=(2\pi^{2}\sigma^{2})^{-\frac{1}{2}}$, while keeping the same real scale factor, then symmetry $\chi=ia$, and hence diffeomorphism invariance, would become broken. In such a case, the Lagrangian density $L\equiv L(\phi,a)$ resulting from $L(\Phi,a)$ is seen (Note that if we let $\Phi$ to be time-dependent, then the kinetic part of the Lagrangian $L(\Phi,a)$ becomes $-[\frac{1}{2}(\frac{\dot{a}}{a^{2}})^{2}\Phi^{2}$ $+(\frac{\dot{a}}{a^{2}})\Phi\frac{\dot{\Phi}}{a}
+\frac{1}{2}(\frac{\dot{\Phi}}{a})^{2}$\]) to describe a typical Higgs model in the isotropic and homogeneous euclidean framework for a charge-$Q$ field $\phi$, a massless gauge field $A\equiv A(\tau)=\frac{TrK}{eQ}$ (with $e$ an arbitrary gauge coupling and $K$ the second fundamental form), and variable $tachyonic$ mass $\mu=\frac{1}{a}$, as now the last term in the potential does not depend on $\phi$ and becomes thereby harmless for the Higgs model. In principle, the Lagrangian could have been written in a form other than the given in (9) by a different use of symmetry $\chi=ia$, allowing the scalar field to become complex afterwards. The reason why the system should spontaneously choose breaking the symmetry from the particular Lagrangian form given by (9) simply is that, in so doing, it will obtain the most stable vacuum configuration.
The integration constant $R_{0}^{2}$ in (9) corresponds to the inclusion of axionic surface terms in the action integral. For a constant real scalar field, such surface terms should produce a generally different integration constant $R'_{0}^{2}$ whose sign is the opposite to that for $R_{0}^{2}$. Therefore, the solutions to the equations of motion and Hamiltonian constraint corresponding to a constant real scalar field $\varphi^{2}=\varphi '_{0}^{2}$ are $$a_{\pm}(\tau)=(m\varphi
'_{0})^{-1}[1\pm(1-2m^{2}R'_{0}^{2})^{\frac{1}{2}}\cosh(2^{\frac{1}{2}}m\varphi
'_{0}\tau)]^{\frac{1}{2}}.$$ Solutions $a_{+}$ and $a_{-}$ lie in the two disconnected, classically allowed regions for which the potential is negative. It is only $a_{+}$ which can be thought of as giving the metric of a wormhole. $a_{-}$ is a singular solution confined to be $\leq a_{-}(0)$, and hence the isotropic manifold provided with a metric given by $a_{-}$ is disconnected from the wormhole manifold and whereby from the two asymptotically flat regions, for any given allowed values of $R'_{0}^{2}$ and $m^{2}$ ($m^{2}R'_{0}^{2}\leq\frac{1}{2}$). The values taken by $a_{-}$ along the time interval (0, $\tau_{0}=(2m^{2}\varphi
'_{0}^{2})^{-\frac{1}{2}}\cosh^{-1}[(1-2m^{2}R'_{0}^{2})^{-\frac{1}{2}}]$) are nonzero and real, inducing the appearance of a real nonzero disconnected region inside the wormhole manifold. Hence, the wormhole manifold will be doubly connected \[9\].
The real classical solution corresponding to the equilibrium minimum of the potential for axionic fields is $\bar{a}=0$, and that for real fields are $\bar{a}_{\pm}=\pm(m\varphi
'_{0})^{-1}$. Therefore, relative to the simply connected inner topology implied by (10), the doubly-connected topology implied by (11) represents an actual pitchfork bifurcation.
We can see now why not all values of $\varphi_{0}$ and $\varphi
'_{0}$ are allowed semiclassically. Although one actually could also gauge the imaginary part of $\phi$ to any value other than $\xi$, since the action $I$ depends on $\varphi_{0}^{-2}$ or $\varphi '_{0}^{-2}$ through solutions (10) and (11), very small values of the constant fields will make this action very large and hence the semiclassical probability $e^{-I}$ becomes vanishingly small. Thus, interpreting $\varphi^{2}$ as a control parameter it turns out that the bifurcation process cannot be reached in a continuous, deterministic way. Moreover, purely real values of $\varphi$ are only compatible with symmetry breaking if the Higgs mechanism is defined in the unitary gauge, and this requires specific boundary conditions for the path integral.
Disregarding the constant term of the potential in (9) to make this potential vanish at its minima, we obtain the same situation as in Fig.1, with $\omega\sim 1$ in Planck units. If it were not for the short intervals where sphalerons are quantically induced, the contribution of the path integral $<0\mid e^{-HT/\hbar}\mid 0>$ to the quantum state of the system would simply be a constant factor. However, if such sphaleron transtions are taken into account, then the contribution of the path integral $<0\mid e^{-HT/\hbar}\mid 0>_{sph}$ will introduce a time-dependent factor like (7), with $\omega\sim 1$, in the full quantum state. Since time separation between the initial and final states cannot be known, one should integrate over $T$ to finally obtain for the full quantum state $$\sum_{j=1}^{\infty}\int_{0}^{\infty}dT\Psi[a,\Psi]\Psi[a',\Psi
']e^{-\frac{1}{2}jK_{sph}T},$$ where the $\Psi '$s are the wave functions for the initial and final states. The full state (12) would give the density matrix for a nonsimply connected wormhole \[10\]. If we take for the small intervals where sphaleron transitions occur a most probable value of the order the Planck time, then $K_{sph}\sim(\hbar G)^{-\frac{1}{2}}$. It follows then that in the limit where either $\hbar\rightarrow 0$ or $G\rightarrow 0$, or both, the path integral vanishes. This suggests that the extra level of quantization introduced by quantum sphalerons can only appear when quantum-gravity effects are considered, or in other words, quantum gravity involves an extra level of quantization which is over and above that is contained in nongravitational quantum theory.
A caveat is worth mentioning finally. It has been pointed out \[8\] that a rotation $t\rightarrow +i\tau$ would imply a repulsive gravitational regime. Nevertheless, a probability functional which is factorizable as a product of equal wave functions \[11\] can no longer represent the ground state if diffeomorphism invariance is preserved, for all eigenenergies are strictly zero \[5\]. However, if diffeomorphism invariance is broken, so that a ground state as (12) becomes well defined, then such a ground state would be below the barrier for $t\rightarrow +i\tau$, in a situation where the Euclidean action is negative and corresponds therefore to a positive gravitational constant and hence to attractive gravity.
$Acknowledgements$. This work was supported by a $CAICYT$ Research Project N’ PB91-0052.
\[1\] A.A. Belavin, A.M. Polyakov, A.S. Schwarz and Yu.S. Tyupkin, Phys. Lett. B59(1975)85.
\[2\] G.’t Hooft, Phys. Rev. D14(1976)3432.
\[3\] N.S. Manton, Phys. Rev. D28(1983)2019.
\[4\] F.R. Klinkhamer and N.S. Manton, Phys. Rev. D30(1984)2212.
\[5\] A.D. Linde, [*Inflation and Quantum Cosmology*]{} (Academic Press, Boston, 1990).
\[6\] S. Coleman, [*The Uses of Instantons*]{} in The Whys of Subnuclear Physics, ed. A. Zichichi (Plenum Press, New York, 1979).
\[7\] G. Gaeta, Phys. Rep. 189(1990)1.
\[8\] For a discussion on the sign of Wick rotation in quantum gravity, see \[5\] and also the contributions of A.D. Linde and S.W. Hawking in [*300 Years of Gravitation*]{} (Cambridge Univ. Press, Cambridge, 1987).
\[9\] P.F. González-Díaz, Phys. Rev. D45(1992)499.
\[10\] P.F. González-Díaz, Nucl. Phys. B351(1991)767.
\[11\] S.W. Hawking, Phys. Rev. D37(1987)904.
LEGEND FOR FIGURE
Fig.1.- Bifurcation itinerary from $\varphi^{2}<0$ to $\varphi^{2}>0$ for the potential in Eqn. (1) for $m=0.5$.
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---
author:
- 'P. Ventura'
- 'F. D’Antona'
date: 'Received ... ; accepted ...'
title: 'Full computation of massive AGB evolution. II. The role of mass loss and cross-sections'
---
Introduction
============
Since the pioneering works by Schwarzschild & Harm (1965, 1967), and Iben (1975, 1976), it is now well known that intermediate mass stars (i.e. stars with initial masses 1 $\leq$ M/$\leq$ 8, hereinafter IMS) soon after the exhaustion of central helium experience a phase of thermal pulses (TPs), during which a CNO burning shell supplies for most of the time the global nuclear energy release; periodically, a He-burning shell is activated in thermally unstable conditions, triggering an expansion of all the outer layers, with the consequent extinction of CNO burning (Lattanzio & Karakas 2001). During the AGB evolution these stars suffer a strong mass loss, which ultimately peels-off all the envelope mass, leaving a carbon-oxygen compact remnant which evolves as a white dwarf. The base of the external convective zone of the most massive IMS may become so hot ($T_{\rm bce} \geq 30\times 10^6$ K) to favor an intense nucleosynthesis (hot bottom burning, HBB), whose results can be directly seen at the surface of the star due to the rapidity of convective motions (e.g. Ventura et al. 2002).
The ejecta of these stars might thus pollute the surrounding medium with material which was at least partially nuclearly processed: this is the reason why this class of objects has been invoked as a possible explanation of the chemical anomalies observed at the surface of giants and turn-off globular clusters stars (see e.g. Gratton et al. 2004), in what is commonly known as the self-enrichment scenario. An early generation of IMS evolved within the first $\sim 100 - 200$ Myr of the cluster life, contaminated the interstellar-medium with gas which would be already nuclearly processed; this gas might have favored the formation of a later generation of stars, which would then show the observed chemical anomalies (Cottrell & Da Costa 1981; D’Antona et al. 1983; Ventura et al. 2001, 2002).
While there is a general agreement that the solution of this problem may be looked for in early AGB pollution, (Gratton et al. 2004), the quantitative agreement between the models and the abundance patterns shown by GC stars is not good (Denissenkov & Herwig 2003; Denissenkov & Weiss 2004). On the other hand, the AGB evolution of these stars is found to be strongly dependent on the convective model which is used to find out the temperature gradient within the external convective zone (Renzini & Voli 1981; Blöcker & Schonberner 1991; Sackmann & Boothroyd 1991; D’Antona & Mazzitelli 1996; Ventura & D’Antona 2005, hereinafter paper I). The chemical content of their ejecta, in particular for some key-elements which are anticorrelated like oxygen and sodium, and magnesium and aluminum, is strongly dependent not only on convection, but also on the assumed mass loss rate and on the nuclear reaction rates. At the moment these uncertainties seriously undermine the predictive power of AGB models, and thus limit the predictions which can be made concerning their role within the framework of the self-enrichment scenario.
We investigate the AGB evolution of initial masses 3 $\leq$ M/$\leq$ 6.5, and focus our attention on their main physical properties, and on the chemical content of their ejecta. In paper I we explored the dependence of the results on the convective model. In this work we complete the exploration by investigating the sensitivity of the results on: 1) the nuclear cross sections: we compare two sets of models calculated by assuming the Angulo et al. (1999) NACRE cross sections and those by Caughlan & Fowler (1988, hereinafter CF88); 2) the mass loss rate.
The evolutionary code
=====================
The stellar evolutions discussed in this paper were calculated by the code ATON2.1, a full description of which can be found in Ventura et al. (1998) (ATON2.0 version). The latest updates of the code, concerning the nuclear network, are given in paper I. The interested reader may find on the afore mentioned papers a detailed description of the numerical structure of the code, and of the macro and micro-physics which is used to simulate the stellar evolutions.
Convection
----------
The code allows us to calculate the temperature gradient within instability regions either by adopting the traditional MLT (Vitense 1953; Böhm-Vitense 1958), or the FST model (Canuto & Mazzitelli 1991; Canuto et al. 1996) for turbulent convection. The interested reader may find in Canuto & Mazzitelli (1991) a detailed description of the physical differences between the two models.
As we shall see, during the AGB evolution a non negligible fraction of the global nuclear release is generated within the convective envelope, therefore it is mandatory to adopt a diffusive approach, treating simultaneously mixing and nuclear burning. We therefore solve for each element the diffusion equation (Cloutman & Eoll 1976):
$$$$
\left( {dX_i\over dt} \right)=\left( {\partial X_i\over \partial t}\right)_{nucl}+
{\partial \over \partial m_r}\left[ (4\pi r^2 \rho)^2 D {\partial X_i
\over \partial m_r}\right] \label{diffeq}
$$$$
stating mass conservation of element $i$. The diffusion coefficient $D$ is taken as $D={1\over 3}ul$, where $u$ is the convective velocity and $l$ is the convective scale length. We allow velocity to decay exponentially starting from the formal convective boundaries as:
$$$$
u=u_b exp \pm \left( {1\over \zeta f_{thick}}ln{P\over P_b}\right)
$$$$
where $u_b$ and $P_b$ are, respectively, turbulent velocity and pressure at the convective boundary, P is the local pressure, $\zeta$ a free parameter connected with the e-folding distance of the decay, and $f_{thick}$ is the thickness of the convective regions in fractions of $H_p$.
A detailed description of the treatment of convective velocities in the proximity of the formal borders of the convective zones (fixed by the Schwarzschild criteria) can be found in Sect.2.2 of Ventura et al.(1998). In the same paper (Sect.4.2) the interested reader may also find an extensive discussion on the extra-mixing determined by the use of a non-zero $\zeta$.
The models presented in this paper adopt the FST convection, and the parameter $\zeta$ is fixed at $\zeta=0.02$. No extra-mixing has been assumed from the base of the convective envelope: therefore the extension of the various dredge-up episodes, and the consequent changes of the surface chemical composition, must be considered as lower limits.
Nuclear network
---------------
The nuclear network includes 30 chemical species up to $^{31}P$ and 64 nuclear reactions. The list of all the reactions included in the nuclear network can be found in paper I. The relevant cross-sections are taken either from Caughlan & Fowler (1988, CF88) or from Angulo et al (1999, NACRE). In the range of temperatures which are of interest here ($7.5 \leq \log(T) \leq 8.2$, which are the typical values at the base of the external convective zone of massive AGBs) the largest differences between the two sets of cross-sections are the following:
- [$^{17}$O destruction by proton fusion is achieved much more easily in the NACRE case; also, contrary to CF88, the reaction $^{17}$O(p,$\alpha$)$^{14}$N is favored with respect to $^{17}$O(p,$\gamma$)$^{18}$F.]{}
- [The cross-sections of the reaction $^{22}$Ne(p,$\gamma$)$^{23}$Na are larger by $\sim$ 3 orders of magnitude in the NACRE case, which makes sodium production much easier. As for sodium burning by proton fusion, the channel leading to the formation of $^{24}$Mg is favored in the NACRE case.]{}
Mass loss
---------
The mass loss rate is calculated according to Blöcker (1995), who modifies the Reimer’s formula in order to simulate the strong mass loss suffered by these stars as they climb along the AGB. The complete expression is: $$$$
\dot M=4.83\times 10^{-9} M^{-2.1} L^{2.7} \dot M_R
$$$$ where $\dot M_R=10^{-13}\eta_R LR/M$ is the canonical Reimer’s rate, and $\eta_R$ is a free parameter directly connected with the mass loss rate.
$\dot M$ was described according to Eq.3, with the parameter $\eta_R=0.02$ for the “standard” case; we then consider evolutions with $\eta_R=0.1$ and $\eta_R=0.2$. In all cases mass loss was applied for all the evolutionary phases.
Model inputs
------------
We evolved models with initial masses $3M_{\odot}
\leq M \leq 6.5M_{\odot}$ starting from the pre-main sequence along the whole TPs phase. Above the upper mass limit, models ignite carbon in the center, skipping the AGB phase. Below the lower limit models do not achieve HBB conditions.
When the envelope mass becomes “small” ($M_e < 1M_{\odot}$) a much higher temporal resolution is required, which renders the computations extremely time-consuming; since the chemical yields are almost unaffected by the following phases, we decided to stop the evolutions when the mass of the envelope falls below $\sim 0.5M_{\odot}$.
We adopted an initial metallicity, $Z$, typical of those globular clusters (GCs) like NGC6752, M3, M13, whose stars show the largest chemical anomalies, i.e. $Z=0.001$ and $Y=0.24$. For all the elements included in our network, we adopted solar-scaled initial abundances. This is to be taken into account if we want to compare the results with observations, as the starting initial mass fractions of abundant elements play a role in the determination of the final yield. E.g., as \[O/Fe\]$\sim +0.3$ in population II stars, this initial abundance will be remembered in the evolution with oxygen depletion. Numerical tests we pursued show that the results for oxygen may be roughly scaled up by the initial enhancement with respect to the solar scaled value. For instance, if the solar scaled model produces a yield with \[O/Fe\]=–0.5, the yield starting from initial \[O/Fe\]=+0.3 would have been \[O/Fe\]=–0.2.
NACRE results
=============
The pre-AGB phase
-----------------
Table \[physics\] summarizes the main physical properties of our models, related to the evolutionary phases preliminary to the AGB evolution. During the main sequence phase the models develop a central convective region which progressively shrinks in mass, with a maximum extension ranging from $\sim$ 0.72for the 3 model up to $\sim$2for the 6.5 model.
The total duration of the H-burning phase (t(H))is a decreasing function of mass: the less massive model, with initial mass M=3, consumes central hydrogen in $\sim 275$ Myr, while the 6.5 model keeps burning hydrogen for 54 Myr (fig. \[times\], top panel).
$$
[c c c c c c c c]{} M & t(H)\^a & M\_[c,H]{}\^b & t(He)\^a & M\_[c,He]{}\^c & M\_[1dup]{}\^d & M\_[2dup]{}\^e & (M\_[2dup]{})\^f\
3.0 & 276 & 0.72 & 56 & 0.28 & 0.85 & 0.78 & 0.01\
3.5 & 195 & 0.93 & 33 & 0.32 & 1.19 & 0.84 & 0.05\
4.0 & 145 & 1.04 & 23 & 0.36 & 1.50 & 0.87 & 0.15\
4.5 & 112 & 1.21 & 16 & 0.44 & 1.86 & 0.90 & 0.26\
5.0 & 90 & 1.40 & 12 & 0.48 & 2.10 & 0.93 & 0.37\
5.5 & 75 & 1.57 & 10 & 0.57 & 2.42 & 0.97 & 0.48\
6.0 & 63 & 1.85 & 7.6 & 0.62 & 2.70 & 1.00 & 0.58\
6.5 & 54 & 2.05 & 6.3 & 0.70 & 2.85 & 1.05 & 0.68\
$$
Times are expressed in Myr.
Maximum extension (in $M_{\odot}$) of the convective core during H-burning.
Maximum extension (in $M_{\odot}$) of the convective core during He-burning.
Mass coordinate (in $M_{\odot}$) of the innermost layer reached during the first dredge-up.
Mass coordinate (in $M_{\odot}$) of the innermost layer reached during the second dredge-up.
Amount (in $M_{\odot}$) of dredged-up material nuclearly processed by CNO burning.
During the first dredge-up, following H-exhaustion, the convective envelope reaches stellar layers which were previously touched by CNO burning via the CN cycle, with the conversion of some $^{12}$C to $^{14}$N: consequently, the surface $^{14}$N is increased by a factor of $\sim 2$, while $^{12}$C decreases from the initial value of X($^{12}$C)=1.73 $\times 10^{-4}$ to X($^{12}$C)=1.3 $\times 10^{-4}$. The lithium surface abundance drops by a factor of $\sim 50$, because surface lithium is mixed within an extended region where lithium was previously destroyed via proton fusion; this drop is dependent on the stellar mass, and a spread of a factor of $\sim 2$ is found among the models.
During the following phase of core helium burning a convective core is formed, again with a dimension increasing with mass: it is $\sim 0.28M_{\odot}$ for the $3M_{\odot}$ model, while it is $\sim 0.7M_{\odot}$ for the $6.5M_{\odot}$ model (see the 5th column of Table \[physics\]). We see from Fig. \[times\] (bottom panel) that the ratio between the He-burning (t(He)) and the MS times is decreasing with mass, ranging from $\sim 20\%$ to slightly higher than $10\%$ for the 6.5M$_{\odot}$ model. Once helium is burnt-out in the stellar core, $3\alpha$ reactions carry on in an intermediate layer, triggering a general expansion of the structure, which eventually extinguishes the CNO burning shell. The general cooling of the star favors the formation of a very deep and extended external convective zone, in what is commonly known as the second dredge-up episode.
From the top panel of fig. \[2dup\] we see that the mass coordinate corresponding to the maximum penetration of the outer envelope is slightly increasing with mass, with a difference of $\sim 0.2$ between the 3 and 6.5$M_{\odot}$ models. The most interesting quantity is however shown in the bottom panel of the same figure, where we report the variation as a function of the initial mass of $\delta M=M_{CNO}-M_{min}$, where $M_{CNO}$ and $M_{min}$ are, respectively, the location of the CNO burning shell immediately before the second dredge-up, and the minimum point (in mass) reached by the base of the outer convective zone. $\delta M$ is therefore a measure of the amount of processed material which is carried to the surface during the second dredge-up. We see that a poor mixing is expected in the 3$M_{\odot}$ model, while in the 6.5$M_{\odot}$ case $\sim 0.7$M$_{\odot}$ of CNO processed material is mixed with the surface layers. During the second dredge-up the surface $^{14}$N is increased by another factor of $\sim 2$, the carbon abundance decreases to X($^{12}$C) $\sim 1.15 \times 10^{-4}$, while lithium is not dramatically affected, because the surface lithium abundance was already heavily lowered during the first dredge-up. At the second dredge-up, the helium, sodium, and oxygen abundances are changed (depending on the stellar mass), as can be seen in the three panels of fig. \[2dupchem\]. This can be understood on the basis of the following considerations:
- [The amount of mixed material during the second dredge-up sensibly increases with the stellar mass (see fig. \[2dup\]).]{}
- [During the second dredge-up the base of the outer envelope reaches layers which were previously touched by full CNO burning, thus explaining the oxygen reduction.]{}
- [The amount of material previously touched by CNO burning and mixed to the surface is larger than in the first dredge-up case, thus the increase of the surface helium abundance is very large, especially in the most massive models.]{}
The AGB phase
-------------
For most of the AGB evolution the global nuclear energy release is generated within a CNO burning shell, which may also overlap, in some cases, with the external convective zone. As hydrogen is consumed, the core mass increases, and the CNO burning takes place at higher temperatures, thus favoring an increase of the stellar luminosity. This is halted by mass loss, which progressively reduces the mass of the envelope, and eventually leads to a general cooling of the outer stellar layers. The base of the convective zone becomes cooler and cooler in the latest evolutionary stages; when the mass of the envelope drops below $\sim$ 1M$_{\odot}$ the temperatures within the whole external zone become so small to prevent any further nucleosynthesis. For each model, we may therefore find out maximum values of both luminosity and $T_{\rm bce}$, which we report as a function of the initial mass of the star in the two panels of fig. \[agbmax\]. We see that even the least massive of our models, i.e. the 3M$_{\odot}$ model, achieves at the base of the external convective zone temperatures so large ($T_{\rm bce} \sim 7.75 \times 10^7$ K) to trigger HBB.
In all the models, shortly after the beginning of the TPs phase, the TDU operates following each TP, changing the surface chemistry. The efficiency $\lambda$ of the TDU increases with the evolution, and is higher in the less massive models. We find that for $M \geq 5M_{\odot}$ a maximum value of $\lambda \sim 0.4$ is attained in the latest evolutionary stages, while a significantly larger value of $\lambda \sim 0.7-0.8$ is reached along the evolutions of the models with masses $3-3.5M_{\odot}$.
The chemical composition of the ejecta of our NACRE models, calculated over the lifetime of the star, is summarized in Table \[ejecta\]. With the only exception of lithium, for the other elements we indicate the logarithm of the ratio between the average chemical abundance of the ejecta and the initial value. Therefore, a value of 0 indicates that the chemical content of the ejecta is the same as the initial chemistry.
The CNO elements
----------------
In the three panels of Fig. \[cnonacre\] we show the variation during the AGB evolution (including also the changes due to the second dredge-up) of the surface abundances of the CNO elements; we chose the stellar mass as abscissa in order to have an idea of the average chemical content of the ejecta of these stars. We see (left-lower panel) that oxygen is depleted in all cases apart from the 3M$_{\odot}$ model, the heaviest depletion being for the largest masses, in agreement with the physical situation present at the base of the external convective zone. We note the apparently anomalous behavior of the 6 and 6.5M$_{\odot}$ models, which have a flatter declining profile of the surface oxygen with mass: this can be understood on the basis of the fact that these latter models are already extremely luminous during the pre-AGB phase, so that a large fraction of the mass is lost when the oxygen abundance is still close to the value left behind by the second dredge-up.
In the right panel we may follow the evolution of the surface abundance of carbon: we note an early phase of destruction in all models, which corresponds to the stage when only the CN cycle is active, followed by a later increase, when the full CNO cycle is activated; in the less massive models we recognize the signature of an efficient third dredge-up.
$$
[c c c c c c c c c c c c c]{} M\_[ZAMS]{} & Y\^a & ((\^7Li))\^b & \[\^[12]{}C\]\^c & \[\^[14]{}N\] & \[\^[16]{}O\] & (C+N+O)\^d & \[\^[23]{}Na\] & \[\^[24]{}Mg\] & \[\^[25]{}Mg\] & \[\^[26]{}Mg\] & \^[25]{}Mg/\^[24]{}Mg & \^[26]{}Mg/\^[24]{}Mg\
3.0 & 0.26 & 2.41 & -0.10 & 1.52 & 0.01 & 3.24 & 1.21 & 0.03 & 0.22 & 0.56 & 0.20 & 0.50\
3.5 & 0.27 & 2.20 & -0.44 & 1.37 & -0.19 & 2.18 & 0.75 &-0.01 & 0.37 & 0.40 & 0.31 & 0.39\
4.0 & 0.29 & 2.06 & -0.56 & 1.28 & -0.39 & 1.70 & 0.46 &-0.28 & 0.60 & 0.41 & 0.98 & 0.73\
4.5 & 0.32 & 1.96 & -0.60 & 1.24 & -0.49 & 1.51 & 0.02 &-0.66 & 0.60 & 0.38 & 2.37 & 1.63\
5.0 & 0.32 & 1.91 & -0.70 & 1.13 & -0.61 & 1.17 &-0.16 &-0.95 & 0.50 & 0.35 & 3.68 & 3.00\
5.5 & 0.32 & 1.94 & -0.80 & 1.04 & -0.59 & 0.99 &-0.37 &-1.13 & 0.53 & 0.31 & 6.13 & 4.18\
6.0 & 0.32 & 1.91 & -0.82 & 1.01 & -0.54 & 0.96 &-0.43 &-1.23 & 0.60 & 0.28 & 8.96 & 4.96\
6.5 & 0.32 & 2.25 & -0.81 & 1.01 & -0.46 & 1.00 &-0.37 &-1.27 & 0.66 & 0.25 & 11.34 & 5.02\
$$
Helium mass fraction.
$\log(\epsilon(^7Li))=\log(^7Li/H)+12.00$
$[A]=\log(X(A)_{ej})-\log(X(A)_{in})$.
Ratio between the average (C+N+O) abundance of the ejecta and the initial (C+N+O) value.
Even in this case we note the peculiar behavior of the 3$M_{\odot}$ model, in which the 3rd dredge-up is highly efficient since the first TPs, so that the surface $^{12}$C abundance increases up to $\log$(X($^{12}$C))$\sim -3.4$; only in a later time, $\sim 200,000$ yr after the beginning of the AGB phase, when $\sim 0.5$M$_{\odot}$ have been lost, HBB occurs, and the $^{12}$C abundance starts to decrease. The 3rd dredge-up after 10 TPs is so efficient that also some $^{16}$O is carried outwards; $^{16}$O reaches a maximum abundance after 15 TPs and then decreases approximately to the initial value: this is the only model for which we find an oxygen content of the ejected material which is larger than the original composition (see the 6th column of tab. \[ejecta\] and the top panel of Fig. \[yield1\] ). The combination of the effects of HBB and of the almost constant surface abundance of $^{16}$O prevents the formation of a carbon star. From this discussion we argue that 3M$_{\odot}$ is approximately the lower limit for models achieving HBB with the full CNO cycle operating during the AGB evolution. No $^{16}$O depletion can be achieved in less massive models.
The carbon, nitrogen and oxygen abundance of the ejecta are reported in columns 4-6 of Table \[ejecta\]. We see that the carbon content of the expelled material is always smaller than the initial value, due to the drop of the surface carbon which follows the first and especially the second dredge-up. The depletion factor is lower the lower is the mass, because in the less massive models more carbon is produced later in the AGB evolution by the 3rd dredge-up. We note that even for the 3M$_{\odot}$ model, despite the early phase of $^{12}$C production at the beginning of the TPs phase (see the right panel of fig. \[cnonacre\]), we find a negative $[^{12}$C\], due to a later phase of $^{12}$C depletion at the base of the convective envelope via proton fusion.
From column 5 of tab. \[ejecta\] we see that the nitrogen abundance of the ejecta is always at least a factor of $\sim 10$ larger than the initial value. Nitrogen is mixed to the surface through the first and second dredge-up, then its surface abundance increases due to HBB, via CN and ON cycling, and may further increase following each third dredge-up episode, via the conversion of additional primary $^{12}$C mixed into the envelope. Also in this case, it is the higher number of TPs and the larger efficiency of 3rd dredge-up episodes the reason of the larger $^{14}$N abundances found in the ejecta of the less massive models.
As for oxygen, with the only exception of the 3M$_{\odot}$ model we find, in agreement with what is shown in the left-lower panel of fig. \[cnonacre\], $[^{16}$O\]$<0$ in all cases (see also the top panel of fig. \[yield1\]). We note that the minimum value of $[^{16}$O\], i.e. $[^{16}$O\]=–0.61, is reached for M=5$M_{\odot}$, because the yield of more massive models is influenced by the very strong mass loss already efficient at the 2nd dredge-up, when no HBB had started yet.
The oxygen isotopes show a similar behavior in all our models. The surface abundance of $^{17}$O is increased during the second dredge-up by $\sim 0.2$ dex; at the very beginning of the AGB evolution $^{17}$O is produced at the base of the external zone due to partial $^{16}$O burning, so that, particularly in the most massive models, its abundance is increased by a factor of $\sim 10$. Later on, when $^{16}$O burning is more efficient, the surface $^{17}$O abundance reaches an equilibrium value, and then decreases as the $^{16}$O. The maximum surface $^{17}$O abundance is log\[X($^{17}$O)\] $\sim -5.5$ for the model with initial mass 6.5M$_{\odot}$, while it is log\[X($^{17}$O)\] $\sim -5.8$ for the 3$M_{\odot}$ model. The $^{18}$O abundance is dramatically decreased during the second dredge-up, passing from log\[X($^{18}$O)\] $\sim -6$ to log\[X($^{18}$O)\] $\sim -10$. At the beginning of the TPs phase some $^{18}$O is produced via proton capture by $^{17}$O, so that its abundance rises up by 2 orders of magnitude for the 6.5M$_{\odot}$ model (and by one order of magnitude in the 3.5M$_{\odot}$ model). Like $^{17}$O, a maximum value is reached, after which the surface $^{18}$O abundance decreases as $^{16}$O is consumed within the envelope.
An important outcome of most of our models is that the global C+N+O abundance of the ejecta is constant within a factor of $\sim 2$. This can be seen in the 7th column of Table. \[ejecta\], where we report the ratio between the global CNO abundance of the ejecta and the initial value. With the only exception of the 3M$_{\odot}$ model, for which the effects of the 3rd dredge-up overwhelm those of HBB, the values of the ratio between the average (C+N+O) abundance of the ejecta and the initial value are always $\leq 2$, being close to 1 for the most massive models.
These results are at odds with recent computations of AGB models of the same metallicity by Fenner et al. (2004), where it was shown that:
- [Only models more massive than 6M$_{\odot}$ achieve surface oxygen depletion.]{}
- [The material expelled by AGBs is for all the computed masses both carbon and nitrogen rich.]{}
- [The C+N+O strongly increases with respect to the initial value.]{}
These findings led the authors to conclude that massive AGBs may hardly have played a relevant role in the pollution of the interstellar medium of GCs, since spectroscopic analysis of NGC6752 stars found an anti correlation between \[C/Fe\] and \[N/Fe\] independent of the luminosity (Grundahl et al. 2002); besides the C+N+O sum is approximately constant in many CGs (Ivans et al. 1999). The different convective model adopted is the reason for the difference between our results and those obtained by Fenner et al. (2004) (see the detailed discussion in paper I), the FST model leading much more easily to efficient HBB which, in turn, triggers larger luminosities, shorter AGB life-times, and a smaller number of 3rd dredge-up episodes.
Sodium nucleosynthesis
----------------------
Fig. \[sodionacre\] shows the behavior of surface sodium. In all cases we see an increase due to the second dredge-up, followed by an early phase of sodium production due to $^{22}$Ne burning during the first TPs. Later on, when the whole Ne-Na cycle is active, the sodium abundance declines. For M $\leq 5$M the third dredge-up favors a later phase of sodium production, via proton capture by $^{22}$Ne mixed into the envelope. The $^{22}$Ne itself is a result of two $\alpha$ captures on the $^{14}$N mixed into the helium intershell at each third dredge up episode.
Particularly in the models with masses M$\leq 4$ a considerable amount of sodium is produced. In the 3M$_{\odot}$ model, following each TP, sodium is produced by $^{22}$Ne burning; during the quiescent CNO burning phases the bottom of the envelope is hot enough to activate the Ne-Na chain, but not to allow the Ne-Na reactions to act as a cycle (Arnould et al. 1999), which favors a large production of sodium.
The bottom panel of fig. \[yield1\] shows the average sodium abundance of the ejected material, as a function of the stellar initial mass. Sodium is produced within the less massive models due to the third dredge-up and to the modest sodium burning, but is destroyed within the massive models, so that the sodium content of the ejecta of these latter is under abundant with respect to the initial value. We see that only the models with initial masses clustering around 4M$_{\odot}$ are able to expell material which is both sodium rich and oxygen poor, and so [*only the abundances in these ejecta would be in agreement with the oxygen-sodium anti correlation*]{} observed within GCs stars (Gratton et al. 2001; Sneden et al. 2004).
Also in regard to sodium, we note the different predictions of our models compared to those by Fenner et al. (2004), who expect extremely large sodium production for all the masses considered here. (see their fig.1 and Fig. \[fenner\]). Within their models the sodium produced is primary, and is produced via $^{22}$Ne burning, this latter being dredged-up from the inner helium layers. In principle, this mechanism could work also in our models (see fig. \[sodionacre\]), but sodium production is made much less efficient due to: [*i)*]{} the smaller number of 3rd dredge-up episodes; [*ii)*]{} the larger temperatures, which favor sodium destruction. We therefore see that it is again the treatment of convection the main reason of the differences found in terms of the sodium content of the ejecta of AGBs. It is interesting to note that, in terms of the self-enrichment scenario, we have the opposite problem compared to the Fenner et al. (2004) models: they produce too much sodium, in great excess with the increase observed in some GCs stars (which is at most of $\sim 0.5$ dex), while in our case, for the most massive models, we destroy it, as also predicted by Denissenkov & Weiss (2004).
Figure \[fenner\] compares our results with those by Fenner et al. (2004) in the plane of oxygen versus sodium abundances, in which we have reported several sets of observational data. Our results should be shifted by +0.3dex in oxygen to be properly compared with the observations. We see that both sets are unable to reproduce the data.
The magnesium and aluminum isotopes
-----------------------------------
Figure \[mgalnacre\] shows the variation of the magnesium and aluminum isotopes along the standard evolutions of 3.5, 4 and 5. As the rate of proton capture on this isotope increases with the temperature at the bottom of the convective envelope, the $^{24}$Mg is more depleted for larger masses. Masses M$>5$ have qualitatively the same behavior, with more efficient $^{24}$Mg destruction. The heaviest destruction is found within the 6.5M$_{\odot}$ model, in which the surface final abundance is lower with respect to the initial value by a factor of $\sim 500$. We see from Tab. \[ejecta\] that the $^{24}$Mg abundance of the ejecta is lower the larger is the initial mass, reaching a minimum value of $[^{24}$Mg\] $\sim -1.3$ for the 6.5M$_{\odot}$ model. The abundance of $^{25}$Mg, on the contrary, in a first stage increases due to the $^{24}$Mg proton capture during the first TPs, and later on its abundance decreases (e.g. in the 5) due to burning to $^{26}$Al. The $^{26}$Al however decays into $^{26}$Mg only on a timescale of 7$\times
10^5$yr, and so this element is a bottleneck for further proton capture on $^{26}$Mg, which leads to $^{27}$Al. A direct path to $^{25}$Mg and $^{26}$Mg is through the third dredge up, as these isotopes are synthesized in the helium shell via capture of $\alpha$ on $^{22}$Ne, and release respectively of a neutron or a gamma. This production mechanism is evident in Figure \[mgalnacre\] for $^{26}$Mg, while it is also clear (especially in the left figure, relative to the 3.5 evolution) that the production of $^{25}$Mg is due to two mechanisms, dredge up and proton capture on $^{24}$Mg. As we do not have a large number of thermal pulses, the $^{27}$Al abundance can not rise by the huge factor (close to 10) shown by Globular Cluster stars (see Grundhal et al. 2002, for the giants of the cluster NGC 6752). Further, the ratios between the magnesium isotopes are not consistent with the results by Yong et al. (2003), which indicate that $^{25}$Mg remains at 10% of the $^{24}$Mg abundance, and $^{26}$Mg reaches at most $\sim 50$%. The observational result both implies a not dramatic burning of $^{24}$Mg, and a mild, if any, increase in $^{25}$Mg and $^{26}$Mg. Notice that, in addition, we have to count into the $^{26}$Mg abundance also the abundance of the unstable isotope $^{26}$Al, and the result is at variance with observations.
In spite of the not good agreement of these abundances with the observation, at least the trend of our models is in the right direction, as the ratios $^{25}$Mg/$^{24}$Mg and $^{26}$Mg/$^{24}$Mg do not exceed $\sim 3$ for masses up to 5. The corresponding models by Fenner et al. (2004), in which the smaller efficiency of convection allows a longer evolutionary phase and many episodes of third dredge up provide ratios larger than 100. Notice also that for elements whose abundances are very small, also the initial abundances and the exact modeling of the thermal pulses may influence strongly the results. The central part of Figure \[mgalnacre\] in fact shows that the evolution of the 4 suffers an anomalous episode of third dredge up, which we are uncertain whether to attribute to numerics or to a real effect. This lonely episode changes the surface abundances of sodium and magnesium in such a way that the resulting yields of the elements having low abundances are affected, although the most abundant yields (e.g. CNO) are not. This requires an additional detailed study before we can reject or accept these results as conclusive for the problem of abundance variations in GCs.
The lithium content of the ejecta
---------------------------------
We conclude this general description with lithium, which is created during the first TPs via the Cameron & Fowler (1971) mechanism, and then destroyed as soon as $^3He$ is extinguished in the envelope. We see from the 3rd column of tab. \[ejecta\] that the lithium content of the ejecta first decreases with increasing mass. In fact, the larger is the mass, the hotter is the base of the convective envelope, the more rapidly $^3$He is destroyed, the shorter is the phase during which the star shows up as lithium-rich; for models more massive than 5M$_{\odot}$, as already discussed for the oxygen content of the ejecta, we have that the mass loss is so strong during the first TPs that a considerable fraction of the mass is lost when lithium has been produced and not yet destroyed. The differences among the lithium abundances of the various models is however within a factor of $\sim 2$, and is about a factor 2 smaller than the average abundance which is observed in population II stars.
The overall chemistry of the ejecta
-----------------------------------
In examining the overall chemical content of the ejecta of our models, there are four common features, which hold independently of mass:
1. [ The lithium content is within a factor 2 of the value observed in population II stars.]{}
2. [The ejecta are helium rich: particularly models with initial mass $M>4M_{\odot}$ during the second dredge-up reach surface helium mass fractions of $Y\sim 0.32$. Helium is also produced during the AGB phase during any interpulse phase, but the consequent overall increase of $Y$ is limited in all cases to $\delta Y\sim 0.005$.]{}
3. [When compared with the initial abundances, we see that the material lost during the evolutions is enhanced in $^{14}N$ by at least a factor of $\sim 10$ and depleted in $^{12}C$ by a factor of $\sim 4$. This trend is consistent with the observations, indicating CN cycled composition and no carbon enhancement (Cohen et al. 2002, Cohen & Melendez 2004). However, the anticorrelation between carbon and nitrogen is not found in the yields as a function of the initial mass (see Table 2).]{}
4. [The (C+N+O) abundance is constant within a factor of $\sim 2$, due to the small number of 3rd dredge-up episodes.]{}
For the other elements, the stellar yields depend sensibly on the stellar mass, both quantitatively and qualitatively. The more massive models pollute the interstellar medium with material which is oxygen and (partially) sodium depleted. $^{24}$Mg is heavily depleted (a factor of $\sim 10$ with respect to the initial value), so that high magnesium isotopic ratios (larger than solar) are expected.
Conversely, within models less massive than $\sim 4.5M_{\odot}$, the temperature at the base of the external zone is such that the Ne-Na cycle is only partially activated, and only in the last AGB phases; consequently, the ejecta of these stars are sodium rich and oxygen poor. The $^{24}$Mg depletion is negligible due to the low temperatures, therefore the magnesium isotopic ratios are reduced, although not at the level which would provide agreement with the relevant observations by Yong et al. (2003). Further, the magnesium - aluminum anti correlation (see e.g. Grundahl et al. 2002) is not fully reproduced, as the $^{27}$Al production is not very efficient.
NACRE vs CF88
=============
The nuclear cross-sections play a delicate role in determining the main physical and chemical properties of the AGB evolutions. From a physical point of view, a variation of the cross-sections of the nuclear reactions mostly contributing to the global energy release might influence the thermal stratification of the star; from a chemical point of view, a change in the cross-sections of those reactions involving key elements like sodium or magnesium, though energetically not relevant, might alter the equilibrium abundances, hence the average chemical content of the ejecta.
We explore the uncertainty of the results connected with the cross-sections of the various reactions included within our network, by performing a detailed comparison between the results presented in the previous section and those obtained with the CF88 release, which are still widely used in modern AGB computations. This was made also to have an idea of the degree of uncertainty of the results connected with the cross-sections of the various reactions included within our network. We calculated a new set of models with the same physical and chemical inputs of the NACRE models, but adopting the CF88 rates for the nuclear cross-sections.
We didn’t find any appreciable difference during the MS evolution for the whole range of masses involved, as the cross-sections of the relevant reactions are the same for both sets of models. For each value of the initial mass, we could verify that the duration of the H-burning phase, the innermost point reached during the first dredge-up and the consequent changes in the surface chemistry are unchanged.
The first differences between the models appear in the duration of the helium burning phase, as can be seen in fig. \[timevol\]. Also the ratio between the times of helium and hydrogen burning are consequently affected. The reason of this difference stands in the rate of the reaction $^{12}$C+$\alpha \rightarrow ^{16}$O, which is larger by a factor of $\sim 1.7$ in the NACRE case. This leads to slightly longer time-scales for helium burning (see the discussion in Imbriani et al. 2001 and Ventura & Castellani 2005).
The AGB evolution of the models is physically very similar, because the global nuclear energy release during the quiescent phase of CNO burning (which, we recall, is for most of the time the only nuclear source active within the star) is dominated by the proton captures by $^{12}$C, $^{13}$C and $^{14}$N nuclei, whose corresponding cross-sections are similar in the two cases. We could verify that the duration of the whole AGB phase for the two sets of models, as well as the temporal evolution of the most relevant physical quantities, are essentially the same.
In the four panels of fig. \[conyie1\] we show the average chemical content of the ejecta, in terms of lithium and CNO abundances. We see that the lithium content is extremely similar for the two sets of models, while the $^{12}$C, $^{14}$N and $^{16}$O abundances are lower in the CF88 models, when compared to NACRE. This difference is due to the cross-sections of the reactions of proton capture by $^{17}$O atoms, which, as already discussed in Sect.2, are much lower in the CF88 case. The three left-panels of fig. \[sezurto\] show, respectively, the variation with temperature of the ratio (CF88/NACRE) between the rates of the reactions $^{17}$O(p,$\gamma)^{18}$F (top panel), $^{17}$O(p,$\alpha)^{14}$N (middle panel), and of the ratio $\sigma(^{17}$O(p,$\gamma)^{18}$F)/$\sigma(^{17}$O(p,$\alpha)^{14}$N) in the two cases. We can see that in the range of temperatures of interest here ($7.5\leq \log T \leq 8$) the reaction $^{17}$O(p,$\alpha)^{14}$N is more efficient in the NACRE case by a factor of a few hundred, while the difference for the reaction $^{17}$O(p,$\gamma)^{18}$F is a factor of $\sim 5$. In the third panel, more important, we can see that the favorite channel of $^{17}$O destruction switches from $^{17}$O(p,$\alpha)^{14}$N to $^{17}$O(p,$\gamma)^{18}$F passing from the NACRE to the CF88 cross-sections.
This, in turn, has two important consequences:
- [In the CF88 models we have a much larger production of the heaviest oxygen isotopes.]{}
- [In the NACRE case, the equilibrium abundance of $^{17}$O is much lower, and the nucleosynthesis favors a return to $^{14}$N rather than a production of $^{18}$F: this explains the differences between the CNO abundances of the ejecta of the two sets of models which can be seen in fig. \[conyie1\].]{}
Turning to heavier elements, we show in the four panels of fig. \[conyie2\] the abundances of $^{23}$Na and $^{24}$Mg of the ejecta of the different models, and the ratio of the magnesium isotopes, $^{25}$Mg/$^{24}$Mg and $^{26}$Mg/$^{24}$Mg. We note from the left-upper panel of fig. \[conyie2\] that sodium production can be achieved efficiently in the NACRE case, while it is not present in the CF88 models. This can be explained very simply on the basis of the cross-section of the reaction $^{22}$Ne(p,$\gamma)^{23}$Na, which, in the range of temperatures relevant in this case, is lower in the CF88 case by at least a factor of $\sim 100$, reaching a maximum difference of a factor of $\sim 2000$ for $\log(T)=7.8$.
In the two panels of fig. \[nane5\] we compare the variation with mass of the $^{22}$Ne and $^{23}$Na surface abundances within two models of initial mass 5M$_{\odot}$ calculated with the NACRE and CF88 nuclear cross sections. We see an early phase of sodium production and neon destruction during the first thermal pulses in the NACRE model, and a later increase of the sodium abundance due the dredging-up of $^{22}$Ne, which is later converted to $^{23}$Na. In the CF88 case the surface $^{22}$Ne never decreases, and sodium is destroyed when the temperatures at the base of the external zone become large enough to activate efficiently the Ne-Na cycle.
The top and middle right panels of fig. \[sezurto\] show the ratio of the cross-sections corresponding to the two channels of sodium destruction ($^{23}$Na(p,$\gamma)^{24}$Mg and $^{23}$Na(p,$\alpha)^{20}$Ne) (always in terms of CF88/NACRE value) and the ratio $\sigma(^{23}$Na(p,$\gamma)^{24}$Mg)/$\sigma(^{23}$Na(p,$\alpha)^{20}$Ne) in the two cases. In the NACRE case sodium is destroyed more easily, and the favorite channel is magnesium production in the relevant range of temperatures; this determines a larger $^{24}$Mg equilibrium abundance, and explains the difference between the models which can be seen in the right-upper panel of fig. \[conyie2\].
Turning to the magnesium isotopic ratios, $^{25}$Mg/$^{24}$Mg is similar in the two cases (see the left-lower panel of fig. \[conyie2\]), while $^{26}$Mg/$^{24}$Mg is lower in the CF88 models; the reason is that the rate of the reaction $^{26}$Mg(p,$\gamma)^{27}$Al is a factor of $\sim 10$ larger in the CF88 case, thus favouring $^{26}$Mg destruction in favor of $^{27}$Al production.
By comparing the AGB models calculated with the two sets of cross-sections we conclude that the physical behavior is essentially the same, because the rates of the reactions mostly contributing to the global energy release are scarcely changed. In terms of nucleosynthesis (and therefore of the average chemical content of the ejecta) we find important variations only for sodium and the heavier isotopes of oxygen. The former is not produced at all in the CF88 models (contrary to the NACRE case) due to the extremely low cross-section of the $^{22}$Ne proton capture reaction; the equilibrium abundances of $^{17}$O and $^{18}$O are much lower in the NACRE case, because of the larger values of both the $^{17}$O proton capture reactions.
The role of mass loss
=====================
The effects of mass loss on the evolution of AGB stars is well documented in the literature (Schönberner 1979): mass loss halts the increase of luminosity, and progressively peels off all the envelope, leaving eventually a central remnant which evolves as a white dwarfs. It determines a general cooling of the structure, therefore reducing the intensity of HBB at the base of the external convective zone. Since the effects of mass loss become evident only when the mass of the envelope is considerably reduced, models with different mass loss rates will differ only in the terminal part of their evolution, while the general physical behavior during the first TPs is unchanged.
The NACRE and CF88 models presented in the previous sections were calculated with the parameter $\eta_R=0.02$, in the Blöcker’s formula. This choice is due to a previous calibration, made on the basis of a detailed comparison between the observed and the theoretical luminosity function of lithium rich AGB stars in the Magellanic Clouds (Ventura et al. 2000). For our models, a value of $0.01 \leq \eta_R \leq 0.02$ for stars with initial mass in the range 3 $\leq$ M/$\leq$ 4.5 is able to reproduce the observed trend of surface lithium vs luminosity which is observed in the Clouds.
We cannot completely rule out the possibility that the parameter $\eta_R$ to be used during the AGB evolution might show a dependency on the metallicity (we recall that the models discussed in this paper have a metallicity which is a factor of 10 lower than the LMC stars) and on the stellar mass, or that the mass loss is heavily influenced by the environment. In order to test the level of uncertainty which is connected with the mass loss, we decided to explore the sensitivity of our results on changes in the value of $\eta_R$, and we discuss it for for two representative examples of our stars.
Massive AGBs
------------
We compare the standard model of initial mass 5M$_{\odot}$ already presented in Sect. 3 (eta002 model) with two models of the same initial mass, computed by assuming, respectively, $\eta_R=0.1$ (eta010 model) and $\eta_R=0.2$ (eta020 model). Fig. \[conmassa5\] shows the variation with time of the mass. Times have been set to 0 at the beginning of the AGB evolution. We see that the total duration of the AGB phase is strongly dependent on $\eta_R$, ranging from $t_{AGB} \sim 73,000$ yr for the eta002 model down to $\sim 27,000$ yr in the eta020 case. The eta002 model reaches a maximum luminosity of $66,000 L_{\odot}$ during the 20th interpulse period, while the eta010 and eta020 models achieve a maximum luminosity of, respectively, $50,000 L_{\odot}$ (13th interpulse period), and $43,500 L_{\odot}$ (10th interpulse period). In conjunction with the maximum luminosity, all the models also attain the largest temperature at the base of the external envelope. The maximum $T_{bce}$ are: $T_{bce}=103 \times 10^6$K (eta002), $T_{bce}=100 \times 10^6$K (eta010) and $T_{bce}=98 \times 10^6$K (eta020). These values are quite similar, especially when compared to the differences in the maximum values of luminosity reached by the three models. This is not surprising, as a main feature of the FST convective model is that, within the most massive AGBs, it leads to a very efficient HBB already during the first TPs: even a strong increase of mass loss cannot prevent the base of the external zone to become extremely hot. On the basis of these results, we may expect a deep nucleosynthesis to take place at the base of the external convective zone even in the eta020 model.
In the four panels of fig. \[confcno5\] we show the evolution of the surface CNO abundances for the three models, plus the variation of the total C+N+O abundance. For each of these elements we show both the variation with time and with mass. In the left-upper panel we show the variation of surface $^{12}$C. In the top of this panel we see that the temporal evolution is very similar, with an early phase of destruction at the beginning of the AGB evolution followed by a later phase of production, when the temperatures at the base of the external zone are sufficient to allow the full CNO cycle to be activated, and the effects of the 3rd dredge-up become more evident. The only difference among the three models is that the AGB evolution is halted earlier for larger values of $\eta_R$. Since for all the models the evolution stops when the carbon abundance was increasing, this acts in favor of a larger $^{12}$C content of the ejecta for lower mass loss rates. In the lower part of this panel we see the evolution of $^{12}$C with the stellar mass. The above effect is partly compensated by the fact that, for larger $\eta_R$, the star looses a not negligible fraction of its mass when the carbon abundance was still unchanged, even before the early phase of destruction at the beginning of AGB. This is the reason why the average $^{12}$C abundance of the ejecta of our models show a maximum difference of $\sim 0.1$ dex, and is therefore consistent with the value $[^{12}$C\]=–0.7 given in Sect. 3.
An analogous discussion can be made for nitrogen, as can be seen in the right-upper panel of fig. \[confcno5\]. The surface $^{14}$N increases in all cases, because nitrogen is created at the base of the external envelope due to HBB and, in the final part of the evolution, also due to the effects of the 3rd dredge-up. Again we note a strong similarity in the temporal evolution, the only difference being that in the large $\eta_R$ models the $^{14}$N content of the ejecta is expected to be lower because the AGB evolution is halted earlier. In reality, at odds with the $^{12}$C case, we expect a larger nitrogen content of the ejecta of the eta002 model because this latter case loses less mass at the very beginning of the AGB evolution, when the $^{14}$N abundance was still unchanged since the second dredge-up (see the lower part of the right-upper panel of fig. \[confcno5\]). The $^{14}$N average content of the ejecta is therefore more dependent on mass loss, ranging from $[^{14}$N\]=1.15 in the eta002 model, down to $[^{14}$N\]=0.83 in the eta020 model; the eta010 model, with $[^{14}$N\]=0.92, shows an intermediate behavior. The global spread of the $[^{14}$N\] value varies at most by a factor of $\sim 2$ if $\eta_R$ varies by one order of magnitude.
As already pointed out, the temperatures at the base of the convective zone are sufficiently large to activate the full CNO cycle in all the models, so that in all cases we have a certain amount of $^{16}$O depletion, as can be seen in the left-lower panel of fig. \[confcno5\]. The trend of the $^{16}$O average content of the ejected material for various $\eta_R$ is straightforward:
- [For larger values of $\eta_R$ a consistent part of the envelope mass ($\Delta M \sim 1M_{\odot}$) is lost when the oxygen abundance is still unchanged.]{}
- [Since the temperature at the base of the envelope reached by the large $\eta_R$’s models is lower, in these cases $^{16}$O is destroyed less heavily, and this leads to higher oxygen equilibrium abundances.]{}
We thus find that a strong oxygen destruction is hardly found within the models with the largest mass loss rates. The $[^{16}$O\] of the ejecta is more dependent than $[^{14}$N\] on the assumed $\eta_R$: we have $[^{16}$O\]=–0.6 for the eta002 model, $[^{16}$O\]=–0.3 in the $\eta_R=0.1$ case, and $[^{16}$O\]=–0.15 for $\eta_R=0.2$.
In the right-bottom panel of Fig. \[confcno5\] we show the total C+N+O abundance. In the eta010 and eta020 models the sum of the CNO abundances is constant, because mass is lost so rapidly that the effects of the 3rd dredge-up are negligible. In the eta002 model, during the last TPs, carbon is efficiently dredged-up, and is later converted to $^{14}$N by HBB; however, the total increase of the C+N+O is within $\sim 0.2$ dex. We may therefore conclude that within the FST framework the most massive AGB models show surface C+N+O abundances which are constant within a factor of $\sim 2$, independently of mass loss.
Turning to heavier nuclei, we focus our attention on sodium. Fig. \[confna5\] shows the variation of the surface sodium abundance as a function of time (top panel) and mass (bottom). An early phase of production, due to proton capture by $^{22}$Ne nuclei, is followed by a phase of sodium destruction when the temperatures at the base of the outer convective zone become large enough that sodium is destroyed by proton capture. A larger mass loss rate acts in favor of larger sodium yield because a large fraction of the stellar mass is lost when sodium is produced, and also because the evolution is halted when the surface sodium has not yet been completely destroyed (bottom panel).
Actually, sodium turns out to be the element most sensitive to variations of the mass loss rate. A larger $\eta_R$ changes completely the situation, in the sense that now we expect the mass expelled by massive AGBs to be sodium rich (with respect to the initial mass fraction) rather than sodium poor. $[^{23}$Na\] linearly increases with $\eta_R$, while a positive sodium yield is not possible at $5M_{\odot}$ with the standard $\eta_R=0.02$ value. The oxygen yield shows a similar behavior, though in this case the convection is so efficient that $[^{16}$O\] is negative in all cases. A simultaneous sodium production and oxygen depletion in the chemistry of the ejecta, in agreement with the observed anti correlation, is possible only for $\eta_R \sim 0.1$.
Turning to magnesium, we find that $^{24}$Mg is depleted in all cases, but the final abundance is a factor of $\sim 500$ lower in the eta002 case, while it is just a factor of 2 lower in the eta020 model. The $^{24}$Mg abundance of the ejecta is a factor of $\sim 10$ lower than the initial value for the eta002 model, while it is lower by only $\sim 0.1$ dex for $\eta_R=0.20$.
The average content of $^{25}$Mg is not strongly dependent on the mass loss rate, because it reaches a maximum value and then declines as $^{24}$Mg is destroyed; in reality, in the eta010 and eta020 models the evolution is completed when the $^{25}$Mg is almost at its maximum value. The net result is that within $\sim 0.1$ dex we find $[^{25}$Mg\]=0.5 for all the models. The situation is different for the heaviest isotope, because the surface $^{26}$Mg increases for the whole evolution. In this case a larger mass loss rate leads to a lower final abundance, so that the yield is lower. For the eta002 model we find $[^{26}$Mg\]=0.35, while $[^{26}$Mg\]=0.15 in the $\eta_R=0.01$ case. The average $^{26}$Mg content of the eta020 model is practically unchanged with respect to the initial value.
In terms of isotopic ratios, lower values of both $^{25}$Mg/$^{24}$Mg and $^{26}$Mg/$^{24}$Mg are expected for larger mass loss rates, because in that case we have a lower depletion of $^{24}$Mg. In agreement with that, we find isotopic ratios $\sim 3$ for $\eta_R=0.02$, $\sim 1$ for $\eta_R=0.1$ and $\sim .4$ for $\eta_R=0.2$, the $^{25}$Mg/$^{24}$Mg ratio being always slightly larger than the $^{26}$Mg/$^{24}$Mg.
The 4M$_{\odot}$ model.
-----------------------
The situation is a bit more complex for lower masses, because in that case the temperature reached by the base of the outer convective zone never exceeds $\sim 10^{8}$ K, therefore they achieve only in a later phase of their AGB evolution the conditions which are necessary to trigger a deep nucleosynthesis within the convective envelope. In these cases, at odds with the most massive models, we expect that a stronger mass loss, triggering an earlier cooling of the structure, may prevent some reactions to occur at all.
We therefore calculated a model with initial mass M=4M$_{\odot}$ with a parameter for mass loss $\eta_R=0.10$ (eta010 model), and we compare it with the model with the same initial mass calculated with $\eta_R=0.02$, presented in Sect. 3. The total duration of the AGB phase for the eta010 model is shorter, as expected: The total mass of the star reduces to $\sim 1.4M_{\odot}$ within $t_{AGB}\sim 87,000$ yr, to be compared to $t_{AGB}\sim 150,000$ yr of the eta002 model.
In fig. \[confphys4\] we compare the variation with time of the luminosity and of the temperature at the base of the envelope of the two models, as a function of the AGB time. We see from the top panel that there is a difference of $\sim 0.2$ dex between the maximum value of the luminosity reached by the two models, while, in terms of temperature, the base of the convective zone of the eta002 model achieves a maximum value of $T_{\rm bce}=95\times 10^6$ K, to be compared to the maximum temperature $T_{\rm bce}=88\times 10^6$ K reached in the $\eta_R=0.10$ case. In terms of the chemical content of the ejecta, we may repeat for $^{12}$C and $^{14}$N the same discussion performed for the 5M$_{\odot}$ model, because the temperatures in this case, though lower, are still sufficient to favor an early phase of $^{12}$C destruction followed by a later phase of production, and a progressive increase of the surface $^{14}$N abundance due both to HBB and to the effects of the 3rd dredge-up. Thus, the $^{12}$C abundance of the ejecta is almost the same for both models, while the $^{14}$N abundance is lower in the eta010 model by a factor of $\sim 2$.
The different values of the temperatures reached at the base of the outer convective zone in the two models lead to a different degree of the oxygen depletion at the base of the envelope, as can be seen in the two panels of fig. \[confo164\], where we show the variation of the surface $^{16}$O with time (top panel) and mass (bottom). We see that both models start to deplete oxygen after $\sim 40,000$ yr, but the depletion is made difficult in the eta010 models by the lower temperatures, so that the final abundance is only $\sim 0.15$ dex lower than the $^{16}$O present in the envelope at the beginning of the TPs phase. In the eta002 model a stronger depletion is achieved. The average oxygen content of the ejecta is $[^{16}$O\]=–0.4 in the eta002 model, while it is $[^{16}$O\]=–0.1 for $\eta_R=0.1$. Even for the $M=4M_{\odot}$ model we find that the C+N+O abundance is constant within a factor of $\sim 2$ for the whole evolution, the eta002 models showing the largest increase due to the higher number of 3rd dredge-up episodes.
The situation concerning sodium is more tricky. The eta002 model, after the initial phase of production, destroys sodium more efficiently due to the larger temperatures reached; yet, the evolution is so long that some sodium is dredged-up later on; as a consequence, the sodium content of the ejecta is almost the same in the two cases, i.e. $[^{23}$Na\]=0.5. The less massive models are therefore efficient sodium producers, independently of the mass loss rate adopted. In terms of the oxygen-sodium anti correlation, the eta002 model in this case is consistent with a simultaneous oxygen depletion and sodium production, while only a poor oxygen depletion is expected for larger mass loss rates.
As for magnesium, the situation is deeply different with respect to the 5M$_{\odot}$ case. The temperatures here are not sufficiently high to favor an efficient magnesium destruction, so that even in the eta002 model $^{24}$Mg is reduced by only a factor of $\sim 2$. For $\eta_R=0.1$ the surface $^{24}$Mg is almost unchanged. Even for the heavier isotopes the production is much lower than in the 5M$_{\odot}$ model. In terms of the isotopic ratios, we find $^{25}$Mg/$^{24}$Mg $\sim 0.9$ and 0.2, respectively, for $\eta_R=0.02$ and $\eta_R=0.1$, while $^{26}$Mg/$^{24}$Mg is 0.7 for $\eta_R=0.02$ and 0.2 for $\eta_R=0.1$.
We may therefore summarize the influence of mass loss on the AGB models as follows:
1. [The carbon content of the ejecta is almost independent of mass loss, while the nitrogen abundance may vary by a factor of $\sim 2$, lower $\eta_R$ models showing the larger enhancement. This holds for all the masses calculated, because at least the CN cycle is always operating.]{}
2. [The oxygen abundance of the ejecta proves to be more sensitive to mass loss; a lower $\eta_R$ favors larger oxygen depletion. For all the masses considered we achieve oxygen depletion for $\eta_R=0.02$, while a poor depletion is expected for larger values of $\eta_R$, particularly for the lowest masses.]{}
3. [The C+N+O sum is in all cases constant within a factor of $\sim 2$, independently of mass and mass loss.]{}
4. [Larger $\eta_R$ favor sodium production in the more massive models, because in that case the AGB evolution is halted when the surface sodium has not yet been destroyed. The less massive models are efficient sodium producers, independently of the mass loss rate.]{}
5. [The isotopic magnesium isotopes keep below unity independently of mass loss for the less massive models. For larger masses we have a steeper dependence on $\eta_R$: we find $^{25}$Mg/$^{24}$Mg $\sim^{26}$Mg/$^{24}$Mg $\sim 3$ for $\eta_R=0.02$, down to $^{25}$Mg/$^{24}$Mg $\sim^{26}$Mg/$^{24}$Mg $\sim 0.4$ for $\eta_R=0.2$.]{}
Which implications for the self-pollution scenario?
===================================================
There is still a strong debate concerning the role which AGBs may have played in the pollution of the interstellar medium of GCs: Fig. \[fenner\] shows that we are far from being able to falsify the hypothesis that the chemical content of their ejecta may account for the chemical anomalies observed in GCs stars (Denissenkov & Herwig 2003; Fenner et al. 2004; Ventura et al. 2002; paper I).
Within the MLT framework for the treatment of convection the most recent work by Fenner et al. (2004) shows that it is hardly possible to reconcile the theoretical findings with the observational scenario, because the expected chemical content of the ejecta show a largely increased value of the global C+N+O abundance, a very poor oxygen depletion, and an extremely large sodium production. These results are all in contrast with the observational evidence. Their findings were confirmed by our AGB models calculated with the MLT convection, presented and extensively discussed in paper I.
If the FST model is used, due to the larger temperatures reached at the bottom of the external envelope, we find on the contrary that the C+N+O is always constant within a factor of $\sim 2$, in agreement with the results of Ivans et al. (1999). Also, oxygen depletion is easily achieved in all models more massive than $3M_{\odot}$.
The FST models in the range $3.5 - 4.5$M$_{\odot}$ with a mass loss rate in agreement with the calibration given in Ventura et al. (2000) pollute the interstellar medium with material having a chemistry qualitatively in agreement with the chemical anomalies observed, that is:
- [The C+N+O sum is almost constant]{}
- [Oxygen and sodium are anticorrelated, in qualitative agreement with the observed trend. From a quantitative point of view a larger sodium content, coupled to an even stronger oxygen depletion would be necessary to match the observational evidence.]{}
- [The magnesium isotopic ratios are well below unity, but the $^{26}$Mg/$^{24}$Mg ratio is larger than the $^{25}$Mg/$^{24}$Mg contrary to observations (Yong et al. 2003). In addition, the magnesium vs. aluminum anti correlation is not reproduced.]{}
In the most massive models, on the contrary, the temperatures are so large that sodium, after an early phase of production at the beginning of the AGB phase, is destroyed; with the standard $\eta_R=0.02$ value for the mass loss rate parameter we expect to have a negative \[Na/Fe\] and a correlation between sodium and oxygen. One further problem in this case would be the extremely high values of the magnesium isotopic ratios, which cluster around $\sim 3$, duo to strong $^{24}$Mg burning. If a larger mass loss rate is adopted, we expect a poor oxygen depletion and sodium destruction, which would be more consistent with the observations. The magnesium isotopic ratios would be in this case below unity, which is also in better agreement with the observed abundances (Yong et al. 2003).
The CF88 models share almost the same properties of the NACRE models, with the only difference of sodium, which can not be produced in this latter case, because of the extremely low values of the $^{22}$Ne proton capture reaction cross-sections.
Conclusions
===========
We present AGB models of intermediate mass in the range $3M_{\odot} \leq M \leq 6.5M_{\odot}$ with metallicity Z=0.001.
This work, together with the results of paper I, explores the role of some parameters of AGB evolution and helps to understand that it is affected by so many uncertainties that it is still implausible to use the results of a unique set of AGB computations to falsify the self–enrichment scenario for globular cluster stars. Nevertheless, these results show trends in the yields which are different from those of other researchers, and may help to find a way for the solution of the self–enrichment problem. In particular, contrary to several recent AGB computations, and thanks to the use of the FST model, we find that convection at the base of the external zone during the quiescent phase of CNO burning is so efficient to lead to extremely high temperatures ($T_{\rm bce} \sim 10^8$K), sufficient to trigger strong HBB. We find that oxygen is depleted in all cases with the only exception of the 3M$_{\odot}$ model.
Our main findings are the following:
1. [ The physical behavior of the models turns out to be independent on the nuclear cross-sections used, the results obtained with the NACRE rates being very similar to CF88: this is due to the fact that the reactions most contributing to the global energy release, i.e. proton captures by carbon and nitrogen nuclei, are the same in the two cases.]{}
2. [ One strong prediction, which holds independently of mass, is that the total C+N+O abundance of the ejecta is almost constant, ad odds with previous investigations.]{}
3. [ Sodium is produced within the NACRE models with M $< 4.5$, while it is destroyed in the more massive stars, due to an efficient action of the Ne-Na cycle. In the CF88 models sodium is systematically destroyed, because of the extremely low values of the $^{22}$Ne proton capture reaction.]{}
4. [ In terms of magnesium isotopes, the $^{25}$Mg/$^{24}$Mg and $^{26}$Mg/$^{24}$Mg ratios are well below unity in the models with M $<$ 4.5M$_{\odot}$, while they reach values approaching $\sim 10$ for M$>$5M$_{\odot}$, due to strong $^{24}$Mg burning.]{}
5. Mass loss influences the global duration of the AGB life; it also determines a general cooling of the structure before some nuclear reactions can be efficiently activated, therefore changing the average chemical content of the ejecta. With very strong mass loss, the HBB nucleosynthesis has no time to be completely established. This affects mainly the oxygen and sodium yields, while it leaves almost unaltered the lithium yield. In the most massive models a stronger mass loss rate (with respect to the standard value adopted, calibrated on slightly less massive models to reproduce the lithium-rich stars luminosity function in the LMC) might lead to ejecta which are sodium rich, and which show low magnesium isotopic ratios; there are two factors behind such findings:
- [i) Massive AGBs achieve large luminosities already during the first TPs; if mass loss is increased during these phases, we have a larger ejection of material which is still sodium rich and not extremely $^{24}$Mg depleted.]{}
- [ii) As a consequence of the reduction of the mass of the envelope, the evolution is halted earlier, before strong sodium and $^{24}$Mg depletion may take place.]{}
For models with masses M$\leq 4.5$M$_{\odot}$ the ejecta are sodium rich and with low magnesium isotopic ratios in any case, with the only difference that, for larger values of $\eta_R$, oxygen is scarcely depleted, as the increase of the temperature at the base of the external envelope is halted before it may reach values sufficiently high to activate efficiently the full CNO cycle.
As a general conclusion, these models show that the predictive power of AGB models is still undermined by many uncertainties. The parameters space, however, has not yet been fully explored (e.g. the role of extra-mixing at the bottom of the envelope) and for this reason we should not discard the hypothesis that massive AGB stars are responsible for the chemical anomalies observed in GCs stars.
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abstract: '[It is by now well known that the wave functions of rational solutions to the KP hierarchy which can be achieved as limits of the pure $n$-soliton solutions satisfy an eigenvalue equation for ordinary differential operators in the spectral parameter. This property is known as “bispectrality” and has proved to be both interesting and useful. In this note, it is shown that all pure soliton solutions of the KP hierarchy (as well as their rational degenerations) satisfy an eigenvalue equation for a non-local operator constructed by composing ordinary differential operators in the spectral parameter with [*translation*]{} operators in the spectral parameter, and therefore have a form of bispectrality as well.]{}'
address: |
Mathematical Sciences Research Institute\
Berkeley, CA 94720
author:
- Alex Kasman
title: Spectral Difference Equations Satisfied by KP Soliton Wavefunctions
---
Introduction
============
The KP Hierarchy and Bispectrality
----------------------------------
Let $\D$ be the vector space spanned over $\C$ by the set $$\{\Delta(j,\lambda)\mid \lambda\in\C, j\in\N \}$$ whose elements differentiate and evaluate functions of the variable $z$: $$\Delta(j,\lambda)[f(z)]:=f^{(j)}(\lambda).$$ The elements of $\D$ are thus *finitely supported distributions* on appropriate spaces of functions in $z$. For lack of a better term, we will continue to call them distributions even though their main use in this paper will be their application to functions of two variables. (Such distributions were called “conditions” in since a KP wave function was specified by requiring that it be in their kernel.) Note that if $c\in\D$ and $f(x,z)$ is sufficiently differentiable in $z$ on the support of $c$, then $\hat f(x)=c[f(x,z)]$ is a function of $x$ alone. Furthermore, note that one may “compose” a distribution with a function of $z$, i.e. given $c\in\D$ and $f(z)$ (sufficiently differentiable on the support of $c$) then there exists a $c':=c\circ f\in\D$ such that $$c'(g(z))=c(f(z)g(z))\qquad\forall g.$$
The subspaces of $\D$ can be used to generate solutions to the KP hierarchy in the following way. Let $C\subset\D$ be an $n$ dimensional subspace with basis $\{c_1,\ldots,c_n\}$. Then, if $K=K_C$ is the unique, monic ordinary differential operator in $x$ of order $n$ having the functions $c_i(e^{xz})$ in its kernel (see ) we define $\L_C=K \frac{\partial}{\partial x} K^{-1}$ and $\psi_C=\frac{1}{z^n}K e^{xz}$. The connection to integrable systems comes from the fact that adding dependence to $C$ on a sequence of variables $t_j$ ($j=1,2,\ldots$) by letting $C(t_j)$ be the space with basis $$\{c_1\circ e^{\sum -t_jz^j},c_2\circ e^{\sum -t_jz^j},\ldots,
c_n\circ e^{\sum -t_jz^j}\}$$ it follows that the “time dependent” pseudo-differential operator $\L=\L(t_j)$ satisfies the equations of the KP hierarchy $$\frac{\partial}{\partial t_j}\L=[(\L^j)_+,\L].$$ The [*wave function*]{} $\psi_C(x,z)$ generates the corresponding subspace of the infinite dimensional grassmannian $Gr$ which parametrizes KP solutions and thus it is not difficult to see that this construction produces precisely those solutions associated to the subgrassmannian $Gr_1\subset Gr$ .
Moreover, the ring $A_C=\{p\in\C[z]|c_i\circ p\in C\ 1\leq i \leq n\}$ is necessarily non-trivial (i.e. contains non-constant polynomials) and the operator $L_p=p(\L)$ is an [*ordinary*]{} differential operator for every $p\in A_C$ and satisfies $$L_p\psi_C(x,z)=p(z) \psi_C(x,z).\label{eigenx}$$ The subject of this paper is the existence of [*additional*]{} eigenvalue equations satisfied by $\psi_C(x,z)$. In particular, we wish to consider the question of whether there exists an operator $\hatLambda $ acting on functions of the variable $z$ such that $$\hatLambda \psi_C(x,z)=\pi(x)\psi_C(x,z)\label{eigenz}$$ where $\pi(x)$ is a non-constant function of $x$. For example, the following theorem is due to G. Wilson in : In other words, for this special class of KP solutions for which the coefficients of $\L$ are rational functions of $x$, the wave function $\psi_C$ satisfies an additional eigenvalue equation of the form where $\hatLambda$ is an ordinary differential operator in $z$ and $\pi(x)$ a non-constant polynomial in $x$.[^1] Together and are an example of [*bispectrality*]{} . The bispectral property is already known to be connected to other questions of physical significance such as the time-band limiting problem in tomography , Huygens’ principle of wave propagation , quantum integrability and, especially in the case described above, the self duality of the Calogero-Moser particle system .
It is known that the only subspaces $C$ for which the corresponding wave function satisfies and with $L_p$ and $\hatLambda $ ordinary differential operators in $x$ and $z$ respectively are those described in Theorem \[Th:wilson\]. However, suppose we allow $\hatLambda$ to involve not only differentiation and multiplication in $z$ but also [*translation*]{} in $z$ and call this more [*general*]{} situation t-bispectrality.[^2] It will be shown below that there are more KP solutions which are bispectral in this sense. In particular, it will be shown that the KP solution associated to *any* subspace $C$ shares its eigenfunction with a ring of translational-differential operators in the spectral parameter.
Notation
--------
Using the shorthand notation $\partial=\frac{\partial}{\partial x}$ any ordinary differential operator in $x$ can be written as $$L=\sum_{i=0}^N f_i(x) \partial^i\qquad (N\in\N).$$ We say that a function of the form $$f(x)=\sum_{i=1}^n p_i(x)e^{\lambda_i x}\qquad \lambda_i\in\C,\
p_i\in\C[x]$$ is a *polynomial-exponential function* and that the quotient of two such functions is *rational-exponential*. This note will be especially concerned with the ring of differential operators with rational-exponential coefficients and especially with the subring having polynomial-exponential coefficients. Similarly, we will write $\partial_z=\frac{\partial}{\partial z}$ but will need to consider only differential operators in $z$ with rational coefficients.
For any $\lambda\in\C$ let $\Trans_{\lambda}=e^{\lambda\partial_z}$ be the translational operator acting on functions of $z$ as $$\Trans_{\lambda}[f(z)]=f(z+\lambda).$$ Then consider the ring of translational-differential operators $\TODOs$ generated by these translational operators and ordinary differential operators in $z$. Any translational-differential operator $\hat
T\in\TODOs$ can be written as $$\genT =\sum_{i=1}^N p_i(z,\partial_z) \Trans_{\lambda_i}$$ where $p_i$ are ordinary differential operators in $z$ with rational coefficients and $N\in\N$. Note that the ring of ordinary differential operators in $z$ with rational coefficients is simply the subring of $\TODOs$ of all elements which can be written as $p \Trans_{0}$ for a differential operator $p$.
Translational Bispectrality of $\C[\partial]$
=============================================
It has been frequently observed that the ring $\mathcal{A}=\C[\partial]$ of constant coefficient differential operators in $x$ is *bispectral* since it has the eigenfunction $e^{xz}$ which it shares with the ring of constant coefficient differential operators in $z$. Here, however, we will consider a more general form of bispectrality for the ring $\mathcal{A}$.
Let $\mathcal{A}'\subset\TODOs$ be the ring of constant coefficicient *translational*-differential operators. Note that for any element $\genT\in\mathcal{A}'$ of the form $$\genT =\sum_{i=1}^N p_i(\partial_z) \Trans_{\lambda_i}$$ one has simply that $$\genT[e^{xz}]=\left(\sum_{i=1}^N p_i(x)e^{\lambda_i x}\right)e^{xz}.$$ In particular, $e^{xz}$ is an eigenfunction for the operator with an eigenvalue which is a polynomial-exponential function of $x$. Consequently, the rings $\mathcal{A}$ and $\mathcal{A}'$ are both bispectral, sharing the common eigenfunction $e^{xz}$.
Let $\mathcal{R}$ be the ring of differential operators in $x$ with polynomial-exponential coefficients and $\mathcal{R'}$ be the ring of translational-differential operators in $z$ with rational coefficients. Note that $\mathcal{R}$ is generated by $\mathcal{A}$ and the eigenvalues of the operators in $\mathcal{A'}$ while $\mathcal{R}'$ is generated by $\mathcal{A}'$ and the eigenvalues of the elements of $\mathcal{A}$. It then follows (see also ) that the map $b:\mathcal{R}\to\mathcal{R}'$ defined by the relationship $$L[e^{xz}]=b(L)[e^{xz}]\qquad \forall L\in\mathcal{R}$$ is an anti-isomorphism of the two rings.
Translational Bispectrality of KP Solitons
==========================================
Let us say that a finite dimensional subspace $C\subset\D$ is [*t-bispectral*]{} if there exists a translational-differential operator $\hatLambda \in \TODOs$ satisfying equation for the corresponding KP wave function $\psi_C(x,z)$. By Theorem \[Th:wilson\] and the fact that the ring of rational coefficient ordinary differential operators in $z$ is contained in $\TODOs$, we know that $C$ is t-bispectral[^3] if it has a basis of point supported distributions. Here we will show that, in fact, all subspaces $C\subset\D$ are $t$-bispectral.
An important object in much of the literature on integrable systems is the “tau function”. The tau function of the KP solution associated to $C$ can be written easily in terms of the basis $\{c_i\}$. In particular, define (cf. ) $$\tau_C(x)=\textup{Wr}\left(c_1(e^{xz}),c_2(e^{xz}),\ldots,c_n(e^{xz})\right)$$ to be the Wronskian determinant of the functions $c_i(e^{xz})$. Similarly, there is a Wronskian formula for the coefficients of the operator $K_C$ since its action on an arbitrary function $f(x)$ is given as: $$K_C(f(x))=\frac{1}{\tau_C(x)} \textup{Wr}\left(c_1(e^{xz}),c_2(e^{xz}),\ldots,c_n(e^{xz}),f(x)\right).\label{eqn:wrformK}$$ Then the coefficients of the differential operator $\bar
K_C:=\tau_C(x)K_C(x,\partial)$ are all polynomials-exponential functions.
[For any $C\subset\D$ there exists a constant coefficient operator $L_0\in\mathcal{A}$ which factors as $$L_0=\bar Q_g\circ\frac{1}{\pi(x)}\circ \bar K_C$$ where $\bar Q_g,\bar K_C\in\mathcal{R}$ and $\pi(x)=g(x)\tau_C(x)\in\mathcal{R}$ is a polynomial-exponential function.]{} Let $\lambda_i\in\C$ ($1\leq i \leq N$) be the support of the distributions in $C$ and $m_i$ be the highest derivative taken at $\lambda_i$ by any element of $C$. Then the polynomial $$q_C(z):=(z-\lambda_i)^{m_i+1}\label{eqn:q}$$ has the property that $c\circ q_C\equiv0$ for any $c\in C$. Let $L_0:=q_C(\partial)$ and consider $L_0[c(e^{xz})]$ for any element $c\in C$. Since $L_0$ is an operator in $x$ alone, it commutes with $c$ and we have $$L_0[c(e^{xz})]=c\left(L_0[e^{xz}]\right)=c\left(q(z)e^{xz}\right)=c\circ
q(e^{xz})=0.$$ So, by the definition of $K_C$, we see that $L_0$ annihilates the kernel of $K_C$ and thus has a right factor of $K_C$. This gives a factorization of the form $L_0=Q\circ K_C$ with $Q$ having rational-exponential coefficients. Then, by choosing a polynomial-exponential function $g(x)$ so that $\bar Q_g:=Q\circ
g(x)\in\mathcal{R}$ we find the desired factorization.
Given this factorization, the t-bispectrality of all $C$’s now follows from Theorem 4.2 in .
[For any subspace $C\subset\D$ the corresponding KP wave function $\psi_C(x,z)$ satisfies the eigenvalue equation $$\hatLambda_g[\psi_C(x,z)]=g(x)\tau_C(x)\psi_C(x,z)$$ where $\hatLambda_g\in\TODOs$ is the translational-differential operator defined by $$\hatLambda_g:=z^{-n}\circ b(\bar K_C)\circ b(\bar Q_g)\circ \frac{z^n}{q_C(z)}$$ with $\bar Q_g$ defined as in Lemma \[lem:factor\].]{} Formally introducing inverses , we determine from Lemma \[lem:factor\] that $$\pi(x):=g(x)\tau_C(x)=\bar K_C \circ L_0^{-1} \circ \bar Q$$ and hence (by applying the anti-involution $b$ to this equation) $$b(\pi(x))=b(\bar Q)\circ \frac{1}{q_C(z)}\circ b(\bar K_C).$$ Then $$\begin{aligned}
\hatLambda_g[\psi_C(x,z)] &=&
z^{-n}\circ b(\bar K_C)\circ b(\bar Q)\circ
\frac{z^n}{q_C(z)}[\frac{1}{z^n \tau_C(x)} \bar K_C e^{xz}]\\
&=& \frac{z^{-n}}{\tau_C(x)} \circ b(\bar K_C)\circ b(\bar Q)\circ
\frac{1}{q_C(z)}[\bar K_C e^{xz}]\\
&=& \frac{z^{-n}}{\tau_C(x)} \circ b(\bar K_C)[\pi(x)e^{xz}]\\
&=& \frac{z^{-n}\pi(x)}{\tau_C(x)} \circ \bar K_C[e^{xz}]\\
&=& \pi(x)\psi_C(x,z)\end{aligned}$$
Note that according to Theorem \[thm:main\], each operator $\hatLambda_g$ satisfies an *intertwining relationship* $$W \circ b(\pi(x)) = \hatLambda_g \circ W$$ with the constant coefficient operator $b(\pi(x))$ where $W=z^{-n}\circ b(\bar K_C)$. As a result we find that:
Examples
========
If we choose $C$ to be the two dimensional space spanned by $c_1=\Delta(1,0)$ and $c_2=\Delta(1,1)$ (a “two-particle” Calogero-Moser type solution) then $$\psi_C(x,z)=(1+\frac{2+x-(2x+x^2)z}{x^2z^2})e^{xz}.$$ In this case the translational-differential operators $\hatLambda$ given by Theorem \[thm:main\] are simply ordinary differential operator. For instance, $$\hatLambda =\partial_z^3+\frac{3}{z-z^2}\partial_z^2-\frac{6z^2-12z+3}{z^3(z-1)^2}\partial
+ \frac{12z-6}{z^2(z-1)^2}$$ which satisfies $\hatLambda \psi_C(x,z)=x^3\psi_C(x,z)$ (as we would expect from earlier results on bispectrality.)
However, if we had chosen instead $c_1=\Delta(0,1)+\Delta(0,-1)$ and $c_2=\Delta(0,2)+\Delta(0,0)$ we would instead have the case of a 2-soliton solution with $$\psi_C(x,z)=(1-\frac{6+(3z-2)e^{2x}+2z-ze^{-2x}}{(e^x+e^{-x})^2z^2})e^{xz}.$$ One finds from the procedure given in the theorem that $$\begin{aligned}
\hatLambda &=& z^{-2}\circ\left(
\left( 20z + 11{z^2} - 8{z^3} +
{z^4} \right)\Trans_{-3}
+
\left( 60 - 68z - {z^2} + 8{z^3} + {z^4} \right) \Trans_{5}
\right.\\
&& +
\left( -36 + 24z + 16{z^2} - 16{z^3} +
4{z^4} \right)\Trans_{-1}
+
\left( -44 - 88z - 8{z^2} + 16{z^3} + 4{z^4} \right) \Trans_{3}\\
&&
\left.\left( -12 - 16z - 2{z^2} +
6{z^4} \right)\Trans_{1}
\right)\circ\frac{z^2}{z^4-2z^3-z^2+2z}\end{aligned}$$ satisfies $\hatLambda [\psi_C(x,z)]=e^{-3x}(1+e^{2x})^4\psi_C(x,z)$.
Conclusions
===========
In addition to being a generalization of the results of on bispectral ordinary differential operators, the present note may be seen as a generalization of in which wave functions of $n$-soliton solutions of the KdV equation are shown to satisfy difference equations in the spectral parameter. The idea that KP solitons might be translationally bispectral was proposed in .
As in , the equations and lead to the well known “ad” relations associated to bispectral pairs. That is, defining the ordinary differential operator $A_m$ in $x$ and the translational-differential operator $\hat A_m$ in $z$ by $$A_m=\textup{ad}_{L_p}^m(\pi(x))
\qquad
\hat A_m=(-1)^m\textup{ad}_{p(z)}^m(\hatLambda)$$ one finds that $A_m\psi_C(x,z)=\hat A_m\psi_C(x,z)$. Similarly, if $$B_m=\textup{ad}_{\pi(x)}^m(L_p)
\qquad
\hat B_m=(-1)^m\textup{ad}_{\hatLambda}^m(p(z))$$ then $B_m\psi_C(x,z)=\hat B_m\psi_C(x,z)$. Note that whenever the order of $B_{m-1}=N>0$ the order of $B_m$ cannot be greater than $N-1$. So, the familiar result that $B_m\equiv0$ and $\hat B_m\equiv0$ for $m>\ord L_p$ holds, which is clearly a strong restriction on the operator $\hatLambda$. However, unlike the case of bispectral ordinary differential operators, one cannot conclude that $A_m\equiv0$ for sufficiently large $m$ since the order of $\hat A_m$ may not be reduced by increasing $m$.
The bispectrality of the rational KP solutions has been shown to have a dynamical significance. In particular, it was shown that the *bispectral involution* is the linearizing map for the classical Calogero-Moser particle system . Moreover, other bispectral KP solutions have been found to have similar properties . This would seem to indicate that it is likely that the bispectrality of KP solitons also has a dynamical significance, as a map between the classical Ruijsenaars and Sutherland systems (cf. ). In fact, such a bispectral relationship between the *quantum* versions of these systems has been recently found in . The dynamical significance of these results will be considered in a separate paper.
[11]{} \#1
G. Wilson, *Bispectral commutative ordinary differential operators*, J. Reine Angew. Math. **442** (1993), 177–204.
[^1]: Moreover, he demonstrated that up to conjugation or change of variables, the operators $L_p$ found in this way are the only bispectral operators which commute with differential operators of relatively prime order, but this fact will not play an important role in the present note.
[^2]: It should be noted that the term “bispectrality” already applies to more general situations than simply differential operators , but in the case of the KP hierarchy I believe only differential bispectrality has thus far been considered.
[^3]: ...and also bispectral in the sense of .
|
---
abstract: 'We report on the performance of a vector apodizing phase plate coronagraph that operates over a wavelength range of $2-5\,\mu$m and is installed in MagAO/Clio2 at the $6.5\,\mathrm{m}$ *Magellan* Clay telescope at Las Campanas Observatory, Chile. The coronagraph manipulates the phase in the pupil to produce three beams yielding two coronagraphic point-spread functions (PSFs) and one faint leakage PSF. The phase pattern is imposed through the inherently achromatic geometric phase, enabled by liquid crystal technology and polarization techniques. The coronagraphic optic is manufactured using a direct-write technique for precise control of the liquid crystal pattern, and multitwist retarders for achromatization. By integrating a linear phase ramp to the coronagraphic phase pattern, two separated coronagraphic PSFs are created with a single pupil-plane optic, which makes it robust and easy to install in existing telescopes. The two coronagraphic PSFs contain a 180$^\circ$ dark hole on each side of a star, and these complementary copies of the star are used to correct the seeing halo close to the star. To characterize the coronagraph, we collected a dataset of a bright ($m_L=0-1$) nearby star with $\sim$1.5 hr of observing time. By rotating and optimally scaling one PSF and subtracting it from the other PSF, we see a contrast improvement by 1.46 magnitudes at $3.5\ \lambda/D$. With regular angular differential imaging at 3.9 $\mu$m, the MagAO vector apodizing phase plate coronagraph delivers a $5\sigma\ \Delta \mathrm{mag}$ contrast of 8.3 ($=10^{-3.3}$) at 2 $\lambda/D$ and 12.2 ($=10^{-4.8}$) at $3.5\ \lambda/D$.'
author:
- 'Gilles P.P.L. Otten, Frans Snik, Matthew A. Kenworthy, Christoph U. Keller, Jared R. Males, Katie M. Morzinski, Laird M. Close, Johanan L. Codona, Philip M. Hinz, Kathryn J. Hornburg, Leandra L. Brickson, Michael J. Escuti'
bibliography:
- 'apj\_otten\_proof.bib'
title: 'On-sky performance analysis of the vector Apodizing Phase Plate coronagraph on MagAO/Clio2'
---
Introduction
============
In direct imaging, the sensitivity for detecting companions close to the star is primarily limited by residual atmospheric [@Racine:99] and quasi-static wavefront variations [@marois:05; @hinkley:07]. These time-varying wavefront errors manifest themselves as irregularities in the diffraction halo around the star (speckles). Coronagraphs reduce the diffraction halo of the star at specific angular scales, and since errors are modulated by diffraction rings, the signal-to-noise ratio (S/N) for companion detection is thus increased. Both pupil- and focal-plane coronagraphs exist and are used on sky with success [@Guyon:06; @Mawet:12]. Many of the latest generation of instruments optimized for high-contrast imaging contain focal-plane coronagraphs, which are typically limited to a raw contrast of $\sim$10$^{-4}$ at small angular separations from the star (a few $\lambda/D$), mostly because of tip/tilt instabilities of the point-spread function (PSF) due to, for example, telescope vibrations and residual seeing effects [@Fusco:14; @Jovanovic:14; @Macintosh:14]. Pupil-plane coronagraphs are inherently impervious to such effects, as their performance is independent of the position of the star on the science detector, and they can be amplitude- [@Carlotti:11] or phase-based [@Codona:04]. One type of pupil-plane coronagraph, called the apodizing phase plate (APP) coronagraph, is located in the pupil plane and modifies the complex field of the incoming wavefront by adjusting only the phase [@Codona:06; @Kenworthy:07]. The flux within the PSF of the telescope is redistributed, resulting in a (e.g., D-shaped) dark region close to the star. Since the apodization is with phase only, the throughput of the APP is higher compared to traditional amplitude apodizers [@Carlotti:13SPIE], and the PSF core only grows slightly in angular size (11.1% for the phase design in this work). Because the APP is located in the pupil plane, it is not only insensitive to residual tip/tilt variations, but also furnishes nodding, chopping, and dithering motions of the telescope or in the instrument, and indeed observations of close binary stars [@Rodigas:15b]. The PSFs of all stars in the image remain suppressed in the dark hole regardless of the shifts on the focal plane. In the infrared, the APP can be combined with conventional nodding motions as a thermal background subtraction technique. Early versions of the APP were realized by diamond-turning a height pattern in a piece of zinc selenide substrate [@Kenworthy:07]. The phase pattern corresponded to the variation in height of the substrate as a function of position in the telescope pupil (i.e., the “classical phase” through optical path differences). As a result of this, the APP was chromatic and suppressed only one side of the star at a time, and the manufacturing was limited to phase solutions with low spatial frequencies.
The vector apodizing phase plate [vAPP, @Snik:12] is an improved version of the APP coronagraph and is designed to yield high-contrast performance across a large wavelength range. In contrast to the regular APP, the phase pattern of the vAPP is encoded in an orientation pattern of the fast axis of a half-wave retarder. Such a device imposes a positive phase pattern upon right-handed circular polarization and a negative phase pattern upon left-circular polarization, through the geometric (or Pancharatnam-Berry) phase [@Pancharatnam:56; @Berry:84; @Mawet:09], with the emergent phase pattern equal to $\pm$twice the fast-axis orientation pattern. This orientation pattern, as well as any other arbitrary pattern, can be embodied by a liquid crystal layer structure, which locally aligns its fast axis to a photo-alignment layer.
The geometric phase is *inherently achromatic*, but leakage terms (which in this case take the shape of the regular PSF) can emerge if the retardance is not exactly half-wave [@Mawet:09; @Snik:12; @Kim:15]. A typical APP phase design is antisymmetric in the pupil function, which results in a D-shaped dark hole next to the star. By splitting the circular polarization states with inverse geometric phase signs in the pupil, the vAPP creates two PSFs with dark holes on either side. By combining multiple self-aligning layers of twisting liquid crystals, it is possible to create retarder structures that have a retardance close to half-wave across a broad wavelength range [up to even more than one octave @Komanduri:13], at wavelength ranges from the ultraviolet (UV) to the thermal infrared (IR). This class of retarders are called multitwist retarders (MTRs). The direct-write manufacturing technique of the alignment layer and hence the MTR liquid crystal orientation pattern [@Miskiewicz:14] gives high control of the phase of the optic and allows the manufacturing of complex phase designs with typically $\sim 10$ micron spatial resolution that were not manufacturable using the diamond-turning techniques of earlier APPs. A vAPP prototype that was optimized for $500-900$ nm was built using both these techniques, and it was characterized in @otten:14.
In this paper we present the first on-sky results of the vAPP installed inside the MagAO/Clio2 [@Close:2010; @Close:13; @Sivanandam:2006; @Morzinski:2014] instrument on the $6.5\,\mathrm{m}$ *Magellan*/Clay telescope at Las Campanas Observatory. We demonstrate the contrast performance at infrared wavelengths at small angular separations from a bright star, and we show how the two coronagraphic PSFs of the vAPP can be combined to suppress speckle noise inside the dark holes.
The vAPP coronagraph for MagAO/Clio2
====================================
The Grating-vAPP principle
--------------------------
The original implementation of the vAPP included a quarter-wave plate and a Wollaston prism to split circular polarization in a truly broadband fashion. Note, however, that leakage terms due to retardance offsets for both the half-wave vAPP optic and the quarter-wave plate limit the contrast performance [@Snik:12]. In @Otten:14spie we introduced a simplified version of the vAPP (grating-vAPP or gvAPP) that includes a linear phase ramp [i.e., a “polarization grating”; @OhEscuti:08; @Packham:10] to impose the circular polarization splitting.
For MagAO/Clio2 we have manufactured an infrared version of such a gvAPP device which has a phase pattern that is composed of two separate patterns: the first is an APP phase pattern optimized for the *Magellan* telescope pupil that produces the coronagraphic PSFs with dark D-shaped holes, and the second is a linear phase ramp that is opposite for the two circular polarization states and provides an angular splitting of the two beams with the opposite coronagraphic phase patterns. This polarization grating splits the two PSFs without the need for a quarter-wave plate and Wollaston prism, which greatly decreases the cost and enhances the ease of installation. As both the modification of the PSF and the splitting direction depend on the handedness of circular polarization following the geometric phase, the grating-vAPP produces two separate coronagraphic PSFs with dark holes on opposite sides, providing continuous coverage around the star. The inclusion of the linear phase also ensures that the leakage term due to the plate not being perfectly half-wave ends up between the two coronagraphic PSFs as a third (unaberrated) PSF. The positioning of the leakage-term PSF in between the coronagraphic PSFs minimizes the impact of any residual non-half-wave behavior of the retarder on the contrast inside the dark holes [@otten:14], and thus it enhances the contrast performance with respect to a coronagraph with a quarter-wave plate and Wollaston prism. This PSF can be used as a photometric and astrometric reference and as an image quality indicator. Both the structure of the coronagraphic PSFs and their splitting angle are not dependent on the retardance of the gvAPP device. Only the brightness ratio of the leakage PSF with respect to the coronagraphic PSFs changes with varying retardance. As the splitting between the coronagraphic PSFs is imposed by a diffractive grating pattern, their separation is a linear function of wavelength. Hence, while the vAPP optic offers high-contrast coronagraphic performance over a broad wavelength range, to produce sharp PSFs without radial smearing, narrowband filters have to be applied throughout the broad wavelength range over which the device is highly efficient. By orienting the dark holes left/right with respect to the up/down splitting, this grating effect can furnish low-resolution spectroscopy of point sources inside either of the dark holes. Using the gvAPP in combination with an integral field spectrograph overcomes the spectral smearing issue altogether, and such a setup can therefore provide snapshot coronagraphic spectroscopy over the entire efficiency bandwidth.
Phase pattern design
--------------------
The phase pattern is determined with a simple, iterative algorithm akin to a Gerchberg–Saxton iteration [@Gerchberg:72; @Fienup:80]. We switch between electric fields in the pupil plane and the focal plane with Fourier transformations and enforce constraints in the corresponding planes. In the pupil plane, the amplitude field amplitude is set to unity inside the telescope aperture and zero everywhere else. In the focal plane, we set the electric field amplitude to zero in the dark hole. This process is repeated hundreds of times until we obtain a phase pattern that achieves the desired contrast. This approach does not guarantee the highest PSF core throughput for a desired contrast, but we found it to perform better than any other design approach that we are aware of.
Since this particular APP design only has a dark hole on one side of the focal plane, the phase pattern in the pupil will be antisymmetric. We use this symmetry to improve the performance of the algorithm. Instead of setting the electric field to zero in the dark hole, we add a scaled and mirrored version to the electric field on the other side of the dark hole. This is motivated by the fact that a one-sided dark hole created by an antisymmetric phase pattern is achieved in the focal plane by symmetric and antisymmetric parts of the electrical field canceling each other in the dark hole and adding to each other on the other side. The scaling enforces energy conservation in the focal plane. A comprehensive description of our design algorithm including applications to symmetric dark holes will be provided in a forthcoming publication by Keller et al. (2017, in preparation). For the optimization in this paper, we define a dark hole from 2 to 7 $\lambda/D$ and with a $180^\circ$ opening angle and a desired normalized intensity of $10^{-5}$. The final design has a PSF core throughput of 40.3% with respect to an unaberrated PSF as the light gets redistributed across the PSF (mostly on the other side from the dark hole).
Coronagraph optic specifications
--------------------------------
The gvAPP optic has a diameter of $25.4$ mm and a thickness of approximately $3.3$ mm and is designed to work with the Clio2 camera with a nominal size of 3.32 mm of the reimaged *Magellan* telescope pupil. The diameter of the vAPP pupil mask was undersized by $100$ microns (from a diameter of 3.32 to 3.22 mm) to create a tolerance against pupil misalignments in the instrument. A $1^\circ$ wedge is added on one side of the coronagraph in order to deflect reflection ghosts. To further suppress ghost reflections and improve the overall transmission, both sides of the optic are broadband antireflection coated with an average transmission between $2$ and $5$ microns of 98.5%. An aluminum aperture mask, matching the *Magellan* pupil, with a pixelated edge (with a pixel size of $11.54$ microns) is deposited on one of the substrates and is sandwiched directly against the retarder layers, manually aligned using a high-power microscope, and fixed in place with an optical adhesive. The phase pattern (the coronagraphic pupil phase pattern plus the grating pattern) is written as an orientation pattern of an alignment layer of “DIC LIA-CO01” by a UV laser with polarization-angle control [@Miskiewicz:14]. The pixel size is $11.54$ microns for both the phase and amplitude pattern. During fabrication, the writing accuracy of the fast axis is calibrated to approximately $2^\circ$, corresponding to a maximum phase error of $4^\circ$, that is, $\sim \lambda/100$. The patterned retarding layer consists of three MTR layers (Merck RMS09-025; see also Table \[tab:tab1\]) and is optimized to produce a retardance $\delta$ that is half-wave to within 0.38 radians for wavelengths between $2$ and $5$ microns, corresponding to a maximum flux leakage from the coronagraphic PSFs to the leakage-term PSF of 3.5%. The design recipe of the MTR is \[$\phi_1=78^\circ$, $d_1=3.5 \,\mu$m, $\phi_2=0^\circ$, $d_2=7.3\,\mu$m, $\phi_3=-78^\circ$, $d_3=3.5\,\mu$m\], where $d_i$ stands for layer thickness, $\phi_i$ for the twist of a layer, and $i$ for the layer number [see @Komanduri:13]. This recipe is used to build our coronagraph with our custom fast-axis pattern and also a test article with the same parameters but a fixed fast axis. The transmission of this test article is measured between crossed linear polarizers with a VIS-NIR spectrometer up to 2800 nm. A model of the MTR is fitted to the observed transmission between crossed polarizers with five free parameters (three thicknesses and two relative twists with respect to the middle layer). The best-fit parameters are \[$\phi_1=81^\circ$, $d_1=3.5\,\mu$m, $\phi_2=0^\circ$, $d_2=7.3\,\mu$m, $\phi_3=-77^\circ$, $d_3=3.9\,\mu$m\], and are used afterward to predict the transmission, retardance, and leakage at wavelengths out to 5000 nm, as shown in Fig. \[fig:retardance\].
{width="50.00000%"} {width="50.00000%"} {width="50.00000%"}
The leakage PSF intensity is derived by measuring the peak ratio of either of the coronagraphic PSFs to the leakage-term PSF in a sequence of unsaturated images. The mean and standard deviation of the ratio in this sequence are $31.47\pm1.07$. This ratio is divided by the theoretical PSF core throughput (i.e., Strehl) of 0.403 to yield the ratio as if the coronagraph were not present. This means that the intensity of the leakage term is $1/78.1 \cdot I_\mathrm{coron}$, where $I_\mathrm{coron}$ is the intensity of the coronagraphic PSF. This value is normalized by the total intensity $(2+1/78.1) \cdot I_\mathrm{coron}$ to yield the fractional leakage intensity (the amount of light that goes into the leakage term). In the completed coronagraph, we measure a leakage-term intensity of 0.636% at $3.94\, \mathrm{microns}$, which corresponds to $\delta =2.98\,\mathrm{rad}$, using this method, which is within the previously defined specifications. While this leakage is slightly larger than the theoretical expectation at that wavelength (0.16%), it is comparable in magnitude to the maximum retardance offset of the curve (see Fig. \[fig:retardance\]c).
The polarization grating pattern spans $17.5$ waves in terms of phase, corresponding to a displacement of $35\ \lambda/D$ between the two coronagraphic PSFs. In this way, both of the coronagraphic PSFs fit on the chip at the longest wavelengths ($M'$ band) while minimizing the contribution of the leakage-term diffraction pattern in the dark holes. The grating creates a splitting angle that is dependent on the wavelength in terms of pixels of separation, and so the PSFs are laterally smeared. For optimal image quality with smearing of at most 1 $\lambda/D$, the filter FWHM needs to be $\frac{\Delta \lambda}{\lambda} \leq 0.06$. Due to the optic’s broadband efficiency, filters can be used anywhere between $2$ and $5$ microns for coronagraphic imaging. Note that even outside the specified wavelength range, the coronagraphic performance is never deteriorated by leakage terms, but the coronagraphic PSFs are less efficient as they lose light to the leakage-term PSF.
After installation inside MagAO/Clio2, we collected pupil image measurements with and without the coronagraph at several IR bands during good sky conditions and with adaptive optics (AO) to obtain accurate on-sky pupil transmission measurements. We determine the transmission of the optic from the ratio of the pupil intensity with and without the coronagraph. The theoretical transmission values are detailed in Table \[tab:tab1\] per layer and compared to the measured transmission. Since the measured retardance is close to half-wave (as expected from the theory), the thickness of the liquid crystal layers cannot deviate significantly from the theoretical value. We therefore set their thicknesses to the fitted values for the MTR recipe, which adds up to $14.7\,\mathrm{microns}$. The absorption properties of the retarder layer were measured in a 900 nanometer thick sample at a wavelength of $4\,\mathrm{microns}$ and extrapolated to the $14.7\,\mathrm{micron}$ thick layer. The absorption coefficient derived from this measurement falls on the high end of the range seen in Fig. 3 of @Packham:10, who measured the transmission of a similar family of liquid crystals. The absorption coefficient of the glue layer is derived from the spectral transmission graph on the Norland Products website[^1] and the known thickness of their sample. The thickness of the glue layer constitutes the largest uncertainty because it was not measured during the manufacturing process. Because the other transmission values are well constrained, we let the thickness of the glue layer vary as a free parameter to match the observed transmission. Our derived glue layer thickness of $50\,\mathrm{microns}$ is not unexpected for glass–glass interface bonding. The breakdown shows that the throughput is primarily limited by the optical adhesive NOA-61. The absorption features of both the optical adhesive and the retarding layer are related to the vibrational modes of chemical bonds with carbon, such as C–C, C–O, C–N and C–H.
The gvAPP coronagraph is located in the pupil stop wheel of Clio2 and oriented with the grating splitting angle perpendicular to the arc traveled by the pupil in the pupil wheel. The wedge splitting angle was oriented perpendicular to the splitting direction. The orientation of the splitting angle corresponds in theory with splitting the PSFs along the short axis of the chip. This leaves a large amount of space along the long axis to nod the PSFs along for background subtraction. From our PSF measurements we see that the orientation of the PSFs on the chip is approximately 26 rotated away from the preferred orientation. This rotation does not interfere with the background subtraction.
Layers Material Thickness 3.9 micron 4.7 micron ($M'$)
-------------------------------- --------------------- ------------- ------------ -------------------
AR-coating $\cdots$ $\cdots$ 0.98 0.99
Substrate with $1^\circ$ wedge CaF$_2$ 0.8 mm 0.99 0.99
Amplitude mask evaporated aluminum 250 nm $\cdots$ $\cdots$
Bonding glue NOA-61 epoxy 50 $\mu$m 0.81 0.81
Substrate CaF$_2$ 1 mm 0.99 0.99
Retarder layers Merck RMS09-025 14.7 $\mu$m 0.85 $\sim0.85$
Alignment layer DIC LIA-CO01 50 nm $\cdots$ $\cdots$
Bonding glue NOA-61 epoxy 50 $\mu$m 0.81 0.81
Substrate CaF$_2$ 1 mm 0.99 0.99
AR-coating $\cdots$ $\cdots$ 0.98 0.99
Theoretical throughput $\cdots$ $\cdots$ 0.53 0.54
Measured throughput $\cdots$ $\cdots$ 0.51 0.54
\[tab:tab1\]
Observations
============
The observations with the vAPP coronagraph at MagAO/Clio2 were taken during 2015 June 6, 07:38:40 - 10:07:34 UT during excellent atmospheric conditions (with only high cirrus clouds). The filter used for these observations is the $3.9$ micron narrowband filter with a width of 90 nm and a central wavelength of $3.94$ microns. This filter was chosen to take advantage of the extremely high Strehl ratio of the adaptive optics system at longer wavelengths ($>$95%), and to make sure the radial smearing ($<0.4\ \lambda/D$) interferes only minimally with the interpretation of the PSF suppression in the dark hole. The plate scale of the detector is 15.85 arcseconds pixel$^{-1}$ [@Morzinski:15]. The target discussed in this paper to assess the contrast performance of the vAPP is an A-type star with an $L'$-band magnitude between 0 and 1. The star was selected to be bright and without a known companion to explore the limits of the coronagraph’s performance. Note that the coronagraphic system with the gvAPP at MagAO/Clio is also fully applicable for fainter stars and has been tested on sky down to magnitude-7 targets. The performance of the adaptive optics system remains invariably high down to $R=7$ magnitude stars [@Close:12]. A total of 287 data-cubes were taken on sky, each with 20 subframes and an exposure time of 1 s each. The dataset has a total on-target exposure time of 5740 s. The derotator was off during the observations, and the observations span a total of $39\fdg45$ of field rotation. To perform background subtraction, data cubes were recorded with the PSF off the chip, centered approximately 10 arcsec to the left from the nominal center of the science camera array, so that no sources and ghosts are seen on the same part of the chip. This background estimation was repeated four times during the sequence. No flats have been applied to the data, and a sky correction was made using the off-target nods.
Figure \[fig:psfs\] shows a comparison between the theoretical and observed PSFs using a median combination of the top 50% best frames in terms of the fitted radius ($1.22\ \lambda/D$) of the leakage-term PSF, acting as a proxy for subframe quality as it expands with increased turbulence. The observed coronagraphic PSFs are saturated in the core and in the first diffraction ring but are corrected to the peak flux consistent with the unsaturated calibration images. While the two PSFs have approximately the same brightness, the two PSF halos inside the dark holes have a slightly different intensity. A potential source of the difference is discussed in Section \[sec:diff\].
Data reduction
--------------
To remove hot, dead, or flaky pixels in each image, we subtracted a median-filtered image with a 3$\times$5 pixel box from the cleaned and centroided image cube to generate an image where the outliers clearly stand out. The $3 \times 5$ box is chosen because the outlying pixels tend to have structure in the direction of the readout and not perpendicular to it. The data points that deviate more than 1000 counts are replaced by the local value of the 3$\times$5 median. Four sets of sky reference frames are taken between on-target observations. The sky frames were median-combined for every one of the four sets. Each of the previously taken frames on target were subtracted by the first consecutive median-combined master sky frame. After the background subtraction, the median of a cosmetically clean part of the chip is subtracted in order to remove any residual background offset. A theoretical diffraction pattern consistent with the geometry of the telescope and wavelength of observation is used as a fiducial. This theoretical PSF is then fit to the central leakage PSF by minimizing the chi-squared residuals between the theoretical PSF and the leakage PSF, with $x$, $y$, radius ($1.22\ \lambda/D$), and intensity as the free parameters of the fit. All images of the data cubes are coregistered by shifting the images to the central pixel of the frame with the previously fit $x$ and $y$ values using a bilinear interpolator. The radius of the leakage PSF fit is used as a subframe quality indicator. The few images (39) that have fitted radii significantly smaller ($< 9$ pixels) or larger ($>12$ pixels) than the diffraction limit are excluded. The best 91% (5200) of the images are selected after sorting the frames by radius from smallest to largest. Both criteria remove the frames where the seeing conditions temporarily worsened or where the AO system lost its lock. After these selections, the images are reordered to original chronological order and binned by four frames, corresponding to 4 s integration time per binned frame to reduce memory consumption and computational time.
Instrumental ghosts due to internal reflection of the refractive optics are present in the image; see Fig. \[fig:psfs\]. Several of these ghosts are typically $10^{-2}-10^{-3}$ in intensity, and their position relative to the central PSF changes as a function of position on the chip. To reduce their influence, the regions of identified ghosts are masked off from any subsequent fitting or stacking process. These ghosts can be removed from the dark holes by setting the rotator to an angle of $30^\circ$.
Rotation, Scaling, and Subtraction
----------------------------------
Each on-target image consists of three PSFs, which we label “$+$” for the upper coronagraphic APP PSF, “$-$” for the lower coronagraphic PSF with the dark region on the opposite side of the star, and “$0$” for the leakage PSF, which is consistent with the PSF obtained with no coronagraph in the optical path and with a flux typically $10^{-2}$ of the other two PSF cores. To suppress the noise contribution of the seeing-driven halo inside the dark holes, we use one coronagraphic PSF as a reference for the other coronagraphic PSF of the same star, and we subtract “$-$” from “$+$.” This PSF subtraction technique avoids self-subtraction of the flux of a potential companion as it is very unlikely to have another companion at the same separation and brightness on the opposite side of the star. A similar approach is taken by @Marois:07 and @Dou:15, who use the (noncoronagraphic) PSF (itself) under rotation as a reference and measure an order-of-magnitude improvement compared to regular LOCI [@lafreniere:07] without rotation. Our approach works by rotating, scaling, and subtracting PSF “$-$” from PSF “$+$” in a three-step process. First, the image is flipped in both dimensions so that “$-$” has the same orientation as “$+$”. We align each “$+$” and “$-$” PSF with the median of all “$+$” PSFs, by performing a cross-correlation on a bright, isolated feature at $10\ \lambda/D$ on the bright side of the PSF. With the obtained centroids, the “$+$” and “$-$” PSFs are subpixel shifted to the frame center with the python routine `scipy.ndimage.interpolation.shift` set to first-order spline interpolation. The “$-$” cube is then multiplied by a fixed amplitude ratio and subtracted from the “$+$” cube. An example of the PSFs before and after subtraction can be seen in Fig. \[fig:drieluik\] for three different scaling factors (0, 1.04, and 0.71). The diffraction structures on the bright side of the PSFs are optimally canceled using an intensity scaling ratio of 1.04. This is consistent with the ratio of the encircled energies of both PSFs. However, with this ratio, the seeing-driven halo in “$-$” is oversubtracting the halo in “+,” resulting in larger amounts of speckle noise in the final combined image. A likely cause for this is that aberrations create pinned speckles on the diffraction structure in the dark holes, but the intensities may be different in the left and right dark holes. Although this diffraction structure ideally has an intensity of $<10^{-5}$ with respect to the PSF core (and therefore is not visible in the left panel of Fig. \[fig:psfs\]), it becomes brighter due to residual seeing or quasi-static aberrations of the telescope and instrument. Because this diffraction structure is fully point-symmetric between the two PSFs, a rotation-subtraction approach with a variable scaling factor always reduces the pinned speckle structure in the halo. A simple simulation shows that with the realistic seeing and AO performance the intensity of this halo is practically balanced, even when an AO loop time lag (3 ms) and strong wind speed ($10\,\mathrm{m\,s}^{-1}$) in the worst-case direction of the dark hole orientations are taken into account. As for (quasi-)static optical aberrations, only odd modes (like trefoil aberration) cause an asymmetry between the holes in the two dark holes, while even modes generate complete symmetric PSF structures. However, to first order, odd aberrations will also merely brighten the symmetric diffraction structure inside the dark holes, just with different intensities. Assuming trefoil is the dominant aberration, we simulate how much trefoil could create a PSF that is still consistent with the observed in terms of the asymmetry between the dark holes. Based on this simulation, we conclude that the RMS error of the trefoil aberration needs to be $\sim$0.04 radians (25 nm at 3.94 microns) to match the observations. We therefore conclude that the vAPP is not only insensitive to tip/tilt errors, but, through the rotation-scaling-subtraction technique, can also generically cope with low-order wavefront errors.\[sec:diff\]
Another option for scaling the two PSFs is to take the intensity ratio that minimizes *the halo noise in time* and applying that to all frames (see also @Marois:06). To determine this ratio, we calculate the standard deviation for the temporal intensity variation in many randomly selected 3$\times$3 pixel patches inside the combined dark hole. Figure \[fig:probes\] shows the standard deviation for various 3$\times$3 patches, which are color coded according to angular separation, as a function of the applied intensity ratio. The vertical lines indicate the ratio at which the noise is minimal on average for a series of $\lambda/D$ bins. As reducing the noise closest to the star is the most important, the value 0.71, which on average minimizes the noise in the bin at $2-3\ \lambda/D$, is used to scale the amplitude of the bottom PSF cube before subtracting it from the top PSF. Figure \[fig:ratiomap\] shows the optimal scaling factor to minimize variance for each pixel inside the combination of dark holes. It is apparent that rotation-subtracting PSFs is only effective close to the star, at the location of the seeing-driven halo. Farther away from the star it is preferred to not perform any subtraction at all, as at the outer parts of the dark holes the noise is uncorrelated (e.g., photon shot noise from the thermal background and read-noise), and therefore subtracting the two images will actually inject noise and consequently increase it with a factor of $\approx \sqrt{2}$. This effect is also the likely cause of the reduced optimal factor to minimize variance (0.71), in comparison to the factor of 1.04 that we found to optimally balance out the intensity structure. A further minimization of the variance in the combined dark hole can be achieved by optimizing the ratio in radial bins as is commonly done with localized optimization of combination of images [LOCI @lafreniere:07] and principal component analysis [PCA @Amara:12; @Soummer:12].
For a scaling ratio of 0.71, we plot in Fig. \[fig:allhisto\] the time series and histograms for three 3$\times$3 pixel patches at four locations inside the dark holes as indicated in Fig. \[fig:drieluik\] before and after the subtraction to see how the rotation-subtraction technique improves the intensity variability. At the location closest to the PSF core (1.8 $\lambda/D$), both the average value and the standard deviation of the intensity are significantly reduced. This effect is seen in both the time series and the histograms. This is particularly evident in cases of worse AO performance (for instance, around the $\#=750$ mark). Moreover, the rotation-subtraction technique produces histograms that are much more Gaussian than before. As discussed already, pixels farther away from the central star obtain a $\sim$$\sqrt{2}$ increase in the noise, as their noise properties are already close to Gaussian and independent.
Results: contrast curve
=======================
The combination of the intrinsic coronagraphic performance of the vAPP coronagraph inside the dark holes and the optimal rotation-subtraction of its two complementary PSFs to subtract the residual seeing-driven halo delivers essential suppression at very small angular separations from the central star to detect and characterize planetary companions. We apply median-filtering and classical angular differential imaging [ADI @Marois:06 without excluding frames based on the angular distance] to further suppress static and quasi-static speckles inside the combined dark holes to reach the ultimate contrast. After rotation-subtracting the two PSFs with the optimal ratio, the median value inside a wedge for $5-7\ \lambda/D$ in the dark hole is subtracted from every pixel in every frame of the data cube. This process is repeated for every frame to remove any residual intensity offsets. The median across the time dimension per pixel is removed from the whole cube to remove any residual static PSF structures. After these steps, the frames are derotated to the sky frame and coadded by taking the mean across the time dimension.
To assess the contrast performance, artificial companions are injected in the original data cube at steps of 0.5 $\lambda/D$ and with steps in magnitude of 1 with the expected amount of sky rotation. The injected sources are a rescaled and translated version of the unsaturated calibration data set and therefore have the correct PSF for each dark hole. The previously described pipeline of optimal rotation-subtraction, median-filtering, and ADI is applied to these data cubes with injected sources of varying contrast ratio. The S/N of these planets is calculated by calculating both the sum of the flux in an aperture with a width of $1\ \lambda/D$ and the noise in the same aperture without the planet added. The standard deviation in this aperture is multiplied by the square root of the number of pixels in the subaperture to obtain the measurement noise on the planet flux, assuming that this noise is Gaussian (which is supported by the results in Fig. \[fig:allhisto\]). The magnitude of the injected point source is rescaled to obtain an $\mathrm{S/N}=5$, and these values are plotted as a contrast curve for 5$\sigma$ point source detection sensitivity versus angular separation in Fig. \[fig:contrast\]. Although this method is necessarily different from the procedure introduced by @Mawet:14, as at small $\lambda/D$ the dark hole is too small to obtain a measure of the standard deviation at neighboring patches, it is fully consistent for pure Gaussian noise. In any case, the contrast performance is clearly validated by the fact that the injected point sources at the corresponding contrast ratios are detected with large S/N, and the numbers are therefore reliable at least within a factor of a few (which is fairly insignificant on a logarithmic scale).
In Fig. \[fig:contrast\] we note that within $4.5\ \lambda/D$, the contrast performance is significantly improved by subtracting the other PSF with a fixed amplitude scaling factor of 0.71. This is most evident at a angular separation of $3.5\ \lambda/D$, where the improvement is 1.46 magnitudes (four-fold improvement) to a $\Delta \mathrm{mag}$ of 12.2, which corresponds to a contrast of $10^{-4.8}$. Beyond $4.5\ \lambda/D$, the contrast performance for the rotation-subtraction technique is degraded, as here the noise is random and uncorrelated, and therefore aggravated after the combination with the second PSF. As previously mentioned, we expect to be able to reduce this effect by optimizing the scaling factor in radial bins, although this also increases the degrees of freedom. The turnover point at $4.5\ \lambda/D$ is dependent on the brightness of the target as it moves inward with fainter targets as the background noise contribution becomes more dominant. For this data set, the turnover point at $4.5\ \lambda/D$ has a $\Delta \mathrm{mag}$ of 12.5, corresponding to a 5$\sigma$ contrast of $10^{-5}$. Like many other reduction methods, our classical ADI approach also removes part of the planet flux in addition to residual speckles in the stellar PSF. To quantify this effect, we retrieve the planet flux after applying the entire data-reduction pipeline to the data and compare it to the injected planets. The efficiency of the ADI algorithm is as low as 29% at 2 $\lambda/D$ and reaches 68% at 7.5 $\lambda/D$. This lower efficiency close to the star is expected as there is less angular displacement of the planet in terms of $\lambda/D$ which leads to more self-subtraction. We overplot in Fig. \[fig:contrast\] the 5$\sigma$ contrast excluding self-subtraction, which would reach down to below $10^{-5}$ for $>$3 $\lambda/D$. This limiting case may be reached by applying more advanced PSF subtraction techniques, like Principal Component Analysis [@Amara:12; @Soummer:12].
Discussion and conclusions {#sec:disc}
==========================
To put this contrast performance of the vAPP coronagraph at MagAO/Clio2 in context, we compare our results to published on-sky contrast curves for different coronagraphic instruments. Such an analysis necessarily uses heterogeneous data sets because of variations in the brightness of the star, the wavelength, the size of the telescope, and the applied data-reduction techniques. It is important to note, however, that close to the star the brightness of the star has little impact as the contrast there is limited by speckle halo noise. Moreover, all published contrast curves are produced for stars that are bright enough that the AO system still has its optimal performance. Furthermore, we put all curves on a $\lambda/D$ scale to account for differences in telescope diameter and observation wavelength, which provides the most honest comparison.
To begin with a related coronagraph, in comparison to the performance of the regular APP at VLT/NACO [@Quanz:10; @Kenworthy:13; @Meshkat:14], the vAPP PSFs do not exhibit any clear diffraction structure close to the star, whereas the VLT APP PSF clearly does. The much-improved manufacturing accuracy of the phase patterns now permits the creation of dark holes that are devoid of diffraction structure down to $10^{-5}$. Moreover, the coronagraphic PSFs of the grating-vAPP are not deteriorated by leakage PSFs, as they form a separate PSF, which actually can be used to one’s advantage as a photometric or astrometric reference.
The MagAO/Clio2 gvAPP coronagraph contrast performance from Fig. \[fig:contrast\] is compared in Fig. \[fig:comparecontrast\] with the following contrast curves: the annular groove phase mask (AGPM) at LBT [@Defrere:14], the vector-vortex coronagraph (VVC) at the 1.5 m well-corrected aperture at Palomar [@Serabyn:10], the GPI first-light results [@Macintosh:14], SPHERE with the apodized Lyot coronagraph (ALC) [@Vigan:15], and the APP at the VLT [@Meshkat:14]. All these published contrast curves are corrected from the published $N\sigma$ to a $5\sigma$ detection limit. We assume the contrast is not limited by photon noise in all cases, and therefore we do not correct the curves for differences in exposure time and telescope diameter. In terms of $\lambda/D$, both the GPI and SPHERE contrast curves tend to reach high contrasts farther away from the star, which is likely due to the fact that they were taken at shorter wavelengths where the sky background is lower than in the $L$ band. Most notably, the vAPP has a much smaller inner working angle (IWA) in combination with better contrast performance at small angular separations than GPI and SPHERE, when measured in $\lambda/D$. The IWA of the SPHERE ALC is restricted by the focal-plane mask to 120 mas. Moreover, all focal-plane coronagraphs are limited in their contrast performance at small $\lambda/D$ due to imperfect tip/correction. The VVC result at the Palomar 1.5 m well-corrected aperture is a bit of an outlier because of the significantly different $D/r_0$ ratio as compared with the other telescopes, but it is included because of its high performance at a $\lambda/D$-sized IWA. The assumption of being speckle limited likely does not hold here as the VVC results were within a factor of two from the photon noise on the background. Nevertheless, we see that the vAPP is a very strong contender or even outperforms the other coronagraphs within $5\ \lambda/D$ with an improvement of up to 2 magnitudes for $2.5-3.5\ \lambda/D$.
The exceptional contrast performance of the vAPP coronagraph is owed to the unique combination of the following properties:
1. Insensitivity to tip/tilt errors that impact focal-plane coronagraphs but not pupil-plane coronagraphs like the vAPP.
2. Deep suppression of the PSF diffraction structure with an accurately manufactured (geometric) phase pattern already at the first diffraction ring down below the seeing-driven halo.
3. Subtraction of the halo in the dark holes by combining both PSFs with a rotation-subtraction technique.
We see that using the second coronagraphic PSF as a PSF reference gives an improvement of $1-1.5$ magnitudes (a factor $2.5-4$ in terms of S/N) at $3-3.5\ \lambda/D$. The PSF subtraction is shown to improve the contrast within $4.5\ \lambda/D$. With a radially optimized subtraction, the degradation of the contrast outside this distance can be reduced. Given a fixed ratio based on optimal contrast close to the star, we achieve a $5\sigma\ \Delta \mathrm{mag}$ contrast of 10.8 ($=10^{-4.3}$) at 2.5 $\lambda/D$, 12.2 ($=10^{-4.8}$) at $3.5\ \lambda/D$, and 12.5 ($=10^{-5.0}$) at $4.5\ \lambda/D$. Using a PCA-based algorithm instead of applying classical ADI, we expect that our performance will be less impacted by self-subtraction and will improve toward the dashed line of Fig. \[fig:contrast\]. Use of a simultaneous reference PSF was also explored by @Dou:15, who used the roughly symmetric PSF itself under rotation to feed a PCA algorithm. Their approach gave an improvement of an order of magnitude in terms of contrast when compared to LOCI. @Rodigas:15b used a close binary star to build their reference PSF library. By having a simultaneous reference within the isoplanatic patch, and with roughly the same optical path through the telescope, a better sensitivity is expected than using the star as its own reference. In their study, a 0.5 magnitude improvement within 1 arcsec from the star was seen as compared to normal ADI. @Rodigas:15b suggest combining their binary differential imaging (BDI) technique with the vAPP coronagraph to reach better contrasts. We can extend this by noting that a double correction can be done by combining BDI and the second vAPP PSF as another reference. In both previous cases, the methods are less impacted by self-subtraction because it is unlikely that a companion exists in the reference library with similar brightness, position angle, and separation. Both papers give us confidence that an advanced PCA-based algorithm can be used to generate a better reference PSF and the contrast can be pushed down even more.
To improve the transmission of the optics, one of the three substrates and consequently one adhesive layer could be eliminated by directly depositing the liquid crystal layer on top of the antireflection-coated substrate and bonding it directly with the substrate with the aluminum mask. This procedure increases the transmission by about 20% but makes manufacturing slightly more difficult and expensive.
Future work
-----------
We have demonstrated a manufacturing technique, based on the geometric phase imposed by to patterned liquid crystals, that allows precise control of the phase pattern and a broadband coronagraphic response that can be optimized at any wavelength range from the UV to the mid-IR. The gvAPP coronagraph is relatively straightforward to manufacture and install at existing telescopes as it consists of a single optic in a pupil plane. Using these capabilities, we are looking into new phase patterns with properties different from that normally expected from classical APP theory. For instance, a vAPP with a dark hole spanning $360^\circ$ per PSF or with integrated holographic wavefront sensing solutions [@Wilby:2016] has been implemented and tested on sky. Future work also includes exploring hybrid coronagraphs, for instance as described by @Ruane:15. Furthermore, we will study the photometric and astrometric stability of the leakage term to assess how well this works as a reference PSF. Following the implementations described by @Snik:14spie, we also intend to explore the dual-beam polarimetric capabilities of the vector APP in the optical lab and on sky. For polarized sources, an increased sensitivity is expected by simultaneously using the coronagraphic capabilities and polarimetric beam-switching.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank the anonymous reviewer for constructive comments that helped improve this paper. This work is part of the research program Instrumentation for the E-ELT, which is partly financed by the Netherlands Organization for Scientific Research (NWO). The MagAO vAPPs were purchased from ImagineOptix with help from the Lucas Foundation and the NASA Origins of Solar Systems program. FS is supported by European Research Council Starting Grant 678194 (FALCONER). KMM’s and LMC’s work is supported by the NASA Exoplanets Research Program (XRP) by cooperative agreement NNX16AD44G. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. This paper includes data gathered with the 6.5 meter *Magellan* Telescopes located at Las Campanas Observatory, Chile.
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[^1]: <https://www.norlandprod.com/adhesives/NOA%2061.html>
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abstract: 'We theoretically study trapped ions that are immersed in an ultracold gas of Rydberg-dressed atoms. By off-resonant coupling on a dipole-forbidden transition, the adiabatic atom-ion potential can be made repulsive. We study the energy exchange between the atoms and a single trapped ion and find that Langevin collisions are inhibited in the ultracold regime for these repulsive interactions. Therefore, the proposed system avoids recently observed ion heating in hybrid atom-ion systems caused by coupling to the ion’s radio frequency trapping field and retains ultracold temperatures even in the presence of excess micromotion.'
author:
- 'T. Secker'
- 'N. Ewald'
- 'J. Joger'
- 'H. Fürst'
- 'T. Feldker'
- 'R. Gerritsma'
title: 'Trapped ions in Rydberg-dressed atomic gases'
---
[*Introduction*]{} – Recent years have seen significant interest in coupling ultracold atomic and ionic systems [@Grier:2009; @Zipkes:2010; @Schmid:2010; @Zipkes:2010b; @Harter:2013] with the purpose of realising quantum simulators [@Bissbort:2013; @Negretti:2014], studying ultracold chemistry [@Rellergert:2011; @Ratschbacher:2012] and collisions or employing ultracold gases to sympathetically cool trapped ions [@Krych:2010; @Krych:2013; @Meir:2016]. It has become clear however, that the time-dependent trapping field of Paul traps limits attainable temperatures in interacting atom-ion systems [@Zipkes:2010; @Schmid:2010; @Nguyen:2012; @Cetina:2012; @Krych:2013; @Chen:2014; @Weckesser:2015; @Meir:2016; @Rouse:2017]. This effect stems from the fast micromotion of ions trapped in radio frequency traps which may add significant energy to the system when short-range (Langevin) collisions with atoms occur.
Here, we theoretically study ionic impurities immersed in a cloud of atoms that are coupled to Rydberg states [@Saffman:2009; @Henkel:2010; @Pupillo:2010; @Honer:2010; @Balewski:2014; @Hofmann:2014]. By selecting a Rydberg state that is coupled to the ground state via a dipole-forbidden transition, we can change the atom-ion interaction such that it becomes repulsive for ultracold atoms. In this situation, the atoms cannot get close enough to the ion for the trap drive to add energy to the system and no excess heating takes place. Instead, the system slowly thermalizes at a rate that is compatible to the rate expected for an ion trapped in a time-independent trap. We also show that in contrast to ground state (attractive) atom-ion systems, which require the mass of the ion to be larger than that of the atom [@Major:1968; @DeVoe:2009; @Zipkes:2011; @Cetina:2012; @Chen:2014; @Weckesser:2015; @Rouse:2017], the proposed scheme retains ultracold temperatures for the combinations $^{87}$Rb/$^{171}$Yb$^+$ and $^{87}$Rb/$^{9}$Be$^+$ alike.
![A trapped ion (blue sphere) in a cloud of atoms (orange spheres), the atomic ground states $|gS\rangle$ are dressed with the Rydberg state $|nS\rangle$ with $n$ the principal quantum number and $\Delta_0$ the laser detuning. By appropriate engineering of the Rydberg laser field, the adiabatic atom-ion potential can be made repulsive such that the atoms cannot get close enough to undergo Langevin collisions (as depicted by the sphere around the ion). To optically shield the ion in three dimensions, two laser fields are needed with linear and circular polarization as explained in the text. []{data-label="fig_blup"}](fig1.pdf){width="8cm"}
The paper is organized as follows: First we will describe how the atom-ion interaction can be engineered to be repulsive by coupling to a Rydberg state. Second, we will show that such an atom-ion potential does not lead to heating when the ion is trapped in a Paul trap and is interacting with a cloud of atoms. For this, we will use a classical description of the atom-ion dynamics [@Cetina:2012; @Meir:2016]. Finally we discuss possible experimental issues and limitations of our proposed system.
[*Dressing on a dipole-forbidden transition*]{} – We are interested in dressing an atom in the ground state $|gS\rangle$ with a Rydberg state $|nS\rangle$ using a single near-resonant laser field, as shown in Fig. \[fig\_blup\]. When there are no electric fields present, such a transition is prohibited. However, the field of a nearby ion may mix $|nS\rangle$ with Rydberg states that can couple to the $|gS\rangle$ state via a laser field. Within the dipole approximation and using perturbation theory we obtain for the Rydberg wave function: $$\label{psiR}
|\psi (\mathbf{R})\rangle \approx |nS\rangle + e E_{\rm{ion}}(\mathbf{R})\sum_k \frac{\langle kP|z|nS\rangle}{\mathcal{E}_{nS}-\mathcal{E}_{kP}}|kP\rangle,$$ where $\mathcal{E}_j$ denotes the energy of state $|j\rangle$, and $E_{\rm{ion}}(\mathbf{R})$ with $\mathbf{R}=\mathbf{r}_{\rm{i}}-\mathbf{r}_{\rm{a}}$, the electric field generated by an ion located at position $\mathbf{r}_{\rm{i}}$, evaluated at the atomic position $\mathbf{r}_{\rm{a}}$. The relative electron-atom core position projected on the electric field direction is denoted by $z$, where we assume a reference frame in which the local electric field $\mathbf{E}_{\rm{ion}}(\mathbf{R})$ always points in the $z$-direction. In this picture, only states with magnetic quantum number $m_L=0$ are mixed into $|\psi (\mathbf{R})\rangle$ by the field. For further simplification we have neglected fine structure effects for now, which is justified when the laser detuning is much larger than the fine structure splitting of the $|kP\rangle$ Rydberg levels, which will usually be the case. Using expression (\[psiR\]) for the Rydberg state, we can estimate the Rabi frequency for one-photon coupling to the ground state $\Omega(\mathbf{R})=e \langle \psi (\mathbf{R})|\mathbf{E}_{\rm{L}}\cdot \mathbf{r}|gS\rangle/\hbar$, where $\mathbf{E}_{\rm{L}}$ denotes the electric field of the laser within the rotating wave approximation and $\mathbf{r}$ the position of the electron in the lab-frame. Combining this equation with eq. (\[psiR\]), we get: $$\Omega(\mathbf{R})\approx \frac{e^2\mathbf{E}_{\rm{L}} \cdot\mathbf{E}_{\rm{ion}}(\mathbf{R})}{\hbar}\sum_k \frac{\langle nS|z|kP\rangle}{\mathcal{E}_{nS}-\mathcal{E}_{kP}}\langle kP|z|gS\rangle.$$ Now we can see what happens if we dress the ground state $|gS\rangle$ with $|\psi (R)\rangle$. For large distances $R\rightarrow \infty$, the transition between $|gS\rangle$ and $|\psi (\mathbf{R})\rangle$ will be dipole-forbidden and no dressing occurs. As the distance decreases, we have that the $|nS\rangle$ state gets polarized by the ion and its adiabatic energy shift is given to lowest order by $V_{nS}(R)=-C_4^{nS}/R^4$ [@Secker:2016]. We can find the dressed potential by looking at the 2-level Hamiltonian in the $|gS\rangle$, $|\psi (\mathbf{R})\rangle$ subspace after performing the rotating wave approximation: $$\label{eq:H3level}
H_{\rm{2-level}}=\left( \begin{array}{cc}
-\frac{C_4^{gS}}{R^4} & \hbar\Omega(\mathbf{R}) \\
\hbar\Omega^*(\mathbf{R}) &-\hbar\Delta_0 -\frac{C_4^{nS}}{R^4} \end{array} \right),$$ where we for now neglected any off-resonant couplings and assumed that the laser is blue-detuned by $\Delta_0$ from the $|gS\rangle \rightarrow |nS\rangle$ transition and that this detuning is much smaller than the level splitting between $|nS\rangle$ and all other Rydberg states. In the dispersive limit, where $\hbar|\Omega(\mathbf{R})|\ll \hbar\Delta_0+C_4^{nS}/R^4$ and $C_4^{gS}\ll C_4^{nS}$, diagonalisation results in the following dressed potential: $$\label{eq_dress_simple}
V_{\rm d}(\mathbf{R})\approx\frac{\hbar^2|\Omega(\mathbf{R})|^2}{\hbar\Delta_0+C_4^{nS}/R^4}-\frac{C_4^{gS}}{R^4}.$$ In order to proceed, we need to evaluate $|\Omega(\mathbf{R})|^2$, which now depends on the atom-ion separation. We have: $|\Omega(\mathbf{R})|^2=|\beta(\theta,\phi)|^2/R^4$, with: $$\label{eq_beta}
\beta (\theta,\phi )=\frac{e^3E_{\rm{L}}^{\parallel}(\theta,\phi)}{4\pi\epsilon_0\hbar}\sum_k \frac{\langle nS|z|kP\rangle\langle kP|z|gS\rangle}{\mathcal{E}_{nS}-\mathcal{E}_{kP}}.$$ Here, $E_{\rm{L}}^{\parallel}(\theta,\phi)=\mathbf{E}_{\rm{L}} \cdot\mathbf{E}_{\rm{ion}}(\mathbf{R})/|\mathbf{E}_{\rm{ion}}(\mathbf{R})|$ denotes the projection of the laser’s electric field onto the electric field of the ion and $\theta$ and $\phi$ denote the azimuthal and polar angle of ${\mathbf R}$ according to the laboratory reference frame, respectively. Combining equations (\[eq\_dress\_simple\] and \[eq\_beta\]), our final result within first order perturbation theory is: $$\label{eq_dress}
V_{\rm d}(\mathbf{R})=\frac{A(\theta,\phi) R_{\rm w}^4}{R^4+R_{\rm w}^4}-\frac{C_4^{gS}}{R^4}.$$ The first term in the potential (\[eq\_dress\]) is repulsive and has a maximum height $A(\theta,\phi)=\frac{\hbar^2|\beta(\theta,\phi)|^2}{C_4^{nS}}$ and width $R_{\rm w}=(C_4^{nS}/(\hbar\Delta_0))^{1/4}$. The angular dependence of $\beta(\theta,\phi)$ is determined by the laser polarization. For linear polarization along the $z$-direction of the laboratory frame for instance, we have $\mathbf{E}^{z}_{\rm{L}}=(0,0,E_{\rm{L}})$ such that $E_{\rm{L}}^{\parallel}\propto \cos \theta$, whereas circular polarization, $\mathbf{E}^{+}_{\rm L}=E_{\rm L}(1,i,0)/\sqrt{2}$ gives us $E_{\rm{L}}^{\parallel}\propto i\sin\theta\cos\phi +\sin\theta\sin\phi$. The latter results in a donut-shaped repulsive potential in the $x,y$-plane and no interaction along the $z$-direction. In order to make the potential repulsive in all directions we may use two laser fields coupling on the transitions $|gS\rangle\rightarrow|nS\rangle$ and $|gS\rangle\rightarrow|n'S\rangle$ and such that $R_{\rm w}=R'_{\rm w}=(C_4^{n'S}/\hbar\Delta_0')^{1/4}$ with $\Delta_0'$ the bare detuning on the $|gS\rangle\rightarrow|n'S\rangle$ transition. Employing the laser fields $E_{\rm{L}}^z$ and $E_{\rm{L}}^+$ and appropriate tuning of the laser powers, such that the Rabi frequencies are equal, results in a spherically symmetric potential $A(\theta,\phi)=A$, as described in more detail in the Appendix \[Ap\_Potential\].
An example of a dressed potential calculated within first order perturbation theory is shown in Fig. \[fig\_dress\], where we used the approximate analytic wavefunctions of Ref. [@Kostelecky:1985] to evaluate the matrix elements in Eq. \[eq\_beta\] and $\Delta_0=2\pi\times$ 1 GHz such that $R_{\rm w}=$ 270 nm. We also show the potential obtained by diagonalization of the Rydberg manifold in the presence of the ion electric field and dressing laser, within the rotating wave approximation, taking spin-orbit coupling and terms up to charge-quadrupole coupling into account. For these calculations, we used the quantum defects found in Ref. [@vanDitzhuijzen:2009] while the Rydberg wavefunctions were obtained by the Numerov method (see Appendix \[sec\_Diagonalisation\]). This more accurate calculation shows good agreement with our approximated potential for atom-ion separations down to about 160 nm. At close atom-ion proximity, the $|19S\rangle$ state gets strongly coupled to other Rydberg states, causing avoided crossings. To properly compute the dressed potential in this regime, terms beyond the charge-dipole and quadrupole interaction as well as charge exchange effects would have to be taken into account. However, for atom temperatures that are much smaller than the repulsive barrier, $T_{\rm a}\ll A/k_{\rm B}$, short-range collisions between the atoms and ion are suppressed. If we assume a Rabi frequency of $\Omega_{19P,m_L=0}=2\pi\times$ 200 MHz on the $|5S\rangle \rightarrow |19P\rangle$ transition in Rb and use that the transition-matrix elements scale as $\langle kP|r|gS\rangle {\mathrel{\vcenter{
\offinterlineskip\halign{\hfil$##$\cr
\propto\cr\noalign{\kern2pt}\sim\cr\noalign{\kern-2pt}}}}}k^{-3/2}$, we can estimate $A/k_{\rm{B}} \sim 27 \mu$K, which is larger than typical ultracold atom temperatures. We can therefore neglect the inner part of the dressed potential and the second term in equation (\[eq\_dress\]) for these parameters in the ultracold regime, and for the remaining simulations in this work we replace $V_{\rm d}(\mathbf{R})$ by the spherically symmetric $\tilde{V}_{\rm d}(R)=\frac{A R_{\rm w}^4}{R^4+R_{\rm w}^4}$.
![The dressed potential $V_{\rm d}(R)$ assuming coupling to the $|19S\rangle$ state of Rb with $\Delta_0=2\pi\times$ 1.0 GHz. The dashed line is calculated from the Rydberg wavefunctions taken from Ref. [@Kostelecky:1985] in first order perturbation theory, within the dipole approximation, and neglecting spin-orbit coupling. Here, we assumed the defects of the $L-1/2$ for the $L$ states and set the Rabi frequency on the $|5S\rangle \rightarrow |19P\rangle$ transition to $\Omega_{19P,m_L=0}=2\pi\times$ 200 MHz by tuning the laser intensity. The black dots are computed by diagonalization of the Rydberg manifold in the presence of the dressing laser and ion, in the rotating wave approximation, including spin-orbit coupling, up to charge-quadrupole order in the atom-ion interaction terms. The laser intensity was set to the same value as for the approximated potential. For both calculations, we took the Rydberg states $n=10..30$ into account as described in more detail in Appendix \[sec\_Diagonalisation\]. []{data-label="fig_dress"}](fig2.pdf){width="8cm"}
[*Thermalization in Paul traps*]{} – To investigate whether the dressed repulsive potential prevents ion heating out of the ultracold regime we have studied classical collisions between an ion and atoms. We assume that we have an ion trapped in a Paul trap generated by the electric fields $\mathbf{E}_{\rm{PT}}(\mathbf{r},t)=\mathbf{E}_{\rm s}(\mathbf{r})+\mathbf{E}_{\rm rf}(\mathbf{r},t)$ with $$\begin{aligned}
\mathbf{E}_{\rm s}(x,y,z)&=&\frac{m_{\rm{i}}\omega_z^2}{e}\left(\alpha_x\frac{x}{2},\alpha_y\frac{y}{2},-z\right)\label{eqS},\\
\mathbf{E}_{\rm rf}(x,y,z,t)&=&\frac{m_{\rm{i}}\Omega_{\rm rf}^2q}{2e}\cos (\Omega_{\rm rf} t)\,\left(x,-y,0\right)\label{eqRF}.\end{aligned}$$ Here, $\Omega_{\rm rf}$ is the trap drive frequency and $q$ is the dynamic stability parameter for an ion of mass $m_{\rm i}$ and charge $e$. The motion of the ion in the transverse $x,y$-direction is given by a slow secular motion of frequency $\omega_{x,y}\approx \frac{\Omega_{\rm rf}}{2}\sqrt{a_{\rm s}+q^2/2}$ with $a_{\rm s}=-\alpha_{x,y}2\omega_z^2/\Omega_{\rm rf}^2$ the static stability parameter, and a fast micromotion of frequency $\Omega_{\rm rf}$. The unitless parameters $\alpha_x+\alpha_y=2$ are introduced to lift the degeneracy in the transverse confinement, as is common in ion trap experiments. The ion dynamics in the $z$-direction is purely harmonic with trap frequency $\omega_z$.
To begin with, we assume that the ion has initially no energy and sits at the center of the Paul trap as in references [@Cetina:2012; @Meir:2016]. The idea behind this approach is that although each individual system may be prepared in the ultracold regime by e.g. laser- and evaporative cooling, it is only when the atoms and ions are brought together that excessive heating is observed [@Meir:2016]. In order to simulate the effect of the atomic cloud we calculate classical trajectories of Rb atoms, with the atoms appearing one-by-one on a sphere of radius 2 $\mu$m around the ion and with a velocity that is randomly picked from a Maxwell-Boltzmann distribution of temperature $T_{\rm a}$. As soon as the atom has left the sphere around the ion at the end of the collision, we introduce a new atom to interact with the ion. We assume the atomic bath to be very large such that its temperature does not change as more collisions occur. After each collision, we obtain the ion’s secular energy by fitting an approximate analytic solution of the equations of ion motion to the numerically obtained orbit (see Appendix \[Ap\_cols\] and reference [@Meir:2016]). We define the ion temperature to be $T_{\rm ion}=\bar{\mathcal{E}}/(3k_{\rm B})$ with $\bar{\mathcal{E}}$ the total average secular energy of the ion. Assuming $^{171}$Yb$^+$ for the ion, we set $\Omega_{\rm rf}=2\pi\times$ 2 MHz and $\omega_x=2\pi\times$ 54 kHz, $\omega_y=2\pi\times$ 51 kHz, $\omega_z=2\pi\times$ 42 kHz.
We first study the case where the atoms are not dressed to a Rydberg state. Thus, we set the atom-ion interaction to $V_{\rm ai}(R)=-C^{gS}_4/R^4+C_6/R^6$, where the first term denotes the ground state atom-ion induced polarization-charge interaction and the second term accounts for some short-range repulsion, which we set to dominate for $R<10$ nm. The results of averaging 100 of these simulation runs can be seen in the inset of Fig. \[fig\_cols\]. From the individual trajectories, it is clear that most atoms fly by the ion without causing much effect. One in a few 100 collisions, however, is of the Langevin type causing significant energy exchange. The ion quickly heats up to temperatures much higher than the atomic cloud temperature of 2 $\mu$K. These observations are in line with the findings of references [@Cetina:2012; @Meir:2016]. If we simulate more collisions, finite size effects start to limit the accuracy of these calculations, as the ionic oscillation amplitude becomes comparable to the radius of the sphere of atomic starting positions. For comparison, we also show the results obtained when assuming that the ion is in a time-independent trap with the same trap frequencies (the secular approximation). Here, we see that the ion slowly thermalizes with the atomic gas.
Now, we will study how these dynamics change for the Rydberg-dressed atoms for which the atom-ion interaction is repulsive. We replace $V_{\rm ai}(R)$ by $\tilde{V}_{\rm d}(R)$ with $A/k_{\rm B} = 27 \mu$K and $R_{\rm w}=$ 270 nm leaving all other parameters in the simulation the same. The results can be seen in Fig. \[fig\_cols\]: the ion thermalizes slowly with the atomic cloud after about 4$\times 10^3$ ($1/e$) collisions. Starting with an initial ion velocity of $0.02(1,1,1)$ m/s, which corresponds to an initial energy of $\sim$ 4 $\mu$K, we can also simulate cooling of the ion towards the temperature of the atomic gas. Up until now, we have assumed ideal experimental conditions in which the minima of $\mathbf{E}_{\rm s}(\mathbf{r})$ and $\mathbf{E}_{\rm rf}(\mathbf{r},t)$ are perfectly overlapped. In practice however, stray electric fields may cause the trap center to not coincide with the radio-frequency null. This causes additional ion motion known as excess micromotion [@Berkeland:1998]. To study its effect, we introduce an additional electric field of $\mathbf{E}_{\rm EMM}=(0.05,0,0)$ V/m, which displaces the potential minimum by $x_{\rm EMM}=$ 243 nm corresponding to an average kinetic energy of $\mathcal{E}_{\rm EMM}/k_{\rm B}=m_{\rm i}q^2\Omega^2x_{\rm EMM}^2/(16k_{\rm B})\approx$ 100 $\mu$K, and run the simulations. For the repulsive case we find that the dynamics are not affected by the excess micromotion, confirming that the repulsive potential prevents excessive heating of the ion in the Paul trap. Indeed, we repeated the simulation assuming a time-independent potential with the same secular trap frequencies and see no differences beyond statistical fluctuations due to the random atom sampling. Hence, the repulsive potential allows us to make the secular approximation in the hybrid atom-ion system for the parameters used. Note that the assumption that the atom and ion cannot undergo short range collisions still holds in this case as long as the amplitude of oscillation $r_{\rm EMM} =qx_{\rm EMM}/2 \sim$ 10 nm $\ll R_{\rm w}$, even though $\mathcal{E}_{\rm EMM}/k_{\rm B} > A/k_{\rm B}$.
Finally, we study the effect of the atom-ion mass ratio for the repulsive case. For attractive atom-ion interactions it is well-known that stable cold trapping can only be achieved as longs as $m_{\rm a} \lesssim m_{\rm i}$, with $m_{\rm a}$ the mass of the atom [@Major:1968; @DeVoe:2009; @Zipkes:2011; @Cetina:2012; @Chen:2014; @Weckesser:2015; @Rouse:2017]. To see how this changes for the repulsive case, we repeated the simulations for the combination $^{87}$Rb/$^{9}$Be$^+$, whose mass ratio lies far outside of the range where thermalization can be expected. As can be seen in Fig. \[fig\_cols\], this combination also remains within the ultracold regime when the interactions are repulsive, thermalizing after about 2$\times 10^3$ ($1/e$) collisions.
![ Collision dynamics for an $^{171}$Yb$^+$ ion in a Rydberg-dressed Rb gas with repulsive interactions and $T_{\rm a}=$ 2 $\mu$K. The ions heats up from $T_{\rm ion}=0$ to about $1.8$ $\mu$K, corresponding to near-thermalization. Excess micromotion due to a DC field of $E_{\rm EMM}=0.05$ V/m does not affect the thermalization dynamics, which is completely equivalent to that expected for an ion trapped in a time-independent trap with the same secular trap frequencies. For the curve starting at around 4 $\mu$K, we gave the ion an initial velocity of $0.02(1,1,1)$ m/s and we simulate cooling towards the atomic gas temperature. The fastest thermalization curve shows the results for the combination $^{87}$Rb/$^{9}$Be$^+$. The inset shows results for ground state atoms, in which the interaction is attractive. Even without excess micromotion, the ion quickly heats up to temperatures higher than the atomic cloud. This is in stark contrast to the case where we make the secular approximation, where we see that slow thermalization would occur. All results represent the average of 50-100 runs. []{data-label="fig_cols"}](fig3.pdf){width="8cm"}
[*Experimental issues*]{} – For the calculations in this paper we have neglected the effect of the ionic trapping fields on the dressing of the atom. This is justified since the trapping field at maximal amplitude is much smaller than the field of the ion at the dressing point: $\rm{Max}(E_{\rm PT}(R_{\rm w},t))/E_{\rm ion}(R_{\rm w})\approx $ $10^{-4}$ ($\approx $ $10^{-5}$ for $^9$Be$^+$), such that we can safely neglect these fields. Far away from the center of the trap, however, the maximal amplitude of the trapping fields become similar to $E_{\rm ion}(R_{\rm w})$. For the numbers used in this work, this occurs only at $R\approx$ 1 mm from the trap center, such that we can also safely neglect the effect of the dressing on the atomic cloud far away from the ion.
For the simulations, we assumed that only a single atom was interacting with the ion at the same time, which would require a density of $\rho_{\rm a}\leq 1/R_{\rm w}^3\approx 10^{19}$ m$^{-3}$. Assuming an atomic density of $\rho_{\rm a}=10^{18}$ m$^{-3}$ and $T_{\rm a}=$ 2 $\mu$K, we have a flux of about 6$\times$10$^5$ atoms/s entering the sphere of $R=2$ $\mu$m, such that each collision in Fig. \[fig\_cols\] amounts to a few $\mu$s. We estimate an upper bound to the photon scattering rate as $\Gamma\leq f \Gamma_{19S}\rho_{\rm a}R_{\rm w}^3\approx 10$ Hz, for $\rho_{\rm a}=10^{18}$ m$^{-3}$ and with $\Gamma_{19S}$ the decay rate of the Rydberg state and $f$ the probability for an atom to be found in the Rydberg state: $f\propto \hbar^2|\Omega(\mathbf{R})|^2/(\hbar\Delta_0+C_4^{nS}/R^4)^2 \leq 10^{-3}$. Therefore, we expect that photon scattering will not affect the collision dynamics considerably. For increasing ion temperatures, there is a probability that the atom makes it over the repulsive barrier in a collision. To investigate this regime, we simulated 2$\times$10$^5$ collisions between atoms with $T_{\rm a}=2\mu$K and $T_{\rm ion}$ about 100 $\mu$K and found that in 146 instances, the atoms approached the ion to distances below $R_{\rm w}/2$, such that they entered the inner regime of the dressed potential. For $T_{\rm ion}$ around 10.6 $\mu$K, the number of atoms entering the inner part of the potential dropped to zero, as described in more detail in Appendix \[Ap\_OverBar\]. We conclude that our scheme works best for ions and atoms that are pre-cooled into the ultracold regime before the systems are merged. Finally, we estimate the probability $P_{\rm tunnel}$ for the atom to tunnel through the potential barrier to be $P_{\rm tunnel}\approx {\rm exp}(-\sqrt{2 \mu_{\rm ai} (A-k_{\rm B} T_{\rm a})/\hbar^2}R_{\rm w})\sim 10^{-9}$, with $\mu_{\rm ai}$ the atom-ion reduced mass, such that we can safely neglect this effect.
[*Conclusions*]{} – We have shown that trapped ions in Rydberg-dressed gases are stable against micromotion assisted heating when we design the dressed atom-ion potential to be repulsive. To engineer this repulsive interaction, a scheme that employs one-photon dressing on a dipole forbidden transition can be used. While Rydberg states with negative polarizability could also be used to engineer repulsive atom-ion interactions, these are not always available or can be hard to spectrally resolve. In contrast, the scheme proposed in this paper works for any alkali atom and avoids close-by spectator states by employing Rydberg states without orbital angular momentum, which generally have large energy separations from other Rydberg states. The scheme may be employed for a wider range of atom-ion mass ratios than is the case for the attractive atom-ion system. The results may be of interest when studying atom-ion interactions in the quantum regime and may find applications in hybrid quantum computation or simulation approaches [@Doerk:2010; @Bissbort:2013; @Secker:2016] and the study of polarons [@Casteels:2011] in these systems.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by the EU via the ERC (Starting Grant 337638) and the Netherlands Organization for Scientific Research (NWO, Vidi Grant 680-47-538) (R.G.). We gratefully acknowledge fruitful discussions with Antonio Negretti and Alexander Gl[ä]{}tzle.
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Spherically symmetric potential {#Ap_Potential}
===============================
We can use two dressing lasers to engineer a spherically symmetric potential, coupling on the transitions $|gS\rangle\rightarrow|nS\rangle$ and $|gS\rangle\rightarrow|n'S\rangle$. After performing the rotating wave approximation, we can write the Hamiltonian in the $|gS\rangle$, $|\psi (\mathbf{R})\rangle$, $|\psi' (\mathbf{R})\rangle$ subspace
$$\label{eq:H3level}
H_{\rm{3-level}}=\left( \begin{array}{ccc}
-\frac{C_4^{gS}}{R^4} & \hbar\Omega(\mathbf{R}) & \hbar\Omega'(\mathbf{R})\\
\hbar\Omega^*(\mathbf{R}) &-\hbar\Delta_0 -\frac{C_4^{nS}}{R^4} & 0\\
\hbar\Omega'^*(\mathbf{R})& 0 & -\hbar\Delta'_0 -\frac{C_4^{n'S}}{R^4}\end{array} \right).$$
To obtain the adiabatic potential, we diagonalise $H_{\rm{3-level}}$ assuming $\hbar|\Omega(\mathbf{R})|\ll \hbar\Delta_0+C_4^{nS}/R^4$, $C_4^{gS}\ll C_4^{nS}$ and $\hbar|\Omega'(\mathbf{R})|\ll \hbar\Delta'_0+C_4^{n'S}/R^4$, $C_4^{gS}\ll C_4^{n'S}$ so that we can expand to second order in $\hbar|\Omega(\mathbf{R})|$ and $\hbar|\Omega'(\mathbf{R})|$ to find:
$$\label{eq_dress3}
V_{3d}(\mathbf{R})\approx\frac{A(\theta,\phi) R_{\rm w}^4}{R^4+R_{\rm w}^4}+\frac{A'(\theta,\phi) {R'}_{\rm w}^4}{R^4+{R'}_{\rm w}^4}-\frac{C_4^{gS}}{R^4}.$$
with $A(\theta,\phi)=\frac{\hbar^2|\beta(\theta,\phi)|^2}{C_4^{nS}}$ and $A'(\theta,\phi)=\frac{\hbar^2|\beta'(\theta,\phi)|^2}{C_4^{n'S}}$. The angular dependence of $\beta$ is expressed as:
$$\begin{aligned}
\label{eq_beta3}
\beta(\theta,\phi)& =\frac{e^3E_{\rm L}^{\parallel}(\theta,\phi)}{4\pi\epsilon_0\hbar}\sum_k \frac{\langle nS|z|kP\rangle\langle kP|z|gS\rangle}{\mathcal{E}_{nS}-\mathcal{E}_{kP}}\\
\beta'(\theta,\phi)& =\frac{e^3{E'}_{\rm L}^{\parallel}(\theta,\phi)}{4\pi\epsilon_0\hbar}\sum_k \frac{\langle n'S|z|kP\rangle\langle kP|z|gS\rangle}{\mathcal{E}_{n'S}-\mathcal{E}_{kP}}.\end{aligned}$$
Where $E_{\rm L}^{\parallel}(\theta,\phi)$ denotes the projection of the laser electric field onto the electric field of the ion. Now we set $\mathbf{E}_{\rm L}=(0,0,E_{\rm L})$ such that $\beta(\theta,\phi)= \beta_0\cos \theta$ and $\mathbf{E}'_L={E'}_L(1,i,0)/\sqrt{2}$ such that $\beta'(\theta,\phi)= \beta_0'( i\sin\theta\cos\phi +\sin\theta\sin\phi)$. In this situation,
$$\label{eq_dress3}
V_{3\rm{d}}(\mathbf{R})\approx\frac{|\beta_0|^2 \cos^2\theta R_{\rm w}^4}{C_4^{nS}(R^4+R_{\rm w}^4)}+\frac{|\beta'_0|^2 \sin^2\theta {R'}_{\rm w}^4}{C_4^{n'S}(R^4+{R'}_{\rm w}^4)}-\frac{C_4^{gS}}{R^4},$$
which results in a spherically symmetric potential if we set $R_{\rm w}=R'_{\rm w}$ and $|\beta_0|^2/C_4^{nS}=|\beta'_0|^2/C_4^{n'S}$, which can be done by tuning $\Delta_0$, $\Delta'_0$ and the laser intensities.
Determination of ion temperature {#Ap_cols}
================================
The explicit time dependence of the electric trapping field obstructs a straight-forward definition of some physical quantities such as energy and temperature. Nonetheless, if the time dependence of the external field is periodic and the underlying equations of motion are linear, we are able to apply Floquet’s theory and the problem can be decomposed into a part which oscillates with the same period as the external field and a part, referred to as the secular part, which is time independent [@Leibfried:2003]. Often useful physical quantities can be constructed from the secular problem and its solution, e.g. in systems of multiple ions energy and temperature can be defined via the secular solutions of the linearized problem. For the combined system of atoms and ions we do not have a closed form for the solution at hand, but in the considered case of low densities the system will evolve as the uncoupled one for time periods in which all atoms are spatially well separated from the ion. During those time periods the interaction forces can be neglected and a measurement of the secular energy of the isolated ion system is possible. The ion’s secular energy $\cal{E}$ can be determined via its secular frequencies $\omega_i$, $i=x,y,z$, and the amplitude of the orbit’s secular component. For a single ion in an oscillating electric field $\mathbf{E}_{\rm PT}(\mathbf{r},t)$, as defined in Eq. 7 of the main text, with appropriately chosen trap parameters the equations of motion can be solved analytically and the solutions take the following approximate form $$\begin{aligned}
x (t) & \approx \left( C_x \cos(\omega_x t) + S_x \sin(\omega_x t) \right) \left( 1 + \frac{q}{2} \cos(\Omega_{\text{rf}} t ) \right) - \frac{\omega_x q}{\Omega_{\text{rf}}} \left( S_x \cos( \omega_x t) - C_x \sin( \omega_x t) \right) \sin( \Omega_{ \text{rf}} t ),\\
y (t) & \approx \left( C_y \cos(\omega_y t) + S_y \sin(\omega_y t) \right) \left( 1 - \frac{q}{2} \cos(\Omega_{\text{rf}} t ) \right) + \frac{\omega_y q}{\Omega_{ \text{rf}}} \left( S_y \cos( \omega_y t) - C_y \sin( \omega_y t) \right) \sin( \Omega_{\text{rf}} t ),\\
z (t) & = \left( C_z \cos(\omega_z t) + S_z \sin(\omega_z t) \right) \, . \phantom{\frac{q}{2}}\end{aligned}$$ Here we neglected correction terms which oscillate faster than the trap drive frequency $\Omega_{\text{rf}}$. Given the ion’s position at two instances of time the above system of linear equations can be inverted to obtain the coefficients $C_i$ and $S_i$ of the secular solution’s cosine and sine part. As a preliminary step the secular frequencies can be determined numerically by fitting the above analytic approximation to a numerically obtained solution. This increases accuracy as compared to the approximate analytical expression for the $\omega_i$ we gave in the main text. Given all this we can now calculate the secular energy $\mathcal{E}=\sum_{i=x,y,z} m_{\rm i} \omega_{i}^2 (C_i^2+S_i^2) / 2 $ of the isolated ion system.\
In the case of the repulsive potential, which we analyzed in the main text, the strong similarity of the thermaliztion process for the full and the secular problem indicates that the ion’s secular temperature for this type of potentials is indeed a meaningful physical observable, in contrast to the attractive case.
Over barrier collisions {#Ap_OverBar}
=======================
In our simulations of the classical collision dynamics there is a non-vanishing probability for the atoms to overcome the repulsive barrier. Since the ion is strongly localized, we estimate this to occur when the amplitude of ion oscillation exceeds $R_{\rm w}$, or $T_{\rm ion}\gtrapprox m_{\rm i}\bar{\omega}^2R_{\rm w}^2/(2k_{\rm B})\approx$ 70 $\mu$K for the parameters used in this work, with $\bar{\omega}$ the average trap frequency. To estimate if those events pose a problem we performed collision simulations where we kept record of the minimal separation $d_{\rm min}$ between atom and ion. More precisely we simulated the interaction of an ion with a bath of $T_a=$2 $\mu$K atoms for 200 collisions. The result of 1000 such simulations for ion temperatures $T_{\rm ion}$ around $10.6$ $\mu$K and $100$ $\mu$K are shown in Fig. \[fig:over\_barrier1\] and \[fig:over\_barrier2\]. For $T_{\rm ion}\approx$100 $\mu$K we found that in 146 instances out 2 $\times$ 10$^5$ times, the atoms approached the ion to distances below $R_{\rm w}/2$, such that they entered the inner regime of the dressed potential. For $T_{\rm ion}\approx$10.6 $\mu$K, the number of atoms entering the inner part of the dressed potential dropped to zero.
Diagonalization of the Rydberg-ion interaction Hamiltonian {#sec_Diagonalisation}
==========================================================
In this section we discuss how we calculated the adiabatic atom-ion potential from a direct-diagonalisational approach. The atom-ion potential can within the adiabatic approximation be obtained from the internal level structure of the atom for different but fixed atom ion separations ${\mathbf R}$. Each of those ${\mathbf R}$-dependent energy levels can be viewed as the effective potential between atom and ion, under the assumption, that the internal state follows the corresponding eigenstate adiabatically [@Secker:2016]. We focus on the potential or channel, which correspond to the atom’s ground state at infinite separation. The relevant shifts of the ground state will with the parameters we choose and for separations larger than $\sim$ 0.2 $\mu$m result from the off-resonant coupling to the Rydberg manifold, as can be seen from the simpler perturbational approach discussed in the main text. Therefore, we neglect the effect of the ion’s electric field on the ground state level for now keeping in mind that the resulting potential will be a good approximation just in the range considered above. For the Rydberg manifold on the other hand the presence of the ion leads to relevant corrections, since it introduces strong couplings between the Rydberg states due to the large polarizability $\alpha\propto n^7$ of the Rydberg states, with $n$ the principal quantum number. Therefore, we model the Hamiltonian representing the atom’s internal structure by a single ground state ${|gS\rangle}$ with energy $\mathcal{E}_{gS}$, which we assume to be unaffected by the ion’s electric field, coupled by the dressing laser field to a set of Rydberg states, which are affected by the ion field. With those considerations the Hamiltonian representing the atom’s internal structure in the presence of ion and dressing laser reads
\[HBO\] H =&\_[gS]{}+\_[Ryd]{}+\_[ia]{}+\_[L]{},\
where $\hat{H}_{gS}$ describes the unperturbed atomic ground state ${|gS\rangle}$ and is given by:
\_[gS]{}=&\_[gS]{} [|gS]{}[gS|]{}.\
$\hat{H}_{\rm Ryd}$ represents a set of unperturbed atomic Rydberg states in diagonal form, which in quantum defect theory [@vanDitzhuijzen:2009] can be represented by a single Rydberg electron and a singly charged atomic core, which represents the inner electrons and the nucleus:
\_[Ryd]{}=&\_[k]{} \_[(k)]{} [|k]{}[k|]{},\
where $k$ represents the tuple of quantum numbers $(n,l,j,m_j)$ ranging over the common hydrogen fine structure states and $\mathcal{E}_{(k)}$ is the corresponding Rydberg energy level with eigenstate $ {|k\rangle}$ in the relative Rydberg electron-core variable. We obtained those eigenenergies and states in coordinate representation employing the Numerov method and quantum defects found in literature [@vanDitzhuijzen:2009].
$\hat{H}_{\rm ia}$ comprises the interaction terms between Rydberg atom and ion in the quantum defect theory approach. It consists of two Coulomb interaction terms $V_C$, a repulsive one between ion and atomic core and an attractive one between ion and Rydberg electron. In addition, we included a spin-orbit like coupling term $V_{SO}^{e-i}$ representing the coupling of the ion’s electric field to the spin of the Rydberg electron [@Secker:2016].
\[eq:atomionpert\]
\_[ia]{}=&V\_C([r]{}\_c-[r]{}\_i)-V\_C([r]{}\_e-[r]{}\_i)+V\_[SO]{}\^[e-i]{}\
=&- + - (()[p]{})\
& (-[r]{}[R]{}+[r]{}\^2)-([r]{}[R]{})\^2-i ([R]{}).\
Here ${\mathbf r}_i$, ${\mathbf r}_c$ and ${\mathbf r}_e$ are the ion, core and Rydberg electron position operators in the laboratory frame respectively, ${\mathbf r}$ and ${\mathbf p}$ are the relative position and relative momentum operator between core and Rydberg electron, ${\hat{\mathbf S}}$ is the spin-1/2 operator of the electron, $e$ denotes the elementary charge, $m_e$ and $m_c$ denote the electron and core mass respectively, $M$ and $\mu$ are the total and reduced mass of the atom respectively and $\epsilon_0$ is the vacuum permittivity. The approximation has been obtained by taylor expanding the terms and substituting the momentum operator in the last term with ${\mathbf p}\approx i 2 \frac{\mu}{ \hbar} [\hat H_{Ryd},{\mathbf r}]$ [@Secker:2016]. The approximation includes terms up to quadrupole order in the Coulombic terms taking account for the non-linearity of the potential over the spatial extent of the Rydberg wavefunctions considered.
The term $\hat{H}_L$ of Eq. \[HBO\] describes the dressing laser field in dipole approximation, which couples the Rydberg states off-resonantly to the ground state ${|gS\rangle}$.
\_L=&\_k e\^[i\_L t]{}\_[(k,gS)]{}[|gS]{}[k|]{}+h.c.\
where $\Omega_{(k,gS)}$ denotes the Rabi frequency of the transition ${|k\rangle}$ to ${|gS\rangle}$ and the sum runs over all Rydberg states $(k)=(n,P,j,m_j)$ for which the transition is dipole allowed.
We proceed in changing to a rotating frame of the ground state with respect to the dressing laser frequency and perform a rotating wave approximation. This affects only $\hat{H}_{gS}$ and $\hat{H}_L$, which transform to
\_[gS]{}\^[’]{}=&(\_[gS]{}+h \_L) [|gS]{}[gS|]{}\
\_L\^[’]{}=&\_k \_[(k,gS)]{}[|gS]{}[k|]{}+h.c.\
To determine the coupling strengths $\Omega_{((n,P,j,m_j),gS)}$ we computed the transition matrixelements with an approximate solution for the ground state wavefunction, which we obtained as in the Rydberg case. We restrict to the case of linear polarization along the direction of the ion electric field and fix the couplings such that they are consistent with the case that neglects the fine structure in the main text, namely $\Omega_{(19P,gS)}=2 \pi\times$ 200 MHz. To this end, we rescaled the couplings such that $\Omega_{((19,P,3/2),gS)}=\Omega_{(19P,gS)} A_{FS}/A$ with $A={\langle P,m_l=0|}\cos(\theta) {|S,m_l=0\rangle}$ and $A_{FS}={\langle P,j=3/2|}\cos(\theta) {|S,j=1/2\rangle}$ the angular component of the dipole operator coupling states without fine structure and with fine structure states respectively. The couplings obtained in this way were found to be in agreement with the scaling law $\Omega_{((n,P,j),gS)}\propto n^*(n,P,j)^{-3/2}$ with $n^*(n,l,j)=n-\delta_{nlj}$ and $\delta_{nlj}$ the quantum defects. Diagonalisation for $n=10..30$ and $\omega_L=(\mathcal{E}_{(19,S,1/2)}-\mathcal{E}_{gS})/h+\Delta$ results in the potential shown in Fig. 2 of the main text.
![Ion temperature $T_{\rm ion}$ vs. minimal atom-ion separation $d_{\rm min}$ for 2 $\times$ 10$^5$ collision simulations of Rb and Yb$^+$ as described in the main text with $T_{\rm ion}$ around 10.6 $\mu$K. The dashed line indicates $R_{\rm w}/2$, where the atoms find themselves within the inner part of the dressed potential.[]{data-label="fig:over_barrier1"}](fig5.pdf){width="8.5cm"}
![Ion temperature $T_{\rm ion}$ vs. minimal atom-ion separation $d_{\rm min}$ for 2 $\times$ 10$^5$ collision simulations of Rb and Yb$^+$ as described in the main text with (a) $T_{\rm ion}$ around 100 $\mu$K. The dashed line indicates $R_{\rm w}/2$, where the atoms find themselves within the inner part of the dressed potential.[]{data-label="fig:over_barrier2"}](fig4.pdf){width="8.5cm"}
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abstract: 'Let $({{\mathcal M}}, Q)$ be a dg manifold. The space of shifted vector fields $({\mathscr{X} }({{\mathcal M}})[-1], L_Q)$ is a Lie algebra object in the homology category $\mathrm{H}({\mathbf{dg}\mathrm{-}\mathbf{mod} })$ of dg modules over $({{\mathcal M}},Q)$, the Atiyah class ${\alpha}_{{{\mathcal M}}}$ being its Lie bracket. The triple $({\mathscr{X} }({{\mathcal M}})[-1], L_Q; \, {\alpha}_{{{\mathcal M}}})$ is also a Lie algebra object in the Gabriel-Zisman homotopy category $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$. In this paper, we describe the universal enveloping algebra of $({\mathscr{X} }({{\mathcal M}})[-1], L_Q; \, {\alpha}_{{{\mathcal M}}})$ and prove that it is a Hopf algebra object in $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$. As an application, we study Fedosov dg Lie algebroids and recover a result of Chen, Stiénon and Xu on the Hopf algebra arising from a Lie pair.'
address:
- Jiahao Cheng
- Zhuo Chen
- Dadi Ni
- 'Department of Mathematics, Tsinghua University'
author:
- Jiahao Cheng
- Zhuo Chen
- Dadi Ni
title: Hopf algebras arising from dg manifolds
---
[^1]
\
Atiyah class, dg manifold, Hopf algebra, HKR-theorem, Fedosov dg manifold.\
Introduction
============
Atiyah classes were introduced by Atiyah [@Atiyah] as the obstruction to the existence of a holomorphic connection on a complex manifold. It was shown by Kapranov [@Kapranov] that the Atiyah class of a complex manifold $X$ endowed the shifted holomorphic tangent bundle $T_{X}[-1]$ with a Lie algebra structure in the derived category of coherent sheaves of ${{\mathcal O}}_X$-modules. This structure plays an important role in the construction of Rozansky-Witten invariants [@Kontsevich1]. In his work on the deformation quantization of Poisson manifolds [@Kontsevich2], Kontsevich indicated a deep link between the Todd genus of complex manifolds and the Duflo element of Lie algebras. See [@L-S-X] for the formality theorem for smooth dg manifolds, which implies the Kontsevich-Duflo theorem for Lie algebras and Kontsevich’s theorem for complex manifolds [@Kontsevich2] under a unified framework. Later, Chen, Stiénon and Xu introduced the Atiyah class of Lie algebroid pairs in [@C-S-X2], which encodes both the Atiyah class of complex manifolds and Molino class [@Molino] of foliations as special cases. Towards a different direction, Metha, Stiénon and Xu [@M-S-X] introduced the Atiyah class and the Todd class of dg manifolds. For a wealth of further investigation, see [@Calaque-VandeBergh; @Costello; @G-G; @Shoikhet].
Atiyah classes form a bridge between complex geometry and Lie theory. In Ramadoss’s work [@Ramadoss], it is proved that the universal enveloping algebra of the Lie algebra object $T_X[-1]$ in $ D^b({{\mathcal O}}_X) $, the derived category of bounded complexes of ${{\mathcal O}}_X$-modules, is the Hochschild cochain complex $(D_{\mathrm{poly}}^{\bullet}, d_H)$. This result played an important role in the study of the Riemann-Roch theorem [@MR2472137], the Chern character [@Ramadoss] and the Rozansky-Witten invariants [@MR2661534].
We are motivated by Ramadoss’s work [@Ramadoss], and Chen-Stiénon-Xu’s [@C-S-X; @C-S-X2], where it is shown that the quotient $L/A[-1]$ of a Lie pair $(L,A)$ is a Lie algebra object in the derived category $D^b(\mathcal{A})$ of the category of $A$-modules, the Atiyah class $${\alpha}_{L/A}\in H^1_{\mathrm{CE}}(A, {\mathrm{Hom }}(L/A \otimes L/A, L/A))\cong \Hom_{D^b(\mathcal{A})}(\Gamma(L/A)[-1]\otimes_{{C^{\infty}}_M}\Gamma(L/A)[-1],\Gamma(L/A)[-1])$$ being its Lie bracket. Moreover, the universal enveloping algebra of the Lie algebra object $L/A[-1]$ is $(D_{\mathrm{poly}}^{\bullet}(L/A), d_H)$, which is also a Hopf algebra object in $D^b(\mathcal{A})$.
In the present paper, we study a dg manifold $({{\mathcal M}}, Q)$, and the associated Atiyah class, which is an element $${\alpha}_{{{\mathcal M}}}\in
{\mathrm{Hom }}_{\mathrm{H}({\mathbf{dg}\mathrm{-}\mathbf{mod} })}({\mathscr{X} }({{\mathcal M}})[-1]\otimes_{{C^{\infty}}_{{{\mathcal M}}}}{\mathscr{X} }({{\mathcal M}})[-1], {\mathscr{X} }({{\mathcal M}})[-1]),$$ where ${\mathscr{X} }({{\mathcal M}})$ is the space of vector fields, and $\mathrm{H}({\mathbf{dg}\mathrm{-}\mathbf{mod} })$ is the homology category of the category ${\mathbf{dg}\mathrm{-}\mathbf{mod} }$ of dg modules over $({{\mathcal M}},Q)$ (see Section \[Section4\] for more explanations of notations). It turns out that $({\mathscr{X} }({{\mathcal M}})[-1],L_Q; \, {\alpha}_{{{\mathcal M}}})$ is a Lie algebra object in the homology category $\mathrm{H}({\mathbf{dg}\mathrm{-}\mathbf{mod} })$, and in the homotopy category $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$ as well, where the Atiyah class ${\alpha}_{{{\mathcal M}}}$ is regarded as a Lie bracket on ${\mathscr{X} }({{\mathcal M}})[-1]$.
It is well known that every ordinary Lie algebra $\mathfrak{g}$ admits a universal enveloping algebra $U(\mathfrak{g})$, which is a Hopf algebra. We are thus led to the natural question: does there exists a universal enveloping algebra for ${\mathscr{X} }({{\mathcal M}})[-1]$ in $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$?
We give a positive answer to this question.
\[maintheoreminintroduction\]
- The natural inclusion map $\theta:~ ({\mathscr{X} }({{\mathcal M}})[-1],L_Q; \, {\alpha}_{{{\mathcal M}}}) \rightarrow ({\mathrm{tot} }{L({\mathscr{D} }_{\mathrm{poly}}^1) }, L_Q+d_H; \, {\textup{\textlbrackdbl ~,~ \textrbrackdbl} })$ is an isomorphism in the homotopy category $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$. Moreover, it is an isomorphism of Lie algebra objects, i.e., the following diagram commutes in $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$: $$\label{maindiagraminintroduction}
\xymatrix{
({\mathscr{X} }({{\mathcal M}})[-1],L_Q) \otimes_{{C^{\infty}}_{{{\mathcal M}}}} ({\mathscr{X} }({{\mathcal M}})[-1],L_Q) \ar[r]^>>>>>>>>>>>{{\alpha}_{{{\mathcal M}}}} \ar[d]_{\theta \otimes \theta} &
({\mathscr{X} }({{\mathcal M}})[-1],L_Q)\ar[d]^{\theta}\\
({\mathrm{tot} }{L({\mathscr{D} }_{\mathrm{poly}}^1) },L_Q+d_H)
\otimes_{{C^{\infty}}_{{{\mathcal M}}}}
({\mathrm{tot} }{L({\mathscr{D} }_{\mathrm{poly}}^1) },L_Q+d_H)\ar[r]^>>>>>>{{\textup{\textlbrackdbl ~,~ \textrbrackdbl} }} & ({\mathrm{tot} }L({\mathscr{D} }_{\mathrm{poly}}^1),L_Q+d_H)\\
}$$
- The dg module $({\mathrm{tot} }{{\mathscr{D} }_{\mathrm{poly}}}, L_Q+d_H)$ is the universal enveloping algebra of the Lie algebra object $({\mathscr{X} }({{\mathcal M}})[-1],L_Q; \,{\alpha}_{{{\mathcal M}}} )$, and a Hopf algebra object, in $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$.
Here $({\mathrm{tot} }{L({\mathscr{D} }_{\mathrm{poly}}^1) }, L_Q+d_H; \, {\textup{\textlbrackdbl ~,~ \textrbrackdbl} })$ is the free Lie algebra object spanned by ${{\mathscr{D} }_{\mathrm{poly}}^1}$ (see Sections \[Section2\] and \[Section3\] for explanations of notations). This result can be regarded as an analogue of Ramadoss’s Theorem 2 in [@Ramadoss], in the context of dg manifolds.
In the commutative diagram , the Atiyah class ${\alpha}_{{{\mathcal M}}}$ on the upper side reflects geometric information of $({{\mathcal M}}, Q)$, while the Lie bracket of lower side is the naive free Lie algebra bracket, which is a pure algebraic operation. Thus Diagram can be regarded as a universal representation of Atiyah classes of dg manifolds.
We recall some results due to Chen-Stiénon-Xu [@C-S-X; @C-S-X2]. As we have mentioned, the Atiyah class ${\alpha}_{L/A}$ of a Lie pair $(L,A)$ defines a Lie bracket on $ L/A [-1]$ in $D^b(\mathcal{A})$. It is further introduced a free Lie algebra object $(L(D_{\mathrm{poly}}^{1}(L/A)), d_H ; \, {\textup{\textlbrackdbl ~,~ \textrbrackdbl} })$, leading to the universal enveloping algebra object $(D_{\mathrm{poly}}^{\bullet}(L/A), d_H)$ of the Lie algebra $( L/A [-1], {\alpha}_{L/A})$ in the derived category $D^b(\mathcal{A})$ of $A$-modules — see the following
[@C-S-X]\[C-S-Xinintroduction\] Let $\beta:~~ \Gamma(L/A)[-1] \rightarrow L(D_{\mathrm{poly}}^{1}(L/A))$ be the natural inclusion map.
- The map $\beta$ is an isomorphism in the derived category $D^b(\mathcal{A})$ of $A$-modules. Moreover, it is an isomorphism of Lie algebra objects, i.e., the following diagram commutes in $D^b(\mathcal{A})$: $$\label{Eqt:olddiagram}
\xymatrix{
\Gamma(L/A)[-1] \otimes_{{C^{\infty}}_{M}} \Gamma(L/A)[-1] \ar[r]^>>>>>{\beta \otimes \beta} \ar[d]_{{\alpha}_{L/A}} & L(D_{\mathrm{poly}}^{1}(L/A))
\otimes_{{C^{\infty}}_{M}} L(D_{\mathrm{poly}}^{1}(L/A)) \ar[d]^{{\textup{\textlbrackdbl ~,~ \textrbrackdbl} }}\\
\Gamma(L/A)[-1] \ar[r]^>>>>>>>>>>>>>>>{\beta} & L(D_{\mathrm{poly}}^{1}(L/A)) \, .\\
}$$
- The complex of $A$-modules $(D^{\bullet}_{\mathrm{poly}}({L/A}), d_H)$ is the universal enveloping algebra object of the Lie algebra object $( L/A [-1], {\alpha}_{L/A})$, and a Hopf algebra object, in $D^b(\mathcal{A})$.
Certainly, Diagram can be regarded as an universal representation of Atiyah classes of Lie pairs. More details and explanations of these facts can be found in Section \[Section6\].
It is therefore natural to ask how the commutative diagrams and are related, and how the Hopf algebras $({\mathrm{tot} }{{\mathscr{D} }_{\mathrm{poly}}}, L_Q+d_H)$ and $(D^{\bullet}_{\mathrm{poly}}({L/A}), d_H)$ are related.
Our answer is based on the notion of Fedosov dg Lie algebroid $({{\mathcal F}}, L_Q)$ associated with a Lie pair $(L,A)$, which was first constructed in Stiénon-Xu [@S-X]. Moreover, Batakidis-Voglaire [@B-V] and Liao-Stiénon-Xu [@L-S-X2] proved a relation of the Lie pair Atiyah class ${\alpha}_{L/A}$ and the Atiyah class of the Fedosov dg Lie algebroid $({{\mathcal F}}, L_Q)$. See Section \[Section6\] for a brief review of these facts. Our main result below, brings together two representations of Atiyah classes of different types — one from dg manifolds, the other from Lie pairs.
\[bigdiagraminintroduction\] The natural inclusion map $$\beta:~~ (\Gamma(L/A)[-1]\otimes_{{C^{\infty}}_M}\Omega(A),d_{\mathrm{CE}}^{L/A}) \rightarrow (L(D_{\mathrm{poly}}^{1}(L/A))\otimes_{{C^{\infty}}_M}\Omega(A), d_{\mathrm{CE}}^{L/A}+d_H)$$ is an isomorphism in the homotopy category $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$ of ${C^{\infty}}_{{{\mathcal M}}}$-modules. Moreover, the following diagram commutes in $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$, $$\label{CUBEDIAGRAMinintroduction}
\begin{tiny}
\xymatrixcolsep{-1.5pc}
\xymatrix{
\otimes_{{C^{\infty}}_{{{\mathcal M}}}}^{2}(\Gamma({{\mathcal F}}[-1],L_Q)\ar[rd]^{(\iota^{\ast})^{\otimes 2}}
\ar[rr]^{\theta^{\otimes 2}}
\ar[dd]^{\alpha_{{{\mathcal F}}}}&&
\otimes_{{C^{\infty}}_{{{\mathcal M}}}}^{2}({\mathrm{tot} }L({\mathscr{D} }_{\mathrm{poly}}^1({{\mathcal F}})),L_Q+d_H )\ar[rd]^{I^{\otimes 2}} \ar[dd]_<<<<<<<{\textup{\textlbrackdbl}\; , \; \textup{\textrbrackdbl}}|(0.5)\hole\\
&\otimes_{\Omega(A)}^{2}(\Gamma(L/A)[-1]{\otimes_{{C^{\infty}}_M}}\Omega(A),d_{\mathrm{CE}}^{L/A})\ar[rr]^>>>>>>>>>{\beta^{\otimes 2}}\ar[dd]_<<<<<<<{\alpha_{L/A}} &&
\otimes_{\Omega(A)}^{2}(L(D_{\mathrm{poly}}^{1}(L/A)){\otimes_{{C^{\infty}}_M}}\Omega(A), d_{\mathrm{CE}}^{L/A}+d_H) \ar[dd]_{\textup{\textlbrackdbl}\; , \; \textup{\textrbrackdbl}}
\\
(\Gamma({{\mathcal F}})[-1],L_Q)\ar[rd]^{\iota^{\ast}}\ar[rr]^>>>>>>>>>>>{\theta}|(0.44)\hole && ({\mathrm{tot} }L({\mathscr{D} }_{\mathrm{poly}}^1({{\mathcal F}})),L_Q+d_H ) \ar[rd]^{I} \hole\\
&(\Gamma(L/A)[-1]{\otimes_{{C^{\infty}}_M}}\Omega(A),d_{\mathrm{CE}}^{L/A})\ar[rr]^>>>>>>>>>>>>>{\beta}&&
(L(D_{\mathrm{poly}}^{1}(L/A)){\otimes_{{C^{\infty}}_M}}\Omega(A), d_{\mathrm{CE}}^{L/A}+d_H) \, , \\
}
\end{tiny}$$ where the maps $\iota^{\ast}$ and $I$ are isomorphisms in $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$.
In this cubic diagram, we simultaneously relate the Atiyah classe ${\alpha}_{{{\mathcal F}}}$ with the Atiyah class ${\alpha}_{L/A}$, and the free Lie algebra $({\mathrm{tot} }L({\mathscr{D} }_{\mathrm{poly}}^1({{\mathcal F}}))$, $L_Q+d_H$; ${\textup{\textlbrackdbl ~,~ \textrbrackdbl} })$ with the free Lie algebra $(L(D_{\mathrm{poly}}^{1}(L/A))$, $d_H$; ${\textup{\textlbrackdbl ~,~ \textrbrackdbl} })$. We also remark that, Theorem \[bigdiagraminintroduction\] implies Theorem \[C-S-Xinintroduction\] (which was stated without proofs in [@C-S-X]).
The paper is structured as follows. In Section \[Section2\], we recall graded manifolds and related notions, and then we introduce polyvector fields and polydifferential operators. In Section \[Section3\], we recall dg manifolds, dg structures on polyvector fields and polydifferential operators, HKR and PBW theorems. In Section \[Section4\], we describe Atiyah classes in dg geometry, which is the main object in this paper. Section \[Section5\] is devoted to our first main result Theorem \[maintheoreminintroduction\]. Section \[Section6\], which begins by recalling Atiyah classes of Lie pairs and Fedosov dg Lie algebroids associated with Lie pairs, is devoted to our second main result Theorem \[bigdiagraminintroduction\]. In Section \[Section7\], we compare Ramadoss’s work with our results. Section \[Section8\] is an appendix for technological formulas.
**Acknowledgments.** We would like to thank Ping Xu and Mathieu Stiénon for fruitful discussions and useful comments.
Graded manifolds, polyvector fields and polydifferentials {#Section2}
=========================================================
Through out the paper, ${{\mathbb K } }$ denotes either the field ${{\mathbb R} }$ or ${{\mathbb C} }$. All vector spaces, algebras, modules etc are over ${{\mathbb K } }$, and ${{\mathbb Z} }$-graded objects. Below is a brief review of the notion of dg manifold (see [@M-S-X] for more about this subject) and many related concepts.
Graded manifolds and related notions {#Sec:gradedstuff}
------------------------------------
By a finite dimensional graded manifold ${{\mathcal M}}$, we mean a pair $(M, {{\mathcal O}}_{{{\mathcal M}}})$, where $M$ is a smooth manifold, and ${{\mathcal O}}_{{{\mathcal M}}}$ is a sheaf of graded commutative algebras over $M$ locally of the form $${{\mathcal O}}_{{{\mathcal M}}}(U)
\simeq C^{\infty}(U, {{\mathbb K } })\otimes_{{{\mathbb K } }}
\hat{S}(V^{\vee})$$ for a finite dimensional graded vector space $V$ over ${{\mathbb K } }$.
We denote by $C^{\infty}_{{{\mathcal M}}}$ the space of global sections of ${{\mathcal O}}_{{{\mathcal M}}}$, by $T_{{{\mathcal M}}}$ the tangent vector bundle over ${{\mathcal M}}$, and by ${\mathscr{X} }({{\mathcal M}})=\mathrm{Der}(C^{\infty}_{{{\mathcal M}}})$ the space of vector fields on ${{\mathcal M}}$. The degree of a homogeneous function $f\in C^{\infty}_{{{\mathcal M}}}$ is denoted by $\widetilde{f}$. Similarly, the degree of a homogeneous vector field $X \in {\mathscr{X} }({{\mathcal M}})$ is denoted by $\widetilde{X}$. This should not be confused with the notation $|X|$, which is reserved for total degrees (see below).\
By a ${C^{\infty}}_{{{\mathcal M}}}$-module, we mean a graded (left) ${C^{\infty}}_{{{\mathcal M}}}$-module, say ${\mathfrak{N}}$. The degree of an element $\xi\in {\mathfrak{N}}$ is denoted by $\widetilde{\xi}$. The tensor product of ${C^{\infty}}_{{{\mathcal M}}}$-modules ${\mathfrak{N}}$ and ${\mathfrak{N}}'$ is denoted by ${\mathfrak{N}}\otimes_{{C^{\infty}}_{{{\mathcal M}}}} {\mathfrak{N}}'$.
By a ${C^{\infty}}_{{{\mathcal M}}}$-complex, we mean a sequence of ${C^{\infty}}_{{{\mathcal M}}}$-modules $\Upsilon=\{\Upsilon^p\}_{p\in {{\mathbb Z} }}$ together with a degree 0 linear operator $\delta: ~\Upsilon^{\bullet}\to \Upsilon^{\bullet+1}$ which squares to zero, and is ${C^{\infty}}_{{{\mathcal M}}}$-linear in the sense that $$\delta(f\xi)=(-1)^{\widetilde{f}}f\delta(\xi), \quad \forall \xi \in \Upsilon^{\bullet}, f\in {C^{\infty}}_{{{\mathcal M}}}.$$
Let $\Upsilon^{p,q}$ be the degree $q$ subspace of $\Upsilon^p$. Thus, $\Upsilon$ is a bigraded object $$\Upsilon=\{ \Upsilon^p \}_{p\in {{\mathbb Z} }} =\Upsilon^{\bullet, \bullet}=\{\oplus_{q\in {{\mathbb Z} }}\Upsilon^{p,q}\}_{p\in {{\mathbb Z} }} .$$ It is convenient to denote such a ${C^{\infty}}_{{{\mathcal M}}}$-complex $(\Upsilon, \delta)$ by a diagram of the form:
$$\begin{array}{*{20}{c}}
{}&{}&{\oplus}&{}&{\oplus}&{} \\
{\cdots}&{\xrightarrow{{\delta}}}&{\Upsilon^{p, q+1}}&{\xrightarrow{{\delta}}}&{\Upsilon^{p+1, q+1}}&{\xrightarrow{{\delta}}}&{\cdots} \\
{}&{}&{ \oplus}&{}&{ \oplus}&{} \\
{\cdots}&{\xrightarrow{{\delta}}}&{\Upsilon^{p,q}}&{\xrightarrow{{\delta}}}&{\Upsilon^{p+1,q}}&{\xrightarrow{{\delta}}}&{\cdots} \\
{}&{}&{\oplus}&{}&{\oplus}&{}
\end{array}$$
For $x \in \Upsilon^{p,q}$, we will refer to $p$ as the horizontal degree, $q$ the vertical degree, $(p,q)$ the bi-degree, and $p+q$ the **total degree**. The total degree is also denoted by $|x|=p+q$.
Note that when $x\in \Upsilon^{p,q}$ is regarded as an element of the graded module $\Upsilon^p=\Upsilon^{p, \bullet}$, we have $\widetilde{x}=q$. We will denote by $\overline{x}$ the element in the graded module $\Upsilon^p [-p]$ that corresponds to the element $x$. Clearly, we have $\widetilde{\overline{x}}= p+\widetilde{x}=p+q=|x|$.
We will call $\delta$ the **horizontal differential** as it only raises the horizontal degree. Note also that ${C^{\infty}}_{{{\mathcal M}}}$-multiplications only affect the vertical degree.
The total space of $\Upsilon=(\Upsilon,\delta)$ is the direct sum $${\mathrm{tot} }\Upsilon=\oplus_{p\in {{\mathbb Z} }} \Upsilon^p[-p] .$$ With respect to the total degree, ${\mathrm{tot} }\Upsilon$ is a ${C^{\infty}}_{{{\mathcal M}}}$-module. In the sequel, for $x\in \Upsilon$, the notation $\overline{x}$ denotes the corresponding element in ${\mathrm{tot} }\Upsilon$.
Moreover, $\delta$ can be considered as a degree $(+1)$ ${C^{\infty}}_{{{\mathcal M}}}$-endomorphism on $ {\mathrm{tot} }\Upsilon$.
We now describe the tensor product $(\Upsilon_1 \widetilde{\otimes} \Upsilon_2, \widetilde{\delta} ) $ of two ${C^{\infty}}_{{{\mathcal M}}}$-complexes $(\Upsilon_1,\delta_1)$ and $(\Upsilon_2,\delta_2)$. Here and in the sequel, $\widetilde\otimes$ is reserved for such a monoidal product.
First, the $n$-th horizontal term of $\Upsilon_1 \widetilde{\otimes} \Upsilon_2$ is the ${C^{\infty}}_{{{\mathcal M}}}$-module $$(\Upsilon_1 \widetilde{\otimes} \Upsilon_2)^n=\oplus_{p+q=n}(\Upsilon_1^p[-p] \otimes_{{C^{\infty}}_{{{\mathcal M}}}}\Upsilon_2^q[-q])[n]\,.$$ We note degree shiftings to the first and second factors. In other words, the ${C^{\infty}}_{{{\mathcal M}}}$-module structure of $(\Upsilon_1 \widetilde{\otimes} \Upsilon_2)^n$ is given by $$f(x\widetilde{\otimes}y):~~=
fx \widetilde{\otimes} y = (-1)^{\widetilde{f}|x|}x \widetilde{\otimes} fy,\quad \forall f\in {C^{\infty}}_{{{\mathcal M}}}, \quad x\widetilde{\otimes} y \in (\Upsilon_1 \widetilde{\otimes} \Upsilon_2)^n.$$
Second, the horizontal differential $\widetilde\delta:~(\Upsilon_1 \widetilde{\otimes} \Upsilon_2)^n\to (\Upsilon_1 \widetilde{\otimes} \Upsilon_2)^{n+1}$ is defined by: $$\widetilde{\delta} (x \widetilde{\otimes} y):~~=\delta_1(x) \widetilde{\otimes} y + (-1)^{|x|}x \widetilde{\otimes} \delta_2(y).$$
The bigrading on $\Upsilon_1 \widetilde{\otimes} \Upsilon_2$ is clear — $(\Upsilon_1 \widetilde{\otimes} \Upsilon_2)^{n,m}$ consists of ${{\mathbb K } }$-linear combinations of elements of the form $x\widetilde\otimes y$, for $x\in \Upsilon_1^{p,r}$ and $g\in\Upsilon_2^{q,s}$ such that $p+q=n, r+s=m$. Moreover, we have $${\mathrm{tot} }(\Upsilon_1 \widetilde{\otimes} \Upsilon_2)=({\mathrm{tot} }\Upsilon_1)\otimes_{{C^{\infty}}_{{{\mathcal M}}}}({\mathrm{tot} }\Upsilon_2).$$
There is also a symmetric product $\widetilde{\odot}$ of ${C^{\infty}}_{{{\mathcal M}}}$-complexes defined in a similar manner.
The Schouten-Nijenhuis algebra ${{\mathscr{T} }_{\mathrm{poly}}}$ of polyvector fields
--------------------------------------------------------------------------------------
In this paper, we will consider a special ${C^{\infty}}_{{{\mathcal M}}}$-complex, denoted by $${{\mathscr{T} }_{\mathrm{poly}}^1}=({{\mathscr{T} }_{\mathrm{poly}}^1},\delta=0),$$ arising from ${\mathscr{X} }({{\mathcal M}})$. Roughly speaking, it is a copy of ${\mathscr{X} }({{\mathcal M}})$ but concentrated in the horizontal degree $(+1)$. In other words, $({{\mathscr{T} }_{\mathrm{poly}}^1})^{1}={\mathscr{X} }({{\mathcal M}})$ and $({{\mathscr{T} }_{\mathrm{poly}}^1})^{n}=0$ for $n\neq 1$.
An element in ${{\mathscr{T} }_{\mathrm{poly}}^1}$ is just some vector field $X\in {\mathscr{X} }({{\mathcal M}})$. The only difference is that it has a double grading $(1,\widetilde{X})$. If we treat $X \in {{\mathscr{T} }_{\mathrm{poly}}^1}$, then $\widetilde{X}$ means the original degree of $X$ as a vector field and ${{|X|}}=1+\widetilde{X}$ the total degree. One could also treat $\overline{X}\in {\mathrm{tot} }{{\mathscr{T} }_{\mathrm{poly}}^1}={\mathscr{X} }({{\mathcal M}})[-1]$.
The ${C^{\infty}}_{{{\mathcal M}}}$-complex of polyvector fields, denoted by ${{\mathscr{T} }_{\mathrm{poly}}}$, is the symmetric algebra of ${{\mathscr{T} }_{\mathrm{poly}}^1}$: $${{\mathscr{T} }_{\mathrm{poly}}}=({{\mathscr{T} }_{\mathrm{poly}}},\delta=0)
:~~= S^{\bullet}({{\mathscr{T} }_{\mathrm{poly}}^1}, 0).
$$ The symmetric product in ${\mathscr{T} }_{\mathrm{poly}}$ is denoted by $\widetilde{\odot}$. Note that the component $
{\mathscr{T} }_{\mathrm{poly}}^{n}=S^n({{\mathscr{T} }_{\mathrm{poly}}^1}),
$ whose elements, called $n$-polyvector fields on ${{\mathcal M}}$, are finite sums of homogeneous elements of the form: $$X=X_1\widetilde{\odot} X_2\widetilde{\odot} \cdots \widetilde{\odot} X_n.$$ Here each vector field $X_i$ is considered as an element in ${{\mathscr{T} }_{\mathrm{poly}}^1}$. The horizontal degree of $X$ is $n$, whereas vertical degree of $X$ is $\widetilde{X}_1+\dots+\widetilde{X}_n$, and the total degree of $X$ is $|X|=n+\widetilde{X}_1+\dots+\widetilde{X}_n$.\
The space ${{\mathscr{T} }_{\mathrm{poly}}}$ admits a canonical Schouten-Nijenhuis algebra structure, also known as a degree $1$ Poisson algebra. See [@B-C-S-X3] for more details.
The Hopf algebra ${{\mathscr{D} }_{\mathrm{poly}}}$ of polydifferential operators {#Dpoly}
---------------------------------------------------------------------------------
The tangent bundle $T_{{{\mathcal M}}}$ is a Lie algebroid over ${{\mathcal M}}$. Its universal enveloping algebra, denoted by ${{\mathcal D}}_{{{\mathcal M}}}$, is called the space of differential operators. Denote by ${\mathscr{D} }_{\mathrm{poly}}^1$ the bigraded object which is a copy of ${{\mathcal D}}_{{{\mathcal M}}}$ and concentrated in horizontal degree $(+1)$. If a differential operator $D \in {{\mathcal D}}_{{{\mathcal M}}}$ is considered as in ${\mathscr{D} }_{\mathrm{poly}}^1$, then it has horizontal degree $(+1)$, vertical degree $\widetilde{D}$ and total degree $|D|=1+\widetilde{D}$. It is convenient to denote $\overline{D}\in {\mathrm{tot} }{\mathscr{D} }_{\mathrm{poly}}^1={{\mathcal D}}_{{{\mathcal M}}}[-1]$.
Elements of the space $
{\mathscr{D} }_{\mathrm{poly}}^n:~~=\widetilde{\otimes}^n {\mathscr{D} }_{\mathrm{poly}}^1
$ are called $n$-polydifferential operators. Denote by ${\mathscr{D} }_{\mathrm{poly}}^{n,m}\subset {\mathscr{D} }_{\mathrm{poly}}^n$ the set of elements of horizontal degree $n$ and vertical degree $m$. In other words, ${\mathscr{D} }_{\mathrm{poly}}^{n,m}$ is ${{\mathbb K } }$-linearly spanned by homogenous elements of the form $$D=D_1\widetilde{\otimes}\cdots \widetilde{\otimes}D_n$$ where $D_i \in {\mathscr{D} }_{\mathrm{poly}}^1$ and $\widetilde{D}_1+ \cdots + \widetilde{D}_n =m$. Note that the total degree $|D|=n+m$.
Following Tamarkin and Tsygan [@Tamarkin-Tsygan], a $n$-polydifferential operator $D\in {\mathscr{D} }_{\mathrm{poly}}^n$ can be identified as a map $$D: \quad \otimes_{{{\mathbb K } }}^n {C^{\infty}}_{{{\mathcal M}}} \rightarrow {C^{\infty}}_{{{\mathcal M}}},$$ such that for each position $i$, and $f_1, \cdots ,f_{i-1}, f_{i+1}, \cdots ,f_n\in {C^{\infty}}_{{{\mathcal M}}}$, the map $${C^{\infty}}_{{{\mathcal M}}}\to {C^{\infty}}_{{{\mathcal M}}},\quad
f \mapsto D(f_1, \cdots f_{i-1}, f, f_{i+1}, \cdots ,f_n)$$ is a differential operator.
In fact, if $D=D_1\widetilde{\otimes} \cdots \widetilde{\otimes}D_n \in {\mathscr{D} }_{\mathrm{poly}}^n$, where $D_i \in {{\mathcal D}}_{{{\mathcal M}}}$, then $D$ viewed as a map $\otimes_{{{\mathbb K } }}^n {C^{\infty}}_{{{\mathcal M}}} \rightarrow {C^{\infty}}_{{{\mathcal M}}}$ is given by $$D(f_1, f_2, ..., f_n):~~=(-1)^{\star}D_1(f_1)\cdots D_n(f_n),$$ where $\star=\sum_{i=1}^{n}\sum_{j=i+1}^{n}(\widetilde{f}_i +1)|D_j|$.
The space of polydifferential operators $${\mathscr{D} }_{\mathrm{poly}}:~~=\oplus_{n \geqslant 0}{\mathscr{D} }_{\mathrm{poly}}^n
=\oplus_{n \geqslant 0, m\in {{\mathbb Z} }}{\mathscr{D} }_{\mathrm{poly}}^{n,m}$$ admits a Hopf algebra structure:
- The multiplication is the cup product $$\cup:~~ {\mathscr{D} }_{\mathrm{poly}} \widetilde{\otimes} {\mathscr{D} }_{\mathrm{poly}}
\rightarrow
{\mathscr{D} }_{\mathrm{poly}}, \qquad
D\cup E :~~ = D\widetilde{\otimes}E.$$ In terms of a map from $\otimes_{{{\mathbb K } }}^{p+q}{C^{\infty}}_{{{\mathcal M}}}$ to ${C^{\infty}}_{{{\mathcal M}}}$, we have $$\label{cupproductformula}
(D\widetilde{\otimes} E)(f_1, \cdots f_{n+m})=
(-1)^{\flat}D(f_1,\cdots ,f_n)E(f_{n+1},\cdots ,f_{n+m}),$$ for $f_1, \cdots , f_{n+m}\in {C^{\infty}}_{{{\mathcal M}}}$, where $\flat=|E|(\widetilde{f}_1+\cdots +\widetilde{f}_n + n)$.
Note that ${\mathscr{D} }_{\mathrm{poly}}^{p,r}\cup {\mathscr{D} }_{\mathrm{poly}}^{q,s} \subset {\mathscr{D} }_{\mathrm{poly}}^{p+q, r+s}$.
- The comultiplication is the shuffle coproduct $$\widetilde{\Delta}:~~ {\mathscr{D} }_{\mathrm{poly}} \rightarrow {\mathscr{D} }_{\mathrm{poly}} \widetilde{\otimes} {\mathscr{D} }_{\mathrm{poly}},$$ $$\label{coproductformulaofDpoly}
\widetilde{\Delta}(D_1\widetilde{\otimes} \cdots \widetilde{\otimes} D_n)=\sum_{\substack{p+q=n, \\ \sigma\in \mathrm{Sh}(p,q)}}\kappa(\sigma)(D_{\sigma(1)}\widetilde{\otimes} \cdots \widetilde{\otimes} D_{\sigma(p)}) \widetilde{\bigotimes}
(D_{\sigma(p+1)}\widetilde{\otimes} \cdots \widetilde{\otimes} D_{\sigma(p+q)}).$$ where $\mathrm{Sh}(p,q)$ stands for the set of $(p,q)$-shuffles and $\kappa(\sigma)$ is the Koszul sign determined by $$\label{Koszulsign}
D_1\widetilde{\odot}\cdots \widetilde{\odot} D_n=\kappa(\sigma)
D_{\sigma(1)}\widetilde{\odot}\cdots \widetilde{\odot} D_{\sigma(n)}.$$ Here $\widetilde{\odot}$ is the symmetric product in $S^{\bullet} ({\mathscr{D} }_{\mathrm{poly}}^1)$.
- The unit is the natural inclusion $\eta: {C^{\infty}}_{{{\mathcal M}}}={\mathscr{D} }_{\mathrm{poly}}^0\hookrightarrow {{\mathscr{D} }_{\mathrm{poly}}}$.
- The counit $\varepsilon: {\mathscr{D} }_{\mathrm{poly}} \twoheadrightarrow {\mathscr{D} }_{\mathrm{poly}}^0={C^{\infty}}_{{{\mathcal M}}}$ is the natural projection.
- The antipode is the map $$t: {\mathscr{D} }_{\mathrm{poly}} \rightarrow {\mathscr{D} }_{\mathrm{poly}}$$ $$t(D_1 \widetilde{\otimes} D_2 \cdots \widetilde{\otimes} D_n)
=
(-1)^{\natural}D_n \widetilde{\otimes} \cdots D_2 \widetilde{\otimes} D_1,$$ where $\natural=\sum_{i=0}^{n-1}|D_{n-i}|(|D_1|+ \cdots +|D_{n-i-1}|)$.
We further describe the ${C^{\infty}}_{{{\mathcal M}}}$-complex structure on ${{\mathscr{D} }_{\mathrm{poly}}}$. First, recall that the Gerstenhaber product of two elements $D\in {\mathscr{D} }_{\mathrm{poly}}^n$ and $E \in {\mathscr{D} }_{\mathrm{poly}}^m$, denoted by $D\circ E \in {\mathscr{D} }_{\mathrm{poly}}^{n+m-1}$, is the operator $$\label{Gerstenhaberproductformula}
(D\circ E)(f_1,\cdots ,f_{n+m-1})=
\sum_{j\geqslant 0}(-1)^{\star_j}
D(f_1,\cdots , f_j,E(f_{j+1},\cdots , f_{j+m}),\cdots ),$$ where $\star_j=(|E|+1)(\widetilde{f}_1+\cdots +\widetilde{f}_j+j)$.
The Gerstenhaber bracket in the Hopf algebra ${\mathscr{D} }_{\mathrm{poly}}$ is defined by $$\label{Gernstenhaberbracket}
[D,E]=D\circ E-(-1)^{(|D|+1)(|E|+1)}E\circ D, \quad D, E \in {\mathscr{D} }_{\mathrm{poly}}.$$ It satisfies the following well known properties (see [@Tamarkin-Tsygan]):
$$[D,E]=-(-1)^{(|D|+1)(|E|+1)}[E,D],$$
$$\label{Jacobi}
\bigl[D,[E,F]\bigr]=\bigl[[D,E],F\bigr]
+(-1)^{(|D|+1)(|E|+1)}\bigl[E,[D,F]\bigr].$$
Consider the special element $${\mathrm{m} }: ~~ =-1\widetilde{\otimes}1 \in {\mathscr{D} }_{\mathrm{poly}}^{2, 0}.$$ As an operator, $$\label{multsign}
{\mathrm{m} }(f_1, f_2)=(-1)^{\widetilde{f}_1}f_1 f_2, \quad f_1, f_2 \in {C^{\infty}}_{{{\mathcal M}}}.$$ The horizontal differential on ${\mathscr{D} }_{\mathrm{poly}}$ is defined and denoted by $$d_H:~~=[{\mathrm{m} }, \, \cdot \,]:~~ {\mathscr{D} }_{\mathrm{poly}}^{n,\bullet} \rightarrow {\mathscr{D} }_{\mathrm{poly}}^{n+1,\bullet}.$$ It is also called the Hochschild coboundary operator. The following proposition is standard.
The pair $({{\mathscr{D} }_{\mathrm{poly}}}, \delta=d_H)$ is a ${C^{\infty}}_{{{\mathcal M}}}$-complex.
We call $({{\mathscr{D} }_{\mathrm{poly}}}, d_H)$ the ${C^{\infty}}_{{{\mathcal M}}}$-complex of polydifferential operators.
Define an operation $$\Delta:~~ {{\mathscr{D} }_{\mathrm{poly}}^1}\rightarrow {\mathscr{D} }_{\mathrm{poly}}^2,$$ $$\label{Delta}
\Delta(D)(f_1, f_2):~~=(-1)^{\widetilde{f_1}+1}D(f_1 f_2) , \qquad \forall D\in {{\mathscr{D} }_{\mathrm{poly}}^1}, \quad f_1, f_2 \in {C^{\infty}}_{{{\mathcal M}}}.$$ We call $\Delta$ the coproduct of ${{\mathscr{D} }_{\mathrm{poly}}^1}$. Note that we have $$\Delta(X)=X \widetilde{\otimes} 1 + (-1)^{\widetilde{X}}1 \widetilde{\otimes} X, \quad \forall X \in {{\mathscr{T} }_{\mathrm{poly}}^1}.$$ A general formula of $\Delta(D)$, for $D\in {{\mathscr{D} }_{\mathrm{poly}}^1}$, is given by Proposition \[Deltaformula\] in Appendix.
\[coproduct\] For all $D\in {\mathscr{D} }_{\mathrm{poly}}^1$, we have $$d_H(D)=(-1)^{|D|}(D\widetilde{\otimes} 1 -\Delta(D) -(-1)^{|D|}1\widetilde{\otimes} D).$$
By definition, we have $$\begin{split}
d_H(D)(f_1, f_2)= &({\mathrm{m} }\circ D-(-1)^{\widetilde{D}}D\circ {\mathrm{m} })(f_1,f_2)\\=& \, (-1)^{\widetilde{D}+\widetilde{f}_1+1}
D(f_1 f_2)
+
(-1)^{\widetilde{D}+\widetilde{f}_1}
(Df_1)f_2 + (-1)^{\widetilde{D}(\widetilde{f}_1 + 1) + \widetilde{f}_1}f_1(Df_2)\\
=& (-1)^{|D|}(D\widetilde{\otimes} 1 -(-1)^{|D|}1 \widetilde{\otimes} D -\Delta(D))(f_1,f_2),
\end{split}$$ where $f_1, f_2 \in {C^{\infty}}_{{{\mathcal M}}}$.
The following lemma is needed.
[@Gerstenhaber; @Tamarkin-Tsygan]\[Hochschildandcupproduct\] For all $D,E\in {\mathscr{D} }_{\mathrm{poly}}$, we have $$d_H(D\widetilde{\otimes} E)=d_H D\widetilde{\otimes} E +(-1)^{|D|}D \widetilde{\otimes} d_H E.$$
A direct consequence of Lemma \[coproduct\] and Lemma \[Hochschildandcupproduct\] is the following
We have $$\begin{split}
d_H(D_1\widetilde{\otimes} \cdots \widetilde{\otimes} D_n)= & \,
(-1)^{\sum_{i=1}^{n}|D_i|}
(
(-1)^{1+\sum_{i=1}^n|D_i|}1\widetilde{\otimes} D_1 \widetilde{\otimes}\cdots \widetilde{\otimes} D_n
+ \\
& \quad - \sum_{i=1}^{n} (-1)^{\sum^{i-1}_{j=1}|D_j|}D_1\widetilde{\otimes} \cdots D_{i-1} \widetilde{\otimes} \Delta(D_i) \widetilde{\otimes} D_{i+1} \cdots D_{n} \\
& \quad\quad + D_1 \widetilde{\otimes}\cdots \widetilde{\otimes} D_n \widetilde{\otimes} 1 ).
\end{split}$$
Dg manifolds {#Section3}
============
Dg manifolds and dg modules
---------------------------
A dg manifold is a pair $({{\mathcal M}}, Q)$, where ${{\mathcal M}}$ is a graded manifold, and $Q\in {\mathscr{X} }({{\mathcal M}})$ is a homological vector field, i.e., a degree $(+1)$ vector field such that $[Q,Q]=2Q^2=0$.\
A dg vector bundle is a vector bundle object in the category of dg manifolds. In other words, a dg vector bundle over $({{\mathcal M}}, Q)$ is a pair $({{\mathcal E}}, L_Q)$, where
- ${{\mathcal E}}$ is a vector bundle over ${{\mathcal M}}$,
- $L_Q:~ \Gamma({{\mathcal E}}) \rightarrow \Gamma({{\mathcal E}})$ is a degree $(+1)$ differential, i.e., $$L_Q^2=0$$ and $$L_Q(fs)=Q(f)s+(-1)^{\widetilde{f}}fL_Q(s)$$ for all $ f \in {C^{\infty}}_{{{\mathcal M}}}$, and $s\in \Gamma({{\mathcal E}})$.
In this note, we abuse the notation $L_Q$ to denote all differentials of dg vector bundles, unless the differential is otherwise specified. For example, the tangent space $ T_{{{\mathcal M}}} $ of $({{\mathcal M}},Q)$ is a dg vector bundle, whose differential $L_{Q}=[Q, \cdot\,]$.
A dg module over $({{\mathcal M}}, Q)$ is a pair $({\mathfrak{N}}, L_Q)$ where
- ${\mathfrak{N}}$ is a ${C^{\infty}}_{{{\mathcal M}}}$-module,
- $L_Q:~ {\mathfrak{N}}\rightarrow {\mathfrak{N}}$ is a degree $(+1)$ differential, i.e., $$L_Q^2=0$$ and $$L_Q(f\xi)=Q(f)\xi+(-1)^{\widetilde{f}}fL_Q(\xi)$$ for all $f\in {C^{\infty}}_{{{\mathcal M}}}$ and $\xi \in {\mathfrak{N}}$.
Again, the notation $L_Q$ is abused to denote all differentials of dg modules, unless stated otherwise. For a dg vector bundle $ ({{\mathcal E}},L_Q) $ over $({{\mathcal M}},Q)$, the section space $ \Gamma({{\mathcal E}}) $ is a dg module over $({{\mathcal M}},Q)$. In particular, $ ({\mathscr{X} }({{\mathcal M}}),L_Q=[Q,\cdot\,]) $ is a dg module.
A morphism of dg modules from $({\mathfrak{N}}, L_{Q})$ to $({\mathfrak{N}}',L_{Q})$ is a ${C^{\infty}}_{{{\mathcal M}}}$-linear and degree $0$ map $f:~ {\mathfrak{N}}\rightarrow {\mathfrak{N}}'$ that satisfies $f\circ L_{Q}=L_{Q} \circ f$. A morphism $f:({\mathfrak{N}}, L_{Q}) \rightarrow ({\mathfrak{N}}', L_{Q})$ is called quasi-isomorphism if it induces isomorphism $H(f): H({\mathfrak{N}}, L_Q) \rightarrow H({\mathfrak{N}}', L_Q)$ of cohomology groups.
In this note, let us fix the dg manifold $({{\mathcal M}}, Q)$. The category of dg modules over $({{\mathcal M}},Q)$ is denoted by ${\mathbf{dg}\mathrm{-}\mathbf{mod} }$. The homology category $\mathrm{H}({\mathbf{dg}\mathrm{-}\mathbf{mod} })$ is the category whose objects are dg modules, and whose morphisms are morphisms of dg modules modulo cochain homotopy [@G-Z; @Keller]. The homotopy category $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$ is the Gabriel-Zisman localization of ${\mathbf{dg}\mathrm{-}\mathbf{mod} }$ by the set of quasi-isomorphisms [@G-Z]. We have a sequence of natural functors between these categories: $${\mathbf{dg}\mathrm{-}\mathbf{mod} }\rightarrow \mathrm{H}({\mathbf{dg}\mathrm{-}\mathbf{mod} }) \rightarrow \Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} }).$$
Dg complexes {#Sec:dgcomplexes}
------------
A dg complex over $({{\mathcal M}}, Q)$ is a ${C^{\infty}}_{{{\mathcal M}}}$-complex $(\Upsilon,\delta)$ such that each $\Upsilon^p$ ($p\in{{\mathbb Z} }$) is endowed with a dg module structure over $({{\mathcal M}},Q)$: $$L_Q:~\Upsilon^{p,\bullet} \rightarrow \Upsilon^{p,\bullet+1}$$ which is anti-commutes with $\delta$, i.e., $$\label{Eqn:LQdeltacompatible}
\delta \circ L_Q + L_Q \circ \delta =0.$$
So, a dg complex is indeed a triple $\Upsilon=(\Upsilon, L_Q, \delta)$. It is convenient to denote such a dg complex by a diagram of double complex: $$\begin{array}{*{20}{c}}
{}& {}&{\uparrow}&{}&{\uparrow}&{} \\
{\cdots}& {\xrightarrow{{\delta}}}&{\Upsilon^{p, q+1}}&{\xrightarrow{{\delta}}}&{\Upsilon^{p+1, q+1}}&{\xrightarrow{{\delta}}}&{\cdots} \\
{}&{}&{L_Q \bigg\uparrow}&{}&{L_Q \bigg\uparrow}&{} \\
{\cdots}&{\xrightarrow{{\delta}}}&{\Upsilon^{p,q}}&{\xrightarrow{{\delta}}}&{\Upsilon^{p+1,q}}&{\xrightarrow{{\delta}} }&{\cdots} \quad. \\
{}&{}&{\uparrow}&{}&{\uparrow}&{}
\end{array}$$ We call $\delta$ the **horizontal differential** and $L_Q$ the **vertical differential**. By the compatibility condition , we have a total complex $({\mathrm{tot} }\Upsilon, L_Q^{{\mathrm{tot} }}=L_Q+\delta)$. With respect to the total grading $({\mathrm{tot} }\Upsilon)^n=\oplus_{p+q=n}\Upsilon^{p,q}$, $({\mathrm{tot} }\Upsilon, L_Q+\delta )$ is a dg module over $({{\mathcal M}},Q)$.
A morphism of dg complexes $\varphi:~ (\Upsilon_1, L_Q, \delta_1) \rightarrow (\Upsilon_2, L_Q, \delta_2)$ is a degree $(0,0)$ morphism of bigraded vector spaces $\varphi:~ \Upsilon_1^{\bullet, \bullet} \rightarrow \Upsilon_2^{\bullet, \bullet}$ such that the $(p,q)$-components $\varphi^{p,q}:~ \Upsilon_1^{p,q} \rightarrow \Upsilon_2^{p,q}$ satisfy
- for each $p \in {{\mathbb Z} }$, $\varphi^{p}=\oplus_{q\in {{\mathbb Z} }}\varphi^{p,q}:~ \Upsilon_1^{p,\bullet} \rightarrow \Upsilon_2^{p,\bullet}$ is a morphisms of dg modules, i.e., $$L_Q \circ \varphi^{p,q}=\varphi^{p,q+1}\circ L_Q,$$
- $\varphi$ commutes with $\delta$: $$\delta_2 \circ \varphi^{p,q}=\varphi^{p+1,q} \circ \delta_1.$$
Clearly, $\varphi$ induces a morphism of dg modules ${\mathrm{tot} }\varphi: ({\mathrm{tot} }\Upsilon_1,L_Q+\delta_1)\to ({\mathrm{tot} }\Upsilon_2,L_Q+\delta_2)$.
Denote the category of dg complexes over $({{\mathcal M}},Q)$ by $Ch({\mathbf{dg}\mathrm{-}\mathbf{mod} })$. A quasi-isomorphism $\varphi:~ (\Upsilon_1, L_Q, \delta_1) \rightarrow (\Upsilon_2, L_Q, \delta_2)$ of dg complexes is a morphism in ${Ch({\mathbf{dg}\mathrm{-}\mathbf{mod}}) }$ that induces a quasi-isomorphism between the corresponding total complexes $({\mathrm{tot} }\Upsilon_1, L_Q+ \delta_1 )$ and $({\mathrm{tot} }\Upsilon_2, L_Q+\delta_2 )$.
The derived category $D({\mathbf{dg}\mathrm{-}\mathbf{mod} })$ of dg complexes over $({{\mathcal M}},Q)$ is the Gabriel-Zisman localization of $Ch({\mathbf{dg}\mathrm{-}\mathbf{mod} })$ by the set of quasi-isomorphisms.
The operation of taking total complex can be regarded as a functor $${\mathrm{tot} }:~~ Ch({\mathbf{dg}\mathrm{-}\mathbf{mod} }) \rightarrow {\mathbf{dg}\mathrm{-}\mathbf{mod} }.$$ We also denote by $${\mathrm{tot} }:~~ D({\mathbf{dg}\mathrm{-}\mathbf{mod} }) \rightarrow \Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$$ the induced functor between the two Gabriel-Zisman localizations of categories.
The category $Ch({\mathbf{dg}\mathrm{-}\mathbf{mod} })$ is a monoidal tensor category. We now explain the tensor product $(\Upsilon_1 \widetilde{\otimes} \Upsilon_2, L_Q, \widetilde{\delta} )$ of two dg complexes $(\Upsilon_1, L_Q,\delta_1)$ and $(\Upsilon_2, L_Q ,\delta_2)$ — The ${C^{\infty}}_{{{\mathcal M}}}$-complex $(\Upsilon_1 \widetilde{\otimes} \Upsilon_2, \widetilde{\delta} )$ is already defined in Section \[Sec:gradedstuff\]; The vertical differential $L_Q$ is defined in analogous to that of $\widetilde{\delta}$: $$L_Q(x \widetilde{\otimes} y ):~~=L_Q(x) \widetilde{\otimes} y + (-1)^{|x|}x \widetilde{\otimes} L_Q(y).$$
It is straightforward to verify that $(\Upsilon_1 \widetilde{\otimes} \Upsilon_2, L_Q, \widetilde{\delta})$ is a dg complex. There is also a symmetric product $\widetilde{\odot}$ on $Ch({\mathbf{dg}\mathrm{-}\mathbf{mod} })$ defined in a similar manner.
The following natural diagram summarizes the relations between all the categories that we introduced. $$\begin{xy}
(0, 20)*+{{\mathbf{dg}\mathrm{-}\mathbf{mod} }}="a";
(30, 20)*+{\mathrm{H}({\mathbf{dg}\mathrm{-}\mathbf{mod} })}="b";
(60,20)*+{\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })}="c";
(0,0)*+{Ch({\mathbf{dg}\mathrm{-}\mathbf{mod} })}="d";
(60,0)*+{D({\mathbf{dg}\mathrm{-}\mathbf{mod} })}="e";
{\ar "a"; "b"};
{\ar "b"; "c"};
{\ar "d"; "e"};
{\ar_{{\mathrm{tot} }} "e";"c"};
{\ar_{{\mathrm{tot} }} "d";"a"};
\end{xy}$$
Dg complex structures on ${{\mathscr{D} }_{\mathrm{poly}}}$ and ${{\mathscr{T} }_{\mathrm{poly}}}$, and the HKR theorem
-----------------------------------------------------------------------------------------------------------------------
We have defined ${C^{\infty}}_{{{\mathcal M}}}$-complex structures on ${{\mathscr{T} }_{\mathrm{poly}}}$ and ${{\mathscr{D} }_{\mathrm{poly}}}$. As we are working on a dg manifold $({{\mathcal M}},Q)$, ${{\mathscr{T} }_{\mathrm{poly}}}$ and ${{\mathscr{D} }_{\mathrm{poly}}}$ are equipped with dg complex structures. In fact, the homological vector field $Q$ can be regarded as an element in $({{\mathscr{T} }_{\mathrm{poly}}^1})^1\subset {\mathscr{D} }_{\mathrm{poly}}^{1,1}$. Both ${{\mathscr{T} }_{\mathrm{poly}}}$ and ${{\mathscr{D} }_{\mathrm{poly}}}$ inherits the natural vertical differential $
L_Q:~~= [Q\, , \, \cdot \,]
$, which can be treated as maps $L_Q: {\mathscr{T} }_{\mathrm{poly}}^{n, \bullet} \rightarrow
{\mathscr{T} }_{\mathrm{poly}}^{n, \bullet+1}$ and $L_Q: {\mathscr{D} }_{\mathrm{poly}}^{n, \bullet} \rightarrow
{\mathscr{D} }_{\mathrm{poly}}^{n, \bullet+1}$.
We call $({{\mathscr{T} }_{\mathrm{poly}}}, L_Q, \delta=0)$ the dg complex of polyvector fields.
\[Dpolyconstruction\] The triple $({\mathscr{D} }_{\mathrm{poly}}, L_Q, d_H)$ is a dg complex.
First, we note the easy fact that $d_H(f)=[m,f]=0$. Then we verify that $d_H$ is ${C^{\infty}}_{{{\mathcal M}}}$-linear: $$d_H(fD)=d_H(f\widetilde{\otimes} D)=d_H(f)\widetilde{\otimes} D+(-1)^{\widetilde{f}}f\widetilde{\otimes} d_HD
=(-1)^{\widetilde{f}}f d_HD,$$ by Lemma \[Hochschildandcupproduct\].
To verify that $d_H$ and $L_Q$ are anti-commutative, we observe that $$d_H\circ L_Q+L_Q \circ d_H
=\bigl[{\mathrm{m} },[Q, \cdot\,]\bigr]+\bigl[Q,[{\mathrm{m} }, \cdot\,]\bigr]
=\bigl[[{\mathrm{m} },Q],\cdot\, \bigr]=0.$$ Here we have used Eqt. (\[Jacobi\]), and the obvious fact that $$[{\mathrm{m} }, \, Q]=-[1\widetilde{\otimes}1, \, Q]=0.$$
The Hochschild-Kostant-Rosenberg map $${\mathrm{hkr} }:~~ ({{\mathscr{T} }_{\mathrm{poly}}}, L_Q, 0) \rightarrow ({{\mathscr{D} }_{\mathrm{poly}}}, L_Q, d_H )$$ is a morphism of dg complexes defined by $${\mathrm{hkr} }(X_1\widetilde{\odot}\cdots \widetilde{\odot} X_n)
:~~=\frac{1}{n!}\sum_{\sigma \in S_{n}}
\kappa(\sigma) X_{\sigma(1)}\widetilde{\otimes}\cdots
\widetilde{\otimes} X_{\sigma(n)},$$ where the vector fields $X_i$ are regarded as elements in ${{\mathscr{T} }_{\mathrm{poly}}^1}$. Here $\kappa(\sigma)$ is the Koszul sign (see Eqt. ).
The following Hochschild-Konstant-Rosenberg type theorem is due to Liao-Stiénon-Xu (see Proposition 4.1 in [@L-S-X]).
\[LSX\] The map ${\mathrm{hkr} }$ is a quasi-isomorphism of dg complexes from $({{\mathscr{T} }_{\mathrm{poly}}},L_Q,0)$ to $({{\mathscr{D} }_{\mathrm{poly}}},L_Q,d_H)$.
The universal enveloping algebra of a free Lie algebra object
-------------------------------------------------------------
A well known fact in Lie theory is that the universal enveloping algebra of a free Lie algebra is the free associate algebra. In this part we describe a similar result in monoidal categories, which is generalized from the work of Ramadoss [@Ramadoss].
If it exists, the universal enveloping algebra of a Lie algebra object $G$ in $Ch({\mathbf{dg}\mathrm{-}\mathbf{mod} })$ (resp. $D({\mathbf{dg}\mathrm{-}\mathbf{mod} })$) is an associative algebra object $U(G)$ in $Ch({\mathbf{dg}\mathrm{-}\mathbf{mod} })$ (resp. $D({\mathbf{dg}\mathrm{-}\mathbf{mod} })$) together with a morphism of Lie algebra objects $i:~~G \rightarrow U(G)$ satisfying the following universal property: given any associative algebra object $K$ and any morphism of Lie algebras $f:~~ G\rightarrow K$ in $Ch({\mathbf{dg}\mathrm{-}\mathbf{mod} })$ (resp. $D({\mathbf{dg}\mathrm{-}\mathbf{mod} })$), there exists a unique morphism of associative algebras $f':~~ U(G) \rightarrow K$ in $Ch({\mathbf{dg}\mathrm{-}\mathbf{mod} })$ (resp. $D({\mathbf{dg}\mathrm{-}\mathbf{mod} })$) such that $f=f'\circ i$.
It can be easily verified that, if exists, $U(G)$ is unique up to isomorphism in $Ch({\mathbf{dg}\mathrm{-}\mathbf{mod} })$ (resp. $D({\mathbf{dg}\mathrm{-}\mathbf{mod} })$).\
Let $V$ be a ${C^{\infty}}_{{{\mathcal M}}}$-module. Denote by $\underline{V}$ the bigraded ${C^{\infty}}_{{{\mathcal M}}}$-module which is a copy of $V$ and concentrates in horizontal degree $(+1)$. One forms the bigraded ${C^{\infty}}_{{{\mathcal M}}}$-module $T^{\bullet}V=\oplus_{p\geqslant 0} \underline{V}^{\widetilde{\otimes}p}$.
Note that in $T^{\bullet}V$, an element $D\in T^i(V)$ has total degree $|D|=\widetilde{D}+i$. The Lie bracket of two elements $D\in T^i(V)$ and $E\in T^j(V)$ is the element $$\label{freeLiebracket}
\textup{\textlbrackdbl}D,E\textup{\textrbrackdbl}=D\widetilde{\otimes} E - (-1)^{|D| |E|}E\widetilde{\otimes} D \in T^{i+j}(V).$$ Thus $T^{\bullet}V$ is a Lie algebra object in the category of bigraded objects.
Denote by $L(V)$ the smallest Lie subalgebra of $T^{\bullet}V$ containing $\underline{V}=T^1 V$. The space $L(V)$ is made of all ${{\mathbb K } }$-linear combinations of elements of the form $\textup{\textlbrackdbl} D_1,\cdots , \textup{\textlbrackdbl}
D_{n-1}, D_{n} \textup{\textrbrackdbl},\cdots \textup{\textrbrackdbl}$ with $D_1 , \cdots ,D_n \in \underline{V}$.
We also denote by ${\textup{\textlbrackdbl ~,~ \textrbrackdbl} }$ the induced Lie bracket on ${\mathrm{tot} }T^{\bullet}V$ (resp. ${\mathrm{tot} }L(V)$), i.e., $$\label{freeLiebracketonTOTALcomplex}
\textup{\textlbrackdbl}\overline{D},\overline{E}\textup{\textrbrackdbl}=\overline{D}\otimes_{{C^{\infty}}_{{{\mathcal M}}}} \overline{E} - (-1)^{|{D}||{E}|}\overline{E}\otimes_{{C^{\infty}}_{{{\mathcal M}}}} \overline{D},$$ for $\overline{D}, \overline{E} \in {\mathrm{tot} }T^{\bullet}V$ (resp. $\overline{D},\overline{ E} \in {\mathrm{tot} }L(V)$).
So far $T^{\bullet}V$ is only a bigraded ${C^{\infty}}_{{{\mathcal M}}}$-module. Now suppose that it is equipped with a dg complex structure of the form $(T^{\bullet}V,L_Q,\delta)$, such that $L_Q$ and $\delta$ are compatible with the tensor product $\widetilde{\otimes}$, i.e., for all $D,E \in T^{\bullet}V$, $$L_Q(D\widetilde{\otimes}E)=L_Q(D)\widetilde{\otimes}E+(-1)^{|D|}D\widetilde{\otimes}L_Q(E),$$ $$\delta(D\widetilde{\otimes}E)=\delta(D)\widetilde{\otimes}E+(-1)^{|D|}D\widetilde{\otimes}\delta(E).$$ In other words, $(T^{\bullet}V, L_Q, \delta)$ is an associative algebra object in $Ch({\mathbf{dg}\mathrm{-}\mathbf{mod} })$.
\[UL\] Let $(T^{\bullet}V, L_Q, \delta)$ be as above. If $\delta(\underline{V})\subset L(V)\cap T^2 V$, then
- The horizontal differential $\delta$ preserves $L(V)$, and thus $(L(V), L_Q, \delta)$ is a dg sub-complex of $(T^{\bullet}V, L_Q, \delta)$. Moreover, $(L(V), L_Q, \delta)$ is a Lie algebra object in the category $Ch({\mathbf{dg}\mathrm{-}\mathbf{mod} })$.
- The universal enveloping algebra $U(L(V),L_Q,\delta) =(T^{\bullet}(V),L_Q,\delta)$, in $Ch({\mathbf{dg}\mathrm{-}\mathbf{mod} })$.
- The universal enveloping algebra $U(L(V),L_Q,\delta)=(T^{\bullet}(V),L_Q,\delta)$, in the derived category $D({\mathbf{dg}\mathrm{-}\mathbf{mod} })$.
This theorem is due to Ramadoss [@Ramadoss]. It can be proved along the same lines of Ramadoss’s approach and thus omitted.
The PBW map
-----------
For a graded ${C^{\infty}}_{{{\mathcal M}}}$-module $V$, we have the Poincaré-Birkhoff-Witt map $${\mathrm{pbw} }:~~ S^{\bullet}(L(V)) \rightarrow U(L(V))=T^{\bullet}V,$$ $${\mathrm{pbw} }(D_1 \widetilde{\odot} \cdots \widetilde{\odot} D_n)
:~~=\frac{1}{n!}\sum_{\sigma \in S_{n}}
\kappa(\sigma) D_{\sigma(1)}\widetilde{\otimes}\cdots
\widetilde{\otimes} D_{\sigma(n)},$$ which is an isomorphism of bigraded ${C^{\infty}}_{{{\mathcal M}}}$-modules. Here $D_i \in L(V)$ are homogenous elements, and $\kappa(\sigma)$ is the Koszul sign determined by the total degrees of $D_i$ (see Eqt. ).
The case $V={{\mathcal D}}_{{{\mathcal M}}}$
--------------------------------------------
Now we consider the case $V={{\mathcal D}}_{{{\mathcal M}}}$, and hence $\underline{V}={\mathscr{D} }_{\mathrm{poly}}^1$. Recall that we have the Hochschild coboundary $d_H=[{\mathrm{m} }\, , \, \cdot\,]\,:~~
{\mathscr{D} }_{\mathrm{poly}}^{\bullet} \rightarrow {\mathscr{D} }_{\mathrm{poly}}^{\bullet+1}$, and the coproduct $\Delta:~~ {\mathscr{D} }_{\mathrm{poly}}^1 \rightarrow {\mathscr{D} }_{\mathrm{poly}}^2$.
The following fact is a direct consequence of Lemma \[coproduct\] and Proposition \[Deltaformula\] in Appendix.
We have $d_H({\mathscr{D} }_{\mathrm{poly}}^1)\subset {L({\mathscr{D} }_{\mathrm{poly}}^1) }\cap {\mathscr{D} }_{\mathrm{poly}}^2$.
By Theorem \[UL\] and the above lemma, we have
\[UL2\]
- The dg complex $({L({\mathscr{D} }_{\mathrm{poly}}^1) }, L_Q, d_H)$ is a Lie algebra object in $Ch({\mathbf{dg}\mathrm{-}\mathbf{mod} })$.
- The universal enveloping algebra $U({L({\mathscr{D} }_{\mathrm{poly}}^1) }, L_Q, d_H)=({{\mathscr{D} }_{\mathrm{poly}}}, L_Q, d_H)$, in $Ch({\mathbf{dg}\mathrm{-}\mathbf{mod} })$.
- The universal enveloping algebra $U({L({\mathscr{D} }_{\mathrm{poly}}^1) }, L_Q, d_H)=({{\mathscr{D} }_{\mathrm{poly}}}, L_Q, d_H)$, in $D({\mathbf{dg}\mathrm{-}\mathbf{mod} })$.
The following theorem is an analogue of [@Ramadoss Theorem $1$]. The proof is essential the same, and thus omitted.
\[R1\] The following diagram commutes in the category $Ch({\mathbf{dg}\mathrm{-}\mathbf{mod} })$: $$\label{R1diagram}
\xymatrix{
S^{\bullet}(({L({\mathscr{D} }_{\mathrm{poly}}^1) }, L_Q, d_H)) \widetilde{\otimes} ({L({\mathscr{D} }_{\mathrm{poly}}^1) }, L_Q, d_H) \ar[r]^<<<<<{\mu \circ \frac{\omega}{1-e^{- \omega}}} \ar[d]_{{\mathrm{pbw} }\otimes {\mathrm{id}}} &
S^{\bullet}(({L({\mathscr{D} }_{\mathrm{poly}}^1) }, L_Q, d_H))\ar[d]^{{\mathrm{pbw} }}\\
({{\mathscr{D} }_{\mathrm{poly}}},L_Q,d_H)
\widetilde{\otimes}
({L({\mathscr{D} }_{\mathrm{poly}}^1) },L_Q,d_H)\ar[r]^<<<<<<<<<<{\cup} & ({{\mathscr{D} }_{\mathrm{poly}}},L_Q,d_H) \, . \\
}$$ Here $\mu$ denotes the natural symmetric product of $S({L({\mathscr{D} }_{\mathrm{poly}}^1) }, L_Q, d_H)$, and the map $$\omega:~~ S^{\bullet}(({L({\mathscr{D} }_{\mathrm{poly}}^1) }, L_Q, d_H)) \widetilde{\otimes} ({L({\mathscr{D} }_{\mathrm{poly}}^1) }, L_Q, d_H) \rightarrow
S^{\bullet}(({L({\mathscr{D} }_{\mathrm{poly}}^1) }, L_Q, d_H)) \widetilde{\otimes} ({L({\mathscr{D} }_{\mathrm{poly}}^1) }, L_Q, d_H)$$ is defined by $$\label{omegaformula}
\omega( (G_1 \widetilde{\odot} \cdots \widetilde{\odot} G_n) \widetilde{\otimes} G):~~=
\sum_{i=1}^n (-1)^{(|G_{i+1}|+ \cdots +|G_n|)|G_i|}(G_1 \widetilde{\odot} \cdots \hat{G}_i \cdots \widetilde{\odot} G_n) \widetilde{\otimes}
\textup{\textlbrackdbl}
G_i, \, G
\textup{\textrbrackdbl}$$ for elements $G_i, G \in {L({\mathscr{D} }_{\mathrm{poly}}^1) }$.
Denote by $\theta:~~({{\mathscr{T} }_{\mathrm{poly}}^1}, L_Q ,0) \rightarrow ({L({\mathscr{D} }_{\mathrm{poly}}^1) }, L_Q,d_H )$ the natural inclusion map.
\[betaquasi-isomorphism\] The map $\theta:~~({{\mathscr{T} }_{\mathrm{poly}}^1}, L_Q ,0
) \rightarrow ({L({\mathscr{D} }_{\mathrm{poly}}^1) }, L_Q,d_H )$ is a quasi-isomorphism in $Ch({\mathbf{dg}\mathrm{-}\mathbf{mod} })$.
It is obvious that the map $${\mathrm{pbw} }:~~ S^{\bullet}(({L({\mathscr{D} }_{\mathrm{poly}}^1) },L_Q,d_H))
\rightarrow U({L({\mathscr{D} }_{\mathrm{poly}}^1) },L_Q,d_H)=( {{\mathscr{D} }_{\mathrm{poly}}},L_Q,d_H )$$ is compatible with $L_Q$ and the Hochschild coboundary $d_H$. Thus the map ${\mathrm{pbw} }$ is an isomorphism of dg complexes.
Recall that we have the HKR quasi-isomorphism: $({{\mathscr{T} }_{\mathrm{poly}}}, L_Q,0) \xrightarrow{{\mathrm{hkr} }} ({{\mathscr{D} }_{\mathrm{poly}}}, L_Q, d_H)$ (see Theorem \[LSX\]). We can decompose $${\mathrm{hkr} }={\mathrm{pbw} }\circ S^{\bullet}(\theta),$$ where $S^{\bullet}(\theta)$ the symmetrization of $\theta$ in the tensor category $(Ch({\mathbf{dg}\mathrm{-}\mathbf{mod} }), \widetilde{\otimes})$: $$S^{\bullet}(\theta):~~
({{\mathscr{T} }_{\mathrm{poly}}}, L_Q, 0)=S^{\bullet}(({{\mathscr{T} }_{\mathrm{poly}}^1}, L_Q,0))
\rightarrow
S^{\bullet}(({L({\mathscr{D} }_{\mathrm{poly}}^1) }, L_Q ,d_H)).$$
As the map ${\mathrm{hkr} }$ is a quasi-isomorphism of dg complexes, the map $S^{\bullet}(\theta)$ is a quasi-isomorphism as well. In particular, the map $\theta$, or the first component of $S^{\bullet}(\theta)$, is a quasi-isomorphism of dg complexes.
Atiyah classes of dg manifolds {#Section4}
==============================
The jet sequence
----------------
Given a graded manifold ${{\mathcal M}}$, consider the space ${{\mathcal D}}_{{{\mathcal M}}}^{\leqslant 1}$ of first-order differential operators on ${{\mathcal M}}$. Namely, ${{\mathcal D}}_{{{\mathcal M}}}^{\leqslant 1}$ consists of those operators on the algebra ${C^{\infty}}_{{{\mathcal M}}}$ that are the sum of a derivation and the multiplication by an element of ${C^{\infty}}_{{{\mathcal M}}}$. Indeed, ${{\mathcal D}}_{{{\mathcal M}}}^{\leqslant 1}$ can be identified to ${\mathscr{X} }({{\mathcal M}})\oplus {C^{\infty}}_{{{\mathcal M}}}$ in a canonical way.
Let ${\mathfrak{N}}$ be a ${C^{\infty}}_{{{\mathcal M}}}$-module. Since ${{\mathcal D}}_{{{\mathcal M}}}^{\leqslant 1}$ is a ${C^{\infty}}_{{{\mathcal M}}}$-bimodule, we can form the space ${{\mathcal D}}_{{{\mathcal M}}}^{\leqslant 1} \underset{{C^{\infty}}_{{{\mathcal M}}}}{\otimes} {\mathfrak{N}}$. Moreover, we have an exact sequence of left graded ${C^{\infty}}_{{{\mathcal M}}}$-modules: $$\label{Short}
0\rightarrow {\mathfrak{N}}\xrightarrow{i}
{{\mathcal D}}_{{{\mathcal M}}}^{\leqslant 1} \underset{{C^{\infty}}_{{{\mathcal M}}}}{\otimes} {\mathfrak{N}}\xrightarrow{j}
{\mathscr{X} }({{\mathcal M}})\otimes_{{C^{\infty}}_{{{\mathcal M}}}}{\mathfrak{N}}\rightarrow 0.$$ Here the maps $i$ and $j$ are defined by: $$\begin{split}
i(\xi) & =1\underset{{C^{\infty}}_{{{\mathcal M}}}}{\otimes}\xi, \\
j(X\underset{{C^{\infty}}_{{{\mathcal M}}}}{\otimes}\xi) & =X\otimes_{{C^{\infty}}_{{{\mathcal M}}}}\xi, \\
j(f\underset{{C^{\infty}}_{{{\mathcal M}}}}{\otimes}\xi) &=0,
\end{split}$$ for all $\xi \in {\mathfrak{N}}$ , $X\in {\mathscr{X} }({{\mathcal M}})$ and $f\in {C^{\infty}}_{{{\mathcal M}}}$.
We call the (first-)jet sequence associated with the ${C^{\infty}}_{{{\mathcal M}}}$-module ${\mathfrak{N}}$.
Now suppose that $({{\mathcal M}},Q)$ is a dg manifold, and $({\mathfrak{N}}, L_Q)$ a dg module over $({{\mathcal M}},Q)$. Then the space ${{\mathcal D}}_{{{\mathcal M}}}^{\leqslant 1} \underset{{C^{\infty}}_{{{\mathcal M}}}}{\otimes} {\mathfrak{N}}$ can be endowed with a dg module structure by setting $$L_Q(D\underset{{C^{\infty}}_{{{\mathcal M}}}}{\otimes} \xi):~~=[Q,D]\underset{{C^{\infty}}_{{{\mathcal M}}}}{\otimes} \xi +(-1)^{\widetilde{D}}D\underset{{C^{\infty}}_{{{\mathcal M}}}}{\otimes} L_Q(\xi),$$ where $D\in {{\mathcal D}}_{{{\mathcal M}}}^{\leqslant 1}$ , $\xi \in {\mathfrak{N}}$. Moreover, the jet sequence becomes an exact sequence in the category ${\mathbf{dg}\mathrm{-}\mathbf{mod} }$ of dg modules over $({{\mathcal M}},Q)$.
Smooth connections and Atiyah classes of dg modules
---------------------------------------------------
A smooth connection $\nabla$ on a ${C^{\infty}}_{{{\mathcal M}}}$-module ${\mathfrak{N}}$ is a degree $0$ map $$\nabla:~~ {\mathscr{X} }({{\mathcal M}})
\otimes_{{{\mathbb K } }} {\mathfrak{N}}\rightarrow {\mathfrak{N}},$$ $$(X, \xi) \mapsto
\nabla_{X}\xi$$ satisfying $$\nabla_{fX}\xi=f\nabla_{X}\xi,$$ $$\nabla_{X}(f\xi)=X(f)\xi+(-1)^{\widetilde{f}\widetilde{X}}f\nabla_{X}\xi,$$ for all $f\in {C^{\infty}}_{{{\mathcal M}}}$, $X \in {\mathscr{X} }({{\mathcal M}})$ and $\xi\in {\mathfrak{N}}$.
The following lemma is immediate.
The set of smooth connections on ${\mathfrak{N}}$ is in one-to-one correspondence with the set of ${C^{\infty}}_{{{\mathcal M}}}$-linear splittings of the jet sequence .
Now suppose that $\nabla$ is a smooth connection on the ${C^{\infty}}_{{{\mathcal M}}}$-module ${\mathfrak{N}}$. If moreover, $({\mathfrak{N}}, L_Q)$ is a dg module over $({{\mathcal M}},Q)$, then one can form a map $${\alpha}_{{\mathfrak{N}}}^{\nabla}:~~=[L_Q,\nabla]:~~{\mathscr{X} }({{\mathcal M}})\otimes_{{{\mathbb K } }}{\mathfrak{N}}\rightarrow {\mathfrak{N}},$$ $${\alpha}_{{\mathfrak{N}}}^{\nabla}(X,\xi)=L_Q(\nabla_X \xi)-\nabla_{[Q,X]}\xi-(-1)^{\widetilde{X}}\nabla_{X}L_Q(\xi).$$ It is easy to verify that ${\alpha}_{{\mathfrak{N}}}^{\nabla}$ is a degree $(+1)$ and ${C^{\infty}}_{{{\mathcal M}}}$-bilinear map. Moreover, it is a cocycle, i.e., $L_Q({\alpha}_{{\mathfrak{N}}}^{\nabla})=0$, or $${\alpha}_{{\mathfrak{N}}}^{\nabla} \in {\mathrm{Hom }}_{{\mathbf{dg}\mathrm{-}\mathbf{mod} }}({\mathscr{X} }({{\mathcal M}})\otimes_{{C^{\infty}}_{{{\mathcal M}}}}{\mathfrak{N}}, {\mathfrak{N}}[1]).$$ Alternatively, we treat $${\alpha}_{{\mathfrak{N}}}^{\nabla} \in {\mathrm{Hom }}_{{\mathbf{dg}\mathrm{-}\mathbf{mod} }}({\mathscr{X} }({{\mathcal M}})[-1]\otimes_{{C^{\infty}}_{{{\mathcal M}}}}{\mathfrak{N}}, {\mathfrak{N}})
\subset {\mathrm{Hom }}^{0}_{{C^{\infty}}_{{{\mathcal M}}}}({\mathscr{X} }({{\mathcal M}})[-1]\otimes_{{C^{\infty}}_{{{\mathcal M}}}}{\mathfrak{N}}, {\mathfrak{N}}).$$ Note that a sign correction is necessary: $$\label{Atiyahcocycleformula}
{\alpha}_{{\mathfrak{N}}}^{\nabla}(\overline{X}, \xi)=
(-1)^{\widetilde{X}}
(
L_Q(\nabla_X \xi)-\nabla_{[Q,X]}\xi-(-1)^{\widetilde{X}}\nabla_{X}L_Q(\xi) ).$$ Here the element $\overline{X} \in {\mathscr{X} }({{\mathcal M}})[-1]$ corresponds to the element $X \in {\mathscr{X} }({{\mathcal M}})$.
Moreover, we have a natural isomorphism: $$H^0({\mathrm{Hom }}^{\bullet}_{{C^{\infty}}_{{{\mathcal M}}}}({\mathscr{X} }({{\mathcal M}})[-1]\otimes_{{C^{\infty}}_{{{\mathcal M}}}}{\mathfrak{N}}, {\mathfrak{N}}), L_Q)
\simeq
{\mathrm{Hom }}_{\mathrm{H}({\mathbf{dg}\mathrm{-}\mathbf{mod} })}({\mathscr{X} }({{\mathcal M}})[-1]\otimes_{{C^{\infty}}_{{{\mathcal M}}}}{\mathfrak{N}}, {\mathfrak{N}}).$$
We call ${\alpha}^{\nabla}_{{\mathfrak{N}}}$ defined in the Atiyah cocycle of the dg module $({\mathfrak{N}}, L_Q)$ with respect to the smooth connection $\nabla$. We call $${\alpha}_{{\mathfrak{N}}}:~~=[{\alpha}^{\nabla}_{{\mathfrak{N}}}] \in
{\mathrm{Hom }}_{\mathrm{H}({\mathbf{dg}\mathrm{-}\mathbf{mod} })}({\mathscr{X} }({{\mathcal M}})[-1]\otimes_{{C^{\infty}}_{{{\mathcal M}}}}{\mathfrak{N}}, {\mathfrak{N}})$$ the Atiyah class of the dg module $({\mathfrak{N}}, L_Q)$.
Thus the Atiyah class ${\alpha}_{{\mathfrak{N}}}$ can be regarded as a morphism $${\alpha}_{{\mathfrak{N}}}: ({\mathscr{X} }({{\mathcal M}})[-1],L_Q)\otimes_{{C^{\infty}}_{{{\mathcal M}}}}({\mathfrak{N}},L_Q) \rightarrow ({\mathfrak{N}},L_Q)$$ in the homology category $\mathrm{H}({\mathbf{dg}\mathrm{-}\mathbf{mod} })$.
In general, it is not guaranteed the existence of a smooth connection on the ${C^{\infty}}_{{{\mathcal M}}}$-module ${\mathfrak{N}}$. So one can not use the definition of Atiyah class as above. Instead, we define the Atiyah class ${\alpha}_{{\mathfrak{N}}}$ of the dg module $({\mathfrak{N}}, L_Q)$ to be the element $${\alpha}_{{\mathfrak{N}}}\in
{\mathrm{Ext }}^1_{{\mathbf{dg}\mathrm{-}\mathbf{mod} }}({\mathscr{X} }({{\mathcal M}})\otimes_{{C^{\infty}}_{{{\mathcal M}}}}{\mathfrak{N}}, {\mathfrak{N}})$$ corresponding to the exact sequence in the category ${\mathbf{dg}\mathrm{-}\mathbf{mod} }$. It is standard that $${\mathrm{Ext }}^1_{{\mathbf{dg}\mathrm{-}\mathbf{mod} }}({\mathscr{X} }({{\mathcal M}})\otimes_{{C^{\infty}}_{{{\mathcal M}}}}{\mathfrak{N}}, {\mathfrak{N}})
\simeq
{\mathrm{Hom }}_{\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })}({\mathscr{X} }({{\mathcal M}})[-1]\otimes_{{C^{\infty}}_{{{\mathcal M}}}}{\mathfrak{N}}, {\mathfrak{N}}).$$ Thus ${\alpha}_{{\mathfrak{N}}}$ can be regarded as an element in the homotopy category ${\mathrm{Hom }}_{\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })}({\mathscr{X} }({{\mathcal M}})[-1]\otimes_{{C^{\infty}}_{{{\mathcal M}}}}{\mathfrak{N}}, {\mathfrak{N}})$ which is represented by the following roof: $$\label{roof}
\begin{xy}
(0,0)*+{{\mathscr{X} }({{\mathcal M}})[-1]\otimes_{{C^{\infty}}_{{{\mathcal M}}}}{\mathfrak{N}}}="a";
(60,0)*+{{\mathfrak{N}}\,\,}="b";
(30,20)*+{Cone(i)}="c";
{\ar@{~>}_{q} "c";"a"};
{\ar^{p} "c";"b"};
{\ar^<<<<<<<<<<<<<<<{{\alpha}_{{\mathfrak{N}}}} "a";"b"};
\end{xy}$$ Here $$Cone(i)=
( ({{\mathcal D}}_{{{\mathcal M}}}^{\leqslant 1})[-1]\underset{{C^{\infty}}_{{{\mathcal M}}}}{\otimes} {{\mathfrak{N}}})
\oplus {\mathfrak{N}}\, ,$$ is indeed the mapping cone of $i$. The left roof $q=j\circ pr_1$ is a quasi-isomorphism in $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$, and the right roof $p=pr_2$, where $pr_1, pr_2$ are respectively the projection to the first and second component.
The following fact is easy to prove.
Suppose that the jet sequence splits in the category of ${C^{\infty}}_{{{\mathcal M}}}$-modules. The following statements are equivalent.
- The jet sequence splits in the category ${\mathbf{dg}\mathrm{-}\mathbf{mod} }$.
- There is a smooth connection $\nabla$ on ${\mathfrak{N}}$ which is compatible with the dg module structures of ${\mathscr{X} }({{\mathcal M}})$ and ${\mathfrak{N}}$, i.e., $L_Q \nabla = \nabla L_Q$.
- The Atiyah class ${\alpha}_{{\mathfrak{N}}}=0$.
\[Atiyahcocyleoftensorproduct\] Let $\nabla^1$ and $\nabla^2$ be, respectively, smooth connections on ${C^{\infty}}_{{{\mathcal M}}}$-modules ${\mathfrak{N}}_1$ and ${\mathfrak{N}}_2$. Then $$\nabla_X (\xi_1 \otimes_{{C^{\infty}}_{{{\mathcal M}}}} \xi_2):~~ =(\nabla^1_X \xi_1)\otimes_{{C^{\infty}}_{{{\mathcal M}}}} \xi_2 +(-1)^{\widetilde{X}\widetilde{\xi}_1} \xi_1 \otimes_{{C^{\infty}}_{{{\mathcal M}}}} \nabla^2_X \xi_2 ,\quad \forall \xi_1 \in {\mathfrak{N}}_1, \, \xi_2 \in {\mathfrak{N}}_2,$$ defines a smooth connection $\nabla$ on ${\mathfrak{N}}_1 \otimes_{{C^{\infty}}_{{{\mathcal M}}}} {\mathfrak{N}}_2$. Moreover, if $({\mathfrak{N}}_1, L_Q)$ and $({\mathfrak{N}}_2, L_Q)$ are dg modules, then the Atiyah cocyle $${\alpha}^{\nabla}_{{\mathfrak{N}}_1 \otimes_{{C^{\infty}}_{{{\mathcal M}}}} {\mathfrak{N}}_2 }:~~ {\mathscr{X} }({{\mathcal M}})[-1] \otimes_{{C^{\infty}}_{{{\mathcal M}}}} ({\mathfrak{N}}_1 \otimes_{{C^{\infty}}_{{{\mathcal M}}}} {\mathfrak{N}}_2) \rightarrow
{\mathfrak{N}}_1 \otimes_{{C^{\infty}}_{{{\mathcal M}}}} {\mathfrak{N}}_2$$ is given by $$\label{Atiyahcocyleoftensorproductformula}
{\alpha}^{\nabla}_{{\mathfrak{N}}_1 \otimes_{{C^{\infty}}_{{{\mathcal M}}}} {\mathfrak{N}}_2 }(\overline{X}, \xi_1 \otimes_{{C^{\infty}}_{{{\mathcal M}}}} \xi_2)=
{\alpha}^{\nabla^1}_{{\mathfrak{N}}_1}(\overline{X} , \xi_1)\otimes_{{C^{\infty}}_{{{\mathcal M}}}} \xi_2 + (-1)^{(\widetilde{X}+1)\widetilde{\xi}_1} \xi_1 \otimes_{{C^{\infty}}_{{{\mathcal M}}}}
{\alpha}^{\nabla^2}_{{\mathfrak{N}}_2}(\overline{X} , \xi_2),$$ for all $X \in {\mathscr{X} }({{\mathcal M}})$, $\xi_1 \in {\mathfrak{N}}_1$ and $\xi_2 \in {\mathfrak{N}}_2$.
The proof is a direct calculation and thus omitted. We also have the following functoriality property of Atiyah classes.
\[functoriality\] If $({\mathfrak{N}}_1, L_Q)$ and $({\mathfrak{N}}_2, L_Q)$ both admit smooth connections, and $\varphi:~~ ({\mathfrak{N}}_1, L_Q)\rightarrow ({\mathfrak{N}}_2, L_Q)$ is a morphism of dg modules, then the following diagram $$\xymatrix{
{\mathscr{X} }({{\mathcal M}})[-1] {\otimes_{{C^{\infty}}_{{{\mathcal M}}}}} {\mathfrak{N}}_1 \ar[r]^{{\mathrm{id}}\otimes \varphi} \ar[d]_{{\alpha}_{{\mathfrak{N}}_1}} & {\mathscr{X} }({{\mathcal M}})[-1]{\otimes_{{C^{\infty}}_{{{\mathcal M}}}}} {\mathfrak{N}}_2 \ar[d]^{{\alpha}_{{\mathfrak{N}}_2}}\\
{\mathfrak{N}}_1 \ar[r]^{\varphi} & {\mathfrak{N}}_2 ~~ \\
}$$ in the homology category $\mathrm{H}({\mathbf{dg}\mathrm{-}\mathbf{mod} })$ is commutative.
Let $\nabla_1$ and $\nabla_2$ be smooth connections on $({\mathfrak{N}}_1, L_Q)$ and $({\mathfrak{N}}_2, L_Q)$ respectively. We only need to show that the diagram in the category ${\mathbf{dg}\mathrm{-}\mathbf{mod} }$: $$\xymatrix{
{\mathscr{X} }({{\mathcal M}})[-1] {\otimes_{{C^{\infty}}_{{{\mathcal M}}}}} {\mathfrak{N}}_1 \ar[r]^{{\mathrm{id}}\otimes \varphi} \ar[d]_{{\alpha}_{{\mathfrak{N}}_1}^{\nabla_1}} & {\mathscr{X} }({{\mathcal M}})[-1] {\otimes_{{C^{\infty}}_{{{\mathcal M}}}}} {\mathfrak{N}}_2 \ar[d]^{{\alpha}_{{\mathfrak{N}}_2}^{\nabla_2}
}\\
{\mathfrak{N}}_1 \ar[r]^{\varphi} & {\mathfrak{N}}_2 \\
}$$ commutes up to a cochain homotopy between dg modules. This can be verified directly and thus is omitted.
Atiyah classes of dg complexes
------------------------------
Now we consider a dg complex $(\Upsilon, L_Q ,\delta)$ over $({{\mathcal M}}, Q)$. Recall that the corresponding total complex $({\mathrm{tot} }\Upsilon, L_Q^{{\mathrm{tot} }})$ is a dg module, where $L_Q^{{\mathrm{tot} }}=\delta + L_Q$.
Let $(\Upsilon,L_Q,\delta)$ be a complex of dg modules. We define the Atiyah class ${\alpha}_{\Upsilon}$ of the dg complex $(\Upsilon,L_Q,\delta)$ to be the Atiyah class of the dg module $({\mathrm{tot} }\Upsilon, L_Q^{{\mathrm{tot} }})$, i.e., $${\alpha}_{\Upsilon}:~~={\alpha}_{{\mathrm{tot} }\Upsilon}
\in
H^0({\mathrm{Hom }}^{\bullet}_{{C^{\infty}}_{{{\mathcal M}}}}({\mathscr{X} }({{\mathcal M}})[-1]\otimes_{{C^{\infty}}_{{{\mathcal M}}}}{\mathrm{tot} }\Upsilon, {\mathrm{tot} }\Upsilon), L_Q^{{\mathrm{tot} }}).$$
\[remark on connections on dg complexes\] Suppose that each component $\Upsilon^{n,\bullet}$ of $(\Upsilon,L_Q,\delta)$ admits a smooth connection $\nabla_n$, then $\nabla^{{\mathrm{tot} }}=\oplus_n \nabla_n$ is a smooth connection on ${\mathrm{tot} }\Upsilon$. In this case, we have: $$\begin{split}
{\alpha}^{\nabla^{{\mathrm{tot} }}}_{({\mathrm{tot} }\Upsilon, L^{{\mathrm{tot} }}_Q)}
&={}
[L_Q^{{\mathrm{tot} }}, \nabla^{{\mathrm{tot} }} ] ={} [L_Q + \delta, \nabla^{{\mathrm{tot} }}]\\
& \quad\quad ={}
( (L_Q\circ \nabla^{{\mathrm{tot} }} - \nabla^{{\mathrm{tot} }}\circ L_Q)+
(\delta\circ \nabla^{{\mathrm{tot} }} - \nabla^{{\mathrm{tot} }}\circ ({\mathrm{id}}\otimes\delta) ) \\
& \quad\quad\quad ={}
\sum_n {\alpha}_{(\Upsilon^{n\bullet}, L_Q)}^{\nabla_n} +
(\delta\circ \nabla_n - \nabla_{n+1}\circ ({\mathrm{id}}\otimes\delta)).
\end{split}$$ We observe that the first part $\sum\limits_n
{\alpha}_{(\Upsilon^{n\bullet}, L_Q)}^{\nabla_n}$ sends the component $({{\mathscr{T} }_{\mathrm{poly}}^1}\widetilde{\otimes} \Upsilon)^{p,q}$ to $\Upsilon^{p-1,q+1}$, while the second part $\sum\limits_n (\delta\circ \nabla_n - \nabla_{n+1}\circ ({\mathrm{id}}\otimes\delta))$ sends $({{\mathscr{T} }_{\mathrm{poly}}^1}\widetilde{\otimes} \Upsilon)^{p,q}$ to $\Upsilon^{p,q}$. Thus the Atiyah cocycle ${\alpha}^{\nabla^{{\mathrm{tot} }}}_{({\mathrm{tot} }\Upsilon, L^{{\mathrm{tot} }}_Q)}$ can NOT be regarded as a morphism of dg complexes ${{\mathscr{T} }_{\mathrm{poly}}^1}\widetilde{\otimes} \Upsilon \rightarrow \Upsilon$.
Lie structures {#Section5}
==============
The Atiyah class as a Lie bracket on ${\mathscr{X} }({{\mathcal M}})[-1]$
-------------------------------------------------------------------------
Let $({{\mathcal M}}, Q)$ be a dg manifold. In Mehta, Stiénon and Xu [@M-S-X], it is shown that the ${{\mathbb K } }$-vector space $ {\mathscr{X} }({{\mathcal M}})[-1] $ admits an $L_{\infty}$-algebra structure. In fact, one can choose a torsion free smooth connection $\nabla$ on ${\mathscr{X} }({{\mathcal M}})$, and construct a sequence of degree $(+1)$ operations: $$\label{Linfinity}
\begin{split}
& R_1=L_Q :~~ {\mathscr{X} }({{{\mathcal M}}})[-1] \rightarrow {\mathscr{X} }({{{\mathcal M}}})[-1], \\
& R_2={\alpha}_{{\mathscr{X} }({{\mathcal M}})[-1]}^{\nabla}\in {\mathrm{Hom }}(S^2({\mathscr{X} }({{{\mathcal M}}})[-1]), \, {\mathscr{X} }({{{\mathcal M}}})[-1]), \\
& \quad \quad \quad \quad\quad \quad \quad \quad \quad \cdots \\
& R_k\in {\mathrm{Hom }}(S^k({\mathscr{X} }({{{\mathcal M}}})[-1]), \, {\mathscr{X} }({{{\mathcal M}}})[-1]), \cdots .
\end{split}$$ Then the operations $\{R_k\}_{k \geqslant 1 }$ turn ${\mathscr{X} }_{{{\mathcal M}}}[-1]$ into an $L_{\infty}$-algebra. Details of this construction can be found in [@M-S-X; @LG-S-X2].
We denote by ${\alpha}_{{{\mathcal M}}}$ the Atiyah class of the dg module $({\mathscr{X} }({{\mathcal M}})[-1],L_Q)$, which can be seen as a morphism $${\alpha}_{{{\mathcal M}}}:~~
({\mathscr{X} }({{{\mathcal M}}})[-1], L_Q)\otimes_{{C^{\infty}}_{{{\mathcal M}}}}({\mathscr{X} }({{{\mathcal M}}})[-1], L_Q)
\rightarrow
({\mathscr{X} }({{{\mathcal M}}})[-1], L_Q)$$ in the homology category $\mathrm{H}({\mathbf{dg}\mathrm{-}\mathbf{mod} })$.
As ${\mathscr{X} }({{{\mathcal M}}})[-1]$ is an $L_{\infty}$-algebra whose binary bracket is the Atiyah cocycle ${\alpha}_{{\mathscr{X} }({{\mathcal M}})[-1]}^{\nabla}$, we have
\[Thm:AtiyahdefinesLiebracket\] The Atiyah class ${\alpha}_{{{\mathcal M}}}$ defines a Lie bracket on $({\mathscr{X} }({{{\mathcal M}}})[-1], L_Q)$ in the homology category $\mathrm{H}({\mathbf{dg}\mathrm{-}\mathbf{mod} })$.
Therefore, ${\alpha}_{{{\mathcal M}}}$ also defines a Lie bracket on $({\mathscr{X} }({{{\mathcal M}}})[-1], L_Q)$ in the homotopy category $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$.
The following fact is analogous to that of Kapranov [@Kapranov], Chen-Stiénon-Xu [@C-S-X2] and Chen-Liu-Xiang [@C-L-X-HHA].
\[Atiyahclassasmodulestructuremap\] Let $({\mathfrak{N}}, L_Q)$ be a dg module. The Atiyah class $${\alpha}_{{\mathfrak{N}}}:~~ {\mathscr{X} }({{\mathcal M}})[-1]\otimes_{{C^{\infty}}_{{{\mathcal M}}}}{\mathfrak{N}}\rightarrow {\mathfrak{N}}$$ turns $({\mathfrak{N}}, L_Q)$ into a Lie algebra module object over the aforesaid Lie algebra object $({\mathscr{X} }({{\mathcal M}})[-1], L_Q )$ in the homology category $\mathrm{H}({\mathbf{dg}\mathrm{-}\mathbf{mod} })$.
Of course, one can also treat $({\mathfrak{N}}, L_Q)$ as a Lie algebra module object over $({\mathscr{X} }({{\mathcal M}})[-1], L_Q )$ in the homotopy category $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$
Canonical Atiyah cocycle of the dg complex $({{\mathscr{D} }_{\mathrm{poly}}}, L_Q, d_H)$
------------------------------------------------------------------------------------------
As a ${C^{\infty}}_{{{\mathcal M}}}$-module, ${{\mathscr{D} }_{\mathrm{poly}}^1}$ admits a canonical connection $$\nabla^{\mathrm{can}}:~~ {\mathscr{X} }({{\mathcal M}})\otimes_{{{\mathbb K } }} {{\mathscr{D} }_{\mathrm{poly}}^1}\rightarrow {{\mathscr{D} }_{\mathrm{poly}}^1},$$ $$\nabla^{\mathrm{can}}_{X}D:~~=X\circ D,$$ for all $X\in {\mathscr{X} }({{\mathcal M}})$ and $D\in {{\mathscr{D} }_{\mathrm{poly}}^1}$. The connection $\nabla^{\mathrm{can}}$ extends naturally to each component ${{\mathscr{D} }_{\mathrm{poly}}}^n$ ($n \in {{\mathbb Z} }$), i.e., $$\nabla^{\mathrm{can}}_{X}(D_1\widetilde{\otimes} \cdots \widetilde{\otimes} D_n):~~=
\sum_{i=1}^{n}(-1)^{\sum_{j=1}^{i-1}|D_j|\widetilde{X}}D_1\widetilde{\otimes} \cdots \widetilde{\otimes}\nabla^{\mathrm{can}}_XD_i \widetilde{\otimes} \cdots \widetilde{\otimes}D_n ,$$ for all $ X\in {\mathscr{X} }({{\mathcal M}})$ and $D_i\in {{\mathscr{D} }_{\mathrm{poly}}^1}$. By Remark \[remark on connections on dg complexes\], the sum of all these connections induces a connection on ${\mathrm{tot} }{{\mathscr{D} }_{\mathrm{poly}}}$, which is also denoted by $\nabla^{\mathrm{can}}$.
It is easy to see that the operator $\nabla^{\mathrm{can}}_X$ preserves ${L({\mathscr{D} }_{\mathrm{poly}}^1) }$, and thus the operator $\nabla^{\mathrm{can}}$ can be regarded as a connection on ${\mathrm{tot} }{L({\mathscr{D} }_{\mathrm{poly}}^1) }$.
Recall the coproduct $\Delta:~~ {{\mathscr{D} }_{\mathrm{poly}}^1}\rightarrow {\mathscr{D} }_{\mathrm{poly}}^2$ defined in Eqt..
\[canonicalconnectionandDelta\] We have $\nabla^{\mathrm{can}}_X \Delta = \Delta \nabla^{\mathrm{can}}_X$ , for all $X \in {\mathscr{X} }({{\mathcal M}})$.
Let $D=X_1 \circ \cdots \circ X_n$, where $X_i \in {{\mathscr{T} }_{\mathrm{poly}}^1}$. On the one hand, by Lemma \[Gerstenhaberproductandcupproduct\] and Eqt. in Appendix, we have $$\label{L.H.S}
\begin{split}
\nabla_{X}\Delta(D)=X \circ \Delta(D)
& =\sum (\pm) X \circ \textup{\textlbrackdbl}X_1 \circ X_{\sigma(2)} \circ \cdots \circ X_{\sigma(p)}, \, X_{\sigma(p+1)} \circ \cdots \circ X_{\sigma(n)}
\textup{\textrbrackdbl} \\
& \quad\quad =\sum (\pm) \textup{\textlbrackdbl}X \circ X_1 \circ X_{\sigma(2)} \circ \cdots \circ X_{\sigma(p)}, \, X_{\sigma(p+1)} \circ \cdots \circ X_{\sigma(n)}
\textup{\textrbrackdbl} \\
& \quad\quad\quad\quad + (\pm)\textup{\textlbrackdbl}
X \circ X_{\sigma(p+1)} \circ \cdots \circ X_{\sigma(n)}, \, X_1 \circ X_{\sigma(2)} \circ \cdots \circ X_{\sigma(p)}\textup{\textrbrackdbl}.
\end{split}$$ Here we denote by $(\pm)$ the appropriate Koszul signs which can be worked out explicitly as explained in Appendix.
On the other hand and by Eqt. again, we have $$\label{R.H.S}
\begin{split}
\Delta(\nabla_{X}D)& =\Delta(X\circ D)= \sum (\pm) \textup{\textlbrackdbl}X \circ X_{\sigma(1)} \circ X_{\sigma(2)} \circ \cdots \circ X_{\sigma(p)}, \, X_{\sigma(p+1)} \circ \cdots \circ X_{\sigma(n)}
\textup{\textrbrackdbl}.
\end{split}$$ By careful comparison of Equations and , we see that they coincide.
Now we consider the Atiyah cocyles of $({{\mathscr{D} }_{\mathrm{poly}}}, L_Q, d_H)$ and $({L({\mathscr{D} }_{\mathrm{poly}}^1) }, L_Q, d_H)$ associated to the connection $\nabla^{\mathrm{can}}$: $$\begin{split}
& {\alpha}_{{\mathscr{D} }_{\mathrm{poly}}}^{\nabla^{\mathrm{can}}}:~~{\mathscr{X} }({{\mathcal M}})[-1]\otimes_{{C^{\infty}}_{{{\mathcal M}}}}{\mathrm{tot} }{\mathscr{D} }_{\mathrm{poly}} \rightarrow {\mathrm{tot} }{\mathscr{D} }_{\mathrm{poly}} \, \\
\text{and \quad } & {\alpha}_{L({\mathscr{D} }_{\mathrm{poly}}^1)}^{\nabla^{\mathrm{can}}}:~~{\mathscr{X} }({{\mathcal M}})[-1]\otimes_{{C^{\infty}}_{{{\mathcal M}}}}{\mathrm{tot} }L({\mathscr{D} }_{\mathrm{poly}}^1) \rightarrow {\mathrm{tot} }L({\mathscr{D} }_{\mathrm{poly}}^1) \,.
\end{split}$$
\[inductionlemma\] The Atiyah cocycle ${\alpha}_{{\mathscr{D} }_{\mathrm{poly}}}^{\nabla^{\mathrm{can}} }$ satisfies $$\label{induction}
{\alpha}_{{\mathscr{D} }_{\mathrm{poly}}}^{\nabla^{\mathrm{can}} }
(\overline{X}, \overline{P_1\widetilde{\otimes} P_2})
=
{\alpha}_{{\mathscr{D} }_{\mathrm{poly}}}^{\nabla^{\mathrm{can}} }
(\overline{X}, \overline{P_1})\otimes_{{C^{\infty}}_{{{\mathcal M}}}}\overline{P_2}
+
(-1)^{(\widetilde{X}+1)|P_1|}
\overline{P_1} \otimes_{{C^{\infty}}_{{{\mathcal M}}}}
{\alpha}_{{\mathscr{D} }_{\mathrm{poly}}}^{\nabla^{\mathrm{can}} }
(\overline{X}, \overline{P_2}),$$ for all $X\in {\mathscr{X} }({{\mathcal M}})$, $P_1,P_2\in {{\mathscr{D} }_{\mathrm{poly}}}$, $\overline{X} \in {\mathscr{X} }({{\mathcal M}})[-1]$ the corresponding element of $X$, and $\overline{P_1} \in {\mathrm{tot} }{{\mathscr{D} }_{\mathrm{poly}}}$ (resp. $\overline{P_2} \in {\mathrm{tot} }{{\mathscr{D} }_{\mathrm{poly}}}$) the corresponding element of $P_1$ (resp. $P_2$).
We denote by $\mathrm{pdt}:~~ {\mathrm{tot} }{\mathscr{D} }_{\mathrm{poly}}{\otimes_{{C^{\infty}}_{{{\mathcal M}}}}} {\mathrm{tot} }{\mathscr{D} }_{\mathrm{poly}} \rightarrow {\mathrm{tot} }{\mathscr{D} }_{\mathrm{poly}}$ the ${C^{\infty}}_{{{\mathcal M}}}$-tensor product, which is a morphism of dg modules.
Let $P_1,P_2\in {{\mathscr{D} }_{\mathrm{poly}}}$. We have $$\begin{split}
{\alpha}_{{\mathscr{D} }_{\mathrm{poly}}}^{\nabla^{\mathrm{can}} }
(\overline{X}, \overline{P_1\widetilde{\otimes} P_2}) & = {\alpha}_{{\mathscr{D} }_{\mathrm{poly}}}^{\nabla^{\mathrm{can}} }
(\overline{X}, \overline{P_1}\otimes_{{C^{\infty}}_{{{\mathcal M}}}} \overline{P_2})\\
& \quad =\mathrm{pdt}(
{\alpha}_{{\mathrm{tot} }{\mathscr{D} }_{\mathrm{poly}}{\otimes_{{C^{\infty}}_{{{\mathcal M}}}}} {\mathrm{tot} }{\mathscr{D} }_{\mathrm{poly}}}^{\nabla^{\mathrm{can}}\otimes {\mathrm{id}}+ {\mathrm{id}}\otimes \nabla^{\mathrm{can}}}(\overline{X}, \overline{P_1}\otimes_{{C^{\infty}}_{{{\mathcal M}}}} \overline{P_2})) \\
& \quad \quad = \mathrm{pdt}({\alpha}_{{\mathrm{tot} }{\mathscr{D} }_{\mathrm{poly}}}^{\nabla^{\mathrm{can}} }
(\overline{X}, \overline{P_1})\otimes_{{C^{\infty}}_{{{\mathcal M}}}}\overline{P_2}
+
(-1)^{(\widetilde{X}+1)|P_1|}
\overline{P_1} \otimes_{{C^{\infty}}_{{{\mathcal M}}}}
{\alpha}_{{\mathrm{tot} }{\mathscr{D} }_{\mathrm{poly}}}^{\nabla^{\mathrm{can}} }
(\overline{X}, \overline{P_2})) \\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad
\text{(by Proposition \ref{Atiyahcocyleoftensorproduct})} \\
& \quad \quad \quad = {\alpha}_{{\mathscr{D} }_{\mathrm{poly}}}^{\nabla^{\mathrm{can}} }
(\overline{X}, \overline{P_1})\otimes_{{C^{\infty}}_{{{\mathcal M}}}}\overline{P_2}
+
(-1)^{(\widetilde{X}+1)|P_1|}
\overline{P_1} \otimes_{{C^{\infty}}_{{{\mathcal M}}}}
{\alpha}_{{\mathscr{D} }_{\mathrm{poly}}}^{\nabla^{\mathrm{can}} }
(\overline{X}, \overline{P_2}).
\end{split}$$
Recall that the free Lie algebra bracket ${\textup{\textlbrackdbl ~,~ \textrbrackdbl} }$ of ${{\mathscr{D} }_{\mathrm{poly}}}$ induces a Lie bracket on ${\mathrm{tot} }{{\mathscr{D} }_{\mathrm{poly}}}$, which is also denoted by ${\textup{\textlbrackdbl ~,~ \textrbrackdbl} }$. The induced bracket ${\textup{\textlbrackdbl ~,~ \textrbrackdbl} }$ also preserves ${\mathrm{tot} }{L({\mathscr{D} }_{\mathrm{poly}}^1) }$.
\[bracket\] The Atiyah cocyle ${\alpha}_{{\mathscr{D} }_{\mathrm{poly}}}^{\nabla^{\mathrm{can}}}$ is given by $${\alpha}_{{\mathscr{D} }_{\mathrm{poly}}}^{\nabla^{\mathrm{can}}}
(\overline{X}, \overline{D_1\widetilde{\otimes} \cdots \widetilde{\otimes} D_n})=\textup{\textlbrackdbl}
\overline{X}, \overline{D_1\widetilde{\otimes} \cdots \widetilde{\otimes} D_n}
\textup{\textrbrackdbl},$$ for all $X \in {\mathscr{X} }({{\mathcal M}})$, $D_i\in {{\mathscr{D} }_{\mathrm{poly}}^1}$, $\overline{X} \in {\mathscr{X} }({{\mathcal M}})[-1]$ the corresponding element of $X$, and $\overline{D_1\widetilde{\otimes} \cdots \widetilde{\otimes} D_n} \in {\mathrm{tot} }{{\mathscr{D} }_{\mathrm{poly}}}$ the corresponding element of $D_1\widetilde{\otimes} \cdots \widetilde{\otimes} D_n$.
By Lemma \[inductionlemma\], we have $$\label{derivation}
\begin{split}
{\alpha}_{{\mathscr{D} }_{\mathrm{poly}}}^{\nabla^{\mathrm{can}} }
(\overline{X}, \overline{D_1 \widetilde{\otimes} \cdots \widetilde{\otimes} D_n} )
& =
{\alpha}_{{\mathscr{D} }_{\mathrm{poly}}}^{\nabla^{\mathrm{can}} }
(\overline{X}, \overline{D_1}) \otimes_{{C^{\infty}}_{{{\mathcal M}}}} \overline{D_2} \otimes_{{C^{\infty}}_{{{\mathcal M}}}} \cdots \otimes_{{C^{\infty}}_{{{\mathcal M}}}} \overline{D_n} +
\cdots +
\\
& \quad (-1)^{(\widetilde{X}+1)(|D_1|+ \cdots +|D_{n-1}|)}
\overline{D_1} \otimes_{{C^{\infty}}_{{{\mathcal M}}}} \cdots \otimes_{{C^{\infty}}_{{{\mathcal M}}}} \overline{D_{n-1}} \otimes_{{C^{\infty}}_{{{\mathcal M}}}}{\alpha}_{{\mathscr{D} }_{\mathrm{poly}}}^{\nabla^{\mathrm{can}} }
(\overline{X}, \overline{D_n}),
\end{split}$$ In other words, ${\alpha}_{{\mathscr{D} }_{\mathrm{poly}}}^{\nabla^{\mathrm{can}}}(\overline{X}, \,\cdot \,)$ is a derivation of degree $\widetilde{\overline{X}}=\widetilde{X}+1$ (with respect to the $\widetilde{\otimes}$ product). Clearly, $\textup{\textlbrackdbl}
\overline{X}, \, \cdot \,\textup{\textrbrackdbl}$ is also a derivation of degree $\widetilde{X}+1$. Thus it suffices to prove the $n=1$ case of the statement, i.e., $$\label{Eqt:toproven=1case}
{\alpha}_{{\mathscr{D} }_{\mathrm{poly}}}^{\nabla^{\mathrm{can}} }(\overline{X}, \overline{D})=\textup{\textlbrackdbl}
\overline{X}, \overline{D}\textup{\textrbrackdbl},$$ for all $X\in {\mathscr{X} }({{\mathcal M}})$ and $D \in {{\mathscr{D} }_{\mathrm{poly}}^1}$.
Let $D\in {{\mathscr{D} }_{\mathrm{poly}}^1}$. By definition, we have $$\begin{split}
{\alpha}_{{\mathscr{D} }_{\mathrm{poly}}}^{\nabla^{\mathrm{can}} }
(\overline{X}, \overline{D})
={} &
(-1)^{\widetilde{X}}
(d_H+L_Q)
\nabla^{\mathrm{can}}_{X}
(D) \\
& \quad -
(-1)^{\widetilde{X}}
\nabla^{\mathrm{can}}
({\mathrm{id}}\otimes(d_H+L_Q)
+
L_Q\otimes {\mathrm{id}})(X \widetilde{\otimes} D) \\
={} &
(-1)^{\widetilde{X}}( d_H\nabla^{\mathrm{can}}_{X}
(D)
-
(-1)^{\widetilde{X}}\nabla^{\mathrm{can}}_{X}
d_H(D) )\\
& \quad + (-1)^{\widetilde{X}}
(L_Q\nabla^{\mathrm{can}}_{X}(D)
-
(-1)^{\widetilde{X}}\nabla^{\mathrm{can}}_{X}
L_Q(D)
-
\nabla^{\mathrm{can}}_{[Q,X]}(D)
).
\end{split}$$ We now examine the last two lines:
- By Lemma \[coproduct\], we have $$\begin{split}
d_H\nabla^{\mathrm{can}}_{X}D & = \,d_H(XD) \\
& \quad = \,(-1)^{\widetilde{X}+|D|}(XD\widetilde{\otimes}1-\Delta(XD)-(-1)^{\widetilde{X}+|D|}1\widetilde{\otimes}XD),
\end{split}$$ and $$\begin{split}
(-1)^{\widetilde{X}}\nabla^{\mathrm{can}}_{X}d_H D
& = (-1)^{\widetilde{X}+|D|}\nabla^{\mathrm{can}}_{X}(D\widetilde{\otimes}1-\Delta(D)-(-1)^{|D|}1\widetilde{\otimes}D) \\
& \quad = (-1)^{\widetilde{X}+|D|}(XD\widetilde{\otimes}1+(-1)^{\widetilde{X}|D|}D\widetilde{\otimes}X-\nabla^{\mathrm{can}}_{X}\Delta(D) \\
& \quad\quad -(-1)^{|D|}X\widetilde{\otimes}D -(-1)^{\widetilde{X}+|D|}1\widetilde{\otimes}XD ).
\end{split}$$ Thus by Lemma \[canonicalconnectionandDelta\], we see that $$\begin{split}
d_H\nabla^{\mathrm{can}}_{X}D
-
(-1)^{\widetilde{X}}\nabla^{\mathrm{can}}_{X}d_H D
& = (-1)^{\widetilde{X}}(X\widetilde{\otimes}D-
(-1)^{(\widetilde{X}+1)|D|}D\widetilde{\otimes}X)\\
& \quad = (-1)^{\widetilde{X}}
\textup{\textlbrackdbl}
\overline{X}, D\textup{\textrbrackdbl}.
\end{split}$$
- By direct computation, we have $$\begin{split}
& \quad \, \, L_Q\nabla^{\mathrm{can}}_{X}
D-
(-1)^{\widetilde{X}}\nabla^{\mathrm{can}}_{X}
L_Q D
-
\nabla^{\mathrm{can}}_{[Q,X]}D \\
& =
[Q,XD]-(-1)^{\widetilde{X}}X[Q,D]-[Q,X]D =0.
\end{split}$$
Now Eqt. is clear.
As an immediate consequence, we have
\[bracket2\] The Atiyah cocyle $\alpha^{\nabla^{\mathrm{can}}}_{{L({\mathscr{D} }_{\mathrm{poly}}^1) }}$ is given by $$\alpha^{\nabla^{\mathrm{can}}}_{{L({\mathscr{D} }_{\mathrm{poly}}^1) }}
(\overline{X}, \text{ }
\textup{\textlbrackdbl} \overline{D_1},\cdots , \textup{\textlbrackdbl}
\overline{D_{n-1}}, \overline{D_{n}} \textup{\textrbrackdbl},\cdots \textup{\textrbrackdbl}
)
=
\textup{\textlbrackdbl} \overline{X}, \textup{\textlbrackdbl} \overline{D_1}, \cdots , \textup{\textlbrackdbl}
\overline{D_{n-1}}, \overline{D_{n}} \textup{\textrbrackdbl},\cdots \textup{\textrbrackdbl}
\textup{\textrbrackdbl},$$ for all $X\in {\mathscr{X} }({{\mathcal M}})$ and $D_i\in {{\mathscr{D} }_{\mathrm{poly}}^1}$.
The main result
---------------
Our main result in this section is the following:
\[maintheorem\]
- The natural inclusion map $\theta:~ ({\mathscr{X} }({{\mathcal M}})[-1],L_Q; \, {\alpha}_{{{\mathcal M}}}) \rightarrow ({\mathrm{tot} }{L({\mathscr{D} }_{\mathrm{poly}}^1) }, L_Q+d_H; \, {\textup{\textlbrackdbl ~,~ \textrbrackdbl} })$ is an isomorphism in the homotopy category $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$. Moreover, it is an isomorphism of Lie algebra objects, i.e., the following diagram commutes in $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$: $$\label{maindiagramII}
\xymatrix{
({\mathscr{X} }({{\mathcal M}})[-1],L_Q) \otimes_{{C^{\infty}}_{{{\mathcal M}}}} ({\mathscr{X} }({{\mathcal M}})[-1],L_Q) \ar[r]^>>>>>>>>>>>{{\alpha}_{{{\mathcal M}}}} \ar[d]_{\theta \otimes \theta} &
({\mathscr{X} }({{\mathcal M}})[-1],L_Q)\ar[d]^{\theta}\\
({\mathrm{tot} }{L({\mathscr{D} }_{\mathrm{poly}}^1) },L_Q+d_H)
\otimes_{{C^{\infty}}_{{{\mathcal M}}}}
({\mathrm{tot} }{L({\mathscr{D} }_{\mathrm{poly}}^1) },L_Q+d_H)\ar[r]^>>>>>>{{\textup{\textlbrackdbl ~,~ \textrbrackdbl} }} & ({\mathrm{tot} }L({\mathscr{D} }_{\mathrm{poly}}^1),L_Q+d_H)\\
}$$
- The dg module $({\mathrm{tot} }{{\mathscr{D} }_{\mathrm{poly}}}, L_Q+d_H)$ is the universal enveloping algebra of the Lie algebra object $({\mathscr{X} }({{\mathcal M}})[-1],L_Q; \,{\alpha}_{{{\mathcal M}}} )$, and a Hopf algebra object, in $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$.
Statement $(1)$ follows from Proposition \[betaquasi-isomorphism\], Proposition \[functoriality\] of functoriality of Atiyah classes, and Corollary \[bracket2\].
For Statement $(2)$, we already defined a Hopf algebra structure on ${\mathscr{D} }_{\mathrm{poly}}$ in Section \[Dpoly\]. As structure maps of ${{\mathscr{D} }_{\mathrm{poly}}}$ are compatible with $L_Q$ and $d_H$, it induces a Hopf algebra structure on the dg module $({\mathrm{tot} }{{\mathscr{D} }_{\mathrm{poly}}}, L_Q+d_H)$. By Corollary \[UL2\], the universal enveloping algebra $U({L({\mathscr{D} }_{\mathrm{poly}}^1) }, L_Q, d_H)=({{\mathscr{D} }_{\mathrm{poly}}}, L_Q, d_H)$, in $D({\mathbf{dg}\mathrm{-}\mathbf{mod} })$. Thus the universal enveloping algebra $U({\mathrm{tot} }{L({\mathscr{D} }_{\mathrm{poly}}^1) }, L_Q + d_H)=({\mathrm{tot} }{{\mathscr{D} }_{\mathrm{poly}}}, L_Q+d_H)$, in $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$.
The following corollaries are direct consequences of Theorem \[R1\] and Theorem \[maintheorem\]. They are, respectively, analogues of Corollary $1$ and Theorem $3$ in [@Ramadoss]. The proofs are essential the same, and thus omitted.
\[Rcorollary1\] The following diagram commutes in the homotopy category $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$: $$\label{Rcorollary1diagram}
\xymatrix{
S^{\bullet}(({\mathscr{X} }({{\mathcal M}})[-1], L_Q)) \otimes_{{C^{\infty}}_{{{\mathcal M}}}} ({\mathscr{X} }({{\mathcal M}})[-1], L_Q) \ar[r]^<<<<<{\mu \circ \frac{\widetilde{\omega}}{1-e^{- \widetilde{\omega}}}} \ar[d]_{{\mathrm{hkr} }\otimes \theta} &
S^{\bullet}(({\mathscr{X} }({{\mathcal M}})[-1], L_Q))\ar[d]^{{\mathrm{hkr} }}\\
({\mathrm{tot} }{{\mathscr{D} }_{\mathrm{poly}}},L_Q+d_H)
\widetilde{\otimes}
({\mathrm{tot} }{L({\mathscr{D} }_{\mathrm{poly}}^1) },L_Q+d_H)\ar[r]^<<<<<{\cup} & ({\mathrm{tot} }{{\mathscr{D} }_{\mathrm{poly}}},L_Q+d_H) \, . \\
}$$ Here $\mu$ denotes the natural symmetric product of $S^{\bullet}(({\mathscr{X} }({{\mathcal M}})[-1], L_Q))$, and the map $$\widetilde{\omega}:~~ S^{\bullet}(({\mathscr{X} }({{\mathcal M}})[-1], L_Q)) \otimes_{{C^{\infty}}_{{{\mathcal M}}}} ({\mathscr{X} }({{\mathcal M}})[-1], L_Q) \rightarrow
S^{\bullet}(({\mathscr{X} }({{\mathcal M}})[-1], L_Q)) \otimes_{{C^{\infty}}_{{{\mathcal M}}}} ({\mathscr{X} }({{\mathcal M}})[-1], L_Q)$$ is defined by $$\label{tildeomegaformula}
\widetilde{\omega}( (\overline{X}_1 \odot_{{C^{\infty}}_{{{\mathcal M}}}} \cdots \odot_{{C^{\infty}}_{{{\mathcal M}}}} \overline{X}_n) \otimes_{{C^{\infty}}_{{{\mathcal M}}}}\overline{X}):~~=
\sum_{i=1}^n (-1)^{\Diamond_i}(\overline{X}_1 \odot_{{C^{\infty}}_{{{\mathcal M}}}} \cdots \widehat{\overline{X}}_i \cdots \odot_{{C^{\infty}}_{{{\mathcal M}}}} \overline{X}_n)\otimes_{{C^{\infty}}_{{{\mathcal M}}}}
{\alpha}_{{{\mathcal M}}}(\overline{X}_i, \, \overline{X})$$ for elements $\overline{X}_i, \overline{X} \in {\mathscr{X} }({{\mathcal M}})[-1]$, where we denote by $\odot_{{C^{\infty}}_{{{\mathcal M}}}}$ the ${C^{\infty}}_{{{\mathcal M}}}$-symmetric tensor product, and $\Diamond_i=(\widetilde{X}_{i+1}+\cdots+\widetilde{X}_{n}+n-i)(\widetilde{X}_{i}+1)$.
\[R3\] For any dg module $({\mathfrak{N}}, L_Q)$, the Atiyah class $${\alpha}_{{\mathfrak{N}}}:~~ {\mathscr{X} }({{\mathcal M}})[-1]\otimes_{{C^{\infty}}_{{{\mathcal M}}}}{\mathfrak{N}}\rightarrow {\mathfrak{N}}$$ can be uniquely lifted to a morphism ${\alpha}_{{\mathfrak{N}}}':~~ ({\mathrm{tot} }{{\mathscr{D} }_{\mathrm{poly}}}, L_Q + d_H)\otimes_{{C^{\infty}}_{{{\mathcal M}}}} ({\mathfrak{N}}, L_Q)\rightarrow ({\mathfrak{N}}, L_Q)$ in the homotopy category $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$: $$\label{lifting}
\begin{xy}
(0,25)*+{{\mathscr{X} }({{\mathcal M}})[-1]\otimes_{{C^{\infty}}_{{{\mathcal M}}}}{\mathfrak{N}}}="a";
(0,0)*+{{\mathrm{tot} }{{\mathscr{D} }_{\mathrm{poly}}}\otimes_{{C^{\infty}}_{{{\mathcal M}}}} {\mathfrak{N}}}="b";
(35,25)*+{{\mathfrak{N}}}="c";
{\ar^{\theta \otimes {\mathrm{id}}} "a";"b"};
{\ar@{-->}_{\exists \mathrm{!} \, {\alpha}_{{\mathfrak{N}}}' } "b";"c"};
{\ar^>>>>>>>>>{{\alpha}_{{\mathfrak{N}}}} "a";"c"};
\end{xy}$$
Hopf algebras arising from Lie pairs {#Section6}
====================================
In this part, we study the relation between two Hopf algebra constructions — one arising from dg manifolds and the other from Lie pairs. Let $M$ be a smooth manifold. Throughout this section, we denote by $R={C^{\infty}}(M, {{\mathbb K } })$, the ring of ${{\mathbb K } }$-valued smooth functions on $M$.
The Atiyah class ${\alpha}_{L/A}$ of a Lie pair $(L,A)$
-------------------------------------------------------
The notion of *Lie pair* is a natural framework encompassing a range of diverse geometric contexts including complex manifolds, foliations, and $\mathfrak{g}$-manifolds (that is, manifolds endowed with an action of a Lie algebra $\mathfrak{g}$). By a *Lie pair*, we mean an inclusion $A \hookrightarrow L$ of Lie ${{\mathbb K } }$-algebroids over a smooth manifold $M$. Recall that a *Lie* ${{\mathbb K } }$-algebroid is a ${{\mathbb K } }$-vector bundle $L \rightarrow M$, whose space of sections is endowed with a Lie bracket ${[\text{ }, \text{ }]}$, together with a bundle map $\rho: ~~ L \rightarrow T_M\otimes_{{{\mathbb R} }} {{\mathbb K } }$ called *anchor* such that $\rho:~~ \Gamma(L) \rightarrow {\mathscr{X} }(M)\otimes_{{{\mathbb R} }}{{\mathbb K } }$ is a morphism of Lie algebras and $$[X, \, fY]=f[X, \,Y]+(\rho(X)f)Y,$$ for all $X, Y \in \Gamma(L)$ and $f\in R$.
Given a Lie pair $(L, A)$, let $B=L/A$ be the quotient vector bundle. First of all, there is a short exact sequence $$\label{Liepairshortexactsequence}
0\rightarrow A \xrightarrow{i} L \xrightarrow{pr} B \rightarrow 0.$$ Choosing a splitting $j:~~ B \rightarrow L$, we can identify $L=A\oplus B$.
Note that $B$ is naturally an $A$-module, i.e., $$\label{Bottconnection}
{\nabla^{\mathrm{Bott}} }_{a} b=pr([a, j({b})]),$$ where $a \in \Gamma(A)$, $b\in \Gamma(B)$. This natural flat $A$-connection on $B$ is known as the Bott connection [@Bott; @C-S-X2].
Recall the definition of Atiyah class of the Lie pair $(L,A)$ introduced in [@C-S-X2]. Let $\nabla:~~ \Gamma(L) \otimes_{{{\mathbb K } }} \Gamma(B) \rightarrow \Gamma(B)$ be a smooth $L$-connection on $B$ which extends the Bott connection. Then there associated a $1$-cocyle ${\alpha}^{\nabla}_{B} \in \Gamma(A^{\vee}\otimes B^{\vee}\otimes{\mathrm{End}}(B))$[^2] in the Chevalley-Eilenberg complex of the Lie algebroid $A$ valued in the $A$-module $B^{\vee}\otimes{\mathrm{End}}(B)$, which is called the Atiyah cocyle of the Lie pair $(L,A)$, given by $$\label{LiepairAtiyahcocycleformula}
{\alpha}^{\nabla}_B(a, b)e:~~={\nabla^{\mathrm{Bott}} }_{a}\nabla_{j(b)}e
-\nabla_{j(b)}{\nabla^{\mathrm{Bott}} }_{a}e-\nabla_{[a,j(b)]}e,$$ for all $a \in \Gamma(A)$, and $b, e \in \Gamma(B)$. The cohomology class $$\label{LiepairAtiyahclassdefinition}
{\alpha}_B=[{\alpha}^{\nabla}_B]\in H^1_{\mathrm{CE}}(A, B^{\vee}\otimes{\mathrm{End}}(B))$$ does not depend on the choice of $j$ and $\nabla$. It is called Atiyah class of the Lie pair $(L,A)$.
It is well known that the Lie algebroid structure on $A $ defines a dg manifold $(A[1], d_{\mathrm{CE}}^{A})$. Here we understand that the space of smooth functions on $A[1]$ is ${C^{\infty}}_{A[1]}: =\Omega(A)= \Gamma({\wedge^{\bullet}A^\vee})$, and $d_{\mathrm{CE}}^{A}: \Omega(A)\to \Omega(A)$ is the standard Chevalley-Eilenberg differential. In what follows, $\Omega(A)$ refers to this particular dg algebra.
Let $(B^{!}, d_{\mathrm{CE}}^{B^{!}})$ be the dg vector bundle over $(A[1],d_{\mathrm{CE}}^{A})$ which is the pull back of $B$: $$\xymatrix{
(B^{!}, d_{\mathrm{CE}}^{B^{!}}) \ar[r] \ar[d] & B \ar[d]\\
(A[1], d_{\mathrm{CE}}^{A}) \ar[r] & M \, .\\
}$$ In other words, $$\Gamma(B^{!})=\Gamma(B)\otimes_{R}\Omega(A)=\Gamma(B \otimes \wedge^{\bullet}A^{\vee}),$$ and $d_{\mathrm{CE}}^{B^{!}}$ is the Chevalley-Eilenberg differential of $A$ with respect to the $A$-module $B$.
As the Lie pair Atiyah class $$\begin{split}
{\alpha}_{B} & \in H^1_{\mathrm{CE}}(A, B^{\vee}\otimes {\mathrm{End}}(B)) \\
& \quad \simeq
{\mathrm{Hom }}_{\mathrm{H}(\Omega(A)\mathbf{-mod})}
((\Gamma(B^{!})[-1], d_{\mathrm{CE}}^{B^{!}})\otimes_{\Omega(A)}
(\Gamma(B^{!})[-1], d_{\mathrm{CE}}^{B^{!}})
, (\Gamma(B^{!})[-1], d_{\mathrm{CE}}^{B^{!}})),
\end{split}$$ it can be regarded as a morphism $${\alpha}_{B}:~~ (\Gamma(B^{!})[-1], d_{\mathrm{CE}}^{B^{!}})\otimes_{\Omega(A)}
(\Gamma(B^{!})[-1], d_{\mathrm{CE}}^{B^{!}})
\rightarrow (\Gamma(B^{!})[-1], d_{\mathrm{CE}}^{B^{!}})$$ in the homology category $\mathrm{H}(\Omega(A)\mathbf{-mod})$ of dg modules over $\Omega(A)$.
The Hopf algebra $(D_{\mathrm{poly}}^{\bullet}(L/A), d_H)$
----------------------------------------------------------
It is known [@Xu] that the universal enveloping algebra $U(L)$ of a Lie algebroid $L$ admits a cocommutative coassociative coproduct $\delta:~~ U(L) \rightarrow U(L)\otimes_R U(L)$, which is defined on generators as follows: $\delta(f)=f\otimes_R 1=1 \otimes_R f, \, \forall f \in R$ and $\delta(p)=p\otimes_R 1 + 1 \otimes_R p, \, \forall p \in \Gamma(L)$.
Now given the Lie pair $(L,A)$, consider the quotient $D_{poly}^1(B):=\frac{U(L)}{U(L)\Gamma(A)}$. It is straightforward to see that the coproduct on $U(L)$ induces a coproduct $\delta:~~ D_{\mathrm{poly}}^1(B) \rightarrow D_{\mathrm{poly}}^1(B)\otimes_R D_{\mathrm{poly}}^1(B)$.
Moreover, Chen, Stiénon and Xu defined a complex $(D_{\mathrm{poly}}^{\bullet}(B), d_H)$ associated with the Lie pair $(L,A)$ [@C-S-X], where $D_{\mathrm{poly}}^n(B)=\otimes_{R}^n D_{\mathrm{poly}}^1(B)$, and $d_H:~~ D_{\mathrm{poly}}^n(B) \rightarrow D_{\mathrm{poly}}^{n+1}(B)$ is the Hochschild differential given by $$\label{HochschildofLiepair}
\begin{split}
d_H(p_1 \otimes_R \cdots \otimes_R p_n)& =1\otimes_R p_1 \cdots \otimes_R p_n - \delta(p_1) \otimes_R p_2 \cdots \otimes_R p_n + p_1 \otimes_R \delta(p_2) \cdots \otimes_R p_n - \cdots \\
& \quad \quad +(-1)^n p_1 \otimes_R \cdots \otimes_R \delta(p_n) +(-1)^{n+1}p_1\otimes_R \cdots \otimes_R p_n \otimes_R 1,
\end{split}$$ for $p_1, \cdots , p_n \in D_{\mathrm{poly}}^1(B)$. The object $(D_{\mathrm{poly}}^{\bullet}(B),d_H)$ is a differential graded associative algebra, the product being the tensor product $\otimes_{R}$. It turns out that $(D_{\mathrm{poly}}^{\bullet}(B),d_H)$ is a Hopf algebra object in the category of complexes of $A$-modules [@C-S-X]. Denote by $(L(D_{\mathrm{poly}}^{1}(B)),d_H; \, {\textup{\textlbrackdbl ~,~ \textrbrackdbl} })$ the free differential graded Lie algebra spanned by $D_{\mathrm{poly}}^{1}(B)$ inside $(D_{\mathrm{poly}}^{\bullet}(B),d_H)$.
The Fedosov dg Lie algebroid $({{\mathcal F}}, L_Q)$
----------------------------------------------------
Given a Lie pair $(L,A)$ and a torsion free $L$-connection $\nabla:~~\Gamma(L)\otimes_{{{\mathbb K } }} \Gamma(B) \rightarrow \Gamma(B)$ which is compatible with the Bott $A$-connection. Stiénon and Xu defined a dg manifold $$({{{\mathcal M}}}, {Q}):~~=(L[1]\oplus B, {d_{L}^{\nabla^{\lightning}}}),$$ which is called the Fedosov dg manifold. There are two ways to construct $({{{\mathcal M}}}, {Q})$, one is by way of Fedosov’s iteration method, the other is by the PBW map. We will briefly recall the second one. The reader is referred to [@S-X Section $2$] for more details of Fedosov dg manifolds (see also [@L-S; @B-S-X]). Let $$\label{exp}
{\mathrm{pbw} }^{\nabla,j}:~~ \Gamma(SB) \rightarrow D_{\mathrm{poly}}^1(B)$$ be the PBW map introduced in [@LG-S-X1; @LG-S-X2]. Here $SB$ denotes the symmetric tensor algebra of $B$. There is a canonical flat $L$-connection $$\label{canonical connection on Dpoly1B}
\begin{split}
\nabla^{\mathrm{can}}:~~ \Gamma(L)\otimes_{{{\mathbb K } }} & D_{\mathrm{poly}}^1(B)\rightarrow D_{\mathrm{poly}}^1(B) \, , \\
& \nabla^{\mathrm{can}}_{l}u:~~ =l \cdot u
\end{split}$$ for all $l\in \Gamma(L)$ and $u\in D_{\mathrm{poly}}^1(B)$. Pulling back $\nabla^{\mathrm{can}}$ via the map ${\mathrm{pbw} }^{\nabla,j}$, we obtain a flat $L$-connection on $\Gamma(SB)$: $$\label{lighting connection on SB}
\begin{split}
\nabla^{\lightning}:~~ & \Gamma(L)\otimes_{{{\mathbb K } }}\Gamma(SB) \rightarrow \Gamma(SB)\, , \\\nabla^{\lightning}_{l}s:~~ = & ({\mathrm{pbw} }^{\nabla,j})^{-1} \circ \nabla^{\mathrm{can}}_{l}\circ {\mathrm{pbw} }^{\nabla,j}(s)
\end{split}$$ for $l \in \Gamma(L)$ and $s \in \Gamma(SB)$. The associated Chevalley-Eilenberg differential on the cochain complex $\Gamma(\hat{S}B^\vee)\otimes_{R}\Omega(L)$ is denoted by ${d_{L}^{\nabla^{\lightning}}}$.
Consider the graded manifold ${{{\mathcal M}}}=L[1]\oplus B$ whose ring of functions ${C^{\infty}}({{{\mathcal M}}})=\Gamma(\hat{S}B^\vee)\otimes_{R}\Omega(L)$. Thus ${Q}:={d_{L}^{\nabla^{\lightning}}}$ defines a homological vector field on ${{{\mathcal M}}}$. The pair $({{{\mathcal M}}},{Q})$ is called the Fedosov dg manifold. We work on $({{{\mathcal M}}},{Q})$ throughout this section.
Clearly, ${{{\mathcal M}}}$ can be thought of as a vector bundle over $L[1]$. We denote by $\pi:~~ {{{\mathcal M}}}\rightarrow L[1]$ the natural projection. It can be verified that $\pi:~({{{\mathcal M}}},{Q})\to (L[1],d_{\mathrm{CE}}^L)$ is a dg map (see [@B-V] and [@S-X] for details). Hence, the integrable distribution $\ker \pi_{\ast} \subset T_{{{{\mathcal M}}}}$ is a dg foliation of $(T_{{{{\mathcal M}}}}, L_{{Q}})$ and thus a dg Lie algebroid over $({{{\mathcal M}}},{Q})$. We call it the Fedosov dg Lie algebroid, and denote it by $({{\mathcal F}}, L_Q)$.
As ${{\mathcal F}}\subset T_{{{{\mathcal M}}}}$ is a dg foliation, we can consider the dg sub-algebra ${{\mathcal D}}_{{{\mathcal M}}}({{\mathcal F}})\subset {{\mathcal D}}_{{{\mathcal M}}}$ of fiberwise differential operators of the fibration $({{\mathcal M}}, Q)\xrightarrow{\pi} (L[1], d_{\mathrm{CE}}^{L}) $, which is the universal enveloping algebra of the dg Lie algebroid $({{\mathcal F}}, L_Q)$. Similarly, we have dg sub-objects ${{\mathscr{T} }_{\mathrm{poly}}}({{\mathcal F}})\subset {{\mathscr{T} }_{\mathrm{poly}}}$, ${\mathscr{D} }_{\mathrm{poly}}({{\mathcal F}})\subset {\mathscr{D} }_{\mathrm{poly}}$, $({\mathrm{tot} }L({\mathscr{D} }_{\mathrm{poly}}^1({{\mathcal F}})), L_Q+d_H; \, {\textup{\textlbrackdbl ~,~ \textrbrackdbl} })\subset ({\mathrm{tot} }L({\mathscr{D} }_{\mathrm{poly}}^1), L_Q+d_H ; \, {\textup{\textlbrackdbl ~,~ \textrbrackdbl} })$ etc.
The Atiyah class ${\alpha}_{{{\mathcal F}}}$
--------------------------------------------
In general, given a dg foliation $({{\mathcal F}}, L_Q)$ on a dg manifold $({{\mathcal M}}, Q)$, one has the notion of ${{\mathcal F}}$-Atiyah class of dg modules $({\mathfrak{N}}, L_Q)$ over $({{\mathcal M}},Q)$. For details, see the notion of Atiyah class of dg Lie algebroids in Mehta-Stiénon-Xu’s paper [@M-S-X]. The definition is completely analogous to the Atiyah class of a dg manifolds with everywhere ${\mathscr{X} }({{\mathcal M}})$ replaced by $\Gamma({{\mathcal F}})$.
Here we consider the particular Fedosov dg Lie algebroid $({{\mathcal F}},L_Q)$ and the particular dg module $(\Gamma({{\mathcal F}}[-1]),L_Q)$ over $({{\mathcal M}},Q)$. In this situation, we have a canonical ${{\mathcal F}}$-connection $\nabla^{{{\mathcal F}}}$ on the $R$-module $\Gamma({{\mathcal F}})$ characterized by the relation (see [@L-S-X2] for more details) $$\nabla^{{{\mathcal F}}}_{X}Y=0,\qquad
\forall X, Y \in \Gamma(B)\subset \Gamma({{\mathcal F}}).$$ The operator $\nabla^{{{\mathcal F}}}$ can be also regarded as a connection on $\Gamma({{\mathcal F}})[-1]$.
Accordingly, the ${{\mathcal F}}$-Atiyah cocycle associated to the ${{\mathcal F}}$-connection $\nabla^{{{\mathcal F}}}$ on $\Gamma({{\mathcal F}})[-1]$ is a dg morphism $${\alpha}^{\nabla^{{{\mathcal F}}}}_{\Gamma({{\mathcal F}}[-1])}:~~ (\Gamma({{\mathcal F}})[-1],L_Q)\otimes_{{C^{\infty}}_{{{{\mathcal M}}}}} (\Gamma({{\mathcal F}})[-1],L_Q)
\rightarrow
(\Gamma({{\mathcal F}})[-1],L_Q)$$ $${\alpha}^{\nabla^{{{\mathcal F}}}}_{\Gamma({{\mathcal F}}[-1])}(\overline{X},\overline{Y})=(-1)^{\widetilde{X}}(L_Q(\nabla_X \overline{Y})-\nabla_{[Q,X]}\overline{Y}-(-1)^{\widetilde{X}}\nabla_{X}L_Q(\overline{Y})).$$ Here the notations $\overline{X}, \overline{Y} \in \Gamma({{\mathcal F}})[-1]$ denote the elements that correspond to $X, Y \in \Gamma({{\mathcal F}})$, respectively.
The ${{\mathcal F}}$-Atiyah class of $(\Gamma({{\mathcal F}})[-1],L_Q)$, denoted simply by ${\alpha}_{{{\mathcal F}}}$, is the morphism in the homology category $\mathrm{H}({\mathbf{dg}\mathrm{-}\mathbf{mod} })$ induced by ${\alpha}^{\nabla^{{{\mathcal F}}}}_{\Gamma({{\mathcal F}})[-1]}$. Here and in the rest of this part, ${\mathbf{dg}\mathrm{-}\mathbf{mod} }$ refers to the category of dg modules over the Fedosov dg manifold $({{{\mathcal M}}}, {Q})$.
Similar to Theorem \[Thm:AtiyahdefinesLiebracket\], the Atiyah class ${\alpha}_{{{\mathcal F}}}$ defines a Lie bracket on $(\Gamma({{\mathcal F}})[-1] , L_Q)$ in $\mathrm{H}({\mathbf{dg}\mathrm{-}\mathbf{mod} })$.
Let $\theta:~~ (\Gamma({{\mathcal F}})[-1],L_Q) \rightarrow ({\mathrm{tot} }L({\mathscr{D} }_{\mathrm{poly}}^1({{\mathcal F}})),L_Q+d_H )$ be the natural inclusion map. We have the following analogue of Theorem \[maintheorem\]:
\[betaforfoliation\] Let $({{\mathcal F}}, L_Q)$ be the Fedosov dg Lie algebroid as above.
- The map $\theta:~~ (\Gamma({{\mathcal F}})[-1],L_Q) \rightarrow ({\mathrm{tot} }L({\mathscr{D} }_{\mathrm{poly}}^1({{\mathcal F}})),L_Q+d_H )$ is an isomorphism in the homotopy category $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$ of dg modules over $({{{\mathcal M}}}, {Q})$. Moreover, it is an isomorphism of Lie algebra objects, i.e., the following diagram commutes in $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$: $$\label{foliationdiagram}
\xymatrix{
(\Gamma({{\mathcal F}})[-1],L_Q) \otimes_{{C^{\infty}}_{{{{\mathcal M}}}}} (\Gamma({{\mathcal F}})[-1],L_Q) \ar[r]^>>>>>>>>>>>>>>>>>{{\alpha}_{{{\mathcal F}}}} \ar[d]_{\theta \otimes \theta} & (\Gamma({{\mathcal F}})[-1],L_Q) \ar[d]^{\theta}\\
({\mathrm{tot} }L({\mathscr{D} }_{\mathrm{poly}}^1({{\mathcal F}})),L_Q+d_H )
\otimes_{{C^{\infty}}_{{{{\mathcal M}}}}} ({\mathrm{tot} }L({\mathscr{D} }_{\mathrm{poly}}^1({{\mathcal F}}),L_Q+d_H ) \ar[r]^>>>>>>{{\textup{\textlbrackdbl ~,~ \textrbrackdbl} }} & ({\mathrm{tot} }L({\mathscr{D} }_{\mathrm{poly}}^1({{\mathcal F}}),L_Q+d_H )\, .\\
}$$
- The dg module $({\mathrm{tot} }{\mathscr{D} }_{\mathrm{poly}}({{\mathcal F}}), L_Q+d_H)$ is the universal enveloping algebra of the Lie algebra object $(\Gamma({{\mathcal F}})[-1],L_Q; \,{\alpha}_{{{\mathcal M}}} )$, and a Hopf algebra object, in $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$.
Relation between ${\alpha}_{L/A}$ and ${\alpha}_{{{\mathcal F}}}$
-----------------------------------------------------------------
We have an obvious embedding map $\iota: A[1]\hookrightarrow {{{\mathcal M}}}$. It can be proved that $\iota:~~ (A[1], d_{\mathrm{CE}}^{A})\hookrightarrow ({{{\mathcal M}}}, {Q})$ is a morphism of dg manifolds as well as a quasi-isomorphism [@B-V; @S-X].
The following lemma is implied from the construction of $({{\mathcal F}}, L_Q)$:
The diagram formed by natural maps $$\xymatrix{
(B^{!}, d_{\mathrm{CE}}^{B^{!}}) \ar[r]^{\iota} \ar[d] & ({{\mathcal F}}, L_Q) \ar[d]\\
(A[1], d_{\mathrm{CE}}^{A}) \ar[r]^{\iota} & ({{{\mathcal M}}}, {Q})\\
}$$ is a pull back diagram of dg vector bundles. In other words, $\iota^{\ast}({{\mathcal F}}, L_Q)=(B^{!}, d_{\mathrm{CE}}^{B^{!}})$ or $$(\Gamma(B^{!}), d_{\mathrm{CE}}^{B^{!}})=(\Gamma({{\mathcal F}}), L_Q)\otimes_{{C^{\infty}}_{{{{\mathcal M}}}}}(\Omega(A),d_{\mathrm{CE}}^{A}).$$
By this fact, we can treat $(\Gamma(B^{!}), d_{\mathrm{CE}}^{B^{!}})$ as a dg module over the Fedosov dg manifold $({{{\mathcal M}}}, {Q})$.
The following fact is taken from [@L-S-X2 Proposition $1.20$].
\[leftface\] The restriction map $\iota^{\ast}:~~ (\Gamma({{\mathcal F}}), L_Q) \rightarrow
(\Gamma(B^{!}), d_{\mathrm{CE}}^{B^{!}})$ is a quasi-isomorphism of dg modules. The induced map $$\begin{split}
& \iota^{\ast}:~~ ({\mathrm{Hom }}_{{C^{\infty}}_{{{{\mathcal M}}}}}^{\bullet}(\Gamma({{\mathcal F}})\otimes_{{C^{\infty}}_{{{{\mathcal M}}}}} \Gamma({{\mathcal F}}), \Gamma({{\mathcal F}})), L_Q) \\
& \quad\quad\quad \rightarrow
( {\mathrm{Hom }}_{R}(\Gamma(B)\otimes_{R}\Gamma(B), \Gamma(B))\otimes_{R}\Omega(A), d^B_{\mathrm{CE}})
\end{split}$$ sends ${\alpha}_{{{\mathcal F}}}^{\nabla^{{{\mathcal F}}}}$ to ${\alpha}_{B}^{\nabla}$. Consequently, the following diagram commutes in the homotopy category $\mathrm{\Pi}({\mathbf{dg}\mathrm{-}\mathbf{mod} })$: $$\xymatrix{
(\Gamma({{\mathcal F}})[-1],L_Q)\otimes_{{C^{\infty}}_{{{{\mathcal M}}}}} (\Gamma({{\mathcal F}})[-1],L_Q)
\ar[r]^>>>>>>{{\alpha}_{{{\mathcal F}}}} \ar[d]_{\iota^{\ast} \otimes \iota^{\ast}} &
(\Gamma({{\mathcal F}})[-1],L_Q) \ar[d]^{\iota^{\ast}}\\
(\Gamma(B^{!})[-1], d_{\mathrm{CE}}^{B^{!}})
\otimes_{\Omega(A)} (\Gamma(B^{!})[-1], d_{\mathrm{CE}}^{B^{!}}) \ar[r]^>>>>>{{\alpha}_{B}} & (\Gamma(B^{!})[-1], d_{\mathrm{CE}}^{B^{!}}) \, . \\
}$$
Relation between ${\mathscr{D} }_{\mathrm{poly}}({{\mathcal F}})$ and $D_{\mathrm{poly}}^{\bullet}(L/A)$
--------------------------------------------------------------------------------------------------------
By the construction of ${{\mathcal F}}$, we have the following facts (see also [@B-S-X Page $13$-$14$]): $$\label{Fexpression}
\begin{split}
\Gamma({{\mathcal F}}) & =\mathrm{Der}_{\Omega(L)}({C^{\infty}}_{{{\mathcal M}}}, {C^{\infty}}_{{{\mathcal M}}})=\mathrm{Der}_{\Omega(L)}(\Gamma(\hat{S}B^{\vee})\otimes_{R}\Omega(L), {C^{\infty}}_{{{\mathcal M}}}) \\
& =\Gamma(B)\otimes_{R}{C^{\infty}}_{{{\mathcal M}}}=\Gamma(B\otimes \wedge^{\bullet}L^{\vee} \otimes \hat{S}B^{\vee}),
\end{split}$$ and $$\label{D(F)expression}
{{\mathcal D}}_{{{\mathcal M}}}({{\mathcal F}})=\Gamma(SB)\otimes_{R}{C^{\infty}}_{{{\mathcal M}}}=\Gamma(SB \otimes \wedge^{\bullet}L^{\vee}\otimes \hat{S}B^{\vee}).$$
As the the PBW map $${\mathrm{pbw} }^{\nabla,j}:~~ \Gamma(SB) \rightarrow D_{\mathrm{poly}}^1(B)$$ is an isomorphism of $L$-modules, the induced map $${\mathrm{pbw} }^{\nabla, j}\otimes_{R} {\mathrm{id}}:~~
{{\mathcal D}}_{{{\mathcal M}}}({{\mathcal F}}) \rightarrow D_{\mathrm{poly}}^1(B) \otimes_{R} {C^{\infty}}_{{{\mathcal M}}}$$ is an isomorphism of graded ${C^{\infty}}_{{{\mathcal M}}}$-modules. The isomorphism ${\mathrm{pbw} }^{\nabla, j}\otimes_{R} {\mathrm{id}}$ transports the differential $L_Q$ on ${{\mathcal D}}_{{{\mathcal M}}}({{\mathcal F}})$ to a differential $d_{\mathrm{can}}$ on $D_{\mathrm{poly}}^1(B) \otimes_{R} {C^{\infty}}_{{{\mathcal M}}}$. Thus we have the isomorphism of dg modules: $$\label{D(F)toD(B)}
{\mathrm{pbw} }^{\nabla, j}\otimes_{R} {\mathrm{id}}:~~
({{\mathcal D}}_{{{\mathcal M}}}({{\mathcal F}}), L_Q) \rightarrow (D_{\mathrm{poly}}^1(B) \otimes_{R} {C^{\infty}}_{{{\mathcal M}}}, d_{\mathrm{can}}).$$
We will use the same notation $d_{\mathrm{CE}}^{B}$ to denote the Chevalley-Eilenberg differential on $D_{\mathrm{poly}}^1(B)\otimes_{R}\Omega(A)$ associated to the $A$-module structure of $D_{\mathrm{poly}}^1(B)$. Let ${\mathrm{id}}\otimes_{R}\,\iota^{\ast}:~~ D_{\mathrm{poly}}^1(B)\otimes_{R}{C^{\infty}}_{{{\mathcal M}}} \rightarrow
D_{\mathrm{poly}}^1(B)\otimes_{R}\Omega(A)$ be the restriction map.
We then define a map $I$ which is composed by the horizontal and the vertical arrow shown as in the the following commutative diagram of dg modules over $({{\mathcal M}}, Q)$ — $$\label{Imap}
\begin{xy}
(0,25)*+{({\mathrm{tot} }{\mathscr{D} }_{\mathrm{poly}}({{\mathcal F}}), L_Q+d_H)}="a";
(70,25)*+{(D_{\mathrm{poly}}^{\bullet}(B)\otimes_{R}{C^{\infty}}_{{{\mathcal M}}}, d_{\mathrm{can}}+ d_H \otimes_{R} {\mathrm{id}})}="b";
(70,0)*+{(D_{\mathrm{poly}}^{\bullet}(B)\otimes_{R}\Omega(A), d_{\mathrm{CE}}^{B} + d_H \otimes_{R} {\mathrm{id}}) \, .}="c";
{\ar^-{{\mathrm{pbw} }^{\nabla, j}\otimes_{R}{\mathrm{id}}} "a";"b"};
{\ar_-{ {\mathrm{id}}\otimes_{R}\,\iota^{\ast} } "b";"c"};
{\ar^>>>>>>>>>>>>>>>>>>>>>>>{I} "a";"c"};
\end{xy}$$
The following facts are taken from [@B-S-X Proposition $3.13$ and Lemma $3.14$]:
- the horizontal map ${{\mathrm{pbw} }^{\nabla, j}\otimes_{R}{\mathrm{id}}}$ is an isomorphism of dg modules;
- the vertical map ${\mathrm{id}}\otimes_{R}\,\iota^{\ast}$ is a quasi-isomorphism of dg modules.
Thus $I$ is a quasi-isomorphism of dg modules.\
The canonical $L$-connection $\nabla^{\mathrm{can}}$ (see Eqt. ) induces a Chevalley-Eilenberg differential $\Breve{d}_{L}^{\, \nabla^{\mathrm{can}}}$ on $D_{\mathrm{poly}}^1(B)\otimes_{R}\Omega(L)$. Note that we can identify $D_{\mathrm{poly}}^1(B)\otimes_{R}{C^{\infty}}_{{{\mathcal M}}}$ with $ (D_{\mathrm{poly}}^1(B)\otimes_{R}\Omega(L))\otimes_{\Omega(L)}{C^{\infty}}_{{{\mathcal M}}}$. However, the differential $d_{\mathrm{can}}$ on $D_{\mathrm{poly}}^1(B)\otimes_{R}{C^{\infty}}_{{{\mathcal M}}}$ does not equal to $\Breve{d}_{L}^{\, \nabla^{\mathrm{can}}}\otimes_{\Omega(L)}{\mathrm{id}}+ {\mathrm{id}}\otimes_{\Omega(L)} Q$ under this identification. The best we can say is that the operations $d_{\mathrm{can}}$ and $\Breve{d}_{L}^{\, \nabla^{\mathrm{can}}}\otimes_{\Omega(L)}{\mathrm{id}}+ {\mathrm{id}}\otimes_{\Omega(L)} Q$ share the same restrictions to the $D_{\mathrm{poly}}^1(B)\otimes_{R}\Omega(A)$-component, which is $d_{\mathrm{CE}}^{B}$. The reader is referred to [@B-S-X Lemma $3.14$] for more details.
The next proposition can be verified directly.
\[rightface\] The following diagram commutes in the homotopy category $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$: $$\label{rightfacediagram}
\xymatrix{
\otimes_{{C^{\infty}}_{{{\mathcal M}}}}^{2}({\mathrm{tot} }L({\mathscr{D} }_{\mathrm{poly}}^1({{\mathcal F}})), L_Q+d_H) \ar[r]^>>>>>>>>>>>{{\textup{\textlbrackdbl ~,~ \textrbrackdbl} }} \ar[d]_{I^{\otimes 2} } &
({\mathrm{tot} }L({\mathscr{D} }_{\mathrm{poly}}^1({{\mathcal F}})), L_Q+d_H) \ar[d]^{I}\\
\otimes_{\Omega(A)}^{2}(L(D_{\mathrm{poly}}^{1}(B))\otimes_{R}\Omega(A), d_{\mathrm{CE}}^{B}+ d_H)
\ar[r]^>>>>>{{\textup{\textlbrackdbl ~,~ \textrbrackdbl} }} & (L(D_{\mathrm{poly}}^{1}(B))\otimes_{R}\Omega(A), d_{\mathrm{CE}}^{B}+ d_H) \, . \\
}$$ Here in the lower left and right conners, the notation $d_H$ means $d_H \otimes_{R} {\mathrm{id}}$.
0.5cm
Conclusion — the big diagram
----------------------------
\
Let $$\beta:~~ (\Gamma(B^{!})[-1], d_{\mathrm{CE}}^{B^{!}})
\rightarrow
(L(D_{\mathrm{poly}}^{1}(B))\otimes_{R}\Omega(A), d_{\mathrm{CE}}^{B}+d_H)$$ be the natural inclusion map.
\[bigdiagram\] The following diagram commutes in the homotopy category $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$:
$$\label{CUBEDIAGRAM}
\begin{tiny}
\xymatrixcolsep{-0.3pc}
\xymatrix{
\otimes_{{C^{\infty}}_{{{\mathcal M}}}}^{2}(\Gamma({{\mathcal F}}[-1],L_Q)\ar[rd]^{(\iota^{\ast})^{\otimes 2}}
\ar[rr]^{\theta^{\otimes 2}}
\ar[dd]^{\alpha_{{{\mathcal F}}}}&&
\otimes_{{C^{\infty}}_{{{\mathcal M}}}}^{2}({\mathrm{tot} }L({\mathscr{D} }_{\mathrm{poly}}^1({{\mathcal F}})),L_Q+d_H )\ar[rd]^{I^{\otimes 2}} \ar[dd]_<<<<<<<{\textup{\textlbrackdbl}\; , \; \textup{\textrbrackdbl}}|(.5)\hole\\
&\otimes_{\Omega(A)}^{2}(\Gamma(B^{!})[-1],d_{\mathrm{CE}}^{B^{!}})\ar[rr]^>>>>>>>>>>>>{\beta^{\otimes 2}}\ar[dd]_<<<<<<<{\alpha_{B}} &&
\otimes_{\Omega(A)}^{2}(L(D_{\mathrm{poly}}^{1}(B)){\otimes_{R}}\Omega(A), d_{\mathrm{CE}}^{B}+d_H) \ar[dd]_{\textup{\textlbrackdbl}\; , \; \textup{\textrbrackdbl}}
\\
(\Gamma({{\mathcal F}})[-1],L_Q)\ar[rd]^{\iota^{\ast}}\ar[rr]^{\theta}|(.43)\hole && ({\mathrm{tot} }L({\mathscr{D} }_{\mathrm{poly}}^1({{\mathcal F}})),L_Q+d_H ) \ar[rd]^{I} \hole\\
&(\Gamma(B^{!})[-1],d_{\mathrm{CE}}^{B^{!}})\ar[rr]^>>>>>>>>>>>>>>>>>{\beta}&&
(L(D_{\mathrm{poly}}^{1}(B)){\otimes_{R}}\Omega(A), d_{\mathrm{CE}}^{B}+d_H) \, . \\
}
\end{tiny}$$
Moreover, the map $\beta$ in the front lower edge is an isomorphism in $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$.
It is obvious that the top face and the bottom face of the above cube diagram are both commutative. The back face commutes due to Theorem \[betaforfoliation\]. The left face commutes due to Proposition \[leftface\], while the right face is due to \[rightface\]. Note that $\iota^{\ast}$ and $I$ are isomorphisms in the category $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$. Thus the front face commutes, and the map $\beta$ in the front lower edge is an isomorphism in $\Pi({\mathbf{dg}\mathrm{-}\mathbf{mod} })$.
As an application of Theorem \[betaforfoliation\] and Theorem \[bigdiagram\], we recover the following result of Chen-Stiénon-Xu [@C-S-X]:
Let $\beta:~~ \Gamma(B[-1]) \rightarrow L(D_{\mathrm{poly}}^{1}(B))$ be the natural inclusion map.
- The map $\beta$ is an isomorphism in the derived category $D^b(\mathcal{A})$ of $A$-modules. Moreover, it is an isomorphism of Lie algebra objects, i.e., the following diagram commutes in $D^b(\mathcal{A})$: $$\xymatrix{
\Gamma(B[-1]) \otimes_{R} \Gamma(B[-1]) \ar[r]^>>>>>{\beta \otimes \beta} \ar[d]_{{\alpha}_{B}} & L(D_{\mathrm{poly}}^{1}(B))
\otimes_{R} L(D_{\mathrm{poly}}^{1}(B)) \ar[d]^{{\textup{\textlbrackdbl ~,~ \textrbrackdbl} }}\\
\Gamma(B[-1]) \ar[r]^>>>>>>>>>>>>>>>{\beta} & L(D_{\mathrm{poly}}^{1}(B)) \, .\\
}$$
- The complex of $A$-modules $(D_{\mathrm{poly}}^{\bullet}(B), d_H)$ is the universal enveloping algebra object of the Lie algebra object $(\Gamma(B[-1]), {\alpha}_{B})$, and a Hopf algebra object, in $D^b(\mathcal{A})$.
Relation with Ramadoss’s work {#Section7}
=============================
Ramadoss studied the algebraic Atiyah classes of algebraic vector bundles over a field ${{\mathbb K } }$ of characteristic zero [@Ramadoss]. We will describe, without proofs, the relation between some of Ramadoss’s results and ours, in the case ${{\mathbb K } }={{\mathbb C} }$.
Let $(X, {{\mathcal O}}_{X})$ be a smooth scheme over ${{\mathbb C} }$, and $E$ an algebraic vector bundle over $X$. The algebraic Atiyah class ${\alpha}(E) \in {\mathrm{Ext }}^1_{{{\mathcal O}}_X}(T_X \otimes_{{{\mathcal O}}_X} E,E)$ is the extension class of the jet sequence in the category of ${{\mathcal O}}_{X}$-module: $$\label{ShortII}
0\rightarrow E
\xrightarrow{i}
D_{X}^{\leqslant 1} \underset{{{\mathcal O}}_{X}}{\otimes}E
\xrightarrow{j}
T_{X}\otimes_{{{\mathcal O}}_{X}}E
\rightarrow 0.$$ We denote by ${\alpha}_{X}$ the algebraic Atiyah class of the algebraic tangent bundle $T_X$. We also have the objects $(D_{\mathrm{poly}}^{\bullet}, d_H; \, \otimes_{{{\mathcal O}}_{X}})$ and $(L(D_{\mathrm{poly}}^1), d_H; \, {\textup{\textlbrackdbl ~,~ \textrbrackdbl} })$ for the scheme $(X, {{\mathcal O}}_X)$ [@Ramadoss], which are algebraic versions of ${{\mathscr{D} }_{\mathrm{poly}}}$ and ${L({\mathscr{D} }_{\mathrm{poly}}^1) }$. The following theorem is obtained by Ramadoss [@Ramadoss Theorem 2]:
\[Ramadossresult\] The diagram $$\label{Ramadossmaindiagram}
\xymatrix{
T_X [-1] \otimes_{{{\mathcal O}}_X} T_X [-1] \ar[r]^>>>>>>>{{\alpha}_{X}} \ar[d]_{\theta \otimes \theta} &
T_X [-1] \ar[d]^{\theta}\\
L(D_{\mathrm{poly}}^1)
\otimes_{{{\mathcal O}}_X}
L(D_{\mathrm{poly}}^1)\ar[r]^>>>>>{{\textup{\textlbrackdbl ~,~ \textrbrackdbl} }} & L(D_{\mathrm{poly}}^1) \, \\
}$$ commutes in the derived category $D^{+}({{\mathcal O}}_X)$ of bounded below complexes of coherent ${{\mathcal O}}_X$-modules, the map $\theta: T_{M}[-1] \rightarrow L(D_{\mathrm{poly}}^1)$ being the natural inclusion map.
Denote by $X^{sm}$ the underlying smooth manifold of $X$, $X^{an}$ the complex manifold. Let $({{\mathcal X}}=T^{0,1}(X^{sm})[1],Q=\overline{ \partial})$ be the dg manifold associated to the Dolbeault resolution of the sheaf of holomorphic functions ${{\mathcal O}}^{an}_X$, i.e., the support of ${{\mathcal X}}$ is $X^{sm}$, and the sheaf of smooth functions $$({C^{\infty}}_{{{\mathcal X}}}, Q):~~=( \Omega_{X^{sm}}^{0,\bullet}\, , \, \overline{\partial}).$$
Let $E\to X$ be an algebraic vector bundle over $X$. Denote by $E^{sm}$ (resp. $E^{an}$) the underlying smooth (resp. holomorphic) vector bundle over $X^{sm}$ (resp. $X^{an}$). Consider the dg vector bundle $({{\mathcal E}}, L_Q) \rightarrow ({{\mathcal X}}, Q)$ associated with the Dolbeault resolution of $E^{an}$. In other words, $\Gamma({{\mathcal E}})=\Omega_{X^{sm}}^{0,\bullet}\otimes_{C^\infty(X^{sm},{{\mathbb C} })}\Gamma(E^{sm})$, and $L_Q=\overline{\partial}$.
The correspondence $(X, {{\mathcal O}}_X)$ to $({{\mathcal X}}, Q)$ can be lifted to a functor from the category $\mathbf{Sch}_{{{\mathbb C} }}$ of smooth schemes over ${{\mathbb C} }$ to the category of dg manifolds $\mathbf{dg-Mfd}$. The correspondence $E\to X$ to $({{\mathcal E}}, L_Q)\to ({{\mathcal X}}, Q)$ can also be lifted to functor $c:~~ D^{+}({{\mathcal O}}_X) \rightarrow \Pi({C^{\infty}}_{{{\mathcal X}}}\mathbf{-mod})$.
We are now ready to state the relation between algebraic Atiyah classes and dg Atiyah classes — they coincide at the lower right corner in the following diagram: $$\begin{xy}
(0,15)*+{{\alpha}(E)\in{\mathrm{Ext }}^1_{{{\mathcal O}}_X}(T_X \otimes_{{{\mathcal O}}_{X}} E,E)}="d";
(60,15)*+{{\alpha}_{{{\mathcal E}}}\in\Hom_{\mathrm{H}({C^{\infty}}_{{{\mathcal X}}})}({\mathscr{X} }({{\mathcal X}})[-1]\otimes_{{C^{\infty}}_{{{\mathcal X}}}} {{\mathcal E}}, {{\mathcal E}})}="a";
(0,0)*+{{\mathrm{Hom }}_{D^{+}({{\mathcal O}}_X)}(T_X[-1]\otimes_{{{\mathcal O}}_{X}}E,E)}="b";
(60, 0)*+{{\mathrm{Hom }}_{\Pi({C^{\infty}}_{{{\mathcal X}}})}({\mathscr{X} }({{\mathcal X}})[-1]\otimes_{{C^{\infty}}_{{{\mathcal X}}}} {{\mathcal E}}, {{\mathcal E}}) \, .}="c";
{\ar "a"; "c"};
{\ar^-{c} "b"; "c"};
{\ar^{\simeq} "d"; "b" };
\end{xy}$$ Here the bottom horizontal map comes from the functor $c:~~ D^{+}({{\mathcal O}}_X) \rightarrow \Pi({C^{\infty}}_{{{\mathcal X}}}\mathbf{-mod})$, and the vertical map on the right hand side comes from the natural functor $\mathrm{H}({C^{\infty}}_{{{\mathcal X}}}\mathbf{-mod}) \rightarrow \Pi({C^{\infty}}_{{{\mathcal X}}}\mathbf{-mod})$.
Moreover, the functor $c:~~ D^{+}({{\mathcal O}}_X) \rightarrow \Pi({C^{\infty}}_{{{\mathcal X}}}\mathbf{-mod})$ sends Ramadoss’s commutative diagram to the following commutative diagram: $$\xymatrix{
({\mathscr{X} }({{\mathcal X}})[-1],L_Q) \otimes_{{C^{\infty}}_{{{\mathcal X}}}} ({\mathscr{X} }({{\mathcal X}})[-1],L_Q) \ar[r]^>>>>>>>>>>>{{\alpha}_{{{\mathcal X}}}} \ar[d]_{\theta \otimes \theta} &
({\mathscr{X} }({{\mathcal X}})[-1],L_Q)\ar[d]^{\theta}\\
({\mathrm{tot} }{L({\mathscr{D} }_{\mathrm{poly}}^1) },L_Q+d_H)
\otimes_{{C^{\infty}}_{{{\mathcal X}}}}
({\mathrm{tot} }{L({\mathscr{D} }_{\mathrm{poly}}^1) },L_Q+d_H)\ar[r]^>>>>>{{\textup{\textlbrackdbl ~,~ \textrbrackdbl} }} & ({\mathrm{tot} }L({\mathscr{D} }_{\mathrm{poly}}^1),L_Q+d_H) \, .\\
}$$ In fact, it is an instance of the commutative diagram where one considers the particular dg manifold $({{\mathcal X}}, Q)$. So, Theorem \[maintheorem\] can be considered as the dg analogue to Theorem \[Ramadossresult\] (Theorem $2$ in [@Ramadoss]).
Also, among our results, Theorem \[R1\], Corollary \[Rcorollary1\] and Corollary \[R3\], are, respectively, analogous to Ramadoss’s Theorem $1$, Corollary $1$ and Theorem $3$ in [@Ramadoss].
Appendix {#Section8}
========
This part is devoted to a technical proposition. Recall that elements in ${{\mathcal D}}_{{{\mathcal M}}}$ are of the form $$\label{Eqt:Dform}D=X_1 \circ \cdots \circ X_n,$$ where $X_1,\cdots,X_n \in {\mathscr{X} }({{\mathcal M}})$. In what follows, we will also treat the above $D$ as in ${\mathscr{D} }_{\mathrm{poly}}^1$, $X_i$ as in ${\mathscr{T} }_{\mathrm{poly}}^1$ and $\circ$ as the Gerstenhaber product (see Eqt. ).
Recall also the coproduct $\Delta$ defined in Eqt. and the commutator ${\textup{\textlbrackdbl ~,~ \textrbrackdbl} }$ defined in Eqt. .
\[Deltaformula\]For $D\in {\mathscr{D} }_{\mathrm{poly}}^1$ given in Eqt. , we have $$\label{Eqt:Deltaexpressioninbrackets}
\Delta(D)~=\sum_{\sigma \in {{\mathbb S } }}
(-1)^{\tau(\sigma)}
\textup{\textlbrackdbl}X_1 \circ X_{\sigma(2)} \circ \cdots \circ X_{\sigma(p)}, \, X_{\sigma(p+1)} \circ \cdots \circ X_{\sigma(n)}
\textup{\textrbrackdbl} \, .$$
Here ${{\mathbb S } }$ is the set of permutations $\sigma:~~\{1, \cdots ,n\} \rightarrow \{1, \cdots ,n\}$ such that $\sigma(1)=1$, $1<\sigma(2)< \cdots < \sigma(p)$ and $\sigma(p+1)< \cdots < \sigma(n)$, where $1\leqslant p\leqslant n$. The number $\tau(\sigma)=\tau_1(\sigma)+\tau_2(\sigma)$, where $\tau_{1}(\sigma)=\widetilde{X}_{\sigma(p+1)}+\cdots + \widetilde{X}_{\sigma(n)}$, and $\tau_2(\sigma)$ is the Koszul sign determined by the ordinary degrees $\widetilde{X}_i$, i.e., $$X_2\odot \cdots \odot X_n = (-1)^{\tau_2(\sigma)} X_{\sigma(2)} \odot \cdots \odot X_{\sigma(n)}.$$ Here $\odot$ is the symmetric product in $S^{\bullet}({\mathscr{X} }({{\mathcal M}}))$.
We need the following lemmas.
\[HochschildcoboundaryandGerstenhaberproduct\] We have $$d_H(D\circ E)=d_H(D)\circ E +(-1)^{|D|-1}D \circ d_H(E), \quad \forall D, E \in {{\mathscr{D} }_{\mathrm{poly}}}.$$
This lemma is proved for the case of Hochschild cohomology complexes of associative algebras in [@Gerstenhaber]. It can be easily generalized to the graded case as above.
\[Gerstenhaberproductandcupproduct\] We have $$X\circ \textup{\textlbrackdbl}
D, \, E
\textup{\textrbrackdbl}=
\textup{\textlbrackdbl}
X \circ D, \, E
\textup{\textrbrackdbl}+
(-1)^{|D||E|+1}
\textup{\textlbrackdbl}
X \circ E, \, D
\textup{\textrbrackdbl},$$ for $X\in {{\mathscr{T} }_{\mathrm{poly}}^1}$ and $D, \,E \in {{\mathscr{D} }_{\mathrm{poly}}}$.
By definitions of the cup product $\widetilde{\otimes}$ in Eqt. and the Gerstenhaber product $\circ$ in Eqt. , it is easy to verify that $$X\circ(D\widetilde{\otimes}E)=(X\circ D)\widetilde{\otimes}E +(-1)^{(|X|+1)|D|}D\widetilde{\otimes}(X\circ E),$$ for $X\in {{\mathscr{T} }_{\mathrm{poly}}^1}$ and $D, \,E \in {{\mathscr{D} }_{\mathrm{poly}}}$. Similarly, we have $$X\circ(E\widetilde{\otimes}D)=(X\circ E)\widetilde{\otimes}D +(-1)^{(|X|+1)|E|}E\widetilde{\otimes}(X\circ D).$$ Therefore, $$\begin{split}
X\circ \textup{\textlbrackdbl}
D, \, E
\textup{\textrbrackdbl} & =X \circ (D\widetilde{\otimes} E- (-1)^{|D||E|}E \widetilde{\otimes}D) \\
& \quad = \textup{\textlbrackdbl}
X \circ D, \, E
\textup{\textrbrackdbl}+
(-1)^{(|X|+1)|D|}
\textup{\textlbrackdbl}
D, \, X \circ E
\textup{\textrbrackdbl} \\
& \quad\quad =
\textup{\textlbrackdbl}
X \circ D, \, E
\textup{\textrbrackdbl}+
(-1)^{(|X|+1)|D|+(|X|+|E|-1)|D|+1}
\textup{\textlbrackdbl}
X\circ E, \, D
\textup{\textrbrackdbl} \\
& \quad\quad\quad =
\textup{\textlbrackdbl}
X \circ D, \, E
\textup{\textrbrackdbl}+
(-1)^{|D||E|+1}
\textup{\textlbrackdbl}
X\circ E, \, D
\textup{\textrbrackdbl}.
\end{split}$$
We now give the
We adopt an inductive approach. For the $n=1$ case, we have $$\begin{split}
\Delta(X_1) & = X_1\widetilde{\otimes}1+(-1)^{\widetilde{X}_1} 1\widetilde{\otimes}X_1 \\
& \quad = X_1\widetilde{\otimes}1-(-1)^{\widetilde{X}_1+1} 1\widetilde{\otimes}X_1 =
\textup{\textlbrackdbl}
X_1, \, 1
\textup{\textrbrackdbl}.
\end{split}$$ Suppose that Eqt. is proved for some $n\geqslant 1$. Consider $D=X_1 \circ \cdots \circ X_{n+1} =X_1 \circ D'$, where $D'=X_2 \circ \cdots \circ X_{n+1}$. Then we have $$\begin{split}
\Delta(D)=\Delta(X_1 \circ D') & =
\textup{\textlbrackdbl}
X_1 \circ D', \, 1
\textup{\textrbrackdbl}
-
(-1)^{\widetilde{X}_1+ \widetilde{D}'+1}
d_H(X_1 \circ D') \quad\quad\quad\quad\,\, \text{(by Lemma \ref{coproduct})}\\
& \quad =
\textup{\textlbrackdbl}
X_1 \circ D', \, 1
\textup{\textrbrackdbl}
+(-1)^{\widetilde{D}'}X_1 \circ d_H(D') \quad\quad\quad\quad\quad\quad \text{(by Lemma \ref{HochschildcoboundaryandGerstenhaberproduct}} \\
& \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad \,\,\text{and $d_H(X_1)=0$} )\\
& \quad\quad =
\textup{\textlbrackdbl}
X_1 \circ D', \, 1
\textup{\textrbrackdbl}
+X_1 \circ
(
\Delta(D')-
\textup{\textlbrackdbl}
D', \, 1
\textup{\textrbrackdbl}
).
\quad \quad \quad \, \text{(by Lemma \ref{coproduct}) }
\end{split}$$ By the induction assumption, we have $$\begin{split}
\Delta(D')- \textup{\textlbrackdbl}
D', \, 1
\textup{\textrbrackdbl} & = \sum_{\sigma' \in {{\mathbb S } }'}
(-1)^{\tau(\sigma')}
\textup{\textlbrackdbl}X_2 \circ X_{\sigma'(3)} \circ \cdots \circ X_{\sigma'(p+1)}, \, X_{\sigma'(p+2)} \circ \cdots \circ X_{\sigma'(n+1)}
\textup{\textrbrackdbl} \, ,
\end{split}$$ where ${{\mathbb S } }'$ is the set of permutations $\sigma':~~\{2, \cdots, n+1\} \rightarrow \{2, \cdots, n+1\}$ such that $\sigma'(2)=2$, $2< \sigma'(3) < \cdots <\sigma'(p+1)$ and $\sigma'(p+2) < \cdots < \sigma'(n+1)$ for some $1 \leqslant p \leqslant n-1$.
By Lemma \[Gerstenhaberproductandcupproduct\], we have $$\begin{split}
& X_1 \circ \textup{\textlbrackdbl}X_2 \circ X_{\sigma'(3)} \circ \cdots \circ X_{\sigma'(p+1)}, \, X_{\sigma'(p+2)} \circ \cdots \circ X_{\sigma'(n+1)}\textup{\textrbrackdbl} \\
& \quad \quad = \textup{\textlbrackdbl}X_1 \circ X_2 \circ X_{\sigma'(3)} \cdots \circ X_{\sigma'(p+1)}, \, X_{\sigma'(p+2)} \circ \cdots \circ X_{\sigma'(n+1)}\textup{\textrbrackdbl} \\
& \quad\quad\quad\quad +(-1)^{(A+1)(B+1)+1}
\textup{\textlbrackdbl}
X_1 \circ X_{\sigma'(p+2)} \circ \cdots \circ X_{\sigma'(n+1)}, \,
X_2 \circ X_{\sigma'(3)} \circ \cdots \circ X_{\sigma'(p+1)}
\textup{\textrbrackdbl},
\end{split}$$ where $A=\widetilde{X}_2 + \widetilde{X}_{\sigma'(3)}+ \cdots +\widetilde{X}_{\sigma'(p+1)}$ and $B= \widetilde{X}_{\sigma'(p+2)}+ \cdots + \widetilde{X}_{\sigma'(n+1)}$.
Thus $$\begin{split}
\Delta(D) & = \textup{\textlbrackdbl}
X_1 \circ D', \, 1
\textup{\textrbrackdbl}+
\sum_{\sigma' \in {{\mathbb S } }'}
(-1)^{\tau(\sigma')}
X_1 \circ \textup{\textlbrackdbl}X_2 \circ X_{\sigma'(3)} \circ \cdots \circ X_{\sigma'(p+1)}, \, X_{\sigma'(p+2)} \circ \cdots \circ X_{\sigma'(n+1)}
\textup{\textrbrackdbl} \\
& \quad =\textup{\textlbrackdbl}
X_1 \circ D', \, 1
\textup{\textrbrackdbl}+
\sum_{\sigma' \in {{\mathbb S } }'}
(-1)^{\tau(\sigma')}
\textup{\textlbrackdbl}X_1 \circ X_2 \circ X_{\sigma'(3)} \circ \cdots \circ X_{\sigma'(p+1)}, \, X_{\sigma'(p+2)} \circ \cdots \circ X_{\sigma'(n+1)}
\textup{\textrbrackdbl}\\
& \quad \quad + \sum_{\sigma' \in {{\mathbb S } }'}
(-1)^{\tau(\sigma')+A+B+AB}
\textup{\textlbrackdbl}
X_1 \circ X_{\sigma'(p+2)} \circ \cdots \circ X_{\sigma'(n+1)}, \,
X_2 \circ X_{\sigma'(3)} \circ \cdots \circ X_{\sigma'(p+1)}
\textup{\textrbrackdbl}.
\end{split}$$
For each $\sigma' \in {{\mathbb S } }' $ as above, we define the following two new permutations of $\{1, \cdots , n+1\}$.
- The permutation $\widetilde{\sigma}':~~ \{1, \cdots, n+1\} \rightarrow \{1, \cdots, n+1\}$ which satisfies $\widetilde{\sigma}'(1)=1$ and $\widetilde{\sigma}'|_{\{2, \cdots, n+1\}}=\sigma'$. It is clear that $\tau(\widetilde{\sigma}')=\tau(\sigma')=\tau_1(\sigma')+\tau_2(\sigma')=B+\tau_2(\sigma')$.
- The permutation $\hat{\sigma}':~~ \{1, \cdots, n+1\} \rightarrow \{1, \cdots, n+1\}$ which satisfies $$\hat{\sigma}'(1)=1,$$ $$\hat{\sigma}'(2)=\sigma'(p+2), \, \cdots, \, \hat{\sigma}'(n+1-p)=\sigma'(n+1),$$ $$\hat{\sigma}'(n+2-p)=2, \, \cdots, \, \hat{\sigma}'(n+1)=\sigma'(p+1).$$ It is clear that $\tau_1(\hat{\sigma}')=A$ and $\tau_2(\hat{\sigma}')=AB+\tau_2(\sigma')$, thus $\tau(\hat{\sigma}')=A+AB+\tau_2(\sigma')$, and $(-1)^{\tau(\sigma')+A+B+AB}=(-1)^{B+\tau_2(\sigma')+A+B+AB}=(-1)^{\tau(\hat{\sigma}')}$.
Combining the above facts, we have $$\begin{split}
\Delta(D) & =\textup{\textlbrackdbl}
X_1 \circ D', \, 1
\textup{\textrbrackdbl}\\
& \quad + \sum_{\{\widetilde{\sigma}':~~ \sigma' \in{{\mathbb S } }'\}}
(-1)^{\tau(\widetilde{\sigma}')}
\textup{\textlbrackdbl}X_1 \circ X_2 \circ X_{\widetilde{\sigma}'(3)} \circ \cdots \circ X_{\widetilde{\sigma}'(p+1)}, \, X_{\widetilde{\sigma}'(p+2)} \circ \cdots \circ X_{\widetilde{\sigma}'(n+1)}
\textup{\textrbrackdbl}\\
& \quad \quad + \sum_{\{\hat{\sigma}':~~ \sigma' \in {{\mathbb S } }' \}}
(-1)^{\tau(\hat{\sigma}')}
\textup{\textlbrackdbl}
X_1 \circ X_{\hat{\sigma}'(2)} \circ \cdots \circ X_{\hat{\sigma}'(n+1-p)}, \,
X_2 \circ X_{\hat{\sigma}'(n+3-p)} \circ \cdots \circ X_{\hat{\sigma}'(n+1)}
\textup{\textrbrackdbl}.
\end{split}$$
Observe that the union $\{ id_{\{1, \cdots, n+1 \}}\}\cup \{\widetilde{\sigma}':~~ \sigma' \in{{\mathbb S } }'\} \cup \{\hat{\sigma}':~~ \sigma' \in {{\mathbb S } }' \}$ is exactly the set of all permutations $\sigma:~~\{1, \cdots ,n+1\} \rightarrow \{1, \cdots ,n+1\}$ such that $\sigma(1)=1$, $1< \sigma(2) < \cdots < \sigma(p)$ and $\sigma(p+1)< \cdots < \sigma(n+1)$ for some $p\geqslant 1$. Thus Eqt. is proved for the $n+1$ case.
[^1]: Research partially supported by NSFC grant 11471179.
[^2]: Here $A^{\vee}$ denotes the ${{\mathbb K } }$-dual of the vector bundle $A$.
|
---
abstract: 'In this paper we compare the results of the MACRO detector at Gran Sasso, that is the largest neutrino telescope operated before the year 2k, with theoretical predictions of the neutrino emission from some promising targets, such as blazars and GRB’s. In particular we propose a new statistical method, justified by the maximum entropy principle, for assessing a model independent upper limit to the differential flux of neutrinos inferred from the measured up-ward going muon flux. This comparison confirms that the acceptance of a detector of up-ward going muons should be of the order of $1\;\mathrm{km}^2$, in order to challenge the present theoretical estimates of possible neutrino production in both galactic and extra-galactic objects.'
bibliography:
- 'vulc01.bib'
- 'waxman.bib'
- 'auriemmag.bib'
- 'halzen.bib'
---
**HIGH ENERGY NEUTRINOS**
GIULIO AURIEMMA
Università degli Studi della Basilicata, Potenza, Italy
and
INFN Sezione di Roma, Rome, Italy
Introduction {#sect:intro}
============
Large detectors of high energy ($>1\;\mathrm{GeV}$) neutrinos, like MACRO in the Gran Sasso Laboratory[@Ambrosio:2000yx] or SuperKamiokande [@Fukuda:1999pp] have been operated in the last decade of the past century, but no extrasolar astrophysical source has been detected. The occasion of this paper is related to the decommissioning of the MACRO detector at Gran Sasso, that is the largest neutrino telescope operated before the year 2k. The proposal for the detector was accepted in November 1984 [@DeMarzo:1984cw]. Physics runs started in March 1989, as soon as the first supermodule of the detector has been built [@Calicchio:1988uw; @Ahlen:1993pe] The full detector [@Ahlen:1993kp] has been run from April, 1994 to December 31, 2000 when it has been decommissioned. A sample of a 1000 up-ward going muons has allowed the measurement of the flux of geophysical neutrinos collected up to September, 1999 [@Ambrosio:1998wu], producing also interesting result [@Ambrosio:2001je]. The purpose of this paper is to compare the upper limits obtained by MACRO with current models of neutrino emission from astrophysical objects. First in the following §\[sect.detec\] we will discuss the physics of neutrino detection in detectors of up-ward going muons. In particular in §\[subsect:neutcs\] we discuss the extrapolation of the differential neutrino cross section to the range of energies of interest for astrophysical neutrino detection, in §\[subsect:conv\] the conversion of neutrinos into muons in the rock and finally in §\[subsect:uplim\] we give the rationale for a new statistical method for assessing a model independent upper limit to the differential flux of neutrinos inferred from the measured up-ward going muon flux. In §\[sect:results\] we present some results obtained from the MACRO survey of neutrino induced up-ward going muons. In §\[subsect:crab\] we present the constraint to the neutrino emission from the Crab nebula, in §\[subsect:neutralino\] that for a possible neutralino decay in the Galactic center, in §\[subsect:blazars\] the constraint to diffuse neutrino background which can be inferred from the cumulative search of coincidences with blazars and in §\[subsect:grbs\] we discuss the same topic for the GRB deriving an upper limit to the diffuse flux of neutrinos correlated with GRB . Finally in §\[sect:concl\] we discuss the prospect for detection of neutrinos in the next future by the planned detectors.
Neutrino detection {#sect.detec}
==================
Extrapolation of neutrino cross section {#subsect:neutcs}
---------------------------------------
The detection process for muon neutrinos and anti neutrinos is the Charged Current (CC) scattering illustrated in Fig. \[fig:parton-mod\]
![The neutrino nucleon deep inelastic scattering in the parton model.[]{data-label="fig:parton-mod"}](auriemma_fig02.eps){width="50.00000%"}
with double differential cross section $$\begin{aligned}
\label{eq:neut1a}
&\displaystyle \frac{d^2\sigma_{\nu (\bar \nu)}}{dx\,dy} =
\frac{G_F^2\,(s-m_N^2)}{2\pi}\,\frac{M_W^4}{(Q^2+M_W^2)^2}
\nonumber \\ &\times
\left\{\dfrac{y^2}{2}\,2 x
F_1^{\nu(\bar \nu)}(x,Q^2)+\left(1-y\right)\,F_2^{\nu(\bar \nu)}(x,Q^2)+
\pm\left(y-\dfrac{y^2}{2}\right)
\,x F_3^{\nu(\bar \nu)}(x,Q^2)\right\}\end{aligned}$$ where $y=1-E_\mu/E_\nu$ is the fraction of neutrino energy lost in the Lab system, $x=Q^2/2m_N(E_\nu-E_\mu)$ the fraction of parton momentum carried by the struck parton, and $Q^2=x y (s-m_N^2)$ the square of the transferred four momentum. The last term is positive for $\nu$’s and negative for $\bar \nu$’s. The three functions $F_1,F_2$ and $F_3$ are the structure functions (SF’s) of the nucleon. In the framework of the quark parton model (QPM) the parton are identified with quarks, and the SF’s can be expressed in terms of the probability density functions (PDF) of each quark types . The PDF’s are the functions $q(x,Q^2)$ where $q=u,d,s,c,b,t$ and the relative anti-quarks, which assign the probability that a quark carries a momentum fraction $x$. In particular we have for an isoscalar target $$\begin{aligned}
F_2^{\nu N(\bar\nu N)} & = & x \left\{u+\bar u+ d+d\bar d+2 s+2c
+2b +2t\right\}\nonumber \\ x F_3 ^{\nu N(\bar\nu N)}& = & x
\left\{u-\bar u+ d-d\bar d\pm 2 s\mp 2c \pm 2b \mp
2t\right\}\nonumber \end{aligned}$$ At very high energies the SF’s $F_2$ and $2\,x\,F_1$, are equal, but if the transverse momentum of the $W$ is not negligible, they are related by the equation $$\label{eq:neut1z}
2\,x\,F_1(x,Q^2)=F_2(x,Q^2)\;\frac{1+4 m_N^2 x^2/Q^2}{1+R(x,Q^2)}$$ where $R(x,Q^2)=\sigma_L/\sigma_T$ is the ratio of the longitudinal to transversal cross section of the $W$.
![Physical integration domain over which Eq.(\[eq:neut1a\]) must be integrated for a given energy of the neutrino.[]{data-label="fig:q2range"}](auriemma_fig03.eps){width="50.00000%"}
The structure functions $F_2$ and $x F_3$ and the function $R$ have been measured experimentally up to $E_\nu\approx 500\;\mathrm{GeV}$ by the CCFR/NuTeV collaboration [@Yang:2001xc; @Yang:2000ju; @Bodek:2001hy; @McDonald:2001vv]. To calculate the neutrino cross sections at higher energies, as we are interested in this paper, the SF’s should be extrapolated. The propagator term in Eq. (\[eq:neut1a\]) ${M_W^4}/{(Q^2+M_W^2)^2}\to 0$ for $Q^2\gg M_W^2$, therefore the cross section at any energy will vanish for $Q^2\gtrsim
\max[xy(s-m_N^2),M_W^2]$. However the double differential cross section of Eq (\[eq:neut1a\]) must be integrated to obtain the differential cross section $d\sigma/dy$ for a given $y$ in the interval $ Q^2/y(s-m_N^2)\le x\le 1$. If the minimum detectable energy for the muon in the detector is $E_{th}$ the lower integration limit is $x_{min}\approx Q^2/2 m_N E_\nu$. In Fig. \[fig:q2range\] we have reported the area of the plane $\log_{10}(1/x),Q^2$ corresponding to $E_\nu=10^6\;\mathrm{GeV}$ where Eq. (\[eq:neut1a\]) must be integrated, compared with the area where the SF’s have been measured.
![Differential neutrino cross section at $E_\nu=425\;\mathrm{GeV}$ measured by CCFR experiment. The fit is the differential cross section obtained in this paper applying CTEQ5 PDF’s.[]{data-label="fig:diffcs-ccfr-425"}](auriemma_fig04.eps){width="50.00000%"}
In the same plot of Fig.\[fig:q2range\], we have reported the lines $W^2=Q^2(1/x-1)=4\,m_h^2$ where $m_h$ are the masses of the heavy quarks $c,b$ and $t$. Above these lines the production of the heavy quarks will give an important contribution to the cross section. In fact it is well established that above $\approx
100\;\mathrm{GeV}$ the contribution of the quark production to the total cross section is of the order of 20% [@Bazarko:1995tt]. From the plot of Fig. \[fig:q2range\] we are forced to conclude that the experimental knowledge of the physics underlying the extrapolation of the cross section is, up to now, rather incomplete. In the frame work of quantum cromodynamics (QCD), the gauge field theory which describes the strong interactions of colored quarks and gluons, the cross sections can be calculated perturbatively in ascending order of the strong coupling constant $\alpha_s$. In practice this corresponds to the solution of a set of partial differential equations $\partial
F_i(x,Q^2)/\partial\ln(Q^2)$ (the DGLAP equations) at fixed $x$, starting from the knowledge of the SF’s $F_i(x,Q^2)$ with $i=1,2,3$ at a given lower value $Q^2_0$ (see e.g. Sterman G. 1999 and references therein). This method has been checked, using the measured differential cross sections. For example we report in Fig.\[fig:diffcs-ccfr-425\] the fit to the measured differential cross section obtained using PDF’s calculated by the CTEQ5 collaboration [@Lai:1999wy], with evolution equations resulting from NLO analysis from a starting scale $Q_0^2=1\;\mathrm{GeV}^2$. It is evident from this figure that the agreement is reasonable but not perfect [@Kretzer:2001mb].
Nevertheless the extrapolation at higher energies of the neutrino cross section is still subjected to a non negligible uncertainty because, as is shown in Fig. \[fig:q2range\], the domain of integration at higher energies extends to smaller and smaller values of $x$. Therefore, even if the evolution of SF’s at fixed $x$ to higher $Q^2$ can be calculated with the DGLAP equations, still we lack of information about the SF’s at low x values. Various estimates, based on different physical assumption [@Frichter:1995mx; @Ralston:1996bb; @Hill:1997iw; @Gluck:1998js; @Kwiecinski:1999bk], as shown in Fig. \[fig:csplot\] disagree by factors up to 2-4 for $E_\nu\gtrsim 10,000\;\mathrm{GeV}$. The experimental point has been obtained by the H1 Collaboration [@Ahmed:1994fa].
![Various extrapolations of the total cross section for isoscalar target and equal mixture of $\nu_\mu$ and $\bar\nu_\mu$. Solid line is the baseline estimate adopted in the following of this paper (see text.). Finely dotted line is based on the small $x$ structure functions measured at HERA, dash-dotted on the dynamical parton model, which is practically coincident with coarsely dotted line, dashed on the Regge theory for $x<10^{-5}$ (see text). []{data-label="fig:csplot"}](auriemma_fig05.eps){width="60.00000%"}
The higher values of the cross section is given by extrapolation of the SF’s for $x< 10^{-5}$ based on the Regge theory [@Donnachie:1998gm; @Berezinsky:2001hf].
In the same Fig. \[fig:csplot\] we have reported the cross section of the reaction $e^- p\to \nu X$ by the H1 experiment at HERA (Ahmed 1994) which seems to be in better agreement with the lower cross section estimate. In the rest of this paper we will take as a baseline extrapolation of the cross section the more conservative one, shown as a solid line in Fig.\[fig:csplot\], based on the CTEQ5 PDF’s.
Evaluation of conversion probability {#subsect:conv}
------------------------------------
The probability $P_{\mu\nu}(E_\nu)$ of detecting a muon with energy $\geq E_{th}$ in the detector, originated by a neutrino with energy $E_\nu$, is given, in the continuous slowing down approximation, by the integral [@Auriemma:1988ag] $$\label{eq:neut2a}
P_{\mu\nu}(E_\nu)=\frac{1}{m_N}\, \int_0^{1-E_{th}/E_\nu}\,\left[a+b\,(1-y)E_\nu\right]^{-1}\,\int_0^1
\frac{d^2\sigma_{\nu}}{dx\,dy}\,dy\,dx$$ where $m_N$ is the mass of the nucleon, $a$ and $b$ are the coefficient of the muon energy loss $-{dE_\mu}/{dz}=a(E_\mu)+b(E_\mu)\,E_\mu$ [@Groom:2000in].
![Conversion probability for various $E_{th}$ values.[]{data-label="fig:pmunu"}](auriemma_fig07.eps){width="60.00000%"}
Model independent upper limits {#subsect:uplim}
------------------------------
The differential up-ward going muon flux in the detector will be $
d\Phi_\mu/dE_\nu=P_{\mu\nu}\, d\Phi_\nu/dE_\nu$ where $d\Phi_\nu/dE_\nu$ is the spectral distribution of the neutrino flux. However what is measured in an up-ward going muons neutrino telescope, like MACRO, is only the integrated muon flux $$\label{eq:neut1d}
\Phi_\mu^{obs}(>E_{th})=\int_{E_{th}}^\infty\,P_{\mu\nu}\,
\frac{d\Phi_\nu}{dE_\nu}\,dE_\nu$$ The conversion of the muon flux into a neutrino flux is subjected to the ambiguities arising from the lack of knowledge of the shape of the neutrino spectrum itself. The usual approach followed in the literature [@Ambrosio:2000yx; @Andres:2001ty] is to give the upper limit to the neutrino fluxes assuming a spectrum $d\Phi_\nu/dE_\nu\propto E_\nu^{-2}$. In this case the integrated neutrino flux can be estimated using an average conversion probability $$\label{eq:uplimz}
\Phi_\nu(>E_{th})= \frac{\Phi_\mu^{obs}(>E_{th})}{\frac{1}{E_{th}}\,
\int_{E_{th}}^\infty\,E_\nu^{-2}\,P_{\mu\nu}\,dE_\nu}$$ We propose here to apply the maximum entropy principle [@Jaynes:1957a; @Jaynes:1957b] to infer the unknown neutrino differential flux, with the less prior assumption on the spectral shape. In fact, even if the maximum entropy principle has been questioned for the lack of solid theoretical foundations, it gives very significant results in several fields of applications of spectral deconvolution [@Smith1983]. According to this principle we assume that the probability distribution function of the energy of the neutrino that could have produced a muon in the MACRO detector will be uniform. In this case we have the ansatz $P_{\mu\nu}\,d\Phi_\nu^{ME}/dE_\nu=\mathrm{const}$. Therefore, normalizing to the integrated neutrino flux of Eq.(\[eq:uplimz\]), we have derived the estimate of the upper limit to the neutrino flux which will be shown in the following.
Results of the MACRO survey {#sect:results}
===========================
![Neutrino sky after MACRO (lower panel). In the upper panel is shown the complementary result of the AMANDA survey.[]{data-label="fig:amanda_Macro"}](auriemma_fig08.eps){width="60.00000%"}
Neutrino emission from the Crab nebula {#subsect:crab}
--------------------------------------
Gamma ray spectrum of the Crab Nebula extends up to 25 TeV. Above $\approx 500\;\mathrm{GeV}$ the spectrum is a simple power law with $\alpha\simeq 2.5$. The total $\gamma$-ray luminosity is above $500\;\mathrm{GeV}$ $L_\gamma\simeq 1.5\times
10^{26}\;\mathrm{erg}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}$. The detection of neutrino emission from the Crab could support the possibility that this high energy emission is due to hadronic interactions. Bednarek & Protheroe (1997) have analyzed the consequences of acceleration of heavy nuclei in the pulsar magnetosphere as a possible mechanism of energetic radiation from the Crab Nebula. The MACRO limit for the Crab Nebula is $$\label{eq:craba}
\Phi_\mu^{Crab}(>1.2\;\mathrm{GeV})\le
2.5\times10^{-14}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}\quad
(90\%\;\mathrm{C.L.})$$
![MACRO Upper Limit (90% C.L.) for neutrino emission from the Crab Nebula[]{data-label="fig:crablim"}](auriemma_fig09.eps){width="60.00000%"}
In Fig. \[fig:crablim\] we compare the limit on the neutrino flux, with two extreme models. The present emission is determined by the trapped high energy proton’s density in the nebula. Therefore the present emission is mainly due to acceleration in the early times after supernova explosion. Magnetic dipole losses determine the pulsar period at present, but the initial period is determined largely by gravitational losses and it has been probably shorter than 10 ms. Hence the two models consider two initial periods: 5 ms, and 10 ms[@Bednarek:1997cn].
Neutralino annihilation in the Galactic Center {#subsect:neutralino}
----------------------------------------------
Intense emission from the Galactic center is predicted if cold dark matter is present there, as in current models of the dark galactic halo [@Berezinsky:1994wv; @Tsiklauri:1998np; @Gondolo:1999ef]. In particular it could be expected, according to Gondolo & Silk (1999) that a massive black hole at the galactic center could redistribute the WIMP’s into a cusp. The effect of this redistribution would be a strong enhancements of the self-annihilation rate of neutralinos. In Fig. \[fig:GC\_annih\] we report the predicted up-ward going muon flux in the two cases.
![Predicted up-ward going muon flux from Galactic Center. The possible enhancement due to the redistribution of neutralinos is shown in the upper panel. []{data-label="fig:GC_annih"}](auriemma_fig10.eps){width="55.00000%"}
After the complete run MACRO [@Ambrosio:2000yx] the 90% C.L. upper limits on up-ward going muons from the Galactic Center is $\Phi_\mu\le
0.34\times10^{-14}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}$. This new limit is reported in Fig. \[fig:GC\_annih\]. It is clear from this figure that the predicted enhancement has not been observed. One possible explanation of this negative result has been given from a subsequent more careful evaluation, in a recent paper, of the dynamics of the cusp formation [@Ullio:2001fb]. The MACRO negative observation supports the possibility that the accreting matter could not spiral fast enough by dynamical friction only. Thus within a Hubble time only a mild enhancement could have taken place.
Diffuse emission from Blazars {#subsect:blazars}
-----------------------------
![MACRO Upper Limit (90% C.L.) for the diffuse neutrino emission from blazars[]{data-label="fig:blazarlim"}](auriemma_fig11.eps){width="60.00000%"}
The second EGRET catalog of high-energy $\gamma$-ray sources [@Thompson:1995] contains 40 high confidence identification of AGN and all appear to be blazars. This is the main reason for which several authors [@Stecker:1996th; @Protheroe:1996uu] have proposed this type of object as potential powerful sources of high energy neutrinos. The proton blazar model [@Nellen:1993dw] both protons and electrons are accelerated and protons interact with synchrotron radiation produced by electrons photoproducing pions that decay into $\gamma$-rays and neutrinos.
The up-ward going muons distribution, shown in Fig.\[fig:amanda\_Macro\] has been searched for angular coincidences in a 3$^\circ$ cone with each of the 181 blazars, listed in a recent catalogue [@Padovani1995], with a declination $\delta\le 40^\circ$. No significant excess has been found for any individual source, with a cumulative average upper limit to the muon flux $\Phi_\mu(\ge 1.2 \mathrm{GeV})\le5.44\times
10^{-16}\;\mathrm{cm}^{-2}\,\mathrm{s}^{-1}$. A bias in favour of northern declinations is clearly present in the sample. In fact 178 out of the 233 objects listed in the catalog, have $\delta >
0^\circ$ and there are no known BL Lacs with $\delta < -57^\circ$. We can estimate, very roughly, that the MACRO survey has covered the declination band $-57^\circ\le \delta\le 40^\circ$, corresponding to $\approx 3.2\,\pi\;\mathrm{sr}$ of sky. From this figure we can estimate the U.L. to the diffuse up-ward going muons from blazars $$\label{eq:blazarsa}
\frac{d\Phi_\mu^{BLac}}{d\Omega}\le 0.98\times
10^{-14}\;\mathrm{cm}^{-2}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1}\quad
(90\%\;\mathrm{C.L.})$$ Applying the maximum entropy method we obtain the upper limit to the diffuse neutrino flux reported in Fig. \[fig:blazarlim\]. In the same plot we have reported the model independent upper bound that can be derived if the UHE cosmic rays are also accelerated in the same source [@Waxman:1998yy]. The higher prediction is obtained, presuming a blazar’s luminosity cosmological evolution following that of QSO’s, which may be described as $f_{QSO}=(1+z)^3$ at redshift $z<1.9$ and constant above. The lower prediction is for no evolution of the blazar’s density.
Search for neutrinos from GRB’s {#subsect:grbs}
-------------------------------
Several mechanisms for production of intense HE and UHE neutrino and $\gamma$ burst associated with the main pulse, in the $0.1\;\mathrm{MeV}$ have been proposed in the literature [@Vietri:1998nm; @Waxman:1998tn; @Guetta:2001cd; @Meszaros:2001ms]. It is somewhat surprising however that a phenomenon that was discovered in the keV-MeV region could be the herald of big activity in the multi-TeraVolt region.
We have searched the catalogues BATSE 3B & 4B [@Paciesas:1999tp] containing 2527 gamma ray bursts from 21 Apr. 1991 to 5 Oct. 1999 [@Ambrosio:2000yx]. They overlap in time with 1085 upward-going muons collected by MACRO during this period. In Fig. \[fig:MACRO\_GRB2\] it is shown a plot of angular distance among the GRB and up-ward going muons in MACRO as a function of arrival time difference.
![Coincidences of up-ward going muons in MACRO with the 2527 gamma ray bursts from 21 Apr. 1991 to 5 Oct. 1999 (see text) .[]{data-label="fig:MACRO_GRB2"}](auriemma_fig12.eps){width="60.00000%"}
We observe that if neutrino are massive, as suggested by observation of neutrino oscillations, the neutrino burst from cosmological GRB will be delayed respect to the observation of the $\gamma$ rays observed by the GRB itself. The time delay will be $$\label{eq:neut7a}
\delta t=\frac{1-\beta}{\beta}(t_z-t_0)$$ where $\beta=\sqrt{E_\nu^2-m^2_\nu c^4}/E_\nu$. For $E_\nu\gg
m_\nu c^2$ we can put ${(1-\beta)}/{\beta}=1/2 (E_\nu/m_\nu
c^2)^{-2}+\mathcal{O}(E_\nu/m_\nu c^2)^{-4}$.
![Time delay of the neutrino burst from the main pulse as a function of energies for different neutrino masses.[]{data-label="fig:timedelay"}](auriemma_fig13.eps){width="60.00000%"}
In Fig. \[fig:timedelay\] we show the expected delay of the massive neutrino signal, for GRB in the range $z=0.5-2$ as a function of the Lorentz factor of the neutrino. In the same figure we have reported the normalized Lorentz factor distribution (for a source spectrum $dN_\nu/dE_\nu\propto E_\nu^{-2}$) of neutrinos that produce upward going muons in the MACRO detector. From this figure we see that a maximum delay of $\approx 1000\;\mathrm{s}$ could be expected for very massive neutrinos ($m_\nu=10\;\mathrm{eV}$).
![MACRO Upper Limit (90% C.L.) for the diffuse neutrino emission from GRB’s.[]{data-label="fig:grblim"}](auriemma_fig14.eps){width="60.00000%"}
Therefore the time window to be searched for should to be expanded up to $\approx 1000$ s after the detection of the GRB for the muon to be detected in MACRO. However according to the current ideas on the production of the GRB explosion, it should not be possible to have the neutrino emission long before the GRB. Therefore we can assume that the coincidences between muons detected in the time window preceding the GRB should be very likely accidental. From the lower panel of Fig. \[fig:MACRO\_GRB2\] we estimate that the accidental coincidences among up-going muons and GRB with $\Delta\theta\le 20^\circ$ will be $0.4$ in the overlap period. From the upper panel of the same figure we could candidate two events as possible true coincidences, one event after 39.4 s from 4B950922 at an angular distance of 17.6$^\circ$ and another very horizontal event in coincidence with the 4B940527 inside 280 s at 14.9$^\circ$. However the cumulative Poissonian probability that those two events are accidental is $7.9\times
10^{-3}$. We assume then that those events are expected accidental, and calculate that the 90% confidence interval for the unobserved true coincident upward-going muons [@Feldman:1998qc] will be $\le 5.41$. The corresponding upper limit to the muon fluence from the average GRB is $ \Phi_\mu(\ge
1.2\;\mathrm{GeV})\le 0.79\times 10^{-9}\;\mathrm{cm}^{-2}\;(90\%
\mathrm{C.L.})$ for neutrino masses $m_\nu\le 10\;\mathrm{eV}$. From this cumulative limit we can derive an upper limit to the diffuse neutrino background from GRB’s, taking into account the fact that the MACRO live time in the period from 21 Apr. 1991 to 5 Oct. 1999 has been $4.62\;\mathrm{y}$, that neutrinos could have been detected for GRB’s with declination $\delta\le 40^\circ$ (corresponding to $3.53\,\pi\;\mathrm{sr}$ of sky) and that the exposure factor for BATSE in this declination range [@Paciesas:1999tp] is $55\%$ we can convert the upper limit for the average burst to a diffuse background $$\label{eq:neut2b}
\frac{d\Phi^{GRB}_\mu}{d\Omega}\le 2.25\times
10^{-15}\;\mathrm{cm}^{-2}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1}\quad
(90\%\;\mathrm{C.L.})$$ We obtain also in this case the upper limit to the diffuse neutrino emission from GRB’s reported in Fig. \[fig:grblim\]. In the same figure we have reported the prediction of diffuse neutrino background produced by a cosmological distribution of GRB’s [@Waxman:1998yy]. Also we have reported the flux expected from the intriguing possibility that before the jet could emerge from the dense stellar progenitor of the GRB, the accelerated protons could interact with the thermal X-ray photon leading to neutrinos with energy $\gtrsim 5\;\mathrm{TeV}$ [@Meszaros:2001ms]. In this model the intensity of diffuse neutrino background depends from the amount of collimation of the jet $\Omega_{jet}/4\pi$, because the $\simeq 1000$ GRB’s observed per year, are only a small fraction of the actual explosions.
Conclusions {#sect:concl}
===========
![A gallery of past, present and future HE neutrino telescopes.[]{data-label="fig:detec_comp"}](auriemma_fig15.eps){width="50.00000%"}
In Fig. \[fig:detec\_comp\] we show a gallery of past, present and future HE neutrino telescopes, compared with a massive detector with 1 km$^3$ of volume. From this figure we have a direct impression of the smallness of the detectors which were operated in the past century, compared with the ones under construction. It is also worth noticing that the preliminary results obtained by Amanda, from April to October, 1997 which are reported in the upper panel of Fig. \[fig:amanda\_Macro\], include 1097 up-ward going muons. A number comparable to the number of events detected by MACRO in $\approx 4.6$ live years. This shows directly that the acceptance of the AMANDA-II detector is already more then a factor 10 greater then the acceptance of MACRO.
In the previous §\[sect:results\] we have practically shown how far was MACRO from effectively testing theoretical production models. From Fig \[fig:crablim\] it can be crudely estimated that an acceptance $\approx 10^{4}\;\mathrm{MACRO}\simeq
1\;\mathrm{km}^2$ will be needed to challenge possibly the predictions for galactic neutrino sources. The situation could be slightly better for the statistical detection of emission from selected classes of objects, like for example the association of up-ward going muons with blazars, where a possible detection seems to be attainable with an acceptance $\approx 0.1\;\mathrm{km}^2$.
|
---
abstract: |
Modern data analytic and machine learning jobs find in the cloud a natural deployment platform to satisfy their notoriously large resource requirements. Yet, to achieve cost efficiency, it is crucial to identify a deployment configuration that satisfies user-defined QoS constraints (e.g., on execution time), while avoiding unnecessary over-provisioning.
This paper introduces [Lynceus]{}, a new approach for the optimization of cloud-based data analytic jobs that improves over state-of-the-art approaches by enabling significant cost savings both in terms of the final recommended configuration and of the optimization process used to recommend configurations.
Unlike existing solutions, [[Lynceus]{}]{} optimizes in a joint fashion both the cloud-related and the application-level parameters. This allows for a reduction of the cost of recommended configurations by up to $3.7\times$ at the 90-th percentile with respect to existing approaches, which treat the optimization of cloud-related and application-level parameters as two independent problems.
Further, [[Lynceus]{}]{} reduces the cost of the optimization process (i.e., the cloud cost incurred for testing configurations) by up to $11\times$. Such an improvement is achieved thanks to two mechanisms: i) a timeout approach which allows to abort the exploration of configurations that are deemed suboptimal, while still extracting useful information to guide future explorations and to improve its predictive model — differently from recent works, which either incur the full cost for testing suboptimal configurations or are unable to extract any knowledge from aborted runs; ii) a *long-sighted and budget-aware* technique that determines which configurations to test by predicting the long-term impact of each exploration — unlike state-of-the-art approaches for the optimization of cloud jobs, which adopt greedy optimization methods.
author:
-
bibliography:
- 'main.bib'
title: |
Lynceus: Cost-efficient Tuning and Provisioning of Data Analytic Jobs\
[^1]
---
cloud computing, machine learning platforms, optimization, virtual machines, Bayesian optimization
Conclusions and future work {#sec:conclusions}
===========================
We presented [[Lynceus]{}]{}, a new tool to provision and tune data analytic jobs. [[Lynceus]{}]{} implements a novel approach that combines cross-layer optimization, budget awareness, long-sightedness, and the ability to cancel sub-optimal sampling while still improving the model. [[Lynceus]{}]{} consistently outperforms state-of-the-art approaches, identifying configurations that are up to $3.7\times$ cheaper — thanks to the joint optimization of cloud and application parameters — and reducing the cost of the optimization process by up to $11\times $ — thanks to its novel optimization method. As a final note, [[Lynceus]{}]{} can be extended to consider multiple constraints (e.g., one may want to enforce that the energy consumed to execute the job is also within a given threshold) and to take into account the costs associated with bootstrapping VMs during the exploration phase. An evaluation of these mechanisms is left for future work.
[^1]:
|
---
abstract: 'In reliability engineering, data about failure events is often scarce. To arrive at meaningful estimates for the reliability of a system, it is therefore often necessary to also include expert information in the analysis, which is straightforward in the Bayesian approach by using an informative prior distribution. A problem called prior-data conflict then can arise: observed data seem very surprising from the viewpoint of the prior, i.e., information from data is in conflict with prior assumptions. Models based on conjugate priors can be insensitive to prior-data conflict, in the sense that the spread of the posterior distribution does not increase in case of such a conflict, thus conveying a false sense of certainty. An approach to mitigate this issue is presented, by considering sets of prior distributions to model limited knowledge on Weibull distributed component lifetimes, treating systems with arbitrary layout using the survival signature. This approach can be seen as a robust Bayesian procedure or imprecise probability method that reflects surprisingly early or late component failures by wider system reliability bounds.'
author:
- |
Gero Walter[^1], Ph.D.\
and Frank P.A. Coolen[^2], Ph.D.
bibliography:
- 'refs.bib'
title: 'ROBUST BAYESIAN RELIABILITY FOR COMPLEX SYSTEMS UNDER PRIOR-DATA CONFLICT'
---
Introduction {#sec:intro}
============
In reliability engineering, a central task is to describe the reliability of a complex system. This is usually done by determining the *reliability function* $R(t)$, in other contexts also known as the *survival function*, giving the probability that the system has not failed by time $t$:
$$\begin{aligned}
{R_\text{sys}}(t) = P(\Tsys \geq t)\,,\end{aligned}$$
where $\Tsys$ is the random variable giving the failure time of the system. Based on the distribution of $\Tsys$, which can also be expressed in terms of the cumulative distribution function $F_\text{sys}(t) = 1 - {R_\text{sys}}(t)$, decisions about, e.g., scheduling of maintenance work can be made.
Often, there is no failure data for the system itself (e.g., if the system is a prototype, or the system is used under unique circumstances), but there is some information about failure times for the components the system is made of. The proposed method allows to analyse systems of arbitrary system structures, i.e., any combination and nesting of series, parallel, k-out-of-n, or bridge-type arrangements, by use of the survival signature [@2012:survsign]. In doing so, the present paper extends similar previous work which focused on a simple parallel system with homogeneous components [@2015:walter]. In this paper, we assume that components can be divided into $K$ different groups, and components within each group $k$ ($k=1, \ldots, K$) can be assumed to be exchangeable, i.e., to follow the same failure time distribution. Components of group $k$ are denoted as *type $k$ components*, and are assumed to be independent from components of other types. Type $k$ component lifetimes $T_i^k$ ($i = 1, \ldots, n_k$) are assumed as Weibull distributed with type-specific parameters, where the shape parameter $\beta_k$ is known and the scale parameter $\lambda_k$ is unknown.
Focusing on a running system, we assume that observations consist solely of the failure times of components in this system up to time $\tnow$, such that the failure times of components that have not failed by $\tnow$ are right-censored, and calculate ${R_\text{sys}}(t \mid t > \tnow) = P(\Tsys \geq t \mid t > \tnow)$, which can be used to determine the remaining useful life of the system (in short RUL, see, e.g., ). The Bayesian approach allows to base estimation of the component failure distributions on both data and further knowledge not given by the data, the latter usually provided in the form of expert knowledge. This knowledge is encoded in form of a so-called prior distribution, which here, as the shape parameter $\beta_k$ is assumed to be known, is a distribution on the scale parameter $\lambda_k$. Expert knowledge is especially important when there is very few data on the components, as only with its help meaningful estimates for the system reliability can be made. Due to the iterative nature of the Bayesian framework, it is also possible to use a set of posteriors based on component test data as the set of priors instead of a purely expert-based set of priors as discussed so far. The component test data may also contain right-censored observations, which can be treated in the same way as those from the running system (see Section \[sec:lambdawithcens\] for details).
The choice of prior distribution to encode given expert knowledge is often debatable, and a specific choice of prior is difficult to justify. A way to deal with this is to employ sensitivity analysis, i.e., studying the effect of different choices of prior distribution on the quantities of interest (in our case, the system reliability function, which, in Bayesian terms, is a posterior predictive distribution). This idea has been explored in systematic sensitivity analysis, or robust Bayesian methods (for an overview on this approach, see, e.g., or ). The work presented here can be seen as belonging to the robust Bayesian approach since it uses sets of priors. However, our focus and interpretation is slightly different, as we consider the result of our procedure, sets of reliability functions, as the proper result, while a robust Bayesian would base his analyses on a single reliability function from the set in case (s)he was able to conclude that quantities of interest are not ‘too sensitive’ to the choice of prior. In contrast, our viewpoint is rooted in the theory of imprecise or interval probability [@1991:walley; @itip], where sets of distributions are used to express the precision of probability statements themselves: the smaller the set, the more precise the probability statement. Indeed, the system reliability function ${R_\text{sys}}(t)$ is a collection of probability statements, and a small set for ${R_\text{sys}}(t)$ will indicate that the reliability of the system can be can quantified quite precisely, while a large set will indicate that available knowledge about $\Tsys$ is rather shaky. In line with imprecise or interval probability methods, the method provides, for each $t$, a lower reliability ${{\underline{R}}_\text{sys}}(t) = {\underline{P}}(T_\text{sys} \geq t)$, and an upper reliability ${{\overline{R}}_\text{sys}}(t) = {\overline{P}}(T_\text{sys} \geq t)$. Sections \[sec:modforsurpr\] and \[sec:robrel\] will explain how these bounds are obtained based on sets of prior distributions on the scale parameters of the component lifetime distributions.
The central merit of the proposed method is that it adequately reflects prior-data conflict (see, e.g., ), i.e. the conflict that can arise between prior assumptions on component lifetimes and observed behaviour of components in the system under study. As will be shown in Section \[sec:weibull\], when taking the standard choice of a conjugate prior, prior-data conflict is ignored, as the spread of the posterior distribution does not increase in case of such a conflict, ultimately conveying a false sense of certainty by communicating that the reliability of a system can be quantified quite precisely when in fact it can not. In contrast, the proposed method will indicate prior-data conflict by wider bounds for ${R_\text{sys}}(t)$. This behaviour is obtained by a specific choice for the set of priors (see and §3.1.4) which leads to larger sets of posterior distributions when prior knowledge and data are in conflict (see Section \[sec:modforsurpr\] for more details). If a prior based on component test data is used, such prior-data conflict sensitivity furthermore allows to uncover a conflict between current observations in the running system and past observations from component tests.
The paper is organized as follows. In Section \[sec:weibull\], a Bayesian analysis for Weibull component lifetimes with fixed shape parameter is described, illustrating the issue of prior-data conflict. Section \[sec:modforsurpr\] then details the use of sets of priors for the scale parameter $\lambda_k$ of the Weibull distribution, showing how this mitigates the prior-data conflict issue for $\lambda_k$. Section \[sec:robrel\] then develops a method to calculate system reliability bounds for a running system based on sets of priors for $\lambda_k$. Section \[sec:elicitation\] discusses elicitation, by giving guidelines on how to choose prior parameter bounds that reflect expert information on components. Section \[sec:examples\] contains examples illustrating the merits of our method, by studying the effect of surprisingly early or late component failures, showing that observations in conflict to prior assumptions indeed lead to more cautious system reliability predictions. Section \[sec:concluding\] concludes the paper by summarizing results and indicating avenues for further research.
Bayesian Analysis of Weibull Lifetimes {#sec:weibull}
======================================
Consider a system with components of $k=1,\ldots,K$ different types; for each type $k$, there are $n_k$ exchangeable components in the system. For each type $k$ component, we assume for its lifetime $T_i^k$ ($i=1,\ldots,n_k$, $k = 1, \ldots, K$) a Weibull distribution with fixed shape parameter $\beta_k > 0$, in short $T_i^k \mid \lambda_k \sim {\operatorname{Wei}}(\beta_k,\lambda_k)$, with density and cdf
$$\begin{aligned}
\label{eq:weibulldens}
f(t_i^k \mid \lambda_k) &= \frac{\beta_k}{\lambda_k} (t_i^k)^{\beta_k-1} e^{-\frac{(t_i^k)^{\beta_k}}{\lambda_k}}\,, \\
\label{eq:weibullcdf}
F(t_i^k \mid \lambda_k) &= 1 - e^{-\frac{(t_i^k)^{\beta_k}}{\lambda_k}} = P(T_i^k \leq t_i^k \mid \lambda_k)\,,\end{aligned}$$
where $\lambda_k > 0$ and $t > 0$.
The shape parameter $\beta_k$ determines whether the hazard rate is increasing ($\beta_k > 1$) or decreasing ($\beta_k < 1$) over time. For $\beta_k=1$, one obtains the Exponential distribution with constant hazard rate as a special case. The value for $\beta_k$ will often be clear from practical considerations.
The scale parameter $\lambda_k$ can be interpreted through the relation
$$\begin{aligned}
{\operatorname{E}}[T_i^k \mid \lambda_k] &= \lambda_k^{1/\beta_k}\, \Gamma(1 + 1/\beta_k)\,.
\label{eq:lambdainterpret}\end{aligned}$$
For encoding expert knowledge about the reliability of the components, one needs to assign a prior distribution over the scale parameter $\lambda_k$. A convenient choice is to use the inverse Gamma distribution, commonly parametrized in terms of the hyperparameters $a_k > 0$ and $b_k > 0$:
$$\begin{aligned}
f(\lambda_k\mid a_k,b_k) &= \frac{(b_k)^{a_k}}{\Gamma(a_k)} \lambda_k^{-a_k -1} e^{-\frac{b_k}{\lambda_k}}
\label{eq:ig-def}\end{aligned}$$
in short, $\lambda_k \mid a_k, b_k \sim {\operatorname{IG}}(a_k,b_k)$. The inverse Gamma is convenient because it is a conjugate prior, i.e., the posterior obtained by Bayes’ rule is again inverse Gamma and thus easily tractable; the prior parameters only need to be updated to obtain the posterior parameters.
In the standard Bayesian approach, one has to fix a prior by choosing values for $a_k$ and $b_k$ to encode specific prior information about component lifetimes. In our imprecise approach, we allow instead these parameters to vary in a set, this is advantageous also because expert knowledge is often vague, and it is difficult for the expert(s) to pin down precise hyperparameter values. For the definition of the hyperparameter set, we use however a parametrization in terms of $\nz > 1$ and $\yz > 0$ instead of $a_k$ and $b_k$, where we drop the index $k$ for the discussion about the prior model in the following, keeping in mind that each component type will have its own specific parameters. We use $\nz = a - 1$ and $\yz = b / \nz$, where $\yz$ can be interpreted as the prior guess for the scale parameter $\lambda$, as ${\operatorname{E}}[\lambda\mid\nz,\yz] = \yz$. This parametrization also makes the nature of the combination of prior information and data through Bayes’ rule more clear: After observing $n$ component lifetimes ${{\bmt}} = (t_1, \ldots, t_n)$, the updated parameters are
$$\begin{aligned}
\nn &= \nz + n\,,
&
\yn &= \frac{\nz \yz + \taut}{\nz + n}\,,
\label{eq:ig-update}\end{aligned}$$
where $\taut = \sum_{i=1}^n (t_i)^\beta$. We thus have
$$\begin{aligned}
\lambda \mid \nz, \yz, {{\bmt}} \sim {\operatorname{IG}}(\nz + n + 1, \nz \yz + \taut). \label{eq:ig-update-alpha}\end{aligned}$$
From the simple update rule , we see that $\yn$ is a weighted average of the prior parameter $\yz$ and the maximum likelihood (ML) estimator $\taut/n$, with weights proportional to $\nz$ and $n$, respectively. $\nz$ can thus be interpreted as a prior strength or pseudocount, indicating how much our prior guess should weigh against the $n$ observations. Furthermore, ${\operatorname{Var}}[\lambda\mid\nz,\yz] = (\yz)^2 / (1 - 1/\nz)$, so for fixed $\yz$, the higher $\nz$, the more probability mass is concentrated around $\yz$. However, the weighted average structure for $\yn$ is behind the problematic behaviour in case of prior-data conflict. Assume that from expert knowledge we expect to have a mean component lifetime of 9 weeks. Using , with $\beta=2$ we obtain $\yz = 103.13$. We choose $\nz = 2$, so our prior guess for the mean component lifetime counts like having two observations with this mean. If we now have a sample of two observations with surprisingly early failure times $t_1 = 1$ and $t_2 = 2$, using we get ${n^{({2})}} = 4$ and ${y^{({2})}} = \frac{1}{4}(2 \cdot 103.13 + 1^2 + 2^2) = 52.82$, so our posterior expectation for the scale parameter $\lambda$ is $52.82$, equivalent to a mean component lifetime of $6.44$ weeks. The posterior standard deviation (sd) for $\lambda$ is $60.99$. Compared to the prior standard deviation of $145.85$, the posterior expresses now more confidence that mean lifetimes are around ${y^{({2})}} = 52.82$ than the prior had about $\yz = 103.13$. This irritating conclusion is illustrated in Figure \[fig:weibull-pdc\]; the posterior cdf is shifted halfway towards the values for $\lambda$ that the two observations suggest (the ML estimator for $\lambda$ would be $2.5$), and is steeper than the prior (so the pdf is more pointed), thus conveying a false sense of certainty about $\lambda$. We would obtain almost the same posterior distribution if we had assumed the mean component lifetime to be 7 weeks (so $\yz = 62.39$), and observed lifetimes $t_1 = 6$, $t_2 = 7$ in line with our expectations. It seems unreasonable to make the same probability statements on component lifetimes in these two fundamentally different scenarios.
Note that this is a general problem in Bayesian analysis with canonical conjugate priors. For such priors, the same update formula applies, and so conflict is averaged out, for details see and .
![Prior and posterior cdf for $\lambda$ given surprising observations; the conflict between prior assumptions and data is averaged out, with a more pointed posterior giving a false sense of certainty.[]{data-label="fig:weibull-pdc"}](fig1){width="80.00000%"}
Models reflecting surprising data {#sec:modforsurpr}
=================================
Despite this issue of ignoring prior-data conflict, the tractability of the update step is a very attractive feature of the conjugate setting. As was shown by (see also , §3.1), it is possible to retain tractability and to have a meaningful reaction to prior-data conflict when using sets of priors generated by varying both $\nz$ and $\yz$. Then, the magnitude of the set of posteriors, and with it the precision of posterior probability statements, will be sensitive to the degree of prior-data conflict, i.e. leading to more cautious probability statements when prior-data conflict occurs.
Instead of a single prior guess $\yz$ for the mean component lifetimes, we will now assume a range of prior guesses $[\yzl, \yzu]$, and also a range $[\nzl, \nzu]$ of pseudocounts, i.e., we now consider the set of priors
$$\begin{aligned}
\MZ := \{ f(\lambda\mid\nz,\yz) \mid \nz \in [\nzl, \nzu], \yz \in [\yzl, \yzu] \}
\label{eq:setofpriors}\end{aligned}$$
to express our prior knowledge about component lifetimes. Each of the priors $f(\lambda\mid\nz,\yz)$ is then updated to the posterior $f(\lambda\mid\nz,\yz,{{\bmt}}) = f(\lambda\mid\nn,\yn)$ by using , such that the set of posteriors $\MN$ can be written as $\MN = \{ f(\lambda\mid\nn,\yn) \mid \nz \in [\nzl, \nzu], \yz \in [\yzl, \yzu] \}$. This procedure of using Bayes’ Rule element by element is seen as self-evident in the robust Bayesian literature, but can be formally justified as being *coherent* (a self-consistency property) in the framework of imprecise probability, where it is known as *Generalized Bayes’ Rule* [@1991:walley §6.4].
Technically, it is crucial to consider a range of pseudocounts $[\nzl, \nzu]$ along with the range of prior guesses $[\yzl, \yzu]$, as only then $\taut/n \not\in [\yzl, \yzu]$ leads to the set of posteriors being larger and hence reflecting prior-data conflict.
Continuing the example from Section \[sec:weibull\] and Figure \[fig:weibull-pdc\], assume now for the mean component lifetimes the range 9 to 11 weeks, this corresponds to $[\yzl,\yzu] = [103.13, 154.06]$. Choosing $[\nzl,\nzu] =[2, 5]$ means to value this information on mean component lifetimes as equivalent to having seen two to five observations. Compare now the set of posteriors obtained from observing $t_1 = 1$, $t_2 = 2$ (as before), see Figure \[fig:setofpost-pdc-nopdc\] (left), and $t_1 = 10$, $t_2 = 11$, see Figure \[fig:setofpost-pdc-nopdc\] (right). There is now a clear difference between the two scenarios of observations in line with expectations and observations in conflict. In the prior-data conflict case, the set of posteriors (blue) is shifted towards the left, but has about the same size as the set of priors (yellow), and so posterior quantification of reliability has the same precision, despite having seen two observations. Instead, in the no conflict case, the set of posteriors is smaller than the set of priors, such that the two observations have increased the precision of reliability statements.
![Set of prior and posterior cdfs for $\lambda$ for two surprising observations $t_1 = 1$, $t_2 = 2$ (left) and two unsurprising observations $t_1 = 10$, $t_2 = 11$ (right).[]{data-label="fig:setofpost-pdc-nopdc"}](fig2){width="\textwidth"}
As each posterior in $\MN$ corresponds to a predictive distribution for $\Tsys$, we will have a set of reliability functions ${R_\text{sys}}(t)$. The derivation of ${R_\text{sys}}(t)$ for systems with $K$ component types and arbitrary layout will be given in Section \[sec:robrel\] below. This will include, in contrast to previous studies using sets of priors of type , the treatment of censored observations. More specifically, we consider the case of *non-informative right censoring*, where the censoring process is independent of the failure process.
Robust Reliability for Complex Systems via the Survival Signature {#sec:robrel}
=================================================================
Consider now a system of arbitrary layout, consisting of components of $K$ types. This system is observed until time $\tnow$, leading to censored observation of lifetimes of components within the system, and we will explain in Section \[sec:lambdawithcens\] how the scale parameter $\lambda_k$ for component type $k$ can be estimated in this situation. Section \[sec:sysrelwithsurvsign\] describes then how the system reliability function can be efficiently obtained using the survival signature. For this, we need the posterior predictive distribution of the number of components that function at times $t > \tnow$, which we will derive in Section \[sec:postpred\]. Finally, in Section \[sec:optimize\] we describe how the lower and upper bound for the system reliability function are obtained when prior parameters $(\nkz,\ykz)$ vary in sets $\PkZ = [\nkzl,\nkzu] \times [\ykzl,\ykzu]$, defining sets $\MkZ$ of prior distributions over $\lambda_k$, as introduced in Section \[sec:modforsurpr\].
Bayesian Estimation of Component Scale Parameter with Right-censored Lifetimes {#sec:lambdawithcens}
------------------------------------------------------------------------------
Consider observing a system until $\tnow$, where the system has $K$ different types of components, and for each type $k$ there are $n_k$ components in the system. Denoting the number of type $k$ components that have failed by $\tnow$ by $e_k$, there are $n_k - e_k$ components still functioning at $\tnow$. We denote the corresponding vector of observations by
$$\begin{aligned}
{{\bmt}}^k_{e_k;n_k} &= \big( \underbrace{t^k_1, \ldots, t^k_{e_k}}_{e_k \text{failure times}},
\underbrace{\tpnow, \ldots, \tpnow}_{n_k-e_k \text{censored obs.}} \big)\,,\end{aligned}$$
where $t^+$ indicates a right-censored observation. According to Bayes’ rule, multiplying the prior density and the likelihood (which accounts for right-censored observations through the cdf terms) gives a term proportional to the density of the posterior distribution for $\lambda_k$:
$$\begin{aligned}
f(\lambda_k\mid\nkz,\ykz,{{\bmt}}^k_{e_k;n_k})
&\propto f(\lambda_k)
\big[ 1- F(\tnow\mid\lambda_k) \big]^{n_k-e_k}
\prod_{i=1}^{e_k} f(t_i^k \mid \lambda_k) \end{aligned}$$
Conjugacy is preserved and we get $\lambda_k\mid\nkz,\ykz,{{\bmt}}^k_{e_k;n_k} \sim {\operatorname{IG}}(\nkn + 1, \nkn\ykn)$, where
$$\begin{aligned}
\nkn + 1 &= \nkz + e_k + 1 \\
\nkn\ykn &= \nkz\ykz + (n_k-e_k) (\tnow)^\beta + \sum_{i=1}^{e_k} (t_i^k)^\beta\end{aligned}$$
are the updated parameters of the inverse gamma distribution.
System Reliability using the Survival Signature {#sec:sysrelwithsurvsign}
-----------------------------------------------
The structure of complex systems can be visualized by reliability block diagrams, an example is given in Figure \[fig:brakesys-layout\]. Components are represented by boxes or nodes, and the system works when a path from the left end to the right exists which passes only through working components. In a system with $n$ components, the state of the components can be expressed by the state vector ${{\bmx}} = (x_1,x_2,\ldots,x_n) \in \{0,1\}^n$, with $x_i=1$ if the $i$th component functions and $x_i=0$ if not. The structure function $\phi : \{0,1\}^n \rightarrow \{0,1\}$, defined for all possible ${{\bmx}}$, takes the value 1 if the system functions and 0 if the system does not function for state vector ${{\bmx}}$ [@BP75]. Most real-life systems are coherent, which means that $\phi({{\bmx}})$ is non-decreasing in any of the components of ${{\bmx}}$, so system functioning cannot be improved by worse performance of one or more of its components. Furthermore, one can usually assume that $\phi(0, \ldots, 0) = 0$ and $\phi(1, \ldots, 1) = 1$.
The survival signature [@2012:survsign] is a summary of the structure function for systems with $K$ groups of exchangeable components. Denoted by $\Phi(l_1,\ldots,l_K)$, with $l_k=0,1,\ldots,n_k$ for $k=1,\ldots,K$, it is defined as the probability for the event that the system functions given that precisely $l_k$ of its $n_k$ components of type $k$ function, for each $k\in \{1,\ldots,K\}$. Essentially, this creates a $K$-dimensional partition for the event $\Tsys > t$, such that ${R_\text{sys}}(t) = P(\Tsys > t)$ can be calculated using the law of total probability,
$$\begin{aligned}
P(\Tsys > t) &= \sum_{l_1=0}^{m_1} \cdots \sum_{l_K=0}^{m_K} P(\Tsys > t \mid C^1_t = l_1,\ldots, C^K_t = l_K)
P\Big( \bigcap_{k=1}^K \{ C^k_t = l_k\} \Big) \nonumber\\
&= \sum_{l_1=0}^{n_1} \cdots \sum_{l_K=0}^{n_K} \Phi(l_1,\ldots,l_K) P\Big( \bigcap_{k=1}^K \{ C^k_t = l_k\} \Big) \nonumber\\
&= \sum_{l_1=0}^{n_1} \cdots \sum_{l_K=0}^{n_K} \Phi(l_1,\ldots,l_K) \prod_{k=1}^K P(C^k_t = l_k)\,,
\label{eq:sysrel-survsign}\end{aligned}$$
where $P(C^k_t = l_k)$ is the (predictive) probability that exactly $l_k$ components of type $k$ function at time $t$, and the last equality holds as we assume that components of different types are independent. Note that for coherent systems, the survival signature $\Phi(l_1,\ldots,l_K)$ is non-decreasing in each $l_k$.
Posterior Predictive Distribution {#sec:postpred}
---------------------------------
In calculating the system reliability using , the component-specific predictive probabilities $P(C^k_t = l_k)$ need to use all information available at time $\tnow$, which, in the Bayesian framework, are given by the posterior predictive distribution $P(C^k_t = l_k\mid\nkz,\ykz, {{\bmt}}^k_{e_k;n_k})$, $l_k = 0, 1, \ldots, n_k-e_k$. (Remember that $e_k$ type $k$ components have failed by $\tnow$, such that there can be at most $n_k-e_k$ working components beyond time $\tnow$.) This posterior predictive distribution is obtained as
$$\begin{aligned}
\lefteqn{P(C^k_t = l_k\mid\nkz,\ykz, {{\bmt}}^k_{e_k;n_k})}\hspace*{5ex} \nonumber\\ &= { n_k - e_k \choose l_k} \int \big[P(T^k > t \mid T^k > \tnow, \lambda_k)\big]^{l_k} \times \nonumber\\ & \hspace*{17ex}
\big[P(T^k \leq t \mid T^k > \tnow, \lambda_k)\big]^{n_k - e_k - l_k}
f(\lambda_k\mid\nkz,\ykz,{{\bmt}}^k_{e_k;n_k}) {\,\mathrm{d}}\lambda_k\,.
\label{eq:postpredtnow}\end{aligned}$$
Now, by the Weibull assumption , one has
$$\begin{aligned}
P(T^k \leq t \mid T^k > \tnow, \lambda_k)
&= \frac{P(\tnow < T^k \leq t \mid\lambda_k)}{P(T^k > \tnow \mid \lambda_k)} \nonumber\\
&= \frac{F(t\mid\lambda_k) - F(\tnow\mid\lambda_k)}{1-F(\tnow\mid\lambda_k)}
= 1 - e^{-\frac{t^{\beta_k} - (\tnow)^{\beta_k}}{\lambda_k}}\,.\end{aligned}$$
With this and the posterior substituted into , this gives
$$\begin{aligned}
\lefteqn{P(C^k_t = l_k\mid\nkz,\ykz, {{\bmt}}^k_{e_k;n_k})}\hspace*{5ex} \nonumber\\
&= { n_k - e_k \choose l_k} \int \Big[ e^{-\frac{t^{\beta_k} - (\tnow)^{\beta_k}}{\lambda_k}}\Big]^{l_k}
\Big[1 - e^{-\frac{t^{\beta_k} - (\tnow)^{\beta_k}}{\lambda_k}}\Big]^{n_k - e_k - l_k}
\times \nonumber\\ & \hspace*{27ex}
\frac{\big(\nkn\ykn\big)^{\nkn + 1}}{\Gamma(\nkn+1)} \lambda_k^{-(\nkn + 1) - 1} e^{-\frac{\nkn\ykn}{\lambda_k}} {\,\mathrm{d}}\lambda_k \nonumber\\
&= { n_k - e_k \choose l_k} \sum_{j=0}^{n_k-e_k-l_k} (-1)^j { n_k - e_k - l_k \choose j} \frac{\big(\nkn\ykn\big)^{\nkn + 1}}{\Gamma(\nkn+1)}
\times \nonumber\\ & \hspace*{13ex}
\int \lambda_k^{-(\nkn + 1) - 1} \exp\Big\{-\frac{(l_k + j) (t^{\beta_k} - (\tnow)^{\beta_k}) + \nkn\ykn}{\lambda_k}\Big\} {\,\mathrm{d}}\lambda_k\,.\end{aligned}$$
The terms remaining under the integral form the core of an inverse gamma distribution with parameters $\nkn + 1$ and $\nkn\ykn + (l_k + j) (t^{\beta_k} - (\tnow)^{\beta_k}))$, allowing to solve the integral using the corresponding normalization constant. We thus have, for $l_k \in \{0,1,\ldots,n_k-e_k\}$,
$$\begin{aligned}
\lefteqn{P(C^k_t = l_k\mid\nkz,\ykz, {{\bmt}}^k_{e_k;n_k})} \nonumber\\
&= { n_k - e_k \choose l_k} \sum_{j=0}^{n_k-e_k-l_k} (-1)^j { n_k - e_k - l_k \choose j}
\left(\frac{\nkn\ykn}{\nkn\ykn + (l_k + j) \big(t^{\beta_k} - (\tnow)^{\beta_k}\big)}\right)^{\nkn + 1} \nonumber\\
&= \sum_{j=0}^{n_k-e_k-l_k} (-1)^j \frac{(n_k - e_k)!}{l_k! j! (n_k - e_k - l_k - j)!}
\left(\frac{\nkn\ykn}{\nkn\ykn + (l_k + j) \big(t^{\beta_k} - (\tnow)^{\beta_k}\big)}\right)^{\nkn + 1} \nonumber\\
&= \sum_{j=0}^{n_k-e_k-l_k} (-1)^j \frac{(n_k - e_k)!}{l_k! j! (n_k - e_k - l_k - j)!} \times \nonumber\\ & \hspace*{10ex}
\left(\frac{\nkz\ykz + \sum_{i=1}^{e_k} (t_i^k)^{\beta_k} + (n_k-e_k) (\tnow)^{\beta_k} } {\nkz\ykz + \sum_{i=1}^{e_k} (t_i^k)^{\beta_k} + (n_k-e_k-l_k-j) (\tnow)^{\beta_k} + (l_k + j) t^{\beta_k} }\right)^{ \nkz + e_k + 1}.
\label{eq:postpred-priorparams}\end{aligned}$$
These posterior predictive probabilities can also be expressed as a cumulative probability mass function (cmf)
$$\begin{aligned}
F(l_k \mid \nkz,\ykz,{{\bmt}}^k_{e_k;n_k}) = P(C^k_t \leq l_k \mid \nkz,\ykz,{{\bmt}}^k_{e_k;n_k})
= \sum_{j=0}^{l_k} P(C^k_t = j \mid \nkz,\ykz,{{\bmt}}^k_{e_k;n_k})\,.
$$
Optimizing over Sets of Parameters {#sec:optimize}
----------------------------------
Together with , allows to calculate the system reliability ${R_\text{sys}}(t\mid t>\tnow)$ for fixed prior parameters $(\nkz, \ykz)$, $k=1, \ldots, K$. In Section \[sec:modforsurpr\], we argued for using sets of priors $\MZ$, which allow for vague and incomplete prior knowledge, and provide prior-data conflict sensitivity. We will thus use, for each component type, a set of priors $\MkZ$ defined by varying $(\nkz,\ykz)$ in a prior parameter set $\PkZ = [\nkzl,\nkzu] \times [\ykzl,\ykzu]$, and the objective is to obtain the bounds
$$\begin{aligned}
{{\underline{R}}_\text{sys}}(t \mid t > \tnow) &= \min_{{\Pi{^{(0)}}_{1}},\ldots,{\Pi{^{(0)}}_{K}}} {R_\text{sys}}\big(t \mid t > \tnow, \cup_{k=1}^K \{\PkZ, {{\bmt}}^k_{e_k;n_k}\}\big)\,,
\label{eq:lrsysdef}\\
{{\overline{R}}_\text{sys}}(t \mid t > \tnow) &= \max_{{\Pi{^{(0)}}_{1}},\ldots,{\Pi{^{(0)}}_{K}}} {R_\text{sys}}\big(t \mid t > \tnow, \cup_{k=1}^K \{\PkZ, {{\bmt}}^k_{e_k;n_k}\}\big)\,,
\label{eq:ursysdef}\end{aligned}$$
where we suppress in notation that ${{\underline{R}}_\text{sys}}(t \mid t > \tnow)$ and ${{\overline{R}}_\text{sys}}(t \mid t > \tnow)$ depend on prior parameter sets and data.
and seem to suggest that a full $2K$-dimensional box-constraint optimization is necessary, but this is not the case. Remember that $\Phi(l_1,\ldots,l_k)$ from is non-decreasing in each of its arguments $l_1,\ldots,l_K$, so if there is stochastic dominance in $F(l_k \mid \nkz,\ykz,{{\bmt}}^k_{e_k;n_k})$, then there is, for each component type $k$, a prior parameter pair in $\PkZ$ that minimizes system reliability, and a prior parameter pair in $\PkZ$ that maximizes system reliability, independently of the other component types. Indeed, stochastic dominance in $F(l_k \mid \nkz,\ykz,{{\bmt}}^k_{e_k;n_k})$ is provided for $\ykz$. To see this, note that $\ykz$ gives the mean expected lifetime for type $k$ components. Thus, higher values for $\ykz$ mean higher expected lifetimes for the components, which in turn increases the probability that many components survive until time $t$, and with it, decreases the propability of few or no components surviving, so in total giving low probability weight for low values of $l_k$, and high probability weight for high values of $l_k$. Therefore, for any fixed value of $\nkz$, the lower bound of $F(l_k \mid \nkz,\ykz,{{\bmt}}^k_{e_k;n_k})$ for all $l_k$ is obtained with $\ykzu$, and the upper bound of $F(l_k \mid \nkz,\ykz,{{\bmt}}^k_{e_k;n_k})$ for all $l_k$ is obtained with $\ykzl$. There is however no corresponding result for $\nkz$, such that different values of $\nkz$ may minimize (or maximize) $F(l_k \mid \nkz,\ykz,{{\bmt}}^k_{e_k;n_k})$ at different $l_k$’s. Therefore, the $\nkz$ values for lower and upper system reliability bounds are obtained by numeric optimization.
$$\begin{aligned}
\label{eq:25}
\lefteqn{P(\Tsys > t\mid\{\nkz,\ykz, {\mathbf{t}}^k_{e_k;n_k}\}_{k=1}^K)} \hspace*{10ex}\nonumber\\
&= \sum_{l_1=0}^{n_1-e_1} \cdots \sum_{l_K=0}^{n_K-e_K} \Phi(l_1,\ldots,l_K) \prod_{k=1}^K
P(C^k_t = l_k\mid\nkz,\ykz, {{\bmt}}^k_{e_k;n_k})\end{aligned}$$
we cannot simply plug in all the ${\underline{P}}(C^k_t = l_k\mid \ldots)$ to get ${\underline{P}}(\Tsys > t\mid\nkz,\ykz, {\mathbf{t}}^k_{e_k;n_k})$, as the ${\underline{P}}(C^k_t = l_k\mid \ldots)$ could correspond to different $(\nkz,\ykz)$ for each $l_k$. (It would give us a lower bound for ${\underline{P}}(\Tsys > t\mid\ldots)$, however, but it might be very coarse.) While it is clear that the lower bounds for $\ykz$ will lead to the lower system survival function, the role of $\nkz$ is not so clear. We therefore resort to numerical optimization of over $\{\nkz, k=1,\ldots,K\}$ to obtain the lower bound for $P(\Tsys > t\mid\{\nkz,\ykz, {\mathbf{t}}^k_{e_k;n_k}\}_{k=1}^K)$. Writing out , one obtains
$$\begin{aligned}
{{\underline{R}}_\text{sys}}(t \mid t > \tnow)
&= \min_{{\Pi{^{(0)}}_{1}},\ldots,{\Pi{^{(0)}}_{K}}} {R_\text{sys}}\big(t \mid t > \tnow, \cup_{k=1}^K \{\PkZ, {{\bmt}}^k_{e_k;n_k}\}\big) \nonumber\\
&= \min_{\substack{{n{^{(0)}}_{1}} \in \left[{{\underline{n}}{^{(0)}}_{1}}, {{\overline{n}}{^{(0)}}_{1}}\right]\\ \vdots\\ {n{^{(0)}}_{K}} \in \left[{{\underline{n}}{^{(0)}}_{K}}, {{\overline{n}}{^{(0)}}_{K}}\right]}}
\sum_{l_1=0}^{n_1-e_1} \cdots \sum_{l_K=0}^{n_K-e_K} \Phi(l_1,\ldots,l_K) \prod_{k=1}^K P(C^k_t = l_k\mid\nkz,\ykzl, {\mathbf{t}}^k_{e_k;n_k})\,,
\label{eq:sysrel-optim-n0}\end{aligned}$$
such that a $K$-dimensional box-constraint optimization is needed to obtain ${{\underline{R}}_\text{sys}}(t \mid t > \tnow)$. The result for ${{\overline{R}}_\text{sys}}(t \mid t > \tnow)$ is completely analogous. Computing time can furthermore be saved by computing only those summation terms for which $\Phi(l_1,\ldots,l_K) > 0$.
$$\begin{aligned}
\lefteqn{P(\Tsys > t\mid\{\nkz,\ykz, {\mathbf{t}}^k_{e_k;n_k}\}_{k=1}^K)} \\
&= \sum_{l_1=0}^{n_1-e_1} \cdots \sum_{l_K=0}^{n_K-e_K} \Phi(l_1,\ldots,l_K) \prod_{k=1}^K
P(C^k_t = l_k\mid\nkz,\ykz, {\mathbf{t}}^k_{e_k;n_k}) \\
&= \sum_{l_1=0}^{n_1-e_1} \cdots \sum_{l_K=0}^{n_K-e_K} \Phi(l_1,\ldots,l_K) \prod_{k=1}^K
\sum_{j=0}^{n_k-e_k-l_k} (-1)^j \frac{(n_k - e_k)!}{l_k! j! (n_k - e_k - l_k - j)!} \times \\ & \hspace*{45ex}
\left(\frac{\nkn\ykn}{\nkn\ykn + (l_k + j) (t^\kappa - (\tnow)^\kappa)}\right)^{\nkn + 1} \\
&= \sum_{l_1=0}^{n_1-e_1} \cdots \sum_{l_K=0}^{n_K-e_K} \Phi(l_1,\ldots,l_K) \prod_{k=1}^K \label{eq:30}
\sum_{j=0}^{n_k-e_k-l_k} (-1)^j \frac{(n_k - e_k)!}{l_k! j! (n_k - e_k - l_k - j)!} \times \\ & \hspace*{17ex}
\left(\frac{\nkz\ykz + \sum_{i=1}^{e_k} (t_i^k)^\kappa + (n_k-e_k) (\tnow)^\kappa } {\nkz\ykz + \sum_{i=1}^{e_k} (t_i^k)^\kappa + (n_k-e_k-l_k-j) (\tnow)^\kappa + (l_k + j) t^\kappa }\right)^{ \nkz + e_k + 1} \end{aligned}$$
We have implemented the method in the statistical computing environment [@R], using box-constraint optimization via option `L-BFGS-B` of `optim`, and intend to release a package containing functions and code to reproduce all results and figures for the examples in Section \[sec:examples\] below.
Elicitation of prior parameter sets {#sec:elicitation}
===================================
To represent expert knowledge on component failure times through bounds for $\ykz$ and $\nkz$, one can refer to the interpretations as given in Section \[sec:weibull\]: $\ykz$ is the prior expected value of $\lambda_k$, where $\lambda_k$ is linked to expected component lifetimes through . $\nkz$ can be seen as pseudocount, indicating how strong expert knowledge is trusted in comparison to a sample of size $n$. Crucially, the approach allows the expert to give ranges $[\ykzl, \ykzu]$ and $[\nkzl, \nkzu]$ instead of requiring a precise answer.
It is also possible to directly link $\nkz$ and $\ykz$ to observed lifetimes using a prior predictive distribution. Dropping the component index $k$ for ease of notation, this is given by
$$\begin{aligned}
f(t\mid \nz, \yz)
&= \int f(t\mid \lambda) f(\lambda\mid\nz,\yz) {\,\mathrm{d}}\lambda \nonumber\\
&= \beta\, t^{\beta - 1}\, (\nz + 1) \frac{(\nz \yz)^{\nz + 1}}{(\nz \yz + t^\beta)^{\nz + 2}} \,.
\label{eq:tpriopred}\end{aligned}$$
Replacing the prior parameters $\nz$ and $\yz$ with their posterior counterparts $\nn$ and $\yn$ as defined in , the effect of virtual observations on , or the corresponding reliability function, can be determined. This allows to determine $\nz$ and $\yz$ through a number of ‘what-if’ scenarios, by asking the expert to state what (s)he would expect to learn from observing certain virtual data.
This strategy is known as pre-posterior analysis, being first advocated by . We recommend to check whether the effects of $\PkZ$ on the inference of interest (this may not always be the full reliability function) reasonably reflect an expert’s beliefs before the data and in case some specific data become available, both data agreeing with initial beliefs and surprising data. Essentially, we advise to do an analysis like in our examples in Section \[sec:examples\] below, using hypothetical data.
To elicit a meaningful prior distribution, or a set of prior distributions, it is important to ask questions which enable experts to stay close to their actual expertise. discussed the possibility of generalizing the usual conjugate prior distributions, for parameters of exponential family models, by including pseudo-data which are right-censored. If the real data set contains such values, then such generalized priors do not lead to more computational complexities, while they can have several advantages. In addition to providing slightly more general classes of prior distributions through an additional hyperparameter, they may enable more realistic elicitation of expert judgements, for example if the expert has no experience with certain components past a specific life time. For more details we refer to , it should be noted that adopting such generalized prior distributions may also provide more flexibility for modelling the effects of prior-data conflict, this is left as a topic for future research.
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Examples {#sec:examples}
========
As illustrative example, consider a simplified automotive brake system with four types of component. The master brake cylinder (M) activates all four wheel brake cylinders (C1 – C4), which in turn actuate a braking pad assembly each (P1 – P4). The hand brake mechanism (H) goes directly to the brake pad assemblies P3 and P4; the car brakes when at least one brake pad assembly is actuated. The system layout is depicted in Figure \[fig:brakesys-layout\], with those components marked that we assume to fail in each of the three cases studied below. The values for $\Phi \not\in \{0,1\}$ for the complete system are given in Table \[tab:brake-survsign\].
M H C P $\Phi$ M H C P $\Phi$
--- --- --- --- -------- -- --- --- --- --- --------
1 0 1 1 0.25 1 0 2 1 0.50
1 0 1 2 0.50 1 0 2 2 0.83
1 0 1 3 0.75 1 0 3 1 0.75
0 1 0 1 0.50 1 1 0 1 0.50
0 1 0 2 0.83 1 1 0 2 0.83
0 1 1 1 0.62 1 1 1 1 0.62
0 1 1 2 0.92 1 1 1 2 0.92
0 1 2 1 0.75 1 1 2 1 0.75
0 1 2 2 0.97 1 1 2 2 0.97
0 1 3 1 0.88 1 1 3 1 0.88
: Survival signature values $\not\in \{0,1\}$ for the simplified automotive brake system depicted in Figure \[fig:brakesys-layout\].[]{data-label="tab:brake-survsign"}
A fixed prior setting, described in Section \[sec:ex-prior\], will be combined with three different data scenarios, where one observes failure times in accordance with prior expectations in the first case (Section \[sec:ex-case1\]), surprisingly early failures in the second case (Section \[sec:ex-case2\]), and surprisingly late failures in the third case (Section \[sec:ex-case3\]). In each case, it is assumed that C2, C3, P2 and P3 fail, only the failure times are varied, investigating the effect of learning about these failures on the component level. Effects on posterior reliability bounds for the running system are then discussed for all three cases in Section \[sec:ex-sysrel\].
Prior assumptions {#sec:ex-prior}
-----------------
The prior assumptions, which one can imagine to be determined by an expert, or by a combination of expert knowledge and component test data, are given by the prior parameter sets $\PkZ$, $k=\text{M}, \text{H}, \text{C}, \text{P}$, as described in Table \[tab:priorparamsets\]. There, ${{\underline{\operatorname{E}}}}[T_i^k]$ and ${{\overline{\operatorname{E}}}}[T_i^k]$ give the lower and upper bound for expected component lifetimes, respectively, which then have been transformed to bounds for the scale parameter using , resulting in $\ykzl$ and $\ykzu$. For example, according to the expert, the mean time to failure for component type M is between 5 and 8 time units, leading to ${{\underline{y}}{^{(0)}}_{\text{M}}} = 75.4$ and ${{\overline{y}}{^{(0)}}_{\text{M}}} = 244.1$, and the expert considers his knowledge on these expected lifetime bounds as having the strength of at least 2 and at most 5 observations.
[crrrrrrr]{} $k$ & $\beta_k$ & ${{\underline{\operatorname{E}}}}[T_i^k]$ & ${{\overline{\operatorname{E}}}}[T_i^k]$ & $\ykzl$ & $\ykzu$ & $\nkzl$ & $\nkzu$\
M & $2.5$ & $5$
------------------------------------------------------------------------
& $ 8$
------------------------------------------------------------------------
& $75.4$ & $244.1$ & $2$
------------------------------------------------------------------------
& $ 5$
------------------------------------------------------------------------
\
H & $1.2$ & $2$
------------------------------------------------------------------------
& $20$
------------------------------------------------------------------------
& $ 2.5$ & $ 39.2$ & $1$
------------------------------------------------------------------------
& $10$
------------------------------------------------------------------------
\
C & $2 $ & $8$
------------------------------------------------------------------------
& $10$
------------------------------------------------------------------------
& $81.5$ & $127.3$ & $1$
------------------------------------------------------------------------
& $ 5$
------------------------------------------------------------------------
\
P & $1.5$ & $3$
------------------------------------------------------------------------
& $ 4$
------------------------------------------------------------------------
& $ 6.1$ & $ 9.3$ & $1$
------------------------------------------------------------------------
& $10$
------------------------------------------------------------------------
\
These prior assumptions for the four component types are visualized in Figure \[fig:brake-comppriors\], showing the sets of reliability functions corresponding to the prior predictive density . The figure thus displays the bounds for the probability that a single component, having been put under risk at time $0$, will have failed by time $t$. The top left graph in Figure \[fig:brake-sysrels\] shows what the prior assumptions on components signify for the system, depicting the prior bounds for the system reliability on a scale of time elapsed since system startup. For example, the prior probability of the system to survive until time $10$ is between $0.03$% and $6.91$%.
![Sets of prior predictive reliability functions for the four component types, illustrating the choice of prior parameter sets $\PkZ$, $k=\text{M}, \text{H}, \text{C}, \text{P}$.[]{data-label="fig:brake-comppriors"}](fig4){width="\textwidth"}
Case 1: failure times as expected {#sec:ex-case1}
---------------------------------
In the first case, we observe $t_1^\text{C} = 6$, $t_2^\text{C} = 7$, $t_1^\text{P} = 3$, $t_2^\text{P} = 4$, and observe the running system until $\tnow = 8$, i.e., $t_1^\text{M} = t_1^\text{H} = t_3^\text{C} = t_4^\text{C} = t_3^\text{P} = t_4^\text{P} = 8^+$. (Note that component failure times are numbered by order, not by component number in the system layout.) These observations correspond more or less to prior expectations, and the corresponding posterior predictive component distributions are given in Figure \[fig:comppost-1\]. In analogue to Figure \[fig:brake-comppriors\], Figure \[fig:comppost-1\] displays the bounds for the probability that a single component, having been put under risk at time $0$, will have failed by time $t$, after having seen these (partly censored) observations. For easy comparisons, Figure \[fig:comppost-1\] also contains the prior bounds from Figure \[fig:brake-comppriors\].
![Sets of posterior predictive reliability functions for the four component types for observations in line with prior expectations (case 1).[]{data-label="fig:comppost-1"}](fig5a){width="\textwidth"}
We see that the graphs for M and H do not change dramatically, as there is only one component of each in the system to learn from. For C, the bounds have considerably narrowed, showing the effect of having seen the four observations $t_1^\text{C} = 6$, $t_2^\text{C} = 7$, $t_3^\text{C} = t_4^\text{C} = 8^+$. The bounds for P have not narrowed as much; this is due to the two right-censored observations $t_3^\text{P} = t_4^\text{P} = 8^+$; from the viewpoint of the prior, surviving past time 8 is already quite unusual.
Case 2: surprisingly early failure times {#sec:ex-case2}
----------------------------------------
For the second case, with $t_1^\text{C} = 1$, $t_2^\text{C} = 2$, $t_1^\text{P} = 0.25$, $t_2^\text{P} = 0.5$ and $\tnow = 2$ (so $t_1^\text{M} = t_1^\text{H} = t_3^\text{C} = t_4^\text{C} = t_3^\text{P} = t_4^\text{P} = 2^+$), we assume to observe surprisingly early failures; the corresponding posterior predictive component distributions are given in Figure \[fig:comppost-2\].
![Sets of posterior predictive reliability functions for the four component types for surprisingly early failures (case 2).[]{data-label="fig:comppost-2"}](fig6a){width="\textwidth"}
Having observed the system only until $t = 2$, the data are not very informative, such that prior and posterior predictive reliability bounds are very similar. For C, the effect of the early failures is however still visible, and posterior imprecision, i.e., the range between lower and upper posterior bound, is notably larger as compared to prior imprecision, and substantially larger that posterior imprecision in case 1. The effect for P is less pronounced, mainly because observing ${{\bmt}}^\text{P} = (0.25, 0.5, 2^+, 2^+)$ for P is less extreme as observing ${{\bmt}}^\text{C} = (1, 2, 2^+, 2^+)$ for C.
Case 3: surprisingly late failure times {#sec:ex-case3}
---------------------------------------
For the third case, we assume to observe surprisingly late failures, namely $t_1^\text{C} = 11$, $t_2^\text{C} = 12$, $t_1^\text{P} = 8$, $t_2^\text{P} = 9$, and $\tnow = 12$ (so $t_1^\text{M} = t_1^\text{H} = t_3^\text{C} = t_4^\text{C} = t_3^\text{P} = t_4^\text{P} = 12^+$); the corresponding posterior predictive component distributions are given in Figure \[fig:comppost-3\].
![Sets of posterior predictive reliability functions for the four component types for surprisingly late failures (case 3).[]{data-label="fig:comppost-3"}](fig7a){width="\textwidth"}
Having observed the system for a much longer time than in case 2, the data contain now much more information, resulting in considerable differences between prior and posterior bounds. The effect of these surprisingly late failures is most prominent for P, with a set considerably shifted to the right and having very wide posterior bounds. The posterior set for C also indicates that, after having seen these late failures, one expects type C components to fail much later. This effect is also visible for M and H, but is weaker for them as there is only one component of each in the system.
Reliability bounds for the running system {#sec:ex-sysrel}
-----------------------------------------
![Sets of prior and posterior system reliability functions for the three cases on a time showing time elapsed since system startup.[]{data-label="fig:brake-sysrels"}](fig8){width="\textwidth"}
Figure \[fig:brake-sysrels\] depicts the set of prior system reliability functions, together with the sets of posterior system reliability function for the three cases, on a scale of elapsed time since system startup. Due to this time scale, posterior system reliability is 1 at $\tnow$, as it is known that the system has survived until $\tnow = 8, 2, 12$ in case 1,2,3, respectively. For all three cases, the posterior bounds drop faster after $\tnow$ than the prior bounds drop after $t = 0$ since the components in the system have aged until $\tnow$ and so are expected to fail sooner. In the case of surprisingly early failures, posterior bounds are mostly within prior bounds, this is due to $\tnow$ being close to $0$ and weakly informative data in this scenario; the posterior bounds are nevertheless wider than those for case 1; posterior bounds are widest for case 3.
![Imprecision of prior and posterior system reliability sets.[]{data-label="fig:brake-sysrels-imprecision"}](fig9){width="\textwidth"}
This most visible in Figure \[fig:brake-sysrels-imprecision\], which shows ${{\overline{R}}_\text{sys}}(t) - {{\underline{R}}_\text{sys}}(t)$, the difference between upper and lower bound of prior and posterior system reliability. The left panel shows imprecision on the scale of elapsed time since system startup like in Figure \[fig:brake-sysrels\]; the right panel shows imprecision on the scale of prospective time instead, indicating how far in the future periods are for which estimation of system reliability is most uncertain. Posterior imprecision is indeed considerably lower in case 1, where failure times were more or less like expected. On the prospective timescale, one can see that periods of heightened uncertainty are closer to the present for the posteriors, while uncertainty is considerably reduced for periods further in the future.
Concluding Remarks {#sec:concluding}
==================
In this paper we presented a robust Bayesian approach to reliability estimation for systems of arbitrary layout, and showed how the use of sets of prior distributions results in increased imprecision, i.e., more cautious probability statements, in case of prior-data conflict (cases 2 and 3 in Section \[sec:examples\]), while giving more precise reliability bounds when prior and data are in agreement (case 1 in Section \[sec:examples\]). The parameters through which prior information is encoded have a clear interpretation and are thus easily elicited, either directly or with help of the prior predictive . Calculation of lower and upper predictive system reliability bounds is tractable, requiring only a simple $K$-dimensional box-constrained optimization in Equation .
We think that increased imprecision is an appropriate tool for mirroring prior-data conflict when considering sets of priors as is done in both the robust Bayesian and imprecise probability framework. We want to emphazise, however, that this tool may be useful already for just highlighting ‘conflict’ between multiple information sources, and that we do not think that the resulting set of posteriors, although it can form a meaningful basis, must necessarily be used for all consequential inferences, as a strict Bayesian would posit. We believe an analyst is free to reconsider any aspect of a model (of which the choice of prior can be seen to form a part) after seeing the data, and so may use our method only for becoming aware of a conflict between prior and data.
The employed robust Bayesian setting provides many further modelling opportunities beyond the explicit reaction to prior-data conflict. These opportunities have not yet been explored and provide a wide field for further research. An example is our currently ongoing investigation into extending the present model to allow also for an appropriate reflection of very strong agreement between prior and data [@ipmu2016].
There are many aspects to further develop in our analysis and modeling. The general approaches used in this paper, namely the use of sets of conjugate priors for component lifetime models and the survival signature to calculate the system reliability, can be used with other parametric component lifetime distributions that form a canonical exponential family, since for such distributions, a canonical conjugate prior using the same canonical parameters $\nz$ and $\yz$ can be constructed, for details see, e.g., , or . Likewise straightforward to implement is, e.g., the analysis of the effect of replacing failed components in the system on system reliability bounds. Criteria for the trade-off between the cost of replacement and the gain in reliability would have to be adapted for the interval output of our model, leading to very interesting research questions.
To estimate the shape parameters $\beta_k$ together with the scale parameters $\lambda_k$, one could follow standard Bayesian approaches and use a finite discrete distribution for $\beta_k$. Developing this together with suitable sets of priors for $\lambda_k$, in particular to show the effect of prior-data conflict, is another interesting challenge for future research.
Another further important aspect in system reliability we have not accounted for yet is the possibility of common-cause failures, i.e., failure events where several components fail at the same time due to a shared or common root cause. This could be done by combining the common-cause failure model approaches of and .
On a more abstract level, the choice for the set of prior parameters (generating the set of prior component failure distributions) as $[\nkzl, \nkzu] \times [\ykzl, \ykzu]$ has the advantage of allowing for easy elicitation and tractable inferences, but it may not be suitable to reflect certain kinds of prior knowledge. Also, as studied in [@Walter2011a] and [@diss §3.1], the shape of the prior parameter set has a crucial influence on model behaviour like the severity of prior-data conflict reaction. As noted in [@diss pp. 66f], more general prior parameter set shapes are possible in principle, but may be more difficult to elicit and make calculations more complex.
Acknowledgements {#acknowledgements .unnumbered}
================
Gero Walter was supported by the DINALOG project CAMPI (“Coordinated Advanced Maintenance and Logistics Planning for the Process Industries”).
\[section:references\]
Notation
========
*The following symbols are used in this paper:*
------------------------ --------------------
$D$ pile diameter (m);
$R$ distance (m); and
$C_{\mathrm{Oh\;no!}}$ fudge factor.
------------------------ --------------------
[^1]: Postdoctoral Researcher, School of Industrial Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands. E-mail: g.m.walter@tue.nl. Corresponding author.
[^2]: Professor, Department of Mathematical Sciences, Durham University, Durham, United Kingdom. E-mail: frank.coolen@durham.ac.uk.
|
---
abstract: 'We introduce a general framework to deterministically construct binary measurement matrices for compressed sensing. The proposed matrices are composed of (circulant) permutation submatrix blocks and zero submatrix blocks, thus making their hardware realization convenient and easy. Firstly, using the famous Johnson bound for binary constant weight codes, we derive a new lower bound for the coherence of binary matrices with uniform column weights. Afterwards, a large class of binary *base matrices* with coherence asymptotically achieving this new bound are presented. Finally, by choosing proper rows and columns from these base matrices, we construct the desired measurement matrices with various sizes and they show empirically comparable performance to that of the corresponding Gaussian matrices.'
address: 'Graduate School at Shenzhen, Tsinghua University'
bibliography:
- 'IEEEabrv.bib'
- 'OptBin.bib'
title: Deterministic Constructions of Binary Measurement Matrices with Various Sizes
---
Compressed sensing, deterministic measurement matrix, coherence, Johnson bound, Welch bound.
Introduction
============
Compressed sensing (CS) [@candes2006a; @donoho2006] is a novel sampling technique that samples sparse signals at a rate far lower than the Nyquist-Shannon rate. Consider a *$k$-sparse* signal $\textit{\textbf{x}}\in\mathbb{R}^{n}$ with at most $k$ nonzero entries, if we make a linear sampling $\textit{\textbf{y}}=A\textbf{\textit{x}}$ of $\textit{\textbf{x}}$ with the *measurement matrix* $A\in \mathbb{R}^{m\times n}$, where $m<n$, then $\textit{\textbf{x}}$ could be recovered by solving an *$\ell_1$-minimization* problem [@Candes2005] or by a greedy algorithm such as *orthogonal matching pursuit* (OMP) [@Tropp2007]. Actually, if $A$ satisfies the *restricted isometry property* (RIP) [@Candes2005] of order $k$ with enough small $0<\delta_k^A<1$, signals with sparsity $O(k)$ can be exactly recovered by $\ell_1$-minimization or OMP [@Foucart2013 pp. 26], where $\delta_k^A$ denotes the *restricted isometry constant* of $A$.
Many random matrices, such as the Gaussian matrices, have been proved to satisfy RIP of order $k$ *with high probability* if $k\leq O(m/\log(n/k))$ [@Baraniuk2008]. However, there is no guarantee that a specific realization of a random matrix works and some random matrices require lots of storage space. In contrast, a deterministic matrix is often generated on the fly and RIP could be verified definitely. Therefore, deterministic measurement matrices are often preferable in practice.
The *coherence* $\mu(A)$ of a deterministic matrix $A$ is often exploited to prove RIP since $\delta_{k}^A \leq (k-1)\mu(A)$ [@Bourgain2011], where $$\label{eq:cohdef}
\mu(A)\triangleq \max_{1\leq i\neq j\leq n}{\frac{|\langle\textit{\textbf{a}}_{i}, \textit{\textbf{a}}_{j}\rangle|}{||\textit{\textbf{a}}_{i}||_{2}||\textit{\textbf{a}}_{j}||_{2}}},$$ $\textit{\textbf{a}}_{1},
\textit{\textbf{a}}_{2}, \ldots, \textit{\textbf{a}}_{n}$ are the $n$ columns of $A$, $\langle \textit{\textbf{a}}_i, \textit{\textbf{a}}_j\rangle \triangleq \textit{\textbf{a}}_i^T \textit{\textbf{a}}_j$ and for any $\textit{\textbf{z}}=(z_{1}, z_{2}, \ldots, z_{m})^T\in\mathbb{R}^{m}$ , $||\textit{\textbf{z}}||_{2}\triangleq \sqrt{\sum_{i=1}^m z_{i}^2}$. Therefore, given $\mu(A)$, $A$ satisfies RIP of order $$\label{eq:riporder}
k <1+\frac{1}{\mu(A)}.$$
Recently, binary deterministic matrices have been introduced into compressed sensing due to their simplicity [@DeVore2007; @Amini2011; @Lu2012; @Dimakis2012; @Tehrani2013; @Li2014a]. For example, let $q$ be a prime power, DeVore proposed a class of binary (before column normalization) $q^2\times q^{r+1}$ matrices satisfying RIP of order $k<q/r+1$, where $1<r<q$ is a constant integer [@DeVore2007]. By using the codewords of orthogonal optical codes as the columns of matrices, Amini *et al.* constructed a class of binary measurement matrices [@Amini2011]. In [@Li2014a], the incidence matrices of several packing designs based on finite geometry are applied into compressed sensing. These matrices have relatively low coherence and show empirically good performance in compressed sensing.
However, many of them are often based on Galois fields (GF), thus having restrictions to the numbers of rows[^1]. Recently, utilizing the parallel structure of Euclidean geometry, we proposed a class of binary measurement matrices with a bit more flexible sizes [@sxxl2012]. In this paper, we introduce more such matrices. In particular, we focus on the binary matrix $H$ with a constant column weight. By viewing the columns of $H$ as codewords of a *constant weight code* [@MacWilliams1979 pp. 523–531], we derive a new lower bound for its coherence $\mu(H)$ with the help of the famous Johnson bound [@Johnson1962], which improves the traditional Welch bound [@Welch1974]. Then we present a subclass of binary (often quasi-cyclic) matrices asymptotically achieving this new bound and some examples from structural low-density parity-check (LDPC) codes [@Gallager1962] are given. Based on these matrices, a general framework is proposed to obtain practical measurement matrices with various sizes. Finally, simulations show that the proposed matrices perform comparably to, sometimes even better than, the corresponding Gaussian matrices.
Main Results
============
Coherence of Binary Matrices
----------------------------
In this part, we analyze the coherence of binary matrices which have uniform column weights $\gamma>1$. Firstly, some preliminaries are presented. For any matrix $H\in\{0,1\}^{m\times n}$, there is a *Tanner graph* $G_{H}$ [@Tanner1981] corresponding to $H$. $G_H$ is a bipartite graph comprised of $n$ variable nodes labelled by the elements of $I=\{1,2,\ldots,n\}$, $m$ check nodes labelled by the elements of $J=\{1,2,\ldots,m\}$, and the edge set $E\subseteq\{(i,j):i\in I, j\in J\}$, where there is an edge $(i,j)\in E$ if and only if $h_{ji} = 1$. The *girth* $g(H)$ of $H$ or $G_H$ is defined as the minimum length of cycles in $G_{H}$. Girth is always an even number not smaller than 4. $H$ is said to be $(\gamma,\rho)$-*regular* if $H$ has uniform column weight $\gamma$ and uniform row weight $\rho$.
A binary matrix $H$ with uniform column weight $\gamma$ can be viewed as a collection of codewords (as columns of $H$) of certain binary constant weight codes. An $(m, d, \gamma)$ constant weight code $\mathcal{C}$ is a set of binary vectors of length $m$, weight $\gamma$ and minimum distance $d$, where $d$ is always an even number. Let $A(m,d,\gamma)$ be the largest number of codewords in any $(m,d,\gamma)$ constant weight codes, $A(m,d,\gamma)$ could be bounded by the famous Johnson bound [@Johnson1962]: $$\label{eq:johnson2}
A(m,2\delta,\gamma)\le \lfloor\frac{m}{\gamma}\lfloor\frac{m-1}{\gamma-1}\cdots\lfloor\frac{m-\gamma+\delta}{\delta}\rfloor\cdots\rfloor\rfloor,$$ where $\lfloor x\rfloor$ denotes the largest integer no larger than $x$.
Traditionally, the coherence of a matrix is bounded by the Welch bound [@Welch1974]: $$\label{eq:welch}
\mu(A)\geq\sqrt{\frac{n-m}{m(n-1)}}.$$ The equality in (\[eq:welch\]) achieves *if and only if* $A$ is an *equiangular tight frame* (ETF), i.e., A should satisfy the following 3 conditions: (a) the columns of $A$ have unit norm, (b) the rows of $A$ are orthogonal with equal norm, and (c) the inner products between any two different columns of $A$ are equal in modulus [@Bandeira2013]. Therefore, for any binary matrix $H$ with uniform column weight $\gamma>1$, the rows of $H$ will not be orthogonal, thus the Welch bound (\[eq:welch\]) could not be achieved.
In the following, we analyze the coherence of binary matrices by the Johnson bound. Consider the binary $m\times n$ matrix $H$ with uniform column weight $\gamma>0$, suppose the maximum inner product of any two columns of $H$ is $\lambda>0$, then $H$ has coherence $\mu(H)=\frac{\lambda}{\gamma}$. In particular, when $H$ has girth $g(H)>4$, any two distinct columns of $H$ have at most one pair of common ‘1’ at the same row, i.e., $\lambda=1$, we have $$\label{eq:cohbing6}
\mu(H)=\frac{1}{\gamma}.$$
By viewing the column vectors of $H$ as the codewords of an $(m,d,\gamma)$ constant weight code $\mathcal{C}$, then $d=2\gamma-2\lambda$. From the Johnson bound (\[eq:johnson2\]), we have the following fact.
\[lem:johnsonanyg\] For any binary matrix $H\in\{0,1\}^{m\times n}$ with uniform column weight $\gamma>1$, maximum inner product $0<\lambda<\gamma$ of any two distinct columns, $m$, $n$, $\gamma$ and $\lambda$ should satisfy: $$\label{eq:mngljohnson2}
n\le \lfloor\frac{m}{\gamma}\lfloor\frac{m-1}{\gamma-1}\cdots\lfloor\frac{m-\lambda}{\gamma-\lambda}\rfloor\cdots\rfloor\rfloor.$$
In particular, when $H$ has girth $g(H)>4$, we can obtain an explicit lower bound for the coherence of $H$.
\[th:john2\] Let $H\in\{0,1\}^{m\times n}$ be a binary matrix with uniform column weight $\gamma>1$, girth $g(H)>4$ and coherence $\mu(H)\ne0$, then $$\label{eq:cohg6}
\mu(H)\ge\frac{2n}{n+\sqrt{n^2+4mn(m-1)}}.$$
When $g(H)>4$ and $\mu(H)\ne 0$, $\lambda=1$. By (\[eq:mngljohnson2\]), $n \le \lfloor\frac{m}{\gamma}\lfloor\frac{m-1}{\gamma-1}\rfloor\rfloor\le \frac{m(m-1)}{\gamma(\gamma-1)}$, (\[eq:cohg6\]) follows since $\mu(H)=\frac{1}{\gamma}$.
By a simple deduction, it is easy to see that (\[eq:cohg6\]) is always tighter than the Welch bound (\[eq:welch\]) if $m<n$. In addition, throughout this paper, we call the binary matrix with uniform column weight $\gamma>0$, girth $g>4$ and coherence $\mu\ne 0$ *(asymptotically) optimal* if the coherence of this matrix (asymptotically) achieves the lower bound (\[eq:cohg6\]).
Similar to the Johnson bound, (\[eq:cohg6\]) could be achieved. For example, let $H$ be the point-line incidence matrix (rows of $H$ corresponding to the points and columns to the lines) of the Euclidean plane $EG(2,q)$, where $q$ is a prime power. $H$ is a $(q,q+1)$-regular matrix with the size $q^2\times (q^2+q)$, $g(H)=6$, and it is easy to verify that (\[eq:cohg6\]) is achieved, see [@sxxl2012] for more details of $H$ and its application to compressed sensing.
A Subclass of Asymptotically Optimal Binary Matrices in Terms of Coherence
--------------------------------------------------------------------------
In this part, we show a subclass of binary matrices with coherence asymptotically achieving the lower bound (\[eq:cohg6\]). Later on, they will be used to obtain the desired measurement matrices with various sizes and empirically good performance.
Consider an ${s^2\times s^2}$ *base matrix* as follows $$\label{eq:basem}
H = [H_{i,j}], \quad 1\leq i,j\leq s,$$ where $s>1$ and $H_{i,j}\in\{0,1\}^{s\times s}$ is either a permutation block or a zero block $\textbf{0}=\{0\}^{s\times s}$. A *permutation block* $B\in\{0,1\}^{s\times s}$ is a square matrix with each row and each column having exactly one element ‘1’. If $B$ is also cyclic, then $B$ is called a *circulant permutation block*. Each $[H_{i,1}, H_{i,2},\ldots,H_{i,s}]$ (or $[H_{1,j}^T, H_{2,j}^T,\ldots,H_{s,j}^T]^T$) of $H$ is called a *row-block* (or *column-block*) of $H$. $H$ satisfies the following two properties.
- (P1) Every column-block of $H$ has exactly $t$ zero blocks, so does each row-block , i.e., $H$ is $(s-t,s-t)$-regular, where $0\le t\ll s$ is a small constant.
- (P2) The girth of $H$ is larger than 4, i.e., $g(H)>4$.
The $s^2\times s^2$ base matrix $H$ has coherence $\mu(H)=\frac{1}{s-t}$. According to Theorem \[th:john2\], the (nonzero) coherence of any $s^2\times s^2$ binary matrix with uniform column weight $\gamma>1$ and girth larger than 4 has the lower bound $\frac{2}{1+\sqrt{4s^2-3}}\rightarrow\frac{1}{s}$ if $s\rightarrow\infty$. Since $t\ge 0$ is a small constant, the coherence of the base matrix $H$ is asymptotically optimal.
In addition, for some submatrices of the base matrices, their coherences are also asymptotically optimal. Let $A(\gamma,s,t)$ be a $\gamma s\times s^2$ submatrix of the base matrix $H$ by simply choosing the first $\gamma$ row-blocks of $H$, i.e., $$\label{eq:gamm}
A(\gamma,s,t)\triangleq [H_{i,j}], \;\;1\leq i\leq \gamma\le s,\;\;1\leq j\leq s.$$
\[rem:asymopt\] Suppose $\gamma=cs$, where $0<c<1$ is a constant such that $cs$ is an integer. When $t=0$, $A(cs,s,0)$ is a $(cs,s)$-regular matrix with coherence $\mu(A(cs,s,0))=\frac{1}{cs}$. According to (\[eq:cohg6\]), for any binary $cs^2\times s^2$ matrix with uniform column weight, girth larger than 4 and nonzero coherence, its coherence has the lower bound $\frac{1}{0.5+\sqrt{c^2s^2+0.25-c}}\rightarrow\frac{1}{cs}$ if $s\rightarrow\infty$. Therefore, the submatrix $A(cs,s,0)$ is also asymptotically optimal in terms of coherence.
In the following, we review several examples of satisfactory base matrices from structured (often quasi-cyclic) LDPC codes.
\[con:add\] Let $H=H(q,q)$, where $q$ is an odd prime and $H(q,q)$ is the binary matrix defined in (7) in [@Liu2013] with $r=q$. Then $H\in\{0,1\}^{q^2\times q^2}$ is a $(q,q)$-regular base matrix with $t=0$.
\[con:bjq\] Let $H = H^{(1)}(q,q,0)$, where $q$ is a prime power and $H^{(1)}(q,q,0)$ is the parity-check matrix of a first class of B–J based LDPC code proposed in [@Ge2006 Section III.A]. Then $H\in\{0,1\}^{q^2\times q^2}$ is a $(q,q)$-regular base matrix with $t=0$.
Let $GF(q) = \{\alpha^{-\infty}= 0, \alpha^0 = 1, \alpha, \ldots, \alpha^{q-2}\}$ be a Galois field with primitive element $\alpha$. Establish a one-to-one *$(q-1)$-fold correspondence* between the elements in $GF(q)$ and the matrices $P\in\{0,1\}^{(q-1)\times (q-1)}$ as follows:
- $0$ is mapped to the zero block $\textbf{0}=\{0\}^{(q-1)\times (q-1)}$;
- $\alpha^i$ is mapped to a circulant permutation block $P_{q-1}^i$, where $0\leq i\leq q-2$, $$\begin{aligned}
\label{eq:mpq}
P_{q-1}\triangleq\left[
\begin{array}{ccccc}
0&1&0&\cdots&0\\
0&0&1&\cdots&0\\
\vdots&\vdots&\vdots&\ddots&\vdots\\
0&0&0&\cdots&1\\
1&0&0&\cdots&0\\
\end{array}
\right]_{(q-1)\times (q-1)},\end{aligned}$$ $P_{q-1}^i$ denotes the $i$-th power of $P_{q-1}$ and $P_{q-1}^0\triangleq I_{q-1}$ is the identity matrix of order $q-1$.
In the following Examples \[con:rslatin\] and \[con:latin\], we obtain the base matrix $H$ by firstly constructing a matrix $L\in GF(q)^{(q-1)\times(q-1)}$ based on the *Latin square* and then replacing each element in $L$ with a circulant permutation block or a zero block $P\in\{0,1\}^{(q-1)\times (q-1)}$.
An Latin square of order $n$ is an $n\times n$ matrix with $n$ distinct symbols, each of which occurrs exactly once in each row and exactly once in each column.
\[con:rslatin\] Let $\beta$ be a nonzero element in $GF(q)$ and $L_{\rm RS}(\beta)$ be the following Reed-Solomon codes based (cyclic) Latin square of order $q-1$ over $GF(q)\setminus\{-\beta\}$: $$\begin{aligned}
L_{\rm RS}(\beta) =
\left[
\begin{array}{cccc}
1-\beta&\alpha-\beta&\ldots&\alpha^{q-2}-\beta\\
\alpha^{q-2}-\beta&1-\beta&\ldots&\alpha^{q-3}-\beta\\
\vdots&\vdots&\ddots&\vdots\\
\alpha-\beta&\alpha^{2}-\beta&\ldots&1-\beta
\end{array}
\right].\end{aligned}$$ Expand $L_{\rm RS}(\beta)$ by replacing each entry with a circulant permutation block or a zero block according to the $(q-1)$-fold correspondence, and then we could get a quasi-cyclic base matrix $H\in\{0,1\}^{(q-1)^2\times (q-1)^2}$ with $t=1$. Note that no matter which nonzero $\beta$ is chosen, there is exactly one $0$ in each row and exactly one $0$ in each column of $L_{\rm RS}(\beta)$. Therefore, the resulting $H$ is a $(q-2,q-2)$-regular matrix. Finally, as there are $q-1$ nonzero elements $\beta\in GF(q)$, there will be $q-1$ such Latin squares $L_{\rm RS}(\beta)$ and thus $q-1$ such base matrices $H$.
\[con:latin\] Let $\beta$ be any nonzero element in $GF(q)$ and $\bar L(\beta)=W$ be the Latin square of order $q$ over $GF(q)$ in [@Zhang2010a Equation (11)] with $\beta=\eta$. Choose the following $(q-1)\times(q-1)$ submatrix $L(\beta)$ of $\bar L(\beta)$, $L(\beta)=$ $$\begin{aligned}
\left[
\begin{array}{cccc}
\beta-1&\beta-\alpha&\ldots&\beta-\alpha^{q-2}\\
\alpha\beta-1&\alpha\beta-\alpha&\ldots&\alpha\beta-\alpha^{q-2}\\
\vdots&\vdots&\ddots&\vdots\\
\alpha^{q-2}\beta-1&\alpha^{q-2}\beta-\alpha&\ldots&\alpha^{q-2}\beta-\alpha^{q-2}
\end{array}
\right].\end{aligned}$$ Expand $L(\beta)$ according to the $(q-1)$-fold correspondence. In this way, we obtain a quasi-cyclic and $(q-2,q-2)$-regular base matrix $H\in\{0,1\}^{(q-1)^2\times (q-1)^2}$ with $t=1$.
\[con:bjq1\] Let $q$ be a prime power. Let $H = H^{(2)}(q-1,q-1,0)$, where $H^{(2)}(q-1,q-1,0)$ is the parity-check matrix of a second class of B–J based LDPC code proposed in [@Ge2006 Section III.B]. Then $H\in\{0,1\}^{(q-1)^2\times (q-1)^2}$ is a quasi-cyclic and $(q-2,q-2)$-regular base matrix with $t=1$.
General Framework of Matrix Constructions
-----------------------------------------
In this part, we give the general framework to deterministically construct binary measurement matrices, see Algorithm \[alg:construction\]. Note that in the second step of Algorithm \[alg:construction\], we choose the $s^2\times s^2$ base matrix $H$ in such a way as to make the resulting $A$ have the smallest coherence. In practice, we often require $m\gg\sqrt{n}$, thus the outputted matrix will have small coherence and empirically good performance. For example, $m$ scales linearly with $n$, i.e., $m=cn$, where $0<c<1$ is a constant. See the following Theorem \[th:coh\] for a formalized explanation.
**Input**: Matrix size $m$ and $n$.\
**Output**: A binary measurement matrix $A\in \{0,1\}^{m\times n}$.\
**Steps**:
\(1) Base matrix construction: construct several classes of $\bar s^2\times \bar s^2$ base matrices satisfying (P1) and (P2) with $t\ll \bar{s}$.
\(2) Base matrix selection: choose an $s^2\times s^2$ matrix $H$ among these base matrices such that $s\ge \sqrt{n}$ and $(m/s-t)$ is as large as possible.
\(3) Extra elements deletion: remove the last $s^2-m$ rows and the last $s^2-n$ columns of $H$ and output the resulting submatrix as $A$.
\[th:coh\] For any measurement matrix $A\in \{0,1\}^{m\times n}$ constructed by Algorithm \[alg:construction\], we have $$\label{eq:coharbm}
\mu(A)\leq \frac{1}{\gamma-t},$$ where $\gamma = \left\lfloor\frac{m}{s}\right\rfloor$, $0\leq t\ll s$ is a fixed integer.
According to (P1), the minimum possible column weight of $A$ is $\gamma-t$. From (P2), the inner product of any two columns of $A$ is at most 1. By (\[eq:cohdef\]), (\[eq:coharbm\]) follows directly.
As stated in Remark \[rem:asymopt\], when $\gamma=\frac{m}{s}=cs$, $n=s^2$ and $t=0$, the binary matrix $A(cs,s,0)$ outputted by Algorithm \[alg:construction\] is asymptotically optimal in terms of coherence. In other cases, the structure (and thus the coherence) of the resulting matrix $A$ with $m=cn$ and $n=s^2$ is very close to that of $A(cs,s,0)$. Moreover, removing columns of a measurement matrix will not deteriorate its empirical performance. Therefore, it is reasonable to conjecture that the measurement matrices obtained by Algorithm \[alg:construction\] will often perform well in practice and this will be verified by the following experimental results.
Experimental Results {#sec:simu}
====================
In the following simulations, for each measurement matrix $A$ and each $k$-sparse signal $\textit{\textbf{x}}$, we conduct an experiment using $M=1000$ Monte Carlo trials. In the $i$-th trial, a relative recovery error $e_i = ||\textit{\textbf{x}}^*-\textit{\textbf{x}}||_2/||\textit{\textbf{x}}||_2$ is computed, where $\textit{\textbf{x}}^*$ denotes the recovered signal. If $e_i\leq 0.001$, we declare this recovery to be “perfect”. Finally, an average percentage of perfect recovery over the $M$ trials is obtained and shown as a point in the figures.
At first, we give an example to show the empirical effectiveness for the base matrix selecting strategy in the second step of Algorithm \[alg:construction\]. Suppose only one class of base matrices are constructed in the first step, such as the base matrices in Example \[con:rslatin\], and now we want to construct a $100\times 300$ binary measurement matrix. Since $\sqrt{300}=17.32$, we can set $q$ to be $19, 23$, or even larger prime power. The OMP recovery performance of the desired measurement matrices obtained by setting $q=19$, $q=23$ and the Gaussian matrix (‘Rnd’) with the same size are shown in Fig. \[fig:schoose\].
![Empirical performance of the $100\times 300$ measurement matrices obtained by Example \[con:rslatin\] by setting $q=19$ and $q=23$ and the corresponding Gaussian random matrix under OMP recovery.[]{data-label="fig:schoose"}](schoose.pdf){width="45.00000%"}
It is clear that the matrix based on Example \[con:rslatin\] with $q=19$ is better than that with $q=23$, which agrees with the base matrix choosing strategy in the second step of Algorithm \[alg:construction\].
In the following, we consider several binary measurement matrices based on the base matrices in Examples \[con:add\]–\[con:bjq1\], see Fig. \[fig:comparesmall\] for the empirical performance of these matrices with small sizes and Fig. \[fig:comparelarge\] for that of matrices with larger sizes.
Let $q=31$ in Example \[con:add\], $q=32$ in Examples \[con:bjq\]–\[con:bjq1\], $\beta=1$ in Examples \[con:rslatin\] and \[con:latin\]. For each Example \[con:add\]–\[con:bjq1\], construct $3$ measurement matrices with sizes $190\times940$, $225\times950$, and $260\times960$ by removing the last extra rows and columns from the $5$ different base matrices. See Fig. \[fig:comparesmall\] for their empirical performance and the corresponding Gaussian matrices.
![Empirical performance of the proposed measurement matrices and the corresponding Gaussian random matrices with sizes $190\times940$, $225\times950$, and $260\times960$ (the three curve bundles from left to right, respectively) under OMP recovery.[]{data-label="fig:comparesmall"}](comparesmall.pdf){width="45.00000%"}
Let $q=61$ in Example \[con:add\] and $q=64$ in Example \[con:bjq\]–\[con:bjq1\]. For each Example \[con:add\]–\[con:bjq1\], construct $3$ matrices with sizes $450\times3500$, $500\times3600$, and $550\times3700$. See Fig. \[fig:comparelarge\] for their empirical performance.
![Empirical performance of the proposed measurement matrices and the corresponding Gaussian matrices with sizes $450\times3500$, $500\times3600$, and $550\times3700$ (the three curve bundles from left to right, respectively) under OMP recovery.[]{data-label="fig:comparelarge"}](comparelarge.pdf){width="45.00000%"}
In Figs. \[fig:comparesmall\] and \[fig:comparelarge\], all of the proposed matrices perform as well as, sometimes even better than, the corresponding Gaussian matrices. In addition, it is easy to see that the matrices from Example \[con:add\] often perform slightly better than those from Example \[con:bjq\]–\[con:bjq1\] due to the specific matrix sizes. Simple computations on the upper bounds of coherence (according to Theorem \[th:coh\]) show that each coherence upper bound of the six matrices obtained by Example \[con:add\] is smaller than (or sometimes equal to) that of other examples. This also agrees with the base matrix selecting strategy in the second step of Algorithm \[alg:construction\].
Conclusions and Discussions
===========================
This paper has introduced a general framework to deterministically construct binary measurement matrices with various sizes and empirically good performance. In particular, some of them are also shown to be asymptotically optimal according to a new lower bound of coherence derived with the help of the famous Johnson bound. Moreover, these matrices are binary, sparse, and mostly quasi-cyclic, which will benefit the hardware implementation.
This paper mainly focuses on binary matrices with girth larger than 4. However, as has been indicated by Lu [@Lu2012], some empirically even better binary matrices lie in the region of girth $g=4$. In addition, a $(q^2+1)\times q(q^2+1)$ binary measurement matrix with uniform column weight $\gamma=q+1$ and $\lambda=2$ (thus girth $g=4$) has been proposed in [@Li2014a]. It is easy to verify that this matrix achieves the Johnson bound and Equation (\[eq:mngljohnson2\]) in this paper and they are also shown to perform empirically well in [@Li2014a]. As a result, it will be interesting to carry out some theoretical analysis explicitly on binary matrices with $g=4$ and construct more such (asymptotically) optimal matrices which may show perhaps better performance in practice.
[^1]: Generally, removing some columns from a matrix will not deteriorate its theoretical (such as coherence and RIP) and empirical performance.
|
---
abstract: '[We reexamine the basics of the holographic principle to see the consequences of discriminating between stable and unstable subnuclear entities. This gives an insight into predominant modes of decay inherent in the lightest glueball, and enables us to propose a particularly promising way for detecting this particle. ]{}'
---
[How to detect the lightest glueball]{}\
[B. P. Kosyakov${}^{a,b}$, E. Yu. Popov${}^a$, and M. A. Vronski[ĭ]{}${}^{a,c}$]{}\
[[${}^a$Russian Federal Nuclear Centre–VNIIEF, Sarov, 607188 Nizhni[ĭ]{} Novgorod Region, Russia;\
${}^b$Moscow Institute of Physics [&]{} Technology, Dolgoprudni[ĭ]{}, 141700 Moscow Region, Russia;\
${}^c$Sarov Institute of Physics [&]{} Technology, Sarov, 607190 Nizhni[ĭ]{} Novgorod Region, Russia.]{}\
[E-mail:]{} ${\rm kosyakov.boris@gmail.com}$ (corresponding author) ]{}
[**Keywords**]{}: glueball, instability, gauge/gravity duality, $\gamma\gamma$ collider
Introduction {#Introduction}
============
The existence of hadrons made of gluons alone, now known as glueballs, was predicted [@FritzschGell-Mann], [@FritzschMinkowski], [@FreundNambu], [@JaffeJohnson] at the dawn of the age of quantum chromodynamics (QCD). However, such particles have not yet been observed with certainty [@Crede], [@Olive]. The lightest glueball, a colour singlet of two gluons, is specified by zero total angular momentum, and positive parity and charge parity, $J^{PC}=0^{++}$. According to lattice and sum rule calculations, this particle has mass in the range of about $1-1.7$ GeV, for a review see [@Close], [@Amsler], [@MathieuKochelevVento], [@Ochs]. Is the lightest glueball stable? The situation with conventional hadrons is various. The lightest meson, $\pi^0$, is unstable, while the lightest baryon, $p$, is stable at least over $3\cdot 10^{33}$ years. This dissimilarity may be attributed to the conserved baryon number responsible for the stability of protons, and the absence of similar conserved quantum numbers from meson states. At first glance the issue is simple. There is a set of hadrons with suitable quantum numbers which is smaller in mass than the lightest glueball, and hence the decay into these hadrons is inevitable. However, the possibility that there exists a conserved topological charge which affords protection against the decay of the lightest glueball must not be ruled out when it is considered that the analytical and topological properties of solutions of pure Yang–Mills theory are still poorly understood. A further argument in support of stability is that glueballs are immune from the weak and electromagnetic interactions because gluons are not involved in these interactions. As to the strong interaction of a glueball with its environment, this is a subtle point. All hadrons are colourless objects. Nucleons are assembled in nuclei due to a residual colour interaction between quarks, similar to the van der Waals force between neutral molecules. This mechanism is held to be valid in the semiclassical picture where quarks are represented by point particles linked together by thin tubes which enclose the total colour flux of the gluon field. In contrast, localized (kink-like) finite-energy solutions of pure Yang–Mills theory are forbidden on the strength of a theorem by Coleman [@Coleman]. Therefore, there is no rendition of a glueball as a classical colourless localized configuration. Accordingly, the idea of the residual colour interaction between a glueball and its environment remains problematic.
Is it possible to clarify this issue without going into details of the non-perturbative glueball structure? With this aim in view, it would be well to make use of gauge/gravity duality, aka AdS/CFT correspondence, or holography [@Maldacena], [@Witten], [@Gubser]; for a full coverage of ideas and methods of gauge/gravity duality see, [*e. g.*]{}, [@Ammon], [@Nastase]. Loosely speaking, this is a doctrine whereby a good part of subnuclear physics in a four-dimensional realm is modelled on physics of black holes and similar objects (black rings, black branes, etc.) in five-dimensional anti-de Sitter space (${\rm AdS}_5$) whose boundary is just this four-dimensional realm. But this understanding of gauge/gravity duality apparently indulges in wishful thinking. In their 2009 paper [@KlebanovMaldacena] Klebanov and Maldacena remind the reader of the physics joke about the spherical cow as an idealization of a real one, and admit that “in the AdS/CFT correspondence, theorists have really found a hyperbolic cow”. To remedy the situation, a major portion of the standard holographic mapping is to be amputated. As a partial implementation of this project the following criterion for discriminating between stable and unstable microscopic systems was offered [@KPV19]: a system is stable when its dual is an extremal black object. For example, the dual of a proton is an extremal rotating charged black hole. The mapping of the physics of extremal black objects to the physics of stable nuclear and subnuclear entities is by far a sound element of gauge/gravity duality.
It may seem that the converse is also true, that is, if a microscopic system is unstable, its gravitational counterpart is an ordinary black hole amenable to Hawking evaporation. The fact that neutral spinless mesons are unstable may count in favour of this statement. The instability of these particles is related to the absence of extremal black objects among their possible counterparts which are typically Schwarzschild black holes. Actually, there is no well-defined mapping between a particular unstable microscopic system and its associated black object in ${\rm AdS}_5$. The lack of such mappings is clarified in Sect. 2 where we seek to determine the limits and scope of the gauge/gravity duality. It transpires here that the lightest glueball is found in the frontier zone separating the domain in which the holographic principle holds and that in which this principle fails. We therefore must handle the holographic principle with extreme care. Reasoning from some basic symmetry properties of gauge/gravity duality we show that the lightest glueball decays into spinning particles. We then reveal predominant modes of decay inherent in the lightest glueball, which enables us to propose in Sect. 3 a convenient way for making the lightest glueball experimentally recognizable.
It is commonly supposed that the glueball field mixes with the quark-antiquark fields, $\left(u{\bar u}+d{\bar d}\right)/\sqrt{2}$ and $s{\bar s}$, to form the experimentally observed spinless isoscalar resonances, such as $f_0(980)$, $f_0(1370)$, $f_0(1500)$, and $f_0(1710)$, whose masses are in the range of about $1 - 2$ GeV. The great majority of current studies pivots on the mixing problem; for a review see [@Close], [@Amsler], [@MathieuKochelevVento], [@Ochs], and references therein. But our concern is with the lightest glueball in its pure states. With this in mind, we invoke recent experimental evidence on light-by-light scattering to justify our proposal of using $\gamma\gamma$ colliders to lop off the mixed aggregates and selects pure states of the lightest glueball.
Critique of pure reason underlying the holography {#Critique}
=================================================
Does the holography indeed bear no relation to unstable microscopic systems and their associated gravitational black objects? One may adduce the following evidence.
It is a central tenet of quantum mechanics that microscopic systems of a given species are identical and indistinguishable. In order for a well-defined holographic mapping to be attained, their gravitational duals must possess identical properties, in particular equal masses. However, given two black holes of equal masses at a fixed instant in a particular Lorentz frame, their masses are found to be different in other frames. Indeed, if two black holes are synchronously evaporated in this Lorentz frame, the synchronization of rates of their evaporation fails in other frames because the simultaneity of separated events is frame-dependent.
We further note that quantum-mechanical reversibility presents a formidable obstacle for gaining a well-defined correspondence between a given unstable microscopic system and its gravitational counterpart. The latter suffers from the Hawking evaporation which is an irreversible process.
A notable geometric feature of Schwarzschild black holes in ${\rm AdS}_5$ is that they have the greatest possible spatial isometry group SO$(4)$. This group is equivalent to SO$(3)\times$SO$(3)$, which corresponds to exact chiral SU$(2)_L\times$ SU$(2)_R$ invariance of QCD with $N_f=2$ flavours. In fact, the SU$(2)_L\times$ SU$(2)_R$ group is spontaneously broken down to the isospin group SU$(2)_V$, and so we may expect that the dual SO$(4)$ symmetry is also broken down to SO$(3)$. Therefore, [Schwarzschild black holes]{} in ${\rm AdS}_5$ are to be amenable to [spontaneous splitting]{} into black objects whose symmetry is limited to SO$(3)$, such as spinning black holes of Myers and Perry [@MP] [^1].
By the duality argument, an unstable neutral spinless particle must decay into spinning particles. However, this rule is unrelated to a large body of real decays. The eloquent counterexample is the decay of neutral kaons into pions.
Yet the rule is valid for a certain set of unstable particle. This set is exceptional in the sense that the vast majority of ways for decaying of such particles are forbidden, so that they are found in the frontier zone separating the sets of stable and unstable particles. To interpret the validity of the rule, one may imagine that the gravitational dual of an exceptional particle is a black hole widely separated from other material objects in ${\rm AdS}_5$. For this well isolated object, the requirement on identity of black holes is expected to be deleted or at least smoothed out. The phenomenological facts that the lightest neutral spinless meson $\pi^0$ – whose weak interactions are suppressed – decays into $\gamma\gamma$, and the lightest scalar particle immune from the electromagnetic and strong interactions, the Higgs boson $H^0$, decays into pairs of heavy fermions ($b{\bar b}$, $\tau{\bar\tau}$) or gauge vector bosons ($W^+W^-$, $ZZ$, $gg$, $\gamma\gamma$) support this argument. The lightest glueball is likely to fall into the same category. As noted above, it is free from the electromagnetic and weak couplings, and the presence of the residual colour interaction between this particle and its environment is open to question. This implies that the lightest glueball shares the exceptionality of the $\pi^0$ and $H^0$.
To be more specific, we assume that there are three predominant ways for the lightest glueball to decay, Figure \[decay\]. Diagram $(a)$ represents the decay mode whose outcome is a pair of truly neutral light-quark vector mesons, $\rho^0\rho^0$, or, alternatively, $\omega\omega$. The masses of these particles are, respectively, $m_{\rho^0}=775$ MeV, and $m_\omega=783$ MeV. Diagram $(b)$ displays the outcome as a photon and a truly neutral vector meson, which may be given by either $\rho^0$, or $\omega$, or $\phi$ ($m_\phi=1019$ MeV). Diagram $(c)$ sketches a two-photon decay mode. The ratio of probabilities of these modes can be roughly estimated as $1:{O}(\alpha):{O}(\alpha^2)$, where $\alpha=\alpha(0)=e^2/4\pi\hbar c\approx 1/137$ stands for the fine structure constant. \[c\]\[c\][glueball]{} \[c\]\[c\][vector meson]{} \[c\]\[c\][photon]{}
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Decay modes for the lightest glueball. Photons, quarks, and gluons are shown as the sine waves, oriented lines, and spirals, respectively. The outgoing vector mesons, composed of quarks and antiquarks, are represented by couples of antiparallel rays.[]{data-label="decay"}](pic_a.eps "fig:"){height="5cm"} ![Decay modes for the lightest glueball. Photons, quarks, and gluons are shown as the sine waves, oriented lines, and spirals, respectively. The outgoing vector mesons, composed of quarks and antiquarks, are represented by couples of antiparallel rays.[]{data-label="decay"}](pic_b.eps "fig:"){height="5cm"} ![Decay modes for the lightest glueball. Photons, quarks, and gluons are shown as the sine waves, oriented lines, and spirals, respectively. The outgoing vector mesons, composed of quarks and antiquarks, are represented by couples of antiparallel rays.[]{data-label="decay"}](pic_c.eps "fig:"){height="5cm"}
(a) (b) (c)
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
This estimate is all that the holography can give for understanding the decay modes of the lightest glueball.
It may appear that an extended analysis of the decay widths for the lightest glueball is also possible. In fact, classical background fields for glueballs were assumed to be the image of dilaton field configurations together with pertinent modes of the gravitational field, forming black holes in ${\rm AdS}_5$ [@Csaki]. This strategy was further developed in ensuing holographic models; see, [*e. g*]{}., [@RinaldiVento18], and references therein. However, these models overlook the Coleman theorem [@Coleman] by which the holographic mappings of this kind may only result in a constant, a combination of the colour singlet vacuum expectation values, $\langle G^a_{\mu\nu}G_a^{\mu\nu}\rangle$, $\langle f^{abc}G_{a\lambda\mu}
G_b^{\mu\nu}G_{c\nu}^{~~\lambda}\rangle,\ldots$, specifying the gluon condensation – the phenomenon related to the QCD trace anomay. We normally take the gauge/gravity duality to mean one-to-one smooth mappings. Therefore, to map a gravitational kink-like configuration to a constant is another way of stating that the holographic image is $\emptyset$.
In a more sophisticated top-down holographic approach, proposed in [@SakaiSugimoto-I], [@SakaiSugimoto-II], and elaborated in [@Hashimoto], [@BruennerParganlijaRebhan], glueballs are taken to be excitations of gauge invariant composite operators in QCD, with their holographic duals being supergravity fluctuations associated with the dynamics of black D4-branes. The coupling of glueballs with ordinary mesons, composed of quarks and antiquarks in the fundamental representation, is realized by D8 and anti-[D8]{} probe branes intersecting the D4-branes. The constructed models are indeed able to predict decay modes and their widths for the lightest glueball, but the predictions vary from one model to another. The lack of definiteness in these predictions reflects the fact that there is no criterion for discriminating between stable and unstable subnuclear systems in this approach.
The examination of our holography inspired inference that the lightest glueball decays into spinning particles can be obscured by mixing effects with isoscalar $q{\bar q}$ mesons. By now, five isoscalar resonances are established: the very broad $f_0(500)$, the $f_0(980)$, the broad $f_0(1370)$, and the comparatively narrow $f_0(1500)$ and $f_0(1710)$ [@Olive]. The $f_0(1370)$ and $f_0(1500)$ decay mostly into pions ($2\pi$ and $4\pi$) while the $f_0(1710)$ decays mainly into $K{\bar K}$ final states. Among those resonances, the $f_0(1710)$ can be suspected to be an unmixed scalar glueball [@Janowski], [@Albaladejo], even though this view was challenged in [@Geng]. Meanwhile our interest is with the lightest glueball in its pure states. If it is granted that this entity can be created, its decay into vector particles should dominate over other decay modes.
It remains to be seen whether pure states of the lightest glueball could be actually produced. A credible speculation is that the depth of quark-gluon plasmas may have a beneficial effect on such a production. Indeed, the formation of a mixed state from the bound states $\vert gg\rangle$ and $\vert q{\bar q}\rangle$ is promoted by the fact that the $\vert gg\rangle$ and $\vert q{\bar q}\rangle$ are close in their masses. For this to happen, the quarks must be dressed, and the $\vert q{\bar q}\rangle$ must be a hadronized state. In contrast, the proximity of masses bears no relation to the $\vert gg\rangle$ and $\vert q{\bar q}\rangle$ formed in quark-gluon plasmas. In this instance, the restored chiral symmetry renders quarks massless while quantum constructions of the $\vert gg\rangle$ are the same in any circumstances whether they refer to the quark-gluon plasma or hadron context. Therefore, it is a lump of quark-gluon plasma formed in a relativistic collision of heavy ions which is expected to create glueballs in their pure states. The idea that the high energy density quark-gluon plasma is a gluon rich environment in which the lowest mass glueballs should be copiously produced is not new, specifically it was advocated in [@KabanaMinkowski], [@KochelevMin], [@Vento].
However, the possible yield of glueballs in these experiments is extemely difficult to identify against the background exhibiting many thousands of tracks. The multiplicities of scalar glueballs in a single central heavy-ion collision is estimated [@Mishustin] by fitting the hadron ratios observed in ${\rm Pb}+{\rm Pb}$ collisions at various energies to be $1.5-4$ glueballs at the Large Hadron Collider (LHC).
This sends us in search of other ways for creating the lightest glueball.
Proposal {#Suggestion}
========
Let us digress for a while and try to settle the issue of reversibility for the set of exceptional particles. As already pointed out in Sect. 2, quantum-mechanical reversibility is not compatible with the apparent irreversibility of Hawking evaporation. This prevents the holography from extending to well-defined mappings between unstable particles and their gravitational counterparts. One way around this difficulty is to restrict our discussion to such completions of the history of a Schwarzschild black hole ${\cal S}$ that ${\cal S}$ splits into extremal spinning black holes, ${\cal S}_{\rm spin}$ and ${\cal A}$, constituting two remnants of ${\cal S}$. The reverse of this splitting is a merger of ${\cal S}_{\rm spin}$ and ${\cal A}$, regaining ${\cal S}$.
With this observation, we take a closer look at the process reverse of that depicted in Figure \[decay\] $(c)$ suggesting a creation of the lightest glueball in a head-on $\gamma\gamma$ collision at a centre-of-mass energy $\sqrt s$ in the range $1-2$ GeV, with the helicity of the $\gamma\gamma$ system being zero [^2]. What is the unique experimental signature of the lightest glueball creation in such collisions? Since the production of two vector mesons $\rho^0\rho^0$, shown in Figure \[decay\] $(a)$, is the holographically predominant way for decaying of the lightest glueball, and taking into account that $\rho^0$ decays into $\pi^+\pi^-$ ($\approx 100$[%]{} fraction, $\Gamma=147.8\pm 0.9$ MeV, [@Olive]), one may expect a drastic increase in the $\pi^+\pi^-\pi^+\pi^-$ yield as $\sqrt s$ approaches the value of the lightest glueball mass.
In order to evaluate the feasibility of this scenario, we invoke recent evidence for the scattering of light by light in quasi-real photon interactions of ultra-peripheral ${\rm Pb}+{\rm Pb}$ collisions, with impact parameters larger than twice the radius of the nuclei, at a nucleon-nucleon centre-of-mass energy $\sqrt{s}=5.02$ TeV by the ATLAS experiment at the LHC [@ATLAS_Collaboration]. The cross section of the process ${\rm Pb}+{\rm Pb}(\gamma\gamma)\to{\rm Pb}^{(\ast)}+{\rm Pb}^{(\ast)}\gamma\gamma$, for diphoton invariant mass greater than 6 GeV, is measured to be $70\pm 24 ({\rm stat}.)\pm 17 ({\rm syst}.)$ nb. This result is in agreement with the Standard Model [@Enterria], where the light-by-light scattering arises, in the leading order of $\alpha$, via one-loop diagrams, Figure \[gamma-gamma\] $(a)$. An important point is that the cross section $\sigma_{\gamma\gamma\to\gamma\gamma}$ for $\sqrt{s}=6$ GeV is not too different from that for $\sqrt{s}=1.7$ GeV [@Jikia], [@Gounaris].
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![$\gamma\gamma$ collisions resulting in: (a) two-photon system; (b) two-gluon system; (c) an aggregate of a glueball and a meson[]{data-label="gamma-gamma"}](fig1.eps "fig:"){height="5cm"} ![$\gamma\gamma$ collisions resulting in: (a) two-photon system; (b) two-gluon system; (c) an aggregate of a glueball and a meson[]{data-label="gamma-gamma"}](fig2.eps "fig:"){height="5cm"} ![$\gamma\gamma$ collisions resulting in: (a) two-photon system; (b) two-gluon system; (c) an aggregate of a glueball and a meson[]{data-label="gamma-gamma"}](fig3.eps "fig:"){height="5cm"}
(a) (b) (c)
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Another piece of information derives from the usual QCD calculations of quarkonium partial widths [@BarbieriGattoKogerler]. The rule for changing two external photon lines by two external gluon lines \[see Figure \[gamma-gamma\], respectively, plots $(a)$ and $(b)$\] refers to the factor $$\frac{9}{8}\frac{\alpha_s^2(m_c)}{\alpha^2}\approx 845\,,
\label
{to_gluon-to_photon}$$ where $\alpha_s(\mu)$ stands for the QCD running coupling constant, which being taken at $\mu$ equal to the mass of charmed quarks $m_c= 1.28$ GeV is $\alpha_s\approx 0.2$. Indeed, this value of $\alpha_s$ can be obtained [@Apelquist] from the ratio $$\frac{\Gamma\left(J/\phi\to {\rm hadrons}\right)}{\Gamma\left(J/\phi\to e^+e^-\right)}=
\frac{5\left(\pi^2-9\right)\alpha_s^3}{18\pi\alpha^2}\,,
\label
{Gamma_to_Gamma}$$ whose experimental value is $\approx 10$.
All things considered, the cross section of the lightest glueball creation in head-on $\gamma\gamma$ collisions at energy region about $\sqrt s=1.7$ GeV is expected to be $\sim 60$ $\mu$b.
Note that it is the lightest glueball in its pure states which will be produced in the proposed experiment. To see this, we compare the probabilities of production of an unmixed scalar glueball and an aggregate of a glueball and a meson \[Figure \[gamma-gamma\], respectively, plots $(b)$ and $(c)$\]: the latter is less than the former by a factor of $\alpha^2_s\approx 0.04$.
To test this prediction, it is attractive to use a photon collider, the most extensively studied prospective device (for a technical design report of the Photon Collider at TESLA see [@B]) having its origin in the conversion of laser photons into high-energy gamma-quanta through the Compton scattering on high-energy electrons [@Ginzburg]. Schematically, the device consists of two beams of electrons moving towards each other to the interaction point ${\bf x}_\ast$. The electrons collide with laser photons at a distance of about $1-5$ mm from ${\bf x}_\ast$. After the scattering, the photons become gamma-quanta with energy comparable to that of the electrons and follow their direction to ${\bf x}_\ast$ where they collide with similar counterpropagating gamma-quanta. The maximum energy $\omega$ of the gamma-quanta is given by $$\omega= \frac{x}{x+1}\,E\,,
\quad
x\approx \frac{4E\omega_0}{m^2}\,,
\label
{omega-max}$$ where $E$ and $\omega_0$ are, respectively, the energy of the electrons and laser photons, and $m$ the electron mass. For example, $E=7.5$ GeV is needed if we are to convert the photon energy $\omega_0=1.17$ eV (Nd: glass laser) into the gamma-quantum energy $\omega=0.85$ GeV.
Using a laser with a flash energy of several joules one can obtain gamma-quanta whose spot size at ${\bf x}_\ast$ will be almost equal to that of the electrons at ${\bf x}_\ast$, and the total luminocity of $\gamma\gamma$ collision will be comparable to the “geometric” luminocity of the electron beams. The energy spectrum of the gamma-quanta becomes most peaked if the initial electrons are longitudinally polarized and the laser photons are circularly polarized. This gives almost a factor of 4 increase of the luminosity in the high-energy peak. The present laser technology has all elements needed for the required photon colliders [@B]. The fact that the Stanford Linear Collider luminocity, $\approx 3\cdot 10^{30} {\rm cm}^2/{\rm s}$, is four orders of magnitude greater than the luminocity, $\approx 5\cdot 10^{26} {\rm cm}^2/{\rm s}$ (corresponding to an integrated luminosity of 480 $\mu{\rm b}^{-1}$), of the light by light scattering event in ${\rm Pb}+{\rm Pb}$ collisions at the LHC [@ATLAS_Collaboration] is an added reason for this statement.
The use of $\gamma\gamma$ collisions for the possible yield of glueballs has already been realized as ingredients of experimental studies on $ee$ and $e{\bar e}$ collisions [@Acciarri], [@Abe], [@Uehara]. However, the physics behind those experiments differs from the physics behind the hunting of the lightest glueball proposed here. Unlike the former, having to do with virtual photons, the latter bears on real photons. According to the concept of vector meson dominance, virtual photons are capable of creating neutral vector mesons, Figure \[virtual\] (i). The most plausible scenario for the collision of such photons relates to the conversion of the created vector mesons into a pair of scalar mesons [^3], represented by planar tree diagrams, Figure \[virtual\] (ii). If higher order diagrams with gluon lines are taken into account, the resonances appearing in the cross section may be attributed to the interposition of a glueball. In fact, this effect is due to a glueball mixed with $q{\bar q}$ states. This comes into particular prominence from Figure \[virtual\] (iii): the glueball history is sandwiched between $q$ and ${\bar q}$ world lines. On the other hand, a real photon cannot spontaneously turn to a massive particle, while a head-on collision of two such photons brings into existence of a massive entity $G$. If the total helicity of the colliding photons is zero, and the centre-of-mass energy $\sqrt{s}$ equals the mass of the lightest glueball, this $G$ proves to be just the lightest glueball. Although the outcome of the array of reactions completed in the above experiments, $\gamma\gamma\to\phi\phi\to K{\bar K}\to 4\pi$, is similar to that of the array of the proposed reactions, $\gamma\gamma\to G\to\rho^0\rho^0\to 4\pi$, the contents of the compared processes seem much different.
\[c\]\[c\][[vector meson]{}]{} \[c\]\[c\][[virtual photon]{}]{} \[c\]\[c\][$K$]{} \[c\]\[c\][$\overline K$]{} -1cm
[p[0.33]{}p[0.33]{}p[0.33]{}]{}
![The strong interaction of photons: (i) a virtual photon becomes a vector meson; (ii) two virtual photons collide to give a $K{\bar K}$ system; (iii) two virtual photons collide to give a $K{\bar K}$ system through the mediation of a glueball.[]{data-label="virtual"}](fig-1.eps){width="38.00000%"}
&
![The strong interaction of photons: (i) a virtual photon becomes a vector meson; (ii) two virtual photons collide to give a $K{\bar K}$ system; (iii) two virtual photons collide to give a $K{\bar K}$ system through the mediation of a glueball.[]{data-label="virtual"}](fig-2.eps){width="38.00000%"}
&
![The strong interaction of photons: (i) a virtual photon becomes a vector meson; (ii) two virtual photons collide to give a $K{\bar K}$ system; (iii) two virtual photons collide to give a $K{\bar K}$ system through the mediation of a glueball.[]{data-label="virtual"}](fig-3.eps){width="38.00000%"}
\
&&
Concluding remarks {#Conclusion}
==================
Glueballs are enigmatic objects. Their properties associated with infrared effects are beyond the control of the conventional perturbation theory. The derivative expansion in chiral perturbations is irrelevant because glueballs are heavier than 1 GeV. Furthermore, nonperturbative semiclassical treatments are hampered by the fact that no rendition of a glueball is available due to the Coleman theorem [@Coleman]. We thus have not the foggiest notion what the size and structure are peculiar to the lightest glueball. Lattice QCD predicts the glueball mass spectrum, but the glueball coupling with ordinary hadrons hitherto eluded reliable analyzing. The existing phenomenological analyses entering into details of the decay of glueballs have to combine the standard QCD calculations with toy models, heuristic considerations, and [*ad hoc*]{} assumptions.
It remains to try to apply the holographic QCD. The previous efforts have been quite successful in reproducing lattice computations of the glueball mass spectrum. This is no great surprise: every meson state is stable in this approach because its decay is suppressed by the factor $1/\sqrt{N_c}$ in the large $N_c$ limit. However, the predictions of predominant decay modes and their widths are suspect; they vary from one model to another. There are two objectional features of those holographic studies. First, certain of the developed models suggest to take the classical background fields for glueballs to be the holographic image of dilaton field configurations together with some modes of the gravitational field, forming black holes – contrary to the Coleman theorem [@Coleman]. Second, there is no criterion for discriminating between stable and unstable subnuclear systems in all these models.
As was made clear in Sect. 2, the gauge/gravity duality bears no relation to unstable microscopic systems and their associated gravitational black objects. Meanwhile some particles are found in the frontier zone separating the domain in which the holographic principle holds and that in which this principle fails. We called these particles exceptional, and adduced arguments that the lightest glueball shares their exceptionality. It was shown that if the behaviour of an exceptional scalar particle is to be consistent with the gauge/gravity duality, it should decay into spinning particles. In particular, this rule requires that the feasible decay modes of the lightest glueball be represented by three options shown in Figure \[decay\].
The experimentally observed spinless isoscalar resonances $f_0(980)$, $f_0(1370)$, $f_0(1500)$, and $f_0(1710)$, whose masses are in the range of about $1 - 2$ GeV, are at times considered as candidates for the lightest glueball. These resonances mostly decay into pairs of scalar (rather than spinning) particles. A plausible explanation is that the glueball field mixes with quark-antiquark fields to form these resonances. However, our concern is with the lightest glueball in its pure states.
The holographically predicted modes of decay of the lightest glueball, Figure \[decay\], served as a guiding principle to advance a promising way for experimental detection of just this particle. The lightest glueball can be produced in head-on $\gamma\gamma$ collisions. The developed technique of $\gamma\gamma$ colliders is the tool capable of detecting the lightest glueball. A clear and unambiguous signal of its creation will be a drastic increase in the $\pi^+\pi^-\pi^+\pi^-$ yield as $\sqrt{s}$ approaches the value of the lightest glueball mass $m_G$. The expected value of $m_G$ is most likely to be different from the established values of masses of the observed resonances.
Acknowledgment {#Acknowledgment .unnumbered}
==============
We are grateful to Professors Wolfgang Ochs and Vicente Vento for useful discussions.
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[^1]: An apparent objection to this statement is that the Schwarzschild geometry is stable against small perturbations, and hence the Schwarzschild black hole splitting is unfeasible in the classical context. However, the case in point is a quantum tunnelling. Recall that particle-antiparticle pairs, ${\cal A}{\bar{\cal A}}$, can be spontaneously created near the black hole horizon. One member of a virtual pair, say ${\bar{\cal A}}$, falls into the black hole while its partner ${\cal A}$ escapes to infinity. We recognize Hawking radiation in those processes of pair creation and the subsequent particle escaping. Clearly, ${\cal A}$ and ${\bar{\cal A}}$ may be spinning particles, and furthermore, they may be spinning black holes. After absorption of ${\bar{\cal A}}$, the Schwarzschild black hole ${\cal S}$ becomes a spinning black hole ${\cal S}_{\rm spin}$, and so the output of this process is two spinning black holes, ${\cal S}_{\rm spin}$ and ${\cal A}$. Of particular interest are scenarios of the final stage of Hawking radiation in which either ${\cal S}_{\rm spin}$ or ${\cal A}$, or both are extremal black holes constituting remnants of the evaporating Schwarzschild black hole. We will return to such scenarios in Sect. 3.
[^2]: This process is justified in the context of a refined gauge/gravity duality. Notice that we are dealing here with a strong-coupling process, rather than a mere conversion $\gamma\gamma\to gg$, Figure \[gamma-gamma\] (b), which can be accounted for in the framework of the conventional perturbation theory.
[^3]: Say into $K{\bar K}$, which is typical for the experiments analyzed in [@Acciarri], [@Abe], [@Uehara].
|
---
abstract: |
We present a new heuristic algorithm finding reset words. The algorithm called CutOff-IBFS is based on a simple idea of inverse breadth-first-search in the power automaton. We perform an experimental investigation of effectiveness compared to other algorithms existing in literature, which yields that our method generally finds a shorter word in an average case and works well in practice.
automata, synchronizing algorithm, reset word
author:
- 'Jakub Kowalski[^1], Marek Szyku[ł]{}a'
bibliography:
- 'bibliography.bib'
title: A New Heuristic Synchronizing Algorithm
---
Introduction
============
Synchronizing automata are important in various fields, such as model-based testing [@BJKLP2005], robotics (for designing so-called part orienters) [@AV2003], bioinformatics (the reset problem) [@BAPLS2003], network theory [@Ka2002], theory of codes [@Ju2008] etc. In many applications it is important to find a reset word as short as possible. Unfortunately the problem of finding a shortest reset word is shown to be $\mathrm{FP^{NP[log]}}$-hard, and the related decision problem is both NP- and coNP-hard [@OM2010] (cf. also [@Be2010] and [@Ma2009; @Ma2011] for approximation hardness and special classes).
Nevertheless there exist many exponential algorithms to deal with this problem [@KRW2012; @RS1993; @Sa2005; @ST2011; @Tr2006] and a lot of polynomial heuristics finding relatively the shortest reset words [@GH2011; @KRW2012; @Ro2009; @Roman2009; @Tr2006]. Recently we developed an exact algorithm based on a bidirectional-breadth-first search, which is currently the fastest algorithm for the problem [@KKS2013]. Our new heuristic algorithm was also used as a part of our exact algorithm.
The algorithm, called CutOff-IBFS, is a heuristic polynomial synchronizing algorithm finding a synchronizing word within the given length. We analyzed some properties of the algorithm and performed an experimental investigation of the algorithm and compared with some others existing in literature. Our method, although simple, seems to be more efficient in practice and generally finds shorter reset words in less time. For simplicity, we describe only the version which find the length of the found word, and later consider a suitable modification to find the word itself.
Description of the Algorithm
============================
Before we run the main part of the algorithm we must run some other algorithm to obtain a reset word and an upper bound on the length of the shortest reset words. This is necessary as we do not have a guarantee that it finds any word in all cases. To have this property it can be easily combined with some other algorithm. We use well-known Eppstein algorithm [@Ep1990] as a preceding algorithm. Then we apply the length of the found word into CutOff-IBFS as $\mathtt{maxlen}$. It is preferred to use a fast preceding algorithm with not greater complexity than CutOff-IBFS, to not increase the overall complexity.
The formal description is given in Algorithm \[alg\_cutoffibfs\]. We start from the list $L$, which contains all singletons of the states. Then in each step we create the set of preimages of the sets from $L$ by each letter. The sets are stored in the trie $T_{ic}$, which allows us to exclude all duplicated sets. Next we create the resulted list $L'$ by taking only the $\mathtt{maxsize}$ largest sets from $T_{ic}$. We repeat the steps until we obtain the complete set or we have done $\mathtt{maxlen}$. In the second case we use the word found by the preceding algorithm as a result.
$A = \langle Q,\Sigma,\delta \rangle$ – an automaton with $n=|Q|$ states and $k =|\Sigma|$ input letters. $\mathtt{maxlen}$ – maximum length of words to be checked. $\mathtt{maxsize}$ – maximum size of the lists. $L \gets$ $T_{ic} \gets$ $S' = \delta(S,a)$ $l$ $L' \gets$ Sort $L'$ by descending cardinalities. Trim $L'$ to $\mathtt{maxsize}$ length, by truncating from the end. $L \gets L'$ “Not found a synchronizing word of length $\leq \mathtt{maxlen}$”
Analysis
--------
The correctness of the algorithm comes from the fact that we can assign a word to each computed set, so that an image of a set by its assigned word is a singleton. The word assigned to the final set $S'$ of size $n$ would be a synchronizing one.
Complexity of the algorithm depends of the given $\mathtt{maxsize}$ parameter. The larger $\mathtt{maxsize}$ is the slower the algorithm works and uses more memory, but it finds shorter words. Complexity of the algorithm depends linearly on $c$ so it is easy to set a desirable trade-off. Also it works fast when it finds a short word, and this is the case for random automata. A trie is used to skip sets already stored in it, keeping insertion time in $O(n)$.
CutOff-IBFS works in $O(lckn+kn^2)$, where $l$ is the length of the found reset word and $c=\mathtt{maxsize} \ge 1$. The space complexity is $O(ckn)$.
Initialization in lines 2–5 takes $O(n^2)$ time ($n$ sets). We assume now the worst case, that the word is not found and the **for** loop in line 6 runs $l$ times. During all the executions except the first we have at most $c$ sets in $L$. Computing preimages of the sets (line 10) by each of the letter can be done in $O(ckn)$ time ($O(n)$ for each preimage). Insertion the preimages into a trie (line 14) is done in $O(ckn)$ time ($O(n)$ for an insertion). Sorting $L$ (line 19) can be done in $O(ckn)$ time by counting sort. During the first execution of the **for** loop there are $n$ sets at the beginning, so it takes $O(kn^2)$ time. The total time complexity is $O(lckn+kn^2)$. Space is required to store the list $L$ and the trie $T_c$ during all steps, which yields $O(ckn)$.
Together with preceding Eppstein algorithm (see [@Ep1990], it works in time $O(kn^2+n^3)$ and $O(kn^2)$ space), and taking $l \in O(n^3)$, the algorithm has $O(ckn^4)$ time complexity. The total space complexity is $O(ckn+kn^2)$.
Inverse breadth-first-search is very effective for some of the most extremal automata (with the longest reset lengths). Similar as our exact algorithm [@KKS2013] works in polynomial time for them, our heuristic algorithm always find the shortest reset word in these cases.
\[th\_cutoffibfs\_slowlysynchro\] When $\mathtt{maxsize} \ge n$, Algorithm \[alg\_cutoffibfs\] always finds the shortest reset word for the Černý automaton $\mathrsfs{C}_n$ and the slowly synchronizing series introduced in $\mathrsfs{D'}_n$,$\mathrsfs{W}_n$,$\mathrsfs{F}_n$,$\mathrsfs{E}_n$,$\mathrsfs{D''}_n$,$\mathrsfs{B}_n$,$\mathrsfs{G}_n$,$\mathrsfs{H}_n$.
Consider the automaton $\mathrsfs{C}_n$. We will show that a set corresponding with the shortest reset word is kept in the list during all executions of the main loop in line 6, that is after $i$ steps there will be a set $S$ which is a preimage of some singleton by the word consisted of the $i$ last letters of the shortest reset word. This is true at the beginning since we have all the singletons in the list.
After the first step (the first execution of the main loop in line 7) there will be a set of size $2$ in the list and the singletons, since it is kept because of the sorting in line 19. After each of the $(n-1)$ next steps there is added a new set of size $2$ and all of them are kept since there are only $n$ such sets after the $n$-th step. In the $(n+1)$-th step a set of size $3$ is created and it is the largest set in the list. We can continue in this way and after each of part of $(n-1)$ steps there will be a step introducing a new set of size greater by $1$, while within a part each step introduces exactly one set of the currently largest size. There are $(n-1)$ steps introducing a larger set and $(n-2)$ parts consisted of $(n-2)$ steps. So in the $((n-1)^2)$-th step there will be introduced the final set of size $n$.
In a similar way one may follow execution of the algorithm for the other automata series to see that a set corresponding with the shortest reset word is kept in the list, and so the algorithm finds the shortest reset word.
Finding a Reset Word
--------------------
The algorithm can also return the found word, not only its length. It however increases space complexity, but the time complexity can be kept. We cannot store complete words together with the sets since they may be long ($l$), so creating a new set would take $O(n+l)$ time due to copying issue. Instead we can store, together with a set, the applied letter and a pointer to the information stored for the preceding set. This additional information costs a constant space for a set and can be computed in a constant time for a preimage set. However we need to preserve this information for each constructed set, so it will take additional $O(cl+n)$ space. This would yield total space complexity $O(c(l+kn)+kn^2)$ space complexity, while keeping time complexity in $O(lckn+kn^2)$.
Technical Improvements
----------------------
We discuss here some technical improvements of the algorithm. They can reduce running time and space but within a constant factor.
First of all, we can skip sorting from the main loop. Instead we can maintain $n$ tries for each possible cardinality of a set. Then we can construct a list by taking sets from these tries from the largest to the smallest.
Another simple improvement comes from the fact, that we can start only from the singletons of states from the sink component, as well as only from the states having in-degree at least $2$ on some letter.
One may also permute the automaton before running the algorithm. Since the sets during computation are usually small, it could be better to have states occurring more frequently to be at the top of tries. It would lead to have smaller heights of the tries and so to faster execution. A simple heuristic method to do that is sorting the states decreasing by their in-degrees.
Empirical Behavior
==================
We have compared some known synchronizing algorithms in terms of the quality (found word lengths) and the execution time. The experiment was performed on $10,000$ uniformly random labeled automata for $n=100,200,\ldots,1000$ (each algorithm worked on the same automata set). Figure \[fig:heur\_mean\] shows the mean length found by the algorithms which is generally the quality measure. Figure \[fig:heur\_time\] shows the average execution time used by the algorithms.
We presented two versions of CutOff-IBFS, with $c=\log n$ and $c=n$. See [@Roman2009; @KRW2012] for the other algorithms. We can see that CutOff-IBFS found shorter reset words than the other algorithms and the $\log n$ version is slower only than two fastest Eppstein and Cycle algorithms.
Since there may be different versions of these algorithms or they may be run with different parameters, which can affect the results of the experiment, we describe here some implementation details. For CutOff-IBFS we included time used by Eppstein algorithm. We used the greedy versions of Eppstein algorithm [@Ep1990] and Cycle algorithm [@Roman2009; @KRW2012], which always selects a pair of states with the shortest synchronizing word. For the tree of the pair automaton we used $n$ as the constant used for marking nodes (this is $y$ constant in Algorithm 2 from [@Ep1990]).
[^1]: Supported in part by Polish MNiSZW grant IP2012 052272
|
---
abstract: 'We compare Einstein-Boltzmann solvers that include modifications to General Relativity and find that, for a wide range of models and parameters, they agree to a high level of precision. We look at three general purpose codes that primarily model general scalar-tensor theories, three codes that model Jordan-Brans-Dicke (JBD) gravity, a code that models $f(R)$ gravity, a code that models covariant Galileons, a code that models Hořava-Lifschitz gravity and two codes that model non-local models of gravity. Comparing predictions of the angular power spectrum of the cosmic microwave background and the power spectrum of dark matter for a suite of different models, we find agreement at the sub-percent level. This means that this suite of Einstein-Boltzmann solvers is now sufficiently accurate for precision constraints on cosmological and gravitational parameters.'
author:
- 'E. Bellini'
- 'A. Barreira'
- 'N. Frusciante'
- 'B. Hu'
- 'S. Peirone'
- 'M. Raveri'
- 'M. Zumalacárregui'
- 'A. Avilez-Lopez'
- 'M. Ballardini'
- 'R. A. Battye'
- 'B. Bolliet'
- 'E. Calabrese'
- 'Y. Dirian'
- 'P. G. Ferreira'
- 'F. Finelli'
- 'Z. Huang'
- 'M. M. Ivanov'
- 'J. Lesgourgues'
- 'B. Li'
- 'N. A. Lima'
- 'F. Pace'
- 'D. Paoletti'
- 'I. Sawicki'
- 'A. Silvestri'
- 'C. Skordis'
- 'C. Umiltà'
- 'F. Vernizzi'
bibliography:
- 'EB\_comparison\_PRD.bib'
title: 'A comparison of Einstein-Boltzmann solvers for testing General Relativity'
---
Introduction
============
Parameter estimation has become an essential part of modern cosmology, e.g. [@Ade:2015xua]. By this we mean the ability to constrain various properties of cosmological models using observational data such as the anisotropies of the cosmic microwave background (CMB), the large scale structure of the galaxy distribution (LSS), the expansion and acceleration rate of the universe and other such quantities. A crucial aspect of this endeavour is to be able to accurately calculate a range of observables from the cosmological models. This is done with Einstein-Boltzmann (EB) solvers, i.e. codes that solve the linearized Einstein and Boltzmann equations on an expanding background [@2003moco.book.....D].
The history of EB solvers is tied to the success of modern theoretical cosmology. Beginning with the seminal work of Peebles and Yu [@1970ApJ...162..815P], Wilson and Silk [@1981ApJ...243...14W], Bond and Efstathiou [@1987MNRAS.226..655B] and Bertschinger and Ma [@1995ApJ...455....7M] these first attempts involved solving coupled set of many thousands of ordinary differential equations in a time consuming, computer intensive manner. A step change occurred with the introduction of the line of sight method and the CMBFAST code [@1996ApJ...469..437S] by Seljak and Zaldarriaga, which sped calculations up by orders of magnitude. Crucial in establishing the reliability of CMBFAST was a cross comparison [@Seljak:2003th] between a handful of EB solvers (including CMBFAST) that showed that it was possible to get agreement to within $0.1\%$. Fast EB solvers have become the norm: CAMB [@2000ApJ...538..473L], DASh [@Kaplinghat:2002mh], CMBEASY [@2005JCAP...10..011D] and CLASS [@Lesgourgues:2011re; @Blas:2011rf] all use the line of sight approach and have been extensively used for cosmological parameters estimation. Of these, CAMB and CLASS are kept up to date and are, by far, the most widely used as part of the modern armoury of cosmological analysis tools.
While CAMB and CLASS were developed to accurately model the standard cosmology – general relativity with a cosmological constant – there has been surge in interest in testing extensions that involve modifications to gravity [@Clifton:2011jh]. Indeed, it has been argued that it should be possible to test general relativity (GR) and constrain the associated gravitational parameters to the same level of precision as with other cosmological parameters. More ambitiously, one hopes that it should be possible to test GR on cosmological scales with the same level of precision as is done on astrophysical scales [@Berti:2015itd]. Two types of codes have been developed for the purpose of achieving this goal: general purpose codes which are either not tied to any specific theory (such as MGCAMB [@Hojjati:2011ix] and ISITGR [@Dossett:2011tn] ) or model a broad class of (scalar-tensor) theories (such as `EFTCAMB` [@Hu:2013twa] and [[hi\_class]{} ]{}[@Zumalacarregui:2016pph]) and specific codes which model targeted theories such as Jordan-Bran-Dicke gravity [@Avilez:2013dxa], Einstein-Aether gravity [@Zuntz:2008zz], $f(R)$ [@Bean:2006up], covariant galileons [@Barreira:2012kk] and others.
The stakes have changed in terms of theoretical precision. Up and coming surveys such as Euclid[^1], LSST[^2], WFIRST[^3], SKA[^4] and Stage 4 CMB[^5] experiments all require sub-percent agreement in theoretical accuracy (cosmic variance is inversely proportional to the angular wavenumber probed, $\ell$, and we expect to at most, reach $\ell\sim$ few$\times 10^3$). While there have been attempts at checking and calibrating existing non-GR N-body codes [@Winther:2015wla], until now the same effort has not been done for non-GR EB solvers with this accuracy in mind. In this paper we attempt to repeat what was done in [@Seljak:2003th; @Lesgourgues:2011rg] with a handful of codes. We will focus on scalar modes, neglecting for simplicity primordial tensor modes and B-modes of the CMB. In particular, we will show that two general purpose codes – `EFTCAMB` and [[hi\_class]{} ]{}– agree with each other to a high level of accuracy. The same level of accuracy is reached with the third general purpose code – COOP; however, the latter code needs further calibration to maintain agreement at sub-Mpc scales. We also show that they agree with a number of other EB solvers for a suite of models such Jordan-Brans-Dicke (JBD), covariant Galileons, $f(R)$ and Hořava-Lifshitz (khronometric) gravity. And we will show that for some models not encompassed by these general purpose codes, i.e. non-local theories of gravity, there is good agreement between existing EB solvers targeting them. This gives us confidence that these codes can be used for precision constraints on general relativity using observables of a linearly perturbed universe.
We structure our paper as follows. In Section \[sec:theories\] we layout the formalism used in constructing the different codes and we summarize the theories used in our comparison. In Section \[sec:codes\] we describe the codes themselves, highlighting their key features and the techniques they involve. In Section \[sec:tests\] we compare the codes in different settings. We begin by comparing the codes for specific models and then choose different families of parametrizations for the free functions in the general purpose codes. In Section \[sec:discussion\] we discuss what we have learnt and what steps to take next in attempts at improving analysis tools for future cosmological surveys.
Formalism and Theories {#sec:theories}
======================
To study cosmological perturbations on large scales, one must expand all relevant cosmological fields to linear order around a homogeneous and isotropic background. By cosmological fields we mean the space time metric, $g_{\mu\nu}$, the various components of the energy density, $\rho_i$ (where $i$ can stand for baryons, dark matter and any other fluid one might consider), the pressure, $P_i$, and momentum, $\theta_i$, as well as the phase space densities of the relativistic components, $f_j$ (where $j$ now stands for photons and neutrinos) as well as any other exotic degree of freedom, (such as, for example, a scalar field, $\phi$, in the case of quintessence theories). One then replaces these linearized fields in the cosmological evolution equations; specifically in the Einstein field equations, the conservation of energy momentum tensor and the Boltzmann equations. One can then evolve the background equations and the linearized evolution equations to figure out how a set of initial perturbations will evolve over time.
The end goal is to be able to calculate a set of spectra. First, the power spectrum of matter fluctuations at conformal time $\tau$ defined by $$\begin{aligned}
\langle \delta^*_{M}(\tau,{\bf k'})\delta_{M}(\tau,{\bf k})\rangle\equiv (2\pi)^3P(k,\tau)\delta^3({\bf k}-{\bf k'}) \;,\end{aligned}$$ where we have expanded the energy density of matter, $\rho_M$ around its mean value, ${\bar \rho}_M$, $\delta_M=(\rho_M-{\bar \rho}_M)/{\bar \rho}_M$, and taken its Fourier transform. Second, the angular power spectrum of CMB anisotropies $$\begin{aligned}
\langle a^*_{\ell'm'}a_{\ell m}\rangle=C^{TT}_\ell\delta_{\ell\ell'}\delta_{mm'} \;,\end{aligned}$$ where we have expanded the anisotropies, $\delta T/T({\hat n})$ in spherical harmonics such that $$\begin{aligned}
\frac{\delta T}{T}({\hat n})=\sum_{\ell m}a_{\ell m}Y_{\ell m}({\hat n})\;.\end{aligned}$$ More generally one should also be able to calculate the angular power spectrum of polarization in the CMB, specifically of the “$E$” mode, $C^{EE}_\ell$, the “$B$” mode, $C^{BB}_\ell$ and the cross-spectra between the “$E$” mode and the temperature anisotropies, $C^{TE}_\ell$, as well as the angular power spectrum of the CMB lensing potential, $C^{\phi\phi}_\ell$. As a by product, one can also calculate “background” quantities such as the history of the Hubble rate, $H(\tau)$, the angular-distance as a function of redshift, $D_A(z)$ and other associated quantities such as the luminosity distance, $D_L(z)$.
To study deviations from general relativity, one needs to consider two main extensions. First one needs to include extra, gravitational degrees of freedom. In this paper we will restrict ourselves to scalar-tensor theories, as these have been the most thoroughly studied, and furthermore we will consider only one extra degree of freedom. This scalar field, and its perturbation, will have an additional evolution equation which is coupled to gravity. Second, there will be modifications to the Einstein field equations and their linearized form will be modified accordingly. How the field equations are modified and how the scalar field evolves depends on the class of theories one is considering. In what follows, we will describe what these modifications mean for different classes of scalar-tensor theories and also theories that evolve restricted scalar degrees of freedom (such as Hořava-Lifshitz and non-local theories of gravity).
The Effective Field Theory of Dark Energy {#sec:eft}
-----------------------------------------
A general approach to study scalar-tensor theories is the so-called Effective Field Theory of dark energy (EFT) [@Gubitosi:2012hu; @Bloomfield:2012ff; @Gleyzes:2013ooa; @Bloomfield:2013efa; @Piazza:2013coa; @Frusciante:2013zop; @Gleyzes:2014rba; @Gleyzes:2015pma; @Perenon:2015sla; @Kase:2014cwa; @Linder:2015rcz; @Frusciante:2016xoj]. Using this approach, it is possible to construct the most general action describing perturbations of single field dark energy (DE) and modified gravity models (MG). This can be done by considering all possible operators that satisfy spatial-diffeomorphism invariance, constructed from the metric in unitary gauge where the time is chosen to coincide with uniform field hypersurfaces. The operators can be ordered in number of perturbations and derivatives. Up to quadratic order in the perturbations, the action is given by $$\begin{aligned}
\label{eq:actionEFT}
S &=& \int d^4x \sqrt{-g} \left \{ \frac{M_{\rm Pl}^2}{2} [1+\Omega(\tau)]R+ \Lambda(\tau) - a^2c(\tau) \delta g^{00}\right.\nonumber \\
&+& \left.\frac{M_2^4 (\tau)}{2} (a^2\delta g^{00})^2 - \frac{\bar{M}_1^3 (\tau)}{2}a^2 \delta g^{00} \delta {K}^\mu_{\phantom{\mu}\mu} - \frac{\bar{M}_2^2 (\tau)}{2} (\delta K^\mu_{\phantom{\mu}\mu})^2 \right.
\nonumber \\
& - &\left. \frac{\bar{M}_3^2 (\tau)}{2} \delta K^\mu_{\phantom{\mu}\nu} \delta K^\nu_{\phantom{\nu}\mu}
+\frac{a^2\hat{M}^2(\tau)}{2}\delta g^{00}\delta R^{(3)} \right.\nonumber \\
& +& \left. m_2^2 (\tau) (g^{\mu \nu} + n^\mu n^\nu) \partial_\mu (a^2g^{00}) \partial_\nu(a^2 g^{00})
+ \ldots \right\}\nonumber\\& +& S_{m} [\chi_i ,g_{\mu \nu}],\nonumber\\\end{aligned}$$ where $R$ is the 4D Ricci scalar and $n^\mu$ denotes the normal to the spatial hypersurfaces; $K_{\mu\nu} = (\delta^\rho_\mu + n^\rho n_\mu) \nabla_\rho n_\nu$ is the extrinsic curvature, $K$ its trace, and $R^{(3)}$ is the 3D Ricci scalar, all defined with respect to the spatial hypersurfaces. Moreover, we have tagged with a $\delta$ all perturbations around the cosmological background. $S_m$ is the matter action describing the usual components of the Universe, which we assume to be minimally and universally coupled to gravity. The ellipsis stand for higher order terms that will not be considered here. The explicit evolution of the perturbation of the scalar field can be obtained by applying the Stückelberg technique to Eq. (\[eq:actionEFT\]) which means restoring the time diffeomorphism invariance by an infinitesimal time coordinate transformation, i.e. $ t \rightarrow t + \, \pi(x^{\mu})$, where $\pi$ is the explicit scalar degree of freedom.
In Eq. (\[eq:actionEFT\]), the functions of time $\Lambda(\tau)$ and $c(\tau)$ can be expressed in terms of $\Omega(\tau)$, the Hubble rate and the matter background energy density and pressure, using the background evolution equations obtained from this action [@Gubitosi:2012hu; @Bloomfield:2012ff; @Gleyzes:2013ooa; @Bloomfield:2013efa]. Then, the general family of scalar-tensor theories is spanned by eight functions of time, i.e. $\Omega(\tau)$, $M_2^4(\tau)$, $M^2_i(\tau)$ (with $i=1,\ldots,3$), ${\hat M}^2(\tau)$, $m^2_2(\tau)$ plus one function describing the background expansion rate as $H\equiv da/(adt)$.[^6] Their time dependence is completely free unless they are constrained to represent some particular theory. Indeed, besides their model independent characterization, a general recipe exists to map specific models in the EFT language [@Gubitosi:2012hu; @Bloomfield:2012ff; @Bloomfield:2013efa; @Gleyzes:2013ooa; @Gleyzes:2014rba; @Frusciante:2015maa; @Frusciante:2016xoj]. In other words, by making specific choices for these EFT functions it is possible to single out a particular class of scalar-tensor theory and its cosmological evolution for a specific set of initial conditions. The number of EFT functions that are involved in the mapping increases proportionally to the complexity of the theory. In particular, linear perturbations in non-minimally coupled theories such as Jordan-Brans-Dicke are described in terms of two independent functions of time, $\Omega(\tau)$ and $H(\tau)$, i.e. by setting $M_2^4=0$, $\bar{M}^2_i=0$ ($i=1,\ldots,3$) and $m^2_2=0$. Increasing the complexity of the theory, perturbations in Horndeski theories [@Horndeski:1974wa; @Deffayet:2009mn] are described by setting $\{\bar{M}^2_2=-\bar{M}^2_3=2\hat{M}^2,m_2^2=0\}$, in which case one is left with four independent functions of time in addition to the usual dependence on $H(\tau)$ [@Gleyzes:2013ooa; @Bloomfield:2013efa]. Moreover, by detuning $2\hat{M}^2$ from $\bar{M}^2_2=-\bar{M}^2_3$ one is considering beyond Horndeski theories [@Gleyzes:2014dya; @Gleyzes:2014qga]. Lorentz violating theories, such as Hořava gravity [@Horava:2008ih; @Horava:2009uw], also fall in this description by assuming $m_2^2\neq 0$.
For practical purposes, it is useful to define a set of dimensionless functions in terms of the original EFT functions as $$\begin{aligned}
\gamma_1&=&\frac{M^4_2}{M_{\rm Pl}^2H_0^2}\,, \
\gamma_2 =\frac{\bar{M}^3_1}{M_{\rm Pl}^2H_0}\,, \
\gamma_3 = \frac{\bar{M}^2_2}{M_{\rm Pl}^2}\,,
\nonumber \\
\gamma_4 &=& \frac{\bar{M}^2_3}{M_{\rm Pl}^2}\,, \ \ \ \ \ \
\gamma_5 =\frac{\hat{M}^2}{M_{\rm Pl}^2}\,, \ \ \ \ \ \
\gamma_6 = \frac{m^2_2}{M_{\rm Pl}^2}\,,\end{aligned}$$ where $H_0$ and $M_{\rm Pl}$ are the Hubble parameter today and the Planck mass respectively.
In this basis, Horndeski gravity corresponds to $\gamma_4=-\gamma_3$, $\gamma_5=\frac{\gamma_3}{2}$ and $\gamma_6=0$. As explained above, this reduces the number of free functions to five, i.e. $\{\Omega,\gamma_1,\gamma_2,\gamma_3\}$ plus a function that fixes the background expansion history. In this limit the EFT approach is equivalent to the $\alpha$ formalism described in the next section. Indeed, a one-to-one map to convert between the two bases is provided in Appendix \[sec:dictionary\].
The Horndeski Action
--------------------
A standard approach to study general scalar-tensor theories is to write down a covariant action by considering explicitly combinations of a metric, $g_{\mu\nu}$, a scalar field, $\phi$, and their derivatives. The result for the most general action leading to second-order equations of motion on any background is the Horndeski action [@Horndeski:1974wa; @Deffayet:2011gz], which reads $$\begin{aligned}
\label{eq:L_horndeski}
S=\int d^4x \sqrt{-g}\sum_{i=2}^5{\cal L}_i[\phi,g_{\mu\nu}]+ S_{m} [\chi_i ,g_{\mu \nu}],\end{aligned}$$ where, as always throughout this paper, we have assumed minimal and universal coupling to matter in $S_m$. The building blocks of the scalar field Lagrangian are $$\begin{aligned}
{\cal L}_2&=& K , \nonumber \\
{\cal L}_3&=& -G_3 \Box\phi , \nonumber \\
{\cal L}_4&=& G_4R+G_{4X}\left\{(\Box \phi)^2-\nabla_\mu\nabla_\nu\phi \nabla^\mu\nabla^\nu\phi\right\} , \nonumber \\
{\cal L}_5&=& G_5G_{\mu\nu}\nabla^\mu\nabla^\nu\phi
-\frac{1}{6}G_{5X}\big\{ (\Box\phi)^3
-3\nabla^\mu\nabla^\nu\phi\nabla_\mu\nabla_\nu\phi\Box\phi
\nonumber \\ & &
+2\nabla^\nu\nabla_\mu\phi \nabla^\alpha\nabla_\nu\phi\nabla^\mu\nabla_\alpha \phi
\big\} \,,
\label{eq:HorndeskiBB}\end{aligned}$$ where $K$ and $G_A$ are functions of $\phi$ and $X\equiv-\nabla^\nu\phi\nabla_\nu\phi/2$, and the subscripts $X$ and $\phi$ denote derivatives. The four functions, $K$ and $G_A$ completely characterize this class of theories.
Horndeski theories are not the most general viable class of theories. Indeed, it is possible to construct scalar-tensor theories with higher-order equations of motion and containing a single scalar degree of freedom, such as the so-called “beyond Horndeski” extension [@Zumalacarregui:2013pma; @Gleyzes:2014dya; @Gleyzes:2014qga]. It was recently realized that higher-order scalar-tensor theories propagating a single scalar mode can be understood as degenerate theories [@Langlois:2015cwa; @Crisostomi:2016czh; @BenAchour:2016fzp].
It is possible to prove that the exact linear dynamics predicted by the full Horndeski action, Eq. (\[eq:L\_horndeski\]), is completely described by specifying five functions of time, the Hubble parameter and [@Bellini:2014fua] $$\begin{aligned}
M^2_*&\equiv&2\left(G_4-2XG_{4X}+XG_{5\phi}-{\dot \phi}HXG_{5X}\right) , \nonumber \\
HM^2_*\alpha_M&\equiv&\frac{d}{dt}M^2_* , \nonumber \\
H^2M^2_*\alpha_K&\equiv&2X\left(K_X+2XK_{XX}-2G_{3\phi}-2XG_{3\phi X}\right) \nonumber \\ & &
+12\dot{\phi}XH\left(G_{3X}+XG_{3XX}-3G_{4\phi X}-2XG_{4\phi XX}\right) \nonumber \\ & &
+12XH^2\left(G_{4X}+8XG_{4XX}+4X^2G_{4XXX}\right)\nonumber \\ & &
-12XH^2\left(G_{5\phi}+5XG_{5\phi X}+2X^2G_{5\phi XX}\right)\nonumber \\ & &
+4\dot{\phi}XH^3\left(3G_{5X}+7XG_{5XX}+2X^2G_{5XXX}\right)\nonumber , \\
HM^2_*\alpha_B&\equiv&2\dot{\phi}\left(XG_{3X}-G_{4\phi}-2XG_{4\phi X}\right) \nonumber \\ & &
+8XH\left(G_{4X}+2XG_{4XX}-G_{5\phi}-XG_{5\phi X}\right) \nonumber \\ & &
+2\dot{\phi}XH^2\left(3G_{5X}+2XG_{5XX}\right) \nonumber , \\
M^2_*\alpha_T&\equiv&2X\left[2G_{4X}-2G_{5\phi}-\left(\ddot{\phi}-\dot{\phi}H\right)G_{5X}\right] \label{eq:alphas}\,,\end{aligned}$$ where dots are derivatives w.r.t. cosmic time $t$ and $H\equiv da/(adt)$.
While the Hubble parameter fixes the expansion history of the universe, the $\alpha_i$ functions appear only at the perturbation level. $M_*^2$ defines an effective Planck mass, which canonically normalize the tensor modes. $\alpha_K$ and $\alpha_B$ (dubbed as *kineticity* and *braiding*) are respectively the standard kinetic term present in simple DE models such as quintessence and the kinetic term arising from a mixing between the scalar field and the metric, which is typical of MG theories as $f(R)$. Finally, $\alpha_T$ has been named *tensor speed excess*, and it is responsible for deviations on the speed of gravitational waves while on the scalar sector it generates anisotropic stress between the gravitational potentials.
It is straightforward to relate the free functions $\{M_*, \alpha_K, \alpha_B,\alpha_T\}$ defined above to the free functions $\{\Omega, \gamma_1, \gamma_2, \gamma_3\}$ used to describe Horndeski theories in the EFT formalism. The mapping between these sets of functions is reported in Appendix \[sec:dictionary\]. For an explicit expression of the functions $\{\Omega, \gamma_1, \gamma_2, \gamma_3\}$ in terms of the original $\{K, G_A\}$ in Eq. (7), we refer the reader to [@Frusciante:2016xoj] (see also [@Gleyzes:2013ooa; @Bloomfield:2013efa]).
Regardless of the basis ($\alpha$s or EFT), it is clear now that there are two possibilities. The first one is to calculate the time dependence of $\alpha_i$ or $\gamma_i$ and the background consistently to reproduce a specific sub-model of Horndeski, the second one is to specify directly their time dependence. Finally, the evolution equation for the extra scalar field and the modifications to the gravitational field equations depend solely on this set of free functions; any cosmology arising from Horndeski gravity can be modelled with an appropriate time dependence for these free functions.
Jordan-Brans-Dicke
------------------
The Jordan-Brans-Dicke (JBD) theory of gravity [@Brans:1961sx], a particular case of the Horndeski theory, is given by the action $$\begin{aligned}
S=\int d^4x \sqrt{-g}\frac{M_{\rm Pl}^2}{2}\left[\phi R-\frac{\omega_{\rm BD}}\phi\nabla_\mu\phi\nabla^\mu\phi-2V\right]+ S_{m} [\chi_i ,g_{\mu \nu}]\,, \nonumber \\\end{aligned}$$ where $V(\phi)$ is a potential term and $\omega_{\rm BD}$ is a free parameter. GR is recovered when $\omega_{\rm BD}\rightarrow \infty$. For our test, we will not consider a generic potential but a cosmological constant instead, $\Lambda$, as the source of dark energy.
In the EFT language, linear perturbations in JBD theories are described by two functions, i.e. the Hubble rate $H(t)$ (or equivalently $c(\tau)$ or $\Lambda(\tau)$) and $$\begin{aligned}
\label{EFT_JBD}
&\Omega(\tau)=\phi-1 \,, \nonumber \\
&\gamma_i(\tau)=0\,.\end{aligned}$$ We can see that in this case there are no terms consisting of purely modified perturbations (i.e. any of the $\gamma_i$).
Alternatively the $\alpha_i(\tau)$ functions read $$\begin{aligned}
\alpha_M(\tau)&=&\frac{d\ln \phi}{d\ln a}, \nonumber \\
\alpha_B(\tau)&=&-\alpha_M, \nonumber \\
\alpha_K(\tau)&=&\omega_{\rm BD}\alpha^2_M, \nonumber \\
\alpha_T(\tau)&=&0. \label{eq:JBDalphas}\end{aligned}$$ As with the EFT basis, one has to consider the Hubble parameter $H(\tau)$ as an additional building function. However, $H(\tau)$ can be written entirely as a function of the $\alpha$s, meaning that the five functions of time needed to describe the full Horndeski theory reduce to two in the JBD case, consistently with the EFT description of the previous paragraph.
In order to fix the above functions one has to solve the background equations to determine the time evolution of $\{H, \phi\}$.
Covariant Galileon {#sec:galileons_theory}
------------------
The covariant Galileon model corresponds to the subclass of scalar-tensor theories of Eq. (\[eq:L\_horndeski\]) that (in the limit of flat spacetime) is invariant under a [*Galilean shift*]{} of the scalar field [@Nicolis:2008in], i.e. $\partial_\mu\phi\rightarrow \partial_\mu\phi+b_\mu$ (where $b_\mu$ is a constant four-vector). The covariant construction of the model presented in [@Deffayet:2009wt] consists in the addition of counter terms that cancel higher-derivative terms that would otherwise be present in the naive covariantization (i.e. simply replacing partial with covariant derivatives; see however [@Gleyzes:2014dya] for why the addition of these counter terms is not strictly necessary). Galilean invariance no longer holds in spacetimes like FRW, but the resulting model is one with a very rich and testable cosmological behaviour. The Horndeski functions in Eqs. (\[eq:HorndeskiBB\]) have this form $$\begin{aligned}
{\cal L}_{2} & = & c_{2}X -\frac{c_{1}M^{3}}{2}\phi \,,\label{eq:L2}\\
{\cal L}_{3} & = & 2\frac{c_{3}}{M^{3}}X\Box\phi\,,\label{eq:L3}\\
{\cal L}_{4} & = & \left(\frac{M_{p}^{2}}{2}+\frac{c_{4}}{M^{6}}X^{2}\right)R
+2\frac{c_{4}}{M^{6}}X\left[\left(\Box\phi\right)^{2}-\phi_{;\mu\nu}\phi^{;\mu\nu}\right]\,,\label{eq:L4}\\
{\cal L}_{5} & = & \frac{c_{5}}{M^{9}}X^{2}G_{\mu\nu}\phi^{;\mu\nu}
-\frac{1}{3}\frac{c_{5}}{M^{9}}X\left[\left(\Box\phi\right)^{3}+2{\phi_{;\mu}}^{\nu}{\phi_{;\nu}}^{\alpha}{\phi_{;\alpha}}^{\mu} \right. \nonumber \\ & & \left. \ \ \ \ -3\phi_{;\mu\nu}\phi^{;\mu\nu}\Box\phi\right]\,,\label{eq:L5}\end{aligned}$$ Here, as usual, we have set $M^{3}=H_{0}^{2}M_{p}$. Note that these definitions are related to Ref. [@Barreira:2014jha] by $c_3^{\rm ours}\to -c_3^{\rm theirs}$ and $c_5^{\rm ours} = 3c_5^{\rm theirs} $. There is some freedom to rescale the field and normalize some of the coefficients. Following Ref. [@Barreira:2014jha] we can choose $c_2<0$ and rescale the field so that $c_2 = -1$ (models with $c_2>0$ have a stable Minkowski limit with $\phi_{,\mu}=0$ and thus no acceleration without a cosmological constant, see e.g. [@Brax:2014vla]). The term proportional to $\phi$ in $\mathcal{L}_2$ is uninteresting, so we will set $c_1 = 0$ from now on. This leaves us with three free parameters, $c_{3,4,5}$.
An analysis of Galileon cosmology was undertaken in [@DeFelice:2010pv; @Barreira:2014jha] identifying some of the key features which we briefly touch upon. The Galileon contribution to the energy density at $a=1$ is [@DeFelice:2010pv] $$\label{eq:Omega_de}
\Omega_{gal}=-\frac{1}{6}\xi^{2}-2c_{3}\xi^{3}+\frac{15}{2}c_{4}\xi^{4}+\frac{7}{3}c_{5}\xi^{5}\,,$$ (defined such that the coefficients are dimensionless) and where $$\xi\equiv \frac{\dot{\phi}H}{M_{\rm Pl} H_{0}^{2}}\,.$$
Given that the theory is shift symmetric, there is an associated Noether current satisfying $\nabla_\mu J^\mu = 0$ [@Deffayet:2010qz]. For a cosmological background $ J^i = 0$, $J^0\equiv n$ and the shift-current decays with the expansion $n \propto a^{-3} \to0$ at late times. The field evolution is thus driven to an attractor where $$\label{eq:attractor}
J^0 \propto -\xi-6c_{3}\xi^{2}+18c_{4}\xi^{3}+5c_{5}\xi^{4} =0\,,$$ i.e. $\xi$ is a constant and the evolution of the background is independent of the initial conditions of the scalar field. Although it has been claimed that background observations favour a non-scaling behaviour of the scalar field [@Neveu:2016gxp], CMB observations (not considered in Ref. [@Neveu:2016gxp]) require that the tracker has been reached before Dark Energy dominates (Fig. 11 of Ref. [@Barreira:2014jha]).[^7] So if only considering the evolution on the attractor, one can use Eqs. (\[eq:Omega\_de\],\[eq:attractor\]) to trade two of the independent $c_i$ for $\xi$ and $\Omega_{gal}$.
It has thus become standard to refer to three models:
1. Cubic: $c_4=c_5=0$, with $c_3$ the only free parameter; choosing $\Omega_{gal}$ determines determes $\xi$. No additional parameters compared to $\Lambda$CDM.
2. Quartic: $c_5 = 0$; $\Omega_{gal}$ and $\xi$ are free parameters. One more parameter than $\Lambda$CDM.
3. Quintic: $c_3, \xi, \Omega_{gal}$ are free parameters. Two extra parameters relative to $\Lambda$CDM.
All of these models are self-accelerating models without a cosmological constant, and hence do not admit a continuous limit to $\Lambda$CDM.
The covariant Galileon model is implemented in `EFTCAMB` and GALCAMB assuming the attractor solution Eq. (\[eq:attractor\]); on the other hand [[hi\_class]{} ]{}solves the the full background equations both on- and off-attractor. The two approaches are equivalent if ones chooses the initial conditions for the scalar field on the attractor, which will be the strategy for the rest of the Galileon comparison. When the attractor solution is considered with the above conventions, the alpha functions read
$$\begin{aligned}
M_*^2\alpha_{\rm K} {\cal E}^4 &=& - \xi^2-12 c_3 \xi^3+54 c_4 \xi^4+20 c_5 \xi^5 \,,\nonumber
\\
M_*^2\alpha_{\rm B} {\cal E}^4 &=& -2 c_3 \xi^3+12 c_4 \xi^4+5 c_5 \xi^5 \,,\nonumber
\\
M_*^2\alpha_{\rm M} {\cal E}^4 &=& 6 c_4 \frac{\dot{{\cal H}}}{{\cal H}^2} \xi^4+4 c_5 \frac{\dot{{\cal H}}}{{\cal H}^2} \xi^5 \,,\nonumber
\\
M_*^2\alpha_{\rm T} {\cal E}^4 &=& 2 c_4 \xi^4+c_5 \xi^5\left(1+\frac{\dot{{\cal H}}}{{\cal H}^2}\right) \,,
\end{aligned}$$
where ${\cal E} = {\cal H}(\tau)/{ H}_0$ is the dimensionless expansion rate with ${\cal H}=aH$ and a dot now denotes a derivative w.r.t. conformal time, $\tau$. With the same conventions, the EFT functions read $$\begin{aligned}
\Omega &=& \frac{a^4 H_0^4 \xi ^4 \left(\mathcal{H}^2 \left(c_4-2 c_5 \xi \right)+2 c_5 \xi \dot{\mathcal{H}}\right)}{2 \mathcal{H}^6} \,,\nonumber\\
\gamma_3 &=&-\frac{a^4 H_0^4 \xi ^4 \left(2 c_4 \mathcal{H}^2+ c_5 \xi \dot{\mathcal{H}}\right)}{\mathcal{H}^6}\,,\nonumber\\
\gamma_2 &=&-\frac{a^3 H_0^3 \xi ^3}{\mathcal{H}^7}\Bigg[ c_5 \xi ^2 \mathcal{H} \ddot{\mathcal{H}}+2 \xi \mathcal{H}^2 \left(4 c_5 \xi -c_4\right) \dot{\mathcal{H}}\nonumber\\ &&+\mathcal{H}^4 \left(\xi \left( c_5 \xi
+14 c_4\right)-2 c_3\right)-6 c_5 \xi ^2 \dot{\mathcal{H}}^2\Bigg]\,, \nonumber\\
\gamma_1 &=&\frac{a^2 H_0^2 \xi ^3}{4 \mathcal{H}^8}\Bigg[2 \xi \mathcal{H}^3 \left(5 c_5 \xi -c_4\right) \ddot{\mathcal{H}} +42 c_5 \xi ^2 \dot{\mathcal{H}}^3 \\&&+\mathcal{H}^4 \left(9 \xi \left(\frac{7}{3} c_5 \xi -2 c_4\right)+2
c_3\right) \dot{\mathcal{H}}\nonumber\\ &&+\xi \mathcal{H}^2 \left( c_5 \xi \dddot{\mathcal{H}}+10 \left(c_4-5 c_5 \xi \right) \dot{\mathcal{H}}^2\right)\nonumber\\ &&-18 c_5 \xi ^2 \mathcal{H} \dot{\mathcal{H}}
\ddot{\mathcal{H}}+4 \mathcal{H}^6 \left(3 \xi \left( c_5 \xi +4 c_4\right)-2 c_3\right)\Bigg]\nonumber\,.\end{aligned}$$
f(R) gravity {#sec:f(R)}
------------
$f(R)$ models of gravity are described by the following Lagrangian in the Jordan frame $$\label{action_fR}
S=\int d^4x \sqrt{-g} \left[R+f(R)\right]+S_m[\chi_i,g_{\mu\nu}] \,,$$ where $f(R)$ is a generic function of the Ricci scalar and the matter fields $\chi_i$ are minimally coupled to gravity. They represent a popular class of scalar-tensor theories which has been extensively studied in the literature [@Song:2006ej; @Bean:2006up; @Amendola:2006kh; @Pogosian:2007sw; @DeFelice:2010aj] and for which N-body simulation codes exist [@Zhao:2010qy; @Baldi:2013iza; @Lombriser:2011zw; @Hammami:2015iwa; @Winther:2015wla]. Depending on the choice of the functional form of $f(R)$, it is possible to design models that obey stability conditions and give a viable cosmology [@Amendola:2006kh; @Sawicki:2007tf; @Pogosian:2007sw]. A well-known example of viable model that also obeys solar system constraints is the one introduced by Hu $\&$ Sawicki in [@Hu:2007nk].
The higher order nature of the theory, offers an alternative way of treating $f(R)$ models, i.e. via the so-called *designer* approach. In the latter, one fixes the expansion history and uses the Friedmann equation as a second-order differential equation for $f[R(a)]$ to reconstruct the $f(R)$ model corresponding to the chosen history [@Song:2006ej; @Pogosian:2007sw]. Generically, for each expansion history, one finds a family of viable models that reproduce it and are commonly labelled by the boundary condition at present time, $f_R^0$. Equivalently, they can be parametrized by the present day value of the function $$\label{ComptonWave}
B=\frac{f_{RR}}{1+f_R}R^\prime\frac{H}{H^\prime}\,,$$ where a prime denotes derivation w.r.t. $\ln{a}$. The smaller the value of $B_0$, the smaller the scale at which the fifth force introduced by $f(R)$ kicks in. As in the JBD case, $f(R)$ models are described in the EFT formalism by two functions [@Gubitosi:2012hu], the Hubble parameter and $$\begin{aligned}
\label{fR_matching}
&& \Omega=f_R\nonumber\\
&&\gamma_i(\tau)=0\,.\end{aligned}$$ This has been used to implement $f(R)$ gravity into `EFTCAMB`, both for the designer models as well as for the Hu-Sawicki one [@Hu:2013twa; @Hu:2016zrh]. Alternatively, they can be described by the Equation of State approach (EoS) implemented in [CLASS\_EOS\_fR]{} [@Battye:2013aaa; @Battye:2015hza].
In this comparison we will focus on designer $f(R)$ models, since our aim is that of comparing the Einstein-Boltzmann solvers at the level of their predictions for linear perturbations.
Hořava-Lifshitz gravity {#sec:HLtheory}
-----------------------
This model was introduced in Ref. [@Horava:2009uw]. It was extended in Ref. [@Blas:2010hb], where it was shown that action for the low-energy *healthy* version of Hořava-Lifshitz gravity is given by $$\begin{aligned}
\label{eq:actionhorava}
\mathcal{S}_{H}&=&\frac{1}{16\pi G_H}\int{}d^4x\sqrt{-g}[ K_{ij}K^{ij}-\lambda K^2 -2 \xi\bar{\Lambda}
\nonumber \\ & +&\xi R^{(3)}+\eta a_i a^i] +S_m[\chi_i,g_{\mu\nu}]\,,\end{aligned}$$ where $\lambda$, $\eta$, and $\xi$ are dimensionless coupling constants, $\bar{\Lambda}$ is the “bare” cosmological constant and $G_H$ is the “bare" gravitational constant related to Newton’s constant via $1/16\pi G_H=M_{\rm Pl}^2/(2\xi-\eta)$ [@Blas:2009qj]. Note that the choice $\lambda=\xi=1,\eta=0$ restores GR. In general, departures from these values lead to the violation of the local Lorentz symmetry of GR and the appearance of a new scalar degree of freedom, known as the *khronon*. It should be pointed out that the model is equivalent to khronometric gravity [@Blas:2009qj], an effective field theory which explicitly operates the khronon.[^8] The correspondence between $\{\lambda, \eta, \xi \}$ and the coupling constants of the khronometric model $\{\alpha, \beta, \lambda\}$ is $$\begin{aligned}
\eta= -\frac{\alpha_{kh} }{\beta_{kh} -1},\quad \xi = -\frac{1}{\beta_{kh} -1},\quad \lambda = -\frac{{\lambda}_{kh} +1}{\beta_{kh} -1}\,,\end{aligned}$$ where the subscript $kh$ is added for clarity.
The parameters $\lambda$, $\eta$, and $\xi$ are subject to various constraints from the absence of the vacuum Cherenkov radiation, Solar system tests, astrophysics, and cosmology [@Elliott:2005va; @Blas:2009qj; @Jacobson:2000xp; @Blas:2010hb; @Yagi:2013ava; @Blas:2016qmn; @Audren:2014hza; @Frusciante:2015maa]. The cosmological consequences of this model have been investigated in Refs. [@ArmendarizPicon:2010rs; @Kobayashi:2010eh; @Blas:2012vn; @Blas:2011en; @Audren:2013dwa], including interesting phenomenological implications for dark matter and dark energy.
The map of the action Eq. (\[eq:actionhorava\]) to the EFT functions [@Frusciante:2015maa] is $$\begin{aligned}
\label{eq:Horava_mapping}
\Omega&=&\frac{\eta}{(2\xi-\eta)}, \nonumber \\
\gamma_4&=& -\frac{2}{(2\xi-\eta)}(1-\xi),\nonumber \\
\gamma_3 &=&-\frac{2}{(2\xi-\eta)}(\xi-\lambda), \nonumber \\
\gamma_6&=&\frac{\eta}{4(2\xi-\eta)},\nonumber \\
\gamma_1&=&\frac{1}{2a^2H_0^2(2\xi-\eta)}(1+2\xi-3\lambda)\left(\dot{{\cal H}}-{\cal H}^2\right) \,,\nonumber\\
\gamma_2&=&\gamma_5=0,\end{aligned}$$ which has been implemented in `EFTCAMB` [@Hu:2014oga].
Non-local gravity {#sec:NL-theory}
-----------------
The non-local theory we consider here is that put forward in [@Maggiore:2014sia] (known as the $\nloc$ model for short), which is described by the action $$\begin{aligned}
\label{eq:NLG-action}
S_{\nloc} = \frac{1}{16\pi G}\int {\rm d}^4 x\sqrt{-g}\left[R - \frac{m^2}{6}R\Box^{-2}R - \mathcal{L}_M\right],\end{aligned}$$ where $\mathcal{L}_M$ is the Lagrange density of minimally coupled matter fields and $\Box^{-1}$ is a formal inverse of the d’Alembert operator $\Box = \nabla^\mu\nabla_\mu$. The latter can be expressed as, $$\begin{aligned}
\label{eq:formalgreen}
(\Box^{-1}A)(x) = A_{\rm hom}(x) - \int{\rm d}^4y\sqrt{-g(y)}G(x, y)A(y),\end{aligned}$$ where $A$ is some scalar function of the spacetime coordinate $x$, and the homogeneous solution $A_{\rm hom}(x)$ and the Green’s function $G(x, y)$ specify the definition of the $\Box^{-1}$ operator. Eq. () is meant to be understood as a toy-model to explore the phenomenology of the $R\Box^{-2}R$ term, while a deeper physical motivation for its origin is still not available (see [@Maggiore:2016gpx] and references therein for works along these lines). In the absence of such a fundamental understanding, different choices for the structure of the $\Box^{-1}$ operator (i.e. different homogeneous solutions and $G(x, y)$) should be regarded as different non-local models altogether, and the mass scale $m$ treated as a free parameter.
In cosmological studies of the $\nloc$ model, it has become common to cast the action of Eq. (\[eq:NLG-action\]) into the following “localized” form $$\begin{aligned}
\label{eq:NLG-action-local}
S_{\nloc, \mathrm{loc}}~ &&= \frac{1}{16\pi G}\int {\rm d}^4 x\sqrt{-g}\left[R - \frac{m^2}{6}R S - \xi_1\left(\Box U + R\right) \right. \nonumber \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left. - \xi_2\left(\Box S + U\right) - \mathcal{L}_m\right],\end{aligned}$$ where $U$ and $S$ are two auxiliary scalar fields and $\xi_1$ and $\xi_2$ are two Lagrange multipliers that enforce the constraints $$\begin{aligned}
\label{eq:u}\Box U &=& - R , \ \ \ \ \ \ \\
\label{eq:s}\Box S &=& - U .\end{aligned}$$ Invoking a given (left) inverse, one can solve the last two equations formally as $$\begin{aligned}
\label{eq:iu} U &=& - \Box^{-1} R , \ \ \ \ \ \ \\
\label{eq:is} S &=& - \Box^{-1} U = \Box^{-2} R.\end{aligned}$$ This allows one to integrate out $U$ and $S$ from the action (as well as $\xi_{1}$ and $\xi_{2}$), thereby recovering the original non-local action. The equations of motion associated with the action of Eq. (\[eq:NLG-action-local\]) are $$\begin{aligned}
&&\label{eq:fe1}G_{\mu\nu} - \frac{m^2}{6}K_{\mu\nu} = {8 \pi G} T_{\mu\nu}, \\
&&\label{eq:fe2}\Box U = -R, \\
&&\label{eq:fe3}\Box S = -U,\end{aligned}$$ with $$\begin{aligned}
\label{eq:fe4}K_{\mu\nu} \equiv 2SG_{\mu\nu} - 2\nabla_\mu\nabla_\nu S - 2\nabla_{(\mu}S\nabla_{\nu)}U \nonumber \\
+ \left(2\Box S + \nabla_\alpha S\nabla^\alpha U - \frac{U^2}{2}\right)g_{\mu\nu}.\end{aligned}$$ An advantage of using Eq. (\[eq:NLG-action-local\]) is that the resulting equations of motion become a set of coupled differential equations, which are comparatively easier to solve than the integro-differential equations of the non-local version of the model. To ensure causality one must impose by hand that the Green’s function used within $\Box^{-1}$ in Eqs. (\[eq:iu\]) and (\[eq:is\]) is of the retarded kind and this condition is naturally satisfied in integrating the localized version forward in time. Further, the quantities $U$ and $S$ should not be regarded as physical propagating scalar degrees of freedom, but instead as mere auxiliary scalar functions that facilitate the calculations. In practice, this means that once the homogeneous solution associated with $\Box^{-1}$ is specified, then the differential equations of the localized problem must be solved with the one compatible choice of initial conditions of the scalar functions. Here, we fix $U$, $S$ and their first derivatives to zero, deep in the radiation dominated regime (this is as was done, for instance, in [@Dirian:2016puz; @2014JCAP...09..031B]; see [@2016arXiv160604349N] for a study of the impact of different initial conditions) which corresponds to choosing vanishing homogeneous solutions for them. Once the initial conditions of the $U$ and $S$ scalars are fixed, then the only remaining free parameter in the model is the mass scale $m$, which effectively replaces the role of $\Lambda$ in $\lcdm$ and can be derived from the condition to render a spatially flat Universe.
Finally, note that the Horndeski Lagrangian is a local theory featuring one propagating scalar degree of freedom, and hence, does not encompass the $\nloc$ model.
The Codes {#sec:codes}
=========
{width="90.00000%"}
There are a number of EB solvers, some of which are described below, developed to explore deviations of GR. While, schematically, we have summarized how to study linear cosmological perturbations, there are a number of subtleties which we will mention now briefly. For a start, there is redundancy (or gauge freedom) in how to parametrize the scalar modes of the linearized metrics; typically EB solvers make a particular choice of gauge – the synchronous gauge – although another common gauge – the Newtonian gauge – is particularly useful in extracting physical understanding of the various effects at play. Also it should be noted that the universe undergoes an elaborate thermal history: it will recombine and subsequently reionize. It is essential to model this evolution accurately as it has a significant effect on the evolution of perturbations. Another key aspect is the use of line of sight methods (mentioned in the introduction) that substantially speed up the numerical computation of the evolution of perturbations by many orders of magnitude; as shown in [@1996ApJ...469..437S] it is possible to obtain an accurate solution of the Boltzmann hierarchy by first solving a truncated form of the lower order moments of the perturbation variables and then judiciously integrating over the appropriate kernel convolved with these lower order moments. All current EB solvers use this approach.
Most (but not all) EB solvers currently being used are modifications of either CAMB or CLASS. This means that they have evolved from very different code bases, are in different languages and use (mostly) different algorithms. This is of tremendous benefit when we compare results in the next section. We should highlight, however, that there are a couple of cases – DASh and COOP – that do not belong to this genealogy.
The codes used in this comparison, along with the models tested, are summarized in Fig. \[fig:code\_theory\_overview\] and Tab. \[tab:code\_theory\_overview\] and the details of each code can be found in the following sections.
------------ ----------------- ----------------- ----- ----------- ---------- ---------- -----------
$\alpha$ EFT JBD Covariant f(R) Hořava Non-Local
Parametrization Parametrization Galileon designer Lifshitz Gravity
`EFTCAMB`
`hi_class`
COOP
GalCAMB
BD-CAMB
DashBD
CLASSig
CLASS-LVDM
NL-CLASS
NL-CAMB
------------ ----------------- ----------------- ----- ----------- ---------- ---------- -----------
EFTCAMB
-------
`EFTCAMB` is an implementation [@Hu:2013twa; @Raveri:2014cka] of the EFT of dark energy into the CAMB [@2000ApJ...538..473L] EB solver (coded in fortran90) which evolves the full set of perturbations (in the synchronous gauge) arising from the action in Eq. (\[eq:actionEFT\]), after a built in module checks for the stability of the model under consideration. The latter includes conditions for the avoidance of ghost and gradient instabilities (both on the scalar and tensor sector), well posedness of the scalar field equation of motion and prevention of exponential growth of DE perturbations. It can treat specific models (such as, Jordan-Brans-Dicke, designer-$f(R)$, Hu-Sawicki f(R), Hořava-Lifshitz gravity, Covariant Galileon and quintessence) through an appropriate choice of the EFT functions. It also accepts phenomenological choices for the time dependence of the EFT functions and of the dark energy equation of state which may not be associated to specific theories.
`EFTCAMB` has been used to place constraints on $f(R)$ gravity [@Hu:2016zrh], Hořava-Lifshitz [@Frusciante:2015maa] and specific dark energy models [@Raveri:2014cka]. It has also been used to explore the interplay between massive neutrinos and dark energy [@Hu:2014sea], the tension between the primary and weak lensing signal in CMB data [@Hu:2015rva] as well as the form and impact of theoretical priors [@Raveri:2017qvt; @Peirone:2017lgi]. An up to date implementation can be downloaded from [http://eftcamb.org/]{}. The JBD EFTCAMB solver is based on $EFTCAMB_{Oct15}$ version, while the others are based on the most recent $EFTCAMB_{Sep17}$ version.
`hi_class`
----------
[[hi\_class]{} ]{}(Horndeski in the Cosmic Linear Anisotropy Solving System) is an implementation of the evolution equations in terms of the $\alpha_i(\tau)$ [@Zumalacarregui:2016pph] as a series of patches to the CLASS EB solver [@Lesgourgues:2011re; @Blas:2011rf] (coded in C). [[hi\_class]{} ]{}solves the modified gravity equations for Horndeski’s theory in the synchronous gauge (CLASS also incorporates the Newtonian gauge) starting in the radiation era, after checking conditions for the stability of the perturbations (both on the scalar and on the tensor sectors). The [[hi\_class]{} ]{}code has been used to place constraints on the $\alpha_i(\tau)$ with current CMB data [@Bellini:2015xja], study relativistic effects on ultra-large scales [@Renk:2016olm], forecast constraints with stage 4 clustering, lensing and CMB data [@Alonso:2016suf] and constraint Galileon Gravity models [@Renk:2017rzu].
The current public version of [[hi\_class]{} ]{}is v1.1 [@Zumalacarregui:2016pph]. The only difference between this version and the first one (v1.0) is that v1.1 incorporates all the parametrizations used in this paper. This guarantees that the results provided in this paper are valid also for v1.0. Lagrangian-based models, such as JBD and Galileons, are still in a private branch of the code and they will be released in the future. The [[hi\_class]{} ]{}code is available from [www.hiclass-code.net](www.hiclass-code.net).
COOP
----
Cosmology Object Oriented Package (COOP) [@Huang:2012mt] is an Einstein-Boltzmann code that solves cosmological perturbations including very general deviations from the $\Lambda$CDM model in terms of the EFT of dark energy parametrization [@Creminelli:2008wc; @Gubitosi:2012hu; @Gleyzes:2013ooa; @Gleyzes:2014rba].
COOP assumes minimal coupling of all matter species and solves the linear cosmological perturbation equations in Newtonian gauge, obtained from the unitary gauge ones by a time transformation $t \to t +\pi $. For the $\Lambda$CDM model, it solves the evolution equation of the spatial metric perturbation and the matter perturbation equations; details are given in Ref. [@Huang:2012mt]. Beyond the $\Lambda$CDM model, COOP additionally evolves the scalar field perturbation $\pi$, using Eqs. (109)–(112) of Ref. [@Gleyzes:2014rba] and verifying the absence of ghost and gradient instability along the evolution. Once the linear perturbations are solved, COOP computes CMB power spectra using a line-of-sight integral approach [@Seljak:1996is; @Hu:1997hp]. Matter power spectra are computed via a gauge transformation from the Newtonian to the CDM rest-frame synchronous gauge. COOP includes also the dynamics of the beyond Horndeski operator and has been used to study the signature of a non-zero $\alpha_{\rm H}$ on the matter power spectrum as well as on the primary and lensing CMB signals [@DAmico:2016ntq]. COOP v1.1 has been used for this comparison. The code and its documentation are available at [www.cita.utoronto.ca/\~zqhuang](www.cita.utoronto.ca/~zqhuang).
Jordan-Brans-Dicke solvers – modified CAMB and DASh
---------------------------------------------------
A systematic study, placing state of the art constraints on Jordan-Brans-Dicke gravity was presented in [@Avilez:2013dxa] using a modified version of CAMB and an altogether different EB Solver – the Davis Anisotropy Shortcut Code (DASh) [@Kaplinghat:2002mh]. DASh was initially written as a modification of CMBFAST [@1996ApJ...469..437S] by separating out the computation of the radiation and matter transfer functions from the computation of the line-of-sight integral. The code in its initial version, precomputed and stored the radiation and matter transfer functions on a grid so that any model was subsequently calculated fast via interpolation between the grid points, supplemented with a number of analytic estimates and fitting functions that speed up the calculation without significant loss of accuracy. Such a speedup allowed the efficient traversal of large multi-dimensional parameter spaces with MCMC methods and made the study of models containing such a large parameter space possible [@Bucher:2004an; @Moodley:2004nz; @Dunkley:2005va].
The use of a grid and semi-analytic techniques was abandoned in later, not publicly available versions of DASh, which returned to the traditional line-of-sight approach of other Boltzmann solvers. It is possible to solve the evolution equations in both synchronous and Newtonian gauge and therefore is amenable to a robust internal validation of the evolution algorithm. Over the last few years a number of gravitational theories, such as the Tensor-Vector-Scalar theory [@Bekenstein:2004ne; @Skordis:2008pq] and the Eddington-Born-Infeld theory [@Banados:2008fj], have been incorporated into the code and has been recently used for cross-checks with CLASS in an extensive study of generalized dark matter [@Thomas:2016iav; @Kopp:2016mhm].
In [@Avilez:2013dxa], the authors used the internal consistency checks within DASh and the cross checks between DASh and a modified version of CAMB to calibrate and validate their results. We will use their modified CAMB code as the baseline against which to compare `EFTCAMB`, [[hi\_class]{} ]{}and CLASSig.
Jordan-Brans-Dicke solvers – CLASSig
------------------------------------
The dedicated Einstein-Boltzmann CLASSig [@Umilta:2015cta] for Jordan-Brans-Dicke (JBD) gravity was used in [@Umilta:2015cta; @Ballardini:2016cvy] to constrain the simplest scalar-tensor dark energy models with a monomial potential with the two Planck product releases and complementary astrophysical and cosmological data. CLASSig is a modified version of CLASS which implements the Einstein equations for JBD gravity at both the background and the linear perturbation levels without any use of approximations. CLASSig adopts a redefinition of the scalar field ($\gamma \sigma^2 = \phi$) which recasts the original JBD theory in the form of induced gravity in which $\sigma$ has a standard kinetic term. CLASSig implements linear fluctuations either in the synchronous and in the longitudinal gauge (although only the synchronous version is maintained updated with CLASS). The implementation and results of the evolution of linear fluctuations has been checked against the quasi-static approximation valid for sub-Hubble scales during the matter dominated stage [@Umilta:2015cta; @Ballardini:2016cvy]. In its original version, the code implements as a boundary condition the consistency between the effective gravitational strength in the Einstein equations at present and the one measured in a Cavendish-like experiment ($\gamma \sigma_0^2 = (1+8 \gamma)/(1+6 \gamma)/(8 \pi G)$, being $G=6.67 \times 10^{-8}$ cm$^3$ g$^{-1}$ s$^{-2}$ the Newton constant) by tuning the potential. For the current comparison, we instead fix as initial condition $\gamma \sigma^2 (a=10^{-15})=1 \,, \dot \sigma (a=10^{-15})= 0$ consistently with the choice used in this paper.
Covariant Galileon – modified CAMB {#sec:galcamb}
----------------------------------
A modified version of CAMB to follow the cosmology of the Galileon models was developed in [@Barreira:2012kk], and subsequently used in cosmological constraints in [@Barreira:2013jma; @Barreira:2014jha]. The code structure is exactly as in default CAMB (gauge conventions, line-of-sight integration methods, etc.), but with the relevant physical quantities modified to include the effect of the scalar field. At the background level, this includes modifying the expansion rate to be that of the Galileon model: this may involve numerically solving for the background evolution, or using the analytic formulae of the so-called tracker evolution (see Sec. \[sec:galileons\_theory\]). At the linear perturbations level, the modifications entail the addition of the Galileon contribution to the perturbed total energy-momentum tensor. More precisely, one works out the density perturbation, heat flux and anisotropic stress of the scalar field, and appropriately adds these contributions to the corresponding variables in default CAMB (due to the gauge choices in CAMB, one does not need to include the pressure perturbation; see [@Barreira:2012kk] for the derivation of the perturbed energy momentum tensor of the Galileon field). In addition to these modifications to the default CAMB variables, in the code one also defines two extra variables to store the evolution of the first and second derivatives of the Galileon field perturbation, which are solved for with the aid of the equation of motion of the scalar field, and enter the determination of the perturbed energy-momentum tensor. Before solving for the perturbations, the code first performs internal stability checks for the absence of ghost and Laplace instabilities, both in the scalar and tensor sectors.
We refer the reader to [@Barreira:2012kk] for more details about the model equations as they are used in this modified version of CAMB. While the latter is not publicly available[^9], we will use this EB solver to compare codes for this class of models.
f(R) gravity code – [CLASS\_EOS\_fR]{}
--------------------------------------
[CLASS\_EOS\_fR]{} implements the Equation of State approach (EoS) [@Battye:2012eu; @Battye:2013aaa; @Battye:2013ida] into the CLASS EB solver [@Blas:2011rf] for a designer $f(R)$ model. In the EoS approach, the $f(R)$ modifications to gravity are recast as an effective dark energy fluid at both the homogeneous and inhomogeneous (linear perturbation) level.
The degrees of freedom of the perturbed dark-sector are the gauge-invariant overdensity and velocity fields, as described in detail in [@Battye:2015hza]. These obey a system of two coupled first-order differential equations, which involve the expressions of the gauge-invariant dark-sector anisotropic stress, $\Pi_{\rm de}$, and entropy perturbation, $\Gamma_{\rm de}$. The expansion of $\Pi_{\rm de}$ and $\Gamma_{\rm de}$ in terms of the other fluid degrees of freedom (including matter) constitute the equations of state at the perturbed level. They are the key quantities of the EoS approach.
The $f(R)$ modifications to gravity manifest themselves in the coefficients that appear in the expressions of $\Pi_{\rm de}$ and $\Gamma_{\rm de}$ in front of the perturbed fluid degrees of freedom, see [@Battye:2015hza] for the exact expressions. At the numerical level, the advantage of this procedure is that the implementation of $f(R)$ modifications to gravity reduces to the addition of two first-order differential equations to the chosen EB code (e.g. CLASS), while none of the other pre-existing equations of motion, for the matter degrees of freedom and gravitational potential, needs to be directly modified since it receives automatically the contribution of the total stress-energy tensor. In the code [CLASS\_EOS\_fR]{}, the effective-dark-energy fluid perturbations are solved from a fixed initial time up to present - the initial time being chosen so that dark energy is negligible compared to matter and radiation.
At this stage, the code [CLASS\_EOS\_fR]{} is operational for $f(R)$ models in both the synchronous and conformal Newtonian gauge. It shall soon be extended to other main classes of models such as Horndeski and Einstein-Aether theories.
A dedicated paper with details of the implementation and theoretical results and discussion is in preparation [@Battye:2017new].\
Hořava-Lifshitz gravity code CLASS-LVDM
---------------------------------------
This code was developed in order to test the model of dark matter with Lorentz violation (LV) proposed in Ref. [@Blas:2012vn]. The code is based on the CLASS code v1.7, and solves the Eqs. (16)–(23) of Ref. [@Audren:2014hza]. The absence of instabilities is achieved by a proper choice of the parameters of LV in gravity and dark matter. All the calculations are performed in the synchronous gauge, and if needed, the results can be easily transformed into the Newtonian gauge. Further details on the numerical procedure can be found in Ref. [@Audren:2013dwa] where a similar model was studied. The code is available at [[`http://github.com/Michalychforever/CLASS_LVDM`](https://github.com/Michalychforever/CLASS_LVDM)]{}.
Compared to the standard CLASS code, one has to additionally specify four new parameters: $\alpha,\beta,\lambda$ - parameters of LV in gravity in the khronometric model, described in Sec.\[sec:HLtheory\], and $Y$ - the parameter controlling the strength of LV in dark matter. For the purposes of this paper we switch off the latter by putting $Y\equiv 0$ and focus only on the gravitational part of khronometric/Hořava-Lifshitz gravity.
The details of differences in the implementation w.r.t. `EFTCAMB` can be found in Appendix \[app:kh\].
Non-local gravity – modified CAMB and CLASS
-------------------------------------------
We compare two EB codes, a modified version of CAMB and a modified version of CLASS, that compute the cosmology of a specific model of non-local gravity modifying the Einstein-Hilbert action by a term $\sim m^2 R \Box^{-2} R$ (see Sec. \[sec:NLG\] for details).
The modified version of CAMB[^10] was developed by the authors of the GalCAMB code, and as a result, the strategy behind the code implementation is in all similar to that already described in Sec. \[sec:galcamb\] for the Galileon model. The strategy and specific equations used for modifying CLASS[^11] are outlined in details in Appendix A of [@Dirian:2016puz] to which we refer the reader for an exhaustive account. In both cases, the equations that end up being coded are those obtained from the localized version of the theory that features two dynamical auxiliary scalar fields (see Sec. \[sec:NL-theory\]). Within both versions, the background evolution is obtained numerically by solving the system comprising the modified Friedmann equations together with the differential equations that govern the evolution of the additional scalar fields. Both implementations include a trial-and-error search of the free parameter $m$ of the model to yield a spatially flat Universe. At the perturbations level, one works out the perturbed energy-momentum tensor of the latter, and then appropriately adds the corresponding contribution to the relevant variables in the default CAMB code, whereas these have been directly put into the linearized Einstein equations in the CLASS version. The resulting equations depend on the perturbed auxiliary fields, as well as their time derivatives, which are solved for with the aid of the equations of motion of the scalar fields. The modified CAMB code was used in [@2014JCAP...09..031B] to display typical signatures in the CMB temperature power spectrum (although [@2014JCAP...09..031B] focuses more on aspects of nonlinear structure formation), whereas the modified CLASS one was used in various observational constraints studies [@Dirian:2014bma; @Dirian:2016puz; @Dirian:2017pwp].
Tests {#sec:tests}
=====
In this section we present the tests that we have performed to compare the codes described in the previous section. Ideally one should compare codes for a wide range of both gravitational and cosmological parameters. If one is to be thorough, this approach can be prohibitive computationally. Furthermore, that is not the way code comparisons have been undertaken in other situations. In practice one chooses a small selection of models and compares the various observables in these cases. This was the approach taken in the original EB code comparisons [@Seljak:2003th] but is also used in, for example, comparisons between N-body codes for $\Lambda$CDM simulations as well as modified gravity theories [@Winther:2015wla]. Therefore, we will follow this approach here: for each theory we will compare different codes for a handful of different parameters.
A crucial feature of the comparisons undertaken in this section is that they always involve at least a comparison between a modified CAMB and a modified CLASS EB solver. This means that we are comparing codes which, at their core, are very different in architecture, language and genesis. For the majority of cases, we will use `EFTCAMB` and [[hi\_class]{} ]{}as the main representatives for either CAMB or CLASS but in one case (non-local gravity) we will compare two independent codes. Another aspect of our comparison is that at least one of the codes for each model is (or will shortly be made) publicly available.
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![**JBD**. Top figure: The relative difference of the $TT$ angular power spectra of the CMB for the same models showed in Fig. \[fig:JBD\]. Every panel corresponds to a model and the comparison of each code in the legend – CLASSig, JBD-CAMB and [[hi\_class]{} ]{}for reference – has been done w.r.t. EFTCAMB. Bottom figure: The same as in the top figure but for the matter power spectrum.[]{data-label="fig:JBD_all"}](ClsJBD_all.pdf "fig:"){width="0.8\columnwidth"}\
![**JBD**. Top figure: The relative difference of the $TT$ angular power spectra of the CMB for the same models showed in Fig. \[fig:JBD\]. Every panel corresponds to a model and the comparison of each code in the legend – CLASSig, JBD-CAMB and [[hi\_class]{} ]{}for reference – has been done w.r.t. EFTCAMB. Bottom figure: The same as in the top figure but for the matter power spectrum.[]{data-label="fig:JBD_all"}](PkJBD_all.pdf "fig:"){width="0.8\columnwidth"}
In our comparisons, we will be aiming for agreement between codes – up to $\ell=3000$ for the CMB spectra and $k=10\, h\, {\rm Mpc}^{-1}$ in the matter power spectrum – such that the relative distance between observables is of order $0.1\%$, with the exception of low-multipoles ($\ell<100$) where we accept differences up to $0.5\%$ since these scales are cosmic variance limited. We consider this as a good agreement, since it is smaller than the cosmic variance limit out to the smallest scales considered, i.e. $0.1\%$ at $\ell=3000$ in the most stringent scenario (see e.g. [@Seljak:2003th]). We shall see that for $\ell\lesssim300$ in the $EE$ spectra the relative difference between codes exceeds the $1\%$ bound. This clearly evades our target agreement, but it is not worrisome. Indeed, on those scales the data are noise dominated and the cosmic variance is larger than $1\%$. It is important to stress here that all the relative differences shown in the following figures are expressed in $[\%]$ units, with the exception of $\delta C_\ell^{TE}$. Since $C_\ell^{TE}$ crosses zero, we decided not to use it and to show the simple difference in $[\mu K^2]$ units instead.
Another crucial aspect has been the calibration of the codes. To do so, we fixed the precision parameters so that all the tests of the following sections (i) had at least the target agreement, and (ii) the speed of each run was still fast enough for MCMC parameter estimation. While the first condition was explained in the previous paragraph, for the latter we established a factor 3-4 as the maximum speed loss w.r.t. the same model run with standard precision parameters. This factor is a rough estimate that assumes that in the next years the CPU speed will increase, but even with the present computing power MCMC analysis with these calibrated codes is already possible. It is important to stress that most of the increased precision parameters are necessary only to improve the agreement in the lensing CMB spectra on small scales, which is by default 1-2 orders of magnitude worse than the other spectra.
We will be parsimonious in the presentation of results. As will become clear, we have undertaken a large number of cross-comparisons and it would be cumbersome to present countless plots (or tables). Therefore, we will limit ourselves to showing a few significant plots that help us illustrate the level of agreement we are obtaining and spell out, in the text, the battery of tests that were undertaken for each class of models. We have found our results (i.e. the precision with which codes agree) to be relatively insensitive to variations of the cosmological parameters.
Before showing the results of our tests, it is useful to stress here that all the precision parameters used by the codes to generate these figures are specified in Appendix \[sec:precision\], while the cosmological parameters for each model are reported in Appendix \[sec:parameters\].
Jordan-Brans-Dicke gravity {#sec:jbd}
--------------------------
We have validated the `EFTCAMB`, [[hi\_class]{} ]{}and CLASSig EB Solvers in two steps. We have first used DASh and the modified CAMB of [@Avilez:2013dxa] to validate `EFTCAMB` with particular caveats. The current implementation of DASh uses an older version of the recombination module [RECFAST]{} – specifically [RECFAST 1.2]{}. We have run `EFTCAMB` with this older recombination module and found that the agreement with DASh is at the sub-percent level. We have confirmed that this is also true in a comparison between `EFTCAMB` and the modified CAMB of [@Avilez:2013dxa]. We note the codes of [@Avilez:2013dxa] have only been cross checked and calibrated out to $\ell=2000$ and for a maximum wavenumber $k_{\rm max}=0.5\, h\, {\rm Mpc}^{-1}$. With the more restricted cross check of the first step in hand, we have then compared `EFTCAMB`, [[hi\_class]{} ]{}and CLASSig with the more up to date recombination module – specifically [RECFAST 1.5]{} – and out to large $\ell$ and $k$. There are two main effects on the perturbation spectrum in JBD gravity: the effect of the scalar field on the background expansion and the interaction of scalar field fluctuations with the other perturbed fields.
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![**Covariant Galileons**. Top figure: The relative difference of the $TT$, $EE$, lensing and $TE$ angular power spectra of the CMB for the same models showed in Fig. \[fig:Galileons\] between GALCAMB and [[hi\_class]{} ]{}(we find the same level of agreement with `EFTCAMB`). Bottom figure: The same as in the top figure but for the matter power spectrum at different redshifts.[]{data-label="fig:Galileons_hiGal"}](ClsGalileons_hiGal.pdf "fig:"){width="0.8\columnwidth"}\
![**Covariant Galileons**. Top figure: The relative difference of the $TT$, $EE$, lensing and $TE$ angular power spectra of the CMB for the same models showed in Fig. \[fig:Galileons\] between GALCAMB and [[hi\_class]{} ]{}(we find the same level of agreement with `EFTCAMB`). Bottom figure: The same as in the top figure but for the matter power spectrum at different redshifts.[]{data-label="fig:Galileons_hiGal"}](PkGalileons_hiGal.pdf "fig:"){width="0.8\columnwidth"}
In Fig. \[fig:JBD\] we show $C_\ell$ and $P(k)$ for a few different values of $\omega_{\rm BD}$ (see Appendix \[sec:JBD\_parameters\] for the cosmological parameters used in this figures) as well as the relative difference for these quantities between [[hi\_class]{} ]{}and `EFTCAMB`. We can clearly see a remarkable agreement between the codes, well within what is required for current and future precision analysis. It is possible to notice that for $\ell\lesssim10^2$ the disagreement in the temperature $C_\ell$ increases for all the models up to $\simeq0.5\%$. As we shall see, this is a common feature when comparing a CAMB-based code with a CLASS-based code, and it is present even for $\Lambda$CDM, i.e. using CAMB and CLASS instead of our modified versions (see e.g. Fig. \[fig:fR\]). Moreover, it has been checked that for $\Lambda$CDM a systematic bias of 1-2 orders of magnitude smaller than the cosmic variance at $\ell<100$ does not affect parameter extraction with present data, see Section 2 of [@Ade:2015xua]. Therefore, even if this issue deserves further investigation for DE/MG models, we believe that a better agreement at those scales is beyond the scope of this paper. The other issue of Fig. \[fig:JBD\], common to all the models we show in this paper, is that the disagreement in the $C^{EE}_\ell$ on very large scales exceeds the $1\%$ bound. As we already mentioned, this is due to the fact that their amplitude approaches zero and then the relative difference is artificially boosted. This is not to worry, since (i) the amplitude of the polarization angular power spectrum is very small on large scales w.r.t. small scales and (ii) we are protected by cosmic variance. Finally, note that the agreement holds even for extremely small values of $\omega_{\rm JBD}$; this is essential if these codes are to be accurately incorporated into any Monte Carlo parameter estimation algorithm.
Similar results can be found in Fig. \[fig:JBD\_all\], where we compare the outputs of BD-CAMB, CLASSig and [[hi\_class]{} ]{}(for reference) with the outputs generated by `EFTCAMB`. For simplicity, we show the result only for $C^{TT}_\ell$ and $P(k)$ at $z=0$, but the other spectra have similar behaviour as in Fig. \[fig:JBD\]. It is possible to note that the level of agreement is well within the $1\%$ requirement for all the codes, validating their outputs even in “extreme” regions of the parameter space.
This is an important first cross check between EB solvers. JBD is a canonical theory, widely studied in many regimes, and at the core of many scalar-tensor theories. It is a simple model to look at in that the background is monotonic and that only a very small subset of gravitational parameters are non-trivial.
Covariant Galileons {#sec:galileons}
-------------------
The Covariant Galileon theory has been implemented in the current version of [[hi\_class]{} ]{}and `EFTCAMB`. Both these codes were compared against the modified CAMB described in Section \[sec:galcamb\], i.e. GALCAMB. The differences in the implementation are that `EFTCAMB` and GALCAMB assume the attractor solution for the evolution of the background scalar field, while [[hi\_class]{} ]{}evolves the full background equations with the possibility of having arbitrary initial conditions. For comparison with other codes, in [[hi\_class]{} ]{}we will set the initial conditions for the background scalar field as if it were on the attractor, to make the two approaches consistent and comparable. As explained in Section \[sec:galileons\_theory\], and unlike in the JBD case (which is not self-accelerating), there is no extra parameter to vary in the case of the cubic Galileon. Once one is on the attractor and one chooses the matter densities, the evolution is completely pinned down. On the contrary, for the quartic and quintic Galileon models, there are one (for the quartic) or two (for the quintic) additional parameters. This implies that care should be had in enforcing the stability conditions (i.e. enforcing ghost-free backgrounds or preventing the existence of gradient instabilities).
In Fig. \[fig:Galileons\] it is possible to see the CMB angular power spectra and the matter power spectrum at different redshifts for two cubic Galileon models, one quartic and one quintic. While the exact values for the parameters used for this comparison are shown in Appendix \[sec:galileon\_parameters\], here it is important to stress that all these models have been chosen to be bad fits to current CMB and expansion history data. From these figures it can be seen that [[hi\_class]{} ]{}and `EFTCAMB` agree to within the required precision. We have checked that they are also completely consistent with GALCAMB, as it is possible to see in Fig. \[fig:Galileons\_hiGal\], where we show the comparison between [[hi\_class]{} ]{}and GALCAMB. As in the case of JBD, we have varied the cosmological and gravitational parameters and found that this agreement is robust.
f(R) Gravity {#sec:fRgravity}
------------
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$f(R)$ gravity has been implemented in both `EFTCAMB` and [CLASS\_EOS\_fR]{} following two independent approaches [^12]. We focus on designer $f(R)$ models that result in a $\Lambda$CDM expansion history and differ from GR at the perturbation level, displaying an enhancement of small scale structure clustering. Once the expansion history has been chosen one has to fix a residual parameter $B_0$, corresponding to the present value of $B$, as in Eq. (\[ComptonWave\]). We focus on two different values of the $B_0$ parameter: at first we compare cosmological predictions for $B_0=1$, a value that has already been excluded by experiments, to make sure no difference between the two codes is hidden by the choice of a small parameter; we then focus on $B_0=0.01$, that is at the boundary of CMB only experimental constraints [@Raveri:2014cka; @Hu:2014sea] and in the range of interest for N-Body simulations. In Fig. \[fig:fR\] it is possible to see that all compared spectra agree within the required precision. Discrepancies in all CMB spectra are consistent with the comparison to other codes and within $0.5\%$. As in the previous cases, we have varied cosmological and gravitational parameters and found that agreement is robust. The matter power spectrum comparison shows some residual difference that reaches approximately $1\%$ on very small scales, $k=10$h/Mpc, for large values of the free parameter, $B_0=1$. The latter value is already largely excluded by CMB only data, and the scales involved are affected by non-linear clustering, hence this discrepancy is not worrisome.
Non-local Gravity {#sec:NLgravity}
-----------------
\[sec:NLG\]
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For the comparison of the two EB solvers of the non-local $RR$ model, we have considered three sets of cosmological parameters values, shown in Appendix \[sec:NL\_parameters\]. Two of them are markedly poor fits to the data ($RR$-2 and $RR$-3 in Fig. \[fig:NLG\], but the other gets closer to what is allowed observationally (called $RR$-1 here). In Fig. \[fig:NLG\], the $\Lambda{\rm CDM}$ predictions shown correspond to the same parameters values as $RR$-1. Recall, the $\Lambda$[CDM]{} and $RR$ models have the same number of free parameters. The corresponding figures show that the level of agreement between these two EB solvers meets the required standards for all spectra, scales and redshifts shown. In fact, the shape of the relative difference curves are similar in between $\Lambda{\rm CDM}$ and the $RR$ models, which suggests that the observed differences (small as they are) are mostly due to intrinsic differences in the default codes (CAMB and CLASS), and less so due to the modifications themselves.
Hořava-Lifshitz Gravity {#sec:HLgravity}
-----------------------
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We now proceed in validating EFTCAMB and CLASS-LVDM for Hořava-Lifshitz gravity. Because of the different implementation of the background solver (see Appendix \[app:kh\] for details), we have limited the comparison to the subset of parameters satisfying the condition $G_{cosmo}=G_N$, eliminating all the differences arising from it. In the top panels of Fig. \[fig:HL\] we compare the TT, EE, lensing and TE power spectra for two different models – HL-A and HL-B – and a reference $\Lambda$CDM model. These are defined by the sets of parameters specified in Appendix \[sec:HL\_parameters\]. As we can see from the plots, the codes agree always within the $1$% precision for TT, EE and TE power spectra. As for the lensing power spectrum we can notice an order $3$% deviation at both small and large scales. Looking more carefully, one can notice that this difference is not a peculiarity of the MG model, but it is already present at the $\Lambda$CDM level (blue line). The differences at large-$\ell$ are common to all the models under investigation. As for the discrepancy at low-$\ell$, the fact that it is present even for $\Lambda$CDM suggests that it is caused by an inaccuracy in CLASS v1.7, which CLASS-LVDM is based on, and not by the modification itself. Indeed, one may observe that this issue is absent in [[hi\_class]{} ]{}based on an updated version of the CLASS code. In the same figure (bottom panels) we show the matter power spectra for the same models. We can see that the two codes agree well up to $k\simeq 0.1\,h\,{\rm Mpc}^{-1}$, always under the $1$% precision. On scales $k\gtrsim 0.1\,h\,{\rm Mpc}^{-1}$ it is possible to notice that the relative differences in $P(k)$ are drastically increasing, both for $\Lambda$CDM and for the MG models. Like for the $C_\ell$ case, this discrepancy is due to the outdated version of the CLASS code (v1.7). For illustrative purposes we decided to cut the matter power spectrum at the value $k=1\,h\,{\rm Mpc}^{-1}$. It should be pointed out that the scales $k\gtrsim 0.1\,h\,{\rm Mpc}^{-1}$ are significantly affected by non-linear clustering, therefore the output of linear Boltzmann codes in this region is of little practical value.
Note that we used the standard CLASS accuracy flags except for lensing, where a more accurate mode has been employed by imposing $\texttt{accurate\_lensing = TRUE}$.
Parametrized Horndeski functions {#sec:alphas}
--------------------------------
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![**Alphas**. Top figure: The relative difference of the $TT$, $EE$, lensing and $TE$ angular power spectra of the CMB for the same models showed in Fig. \[fig:Alphas\] between COOP and `EFTCAMB` (we find the same level of agreement with [[hi\_class]{} ]{}). Bottom figure: The same as in the top figure but for the matter power spectrum at different redshifts.[]{data-label="fig:Alphas_eftCOOP"}](ClsAlphas_eftCOOP.pdf "fig:"){width="0.8\columnwidth"}\
![**Alphas**. Top figure: The relative difference of the $TT$, $EE$, lensing and $TE$ angular power spectra of the CMB for the same models showed in Fig. \[fig:Alphas\] between COOP and `EFTCAMB` (we find the same level of agreement with [[hi\_class]{} ]{}). Bottom figure: The same as in the top figure but for the matter power spectrum at different redshifts.[]{data-label="fig:Alphas_eftCOOP"}](PkAlphas_eftCOOP.pdf "fig:"){width="0.8\columnwidth"}
Up to this point we have considered a specific set of theories which, albeit representative, only involve a very restricted set of possible time evolution for either the Horndeski or EFT functions. This means that either some of the free functions are set to zero or a lower dimensional subspace of the full function space is explored (see Eq. (\[eq:JBDalphas\]) for a good example). We now need to explore a wider choice of theories and time evolutions.
Ideally, we should somehow explore and compare the full parameter space described by the time dependent functions $\{\alpha_i(\tau),\,w_{\rm DE}(\tau)\}$. This is obviously impossible, but also unnecessary for our purposes. Indeed, the only modifications introduced by COOP, `EFTCAMB` and [[hi\_class]{} ]{}are at the level of the Einstein and scalar field equations. Therefore, it is sufficient to use a parametrization that is capable of capturing all the terms present there. Checking that for particular parametrizations, such as rapidly varying time dependent functions, the three codes agree would in practice correspond to a check on the differential equations solvers of each code, and this is beyond the scope of this work.
The guiding principle in choosing a particular parametrization has been to recover standard gravity at early times, to preserve the physics of the CMB and to ensure a quasi-standard evolution until recent times, i.e. approximately until the onset of dark energy. For example a parametrization closely related with this principle, which has been used in both data analysis [@Bellini:2015xja] and forecasts [@Alonso:2016suf], takes the form $$\begin{aligned}
w_{\rm DE} &= w_0 + (1-a) w_a \nonumber\\
\alpha_i &= c_i\Omega_{\rm DE}\,.\end{aligned}$$ Even if this parametrization is capable of turning on all the possible freedom of Horndeski theories up to linear level, it may be not sufficient. Indeed, the system of equations for the evolution of the perturbations contains both $\{\alpha_i(\tau),\,w_{\rm DE}(\tau)\}$ and their time derivatives. Thus, we have extended this parametrization to be able to modulate the magnitude of the derivatives of these functions. The simplest choice is then $$\begin{aligned}
\label{eq:alpha_par}
w_{\rm DE} &= w_0 + (1-a) w_a \nonumber\\
M^2_*&=1+\delta M^2_0 a^{\eta_0} \nonumber \\
\alpha_i&=\alpha^0_ia^{\eta_i}\,,\end{aligned}$$ where $i$ stands for ${K,\, B,\, T}$. The translation from the $\alpha_i$ functions to the EFT functions is provided in Appendix \[sec:dictionary\].
In Fig. \[fig:Alphas\], we show the lensed temperature $C_\ell$ and the matter power spectrum $P(k)$ calculated at different redshifts for few different values of $\{w_0,\,w_a\}$, $\delta M^2_0$, $\alpha^0_i$ and $\eta_i$ (see Appendix \[sec:alpha\_parameters\] for the list of values used in this comparison). The cosmological parameters are the same for each curve in the plots. The models shown in the figures were built so as to isolate the effect of each $\alpha_i$. Considering the fact that $\alpha_K$ and $\alpha_T$ alone are known to have a small effect on the observables, e.g. [@Bloomfield:2012ff; @Piazza:2013pua; @Perenon:2015sla; @Bellini:2015xja; @Alonso:2016suf], we have always combined them with other functions (either $\alpha_i$ or $w_{\rm DE}$). The $\alpha_{K,\,B,\,M,\,T}\,+\,w$ model (green dotted line) contains all the possible modifications that a Horndeski-like theory can produce. We should stress that the values used here were chosen specifically to have large deviations w.r.t. the reference $\Lambda$CDM model and w.r.t. each other. During the comparison process many more models were explored, both close to $\Lambda$CDM and unrealistically far from it.
An additional requirement to accept models for this comparison was that they were not sensitive to the specific initial conditions (ICs) set for the perturbations: The codes are set up to start with and evolve superhorizon adiabatic ICs, as predicted by standard inflation. Typically, in models which go back to GR quickly enough at early times, the other, isocurvature, modes decay with respect to the adiabatic mode, so it is irrelevant what the initial condition for the scalar field is, since it will reach the required adiabatic mode quickly.
However there are situations, typically when the modification of gravity does not decrease rapidly enough to the past, in which the isocurvature modes do not decay quickly enough (or even grow), and then it is very important that the correct, or at least equivalent, ICs be chosen.
The codes currently have different methods of setting ICs, which is irrelevant when the isocurvature modes decay rapidly enough, but can be important when they are not. We thus have to ensure that we are in a situation where the adiabatic ICs are an attractor for perturbations during radiation domination. The issue of setting the correct ICs for dark-energy perturbations is still an open problem and it will be addressed in future versions of the codes under consideration.
In all the cases we explored, except the ones sensitive to initial conditions as explained above, the results shown in Fig. \[fig:Alphas\] holds. The comparison between `EFTCAMB` and [[hi\_class]{} ]{}shows a remarkable agreement, well below the $1\%$ level. It is possible to notice that the $\alpha_{K,\,T}\,+\,w$ and $\alpha_{K,\,B,\,M,\,T}\,+\,w$ models have relative differences slightly larger than the other models for the $EE$ and $TE$ CMB spectra. While it is difficult to identify one of the $\alpha_i$ or $w$ as the responsible for these deviations, we found that improving the precision parameters of each code solves this issue. This indicates that these two models are particularly complicated and they need increased precision parameters to reach the agreement of the other models. For this particular parametrization, a third code has been tested, i.e. COOP. The agreement between COOP and EFTCAMB is shown in Fig. \[fig:Alphas\_eftCOOP\]. It can be noted that, even if the relative differences in CMB spectra remain below the $1\%$ level, they blow up in the matter power spectrum up to $2-3\%$ on small scales. This seems to be an effect of the accuracy of COOP. Indeed, while COOP is calibrated to get a good agreement on large scales, it lacks of precision for $k\gtrsim1hMpc^{-1}$.
Parametrized EFT functions {#sec:gammas}
--------------------------
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The results presented in the previous section are able alone to establish the agreement between the three codes under consideration. However, while COOP and [[hi\_class]{} ]{}were built using the $\alpha_i$ basis, `EFTCAMB` was built using the EFT approach described in Sec. \[sec:eft\]. As such, the structure of this code is based on $\{\Omega,\, \gamma_i\}$ functions. In case `EFTCAMB` is to be used with the $\alpha$ basis, as in the previous section, there is a built-in module which translates the $\alpha_i$ into the EFT basis before solving for the perturbations. Correspondingly [[hi\_class]{} ]{}needs to translate the $\{\Omega,\, \gamma_i\}$ functions into its preferred $\alpha_i$ basis, in order to be used for the comparison.
Let us note that when simple parametrizations are chosen, the two different bases explore different regions of the parameter space. As an example, consider a parametrization where $\alpha_B\propto a$. Using the conversion relations in Appendix \[sec:dictionary\], it is possible to show that (if $\Omega=0$) $\gamma_2\propto\mathcal{H}$, which scales as $a$ during dark-energy domination, as $a^{-1/2}$ during matter domination and as $a^{-1}$ during radiation domination.
Thus, we have also compared `EFTCAMB` and [[hi\_class]{} ]{}with a particular parametrization of the $\{w_{\rm DE},\,\Omega,\,\gamma_i\}$ functions. In the same spirit as in Eqs. (\[eq:alpha\_par\]), we choose $$\begin{aligned}
\label{eq:gamma_par}
w_{\rm DE} &= w_0 + (1-a) w_a \nonumber\\
\Omega&=\Omega_0 a^{\beta_0} \nonumber \\
\gamma_i&=\gamma^0_ia^{\beta_i}\,,\end{aligned}$$ where $i$ stands for ${1,\, 2,\, 3}$.
In Fig. \[fig:Gammas\], we show the $TT$, $EE$, $TE$, lensing $C_\ell$’s and the matter power spectrum $P(k)$ calculated at different redshifts for a selection of different values of $\{w_0,\,w_a\}$, $\Omega_0$, $\gamma_i^0$ and $\beta_i$. The exact parameters used in these figures are shown in Appendix \[sec:gamma\_parameters\], and the cosmological parameters used to obtain all the curves are the same. On top of a $\Lambda$CDM reference model, the model $\Omega$ (dark blue line) represents the model used in the analysis of current data [@Ade:2015rim]. The other models were built to have an increasingly number of $\gamma_i$ functions and different imprints on the observables. Finally, the $\Omega\,+\,\gamma_{1,\,2,\,3}\,+\,w$ model (green dotted line) turns on all possible modifications at the same time. As in the previous section, this last model shows how model dependent are the precision parameters, having deviations in the EE and TE CMB spectra slightly larger than the other models. Within this parametrization, after neglecting all the models sensitive to the initial conditions as described in the previous section, the disagreement between `EFTCAMB` and [[hi\_class]{} ]{}is within our target accuracy even for the “extreme” models shown in the figures.
Discussion {#sec:discussion}
==========
In this paper we have shown that two general purpose publicly available EB solvers – `EFTCAMB` and [[hi\_class]{} ]{}– are sufficiently accurate and reliable to be used to study a range of scalar-tensor theories. The third general purpose code – COOP – has the required precision for large scales, i.e. $k\lesssim1hMpc^{-1}$, but it needs to be calibrated to give accurate predictions on smaller scales. We have done this analysis by comparing these three codes to each other and to six other EB solvers that target specific theories – DASh, BD-CAMB and CLASSig for JBD, GalCAMB for Galileons, [CLASS\_EOS\_fR]{} for $f(R)$ and HL-CLASS for Hořava-Lifshitz. On top of that, we have shown that two EB solvers – RR-CAMB and RR-CLASS – agree very well when compared to each other for non-local gravity models. While the general principle behind these codes are similar, the implementation is sufficiently different that we believe this is a compelling validation of their accuracy. As such they are fit for purpose if we wish to analyse up and coming cosmological surveys.
We have chosen the precision, or accuracy, settings on the codes being compared such that they could be used efficiently in a Monte Carlo Markov Chain (MCMC) analysis. It is possible to get [*even better*]{} agreement between the codes by boosting the precision settings. This would be done, of course, at a great loss of speed which might make the codes unusable for statistical analysis. We believe that the speed and accuracy we have achieved in this paper is a good, practical compromise. We want to emphasize here that the choice of the precision parameters is very model dependent. Indeed, for some particular configurations we had to increase somewhat the default precision to obtain agreement at the sub-percent level. If one uses the default precision parameters provided with each EB solver she might not get exactly the same agreement we have obtained in this paper. For the models we have considered, we have verified that the disagreement between the different codes was never worse than $1\%$, but it remains the responsibility of the user to verify that the precision parameters chosen are sufficient in order to obtain the accuracy desired.
Of course, there is always more to be done. We have compared these codes at specific points in model and parameter space and our hope is that they should be sufficiently stable that this comparison can be extrapolated to other models and parameters. A possibility of taking what we have done a step further is to undertake parallel MCMC analysis with the codes being compared. [^13] This would fully explore the relevant parameter space and would strengthen the validation process we have undertaken in this paper. Furthermore, both `EFTCAMB` and [[hi\_class]{} ]{}will inevitably be extended to theories beyond scalar-tensor [@Lagos:2016wyv; @Lagos:2016gep]. The same level of rigour will need to be enforced once the range of model space is enlarged.
EB solvers can only tackle linear cosmological perturbations. There are attempts at venturing into the mildly non-linear regime using approximation schemes such as the halo-model, perturbation theory and effective field theory of large scale structure. All attempts at doing so with the level of accuracy required by future data have focus on the standard model. There have been preliminary attempts at doing so for theories beyond GR but, it is fair to say, accurate calculations are still in their infancy. Additional complications that need to be considered when exploring this regime will be the effects of baryons, neutrinos and, more specifically, the effects of gravitational screening which can greatly modify the naive predictions arising from linear theory (a crude attempt at incorporating screening was proposed in [@Alonso:2016suf; @Fasiello:2017bot]).
Finally, we want to emphasize that this paper is not meant to be a *passepartout* to justify every kind of analysis with the codes presented here. They should not be used blindly, and we do not guarantee that all the models implemented in each version of the codes investigated here are free from bugs and reliable. When we introduced in Sec. \[sec:codes\] the publicly available codes we referred to a specific version, and our analysis only validates the accuracy of that version. On top of that one has to bear in mind that, even if we are quite confident that the system of equations (linearized Einstein plus scalar field equations) implemented in each code is bug free, these codes have been tested using a limited number of models. This implies that other built-in models may not be correctly implemented. So, if one wants to use one of the codes analyzed here has to follow the following steps:
1. If the version of the code is not the same as the one studied here, check that it gives the same results as this version for the same models (unless this is guaranteed by the developers of the code);
2. If the model that one wants to analyze has not been studied here, check that the map to convert the parameters of the models into the basis used by the code (e.g. $\alpha_i$ or $\gamma_i$) has been correctly implemented. Since the equations of motion are the same as used in this analysis, this is the most probable place where to find bugs, if any;
3. Check that, for the model, adiabatic initial conditions are an attractor at superhorizon scales during radiation domination. If not, implement the correct initial conditons, to ensure that the addition of dark-energy isocurvature modes does not spoil predictions at late times;
4. Check that the precision parameters used are sufficient to get the desired accuracy. This is very model dependent and can be done with an internal test. It is sufficient to improve them and check that the changes in the output are negligible;
5. Check for a few models that the output is realistic. It can be useful to have some known limit in the parameter space to compare with.
We believe that, with this comparison, we have placed the cosmological analysis of gravitational degrees of freedom on a robust footing. With the tools discussed in hand, we are confident that it will be possible to obtain reliable, precision constraints on general relativity with up and coming surveys.
Acknowledgements {#acknowledgements .unnumbered}
================
EB and PGF are supported by ERC H2020 693024 GravityLS project, the Beecroft Trust and STFC. EC is supported by an STFC Rutherford Fellowship. BH is partially supported by the Chinese National Youth Thousand Talents Program, the Fundamental Research Funds for the Central Universities under the reference No. 310421107 and Beijing Normal University Grant under the reference No. 312232102. The work of MI was partly supported by the Swiss National Science Foundation and by the RFBR grant 17-02-01008. The research of NF is supported by Fundação para a Ciência e a Tecnologia (FCT) through national funds (UID/FIS/04434/2013) and by FEDER through COMPETE2020 (POCI-01-0145-FEDER-007672). NF, SP and AS acknowledge the COST Action (CANTATA/CA15117), supported by COST (European Cooperation in Science and Technology). The work of IS and CS is supported by European Structural and Investment Funds and the Czech Ministry of Education, Youth and Sports (Project CoGraDS – CZ.02.1.01/0.0/0.0/15\_003/0000437). The support by the “ASI/INAF Agreement 2014-024-R.0 for the Planck LFI Activity of Phase E2” is acknowledged by MB, FF and DP. MB, FF and DP also acknowledge financial contribution from the agreement ASI n.I/023/12/0 “Attività relative alla fase B2/C per la missione Euclid”. MR is supported by U.S. Dept. of Energy contract DE-FG02-13ER41958. The work of NAL is supported by the DFG through the Transregional Research Center TRR33 The Dark Universe. Numerical work presented in this publication used the Baobab cluster of the University of Geneva. YD is supported by the Fonds National Suisse. UC has been supported within the Labex ILP (reference ANR-10-LABX-63) part of the Idex SUPER, and received financial state aid managed by the Agence Nationale de la Recherche, as part of the programme Investissements d’avenir under the reference ANR-11-IDEX-0004-02. SP and AS acknowledge support from The Netherlands Organization for Scientific Research (NWO/OCW), and from the D-ITP consortium, a program of the Netherlands Organisation for Scientific Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW). MB acknowledge the support from the South African SKA Project. MZ is supported by the Marie Sklodowska-Curie Global FellowshipProject NLO-CO. FV acknowledges financial support from “Programme National de Cosmologie and Galaxies” (PNCG) of CNRS/INSU, France and the French Agence Nationale de la Recherche under Grant ANR- 12-BS05-0002. FP acknowledges support from the post-doctoral STFC grant R120562 ’Astrophysics and Cosmology Research within the JBCA 2017-2020’.
Relation between EFT functions and $\alpha$’s {#sec:dictionary}
=============================================
In this Appendix we report the mapping between the EFT functions and the $\alpha$ bases for Horndeski theories: $$\begin{aligned}
\Omega(a) &=& - 1 + \left(1 + \alpha_T \right)\frac{M_*^2}{M_{\rm Pl}^2}\,, \nonumber \\
\gamma_1(a) &=& \frac{1}{4a^2H_0^2M_{\rm Pl}^2}\left[\alpha_KM_*^2{\cal H}^2-2a^2c\right]\,, \nonumber \\
\gamma_2(a) &=& -\frac{{\cal H}}{aH_0}\left[\alpha_B \frac{M_*^2}{M_{\rm Pl}^2}+\Omega^\prime\right] \,, \nonumber \\
\gamma_3(a) &=& -\alpha_T \frac{M_*^2}{M_{\rm Pl}^2} \,, \nonumber \\
\gamma_4(a) &=& -\gamma_3\,, \nonumber\\
\gamma_5(a)&=&\frac{\gamma_3}{2} \nonumber\\
\gamma_6(a)&=&0,\end{aligned}$$ where $$\begin{aligned}
\frac{a^2 c(a)}{M_{\rm Pl}^2} =& \mathcal{H}\left(\mathcal{H}- \mathcal{H}^\prime\right) \left(1+\Omega +\frac{\Omega^\prime}{2} \right)\nonumber\\ &-\frac{\mathcal{H}^2}{2}(\Omega^{\prime\prime}-\Omega^{\prime}) -\frac{a^2 (\rho_m + p_m)}{2M_{\rm Pl}^2}\,,\end{aligned}$$ $\alpha_M= \left(\ln M_*^2\right)^\prime$ and primes are derivatives w.r.t. $\ln a$. Note that the above $\alpha_B=-2\alpha_B^{\texttt{EFTCAMB}}$.
Model parameters in Plots {#sec:parameters}
=========================
Here we list all the cosmological parameters used in this paper. For each theory we use the parameters name and the notation that can be found in Section \[sec:theories\].
JBD {#sec:JBD_parameters}
---
In Section \[sec:jbd\], we kept fixed the following cosmological parameters:
- $\Omega_b h^2 = 0.02222$
- $\Omega_c h^2 = 0.11942$
- $A_s = 2.3\times 10^{-9}$
- $n_s = 0.9624$
- $\tau_{\rm reio} = 0.09$
and we varied\
[| C[1cm]{} | C[1.7cm]{} | C[1.7cm]{} | C[1.7cm]{} | C[1.7cm]{} |]{} & $\omega_{BD}=10$ & $\omega_{BD}=50$ & $\omega_{BD}=100$ & $\omega_{BD}=1000$\
$\omega_{BD}$ & $10$ & $50$ & $100$ & $1000$\
$H_0$ & $44.31$ & $61.43$ & $64.22$ & $66.90$\
Covariant Galileons {#sec:galileon_parameters}
-------------------
In Section \[sec:galileons\], for the Galileon models we varied all the cosmological parameters (a “D” in parenthesis indicates that we used that parameter as derived):\
[| C[1cm]{} | C[1.7cm]{} | C[1.7cm]{} | C[1.7cm]{} | C[1.7cm]{} |]{} & Cubic Galileon A & Cubic Galileon B & Quartic Galileon & Quintic Galileon\
$H_0$ & $75.55$ & $45$ & $55$ & $55$\
$\Omega_b h^2$ & $0.02173$ & $0.01575$ & $0.02175$ & $0.02202$\
$\Omega_c h^2$ & $0.124$ & $0.100$ & $0.100$ & $0.100$\
$A_s$ & $2.05\times 10^{-9}$ & $2.16\times 10^{-9}$ & $2.16\times 10^{-9}$ & $2.09\times 10^{-9}$\
$n_s$ & $0.955$ & $0.980$ & $0.980$ & $0.954$\
$\tau_{\rm reio}$ & $0.052$ & $0.088$ & $0.088$ & $0.062$\
$\xi$ & $-2.11$ (D) & $-1.60$ (D) & $2.65$ & $1.4$\
$c_3$ & $0.079$ (D) & $0.104$ (D) & $-0.124$ (D) & $0.2$\
$c_4$ & - & - & $-7.74\times 10^{-3}$ (D)& $0.125$ (D)\
$c_5$ & - & - & - & $-0.125$ (D)\
f(R) {#sec:fR_parameters}
----
In Section \[sec:fRgravity\] we kept fixed the standard cosmological parameters to these values
- $H_0=69$
- $\Omega_b h^2 = 0.022032$
- $\Omega_c h^2 = 0.12038$
- $A_s = 2.3\times 10^{-9}$
- $n_s = 0.96$
- $\tau_{\rm reio} = 0.09$
while we varied the additional parameters\
[| C[1cm]{} | C[1.7cm]{} | C[1.7cm]{} | C[1.7cm]{} |]{} & $\Lambda$CDM & fR-1 & fR-2\
$B_0$ & $0$ & $1$ & $0.01$\
Non-Local Gravity {#sec:NL_parameters}
-----------------
In Section \[sec:NLgravity\] we varied all the cosmological parameters (a “D” in parenthesis indicates that we used that parameter as derived):
[| C[1cm]{} | C[1.7cm]{} | C[1.7cm]{} | C[1.7cm]{} |]{} & RR-1 & RR-2 & RR-3\
$H_0$ & $67$ & $55$ & $55$\
$\Omega_b h^2$ & $0.0222$ & $0.0222$ & $0.0222$\
$\Omega_c h^2$ & $0.118$ & $0.100$ & $0.120$\
$A_s$ & $2.21\times 10^{-9}$ & $2.51\times 10^{-9}$ & $1.81\times 10^{-9}$\
$n_s$ & $0.96$ & $0.93$ & $0.98$\
$\tau_{\rm reio}$ & $0.09$ & $0.06$ & $0.12$\
$m^2$ & $4.06\times 10^{-9}$ (D) & $2.51\times 10^{-9}$ (D) & $2.18\times 10^{-9}$ (D)\
The $\Lambda$CDM model has the same parameters as RR-1, but with a cosmological constant instead of the Non-Local parameter $m$.
Hořava-Lifshitz gravity {#sec:HL_parameters}
-----------------------
In Section \[sec:HLgravity\] we used the same standard cosmological parameters shown in Section \[sec:fR\_parameters\] and we varied the additional parameters\
[| C[1cm]{} | C[1.7cm]{} | C[1.7cm]{} | C[1.7cm]{} |]{} & $\Lambda$CDM & HL-A & HL-B\
$\lambda$ & $1$ & $1.2$ & $ 1.02807 $\
$\xi$ & $1$ & $1.3333$ & $1.05263 $\
$\eta$ & $0$ & $0.0666$ & $0.0210526 $\
Parametrized Horndeski functions {#sec:alpha_parameters}
--------------------------------
In Section \[sec:alphas\] we used the same standard cosmological parameters shown in Section \[sec:fR\_parameters\] and we varied the MG parameters (note that here $w_0=-1+\delta w_0$)
[| C[1cm]{} | C[1.7cm]{} | C[1.7cm]{} | C[1.7cm]{} | C[1.7cm]{} |]{} & $\alpha_{K,B}$ & $\alpha_{K,M}$ & $\alpha_{K,T}+w$ & $\alpha_{K,B,M,T}+w$\
$\delta w_0$ & - & - & $0.9$ & $-0.5$\
$w_a$ & - & - & $-1.2$ & $1$\
$\delta M_0^2$ & - & $2$ & - & $3$\
$\eta_0$ & - & $1.6$ & - & $1$\
$\alpha_K^0$ & $1$ & $1$ & $1$ & $1$\
$\eta_K$ & $1$ & $1$ & $1$ & $1$\
$\alpha_B^0$ & $1.8$ & - & - & $1.8$\
$\eta_B$ & $1.5$ & - & - & $1.5$\
$\alpha_T^0$ & - & - & $-0.9$ & $-0.6$\
$\eta_T$ & - & - & $1$ & $1$\
Parametrized EFT functions {#sec:gamma_parameters}
--------------------------
In Section \[sec:gammas\] we used the same standard cosmological parameters shown in Section \[sec:fR\_parameters\] and we varied the MG parameters (note that here $w_0=-1+\delta w_0$)
[| C[1cm]{} | C[1.7cm]{} | C[1.7cm]{} | C[1.7cm]{} | C[1.7cm]{} |]{} & $\Omega$ & $\Omega+\gamma_1$ & $\Omega+\gamma_{1,2}$ & $\Omega+\gamma_{3}+w$\
$\delta w_0$ & - & - & - & $0.9$\
$w_a$ & - & - & - & $-1.2$\
$\Omega_0$ & $2$ & $1$ & $2$ & $2$\
$\beta_0$ & $1$ & $0.4$ & $1.5$ & $1$\
$\gamma_1^0$ & - & $1$ & $1$ & -\
$\beta_1$ & - & $1$ & $1$ & -\
$\gamma_2^0$ & - & - & $-4.8$ & -\
$\beta_2$ & - & - & $0$ & -\
$\gamma_3^0$ & - & - & - & $2$\
$\beta_3$ & - & - & - & $1$\
Precision parameters in Plots {#sec:precision}
=============================
In order to improve the accuracy of the results, keeping in mind that the CPU-time should remain acceptable for MCMC runs, we changed the default values for some precision parameter
- CAMB-based codes
get_transfer = T
transfer_high_precision = T
high_accuracy_default = T
k_eta_max_scalar = 80000
do_late_rad_truncation = F
accuracy_boost = 1
l_accuracy_boost = 1
l_sample_boost = 1
l_max_scalar = 10000
accurate_polarization = T
accurate_reionization = T
lensing_method = 1
massive_nu_approx = 0
use_spline_template = T
accurate_BB = F
EFTCAMB_turn_on_time = 1e-10
- CLASS-based codes (except for CLASS-LVDM)
l_max_scalars = 5000
P_k_max_h/Mpc = 12.
perturb_sampling_stepsize = 0.010
l_logstep=1.026
l_linstep=25
l_switch_limber = 20
k_per_decade_for_pk = 200
accurate_lensing = 1
delta_l_max = 1000
k_max_tau0_over_l_max=8
Hořava-Lifshitz gravity comparison {#app:kh}
==================================
In this Appendix we illustrate the differences in the approaches used to implement Hořava-Lifshitz gravity in CLASS-LVDM and `EFTCAMB`.
\(1) The first key difference between CLASS-LVDM and `EFTCAMB` is the treatment of the background. It is well known that the only effect of Hořava-Lifshitz (khrononmetric) gravity on the homogeneous and isotropic universe is the rescaling of the gravitational constant in the Friedman equation. CLASS-LVDM uses the rescaled background densities defined via the Friedman equation $$\label{eq:classback}
H^2 = \sum_i \tilde{\rho}_i = H_0^2\sum_{i} \tilde\Omega_i(z) \,,$$ in which way the densities $\tilde{\rho}_i$ (correspondingly, $\tilde\Omega_i(z=0)$ - subject to input in CLASS-LVDM) are rescaled by $G_{cos}/G_N$, and the flatness condition $\sum_{all\,species}\tilde\Omega_i(z=0)=1$ is satisfied automatically.
On the other hand, `EFTCAMB` uses the background densities defined via the gravitational constant in the Newtonian limit, which is generically different from that appearing in the Friedman equation. To be more precise, `EFTCAMB` solves the following Friedman equation $$\label{eq:cambback}
H^2 = H_0^2\frac{G_{cos}}{G_{N} }\left(\sum_{dm,\,b,\,\gamma,\,\nu} \Omega_i(z) + \left[\Omega_{DE}^0 + \frac{G_N}{G_{cos}} -1\right] \right)\,,$$ where $\Omega_{DE}^0$ is the present time DE density parameter. The fractional densities $\Omega_i(z=0)$ (subject to input in `EFTCAMB`) are therefore the “bare" parameters. The modification in the effective $\Omega_{DE}^0$ in the square brackets of is dictated by the requirement that the flatness condition ($\sum_{all\,species}\Omega_i =1 $) be satisfied at redshift zero [@Frusciante:2015maa].[^14]
To sum up, the background evolution in both codes is intrinsically different in the case $G_N\neq G_{cos}$, which is why for the purposes of this paper we focused only on the parameters for which $G_N = G_{cos}$.
\(2) The second difference is the definition of the matter power spectrum. As explained in Refs. [@Audren:2013dwa; @Audren:2014hza], in order to match the observations, the power spectrum in CLASS-LVDM is rescaled by the factor $(G_{cosm}/G_N)^2$. This is to be contrasted with `EFTCAMB`, which uses the standard definition. Within our convention to study only the case $G_N = G_{cos}$, this difference becomes irrelevant.
\(3) The third difference is in the normalization of the primordial power spectrum. In order to isolate the LV effects from the standard cosmological parameters, in the CLASS-LVDM code by default the initial power spectrum of metric perturbations is normalized in a way to match the $\Lambda$CDM one for the same choice of $A_s$ regardless of values of the LV parameters. This is not the same in `EFTCAMB`, where additionally to the background densities the initial power spectrum also bears the dependence on the the extra parameters of Hořava/khrononmetric gravity. Qualitatively, there is no difference between these two approaches. For the purposes of this paper for each set of parameters we normalized the initial power spectra to the same value in both codes.
\(4) The fourth difference is in the initial conditions. CLASS-LVDM assumes the initial conditions for the khronon field corresponding to the adiabatic mode [@Audren:2013dwa; @Audren:2014hza]. On the other hand, `EFTCAMB` assumes for the initial conditions that DE perturbations are sourced by matter perturbations at a sufficiently early time so that the theory is close to General Relativity [@Hu:2014oga]. In order to take into account the difference in the initial conditions, only for this comparison in both codes we set the initial conditions as $$\label{eq:piIC}
\pi(\tau_0)=0 \quad \text{and} \quad \dot\pi(\tau_0)=0\,,$$ where $\pi$ is the extra scalar degree of freedom (i.e. khronon). It is important to note that this choice correspond to an isocurvature mode that totally compensates the adiabatic one at the initial time.
[^1]: [https://www.euclid-ec.org/]{}
[^2]: https://www.lsst.org/
[^3]: https://wfirst.gsfc.nasa.gov/
[^4]: http://skatelescope.org/
[^5]: https://cmb-s4.org/
[^6]: Note that $H$ does not completely fix the evolution of all the background quantities; it must be augmented by the evolution of the matter species encoded in $S_m$.
[^7]: Note that if inflation occurred it would set the field very near the attractor by the early radiation era [@Deffayet:2010qz; @Renk:2017rzu].
[^8]: In turn, khronometric gravity is a variant of Einstein–Aether gravity [@Jacobson:2000xp], an effective field theory describing the effects of Lorentz invariance violation. It should be pointed out that these models have identical scalar and tensor sectors.
[^9]: It will nonetheless be made available by the authors upon request.
[^10]: This version of the CAMB code for the $\nloc$ model is not publicly available, but it will be shared by the authors upon request.
[^11]: The code is publicly available, see [@gitnonlocal] for the link.
[^12]: Note that, even though $f(R)$ gravity is a sub-class of Horndeski theories, it has not been implemented in the current version of [[hi\_class]{} ]{}.
[^13]: MCMC parameter extraction has been performed on the same covariant Galileon models. The results found using modified CAMB [@Barreira:2014jha] and Planck 2013 data are fully consistent with those obtained with [[hi\_class]{} ]{}using Planck 2015 [@Renk:2016olm].
[^14]: Note that one could redefine $\Omega_{DE}(z)$ to absorb all the modifications due to the rescaling of the gravitational constant into it. This would lead to a $\Omega_{DE}(z)$ dependent on the HL parameters plus a standard gravitational constant. This different convention would lead to the same cosmology as with our definitions (if all “bare” fractional densities are suitably chosen), since the two descriptions are equivalent.
|
---
abstract: 'We theoretically investigate the equation of state and Tan’s contact of a non-degenerate three dimensional Bose gas near a broad Feshbach resonance, within the framework of large-$N$ expansion. Our results agree with the path-integral Monte Carlo simulations in the weak-coupling limit and recover the second-order virial expansion predictions at strong interactions and high temperatures. At resonance, we find that the chemical potential and energy are significantly enhanced by the strong repulsion, while the entropy does not change significantly. With increasing temperature, the two-body contact initially increases and then decreases like $T^{-1}$ at large temperature, and therefore exhibits a peak structure at about $4T_{c0}$, where $T_{c0}$ is the Bose-Einstein condensation temperature of an ideal, non-interacting Bose gas. These results may be experimentally examined with a non-degenerate unitary Bose gas, where the three-body recombination rate is substantially reduced. In particular, the non-monotonic temperature dependence of the two-body contact could be inferred from the momentum distribution measurement.'
author:
- 'Xia-Ji Liu$^{1}$, Brendan Mulkerin$^{1}$, Lianyi He$^{2}$, and Hui Hu$^{1}$'
title: Equation of state and contact of a strongly interacting Bose gas in the normal state
---
Introduction
============
Understanding strongly interacting Bose gases in three dimensions is a notoriously difficult quest [@Griffin2009; @Griffin1996; @Shi1998; @Liu2004; @Kita2009; @Cooper2010; @Cooper2011; @Zhang2013; @Yukalov2014]. Theoretical studies of these systems have been hindered by the absence of controllable theoretical approaches that can be used to describe their properties within certain errors. Although a formal field-theoretical description of weakly interacting Bose gases was developed more than half a century ago by Lee, Huang and Yang [@Lee1957a; @Lee1957b] and later by Beliaev [@Beliaev1958] based on the ground-breaking Bogoliubov theory [@Bogoliubov1947]. This theory is only applicable in the limit of a small interaction parameter, the so-called gas parameter $na_{s}^{3}\ll1$ - where $n$ is the density and $a_{s}>0$ is the $s$-wave scattering length - as a result of the perturbative expansion. When the gas parameter is extrapolated to infinity, each term appearing in the perturbative field-theoretical description diverges. To the best of our knowledge, a resummation of these divergent terms remains unknown, even in an approximate manner.
Experimental studies, on the other hand, have been hampered by atom losses from inelastic collisions. Unlike a strongly interacting Fermi gas, where the atom loss rate due to three-body recombinations into deeply bound diatomic molecules is greatly suppressed by the Pauli exclusion principle [@Bloch2008], at low temperatures an interacting Bose gas has a three-body loss rate proportional to $a_{s}^{4}$ (i.e., the loss coefficient $\mathcal{L}_{3}\sim\hbar a_{s}^{4}/m$ [@Fedichev1996; @Weber2003]), which grows dramatically when $a_{s}$ is increased. Even in the absence of inelastic collisions, for a strongly interacting Bose gas, the possibility of recombination into deeply bound Efimov trimers [@Efimov1970] indicates that the system can be at best metastable.
Due to these realistic problems, experimental studies of a strongly interacting atomic Bose gas near a broad Feshbach resonance have only been carried out very recently [@Papp2008; @Navon2011; @Rem2013; @Fletcher2013; @Makotyn2014]. The stability or lifetime of a unitary Bose gas with infinitely large scattering length was investigated with $^{7}$Li [@Rem2013] and $^{39}$K atoms [@Fletcher2013] in the non-degenerate regime. It was found that there is a *low-recombination* regime at high temperatures and low densities, in which the loss coefficient saturates at $L_{3}\sim\hbar\lambda_{dB}^{4}/m\propto1/T^{2}$, as predicted [@DIncao2004]. Here, at high temperatures the thermal de Broglie wavelength $\lambda_{dB}=[2\pi\hbar^{2}/(mk_{B}T)]^{1/2}$ replaces the role of the $s$-wave scattering length $a_{s}$. The momentum distribution of a quantum-degenerate unitary Bose gas was also measured with $^{85}$Rb atoms [@Makotyn2014]. These rapid experimental advances have trigged a number of interesting theoretical investigations on the unitary Bose gas [@Cowell2002; @Song2009; @Lee2010; @Diederix2011; @Borzov2012; @Li2012; @Yin2013; @Piatecki2014; @Smith2014; @Skyes2014; @Jiang2014; @Rossi2014; @Laurent2014; @Rossi2015; @Ancilotto2015], focusing particularly on the universal Bertsch parameter $\xi$, the condensate fraction $n_{0}$ at zero temperature and quenching dynamics. The predictions however are very different with each other, due to the absence of an efficient theoretical framework to handle the intrinsic strong correlations of a metastable unitary Bose gas.
In this work, we aim to develop a non-perturbative, controllable theory of a strongly interacting Bose gas in its normal state, with an emphasis on the high-temperature low-recombination regime in which our theoretical predictions might be efficiently tested in future experiments. Our description is built on an earlier innovative theoretical work by Li and Ho [@Li2012], who treated a repulsive Bose gas as a metastable upper branch (defined later) of an interacting Bose gas across a broad Feshbach resonance. By appropriately re-defining the upper branch prescription [@He2015] and using a non-perturbative large-$N$ expansion approach to remove the unphysical non-linear effect in pair fluctuations [@Nikolic2007; @Veillette2007; @Enss2012], we overcome the large mechanically unstable area encountered earlier at low temperatures [@Li2012] and therefore make Li and Ho’s idea practically useful at arbitrary temperatures in the normal state and arbitrary interaction strengths. Our improved theory is able to reproduce the path-integral Monte-Carlo results at weak couplings [@Pilati2006] and the virial expansion at high temperatures [@Liu2009; @Liu2010a; @Liu2010b; @Castin2013; @Liu2013]. In the strongly interacting unitary limit, we calculate the equation of state and Tan’s two-body contact [@Tan2008] as a function of temperature. An interesting non-monotonic temperature dependence of the contact is predicted and is to be compared with future experimental measurements of the momentum distribution.
The rest of the paper is organized as follows. In the next section, we briefly introduce Li and Ho’s idea of the upper branch Bose gas and present the generalized Nozières-Schmitt-Rink (NSR) method. The upper branch is then appropriately defined through an in-medium phase shift. The large-$N$ expansion approach is adopted in order to overcome the unphysical strong pair fluctuations at large interaction strengths. In Sec. III, we first present the results for weakly interacting Bose gases and compare them with the available path-integral Monte Carlo simulations. We then discuss the equation of state and Tan’s two-body contact in the unitary limit. At sufficiently high temperatures, the results are compared with the virial expansion predictions. Finally, Sec. IV is devoted to the conclusions and outlooks.
Generalized Nozières-Schmitt-Rink approach
==========================================
A three-dimensional (3D) interacting Bose gas can be described by the imaginary-time action [@Popov1987] $$\mathcal{S}=\int d\tau d\mathbf{x}\left[\bar{\psi}\left(\partial_{\tau}-\frac{\hbar^{2}}{2m}\nabla^{2}-\mu\right)\psi+\frac{U_{0}}{2}\bar{\psi}^{2}\psi^{2}\right],\label{eq:action}$$ where $\bar{\psi}(x)$, $\psi(x)$ are $c$-number fields representing the creation and annihilation operators of bosonic atoms of equal mass $m$ at a space-time $x=(\mathbf{x},\tau)$. The imaginary time $\tau$ runs from 0 to the inverse temperature $\beta=1/(k_{B}T)$ and $\mu$ is the chemical potential. The interatomic contact interaction is parameterized by the bare strength $U_{0}<0$, which has to be regularized by the two-particle $s$-wave scattering length $a_{s}$ via the relation, $$\frac{1}{U_{0}}=\frac{m}{4\pi\hbar^{2}a_{s}}-\frac{1}{V}\sum_{\mathbf{k}}\frac{1}{2\epsilon_{\mathbf{k}}},$$ where $V$ is the volume of the system and $\epsilon_{\mathbf{k}}\equiv\hbar^{2}\mathbf{k}^{2}/(2m)$ is the free-particle dispersion (i.e., kinetic energy). In experiment, the scattering length $a_{s}$ can be conveniently tuned by a magnetic field across a Feshbach resonance, to arbitrary values [@Bloch2008].
It should be noted that in our model action, Eq. (\[eq:action\]), the contact interaction is always attractive ($U_{0}<0$), although the scattering length can change sign across the Feshbach resonance. This implies the pairing instability of two bosons and therefore the ground state of the system would be a mixture of pairs and of the remaining unpaired bosonic atoms [@Koetsier2009], similar to what happens for an interacting Fermi gas at the crossover from Bardeen-Cooper-Schrieffer (BCS) superfluids to Bose-Einstein condensates (BEC) [@Eagles1969; @Leggett1980; @NSR1985; @SadeMelo1993; @Hu2006]. In the normal state, such a mixture can be described by using the seminal NSR approach [@NSR1985; @SadeMelo1993; @Hu2006].
Following the earlier work by Koetsier and co-workers [@Koetsier2009], we introduce a pairing field $$\phi\left(\mathbf{x},\tau\right)=U_{0}\psi\left(\mathbf{x},\tau\right)\psi\left(\mathbf{x},\tau\right)$$ and decouple the interatomic interaction via the standard Hubbard-Stratonovich transformation, with which the atomic fields appear quadratically and therefore can be formally integrated out. This leads to an effective action for the pairing field and, at the level of Gaussian pair fluctuations, results in the following grand thermodynamic potential: $$\begin{aligned}
\Omega & = & \Omega_{0}+\delta\Omega,\label{eq:Omega}\\
\Omega_{0} & = & k_{B}T\sum_{\mathbf{k}}\ln\left(1-e^{-\beta\xi_{\mathbf{k}}}\right),\label{eq:Omega0}\\
\delta\Omega & = & k_{B}T\sum_{\mathbf{q},i\nu_{l}}\ln\left[-\Gamma^{-1}\left(\mathbf{q},i\nu_{l}\right)\right],\label{eq:OmegaParisFirst}\end{aligned}$$ where $\xi_{\mathbf{k}}=\varepsilon_{{\bf k}}-\mu$. The last equation is the contribution from pairs of bosons, which is characterized by the two-particle vertex function (or the effective Green function of pairs) $\Gamma\left(\mathbf{q},i\nu_{l}\right)$ with bosonic Matsubara frequencies $\nu_{l}=2\pi lT$ ($l=0,\pm1,\pm2,\cdots$) [@Koetsier2009], $$\Gamma^{-1}=\frac{m}{4\pi\hbar^{2}a_{s}}-\sum_{\mathbf{k}}\left[\frac{\gamma_{B}\left(q,k\right)}{i\nu_{l}-\xi_{\mathbf{q}/2+\mathbf{k}}-\xi_{\mathbf{q}/2-\mathbf{k}}}+\frac{1}{2\varepsilon_{\mathbf{k}}}\right].$$ Here $n_{B}(x)=1/(e^{\beta x}-1)$ is the Bose-Einstein distribution function and the factor $\gamma_{B}(\mathbf{q},\mathbf{k})\equiv1+n_{B}(\xi_{\mathbf{q}/2+\mathbf{k}})+n_{B}(\xi_{\mathbf{q}/2-\mathbf{k}})$ takes into account (in-medium) Bose enhancement of pair fluctuations. By further converting the summation over Matsubara frequencies in Eq. (\[eq:OmegaParisFirst\]) into an integral over real frequency and introducing an in-medium two-particle phase shift [@NSR1985; @SadeMelo1993; @Hu2006] $$\delta\left(\mathbf{q},\omega\right)\equiv-{\rm Imln\left[-\Gamma^{-1}\left(\mathbf{q},\omega+i0^{+}\right)\right]},\label{eq:ps}$$ the contribution to thermodynamic potential from the bosonic pairs can be rewritten as $$\delta\Omega=-\frac{1}{\pi}\sum_{\mathbf{q}}\int_{-\infty}^{+\infty}d\omega\frac{1}{e^{\beta\omega}-1}\delta\left(\mathbf{q},\omega\right).\label{eq:OmegaPairs}$$ To make the above integral meaningful, it is easy to see that, the phase shift at zero frequency $\omega=0$ should vanish identically for any momentum $\mathbf{q}$ because of the Bose-Einstein distribution function. This is the so-called Thouless criterion, which is used to determine the onset of pairing superfluidity.
We note that, within the NSR approach, the only parameter in the imaginary-time action - the chemical potential $\mu$ - is to be determined by using the number equation, $$n=-\frac{1}{V}\frac{\partial\left(\Omega_{0}+\delta\Omega\right)}{\partial\mu}\equiv n_{0}+\delta n,\label{eq:numberEQ}$$ where $n$ is the number density of the system, consisting of both the densities of atoms $n_{0}$ and of pairs $\delta n$.
For an attractive Bose gas near broad Feshbach resonances, Eq. (\[eq:Omega\]) or Eq. (\[eq:numberEQ\]) physically describes an ideal, non-interacting mixture of bosonic atoms with destiny $n_{0}$ and pairs with density $\delta n>0$. With increasing strength of attractive interactions, the contribution from pairs, Eq. (\[eq:OmegaPairs\]), becomes more and more significant. As a result, the chemical potential decreases to the half of the binding energy, $\mu\rightarrow-\hbar^{2}/(2ma_{s}^{2})$, as required by the Thouless criterion $\delta\left(\mathbf{q},\omega\right)=0$ [@Koetsier2009].
![(Color online). (a) The in-medium phase shifts for an attractive Bose gas $\delta_{\textrm{att}}(q=0,\omega)$ at the gas parameter $na_{s}^{3}=1$. The real and imaginary parts of the negative inverse of the two-particle vertex function, $-\Gamma^{-1}(q=0,\omega)$ are also shown. (b) The corresponding in-medium phase shifts $\delta_{\textrm{rep}}(q,\omega)$ in the meta-stable upper branch at different momenta $q=0$ (black solid line), $k_{F}$ (red dashed line) and $2k_{F}$ (blue dot-dashed line). The phase shifts $\delta_{\textrm{rep}}(q=0,\omega)$ at different gas parameters $na_{s}^{3}=0.01$ and $na_{s}^{3}=-1$ are shown by gray cross and green stars, respectively. For all the plots, the temperature is fixed at $T=2T_{c0}$, where $T_{c0}\simeq0.436T_{F}$ is the condensation temperature of an ideal Bose gas. The chemical potential $\mu$ is fixed to that of an ideal Bose gas at the same temperature, i.e., $\mu=\mu^{(0)}(2T_{c0})\simeq-0.358\varepsilon_{F}$.[]{data-label="fig1"}](fig1){width="48.00000%"}
In-medium phase shift for the ground state
------------------------------------------
In Fig. \[fig1\](a), we show the typical behavior of the inverse vertex function $\Gamma^{-1}$ and of the in-medium phase shift $\delta_{\textrm{att}}$ for an *attractive* Bose gas with the gas parameter $na_{s}^{3}=1$ at $T=2T_{c0}$, where $T_{c0}\simeq0.436T_{F}$ is the condensation temperature of an ideal Bose gas, measured in units of Fermi temperature $T_{F}\equiv\hbar^{2}(3\pi^{2}n)^{2/3}/(2mk_{B})\equiv\varepsilon_{F}/k_{B}$. The phase shift jumps from zero to $\pi$ at the threshold frequency $\omega_{b}(\mathbf{q})$, which signals the existence of bound states. Upon increasing the frequency beyond the scattering threshold, $$\omega_{s}\left(\mathbf{q}\right)=\frac{\hbar^{2}\mathbf{q}^{2}}{4m}-2\mu,$$ where the imaginary part of the vertex function becomes nonzero, the phase shift decreases towards $\pi/2$ as $\omega\rightarrow+\infty$. Therefore, there are two contributions to the phase shift, originating from the bound states (at $\omega_{b}(\mathbf{q})\leq\omega<\omega_{s}(\mathbf{q})$) and from the scattering states (i.e., $\omega\geq\omega_{s}(\mathbf{q})$), respectively. It is clear from Fig. \[fig1\](a) that the phase shift, as an illustrated example, does not satisfy the constraint $\delta\left(\mathbf{q},\omega=0\right)=0$. This is because we have used an artificially large chemical potential, larger than the actual chemical potential, which has to be solved self-consistently by using the number equation (\[eq:numberEQ\]) for an attractive Bose gas.
In-medium phase shift for the upper branch
------------------------------------------
It is interesting that although we are dealing with an attractive Bose gas, we may also obtain useful information about a repulsively interacting Bose gas, by treating it as a metastable upper branch of the attractive system. This idea may be understood from the fact that there is an ambiguity in calculating the in-medium phase shift Eq. (\[eq:ps\]), as it involves a multi-valued $\ln(x)$ function. By appropriately choosing different branch cuts, one thus may access excited many-body states, in addition to the ground state of the system.
To the best of our knowledge, the proper choice of in-medium phase shift was first emphasized by Engelbrecht and Randeria in the study of a weakly interacting repulsive Fermi gas in two dimensions in 1992 [@Engelbrecht1992]. However, at that time, the connection between attractive and repulsive systems was not realized and the concept of the upper branch was not established. The meaning of the upper branch was only clarified in 2011 by Shenoy and Ho, who claimed that by excluding the contribution from the paired molecular states in calculating the thermodynamics of the system, one could access the upper branch of an attractive Fermi gas [@Shenoy2011]. This excluded molecular pole approximation (EMPA) immediately implies that for the upper branch, the lower boundary of the frequency integral in Eq. (\[eq:OmegaPairs\]) should be modified to $\omega_{s}(\mathbf{q})$, leading to $$\delta\Omega=-\frac{1}{\pi}\sum_{\mathbf{q}}\int_{\omega_{s}\left(\mathbf{q}\right)}^{+\infty}d\omega\frac{1}{e^{\beta\omega}-1}\delta_{\textrm{rep}}\left(\mathbf{q},\omega\right).\label{eq:OmegaPairsRep}$$ This expression was later applied by Li and Ho to a strongly interacting Bose gas [@Li2012]. However, despite the clarification of the concept of the upper branch, in those two studies (i.e., Refs. [@Li2012] and [@Shenoy2011]), the ambiguity in the calculation of the phase shift $\delta_{\textrm{rep}}(\mathbf{q},\omega)$ was not carefully treated. The phase shift of the upper branch was directly calculated by using $$\delta_{\textrm{rep}}^{\textrm{HO}}\left(\mathbf{q},\omega\geq\omega_{s}\left(\mathbf{q}\right)\right)=-\arctan\left[\frac{\textrm{Im\ensuremath{\Gamma^{-1}\left(\mathbf{q},\omega\right)}}}{\textrm{Re\ensuremath{\Gamma^{-1}\left(\mathbf{q},\omega\right)}}}\right]\label{eq:psRepHO}$$ without the explanation for the branch cut. Here, the function $\arctan(x)$ is the usual inverse tangent function that takes values in the first and fourth quadrant $(-\pi/2,+\pi/2)$ [@note1] and we have used the superscript “HO” to indicate the prescription given by Ho and co-workers.
It turns out that a more appropriate phase shift for the upper branch can be physically defined by the prescription $$\delta_{\textrm{rep}}\left(\mathbf{q},\omega\right)=\left[\delta_{\textrm{att}}\left(\mathbf{q},\omega\right)-\pi\right]\Theta\left[\omega-\omega_{s}\left(\mathbf{q}\right)\right],\label{eq:psRep}$$ which can be shown from the viewpoint of the virial expansion [@He2015]. The $\pi$-shift in the above equation can be easily understood from the standard scattering theory: when a two-body bound state emerges, the two-particle phase shift associated with the density of states should increase by $\pi$. The prescription Eq. (\[eq:psRep\]) is therefore simply the many-body generalization of the two-particle phase shift in the absence of bound states. It should be viewed as a physical realization of the EMPA approximation proposed by Ho and co-workers.
For a weakly interacting Bose gas (i.e., $a_{s}\rightarrow0^{+}$), The two prescriptions for the upper branch phase shift, shown in Eqs. (\[eq:psRepHO\]) and (\[eq:psRep\]), agree with each other, as a result of the large value of $\textrm{Re\ensuremath{\Gamma^{-1}\left(\mathbf{q},\omega\right)}}$. Towards the strongly interacting limit $a_{s}\rightarrow+\infty$, however, the two prescriptions differ significantly. In particular, on the BCS side with a negative scattering length, the phase shift $\delta_{\textrm{rep}}^{\textrm{HO}}(\mathbf{q},\omega)$ coincides with the phase shift of the ground state branch, $\delta_{\textrm{att}}(\mathbf{q},\omega)$. As a result, by changing the scattering length and crossing the Feshbach resonance from below, there is a sudden branch switch from the upper branch to the ground state branch [@Li2012]. This branch-switching effect and the related violation of exact Tan’s relations [@Li2012] are absent when the more physical prescription Eq. (\[eq:psRep\]) is used.
In Fig. \[fig1\](b), we show the in-medium phase shift for the upper branch, obtained by performing Eq. (\[eq:psRep\]) for the attractive phase shift shown in Fig. \[fig1\](a). The Thouless criterion $\delta\left(\mathbf{q},\omega=0\right)=0$ is now strictly satisfied. Moreover, for a positive frequency the phase shift becomes negative. This leads to a positive fluctuation thermodynamic potential $\delta\Omega>0$ and a negative pair density $\delta n<0$. As analyzed by Engelbrecht and Randeria [@Engelbrecht1992], the fact $\delta\Omega>0$ implies that the ground-state energy increases due to the interactions, as it should be for a repulsive system. A negative pairing density is also consistent with a repulsive interaction, which, for a specific atom, will expel other atoms away from its position, and therefore make the effective number density around its position smaller.
In Fig. \[fig1\](b), we also report the upper branch phase shift at the gas parameter $na_{s}^{3}=0.01$ (grey crosses) and $na_{s}^{3}=-1$ (green stars). It is worth noting that the negative value of the gas parameter (i.e., on the BCS side above the Feshbach resonance) actually means stronger repulsions between atoms, as indicated by the large absolute value of the phase shift. In contrast, for a positive gas parameter, the interaction effect becomes weaker with decreasing the gas parameter.
Large-$N$ expansion
-------------------
The generalized NSR approach was used earlier by Li and Ho to investigate a strongly interacting Bose gas near unitarity [@Li2012]. A large mechanically unstable area was found when the temperature of the system is below $T<5T_{c0}\sim2T_{F}$, which renders the approach useful only at extremely high temperatures. Here, we show that the mechanical instability is artificial and caused by the inappropriate treatment for the strong pair/density fluctuations in the NSR approach. It can be cured by the so-called large-$N$ expansion technique [@Nikolic2007; @Veillette2007; @Enss2012].
In the large-$N$ expansion, we assign an additional flavor degree of freedom to bosonic atoms ($i,j=1,\cdots,N$) and thereby extend the model action to, $$\begin{aligned}
\mathcal{\tilde{S}} & = & \int d\tau d\mathbf{x}\left[\sum_{i=1}^{N}\bar{\psi_{i}}\left(\partial_{\tau}-\frac{\hbar^{2}}{2m}\nabla^{2}-\mu\right)\psi_{i}\right.\nonumber \\
& & \left.+\frac{U_{0}}{2N}\sum_{i,j=1}^{N}\bar{\psi}_{i}^{2}\left(\mathbf{x},\tau\right)\psi_{j}^{2}\left(\mathbf{x},\tau\right)\right].\label{eq:actionLargeN}\end{aligned}$$ By introducing a pairing field $$\tilde{\phi}\left(\mathbf{x},\tau\right)=\frac{U_{0}}{N}\sum_{i=1}^{N}\psi_{i}\left(\mathbf{x},\tau\right)\psi_{i}\left(\mathbf{x},\tau\right)$$ and again decoupling the interatomic interaction via the standard Hubbard-Stratonovich transformation, we integrate out the atomic fields and obtain the grand thermodynamic potential per flavor $$\frac{\tilde{\Omega}}{N}=\Omega_{0}+\frac{1}{N}\delta\Omega+O(\frac{1}{N^{2}}),\label{eq:OmegaLargeN}$$ up to the first non-trivial order of $O(1/N)$ [@Nikolic2007; @Veillette2007; @Enss2012]. Here, for the metastable upper branch, $\Omega_{0}$ and $\delta\Omega$ are given by Eqs. (\[eq:Omega0\]) and (\[eq:OmegaPairsRep\]), respectively.
It is clear that in the large-$N$ expansion we have introduced an artificial small parameter $1/N$, which can be used to control the accuracy of the theory of strongly interacting Bose gases. The NSR approach, which is based on the summation of infinite ladder diagrams [@NSR1985; @Hu2006], should be understood as an approximate theory obtained by directly setting $N=1$. However, such a procedure cannot be justified *a priori* in the strongly interacting regime, as the controllable parameter $1/N$ is already at the order of unity. Indeed, the appearance of the large mechanically unstable regime at low temperatures, mentioned at the beginning of this subsection, is precisely an indication of the breakdown of the procedure of directly setting $N=1$. A more reasonable treatment is to first solve the thermodynamics of a $N$-flavor system with $N\gg1$ and then *linearly* extrapolate all the desired physical quantities - as a function of $1/N$ - to the limit of $N=1$. This large-$N$ expansion idea has been successfully applied to a strongly interacting two-component Fermi gas in the unitary limit [@Nikolic2007; @Veillette2007]. The equation of state and the Tan contact near the quantum critical point $\mu=0$ was then accurately predicted [@Enss2012]. In this work, we anticipate that the same large-$N$ expansion technique could also lead to very useful information for a unitary Bose gas in the quantum critical region.
![(Color online) Energy as a function of the artificial controlling parameter $1/N$ at $T=2T_{c0}$ (black squares for $na_{s}^{3}=10^{-6}$ and red circles for $na_{s}^{3}=1$) and $T=10T_{c0}$ (blue crosses with $na_{s}^{3}=1$). The energy is measured in units of the energy $E_{0}$ of a non-interacting Bose gas at the same temperature. The lines are linear fits to the small $1/N$ data. Note that, at low temperatures and strong interactions (i.e., red circles), we are not able to find solutions for $N=1$ because of strong correlations.[]{data-label="fig2"}](fig2){width="48.00000%"}
In Fig. \[fig2\], we show the $1/N$-dependence of the total energy of an interacting Bose gas at different gas parameters and temperatures, obtained by solving the coupled equations Eqs. (\[eq:Omega0\]), (\[eq:OmegaPairsRep\]) and (\[eq:OmegaLargeN\]), and subject to the number equation $\tilde{n}/N\equiv n=n_{0}+\delta n/N$ for the number density per flavor $\tilde{n}/N$. At weak interactions (black squares) or high temperatures (blue crosses), roughly the energy changes linearly as a function of $1/N$. The linear extrapolation approximation used in the large-$N$ expansion therefore does not make significant difference. However, for a strongly interacting Bose gas at relatively low temperatures (red circles), the dependence is highly non-linear. In particular, we are not able to find physical solutions when the number of flavors $N\leq2$. Therefore, it becomes crucial to keep only the linear term in the $1/N$ expansion, which provides the first non-trivial and non-pertrubative knowledge about a strongly correlated many-body state.
Results and discussions
=======================
![(Color online) The energy (a) and pressure (b) as a function of the gas parameter $na_{s}^{3}$ at $T=2T_{c0}$ (black solid lines) and $T=4T_{c0}$ (red dashed lines), normalized respectively by their corresponding results of an ideal, non-interacting Bose gas at the same temperature. The results from a path-integral Monte-Carlo calculations are also shown [@Pilati2006], with squares for hard-sphere potential and circles for soft-sphere potential.[]{data-label="fig3"}](fig3){width="48.00000%"}
In this section, we present our large-$N$ results, calculated by the linear extrapolation towards the limit $1/N=1$. In practice, we solve the generalized NSR approach with $N=50-100$ for the chemical potential $\mu(N)$ and the energy $E(N)$, and then expand them around the corresponding non-interacting values $\mu_{0}$ and $E_{0}$, $$\begin{aligned}
\mu\left(N\right) & = & \mu_{0}+\delta\mu/N+O\left(1/N^{2}\right),\\
E\left(N\right) & = & E_{0}+\delta E/N+O\left(1/N^{2}\right),\end{aligned}$$ to extract the corrections $\delta\mu$ and $\delta E$. This leads to the large-$N$ expansion results $\mu=\mu_{0}+\delta\mu$ and $E=E_{0}+\delta E$.
Crossover to strong repulsions
------------------------------
In Figs. \[fig3\](a) and \[fig3\](b), we present respectively the energy and pressure of an interacting Bose gas at two temperatures $T=2T_{c0}$ (black solid lines) and $T=4T_{c0}$ (red dashed lines), as a function of the gas parameter $na_{s}^{3}$, or $k_{F}a_{s}$ if we convert the number density $n$ to a Fermi wavevector $k_{F}=(3\pi^{2}n)^{1/3}$. The large-$N$ expansion results are compared with available path-integral Monte Carlo calculations for a hard-sphere (squares) and soft-sphere potential (circles) [@Pilati2006]. For weak interactions (i.e., $na_{s}^{3}=10^{-6}$ and $10^{-4}$ or $k_{F}a_{s}<0.2$), our predictions agree well with the *ab-initio* simulations. For strong interactions with strength $k_{F}a_{s}\sim0.8$, there is a significant difference. This is due to the effect of non-negligible effective range of interactions $r_{0}$ used in the Monte Carlo simulations (i.e., $\left|k_{F}r_{0}\right|\sim1$), which leads to a sizable correction to the energy and pressure. In our calculations with a contact interaction, the range of interactions is strictly zero.
![(Color online) Two-body contact $\mathcal{I}_{2}$ as a function of the gas parameter $na_{s}^{3}$ at $T=2T_{c0}$ (black solid line) and $T=4T_{c0}$ (red dashed line). The result from the zero-temperature Bogoliubov theory is shown by the blue dot-dashed line.[]{data-label="fig4"}](fig4){width="48.00000%"}
![(Color online) Temperature dependence of the chemical potential (a), energy (b) and entropy (c) of a unitary Bose gas. For comparison, we show the second-order virial expansion predictions by red empty circles and the ideal gas results by dot-dashed lines. The latest QMC results at zero temperature are also plotted by using stars in brown [@Rossi2014]. []{data-label="fig5"}](fig5){width="48.00000%"}
In Fig. \[fig4\], we show the evolution of the corresponding Tan’s contact with increasing gas parameter. Tan’s contact measures the density of pairs at short distance and determines the exact large-momentum or high-frequency behavior of various physical observables [@Tan2008]. It therefore serves as an important quantity to characterize a strongly interacting many-body system. In particular, experimentally it can be measured from the momentum distribution [@Makotyn2014; @Wild2012], which takes a $k^{-4}$ tail in the short-wavelength limit, i.e., $n(\mathbf{k})\rightarrow\mathscr{\mathcal{I}}/\mathbf{k}^{4}$. At finite temperatures, the contact can be conveniently calculated by using the adiabatic relation [@Tan2008; @Hu2011NJP]: $$\mathcal{I}_{2}=-\frac{4\pi m}{\hbar^{2}}\left[\frac{\partial\Omega}{\partial a_{s}^{-1}}\right]_{T,\mu}.\label{eq:TanRelation}$$ We have used the subscript “$2$” to emphasize the fact that in our calculations we do not consider the three-body Efimov physics and the associated inelastic collisions. These effects are instead captured by a three-body contact $\mathcal{I}_{3}$, which can be defined through an adiabatic relation for a three-body parameter. We refer to Ref. [@Smith2014] for more detailed discussions.
The two-body contact is an increasing function of the interaction strength. At small gas parameters, our results are in good agreement with the weak-coupling predictions of a zero-temperature Bogoliubov theory (thin blue dot-dashed line) [@Schakel2010], $$\mathcal{I}_{\textrm{bog}}=\left(4\pi na_{s}\right)^{2}\left[1+\frac{64}{3\sqrt{\pi}}\sqrt{na_{s}^{3}}\right].$$ The slight increase in our large-$N$ expansion results is due to the finite temperature effect. At large gas parameters $na_{s}^{3}$, the contact tends to saturate to a universal value that depends only on the temperature, as it should be.
Unitary Bose gases
------------------
We are now in position to discuss the universal thermodynamics of a unitary Bose gas. In Fig. \[fig5\], we present the chemical potential, energy and entropy, as a function of temperature. For comparison, we also plot in dot-dashed lines the temperature dependence of an ideal, non-interacting Bose gas. For the chemical potential and energy, our results lie systematically above the non-interacting results, clearly indicating the consequence of strong repulsions. They tends to converge to the zero temperature quantum Monte Carlo predictions (brown stars) with decreasing the temperature [@Rossi2014]. In contrast, the entropy seems to be less affected by strong repulsions. The insensitivity of entropy on the interatomic interactions was also previously found for a unitary Fermi gas [@Hu2008; @Hu2010; @Hu2011PRA].
At high temperatures with a small fugacity $z=e^{\beta\mu}\ll1$, we may use the virial expansion theory to study the universal thermodynamics [@Liu2013]. For a unitary Bose gas, the virial expansion of the grand thermodynamic potential takes the form, $$\begin{aligned}
\Omega & = & \Omega_{0}-\frac{k_{B}T}{\lambda_{dB}^{3}}\left(z^{2}\Delta b_{2}+z^{3}\Delta b_{3}+\cdots\right),\label{eq:OmegaVE}\end{aligned}$$ where $\lambda_{dB}\equiv[2\pi\hbar^{2}/(mk_{B}T)]^{1/2}$ is the thermal de-Broglie wavelength, and $\Delta b_{2}=-\sqrt{2}$ is the second-order virial coefficient for strong repulsions [@Castin2013; @note2], which can be easily calculated by using Beth-Uhlenbeck formalism [@Beth1937]. Up to the second order, we can solve Eq. (\[eq:OmegaVE\]) together with the number equation Eq. (\[eq:numberEQ\]). For the fugacity, we find that $$\begin{aligned}
z & \simeq & \sqrt{\frac{16}{9\pi}}\left(\frac{T_{F}}{T}\right)^{3/2}\label{eq:zVE}\end{aligned}$$ at $T\gg T_{F}$. The virial predictions for the equation of state are shown in Fig. \[fig5\] by red circles and agree well with the large-$N$ expansion results at high temperatures.
![(Color online) Temperature dependence of the two-body contact $\mathcal{I}_{2}$ in the unitary limit. The prediction from a second-order virial expansion, Eq. (\[eq:contactVE\]), is shown by red empty circles. At zero temperature, the brown star and green solid circle refer to the latest QMC result [@Rossi2014] and the number extracted from the recent measurement at JILA for a trapped unitary Bose gas [@Makotyn2014; @Smith2014]. The blue cross indicates the zero-temperature contact of a unitary Fermi gas, measured very precisely by using Bragg spectroscopy [@Hoinka2013].[]{data-label="fig6"}](fig6){width="48.00000%"}
In the unitary limit, we may calculate the universal contact by using the adiabatic relation, Eq. (\[eq:TanRelation\]), shown in Fig. \[fig6\]. With increasing temperature, the contact initially increases and then decreases, giving rise to a peak structure at the temperature $T\sim4T_{c0}$. The decrease of the contact at high temperatures can be well understood by using the virial expansion theory for the contact. As a direct consequence of the adiabatic relation, we have the expansion, $$\begin{aligned}
\mathcal{I}_{2} & = & \frac{8\pi m}{\hbar^{2}}\frac{k_{B}T}{\lambda_{dB}^{2}}\left(z^{2}c_{2}+z^{3}c_{3}+\cdots\right),\end{aligned}$$ where $c_{n}=\lambda_{dB}^{-1}(\partial\Delta b_{n}/\partial a_{s}^{-1})$ is the so-called contact coefficient [@Hu2011NJP]. For a unitary Bose gas, by using Beth-Uhlenbeck formalism it is easy to show that $c_{2}=2/\pi$. Using Eq. (\[eq:zVE\]), we then find $$\begin{aligned}
\frac{I_{2}}{nk_{F}} & \simeq & \frac{64}{3}\left(\frac{T_{F}}{T}\right).\label{eq:contactVE}\end{aligned}$$ Thus, at high temperatures the contact decreases as $T^{-1}$. There is no apparent physical explanation for the increase of the contact at low temperatures. However, we notice that with decreasing temperature toward zero temperature, our large-$N$ expansion result seems to be consistent with the zero-temperature value predicted by the latest quantum Monte Carlo simulation [@Rossi2014].
For comparison, we show the experimental data of the contact [@Makotyn2014], analyzed by Smith and co-workers (green solid circle) [@Smith2014] . The significant discrepancy between experiment and our large-$N$ theory should be largely due to the unknown temperature in the experiment, as the Bose cloud could be significantly heated by atom losses [@Makotyn2014]. We also show the zero-temperature contact of a unitary Fermi gas (blue cross), which has been both calculated and measured very accurately [@Hoinka2013]. It is interesting that both the unitary Bose and Fermi gases have similar contact at zero temperature, indicating that a 3D Bose gas may also have the tendency of being fermionized at strong repulsions, analogous to a Bose gas in one dimension.
Conclusions
===========
In summary, based on the upper branch idea and large-$N$ expansion technique, we have developed a unified theory for a normal, strongly interacting Bose gas. The theory reproduces the path-integral Monte Carlo simulation in the weak-coupling limit [@Rossi2014]. While at high temperatures, it nicely recovers the known results from a quantum virial expansion calculation [@Castin2013; @Liu2013]. Thus, we anticipate that the universal thermodynamics predicted by our theory could be qualitatively reliable. A useful check may be provided by experimentally measuring the finite-temperature contact of a unitary Bose gas through momentum distribution or momentum-resolved radio-frequency spectroscopy [@Makotyn2014].
Our results complement the earlier studies of a condensed strongly interacting Bose gas. It is worth nothing that our theoretical framework can naturally be extended to include the condensation (i.e., $T<T_{c0}$) by using a generalized Nozières-Schmitt-Rink approach below the superfluid transition temperature [@Hu2006]. This extension will be addressed in a future investigation.
X.-J.L. and H.H. acknowledge the support from the ARC Discovery Projects (Grant Nos. FT140100003, DP140100637, FT130100815 and DP140103231) and the National Key Basic Research Special Foundation of China (NKBRSFC-China) (Grant No. 2011CB921502). L.H. was supported by the US Department of Energy Nuclear Physics Office (Contract No. DOE-AC02-05CH11231).
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|
---
abstract: 'We formulate and validate a finite element approach to the propagation of a slowly decaying electromagnetic wave, called surface plasmon-polariton, excited along a conducting sheet, e.g., a single-layer graphene sheet, by an electric Hertzian dipole. By using a suitably rescaled form of time-harmonic Maxwell’s equations, we derive a variational formulation that enables a direct numerical treatment of the associated class of boundary value problems by appropriate curl-conforming finite elements. The conducting sheet is modeled as an idealized hypersurface with an effective electric conductivity. The requisite weak discontinuity for the tangential magnetic field across the hypersurface can be incorporated naturally into the variational formulation. We carry out numerical simulations for an infinite sheet with constant isotropic conductivity embedded in two spatial dimensions; and validate our numerics against the closed-form exact solution obtained by the Fourier transform in the tangential coordinate. Numerical aspects of our treatment such as an absorbing perfectly matched layer, as well as local refinement and a posteriori error control are discussed.'
address:
- 'School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, USA.'
- 'Department of Mathematics, and Institute for Physical Science and Technology, and Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, Maryland 20742, USA.'
author:
- Matthias Maier
- Dionisios Margetis
- Mitchell Luskin
bibliography:
- 'maiermargetisluskin.bib'
title: ' Dipole excitation of surface plasmon on a conducting sheet: finite element approximation and validation '
---
Time-harmonic Maxwell’s equations, finite element method, surface plasmon-polariton, weak discontinuity on hypersurface 65N30,78M10,78M30,78A45
Introduction {#sec:Intro}
============
The manipulation of the electronic structure of low-dimensional materials has recently been the subject of active research with applications in spintronics, quantum information processing, and energy storage [@castroneto09; @pesin12; @trauzettel07; @green12; @geim13; @torres-book; @zhang-book]. In particular, the electric conductivity of atomically thick materials such as graphene and black phosphorous yields an effective complex permittivity with a [*negative*]{} real part in the infrared spectrum [@zhang-book; @low14]. This feature allows for the propagation of slowly decaying electromagnetic waves, called [*surface plasmons-polaritons*]{} (SPPs), that are confined near the material interface with wavelengths much shorter than the wavelength of the free-space radiation [@pitarke07; @zhang-book]. These SPPs are promising ingredients in the design of ultrafast photonic circuits [@pitarke07].
Experimental efforts to generate SPPs focus on the requisite phase matching between waves sustained in free space and the material of interest [@bludov13]. This matching is enabled by the excitation of sufficiently large wave numbers tangential to the associated interface. A technique to achieve this goal is to place a resonant antenna on a graphene sheet [@gonzalez14; @liu12]. The prediction of the resulting waves relies on solving an intricate boundary value problem for the time-harmonic Maxwell equations.
Our purpose with this paper is three-fold. First, we aim to develop a general variational framework for the numerical treatment of electromagnetic-wave propagation in conducting materials modeled as hypersurfaces in a Euclidean space of arbitrary dimension. These hypersurfaces are irradiated by fields produced by external, compactly supported current-carrying sources operating at a fixed, yet arbitrary, frequency. Second, we validate this framework by comparison of finite element numerical simulations to an exact solution of Maxwell’s equations in the simplified case when the conducting sheet is infinite and embedded in a two-dimensional (2D) space in the presence of a Hertzian electric dipole directed vertically to the sheet. Third, we demonstrate that our numerical approach is capable of efficiently resolving SPP structures. To this end, goal-oriented adaptivity for local mesh refinement and a perfectly matched layer are tailored to the fine structure of SPPs.
This work has been broadly motivated by the growing urge to engineer microscopic details of low-dimensional conducting materials in order to establish desired optical properties at larger scales [@pitarke07]. A key objective is to elucidate how the electric conductivity of the material affects characteristics of electromagnetic wave propagation [@cheng13]. There is a compelling need for controllable numerical schemes which, placed on firm mathematical grounds, can reliably describe the SPPs in a variety of geometries. Our paper offers a systematic approach to solving this problem. In particular:
- We formulate a variational framework suitable for the finite element treatment of electromagnetic wave propagation along conducting sheets embedded in spaces of arbitrary dimensions in the presence of external sources (Section \[subsec:variational\]).
- We discretize our variational formulation by use of appropriate curl-conforming Nédélec-elements. This scheme is implemented in a modern C++ framework [@dealii83] and accounts for the fine scale of the SPP as well as for the requisite radiation condition at infinity via a [ *perfectly matched layer*]{} (PML) (Section \[sec:numerics\]). Our approach does not require regularization of the conducting sheet.
- We validate our numerical treatment by comparison of numerics to a tractable exact solution in the simplified case with a vertical electric dipole radiating over an infinite, planar conducting sheet; for convenience, we consider an ambient 2D Euclidean space. In particular, we numerically single out the SPP; and also compare it to the slowly-varying radiation field (see Sections \[sec:analytical\] and \[sec:computationalresults\]).
The extensive literature in plasmonics attests to the rich variety of computational methods and tools; see, e.g., [@gallinet2015; @yeh-book]. For example, an approach is to model graphene as a region with finite thickness (as opposed to a boundary) [@koppens2011]. In the present work, we demonstrate the ability of curl-conforming Nédélec-elements to accurately capture the fine scale of the SPPs, by replacing the conducting sheet with a set of boundary conditions on a hypersurface.
Note that our setting and objectives in this work, focusing on the propagation of SPPs along low-dimensional conducting materials, are distinctly different from the modeling and computation of the interaction of plasmonic nanoparticles with electromagnetic fields [@chau2011; @edel2016]. Our focus on the computation of SPPs along low-dimensional conducting materials and its validation against recently derived analytic solutions [@margetis15] is also distinct from the more classical study of surface plasmons on bulk materials [@raether86]. However, our formulation and general approach is also applicable in this setting.
Throughout the paper, we assume that the reader is familiar with the fundamentals of classical electromagnetic wave theory; for extensive and comprehensive treatments of this subject, see, e.g., [@king92; @muller69; @schwartz72].
Motivation: Surface plasmonics {#subsec:SPP-intro}
------------------------------
Atomically thick conducting materials such as graphene, black phosphorus, and van der Waals heterostructures have been the focus of intensive studies [@geim13; @pitarke07]. The dispersion relations of these structures for electromagnetic wave propagation have novel features. The implications of this dispersion at the infrared spectrum is a theme of essence in surface plasmonics [@samaier07; @zhang-book]. Specifically, in the terahertz frequency range, the effective electric conductivity, $\sigma^\Sigma$, emerging from the coordinated motion of quasi-free electrons, can have an appreciable imaginary part. Furthermore, it has been predicted via numerical simulations that the decoration of graphene by chains of organic molecules may result in a dramatic alteration of $\sigma^\Sigma$ [@cheng13; @dewapriya15]. This prediction paves the way to unconventional means of controlling electronic transport.
From the viewpoint of Maxwell’s equations, the effective dielectric permittivity of a conducting sheet may have a negative real part. The resulting [*metamaterial*]{} has optical properties different from those of a conventional conductor [@zhang-book]. In particular, electromagnetic waves of transverse-magnetic (TM) polarization possibly propagating through the atomically thick material are characterized by a dispersion relation that allows for transmitted wave numbers much larger than the free-space wave number, $k$. For an isotropic and homogeneous ambient space, with wave number $k=\omega\sqrt{\tilde{\varepsilon}\mu}$ and scalar and ${{\boldsymbol{x}}}$-independent $\mu$ and $\tilde{\varepsilon}$, and an isotropic and homogeneous conducting sheet, the condition $|\omega\mu\sigma^\Sigma|\ll |k|$ yields the simplified dispersion relation [@samaier07; @hanson08; @raether86] $$\label{eq:simple_disp-reln}
\sqrt{k^2-k_x^2}\approx -\left(\frac{2k}{\omega\mu\sigma^\Sigma}\right)\,k$$ for TM waves; $k_x$ denotes the wave number tangential to the material interface. Hence, if $k$ is positive, has an admissible solution, $k_x$, provided $\text{Im}\sigma^\Sigma >0$ under an assumed $e^{-i\omega t}$ time dependence. Note that $|k_x|\gg |k|$.
The excitation of SPPs on a homogeneous conducting sheet cannot be achieved by direct illumination of the material by an incident plane wave. There is intrinsic need for phase matching between the waves propagating in different materials, e.g., air and conducting sheet [@bludov13; @samaier07]. An indirect means of establishing this matching is to add metal contacts to the interface [@satou07]. More generally, it is plausible to prescribe current-carrying sources of compact support that optimize attributes of the SPP by variation of the frequency or size of source or the esheet conductivity [@cheng13; @liu12; @gonzalez14].
Our approach {#subsec:main_results}
------------
A jump condition created by an electric conductivity on an interface is a key ingredient in the modeling of thin conducting materials such as graphene, black phosphorus, and a variety of heterostructures resulting from stacking a few distinct crystalline sheets on top of each other. One of our tasks with this work is to construct a variational formulation, well-suited for the finite element method, that naturally incorporates such a jump condition in the presence of external sources.
The variational formulation is implemented by utilizing an appropriate curl-conforming finite element space that only enforces the continuity of the tangential components across elements. The use of *higher-order conforming* elements is well suited for the numerical problem at hand. The weak discontinuity across the interface can be aligned with the triangulation and the regularity of the solution away from the interface leads to high convergence rates. For overcoming the two-scale character with much finer SPP structures close to the interface, an adaptive, local refinement strategy based on a posteriori error estimates is used. The a posteriori error estimates are computed by solving an adjoint problem (*dual weighted residual method*) [@becker1996a] and lead to optimally refined meshes.
Notably, our approach does not require the regularization of the conducting sheet by a layer with artificial thickness. Instead, the sheet can be directly approximated as a lower-dimensional interface. Further, we treat the full scattering problem with an incident wave generated by a Hertzian dipole source instead of merely solving the associated eigenvalue problem for the SPP.
For validation of our treatment, the finite element computations stemming from our approach are compared to the exact solution of Maxwell’s equations for a vertical electric dipole over an isotropic and homogeneous conducting sheet in 2D. In this case, all field components are expressed via 1D Fourier integrals and, thus, are amenable to accurate numerical integration. By this formalism, the SPP is defined as the contribution from a simple pole in the Fourier domain. This contribution is to be contrasted to the slowly-varying radiation field. Our numerics indicate that the SPP dominates the scattered field at distances of the order of the free-space wavelength from the dipole source, in agreement with analytical estimates from the exact solution.
Related work {#subsec:past}
------------
Electromagnetic wave propagation along boundaries, especially the boundary separating air and earth or sea, has been the subject of studies for over a century. A review can be found in [@king92].
This insight is valuable yet insufficient for plasmonic applications related to low-dimensional materials. It is compelling to consider implications of the [*metamaterial*]{} character of [*atomically*]{} thick conducting sheets in the terahertz frequency range [@bludov13]. In particular, in the presence of an electric Hertzian dipole source, boundary condition with $\text{Im}\sigma^\Sigma >0$ can result in a SPP [@bludov13], to be contrasted to surface waves in radio-frequencies which have wave numbers nearly equal to the free-space one [@king92].
In the last few decades, several groups have been studying implications of surface plasmonics; for a (definitely non-exhaustive) sample of related works, see [@bludov13; @hanson08; @hanson11; @liu12; @margetis15; @nikitin11; @raether86; @samaier07; @satou07]. For instance, in [@bludov13] the authors review macroscopic properties of the electric conductivity of graphene, derive dispersion relations for electromagnetic plane waves in inhomogeneous structures, and discuss methods for exciting SPPs; see also the integral-equation approach in [@satou07]. On the other hand, the problem of a radiating dipole source near a graphene sheet is semi-analytically addressed in [@hanson08; @hanson11]. In the same vein, in [@nikitin11] the authors numerically study the field produced by dipoles near a graphene sheet, recognizing a region where the scattered field may be significant. Most recently, two of us derived closed-form analytical expressions for the electromagnetic field when the dipole source and observation point lie on the sheet [@margetis15].
In the aforementioned works, the rigorous numerical treatment of Maxwell’s equations is [*not*]{} of primary concern. Our work here aims to build a framework that places the finite element treatment of a variational boundary value problem on firmer mathematical grounds. This opens up the possibility of numerically studying SPPs in experimentally accessible geometries in an error-controllable fashion. The finite element treatment of Maxwell’s equations is a well-established area of research [@brenner13; @brenner16; @monk03; @nedelec86; @nedelec01]. In particular, our work is related to an adaptive finite element framework for Maxwell’s equations [@brenner13; @brenner16] that uses a residual-based a posteriori error estimator for adaptive local refinement of a triangular mesh. Further, we point out that the well-posedness of our variational approach (see Section \[sec:variational\]) is closely connected to the question of well-posedness of time-harmonic Maxwell’s equations with sign-changing dielectric permittivity [@bendhia14]. The reason for this connection is that the jump condition for a field component across a boundary with a complex-valued conductivity (that we use to model graphene) can be understood as the zero-thickness limit of a *bulk graphene* region with a negative dielectric permittivity.
Open problems {#subsec:limitations}
-------------
Our work here focuses on the development of a reasonably general variational framework. We validate this formulation by comparison of the ensuing finite element numerical computations to the exact solution of Maxwell’s equations in a relatively simple yet nontrivial geometry in 2D. We deem these tasks as necessary first steps in establishing the proposed numerical framework; these steps should precede applications to more complicated cases of physical interest.
Therefore, our work admits several extensions and leaves a few pending issues, from the viewpoints of both analysis and applications. For example, we have not made attempts to fully characterize error estimates following from our treatment. The subtleties related to the possible spatial variation or anisotropy of surface conductivity, $\sigma^\Sigma$, lie beyond our present scope. The numerics for a current-carrying source over a conducting film in 3D [@hanson08; @margetis15] have not been carried out, because of the expensive computations involved. The more elaborate yet experimentally accessible case with a receiving antenna lying on the material interface [@gonzalez14], where the current distribution on the source forms part of the solution, is a promising topic of near-future investigation.
Outline of paper {#subsec:outline}
----------------
The remainder of our paper is organized as follows. In Section \[sec:variational\], we provide the desired variational characterization for boundary value problem –. Section \[sec:analytical\] focuses on the derivation of an exact solution for – in 2D, assuming that the external current-carrying source is a vertical electric dipole and the conducting sheet is homogeneous and isotropic. In Section \[sec:numerics\], we describe the discretization of our variational formulation in the context of finite elements; in particular, we discuss the error control by our treatment (Section \[sec:aposteriori\]). Section \[sec:computationalresults\] present computational results stemming from our approach for an infinite conducting sheet in 2D, along with comparisons with an exact solution. Finally, Section \[sec:conclusion\] concludes our paper with a summary of our results and an outlook.
Variational formulation {#sec:variational}
=======================
In this section, we derive a variational formulation for the time-harmonic Maxwell equations with an interface jump condition. We introduce a slightly modified rescaling of the associated equations to dimensionless forms that are best suited for the numerical observation of the SPP in our treatment. The interface jump condition (\[eq:jumpcondition\]) enters the variational formulation in the form of a *weak discontinuity* (with the second jump-condition for ${{\boldsymbol{E}}}$ being naturally encoded in the [*ansatz*]{} space).
Preliminaries: Boundary value problem {#subsec:Prelim}
-------------------------------------
Next, we formulate the corresponding boundary value problem for the conducting sheet, emphasizing the discontinuity of the magnetic field across the sheet. The starting point of our analysis is the strong form of Maxwell’s equations for the time-harmonic electromagnetic field, $({{\boldsymbol{\mathcal{E}}}}({{\boldsymbol{x}}},t), {{\boldsymbol{\mathcal{B}}}}({{\boldsymbol{x}}},t))=\text{Re}\,\big\{e^{-i\omega t}({{\boldsymbol{E}}}({{\boldsymbol{x}}}),
{{\boldsymbol{B}}}({{\boldsymbol{x}}}))\big\}$, viz., [@schwartz72] $$\begin{aligned}
\begin{cases}
\begin{aligned}
-i\omega{{\boldsymbol{B}}}+\nabla\times{{\boldsymbol{E}}}\;&=\; -{{\boldsymbol{M}}_a}~,
\\[0.1em]
\nabla\cdot{{\boldsymbol{B}}}\;&=\; \frac 1{i\omega}
\nabla\cdot{{\boldsymbol{M}}_a}~,
\\[0.1em]
i\omega\tilde{\varepsilon}{{\boldsymbol{E}}}+\nabla\times\big(\mu^{-1}{{\boldsymbol{B}}}\big) \;&=\; {{\boldsymbol{J}}_{a}}~,
\\[0.1em]
\nabla\cdot\big(\tilde{\varepsilon}{{\boldsymbol{E}}}\big) \;&=\; \frac 1{i\omega}
\nabla\cdot{{\boldsymbol{J}}_{a}}~.
\end{aligned}
\end{cases}
\label{eq:timeharmonicmaxwell}\end{aligned}$$ A few comments on are in order. The (constant) parameter $\omega$ is the temporal angular frequency ($\omega
>0$). We assume that all material parameters are time independent; furthermore, the time-independent, externally applied electric- and magnetic-current densities, ${{\boldsymbol{J}}_{a}}({{\boldsymbol{x}}})$ and ${{\boldsymbol{M}}_a}({{\boldsymbol{x}}})$, respectively, arise from the time-harmonic densities ${{\boldsymbol{\mathcal{J}}}_a}({{\boldsymbol{x}}},t)=\text{Re}\,\big\{e^{-i\omega t}{{\boldsymbol{J}}_{a}}({{\boldsymbol{x}}})\big\}$ and ${{\boldsymbol{\mathcal{M}}}_a}({{\boldsymbol{x}}},t)=\text{Re}\,\big\{e^{-i\omega t}{{\boldsymbol{M}}_a}({{\boldsymbol{x}}})\big\}$. The second-rank tensors $\mu({{\boldsymbol{x}}})$ and $\tilde{\varepsilon}({{\boldsymbol{x}}})$ represent the effective magnetic permeability and complex permittivity of the corresponding medium; the latter is $\tilde{\varepsilon}({{\boldsymbol{x}}})={\varepsilon}({{\boldsymbol{x}}})+i \sigma({{\boldsymbol{x}}})/\omega$, where ${\varepsilon}({{\boldsymbol{x}}})$ and $\sigma({{\boldsymbol{x}}})$ are the (second-rank tensorial) dielectric permittivity and conductivity. We assume that $({{\boldsymbol{E}}}, {{\boldsymbol{B}}})$, $({{\boldsymbol{J}}_{a}}, {{\boldsymbol{M}}_a})$ and ($\tilde{\varepsilon}, \mu)$ in are ${{\boldsymbol{x}}}$ dependent with some (weak) regularity of the fields to ensure unique solvability, as discussed in Section \[sec:variational\].
Equations , interpreted in the strong sense, hold in appropriate unbounded regions of the $n$-dimensional Euclidean space, $\mathbb{R}^n$ ($n=2,3$), excluding the set of points comprising the conducting sheet. We now turn our attention to the requisite boundary conditions along the sheet. This is modeled as an idealized, oriented hypersurface $\Sigma$, $\Sigma\subset \mathbb{R}^n$, with unit normal ${\boldsymbol{\nu}}$ and effective surface conductivity $\sigma^\Sigma({{\boldsymbol{x}}})$ [@bludov13; @hanson08; @hanson11]. This consideration amounts to a jump condition in the tangential component of the magnetic field while the tangential electric field is continuous, viz., [@bludov13] $$\begin{aligned}
\begin{cases}
\begin{aligned}
{\boldsymbol{\nu}}\times\big\{\big(\mu^{-1}{{\boldsymbol{B}}}\big)^+-\big(\mu^{-1}{{\boldsymbol{B}}}\big)^-\big\}
\Big|_{\Sigma}
\;&=\;
\sigma^\Sigma({{\boldsymbol{x}}})\,\big\{({\boldsymbol{\nu}} \times{{\boldsymbol{E}}})\times{\boldsymbol{\nu}}\big\}
\Big|_{\Sigma}~,
\\[0.2em]
{\boldsymbol{\nu}}\times\big\{{{\boldsymbol{E}}}^+-{{\boldsymbol{E}}}^-\big\}\Big|_{\Sigma}
\;&=\; 0~,
\end{aligned}
\end{cases}
\label{eq:jumpcondition}\end{aligned}$$ where ${{\boldsymbol{\mathcal{F}}}}^{\pm}$ (${{\boldsymbol{\mathcal{F}}}}= {{\boldsymbol{E}}}, {{\boldsymbol{B}}}$) is the restriction of the vector-valued solution to either side ($\pm$) of the hypersurface. The surface conductivity, $\sigma^{\Sigma}({{\boldsymbol{x}}})$, is in principle a second-rank tensor and is responsible for the creation of the SPP under the appropriate source $({{\boldsymbol{J}}_{a}}, {{\boldsymbol{M}}_a})$ [@bludov13]; see section \[subsec:SPP-intro\]. At the terahertz frequency range in doped graphene, for example, it is possible that the jump in the tangential component of the magnetic field is small compared to the magnitude of the field itself [@bludov13]. For the appropriate polarization and imaginary part of $\sigma^\Sigma$, this feature may yield a surface wave, the SPP, with a wavelength of the order of a few microns, much smaller than the free-space wavelength [@bludov13; @hanson08; @margetis15].
In addition, the electromagnetic field $({{\boldsymbol{E}}}, {{\boldsymbol{B}}})$ must satisfy the Silver-Müller radiation condition, an extension of the Sommerfeld radiation condition, if the ambient (unbounded) medium is isotropic [@muller69]. This amounts to the requirement that ${{\boldsymbol{\mathcal{F}}}}$ $({{\boldsymbol{\mathcal{F}}}}= {{\boldsymbol{E}}}, {{\boldsymbol{B}}})$ approach a spherical wave uniformly in the radial direction as $|{{\boldsymbol{x}}}|\to
\infty$ for points at infinity and away from the conducting sheet. We need to impose $$\label{eq:S-M_cond}
\lim_{|{{\boldsymbol{x}}}|\to\infty}\{{{\boldsymbol{B}}}\times {{\boldsymbol{x}}}-c^{-1}|{{\boldsymbol{x}}}|\,{{\boldsymbol{E}}}\}=0~,\
\lim_{|{{\boldsymbol{x}}}|\to\infty}\{{{\boldsymbol{E}}}\times {{\boldsymbol{x}}}+ c|{{\boldsymbol{x}}}|\, {{\boldsymbol{B}}}\}=0~,\ x\notin \Sigma~;$$ $c$ is the speed of light in the respective medium. In the formulation of our numerical scheme, we avoid making explicit use of condition by using appropriate boundary conditions together with a PML, which eliminates reflection from infinity.
Rescaling {#sec:rescaling}
---------
Loosely following Colton and Kress [@colton83], as well as Monk [@monk03], we introduce a rescaling for the time-harmonic Maxwell equations (\[eq:timeharmonicmaxwell\]). The key differences of our formulation from the above treatments [@colton83; @monk03] are:
- The additional rescaling of every length scale in our problem by the free-space wavelength $2\pi k_0^{-1}:=2\pi
(\omega\sqrt{{\varepsilon}_0\mu_0})^{-1}$, where $\epsilon_0$ and $\mu_0$ denote the [*vacuum*]{} dielectric permittivity and magnetic permeability, respectively. This rescaling recognizes that the typical length scale of the SPP is one to two orders of magnitude smaller than the corresponding free-space wavelength [@bludov13]; consequently, $1/k_0$ is the appropriate macroscopic length scale.
- The rescaling of ${{\boldsymbol{E}}}$, ${{\boldsymbol{B}}}$, ${{\boldsymbol{J}}_{a}}$, and ${{\boldsymbol{M}}_a}$ by a *typical electric current strength*, $J_0$. In our case, $J_0$ is the strength of the prescribed dipole source at location ${\boldsymbol{a}}$ in the ${\boldsymbol{e}}_i$ direction in Cartesian coordinates: $$\begin{aligned}
{{\boldsymbol{J}}_{a}}=J_0\,{\boldsymbol{e}}_i\,\delta({{\boldsymbol{x}}}-{\boldsymbol{a}}).
\end{aligned}$$
Accordingly, we rescale $\mu$ and $\tilde {\varepsilon}$ by $\mu_0$ and ${\varepsilon}_0$, respectively; cf. : $$\begin{aligned}
{5}
\label{eq:rescaling}
\mu
&\quad\longrightarrow\quad&
{\mu_{r}\,}&=\frac1{\mu_0}\mu,
&\qquad&
{\tilde{\varepsilon}}
&\quad\longrightarrow\quad&
{\tilde\varepsilon_{r}\,}&=&\;
\frac1{{\varepsilon}_0}\tilde{\varepsilon}.
\intertext{Furthermore, by use of the free-space wave number,
$k_0=\omega\sqrt{{\varepsilon}_0\mu_0},$ and the dipole strength, $J_0$, the
rescaling of the vector fields and coordinates is carried out:}
{{\boldsymbol{x}}}&\quad\longrightarrow\quad&
\hat{{\boldsymbol{x}}}&= k_0\,{{\boldsymbol{x}}},
&\qquad&
\nabla
&\quad\longrightarrow\quad&
\hat\nabla &=&\; \frac1{k_0}\,\nabla,
\\[0.1em]
{{\boldsymbol{J}}_{a}}&\quad\longrightarrow\quad&
{{\boldsymbol{\hat J}}_{a}}&= \frac1{J_0}\,{{\boldsymbol{J}}_{a}},
&\qquad&
{{\boldsymbol{M}}_a}&\quad\longrightarrow\quad&
{{\boldsymbol{\hat M}}_a}&=&\; \frac {k_0}{\omega\mu_0\,J_0}\,{{\boldsymbol{M}}_a},
\\[0.1em]
{{\boldsymbol{E}}}&\quad\longrightarrow\quad&
{{\boldsymbol{\hat E}}}&= \frac{k_0^2}{\omega\mu_0\,J_0}\,{{\boldsymbol{E}}},
&\qquad&
{{\boldsymbol{B}}}&\quad\longrightarrow\quad&
{{\boldsymbol{\hat B}}}&=&\; \frac {k_0}{J_0}\,
\mu^{-1}
{{\boldsymbol{B}}}.\end{aligned}$$ In addition, the interface conductivity is rescaled as follows: $$\begin{aligned}
\sigma^\Sigma
\quad\longrightarrow\quad
{\sigma^\Sigma_r}= \sqrt{\frac{\mu_0}{{\varepsilon}_0}}\,\sigma^\Sigma.\end{aligned}$$ Finally, rescaling time-harmonic Maxwell’s equations (\[eq:timeharmonicmaxwell\]) results in the following system: $$\begin{aligned}
\begin{cases}
\begin{aligned}
-i{\mu_{r}\,}{{\boldsymbol{\hat B}}}+\hat\nabla\times{{\boldsymbol{\hat E}}}\;&=\; -{{\boldsymbol{\hat M}}_a},
\\[0.1em]
\hat\nabla\cdot\big({\mu_{r}\,}{{\boldsymbol{\hat B}}}\big) \;&=\; \frac 1{i}
\hat\nabla\cdot{{\boldsymbol{\hat M}}_a},
\\[0.1em]
-i{\tilde\varepsilon_{r}\,}{{\boldsymbol{\hat E}}}-\hat\nabla\times{{\boldsymbol{\hat B}}}\;&=\; -{{\boldsymbol{\hat J}}_{a}}.
\\[0.1em]
\hat\nabla\cdot\big({\tilde\varepsilon_{r}\,}{{\boldsymbol{\hat E}}}\big) \;&=\;\frac1i \nabla\cdot{{\boldsymbol{\hat J}}_{a}}.
\end{aligned}
\end{cases}
\label{eq:timeharmonicmaxwellrescaled}\end{aligned}$$ To lighten the notation, the hat ($\,\hat\ \,$) on top of a rescaled quantity will be omitted in the remainder of this paper. (This simplification avoids confusion of the rescaled quantities with the Fourier transforms of fields invoked in Section \[sec:analytical\]).
Variational statement {#subsec:variational}
---------------------
![ Schematic of the computational domain, $\Omega$, with boundary $\partial\Omega$ and outer normal ${\boldsymbol{n}}$. An electric Hertzian dipole, ${{\boldsymbol{J}}_{a}}$, is situated above a prescribed hypersurface, $\Sigma$. []{data-label="fig:domain"}](maiermargetisluskin-figure0.pdf)
Let $\Omega\subset{\mathbb{R}}^n$ ($n=2,3$) be a simply connected and bounded domain with Lipschitz-continuous and piecewise smooth boundary, $\partial\Omega$. Further, let $\Sigma$ be an oriented, Lipschitz-continuous, piecewise smooth hypersurface. Fix a normal field ${{\boldsymbol{\nu}}}$ on $\Sigma$ and let ${\boldsymbol{n}}$ denote the outer normal vector on $\partial\Omega$ ; see Figure \[fig:domain\]. Substituting ${{\boldsymbol{B}}}$ from the first equation of (\[eq:timeharmonicmaxwellrescaled\]) into the third equation yields $$\begin{aligned}
\nabla\times\big({\mu_{r}^{-1}\,}\nabla\times{{\boldsymbol{E}}}\big)
-{\tilde\varepsilon_{r}\,}{{\boldsymbol{E}}}\;=\; i\,{{\boldsymbol{J}}_{a}}- \nabla\times\big({\mu_{r}^{-1}\,}{{\boldsymbol{M}}_a}\big).
\label{eq:2ndorderequation}\end{aligned}$$ Multiplying (\[eq:2ndorderequation\]) by the complex conjugate, $\bar{{\boldsymbol{\varphi}}}$, of a smooth test function ${{\boldsymbol{\varphi}}}$ and integrating by parts in $\Omega\setminus\Sigma$ lead to $$\begin{gathered}
\int_\Omega ({\mu_{r}^{-1}\,}\nabla\times{{\boldsymbol{E}}})\cdot(\nabla\times\bar{{\boldsymbol{\varphi}}}){\,{\mathrm d}x}\;\;-\;
\int_\Omega {\tilde\varepsilon_{r}\,}{{\boldsymbol{E}}}\cdot\bar{{\boldsymbol{\varphi}}}{\,{\mathrm d}x}\\
-\int_\Sigma\big[{{\boldsymbol{\nu}}}\times({\mu_{r}^{-1}\,}\nabla\times{{\boldsymbol{E}}}+{\mu_{r}^{-1}\,}{{\boldsymbol{M}}_a})\big]_\Sigma
\cdot\bar{{\boldsymbol{\varphi}}}_T{\,{\mathrm d}o_x}\\
+\;
\int_{\partial\Omega}\big({{\boldsymbol{\nu}}}\times({\mu_{r}^{-1}\,}\nabla\times{{\boldsymbol{E}}}+{\mu_{r}^{-1}\,}{{\boldsymbol{M}}_a})\big)
\cdot\bar{{\boldsymbol{\varphi}}}_T{\,{\mathrm d}o_x}\\
=\;\;i\int_\Omega{{\boldsymbol{J}}_{a}}\cdot\bar{{\boldsymbol{\varphi}}}{\,{\mathrm d}x}-
\int_\Omega{\mu_{r}^{-1}\,}{{\boldsymbol{M}}_a}\cdot\nabla\times\bar{{\boldsymbol{\varphi}}}{\,{\mathrm d}x},
\label{eq:integratedbyparts}\end{gathered}$$ where the subscript $T$ above denotes the tangential part of the respective vector, ${{\boldsymbol{\mathcal{F}}}}_T = ({{\boldsymbol{\nu}}}\times{{\boldsymbol{\mathcal{F}}}})\times{{\boldsymbol{\nu}}}$, and $[\,.\,]_\Sigma$ denotes the jump over $\Sigma$ with respect to ${{\boldsymbol{\nu}}}$, viz., $$\begin{aligned}
\big[{{\boldsymbol{\mathcal{F}}}}\big]_\Sigma({{\boldsymbol{x}}})
:=\lim_{s\searrow0}\big({{\boldsymbol{\mathcal{F}}}}({{\boldsymbol{x}}}+s{{\boldsymbol{\nu}}})-{{\boldsymbol{\mathcal{F}}}}({{\boldsymbol{x}}}-s{{\boldsymbol{\nu}}})\big)\qquad ({{\boldsymbol{x}}}\in\Sigma).\end{aligned}$$ For the computational domain $\Omega$, an *absorbing boundary condition* at $\partial\Omega$ is imposed: $$\begin{aligned}
{{\boldsymbol{\nu}}}\times{{\boldsymbol{B}}}+\sqrt{{\mu_{r}^{-1}\,}{\tilde\varepsilon_{r}\,}}{{\boldsymbol{E}}}_T = 0\qquad ({{\boldsymbol{x}}}\in\partial\Omega).
\label{eq:absorbingbc}\end{aligned}$$ The last boundary condition is obtained by using a first-order approximation of the Silver-Müller radiation condition, equation , truncated at $\partial\Omega$ [@jin08]. We will occasionally refer to as the *first-order* absorbing boundary condition. We assume that ${\mu_{r}^{-1}\,}$ and ${\tilde\varepsilon_{r}\,}$ are scalar functions such that the square root in (\[eq:absorbingbc\]) is well defined. In our numerical computation, we combine the above absorbing boundary condition with a PML; see Section \[sub:pollutionandpml\].
An advantage of variational formulation (\[eq:integratedbyparts\]) is that the jump condition over the conducting sheet can be expressed as a *weak discontinuity*. Rewriting jump condition (\[eq:jumpcondition\]) as well as absorbing boundary condition (\[eq:absorbingbc\]) in terms of ${{\boldsymbol{B}}}$ and ${{\boldsymbol{M}}_a}$ by utilizing the first equation of (\[eq:timeharmonicmaxwellrescaled\]) yields $$\begin{aligned}
\big[{{\boldsymbol{\nu}}}\times({\mu_{r}^{-1}\,}\nabla\times{{\boldsymbol{E}}}+{\mu_{r}^{-1}\,}{{\boldsymbol{M}}_a})\big]_\Sigma &=
i\,{\sigma^\Sigma_r}{{\boldsymbol{E}}}_T\quad\text{on }\Sigma, \label{eq:jumpconditionrescaled}
\\[0.2em]
{{\boldsymbol{\nu}}}\times({\mu_{r}^{-1}\,}\nabla\times{{\boldsymbol{E}}}+{\mu_{r}^{-1}\,}{{\boldsymbol{M}}_a}) &= -i\sqrt{{\mu_{r}^{-1}\,}{\tilde\varepsilon_{r}\,}}\,{{\boldsymbol{E}}}_T
\quad\text{on }\partial\Omega.\end{aligned}$$ The last two relations allow us to enforce the jump and boundary conditions weakly by simply substituting them into (\[eq:integratedbyparts\]). We summarize our main result below.
\[prop:existence\]
In order to ensure unique solvability, we assume that ${\sigma^\Sigma_r}\in
L^\infty(\Sigma)^{3\times 3}$ is matrix-valued and symmetric, with semi-definite real and complex part. Further, let ${\tilde\varepsilon_{r}\,}$ be a smooth scalar function and ${\mu_{r}^{-1}\,}$ be a constant scalar such that
- $\text{Im}\,\big({\tilde\varepsilon_{r}\,})=0$, or $\text{Im}\,\big({\tilde\varepsilon_{r}\,})\ge c>0$ in $\Omega$,
- $\sqrt{{\mu_{r}^{-1}\,}{\tilde\varepsilon_{r}\,}}$ is real-valued and strictly positive on $\partial\Omega$.
Define a Hilbert space (cf. [@monk03 Th.4.1]) $$\begin{aligned}
{\boldsymbol{X}}(\Omega)=\Big\{{{\boldsymbol{\varphi}}}\in {\boldsymbol{H}}(\text{curl};\Omega)\;:\;
{{\boldsymbol{\varphi}}}_T\big|_\Sigma\in L^2(\Sigma)^3,\;
{{\boldsymbol{\varphi}}}_T\big|_{\partial\Omega}\in L^2(\partial\Omega)^3
\Big\}
\end{aligned}$$ equipped with the norm $\|{{\boldsymbol{\varphi}}}\|^2_{{\boldsymbol{X}}}=\|\nabla\times{{\boldsymbol{\varphi}}}\|^2+
\|{{\boldsymbol{\varphi}}}_T\|_{L^2(\Sigma)}^2 + \|{{\boldsymbol{\varphi}}}_T\|_{L^2(\Omega)}^2$. In the above, ${\boldsymbol{H}}(\text{curl}; \Omega)$ denotes the space of vector-valued, measurable and square integrable functions whose (distributive) curl admits a representation by a square integrable function.
The rescaled, weak formulation of (\[eq:timeharmonicmaxwell\]) with jump condition (\[eq:jumpcondition\]) and absorbing boundary condition (\[eq:absorbingbc\]) can be stated as follows: Find ${{\boldsymbol{E}}}\in{\boldsymbol{X}}(\Omega)$, such that $$\begin{aligned}
A({{\boldsymbol{E}}},{{\boldsymbol{\varphi}}}) = F({{\boldsymbol{\varphi}}}),
\label{eq:variationalformulation}
\end{aligned}$$ for all ${{\boldsymbol{\varphi}}}\in{\boldsymbol{X}}(\Omega)$. It admits a unique solution. The sesquilinear form and the right-hand side are given by $$\begin{aligned}
A({{\boldsymbol{E}}},{{\boldsymbol{\varphi}}}) &:=
\int_\Omega ({\mu_{r}^{-1}\,}\nabla\times{{\boldsymbol{E}}})\cdot(\nabla\times\bar{{\boldsymbol{\varphi}}}){\,{\mathrm d}x}\;\;-\;
\int_\Omega{\tilde\varepsilon_{r}\,}{{\boldsymbol{E}}}\cdot\bar{{\boldsymbol{\varphi}}}{\,{\mathrm d}x}\\\notag
&\qquad
-\, i\,
\int_{\Sigma}({\sigma^\Sigma_r}{{\boldsymbol{E}}}_T)\cdot\bar{{\boldsymbol{\varphi}}}_T{\,{\mathrm d}o_x}\;\;-\; i\,
\int_{\partial\Omega}\sqrt{{\mu_{r}^{-1}\,}{\tilde\varepsilon_{r}\,}}{{\boldsymbol{E}}}_T\cdot\bar{{\boldsymbol{\varphi}}}_T{\,{\mathrm d}o_x},
\\[0.1em]
F({{\boldsymbol{\varphi}}})
&:=\;\;
i\int_\Omega{{\boldsymbol{J}}_{a}}\cdot\bar{{\boldsymbol{\varphi}}}{\,{\mathrm d}x}-
\int_\Omega{\mu_{r}^{-1}\,}{{\boldsymbol{M}}_a}\cdot\nabla\times\bar{{\boldsymbol{\varphi}}}{\,{\mathrm d}x}.
\label{eq:variationalformulation2}
\end{aligned}$$
The existence result for time-harmonic Maxwells equations with an absorbing boundary condition [@colton83; @colton98; @monk03] can be applied almost directly to problem (\[eq:variationalformulation\]); the additional interface integral in $A({{\boldsymbol{E}}},{{\boldsymbol{\varphi}}})$, $$\begin{aligned}
-\,i\,\int_\Sigma{\sigma^\Sigma_r}{{\boldsymbol{E}}}_T\cdot\bar{{\boldsymbol{\varphi}}}_T{\,{\mathrm d}o_x}.
\end{aligned}$$ requires a careful discussion. For this we split the integral into two contributions, $$\begin{aligned}
\label{eq:surface-int}
-\,i\,\int_\Sigma{\sigma^\Sigma_r}{{\boldsymbol{E}}}_T\cdot\bar{{\boldsymbol{\varphi}}}_T{\,{\mathrm d}o_x}=
-\,i\,\int_\Sigma\text{Re}\,\big({\sigma^\Sigma_r}\big){{\boldsymbol{E}}}_T\cdot\bar{{\boldsymbol{\varphi}}}_T{\,{\mathrm d}o_x}+
\int_\Sigma\text{Im}\,\big({\sigma^\Sigma_r}\big){{\boldsymbol{E}}}_T\cdot\bar{{\boldsymbol{\varphi}}}_T{\,{\mathrm d}o_x}.
\end{aligned}$$ The first term on the right-hand side of can be treated similarly to the absorbing boundary condition on $\partial\Omega$ (cf. [@monk03 Sec.4.5]). For the second term in , involving $\text{Im}\,\big({\sigma^\Sigma_r}\big)$, it holds that $$\begin{aligned}
\int_\Sigma\text{Im}\,\big({\sigma^\Sigma_r}\big){{\boldsymbol{E}}}_T\cdot\bar{{\boldsymbol{E}}}_T{\,{\mathrm d}o_x}\ge 0.
\end{aligned}$$ Thus, this term does not affect any proof based on showing coercivity of a modified sesquilinear form and using Fredholm’s alternative on a compact perturbation.
In order to prove uniqueness we follow the strategy in [@monk03 Sec.4.6]. First, test the homogeneous equation $A({{\boldsymbol{e}}},{{\boldsymbol{\varphi}}}) = 0$ with ${{\boldsymbol{e}}}$ itself and take the imaginary part, $$\begin{aligned}
\int_\Omega(\text{Im}({\tilde\varepsilon_{r}\,})\,{{\boldsymbol{e}}}\big)\cdot\bar{{\boldsymbol{e}}}{\,{\mathrm d}x}+
\int_{\Sigma}(\text{Im} ({\sigma^\Sigma_r})\,{{\boldsymbol{e}}}_T)\cdot\bar{{\boldsymbol{e}}}_T{\,{\mathrm d}o_x}+
\int_{\partial\Omega}\sqrt{{\mu_{r}^{-1}\,}{\tilde\varepsilon_{r}\,}}{{\boldsymbol{e}}}_T\cdot\bar{{\boldsymbol{e}}}_T{\,{\mathrm d}o_x}= 0.
\end{aligned}$$ This immediately implies ${{\boldsymbol{e}}}_T=0$ on $\partial\Omega$ and $\Sigma$. The (nontrivial) conclusion ${{\boldsymbol{e}}}=0$ now follows verbatim by the proof of [@monk03 Th.4.12].
The following remarks are in order.
\[remark:2dmodel\] The 3D problem description given by (\[eq:variationalformulation\]) readily translates into a corresponding problem in 2D: Assume that the interface $\Sigma$, the electric field ${{\boldsymbol{E}}}$, the permeability ${\mu_{r}\,}$, as well as the permittivity ${\tilde\varepsilon_{r}\,}$ and surface conductivity ${\sigma^\Sigma_r}$ are translation- and mirror-invariant in the $z$-direction. Thus, the term $\nabla\times{{\boldsymbol{E}}}$ and, consequently, the ${{\boldsymbol{B}}}$ field have nonzero components only in the $z$-direction. Hence, we can rewrite (\[eq:variationalformulation\]) with the 2D curl $$\begin{aligned}
\begin{aligned}
\nabla\times{{\boldsymbol{\mathcal{F}}}}:= \partial_x\mathcal{F}_y - \partial_y\mathcal{F}_x.
\end{aligned}
\end{aligned}$$
In the 2D version of Maxwell’s equations with a vertical electric dipole (see Section \[sec:analytical\]), the magnetic field, ${{\boldsymbol{B}}}$, given by $$\begin{aligned}
{{\boldsymbol{B}}}={\mu_{r}^{-1}\,}\big({{\boldsymbol{M}}_a}+\nabla\times{{\boldsymbol{E}}}\big),\qquad {{\boldsymbol{M}}_a}\equiv 0~,
\end{aligned}$$ only has a $z$-component when viewed as a vector field in $\mathbb{R}^3$. Thus, this field is parallel to the hypersurface $\Sigma$ and orthogonal to the computational domain, $\Omega$ (which is part of the $xy$-plane). Consequently, the SPP in this 2D setting, and in the corresponding numerical simulation, has the desired TM polarization [@bludov13].
Exact solution for 2D infinite conducting sheet {#sec:analytical}
===============================================
In this section, we derive an exact solution to the three-dimensional (3D) version of boundary value problem – in the case with a 2D vertical electric dipole radiating at frequency $\omega$ over an infinite conducting sheet embedded in an isotropic and homogeneous space of (scalar) magnetic permeability $\mu$; see Figure \[fig:geometry\]. For physical clarity, we invoke the vector-valued electromagnetic field without rescaling, unless we state otherwise.
The sheet separates the space into region 1 ($\{y>0\}$) with wave number $k_1$ and region 2 ($\{y<0\}$) with wave number $k_2$; $k_j^2=\omega^2
\tilde\epsilon_j\mu$ where $\tilde\epsilon_j$ is the complex permittivity ($j=1,\,2$). We assume that the surface conductivity, $\sigma^\Sigma$, of the sheet is a scalar constant. Note that we assign different complex permittivities to each half-space (regions 1, 2). In the end of this section, we set them equal.
The dipole has current density ${{\boldsymbol{J}}_{a}}= J_0\,\delta({{\boldsymbol{x}}}-{\boldsymbol{a}})\,e_y$ where ${{\boldsymbol{x}}}=x\,e_x+y\,e_y$ and ${\boldsymbol{a}}= a\,e_y$; $e_s$ is a unit Cartesian vector ($s= x, y$). Now define the Fourier transform, $\widehat{{{\boldsymbol{F}}}}_j(\xi, y)$, of the vector-valued field ${{\boldsymbol{F}}}_j(x, y)$ (${{\boldsymbol{F}}}={{\boldsymbol{E}}},\,{{\boldsymbol{B}}}$) in region $j$ through the integral formula $$\label{eq:EB-FT}
{{\boldsymbol{F}}}_j(x,y)=\frac{1}{2\pi}\int_{\mathbb{R}}d\xi\ \widehat{{{\boldsymbol{F}}}}_j(\xi, y)\,
e^{i \xi x}~.$$ The transformation of 3D Maxwell’s equations in region $j$ yields $$\begin{aligned}
& -i\xi \widehat{E}_{jy}+\frac{\partial \widehat{E}_{jx}}{\partial y}
=-i\omega \widehat{B}_{jz}~,\label{eq:maxwell_1}\\
& -\frac{\partial\widehat{B}_{jz}}{\partial y}
=\frac{ik_j^2}{\omega}\widehat{E}_{jx}~,\ -i\xi\widehat{B}_{jz}
=-\frac{ik_j^2}{\omega}\widehat{E}_{jy}+\mu\, \delta (y-a) ~,
\label{eq:maxwell_2}\end{aligned}$$ where we set $E_{jz}\equiv 0$, $B_{jx}\equiv 0,$ and $B_{jy}\equiv 0$ by symmetry; $F_{jz}$ ($F= E,\,B$) is the component perpendicular to the $xy$-plane.
![Schematic of a vertical electric dipole at distance $a$ from conducting sheet $\Sigma$ in 2D. The dipole has current density ${{\boldsymbol{J}}_{a}}=J_0 \delta({{\boldsymbol{x}}}-{\boldsymbol{a}})\,e_y$, where ${{\boldsymbol{x}}}= x e_x+ y e_y$ and ${\boldsymbol{a}}= a\,e_y$. The sheet lies in $y=0$, separates the space into region $1$ (half space $\{y>0\}$) with wave number $k_1$ and region $2$ ($\{y<0\}$) with wave number $k_2$, and has surface electric conductivity $\sigma^\Sigma$.[]{data-label="fig:geometry"}](maiermargetisluskin-figure1.pdf)
Equations and furnish the differential equation $$\frac{\partial^2\widehat{B}_{jz}}{\partial y^2}+\beta_j^2\,\widehat{B}_{jz}
=-i\xi \mu\,\delta(y-a)~,\quad \beta_j^2:=k_j^2-\xi^2~,$$ which has solution $$\label{eq:soln-Bz}
\widehat{B}_{jz}(\xi,y)=\left\{\begin{array}{lr}
\displaystyle{C_{>}\,e^{i\beta_1 y}-\frac{\xi\mu}{2\beta_1}\,
e^{i\beta_1|y-a|}}~,&\ \mbox{if}\ j=1\ (y>0)~;\\
C_{<}\, e^{-i\beta_2 y}~,&\ \mbox{if}\ j=2\ (y<0)~.\\
\end{array}\right.$$ This solution is consistent with radiation condition provided $$\label{eq:beta-cond}
\text{Im}\beta_j(\xi)>0\quad (j=1,\,2)~.$$
Furthermore, we apply conditions in order to determine integration constants $C_>$ and $C_<$. Specifically, we impose $$\big(\widehat{B}_{1z}-\widehat{B}_{2z}\big)\bigl|_{y=0}=\mu\,\sigma^\Sigma
\widehat{E}_{1x}\bigl|_{y=0}~,\quad
\big(\widehat{E}_{1x}-\widehat{E}_{2x}\big)\bigl|_{y=0}=0~.$$ Accordingly, by and we find $$C_>=-\frac{\xi\mu}{2\beta_1}\,\frac{k_2^2\beta_1-k_1^2\beta_2+
\omega\mu\sigma^\Sigma \beta_1\beta_2}{k_2^2\beta_1+k_1^2\beta_2+
\omega\mu\sigma^\Sigma \beta_1\beta_2 }e^{i\beta_1 a}~,\
C_<=-\frac{\mu k_2^2\xi\,e^{i\beta_1a}}{k_2^2\beta_1+k_1^2\beta_2+
\omega\mu\sigma^\Sigma\beta_1\beta_2}~.$$
We can now write down all field components in view of –. In particular: $$\begin{aligned}
E_{1x}(x,y)&=\frac{\omega\mu}{4\pi k_1^2}\int_{-\infty}^\infty d\xi\
\xi\Biggl[\frac{k_2^2\beta_1-k_1^2\beta_2+\omega\mu\sigma^\Sigma
\beta_1\beta_2}{k_2^2\beta_1+k_1^2\beta_2+\omega\mu\sigma^\Sigma
\beta_1\beta_2 } e^{i\beta_1(y+a)}
\label{eq:E1x-ex}\\\nonumber
&\qquad +\text{sgn}(y-a)\, e^{i\beta_1 |y-a|}\Biggr]e^{i\xi x}
\quad (y>0)~;
\\
E_{2x}(x,y)&=-\frac{\omega\mu}{2\pi}\int_{-\infty}^\infty d\xi\
\xi\beta_2\,\frac{e^{i\beta_1a-i\beta_2y}e^{i\xi x}}
{k_2^2\beta_1+k_1^2\beta_2+\omega\mu\sigma^\Sigma\beta_1\beta_2}\quad (y<0)~.
\label{eq:Ex-ex}\end{aligned}$$ In the above, $\text{sgn}(y)=1$ if $y>0$ and $\text{sgn}(y)=-1$ if $y<0$.
Note that all field components have Fourier transforms defined in the $\xi$-plane with: (i) branch points at $\xi=\pm k_j$ ($j=1, 2$); and (ii) simple poles at points where $$\label{eq:SPP-dispersion}
k_2^2\beta_1(\xi)+k_1^2\beta_2(\xi)+
\omega\mu\sigma^\Sigma\beta_1(\xi)\beta_2(\xi)=0~,$$ under condition which defines the appropriate branch of the multiple-valued $\beta_j(\xi)$. Equation is the dispersion relation for the SPP [@raether86; @margetis15]. In particular, if $k_1=k_2=k$ and $|\omega\mu\sigma^\Sigma|\ll |k|$, reduces to with $\xi=k_x$.
This observation motivates the following definition within the 2D model [@margetis15].
For an infinite conducting sheet, the SPP is identified with the part of the electromagnetic field equal to the contribution to the Fourier integrals of the simple pole $\xi=k_m$, $\text{Im} k_m>0$, that solves under .\[def:SPP\]
For the sake of completeness, we conclude this section by focusing on the case with $k_1=k_2=k$ (identical half-spaces). In particular, we provide explicit expressions for two distinct physical contributions to the $x$-component of the electric field on the sheet ($y=0$), which is continuous across the sheet. After suitable rescaling of the variables and parameters (Section \[sec:rescaling\]), by the pole contribution to $E_{x}:=E_{1x}(x,0)=E_{2x}(x,0)$ takes the form $$\begin{aligned}
\label{eq:polecontributionrescaled}
E_{x,\text{p}}
= -2i\frac{{\mu_{r}\,}{\tilde\varepsilon_{r}\,}}{({\sigma^\Sigma_r})^2}\exp\big[i{k_{m,r}}x - (2i/{\sigma^\Sigma_r})\,a_r]~,\end{aligned}$$ where ${k_{m,r}}=k_m/k_0$ and $a_r=k_0 a$; cf. . Equation expresses the SPP pertaining to the tangential electric field (see Definition \[def:SPP\]).
A separate physical contribution expresses the radiation part of the scattered field, ${{\boldsymbol{E}}}^{\rm sc}={{\boldsymbol{E}}}-{{\boldsymbol{E}}}^{\rm pr}$, where ${{\boldsymbol{E}}}^{\rm pr}$ is the (primary) dipole field in the absence of the sheet. To single out this contribution for the $x$-component, $E_x$, we choose to integrate in the $\xi$ (Fourier) variable around the branch cut emanating from the branch point $\xi=k$ after removal of the primary field component. The result reads $$\begin{gathered}
\label{eq:branchcutcontributionrescaled}
E_{x,\text{bc}}^{\text{sc}} =
\frac1{4\pi}\frac{1}{{\sigma^\Sigma_r}}
\Bigg\{
\int_0^1\text{d}\xi\ \xi\sqrt{1-\xi^2}\,e^{i\sqrt{{\mu_{r}\,}{\tilde\varepsilon_{r}\,}}x\,\xi}
\,\frac1{\xi^2+4\frac{{\mu_{r}\,}{\tilde\varepsilon_{r}\,}}{({\sigma^\Sigma_r})^2}-1}
\\
\cdot\Big(4\sqrt{{\mu_{r}\,}{\tilde\varepsilon_{r}\,}}\cos\big(\sqrt{{\mu_{r}\,}{\tilde\varepsilon_{r}\,}}a\sqrt{1-\xi^2}\big)-
2i{\sigma^\Sigma_r}\sqrt{1-\xi^2}\sin\big(\sqrt{{\mu_{r}\,}{\tilde\varepsilon_{r}\,}}a\sqrt{1-\xi^2}\big)\Big)
\\
-
\int_0^\infty\text{d}\varsigma\ \varsigma\sqrt{1+\varsigma^2}\,e^{i\sqrt{{\mu_{r}\,}{\tilde\varepsilon_{r}\,}}x\,\varsigma}
\,\frac1{\varsigma^2-4\frac{{\mu_{r}\,}{\tilde\varepsilon_{r}\,}}{({\sigma^\Sigma_r})^2}-1}
\\
\cdot\Big(4\sqrt{{\mu_{r}\,}{\tilde\varepsilon_{r}\,}}\cos\big(\sqrt{{\mu_{r}\,}{\tilde\varepsilon_{r}\,}}a\sqrt{1+\varsigma^2}\big)-
2i{\sigma^\Sigma_r}\sqrt{1+\varsigma^2}\sin\big(\sqrt{{\mu_{r}\,}{\tilde\varepsilon_{r}\,}}a\sqrt{1+\varsigma^2}\big)\Big)
\Bigg\}.\end{gathered}$$
Figure \[fig:pole-vs-branchcut\] shows a comparison of the two contributions, $ E_{x,\text{p}}$ and $E_{x,\text{bc}}^{\text{sc}}$, for a typical choice of parameters ($a=1.0$, $\sigma^\Sigma=2.00\times10^{-3}+0.200i$) by use of formulas and with ${\mu_{r}\,}=1$ and ${\tilde\varepsilon_{r}\,}=1$. We observe that the SSP (pole contribution) dominates in the (rescaled) range $10\le x\le 25$ but is eventually dominated by the branch-cut contribution which has the (slower) algebraic decay.
Numerics: Discretization scheme {#sec:numerics}
===============================
In this section, variational formulation (\[eq:variationalformulation\]) is discretized on a non-uniform quadrilateral mesh with higher-order, curl-conforming Nédélec elements [@nedelec86; @nedelec01; @bokil15]. Such a conforming discretization is an ideal choice for the problem at hand. The interface with the (weak) discontinuity can be aligned with the mesh and away from it; and the regularity of the solution leads to high convergence rates. Key ingredient of our treatment is the use of an appropriately defined PML (Section \[sec:pml\]), as well as local mesh refinement based on a posteriori error estimates (Section \[sec:aposteriori\]).
Let $X_h(\Omega)\subset X(\Omega)$ be a finite element subspace spanned by Nédélec elements. We will in particular use second-order Nédélec elements in the numerical computations. Then, under the assumption of a sufficiently refined initial mesh, the variational equation to find $E_h\in X_h(\Omega)$ such that $$\begin{aligned}
A({{\boldsymbol{E}}}_h,{{\boldsymbol{\varphi}}}) = F({{\boldsymbol{\varphi}}}),
\quad\forall{{\boldsymbol{\varphi}}}\in X_h(\Omega),\end{aligned}$$ is uniquely solvable [@monk03 Section 7.2].
From an approximation theory point of view, we expect the convergence $$\begin{aligned}
\big\|{{\boldsymbol{E}}}-{{\boldsymbol{E}}}_h\big\|_X\sim\mathcal{O}(h^2)=\mathcal{O}(\#\text{dofs})
\label{eq:convergenceorder}
\end{aligned}$$ for second-order Nédélec elements and under the conditions of Proposition \[prop:existence\], i.e., quadratic convergence in terms of a uniform refinement parameter $h$, or linear convergence in the number of degrees of freedom. This is evidenced by our numerical results presented in Section \[sec:computationalresults\]. We refrain from stating a rigorous convergence result at this point because the use of non-uniform, locally refined meshes with an additional approximation of a curved boundary significantly complicates the classical approximation theory (see [@monk03 Section 7.3] and references therein).
The interface jump condition on the sheet introduces a pronounced two-scale character to the problem that needs special numerical treatment. Depending on the surface electric conductivity, $\sigma^\Sigma$, of the sheet, the observed SPP may have a rescaled wave number, ${k_{m,r}}$, that is one to two orders of magnitude higher than that produced by the dipole radiation in free space (which has rescaled wave number $k_r=1$). This fact has important consequences with respect to the refinement strategy and boundary conditions. In particular:
- First-order absorbing boundary conditions alone are in principle not well suited for conducting sheets sustaining SPPs. These boundary conditions lead to a significant suppression of the SPP amplitude. This is especially an issue for configurations where the SPP given by has a significantly smaller amplitude than the branch-cut contribution .
- A much finer minimal mesh refinement is necessary near the interface $\Sigma$ in order to resolve the highly oscillatory SPP. In addition, failure to sufficiently resolve the interface in the *whole* computational domain results in a suppression of the SPP due to non-local cancellation effects (*pollution effect*); see Section \[sub:pollutionandpml\] for computational examples.
One of the major advantages of a finite element approach for discretizing variational formulation (\[eq:variationalformulation\]) is the fact that no regularization of the interface by a layer with artificial thickness is necessary. Instead, the sheet can be *directly approximated* as a lower-dimensional interface.
In the remainder of this section, we discuss our choice of a PML to remedy the negative effect of the absorbing boundary condition on the SPP amplitude. Further, a strategy for adaptivity and *local mesh refinement* is introduced, which is based on *a posteriori* error control combined with a fixed (*a priori*) local refinement of the interface.
Perfectly matched layer {#sec:pml}
-----------------------
In this subsection, we carry out a construction of a PML [@berenger94; @chew94; @zhao96] for the rescaled Maxwell equations with a jump condition. The concept of a PML was pioneered by Bérenger [@berenger94]. It is essentially a layer with modified material parameters (${\tilde\varepsilon_{r}\,}$, ${\mu_{r}\,}$) placed near the boundary such that all outgoing electromagnetic waves decay exponentially with no “artificial” reflection due to truncation of the domain. The PML is an indispensable tool for truncating unbounded domains for wave equations and often used in the numerical approximation of scattering problems [@monk03; @bao10; @chew94; @berenger94; @zhao96].
We use an approach for a PML for time-harmonic Maxwell’s equations outlined by Chew and Weedon [@chew94], as well as Monk [@monk03]. The idea is to use a formal change of coordinates from the computational domain $\Omega\subset{\mathbb{R}}^3$ with real-valued coordinates to a domain $\acute\Omega\subset\{z\in\mathbb{C}\::\:\text{Im\,}z\ge0\}^3$ with complex-valued coordinates and non-negative imaginary part [@monk03]; and transform back to the real-valued domain. This results in modified material parameters ${\tilde\varepsilon_{r}\,}$, ${\mu_{r}\,}$ and ${\sigma^\Sigma_r}$ within the PML.
![The computational domain $\Omega$ with a vertical dipole ${{\boldsymbol{J}}_{a}}$ situated above a planar conducting sheet $\Sigma$ located on the $x$-axis. []{data-label="fig:graphenesheet"}](maiermargetisluskin-figure4.pdf)
We assume that a PML is imposed as a concentric spherical shell in a small outer region near the boundary $\partial\Omega$; see Figure \[fig:graphenesheet\]. The transformation is chosen to be a function of distance $\rho$ in radial (${\boldsymbol{e}}_r$-) direction from the origin. Furthermore, assume that the normal field ${{\boldsymbol{\nu}}}$ of $\Sigma$ is orthogonal to ${\boldsymbol{e}}_r$ (for ${{\boldsymbol{x}}}\in \Sigma \cap {\rm (PML)}$) and that ${{\boldsymbol{J}}_{a}}\equiv0$ and ${{\boldsymbol{M}}_a}\equiv0$ within the PML. Introduce a change of coordinates $\Omega\to \acute\Omega$ with $$\begin{aligned}
{{\boldsymbol{x}}}\quad\mapsto\quad \acute {{\boldsymbol{x}}}=
\begin{cases}
\begin{aligned}
&{{\boldsymbol{x}}}+ i {\boldsymbol{e}}_r \int_\rho^r s(\tau)\,\text{d}\tau
&\quad &\text{if }r\ge \rho,
\\[0.1em]
&{{\boldsymbol{x}}}& &\text{otherwise},
\end{aligned}
\end{cases}
\label{eq:pmltransformation}\end{aligned}$$ with $r={\boldsymbol{e}}_r\cdot{{\boldsymbol{x}}}$ and an appropriately chosen, nonnegative scaling function $s(\tau)$ [@monk03]. This prescription leads to a modified system of Maxwell’s equations defined on $\acute\Omega$, which takes the following (rescaled) form within the PML: $$\begin{aligned}
\begin{cases}
\begin{aligned}
\acute\nabla\times\big({\mu_{r}^{-1}\,}\acute\nabla\times{{\boldsymbol{\acute E}}}\big) -{\tilde\varepsilon_{r}\,}{{\boldsymbol{\acute E}}}\;&=\;
0,
\\[0.1em]
\big[{{\boldsymbol{\nu}}}\times({\mu_{r}^{-1}\,}\acute\nabla\times{{\boldsymbol{\acute E}}})\big]_\Sigma
\;&=\;
i\,{\sigma^\Sigma_r}{{\boldsymbol{\acute E}}}_T\quad\text{on }\Sigma,
\end{aligned}
\end{cases}
\label{eq:pmlsystemcomplex}\end{aligned}$$ where the accent on top of ${{\boldsymbol{E}}}$ and $\nabla$ indicates the dependence on as well as the differentiation with respect to $\acute{{\boldsymbol{x}}}$. Next, we transform (\[eq:pmlsystemcomplex\]) back from $\acute\Omega$ to $\Omega$ with the help of diffeomorphism (\[eq:pmltransformation\]). It follows that [@monk03] $$\begin{aligned}
\acute\nabla\times\acute{{\boldsymbol{\mathcal{F}}}}= A(\nabla\times B{{\boldsymbol{\mathcal{F}}}}),
\qquad
{{\boldsymbol{\nu}}}\times\acute{{\boldsymbol{\mathcal{F}}}}= C({{\boldsymbol{\nu}}}\times B{{\boldsymbol{\mathcal{F}}}}),\end{aligned}$$ for ${{\boldsymbol{\nu}}}$ orthogonal to ${\boldsymbol{e}}_r$. In the above, we introduce the $3\times3$ matrices $$\begin{gathered}
A=T^{-1}_{{\boldsymbol{e}}_x{\boldsymbol{e}}_r}\text{diag}\,
\Big(\frac{1}{\bar d^2},\frac{1}{d\bar d},\frac{1}{d\bar d}\Big)
T_{{\boldsymbol{e}}_x{\boldsymbol{e}}_r},
\quad
B=T^{-1}_{{\boldsymbol{e}}_x{\boldsymbol{e}}_r}\text{diag}\,\big(d,\bar d,\bar d\big)
T_{{\boldsymbol{e}}_x{\boldsymbol{e}}_r},
\\\notag
C=T^{-1}_{{\boldsymbol{e}}_x{\boldsymbol{e}}_r}\text{diag}\,\Big(\frac{1}{\bar d},\frac{1}{\bar d},
\frac{1}{d}\Big)
T_{{\boldsymbol{e}}_x{\boldsymbol{e}}_r},\end{gathered}$$ where $$\begin{aligned}
\label{eq:d}
d=1+i\,s(r), \quad
\bar d=1+i/r\int_\rho^r s(\tau)\,\text{d}\tau,\end{aligned}$$ and $T_{{\boldsymbol{e}}_x{\boldsymbol{e}}_r}$ is the rotation matrix that rotates ${\boldsymbol{e}}_r$ onto ${\boldsymbol{e}}_x$. Thus, applying the rescaling $$\begin{aligned}
\begin{cases}
\begin{aligned}
{\mu_{r}^{-1}\,}&\quad\longrightarrow\quad
\breve\mu_r^{-1} = B{\mu_{r}^{-1}\,}A,
\\[0.1em]
{\tilde\varepsilon_{r}\,}&\quad\longrightarrow\quad
\breve{\varepsilon}_r =\; A^{-1}{\tilde\varepsilon_{r}\,}B^{-1},
\\[0.1em]
{\sigma^\Sigma_r}&\quad\longrightarrow\quad
{{\breve\sigma}^\Sigma_r}= C^{-1}{\sigma^\Sigma_r}B^{-1},
\end{aligned}
\end{cases}
\label{eq:pmlcoefficients}\end{aligned}$$ to (\[eq:pmlsystemcomplex\]) leads to the system (with $\breve{{\boldsymbol{E}}}:= B\,\acute{{\boldsymbol{E}}}$) $$\begin{aligned}
\begin{cases}
\begin{aligned}
\nabla\times\big(\breve\mu_r^{-1}\nabla\times\breve{{\boldsymbol{E}}}\big)
-\breve{\varepsilon}_r\breve{{\boldsymbol{E}}}\;&=\;
0,
\\[0.2em]
\big[{{\boldsymbol{\nu}}}\times(\breve\mu_r^{-1}\nabla\times\breve{{\boldsymbol{E}}})\big]_\Sigma
\;&=\;
i\,{{\breve\sigma}^\Sigma_r}\breve{{\boldsymbol{E}}}_T\quad\text{on }\Sigma,
\end{aligned}
\end{cases}
\label{eq:maxwellpml}\end{aligned}$$ within the *real-valued* domain $\Omega$. Note that outside of the PML matrices $A$, $B$, and $C$ simply reduce to the unit matrix. Thus, ${{\boldsymbol{E}}}=\breve{{\boldsymbol{E}}}$ and is identical to with jump condition . The modified equations for the PML can be implemented by suitably *replacing* ${\tilde\varepsilon_{r}\,}$, ${\mu_{r}^{-1}\,}$, and ${\sigma^\Sigma_r}$ by their counterparts according to (\[eq:pmlcoefficients\]) within the PML. For the 2D model discussed in Remark \[remark:2dmodel\], the scalar ${\mu_{r}^{-1}\,}$ is transformed via $$\begin{aligned}
{\mu_{r}^{-1}\,}\quad\longrightarrow\quad
\breve\mu_r^{-1} = \frac{{\mu_{r}^{-1}\,}}d,\end{aligned}$$ where $d$ is given by .
A posteriori error estimation and local refinement {#sec:aposteriori}
--------------------------------------------------
One of the computationally challenging aspects of the numerical simulation is the *two-scale* behavior of problem . A much finer minimal mesh refinement is necessary near the interface $\Sigma$ in order to resolve the highly oscillatory SPP. In this subsection we give a brief overview of a local mesh adaptation strategy based on *goal-oriented* a posteriori error estimation. This leads to a substantial saving in computational costs (see Section \[sec:computationalresults\]), due to local refinement, while maintaining an optimal convergence order in a quantity of interest[@becker2001]. We focus primarily on aspects of implementation—a full, detailed discussion is beyond the scope of this paper and will be the subject of future work.
An efficient method for *a posteriori* error control is the *dual weighted residual* (DWR) method developed by Becker and Rannacher [@becker1996a; @becker1996b; @becker2001]. This method constructs estimates of local error contributions in terms of a target functional ${\mathcal{J}}$ with the help of a “dual problem”. More precisely, let ${\mathcal{J}}({{\boldsymbol{E}}})$ be a quantity of interest given by a possibly non-linear Gâteaux-differentiable function, viz., $$\begin{aligned}
{\mathcal{J}}\,:\,{\boldsymbol{H}}(\text{curl};\Omega) \to{\mathbb{C}}.\end{aligned}$$ The corresponding dual problem to (\[eq:variationalformulation\]) is to find a solution ${{\boldsymbol{Z}}}\in{\boldsymbol{H}}(\text{curl};\Omega)$ such that $$\begin{aligned}
A({{\boldsymbol{\varphi}}},{{\boldsymbol{Z}}}) =\;\; \text{D}_{{{\boldsymbol{E}}}}\mathcal{J}({{\boldsymbol{E}}})[{{\boldsymbol{\varphi}}}]
\label{eq:dualproblem}\end{aligned}$$ for all ${{\boldsymbol{\varphi}}}\in{\boldsymbol{H}}(\text{curl};\Omega)$. Here, $\text{D}_{{{\boldsymbol{E}}}}.[{{\boldsymbol{\varphi}}}]$ denotes the Gâteaux derivative in direction ${{\boldsymbol{\varphi}}}$ with respect to ${{\boldsymbol{E}}}$. The following result can be directly applied to variational problem .
Let ${{\boldsymbol{E}}}$ and ${{\boldsymbol{Z}}}$ be the solution of (\[eq:variationalformulation\]) and (\[eq:dualproblem\]), respectively. Let ${{\boldsymbol{E}}}_H$ and ${{\boldsymbol{Z}}}_H$ be finite element approximations of the primal and dual solution associated with a discretization ${\mathbb{T}_H}(\Omega)$ of $\Omega$. Then, up to a term $R$ of higher order: $$\begin{gathered}
\big|\mathcal{J}({{\boldsymbol{E}}})-\mathcal{J}({{\boldsymbol{U}}})\big|
\le \sum_{Q\in\,{\mathbb{T}_H}}\eta_Q
+R,\quad\text{with}
\\
\eta_Q:=
\frac 12\,\Big|
\rho_Q({{\boldsymbol{E}}}_H,{{\boldsymbol{Z}}}-{{\boldsymbol{Z}}}_H) +
\rho^\ast_Q({{\boldsymbol{Z}}}_H,{{\boldsymbol{E}}}-{{\boldsymbol{E}}}_H)
\Big|.
\label{eq:localindicator}
\end{gathered}$$
Here, $\rho_Q$ and $\rho^\ast_Q$ denote the primal and dual cell-wise residual, respectively, associated with variational equations (\[eq:variationalformulation\]) and (\[eq:dualproblem\]): $$\begin{aligned}
\label{eq:rhoQ}
\rho_Q({{\boldsymbol{E}}}_H,{{\boldsymbol{Z}}}-{{\boldsymbol{Z}}}_H) &=
F\big(({{\boldsymbol{Z}}}-{{\boldsymbol{Z}}}_H)\chi_Q\big) - A\big({{\boldsymbol{E}}}_H,({{\boldsymbol{Z}}}-{{\boldsymbol{Z}}}_H)\chi_Q\big),
\\[0.1em]
\label{eq:rhoastQ}
\rho^\ast_Q({{\boldsymbol{Z}}}_H,{{\boldsymbol{E}}}-{{\boldsymbol{E}}}_H) &=
\text{D}_{{{\boldsymbol{E}}}}\mathcal{J}({{\boldsymbol{E}}})[({{\boldsymbol{E}}}-{{\boldsymbol{E}}}_H)\chi_Q] -
A\big(({{\boldsymbol{E}}}-{{\boldsymbol{E}}}_H)\chi_Q,{{\boldsymbol{E}}}_H\big),\end{aligned}$$ with the indicator function $\chi_Q$ that is 1 on the cell $Q$, and 0 otherwise. The *local error indicator* $\eta_Q$ given by can now be approximated and used in a local refinement strategy [@becker2001].
Our goal is an optimal local refinement for the numerical observation of SPPs on the conducting sheet $\Sigma$. In principle, a number of choices for the quantity of interest, $\mathcal{J}({{\boldsymbol{E}}})$, are possible in order to achieve this goal. Here, we choose the quantity $$\begin{aligned}
\mathcal{J}({{\boldsymbol{E}}}):=\int_\Omega\varpi({{\boldsymbol{x}}})\,\big\|\nabla\times{{\boldsymbol{E}}}\,\big\|^2
\rm{d}{{\boldsymbol{x}}},
\label{eq:quantityofinterest}\end{aligned}$$ with an appropriate (essentially bounded), non-negative weighting function $\varpi({{\boldsymbol{x}}})$ that localizes the integral around the interface $\Sigma$; see Section \[sec:computationalresults\] for the concrete choice of $\varpi$ for our simulations. Choice for the quantity of interest leads to a localized right-hand side $\mathcal{J}$ of the dual problem that is sensitive to the highly oscillatory SPP associated with the electric field, $E$. Consequently, the *weight* ${{\boldsymbol{Z}}}-{{\boldsymbol{Z}}}_H$ in residual is generally large near the interface and at points where the influence of the solution on quantity is high.
A purely residual-based error estimator on the other hand corresponds to a uniform weight distribution. The DWR method (with our choice of right hand side) will lead to a more localized refinement.
In order to guarantee a uniform refinement over the interface $\Sigma$, the local refinement strategy is augmented by *additionally* selecting all cells $Q$ for refinement that fulfill $$\begin{aligned}
1-(1/2)^{\text{\#cycle}-1}
\;\le\;
\varpi(x_Q)/\max(\varpi),\end{aligned}$$ where $x_Q$ denotes the center of $Q$.
A classical approach to evaluate (\[eq:rhoQ\]) and (\[eq:rhoastQ\]) is to use a higher-order approximation for ${{\boldsymbol{Z}}}$ and ${{\boldsymbol{E}}}$ and transform $\rho_Q$ and $\rho^\ast_Q$ into a strong residual form [@becker2001]. We follow a different strategy that does not involve solving higher-order solutions. Instead of a higher-order approximation of ${{\boldsymbol{Z}}},$ we use a patch-wise projection $\pi_{2H}^{(2)}{{\boldsymbol{Z}}}_H$ to a higher-order space on a coarser mesh level [@richter2006], viz., $$\begin{aligned}
\label{eq:projection-est}
{{\boldsymbol{Z}}}-{{\boldsymbol{Z}}}_H \approx \pi_{2H}^{(2)}{{\boldsymbol{Z}}}_H - {{\boldsymbol{Z}}}_H,
\quad
{{\boldsymbol{E}}}-{{\boldsymbol{E}}}_H \approx \pi_{2H}^{(2)}{{\boldsymbol{E}}}_H - {{\boldsymbol{E}}}_H,
\end{aligned}$$ in combination with the (variational) residuals and .
Error estimator also has a form with only the primal residual $\rho(\,.\,)(\,.\,)$ appearing. However, the mixed form of the above error estimator should be used here to ensure adequate, simultaneous refinement not just only with respect to a localized SPP (on the interface), but also with respect to a singular right-hand side ${{\boldsymbol{J}}_{a}}$ (modeling of a Hertzian dipole).
Numerical results {#sec:computationalresults}
=================
In this section, we present a number of computational results pertaining to the excitation by a vertical electric dipole of an SPP on a 2D conducting sheet. First, we construct a PML which involves use of a certain parameter, $s_0$. Second, we demonstrate the necessity for a PML and carry out a study for the suitable choice of $s_0$. Third, we compare the analytical results of Section \[sec:analytical\] to (direct) numerical simulations based on our finite element formulation of Section \[sec:numerics\]. All numerical computations are carried out with the `C++` finite element toolkit `deal.II` [@dealii82; @dealii83]. We use the direct solver `UMFPACK` [@davis11; @suitesparse4] for the resulting linear system of equations.
Setup
-----
In order to carry out the numerical simulations, we consider a vertical electric dipole positioned at ${\boldsymbol{a}} = (0,0.75)$ and ${\boldsymbol{a}} = (0,1)$ above the conducting sheet. Recall that the corresponding current densities are $$\begin{aligned}
{{\boldsymbol{J}}_{a}}=\begin{pmatrix}0\\J_0\end{pmatrix}\,\delta({{\boldsymbol{x}}}- {\boldsymbol{a}}),
\qquad
{{\boldsymbol{M}}_a}=0.\end{aligned}$$ The conducting sheet has surface conductivity ${\sigma^\Sigma_r}=(\sigma_1+i\sigma_2)I$ in tensor form ($I$: unit second-rank tensor). Below, we carry out a parameter study with different values of constant scalar ${\sigma^\Sigma_r}$.
The computational domain $\Omega$ is chosen to be the ball with radius $R=8\pi$, which is 8 times the free-space wavelength of the dipole radiation. A PML is used in the outer region at distance $\rho>0.8\,R$ from the origin; see Section \[sec:pml\]. Following Monk [@monk03], the (nonnegative) scaling function $s(\rho)$ is chosen to be $$\begin{aligned}
s(\rho) = s_0 \frac{(\rho - 0.8 R)^2}{(R-0.8 R)^2},\end{aligned}$$ where $s_0$ is a free parameter; the suitable choice of $s_0$ is discussed in Section \[sub:pollutionandpml\].
We use the adaptive refinement strategy that was outlined in Section \[sec:aposteriori\] with the quantity of interest (\[eq:quantityofinterest\]) and the choice $$\begin{aligned}
\varpi({{\boldsymbol{x}}}) =
\begin{cases}
\begin{aligned}
&\cos^2\Big(\frac\pi2\,\frac{y}{d_\varpi}\Big),
&\quad&\text{if }|y|\le d_\varpi,
\\[0.2em]
&0 &\quad&\text{otherwise}.
\quad
\end{aligned}
\end{cases}
\label{eq:interfacereg}\end{aligned}$$ The parameter $d_\varpi$ controls the width of the region around the interface that is uniformly refined and is chosen to be $d_\varpi=1.5625$, a few multiples of the SPP wavelength $2\pi/(\text{Re}\,{k_{m,r}})$. The current density of the dipole source is regularized according to $$\begin{aligned}
\delta({{\boldsymbol{x}}}-{\boldsymbol{a}})\approx
\begin{cases}
\begin{aligned}
&\Big(\frac{\pi}{2}-\frac{2}{\pi}\Big)^{-1} d^{-2}
\cos^2\Big(\frac\pi2\,\frac{\|{{\boldsymbol{x}}}- {\boldsymbol{a}}\|}{d}\Big)
& &\text{for }\|{{\boldsymbol{x}}}-{\boldsymbol{a}}\|<d,
\\[0.2em]
&0 & &\text{otherwise,}
\end{aligned}
\end{cases}
\label{eq:dipolereg}\end{aligned}$$ with $d=0.15625$. The regularization parameter $d$ should be chosen as small as possible and has to be scaled with the initial mesh size such that ${{\boldsymbol{J}}_{a}}$ is always well integrated numerically. Figure \[fig:graphenesheet\] summarizes the aforementioned setup.
On perfectly matched layer {#sub:pollutionandpml}
--------------------------
In this subsection, we demonstrate the necessity for a PML. In our numerical setup, the challenging part of a direct finite element simulation is the two-scale character that the electromagnetic wave exhibits in the spatial resolution. Recall that the desired SPP has a wavelength much smaller than the one manifested by the dipole free-space radiation field.
In particular, we are interested in observing SPPs with an associated wave number $\text{Re}\,{k_{m,r}}\approx 10-100$, compared to the (rescaled) free-space wave number $k_r=1$. Moreover, the amplitude of the SPP scales exponentially with the distance, $a$, of the dipole from the interface; hence, certain configurations of physical appeal exhibit a ratio in amplitude of about 1:10 to 1:1000 between the SPPs and the dipole free-space radiation field. It turns out that absorbing boundary condition that we use–although it is a first-order absorbing boundary condition–is *not* well suited for the numerical study of the SPP. A significant suppression of the SPP amplitude can be evident.
In order to study the influence of the absorbing boundary condition on the SPP, we perform a series of numerical simulations of the geometry given in Figure \[fig:graphenesheet\], for different values of $s_0$ which controls the absorption strength of the PML; specifically, we use $s_0=0$, $0.25$, $0.5$, $1.0$, $2.0$, $4.0$, and $8.0$, respectively. The material parameters are set to ${\mu_{r}\,}={\tilde\varepsilon_{r}\,}=I$. We choose ${\boldsymbol{a}}=(0,1)$ for the position vector of the dipole, with sheet of relatively small, purely imaginary surface conductivity, i.e., $$\begin{aligned}
{\sigma^\Sigma_r}= i\sigma_2 = 0.15\,i.\end{aligned}$$ This value corresponds to a relatively weak SPP, i.e., with a relatively small amplitude, that has a purely real-valued wave number, ${k_{m,r}}\approx
13.33$. This can be understood as follows: Because $\text{Re}\sigma^\Sigma_r=0$, the sheet does not cause any dissipation and, thus, the corresponding SPP does not exhibit any decay.
In order to examine the influence of different boundary conditions and choices of parameters in some detail, we extract the $x$-component of the scattered electric field, $E_x^{\rm sc}=E_x-E_x^{\rm pr}$, on the interface $\Sigma$, and plot this component as a function of $x$, where ${{\boldsymbol{E}}}^{\rm
pr}$ is the (primary) dipole field in the absence of the sheet; see Figure \[fig:pmltest\]. Boundary condition without PML ($s_0=0$) has a strong influence on the observed scattered field. While branch-cut contribution (\[eq:branchcutcontributionrescaled\]) of the scattered field can be observed in the numerical simulation, pole contribution (\[eq:polecontributionrescaled\]), which is responsible for the fast oscillation with wave number ${k_{m,r}}$, is suppressed (Figure \[fig:pmltest\]a). This property can be explained by the fact that boundary condition (\[eq:absorbingbc\]), viz., $$\begin{aligned}
{\boldsymbol{n}}\times{{\boldsymbol{B}}}+\sqrt{{\mu_{r}^{-1}\,}{\tilde\varepsilon_{r}\,}}{{\boldsymbol{E}}}_T = 0,\end{aligned}$$ is only applicable to dipole radiation with a wave number $k_r=1$. In the case where the pole contribution is characterized, e.g., by ${k_{m,r}}\approx13.3$, this boundary condition causes a reflection that results in a suppression of the fast spatial oscillation of the SPP. In light of the parameter study (Fig. \[fig:pmltest\]), we choose a PML with strength $s_0=2$ for all subsequent computations. This is a balanced choice between a PML that is strong enough to minimize the influence of the boundary ($s_0\ge2$) and the unwanted influence of the PML ($s_0\le2$); cf. Figure \[fig:pmltest\]c.
Comparison of numerical results to analytical solution {#sub:comparison}
------------------------------------------------------
In this subsection, we compare our numerical results to the analytical solution of Section \[sec:analytical\]. In particular, we expect to observe the SPP described by (\[eq:polecontributionrescaled\]). The (complex-valued) wave number ${k_{m,r}}$ associated with this SPP scales with the surface conductivity ${\sigma^\Sigma_r}$ as follows (given in rescaled quantities, as explained in Section \[sec:rescaling\]): $$\begin{aligned}
{k_{m,r}}=
\sqrt{{\mu_{r}\,}{\tilde\varepsilon_{r}\,}-\frac{4{\mu_{r}\,}^2{\tilde\varepsilon_{r}\,}^2}{({\sigma^\Sigma_r})^2}};
\qquad
{k_{m,r}}\approx
\frac{2i\,{\mu_{r}\,}{\tilde\varepsilon_{r}\,}}{\sigma_r^\Sigma}
\quad
\text{if}\quad |2\sqrt{{\mu_{r}\,}{\tilde\varepsilon_{r}\,}}|\gg|\sigma_r^\Sigma|.
\label{eq:polaritonscalingrescaled}\end{aligned}$$ In order to test our numerical method (Section \[sec:numerics\]) against the analytical results (Section \[sec:analytical\]), we carry out a parameter study for a variety of different values of ${\sigma^\Sigma_r}$; see Table \[tab:sigma\].
[c c]{} Surface conductivity ${\sigma^\Sigma_r}$ & Predicted ${k_{m,r}}$\
$2.56\times10^{-4}+0.160i$ & $12.5+0.02i$\
$1.78\times10^{-4}+0.133i$ & $15.0+0.02i$\
$1.28\times10^{-3}+0.160i$ & $12.5+0.1i$\
$8.89\times10^{-4}+0.133i$ & $15.0+0.1i$\
Our computations are performed for dipole elevation distances $a=0.75,
1.00$. For each choice of parameters, we start with a relatively coarse mesh (with 10k degrees of freedom) for the numerical simulation and run 6 mesh-adaptation cycles (with approximately 1.6M degrees of freedom on the finest mesh.) Figure \[fig:scatteredfield\] shows the $x$-directed scattered field for ${\sigma^\Sigma_r}=4.0\times10^{-4}+0.2i$ (${k_{m}}\approx10.0+0.02i$), and ${\sigma^\Sigma_r}=2.0\times10^{-3}+0.2i$ (${k_{m}}\approx10.0+0.1i$). For this choice of parameters, a relatively strong pole contribution can be observed as shown in Figure \[fig:scatteredfield\](b). Figure \[fig:localrefinement\] shows the locally refined mesh after the final refinement cycle.
For a comparison of our numerical method to the analytical results of Section \[sec:analytical\], the pole contribution (\[eq:polecontributionrescaled\]) and branch-cut contribution (\[eq:branchcutcontributionrescaled\]) are computed numerically. For this purpose, we use a summed trapezoidal rule to evaluate the two integrals involved in the branch-cut contribution. For the improper integral over $(0, \infty)$, we further exploit an exponential decay of the integrand (as a function of the integration variable, $\varsigma$) and introduce a cutoff at $\varsigma\approx 1/\sqrt{hx}$, where $h$ is the interval size of the summed trapezoidal rule. This $h$ is chosen adaptively in an iterative cycle such that the relative change in the value of the integral between $2h$ and $h$ is less than $0.5\,\%$.
In Figures \[fig:parameterstudyinterface\] and \[fig:parameterstudyinterface2\], the analytical and numerical results are compared graphically for dipole elevation distances $a=0.75, 1.00$ and ${k_{m,r}}=12.5+0.02 i, 15.0+0.1 i$. The real and imaginary parts of the scattered electric field in the $x$-direction, $E_x^{\rm sc}$, are plotted as a function of position $x$ on the interface, $\Sigma$. It is evident that the numerical and analytical results are in excellent agreement outside the PML ($0\le x\le 20$).
Table \[tab:comparison\] summarizes the parameter study quantitatively. The $L^2$-error between numerical and analytical result is computed outside of the PML on the interface $\Sigma$. The SPP is generally well approximated after the 4th cycle with a resolution of around 100 thousand degrees of freedom. According to , we expect a convergence order of $\|{{\boldsymbol{E}}}_{h,T}-{{\boldsymbol{E}}}_T\|_{L^2(\Omega)}\sim\mathcal{O}(\#\text{dofs})$. Indeed, we observe a linear convergence of the error with respect to the number of refinement steps, and thus, a linear convergence in number of degrees of freedom.
Conclusion and outlook {#sec:conclusion}
======================
In this paper, we developed a variational framework for the numerical simulation of electromagnetic SPPs excited by current-carrying sources along an infinitely thin conducting sheet, e.g., single-layer graphene. The sheet is modeled by an idealized, oriented hypersurface. The effect of the induced surface current of the sheet was taken into account in the corresponding boundary value problem of Maxwell’s equations via jump condition for the tangential component of the magnetic field; this jump is proportional to the surface current density. The conductivity of the sheet is a parameter that controls the strength of this discontinuity.
One of the main advantages offered by our approach is the natural incorporation of the jump condition in a variational formulation as a weak discontinuity, *without* regularization of the interface by a layer of finite thickness. We tested our numerical treatment against analytical predictions in the case with a vertical dipole radiating over an infinite conducting sheet in 2D, and observed excellent agreement. In our numerical treatment, a linear asymptotic reduction rate could be observed for all testcases. Notably, the use of an adaptive local refinement procedure within our approach achieved significant economy in the total number of degrees of freedom in comparison to uniform mesh refinement.
Our numerics admit several generalizations and extensions. For instance, the treatment of the jump condition as a weak discontinuity over an interface is *not* limited to a simple (lower-dimensional) hyperplane; it can be generalized to reasonably arbitrary hypersurfaces. In fact, technically speaking, our variational framework can be readily used without modifications to model any geometry that is meshable by quadrilaterals. This flexibility should enable efficient numerical simulations of complex geometries, e.g., waveguides that contain a few graphene layers [@yeh-book]. Although our numerical results focused on 2D thus far, our underlying choice of local adaptivity can lead to a significant reduction of computational cost in higher spatial dimension.
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---
abstract: 'We describe an experimental and computational investigation of the ordered and disordered phases of a vibrating thin, dense granular layer composed of identical metal spheres. We compare the results from spheres with different amounts of inelasticity and show that inelasticity has a strong effect on the phase diagram. We also report the melting of an ordered phase to a homogeneous disordered liquid phase at high vibration amplitude or at large inelasticities. Our results show that dissipation has a strong effect on ordering and that in this system ordered phases are absent entirely in highly inelastic materials.'
author:
- Francisco Vega Reyes
- 'Jeffrey S. Urbach'
date: 'November 4, 2008'
title: The effect of inelasticity on the phase transitions of a thin vibrated granular layer
---
Ordered phases observed in non-cohesive granular media demonstrate both profound similarities to and differences from those observed in elastic systems, where the results of equilibrium statistical mechanics apply [@reis06; @SprBook; @JCP; @clerc08; @watanabe08]. Controlled experiments on simple model systems are necessary to develop and test extensions of statistical mechanics to non-equilibrium systems [@aranson06]. In previous research [@PRLord; @PREcub; @JCP; @PRLmelt], we have reported the existence of ordered phases in vibrated dense thin layers of identical metal spheres. For monolayers at high densities, hexagonal ordering is observed [@PRLord], and recent results from our group [@PRLmelt] and, in a somewhat different setup, Reis et al. [@reis06], show that the crystallization of the ordered phase can be directly mapped onto the analogous 2D equilibrium system. In the presence of a confining lid, we have reported a complex phase diagram that is closely related to that observed in similarly confined equilibrium colloidal systems, including two-layer crystals with square or hexagonal symmetry [@JCP; @PREcub]. In a recent study Clerc et al. [@clerc08] extended this work to quasi-one-dimensional systems, and showed that the transition was mediated by traveling waves and was triggered by negative compressibility.
In this paper we present the experimental phase diagram for the confined granular layer in more detail than reported previously, for both stainless steel and brass spheres. We report the existence of a melting transition of the ordered phase as the vibration amplitude is increased, and show that similar behavior is observed in computer simulations of inelastic spheres. The melting of the crystalline phase occurs significantly earlier in the system of brass spheres, and computer simulations show that the ordered phase disappears entirely in the presence of high inelasticity. This result demonstrates that dissipation can dominate over geometric packing effects for sufficiently inelastic spheres.
![Schematic side view of the experimental setup, consisting of two horizontal planes, separated by a gap of $h=1.75~\sigma$. Both walls vibrate together sinusoidally in the vertical direction.[]{data-label="setup"}](setup2.pdf){height="1.5cm"}
{width="15cm"}
The experimental system consists of nearly identical metallic spheres confined in a gap which lies horizontally between a circular plate and a lid (Fig. \[setup\]). In this paper we report results with stainless steel and brass spheres with diameter $\sigma=1.5875\pm0.0032$ mm ($1/16''$). The plate diameter is $168$ mm, or equivalently, $d_p=112~\sigma$, and the gap spacing is $h=2.78$ mm $=1.75~\sigma$ for all experiments. We characterize the system’s density with the 2D density $\rho$, defined as $\rho=N/N_{max}$, where $N_{max}=11377$ is the maximum number of balls that can fit in a hexagonally packed monolayer of balls on the plate at rest. In the experiments reported here the frequency of the sinusoidal plate vibration, provided by an electromagnetic shaker, was fixed at $\nu=60$ Hz. In what follows we report the amplitude of the oscillation, $A$, as the reduced acceleration $\Gamma=A\omega^2/g$, where $\omega=2\pi \nu$ and $g$ is the acceleration due to gravity. The acceleration is measured by a fast response accelerometer mounted on the bottom plate and maintained at a constant value with a computer-based feedback loop.
Under experimental conditions similar to our setup, it has been shown that the particle-particle collisions can be accurately described by a model characterized by three coefficients: $e$, which characterizes the incomplete restitution in the normal component of the relative particle velocities in the collision; $\beta_0$, which is the tangential coefficient of restitution for non-sliding collisions; and $\mu$, which is a frictional coefficient for sliding collisions [@LougeModel]. Brass and steel spheres differ primarily in their coefficients of normal restitution $e$: for steel, $e=0.95$; and for brass, $e=0.77$ [@Impact]. Therefore, brass balls lose approximately 4 times more energy per unit mass in particle-particle collisions, compared to steel balls, since the kinetic energy lost by the normal component of relative velocities is proportional to $1-e^2$ [@Goldhirsch] ($1-e^2=0.0975$ for steel whereas $1-e^2=0.4071$ for brass). Thus, the system of brass spheres is significantly further from equilibrium than the comparable system of stainless steel spheres.
In order to map out an experimental phase diagram, we started from an acceleration close to 1 $g$ and increased the amplitude in small increments $\Delta\Gamma\sim 0.025~g$. After each increment, we waited until there was no discernible evolution of the system, and then waited several minutes longer to ensure that a steady state had been reached. The phases present were then determined by visual inspection. This procedure was repeated for a range of densities, for both steel and brass spheres. The results are summarized in Fig. \[expfase\]. Figs. \[expfase\]a and b show the phase diagrams obtained for steel and brass, respectively, while panels (c) and (d) show expanded views of the region below $\Gamma=2$.
At the lowest amplitudes, we observe a ‘collapse’ of motionless, hexagonal close-packed spheres [@PRLord]. At slightly higher acceleration, the spheres order into a fluctuating hexagonally ordered singly layer [@PRLmelt]. As the acceleration is increased further, the hexagonal phase melts into a homogeneous liquid. If we continue to increase input acceleration, small clusters, denser than the surrounding fluid, begin to appear, initially unstable in time. With further increase of input acceleration one of the clusters becomes more stable and nucleates a two-layer ordered phase with square symmetry [@PREcub]. These phase diagrams are consistent with previous reports, but are more comprehensive and represent the first direct comparison between different types of particles.
This experimental phase diagram shows two notable new features. The first one is that for some densities the square phase melts at high vibration amplitude, and the second one is that for some parameter ranges the ordered square phase present in steel spheres is completely absent in brass. (For instance the square phase forms for steel spheres in the range of density $\rho=0.80-0.815$, while it is not present for brass spheres for densities below $\rho=0.82$ for any vibration amplitude). While much of the phase behavior of this system can be understood by analogy with equilibrium colloidal systems [@PREcub; @JCP], the meting of the square phase due to increase of acceleration and/or inelasticity reported here are purely nonequilibrium effects. We have speculated that some of the behavior we have observed upon increasing acceleration may be due to layer compression [@PREcub; @JCP]. It is not clear why this effect would be more pronounced in more inelastic material, except that it is clearly somehow associated with deviations from elastic behavior. Computer simulations by Clerc et al. [@clerc08] of a confined 2D (hard disk) system showed that increasing inelasticity reduced the liquid-solid coexistence region, consistent with the results reported here.
![The area of the square phase, $A_c$, divided by the plate area, $A_p$, in experiments with densities $\rho=0.82$ (triangles) and $\rho=0.86$ (squares) for both brass (open symbols) and steel balls (solid symbols), and MD simulations ($\rho=0.90$ and $\gamma_n=200~s^{-1}$ ($\times$) and $\gamma_n=262.5~s^{-1}$ ($+$), $N=2000$) as a function of input acceleration.[]{data-label="tam"}](fig4v4.pdf){height="4.5cm"}
To further investigate the dependence of the phase coexistence on acceleration and inelasticity, we have measured the area occupied by the square phase as a function of input acceleration, from its appearance up to its disappearance, for both brass and steel spheres. The results for two different densities are shown in Fig. \[tam\]. In all cases the freezing transition is abrupt, consistent with previous observations [@PREcub], while the disappearance is gradual. For both density values, the square phase maximum size is lower for the more inelastic brass spheres and it is observed over a smaller range of accelerations.
In order to confirm that the inevitable imperfections in the experimental setup are not responsible for the features observed in the phase map and to more thoroughly study the effect of varying inelasticity, we have also analyzed a series of molecular dynamics simulations (MD), using the same procedure as in [@PREcub]. The collisions are characterized by three forces: two normal forces (one elastic restoring force, proportional to particle overlap, and one frictional dissipative force, proportional to relative normal velocity) and a tangential force (inelastic). We have used values of parameters in the simulation that mimic the behavior of stainless steel balls (see [@ChenMD] for details), and then vary the normal frictional force by varying the constant of proportionality, $\gamma_n$, to investigate the effect of increased inelasticity. The interactions included in the model do not capture the full complexity of real inelastic collisions, but reproduce the dominant effects of vibration, collisions, and dissipation, and show all of the same qualitative features of the experiment.
The simulations followed the same sequence as the laboratory experiments. First, an initial state is prepared by placing spheres at random positions, with the constraint that particles do not overlap. Particles are assigned random velocities chosen from a uniform distribution. The simulation then runs at constant frequency and amplitude until a steady state is reached (as indicated by a constant granular temperature), which typically takes a few seconds of simulated time. Then the acceleration was increased in small increments, $\Delta\Gamma=0.025~g$, keeping the frequency constant, with sufficient delay between each increase to ensure that a steady state was again reached. As reported previously [@PREcub], the simulations reproduce the ordered phases observed experimentally. The crosses in Fig. \[tam\]a show the size of the square phase as a function of acceleration for one value of the normal dissipative force parameter, $\gamma_n=200~s^{-1}$, and the plusses show the size for a larger value, $\gamma_n=262.5~s^{-1}$, for a simulation with $N=2000$ particles.
![Renderings of MD simulations snapshots, for different values of the vibration amplitude. In (*a*), with $\Gamma=1.85$, there is a stable ordered two-layer square phase coexisting with the disordered phase. (*b*) A snapshot $\sim 3~$s after a sudden acceleration increase to $\Gamma=3.5$, showing the shrinking unstable crystal. (*c*) The steady state for $\Gamma=3.5$, where the only present phase is a homogeneous liquid. ($\gamma_n=200$ s$^{-1}$, $\rho=0.895$, $N=2000$).[]{data-label="simfase"}](evgamma3.pdf){height="11.75cm"}
The simulation reproduces the behavior of the experiments: the ordered phase disappears for large acceleration amplitudes, and the region of stability is smaller for more inelastic spheres. An example of the evolution upon increasing acceleration is shown in Fig. \[simfase\]. Fig. \[simfase\]a shows a rendering of the sphere positions for $\Gamma=1.85$, where the square phase is stable. Upon sudden increase in acceleration amplitude to $\Gamma=3.5$, the crystal shrinks (Fig. \[simfase\]b) and then disappears entirely, leaving only a disordered state (Fig. \[simfase\]c).
![(*a*) Critical acceleration values from MD simulations for square phase nucleation (solid symbols) and melting (open symbols), for varying $\gamma_n$, for simulations with $N=2000$ (triangles) and $N= 6000$ (squares) (*b*) Maximum value of $\gamma_n$ for existence of the square phase in MD simulations with different values $N$. ($\rho=0.895$, $h=1.75~\sigma$, $\nu=60$ Hz.)[]{data-label="Gg"}](Gg.pdf){height="8.5cm"}
Figure \[Gg\]a shows the critical value of the acceleration for the formation of the square phase at $\rho=0.895$ upon increasing acceleration (solid symbols) and for its melting (open symbols) as a function of inelasticity, for $N=2000$ (triangles) and $N=6000$ (squares). For both values of $N$, the region of stability of the crystal decreases as the inelasticity increases, as does the maximum size of the crystal (data not shown). Above a critical value of $\gamma_n$ ($\gamma_n^{crit}$), the crystal does not form for any vibration amplitude. Figure \[Gg\]a also shows that the crystallization transition depends on the size of the simulated system. Finite-size effects are well known in constant volume simulations of equilibrium transitions [@wedekind06], and arise because the formation of a dense phase exceeding the critical nucleation size results in a finite reduction in the density of the coexisting disordered phase. The appearance of finite-size effects in the granular crystallization suggests that something analogous to a strong surface tension is present, despite the absence of attractive interactions. The origin of this effect is unknown, but we note that the granular temperature in the crystal is higher near the edges than in the center [@PREcub], and this ‘boundary layer’ may contribute to an effective surface tension.
In Fig. \[Gg\]b we plot the value of $\gamma_n^{crit}$ as a function of system size. The results clearly show that the finite-size effects become negligible for sufficiently large systems, and that there is a critical value of inelasticity above which the ordered phase is suppressed for any system size. This surprising result provides a vivid demonstration that free-volume considerations that explain the transitions observed in confined equilibrium hard-sphere systems are not sufficient to describe the behavior of inelastic spheres, despite the strong similarities. We have observed previously that the granular temperature is significantly lower in the crystal than in the coexisting liquid [@PREcub], and suggested that this difference may account for the presence of a coexistence region significantly larger than that found in equilibrium systems [@JCP]. To investigate whether the absence of equipartition was also playing a role in the suppression of the crystalline phase, we measured the granular temperature in the ordered and liquid phases in the MD simulations reported here. The temperatures of the phases do change with the dissipation, but we observed no obvious anomalies near $\gamma_n^{crit}$. We did find that the density of the crystal increases with increasing dissipation, and that close $\gamma_n^{crit}$ the crystal is nearly close-packed.
Finally, we investigated whether the melting transition is described by the Lindemann ratio [@lindemann], the ratio of the rms fluctuation in particle positions in the solid phase to the lattice spacing. Equilibrium solids typically melt when this ratio exceeds about 0.15, and the melting of a granular monolayer was found be be consistent with this criterion [@reis06]. We find that the Lindemann ratio at the melting line (the open squares in Fig. \[Gg\]a) ranges from about 0.3 at very low inelasticity ($\gamma_n=0.1$) to 0.2 at $\gamma_n=350$. Due to the small size of the crystal close to the melting transition it is difficult to unambiguously measure the amplitude of the positional fluctuations, so these preliminary results will need to be confirmed with larger scale simulations. We have observed previously that the diffusion coefficient in the gas phase is larger for more inelastic particles [@JCP], presumably as a result of increased correlations. The rms fluctuations in the solid phase also increases with increasing inelasticity for fixed acceleration (data not shown). However, the amplitude of the fluctuations decrease upon decreasing acceleration, and along the melting line the decrease from the acceleration is greater than the increase from the inelasticity, resulting in a reduction of the Lindemann ratio. These results suggest that the effects of forcing and/or dissipation on the ordering transition result in a complexity that is not present in equilibrium systems, where the Lindemann ratio usually provides a reliable guide to the transition point.
These results demonstrate that the phase behavior of steady states of inelastic spheres is considerably different than their equilibrium analogs. The extension of the ideas of equilibrium statistical mechanics to nonequilbrium systems remains an ongoing challenge, and the insights gained from careful comparisons of closely analogous model systems are necessary to develop and test new approaches. Our observation that ordering is suppressed by inelasticity for identical inelastic spheres shows that quantitative changes in the strength of dissipation can qualitatively alter the phase diagram of even the simplest granular media.
This work was supported by NASA under award number NNC04GA63G. One of the authors (F. V. R.) also acknowledges financial support from the Spanish Ministry of Education and Science through “Juan de la Cierva” research program.
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---
abstract: 'Biometric-based identification has drawn a lot of attention in the recent years. Among all biometrics, palmprint is known to possess a rich set of features. In this paper we have proposed to use DCT-based features in parallel with wavelet-based ones for palmprint identification. PCA is applied to the features to reduce their dimensionality and the majority voting algorithm is used to perform classification. The features introduced here result in a near-perfectly accurate identification. This method is tested on a well-known multispectral palmprint database and an accuracy rate of 99.97-100% is achieved, outperforming all previous methods in similar conditions.'
address: 'Electrical and Computer Engineering Department, New York University, USA.'
title: 'O T P J W-DCT F M P R'
---
Introduction {#sec:intro}
============
To personalize an experience or make an application more secure, we may need to be able to distinguish a person from others. To do so, many alternatives are available, such as keys, passwords and cards. The most secure options so far, however, are biometric features. They are divided into behavioral features that the person can uniquely create (signatures or walking rhythm), and physiological characteristics (fingerprints and iris pattern). Many works revolve around identification, verification and categorization of such data including but not limited to fingerprints [@Fingerprint], palmprints [@Palm], faces [@Face] and iris patterns [@Iris].
Palmprint is among the most popular biometrics due to the many features it possesses and its stability over time. To use palmprints to such end, two widespread methods exist; either transforming the images into another domain like Fourier, DCT, wavelet or Gabor; or attempting to extract the lines and the geometrical characteristics from the palms. Many transform-based approaches exist, such as [@Gabor], in which Zeng utilized two-dimensional Gabor-based features and a nearest-neighbor classifier for palmprint recognition, [@wavelet] in which Wu presented a wavelet-based approach for palmprint recognition and used wavelet energy distribution as a discriminant for the recognition process and [@PCA+GWR] in which Ekinci proposed a wavelet representation approach followed by kernel PCA for palmprint recognition. Among notable line-based approaches is [@crease] where Cook proposed an automated flexion crease identification algorithm using image seams and KD-tree nearest-neighbor searching which results in a very high recognition accuracy.
There have also been notable developments in the more recent works. In [@hol], Jia proposed a new descriptor for palmprint recognition called histogram of oriented lines (HOL) which is inspired by the histogram of oriented gradients (HOG) descriptors. The work presented in [@quaternion] by Xu involves a quaternion principal component analysis approach for multispectral palmprint recognition with a high accuracy. In [@palm1], Minaee proposed to use a set of statistical and wavelet features to perform the identification task. In [@texture], Minaee proposed a set of textural features derived from the co-occurrence matrices of palmprint blocks and with the use of majority voting, achieved a highly accurate identification.
Most of the palmprint recognition systems consist of four general steps: image acquisition, preprocessing, feature extraction and template matching. These steps are shown in the block diagram in Figure 1.
Images can be acquired by different devices, such as CCD cameras, digital cameras and scanners. In our work, we have used the multispectral palmprint database which is provided by Polytechnic University of Hong Kong [@verification], [@database]. Four sample palmprints from this dataset are shown in Figure 2.
In the feature extraction step, we have used a combined set of DCT and wavelet features. PCA is applied to the features to reduce their dimensionality. In spite of the simplicity of these features, they prove to be quite effective for multispectral palmprint recognition. After feature extraction, we have used the majority voting scheme and minimum distance classifier to match and identify palmprints. The proposed algorithm is very fast and can be implemented in electronic devices in conjunction with energy-efficient algorithms [@hoseini1], [@hoseini2].
The following sections of this paper explain what and how features are used in this classification. Section \[SectionII\] describes the proposed set of features. Section \[SectionIII\] contains an explanation of our classification technique. Results of our experiments and comparisons with other works are in Section \[SectionIV\] and the conclusion is in Section \[SectionV\].
Features {#SectionII}
========
Feature extraction is a primary step in data analysis, and the information that features provide is correlated with the accuracy of the algorithm. Highly discriminating features usually have a large variance across different classes of target values and a small variance across samples of each class. There are many approaches used for feature extraction [@feature]. One approach is to attempt to automatically derive the useful set of features from a set of training data by projecting it onto discriminative components such as PCA and ICA. The other approach is to use hand-crafted features such as SIFT and HOG (Histogram of Oriented Gradients), or features from transform domain such as wavelet [@palm1]. Geometric features are also popular in many medical applications [@chro]. Sparse representation has also been used for extracting features in image classification task [@hojat1], [@hojat2]. One can also use dictionary learning framework to learn a good set of features from a set of training data [@joneidi1]-[@joneidi2]. Here a combined set of DCT- and wavelet-based features is used to perform multispectral palmprint identification. These features are extracted from small patches of each image and subsequently, features of different patches are concatenated to form the final feature matrix of each image. PCA can also be applied to the features for dimensionality reduction.
DCT Domain Features
-------------------
Discrete cosine transform (DCT) has many applications in various areas of image processing including compression and denoising [@dct]. Because of its energy compaction property, most of the image information tends to be concentrated in a few DCT coefficients and makes it favorable for image compression applications.
Suppose we have a 2D discrete function $f(m,n)$ of size $M\times N$. Its 2D DCT is defined as: $$\begin{gathered}
F(u,v)= \alpha_u \alpha_v \sum_{m=0}^{M-1} \sum_{n=0}^{N-1} f(m,n) cos(\frac{\pi(2m+1)u}{2M})cos(\frac{\pi(2n+1)v}{2N})\end{gathered}$$ where $0 \leq u <M-1$, $0 \leq v <N-1$ and: $$\alpha_u=\begin{cases}
\sqrt{1/M} & \text{if $u=0$}\\
\sqrt{2/M} & \text{otherwise}
\end{cases} \ , \ \ \alpha_v=\begin{cases}
\sqrt{1/N} & \text{if $v=0$}\\
\sqrt{2/N} & \text{otherwise}
\end{cases}$$
To extract DCT features from palmprints, every palmprint is divided into non-overlapping blocks of size 16$\times$16 and the 2D DCT of each block is computed. As we know, for most of camera-captured images, the majority of the energy is contained in the upper right subset of DCT coefficients. Because of that, the first 9 DCT coefficients in the zig-zag order are selected as DCT features. These features are shown in the following matrix.
$$\begin{bmatrix}
f_0 & f_1 & f_5 & f_6 & \cdots \\
f_2 & f_4 & f_7 & ~ & ~\\
f_3 & f_{8} & \ddots & ~ & ~\\
f_{9} & ~ & ~ & ~ & ~ \\
\vdots &~&~&~&~
\end{bmatrix}_{16\times16}$$ One can also keep more than 9 DCT coefficients or can also make use of all DCT coefficients. However, based on our experiments, using more than 9 coefficients does not provide us with significant improvement.
Wavelet Domain Features
-----------------------
Wavelet is a very popular tool for a variety of signal processing applications such as signal denoising, signal recovery and image compression [@mallat]. Perhaps JPEG2000 [@jpeg] is one of the most notable examples of wavelet applications. In our feature extraction procedure, the images are first divided into $16\times 16$ non-overlapping blocks. Then the 2D-wavelet decomposition is performed up to three stages, and in the end, 10 sub-bands are produced. The energy of wavelet coefficients in these subbands are used as the wavelet features (the LL subband of last stage is not used here). The summary of our wavelet feature extraction stage is presented in the following algorithm:
1. Divide each palm image into $16\times 16 $ non-overlapping blocks;
2. Decompose each block up to 3 levels using Daubechies 2 wavelet transform;
3. Compute the energy of each subband and treat it as a feature.
After computation, there will be a total of 18 different features (9 DCT plus 9 wavelet) for each block which can be combined in a vector together: $\textbf{f}=(f_1,f_2,...,f_{18})^\intercal$. It is necessary to find the above features for each image block. If each palm image has a size of $W \times H$, the total number of non-overlapping blocks of size $16\times 16 $ will be: $$\begin{gathered}
M=\frac{W \times H}{256}\end{gathered}$$ Therefore there are $M$ such feature vectors, $\textbf{f}^{(m)}$. Similarly, they can be put in the columns of a matrix to produce the feature matrix of that palmprint, $\textbf{F}$: $$\begin{gathered}
\textbf{F}=[\textbf{f}^{(1)} ~ \textbf{f}^{(2)} \cdots ~\textbf{f}^{(M)}]\end{gathered}$$ There are a total of 1152 features for each palmprint image. Using all of the 1152 features may not be efficient for some applications. In those cases, dimensionality reduction techniques can be used to reduce the complexity.
Principal Component Analysis
----------------------------
Principal component analysis (PCA) is a powerful algorithm used for dimensionality reduction [@PCA]. Given a set of correlated variables, PCA transforms them into another domain such the transformed variables are linearly uncorrelated. This set of linearly uncorrelated variables are called principal components. PCA is usually defined in a way that the first principal component has the largest possible variance, the second one has the second largest variance and so on. Therefore after applying PCA, we can keep a subset of principal components with the largest variance to reduce the dimensionality. PCA has a lot of applications in computer vision and neuroscience. Eigenface is one representative application of PCA in computer vision, where PCA is used for face recognition.
Without going into too much detail, let us assume we have a dataset of $N$ palmprint images and $\{f_1,f_2,...,f_N\}$ denote their features. Also let us assume that each feature has dimensionality of $d$. To apply PCA, we first need to remove the mean value of the features as $z_i= f_i- \bar{f}$ where $\bar{f}= \frac{1}{N} \sum_{i=1}^m f_i$. Then the covariance matrix of the centered images is calculated: $$\begin{gathered}
C= \sum_{i=1}^m z_i z_i^T \end{gathered}$$ Next the eigenvalues $\lambda_k$ and eigenvectors $\nu_k$ of the covariance matrix $C$ are computed. Suppose $\lambda_k$’s are ordered based on their values. Then each $z_i$ can be written as $z_i= \sum_{i=1}^d \alpha_i \nu_i$. By keeping the first $k (\ll d)$ terms in this summation, we can reduce the dimensionality of the data by a factor of $\frac{k}{d}$ and derive new feature representation as $\hat{z_i} $. By keeping $k$ principal components, the percentage of retained energy will be equal to $\frac{\sum_{i=1}^k \lambda_i}{\sum_{i=1}^d \lambda_i }$.
Majority Voting Classifier {#SectionIII}
==========================
After the features are extracted, a classifier is required to match the most similar image in the data set to the test subject. There are different classification algorithms that can be used. Some of the most widely used include minimum-distance classifier, neural networks and support vector machines. These algorithms usually have some parameters which need to be tuned. The parameter tuning is usually done by minimizing a cost function on the training set. If the dataset is large enough, the cost function is basically the training error. However if the data set is small, the cost function should have two terms: one term tries to minimize the error; and the other term tries to minimize the risk of over-fitting. One such a work is studied in [@mtbi].
Here we have used the majority voting algorithm. It is performed by individual predictions by every feature followed by adding all the votes to determine the outcome. One can also use weighted majority voting where each feature is given a weight in the voting process. The weight of each feature is usually related to the single feature accuracy in the classification task; the more it can successfully predict on its own, the greater weight it is given. Here we have assigned similar weights to all features to make the algorithm parameters independent of the dataset.
In our classifier, first the training images’ features are extracted. Then, the features of the test sample are extracted and the algorithm searches for a training image which has the minimum distance from the test image. Each time one feature is used to select a training sample with the minimum distance and that sample is given one unit of score and this procedure should be performed for all features. In the end, the training sample with the highest score is selected as the most similar sample to the test subject.
Let us denote the $i$-th feature of the test sample by $\textbf{f}_i^{(t)}$, the predicted match for the test sample using this feature will be: $$k^*(i)={\arg\!\min}_k \| \textbf{f}_i^{(t)}-\textbf{f}_i^{(k)} \|_2$$ where $\textbf{f}_i^{(k)}$ is $i$-th feature of the $k$-th person in the training data.
Let us denote the score of the $j$-th person based on $\textbf{f}_i$ by $S_j(i)$. $S_j(i)$ is equal to $\textrm{I}(j={\arg\!\min}_k\|\textbf{f}_i^{(t)}-\textbf{f}_i^{(k)}\|)$, where $\textrm{I}(x)$ denotes the indicator function. Then the total score of the $j$-th training sample using all the spectra is found by the following formula: $$S_j=\sum_{All~spectra}\sum_{i=1}^{i_{max}}{\textrm{I}(j={\arg\!\min}_k|\textbf{f}_i^{(t)}-\textbf{f}_i^{(k)}|)}$$ Finishing the calculations, $j^*$ or the matched training sample will be: $$\begin{gathered}
j^*={\arg\!\max}_j \big[ S_j\big]= {\arg\!\max}_j \big[\sum_{All~spectra}\sum_{i}S_j(i) \big]\end{gathered}$$
Results {#SectionIV}
=======
We have tested the proposed algorithm on the PolyU multisprectral palmprint database [@database] which has 6000 palmprints sampled from 500 persons (12 samples for each person). Each palmprint is taken under four different lights in two days resulting in a total of 24000 images. Each image is preprocessed and its ROI is extracted (with a size of 128 $\times$ 128). Images are acquired using four CCD cameras to take four images from each palmprint under four distinct lights: blue, green, red and near-infrared (NIR).
Before presenting the results, let us discuss briefly about the parameters of our model. 18 features are derived locally from blocks of size $16 \times 16$ (18 features for each block). Features of different blocks are concatenated resulting in a total of 1152 features for each image. For wavelet transform, Daubichies 2 is used. The recognition task is conducted using both majority voting and minimum distance classifier. Based on our experiment, majority voting algorithm achieves higher accuracy rate than minimum distance classifier and its result is used for comparison with other previous works.
We have studied the palmprint identification task for two different scenarios. In the first scenario, we have applied PCA to reduce the dimensionality of the feature space and used minimum distance classifier to perform template matching. The recognition accuracy for different number of PCA features is shown in Figure 3. As it can be seen, even by using 100 PCA features we are able to achieve very high accuracy rate.
In the second scenario, we have used all 1152 features followed by weighted majority voting to perform palmprint recognition. Using all features enables us to achieve highly accurate results. The recognition rate for different fractions of training and testing data is shown in Table \[TblRes1\]. For instance, in the case that the fraction of training sample is 4/12, we have used 2000 palmprints as training and the remaining 4000 ones as test samples.
\[h\]
[|m[1.4cm]{}|m[1.5cm]{}|m[1.5cm]{}|m[1.5cm]{}|]{} & & & [6/12]{}\
& 99.97% & 100% & 100%\
\[TblRes1\]
Table \[TblComp\] provides a comparison of the results of our work and those of three other highly accurate schemes. The reported result for the proposed scheme corresponds to the case where all features are used and majority voting algorithm is employed for template matching. It can clearly be observed that the proposed method can perform better than the others which can be the result of the compatibility of the proposed features in this procedure.
\[h\]
[|m[6cm]{}|m[2cm]{}|]{} Palmprint Recognition Schemes & Recognition Rate\
K-PCA+GWR [@PCA+GWR] & 95.17%\
Quaternion principal component analysis [@quaternion] & 98.13%\
Histogram of Oriented Lines [@hol] & 99.97%\
Proposed scheme using majority voting algorithm & 100%\
\[TblComp\]
Conclusion {#SectionV}
==========
This paper proposed a set of joint wavelet-DCT features for palmprint recognition. These features are extracted from non-overlapping sub-images so that they capture the local information of palmprints. These features are sensitive to the small differences between different palmprints. Therefore they are able to discriminate different palms with very similar patterns. After the features are extracted, PCA is applied for dimensionality reduction and majority voting algorithm is used to match each template to the most similar palmprint. The proposed algorithm has significant advantages over the previous popular approaches. Firstly, the proposed features here are very simple to extract and the algorithm is very fast to compute. Secondly, it has a very high accuracy rate for small fractions of training samples. The same framework can be applied to other recognition tasks, such as fingerprint recognition and iris recognition.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank the Hong Kong Polytechnic University (PolyU) for sharing their multisprectral palmprint database with us.
\[sec:ref\]
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http://www.comp.polyu.edu.hk/˜biometrics/MultispectralPalmprint
|
---
abstract: 'By applying a magnetic field perpendicular to GaAs/AlGaAs two-dimensional electron systems, we study the low-field Landau quantization when the thermal damping is reduced with decreasing the temperature. Magneto-oscillations following Shubnikov-de Haas (SdH) formula are observed even when their amplitudes are so large that the deviation to such a formula is expected. Our experimental results show the importance of the positive magneto-resistance to the extension of SdH formula under the damping induced by the disorder.'
address:
- '$^{1}$Department of Physics, National Taiwan University, Taipei, Taiwan, R.O.C.'
- '$^{2}$Department of Materials Science and Optoelectronic Engineering, National Sun Yat-sen University, Kaohsiung 804, Taiwan, R.O.C.'
- '$^{3}$Center for Nanoscience and Nanotechnology, National Sun Yat-sen University, Kaohsiung 804, Taiwan, R.O.C. '
- '$^{4}$National Measurement Laboratory, Center for Measurement Standards, Industrial Technology Research Institute, Hsinchu, Taiwan, R.O.C.'
- '$^{5}$Institute of Materials Science and Engineering, National Sun Yat-sen University, Kaohsiung 804, Taiwan, R.O.C.'
- '$^{6}$Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, United Kingdom'
- '$^{7}$School of Physics, University of New South Wales, Sydney 2052, Australia'
author:
- 'Jing-Han Chen$^{1}$, D. R. Hang$^{2,3}$, C. F. Huang$^{4}$, Tsai-Yu Huang$^{1}$, Jyun-Ying Lin$^{1}$, S. H. Lo$^{5}$, J. C. Hsiao$^{4}$, Ming-Gu Lin$^{1}$, M.Y. Simmons$^{6,7}$, D.A. Ritchie$^{6}$, and C.-T. Liang$^{1}$'
title: 'Experimental Studies of Low-field Landau Quantization in Two-dimensional Electron Systems in GaAs/AlGaAs Heterostructures'
---
By applying a magnetic field $B$ perpendicular to the two-dimensional electron systems (2DESs) in semiconductor heterostructures, we can observe the magneto-oscillations in the longitudinal resistivity $\rho _{xx}$ with decreasing the temperature $T$ because of Landau quantization. Such oscillations are expected to follow Shubnikov-de Haas (SdH) formula [@Coleridge; @SdH] $$\begin{aligned}
\rho _{xx} \sim \rho _{xx,B=0} + \Delta \rho _{xx} \mathrm{cos} [ \pi ( \nu - 1 ) ]\end{aligned}$$ with the oscillating amplitude [@GaN_I-QH] $$\begin{aligned}
\Delta \rho _{xx} = 4 \rho _{0} \frac{2 \pi ^{2} k _{B} T m ^{*} / \hbar e B }{\mathrm{sinh}(2 \pi ^{2} k _{B} T m ^{*} / \hbar e B)} e ^{ - \pi m ^{*} / e \tau _{q} B} \end{aligned}$$ at low $B$ when the spin-splitting is unresolved. Here $\nu$ is the filling factor, $\rho _{xx, B=0}$ is the value of $\rho _{xx}$ at $B$=0, $\tau _{q}$ is the quantum lifetime, $k _{B}$ and $m^{*}$ represent the Boltzmann constant and effective mass, $\hbar$ equals Plank constant $h$ divided by $2 \pi$, and $\rho _{0}$ is a constant. It is expected that $\rho _{xx, B=0}$ is independent of the temperature, and the ratio $\rho _{0} / \rho _{xx, B=0} \sim 1$ [@Coleridge; @GaN_I-QH]. It is well-established how to obtain the carrier concentration $n$, quantum lifetime, and effective mass from the last (oscillating) term of Eq. (1). On the other hand, Landau quantization results in the integer quantum Hall effect (IQHE) [@Klitzing] with increasing $B$ when the high-field localization [@high-field-localization] becomes important. Such an effect is characterized by a series of plateaus in the Hall resistivity $\rho _{xy}$ when $\rho _{xx}$ approaches zero. The Hall plateaus are so accurate that the IQHE has been used to maintain the resistance standard. [@Delahaye] By studying the magnetic-field-induced transitions in the IQHE, we can investigate the renormalization-group theory [@Pruisken] and modular symmetry [@Shahar; @Dolan].
The localization arising from the quantum interference is very important to standard theories for the IQHE. [@Jiang; @Kivelson; @Murzin] While these theories are successful at high $B$, they are inappropriate for the low-field Landau quantization in most 2DESs because the localization length may become much longer than the effective size with decreasing $B$. [@Jiang] Actually the quantum interference is ignored in SdH formula, which can be obtained from a semiclassical approach. [@Coleridge; @Coleridge2] Therefore, we shall investigate the low-field Landau quantization to understand the crossover from the semiclassical regime to the IQHE with increasing $B$. To explain the appearance of Hall plateaus when Eq. (1) is still applicable for $\rho _{xx}$, it has been shown that the quantum interference is more robust in $\rho _{xy}$ than that in $\rho _{xx}$ in such a crossover. [@Coleridge; @Hang] Actually, the IQHE can also be explained by fixing the chemical potential without considering the high-field localization. [@Tobias*other*Mahan; @Tobias2].
According to Eq. (2), the oscillating amplitude $\Delta \rho _{xx}$ increases with increasing the magnetic field $B$. Deviations to SdH formula are expected when the amplitude becomes so large that $\Delta \rho _{xx} \sim \rho _{xx, B=0}$ and thus the minimum of $\rho _{xx}$ approaches 0, which is an important feature of the IQHE. [@Coleridge2; @Hang] However, our group showed in Ref. [@Hang] that Eq. (2) can be applicable even when $\Delta \rho _{xx} > \rho _{xx, B=0}$. The extension of Eq. (2) can be due to the thermal damping [@Martin], and it is important to incorporate the positive magneto-resistance into Eq. (1) for the positivity of $\rho _{xx}$. [@Hang] In fact, it has been predicted [@Martin] that the extension of SdH formula not only under the thermal damping, but also under the disorder effects determining the Dingle factor $\mathrm{exp} (- \pi m^{*} / e \tau _{q} B)$. To further investigate disorder effects in the low-field Landau quantization, in this paper we probe SdH formula when the thermal damping is reduced by decreasing the temperature $T$. Two GaAs/AlGaAs samples are used for this study. For convenience, we denote them as samples A and B, respectively. The samples are made into Hall patterns by the standard lithography. We study the magneto-transport properties of these two samples by the superconducting magnet and top-loading He$^{3}$ system. Magneto-oscillations following Eq. (2) are observed in these two samples, and the typical IQHE appears with increasing $B$. From SdH oscillations, the carrier concentration $n= 2.8 \times 10 ^{15} \text{ } m^{-2}$ and $ 3.5 \times 10 ^{15} \text{ } m ^{-2}$ for samples A and B, respectively. The scattering mobility $\mu _{c}$ obtained from $\rho _{xx, B=0} = 1/ n e \mu _{c} $ is $5.7 \times 10 ^{2} \text{ } m^{2}/V$-$s$ for sample A and is $44~m ^{2}/V$-$s$ for sample B.
For convenience, in the following we focus on sample A first. Figure 1 shows the low-field curves of $\rho _{xx}$ observed in such a sample. A series of oscillations appear in Fig. 1 with increasing the perpendicular magnetic field $B$. The oscillating amplitude should follow $$\begin{aligned}
\mathrm{ln} \frac{ \Delta \rho _{xx} }{ X / sinh X } = ln ( 4 \rho _{0}) + \pi m ^{*} / e \tau _{q} B\end{aligned}$$ with $X=2 \pi ^{2} k _{B} T m ^{*} / \hbar e B $ if Eq. (2) holds true. It is known from the above equation that the curves of $ln \frac{\Delta \rho _{xx}}{X/sinhX}-1/B$ at different temperatures collapse into a straight line in the Dingle diagram if SdH formula is applicable. To avoid effects due to the exchange enhanced spin gaps [@Leadley] and asymmetric spin-resolved oscillations [@Ando], we construct such a diagram by considering the spin-degenerate oscillations, which can survive at higher $B$ with increasing $T$. By taking $m^{*}=0.067 m_{0}$ to calculate $X/sinhX$, as shown in Fig. 2, the logarithmic values of $\Delta \rho _{xx} / (X / sinh X) $ for the spin-degenerate oscillations collapse well into a straight line with respect to $1/B$. The slope yields $\mu _{q} = 3.6 \text{ } m ^{2}/ V$-$s$. The good collapse indicates the validity of Eq. (2) and SdH theory.
The oscillating amplitude $\Delta \rho _{xx}$ at $B>0.35$ T, in fact, is larger than the zero-field longitudinal resistivity $\rho _{xx, B=0}$ with increasing $T$. When $\Delta \rho _{xx} > \rho _{xx, B=0}$, the minimum of $\rho _{xx}$ become negative according to Eq. (1) and thus the deviations to SdH formula is expected. However, Eq. (2) still holds when $B>0.35$ T until the spin-splitting becomes resolved. It has been shown in Ref. [@Hang] that the positive magneto-resistance is important to the validity of Eq. (2) when $\Delta \rho _{xx} > \rho _{xx, B=0}$. As shown in the inset of Fig. 2, the positive magnetoresistance becomes apparent after taking the average with respect to the magneto-oscillations to obtain the non-oscillatory background. Hence the experimental results support the importance of the positive magneto-resistance to refine Eq. (1). The extension of Eq. (2) under $\Delta \rho _{xx} > \rho _{xx, B=0}$ in Ref. [@Hang] can be due to the thermal-damping factor $X/sinhX$. At $T=0.27 $ K, however, the damping term $X / sinhX > 0.9$ and is close to the zero-temperature value 1 when $B > 0.35$ T. Thus the extension of SdH formula cannot be fully due to the thermal damping in our study. We note that such a formula can also survive when the Dingle factor $\mathrm{exp} (- \pi m^{*} /e \tau _{q} B)$, which represents the disorder, induces strong damping. [@Martin] Such a factor is smaller than 0.27 for the spin-degenerate oscillations at $T=0.27$ K, so it can result in the remarkable damping effect. For sample A, the Dingle factor is significant in comparison with the thermal damping when $T<1$ K. Therefore, our observations support importance of the disorder effects to extension of SdH formula when there exists the positive magneto-resistance.
As mentioned above, we construct the Dingle diagram by only considering the spin-degenerate oscillations to avoid exchange and asymmetric effects. Deviations to SdH theory are expected at the onset of spin splitting although the SdH theory is applicable when such a splitting is fully resolved. [@spin-resolved*beating]. At $T=0.27$ K, the effects of the spin-splitting in sample A appear when $B>0.5$ T, where Eq. (2) becomes invalid. With increasing the temperature $T$, the spin-degenerate oscillations can survive at larger $B$ and still follow Eq. (2) when $B=0.5 \sim 1$ T. While the thermal effects in Ref. [@Hang] result in the extension of SdH formula by the damping factor $X/sinhX$, our study show that the thermal effect can also suppress the spin-splitting to extend such a formula.
Refinements to SdH theory are discussed in the literature. The value of $\rho _{0}$, which can be obtained from the the intercept of $\mathrm{ln}\frac{\Delta \rho _{xx}}{X/sinhX}-1/B$ at $1/B \rightarrow 0$ according to Eq. (3), can deviate from $\rho _{xx, B=0}$ although $\rho _{0}/ \rho _{xx, B=0} \sim 1$ is expected in the conventional SdH theory. [@GaN_I-QH] In our study, the ratio $ \rho _{0}/ \rho _{xx,B=0} = 3.6$ for sample A. From the reports on high-disorder 2DESs, in fact, it is not always appropriate to relate the constant $\rho _{0}$ to $\rho _{xx, B=0}$ because the zero-field value of $\rho _{xx}$ can depend on the temperature rather than being a constant. [@GaN_I-QH]
A quantum Hall state is characterized by $\rho _{xx} =0$ in addition to the quantized Hall plateau. In sample A, the spin-splitting is resolved before the appearance of the zero longitudinal resistivity as the field $B$ increases. Thus we cannot probe Eq. (2) with well-developed quantum Hall states in such a sample. In sample B, as shown in the inset of Fig. 3, the minimum of $\rho _{xx}$ approach 0 at low temperatures as $B > B _{1}$ when the spin-splitting is still unresolved. Therefore, we can probe SdH formula under the appearance of zero longitudinal resistivity by investigating such a sample. The Dingle diagram of sample B is shown in Fig. 3. By taking $m ^{*} =0.064 \text{ } m _{0} $, we can approximate $\mathrm{ln} \frac{ \Delta \rho _{xx} }{ X/sinhX}-1/B$ by a straight dash line even when $B > B_{1}$. The slope of the straight line yields $\mu _{q} = 2.7 m ^{2}/V$-$s$, and the Dingle term provides stronger damping effects than the thermal factor $X/sinh X$ does at low temperatures as $B>B _{1}$. Our observations indicate the importance of the disorder effects to the coexistence of quantum Hall states and SdH formula.
Both the quantum and scattering mobilities of sample B are lower than those of sample A, so the disorder strength should be stronger in the former sample. We note that the disorder may destruct the spin gaps, and hence the spin-splitting is resolved at larger magnetic field in sample B than in sample A. When the minimum of $\rho _{xx} \rightarrow 0$, the oscillating amplitude $\Delta \rho _{xx}$ is determined by the peak values of $\rho _{xx}$. In addition, the non-oscillatory background can be approximated by the half of the envelope function for the peaks of oscillations. Hence the validity of Eq. (2) for sample B when $B>B_{1}$ indicates that SdH formula can provide good estimations to both the peak values and non-oscillatory background of $\rho _{xx}$ under suitable conditions.
In conclusion, the low-field Landau quantization is studied by probing the crossover from semiclassical SdH regime to the integer quantum Hall effect. Our experimental results support the extension of SdH formula, to which we shall include the positive magnetoresistance, under the damping due to the disorder effects. In addition, the thermal effects can suppress the spin-splitting for the extension. When the minimum of $\rho _{xx}$ approaches zero, such a formula may provide estimations to both the peak values and the positive-magnetoresistance background.
This work is supported by NSC, Taiwan. C.T.L. acknowledges financial support from NSC 94-2112-M-002-037. D.R.H. acknowledges support from NSC 94-2112-M-110-009, ACORC and Aim for the Top University Plan, Taiwan. The work undertaken at Cambridge was funded by the EPSRC, UK. C.T.L. thanks Tina Liang and Valen Liang for their support.
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Figure Captions
Fig. 1 The low-field curves of the longitudinal resistivity of sample A at $T=$0.27, 0.37, 0.47, 0.57, 0.67, 0.82, 0.97, 1.1, 1.3, and 1.4 K, respectively.
Fig. 2 The Dingle diagram for sample A. The straight line is the best fit to $\mathrm{ln} \Delta \rho
_{xx} - 1/B$. The dash line in the inset shows the non-oscillatory background of the curve of the longitudinal resistivity (the solid line) at $T=0.27$ K.
Fig. 3 The Dingle diagram for sample B. The straight line is the best fitting to $\mathrm{ln} \Delta \rho
_{xx} - 1/B$. The inset shows the low-field curves of the longitudinal resistivity at different temperatures for sample B.
|
---
abstract: 'Does gravity influence local measurements? Freely falling atoms are ideal test masses to search for equivalence principle violation. We use a dual-species atom interferometer with $2\,\text{s}$ of free-fall time to measure the relative acceleration between $^{85}$Rb and $^{87}$Rb. Systematic errors arising from kinematic differences between the isotopes are suppressed by calibrating the angles and frequencies of the interferometry beams. We find $\eta = [1.6\; \pm\; 1.8\; \text{(stat)}\; \pm \; 3.4 \; \text{(sys)}] \times 10^{-12}$, consistent with zero violation. With a resolution of up to $1.4 \times 10^{-11} \, g$ per shot, we demonstrate the highest sensitivity to $\eta$ obtained in a laboratory setting.'
author:
- Peter Asenbaum
- Chris Overstreet
- Minjeong Kim
- Joseph Curti
- 'Mark A. Kasevich'
bibliography:
- 'EP-bib.bib'
date: 'May 23, 2020'
title: 'Atom-interferometric test of the equivalence principle at the $10^{-12}$ level'
---
[^1]
[^2]
The equivalence principle (EP), which posits that all gravitational effects disappear locally [@Pauli1921], is the foundation of general relativity [@Weinberg1972] and other geometric theories of gravity. Most theoretical unification attempts that couple gravity to the Standard Model lead to EP violations [@Damour1996]. In addition, tests of the equivalence principle search for perturbations of geometric gravity and are sensitive to exotic interactions [@Hees2018] that couple differently to the test masses. These tests are complementary to searches for large-scale variations of unknown fields [@Graham2016] and are carried out with local probes that can be precisely controlled.
EP tests are often characterized by the Eötvös parameter $\eta$, which is the relative acceleration of the test masses divided by the average acceleration between the test masses and the nearby gravitational source. With classical accelerometers, EP violation has been constrained to $\eta < 1.8 \times 10^{-13}$ by torsion balances in a laboratory setting [@Schlamminger2008] and to $\eta < 1.3 \times 10^{-14}$ by the concluded space mission MICROSCOPE [@Touboul2017].
We perform an equivalence principle test by interferometrically measuring the relative acceleration of freely falling clouds of atoms. Atom clouds are ideal test masses because they spend $99.9\%$ of the interrogation time in free fall and the remainder in precisely controlled interactions with the interferometry lasers. Compared to classical tests, atom-interferometric (AI) EP tests are influenced by different sources of systematic error. In addition, AI EP tests can be performed between isotopes that differ only in neutron number, and quantum tests are especially sensitive to particular violation mechanisms [@Goklu2008]. However, previous AI EP tests [@Bonnin2013; @Schlippert2014; @Tarallo2014; @Zhou2015] have been limited to $\eta < 3 \times 10^{-8}$ in dual-species comparisons [@Zhou2015] and $\eta < 1.4 \times 10^{-9}$ in comparisons between ground states of a single species [@Rosi2017b], largely due to a lack of sensitivity compared to classical experiments.
In this work, we report an atom-interferometric test of the equivalence principle between $^{85}$Rb and $^{87}$Rb with $\eta = [1.6\; \pm\; 1.8\; \text{(stat)}\; \pm \; 3.4 \; \text{(sys)}] \times 10^{-12}$, consistent with zero violation at the $10^{-12}$ level. This result improves by four orders of magnitude on the best previous dual-species EP test with atoms [@Zhou2015]. We achieve high sensitivity by utilizing a long interferometer time $T$ and a large momentum splitting between interferometer arms. At a resolution of $1.4 \times 10^{-11} \, g$ per shot, the interferometer attains the highest sensitivity to $\eta$ of any laboratory experiment to date [@Schlamminger2008].
The relative acceleration between $^{85}$Rb and $^{87}$Rb is measured with a dual-species atom interferometer. The experimental apparatus is described in [@Overstreet2018]. We prepare ultra-cold clouds of $^{85}$Rb and $^{87}$Rb by evaporative cooling in a magnetic trap. The subsequent magnetic lensing sequence lowers the kinetic energies to $25\, \text{nK}$ but introduces a $1.8\, \text{mm}$ vertical offset between the two isotopes. The other kinematic degrees of freedom (DoFs) remain matched. The clouds are then trapped in a vertical 1D optical lattice and accelerated to $13\, \text{m/s}$ in $20\, \text{ms}$. This laser lattice launch transfers an even number of photon recoil momenta $\hbar k$ to the atoms. The final velocity, and therefore the launch height $(\sim 8.5\, \text{m})$, is carefully chosen to match the vertical velocities of the two isotopes. To spatially overlap the clouds, we apply species-selective Raman transitions that kick the two isotopes in opposite directions. After a $77\, \text{ms}$ drift time and removal of untransferred atoms, the Raman transitions are reversed, and the clouds are overlapped to better than $65\, \mu$m. The Raman pulses also provide velocity selection, and the detunings of the Raman pulses allow the average vertical velocity of each isotope to be individually controlled.
The interferometer beamsplitters consist of sequences of two-photon Bragg transitions [@Overstreet2018] that transfer $4\hbar k$, $8\hbar k$, or $12 \hbar k$ momentum. These pulses split the cloud symmetrically in the vertical direction. The symmetric interferometer geometry guarantees that the midpoint trajectory [@Antoine2003] of each cloud remains essentially unperturbed. The interferometer duration $2T$ is $1910\, \text{ms}$. After a total drift time of $2.5\, \text{s}$, the output ports (separated by $2\hbar k$ momentum) are imaged with two orthogonal CCD cameras along the horizontal directions. One isotope is imaged with a time delay of $1\,\text{ms}$ so that the two species can be individually resolved. The phase of each interferometer is given by the population ratio of its output ports. Fig. \[Fig:1\] shows a schematic of the interferometer sequence.
In an EP test configuration, the differential phase between $^{85}$Rb and $^{87}$Rb is close to zero. To distinguish small positive from small negative differential phases, a precise phase offset is needed. By adjusting the angles of the interferometer beams, we imprint a horizontal phase gradient so that each image contains a full interference fringe. This “detection fringe” is highly common to both isotopes and allows the contrast and phase of each interferometer to be extracted from a single shot. Fig. \[Fig:1\](b) shows a fluorescence image in which the detection fringe is visible.
![(a) Simultaneous $^{85}$Rb and $^{87}$Rb interferometer. In pulse zone 1 ($t=0$), each atom cloud is split into two interferometer paths with $\hbar \mathbf{k}_1$ momentum difference. In pulse zone 2 ($t=T$), the paths are reflected toward each other with wavevector $\mathbf{k}_2$. In pulse zone 3 ($t=2T$), the paths are recombined and interfered with wavevector $\mathbf{k}_3$. The wavevectors $\mathbf{k}_1$, $\mathbf{k}_2$, and $\mathbf{k}_3$ differ slightly in orientation and length to create a tailored phase response to kinematic initial conditions. The midpoint trajectory of each isotope remains unperturbed throughout the interferometer. (b) Single fluorescence image of $^{85}$Rb and $^{87}$Rb output ports ($0\hbar k$ and $-2\hbar k$) with 8$\hbar k$ beamsplitters. The detection fringe allows precise single-shot phase extraction.[]{data-label="Fig:1"}](epfig1v7.pdf){width="\linewidth"}
The differential phase is proportional to the relative acceleration between the atoms. We achieve a single-shot differential phase resolution of $8\, \text{mrad}$ in an $8 \hbar k$ interferometer, which corresponds to a relative acceleration sensitivity of $1.4 \times 10^{-11}\, g$ per shot with duty cycle $15\, \text{s}$. In each data run, the initial beamsplitter direction, number of photon recoils per beamsplitter (4, 8, or 12), detection fringe direction, and imaging order are permuted to suppress systematic errors. A full run consists of about $20$ shots in each configuration ($480$ shots total). The statistical sensitivity is derived from three full runs taken on three separate days. Throughout the data-taking and analysis process, the EP result was blinded by the addition of an unknown offset to each differential phase measurement.
Systematic errors arise from effects that shift the $^{85}$Rb interferometer phase relative to the $^{87}$Rb phase. In our experiment, there are three significant sources of systematic error: differences in kinematic DoFs, differences in the interaction with the electromagnetic field, and imaging errors. A summary of the systematic errors is presented in Table \[table\]. The most significant systematic effects are described in the text below, and all indicated errors are discussed in the supplement.
The relevant kinematic DoFs are the initial position and velocity of each species in the vertical and horizontal directions. Each species is coupled by its kinematic DoFs to gravity and to the wavefront of the interferometer beams. In order to reduce the associated systematic errors, we minimize differences in the kinematic DoFs and suppress the sensitivity of the interferometer to them. The phase sensitivity to the vertical DoFs can be minimized by adjusting the frequency of the interferometry lasers between each pulse zone, and the phase sensitivity to the horizontal DoFs can be minimized by adjusting the angles of the lasers.
To be concrete, suppose that the average frequency of the interferometer lasers during pulse zone 2 is $f$. Changing the average frequency of pulse zone 1 to $f + \Delta f_1$ adds a phase shift $2 \pi /c \; n\Delta f_1 z$, where $n$ is the number of photon recoils and $z$ is the initial vertical position. Similarly, if the average frequency of pulse zone 3 is $f + \Delta f_3$, the added phase shift $2 \pi /c\; n \Delta f_3 (z + 2 v_{z} T)$ depends on the initial vertical velocity $v_z$. Gravity gradients cause systematic errors that are proportional to the vertical initial conditions [@Asenbaum2017; @Overstreet2018]. By choosing the appropriate combination of $\Delta f_1$ and $\Delta f_3$, we can simultaneously minimize the phase sensitivity to $z$ and $v_z$, realizing a generalized version of the compensation technique reported in [@Overstreet2018]. To calibrate the pulse zone frequencies, we vertically displace the two isotopes in each DoF and choose $\Delta f_1$ and $\Delta f_3$ to minimize the dependence of the differential phase on the displacement.
An analogous technique can be used to suppress the sensitivity of the interferometer to horizontal DoFs. Suppose that the beam angle during pulse zone 2 is $\theta_2 = 0$. Setting the angle of pulse zone 1 to $\theta_1$ in the $xz$ plane adds the phase shift $k \theta_1 x$, where $x$ is the initial horizontal position. Likewise, the angle $\theta_3$ in pulse zone 3 adds the phase shift $k \theta_3 (x + 2 v_x T)$, where $v_x$ is the initial horizontal velocity. An appropriate choice of angles in each horizontal direction provides complete compensation of linear phase gradients from the interferometer wavefront. Such phase gradients arise due to the rotation of the earth [@Sugarbaker2013].
We control the angle of the interferometer lasers in each pulse zone by setting the angle of the mirror that retroreflects them. This angle is adjusted during the interferometer to undo the velocity-dependent phase that would otherwise be imprinted by the earth’s rotation. To calibrate the rotation rate, we add and subtract an additional velocity-dependent phase to an $^{87}$Rb interferometer and null the fringe frequency difference [@Sugarbaker2013]. This procedure suppresses phase shifts proportional to horizontal velocity by a factor of $1000$. For the EP test, we imprint a horizontal phase that is proportional to the detected position (a combination of initial position and initial velocity). The use of a detection fringe for phase readout avoids systematic errors from initial horizontal displacements between the isotopes.
![Compensation method for kinematic DoFs. Changing the angle (frequency) of pulse zone 1 creates a horizontal (vertical) position-dependent phase. Changing the angle (frequency) of pulse zone 1 by $-\theta$ ($-\Delta f$) and pulse zone 3 by $\theta$ ($\Delta f$) creates a horizontal (vertical) velocity-dependent phase. The angle and frequency steps are calibrated to eliminate phase sensitivity to kinematic DoFs.[]{data-label="fig:fringes"}](fringesv7.pdf){width="\linewidth"}
These compensation techniques allow us to suppress all phase shifts that arise from linear horizontal or vertical phase gradients. However, the atom clouds also have a finite width that can couple to higher-order horizontal wavefront perturbations. To bound this effect, we correlate the cloud width with the differential phase and measure the phase difference between the middle and the edge of each cloud.
To extract the relative acceleration between $^{85}$Rb and $^{87}$Rb, we compare the differential phase of interferometers with beamsplitter momentum $n \hbar k$, where $n \in \{4, 8, 12\}$. The relative acceleration is given by the linear dependence of the differential phase on $n$. We vary $n$ by adding additional $2\hbar k$ pulses to each pulse zone. This approach eliminates systematic errors that arise from the finite duration and detuning of the initial and final $\pi/2$ beamsplitter pulses [@Antoine2006; @Borde1999], which are the same for all $n$.
![EP data (1481 shots). Left: time series of the interferometer phase difference $\phi_{85}-\phi_{87}$, color-coded by interferometer order ($4\hbar k$ red, $8\hbar k$ gray, $12\hbar k$ blue). Hollow (solid) points represent measurements with initial beamsplitter direction down (up). Right: histogram of the phase difference for each interferometer order.[]{data-label="fig:data"}](epdatafigunblindedv3.pdf){width="\linewidth"}
The direction of the initial and final beamsplitters determines the momenta of the interferometer output ports (either $\{2\hbar k, 0\hbar k\}$ or $\{0\hbar k, -2\hbar k\}$ with respect to the launched clouds). We switch the beamsplitter direction to identify phase shifts that scale with $k^2$. Such phase shifts arise due to parasitic recoil interferometers [@Altin2013] that are caused by imperfect transfer efficiency. The phase of a recoil-sensitive interferometer scales as $\hbar k^2 T/m$, and the dependence on the mass $m$ creates a systematic error in the EP measurement. When the beamsplitter direction is reversed, the recoil phase shifts change sign relative to the acceleration-induced phase shift, allowing the two effects to be distinguished.
Electromagnetic interactions cause significant systematic errors in two ways. First, a differential acceleration arises from off-resonant forces (AC-Stark shifts) induced by the interferometry lasers. This effect is reduced by using an optical spectrum that suppresses the shifts of the $^{87}$Rb $F = 2$ and $^{85}$Rb $F = 3$ states. To lowest order in the wavepacket separation, the differential Stark shift $\Delta \phi_S$ in our interferometer geometry is given by $\Delta \phi_S = \left(f_{85}/m_{85} -f_{87}/m_{87} \right) 2 \pi (n - 1) \beta \left(n \hbar k T \right)$ where $\beta$ is the fractional intensity gradient at the height of the middle pulse zone and $f_i$ is a factor that characterizes the Stark shift suppression of isotope $i$. The interferometry beams are retroreflected at an angle of $0.5\, \text{mrad}$ to avoid etaloning, which creates a fractional intensity gradient $\beta = 1 \times 10^{-3}/\text{(5 cm)}$.
Our optical spectrum is designed to achieve $f_i \sim 10^{-2}$ for both isotopes simultaneously. The $780\; \text{nm}$ spectrum of each interferometry beam is created by frequency doubling a $1560\, \text{nm}$ laser that is phase modulated at $30 \,\text{GHz}$. After doubling, the spectrum consists of two strong sidebands and a highly suppressed carrier with relative intensity $\sim 10^{-3}$. The carrier frequency is positioned between the $^{87}$Rb $F=2 \longrightarrow F'$ transitions and the $^{85}$Rb $F=3 \longrightarrow F'$ transitions, which are separated by $1\, \text{GHz}$. The blue-detuned sidebands are used to drive Bragg transitions, and the red-detuned sidebands compensate the optical forces from the blue-detuned sidebands [@Kovachy2015a]. To bound the magnitude of the residual AC-Stark effect, we add additional off-resonant pulses to an $8\hbar k$ interferometer. We observe no statistically significant differential phase shift with these additional pulses, which implies that residual AC-Stark effect induces a differential acceleration of below $2.7 \times 10^{-12}\, g$.
Second, the two isotopes are differentially accelerated by magnetic forces. This effect is reduced by creating a nearly uniform magnetic field in the interferometry region via a solenoid coil and three layers of magnetic shielding [@Dickerson2012]. The atoms enter the shielded region in the magnetically sensitive states ${\ensuremath{\left|F=3, m_F=3\right>}}$ ($^{85}$Rb) and ${\ensuremath{\left|F=2, m_F=2\right>}}$ ($^{87}$Rb). A series of microwave pulses transfers the atoms to the magnetically insensitive states ${\ensuremath{\left|F=3, m_F=0\right>}}$ ($^{85}$Rb) and ${\ensuremath{\left|F=2, m_F=0\right>}}$ ($^{87}$Rb). Nevertheless, the second-order Zeeman effect causes a phase shift $\phi_i = -2(\hbar/m_i) \alpha_i B (\partial_z B) k T^2$ of each isotope, where $\alpha_i$ is the second-order Zeeman coefficient of isotope $i$, $B$ is the magnetic field magnitude, $\partial_z B$ is the vertical magnetic field gradient, and $m_i$ is the mass of isotope $i$. To measure $\partial_z B$, we compare the phase of a $^{87}$Rb interferometer in state ${\ensuremath{\left|F=2, m_F=1\right>}}$ with the phase of an $^{87}$Rb interferometer in state ${\ensuremath{\left|F=2, m_F=0\right>}}$. At $B = 41\, \text{mG}$, the ${\ensuremath{\left|F=2, m_F=1\right>}}$ interferometer has an increased sensitivity to magnetic field gradients by five orders of magnitude. The magnetic field gradient in the interferometry region averages to $(-0.41 \pm 0.036)\, \text{mG/m}$, which corresponds to a differential acceleration of $(5.9 \pm 0.5) \times 10^{-12}\, g$ in the EP measurement.
The phase of each interferometer is encoded in the spatial position of the imaged detection fringe. Therefore, imaging differences between the isotopes can give rise to systematic errors. To image both species in a single CCD frame, the fluorescence light for one species is delayed by $1\,\text{ms}$, during which the two isotopes drift apart by $1\, \text{cm}$. The CCD axis is misaligned with respect to the drift direction by $4\,\text{mrad}$, which causes a differential phase shift of $40\,\text{mrad}$. To eliminate this phase shift, we rotate the camera images in software. We also switch the imaging order and reverse the direction in which the detection fringe is imprinted, each of which changes the sign of imaging-related phase shifts relative to the EP signal. The imaging order and detection fringe direction can each be reversed with fidelity $> 0.99$. Together, these reversals ensure that imaging effects do not contribute significantly to the systematic error.
Parameter Shift Uncertainty
--------------------- ------- -------------
Total kinematic 1.5 2.0
$\Delta z$ 1.0
$\Delta v_z$ 1.5 0.7
$\Delta x$ 0.04
$\Delta v_x$ 0.04
$\Delta y$ 0.2
$\Delta v_y$ 0.2
Width 1.6
AC-Stark shift 2.7
Magnetic gradient -5.9 0.5
Pulse timing 0.04
Blackbody radiation 0.01
Total systematic -4.4 3.4
Statistical 1.8
: Error budget in units of $10^{-12}\,g$. The parameter $\Delta z$ ($\Delta v_z$) includes all errors that are linearly proportional to the initial vertical position (velocity) difference between the two isotopes. Likewise, $\Delta x$ ($\Delta v_x$) includes all errors proportional to the initial position (velocity) difference in the detection fringe direction, and $\Delta y$ ($\Delta v_y$) includes all errors proportional to the initial position (velocity) difference in the orthogonal horizontal direction. See main text and supplement for descriptions of other systematic errors. All uncertainties are $1\sigma$.[]{data-label="table"}
We have tested the equivalence principle between $^{85}$Rb and $^{87}$Rb at the level of $10^{-12}\, g$. The result is consistent with $\eta = 0$, which places generic constraints on new interactions that would differentially accelerate the two isotopes. The systematic uncertainty is primarily limited by the AC-Stark shift, which can be reduced by using a laser system with larger single-photon detuning and higher power. Such a system could also allow the momentum transfer in each pulse zone to be increased, improving the single-shot sensitivity and reducing the time required to characterize systematic errors. The shift due to the magnetic gradient can be reduced by tailoring the current through each segment of the solenoid within the magnetic shield. By demonstrating that the high sensitivity of large-area atom interferometers can be utilized in precision measurement applications, this work provides a proof of concept for future AI EP tests [@Aguilera2014] and gravitational wave detectors [@Canuel2018; @Coleman2018].
We acknowledge funding from the Defense Threat Reduction Agency, the Office of Naval Research, and the Vannevar Bush Faculty Fellowship program. We thank Robin Corgier, Salvador Gomez, Jason Hogan, Tim Kovachy, Remy Notermans, and Stefan Seckmeyer for their assistance with this work.
[^1]: These authors contributed equally to this work.
[^2]: These authors contributed equally to this work.
|
---
abstract: 'In this paper, two distributed multi-proximal primal-dual algorithms are proposed to deal with a class of distributed nonsmooth resource allocation problems. In these problems, the global cost function is the summation of local convex and nonsmooth cost functions, each of which consists of one twice differentiable function and multiple nonsmooth functions. Communication graphs of underling multi-agent systems are directed and strongly connected but not necessarily weighted-balanced. The multi-proximal splitting is designed to deal with the difficulty caused by the unproximable property of the summation of those nonsmooth functions. Moreover, it can also guarantee the smoothness of proposed algorithms. Auxiliary variables in the multi-proximal splitting are introduced to estimate subgradients of nonsmooth functions. Theoretically, the convergence analysis is conducted by employing Lyapunov stability theory and integral input-to-state stability (iISS) theory with respect to set. It shows that proposed algorithms can make states converge to the optimal point that satisfies resource allocation conditions.'
address:
- 'Peng Cheng Laboratory, China'
- 'The Key Laboratory of Intelligent Control and Decision of Complex Systems, Beijing Institute of Technology, China'
- 'Graduate School at Shenzhen, Tsinghua University, China'
- 'Center of Applied Optimization (CAO), Industrial and Systems Engineering, University of Florida, USA'
author:
- Yue Wei
- Chengsi Shang
- Hao Fang
- Xianlin Zeng
- Lihua Dou
- Panos Pardalos
title: 'Solving A Class of Nonsmooth Resource Allocation Problems with Directed Graphs though Distributed Smooth Multi-Proximal Algorithms'
---
,
,
,
,
,
Distributed resource allocation, nonsmooth cost function, directed graph, splitting method.
Introduction
============
In this paper, we consider a class of distributed nonsmooth convex resource allocation problems with directed graphs. A wide range of problems in the field of coordination of multi-agent systems [@ZY2017]-[@JC2018], economic dispatch of power systems [@Cortes2016] and machine learning belong to this class of problems. As examples, in distributed constrained coordination of multi-agent systems with directed graphs, the local cost function of agent $i$ usually consists of a smooth function and multiple nonsmooth functions standing for different constraints and tasks. Moreover, multi-agent systems are required to maintain some configurations described by resource allocation conditions. When considering a classical machine learning problem - the fused LASSO problem [@Lasso] - with constraints and directed graphs, the least squares loss is smooth. The $l_{1}$ penalty and indicator functions of local constraints in this problem are usually nonsmooth. Then resource allocation conditions are employed here as global constraints. As common features, each global cost function in these problems is summed up by local cost functions, and each local cost function consists of a smooth convex function and multiple nonsmooth convex functions. Even though nonsmooth functions are proximable, their summation might not be, where a function being proximable means that the proximal operator of this function has a closed or semi-closed form solution and is computationally easy to evaluate [@CTOS3]. Besides, connected graphs of these problems are directed and maybe weight-unbalanced, where a directed graph being weight-unbalanced means that the in-degree and out-degree of some nodes in this graph are unequal. The difficulty of these problems is to tackle nonsmooth cost functions and directed connecting graphs simultaneously. Due to important applications and challenges mentioned above, these problems have attracted increasing attentions.
Literature review {#literature-review .unnumbered}
-----------------
Communication between agents in multi-agent systems has attracted much attention due to the importance of information exchange. Recently, continuous-time distributed algorithms for resource allocation problems have been widely investigated with different kinds of connected graphs [@YP2016]-[@YWW]. For undirected graphs, [@YP2016] designed an initialization-free distributed algorithm for distributed resource allocation problems. [@YH2019] proposed a new distributed private-guaranteed algorithm to solve economic dispatch problems with undirected graphs. As to directed graphs, [@SP] proposed a continuous-time algorithm via singular perturbation for distributed resource allocation problems. While [@SP] did not consider local constraints. In [@SP1], a distributed projection-based algorithm was designed to deal with distributed resource allocation problems with weight-balanced graphs. [@DZ2019] investigated constrained nonsmooth resource allocation problems via a distributed algorithm, which can solve resource allocation problems with strongly convex cost functions and weight-balanced digraphs, as well as resource allocation problems with strictly convex cost functions and connected undirected graphs. For distributed resource allocation problems with weight-unbalanced graphs, [@YWW] proposed a distributed adaptive algorithm to achieve the optimal solution. While this algorithm fails to solve resource allocation problems with local constraints and weight-unbalanced graphs simultaneously.
Nonsmoothness is a natural property of many resource allocation problems in real-world science and engineering areas. Two important categories of existing algorithms for solving distributed nonsmooth optimization and resource allocation problems are shown here. The first category is subgradient-based algorithms proposed in [@DCO1]-[@DCO5], whose convergence was proven based on nonsmooth analysis [@NCO]. [@Xie1] designed a distributed continuous-time projected algorithm to deal with distributed constrained nonsmooth optimization problems. [@DOC1] investigated the distributed nonsmooth constrained optimization problem with distributed projection-based saddle-point subgradient algorithms. While the discontinuous subgradient of cost function is directly employed in aforementioned algorithms, which may cause vibrations of systems. The second category includes distributed smooth algorithms [@DCFO1]-[@OFW] which employed splitting method [@PM]. Most existing works of distributed smooth algorithms only consider one or two proximal operators [@CTOS3; @OFW] in their algorithms. They can not directly solve the nonsmooth resource allocation optimization problem where multiple nonsmooth functions are contained in each local cost function, since summation of multiple proximable nonsmooth functions may not be proximable. [@Dhingra] designed a proximal augmented Lagrangian and achieved continuous-time primal-dual dynamics to solve nonsmooth optimization problems. However, more extension works are needed to solve distributed nonsmooth resource allocation optimization problems with multiple nonsmooth functions and directed graphs.
Contribution {#contribution .unnumbered}
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In this paper, two smooth primal-dual algorithms are proposed for a class of distributed nonsmooth convex resource allocation problems with directed graphs. A distributed estimator of the left eigenvector associated with zero eigenvalue of Laplacian matrix of the directed graph is considered in the second algorithm. The global cost function in these problems is a summation of local cost functions, and each of them consists of a smooth convex function and multiple nonsmooth convex functions. Although each nonsmooth function is proximable, their summation might not be. Contributions of this paper are summarized as follows.
**(i)** This paper explores a class of nonsmooth resource allocation problems with directed graphs. Compared with [@YP2016]-[@SP3], this paper considers resource allocation problems with weight-unbalanced graphs. In contract to [@YWW], smooth algorithms are designed for nonsmooth resource allocation problems with local constraints.
**(ii)** Distributed smooth primal-dual algorithms employing multi-proximal splitting are proposed in this paper. The multi-proximal splitting is used to deal with the unproximable property of the summation of nonsmooth functions and ensure smoothness of proposed algorithms.
**(iii)** A Lyapunov function and an iISS-Lyapunov function with respect to the set of equilibria are designed. Then the convergence and correctness of proposed algorithms are proved by using Lyapunov stability theory and iISS theory, which provides novel insights into analysis of the asymptotically convergent system with inputs by employing iISS theory with respect to set.
Organization {#organization .unnumbered}
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The rest of this paper is organized as follows. In Section II, some basic definitions of graph theory, proximal operator and iISS theory are presented. Section III shows the nonsmooth resource allocation problem with directed graph. In Section IV, we propose two distributed multi-proximal splitting based smooth continuous-time primal-dual algorithms with and without left eigenvector estimator, respectively. Then proofs for the convergence and correctness of these algorithms are also presented. In Section V, simulations show the effectiveness of our proposed algorithm. Finally, Section VI concludes this paper.
Mathematical Preliminaries
==========================
In this section, we introduce necessary notations, definitions and preliminaries about graph theory, proximal operator and integral input-to-state stability (iISS).
Graph Theory
------------
A weighted directed graph $\mathcal{G}$ is denoted by $\mathcal{G(V,E,A)}$, where $\mathcal{V} = \lbrace 1, \dots, n \rbrace$ is a set of nodes, $\mathcal{E}$ is a set of edges, and $\mathcal{A} = [a_{ij}] \in \mathbb{R}^{n \times n}$ is a weighted adjacency matrix. An edge $e_{ij} \in \mathcal{E}$ indicates that agent $i$ can receive information from agent $j$. If $e_{ij} \in \mathcal{E}$, then $a_{ij} > 0$; otherwise, $a_{ij} = 0$. Moreover, $a_{ii} = 0, i \in \mathcal{I}$. Agent $j \in \mathcal{N}_{i}$ denotes agent $j$ is a neighbour of agent $i$. The in-degree and out-degree of agent $i$ are $d^{in}_{i} = \sum_{j=1}^{n} a_{ij}$ and $d^{out}_{i} = \sum_{j=1}^{n} a_{ji}$, respectively. The Laplacian matrix is $L_{n} = D^{in} - \mathcal{A}$, where $D^{in} \in \mathbb{R}^{n \times n}$ is diagonal with $D^{in}_{ii} = \sum^{n}_{j=1} a_{ij}$, $i \in \lbrace 1, \dots, n \rbrace$. We use $\Vert \cdot \Vert $ to indicate Euclidean norm. Let $\mathbb{R}$ denote the set of real numbers. $\mathbb{R}^{+}$ denotes the set of positive real numbers. $diag \lbrace b_{1}, \cdots, b_{n} \rbrace \in \mathbb{R}^{n \times n}$ is denoted as the diagonal matrix, whose $i$-th diagonal element is $b_{i} \in \mathbb{R}$ for $i \in \lbrace 1, \cdots, n \rbrace$. $I_{n}$ is the $n$-dimensional identity matrix. Let $\textbf{0}_{n} \in \mathbb{R}^{n}$ denote the vector of all zeros. $O_{n}$ is the $n$-dimensional null matrix, which means that every element in $O_{n}$ is zero. $(\cdot)^{T}$ denotes transpose of matrix.
([@LH+HL]) Assume that graph $\mathcal{G}$ is strongly connected with the Laplacian matrix $L_{n}$. Then:
$(1)$ There is a positive left eigenvector $h = (h_{1}, h_{2}$, $\cdots,h_{n})^{T}$ associated with the zero eigenvalue such that $h^{T}L = \textbf{0}^{T}_{n}$ and $\sum_{i = 1}^{n} h_{i} = 1$.
$(2)$ $\min_{\textbf{1}^{T}_{n}x=0} x^{T}\textbf{L} x \geq \lambda_{2}(\textbf{L}) \Vert x \Vert^{2}$, where $\textbf{L} = (HL + L^{T}H)/2$ with $H = diag(h_{1}, h_{2}, \cdots , h_{n})$ and $\lambda_{2}(\textbf{L})$ being its second smallest eigenvalue. \[L2.1\]
Proximal Operator
-----------------
Let $f(\delta)$ be a lower semi-continuous convex function for $\delta \in \mathbb{R}^{r}$. Then the proximal operator $prox_{f}[\theta]$ of $f(\delta)$ at $\theta \in \mathbb{R}^{r}$ is $$prox_{f}[\theta] = \arg \min_{\delta} \lbrace f(\delta) + \frac{1}{2} \Vert \delta - \theta \Vert^{2} \rbrace.$$
Let $ \partial f(\delta)$ denote the subdifferential of $f(\delta)$. If $f(\delta)$ is convex, then $\partial f(\delta)$ is monotone, that is, $(\zeta_{\delta_{1}} - \zeta_{\delta_{2}})^{T}(\delta_{1} - \delta_{2}) \geq 0$ for all $\delta_{1} \in \mathbb{R}^{r}, \delta_{2} \in \mathbb{R}^{r}$, $\zeta_{\delta_{1}} \in \partial f(\delta_{1})$, and $\zeta_{\delta_{2}} \in \partial f(\delta_{2})$. $\delta =
prox_{f} [\theta]$ is equivalent to $$\theta - \delta \in \partial f(\delta).
\label{Proximal Property}$$
Integral Input-to-State Stability with respect to set
-----------------------------------------------------
Consider the system $$\label{system}
\dot{x} = f(x,u), x(0)=x_{0}, t \geq 0,$$ where $x \in \mathbb{R}^{n}$ and $u \in \mathbb{R}^{m}$. Inputs are measurable and locally essentially bounded functions $u: \mathbb{R}_{\geq 0} \to \mathbb{R}^{m}$, and $f: \mathbb{R}^{n}\times\mathbb{R}^{m} \to \mathbb{R}^{n}$ is assumed to be locally Lipschitz continuous. Equilibria of system consist a closed set $\mathcal{M}$. For each $\xi \in \mathbb{R}^{n}$, the point-to-set distance from $\xi$ to $\mathcal{M}$ is denoted by $$\begin{gathered}
\Vert \xi \Vert_{\mathcal{M}} \triangleq d(\xi,\mathcal{M})=\inf\lbrace \Vert \xi -\psi \Vert, \psi \in \mathcal{M} \rbrace\end{gathered}$$ In particular, $\Vert \xi \Vert_{\lbrace 0 \rbrace} = \Vert \xi \Vert$. Let $\mathcal{K}$ denote the class of functions $a(x): [0,\infty) \to [0,\infty)$ which are strictly increasing, continuous and $a(0)=0$; $\mathcal{K}_{\infty}$ denotes the class of functions $a(x): [0,\infty) \to [0,\infty)$ which are a subset of $\mathcal{K}$ functions that $\lim_{x\to\infty} a(x)$ $\to \infty$; $\mathcal{L}$ is the set of functions $a(x):[0,+\infty) \to [0,+\infty)$ which are continuous, decreasing and $\lim_{x\to+\infty}a(x)=0$; $\mathcal{KL}$ is the class of functions $a(x,y):[0,\infty)^{2} \to [0,\infty)$ where $a(x,y)$ belongs to class $\mathcal{K}$ with respect to $x:[0,\infty)$ and to class $\mathcal{L}$ with respect to $y:[0,\infty)$ [@K]. A positive definite function $a(x):[0,\infty) \to [0,\infty)$ is one that $a(0)=0$ and $a(x)>0$ when $x>0$. A function $V(x) \in \mathbb{R}$ is semiproper if and only if for each $r$ in the range of $V(x)$, the sublevel set $\lbrace x \vert V(x) \leq r \rbrace$ is compact. A positive definite function with respect to $\mathcal{M}$ is one that is zero at $\mathcal{M}$ and positive otherwise [@iISS; @ISS_Lin]. A nonempty set $\mathcal{M}$ is 0-invariant for system if the solution starting from $\mathcal{M}$ is defined for all $t \geq 0$ and stays in $\mathcal{M}$ when $u \equiv \textbf{0}_{m}$. System is said to be forward complete if the solution $x(t,x_{0},u)$ is defined for all $t>0$ [@In_Lya].
Define $DV(x) = [\frac{\partial V(x)}{\partial x}]^{T}$. Then definitions of integral input-to-state stability (iISS) and iISS-Lyapunov function with respect to a closed and 0-invariant set $\mathcal{M}$ are given below.
\[D1\] System is **Integral Input-to-State Stability (iISS) with respect to a closed and 0-invariant set $\mathcal{M}$** if system is forward complete and there exist functions $a_{1} \in \mathcal{K}_{\infty}$, $a_{2} \in \mathcal{KL}$ and $a_{3} \in \mathcal{K}$, such that $$a_{1}(\Vert x(t,x_{0},u) \Vert_{\mathcal{M}}) \!\leq\! a_{2}(\Vert x_{0} \Vert_{\mathcal{M}}, t) + \!\!\!\int^{t}_{0}\!\!\!\! a_{3}(\Vert u(s) \Vert)ds.\label{IISS}$$
\[D3\] A continuously differentiable function $V$ is called an **iISS-Lyapunov function with respect to a closed and 0-invariant set $\mathcal{M}$** for system if system is forward complete and there exist functions $a_{4}, a_{5} \in \mathcal{K}_{\infty}$ and a continuous positive definite function $a_{6}$, and $a_{7} \in \mathcal{K}$ such that $$a_{4}(\Vert x \Vert_{\mathcal{M}}) \leq V(x) \leq a_{5}(\Vert x \Vert_{\mathcal{M}}),$$ and $$DV(x)f(x,u) \leq - a_{6}(\Vert x \Vert_{\mathcal{M}}) + a_{7}(\Vert u \Vert)$$ for all $x \in \mathbb{R}^{n}$ and all $u \in \mathbb{R}^{m}$.
Note that $V$ in Definition \[D3\] is positive definite and proper (i.e., radially unbounded) with respect to $\mathcal{M}$. If the 0-input system $\dot{x} = f(x,\textbf{0}_{m}) \label{0system}$ is **globally asymptotically stable** (GAS) with respect to $\mathcal{M}$, the system is to be said 0-GAS with respect to $\mathcal{M}$.
Similar to definitions of dissipation and zero-output dissipation in [@iISS2], here we introduce concepts of dissipation and zero-output dissipation with respect to $\mathcal{M}$.
\[D2\] The system with output p$: \mathbb{R}^{n} \to \mathbb{R}^{r}$ is **dissipative with respect to a closed and 0-invariant set $\mathcal{M}$** if system is forward complete and there exists a continuously differentiable, proper, and positive definite function $V$ with respect to $\mathcal{M}$, together with a continuous positive definite function $a_{8}$ and a function $a_{9} \in \mathcal{K}$, such that $$DV(x) f(x,u) \leq - a_{8}(\Vert p(x) \Vert) + a_{9}(\Vert u \Vert)\label{Dissipative}$$ for all $x \in \mathbb{R}^{n}$ and all $u \in \mathbb{R}^{m}$. Moreover, if holds with $p = \textbf{0}_{r}$, i.e., if there exist a proper and positive definite $V$ with respect to $\mathcal{M}$, and an $a_{9} \in \mathcal{K}$, such that $$DV(x) f(x,u) \leq a_{9}(\Vert u \Vert)\label{Dissipative0}$$ holds for all $x \in \mathbb{R}^{n}$ and all $u \in \mathbb{R}^{m}$, we say that the system is **zero-output dissipative (ZOD) with respect to $\mathcal{M}$**.
Consider a system $$\dot{x}(t) = J(x(t)), x(0) = x_{0}, t \geq 0 \label{syt}$$ where $J: \mathbb{R}^{n} \to \mathbb{R}^{n}$ is Lipschitz continuous. The following result is a special case of Theorem 3.1 in [@OE].
\[Aequilibrium\] Let $\mathcal{D}$ be a compact, positive invariant set with respect to system , $V: \mathbb{R}^{n} \to \mathbb{R}$ be a continuously differentiable function, and $x(\cdot) \in \mathbb{R}^{q}$ be a solution of with $x(0) = x_{0} \in \mathcal{D}$. Assume $\dot{V}(x) \leq 0$, $\forall x \in \mathcal{D}$, and define $\mathcal{Z} = \lbrace x \in \mathcal{D}: \dot{V}(x) = 0 \rbrace$. If every point in the largest invariant subset $\mathcal{M}$ of $\bar{\mathcal{Z}}\cap\mathcal{D}$ is Lyapunov stable, where $\bar{\mathcal{Z}}$ is the closure of $\mathcal{Z} \subset \mathbb{R}^{n}$, then system converges to one of its equilibria.
Problem Description
===================
In this section, the resource allocation problem with a directed graph is formulated. We consider a network of $n$ agents with first-order dynamics, interacting over a graph $\mathcal{G}$. The nonsmooth resource allocation problem is given as $$\begin{gathered}
\min_{x \in \mathbb{R}^{nq}} F(x), \ \ s.t. \sum_{i=1}^{n} x_{i} = \sum_{i=1}^{n} d_{i}, \label{Problem 2}\end{gathered}$$ where $F(x) = \sum_{j=0}^{m} F^{j}(x) = \sum_{i=1}^{n} f_{i}(x_{i})$, $f_{i}(x_{i}) = \sum_{j=0}^{m}f^{j}_{i}(x_{i})$, and $F^{j}(x) = \sum_{i=1}^{n} f^{j}_{i}(x_{i})$, $j \in \lbrace 0, 1, \cdots$,$m \rbrace$, $m \geq 2$. Note that $x_{i} \in \mathbb{R}^{q}$ is the state of $i$-th agent and $x = [ x_{1}^{T}, x_{2}^{T}, \cdots, x_{n}^{T} ]^{T} \in \mathbb{R}^{nq}$.
For each agent $i \in \lbrace 1, \cdots, n \rbrace$, there are $m+1$ function $f_{i}^{0}, \cdots, f_{i}^{m} : \mathbb{R}^{q} \to \mathbb{R}$, contained in the local cost function $f_{i}(x_{i}): \mathbb{R}^{q} \to \mathbb{R}$, where $f^{0}_{i}$ is a smooth convex function, $f^{j}_{i}$ is a nonsmooth convex function for $j \in \lbrace 1,\cdots,m\rbrace$. Each agent $i$ only has the information about $f_{i}^{j}$ for $j \in \lbrace 0, 1, \cdots, m \rbrace$. The constraint presented in indicates that all solutions must achieve resource allocation conditions $\sum_{i=1}^{n} x_{i} = \sum_{i=1}^{n} d_{i}$. Each agent only exchanges information with its neighbours in a fully distributed manner.
Assumptions below are made for the wellposedness of the problem in this section.
$f^{0}_{i}$ is twice continuously differentiable and strongly convex for all $i \in \lbrace 1, \cdots, n \rbrace$, which means that there exists a constant $c > 0$ such that for agent $i$, $$(\nabla f^{0}_{i}(\vartheta_{1}) - \nabla f^{0}_{i}(\vartheta_{2}))^{T}(\vartheta_{1} - \vartheta_{2}) \geq c \Vert \vartheta_{1} - \vartheta_{2} \Vert^{2},
\label{Strongly Convex}$$ where $\vartheta_{1} \in \mathbb{R}^{q}$, $\vartheta_{2} \in \mathbb{R}^{q}$, $\vartheta_{1} \neq \vartheta_{2}$. Without loss of generality, we assume $c > m-1$. \[A1\]
Each $f^{j}_{i}$ is (nonsmooth) lower semi-continuous closed proper convex functions for all $i \in \lbrace 1, \cdots, n \rbrace$, $j \in \lbrace 1, \cdots, m \rbrace$, and it is proximable. \[A2\]
The weighted graph $\mathcal{G}$ is directed and strongly connected. \[A3\]
There exists at least one feasible point to problem . \[A4\]
The condition $c>m-1$ in Assumption \[A1\] is mild. When $0 < c \leq m-1$, there always exists a function $f^{0'}_{i}(x) = K f^{0}_{i}(x)$ for agent $i$ with $ K >\frac{m - 1}{c}$ such that $ (\nabla \! f^{0'}_{i}(\vartheta_{1}) - \nabla \! f^{0'}_{i}(\vartheta_{2}))^{T}\!(\vartheta_{1} - \vartheta_{2})\! \geq Kc \Vert \vartheta_{1} - \vartheta_{2} \Vert^{2} \! > \! (m - 1) \Vert \vartheta_{1} - \vartheta_{2} \Vert^{2}$. $\hfill$ $\blacklozenge$
Then, we arrive at the following lemma by the Karush-Kuhn-Tucker (KKT) condition of convex optimization problems.
Under Assumptions $\ref{A1}$-$\ref{A4}$, a feasible point $x^{*} \in \mathbb{R}^{nq}$ is a solution of problem if and only if there exist $x^{*} \in \mathbb{R}^{nq}$, a constant $v_{0} \in \mathbb{R}^{q}$, and $v^{*} \in \mathbb{R}^{nq}$ such that
$$\begin{aligned}
& \textbf{0}_{nq} \in \nabla F^{0}(x^{*}) + \sum_{j=1}^{m} \partial F^{j}(x^{*}) - v^{*}, \label{KKT1}\\
& \sum_{i=1}^{n} x_{i}^{*} = \sum_{i=1}^{n} d_{i}, v_{i}^{*} = v_{0} \text{ for } i \in \lbrace 1,\cdots,n \rbrace, \label{KKT2}\end{aligned}$$
\[KKT\]
where $v = [ v_{1}^{T}, v_{2}^{T}, \cdots, v_{n}^{T} ]^{T}$ is the Lagrange multiplier, $\nabla F^{0}(x) = [(\nabla f^{0}_{1}(x_{1}))^{T}\!\!$, $(\nabla f^{0}_{2}(x_{2}))^{T} \!\!, \cdots$, $(\nabla f^{0}_{n}(x_{n}))^{T} ]^{T}\!\!$, and $\partial F^{j}(x) \!=\! [(\partial f^{j}_{1}(x_{1}))^{T}\!$, $(\partial f^{j}_{2}(x_{2}))^{T}, \cdots$, $(\partial f^{j}_{n}(x_{n}))^{T} ]^{T}$ for $j \in \lbrace 1, \cdots, m \rbrace$. \[LKKT\]
The proof of Lemma \[LKKT\] is omitted since it is a trivial extension of the proof for Theorem 3.25 in [@NO].
Distributed Algorithms with Multi-Proximal Operator
===================================================
The purpose of this section is to design two continuous-time distributed algorithms based on multi-proximal splitting to solve the nonsmooth resource allocation problem for two cases that with known left eigenvector $h$ and with a distributed estimator of left eigenvector $h$, respectively.
In order to tackle the difficulty caused by the unproximable property of $\sum_{j=1}^{m}f^{j}_{i}(x_{i})$ for each agent $i$, here we introduce a class of auxiliary variables $z^{j}(t) \in \mathbb{R}^{nq}$ for $j \in \lbrace 1, \cdots, m-1 \rbrace$ combined with a constant parameter $\gamma \in \mathbb{R}^{+}$ such that there exist feasible points $z^{j*}$ splitting (\[KKT1\]) as
$$\begin{aligned}
- \nabla F^{0}(x^{*}) + v^{*} + \gamma \sum_{j=1}^{m-1} z^{j*} \in \partial F^{m}(x^{*}), \label{z1}\\
- \gamma z^{j*} \in \partial F^{j}(x^{*}), j \in \lbrace 1,\cdots,m-1 \rbrace . \label{z2}\end{aligned}$$
\[z\]
According to the property of proximal operator, we can transfer as $$\begin{gathered}
\begin{split}
& x^{*} = Prox_{F^{m}}[x^{*} - \nabla F^{0}(x^{*}) + v^{*} + \gamma \sum_{j=1}^{m-1} z^{j*}], \\
& x^{*} = Prox_{F^{j}}[x^{*} - \gamma z^{j*}], j \in \lbrace 1,\cdots,m-1 \rbrace, \label{ABC}
\end{split}\end{gathered}$$ where for any $\xi = [\xi_{1}^{T}, \xi_{2}^{T}, \cdots, \xi_{n}^{T}]^{T} \in \mathbb{R}^{nq}$, $\xi_{i} \in \mathbb{R}^{q}$, $i \in \lbrace 1, \cdots, n \rbrace$, $Prox_{F^{j}}[\xi] = [(prox_{f^{j}_{1}}[\xi_{1}])^{T}, (prox_{f^{j}_{2}}[\xi_{2}])^{T}$, $\cdots, (prox_{f^{j}_{n}}[\xi_{n}])^{T}]^{T}$. $x^{*}$ and $v^{*}$ are defined in . From , it is clear that $-\gamma z^{j*}$ is presented to estimate a subgradient in $\partial F^{j}(x^{*})$ for $j \in \lbrace 1, \cdots, m-1 \rbrace$.
Algorithm Design with Known Left Eigenvector h
----------------------------------------------
In this subsection, we present a distributed smooth multi-proximal primal-dual algorithm for solving problem ($\ref{Problem 2}$) with the information of left eigenvector $h$.
According to and , we propose a smooth algorithm as $$\begin{aligned}
\dot{x}_{i}(t) & = & prox_{f^{m}_{i}} \!\Big[\! x_{i}(t) \!-\! \nabla \!f^{0}_{i}\!(x_{i}(t)) \!+\! v_{i}(t) \!+\! \gamma\! \sum_{j=1}^{m-1}\!\! z_{i}^{j}(t) \Big] \!\!-\! x_{i}(t), \notag \\
\dot{z}_{i}^{j}(t) & = & prox_{f^{j}_{i}}[x_{i}(t) - \gamma z_{i}^{j}(t)] - x_{i}(t), \notag \\
\dot{v}_{i}(t) & = & - h_{i}^{-1}(x_{i}(t) - d_{i}) \!-\! \alpha \sum_{k \in \mathcal{N}_{i}} a_{ik} (v_{i}(t) - v_{k}(t))\! -\! w_{i}(t), \notag \\
\dot{w}_{i}(t) & = & \alpha \sum_{k \in \mathcal{N}_{i}} a_{ik} (v_{i}(t) - v_{k}(t)), \quad w_{i}(0)=\textbf{0}_{q},
\label{Algorithm 11}\end{aligned}$$ where $t \geq 0$, $0 < \gamma < \frac{1}{m-1}$, $i \in \lbrace 1, \cdots, n \rbrace$, and $j \in \lbrace 1,\cdots,m-1 \rbrace$.
Because all proximal operators $prox_{f^{j}_{i}}(\cdot)$ for $i \in \lbrace 1, \cdots, n \rbrace$ and $ j \in \lbrace 1, \cdots, m-1 \rbrace$ are continuous and nonexpansive, the proposed algorithm is locally Lipschitz continuous even though each $f^{j}_{i}(x_{i})$ in problem is nonsmooth, which means that the **smoothness** of algorithm is guaranteed. $\hfill$ $\blacklozenge$
Algorithm can be written in a compact form as
$$\begin{aligned}
\dot{x}(t) = & Prox_{F^{m}} \Big[ x(t) \!\!-\!\! \nabla F^{0}(x(t)) \!\!+\!\! v(t) \notag \\
& +\!\! \gamma \sum_{j = 1}^{m-1} z^{j}(t) \Big] \!\!-\!\! x(t), \label{another proximal} \\
\dot{z}^{j}(t) = & Prox_{F^{j}}[x(t) - \gamma z^{j}(t)] - x(t), \label{estimator} \\
\dot{v}(t) = & - H^{-1}_{nq}(x(t) - d) - \alpha L_{nq} v(t) - w(t), \\
\dot{w}(t) = & \alpha L_{nq} v(t), \quad w(0)=\textbf{0}_{nq}, \label{lagrangian}\end{aligned}$$
\[Algorithm 21\]
where $j \in \lbrace 1,\cdots,m-1 \rbrace$, $H_{nq} = diag \lbrace h_{1}, \cdots, h_{n} \rbrace \otimes I_{q}$, $d=[d_{1}^{T}, \cdots, \d_{n}^{T}]^{T} \in \mathbb{R}^{nq}$, and $L_{nq}= L_{n} \otimes I_{q}$. The matrix $L_{n} \otimes I_{q}$ is the Kronecker product of matrices $L_{n}$ and $I_{q}$.
From , it is shown that $-\gamma z^{j}$ is the proximal-based estimator of a subgradient in $\partial F^{j}(x)$ for $j \in \lbrace 1, \cdots, m-1 \rbrace$. With the help of estimator $- \gamma z^{j}$, the corresponding proximal operator , which employs the information of $-\gamma z^{j}$ instead of $\partial F^{j}(x)$ for $j \in \lbrace 1, \cdots, m-1 \rbrace$, is presented to tackle the difficulty caused by the unproximable property of $\sum_{j=1}^{m-1} F^{j}(x)$. The scheme combined by and is called the **multi-proximal splitting**, which may be viewed as an extension of three operator splitting shown in [@OFW]. $\hfill$ $\blacklozenge$
Under Assumptions $\ref{A1}$-$\ref{A4}$, if $(x^{*}$, $z^{*}$, $v^{*}$, $w^{*}) \in (\mathbb{R}^{nq}, \mathbb{R}^{(m-1)nq}, \mathbb{R}^{nq}, \mathbb{R}^{nq})$ is an equilibrium of algorithm and $(\textbf{1}_{n}\otimes I_{q})^{T}H_{nq}w^{*}=\textbf{0}_{q}$, then $x^{*}$ is a solution of problem , where $z = [(z^{1})^{T}, \cdots, (z^{2})^{T}]^{T}$. \[KA1\]
If $(x^{*}, z^{*}, v^{*}, w^{*})$ is an equilibrium of algorithm , then according to the property of proximal operator and algorithm , it yields that for $j \in \lbrace 1,\cdots,m -1 \rbrace$,
$$\begin{aligned}
- \nabla F^{0}(x^{*}) + v^{*} + \gamma \sum_{j=1}^{m-1} z^{j*} \!\!\in & \partial F^{m}(x^{*}),
\label{P11} \\
- \gamma z^{j*} \!\!\in & \partial F^{j}(x^{*}), \label{P21} \\
- H^{-1}_{nq}(x^{*} - d) - \alpha L_{nq}v^{*} - w^{*} \!\!= & \textbf{0}_{nq},
\label{P31} \\
\alpha L_{nq} v^{*} \!\!= & \textbf{0}_{nq}. \label{P41} \end{aligned}$$
From , and , there exists a $v^{0} \in \mathbb{R}^{q}$ such that $$\begin{gathered}
\label{L4A}
\begin{split}
\textbf{0}_{nq} \in & - \nabla F^{0}(x^{*}) - \sum_{j=1}^{m-1} \partial F^{j}(x^{*}) + v^{*}, \\
v^{*} = & \textbf{1}_{n} \otimes v^{0}.
\end{split}\end{gathered}$$
Summing and yields that $-(x^{*} - d) - H_{nq}w^{*} = \textbf{0}_{nq}$, which means that $$\label{L4B}
\sum_{i = 1}^{n} (x^{*}_{i} \!\!-\! d_{i})\! =\! -\!\! \sum_{i = 1}^{n} \! h_{i}I_{q}w^{*}_{i}\!\! =\!\! -(\textbf{1}_{n} \otimes \! I_{q})^{T}\!H_{nq}w^{*} \!\! = \textbf{0}_{q}.$$
Considering together with and according to Lemma $\ref{LKKT}$, $x^{*}$ is a solution of problem . $\hfill$ $\blacksquare$
Then we state the convergence result of the proposed distributed algorithm . Let $(x^{*}, z^{*}, v^{*}, w^{*})$ be an equilibrium of algorithm . Define a Lyapunov candidate $V(x,z,v,w) = V_{1}(x,z) + V_{2}(x) + V_{3}(v,w)$, where $$\begin{aligned}
\label{VV123}
& & V_{1}(x,z) = (\eta \! + \! 1) [\frac{1}{2} \Vert \bar{x}^{*} \Vert^{2} \!+ \! \frac{1}{2} \gamma \sum_{j=1}^{m-1} (\Vert \bar{z}^{j*} \Vert^{2} \! - \! 2 (\bar{x}^{*})^{T}\bar{z}^{j*})], \notag \\
& & V_{2}(x) = (\eta \! + \! 1) [F^{0}(x) \! - \! F^{0}(x^{*}) - (\bar{x}^{*})^{T} \nabla F^{0}(x^{*})], \\
& & V_{3}(v,w) = \frac{\eta}{2} (\bar{v}^{*})^{T}H_{nq}\bar{v}^{*} + \frac{1}{2} (\bar{v}^{*} + \bar{w}^{*})^{T}H_{nq}(\bar{v}^{*} + \bar{w}^{*}), \notag \end{aligned}$$ and $\eta > 0$, $\bar{x}^{*} \triangleq x - x^{*}$, $\bar{z}^{j*} \triangleq z^{j} - z^{j*}$, $\bar{v}^{*} \triangleq v - v^{*}$, $\bar{w}^{*} \triangleq w - w^{*}$.
By analysing the convergence of , the main theorem of this subsection is obtained as below.
Consider algorithm . Suppose Assumptions \[A1\]-\[A4\] hold. If following inequalities $$\begin{gathered}
\alpha > \frac{(\eta + 1)^{2}}{\eta \lambda_{2}(\textbf{L}_{nq})}, \text{ } \eta > \max \lbrace \frac{1}{b_{2}h^{*}} - 1, 0 \rbrace \label{IE}\end{gathered}$$ hold, where $b_{2} = c - \frac{1}{2}(1 + \gamma)(m - 1) \frac{1}{\beta}$, $\frac{(1 + \gamma)(m - 1)}{2c} < \beta < \frac{2}{1 + \gamma}$, $h^{*} = \min_{i \in \mathcal{I}} \lbrace h_{1}, \cdots, h_{n} \rbrace$, then the trajectory of $x(t)$ converges, and $\lim_{t \to \infty} x(t)$ is the solution of problem . \[Thm1\]
It can be easily verified that $V(x^{*}, z^{*}, v^{*}, w^{*})$ $= 0$. Next, we will show that $V(x,z,v,w) > 0$ for all $(x,z,v,w) \neq (x^{*}, z^{*}, v^{*}, w^{*})$.
Since all $f^{0}_{i}(x)$ for $i \in \lbrace 1,\cdots, n \rbrace$ are convex, then $F^{0}(x) - F^{0}(x^{*}) - (\bar{x}^{*})^{T} \nabla F^{0}(x^{*}) \geq 0$. Hence $ V_{2}(x) \geq 0 $.
Since $0 < \gamma < \frac{1}{m-1}$, $$\begin{aligned}
V_{1}(x,z) & = & \frac{\eta+1}{2} \sum_{j=1}^{m-1} \left[ \Vert (\frac{1}{m-1})^{\frac{1}{2}}\bar{x}^{*} - \gamma (m-1)^{\frac{1}{2}} \bar{z}^{j*}\Vert^{2}\right. \notag \\
& & + \gamma (1 - \gamma(m-1))\Vert \bar{z}^{j*}\Vert^{2} \biggr] \geq 0. \label{V1}\end{aligned}$$
Since $V_{2}(x) \geq 0$, $V(x,z,v,w) \geq V_{1}(x,z) + V_{3}(v,w) \geq 0 $. Clearly $V(x,z,v,w)$ is positive definite, radically unbounded, $V(x,z,v,w) \geq 0$ and is zero if and only if $(x,z,v,w) = (x^{*},z^{*},v^{*},w^{*})$.
It follows from algorithm that $$\begin{gathered}
\label{x-x*}
\begin{split}
x + \dot{x} = & Prox_{F^{m}}[x - \nabla F^{0}(x) + v + \sum_{j=1}^{m-1} \gamma z^{j}], \\
x^{*} = & Prox_{F^{m}} [x^{*} \! - \! \nabla F^{0}(x^{*}) + v^{*} \! + \! \sum_{j=1}^{m-1} \gamma z^{j*}], \\
x + \dot{z}^{j} = & Prox_{F^{j}}[x - \gamma z^{j}], j \in \lbrace 1, \cdots, m-1 \rbrace, \\
x^{*} = & Prox_{F^{j}} [x^{*} - \gamma z^{j*}], j \in \lbrace 1, \cdots, m-1 \rbrace.
\end{split}\end{gathered}$$
Since $f^{j}_{i}(\cdot)$ is convex, $\partial f^{j}_{i}(\cdot)$ is monotone for agent $i \in \lbrace 1, \cdots, n \rbrace$, where $j \in \lbrace 1, \cdots, m-1 \rbrace$. According to the property of proximal operator, it follows from that for $j \in \lbrace 1, \cdots, m-1 \rbrace$, $$\begin{gathered}
\begin{split}
(\gamma \sum_{j=1}^{m-1} \bar{z}^{j*}\!\! - \!\!\nabla F^{0}(\bar{x}^{*}) + \bar{v}^{*} \!\!-\! \dot{x})^{T} (\bar{x}^{*}\!\! + \!\dot{x}) \geq & 0,\\
(-\gamma \bar{z}^{j*} \!\!-\!\! \dot{z}^{j})^{T}(\bar{x}^{*} \!\!+\! \dot{z}^{j}) \geq & 0,
\end{split}\label{Prox_pro}\end{gathered}$$ where $\nabla F^{0}(\bar{x}^{*}) \triangleq \nabla F^{0}(x) - \nabla F^{0}(x^{*})$.
From , it can be shown that for $j \in \lbrace 1, \cdots, m-1 \rbrace$, $$\begin{gathered}
\label{Proxmal relation1}
\begin{split}
& \gamma \sum_{j=1}^{m-1} [(\bar{z}^{j*})^{T}\! \bar{x}^{*}] \!\!-\!\! (\nabla \! F^{0}\! (\bar{x}^{*}))^{\!T}\bar{x}^{*} \!\!+\!\! (\bar{v}^{*})^{\!T} \bar{x}^{*} \!+\! (\bar{v}^{*}\!)^{T} \! \dot{x} \\
+ & \gamma \! \sum_{j=1}^{m-1} \! [(\bar{z}^{j*})^{T} \dot{x}] \!-\! (\nabla \! F^{0}\!(\bar{x}^{*}))^{T} \!\! \dot{x} \!-\! (\bar{x}^{*})^{T} \!\!\dot{x} \!-\! \Vert \dot{x} \Vert^{2} \! \geq \! 0,
\end{split}\end{gathered}$$ and $$\begin{gathered}
\label{Proxmal relation2}
- \gamma (\bar{z}^{j*})^{T} \bar{x}^{*} - (\bar{x}^{*})^{T} \dot{z}_{j} - \gamma (\bar{z}^{j*})^{T} \dot{z}_{j} - \Vert \dot{z}_{j} \Vert^{2} \geq 0.\end{gathered}$$
The derivative of Lyapunov candidate $V(x,z,v,w)$ along the trajectory of algorithm satisfies $$\begin{gathered}
\begin{split}
& \dot{V}(x,z,v,w) \\
= & (\eta+1)(\bar{x}^{*})^{T} \! \dot{x} \! + \! \gamma (\eta+1)\sum_{j=1}^{m-1} (\bar{z}^{j*})^{T} \! \dot{z} \\
& \! - \! \gamma (\eta+1)\sum_{j=1}^{m-1} ((\bar{x}^{*})^{T} \! \dot{z}^{j} \! + \! (\bar{z}^{j*})^{T} \! \dot{x}) \\
& + (\eta+1)(\nabla F^{0}(\bar{x}^{*}))^{T} \dot{x} + \dot{V}_{3}(v,w), \label{dot_V1}
\end{split}\end{gathered}$$ where $$\begin{gathered}
\label{DV3}
\begin{split}
& \dot{V}_{3}(v,w)\\
\leq & - (\eta + 1)(\bar{v}^{*})^{T}\bar{x}^{*} - (\bar{w}^{*})^{T}\bar{x}^{*} - (\bar{w}^{*})^{T}H_{nq}\bar{w}^{*} \\
& - (\eta + 1)(\bar{v}^{*})^{T}H_{nq}\bar{w}^{*} - \alpha \eta (\bar{v}^{*})^{T}\textbf{L}_{nq}\bar{v}^{*},
\end{split}\end{gathered}$$ and $\textbf{L}_{nq} = (H_{nq}L_{nq} + L_{nq}^{T}H_{nq})/2$.
According to -, it follows that $$\begin{gathered}
\begin{split}
& \dot{V}(x,z,v,w) \\
\leq & - (\eta + 1) \Vert \dot{x} \Vert^{2} - (\eta + 1) \sum_{j=1}^{m-1} \Vert \dot{z}^{j} \Vert^{2} \\
& - (\eta + 1)(1 + \gamma) \sum_{j=1}^{m-1}(\bar{x}^{*})^{T} \dot{z}^{j} - (\bar{w}^{*})^{T}H_{nq}\bar{w}^{*} \\
& - (\eta + 1) (\nabla F^{0}(\bar{x}^{*}))^{T} \! \bar{x}^{*} + (\eta + 1)(\bar{v}^{*})^{T}\dot{x} \\
& - (\bar{w}^{*})^{T}\bar{x}^{*} - \! \alpha \eta (\bar{v}^{*})^{T}\textbf{L}_{nq}\bar{v}^{*} - (\eta + 1)(\bar{v}^{*})^{T}\bar{w}^{*}.
\end{split}\end{gathered}$$
Then according to Assumption \[A1\], there exists a parameter $\beta > 0$ such that $$\begin{gathered}
\begin{split}
& (1 + \gamma) \sum_{j=1}^{m-1} (\bar{x}^{*})^{T}\!\dot{z}^{j} \\
\geq & -\! \frac{1}{2}(1\! + \gamma) \beta \sum_{j=1}^{m-1} \Vert \dot{z}^{j} \Vert^{2} - \frac{(1 + \gamma)(m-1)}{2\beta} \Vert \bar{x}^{*} \Vert^{2}.
\end{split}\end{gathered}$$
Hence we have the conclusion that $$\begin{gathered}
\label{DV}
\begin{split}
& \dot{V}(x,z,v,w) \\
\leq & -\! (\eta \! + \! 1) \Vert \dot{x} \Vert^{2} - (\eta + 1) b_{1} \sum_{j=1}^{m-1} \Vert \dot{z}^{j} \Vert^{2} - (\bar{w}^{*})^{T}\bar{x}^{*} \\
& -\! \alpha \eta (\bar{v}^{*}\!)^{T}\textbf{L}_{nq}\bar{v}^{*} \!\!-\! (\eta \! + \! 1)b_{2}\Vert \bar{x}^{*} \Vert^{2} \!\!-\! (\bar{w}^{*}\!)^{T} \! H_{nq}\bar{w}^{*} \\
& + (\eta + 1)(\bar{v}^{*})^{T}\dot{x} - (\eta + 1)(\bar{v}^{*})^{T}H_{nq}\bar{w}^{*},
\end{split}\end{gathered}$$ where $b_{1} = 1 - \frac{1}{2}(1 + \gamma) \beta$ and $b_{2} = c - \frac{1}{2}(1 + \gamma)(m-1) \frac{1}{\beta}$.
In order to illustrate that there always exists a $\beta > 0$ such that $b_{1} > 0$ and $b_{2} > 0$, here we define a function $B(\gamma)$ of $\gamma$ that $B(\gamma) = \frac{2}{\gamma + 1} - \frac{(\gamma + 1)(m - 1)}{2c}$. The derivative of $B(\gamma)$ is shown as $$\begin{gathered}
\frac{d B(\gamma)}{d \gamma} = -\frac{2}{(\gamma + 1)^{2}} - \frac{m - 1}{2c} < 0. \label{Beta}\end{gathered}$$
Note that $0 < \gamma < \frac{1}{m - 1} \leq 1$ and $c > m - 1$. According to , we have $B_{min}(\gamma) > B(1) = 1 - \frac{m - 1}{c} > 0$.
As the result, there exists a $\beta$ such that $$\frac{(1 + \gamma)(m - 1)}{2c} < \beta < \frac{2}{1 + \gamma}, \label{Beta1}$$ which means that $b_{1} = 1 - \frac{1}{2}(1 + \gamma) \beta > 0$, and $b_{2} = c - \frac{1}{2}(1 + \gamma)(m - 1) \frac{1}{\beta} > 0$.
In light of the above analysis and using the inequality $x^{T}y \leq \frac{1}{2\tau} \Vert x \Vert^{2} + \frac{\tau}{2} \Vert y \Vert^{2}$, equation can be written as $$\begin{gathered}
\begin{split}
\dot{V}(x,z,v,w) \leq & \! - \epsilon_{1} \Vert \dot{x} \Vert^{2} \! - \epsilon_{2} \sum_{j=1}^{m-1} \Vert \dot{z}^{j} \Vert^{2} - \epsilon_{3}\Vert \bar{x}^{*} \Vert^{2} \\
& \! - \epsilon_{4} \Vert \bar{v}^{*} \Vert^{2} \! - \epsilon_{5} (\bar{w}^{*})^{T}H_{nq}\bar{w}^{*},
\end{split}\end{gathered}$$ where $\epsilon_{1} = \eta + \frac{1}{2}$, $\epsilon_{2} = (\eta + 1) b_{1}$, $\epsilon_{3} = (\eta + 1)b_{2}- \frac{1}{h^{*}}$, $\epsilon_{4} = \alpha \eta \lambda_{2}(\textbf{L}_{nq}) - (\eta + 1)^{2}$, and $\epsilon_{5} = \frac{1}{4}$.
According to , it follows that $\epsilon_{k} > 0$ for $k \in \lbrace 1,2,3,4,5 \rbrace$. Additionally, since $V(x,z,v,w)$ is positive-definite, radically unbounded, lower bounded, $(x^{*},z^{*},$ $v^{*}, w^{*})$ is Lyapunov stable. It follows from the LaSalle invariant principle and Lemma \[Aequilibrium\] that $(x(t), z(t), v(t), w(t))$ converges to an equilibrium of algorithm in the largest invariant set $\mathcal{M}$ in $ E = \lbrace (x,z,v,w) \vert x=x^{*}, v=v^{*},w=w^{*}, -\gamma z^{j} \in \partial F^{j}(x^{*})$ for $j \in \lbrace 1, \cdots, m-1\rbrace \rbrace$. Since $(h \otimes I_{q})^{T} L_{nq} = (\textbf{0}_{n} \otimes I_{q})^{T}$, $\sum_{i = 1}^{n} h_{i}I_{q} \dot{w}_{i}(t) = (\textbf{1}_{n}\otimes I_{q})^{T}H_{nq} L_{nq} v(t) = \textbf{0}_{q} $. With $w(0) = \textbf{0}_{nq}$, it shows that $\sum_{i = 1}^{n} h_{i}I_{q} w_{i}(t) = (\textbf{1}_{n}\otimes I_{q})^{T}H_{nq}w^{*} = \textbf{0}_{q}$. According to Lemma \[KA1\], $x^{*}$ is a solution of problem . $\hfill$ $\blacksquare$
Algorithm Design with Distributed Estimator of Left Eigenvector h
-----------------------------------------------------------------
However, the left eigenvector $h$ corresponding to $\lambda_{1}(L_{nq})=0$ may not be known by any single agent, since $h$ is a global variable for multi-agent systems. In this subsection, we present a distributed smooth multi-proximal primal-dual algorithm for solving the problem with a distributed estimator of left eigenvector $h$.
Similar to algorithm , according to and , we propose a smooth algorithm as $$\begin{aligned}
\label{Algorithm 2}
\dot{x}(t) & = & Prox_{F^{m}}[x(t) \!\!-\!\! \nabla F^{0}(x(t)) \!\!+\!\! v(t) \!\!+\!\! \gamma \sum_{j = 1}^{m-1} z^{j}(t)] \!\!-\!\! x(t), \notag \\
\dot{z}^{j}(t) & = & Prox_{F^{j}}[x(t) - \gamma z^{j}(t)] - x(t), \notag \\
\dot{v}(t) & = & - Y^{-1}(t)(x(t) - d) - \alpha L_{nq} v(t) - w(t), \\
\dot{w}(t) & = & \alpha L_{nq} v(t), \quad w(0)=\textbf{0}_{nq}, \notag \\
\dot{y}(t) & = & - L_{nn} y(t), \quad y(0)=[I_{n}^{1},\cdots,I_{n}^{n}]^{T} \in \mathbb{R}^{nn}, \notag\end{aligned}$$ where $j \in \lbrace 1, \cdots, m-1 \rbrace$, $Y = diag \lbrace y_{1}^{1}, \cdots, y_{n}^{n} \rbrace \otimes I_{q}$, $L_{nn}=L_{n}\otimes I_{n}$, and $I_{n}^{i}$ is the $i$-th row of $I_{n}$.
When the directed graph $\mathcal{G}$ of problem is weight-balanced, it follows that $h_{i}=h_{j}, i,j \in \mathcal{V}$. While $\mathcal{G}$ is usually weight-unbalanced, hence a distributed estimator of $h$ is required for problem . Variable $y$ in algorithm is designed to obtain the estimated value of $h$. Lemma \[KA2\] combined with Theorem 2 will show that $y_{i}^{i*} = h_{i}$, where $y_{i}^{i*} = \lim\limits_{t \to \infty} y_{i}^{i}(t)$ for $i \in \lbrace 1, \cdots, n \rbrace$.
Under Assumptions $\ref{A1}$-$\ref{A4}$, if $(x^{*}, z^{*}, v^{*},$ $w^{*}, y^{*}) \in (\mathbb{R}^{nq}, \mathbb{R}^{(m-1)nq}, \mathbb{R}^{nq}, \mathbb{R}^{nq}, \mathbb{R}^{nn})$ is an equilibrium of algorithm , $(\textbf{1}_{n}\otimes I_{q})^{T}H_{nq}w^{*}=\textbf{0}_{q}$ and $y^{*} = \textbf{1}_{n} \otimes h$, then $x^{*}$ is a solution of problem . \[KA2\]
If $(x^{*}, z^{*}, v^{*}, w^{*}, y^{*})$ is an equilibrium of algorithm , similar to the proof of Lemma \[KA1\], it can be shown that there exists a $v^{0} \in \mathbb{R}^{q}$ such that $$\begin{gathered}
\begin{split}
\textbf{0}_{nq} \in & - \nabla F^{0}(x^{*}) - \sum_{j=1}^{m-1} \partial F^{j}(x^{*}) + v^{*}, \\
v^{*} = & \textbf{1}_{n} \otimes v^{0},
\end{split}\end{gathered}$$ and
\[PPY\] $$\begin{aligned}
%- \nabla F^{0}(x^{*}) + v^{*} + \gamma \sum_{j=1}^{m-1} z^{j*} \in & \partial F^{m}(x^{*}),
%\label{P111} \\
%- \gamma z^{j*} \in \partial F^{j}(x^{*}), j = & 1,\cdots,m-1 \label{P211} \\
- Y^{-1*}(x^{*} - d) - \alpha L_{nq}v^{*} - w^{*} = & \textbf{0}_{nq},
\label{P311} \\
\alpha L_{nq} v^{*} = \textbf{0}_{nq}, L_{nn} y^{*} = & \textbf{0}_{nq}. \label{P411} \end{aligned}$$
Adding and yields that $-(x^{*} - d) - Y^{*}w^{*} = \textbf{0}_{nq}$, which means that $$\begin{gathered}
\begin{split}
\sum_{i = 1}^{n} (x^{*}_{i} - d_{i}) = & -\sum_{i = 1}^{n} y_{i}^{i*}I_{q} w_{i}(t) = -\sum_{i = 1}^{n} h_{i}I_{q}w^{*}_{i} \\
= & -(\textbf{1}_{n}\otimes I_{q})^{T}H_{nq}w^{*} = \textbf{0}_{q},
\end{split}\end{gathered}$$ where $y_{i}^{i*}$ is the $[(i-1)q+i]$-th element of $y^{*}$. According to Lemma $\ref{LKKT}$, $x^{*}$ is a solution of problem . $\hfill$ $\blacksquare$
Next, we will state the convergence result of the proposed distributed algorithm .
Firstly, some lemmas should be given to obtain the final result.
\[Diiss\] Assume system can be written as $$\label{system+}
\dot{x} = f(x,u) = g(x) + u.$$ If system is forward complete, 0-GAS with respect to a closed and 0-invariant set $\mathcal{M}$, ZOD with respect to $\mathcal{M}$ with a positive definite function $W_{1}$ that $$\begin{gathered}
\label{Dissipative0+}
\begin{split}
a_{10}(\Vert x \Vert_{\mathcal{M}}) \leq W_{1}(x) \leq a_{11}(\Vert x \Vert_{\mathcal{M}})\\
DW_{1}(x) f(x,u) \leq a_{12}(\Vert u(t) \Vert),
\end{split}\end{gathered}$$ for $a_{10}, a_{11} \in \mathcal{K}_{\infty}$ and $a_{12} \in \mathcal{K}$, then system is iISS with respect to $\mathcal{M}$ with $a_{13} \in \mathcal{KL}$ such that $$\begin{aligned}
& & a_{10}(\Vert x(t,x_{0},u) \Vert_{\mathcal{M}}) \notag \\
& \leq & a_{13}(\Vert x_{0} \Vert_{\mathcal{M}}, t) + \int^{t}_{0} 2 \left( a_{12}(\Vert u(s) \Vert) + \Vert u(s) \Vert \right) ds. \label{IISS+}\end{aligned}$$ Moreover, if $a_{12}(\Vert u(t) \Vert) = k \Vert u(t) \Vert^{2}$, where $k \in \mathbb{R}^{+}$, $u(t)$ is exponentially convergent to zero, then system converges to $\mathcal{M}$.
If system is forward complete and 0-GAS with respect to $\mathcal{M}$, then by Theorem 2.8 and Remark 4.1 in [@In_Lya], there exists a smooth function $W_{2}: \mathbb{R}^{n} \to \mathbb{R}$ and functions $a_{14},a_{15},a_{16} \in \mathcal{K}_{\infty}$ such that $$\begin{gathered}
\label{CL+}
\begin{split}
a_{14}(\Vert x \Vert_{\mathcal{M}}) \leq W_{2}(x) \leq a_{15}(\Vert x \Vert_{\mathcal{M}})\\
DW_{2}(x) f(x,0) \leq - a_{16}(\Vert x \Vert_{\mathcal{M}}).
\end{split}\end{gathered}$$ Then according to , proof of Lemma IV.10 and Proposition II.5 in [@iISS2], there exists an iISS Lyapunov function $W_{3}$ with respect to $\mathcal{M}$ such that $$W_{3}(x) = W_{1}(x) + \pi (W_{2}(x)), \label{CLF++}$$ where $\pi(r) \triangleq \int^{r}_{0} \frac{ds}{1+\kappa(a_{14}^{-1}(s))}$, and $\kappa(r) \triangleq r$ $+ \max_{ \Vert x \Vert_{\mathcal{M}} \leq r}$ $ \lbrace \Vert DW_{2}(x) \Vert\rbrace$.
From , and , we have the conclusion that $$\begin{gathered}
\label{Dissipative0++}
\begin{split}
& DW_{3}(x) f(x,u) \\
\leq & - \rho(W_{3}(x)) + (\Vert u(t) \Vert + a_{12}(\Vert u(t) \Vert)),
\end{split}\end{gathered}$$ where $\rho$ is a positive definite function. Then according to and Corollary IV.3 in [@iISS2], there exist an $a_{17} \in \mathcal{KL}$ such that $$\begin{gathered}
\begin{split}
& W_{3}(x(t)) \\
\leq & a_{17}(W_{3}(x_{0}),t) + \int^{t}_{0} \! \! 2 (\Vert u(\tau) \Vert + a_{12}(\Vert u(\tau) \Vert)) d\tau.
\end{split}\end{gathered}$$ Since $a_{10}(\Vert x \Vert_{\mathcal{M}}) \leq W_{1}(x) \leq W_{3}(x) \leq W_{1}(x)+W_{2}(x)$, equation holds.
Let $U(t)=\int^{\infty}_{t} 2(k\Vert u(\tau) \Vert^{2} + \Vert u(\tau) \Vert )d\tau$ for $t \geq 0$. Since $u(t) $ is exponentially convergent to zero, $U(t) \leq M_{U}$ for a $M_{U} \in \mathbb{R}^{+}$, $U(t)$ is decreasing, and $\lim_{t\to\infty}U(t)=0$. From , for $t \geq 0$, it follows that $$\begin{gathered}
\begin{split}
\Vert x(t) \Vert_{\mathcal{M}} \leq a_{10}^{-1}(a_{13}(\Vert x(0) \Vert_{\mathcal{M}},0)+M_{U}) \triangleq M_{X}.
\end{split}\end{gathered}$$ For any $\varepsilon > 0$, choose $T_{U} \geq 0$ and $T_{X} \geq 0$ such that $U(T_{U}) \leq a_{10}(\varepsilon)/2$ and $a_{13}(M_{X},T_{X}) \leq a_{10}(\varepsilon)/2$. Let $T \triangleq T_{X} + T_{U}$. Then from , for any $t \geq T$, $$\begin{gathered}
\begin{split}
& a_{10}(\Vert x(t) \Vert_{\mathcal{M}})\\
\leq & a_{13}(\Vert x(T_{U}) \Vert_{\!\mathcal{M}},t\!-\!T_{U})\!+\!\!\!\int^{t}_{T_{U}}\!\! \!\!2(k\Vert u(\tau) \Vert^{2}\!\! + \!\!\Vert u(\tau) \Vert )d\tau \\
\leq & a_{13}(M_{X},T_{X}+(t-T))+U(T_{U})\\
\leq & a_{13}(M_{X},T_{X})+U(T_{U}) \leq a_{10}(\varepsilon),
\end{split}\end{gathered}$$ which means that $\Vert x(t) \Vert_{\mathcal{M}} \leq \varepsilon$ for all $t \geq T$. Hence system converges to $\mathcal{M}$. $\hfill$ $\blacksquare$
Let $\mathcal{M}_{Yj} = \lbrace [(\varphi_{1}x^{*} - \varphi_{2}z^{j*})^{T}, (\varphi_{3}z^{j*})^{T}]^{T} \vert (x^{*},z^{*},v^{*},w^{*},$ $y^{*}) $ $\in \mathcal{M}_{Y}\rbrace$ for $j \in \lbrace 1, \cdots, m-1 \rbrace$ and $\varphi_{k} \in \mathbb{R}$ for $k \in \lbrace 1, 2, 3 \rbrace$, where $\mathcal{M}_{Y}$ is the largest invariant set in $ E = \lbrace (x,z,v,w,y) \vert x=x^{*}, v=v^{*},w=w^{*}, y=y^{*}, -\gamma z^{j} \in \partial F^{j}(x^{*})$ for $j \in \lbrace 1, \cdots, m-1\rbrace \rbrace$. Then it follows that for any $\xi \in \mathbb{R}^{2nq}$, $\xi \in \mathcal{M}_{Yj}$ if and only if $\xi = [(\varphi_{1}x^{*} - \varphi_{2}z^{j*})^{T}, (\varphi_{3}z^{j*})^{T}]^{T}$ for $(x^{*},z^{*},v^{*},w^{*},y^{*}) \in \mathcal{M}_{Y}$ and $j \in \lbrace 1, \cdots, m-1 \rbrace$.
\[x-z\] Consider algorithm . For $\xi$ and $\mathcal{M}_{Yj}$ with $j \in \lbrace 1, \cdots, m-1 \rbrace$, it follows that
- For each $\xi \in \mathbb{R}^{2nq}$, there exists a unique $x \in \mathbb{R}^{nq}$ and $z^{j} \in \mathbb{R}^{nq}$ such that $\xi = [(\varphi_{1}x - \varphi_{2}z^{j})^{T}, (\varphi_{3}z^{j})^{T}]^{T}$.
- Let $P_{\mathcal{M}_{Yj}}(\xi) \triangleq \arg \min_{\psi \in \mathcal{M}_{Yj}} \lbrace \Vert \xi - \psi \Vert^{2} \rbrace$. Then $P_{\mathcal{M}_{Yj}}(\xi)=[(\varphi_{1}x^{*} - \varphi_{2}z^{j*})^{T}, (\varphi_{3}z^{j*})^{T}]^{T}$ for some $(x^{*},z^{*},v^{*},w^{*},y^{*}) \in \mathcal{M}_{Y}$.
- For each $\xi \in \mathbb{R}^{2nq}$, there exists an $(x^{*},z^{*},v^{*},w^{*},y^{*}) \in \mathcal{M}_{Y}$ such that $\Vert \xi \Vert^{2}_{\mathcal{M}_{Yj}} = \Vert \varphi_{1} \bar{x}^{*} - \varphi_{2} \bar{z}^{j*} \Vert^{2} + \Vert \varphi_{3} \bar{z}^{j*} \Vert^{2}$.
- Let $V(\xi) = \frac{1}{2}\Vert \xi \Vert^{2}_{\mathcal{M}_{Yj}}$. For each $\xi \in \mathbb{R}^{2nq}$, there exist an $(x^{*},z^{*},v^{*},w^{*},y^{*}) \in \mathcal{M}_{Y}$ such that $\nabla V(\xi) = [(\varphi_{1}\bar{x}^{*} - \varphi_{2} \bar{z}^{j*})^{T}, (\varphi_{3} \bar{z}^{j*})^{T}]^{T}$.
Obviously $(i)$ is true. It follows from $(i)$ that there exists a unique $(x^{*},z^{*},v^{*},w^{*},y^{*})$ such that $P_{\mathcal{M}_{Yj}}(\xi)=[(\varphi_{1}x^{*} - \varphi_{2}z^{j*})^{T}, (\varphi_{3}z^{j*})^{T}]^{T}$. By definition of $P_{\mathcal{M}_{Yj}}(\xi)$, $[(\varphi_{1}x^{*} - \varphi_{2}z^{j*})^{T}, (\varphi_{3}z^{j*})^{T}]^{T} \in \mathcal{M}_{Yj}$, which means that $(x^{*},z^{*},v^{*},w^{*},y^{*}) \in \mathcal{M}_{Y}$. Thus $(ii)$ is proved.
Note that $\Vert \xi \Vert^{2}_{\mathcal{M}_{Yj}} = \Vert \xi - P_{\mathcal{M}_{Yj}}(\xi) \Vert^{2}$. Then according to $(i)$ and $(ii)$, it shows that $\Vert \xi \Vert^{2}_{\mathcal{M}_{Yj}} = \Vert [((\varphi_{1}x - \varphi_{2}z^{j})-(\varphi_{1}x^{*} - \varphi_{2}z^{j*}))^{T}, (\varphi_{3}z^{j} - \varphi_{3}z^{j*})^{T}]^{T} \Vert^{2}=\Vert (\varphi_{1}x - \varphi_{1}x^{*})-(\varphi_{2}z^{j} - \varphi_{2}z^{j*}) \Vert^{2} + \Vert \varphi_{3}z^{j} - \varphi_{3}z^{j*} \Vert^{2}$. Hence $(iii)$ is proved.
Similarly to the analysis of $(iii)$, there holds that $\nabla V(\xi) = \frac{1}{2} \nabla \Vert \xi - P_{\mathcal{M}_{Yj}}(\xi) \Vert^{2} = \xi - P_{\mathcal{M}_{Yj}}(\xi)$. Then according to $(ii)$, it is shown that $\xi - P_{\mathcal{M}_{Yj}}(\xi) = [(\varphi_{1} \bar{x}^{*} - \varphi_{2} \bar{z}^{j*})^{T}, (\varphi_{3} \bar{z}^{j*})^{T}]^{T}$. This completes the proof of $(iv)$. $\hfill$ $\blacksquare$
Then, the main theorem of this subsection is given below.
Consider algorithm . Suppose Assumptions $\ref{A1}$-$\ref{A4}$ hold. If inequalities hold, then the trajectory of $x(t)$ converges, and $\lim\limits_{t \to \infty} x(t)$ is the solution of problem . \[Thm2\]
Define $\phi = col(x,z,v,w)$. The first-order system controlled by (\[Algorithm 2\]) can be considered as $$\begin{gathered}
\label{G123}
\dot{\phi} = g_{1}(\phi) + g_{2}(\phi,y) + g_{3}(y),\end{gathered}$$ where $g_{1}(\phi)=col(\dot{x}$,$\dot{z}$,$G_{1}$,$\dot{w})$, $G_{1} = - H^{-1}_{nq}(x - d) - \alpha L_{nq} v - w$, $g_{2}(\phi,y) = col(\textbf{0}_{nq},\textbf{0}_{(m-1)nq},G_{2},\textbf{0}_{nq})$, $G_{2} = (H^{-1}_{nq} \!\!-\!\! Y^{-1})\bar{x}^{*}$, $g_{3}(y) = col(\textbf{0}_{nq},\textbf{0}_{(m-1)nq},u,\textbf{0}_{nq})$, and $u(t) = (H^{-1}_{nq} \!\!-\!\! Y^{-1}(t))(x^{*} \!\!-\!\! d)$.
**i)** Firstly, with only the first part in , we consider the system $$\dot{\phi} = g_{1}(\phi). \label{G1}$$ From Theorem \[Thm1\], it is clear that under system , $(x(t), z(t), v(t), w(t))$ converges to the largest invariant set $\mathcal{M}$ in $ E = \lbrace (x,z,v,w) \vert x=x^{*}, v=v^{*},w=w^{*}, -\gamma z^{j} \in \partial F^{j}(x^{*})$ for $j \in \lbrace 1, \cdots, m-1\rbrace \rbrace$.
**ii)** Consider the system $$\dot{\phi} = g_{1}(\phi) + g_{2}(\phi, y),
\label{G12}$$ where $[(\phi^{*})^{T},(y^{*})^{T}]^{T}$ is an equilibrium of algorithm , and $g_{2}(\phi, y)$ satisfies that $g_{2}(\phi^{*},y^{*}) = \textbf{0}$.
From and the Lyapunov candidate $V_{Y}(x,z,v,w,y)=V(x,z,v,w)+V_{4}(y)$, where $V_{4}(y) = \frac{1}{2} \Vert \bar{y}^{*} \Vert^{2}$ and $\bar{y}^{*} \triangleq y - y^{*}$, it yields that $$\begin{gathered}
\label{DVG12}
\begin{split}
& \dot{V}_{Y}(x,z,v,w,y)\\
\leq & \! - \epsilon_{1} \Vert \dot{x} \Vert^{2} \!\!-\! \epsilon_{2} \sum_{j=1}^{m-1} \Vert \dot{z}^{j} \Vert^{2} \!\! - \! \epsilon_{3}\Vert \bar{x}^{*} \Vert^{2} \!\! - \! \epsilon_{4} \Vert \bar{v}^{*} \! \Vert^{2}\\
& - \epsilon_{5} (\bar{w}^{*})^{T} \! H_{nq}\bar{w}^{*} - \frac{1}{2}(\bar{y}^{*})^{T}(L_{nn}+L_{nn}^{T})\bar{y}^{*} \\
& + DV_{Y},\\
\end{split}\end{gathered}$$ where $$\begin{gathered}
\label{DVG2}
\begin{split}
& DV_{Y} = \frac{\partial V_{3}(v,w)}{\partial v} G_{2} \\
= & (\eta+1) (\bar{v}^{*})^{T}Qx^{-*} + (\bar{w}^{*})^{T}Qx^{-*} \\
\leq & \zeta_{1} (\bar{v}^{*})^{T}Qv^{-*} + \zeta_{2} (\bar{x}^{*})^{T}Qx^{-*} + \zeta_{3} (\bar{w}^{*})^{T}Qw^{-*}\\
\leq & \rho(t) \left[ \zeta_{1} \Vert \bar{v}^{*} \Vert^{2} + \zeta_{2} \Vert \bar{x}^{*} \Vert^{2} + \zeta_{3} \Vert \bar{w}^{*} \Vert^{2} \right]\\
\end{split}\end{gathered}$$ and $Q = I_{nq} - H_{nq}Y^{-1}$, $\zeta_{1} = \frac{\eta+1}{2}$, $\zeta_{2} = \frac{\eta}{2}+1$, $\zeta_{3} = \frac{1}{2}$, $\rho(t) = \max_{i\in I}\vert 1 - h_{i}(y_{i}^{i}(t))^{-1} \vert$.
Since $y(t) = e^{-L_{nn}t}y(0)$ and $y(0)=[I_{n}^{1},\cdots,I_{n}^{n}]^{T}$ from , it is shown that $\lim\limits_{t \to \infty} y(t) = \textbf{1}(h^{T} \otimes I_{q}) y(0) = \textbf{1}_{n} \otimes h$. Therefore, $y^{*} = \textbf{1}_{n} \otimes h$. Then according to Lemma 2.6 in [@RWB], $y(t)$ is exponentially convergent to $\textbf{1}_{n}\otimes h$, and $y_{i}^{i}(t) > 0$ for all $i \in \lbrace 1,\cdots, n\rbrace$ and $t > 0$. As the result, $\rho(t)$ and $u(t)$ are both exponentially convergent to zero.
With and , it is followed that $$\begin{gathered}
\begin{split}
& \dot{V}_{Y}(x,z,v,w,y)\\
\leq & - \epsilon_{1} \Vert \dot{x} \Vert^{2} - \epsilon_{2} \sum_{j=1}^{m-1} \Vert \dot{z}^{j} \Vert^{2} - l_{1}\Vert \bar{x}^{*} \Vert^{2}\\
& - l_{2} \Vert \bar{v}^{*} \Vert^{2} - l_{3} (\bar{w}^{*})^{T}H_{nq}\bar{w}^{*},\\
\end{split}\end{gathered}$$ where $l_{1}=\epsilon_{3}-\rho(t)\zeta_{2}$, $l_{2}=\epsilon_{4}-\rho(t)\zeta_{1}$, $l_{3}=\epsilon_{5}-\rho(t)\frac{\zeta_{3}}{h^{*}}$.
Since $\rho(t) \to 0$ when $t \to \infty$, there exists $T_{0} > 0$ that when $t > T_{0}$, $ l_{1} \geq \frac{1}{2}\epsilon_{3}, l_{2} \geq \frac{1}{2}\epsilon_{4} , l_{3} \geq \frac{1}{2}\epsilon_{5}$. Therefore $\dot{V}_{Y}(x,z,v,w,y) \leq 0$ when $t > T_{0}$.
When $t \leq T_{0}$, since $0 < \rho(t) < 1$, $$\begin{gathered}
\begin{split}
& \dot{V}_{Y}(x,z,v,w,y) \\
\leq & \zeta_{2}\Vert \bar{x}^{*} \Vert^{2} + \zeta_{1} \Vert \bar{v}^{*} \Vert^{2} + \zeta_{3} \Vert \bar{w}^{*}\Vert^{2}\\
\leq & \zeta_{2}(\Vert \bar{v}^{*} \Vert^{2}+\Vert \bar{w}^{*}\Vert^{2}) + \zeta_{2} \Vert \bar{x}^{*} \Vert^{2} \\
\leq & \zeta_{2}(\Vert \bar{v}^{*} \Vert^{2}+\Vert \bar{w}^{*}\Vert^{2}) + \iota_{1} V_{1}(x,z), \label{DVT}
\end{split}\end{gathered}$$ where $\iota_{1} = \frac{\eta+2}{(\eta+1)[1-(m-1)\gamma]}$.
Note that $$\begin{gathered}
\label{V3T}
\begin{split}
& V_{3}(v,w) \\
\geq & \frac{\eta+1}{2h_{max}}\Vert \bar{v}^{*} \Vert^{2} + \frac{1}{2h_{max}} \Vert \bar{w}^{*} \Vert^{2} + \frac{1}{h_{max}}(\bar{v}^{*})^{T}\bar{w}^{*} \\
\geq & \iota_{2}\Vert \bar{v}^{*} \Vert^{2} + \iota_{3} \Vert \bar{w}^{*} \Vert^{2} \\
\geq & \min \lbrace \iota_{2}, \iota_{3} \rbrace (\Vert \bar{v}^{*} \Vert^{2} + \Vert \bar{w}^{*} \Vert^{2}),
\end{split}\end{gathered}$$ where $h_{max} = max_{i \in I} \lbrace h_{1}, \cdots, h_{n} \rbrace$, $1< r < \eta + 1$, $\iota_{2} = \frac{\eta+1}{2h_{max}} - \frac{1}{2h_{max} r}$, $\iota_{3} = \frac{1}{2h_{max}} - \frac{r}{2}$.
From and , it is shown that $$\begin{gathered}
\label{DVTK}
\begin{split}
& \dot{V}_{Y}(x,z,v,w,y) \\
\leq & \frac{\zeta_{2}}{\min \lbrace \iota_{2}, \iota_{3} \rbrace} V_{3}(x,z,v,w) + \iota_{1} V_{1}(x,z) \\
\leq & \kappa_{1} V_{Y}(x,z,v,w,y),
\end{split}\end{gathered}$$ where $\kappa_{1} = \max \lbrace \frac{\zeta_{2}}{\min \lbrace \iota_{2}, \iota_{3} \rbrace}, \iota_{1} \rbrace$.
According to , when $t=T_{0}$, $$V_{Y}(T_{0}) \leq e^{\kappa T_{0}} V_{Y}(0).
\label{VG12}$$
To sum up, $V_{Y}(x,z,v,w,y)$ is proper and $\dot{V}_{Y}(x,z,v,w,y)$ $\leq 0$ when $t > T_{0}$. Then according to the LaSalle invariant principle and Lemma \[Aequilibrium\], system converges to the largest invariant set $\mathcal{M}_{Y}$ in $ E = \lbrace (x,z,v,w,y) \vert x=x^{*}, v=v^{*},w=w^{*}, y=y^{*}, -\gamma z^{j} \in \partial F^{j}(x^{*})$ for $j \in \lbrace 1, \cdots, m-1\rbrace \rbrace$, which also means that system is GAS to $\mathcal{M}_{Y}$.
**iii)** Now consider the complete system . Clearly $\mathcal{M}_{Y}$ is a closed, 0-invariant set for system . Similar to , there exist $\kappa_{2} > 0$ and $\nu_{u} >0$ such that $$\begin{gathered}
\label{VG123}
\begin{split}
& \forall t \leq T_{0}:\quad \dot{V}_{Y}(t) \leq \kappa_{2} V_{Y}(t) + \nu_{u} \Vert u(t) \Vert^{2},\\
& \forall t > T_{0}:\quad \dot{V}_{Y}(t) \leq \nu_{u} \Vert u(t) \Vert^{2}.
\end{split}\end{gathered}$$ As a result, $V_{Y}(t)$ is bounded for all $t < +\infty$. Since $V_{Y}(t)$ is proper, system is forward complete. Moreover, note that $T_{0} \to 0$ and $\Vert u(t) \Vert \to 0$ when $y \to y^{*}$. Therefore, according to , each $(x^{*},z^{*},v^{*},w^{*},y^{*}) \in \mathcal{M}_{Y}$ is Lyapunov stable.
Then, we define an iISS-Lyapunov candidate $V_{\mathcal{M}_{Y}}(x,z,$ $v,w,y) = V_{1\mathcal{M}_{Y}}(x,z)+V_{2}(x)+V_{3}(v,w)+V_{4}(y)$ with respect to $\mathcal{M}_{Y}$, where $$\begin{gathered}
V_{\!1\!\mathcal{M}_{\!Y}}\!(\!x,\!z\!) \!\!=\!\! \frac{\eta \!+\! 1}{2} \!\!\sum_{j=1}^{m-1}\!\! \Vert [ (\varphi_{1} x \!-\! \varphi_{2} z^{j})^{\!T}\!\!, (\varphi_{3} z^{j})^{\!T}]^{\!T}\!\Vert^{2}_{\!\mathcal{M}_{\!Yj}},\end{gathered}$$ and $\varphi_{1}=(\frac{1}{m-1})^{\frac{1}{2}}$, $\varphi_{2}=\gamma (m-1)^{\frac{1}{2}}$, $\varphi_{3}=[\gamma (1 - \gamma(m-1))]^{\frac{1}{2}}$, $\mathcal{M}_{Yj} \!\! \triangleq \!\! \lbrace [ (\varphi_{1} x^{*} - \varphi_{2} z^{j*})^{T} , (\varphi_{3} z^{j*})^{T} ]^{T}, (x^{*},z^{j*},v^{*}$, $w^{*},y^{*}) \in \mathcal{M}_{Y} \rbrace$ for $j \in \lbrace 1, \cdots, m-1 \rbrace$.
According to Lemma \[x-z\] and proof of Theorem \[Thm1\], when $t > T_{0}$, it follows that $$\begin{gathered}
\begin{split}
& \dot{V}_{\mathcal{M}_{Y}}(x,z,v,w,y) \\
\leq & - l_{2} \Vert \bar{v}^{*} \Vert^{2} - l_{3} (\bar{w}^{*})^{T}H_{nq}\bar{w}^{*} \\
& + \left[ (\bar{v}^{*} + \bar{w}^{*})^{T}H_{nq} + \eta(\bar{v}^{*})^{T}H_{nq} \right] u(t) \\
\leq & - \iota_{4} \Vert \bar{v}^{*} \Vert^{2} - \iota_{5}(\bar{w}^{*})^{T}H_{nq}\bar{w}^{*} + \iota_{6} \Vert u(t) \Vert^{2} \\
\leq & \iota_{6} \Vert u(t) \Vert^{2}, \label{0P}
\end{split}\end{gathered}$$ where $\iota_{4} = \frac{1}{2}\epsilon_{3} - \frac{(\eta+1)\tau_{1}}{2h^{*}}$, $\iota_{5} = \frac{1}{2}\epsilon_{4} - \frac{\tau_{2}}{2}$ and $\iota_{6} = \frac{\eta+1}{2\tau_{1}} + \frac{1}{2\tau_{2}}$. Note that $\iota_{4} > 0$ and $\iota_{5} > 0$ always hold, since $\tau_{1}$ and $\tau_{2}$ can be chosen arbitrarily small.
From and Definition $\ref{D2}$, it is clear that system is ZOD with respect to $\mathcal{M}_{Y}$ when $t > T_{0}$. Remind that $u(t)$ is exponentially convergent to zero. Since system is 0-GAS with respect to $\mathcal{M}_{Y}$, then according to Lemma \[Diiss\], $\phi_{Y}(t,u(t))$ converges to $\mathcal{M}_{Y}$. Note that each $(x^{*},z^{*},v^{*},w^{*},y^{*}) \in \mathcal{M}_{Y}$ is Lyapunov stable, then according to Lemma \[Aequilibrium\], system converges to one of its equilibria in $\mathcal{M}_{Y}$. Similar to the analysis in proof of Theorem $\ref{Thm1}$, it is clear that $(\textbf{1}_{n}\otimes I_{q})^{T}H_{nq}w^{*} = \textbf{0}_{q}$. Then according to Lemma $\ref{KA2}$, $x^{*}$ is the solution of problem (\[Problem 2\]). This completes the proof. $\hfill$ $\blacksquare$
For $\xi = [(\varphi_{1} x - \varphi_{2} z^{j})^{T} , (\varphi_{3} z^{j})^{T} ]^{T}$ with $j \in \lbrace 1, \cdots, m-1\rbrace$, let $P_{\mathcal{M}_{Yj}}(\xi)=[(\varphi_{1} \hat{x}^{*} - \varphi_{2} \hat{z}^{j*})^{T}, (\varphi_{3} \hat{z}^{j*})^{T} ]^{T}$, $P_{\mathcal{M}_{Yx}}(x) = \tilde{x}^{*}$, and $P_{\mathcal{M}_{Yz^{j}}}(z^{j}) = \tilde{z}^{j*}$, where $\mathcal{M}_{Yx} \triangleq \lbrace x^{*} \vert (x^{*},z^{*},v^{*},w^{*},y^{*}) \in \mathcal{M}_{Y} \rbrace$ and $\mathcal{M}_{Yz^{j}} \triangleq \lbrace z^{j*} \vert (x^{*},z^{*},v^{*},w^{*},y^{*}) \in \mathcal{M}_{Y} \rbrace$. Hence $\hat{x}^{*} = \tilde{x}^{*} = x^{*}$ and usually $\hat{z}^{j*} \neq \tilde{z}^{j*}$. While it is true that $\hat{z}^{j*} \in \mathcal{M}_{Yz^{j}}$, hence $\dot{V}_{\mathcal{M}_{Y}}(x,z,v,w,y)$ can be deduced based on the analysis of $\dot{V}_{Y}(x,z,v,w,y)$ in proof of Theorem \[Thm1\]. $\hfill$ $\blacklozenge$
In proof of Theorem $\ref{Thm2}$, the first-order system controlled by had been separated to three parts. Since the existence of estimation error between $y$ and $h$, the Lyapunov function $V_{Y}(x,z,v,w,y)$ of system may increase before $T_{0}$. Then we proved that the Lyapunov function $V_{Y}(x,z,v,w,y)$ of system is bounded when $t \leq T_{0}$ and $\dot{V}_{Y}(x,z,v,w,y) \leq 0$ when $t > T_{0}$. Finally, with the help of iISS theory with respect to set, it is proved that system is asymptotically convergent to its equilibria in $\mathcal{M}_{Y}$, which provides new ideas about stability analysis of asymptotically convergent system with exponentially convergent inputs. $\hfill$ $\blacklozenge$
Simulations
===========
In this section, simulations are performed to validate the proposed algorithm . Consider the fused LASSO problem with four agents moving in a 2-D space with first-order dynamics as $$\begin{gathered}
\min_{x \in \mathbb{R}^{8}} F(x), \ \ s.t. \sum_{i=1}^{4} x_{i} = \sum_{i=1}^{4} d_{i},\end{gathered}$$ where $x_{i} = [ x^{1}_{i}, x^{2}_{i} ]^{T} \in \mathbb{R}^{2}, i \in \lbrace 1, 2, 3, 4 \rbrace$, $F(x) = \sum_{j=0}^{3} f^{j}(x) = 2\Vert x - s \Vert^{2} + \iota(x) + \Vert x - p \Vert_{1} + \Vert Dx \Vert_{1}$, $\iota(x) = \begin{cases}
0, & \mbox{if } x \in \Omega\\
\infty, & \mbox{if } x \notin \Omega
\end{cases}$, and $$D =
\begin{bmatrix}
1 & -1 & & & & & &\\
& & 1 & -1 & & & &\\
& & & & \cdots & \cdots & & \\
& & & & & & 1 & -1\\
\end{bmatrix} \in \mathbb{R}^{8 \times 8}.$$ The local cost function $f_{i}(x_{i})$ for agent $i$ is consisted by $$\begin{array}{l}
f^{0}_{i}(x_{i}) = 2\Vert x_{i} - s_{i}\Vert^{2},\\
f^{1}_{i}(x_{i}) = \Vert x_{i} - p_{i} \Vert_{1},\\
f^{2}_{i}(x_{i}) = \Vert x_{i}^{1} - x_{i}^{2} \Vert_{1},\\
f^{3}_{i}(x_{i}) =
\begin{cases}
0, & \mbox{if } x_{i} \in \Omega_{i}\\
\infty, & \mbox{if } x_{i} \notin \Omega_{i}
\end{cases},\\
\end{array}
\label{Simulation_f}$$ where $s_{i} = [s^{1}_{i}, s^{2}_{i}]^{T} = [i - 2.5, 0]^{T}$, $p_{i} = [p^{1}_{i}, p^{2}_{i}]^{T} = [0, i - 2.5]^{T}$ and $\Omega_{i} = \lbrace \delta \in \mathbb{R}^{2} \vert \Vert \delta - x_{i}(0) \Vert^{2} \leq 64\rbrace$. Then $f^{0}_{i}(x_{i})$, $f^{1}_{i}(x_{i})$, $f^{2}_{i}(x_{i})$ and $f^{3}_{i}(x_{i})$ represent respectively the quadratic objective, the $l_{1}$ penalty with an anchor $p_{i}$, another $l_{1}$ penalty associated with the matrix $D$, and the indicator function of the constraint set $x_{i} \in \Omega_{i}$ for each agent $i$. Resource allocation conditions are described as $d_{1}=[2,-1]^{T}$, $d_{2}=[-1,1]^{T}$, $d_{3}=[-1,-1]^{T}$ and $d_{4}=[2,2]^{T}$.
Based on ($\ref{Simulation_f}$), the gradient of $f^{0}_{i}$ and proximal operators of $f^{1}_{i}$, $f^{2}_{i}$ and $f^{3}_{i}$ for agent $i$ are shown as $$\begin{gathered}
\begin{split}
\nabla f^{0}_{i}(x_{i}) = & [ 4(x^{1}_{i} - s^{1}_{i}), 4(x^{2}_{i} - s^{2}_{i}) ]^{T}, \\
prox_{f^{1}_{i}}[\eta_{1}] = & [\phi(\eta_{1}^{1}, p_{i}^{1}), \phi(\eta_{1}^{2}, p_{i}^{2})]^{T}, \\
prox_{f^{2}_{i}}[\eta_{2}] = & [\phi(\eta_{2}^{1}, \eta_{2}^{2}), \phi(\eta_{2}^{2}, \eta_{2}^{1})]^{T}, \\
prox_{f^{3}_{i}}[\eta_{3}] = & \arg \min_{\delta \in \Omega_{i} } \Vert \delta - \eta_{3} \Vert^{2}, \\
\end{split}\end{gathered}$$ where $\eta_{j} \in \mathbb{R}^{2}$, $j \in \lbrace 1, 2 ,3 \rbrace$. For $\xi_{1} \in \mathbb{R}$ and $ \xi_{2} \in \mathbb{R}$, the function $\phi(\xi_{1},\xi_{2})$ is defined as follows $$\begin{aligned}
\phi(\xi_{1},\xi_{2}) & = &
\begin{cases}
\xi_{1}-1, & \mbox{if } \xi_{1} > \xi_{2}+1 \\
\xi_{2}, & \mbox{if } \xi_{2}-1 \leq \xi_{1} \leq \xi_{2}+1 \\
\xi_{1}+1, & \mbox{if } \xi_{1} < \xi_{2}-1
\end{cases}.\end{aligned}$$
Note that the proximal operator of $f^{1}_{i}(x_{i})+f^{2}_{i}(x_{i})+f^{3}_{i}(x_{i})$, e.i., $ prox_{(f^{1}_{i}+f^{2}_{i} + f^{3}_{i})}[\eta_{4}] = \arg \min_{\delta \in \Omega_{i}} \lbrace \Vert \delta \! - \! p_{i} \Vert_{1} + \Vert \delta^{1} \! - \! \delta^{2} \Vert_{1} + \frac{1}{2} \Vert \delta - \eta_{4} \Vert^{2} \rbrace$ is not proximable, where $\eta_{4} \in \mathbb{R}^{2}$. Hence proximal algorithms [@DCFO1]-[@OFW] may not fit for this problem.
The Laplacian matrix of weight-unbalanced directed graph $\mathcal{G}$ is given as $$L_{4} =
\begin{bmatrix}
1 & 0 & 0 & -1 \\
-1 & 2 & -1 & 0 \\
0 & -1 & 1 & 0 \\
0 & 0 & -1 & 1 \\
\end{bmatrix}$$
We set $\alpha = 5$ and $\gamma = 0.2$ as coefficients in algorithm . Initial positions of agents 1, 2, 3, and 4 are set as $x_{1}(0) = [-4, 5.5]^{T}$, $x_{2}(0) = [6, 5]^{T}$, $x_{3}(0) = [5, -3.5]^{T}$, and $x_{4}(0) = [-5, -5]^{T}$. We set initial values for Lagrangian multipliers $v_{i}$ and auxiliary variables $z_{i}^{1}, z_{i}^{2}, w_{i}$ for $i \in \lbrace 1, 2, 3, 4 \rbrace$ as zeros.
Motions of system versus time and trajectories of $\sum_{i=1}^{4} x^{1}_{i}$ and $\sum_{i=1}^{4} x^{2}_{i}$ with algorithm ($\ref{Algorithm 2}$) are shown in Fig.$\ref{Fig.1}$, which show that resource allocation conditions $\sum_{i=1}^{4} x^{1*}_{i} = \sum_{i=1}^{4} d_{i}^{1} = 5$ and $\sum_{i=1}^{4} x^{2*}_{i} = \sum_{i=1}^{4} d_{i}^{2} = 1$ are satisfied. Fig.$\ref{Fig.2}$ gives trajectories of $x_{i}(t)$ for $i \in \lbrace 1, 2, 3, 4 \rbrace$. Fig.$\ref{Fig.3}$ shows the trajectory of $F(x)$, which proves that the global cost function is minimized. It can be seen from Fig.$\ref{Fig.1}$-Fig.$\ref{Fig.3}$ that all agents converge to the optimal solution which minimizes the global cost function and satisfies resource allocation conditions. Fig.$\ref{Fig.4}$ - Fig.$\ref{Fig.7}$ show trajectories of Lagrange multipliers $v_{i}(t)$ and auxiliary variables $z_{i}^{1}, z_{i}^{2}, w_{i}$ for $i \in \lbrace 1, 2, 3, 4 \rbrace$ respectively, which also verify the boundedness of system steered by algorithm .
![The trajectory of $F(x)$ with algorithm ($\ref{Algorithm 2}$)[]{data-label="Fig.3"}](f.eps){width="20.00000%"}
Conclusion
==========
In this paper, a class of nonsmooth resource allocation problems with directed graphs was solved via two distributed multi-proximal operator based primal-dual algorithms. The second algorithm considered a distributed estimator of the left eigenvector $h$ corresponding to $\lambda_{1}(L_{nq}) = 0$. These two algorithms were smoothed thanks to the multi-proximal splitting. Moreover, the design of the second proposed algorithm can also give a new viewpoint to tackle many widely studied distributed constrained resource allocation problems. Future extensions will involve considering nonsmooth resource allocation problems with switching topologies and more complex communication situations such as time delay and packet losses.
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---
abstract: 'We show that for a large class of finite groups $G$, the number of Galois extensions $E/{\mathbb{Q}}$ of group $G$ and discriminant $|d_E|\leq y$ grows like a power of $y$ (for some specified exponent). The groups $G$ are the regular Galois groups over ${\mathbb{Q}}$ and the extensions $E/{\mathbb{Q}}$ that we count are obtained by specialization from a given regular Galois extension $F/{\mathbb{Q}}(T)$. The extensions $E/{\mathbb{Q}}$ can further be prescribed any unramified local behavior at each suitably large prime $p\leq \log (y)/\delta$ for some $\delta\geq 1$. This result is a step toward the Malle conjecture on the number of Galois extensions of given group and bounded discriminant. The local conditions further make it a notable constraint on regular Galois groups over ${\mathbb{Q}}$. The method uses the notion of self-twisted cover that we introduce.'
address: 'Laboratoire Paul Painlevé, Mathématiques, Université Lille 1, 59655 Villeneuve d’Ascq Cedex, France'
author:
- Pierre Dèbes
bibliography:
- 'LocalMalle.bib'
title: 'On the Malle conjecture and the self-twisted cover'
---
Main results {#sec:main_results}
============
Given a finite group $G$ and a real number $y>0$, there are only finitely many Galois extensions $E/{\mathbb{Q}}$ (inside a fixed algebraic closure $\overline {\mathbb{Q}}$ of ${\mathbb{Q}}$) of group $G$ and discriminant $|d_E|\leq y$ (Hermite’s theorem). Estimating their number $N(G,y)$ is a classical topic (§\[ssec:malle\]). Here we consider the extensions $E/{\mathbb{Q}}$ obtained by specialization from a given Galois function field extension $F/{\mathbb{Q}}(T)$ of group $G$ (§\[ssec:contribution\]). We obtain estimates for the number of those which satisfy the above group and ramification conditions. Our lower bound (obviously also a lower bound for $N(G,y)$) already has the conjectural growth for $N(G,y)$: a power of $y$ (§\[ssec:main-statement\]). Furthermore the extensions $E/{\mathbb{Q}}$ we produce satisfy some additional local conditions at a finite – but growing with $y$ – set of primes. This provides noteworthy constraints (though unknown yet not to be non-vacuous) on regular Galois groups over ${\mathbb{Q}}$, related to analytic issues around the Tchebotarev theorem (§\[ssec:grunwald-tchebotarev\]). The role of the [*self-twisted cover*]{} from the title is explained in §\[ssec:role-self-twisted\].
The Malle conjecture \[ssec:malle\][is a classical landmark in this context]{}
------------------------------------------------------------------------------
It predicts that for some constant $a(G) \in ]0,1]$, specifically defined by Malle (recalled in §\[ssec:main-statement\]), and for all $\varepsilon >0$,
[(\*)]{} $c_1(G) \hskip 2pt y^{a(G)} \leq N(G,y) \leq c_2(G,\varepsilon) \hskip 2pt y^{a(G)+\varepsilon} \hskip 5mm \hbox{\rm for all $y>y_0(G,\varepsilon)$}$ 1,2mm for some positive constants $c_1(G)$, $c_2(G,\varepsilon)$ and $y_0(G,\varepsilon)$ [@Ma]. A more precise asymptotic for $N(G,Y)$ (as $y\rightarrow \infty$) is even offered in [@Ma2], namely $N(G,y) \sim c(G) \hskip 2pt y^{a(G)} \hskip 2pt (\log(y))^{b(G)}$, for some other specified constant $b(G)\geq 0$, and an another (unspecified) constant $c(G)>0$. The lower bound in (\*) is a strong statement; it implies in particular that $G$ is the Galois group of at least one extension $E/{\mathbb{Q}}$, which is an open question for many groups – the so-called [*Inverse Galois Problem*]{}. Relying on the Shafarevich theorem solving the IGP for solvable groups, Klüners and Malle proved the conjecture (\*) for nilpotent groups [@kluners-malle]. Klüners also established the lower bound for dihedral groups of order $2p$ with $p$ an odd prime [@kluners]. As to the more precise asymptotic for $N(G,y)$, it has only been proved for abelian groups [@wright], [@maki], for $S_3$ [@belabas-fouvry] [@bhargava-wood] and for generalized quaternion groups [@Klueners-habilitation ch.7, satz 7.6].
We point out that there is a more general form of the conjecture for [not necessarily Galois extensions]{} $E/{\mathbb{Q}}$ for which there are further significant results, notably in the case the group (of the Galois closure) is $S_n$ (with $n=[E:{\mathbb{Q}}]$): results of Davenport-Heilbronn [@davenport-heilbronn] and Datskovsky-Wright [@datskovsky-wright] ($n=3$), Bhargava [@barghava-S4], [@barghava-S5] ($n=4,5$), Ellenberg-Venkatesh [@MR2199231] (upper bounds). There is also a counter-example to this more general form of the conjecture [@kluners2]. Finally there are quite interesting investigations on analogs of the problem over function fields of finite fields [@MR2159381], [@MR2827801], [@ellenberg-et-al].
Our specialization approach {#ssec:contribution}
---------------------------
Besides solvable groups, there is another classical class of finite groups known to be Galois groups over ${\mathbb{Q}}$: those groups $G$ which are [*regular Galois groups*]{} over ${\mathbb{Q}}$, [*i.e.*]{}, such that there exists a Galois extension $F/{\mathbb{Q}}(T)$ of group $G$ with $F\cap \overline {\mathbb{Q}}= {\mathbb{Q}}$. In addition to abelian and dihedral groups, this class includes many non solvable groups, [*e.g.*]{} all symmetric and alternating groups and many simple groups. We obtain for all these groups a lower bound in $y^\alpha$ for $N(G,y)$, as predicted by the Malle estimate (\*). To our knowledge, this is a new step toward the conjecture. Expectedly our exponent $\alpha$ is smaller than its Malle counterpart $a(G)$: our approach only takes into account those extensions which are specializations of a geometric extension $F/{\mathbb{Q}}(T)$. Our result is rather a specialization result (as presented in §\[sec1\]). Still it is interesting to already get the right growth for $N(G,y)$ from a single extension $F/{\mathbb{Q}}(T)$.
In this geometric situation, Hilbert’s irreducibility theorem classically produces “many” $t_0 \in {{\mathbb{Q}}}$ such that the corresponding specialized extensions $F_{t_0}/{{\mathbb{Q}}}$ are Galois extensions of group $G$. Beyond making more precise these “many $t_0 \in {\mathbb{Q}}$” and controlling the corresponding discriminants, our goal requires a further step which is to show that many of these extensions are distinct. It is for this part that the [*self-twisted cover*]{}, a novel tool that we construct in §\[sec:self-twisted\], will be used (we say more in §\[ssec:role-self-twisted\]).
Statement of the main result {#ssec:main-statement}
----------------------------
In addition to being of group $G$ , we will be able to prescribe their local behavior at many primes to the Galois extensions $E/{\mathbb{Q}}$ that we will produce. The following notation helps phrase these “local conditions”.
Given a finite group $G$, a finite set $S$ of primes and for each $p\in S$, a subset ${\mathcal F}_p\subset G$ consisting of a non-empty union of conjugacy classes of $G$, the collection ${\mathcal F}= ({\mathcal F}_p)_{p\in S}$ is called a [*Frobenius data*]{} for $G$ on $S$. The number of Galois extensions $E/{\mathbb{Q}}$ of group $G$, of discriminant $|d_E| \leq y$ and which are unramified with Frobenius ${\rm Frob}_p(E/{\mathbb{Q}}) \in {\mathcal F}_p$ ($p\in S$) is denoted by $N(G,y,{\mathcal F})$.
The parameter $\delta(G)$ that appears below in the definition of our exponent is the [*minimal affine branching index*]{} of regular realizations of $G$ over ${\mathbb{Q}}$, [*i.e.*]{}, the minimal degree of the discriminant $\Delta_P(T)$ of a polynomial $P\in {\mathbb{Q}}[T,Y]$, monic in $Y$, such that ${\mathbb{Q}}(T)[Y]/\langle P\rangle$ is a regular Galois extension of ${\mathbb{Q}}(T)$ of group $G$.
\[thm:main\] Let $G$ be a regular Galois group over ${\mathbb{Q}}$, non trivial. There exists a constant $p_0(G)$ with the following property. For every $\delta >\delta(G)$, for every suitably large $y$ (depending on $G$ and $\delta$) and every Frobenius data ${{\mathcal F}_y}$ on ${\mathcal S}_y=
\{p_0(G) < p \leq \log (y)/\delta\}$, we have $$N(G,y,{\mathcal F}_y) \geq \hskip 2pt y^{\alpha(G,\delta)}\hskip 8mm \hbox{\it with}\hskip 3mm \alpha(G,\delta)= (1-1/|G|)/\delta.$$
Furthermore, the desired extensions $E/{\mathbb{Q}}$ can be obtained by specializing some regular realization $F/{\mathbb{Q}}(T)$ of $G$.
§\[sec1\] says more on $\delta(G)$. If a regular realization $F/{\mathbb{Q}}(T)$ of $G$ is given by a polynomial $P\in {\mathbb{Q}}[T,Y]$, monic in $Y$, then $\delta(G) < 2\hskip 1pt |G| \deg_T(P)$. One can then take $\delta=2 |G| \deg_T(P)$ in theorem \[thm:main\] or the more intrinsic value $\delta = 3 \hskip 1pt r \hskip 1pt |G|^3 \log|G|$ with $r$ the branch point number of $F/{\mathbb{Q}}(T)$. By comparison, Malle’s exponent is $a(G)= (|G| (1- 1/\ell))^{-1}$ where $\ell$ is the smallest prime divisor of $|G|$; inequality $a(G) \geq 1/\delta(G)$ is proved in general in lemma \[lem:exponent\], following a suggestion of G. Malle.
A more precise form of theorem \[thm:main\] is stated in §\[sec1\] which starts from any given regular realization $F/{\mathbb{Q}}(T)$ of $G$ and which shows other features of our result: ramification can also be prescribed at any finitely many suitably large primes under the assumption that $F/{\mathbb{Q}}(T)$ has at least one ${\mathbb{Q}}$-rational branch point; the exponent $\alpha(G,\delta)$ can be replaced by $1/\delta$ under Lang’s conjecture; the Hilbert irreducibility aspect is expanded; and there is an upper bound part; see theorem \[thm:main-plus\].
Extending theorem \[thm:main\] (and theorem \[thm:main-plus\]) to arbitrary number fields (instead of ${\mathbb{Q}}$) seems to present no major obstacles, only requiring to extend some ingredients we use, which are only available in the literature over ${\mathbb{Q}}$ but should hold over number fields. As each finite group is known to be a regular Galois group over some suitably big number field, we could deduce that the same is true for the lower bound part in the Malle conjecture: [*given any finite group, there is a number field $k_0$ such that a lower bound like in [(\*)]{} (appropriately generalized) holds over every number field containing $k_0$.*]{}
On the local conditions {#ssec:grunwald-tchebotarev}
-----------------------
\(a) Regarding this aspect, theorem \[thm:main\] improves on our previous work, with N. Ghazi, about the [Grunwald problem]{}. From [@DEGha], if $G$ is a regular Galois group over ${\mathbb{Q}}$, then every [*unramified Grunwald problem*]{} for $G$ at some finite set ${\mathcal S}$ of primes $p\geq p_0(G)$ can be solved, [*i.e.*]{} every collection of unramified extensions $E^p/{\mathbb{Q}}_p$ of group $H_p\subset G$ ($p\in {\mathcal S}$) is induced by some Galois extension $E/{\mathbb{Q}}$ of group $G$. Theorem \[thm:main\] does more: it provides, for every given discriminant size, a big number of such extensions $E/{\mathbb{Q}}$, a number that grows as in Malle’s predictions.
Malle had suggested that his estimates should hold with some local conditions [@Ma2 Remark 1.2]. However, unlike his, ours have a set of primes, ${\mathcal S}_y$, which grows with $y$. We focus below on this.
\(b) First we note that the set of primes where the local behavior can be prescribed as in theorem \[thm:main\] cannot be expected to be much bigger than the set ${\mathcal S}_y$: 0,5mm
- indeed, that every possible Frobenius data on ${\mathcal S}_y$ occurs in at least one extension $E/{\mathbb{Q}}$ counted by $N(G,y)$ already gives $N(G,y)\geq c^{u(y)}$, with $c$ the number of conjugacy classes of $G$ and $u(y)$ the number of primes in ${\mathcal S}_y=\{p_0(G) < p \leq \log (y)/\delta\}$. Now $c^{u(y)}$ compares to the conjectural upper bounds for $N(G,y)$: $\log c^{u(y)} \sim\log (y)/ \log(\log y)$ and $\log(y^{a(G)+\varepsilon}) \sim \log y$ (up to multiplicative constants). 0,5mm
- the restriction that the primes $p$ be suitably large ($p>p_0(G)$) cannot be removed either as the famous Wang’s counter-example shows [@wang]: no Galois extension $E/{\mathbb{Q}}$ of group ${\mathbb{Z}}/8{\mathbb{Z}}$ is unramified at $2$ with Frobenius of order $8$. Other counter-examples with other primes than $2$ have been recently produced by Neftin [@Neftin].
\(c) There is a further connection of our result with the Tchebotarev density theorem. The following definition helps explain it.
\[def:tche\] Given a real number $\ell \geq 0$, we say that a finite group $G$ is of [*Tchebotarev exponent*]{} $\leq \ell$, which we write ${\rm tch}(G)\leq \ell$, if there exist real numbers $m,\delta >0$ such that for every $x>m$ and every [*Frobenius data*]{} ${\mathcal F}_x= ({\mathcal F}_p)_{m<p\leq x}$ for $G$, there exists [*at least one*]{} Galois extension $E/{\mathbb{Q}}$ of group $G$ such that these two conditions hold:
1\. for each $m<p\leq x$, $E/{\mathbb{Q}}$ is unramified and ${\rm Frob}_p(E/{\mathbb{Q}}) \in {\mathcal F}_p$,
2\. $\log |d_E| \leq \delta x^\ell$.
Fix $\delta >\delta(G)$ and $m$ suitably large (in particular $m\geq p_0(G)$). Theorem \[thm:main\] for $y=e^{\delta x}$ with $x>m$ provides many[^1] extensions $E/{\mathbb{Q}}$ satisfying conditions of definition \[def:tche\] with $\ell = 1$.
\[cor:tch(G)\] If a finite group $G$ is a regular Galois group over ${\mathbb{Q}}$, then ${\rm tch}(G) \leq 1$.
On the other hand there is a universal lower bound for ${\rm tch}(G)$. Some famous estimates on the Tchebotarev theorem [@LaMoOd] (see also [@Lagarias-Odlyzko], [@Se_Cebotarev]) show that, under the General Riemann Hypothesis, for every finite group $G$, we have $${\rm tch}(G) > (1/2) - \varepsilon,\ \hbox{\rm for every}\ \varepsilon >0.\ \footnote{\cite{LaMoOd} also has an unconditional conclusion, which, using our terminology, leads to ${\rm tch}(G) \geq (\log \log x)/(2\log x)$ (with definition \ref{def:tche} extended to allow $\ell$ to be a function of $x$).} \leqno\hbox{\rm (**)}$$ (More precisely, they show that if a Galois extension $E/{\mathbb{Q}}$ is of group $G$ and $\log|d_E| \leq x^{1/2}/\log x$, there are at least $\pi(x) - 2x/(|G| \log x)$ non totally split primes $p\leq x$ in $E/{\mathbb{Q}}$ (with $\pi(x)$ is the number of primes $p\leq x$). As $\pi(x) - 2x/(|G| \log x) \rightarrow +\infty$, the trivial totally split behavior — ${\mathcal F}_p = \{1\}$ for each $m < p \leq x$ — does not occur if $x\gg 1$).
-1,5mm Corollary \[cor:tch(G)\] raises the question of whether ${\rm tch}(G)>1$ for some group $G$, in which case $G$ could not be a regular Galois group over ${\mathbb{Q}}$. Such a group may not exist (if the so-called Regular Inverse Galois Problem is true), while at the other extreme it cannot be ruled out at the moment that ${\rm tch}(G) =\infty$ for some group $G$. Many possibilities exist in between for Galois groups $G$ over ${\mathbb{Q}}$: that realizations exist that satisfy the local conditions of definition \[def:tche\] (1) or not, that the corresponding discriminants can be bounded as in definition \[def:tche\] (2), for some $\ell \in [1/2,\infty[$ or not. Somehow the Tchebotarev exponent provides a measure of the gap (possibly empty) between the [*classical*]{} and [*regular*]{} Inverse Galois Problems.
Gaining information on Tchebotarev exponents however seems difficult. Even for $G={\mathbb{Z}}/2{\mathbb{Z}}$ and the case of the totally split behavior, for which the problem amounts to bounding the least square-free integer $d_m(x)$ that is a quadratic residue modulo each prime $m<p\leq x$. Chan-ging $1/2$ to $1$ in (\*\*), the remaining possible improvement (as ${\mathbb{Z}}/2{\mathbb{Z}}$ is a regular Galois group over ${\mathbb{Q}}$), is plausible as some easy heuristics show but relates to deep questions in number theory ([*e.g.*]{} [@Se_Cebotarev §2.5]).
Role of the self-twisted cover {#ssec:role-self-twisted}
------------------------------
Our method starts with a regular realization $F/{\mathbb{Q}}(T)$ of $G$. The extensions $E/{\mathbb{Q}}$ that we wish to produce will be specializations $F_{t_0}/{\mathbb{Q}}$ of it at some integers $t_0$. A key tool is the twisting lemma from [@DEGha], which reduces the search of specializations of a given type to that of rational points on a certain twisted cover. We use it twice, first over ${\mathbb{Q}}_p$ as in [@DEGha], to construct specializations $F_{t_0}/{\mathbb{Q}}$ with a specified local behavior. A main ingredient for this first stage is the Lang-Weil estimate for the number of rational points on a curve over a finite field. We obtain many good specialisations $t_0\in {\mathbb{Z}}$ and a lower bound for their number.
The next question is to bound the number of the corresponding specializations $F_{t_0}/{\mathbb{Q}}$ that are non-isomorphic. First we reduce it to counting integral points of a given size on certain twisted covers. This is our second use of the twisting lemma, over ${\mathbb{Q}}$ this time. For the count of the integral points, we use a result of Walkowiak [@Wa][^2], based on a method of Heath-Brown [@HB]. However the bounds from [@Wa] involve the height of the defining polynomials, which here depend on the specializations $F_{t_0}/{\mathbb{Q}}$. We have to control the dependence in $t_0$. This is where enters the self-twisted cover, which as we will see, is a family of covers, depending only on the original extension $F/{\mathbb{Q}}(T)$ and which has all the twisted covers among its fibers. As a result, a bound of the form $c_1 t_0^{c_2}$ for the height of the polynomials above will follow with $c_1$ and $c_2$ depending only on $F/{\mathbb{Q}}(T)$.
In §2 below we present theorem \[thm:main-plus\], the more precise version of theorem \[thm:main\]. §3 is the construction of the self-twisted cover. §4 gives the proof of theorem \[thm:main-plus\].
[**Acknowledgement.**]{} I am grateful to G. Malle and J. Klüners for their interest in the paper and some valuable suggestions.
-1300
The specialization version of the result {#sec1}
========================================
Basics {#ssec:basics}
------
Given a field $k$, we work without distinction with a regular extension $F/k(T)$ or with the associated $k$-cover $f:X\rightarrow {\mathbb{P}}^1$: $f$ is the normalization of ${\mathbb{P}}^1_k$ in $F$ and $F$ is the function field $k(X)$ of $X$. We assume that $k$ is of characteristic $0$; $k={\mathbb{Q}}$ in most of the paper.
Recall that a [*(regular) $k$-cover of ${\mathbb{P}}^1$*]{} is a finite and generically unramified morphism $f:X \rightarrow {\mathbb{P}}^1$ defined over $k$ with $X$ a normal and geometrically irreducible variety. The $k$-cover $f:X \rightarrow {\mathbb{P}}^1$ is said to be [*Galois*]{} if the field extension $k(X)/k(T)$ is Galois; if in addition $f:X\rightarrow {\mathbb{P}}^1$ is given together with an isomorphism $G\rightarrow \Gal(k(X)/k(T))$, it is called a (regular) $k$-G-[*Galois cover*]{} of group $G$.
By [*group*]{} and [*branch point set*]{} of a $k$-cover $f$, we mean those of the $\overline k$-cover $f\otimes_k\overline k$: the group of a $\overline k$-cover $X\rightarrow {\mathbb{P}}^1$ is the Galois group of the Galois closure of the extension $\overline k(X)/\overline k(T)$. The branch point set of $f\otimes_k\overline k$ is the (finite) set of points $t\in {\mathbb{P}}^1(\overline k)$ such that the associated discrete valuations are ramified in the extension $\overline k(X)/\overline k(T)$.
Given a regular Galois extension $F/k(T)$ and $t_0\in {\mathbb P}^1(k)$, the [*specialization of $F/{\mathbb{Q}}(T)$ at $t_0$*]{} is the residue extension of an (arbitrary) prime above $\langle T-t_0\rangle$ in the integral closure of ${\mathbb{Q}}[T]_{\langle T-t_0\rangle}$ in $F$ (as usual use ${\mathbb{Q}}[1/T]_{\langle 1/T\rangle}$ instead if $t_0=\infty$). We denote it by $F_{t_0}/k$.
Given a regular Galois extension $F/k(T)$, we say a prime $p$ is [*good*]{} for $F/{\mathbb{Q}}(T)$ if $p\not|\hskip 1mm |G|$, the branch divisor ${\bf t}=\{t_1,\ldots,t_r\}$ is étale at $p$ and there is no vertical ramification at $p$; and it is [*bad*]{} otherwise. We refer to [@DEGha] for the precise definitions. We only use here the standard fact that there are only finitely many bad primes.
The minimal affine branching index $\delta(G)$ {#ssec:exponent}
----------------------------------------------
Given a regular extension $F/{\mathbb{Q}}[T]$, we call the irreducible polynomial $P(T,Y)$ of a primitive element, integral over ${\mathbb{Z}}[T]$, an [*integral affine model*]{} of $F/{\mathbb{Q}}(T)$; $P(T,Y)\in {\mathbb{Z}}[T]$ and is monic in $Y$. Denote the discriminant of $P$ relative to $Y$ by $\Delta_P(T)\in {\mathbb{Z}}[T]$ and its degree by $\delta_P$. The minimal degree $\delta_P$ obtained in this manner is called the [*minimal affine branching index of $F/{\mathbb{Q}}(T)$*]{} and denoted by $\delta_{F/{\mathbb{Q}}(T)}$. For any integral affine model $P(T,Y)$ of $F/{\mathbb{Q}}(T)$, we have $$\delta_{F/{\mathbb{Q}}(T)} \leq \delta_P < 2 |G| \deg_T(P).$$
If $G$ is a regular Galois group over ${\mathbb{Q}}$, the parameter $\delta(G)$ involved in theorem \[thm:main\] is the minimum of all $\delta_{F/{\mathbb{Q}}[T]}$ with $F/{\mathbb{Q}}[T]$ running over all regular realizations of $G$.
-1200
\[lem:exponent\] Let $G$ be a non trivial regular Galois group over ${\mathbb{Q}}$.
[(a)]{} If $F/{\mathbb{Q}}(T)$ is a regular realization of $G$ with $r$ branch points and $g$ is the genus of $F$, then $$\delta(G) < 3 \hskip 1pt (2g+1) |G|^2 \log|G| \leq 3 \hskip 1pt r \hskip 1pt |G|^3 \log|G|.$$
[(b)]{} Furthermore we have $\displaystyle \delta(G) \geq 1/a(G).$
\(a) The first inequality follows from a result of Sadi [@SadiT §2.2] which provides an affine model $P(T,Y)$ of $F/{\mathbb{Q}}(T)$ such that $$\deg_T(P) \leq (2g+1) |G| \log|G| /\log 2.$$ The second inequality follows from the Riemann-Hurwitz formula.
\(b) Let $F/{\mathbb{Q}}(T)$ be a regular realization of $G$, $d_F\in {\mathbb{Q}}[T]$ be the absolute discriminant of $F/{\mathbb{Q}}(T)$ (the discriminant of a ${\mathbb{Q}}[T]$-basis of the integral closure of ${\mathbb{Q}}[T]$ in $F$) and $P(T,Y)$ be an integral affine model of $F/{\mathbb{Q}}(T)$. Inequality (b) follows from the following ones: $$\delta_P \geq \deg(d_F) \geq |G| (1- 1/\ell) = a(G).$$
where $\ell$ is as before the smallest prime divisor of $|G|$.
The first inequality $\deg(d_F) \leq \delta_P$ is standard. Classically the polynomial $d_F$ is a generator of the ideal $N_{F/{\mathbb{Q}}(T)}({\mathcal D}_{F/{\mathbb{Q}}(T)})$ where ${\mathcal D}_{F/{\mathbb{Q}}(T)}$ is the different and $N_{F/{\mathbb{Q}}(T)}$ is the norm relative to the extension $F/{\mathbb{Q}}(T)$ [@SeCorps_locaux III, §3]. From [@SeCorps_locaux III, §6], in the prime ideal decomposition ${\mathcal D}_{F/{\mathbb{Q}}(T)} =\prod_{\mathcal P} \mathcal P^{u_{\mathcal P}}$, we have $u_{\mathcal P}\geq e_{{\mathcal P}}-1$ for each prime ${\mathcal P}$, where $e_{\mathcal P} = e_{\frak p}$ is the corresponding ramification index, which only depends on the prime ${\frak p}$ below ${\mathcal P}$. The following inequalities, where $f_{{\mathcal P}}$ denotes the residue degree of ${\mathcal P}$, finish the proof:
$ \deg(d_F) \geq \sum_{{\frak p}} \sum_{{\mathcal P} | {\frak p}} f_{\mathcal P}(e_{\mathcal P} - 1) = \sum_{{\frak p}} |G| - |G|/e_{{\frak p}} \geq |G| (1-1/\ell)$
Our parameter $\delta (G)$ can also be compared to the minimum, say $\rho(G)$, of all branch point numbers $r$ of regular realizations $F/{\mathbb{Q}}(T)$ of $G$: for such an extension $F/{\mathbb{Q}}(T)$ we have $\deg(d_F)\geq r-1$ whence $\delta(G) \geq \rho(G) - 1$. But the inequality $a(G) \geq 1/(\rho(G)-1)$ does not hold in general. For example the symmetric group $S_n$ can be regularly realized over ${\mathbb{Q}}$ with $3$ branch points so $\rho(S_n)= 3$ while $a(S_n) = 2/n!$ The analog of theorem \[thm:main\] with $r-1$ replacing $\delta_{F/{\mathbb{Q}}(T)}$ is false if the upper bound part of Malle’s conjecture is true.
The specialization result {#ssec:generalization}
-------------------------
Theorem \[thm:main-plus\] is a more precise version of our main result. It gives explicit estimates from which the asymptotic estimates of theorem \[thm:main\] can easily be deduced (as explained in §\[main-plus-implies-main\]). Another difference is that it starts from a given regular Galois extension $F/{\mathbb{Q}}(T)$ and the extensions $E/{\mathbb{Q}}$ we count are specializations of it. Below are the necessary additional notation and data.
### Notation
The following notation is used throughout the paper:
- for a Frobenius data ${{\mathcal F}}$ on a set of primes $S$, the product of all ratios $|{\mathcal F}_p|/{|G|}$ with $p\in S$, [*i.e.*]{} the [*density*]{} of ${{\mathcal F}}$, is denoted by $\chi({\mathcal F})$,
- for a finite set $S$ of primes, set $\Pi(S)=\prod_{p\in S}p$,
- we also use the classical functions $\pi(x)$ and ${\Pi}(x)$ to denote respectively the number of primes $\leq x$ and the product of all primes $\leq x$. We have the classical asymptotics at $\infty$: $\pi(x) \sim x/\log(x)$ and $\log \Pi(x) \sim x$,
- the height of a polynomial $F$ with coefficients in ${\mathbb{Q}}$ is the maximum of the absolute values of its coefficients and is denoted by $H(F)$.
### Data {#ssec:data}
Fix the following for the rest of the paper:
- $G$ is a non trivial finite group,
- $F/{\mathbb{Q}}(T)$ is a regular Galois extension of group $G$, - $f:X\rightarrow {\mathbb{P}}^1$ is the corresponding ${\mathbb{Q}}$-cover,
- ${\mathbf t} = \{t_1,\ldots,t_r\} \subset {\mathbb{P}}^1(\overline {\mathbb{Q}})$ is the branch point set of $F/{\mathbb{Q}}(T)$,
- $g$ is the genus of the curve $X$,
- $p_0(F/{\mathbb{Q}}(T))$ is the prime defined as follows. Let $p_{-1}$ be the biggest prime $p$ such $p$ is bad for $F/{\mathbb{Q}}(T)$ or $p< r^2 |G|^2$. Then $p_0(F/{\mathbb{Q}}(T))$ is the smallest prime $p$ such that the interval $]p_{-1},p_0]$ contains as many primes as there are non-trivial conjugacy classes of $G$. For the rest of the paper we fix a prime $p_0 \geq p_0(F/{\mathbb{Q}}(T))$.
- $\delta_{F/{\mathbb{Q}}(T)}$ or $\delta_F$ for short is the minimal affine branching index of $F/{\mathbb{Q}}(T)$,
- $P(T,Y)$ is an integral affine model of $F/{\mathbb{Q}}(T)$ such that $\delta_P=\delta_F$ (with $\delta_P$ the degree of the discriminant $\Delta_P$) and which is primitive, [*i.e.*]{} has relatively prime coefficients (an assumption which one can always reduce to),
- $S$ is a finite set of primes subject to these conditions:
\(a) if no branch point of $f$ is in ${\mathbb{Z}}$ then $S=\emptyset$.
\(b) if at least one of the branch points of $f$, say $t_1$ is in ${\mathbb{Z}}$, then $S$ is a finite set of good primes $p$, not dividing $t_1$ and not in $]p_{-1},p_0]$.
(If at least one branch point is ${\mathbb{Q}}$-rational, one can reduce to the assumption in (b) [*via*]{} a simple change of the variable $T$). - for $x> p_0$, $S_x$ is the set of primes $p$ such that $p_0<p\leq x$ and $p\notin S$.
- for technical reasons we change the condition “$|d_E| \leq y$” from theorem \[thm:main\] to the more complicated one “$|d_E| \leq \rho(x)$” where $$\rho(x) = (1+\delta_P) H(\Delta_P) [\Pi(S) \hskip 1pt \Pi(x)]^{\delta_P}$$ -1mm (we have $\log \rho(x) \sim \delta_P x$ and so a simple change of variable leads back to the original condition).
### Statement {#ssec:statement}
For $x> p_0$ let ${\mathcal F}_x$ be a Frobenius data on ${S}_x$. Theorem \[thm:main-plus\] is about the number $N_F(x,S,{\mathcal F}_x)$ of distinct specializations $F_{t_0}/{\mathbb{Q}}$ at points $t_0\in {\mathbb{Z}}$ that satisfy
[(i)]{} $\Gal(F_{t_0}/{\mathbb{Q}}) = G$,
[(ii)]{} for each $p\in {S}_x$, $F_{t_0}/{\mathbb{Q}}$ is unramified and ${\rm Frob}_p(F_{t_0}/{\mathbb{Q}}) \in {\mathcal F}_p$,
[(iii)]{} for each $p\in S$, $F_{t_0}/{\mathbb{Q}}$ is ramified at $p$,
[(iv)]{} $|d_{F_{t_0}}| \leq \rho(x)$.
\[thm:main-plus\]
[(a)]{} There exist constants $C_1,C_2,C_3,C_4$ only depending on $P(T,Y)$ such that for every $x>p_0$, we have $$N_F(x,S,{\mathcal F}_x) \geq \hskip 2pt C_{1} \hskip 2pt \frac{\chi({\mathcal F}_x)}{\Pi(S)^2} \hskip 4pt \frac{{\Pi}(x)^{1-1/|G|}}{(\log {\Pi}(x))^{C_2} \hskip 2pt C_3^{\pi(x)}} - C_4$$ so $\log(N_F(x,S,{\mathcal F}_x))$ is bigger than a function $\lambda(x) \sim (1-1/|G|)\hskip 1pt x$.
1,5mm [(b)]{} Furthermore the specializations $F_{t_0}/{\mathbb{Q}}$ counted by the lower bound can be taken to be specializations at integers $t_0 \in [1, \Pi({S})\hskip 2pt \Pi(x)]$. 1,5mm
[(c)]{} Under Lang’s conjecture on rational points on a variety of general type and if $g\geq 2$, we have this better inequality $$N_F(x,S,{\mathcal F}_x) \geq \hskip 2pt C_{5} \hskip 2pt \frac{\chi({\mathcal F}_x)}{\Pi(S)^2} \hskip 4pt \frac{{\Pi}(x)}{C_6^{\pi(x)}} - C_7$$ for constants $C_5,C_6,C_7$ only depending on $P$, so then $\log(N_F(x,S,{\mathcal F}_x))$ is bigger than a function $\lambda(x) \sim x$. 1,5mm
[(d)]{} We have the following upper bound for the number ${\mathcal N}_F(x,{\mathcal F}_x)$ of integers $t_0 \in [1,\Pi({S_x)}]$ such that condition [(ii)]{} above holds: $${\mathcal N}_F(x,{\mathcal F}_x) \leq \hskip 2pt \chi({\mathcal F}_x) \hskip 1mm \frac{\Pi({S_x)}}{\beta} \hskip 1mm \left( 2 - \lambda\right)^{|S_x|}$$ where $\lambda = (r|G|-1)/r^2 |G|^2\in ]0,1/4]$ and $\beta$ depends only on $F/{\mathbb{Q}}(T)$.
### Proof of theorem \[thm:main\] assuming theorem \[thm:main-plus\] {#main-plus-implies-main}
Pick a regular realization $F/{\mathbb{Q}}(T)$ of $G$ and an integral affine model $P(T,Y)$ such that $\delta_P=\delta_F=\delta(G)$. Set $p_0(G) = p_0(F/{\mathbb{Q}}(T))$. Fix $\delta > \delta(G)$ and set $\delta^- = (\delta+\delta(G))/2$. Let $y>0$ and $x= \log (y)/\delta^-$.
As $\delta^-<\delta$ we have ${\mathcal S}_y \subset {S}_x$. Complete the given Frobenius data ${\mathcal F}_y$ on ${\mathcal S}_y$ in an arbitrary way to make it a Frobenius data ${\mathcal F}_x$ on $S_x$. Apply theorem \[thm:main-plus\] to ${\mathcal F}_x$ and $S=\emptyset$. As $\log \rho(x)\sim (\delta_P/\delta^-) \log y$, if $y$ is suitably large, $\rho(x) \leq y$. It follows that $N(G,y,{\mathcal F}_y) \geq N_F(x,\emptyset ,{\mathcal F}_x)$ and so $N(G,y,{\mathcal F}_y)$ can be bounded from below by the right-hand side term of the inequality from theorem \[thm:main-plus\] (a) with $x= \log (y)/\delta^-$. The logarithm of this term is asymptotic to $(1-1/|G|) \log (y) /\delta^-$. Conclude that for suitably large $y$, this term is bigger than $y^{(1-1/|G|)/\delta}$.
### Remarks
\(a) In the situation $S\not=\emptyset$, for which it is possible to prescribe ramification at some primes, the assumption that at least one branch point is ${\mathbb{Q}}$-rational cannot be removed, as explained in Legrand’s paper [@Legr1], which is devoted to this situation. Many groups have a regular realization $F/{\mathbb{Q}}(T)$ satisfying this assumption, although being of even order is a necessary condition [@Legr1 §3.2]: abelian groups of even order, symmetric groups $S_n$ ($n\geq 2$), alternating groups $A_n$ ($n\geq 4$), many simple groups (including the Monster), etc.
For these groups, theorem \[thm:main-plus\] leads to a generalized version of the inequality from theorem \[thm:main\] where the left-hand term $N(G,y,{\mathcal F}_y)$ is replaced by the (smaller) number, say $N(G,S,y,{\mathcal F}_y)$, of extensions $E/{\mathbb{Q}}$ which, in addition to the conditions prescribed in theorem \[thm:main\], are required to ramify at every prime from a finite set $S$ of suitably big primes (and where the set ${\mathcal S}_y$ is of course replaced by ${\mathcal S}_y\setminus S$).
\(b) The upper bound in theorem \[thm:main-plus\] (d) concerns extensions $E/{\mathbb{Q}}$ that are specializations of a given regular extension $F/{\mathbb{Q}}(T)$ (at integers $t_0$) and so does not directly lead to upper bounds for $N(G,y,{\mathcal F}_y)$. A natural hypothesis to make in this context is that $G$ has a generic extension $F/{\mathbb{Q}}(T)$ (or more generally a parametric extension, as defined in [@Legr1]): indeed then all Galois extensions $E/{\mathbb{Q}}$ of group $G$ are specializations of $F/{\mathbb{Q}}(T)$ (at points $t_0\in {\mathbb{Q}}$). But only the four groups $\{1\}$, ${\mathbb{Z}}/2{\mathbb{Z}}$, ${\mathbb{Z}}/3{\mathbb{Z}}$, $S_3$ have a generic extension $F/{\mathbb{Q}}(T)$ [@JLY p.194].
### An application of the upper bound part from theorem \[thm:main-plus\]
For every $x > p_0$, let ${\mathcal N}_{\rm tot.split}(x)$ be the set of all integers $t_0\geq 1$ such that the specialization $F_{t_0}/{\mathbb{Q}}$ is totally split at each prime $p_0 <p\leq x$, [*i.e.*]{} which satisfy condition (ii) from theorem \[thm:main-plus\] with ${\mathcal F}_p$ taken to be the trivial conjugacy class for each $p\in S_x$.
For every $x > p_0$, ${\mathcal N}_{\rm tot.split}(x)$ is a union of (many) cosets modulo $\Pi({S_x})$ but its density decreases to $0$ as $x\rightarrow +\infty$.
We anticipate on §\[sec:proof\] to say that ${\mathcal N}_{\rm tot.split}(x)$ is a union of cosets modulo $\Pi({S_x})$ (see proposition \[prop:first-part\] (c)) and focus on the density part of the statement. Every integer $t_0\in {\mathcal N}_{\rm tot.split}(x)$ writes $t_0 = u + k \hskip 2pt \Pi({S_x})$ with $u$ one of the elements in $[1,\Pi({S_x})]$ counted by ${\mathcal N}_F(x,{\mathcal F}_x)$ and $k\in {\mathbb{Z}}$. Let $N\geq 1$ be any integer. If $1\leq t_0 \leq N$, then $k \leq N/\Pi({S_x})$. It follows then from theorem \[thm:main-plus\] (d) that $$\left|{\mathcal N}_{\rm tot.split}(x) \cap [1,N]\right| \leq \frac{N}{\Pi({S_x})} \times {\mathcal N}_F(x,{\mathcal F}_x) \leq \hskip 2pt \frac{N}{\beta} \times \left( \frac{2-\lambda}{|G|}\right)^{|S_x|}$$
which divided by $N$ tends to $0$ as $x\rightarrow +\infty$.
Similar density conclusions can be obtained for other local behaviors for which the sets ${\mathcal F}_p$ are not too big compared to $G$.
The self-twisted cover {#sec:self-twisted}
======================
In §\[sec:twisting\], we recall the twisting operation on covers and the twisting lemma (§\[ssec:twisting\_lemma\]) and explain the motivation for introducing the self-twisted cover (§\[ssec:motivation\]). §\[ssec:self-twisted\] is devoted to its construction. Covers are viewed here as fundamental group representations. The correspondence is briefly recalled in §\[ssec:fund-group-basics\].
Twisting G-Galois covers {#sec:twisting}
------------------------
For the material of this subsection, we refer to [@DEGha].
### Fundamental groups representations of covers {#ssec:fund-group-basics}
Given a field $k$, denote its absolute Galois group by $\Gabs_k$.
If $E/k$ is a Galois extension of group $G$, an epimorphism $\varphi: \Gabs_k\rightarrow G$ such that $E$ is the fixed field of ${\rm ker}(\varphi)$ in $\overline k$ is a called a [*$\Gabs_k$-representation*]{} of $E/k$.
Given a finite subset ${\mathbf t}\subset {\mathbb{P}}^1(\overline k)$ invariant under $\Gabs_k$, the $k$-fundamental group of ${\mathbb{P}}^1\setminus {\mathbf t}$ is denoted by $\pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_k$; here $t$ denotes the fixed [*base point*]{}, which corresponds to choosing an embedding of $k(T)$ in an algebraically closed field $\Omega$. The $\overline k$-fundamental group $\pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{\overline k}$ is defined as the Galois group of the maximal algebraic extension $\Omega_{{\mathbf t},k}/\overline k(T)$ (inside $\Omega$) unramified above ${\mathbb{P}}^1\setminus {\mathbf t}$ and the $k$-fundamental group $\pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{k}$ as the group of the Galois extension $\Omega_{{\mathbf t},k}/k(T)$.
Degree $d$ $k$-covers of ${\mathbb{P}}^1$ (resp. $k$-G-Galois covers of ${\mathbb{P}}^1$ of group $G$) with branch points in $\mathbf t$ correspond to transitive homomorphisms $\pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{k} \rightarrow S_d$ (resp. to epimorphisms $\pi_1({\mathbb{P}}^1\setminus {\mathbf t})_{k},t) \rightarrow G$), with the extra regularity condition that the restriction of $\phi$ to $\pi_1(B\setminus D, t)_{\overline k}$ remains transitive (resp. remains onto). These corresponding homomorphisms are called the [*fundamental group representations*]{} (or [*$\pi_1$-representations*]{} for short) of the cover $f$ (resp the G-cover $f$).
Each $k$-rational point $t_0\in {\mathbb{P}}^1(k)\setminus {\mathbf t}$ provides a section ${\sf s}_{t_0}: \Gabs_k\rightarrow \pi_1({\mathbb{P}}^1\setminus {\mathbf t},t)_{k}$ to the exact sequence $$1\rightarrow \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{\overline k} \rightarrow \pi_1({\mathbb{P}}^1\setminus {\mathbf t},t)_k \rightarrow \Gabs_k \rightarrow 1$$ which is uniquely defined up to conjugation by an element in the fundamental group $\pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{\overline k}$.
If $\phi: \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_k \rightarrow G$ represents a $k$-G-Galois cover $f:X\rightarrow {\mathbb{P}}^1$, the morphism $\phi \circ {\sf s}_{t_0}:\Gabs_k \rightarrow G$ is the [*specialization representation*]{} of $\phi$ at $t_0$. The fixed field in $\overline k$ of ${\rm ker}(\phi \circ {\sf s}_{t_0})$ is the specialization $k(X)_{t_0}/k(T)$ of $k(X)/k(T)$ at $t_0$ (defined in §\[ssec:basics\]).
### The twisting lemma {#ssec:twisting_lemma}
We recall how a regular $k$-G-Galois cover $f:X\rightarrow {\mathbb{P}}^1$ of group $G$ can be twisted by some Galois extension $E/k$ of group $H\subset G$. Formally this is done in terms of the associated $\pi_1$- and $\Gabs_k$- representations. Let $\phi: \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_k \rightarrow G$ be a $\pi_1$-representation of the regular $k$-G-cover $f:X\rightarrow {\mathbb{P}}^1$ and $\varphi:\Gabs_k \rightarrow G$ be a $\Gabs_k$-representation of the Galois extension $E/k$.
Denote the right-regular (resp. left-regular) representation of $G$ by $\delta: G\rightarrow S_d$ (resp. by $\gamma: G\rightarrow S_d$) where $d=|G|$. Define $\varphi^\ast:\Gabs_k \rightarrow G$ by $\varphi^\ast (g) = \varphi (g) ^{-1}$. Consider the map $\widetilde \phi^\varphi:
\pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_k \rightarrow S_d$ defined by the following formula, where $r$ is the restriction map $\pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_k \rightarrow \Gabs_k$ and $\times$ is the multiplication in the symmetric group $S_d$: $$\widetilde \phi^\varphi(\theta) =
\gamma \phi(\theta) \times \delta \varphi^\ast r (\theta) \hskip 6mm (\theta\in \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_k).$$ The map $\widetilde \phi^\varphi$ is a group homomorphism with the same restriction on $\pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{\overline k}$ as $\phi$. It is called the [*twisted representation*]{} of $\phi$ by $\varphi$. The associated regular $k$-cover is a $k$-model of the cover $f\otimes_k{\overline k}$. It is denoted by $\widetilde f^\varphi:
\widetilde X^\varphi \rightarrow {\mathbb{P}}^1$ and called the [*twisted cover*]{} of $f$ by $\varphi$. The following statement is the main property of the twisted cover.
\[prop:twisted cover\] Let $t_0\in {\mathbb{P}}^1(k)\setminus {\mathbf t}$. Then the specialization representation $\phi \circ {\sf s}_{t_0}:\Gabs_k \rightarrow G$ is conjugate in $G$ to $\varphi:\Gabs_k \rightarrow G$ if and only if there exists $x_0\in \widetilde X^\varphi(k)$ such that $\widetilde f^\varphi(x_0)=t_0$.
### The motivation for the self-twisted cover {#ssec:motivation}
As explained in §\[ssec:role-self-twisted\], we will have to control the height of some polynomials defining some twisted covers. These twisted covers are obtained by twisting the given G-Galois cover $f:X\rightarrow {\mathbb{P}}^1$ by its own specializations ${\mathbb{Q}}(X)_{u_0}/{\mathbb{Q}}$ ($u_0\in {\mathbb{Q}}$); we call them the [*fiber-twisted covers*]{}. §\[ssec:self-twisted\] shows that the fiber-twisted covers are all members of an algebraic family of covers: the [*self-twisted cover*]{}. The practical use for the end of the paper is the following result. It is a consequence of lemma \[lemma:spec-self-twist\].
\[prop:self-twist\] Given a regular $k$-G-cover $f:X\rightarrow {\mathbb{P}}^1$, there exists a polynomial $\widetilde P(U,T,Y) \in k[U,T,Y]$ irreducible in $\overline{k(U)}(T)[Y]$, monic in $Y$, and a finite set ${\mathcal E}\subset k$ such that for every $u_0\in k\setminus {\mathcal E}$,
[(a)]{} $\widetilde P(u_0,T,Y)$ is irreducible in $\overline{k}(T)[Y]$,
[(b)]{} $\widetilde P(u_0,T,Y)$ is an integral affine model of the fiber-twisted cover of $f$ at $u_0$, [(c)]{} the genus of the curve $\widetilde P(u_0,t,y)=0$ is $g$.
The self-twisted cover {#ssec:self-twisted}
----------------------
Let $U$ be a new indeterminate (algebraically independent from $T$ and $Y$). Fix an algebraically closed field $\Omega$ containing $k(T,U)$, which we will use as a common base point $t$ for all fundamental groups involved. The algebraic closures of $k(T,U)$, $k(T)$, $k(U)$ and $k$ should be understood as the ones inside $\Omega$.
### A $\pi_1$-representation of $f\otimes_kk(U)$
As the compositum $\Omega_{{\mathbf t},k} \cdot\overline{k(U)}$ is contained in $\Omega_{{\mathbf t},k(U)}$, there is a restriction morphism $${\rm res}_{k(U)/k}: \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{{k(U)}} \rightarrow \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{k},$$ which induces a map between the geometric parts of the fundamental groups: $${\rm res}_{\overline{k(U)}/\overline k}: \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{\overline{k(U)}}\rightarrow \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{\overline k}$$ We also use the notation ${\rm res}_{\overline{k(U)}/\overline k}$ for the map $\Gabs_{k(U)} \rightarrow \Gabs_k$ induced on the absolute Galois groups.
${\rm res}_{k(U)/k}: \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{{k(U)}} \rightarrow \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{k}$ is surjective and ${\rm res}_{\overline{k(U)}/\overline k}: \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{\overline{k(U)}}\rightarrow \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{\overline k}$ is an isomorphism.
Every $\sigma \in \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{k}$ extends to an element of $\Gabs_{k(T)}$, which extends naturally to an automorphism of $\overline{k(T)}(U)$ fixing $U$ (and $k(T)$), which in turn extends to an element $\tilde \sigma \in \Gabs_{{k}(T,U)}$. As ${\mathbf t}$ is $\Gabs_k$-invariant, $\tilde \sigma$ permutes the extensions $F/{k(U)}(T)$ that are unramified above ${\mathbb{P}}^1\setminus \mathbf t$. Conclude that $\tilde \sigma$ factors through $\pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{{k(U)}}$ to provide a preimage of $\sigma$ [*via*]{} the map ${\rm res}_{k(U)/k}$, as desired in the first statement. To show that ${\rm res}_{\overline{k(U)}/\overline k}:\pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{\overline{k(U)}}\rightarrow \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{\overline k}$ is surjective, it suffices to show that the following morphism is: $$\Gal(\Omega_{{\mathbf t},k} \cdot \overline{k(U)} / \overline{k(U)}(T)) \rightarrow \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{\overline k}=\Gal(\Omega_{{\mathbf t},k} / \overline{k}(T)).$$ This morphism is in fact an isomorphism: indeed extending the base field from $\overline k$ to $\overline{k(U)}$ (over which $T$ is transcendental) does not change the group of regular Galois extensions.
As $k$ is of characteristic $0$, the morphism ${\rm res}_{\overline{k(U)}/\overline k}: \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{\overline{k(U)}}\rightarrow \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{\overline k}$ is even an isomorphism. More precisely, it follows from [@Serre-topics theorem 6.3.3] that $\Omega_{{\mathbf t},k(U)}= \Omega_{{\mathbf t},k} \cdot \overline{k(U)}$.
Set $\phi \otimes_k k(U) = \phi \circ {\rm res}_{{k(U)}/k}$. The epimorphism $$\phi \otimes_k k(U): \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{k(U)} \rightarrow G$$ is a $\pi_1$-representation of the regular G-Galois cover $f\otimes_kk(U)$.
### A $\Gabs_{k(U)}$-representation
Composing $\phi \otimes_k k(U)$ with the section ${\sf s}_{U}: \Gabs_{k(U)} \rightarrow \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{k(U)}$ associated with the point $U\in {\mathbb{P}}^1(k(U))$ provides a $\Gabs_{k(U)}$-representation $$\phi_{U}: \Gabs_{k(U)} \rightarrow G$$ which is the specialization representation of $\phi \otimes_k k(U)$ at $t=U$. It corresponds to the generic fiber of $F/k(T)$. Denote it by $F_U/k(U)$.
### The self-twisted cover {#the-self-twisted-cover}
Twist the representation $\phi \otimes_k k(U)$ by the epimorphism $\phi_U$ to get the [*self-twisted representation*]{} $$\widetilde{\phi \otimes_k k(U)}^{\phi_U}: \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{k(U)} \rightarrow S_d.$$ We call the corresponding cover $$\widetilde{f\otimes_k k(U)}^{\phi_U}: \widetilde{X \otimes_k k(U)}^{\phi_U}\rightarrow {\mathbb{P}}^1_{k(U)}$$ the [*self-twisted cover*]{} of $f$.
### The fiber-twisted cover at $t_0$
Let $t_0\in {\mathbb{P}}^1(k)\setminus{\mathbf t}$. Twist the representation $\phi$ by the specialization representation $\phi \circ {\sf s}_{t_0}:\Gabs_k \rightarrow G$ to get the twisted representation $$\widetilde{\phi}^{\phi \hskip 1pt {\sf s}_{t_0}}: \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{k} \rightarrow S_d$$ which corresponds to a cover $$\widetilde{f}^{\phi \hskip 1pt {\sf s}_{t_0}}: \widetilde{X}^{\phi \hskip 1pt {\sf s}_{t_0}}\rightarrow {\mathbb{P}}^1_{k}.$$ We call them respectively the [*fiber-twisted representation*]{} and the [*fiber-twisted cover*]{} at $t_0$.
### Description of the self-twisted cover
Set $\Psi_U= \widetilde{\phi \otimes_k k(U)}^{\phi_U}$. For every $\Theta\in \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{k(U)}$, we have $$\Psi_U (\theta) = \gamma(((\phi \otimes_k k(U)) (\Theta)) \times \delta (\phi_U(R(\Theta))^{-1})$$ where $R: \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{k(U)} \rightarrow \Gabs_{k(U)}$ is the natural surjection. The element $\Theta$ uniquely writes $ \Theta = \chi \hskip 2pt {\sf s}_U(\sigma)$ with $\chi \in \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{\overline{k(U)}}$ and $\sigma\in \Gabs_{k(U)}$. Whence $$(\phi \otimes_k k(U)) (\Theta)) = (\phi \otimes_k k(U)) (\chi) \hskip 2pt (\phi \otimes_k k(U)) ({\sf s}_U(\sigma))$$ and, using that $\phi_U = \phi \otimes_k k(U) \circ {\sf s}_{U}$, $$\phi_U(R(\Theta)) = (\phi \otimes_k k(U))({\sf s}_{U}(\sigma)).$$ Finally we obtain the following formula, where, by ${\rm conj}(g)$ ($g\in G$), we denote the permutation of $G$ induced by the conjugation $x\rightarrow gxg^{-1}$: $$\Psi_U (\theta) = \gamma((\phi \otimes_k k(U)) (\chi)) \times {\rm conj}({(\phi \otimes_k k(U))({\sf s}_{U}(\sigma))}).$$ Denote the field extension corresponding to the $\pi_1$-representation $\Psi_U$ by $\widetilde{F{k(U)}}^{\phi_U}/k(U)(T)$. The field $\widetilde{F{k(U)}}^{\phi_U}$ is the fixed field in $\Omega_{{\mathbf t},k(U)}$ of the subgroup $\Gamma_U\subset \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{{k(U)}}$ of all elements $\Theta$ such that $\Psi_U (\theta)$ fixes the neutral element of $G$ [^3]. We obtain $$\Gamma_U = {\rm ker}(\phi \otimes_k k(U)) \cdot {\sf s}_U(\Gabs_{k(U)})$$ and $\widetilde{F{k(U)}}^{\phi_U}$ is the fixed field in $F\hskip 1pt k(U)$ of all elements in ${\sf s}_U(\Gabs_{k(U)})$.
### Description of the fiber-twisted covers
Let $t_0\in {\mathbb{P}}^1(k)\setminus{\mathbf t}$ and set $\phi_{t_0}= \phi \circ {\sf s}_{t_0}$ and $\Psi_{t_0} = \widetilde{\phi}^{\phi_{t_0}}$. Every element $\theta\in \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{k}$ uniquely writes $\theta = x\hskip 2pt {\sf s}_{t_0}(\tau)$ with $x \in \pi_1({\mathbb{P}}^1\setminus {\mathbf t}, t)_{\overline{k}}$ and $\tau \in \Gabs_{k}$. Proceeding exactly as above but with $U$ replaced by $t_0$, $\phi \otimes_k k(U)$ by $\phi$ and $\Theta = \chi \hskip 2pt {\sf s}_U(\sigma)$ by $\theta = x\hskip 2pt {\sf s}_{t_0}(\tau)$, we obtain that $$\Psi_{t_0} (\theta) = \gamma(\phi(x)) \times {\rm conj}({\phi({\sf s}_{t_0}(\tau))})$$ and if $\widetilde F^{\phi_{t_0}}/k(T)$ is the field extension corresponding to the $\pi_1$-representation $\Psi_{t_0}$, $\widetilde F^{\phi_{t_0}}$ is the fixed field in $F$ of all elements in ${\sf s}_{t_0}(\Gabs_{k})$.
### Comparison
\[lemma:spec-self-twist\] There is a finite subset ${\mathcal E}\subset k$ such that for each $t_0\in k\setminus {\mathcal E}$, the fiber-twisted cover $\widetilde{f}^{\phi \hskip 1pt {\sf s}_{t_0}}: \widetilde{X}^{\phi \hskip 1pt {\sf s}_{t_0}}\rightarrow {\mathbb{P}}^1_{k}$ is $k$-isomorphic to the specialization of the self-twisted cover $\widetilde{f\otimes_k k(U)}^{\phi_U}: \widetilde{X \otimes_k k(U)}^{\phi_U}\rightarrow {\mathbb{P}}^1_{k(U)}$ at $U=t_0$.
Set $d=|G|$. By construction, the extension $\widetilde{F{k(U)}}^{F_U}/k(U)(T)$ is regular over $k(U)$. From the Bertini-Noether theorem, for every $t_0\in k$ but in a finite subset ${\mathcal E}$, which we possibly enlarge to contain the branch point set ${\mathbf t}$, the specialized extension at $U=t_0$ is regular over $k$ and is of degree $$[\widetilde{F{k(U)}}^{F_U}:k(U)(T)]=[Fk(U):k(U)(T)] = [F:k(T)]=d.$$ Up to enlarging again ${\mathcal E}$, one may also assume that the genus of this specialization is the same as the genus of the function field $\widetilde{F{k(U)}}$, which equals $g$, the genus of $F$. The rest of the proof shows that this specialization is the extension $\widetilde F^{F_{t_0}}/k(T)$.
Set $d=|G|$. Pick primitive elements ${\mathcal Y}$ and $\widetilde{\mathcal Y}_U$ of the two extensions $F/k(T)$ and $\widetilde{F{k(U)}}^{\phi_U}/k(U)(T)$, integral over $k[T]$ and $k[U,T]$ respectively. As $\widetilde{F{k(U)}}^{\phi_U} \subset Fk(U)$, one can write $$\widetilde{\mathcal Y}_U= \sum_{i=0}^{d-1} a_i(U) {\mathcal Y}^i$$ with $a_0(U),\ldots,a_{d-1}(U) \in k(U)$. Enlarge the set ${\mathcal E}$ to contain the poles of $a_0(U),\ldots,a_{d-1}(U)$. Fix $t_0\in k\setminus {\mathcal E}$. Consider the corresponding specialization $\widetilde{\mathcal Y}_{t_0} = \sum_{i=0}^{d-1} a_i(t_0) {\mathcal Y}^i$. The associated extension $k(T, \widetilde{\mathcal Y}_{t_0})/k(T)$ is the specialization of $\widetilde{F{k(U)}}^{\phi_U}/k(U)(T)$ at $U=t_0$. By construction $\widetilde{\mathcal Y}_{t_0} \in F$. The last paragraph of the proof below shows that $\widetilde{\mathcal Y}_{t_0}$ is fixed by all elements in ${\sf s}_{t_0}(\Gabs_{k})$. We will then be able to conclude that $k(T, \widetilde{\mathcal Y}_{t_0}) \subset \widetilde F^{\phi_{t_0}}$ and finally that these two fields are equal since $[k(T, \widetilde{\mathcal Y}_{t_0}):k(T)] = [\widetilde F^{\phi_{t_0}}:k(T)]=d$. As $U\notin {\mathbf t}$, there exists an embedding $$\widetilde{F{k(U)}}^{\phi_U} \rightarrow \overline{k(U)} ((T-U))$$ which maps ${\mathcal Y}_U$ to a formal power series $$\widetilde{\mathcal Y}_U = \sum_{n=0}^\infty b_n(U) (T-U)^n \hskip 3mm \hbox{with $b_n(U)\in \overline{k(U)}$ ($n\geq 0$)}.$$ Furthermore, $\widetilde{F{k(U)}}^{\phi_U}$ is fixed by all elements ${\sf s}_U(\sigma) \in {\sf s}_U(\Gabs_{k(U)})$, which, by definition of ${\sf s}_U$, act [*via*]{} the action of $\sigma \in \Gabs_{k(U)}$ on the coefficients $b_n(U)$; conclude that $b_n(U)\in {k(U)}$ ($n\geq 0$). Finally from the Eisenstein theorem[^4], there exists a polynomial $E(U) \in k[U]$ such that $E(U)^{n+1} \hskip 2pt b_n(U)\in k[U]$ for every $n\geq 0$. Enlarge again the set ${\mathcal E}$ to contain the roots of $E(U)$. For $t_0 \in k\setminus {\mathcal E}$, specializing $U$ to $t_0$ in the displayed formula above produces $\widetilde{\mathcal Y}_{t_0}$ as a formal power series in $k[[T-t_0]]$, which amounts to saying that, up to some $k$-isomorphism, $\widetilde{\mathcal Y}_{t_0}$ and so $\widetilde F^{\phi_{t_0}}$ are fixed by all elements in ${\sf s}_{t_0}(\Gabs_{k})$.
Let $\widetilde P(U,T,Y) \in k[U,T,Y]$ be the irreducible polynomial of $\widetilde{\mathcal Y}_U$ over $k[U,T]$. Theorem \[prop:self-twist\] holds for this polynomial $\widetilde P(U,T,Y)$ (up to enlarging again the finite set ${\mathcal E}$). When $k={\mathbb{Q}}$ we may and will choose the element $\widetilde{\mathcal Y}_U$ integral over ${\mathbb{Z}}[T,Y]$ (and not just ${\mathbb{Q}}(T,Y)$) so that $\widetilde P(U,T,Y)$ lies in ${\mathbb{Z}}[U,T,Y]$ and will assume further that the coefficients of $\widetilde P(U,T,Y)$ are relatively prime.
Proof of theorem \[thm:main-plus\] {#sec:proof}
==================================
We retain the notation and data introduced in §\[ssec:basics\].
Fix a real number $x> p_0$ and a Frobenius data ${\mathcal F}_x$ on ${S}_x$.
Fix also a subset $S_0$ of primes $p\in ]p_{-1},p_0]$, with as many elements as there are non trivial conjugacy classes in $G$. Associate then in a one-one way a non-trivial conjugacy class ${\mathcal F}_p$ to each prime $p\in S_0$. Set $S_{0x} = S_0\cup S_x$ and denote the Frobenius data $({\mathcal F}_p)_{p\in S_{0x}}$ by ${\mathcal F}_{0x}$.
First part: many good specializations $t_0\in {\mathbb{Z}}$
-----------------------------------------------------------
The goal of the first part is proposition \[prop:first-part\] which shows that there are “many” $t_0\in {\mathbb{Z}}$ such that conditions (i)-(iv) of theorem \[thm:main-plus\] are satisfied. The goal of the second part will be to show that there are “many” distinct corresponding extensions $F_{t_0}/{\mathbb{Q}}$.
We use the method of [@DEGha] for this first part. We re-explain it in the special context of this paper and make the adjustments that we will need for the rest of the proof. We refer to [@DEGha] for more details on the main arguments and for references. Working over number fields and even over ${\mathbb{Q}}$, we can give improved quantitative conclusions (compared to the existence statements of [@DEGha]). As in [@DEGha], there is first a local stage followed by a globalization argument.
### Local stage
Below, given $t_0\in {\mathbb{Q}}_p$ we say that $t_0\notin {\mathbf t}$ modulo $p$ if $t_0$ does not meet any of the branch points of $F/{\mathbb{Q}}(T)$ modulo $p$ [^5].
\[prop:degha-method\] Given our regular ${\mathbb{Q}}$-G-Galois cover $f:X\rightarrow {\mathbb{P}}^1$, a prime $p$ and a subset ${\mathcal F}_p\subset G$ consisting of a non-empty union of conjugacy classes of $G$, consider the set $${\mathcal T}({\mathcal F}_p) = \left\{t_0\in {\mathbb{Z}}\hskip 2mm \left|
\begin{matrix}
& \hbox{\rm $t_0\notin {\mathbf t}$ modulo $p$} \hfill \cr
& {\rm Frob}_p(F_{t_0}/{\mathbb{Q}}) \in {\mathcal F_p} \hfill \cr
\end{matrix}\right.
\right\}.$$ If $p$ is a good prime for $f$, the set ${\mathcal T}({\mathcal F}_p)$ is the union of cosets modulo $p$. Furthermore, the number $\nu({\mathcal F}_p)$ of these cosets satisfies $$\nu({\mathcal F}_p) \geq \frac{|{\mathcal F}_p|}{|G|} \times (p+1-2g\sqrt{p} -|G|(r+1))$$
$$\hbox{\it and}\hskip 5mm \nu({\mathcal F}_p) \leq \frac{|{\mathcal F}_p|}{|G|} \times (p+1+2g\sqrt{p} ).$$
We follow the method from [@DEGha]. Similar estimates though not in this explicit form can also be found in [@ekedahl].
We may and will assume that the subset ${\mathcal F}_p$ consists of a single conjugacy class.
Set $f_p=f\otimes_{{\mathbb{Q}}}{\mathbb{Q}}_p$ and denote the corresponding $\pi_1$-representation by $\phi_p: \pi_1({\mathbb{P}}^1\setminus{\mathbf t},t)_{{\mathbb{Q}}_p} \rightarrow G$. Pick an element $g_p\in {\mathcal F}_p$ and consider the unique unramified epimorphism $\varphi_p:\Gabs_{{\mathbb{Q}}_p} \rightarrow \langle g_p \rangle$ that sends the Frobenius of ${\mathbb{Q}}_p$ to $g_p$.
The condition “$t_0\notin {\mathbf t}$ modulo $p$” implies that $p$ is unramified in the specialization $F_{t_0}/{\mathbb{Q}}$. Then $t_0\in {\mathcal T}({\mathcal F}_p)$ if and only if the specialization representation $\phi \circ{\sf s}_{t_0}: \Gabs_{\mathbb{Q}}\rightarrow G$ of $F/{\mathbb{Q}}(T)$ at $t_0$ is conjugate in $G$ to $\varphi_p:\Gabs_{{\mathbb{Q}}_p} \rightarrow \langle g_p \rangle$. From the twisting lemma \[prop:twisted cover\], this is equivalent to the existence of a $k$-rational point above $t_0$ in the covering space of the twisted cover ${\widetilde{f_p}}^{\varphi_p}: {\widetilde{X_p}}^{\varphi_p} \rightarrow {\mathbb{P}}^1$. As $p$ is a good prime, this last cover has good reduction; denote the special fiber by ${\widetilde{\sf f}}_p: {\widetilde{\sf X}}_p \rightarrow {\mathbb{P}}^1_{{\mathbb{F}}_p}$. The last existence condition is then equivalent to the existence of some point $\overline x\in {\widetilde{\sf X}}_p({\mathbb{F}}_p)$ above the coset $\overline{t_0} \in {\mathbb{P}}^1({\mathbb{F}}_p)$ of $t_0$ modulo $p$: the direct part is clear while the converse follows from Hensel’s lemma.
From the Lang-Weil bound, the number of ${\mathbb{F}}_p$-rational points on ${\widetilde{\sf X}}_p$ is $\geq p+1 - 2g\sqrt{p}$. Removing the points that lie above a branch point or the point at infinity leads to the announced first estimate, a final observation for this calculation being that for $t_0 \notin {\mathbf t}$ modulo $p$, the number of ${\mathbb{F}}_p$-rational points $\overline x\in {\widetilde{\sf X}}_p({\mathbb{F}}_p)$ above $\overline{t_0}$ is $|{\rm Cen}_G(g_p)| = |G|/|{\mathcal F}_p|$: this number is indeed the same as the number of $\omega\in G$ such that $\phi \circ{\sf s}_{t_0} = \omega \varphi_p \omega^{-1}$ (as the proof of the twisting lemma in [@DEGha] shows). Using the upper bound part of Lang-Weil leads to the second estimate.
If in addition $p\geq r^2 |G|^2$ (in particular if $p\in S_{0x}$), then the right-hand side term in the inequality of proposition \[prop:degha-method\] is $>0$ (use that $g < r|G|/2 -1$ if $|G|>1$, which follows from Riemann-Hurwitz).
\[prop:francois\] Assume the branch point $t_1$ of the ${\mathbb{Q}}$-G-Galois cover $f:X\rightarrow {\mathbb{P}}^1$ is in ${\mathbb{Z}}$. Given a prime $p$, consider the set $${\mathcal T}(\hbox{\rm ra}/p) = \left\{t_0\in {\mathbb{Z}}\hskip 1mm \left| \hskip 1mm F_{t_0}/{\mathbb{Q}}\ \hbox{\rm is ramified at}\ p \right.
\right\}.$$ If $p$ is a good prime for $f$, the set ${\mathcal T}(\hbox{\rm ra}/p)$ contains the coset of $t_1+p\in {\mathbb{Z}}$ modulo $p^2$.
Let $t_0\in {\mathbb{Z}}$ such that $t_0 \equiv t_1+p$ modulo $p^2$. Then $t_0-t_1$ is of $p$-adic valuation $1$. As $p$ is good, it follows that $F_{t_0}/{\mathbb{Q}}$ is ramified at $p$. This last conclusion is part of the “Grothendieck-Beckmann theorem” for which we refer to [@SGA] and [@Beckmann proposition 4.2]; see also [@Legr1] where this result is discussed together with further developments in the spirit of proposition \[prop:francois\].
### Globalization
Set $$\left\{\begin{matrix}
& \displaystyle \beta = \Pi({S_0}) \hfill \cr
& \displaystyle B(x)= \beta \hskip 2pt \Pi({S})^2 \hskip 1pt \Pi({S_x}) \hfill \cr
\end{matrix} \right.$$
and consider the intersection $$\bigcap_{p\in S_{0x}} {\mathcal T}({\mathcal F}_p) \cap \bigcap_{p\in S} {\mathcal T}(\hbox{\rm ra}/p).$$
From proposition \[prop:degha-method\], proposition \[prop:francois\] and the Chinese remainder theorem, this set contains $${\mathcal N}(S,{\mathcal F}_{0x})=\prod_{p\in S_{0x}} \nu({\mathcal F}_p)$$ $B(x)$. Denote the set of their representatives in $[1,B(x)]$ by ${\mathcal T}(S,{\mathcal F}_{0x})$; the cardinality of this set is ${\mathcal N}(S,{\mathcal F}_{0x})$.
\[prop:first-part\] [(a)]{} For every integer $t_0\in {\mathcal T}(S,{\mathcal F}_{0x})$, the extension $F_{t_0}/{\mathbb{Q}}$ satisfies the four conditions [(i)-(iv)]{} from theorem \[thm:main-plus\], with [(ii)]{} even replaced by the following sharper version [(ii+)]{} of [(ii)]{}, that is [(i)]{} $\Gal(F_{t_0}/{\mathbb{Q}})=G$,
[(ii+)]{} $F_{t_0}/{\mathbb{Q}}$ is unramified and ${\rm Frob}_p(F_{t_0}/{\mathbb{Q}}) \in {\mathcal F_p}$ for every $p\in S_{0x}$ [(and not just for every $p\in S_x$)]{},
[(iii)]{} $F_{t_0}/{\mathbb{Q}}$ is ramified at $p$ for every $p\in S$,
[(iv)]{} $| d_{F_{t_0}} | \leq \rho(x)$. -4mm
$$\hbox{\it We have} \hskip 5mm {\mathcal N}(S,{\mathcal F}_{0x}) \geq \hskip 2pt \chi({\mathcal F}_x)\hskip 1mm \frac{B(x)}{\beta \hskip 1pt \Pi({S})^2} \left( \frac{1}{2r|G|}\right)^{|S_x|} \hskip 22mm\leqno(\hbox{\rm b})$$
[(c)]{} The set of integers $t_0\in {\mathbb{Z}}$ such that for each $p\in S_x$, $F_{t_0}/{\mathbb{Q}}$ is unramified and ${\rm Frob}_p(F_{t_0}/{\mathbb{Q}}) \in {\mathcal F_p}$ consists of cosets modulo $\Pi(S_x)$ and the set ${\mathcal T}(\emptyset,{\mathcal F}_x)$ of their representatives in $[1,\Pi({S_x})]$ is of cardinality $${\mathcal N}(\emptyset,{\mathcal F}_x) = \prod_{p\in S_x} \nu({\mathcal F}_p) \leq \hskip 2pt \chi({\mathcal F}_x) \hskip 1mm \frac{\Pi(S_x)}{\beta} \hskip 1mm \left( 2 - \lambda\right)^{|S_x|}$$ where $\lambda = (r|G|-1)/r^2 |G|^2$.
Conclusion (c) proves conclusion (d) of theorem \[thm:main-plus\].
\(a) Fix $t_0\in {\mathcal T}(S,{\mathcal F}_{0x})$ (or more generally congruent modulo $B(x)$ to some element in ${\mathcal T}(S,{\mathcal F}_{0x})$).
Conditions (ii+), (iii) hold by definition of the sets ${\mathcal T}({\mathcal F}_p)$ and ${\mathcal T}(\hbox{\rm ra}/p)$.
A classical argument then shows that (i) follows from (ii+): indeed because of the Frobenius condition at the primes $p\in S_0$, the subgroup $\Gal(F_{t_0}/{\mathbb{Q}})\subset G$ meets every conjugacy class of $G$; from a lemma of Jordan [@jordan], it is all of $G$.
From (i), the polynomial $P(t_0,Y)$ is irreducible in ${\mathbb{Q}}[Y]$. As it is monic and with integral coefficients, its discriminant, which is $\Delta_P(t_0)$, is a multiple of the absolute discriminant $d_{F_{t_0}}$ of the extension $F_{t_0}/{\mathbb{Q}}$. Conjoined with $1\leq t_0 \leq B(x)$, this leads to $$|d_{F_{t_0}}| \leq (1+\delta_P) \hskip 2pt H(\Delta_P) \hskip 2pt B(x)^{\delta_P}$$ and conclusion (iv) follows from the definition of $\rho(x)$ (given in §\[ssec:data\]).
\(b) Using proposition \[prop:degha-method\], we obtain $$\begin{matrix}
{\mathcal N}(S,{\mathcal F}_{0x}) & \displaystyle \geq \prod_{y\in S_x}\frac{|{\mathcal F}_p|}{|G|} \times (p+1-2g\sqrt{p} -|G|(r+1))\hfill \cr
&\displaystyle \geq \chi ({\mathcal F}_x) \times \prod_{p\in S_x} p \times \prod_{p\in S_x} \left( 1+ \frac{1}{p} -\frac{2g}{\sqrt{p} }- \frac{(r+1)|G|}{p}\right) \hfill \cr
\end{matrix}$$ Using again that $g < r|G|/2 -1$ (if $|G|>1$) and that $p\geq r^2 |G|^2$ for each $p\in S_x$, we have $$\begin{matrix}
\displaystyle 1+ \frac{1}{p} -\frac{2g}{\sqrt{p} }- \frac{|G|(r+1)}{p} & \displaystyle > \hskip 2mm 1 - \frac{r|G|-2}{r|G|} - \frac{(r+1)|G|}{r^2 |G|^2} \hfill\cr
& \displaystyle = \hskip 2mm \frac{2}{r|G|} - \frac{(r+1)|G|}{r^2 |G|^2} \hfill \cr
& \displaystyle = \hskip 2mm \frac{(r-1)|G|}{r^2 |G|^2} \hskip 2mm \geq \hskip 2mm \frac{1}{2r|G|}\hfill \cr
\end{matrix}$$ which yields the announced first estimate.
\(c) Here we use the conclusion from proposition \[prop:degha-method\] that for each $p\in S_x$, the set ${\mathcal T}({\mathcal F}_p)$ consists exactly of $\nu({\mathcal F}_p)$ cosets modulo $p$. Combined with the Chinese remainder, this gives that the set ${\mathcal T}(\emptyset,{\mathcal F}_x)$ consists of exactly ${\mathcal N}(\emptyset,{\mathcal F}_x) = \prod_{p\in S_x} \nu({\mathcal F}_p)$ elements. Proceed then similarly as in (b) but using the upper bound part of proposition \[prop:degha-method\] to obtain the desired estimate.
Consider the situation with $S = \emptyset$ and allowing no local condition at some primes $p\in S_x$ — no restriction on ${\rm Frob}_p(F_{t_0}/{\mathbb{Q}})$ and no unramified condition —. We have $\nu({\mathcal F}_p) = p$ for such primes and obtain this generalized lower bound: if $S_x^\prime\subset S_x$ is the subset of primes where there [*is*]{} a local condition, then $${\mathcal N}(\emptyset,{\mathcal F}_{0x}) \geq \hskip 2pt \chi({\mathcal F}_x)\hskip 1mm \frac{B(x)}{\beta} \left( \frac{1}{2r|G|}\right)^{|S_x^{\prime}|} .$$ In particular, the number of integers $t_0 \in [1,B(x)]$ where conditions [(i)]{} $\Gal(F_{t_0}/{\mathbb{Q}})=G$ and [(iv)]{} $| d_{F_{t_0}} | \leq \rho(x)$ hold ([*i.e.*]{} no local condition at any prime) is $ \geq \hskip 2pt {B(x)}/{\beta} $.
Second part: many good specializations $F_{t_0}/{\mathbb{Q}}$
-------------------------------------------------------------
### Reduction to counting integral points on curves
We will now estimate the number, say ${N}(S,{\mathcal F}_{0x})$, of distinct specializations $F_{t_0}/{\mathbb{Q}}$ with $t_0\in {\mathcal T}(S,{\mathcal F}_{0x})$. We will give a lower bound for the number of non conjugate specialization representations $\phi \circ {\sf s}_{t_0}:\Gabs_{{\mathbb{Q}}} \rightarrow G$ with $t_0\in {\mathcal T}(S,{\mathcal F}_{0x})$. Given two such representations, the associated field extensions are equal if and only if the representations have the same kernel, or, equivalently, if they differ by some automorphism of $G$. Dividing the previous bound by $|{\rm Aut}(G)|$ will thus yield the desired bound for ${N}(S,{\mathcal F}_{0x})$.
Consider the polynomial $\widetilde P(U,T,Y) \in {\mathbb{Z}}[U,T,Y]$ given by theorem \[prop:self-twist\] and its discriminant $\Delta_{\widetilde P} \in {\mathbb{Z}}[U,T]$ (relative to $Y$). As $\widetilde P(U,T,Y)$ is irreducible in ${\mathbb{Q}}(U,T)[Y]$, $\Delta_{\widetilde P}(U,T)\not= 0$. Write it as a polynomial in $T$ of degree $N$ and denote its leading coefficient by $\Delta_{\widetilde P,N} (U)$; we have $\Delta_{\widetilde P,N} (U)\in {\mathbb{Z}}[U]$ and $\Delta_{\widetilde P,N} (U)\not= 0$.
Drop from the set ${\mathcal T}(S,{\mathcal F}_{0x})$ the finitely many integers $u_0$ for which $\widetilde \Delta_{\widetilde P,N} (u_0)=0$ or which are in the exceptional set ${\mathcal E}$ from theorem \[prop:self-twist\]. Denote the resulting set by ${\mathcal T}(S,{\mathcal F}_{0x})^\prime$ and the number of dropped elements by ${\rm E}$.
Fix $u_0\in {\mathcal T}(S,{\mathcal F}_{0x})^\prime$ and consider the fiber-twisted cover at $u_0$: $$\widetilde{f}^{\phi \hskip 1pt {\sf s}_{u_0}}: \widetilde{X}^{\phi \hskip 1pt {\sf s}_{u_0}}\rightarrow {\mathbb{P}}^1_{{\mathbb{Q}}}.$$ Let $t_0\in {\mathcal T}(S,{\mathcal F}_{0x})^\prime$. From the twisting lemma \[prop:twisted cover\], the two representations $\phi \circ {\sf s}_{u_0}$ and $\phi \circ {\sf s}_{t_0}$ are conjugate in $G$ if and only if there exists $x_0\in \widetilde{X}^{\phi \hskip 1pt {\sf s}_{u_0}}({\mathbb{Q}})$ such that $\widetilde{f}^{\phi \hskip 1pt {\sf s}_{u_0}}(x_0)=t_0$.
We have $\Delta_{\widetilde P}(u_0,t_0) \not=0$ except for at most $N$ integers $t_0$. For the non-exceptional $t_0$, the polynomial $\widetilde P(u_0,t_0,Y)$ has only distinct roots $y \in \overline {\mathbb{Q}}$ and, using theorem \[prop:self-twist\], the corresponding points $(t_0,y)$ on the affine curve $\widetilde P(u_0,t,y)=0$ exactly correspond to the points $x$ on the smooth projective curve $\widetilde{X}^{\phi \hskip 1pt {\sf s}_{u_0}}$ above $t_0$. Furthermore, in this correspondence, the ${\mathbb{Q}}$-rational points $x$ correspond to the couples $(t_0,y)$ with $y\in {\mathbb{Q}}$. Conclude that up to some term $\leq N$, the number of $t_0$ for which $\phi \circ {\sf s}_{u_0}$ and $\phi \circ {\sf s}_{t_0}$ are conjugate in $G$ is equal to the number of ${\mathbb{Q}}$-rational points $(t_0,y)$ on the affine curve $\widetilde P(u_0,t,y)=0$.
Note further that such a ${\mathbb{Q}}$-rational point $(t_0,y)$ has necessarily integral coordinates as $t_0\in {\mathbb{Z}}$ and $\widetilde P(u_0,T,Y)\in {\mathbb{Z}}[T,Y]$ and is monic in $Y$. Therefore we are reduced to counting the integers $t_0\in [1,B(x)]$ such that there is an integral point $(t_0,y)\in {\mathbb{Z}}^2$ on the curve $\widetilde P(u_0,t,y)=0$.
### Diophantine estimates
The constants $c_i$, $i>0$ that will enter depend only on the polynomial $P(T,Y)$.
The curve $\widetilde P(u_0,t,y)=0$ is of genus $g$ (theorem \[prop:self-twist\] (c)) and we have $$\left\{
\begin{matrix}
& \deg(\widetilde P(u_0,T,Y)) \leq \deg(\widetilde P(U,T,Y)) = c_1 \hfill \cr
& \deg_Y(\widetilde P(u_0,T,Y)) = \deg_Y (\widetilde P(U,T,Y)) = |G| \hfill \cr
& H(\widetilde P(u_0,T,Y)) \leq c_2 u_0^{c_3} \leq c_2 B(x)^{c_3} \hfill \cr
\end{matrix}
\right.$$
For real numbers $\gamma ,D, H, B \geq 0$ and $d_Y\geq 2$, consider all polynomials $F\in {\mathbb{Z}}[T,Y]$, primitive, monic in $Y$, irreducible in $\overline {\mathbb{Q}}(T)[Y]$, such that $\deg_Y(F) = d_Y$, of total degree $\leq D$, of height $\leq H$ and such that the affine curve $P(t,y)=0$ is of genus $\leq \gamma$. For each such polynomial, the number of integers $t\in [1,B]$ such that there exists $y\in {\mathbb{Z}}$ such that $F(t,y)=0$ is a finite set. Denote by $Z(\gamma,D, d_Y, H, B)$ the maximal cardinality of all these finite sets.
Using the diophantine parameter $Z(\gamma,D, d_Y, H, B)$, conclude that the number of $t_0\in {\mathcal T}(S,{\mathcal F}_{0x})^\prime$ such that the two representations $\phi \circ {\sf s}_{u_0}$ and $\phi \circ {\sf s}_{t_0}$ are conjugate in $G$ is less than or equal to $$Z(g,c_1,|G|,c_2 B(x)^{c_3},B(x)).$$
Thus we obtain $${N}(S,{\mathcal F}_{0x}) \geq \frac{{\mathcal N}(S,{\mathcal F}_{0x})-\hbox{\rm E}}{|{\rm Aut}(G)| \hskip 3pt [Z(g,c_1,|G|, c_2 B(x)^{c_3},B(x))+N]}$$
Next take into account proposition \[prop:first-part\] and note that $N_F(x,S,{\mathcal F}_x) \geq {N}(S,{\mathcal F}_{0x})$ to write $$N_F(x,S,{\mathcal F}_x) \geq \frac{\displaystyle \frac{\chi({\mathcal F}_x) \hskip 1mm (2r|G|)^{-|S_x|}\hskip 1mm B(x)}{\beta \hskip 2pt \Pi({S)}^{2}}-\hbox{\rm E}}{|{\rm Aut}(G)| \hskip 3pt [Z(g,c_1,|G|,c_2 B(x)^{c_3},B(x))+N]}$$
Assume that the genus $g$ of $X$ is $\geq 2$ and that Lang’s conjecture holds. This conjecture is that if $V$ is a variety of general type defined over a number field $K$ then the set $V(K)$ of $K$-rational points is not Zariski-dense in $V$ [@lang]. We will use it through the following consequence proved by Caporaso, Harris and Mazur [@charm]: they showed that Lang’s conjecture implies that for every number field $K$ and every integer $g\geq 2$ there exists a finite integer $B(g,K)$ such that ${\rm card} (C(K)) \leq B(g,K)$ for every curve $C$ of genus $g$ defined over $K$.
Under this conjecture we obtain $$Z(g,c_1,|G|,c_2B(x)^{c_3},B(x)) + N \leq c_4.$$
In the general case $g\geq 0$ we use an unconditional result of Walkowiak [@Wa §2.4] which shows that if $d_Y \geq 2$ then $$Z(\gamma,D, d_Y, H, B) \leq a_1 D^{a_2} (\log H^+)^{a_3} B^{1/d_Y} (\log B)^{a_4}$$ where $H^+ = \max(H,e^e)$ and $a_1, \ldots, a_4$ are absolute constants. See §\[ssec:walkowiak\] for more on this result. We deduce $$Z(g,c_1,|G|,c_2B(x)^{c_3},B(x)) + N \leq c_5 B(x)^{1/|G|} \log(B(x)^{c_{6}}.$$ Conclude that unconditionally: $$N_F(x,S,{\mathcal F}_x) \geq \hskip 2pt c_7 \hskip 2pt \frac{\chi({\mathcal F}_x)}{\Pi({S})^{2}} \hskip 4pt \frac{B(x)^{1-1/|G|}}{(\log B(x))^{c_9} \hskip 2pt c_8^{|S_x|}} - c_{10}$$
and, under Lang’s conjecture: $$N_F(x,S,{\mathcal F}_x) \geq \hskip 2pt c_7 \hskip 2pt \frac{\chi({\mathcal F}_x)}{\Pi({S})^{2}} \hskip 4pt \frac{B(x)}{c_8^{|S_x|}} - c_{10}.$$
Note that $c_{11} \Pi(x)\leq B(x) \leq \Pi(x) \Pi(S)^2$, that $|S_x| \leq \pi(x)$ and $0<c_8<1$ to obtain the estimates of theorem \[thm:main-plus\] (a) and (c). Theorem \[thm:main-plus\] (b) follows from the containments ${\mathcal T}(S,{\mathcal F}_{0x}) \subset [1,B(x)] \subset [1,\Pi(S)\hskip 1pt \Pi (x)]$.
### Walkowiak’s result {#ssec:walkowiak}
Let $F\in {\mathbb{Z}}[T,Y]$ be a polynomial, irreducible in ${\mathbb{Z}}[T,Y]$. Set $D=\deg(F)$ and $H^+=\max(H(F),e^e)$. The result we use in the proof above is the following.
\[thm:walkowiak\] Assume $\deg_YF \geq 2$. There exist absolute constants $a_1, \ldots, a_4$ such that for every real number $B>0$, the number of integers $t_0\in [1,B]$ such that $F(t_0,Y)$ has a root in ${\mathbb{Z}}$ is less than $$a_1 D^{a_2} (\log H^+)^{a_3} B^{1/\deg_Y(F)} (\log B)^{a_4}.$$
This result is proved in [@Wa §2.4] but with $B^{1/2}$ instead of $B^{1/\deg_Y(F)}$. We explain here how to modify Walkowiak’s arguments to obtain the better exponent $1/\deg_Y(F)$. The only change to make is in the final stage of the proof in [@Wa §2.2-2.3].
Walkowiak’s central result is the following bound for the number $N(F,B)$ of $(t,y)\in {\mathbb{Z}}^2$ such that $\max(|t|,|y|)\leq B$ and $F(t,y)=0$: $$N(F,B) \leq 2^{36} D^5 \log^3(1250d^{11} B^{5D-1}) \log^2(B)\hskip 2pt B^{1/D}.$$ To prove theorem \[thm:walkowiak\], his basic idea is to use Liouville’s inequality to get upper bounds $|y|\leq B^\prime$ for roots $y\in {\mathbb{Z}}$ of polynomials $F(t_0,Y)$ with $t_0\in [1,B]$; the bound above for $N(F,B)$ with $B$ taken to be $B^\prime$ provides then a bound for the desired set. The main terms that appear in the resulting bound come from $(B^\prime)^{1/D}$. They may be too big however in some cases and Walkowiak uses a trick to obtain his final bound in $B^{1/2}$. In order to obtain $B^{1/n}$ instead, Walkowiak’s trick should be modified as follows.
Set $L_1=\log(H^+)$, $L_2=\log(\log(H^+))$, $m=\deg_TF$ and $n=\deg_YF$; one may assume $m>0$. Let $t_0\in [1,B]$ such that $F(t_0,Y)$ has a root $y\in{\mathbb{Z}}$. Liouville’s inequality gives $$|y| \leq 2(m+1) H^+ B^m = B^\prime.$$ The main terms in $(B^\prime)^{1/D}$ are $(H^+)^{1/D}$ and $(B^m)^{1/D}$. [*Case 1*]{}: $ mnL_1/L_2\leq D$. On the one hand, we have $1/D \leq L_2/L_1$ and so $(H^+)^{1/D} \leq (H^+)^{L_2/L_1}=\log (H^+)$. On the other hand $m/D \leq 1/n$ and so $B^{m/D}\leq B^{1/n}$. The upper bound for $N(F,B^\prime)$ is indeed as announced in the statement of theorem \[thm:walkowiak\].
[*Case 2*]{}: $ mnL_1/L_2> D$. Set $E=[mnL_1/L_2]+1$ and consider the polynomial $G\in {\mathbb{Z}}[T,Y]$ defined by $G(T,Y)=F(T,T^E+Y)$. For $y^\prime = y-t_0^E$ we have $G(t_0,y^\prime) = 0$ and $$|y^\prime| \leq 2(m+1) H^+ B^m + B^E \leq 2(m+1) H^+ B^E = B''.$$ Use then the upper bound for $N(F,B)$ with $F$ and $B$ respectively taken to be $G$ and $B''$. As $\deg_YG=\deg_YF=n$ and $nE\leq \deg G\leq nE+m$, the main terms are in this case $$(H^+)^{1/\deg(G)} \leq (H^+)^{1/nE} \leq (H^+)^{L_1/L_2} = \log(H^+)$$ $$\hbox{\rm and} \hskip 5mm B^{E/\deg G} \leq B^{1/n}.$$
Again the upper bound for $N(G,B'')$ is as announced.
[^1]: at least $e^{\gamma x}$ with $\gamma=1-1/|G|$ (for $x\gg 1$).
[^2]: We need to slightly improve Walkowiak’s result to get the right exponent $\alpha(G,\delta)$ in theorem \[thm:main\]; see §\[ssec:walkowiak\].
[^3]: Taking any other element of $G$ gives the same field up to $k(U)(T)$-isomorphism.
[^4]: This classical result is often stated for formal power series $\sum_{n\geq 0} b_n T^n$, algebraic over ${\mathbb{Q}}(T)$ and with coefficients $b_n \in \overline {\mathbb{Q}}$, but is true in a bigger generality including the situation where ${\mathbb{Q}}$ and ${\mathbb{Z}}$ are respectively replaced by $k(U)$ and $k[U]$. For example, the proof given in [@dwork-robba] easily extends to this situation.
[^5]: Recall that for two points $t,t^\prime \in \overline{{\mathbb{Q}}_p} \cup \{\infty\}$, meeting modulo $p$ means that $|t|_{\overline p} \le 1$, $|t^\prime|_{\overline p} \le 1$ and $|t-t^\prime|_{\overline p} < 1$, or else $|t|_{\overline p} \ge 1$, $|t^\prime|_{\overline p} \ge 1$ and $|t^{-1}-(t^\prime)^{-1}|_{\overline p} < 1$, where $\overline p$ is some arbitrary prolongation of the $p$-adic absolute value $v$ to $\overline{{\mathbb{Q}}_p}$.
|
---
abstract: |
A distortion-noise profile is a function indicating the maximum allowed source distortion value for each noise level in the channel. In this paper, the minimum energy required to achieve a distortion noise profile is studied for Gaussian sources which are transmitted robustly over Gaussian channels. We provide improved lower and upper bounds for the minimum energy behavior of the square-law profile using a family of lower bounds and our proposed coding scheme.
*Index Terms*–Distortion-noise profile, fidelity-quality profile, energy-distortion tradeoff, energy-limited transmission, joint source-channel coding.
author:
- |
Mohammadamin Baniasadi, and Ertem Tuncel\
Department of Electrical and Computer Engineering\
University of California, Riverside, CA\
Email: mohammadamin.baniasadi@email.ucr.edu, ertem.tuncel@ucr.edu\
title: 'Minimum Energy Analysis for Robust Gaussian Joint Source-Channel Coding with a Square-Law Profile '
---
INTRODUCTION
============
Most of emerging wireless applications, such as Internet of things (IoT) and multimedia streaming require lossy transmission of source signals over noisy channels, which is in general a joint source-channel coding (JSSC) problem. Shannon proved the separation theorem which states that in point-to-point scenarios, it is optimal to separate source and channel coding problems. However, in many problems, the optimality of separation breaks down, since JSCC can exploit source correlation to generate correlated channel inputs despite the distributed nature of the encoders, potentially improving the overall performance [@c1]-[@c5].
We consider lossy transmission of a Gaussian source over an additive white Gaussian noise (AWGN) channel, where the channel input constraint is not on power and bandwidth, but on energy per source symbol. This approach has drawn much attention recently, see e.g., [@c6; @c7; @c8; @c9] as a few references. Part of the appeal is the simplifications to both achievable schemes and converses as the bandwidth expansion factor approaches infinity \[8\]. We assume there is no feedback.
It is well-known (for example, see [@c6]) that the minimum distortion that can be achieved with energy $E$ when the channel noise variance $N$ is fixed, is given by $$\begin{aligned}
\label{D1}
D=\exp(-\frac{E}{N}).\end{aligned}$$ In this paper, a robust setting is considered in which the transmitter does not know $N$, while it is known at the receiver, and it can have any value in the interval (0,$\infty$). The system is to be designed to fulfill with a distortion-noise profile ${\cal D}(N)$ so that it achieves $$\begin{aligned}
D\leq \mathcal{D}(N)\nonumber\end{aligned}$$ for all $0<N<\infty$, while minimizing its energy use. This wide spectrum of noise variances is taken into consideration to account for the scenarios in which absolutely nothing is known about the noise level. For instance, the channel could be suffering occasional interferences of unknown power ($N>0$), although it may be originally of very high quality ($N \approx 0$). There are a wide range of applications in which noise variances are not known. For instance, we can point military situation, indoor fires and emergency conditions.
In [@c10], it is shown that for the inversely linear profile, uncoded transmission is optimal. Furthermore,it is represented that exponential profiles are not achievable with finite energy. Then, the square-law profile is studied which is somehow combination of linear and exponential profiles and lower and upper bounds have been derived for the minimum achievable energy of the square-law profile. In this paper, we derive improved lower and upper bounds for the minimum energy, and show that the gap between our lower and upper bounds is significantly reduced compared to [@c10]. Improving lower and upper bounds and making them as tight as possible helps us to design better systems in practical scenarios by comparing the amount of energy with these improved theoretical bounds.
A similar universal coding scenario in the literature is given in [@c11], where a maximum regret approach for compound channels is proposed. The objective in their problem is to minimize the maximum ratio of the capacity to the achieved rate at any noise level. There are other related works including [@c12; @c13], and [@c14].
The rest of the paper is organized as follows. The next section is devoted to notation and preliminaries. In Section III, previous work on lower and upper bounds for the minimum energy is reviewed. In Section IV, we present our main results, which are improved lower and upper bounds for the square-law profile. Finally, in Section V we conclude our work and discuss future work.
Notation and Preliminaries
==========================
Suppose that $X^n$ is an i.i.d unit-variance Gaussian source which is transmitted over an AWGN channel $V^m=U^m+W^m$, where $U^m$ is the channel input, $W^m\sim \mathcal{N} (\mathbf{0},N\mathbf{I}_m)$ is the noise, and $V^m$ is the observation at the receiver. We define bandwidth expansion factor $\kappa=\frac{m}{n}$ which can be arbitrarily large, while the energy per source symbol is limited by $$\begin{aligned}
\frac{1}{n}E(||U^m||^2)\leq E.\end{aligned}$$ The achieved distortion per source symbol is measured as $$\begin{aligned}
D=\frac{1}{n}E(||X^n-\hat{X}^n||^2)\end{aligned}$$ while $\hat{X}^n$ is the reconstruction at the receiver.
A pair of distortion-noise profile $\mathcal{D}(N)$ and energy level $E$ is said to be *achievable* if for every $\epsilon>0$, there exists large enough $(m,n)$, an encoder $$\begin{aligned}
f^{m,n}: R^n \to R^m,\nonumber\end{aligned}$$ and decoders $$\begin{aligned}
g_N^{m,n}: R^m \to R^n\nonumber\end{aligned}$$ for every $0<N<\infty $, such that $$\begin{aligned}
\frac{1}{n}E\{||f^{m,n}(X^n)||^2\}\leq E+\epsilon\nonumber\end{aligned}$$ and $$\begin{aligned}
\frac{1}{n}E\{||X^n-g_N^{m,n}(f^{m,n}(X^n)+W_N^m)||^2\}\leq \mathcal{D}(N)+\epsilon\nonumber\end{aligned}$$ for all $N$, with $W_N^m$ being the i.i.d. channel noise with variance $N$.
For given $\mathcal{D}$, the main quantity of interest would be $$\begin{aligned}
E_{min}(\mathcal{D})=\inf \{E:(\mathcal{D},E) \ \textrm{achievable}\}\nonumber\end{aligned}$$ with the understanding that $E_{min}(\mathcal{D})=\infty $ if there is no finite $E$ for which $(\mathcal{D},E)$ is achievable.
In the sequel, it will prove more convenient to use the notation $F=\frac{1}{D}$ and $Q=\frac{1}{N}$, where $F$ and $Q$ standing for signal *fidelity* and channel *quality*, respectively as in [@c10]. For any $\mathcal{D}(N)$, we define the corresponding *fidelity-quality profile* as $$\begin{aligned}
\mathcal{F}(Q)=\frac{1}{\mathcal{D}(\frac{1}{Q})}\nonumber\end{aligned}$$ and state that $(\mathcal{F},E)$ is achievable if and only if $(\mathcal{D},E)$ is achievable according to *Definition 1*. $E_{min}(\mathcal{F})$ is similarly defined.
Previous Work
=============
A Family of Lower Bounds on $E_{min}(\mathcal{D})$
--------------------------------------------------
In [@c10], the authors used the connection between the problem and lossy transmission of Gaussian sources over Gaussian broadcast channels where the power per channel symbol is limited and the bandwidth expansion factor $\kappa$ is fixed. More specifically, they employed the converse result by Tian *et al.* [@c15], which is a generalization of the 2-receiver outer bound shown by Reznic *et al.* [@c16] to $K$ receivers, and proved the following lemma.
\[Lemma1\] For any $K$, $\tau_1 \ge \tau_2 \ge ...\ge \tau_{K-1} \ge \tau_{K}=0$, and $N_1 \ge N_2 \ge ... \ge N_{K}\ge N_{K+1}=0$, $$\begin{aligned}
\label{eqtn:lemma1}
E_{min}(\mathcal{D})\!\!\!&\ge & \!\!\!N_1 \log \frac{1+\tau_1}{\mathcal{D}(N_1)+\tau_1}\nonumber\\
\!\!\!&& \!\!\!+\sum_{k=2}^{K} N_k \log \frac{(1+\tau_k)(\mathcal{D}(N_k)+\tau_{k-1})}{(1+\tau_{k-1})(\mathcal{D}(N_k)+\tau_{k})}.\end{aligned}$$
Square-Law Fidelity Quality Profiles
------------------------------------
In [@c10], the authors focused on $\mathcal{F}(Q)=1+\alpha Q^2$ for some $\alpha >0$ and analyzed the lower and upper bounds for $E_{min}{(\mathcal{F})}$.
### Lower Bound for $E_{min}(\mathcal{F})$
Invoking *Lemma 1* by properly choosing $\tau_k$ and $N_k$ in (\[eqtn:lemma1\]), the following theorem was obtained in [@c10].
For a fidelity-quality profile $\mathcal{F}(Q)=1+\alpha Q^2$, the minimum required energy is lower-bounded as $$\begin{aligned}
E_{min}(\mathcal{F}) \ge c \ \sqrt[]{\alpha}\nonumber\end{aligned}$$ with $$\begin{aligned}
c=\sum_{k=1}^{\infty}\frac{1}{\sqrt[]{4^k\exp(k)-1}}\approx 0.4507. \nonumber\end{aligned}$$
### Upper Bound for $E_{min}(\mathcal{F})$
Using a scheme first sending the source uncoded, and leveraging the received output as side information for the subsequent digital rounds sending indices of an infinite-layer quantizer, an upper bound for the minimum energy was presented in the following theorem in [@c10].
The minimum required energy for profile $\mathcal{F}(Q)=1+\alpha Q^2$ is upper-bounded as $$\begin{aligned}
E_{min}(\mathcal{F})\leq d \ \sqrt[]{\alpha}\nonumber\end{aligned}$$ with $$\begin{aligned}
d=2 \ \sqrt[]{\log3-Li_2(-2)}\approx 3.1846 \nonumber\end{aligned}$$ where $Li_2(.)$ is the polylogarithm of order 2 defined as $$\begin{aligned}
Li_2(z)=-\int_{0}^{1} \frac{\log(1-zu)}{u}du.\nonumber\end{aligned}$$
Our Main Results
================
Our main contributions in this paper are tighter lower and upper bounds to the energy for the profile ${\cal F}(Q)=1+\alpha Q^2$.
Lower Bound for $E_{min}(\mathcal{F})$
---------------------------------------
We begin with lower bounding $E_{\min}({\cal F})$ by the following theorem.
For a fidelity-quality profile ${\cal F}(Q) = 1 + \alpha Q^2$, the minimum required energy is lower-bounded as $$E_{\min}({\cal F})\geq 0.9057\sqrt{\alpha}.$$
A lower bound on $E_{min}(\mathcal{D})$ follows from (\[D1\]). Since for any fixed $N_0$ and $D_0$ the expended energy cannot be lower than $N_0\log\frac{1}{D_0}$, the lower bound is obtained given by $$\begin{aligned}
E_{min}(\mathcal{D})\ge \sup_{N>0} N \log \frac{1}{\mathcal{D}(N)}\end{aligned}$$ or equivalently by $$\begin{aligned}
\label{lower1}
E_{min}(\mathcal{F})&\ge \sup_{Q>0} \frac{\log \mathcal{F}(Q)}{Q}\nonumber\\
&=\sup_{Q>0} \frac{\log(1+\alpha Q^2)}{Q}\nonumber\\
&=\bigg(\sup_{q>0} \frac{\log(1+ q^2)}{q}\bigg)\sqrt{\alpha}
%&=0.8047\sqrt\alpha.\end{aligned}$$ where $q= \sqrt {\alpha} Q$. By solving (\[lower1\]) numerically, optimal value of $q$ is $q^*=2.01$ and thus $$\begin{aligned}
\label{lbound}
E_{min}(\mathcal{F})\ge 0.8047\sqrt\alpha.\end{aligned}$$ Note that (\[lower1\]) is the special case of *Lemma 1* where $K=1$. Thus, it is reasonable to expect an even better lower bound by increasing $K$. By setting $K=2$ and $\tau_1 = \tau \ge \tau_2=0$, the lower bound is achieved as [$$\begin{aligned}
\label{lower2}
E_{min}(\mathcal{F})&\ge \sup_{Q_2>Q_1>0,\tau>0} \Bigg[\frac{\log (1+\frac{\alpha Q^2_1}{1+\tau(1+\alpha Q^2_1)})}{Q_1}+ \frac{\log (1+\frac{\alpha \tau Q^2_2}{1+\tau})}{Q_2} \Bigg]\nonumber\\
&\ge \Bigg(\sup_{q_2>q_1>0,\tau>0} \bigg[\frac{\log (1+\frac{q^2_1}{1+\tau(1+q^2_1)})}{q_1}+ \frac{\log (1+\frac{ \tau q^2_2}{1+\tau})}{q_2} \bigg]\Bigg)\sqrt{\alpha}\end{aligned}$$]{} where $q_1=\sqrt{\alpha}Q_1$ and $q_2=\sqrt{\alpha}Q_2$, respectively.
In order to compute the supremum in (\[lower2\]), we use the gradient ascent algorithm. As the initial point, we set $q_1=2.01$ and $\tau=0$ which give us the same lower bound (\[lbound\]) for any arbitrary choice of $q_2$. Starting from this initial point (together with the arbitrary choice $q_2=3$), the algorithm converged to $q^*_1=1.5496$, $q^*_2=5.6679$, $\tau^*=0.1285$, and the corresponding lower bound is achieved as $$\begin{aligned}
\label{tightlowerbound}
E_{min}(\mathcal{F})\ge 0.9057\sqrt\alpha.\end{aligned}$$
Comparing *Theorem 1* with *Theorem 3* shows that the lower bound is tightened significantly.
Upper Bound for $E_{min}(\mathcal{F})$
--------------------------------------
To upper bound $E_{min}(\mathcal{F})$, we introduce a $K$-layer coding scheme which has a $K$-layer quantizer and sends the quantization indices using Wyner-Ziv coding, where the $k$th quantization index is to be decoded whenever $N \leq N_k$ for some predetermined $N_1 \geq N_2 \geq \ldots \geq N_K$. However, instead of relying on only one uncoded transmission of the source $X^n$ as the generator of the side information at the receiver, we also send quantization errors uncoded after each layer of quantization. In other words, we have $K$ layers of uncoded transmission while in [@c10] the authors only had the uncoded transmission in first layer.
It is not immediately obvious that this strategy will reduce the total expended energy, because even though the energy needed to convey quantization indices will be reduced because of a richer set of available side information, transmission of the quantization errors themselves consumes additional energy. However, as we show here, the minimum energy needed is indeed reduced compared to the scheme in [@c10].
\[table1\]
Noise interval $N>N_1$ $N_1\geq N>N_2$ $N_2\geq N>N_3$ $...$ $N_{K}\geq N>N_{K+1}$
---------------- ------------------------------ ------------------------------ ------------------------------ ------- ----------------------------------
$-$ $\hat{S}^n_1$ $\hat{S}^n_1$ $\hat{S}^n_1$
$\hat{S}^n_2$ .
$...$ .
.
$\hat{S}^n_{K}$
$\sqrt{A_0} S^n_0+W^n_{0,N}$ $\sqrt{A_0} S^n_1+W^n_{0,N}$ $\sqrt{A_0} S^n_2+W^n_{0,N}$ $\sqrt{A_0} S^n_{K}+W^n_{0,N}$
$\sqrt{A_1} S^n_1+W^n_{1,N}$ $\sqrt{A_1} S^n_2+W^n_{1,N}$ .
$\sqrt{A_2} S^n_2+W^n_{2,N}$ .
$...$ .
.
.
$\sqrt{A_{K}} S^n_{K}+W^n_{K,N}$
The source $X^n$ is successively quantized into source codewords $\hat{S}^n_k$ for $k=1,\ldots,K$, where the underlying single-letter characterization satisfies $$\begin{aligned}
S_k=\hat{S}_{k+1}+S_{k+1}\nonumber\end{aligned}$$ with $S_0=X$ and $\hat{S}_{k+1}\perp S_{k+1}$. Each $S_k^n$ for $k=0,1,\ldots,K$ is then sent in an uncoded fashion, i.e., as $\sqrt{A_k}S_k^n$. For any noise variance $0<N<\infty$ , the received signals will then be given by $$Y^n_{i,N}=\sqrt{A_i} S_i^n +W^n_{i,N}$$ for $i=0,\ldots,K$. When $N>N_1$, the $X^n$ will be estimated only by utilizing $Y^n_{0,N}$. On the other hand, when $N_{k+1}<N\leq N_k$, for $k=1,2,\ldots,K$, since the first $k$ layers of quantization indices will already be decoded, the estimation can rely on all $$\begin{aligned}
\tilde{Y}^n_{i,N}=\sqrt{A_i} S^n_k+W^n_{i,N}\nonumber\end{aligned}$$ as *effective* side information, as all $\hat{S}^n_i$ for $i=1,...,k$ can be subtracted from $X^n$. The utilization of information in our coding scheme is summarized in TABLE I.
Now, to be able to decode $\hat{S_{k}}$ whenever $N \leq N_k$, it suffices to use a binning rate of $$\begin{aligned}
\label{rk}
R_k &=& I(S_{k-1};\hat{S}_k|\tilde{Y}_{0,N_k},\tilde{Y}_{1,N_k},...,\tilde{Y}_{k-1,N_k})\nonumber\\
&=& I(S_{k-1};\hat{S}_k)-I(\tilde{Y}_{0,N_k},\tilde{Y}_{1,N_k},...,\tilde{Y}_{k-1,N_k};\hat{S}_k)\nonumber\\
&=& h(S_{k-1})-h(S_{k})\nonumber\\
&&-h(\tilde{Y}_{0,N_k},\tilde{Y}_{1,N_k},...,\tilde{Y}_{k-1,N_k})\nonumber\\
&&+h(\tilde{Y}_{0,N_k},\tilde{Y}_{1,N_k},...,\tilde{Y}_{k-1,N_k}|\hat{S}_k)\nonumber\\
&=&\frac{1}{2}\log\frac{\sigma^2_{S_{k-1}}}{\sigma^2_{S_{k}}}-\frac{1}{2}\log\frac{\det\mathbf{\Sigma}_{\mathbf{Y}_k}}{\det\mathbf{\Sigma}_{\mathbf{Y}_k|\hat{S}_k}}\end{aligned}$$ where $$\mathbf{\Sigma}_{\mathbf{Y}_k} = \mathbf{A}_k\mathbf{\Sigma}_{\mathbf{Z}_k}\mathbf{A}_k^T$$ with $$\begin{aligned}
\mathbf{A}_k=
\begin{bmatrix}
\sqrt[]{A_0} & 1 & 0 & 0 & . & . & . & 0 \\
\sqrt[]{A_1} & 0 & 1 & 0 & . & . & . & 0 \\
. & . & . & 1 & . & . & . & 0 \\
. & . & . & . & 1 & . & . & 0 \\
. & . & . & . & . & 1 & . & 0 \\
\sqrt[]{A_{k-1}} & 0 & 0 & 0 & . & . & . & 1 \\
\end{bmatrix}\nonumber\end{aligned}$$ and $$\begin{aligned}
\mathbf{Z}_k=
\begin{bmatrix}
S_{k-1} \\
W_{0,N_k} \\
W_{1,N_k}\\
\vdots\\
W_{k-1,N_k}
\end{bmatrix} \; .\nonumber\end{aligned}$$ Similarly, $$\mathbf{\Sigma}_{\mathbf{Y}_k|\hat{S}_k} = \mathbf{A}_k\mathbf{\Sigma}_{\mathbf{\tilde{Z}}_k}\mathbf{A}_k^T$$ with $$\begin{aligned}
\mathbf{\tilde{Z}}_k=
\begin{bmatrix}
S_{k} \\
W_{0,N_k} \\
W_{1,N_k}\\
\vdots \\
W_{k-1,N_k}
\end{bmatrix} \; .\nonumber\end{aligned}$$ Since the source and channel noise are independent, both $\mathbf{\Sigma}_{\mathbf{Z}_k}$ and $\mathbf{\Sigma}_{\mathbf{\tilde{Z}}_k}$ are diagonal, and that makes the computation of $\mathbf{\Sigma}_{\mathbf{Y}_k}$ and $\mathbf{\Sigma}_{\mathbf{Y}_k|\hat{S}_k} $ easy. Specifically, defining the $k\times k$ matrix $$\mathbf{G}_k=
\begin{bmatrix}
1 & 0 & . & . & . & 0\\
0 & 0 & 0 & . & . & 0 \\
. & 0 & 0 & 0 & . & 0 \\
. & . & 0 & . & 0 & 0 \\
. & . & . & . & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
\end{bmatrix} ,$$ one can write $$\mathbf{\Sigma}_{\mathbf{Z}_k}= N_k \mathbf{I}_{k}+(\sigma^2_{S_{k-1}}-N_k)\mathbf{G}_k \; .$$ and $$\mathbf{\Sigma}_{\mathbf{\tilde{Z}}_k}= N_k \mathbf{I}_{k}+(\sigma^2_{S_k}-N_k)\mathbf{G}_k \; .$$ We then have $$\begin{aligned}
\mathbf{\Sigma}_{\mathbf{Y}_k}&=\mathbf{A}_k\bigg(N_k \mathbf{I}_{k}+(\sigma^2_{S_{k-1}}-N_k)\mathbf{G}_k\bigg)\mathbf{A}_k^T\nonumber\\&=N_k\mathbf{A}_k\mathbf{I}_{k}\mathbf{A}_k^T+(\sigma^2_{S_{k-1}}-N_k)\mathbf{A}_k\mathbf{G}_k\mathbf{A}_k^T\nonumber\\
&=N_k\mathbf{A}_k\mathbf{A}_k^T+(\sigma^2_{S_{k-1}}-N_k) \mathbf{a}_k \mathbf{a}_k^T\nonumber \\
&=N_k( \mathbf{a}_k \mathbf{a}_k^T+\mathbf{I}_k)+(\sigma^2_{S_{k-1}}-N_k) \mathbf{a}_k \mathbf{a}_k^T \nonumber \\
\label{covy}
&=N_k \mathbf{I}_k+\sigma^2_{S_{k-1}} \mathbf{a}_k \mathbf{a}_k^T \; \end{aligned}$$ where $\mathbf{a}_k$ is the first column of matrix $\mathbf{A}_k$. Similarly, $$\begin{aligned}
\label{covys}
\mathbf{\Sigma}_{\mathbf{Y}_k|\hat{S}_k}=N_k\mathbf{I}_k+\sigma^2_{S_k} \mathbf{a}_k \mathbf{a}_k^T.\end{aligned}$$ By substituting (\[covy\]) and (\[covys\]) in (\[rk\]), we then get $$\begin{aligned}
\label{Rdet}
R_k=\frac{1}{2}\log\frac{\sigma^2_{S_{k-1}}}{\sigma^2_{S_{k}}}-\frac{1}{2}\log\frac{\det(N_{k}\mathbf{I}_{k}+\sigma^2_{S_{k-1}}\mathbf{a}_k \mathbf{a}_k^T)}{\det(N_{k}\mathbf{I}_{k}+\sigma^2_{S_{k}}\mathbf{a}_k \mathbf{a}_k^T)}.\end{aligned}$$ Using the Matrix Determinant Lemma [@c17], which states for arbitrary invertible $\mathbf{M}$ and column vectors $\mathbf{u}$ and $\mathbf{v}$ that $$\det(\mathbf{M}+\mathbf{u}\mathbf{v}^T) = \det(\mathbf{M}) \cdot (1+\mathbf{v}^T\mathbf{M}^{-1}\mathbf{u})$$ We can write $$\begin{aligned}
R_k&=\frac{1}{2}\log\frac{\sigma^2_{S_{k-1}}}{\sigma^2_{S_{k}}}-\frac{1}{2}\log\frac{(1+\frac{\sigma^2_{S_{k-1}}}{N_k}\mathbf{a}_k^T\mathbf{a}_k)}{(1+\frac{\sigma^2_{S_{k}}}{N_k}\mathbf{a}_k^T\mathbf{a}_k)}\nonumber\\
&=\frac{1}{2}\log\frac{\sigma^2_{S_{k-1}}(1+\frac{\sigma^2_{S_{k}}}{N_k}\mathbf{a}_k^T\mathbf{a}_k)}{\sigma^2_{S_k}(1+\frac{\sigma^2_{S_{k-1}}}{N_k}\mathbf{a}_k^T\mathbf{a}_k)} \nonumber \\
&=\frac{1}{2}\log\frac{\beta_k+Q_kA_{k,\rm{total}}}{\beta_{k-1}+Q_kA_{k,\rm{total}}} \end{aligned}$$ where $\beta_k=\frac{1}{\sigma^2_{S_{k}}}$, $Q_k=\frac{1}{N_k}$, and $$A_{k,\rm{total}}\stackrel{\Delta}{=}A_0+A_1+...+A_{k-1} \; .$$
For the digital message, we use the channel with infinite bandwidth and energy $B_k$. Therefore, the rate must not exceed the channel capacity under the noise level $N_k$, i.e., $$\frac{1}{2}\log\frac{\beta_k+Q_kA_{k,\rm{total}}}{\beta_{k-1}+Q_kA_{k,\rm{total}}} \leq C_k=\frac{B_kQ_k}{2}$$ or equivalently, $$\frac{\beta_k+Q_kA_{k,\rm{total}}}{\beta_{k-1}+Q_kA_{k,\rm{total}}} \leq \exp(B_kQ_k) \; .$$
When $N_{k+1}<N\leq N_k$, or equivalently when $Q_k\leq Q< Q_{k+1}$, the MMSE estimation boils down to estimating $\tilde {S}_k^n$ using all the available effective side information, that is $$\tilde {S}_k^n=\sum_{i=0}^{k}\lambda_i\tilde{Y}_{i,N}$$ with appropriate $\lambda_i$ for $i=0,1,\ldots,k$. Thus, the resultant distortion can be calculated with the help of the Sherman-Morrison-Woodbury identity [@c17] as $$\begin{aligned}
\label{D}
D&=\sigma^2_{S_k}-(\sigma^2_{S_k})^2\mathbf{a}_{k+1}^T\left(N \mathbf{I}_{k+1}+\sigma^2_{S_k} \mathbf{a}_{k+1} \mathbf{a}_{k+1}^T\right)^{-1}\mathbf{a}_{k+1} \nonumber \\
&=\sigma^2_{S_k}-(\sigma^2_{S_k})^2\mathbf{a}_{k+1}^T\left[Q\mathbf{I}_{k+1} - \frac{Q^2 \mathbf{a}_{k+1}\mathbf{a}_{k+1}^T}{\beta_k + Q \mathbf{a}_{k+1}^T\mathbf{a}_{k+1}}\right]\mathbf{a}_{k+1} \nonumber \\
&=\sigma^2_{S_k}-(\sigma^2_{S_k})^2Q A_{k+1,\rm{total}}\left[1 - \frac{Q A_{k+1,\rm{total}}}{\beta_k + Q A_{k+1,\rm{total}}}\right] \nonumber \\
&=\sigma^2_{S_k}-\sigma^2_{S_k}Q A_{k+1,\rm{total}}\left[\frac{1}{\beta_k + Q A_{k+1,\rm{total}}}\right] \nonumber \\
&=\frac{1}{\beta_k + Q A_{k+1,\rm{total}}}.\end{aligned}$$ Equivalently, the fidelity can be written as $$\begin{aligned}
\label{F}
F(Q)=\beta_k + Q A_{k+1,\rm{total}} \; .\end{aligned}$$ Therefore, $F(Q)$ is an “inclined” staircase function with changing slope $A_{k+1,\rm{total}}$ within each $Q_k\leq Q< Q_{k+1}$ as shown in Fig. \[figr:UpperBound\]. The beauty of the work is that we deal with linear segments. Thus, our analysis is easily understandable. Please note that Fig. \[figr:UpperBound\] is different with the figure presented in [@c10]. The slope of inclined staircase function is fixed and equal to $E_0$ in [@c10], which is a special case of our work by letting $A_0=E_0$ and $A_i=0$ for $i=1,2,...$ . We are now ready to propose an upper bound on $E_{min}({\cal F})$.
![\[figr:UpperBound\]The fidelity-quality tradeoff is always above the profile $F(Q)$, coinciding with it at the jump points $Q_k$.](Figure){width="50.00000%"}
The minimum required energy for profile $F(Q)=1+\alpha Q^2$ is upper-bounded as $$\begin{aligned}
E_{min}(F)\leq e\ \sqrt[]{\alpha}\nonumber\end{aligned}$$ with $e\approx 2.3203$.
We will use the scheme described above such that for any $0=Q_0<Q_1<Q_2<...,$ the energy $A_k$ and the source coding parameters $1=\beta_0<\beta_1<\beta_2<...$ will be chosen such that the fidelity-quality tradeoff in (\[F\]) is always above the profile $F(Q)$, coinciding with it at the jump points $Q_k$, as shown in Fig. \[figr:UpperBound\]. In other words, $$\begin{aligned}
A_{k,\rm{total}}Q_k+\beta_{k-1}=1+\alpha Q^2_k\end{aligned}$$ for all $k=1,2,...$ .
Thus, we obtain $$\begin{aligned}
\label{betak}
\beta_{k-1}&=1+\alpha Q^2_k-A_{k,\rm{total}} Q_k\nonumber\\
&=1+\alpha Q^2_k-(A_0+A_1+...+A_{k-1})Q_k.\end{aligned}$$ The requirement that $\beta_{k}$ is increasing in $k$ leads to the following constraint $$\begin{aligned}
\label{Ak_constraint}
A_0&=\alpha Q_1 \nonumber\\
A_k&<\frac{\alpha (Q^2_{k+1}-Q^2_k)- A_{k,\rm{total}}(Q_{k+1}-Q_k)}{Q_{k+1}},\end{aligned}$$ for all $ k\geq 1$.
\[lemma2\] For fixed $\alpha$, the choice $Q_k=k\Delta$, $A_0=\alpha\Delta$ and $A_k=d^k\alpha\Delta$ for $k\geq 1$ satisfies (\[Ak\_constraint\]) for any $0<d<1$ and $\Delta >0$.
Substituting $A_k$ and $Q_k$ in (\[Ak\_constraint\]) yields: $$\begin{aligned}
\label{dk}
d^k<\frac{(2k+1)-(\frac{d^k-1}{d-1})}{k+1}.\end{aligned}$$ We use induction to prove (\[dk\]). For $k=1$, (\[dk\]) reduces to $d<1$ which is true. Substituting $k=l$, we get to the following: $$\begin{aligned}
\label{dn}
d^l(l+1)<(2l+1)-(d^{l-1}+...+d+1)\end{aligned}$$ We assume (\[dn\]) is true. Now we substitute $k=l+1$ in (\[dk\]) and have the following: $$\begin{aligned}
\label{dn+1}
d^{l+1}(l+2)<(2l+3)-(d^l+d^{l-1}+...+d+1).\end{aligned}$$ In order to complete the proof, we show (\[dn+1\]) is true as follows. First, we multiply both sides of (\[dn\]) with $d$ and then add $d^{l+1}$ to both sides, yielding $$\begin{aligned}
\label{dnn}
d^{l+1}(l+2)<(2l+1)d+d^{l+1}-(d^l+...+d).\end{aligned}$$ Now, it suffices to show the right hand side of (\[dnn\]) is less than or equal to the right hand side of (\[dn+1\]), which is the same as $$\begin{aligned}
\label{dinequal}
d^{l+1}+(2l+1)d\leq 2l+2.\end{aligned}$$ Since $0<d<1$, we have $(2l+1)d<(2l+1)$ and $d^{l+1}<1$. Thus, (\[dinequal\]) is valid and the proof of *Lemma 2* is complete.
By substituting $A_k$ and $Q_k$ in (\[betak\]), we get $$\begin{aligned}
\beta_{k-1}&=1+\alpha k^2\Delta^2-k\alpha\Delta^2\bigg(\frac{1-d^k}{1-d}\bigg).\nonumber
%&=1+\alpha k^2\Delta^2-k\alpha\Delta^2(1+d+...+d^{k-1})\nonumber\end{aligned}$$ Letting $K\to\infty$, the total uncoded energy becomes [$$\begin{aligned}
E_{unc}&=\sum_{k=0}^{\infty}\frac{A_k}{\beta_k}\nonumber\\
&=\sum_{k=0}^{\infty}\frac{d^k\alpha\Delta}{1+\alpha (k+1)^2\Delta^2-(k+1)\alpha\Delta^2\bigg(\frac{1-d^{k+1}}{1-d}\bigg)}.\end{aligned}$$]{}
\[table2\]
[|l||\*[5]{}[c|]{}]{} &&& &&\
Lower bound of [@c10] & 0.4507 & 1.4252 & 4.5070 & 14.2524 & 45.0700\
Our lower bound & 0.9057 & 2.8641 & 9.0570 & 28.6407 & 90.5700\
Upper bound of [@c10] & 3.1846 & 10.0706 & 31.8460 & 100.7059 & 318.4600\
Our upper bound & 2.3203 & 7.3374 & 23.2030 & 73.3743 & 232.0300\
The lower bound improvement & 0.4550 & 1.4388 & 4.5500 & 14.3884 & 45.5000\
The upper bound improvement & 0.8643 & 2.7332 & 8.6430 & 27.3316 & 86.4300\
On the other hand, the total digital energy is [$$\begin{aligned}
E_{dig}&=\sum_{k=1}^{\infty}B_k\nonumber\\
&=\sum_{k=1}^{\infty}\frac{1}{Q_k}\log\Bigg(1+\frac{\beta_{k}-\beta_{k-1}}{1+\alpha Q^2_{k}}\Bigg)\nonumber\\
&=\sum_{k=1}^{\infty}\frac{1}{k\Delta}\log\Bigg(1+\frac{\alpha\Delta^2\bigg(2k+1-kd^k-(\frac{1-d^{k+1}}{1-d})\bigg)}{1+\alpha k^2\Delta^2}\Bigg).\end{aligned}$$]{}
Denoting $\Delta=\frac{c}{\sqrt \alpha}$, we then get [$$\begin{aligned}
\label{etot}
&E_{total}=E_{unc}+E_{dig}\nonumber\\
&\leq\sqrt{\alpha}\sum_{k=0}^{\infty}\frac{cd^k}{1+c^2 (k+1)^2-(k+1)c^2\bigg(\frac{1-d^{k+1}}{1-d}\bigg)}\nonumber\\
&+\sqrt{\alpha}\sum_{k=1}^{\infty}\frac{1}{kc}\log\Bigg(1+\frac{c^2\bigg(2k+1-kd^k-(\frac{1-d^{k+1}}{1-d})\bigg)}{1+k^2c^2}\Bigg). \end{aligned}$$]{}
In order to minimize the upper bound on total energy, we solve (\[etot\]) numerically for different values of $c$ and $d$. For optimal values $d^*=0.999$ and $c^*=0.00137$, the upper bound yields $$\begin{aligned}
E_{total}\leq 2.3203\sqrt{\alpha}.\nonumber\end{aligned}$$
Comparing *Theorem 2* with *Theorem 4*, we notice that the upper bound is improved significantly. Note that by setting $d=0$ and $A_0=E_0$ in our work, the method in [@c10] is achieved exactly. This is expected, as [@c10] is a special case of our work. We compare our lower and upper bounds with the bounds in [@c10] for some values of $\alpha$ and show our improvements in TABLE 2.
Conclusions and Future Work
===========================
Minimum energy required to achieve a distortion-noise profile, i.e., a function indicating the maximum allowed distortion value for each noise level, is studied for robust transmission of Gaussian sources over Gaussian channels. In order to analyze the minimum energy behavior for the square-law distortion noise profile, the lower and upper bounds were proposed by our coding scheme. We improved both upper and lower bounds significantly. For future, we are interested to study the distortion-noise profile problem in Multiple Access Channels (MAC). In MAC, instead of having one distortion function, we deal with at least two distortion functions and distortion regions.
[99]{}
A. Lapidoth and S. Tinguely, “Sending a bivariate Gaussian over a Gaussian MAC," *IEEE Transactions on Information Theory*, vol. 56, no. 6, pp. 2714 - 2752, Jun. 2010. P. Minero, S. Lim, and Y.-H. Kim, “Joint source-channel coding via hybrid coding," *IEEE International Symposium on Information Theory Proceedings (ISIT)*, pp. 781-785, Jul. 2011. W. Liu and B. Chen, “Interference channels with arbitrarily correlated sources," *IEEE Transactions on Information Theory*, vol. 57, no. 12, pp. 8027-8037, Dec. 2011. M. P. Wilson, K. Narayanan, and G. Caire, “Joint source channel coding with side information using hybrid digital analog codes," *IEEE Transactions on Information Theory*, vol. 56, no. 10, pp. 4922–4940, Oct. 2010. I. Estella and D. Gunduz, “Hybrid digital-analog transmission for the Gaussian one-helper problem," *IEEE Global Telecommunications Conference (GLOBECOM 2010)*, pp. 1 -5, Dec. 2010. A. Jain, D. Gunduz, S. R. Kulkarni, H. V. Poor, and S. Verdú, “Energy-distortion tradeoffs in Gaussian joint source-channel coding problems," *IEEE Transactions on Information Theory*, vol. 58, no. 5, pp. 3153-3168, May 2012. N. Jiang, Y. Yang, A. Host-Madsen, and Z. Xiong, “On the minimum energy of sending correlated sources over the Gaussian MAC," *IEEE Transactions on Information Theory*, vol. 60, no. 10, pp. 6254-6275, Aug. 2014. E. Koken and E. Tuncel, “On the energy-distortion tradeoff for the Gaussian broadcast problem," *IEEE International Symposium on Information Theory*, Barcelona, Spain, Jul. 2016. E. Koken, D. Gunduz, and E. Tuncel, “Energy-distortion exponents in lossy transmission of Gaussian sources over Gaussian channels," *IEEE Transactions on Information Theory*, vol. 63, no. 2, pp. 1227-1236, Feb. 2017. E. Koken and E. Tuncel, “On minimum energy for robust Gaussian joint source-channel coding with a distortion-noise profile," *IEEE International Symposium on Information Theory*, Aachen, Germany, June 2017. K. Woyach, K. Harrison, G. Ranade, and A. Sahai, “Comments on unknown channels," *IEEE Information Theory Workshop (ITW)*, pp. 172-176, Sep. 2012. K. Eswaran, A. D. Sarwate, A. Sahai, and M. Gastpar, “Using zero-rate feedback on binary additive channels with individual noise sequences," *IEEE International Symposium on Information Theory*, Nice, France, Jun. 2007. V. Misra and T. Weissman, “The porosity of additive noise sequences," *IEEE International Symposium on Information Theory*, Istanbul, Turkey, Jul. 2012. Y. Lomnitz and M. Feder, “Communication over individual channels," *IEEE Transactions on Information Theory*, vol. 57, no. 11, pp. 7333-7358, Nov. 2011. C. Tian, S. Diggavi, S. Shamai, “Approximate characterization for the Gaussian source broadcast distortion region," *IEEE Transactions on Information Theory*, vol. 57, no. 8, pp. 124-136, Jan. 2011. Z. Reznic, M. Feder, and R. Zamir, “Distortion bounds for broadcasting with bandwidth expansion," *IEEE Transactions on Information Theory*, vol. 52, no. 8, pp. 3778-3788, Aug. 2006. J. Ding and A. Zhou, “Eigenvalues of rank-one updated matrices with some applications," *Applied Mathematics Letters*, vol. 20, no. 12, pp. 1223-1226, 2007.
|
---
author:
- |
Pedro Resende\
*Departamento de Matem[á]{}tica, Instituto Superior T[é]{}cnico,\
*Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal**
date:
title: 'A note on infinitely distributive inverse semigroups[^1]'
---
By an *infinitely distributive* inverse semigroup will be meant an inverse semigroup $S$ such that for every subset $X\subseteq S$ and every $s\in S$, if ${\bigvee}X$ exists then so does ${\bigvee}(sX)$, and furthermore ${\bigvee}(sX)=s{\bigvee}X$.
One important aspect is that the infinite distributivity of $E(S)$ implies that of $S$; that is, if the multiplication of $E(S)$ distributes over all the joins that exist in $E(S)$ then $S$ is infinitely distributive. This can be seen in Proposition 20, page 28, of Lawson’s book [@Lawson]. Although the statement of the proposition mentions only joins of nonempty sets, the proof applies equally to any subset.
The aim of this note is to present a proof of an analogous property for binary meets instead of multiplication; that is, we show that for any infinitely distributive inverse semigroup the existing binary meets distribute over all the joins that exist.
A useful consequence of this lies in the possibility of constructing, from infinitely distributive inverse semigroups, certain quantales that are also locales (due to the stability of the existing joins both with respect to the multiplication and the binary meets), yielding a direct connection to étale groupoids via the results of [@Resende]. The consequences of this include an algebraic construction of “groupoids of germs” from certain inverse semigroups, such as pseudogroups, and will be developed elsewhere.
Let $S$ be an inverse semigroup, and let $x,y\in S$ be such that the meet $x\wedge y$ exists. Then the join $$f={\bigvee}\{g\in {{\downarrow}}(xx^{-1}yy^{-1}){\mid}gx=gy\}$$ exists, and we have $x\wedge y=fx=fy$.
Consider the set $Z$ of lower bounds of $x$ and $y$, $$Z=\{z\in S{\mid}z\le x,\ z\le y\}\;,$$ whose join is $x\wedge y$. By [@Lawson Prop. 17, p. 27], the (nonempty) set $$F=\{zz^{-1}{\mid}z\in Z\}$$ has a join $f={\bigvee}F$ that coincides with $({\bigvee}Z)({\bigvee}Z)^{-1}=(x\wedge y)(x\wedge y)^{-1}$. Hence, $x\wedge y=fx=fy$. The lemma now follows from the fact that the elements $zz^{-1}$ with $z\in Z$ are precisely the idempotents $g\in{{\downarrow}}(xx^{-1}yy^{-1})$ such that $gx=gy$.
Under the assumption of infinite distributivity we have a converse:
Let $S$ be an infinitely distributive inverse semigroup, and let $x,y\in S$ be such that the join $$f={\bigvee}\{g\in {{\downarrow}}(xx^{-1}yy^{-1}){\mid}gx=gy\}$$ exists. Then the meet $x\wedge y$ exists, and we have $x\wedge y=fx=fy$.
By [@Lawson Prop. 20, p. 28], the join $${\bigvee}\{gx{\mid}g\le xx^{-1}yy^{-1},\ gx=gy\}$$ exists and it equals $fx$. Similarly, the join $${\bigvee}\{gy{\mid}g\le xx^{-1}yy^{-1},\ gx=gy\}$$ exists and it equals $fy$. But the two sets of which we are taking joins are the same due to the condition $gx=gy$, and thus $fx=fy$. The element $fx$ is therefore a lower bound of both $x$ and $y$. Let $z$ be another such lower bound. Then $z=zz^{-1}x=zz^{-1}y$, and thus $zz^{-1}\le f$, which implies $z\le fx$. Hence, $fx$ is the greatest lower bound of $x$ and $y$.
Let $S$ be an infinitely distributive inverse semigroup, let $x\in S$, and let $(y_i)$ be a family of elements of $S$. Assume that the join ${\bigvee}_i y_i$ exists, and that the meet $x\wedge{\bigvee}_i y_i$ exists. Then, for all $i$ the meet $x\wedge y_i$ exists, the join ${\bigvee}_i(x\wedge y_i)$ exists, and we have $$x\wedge{\bigvee}_i y_i = {\bigvee}_i(x\wedge y_i)\;.$$
Let us write $y$ for ${\bigvee}_i y_i$, $e_i$ for $y_i y_i^{-1}$, and let $f$ be the idempotent $$f={\bigvee}\{g\in {{\downarrow}}(xx^{-1}yy^{-1}){\mid}gx=gy\}\;,$$ which exists, by the first lemma. Furthermore, also by the first lemma, we have $x\wedge y=fx=fy$. We shall prove that $x\wedge y_i$ exists for each $i$, and that it equals $e_i (x\wedge y)=e_i fx=e_i fy$. By the second lemma, it suffices to show that for each $i$ the join $$f_i={\bigvee}\{g\in {{\downarrow}}(xx^{-1}y_i y_i^{-1}){\mid}gx=gy_i\}$$ exists and equals $e_i f$. Consider $g\in {{\downarrow}}(xx^{-1}y_i y_i^{-1})={{\downarrow}}(xx^{-1} e_i)$ such that $gx=gy_i$. The condition $g\in{{\downarrow}}(xx^{-1}e_i)$ implies that $g\le e_i$, and thus $g=g e_i$. Hence, since $y_i=e_i y$, the condition $gx=gy_i$ implies $gx=ge_i y=gy$, and thus $g\le f$ because furthermore $g\in{{\downarrow}}(xx^{-1}yy^{-1})$. Hence, we have both $g\le e_i$ and $g\le f$, [i.e.]{}, $g\le e_i f$, meaning that $e_i f$ is an upper bound of the set $$X=\{g\in {{\downarrow}}(xx^{-1}e_i){\mid}gx=gy_i\}\;.$$ In order to see that it is the least upper bound it suffices to check that $e_i f$ belongs to $X$, which is immediate: first, $e_i f\le xx^{-1}$ because $f\le xx^{-1}$, and thus $e_i f\in{{\downarrow}}( xx^{-1} e_i)$; secondly, $$(e_i f) x= (e_i e_i f) y = (e_i f e_i) y = (e_i f)(e_i y) = (e_i f) y_i\;.$$ Hence, $e_if\in X$, and thus $e_i fx=x\wedge y_i$. In addition, the join ${\bigvee}_i e_i$ exists and it equals $yy^{-1}$, by [@Lawson Prop. 17, p. 27], and thus, using infinite distributivity and the fact that $f\le yy^{-1}$, we obtain $$x\wedge y=fx=yy^{-1} fx=({\bigvee}_i e_i) fx={\bigvee}_i (e_i f x)={\bigvee}_i(x\wedge y_i)\;. \qed$$
[1]{}
M.V. Lawson, Inverse Semigroups — The Theory of Partial Symmetries, World Scientific, 1998.
P. Resende, Étale groupoids and their quantales; arXiv:math/0412478.
[^1]: Research supported in part by FEDER and FCT through CAMGSD.
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address: |
Stanford University\
Department of Mathematics\
450 Serra Mall, Building 380\
Stanford, CA 94305-2125, U.S.A.
author:
- Kannan SOUNDARARAJAN
date: Juin 2016
subtitle: 'after Matom[" a]{}ki and Radziwi[łł]{}'
title: The Liouville function in short intervals
---
[*Résumé*]{}:[^1][^2] [^3] La fonction de Liouville $\lambda$ est une fonction complètement multiplicative à valeur $\lambda(n) = +1$ \[resp. $-1$\] si $n$ admet un nombre pair \[resp. impair\] de facteurs premiers, comptés avec multiplicité. On s’attend à ce qu’elle se comporte comme une collection aléatoire de signes, $+1$ et $-1$ étant équiprobables. Par exemple, une conjecture célèbre de Chowla dit que les valeurs $\lambda(n)$ et $\lambda(n+1)$ (plus généralement en arguments translatés par $k$ entiers distincts fixes) ont corrélation nulle. Selon une autre croyance répandue, presque tous les intervalles de longueur divergeant vers l’infini devraient donner à peu près le même nombre de valeurs $+1$ et $-1$ de $\lambda$. Récemment Matomäki et Radziwiłł ont établi que cette croyance était en effet vraie, et de plus établi une variante d’un tel résultat pour une classe générale de fonctions multiplicatives. Leur collaboration ultérieure avec Tao a conduit ensuite à la démonstration des versions moyennisées de la conjecture de Chowla, ainsi qu’à celle de l’existence de nouveaux comportements de signes de la fonction de Liouville. Enfin un dernier travail de Tao vérifie une version logarithmique de ladite conjecture et, de là, résout la conjecture de la discrépance d’Erd[ő]{}s. Dans ce Séminaire je vais exposer quelques-unes des idées ma[î]{}tresses sous-jacentes au travail de Matomäki et Radziwiłł.
Introduction
============
The Liouville function $\lambda$ is defined by setting $\lambda(n) =1$ if $n$ is composed of an even number of prime factors (counted with multiplicity) and $-1$ if $n$ is composed of an odd number of prime factors. Thus, it is a completely multiplicative function taking the value $-1$ at all primes $p$. The Liouville function is closely related to the M[" o]{}bius function $\mu$, which equals $\lambda$ on square-free integers, and which equals $0$ on integers that are divisible by the square of a prime.
The Liouville function takes the values $1$ and $-1$ with about equal frequency: as $x\to \infty$ $$\label{1.1}
\sum_{n\le x} \lambda(n) = o(x),$$ and this statement (or the closely related estimate $\sum_{n\le x} \mu(n) = o(x)$) is equivalent to the prime number theorem. Much more is expected to be true, and the sequence of values of $\lambda(n)$ should appear more or less like a random sequence of $\pm 1$. For example, one expects that the sum in has “square-root cancelation": for any $\epsilon >0$ $$\label{1.2}
\sum_{n\le x} \lambda(n) = O(x^{\frac 12 +\epsilon}),$$ and this bound is equivalent to the Riemann Hypothesis (for a more precise version of this equivalence see [@S]). In particular, the Riemann Hypothesis implies that $$\label{1.3}
\sum_{x< n \le x+ h} \lambda(n) = o(h), \qquad \text{provided } h >x^{\frac 12 +\epsilon},$$ and a refinement of this, due to Maier and Montgomery [@MM], permits the range $h > x^{1/2}(\log x)^A$ for a suitable constant $A$. Unconditionally, Motohashi [@Mo] and Ramachandra [@Ra] showed independently that $$\label{1.4}
\sum_{x < n \le x+h} \lambda(n) = o(h), \qquad \text{provided } h> x^{\frac 7{12}+\epsilon}.$$ The analogy with random $\pm 1$ sequences would suggest cancelation in every short interval as soon as $h> x^{\epsilon}$ (perhaps even $h \ge (\log x)^{1+\delta}$ is sufficient). Instead of asking for cancelation in every short interval, if we are content with results that hold for almost all short intervals, then more is known. Assuming the Riemann Hypothesis, Gao [@G] established that if $h \ge (\log X)^{A}$ for a suitable (large) constant $A$, then $$\label{1.5}
\int_X^{2X} \Big| \sum_{x<n\le x+h} \lambda(n) \Big|^2 dx = o(X h^2),$$ so that almost all intervals $[x,x+h]$ with $X\le x\le 2X$ exhibit cancelation in the values of $\lambda(n)$. Unconditionally one can use zero density results to show that almost all intervals have substantial cancelation if $h> X^{1/6+\epsilon}$. To be precise, Gao’s result (as well as the results in [@S], [@MM], [@Mo], [@Ra]) was established for the M[" o]{}bius function, but only minor changes are needed to cover the Liouville function.
The results described above closely parallel results about the distribution of prime numbers. We have already mentioned that is equivalent to the prime number theorem: $$\label{1.6}
\psi(x) = \sum_{n\le x} \Lambda(n) = x+o(x),$$ where $\Lambda(n)$, the von Mangoldt function, equals $\log p$ if $n>1$ is a power of the prime $p$, and $0$ otherwise. Similarly, in analogy with , a classical equivalent formulation of the Riemann Hypothesis states that $$\label{1.7}
\psi(x) = x+ O\big(x^{\frac 12} (\log x)^2 \big),$$ so that a more precise version of holds $$\label{1.8}
\psi(x+h) - \psi(x)= \sum_{x < n\le x+h} \Lambda(n) = h+ o(h), \qquad \text{provided } h> x^{1/2}(\log x)^{2+\epsilon}.$$ Analogously to , Huxley [@H] (building on a number of previous results) showed unconditionally that $$\label{1.9}
\sum_{x<n\le x+h} \Lambda(n) \sim h, \qquad \text{provided } h> x^{\frac 7{12}+\epsilon}.$$ Finally, Selberg [@Se] established that if the Riemann hypothesis holds, and $h \ge (\log X)^{2+\epsilon}$ then $$\label{1.10}
\int_X^{2X} \Big| \sum_{x<n\le x+h} \Lambda(n) - h \Big|^2 dx = o(Xh^2),$$ so that almost all such short intervals contain the right number of primes. Unconditionally one can use Huxley’s zero density estimates to show that almost all intervals of length $h>X^{1/6+\epsilon}$ contain the right number of primes.
The results on primes invariably preceded their analogues for the Liouville (or M[" o]{}bius) function, and often there were some extra complications in the latter case. For example, the work of Gao is much more involved than Selberg’s estimate , and the corresponding range in is a little weaker. Although there has been dramatic recent progress in sieve theory and understanding gaps between primes, the estimates , and have not been substantially improved for a long time. So it came as a great surprise when Matom[" a]{}ki and Radziwi[ł]{}[ł]{} established that the Liouville function exhibits cancelation in almost all short intervals, as soon as the length of the interval tends to infinity — that is, obtaining qualitatively a definitive version of unconditionally!
\[Matom[" a]{}ki and Radziwi[ł]{}[ł]{} [@MR2]\] \[thm1\] For any $\epsilon >0$ there exists $H(\epsilon)$ such that for all $H(\epsilon) < h\le X$ we have $$\int_X^{2X} \Big| \sum_{x<n\le x+h} \lambda(n) \Big|^2 dx \le \epsilon Xh^2.$$ Consequently, for $H(\epsilon) < h\le X$ one has $$\Big| \sum_{x < n\le x+h} \lambda(n) \Big| \le \epsilon^{\frac 13}h,$$ except for at most $\epsilon^{\frac 13} X$ integers $x$ between $X$ and $2X$.
As mentioned earlier, the sequence $\lambda(n)$ is expected to resemble a random $\pm 1$ sequence, and the expected square-root cancelation in the interval $[1,x]$ and cancelation in short intervals $[x,x+h]$ reflect the corresponding cancelations in random $\pm 1$ sequences. Another natural way to capture the apparent randomness of $\lambda(n)$ is to fix a pattern of consecutive signs $\epsilon_1$, $\ldots$, $\epsilon_k$ (each $\epsilon_j$ being $\pm 1$) and ask for the number of $n$ such that $\lambda(n+j) = \epsilon_j$ for each $1\le j\le k$. If the Liouville function behaved randomly, then one would expect that the density of $n$ with this sign pattern should be $1/2^k$.
\[Chowla [@C]\] Let $k\ge 1$ be an integer, and let $\epsilon_j = \pm 1$ for $1\le j\le k$. Then as $N \to \infty$ $$\label{C1.1}
|\{ n\le N: \lambda(n+j) = \epsilon_j \text{ for all } 1\le j\le k\}| = \Big( \frac{1}{2^k} + o(1) \Big) N.$$ Moreover, if $h_1$, $\ldots$, $h_k$ are any $k$ distinct integers then, as $N \to \infty$, $$\label{C1.2}
\sum_{n\le N} \lambda(n+h_1)\lambda(n+h_2)\cdots \lambda(n+h_k) = o(N).$$
Observe that $\prod_{j=1}^{k} (1+\epsilon_j \lambda(n+j)) =2^k$ if $\lambda(n+j) =\epsilon_j$, and $0$ otherwise. Expanding this product out, and summing over $n$, it follows that implies . It is also clear that follows if holds for all $k$.
The Chowla conjectures resemble the Hardy-Littlewood conjectures on prime $k$-tuples, and little is known in their direction. The prime number theorem, in its equivalent form , shows that $\lambda(n)=1$ and $-1$ about equally often, so that holds for $k=1$. When $k=2$, there are four possible patterns of signs for $\lambda(n+1)$ and $\lambda(n+2)$, and as a consequence of Theorem \[thm1\] it follows that each of these patterns appears a positive proportion of the time. For $k=3$, Hildebrand [@Hi] was able to show that all eight patterns of three consecutive signs occur infinitely often. By combining Hildebrand’s ideas with the work in [@MR2], Matom[" a]{}ki, Radziwi[ł]{}[ł]{}, and Tao [@MRT2] have shown that all eight patterns appear a positive proportion of the time. It is still unknown whether all sixteen four term patterns of signs appear infinitely often (see [@BE] for some related work).
\[thm2\] For any of the eight choices of $\epsilon_1$, $\epsilon_2$, $\epsilon_3$ all $\pm 1$ we have $$\liminf_{N \to \infty} \frac{1}{N} |\{ n\le N: \lambda(n+j)=\epsilon_j, \ \ j=1, 2, 3 \}| > 0.$$
We turn now to , which is currently open even in the simplest case of showing $\sum_{n\le N} \lambda(n) \lambda(n+1) = o(N)$. By refining the ideas in [@MR2], Matom[" a]{}ki, Radziwi[łł]{} and Tao [@MRT1] showed that a version of Chowla’s conjecture holds if we permit a small averaging over the parameters $h_1$, $\ldots$, $h_k$.
\[thm3\] Let $k$ be a natural number, and let $\epsilon > 0$ be given. There exists $h(\epsilon,k)$ such that for all $x\ge h \ge h(\epsilon,k)$ we have $$\sum_{1\le h_1, \ldots, h_k \le h } \Big| \sum_{n\le x} \lambda(n+h_1) \cdots \lambda(n+h_k) \Big| \le \epsilon h^k x.$$
Building on the ideas in [@MR2] and [@MRT2], and introducing further new ideas, Tao [@T1] has established a logarithmic version of Chowla’s conjecture in the case $k=2$. A lovely and easily stated consequence of Tao’s work is $$\label{Tao1}
\sum_{n\le x} \frac{\lambda(n)\lambda(n+1)}{n} = o(\log x).$$ Results such as , together with their extensions to general multiplicative functions bounded by $1$, form a crucial part of Tao’s remarkable resolution of the Erd[ő]{}s discrepancy problem [@T2]: If $f$ is any function from the positive integers to $\{-1, +1\}$ then $$\sup_{d, n } \Big | \sum_{j=1}^{n} f(jd) \Big| = \infty.$$
While we have so far confined ourselves to the Liouville function, the work of Matom[" a]{}ki and Radziwi[łł]{} applies more broadly to general classes of multiplicative functions. For example, Theorem \[thm1\] holds in the following more general form. Let $f$ be a multiplicative function with $-1\le f(n) \le 1$ for all $n$. For any $\epsilon >0$ there exists $H(\epsilon)$ such that if $H(\epsilon) < h\le X$ then $$\label{Mult1}
\Big| \sum_{x<n\le x+h} f(n) - \frac{h}{X} \sum_{X \le n \le 2X} f(n) \Big| \le \epsilon h,$$ for all but $\epsilon X$ integers $x$ between $X$ and $2X$. In other words, for almost all intervals of length $h$, the local average of $f$ in the short interval $[x,x+h]$ is close to the global average of $f$ between $X$ and $2X$. We should point out that this result holds uniformly for all multiplicative functions $f$ with $-1\le f(n)\le 1$ — that is, the quantity $H(\epsilon)$ depends only on $\epsilon$ and is independent of $f$. A still more general formulation (needed for Theorem \[thm3\]) may be found in Appendix 1 of [@MRT1].
The work of Matom[" a]{}ki and Radziwi[łł]{} permits a number of elegant corollaries, and we highlight two such results; see Section 8 for a brief discussion of their proofs.
\[Matom[" a]{}ki and Radziwi[łł]{} [@MR2]\] \[cor2\] For every $\epsilon>0$, there exists a constant $C(\epsilon)$ such that for all large $N$, the interval $[N,N+C(\epsilon) \sqrt{N}]$ contains an integer all of whose prime factors are below $N^{\epsilon}$.
Integers without large prime factors (called [*smooth*]{} or [*friable*]{} integers) have been extensively studied, and the existence of smooth numbers in short intervals is of interest in understanding the complexity of factoring algorithms. Previously Corollary \[cor2\] was only known conditionally on the Riemann hypothesis (see [@So2]). Further, shows that almost all intervals with length tending to infinity contain the right density of smooth numbers (see Corollary 6 of [@MR2]).
\[cor3\] Let $f$ be a real valued multiplicative function such that (i) $f(p)<0$ for some prime $p$, and (ii) $f(n)\neq 0$ for a positive proportion of integers $n$. Then for all large $N$ the non-zero values of $f(n)$ with $n\le N$ exhibit a positive proportion of sign changes: precisely, for some $\delta >0$ and all large $N$, there are $K \ge \delta N$ integers $1\le n_1 < n_2 < \ldots < n_K \le N$ such that $f(n_j) f(n_{j+1}) <0$ for all $1\le j\le K-1$.
The conditions (i) and (ii) imposed in Corollary \[cor3\] are plainly necessary for $f$ to have a positive proportion of sign changes. For the Liouville function, which is never zero, Corollary \[cor3\] says that $\lambda(n) = -\lambda(n+1)$ for a positive proportion of values $n$; of course this fact is also a special case of Theorem \[thm2\]. Even for the M[" o]{}bius function, Corollary \[cor3\] is new, and improves upon the earlier work of Harman, Pintz and Wolke [@HPW]; for general multiplicative functions, it improves upon the earlier work of Hildebrand [@Hi2] and Croot [@Cr]. Corollary \[cor3\] also applies to the Hecke eigenvalues of holomorphic newforms, where Matom[" a]{}ki and Radziwi[łł]{} [@MR1] had recently established such a result by different means. The sign changes of Hecke eigenvalues are related to the location of “real zeros" of the newform $f(z)$ (see [@GhSa]), and this link formed the initial impetus for the work of Matom[" a]{}ki and Radziwi[łł]{}.
The rest of this article will give a sketch of some of the ideas behind Theorem \[thm1\]; the reader may also find it useful to consult [@MR3; @T3]. For ease of exposition, in our description of the results of Matom[" a]{}ki and Radziwi[łł]{} we have chosen to give a qualitative sense of their work. In fact Matom[" a]{}ki and Radziwi[łł]{} establish Theorem \[thm1\] in the stronger quantitative form (for any $2\le h\le X$) $$\Big| \sum_{x< n \le x+ h} \lambda(n) \Big| \ll \frac{h}{(\log h)^{\delta}}$$ except for at most $X (\log h)^{-\delta}$ integers $x \in [X,2X]$ – here $\delta$ is a small positive constant, which may be taken as $1/200$ for example. The limit of their technique would be a saving of about $1/\log h$. In this context, the Riemann hypothesis arguments would permit better quantifications: for example, Selberg estimates the quantity in as $O(Xh (\log X)^2)$, and similarly Gao’s work shows that the variance in is $O(Xh (\log X)^A)$ for a suitable constant $A$. Thus for a restricted range of $h$, the conditional results exhibit almost a square-root cancelation. As $h$ tends to infinity, one expects that the sum of the Liouville function in a randomly chosen interval of length $h$ should be distributed approximately like a normal random variable with mean zero and variance $h$; see [@GC], and [@Ng] in the nearly identical context of the M[" o]{}bius function, and [@MS] for analogous conjectures on primes in short intervals.
[**Acknowledgments.**]{} I am partly supported through a grant from the National Science Foundation (NSF), and a Simons Investigator grant from the Simons Foundation. I am grateful to Zeb Brady, Alexandra Florea, Andrew Granville, Emmanuel Kowalski, Robert Lemke Oliver, Kaisa Matom[" a]{}ki, Maksym Radziwi[łł]{}, Ho Chung Siu, and Frank Thorne for helpful remarks.
Preliminaries
=============
General Plancherel bounds
-------------------------
Qualitatively there is no difference between the $L^2$-estimate stated in Theorem \[thm1\] and the $L^1$-estimate $$\int_X^{2X} \Big| \sum_{x< n\le x+h} \lambda(n) \Big| dx \le \epsilon X h.$$ However, the $L^2$ formulation has the advantage that we can use the Plancherel formula to transform the problem to understanding Dirichlet polynomials. We begin by formulating this generally.
\[lem1\] Let $a(n)$ (for $n=1, 2, 3 \ldots$) denote a sequence of complex numbers and we suppose that $a(n) =0$ for large enough $n$. Define the associated Dirichlet polynomial $$\label{2.1}
A(y) = \sum_{n} a(n) n^{iy}.$$ Let $T \ge 1$ be a real number. Then $$\int_{0}^{\infty} \Big| \sum_{x e^{-1/T}< n \le x e^{1/T}} a(n) \Big|^2 \frac{dx}{x} =
\frac{2}{\pi} \int_{-\infty}^{\infty} |A(y)|^2 \Big(\frac{\sin(y/T)}{y}\Big)^2 dy.$$
For any real number $x$ put $$f(x) = \sum_{e^{x -1/T} \le n \le e^{x+1/T} } a(n),$$ so that its Fourier transform ${\hat f}(\xi)$ is given by $${\hat f}(\xi) = \int_{-\infty}^{\infty} f(x) e^{-ix\xi} dx = \sum_{n} a(n) \int_{\log n -1/T}^{\log n+1/T} e^{-ix\xi} dx
= A(-\xi) \Big(\frac{2\sin(\xi/T)}{\xi}\Big).$$ The left side of the identity of the lemma is $\int_{-\infty}^{\infty} |f(x)|^2 dx$, and the right side is $\frac{1}{2\pi} \int_{-\infty}^{\infty} |{\hat f}(\xi)|^2 d\xi$, so that by Plancherel the stated identity holds.
In Lemma \[lem1\] we have considered the sequence $a(n)$ in “multiplicatively" short intervals $[xe^{-1/T}, xe^{1/T}]$ which is best suited for applying Plancherel, whereas in Theorem \[thm1\] we are interested in “additively" short intervals $[x,x+h]$. A simple technical device (introduced by Saffari and Vaughan [@SV]) allows one to pass from the multiplicative situation to the additive one.
\[lem2\] Let $X$ be large, and let $a(n)$ and $A(y)$ be as in Lemma \[lem1\], and suppose that $a(n) =0$ unless $c_1 X \le n \le c_2 X$ for some positive constants $c_1$ and $c_2$. Let $h$ be a real number with $1\le h\le c_1X/10$. Then $$\int_0^{\infty} \Big| \sum_{x< n\le x+h} a(n) \Big|^2 dx \ll \frac{c_2^2}{c_1} X \int_{-\infty}^{\infty} |A(y)|^2 \min
\Big( \frac{h^2}{c_1^2X^2}, \frac 1{y^2}\Big) dy.$$
Temporarily define ${\mathcal A}(x) = \sum_{n\le x} a(n)$. Note that for any $\nu \in [2h,3h]$ $$\int_{0}^{\infty} |{\mathcal A}(x+h) -{\mathcal A}(x)|^2 dx \le 2 \int_0^{\infty} (|{\mathcal A}(x+\nu) - {\mathcal A}(x)|^2 + |{\mathcal A}(x+h)-{\mathcal A}(x+\nu)|^2) dx.$$ Integrate this over all $2h \le \nu \le 3h$, obtaining that $h$ times the left side above is $$\begin{aligned}
\label{2.21}
&\ll \int_{2h}^{3h} \int_0^{\infty} |{\mathcal A}(x+\nu ) -{\mathcal A}(x)|^2 dx \ d\nu + \int_{2h}^{3h} \int_0^{\infty} |{\mathcal A}(x+\nu-h) -{\mathcal A}(x)|^2 dx \ d\nu \nonumber\\
&\ll \int_{c_1X/2}^{c_2X} \int_{h}^{3h} |{\mathcal A}(x+\nu) - {\mathcal A}(x)|^2 d\nu \ dx . \end{aligned}$$ Now in the inner integral over $\nu$ we substitute $\nu =\delta x$, so that $\delta$ lies between $h/(c_2X)$ and $6 h/(c_1X)$. It follows that the quantity in is $$\begin{aligned}
& \ll \int_{c_1X/2}^{c_2X} \int_{h/(c_2X)}^{6h/(c_1X)} |{\mathcal A}(x(1+\delta)) -{\mathcal A}(x)|^2 x d\delta \ dx \\
&= \int_{h/(c_2X)}^{6h/(c_1X)} \int_{c_1X/2}^{c_2X} |{\mathcal A}(x(1+\delta)) -{\mathcal A}(x)|^2 x dx \ d\delta \\
%&\ll h \max_{h/(c_2 X) \le \delta \le 6h/(c_1 X)} \int_{c_1X/2}^{c_2X} |{\mathcal A}(x(1+\delta)) -{\mathcal A}(x)|^2 dx
%\\
&\ll \frac{c_2^2 hX}{c_1} \max_{h/(c_2 X) \le \delta \le 6h/(c_1 X)} \int_{c_1X/2}^{c_2X} |{\mathcal A}(x(1+\delta)) -{\mathcal A}(x)|^2 \frac{dx}{x},\end{aligned}$$ and now, appealing to Lemma \[lem1\] (with $T=2/\log (1+\delta)$ and noting that $(\sin(y/T)/y)^2 \ll \min(1/T^2,1/y^2)$), the stated result follows.
The Vinogradov-Korobov zero-free region
---------------------------------------
The Vinogradov-Korobov zero-free region establishes that $\zeta(\sigma+it) \neq 0$ in the region $$\sigma \ge 1 - C (\log (3+|t|))^{-2/3} (\log \log (3+|t|))^{-1/3},$$ for a suitable positive constant $C$. Moreover, one can obtain good bounds for $1/\zeta(s)$ in this region; see Theorem 8.29 of [@IK].
\[lem3\] For any $\delta >0$, uniformly in $t$ we have $$\sum_{n\le x} \lambda(n) n^{it} \ll x \exp\Big( - \frac{\log x}{(\log (x+|t|))^{\frac 23 +\delta}} \Big),$$ and $$\sum_{p\le x} p^{it} \ll \frac{\pi(x)}{1+|t|} + x \exp\Big( - \frac{\log x}{(\log (x+|t|))^{\frac 23 +\delta}} \Big).$$
Perron’s formula shows that, with $c=1+1/\log x$, $$\sum_{n\le x} \lambda(n)n^{it} = \frac{1}{2\pi i} \int_{c-ix}^{c+ix} \frac{\zeta(2w-2it)}{\zeta(w-it)} \frac{x^w}{w} dw + O(x^{\epsilon}).$$ Move the line of integration to Re$(w) = 1- (\log (x+|t|))^{-2/3-\delta}$, staying within the zero-free region for $\zeta(w-it)$. Using the bounds in Theorem 8.29 of [@IK], the first statement of the lemma follows. The second is similar.
For reference, let us note that the Riemann hypothesis gives uniformly $$\label{2.2}
\sum_{p\le x} p^{it} \ll \frac{\pi(x)}{1+|t|} + x^{1/2} \log (x+|t|).$$
Mean values of Dirichlet polynomials
------------------------------------
\[lem4\] For any complex numbers $a(n)$ we have $$\int_{-T}^{T} \Big| \sum_{n\le N} a(n) n^{it} \Big|^2 dt \ll (T+N) \sum_{n\le N} |a(n)|^2.$$
This mean value theorem for Dirichlet polynomials can be readily derived from the Plancherel bound Lemma \[lem1\], or see Theorem 9.1 of [@IK].
We shall draw upon Lemma \[lem4\] many times; one important way in which it is useful is to bound the measure of the set on which a Dirichlet polynomial over the primes can be large.
\[lem5\] Let $T$ be large, and $2\le P\le T$. Let $a(p)$ be any sequence of complex numbers, defined on primes $p$, with $|a(p)| \le 1$. Let $V \ge 3$ be a real number and let ${\mathcal E}$ denote the set of values $|t|\le T$ such that $|\sum_{p\le P} a(p) p^{it}| \ge \pi(P)/V$. Then $$|{\mathcal E}| \ll (V^2 \log T)^{1+ (\log T)/(\log P)}.$$
Let $k= \lceil (\log T)/(\log P) \rceil$ so that $P^k \ge T$. Write $$\Big( \sum_{p\le P} a(p)p^{it}\Big)^k = \sum_{n\le P^{k}} a_k(n) n^{it}.$$ Note that $|a_k(n)| \le k!$ and that $$\sum_{n\le P^k} |a_k(n)| \le \Big( \sum_{p\le P} |a(p)| \Big)^k \le \pi(P)^k.$$ Therefore, using Lemma \[lem4\], we obtain $$|{\mathcal E}| \Big( \frac{\pi(P)}{V}\Big)^{2k} \le \int_{-T}^{T} \Big| \sum_{p\le P} a(p)p^{it} \Big|^{2k} dt
\ll (T+P^k) \sum_{n\le P^k} |a_k(n)|^2 \ll k! P^k\pi(P)^{k}.$$ The lemma follows from the prime number theorem and Stirling’s formula.
The Hal[' a]{}sz-Montgomery bound
---------------------------------
The mean value theorem of Lemma \[lem4\] gives a satisfactory bound when averaging over all $|t|\le T$. We shall encounter averages of Dirichlet polynomials restricted to certain small exceptional sets of values $t\in [-T,T]$. In such situations, an idea going back to Hal[' a]{}sz and Montgomery, developed in connection with zero-density results, is extremely useful (see Theorem 7.8 of [@Mon], or Theorem 9.6 of [@IK]).
\[lem6\] Let $T$ be large, and ${\mathcal E}$ be a measurable subset of $[-T,T]$. Then for any complex numbers $a(n)$ $$\int_{\mathcal E} \Big| \sum_{n\le N} a(n) n^{it} \Big|^2 dt \ll (N+ |{\mathcal E}|T^{\frac 12} \log T ) \sum_{n\le N} |a(n)|^2.$$
Let $I$ denote the integral to be estimated, and let $A(t) = \sum_{n\le N} a(n) n^{it}$. Then $$I = \int_{\mathcal E} \sum_{n\le N} \overline{a(n)} n^{-it} A(t) dt \le \sum_{n\le N} |a(n)| \Big| \int_{\mathcal E} A(t) n^{-it} dt \Big|.$$ Using Cauchy-Schwarz we obtain $$\label{2.61}
I^2 \le \Big( \sum_{n\le N} |a(n)|^2 \Big) \Big( \sum_{n \le 2N} \Big(2- \frac{n}{N} \Big) \Big| \int_{\mathcal E} A(t) n^{-it} dt \Big|^2 \Big),$$ where we have taken advantage of positivity to smooth the sum over $n$ in the second sum a little. Expanding out the integral, the second term in is bounded by $$\label{2.62}
\int_{t_1, t_2 \in {\mathcal E}} A(t_1) \overline{A(t_2)} \sum_{n\le 2N} \Big( 2 - \frac{n}{N} \Big) n^{i(t_2 - t_1)} dt_1 \ dt_2.$$ Now a simple argument (akin to the P[' o]{}lya-Vinogradov inequality) shows that $$\label{2.63}
\sum_{n\le 2N} \Big(2-\frac{n}{N} \Big) n^{it} \ll \frac{N}{1+|t|^2} + (1+|t|)^{1/2} \log (2+|t|);$$ here the smoothing in $n$ allows us to save $1+|t|^2$ in the first term, while the unsmoothed sum would have $N/(1+|t|)$ instead (see the proof of Theorem 7.8 in [@Mon]). Using this, and bounding $|A(t_1)A(t_2)|$ by $|A(t_1)|^2 + |A(t_2)|^2$, we see that the second term in is $$\ll \int_{t_1 \in {\mathcal E}} |A(t_1)|^2 \Big( \int_{t_2 \in {\mathcal E} } \Big( \frac{N}{1+|t_1-t_2|^2} + T^{1/2 } \log T \Big) dt_2 \Big) dt_1
\ll \big( N + |{\mathcal E}| T^{1/2} \log T \big) I.$$ Inserting this in , the lemma follows.
A first attack on Theorem 1.1
=============================
In this section we establish Theorem \[thm1\] in the restricted range $h \ge \exp((\log X)^{3/4})$. This already includes the range $h>X^{\epsilon}$ for any $\epsilon> 0$, and moreover the proof is simple, depending only on Lemmas \[lem2\], \[lem3\] and \[lem4\]. Since a large interval may be broken down into several smaller intervals, we may assume that $h\le \sqrt{X}$.
Let ${\mathcal P}$ denote the set of primes in the interval from $\exp((\log h)^{9/10})$ to $h$. Let us further partition the primes in ${\mathcal P}$ into dyadic intervals. Thus, let ${\mathcal P}_{j}$ denote the primes in ${\mathcal P}$ lying between $P_{j}= 2^j \exp((\log h)^{9/10})$ and $P_{j+1} =2^{j+1} \exp((\log h)^{9/10})$. Here $j$ runs from $0$ to $J=\lfloor (\log h - (\log h)^{9/10})/\log 2\rfloor$. This choice of ${\mathcal P}$ was made with two requirements in mind: all elements in ${\mathcal P}$ are below $h$, and all are larger than $\exp((\log h)^{9/10})$ which is larger than $\exp((\log X)^{27/40})$, and note that $27/40$ is a little larger than $2/3$ (anticipating an application of Lemma \[lem3\]).
Now define a sequence $a(n)$ by setting $$\label{3.1}
A(y) = \sum_{n} a(n) n^{iy} = \sum_{j} \sum_{p \in {\mathcal P}_{j}} \sum_{X/P_{j+1} \le m \le 2X/P_j } \lambda(p) p^{iy} \lambda(m) m^{iy}.$$ In other words, $a(n) = 0$ unless $X/2 \le n \le 4X$, and in the range $X\le n \le 2X$ we have $$\label{3.2}
a(n) = \lambda(n) \omega_{\mathcal P}(n), \qquad \text{where} \qquad \omega_{\mathcal P}(n) = \sum_{\substack{p \in {\mathcal P} \\ p|n }} 1. % \qquad \text{when } X \le n \le 2X.$$
Tur[' a]{}n’s proof of the Hardy-Ramanujan theorem can easily be adapted to show that for $n\in [X,2X]$ the quantity $\omega_{\mathcal P}(n)$ is usually close to its average which is about $W({\mathcal P}) = \sum_{p\in {\mathcal P}} 1/p \sim (1/10) \log \log h$. Precisely, $$\label{3.3}
\sum_{X\le n\le 2X} \big(\omega_{\mathcal P}(n) - W({\mathcal P}) \big)^2 \ll X W({\mathcal P}) \ll X \log \log h.$$ Moreover, note that for all $X/2 \le n \le 4X$ one has $|a(n)| \le \omega_{\mathcal P}(n)$ and so $$\label{3.4}
\sum_{n} |a(n)|^2 \ll X W({\mathcal P})^2.$$
Now $$\begin{aligned}
W({\mathcal P})^2 \int_X^{2X} \Big( \sum_{x< n\le x+h} \lambda(n)\Big)^2 dx
&\ll \int_{X}^{2X} \Big( \sum_{x< n< x+h} \lambda(n) \omega_{\mathcal P}(n) \Big)^2 dx \\
&+
\int_{X}^{2X} \Big(\sum_{x<n \le x+h} \lambda(n) (\omega_{\mathcal P}(n) - W({\mathcal P}) ) \Big)^2 dx, \end{aligned}$$ and the Cauchy-Schwarz inequality and show that the second term above is $$\ll \int_X^{2X} h \sum_{x<n \le x+h} (\omega_{\mathcal P}(n) - W({\mathcal P}))^2 dx \ll Xh^2 W({\mathcal P} ).$$ Combining this with Lemma \[lem2\] we conclude that $$\label{3.5}
\int_X^{2X} \Big( \sum_{x< n\le x+h} \lambda(n)\Big)^2 dx
\ll \frac{X}{W({\mathcal P})^2} \int_{-\infty}^{\infty} |A(y)|^2 \min \Big( \frac{h^2}{X^2},
\frac{1}{y^2} \Big) dy + \frac{Xh^2}{W({\mathcal P})}.$$
It remains now to estimate the integral in . First we dispense with the $|y| \ge X$ contribution to the integral, which will be negligible. Indeed by splitting into dyadic ranges $2^k X \le |y| \le 2^{k+1}X$ and using Lemma \[lem4\] we obtain $$\label{3.6}
\int_{|y|> X }|A(y)|^2 \frac{dy}{y^2} \ll \frac{1}{X} \sum_{X/2 \le n \le 4X} a(n)^2 \ll W({\mathcal P})^2.$$
Now consider the range $|y|\le X$. From the definition and Cauchy-Schwarz we see that (note $\lambda(p) =-1$) $$\begin{aligned}
|A(y)|^2 &\le \Big( \sum_{j=0}^{J} \frac{1}{\log P_j} \Big) \Big( \sum_{j=0}^{J} \log P_j \Big| \sum_{p\in {\mathcal P}_j} p^{iy} \Big|^2 \Big|
\sum_{X/P_{j+1} \le m \le 2 X/P_{j} } \lambda(m) m^{iy} \Big|^2\Big). \\\end{aligned}$$ Thus, setting $$\label{3.65}
I_j = (\log P_j)^2 \int_{-X}^{X} \Big| \sum_{p\in {\mathcal P}_j} p^{iy} \Big|^2 \Big |\sum_{X/P_{j+1} \le m \le 2X/P_j} \lambda(m)m^{iy} \Big|^2 \min \Big( \frac{h^2}{X^2}, \frac{1}{y^2}\Big) dy,$$ and noting that $\sum_j 1/ \log P_j \ll W({\mathcal P})$, we obtain $$\begin{aligned}
\label{3.7}
\int_{-X}^{X} |A(y)|^2 \min \Big( \frac{h^2}{X^2},
\frac{1}{y^2} \Big) dy
\ll W({\mathcal P}) \sum_{j=0}^{J} \frac{1}{\log P_j} I_j \ll W({\mathcal P})^2 \max_{0\le j\le J} I_j.
%(\log P_j)^2 \int_{-X}^{X} \Big| \sum_{p\in {\mathcal P}_j}
%p^{iy} \Big|^2 \Big| \sum_{X/P_{j+1} \le m\le 2X/P_j} \lambda(m)m^{iy} \Big|^2 \min \Big( \frac{h^2}{X^2},
%\frac{1}{y^2} \Big) dy . \end{aligned}$$
To estimate $I_j$, we now invoke Lemma \[lem3\]. As noted earlier, our assumption that $h\ge \exp((\log X)^{3/4})$ gives $\log P_j \ge (\log h)^{9/10} \ge (\log X)^{{27}/{40}}$. Thus for $|y|\le X$, Lemma \[lem3\] shows that $$\sum_{p \in {\mathcal P}_j} p^{iy} \ll \frac{P_j}{\log P_j} \frac{1}{1+|y|} + P_j \exp\big( - (\log X)^{\frac{27}{40}-\frac 23-\delta}\big) \ll
\frac{P_j}{\log P_j} \Big( \frac{1}{1+|y|} + \frac{1}{\log P_j} \Big),$$ say. Using this bound for $X\ge |y| \ge \log P_j$, we see that this portion of the integral contributes to $I_j$ an amount $$\ll \frac{P_j^2}{(\log P_j)^2} \int_{\log P_j \le |y|\le X} \Big| \sum_{X/P_{j+1} \le m \le 2X/P_j} \lambda(m) m^{iy} \Big|^2 \min \Big( \frac{h^2}{X^2}, \frac{1}{y^2} \Big) dy .$$ Split the integral into ranges $|y| \le X/h$, and $2^k X/h \le |y| \le 2^{k+1} X/h$ (for $k=0$, $\ldots$, $\lfloor (\log h)/\log 2\rfloor$) and use Lemma \[lem4\]. Since $X/P_j \gg X/h$, this shows that the quantity above is $$\label{3.8}
\ll \frac{P_j^2}{(\log P_j)^2} \Big( \frac{h^2}{X^2} \frac{X^2}{P_j^2} +\sum_k \frac{h^2}{2^{2k} X^2} \Big( 2^k \frac{X}{h} + \frac{X}{P_j}\Big) \frac{X}{P_j} \Big) \ll \frac{h^2}{(\log P_j)^2}.$$ Finally, if $|y| \le \log P_j$, then Lemma \[lem3\] gives $$\label{3.9}
\sum_{X/P_{j+1} \le m \le 2X/P_j} \lambda(m) m^{iy} \ll \frac{X}{P_j} (\log X)^{-10},$$ say, so that bounding $\sum_{p\in {\mathcal P}_j} p^{iy}$ trivially by $\ll P_j/(\log P_j)$ we see that this portion of the integral contributes to $I_j$ an amount $$\label{3.10}
\ll (\log P_j)^2 \frac{P_j^2}{(\log P_j)^2} \frac{X^2}{P_j^2} (\log X)^{-20} \frac{h^2}{X^2} (\log P_j) \ll h^2 (\log X)^{-19}.$$ Combining this with , we obtain that $I_j \ll h^2/(\log P_j)^2$, and so from it follows that $$\int_{-X}^{X} |A(y)|^2 \min \Big( \frac{h^2}{X^2}, \frac{1}{y^2} \Big) dy \ll W({\mathcal P})^2 \max_j \frac{h^2}{(\log P_j)^2} \ll W({\mathcal P})^2 \frac{h^2}{(\log h)^{9/5}}.$$ Using this and in , we conclude that $$\int_X^{2X} \Big( \sum_{x< n\le x+h} \lambda(n)\Big)^2 dx \ll Xh^2 \Big( \frac{1}{(\log h)^{9/5}} + \frac{1}{W({\mathcal P})} +\frac{1}{h^2}\Big) \ll \frac{Xh^2}{\log \log h}.$$ This proves Theorem \[thm1\] in the range $h\ge \exp((\log X)^{ 3/4})$.
There are two limitations in this argument. In order to use the mean value theorem (Lemma \[lem4\]) effectively we need to restrict the primes in ${\mathcal P}$ to lie below $h$, so that the Dirichlet polynomial over $m$ has length at least $X/h$. Secondly, in order to apply Lemma \[lem3\] to bound the sum over $p\in{\mathcal P}_j$, we are forced to have $P_j > \exp((\log X)^{2/3+ \delta})$ and this motivated our choice of ${\mathcal P}$. If we appealed to the Riemann Hypothesis bound instead of Lemma \[lem3\], then the second limitation can be relaxed, and the argument presented above would establish Theorem \[thm1\] in the wider range $h\ge \exp(10 (\log \log X)^{10/9})$. In the next section, we shall obtain such a range unconditionally.
Theorem 1.1 – Round two
=======================
We now refine the argument of the previous section, adding another ingredient which will permit us to obtain Theorem \[thm1\] in the substantially wider region $h \ge \exp( 10(\log \log X)^{10/9})$. Now we shall also need Lemmas \[lem5\] and \[lem6\].
Let us suppose that $h\le \exp((\log X)^{3/4})$, and let ${\mathcal P}$ and ${\mathcal P}_j$ be as in the previous section. Now we introduce a set of large primes ${\mathcal Q}$ consisting of the primes in the interval from $\exp((\log X)^{4/5})$ to $\exp((\log X)^{9/10})$. As with ${\mathcal P}$, let us also decompose ${\mathcal Q}$ into dyadic intervals with ${\mathcal Q}_k$ denoting the primes in ${\mathcal Q}$ lying between $Q_k=2^k \exp((\log X)^{4/5})$ and $Q_{k+1} = 2^{k+1}
\exp((\log X)^{4/5})$, where $k$ runs from $0$ to $K \sim (\log X)^{9/10}/\log 2$.
In place of we now define the sequence $a(n)$ by setting $$\label{4.1}
A(y) = \sum_{n} a(n) n^{iy} = \sum_{j} \Big( \sum_{p \in {\mathcal P}_j} \lambda(p)p^{iy} \Big) A_j(y),$$ where $$\label{4.11}
A_j(y) = \sum_{k} \sum_{\substack{ q \in {\mathcal Q}_k} } \sum_{X/(P_{j+1}Q_{k+1}) \le m \le 2X/(P_jQ_k)}
\lambda(q)q^{iy} \lambda(m) m^{iy} .$$ Now $a(n)=0$ unless $X/4\le n \le 8X$, and in the range $X\le n\le 2X$ we have $a(n) = \lambda(n) \omega_{\mathcal P}(n) \omega_{\mathcal Q}(n)$, where $\omega_{\mathcal P}(n)$ is as before, and $\omega_{\mathcal Q}$ analogously counts the number of prime factors of $n$ in ${\mathcal Q}$.
As noted already in , a typical number in $X$ to $2X$ will have $\omega_{\mathcal P}(n) \sim W({\mathcal P})$, and similarly will have $\omega_{\mathcal Q}(n) \sim W({\mathcal Q}) = \sum_{q\in {\mathcal Q}} 1/q \sim (1/10) \log \log X$. Precisely, we have $$\sum_{X \le n\le 2X} \big(\omega_{\mathcal P}(n) \omega_{\mathcal Q}(n) - W({\mathcal P}) W({\mathcal Q} ) \big)^2 \ll X W({\mathcal P})^2 W({\mathcal Q})^2 \Big( \frac{1}{W({\mathcal P})} + \frac{1}{W({\mathcal Q})}\Big).$$
Now set (analogously to ) $$\label{4.2}
I_j = (\log P_j)^2 \int_{-X}^{X} \Big| \sum_{p\in {\mathcal P}_j} p^{iy} \Big|^2 |A_j(y)|^2 \min \Big( \frac{h^2}{X^2}, \frac{1}{y^2} \Big) dy.$$ Then arguing exactly as in , , and , we find that $$\label{4.3}
\int_X^{2X} \Big( \sum_{x< n\le x+h} \lambda(n)\Big)^2 dx \ll \frac{X}{W({\mathcal Q})^2} \max_{j} I_{j} + \frac{Xh^2}{\log \log h},$$ so that our problem has now boiled down to finding a non-trivial estimate for $I_j$.
In the range $|y| \le \log P_j$, we may use a modified version of the bounds in and to see that the contribution of this portion of the integral to $I_{j,k}$ is $\ll h^2 (\log X)^{-19}$, which is negligible. It remains now to bound the integral in in the range $\log P_j \le |y|\le X$.
Note that we may not be able to use Lemma \[lem3\] to bound $\sum_{p \in {\mathcal P}_j} p^{iy}$ since the range for $p$ might lie below $\exp((\log X)^{ 2/3 +\delta})$. Define $$\label{4.4}
{\mathcal E}_j = \Big\{ y: \ \ \log P_j \le |y| \le X, \ \ \ \Big| \sum_{p \in {\mathcal P}_j} p^{iy} \Big| \ge \frac{P_j}{(\log P_j)^2} \Big \},$$ which denotes the exceptional set on which the sum over $p \in {\mathcal P}_j$ does not exhibit much cancelation. To bound the integral in in the range $\log P_j \le |y| \le X$, let us distinguish the cases when $y$ belongs to the exceptional set ${\mathcal E}_j$, and when it does not. Consider the latter case first, where by the definition of ${\mathcal E}_j$ the sum over $p \in {\mathcal P}_j$ does have some cancelation. So this case contributes to $$\ll \frac{P_j^2}{(\log P_j)^2} \int_{-X}^{X} |A_j(y)|^2 \min \Big( \frac{h^2}{X^2}, \frac{1}{y^2} \Big) dy.$$ The integral above is the mean value of a Dirichlet polynomial of size about $X/P_j$, which is larger than $X/h$. Therefore applying Lemma \[lem4\] (as in our estimate ) we obtain that the above is $$\ll \frac{P_j^2}{(\log P_j)^2}
\frac{h^2}{X^2} \frac{X}{P_j} \sum_{X/(2P_j) \le n \le 4X/P_j} \Big(\sum_{\substack{ q |n \\ q\in{\mathcal Q}} } 1 \Big)^2
\ll \frac{h^2}{(\log P_j)^2} W({\mathcal Q})^2.$$ Thus the contribution of this case to is small as desired.
Finally we need to bound the contribution of the exceptional values $y \in {\mathcal E}_j$: upon bounding the sum over $p\in {\mathcal P_j}$ trivially, this contribution to $I_j$ is $$\label{4.5}
\ll P_j^2 \int_{{\mathcal E}_j} |A_j(y)|^2 \min \Big( \frac{h^2}{X^2}, \frac{1}{y^2} \Big) dy \ll P_j^2
\frac{h^2}{X^2} \int_{{\mathcal E}_j} |A_j(y)|^2 dy.$$ Now recall the definition of $A_j(y)$ in , and use Cauchy-Schwarz on the sum over $k$ (as in or ) to obtain that the quantity in above is $$\label{4.6}
\ll P_j^2W({\mathcal Q})^2 \frac{h^2}{X^2} \max_k \ (\log Q_k)^2 \int_{{\mathcal E}_j} \Big| \sum_{q \in {\mathcal Q}_k} q^{iy} \Big|^2 \Big| \sum_{X/(P_{j+1}Q_{k+1}) \le m \le 2X/(P_jQ_k)} \lambda(m) m^{iy} \Big|^2 dy.% \min \Big( \frac{h^2}{X^2}, \frac{1}{y^2} \Big) dy.$$ Since $\log Q_k \ge (\log X)^{4/5}$ (note $4/5$ is bigger than $2/3+\delta$), in the range $X\ge |y| \ge \log P_j$ we can use Lemma \[lem3\] to obtain $$\label{4.7}
\sum_{q \in {\mathcal Q}_k} q^{iy} \ll \frac{\pi(Q_{k+1})}{\log P_j} \ll \frac{1}{\log P_j} \frac{Q_k}{\log Q_k},$$ which represents a saving of $1/\log P_j$ over the trivial bound $Q_k/\log Q_k$. Using this in and substituting that back in , we see that the contribution of the exceptional $y\in {\mathcal E}_j$ to $I_j$ is $$\label{4.8}
\ll \frac{P_j^2W({\mathcal Q})^2}{(\log P_j)^2} \frac{h^2}{X^2} \max_{k} Q_k^2 \int_{{\mathcal E}_j} \Big| \sum_{X/(P_{j+1}Q_{k+1}) \le m \le 2X/(P_jQ_k)} \lambda(m) m^{iy} \Big|^2 dy.% \min \Big( \frac{h^2}{X^2}, \frac{1}{y^2} \Big) dy.$$
It is at this stage that we invoke Lemmas \[lem5\] and \[lem6\]. We are assuming that $\exp(10 (\log \log X)^{10/9}) \le h \le \exp((\log X)^{3/4})$, so that $(\log X)^{7} \le P_j \le X^{\epsilon}$. Appealing to Lemma \[lem5\], it follows that $|{\mathcal E}_j| \ll X^{3/7+\epsilon}$. Using now the bound of Lemma \[lem6\], we conclude that the quantity in is $$\label{4.9}
\ll \frac{P_j^2W({\mathcal Q})^2}{(\log P_j)^2} \frac{h^2}{X^2} \max_{k} Q_k^2 \Big( \frac{X}{P_jQ_k} + X^{3/7+\epsilon} X^{1/2+\epsilon} \Big) \frac{X}{P_jQ_k} \ll \frac{h^2W({\mathcal Q})^2}{(\log P_j)^2}.$$ Inserting these estimates back in , we obtain finally that $$\int_X^{2X} \Big( \sum_{x < n \le x+h} \lambda(n) \Big)^2 dx \ll \frac{Xh^2}{\log \log h},$$ which establishes Theorem \[thm1\] in this range of $h$.
The limitation in this argument comes from the last step where in applying the Hal[' a]{}sz-Montgomery Lemma \[lem6\] we need the measure of the exceptional set ${\mathcal E}_j $ to be a bit smaller than $X^{1/2}$, and to achieve this we needed $P_j$ to be larger than a suitable power of $\log X$.
Once more unto the breach
=========================
Now we add one more ingredient to the argument developed in the preceding two sections, and this will permit us to obtain Theorem \[thm1\] in the range $h> \exp( (\log \log \log X)^2)$. Moreover once this argument is in place, we hope it will be clear that a more elaborate iterative argument should lead to Matom[" a]{}ki and Radziwi[łł]{}’s result; we shall briefly sketch their argument, where the details are arranged differently, in the next section. Assume below that $h \le \exp(10 (\log \log X)^{10/9})$.
Let ${\mathcal P}$ and ${\mathcal Q}$ be as in the previous section. Let ${\mathcal P}^{(1)}$ denote the set of primes lying between $\exp(\exp(\frac{1}{100} (\log h)^{9/10})) $ and $\exp(\exp(\frac{1}{30} (\log h)^{9/10}))$, so that this set is intermediate between ${\mathcal P}$ and ${\mathcal Q}$. Again split up ${\mathcal P}^{(1)}$ into dyadic blocks, which we shall index as ${\mathcal P}^{(1)}_{j_1}$. In place of we now define the sequence $a(n)$ by setting $$\label{5.1}
A(y) =\sum_{n} a(n) n^{iy} = \sum_{j} \Big(\sum_{p \in {\mathcal P}_j} \lambda(p) p^{iy}\Big) A_j(y),$$ with $$\label{5.2}
A_j(y) = \sum_{j_1} \Big( \sum_{p_1 \in {\mathcal P}^{(1)}_{j_1}} \lambda(p_1)p_1^{iy} \Big) A_{j,j_1}(y),$$ where, with $M_{j,j_1,k} = X/(P_{j}P^{(1)}_{j_1} Q_{k})$, $$\label{5.3}
A_{j,j_1} (y) = \sum_{k} \sum_{q\in {\mathcal Q}_k} \sum_{M_{j,j_1,k}/8 \le m \le 2M_{j,j_1,k}} \lambda(q) q^{iy} \lambda(m)m^{iy} .$$ Thus $a(n)$ is zero unless $n$ lies in $[X/8,16X]$ and on $[X,2X]$ we have $a(n) = \lambda(n) \omega_{\mathcal P}(n) \omega_{{\mathcal P}^{(1)}}(n)
\omega_{\mathcal Q}(n)$.
Now arguing as in and we obtain $$\label{5.4}
\int_X^{2X} \Big( \sum_{x < n\le x+h} \lambda(n) \Big)^2 dx \ll \frac{X}{W({\mathcal Q})^2 W({\mathcal P}^{(1)})^2} \max_j I_j + \frac{Xh^2}{\log \log h},$$ where $$\label{5.5}
I_j = (\log P_j)^2 \int_{-X}^{X} \Big| \sum_{p \in {\mathcal P}_j} p^{iy} \Big|^2 |A_j(y)|^2 \min \Big( \frac{h^2}{X^2}, \frac{1}{y^2} \Big) dy.$$ As before the small portion of the integral with $|y| \le \log P_j$ can be estimated trivially. Further if the sum over $p\in {\mathcal P}_j$ exhibited some cancelation, then the argument of Section 3 applies and produces the desired savings (we also saw this in Section 4 when dealing with $y$ not in the exceptional set ${\mathcal E}_j$).
So now consider the exceptional set ${\mathcal E}_j$ (exactly as in ) consisting of $y$ with $\log P_j \le |y| \le X$ and $|\sum_{p\in {\mathcal P}_j} p^{iy} | \ge P_j/(\log P_j)^2$, and we must bound the contribution to $I_j$ from $y\in {\mathcal E}_j$. As we remarked at the end of Section 4, in the range of $h$ considered here we are not able to guarantee that the measure of ${\mathcal E}_j$ is below $X^{1/2-\delta}$, which would have permitted an application of Lemma \[lem6\] (as in Section 4). Using a Cauchy-Schwarz argument (similar to the ones leading to , or , or ), we may bound the contribution to $I_j$ from $y \in {\mathcal E}_j$ by $$\label{5.6}
\ll W({\mathcal P}^{(1)})^2 \max_{j_1} (\log P_{j})^2 (\log P^{(1)}_{j_1})^2 I(j,j_1),$$ say, with $$\label{5.7}
I(j,j_1) = \int_{{\mathcal E}_j} \Big| \sum_{p\in {\mathcal P}_j} p^{iy} \Big|^2 \Big| \sum_{p_1 \in {\mathcal P}^{(1)}_{j_1}} p_1^{iy} \Big|^2 |A_{j,j_1}(y)|^2 \min \Big(\frac{h^2}{X^2},\frac{1}{y^2} \Big) dy.$$
Now ${\mathcal P}^{(1)}$ is a suitably large interval (the lower end point is larger than $(\log X)^{100}$ say), so that one can use Lemma \[lem5\] to show that the measure of the set of $y \in [-X,X]$ with $|\sum_{p_1 \in {\mathcal P}^{(1)}_{j_1}} p_1^{iy} | \ge (P^{(1)}_{j_1})^{9/10}$ is at most $X^{1/3}$. For these exceptionally large values of the sum over $p_1$, we bound the sums over $p \in {\mathcal P}_j$ and $p_1\in {\mathcal P}^{(1)}_{j_1}$ trivially and argue as in Section 4, –. This argument shows that the contribution of large values of the sum over $p_1$ to is acceptably small.
We finally come to the new argument of this section: namely, in dealing with the portion of the integral $I(j,j_1)$ where the sum over $p$ is large (since $y\in {\mathcal E}_j$) but the sum over $p_1$ exhibits some cancelation. Bounding the sum over $p_1$ by $\le (P^{(1)}_{j_1})^{9/10}$, we must handle $$\label{5.8}
(P^{(1)}_{j_1})^{9/5} \int_{{\mathcal E}_j} \Big| \sum_{p \in {\mathcal P}_j} p^{iy} \Big|^2 | A_{j, j_1} (y)|^2 \min \Big( \frac{h^2}{X^2}, \frac{1}{y^2} \Big) dy.$$ Above we must estimate the mean square of a Dirichlet polynomial of length about $X/P^{(1)}_{j_1}$; the set ${\mathcal E}_j$ may not be small enough to use Lemma \[lem6\] effectively, and the length of the Dirichlet polynomial is small compared to $X/h$, so that there is also some loss in using Lemma \[lem4\]. The way out is to bound by $$\label{5.9} (P^{(1)}_{j_1})^{9/5} \int_{-X}^{X} \Big| \sum_{p \in {\mathcal P}_j} p^{iy} \Big|^{2+2\ell} \Big(\frac{P_j}{(\log P_j)^2}\Big)^{-2\ell} | A_{j, j_1} (y)|^2 \min \Big( \frac{h^2}{X^2}, \frac{1}{y^2} \Big) dy;$$ here $\ell$ is any natural number, and the inequality holds because on ${\mathcal E}_j$ the sum over $p\in {\mathcal P}_j$ is $\ge P_j/(\log P_j)^2$ by assumption. We choose $\ell = \lceil (\log P^{(1)}_{j_1})/\log P_j \rceil$. Now in , we must estimate the mean square of the Dirichlet polynomial $(\sum_{p\in {\mathcal P_j}} p^{iy})^{1+\ell} A_{j,j_1}(y)$, and by our choice for $\ell$ this Dirichlet polynomial has length at least $X$, permitting an efficient use of Lemma \[lem4\]. With a little effort, Lemma \[lem4\] can be used to bound by (we have been a little wasteful in some estimates below) $$\begin{aligned}
\label{5.10}
&\ll (P^{(1)}_{j_1})^{9/5} \Big(\frac{P_j}{(\log P_j)^2}\Big)^{-2\ell} \frac{h^2}{X^2} W({\mathcal Q})^2 (\ell+1)! \Big( \frac{(2P_j)^{\ell}X}{P^{(1)}_{j_1}} \Big)^2 \nonumber \\
&\ll W({\mathcal Q})^2 h^2 (P^{(1)}_{j_1})^{-1/5} (\ell \log P_j)^{4\ell} \ll W({\mathcal Q})^2 h^2 (P^{(1)}_{j_1})^{-1/15}, \end{aligned}$$ where at the last step we used $\log \log P^{(1)}_{j_1} \le (1/30) \log P_j$. This contribution to is once again acceptably small (having saved a small power of $P^{(1)}_{j_1}$), and completes the proof of Theorem \[thm1\] in this range of $h$.
At this stage, all the ingredients in the proof of Theorem \[thm1\] are at hand, and one can begin to see an iterative argument that would remove even the very weak hypothesis on $h$ made in this section!
Sketch of Matom[" a]{}ki and Radziwi[łł]{}’s argument for Theorem 1.1
=====================================================================
In the previous three sections, we have described some of the key ideas developed in [@MR2]. The argument given in [@MR2] arranges the details differently, in order to achieve quantitatively better results: our version saved a modest $\log \log h$ over the trivial bound, and [@MR2] saves a small power of $\log h$.
Instead of considering $a(n)$ being $\lambda(n)$ weighted by the number of primes in various intervals (as in Sections 3, 4, 5), Matom[" a]{}ki and Radziwi[łł]{} deal with $a(n)$ being $\lambda(n)$ when $n$ is restricted to integers with at least one prime factor in carefully chosen intervals (and $a(n)=0$ otherwise). To illustrate, we revisit the argument in Section 3, and let ${\mathcal P}$ be the interval defined there. Let ${\mathcal S}$ denote the set of integers $n \in [1,2X]$ with $n$ having at least one prime factor in ${\mathcal P}$. A simple sieve argument shows that there are $\ll X/(\log h)^{1/10}$ numbers $n\in [X,2X]$ that are not in ${\mathcal S}$. Therefore $$\label{6.1}
\int_X^{2X} \Big( \sum_{x< n\le x+h} \lambda(n) \Big)^2 dx \ll
\int_X^{2X} \Big( \Big(\sum_{\substack{ x< n\le x+h \\ n\in {\mathcal S}} } \lambda(n) \Big)^2
+ h \sum_{\substack{x< n\le x+h \\ n\notin{\mathcal S}} } 1 \Big) dx,$$ and the second term is $O(Xh^2/(\log h)^{1/10})$. Now we use Lemma \[lem2\] to transform the problem of estimating the first sum above to that of bounding the Dirichlet polynomial $$\label{6.2}
A(y) = \sum_{\substack{X <n \le 2X \\ n\in{\mathcal S}} } \lambda(n) n^{iy} .$$ To proceed further, we need to be able to factor the Dirichlet polynomial $A$: this can be done by means of the approximate identity $$\label{6.3}
A(y) \approx \sum_{p \in {\mathcal P}} \sum_{ \substack{m \\ pm \in [X,2X] \\ pm \in {\mathcal S}}} \frac{\lambda(m) m^{iy} }{\omega_{\mathcal P}(m) +1} \lambda(p)p^{iy} .$$ (The approximate identity above fails to be exact because $n$ might have repeated prime factors from ${\mathcal P}$, but this difference is of no importance.) Now above we can use a standard Fourier analytic technique to separate the variables $m$ and $p$, and in this fashion make $p$ and $m$ range over suitable dyadic intervals. Alternatively one can divide the sum over ${\mathcal P}$ into many short intervals, and for each such short interval the corresponding range for $m$ may be well approximated by a suitable interval; this is the approach taken in [@MR2]. In either case, we obtain a factorization of $A(y)$ very much like what we had in Section 3, and now the argument can follow as before. Note that in the first step we now have a loss of only $O(Xh^2/(\log h)^{1/10})$ which is substantially better than our previous argument in where we had the bigger error term $O(Xh^2/ \log \log h)$.
Jumping to the argument in Section 5, we can take ${\mathcal S}$ to be the set of integers $n \in [1,2X]$ having at least one prime factor in each of the intervals ${\mathcal P}$, ${\mathcal P}^{(1)}$, and ${\mathcal Q}$. Once again the sieve shows that there are $\ll X/(\log h)^{1/10}$ integers in $[X,2X]$ that are not in ${\mathcal S}$. We start with the expression , and perform a dyadic decomposition of $p\in {\mathcal P}$. If for each $j$ the sum $\sum_{p \in {\mathcal P}_j} p^{iy}$ exhibits cancelation, then using Lemma \[lem4\] and we obtain a suitable bound. If on the other hand for some $j$ the sum over $p\in {\mathcal P}_j$ is large, then we decompose the corresponding Dirichlet polynomial $A_j(y)$ using the primes in ${\mathcal P}^{(1)}$: $$\begin{aligned}
\label{6.4}
A_j(y) &= \sum_{\substack{ m \in [X/P_{j+1}, X/P_j] \\ m \in {\mathcal S}^{(1)} } } \frac{\lambda(m)m^{iy}}{\omega_{\mathcal P}(m) + 1}
\nonumber\\
&\approx \sum_{p_1 \in {\mathcal P}^{(1)}} \lambda(p_1) p_1^{iy} \sum_{\substack{ m \\ mp_1 \in [X/P_{j+1}, X/P_j] \\ mp_1 \in {\mathcal S}^{(1)} } }
\frac{\lambda(m)m^{iy}}{(\omega_{\mathcal P}(m)+1)( \omega_{{\mathcal P}^{(1)}} (m)+1)},\end{aligned}$$ where ${\mathcal S}^{(1)}$ denotes the integers in $[1,2X]$ with at least one prime factor in ${\mathcal P}^{(1)}$ and one in ${\mathcal Q}$. Once again we can split up the primes in ${\mathcal P}^{(1)}$ into dyadic blocks, and separate variables. If now the sum over $p_1 \in {\mathcal P}_{j_1}^{(1)}$ always has some cancelation, then we can argue using an appropriately large moment of the sum over $p\in {\mathcal P}_j$ as in –. If for some $j_1$, the sum over $p_1 \in {\mathcal P}_{j_1}^{(1)}$ is large, then we exploit the fact that this set has small measure, and argue as in –. In short the decompositions and give the same flexibility as the factorized expressions and that we used in Section 5.
The argument in [@MR2] generalizes the approach described in the previous paragraph. Matom[" a]{}ki and Radziwi[łł]{} define a sequence of increasing ranges of primes, starting with ${\mathcal P}={\mathcal P}^{(0)}$ (as in our exposition), and proceeding with ${\mathcal P}^{(1)}$, $\ldots$, ${\mathcal P}^{(L)}$ with the last interval getting up to primes of size $\exp(\sqrt{\log X})$, and a final interval ${\mathcal Q}$ (again as in our exposition). Then one restricts to integers having at least one prime factor in each of these intervals. The corresponding Dirichlet series admits many flexible factorizations as in and . Start with the decomposition , and split into dyadic blocks. If $y$ is such that for all dyadic blocks ${\mathcal P}_j ={\mathcal P}^{(0)}_j$ one has cancelation in $p^{iy}$, then Lemma \[lem4\] leads to a suitable bound. Otherwise we proceed to a decomposition as in , and see whether for every dyadic block in ${\mathcal P}^{(1)}$ the corresponding sum has cancelation. If that is the case, then a moment argument as in – works. Else, we must have some dyadic block in ${\mathcal P}^{(1)}$ with a large contribution, and we now proceed to a decomposition involving ${\mathcal P}^{(2)}$. Ultimately we arrive at a dyadic interval in ${\mathcal P}^{(L)}$ which makes a large contribution, and now we use that this happens very rarely and argue as in –. The structure of the proof may be likened to a ladder – a large contribution to a dyadic interval in ${\mathcal P}^{(j)}$ is used to force a large contribution to a dyadic interval in ${\mathcal P}^{(j+1)}$ – and one must choose the intervals ${\mathcal P}^{(j)}$ so that the rungs of the ladder are neither too close nor too far apart. Fortunately the method is robust and a wide range of choices for ${\mathcal P}^{(j)}$ work. We end our sketch of the proof of Theorem \[thm1\] here, referring to [@MR2] for further details of the proof, and noting that somewhat related iterated decompositions of Dirichlet polynomials arose recently in connection with moments of $L$-functions (see [@Har], [@RS]).
Generalizations for multiplicative functions
============================================
As mentioned in , the work of Matom[" a]{}ki and Radziwi[łł]{} establishes short interval results for general multiplicative functions $f$ with $-1\le f(n) \le 1$ for all $n$. Our treatment so far has been specific to the Liouville function; for example we have freely used the bounds of Lemma \[lem3\] which do not apply in the general situation. In this section we discuss an important special class of multiplicative functions (those that are “unpretentious"), and give a brief indication of the changes to the arguments that are needed. There is one notable extra ingredient that we need – an analogue of the Hal[' a]{}sz-Montgomery Lemma for primes (see Lemma \[lem7.1\] below).
A beautiful theorem of Hal[' a]{}sz [@Hal] (extending earlier work of Wirsing) shows that mean values of bounded complex valued multiplicative functions $f$ are small unless $f$ pretends to be the function $n^{it}$ for a suitably small value of $t$. When the multiplicative function is real valued, one can show that the mean value is small unless $f$ pretends to be the function $1$: this means that $\sum_{p\le x} (1-f(p))/p$ is small. There is an extensive literature around Hal[' a]{}sz’s theorem and its consequences; see for example [@GS; @HT; @Mon2; @Ten]. Let us state one such result precisely: suppose $f$ is a completely multiplicative function taking values in the interval $[-1,1]$, and suppose that $$\label{7.1}
\sum_{p\le X} \frac{1-f(p)}{p} \ge \delta \log \log X$$ for some positive constant $\delta$. Then uniformly for all $|t|\le X$ and all $\sqrt{X} \le x \le X^2$ we have $$\label{7.2}
\sum_{n\le x} f(n) n^{it} \ll \frac{x}{(\log x)^{\delta_1}},$$ for a suitable constant $\delta_1$ depending only on $\delta$.
Now let us consider the analogue of Theorem \[thm1\] for such a completely multiplicative function $f$, in the simplest setting of short intervals of length $\sqrt{X} \ge h \ge \exp((\log X)^{17/18})$ (a range similar to that considered in Section 3). In this range we wish to show that $$\label{7.3}
\int_X^{2X} \Big( \sum_{x < n\le x+h} f(n) \Big)^2 dx = o (Xh^2),$$ which establishes for almost all short intervals in this particular situation.
Let ${\mathcal P}$ denote the primes in $\exp((\log h)^{9/10})$ to $h$, as in Section 3, and break it up into dyadic blocks ${\mathcal P}_j$ like before. Analogously to , we define the Dirichlet series $$\label{7.4}
A(y) = \sum_{n} a(n) n^{iy} = \sum_j \sum_{p\in {\mathcal P}_j} \sum_{X/P_{j+1} \le m \le 2X/P_j} f(p)p^{iy} f(m)m^{iy},$$ so that $a(n)$ is zero unless $X/2\le n\le 4X$ and in the range $X\le n\le 2X$ we have $a(n) = f(n) \omega_{\mathcal P}(n)$; all exactly as in . Now arguing as in – we obtain that $$\label{7.5}
\int_X^{2X} \Big( \sum_{x< n\le x+h} f(n) \Big)^2 dx \ll X \max_j I_j + \frac{Xh^2}{\log \log h},$$ where $$\label{7.6}
I_j = (\log P_j)^2 \int_{-X}^{X} \Big| \sum_{p\in {\mathcal P}_j } f(p) p^{iy} \Big|^2 \Big| \sum_{X/P_{j+1} \le m \le 2X/P_j} f(m)m^{iy} \Big|^2 \min
\Big(\frac{h^2}{X^2},\frac{1}{y^2}\Big) dy.$$
Since $f$ is essentially arbitrary, we can no longer use Lemma \[lem3\] to bound the sum over $p$ above. The argument splits into two cases depending on whether the sum over $p\in {\mathcal P}_j$ is large or not. Let $$\label{7.7}
{\mathcal E}_j = \Big\{ y: \ |y|\le X, \ \ \Big| \sum_{p\in {\mathcal P}_j} f(p)p^{iy} \Big| \ge \frac{P_j}{(\log P_j)^2} \Big\},$$ denote the exceptional set on which the sum over $p$ is large. On the complement of ${\mathcal E}_j$, it is simple to estimate the contribution to $I_j$: namely, using Lemma \[lem4\], we may bound this contribution by $$\ll \frac{P_j^2}{(\log P_j)^2} \int_{-X}^{X} \Big| \sum_{X/P_{j+1} \le m \le 2X/P_j} f(m)m^{iy} \Big|^2 \min
\Big(\frac{h^2}{X^2},\frac{1}{y^2}\Big) dy \ll \frac{h^2}{(\log P_j)^2},$$ which is acceptably small in .
It remains to estimate the contribution to $I_j$ from the exceptional set ${\mathcal E}_j$. Here we invoke the bound , so that the desired contribution is $$\label{7.8}
\ll \frac{X^2 (\log P_j)^2}{(\log X)^{2\delta_1} P_j^2} \int_{{\mathcal E}_j} \Big| \sum_{p \in {\mathcal P}_j} f(p) p^{iy} \Big|^2 \min\Big(\frac{h^2}{X^2},\frac{1}{y^2} \Big) dy.$$ Since $h \ge \exp((\log X)^{17/18})$ we have $P_j \ge \exp((\log h)^{9/10}) \ge \exp((\log X)^{17/20})$, and an application of Lemma \[lem5\] shows that the measure of ${\mathcal E}_j$ is $\ll \exp((\log X)^{1/6})$. This is extremely small, and it is tempting to use the Hal[' a]{}sz-Montgomery Lemma \[lem6\] to estimate . However this gives an estimate too large by a factor of $\log P_j$, since Lemma \[lem6\] does not take into account that the Dirichlet polynomial in is supported only on the primes. This brings us to the final key ingredient in [@MR2] – a version of the Hal[' a]{}sz-Montgomery Lemma for prime Dirichlet polynomials.
\[lem7.1\] Let $T$ be large, and ${\mathcal E}$ be a measurable subset of $[-T,T]$. Then for any complex numbers $x(p)$ and any $\epsilon>0$, $$\int_{\mathcal E} \Big| \sum_{p\le P} x(p) p^{it} \Big|^2 dt \ll \Big(\frac{P}{\log P} + |{\mathcal E}| P\exp\Big( - \frac{\log P}{(\log (T+P))^{2/3+\epsilon}} \Big) \Big)\sum_{p\le P} |x(p)|^2.$$
We follow the strategy of Lemma \[lem6\]. Put $P(t) =\sum_{p\le P} x(p)p^{it}$, and let $I$ denote the integral to be estimated. Then using Cauchy-Schwarz as in , we obtain $$\label{7.9}
I^2 \le \Big( \sum_{p\le P} |x(p)|^2 \Big) \Big( \sum_{p\le 2P} \Big(2-\frac{p}{P}\Big) \Big| \int_{{\mathcal E}} P(t) p^{-it} dt \Big|^2\Big).$$ Now expanding out the integral above, as in , the second term of is bounded by $$\int_{t_1, t_2 \in {\mathcal E}} P(t_1) \overline{P(t_2)} \sum_{p\le 2P} \Big(2- \frac{p}{P}\Big) p^{i(t_2-t_1)} dt_1 dt_2.$$ Now in place of , we can argue as in Lemma \[lem3\] to obtain $$\sum_{p\le 2P} \Big(2- \frac{p}{P}\Big) p^{it} \ll \frac{\pi(P)}{1+|t|^2} + P \exp\Big( -\frac{(\log P)}{(\log (T+P))^{2/3+\epsilon}}\Big),$$ where once again the small smoothing in the sum over $p$ produces the saving of $1+|t|^2$ in the first term. Inserting this bound in , and proceeding as in the proof of Lemma \[lem6\], we readily obtain our lemma.
Returning to our proof, applying Lemma \[lem7.1\] we see that the quantity in may be bounded by $$\ll \frac{X^2(\log P_j)^2}{(\log X)^{2\delta_1} P_j^2} \Big( \frac{P_j}{\log P_j} + P_j \exp\Big((\log X)^{1/6} - \frac{(\log X)^{17/20}}{(\log X)^{2/3+\epsilon}}\Big) \Big)
\frac{P_j}{\log P_j} \ll % \frac{P_j^2}{(\log P_j)^2} \frac{h^2}{X^2} \ll
\frac{h^2}{(\log X)^{2\delta_1}}.$$ Thus the contribution of $y \in {\mathcal E}_j$ to $I_j$ is also acceptably small, and therefore follows.
Sketch of the corollaries
=========================
We discuss briefly the proofs of Corollaries \[cor2\] and \[cor3\], starting with Corollary \[cor2\]. The indicator function of smooth numbers is multiplicative, and so Matom[" a]{}ki and Radziwi[łł]{}’s general result for multiplicative functions (see the discussion around ) shows the following: For any $\epsilon>0$ there exists $H(\epsilon)$ such that for large enough $N$ the set $${\mathcal E} = \{ x \in [\sqrt{N}/2, 2\sqrt{N}]: \ \ \text{ the interval } [x,x+H(\epsilon)] \text{ contains no } N^{\epsilon}\text{-smooth number}\},$$ has measure $|{\mathcal E}| \le \epsilon \sqrt{N}$. Now if for some $x\in [\sqrt{N},2\sqrt{N}]$ we have $x\notin {\mathcal E}$ and also $N/x \notin {\mathcal E}$, then we would be able to find $N^{\epsilon}$-smooth numbers in $[x,x+H(\epsilon)]$ and also in $[N/x,N/x+H(\epsilon)]$ and their product would be in $[N,N+4H(\epsilon)\sqrt{N}]$. Thus if Corollary \[cor2\] fails, we must have (with $\chi_{\mathcal E}$ denoting the indicator function of ${\mathcal E}$) $$\sqrt{N} \le \int_{\sqrt{N}}^{2\sqrt{N}} (\chi_{\mathcal E}(x) + \chi_{\mathcal E}(N/x)) dx \le 4|{\mathcal E}| \le 4\epsilon \sqrt{N},$$ which is a contradiction.
Now let us turn to Corollary \[cor3\]. First we recall a beautiful result of Wirsing (see [@GS], or [@Ten]), establishing a conjecture of Erd[ő]{}s, which shows that if $f$ is any real valued multiplicative function with $-1\le f(n) \le 1$ then $$\lim_{N \to \infty} \frac{1}{N} \sum_{n\le N} f(n) = \prod_{p} \Big(1-\frac{1}{p}\Big)\Big(1+\frac{f(p)}{p}+\frac{f(p^2)}{p^2} +\ldots \Big).$$ The product above is zero if $\sum_p (1-f(p))/p$ diverges (this is the difficult part of Wirsing’s theorem), and is strictly positive otherwise.
In Corollary \[cor3\], we are only interested in the sign of $f$ and so we may assume that $f$ only takes the values $0$, $\pm 1$. Wirsing’s theorem applied to $|f|$ shows that condition (ii) of the corollary is equivalent to $\sum_{p, f(p)=0} 1/p <\infty$, and further the condition may be restated as $$\lim_{N\to \infty} \frac{1}{N} \sum_{n\le N} |f(n)| =\alpha > 0.$$ Now applying Wirsing’s theorem to $f$, it follows that $$\lim_{N \to \infty} \frac{1}{N} \sum_{n\le N} f(n) = \beta$$ exists, and since $f(p) <0$ for some $p$ by condition (i), we also know that $0\le \beta < \alpha$. From we may see that if $h$ is large enough then for all but $\epsilon N$ integers $x \in [1,N]$ we must have $$\sum_{x<n\le x+h} f(n) \le (\beta+\epsilon)h, \text{ and } \sum_{x< n \le x+h} |f(n)| \ge (\alpha-\epsilon) h.$$ Since $\alpha > \beta$, if $\epsilon$ is small enough, this shows that for large enough $h$ (depending on $\epsilon$ and $f$) many intervals $[x,x+h]$ contain sign changes of $f$, which gives Corollary \[cor3\].
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[^1]: Je sais gré au Prof. Tokieda d’avoir bien voulu traduire ce résumé en fran[ç]{}ais.
[^2]: Je remercie Tokieda d’avoir traduit la note ci-dessus en fran[ç]{}ais.
[^3]: Je remercie Tokieda d’avoir traduit la note ci-dessus en fran[ç]{}ais.
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abstract: 'We find that the total spectrum of electron states in a bounded 2D electron gas with spin-orbit interaction contains two types of evanescent states lying in different energy ranges. The first-type states fill in a gap, which opens in the band of propagating spin-splitted states if tangential momentum is nonzero. They are described by a pure imaginary wavevector. The states of second type lie in the forbidden band. They are described by a complex wavevector. These states give rise to unusual features of the electron transmission through a lateral potential barrier with spin-orbit interaction, such as an oscillatory dependence of the tunneling coefficient on the barrier width and electron energy. But of most interest is the spin polarization of an unpolarized incident electron flow. Particularly, the transmitted electron current acquires spin polarization even if the distribution function of incident electrons is symmetric with respect to the transverse momentum. The polarization efficiency is an oscillatory function of the barrier width. Spin filtering is most effective, if the Fermi energy is close to the barrier height.'
author:
- 'Vladimir A. Sablikov and Yurii Ya. Tkach'
title: 'Evanescent states in 2D electron systems with spin-orbit interaction and spin-dependent transmission through a barrier'
---
Introduction
============
The spin-orbit interaction (SOI) in low-dimensional structures attracts a great deal of interest since it opens up the possibility to manipulate the electron spin in nonmagnetic structures using electrical means. [@Awschalom; @Zutic] In this view, semiconductor heterostructures with 2D electrons are very promising since the Rashba SOI is effectively controlled [@Nitta; @Matsuyama; @Schmult] by varying applied bias or gate voltages. In recent years, predominant interest was paid to effects appearing when the SOI modifies propagating electron modes with energy above the conduction band bottom. Suffice it to mention the spin-Hall effect, [@Sinova; @Chalaev; @Schliemann] or spin manipulation in strained semiconductors [@Kato]. In this paper we show that interesting effects of the SOI arise also when the electron energy is lower than or near to the conduction band bottom and evanescent states are involved. These states determine electron tunneling. They are important in 2D structures with laterally inhomogeneous potential landscape. We find that such structures can effectively polarize the transmitted electron current.
3D tunnel structures, in which spin polarization arises due to the SOI, were considered in a number of recent works. Zakharova et al [@Zakharova] studied the interband tunneling, Voskoboynikov et al [@Voskoboynikov] considered a tunnel structures with the Rashba SOI at the interfaces. In these cases the electron flow acquires a spin polarization if the structure is asymmetric. In symmetric tunnel structures the spin polarization arises, if the barrier material is noncentrosymmetrical. [@Perel] The polarization mechanism, proposed by Perel’ et al [@Perel; @Tarasenko], consists in a spin-dependent renormalization of the electron effective mass owing to the Dresselhaus SOI in the barrier. However, all these structures have a common property restricting their capability to generate spin polarization. The polarization is absent if the electron current is perpendicular to the barrier. In other words, for the spin polarization to appear the momentum distribution function of incident electrons must be asymmetric with respect to the momentum component parallel to the barrier.
The effective mass renormalization occurs if the Hamiltonian of the SOI is quadratic in longitudinal momentum. However, the dispersion relation of electrons in the presence of the SOI is generally much more complicated and therefore a more careful analysis of the complex band structure and evanescent states should be carried out to study the spin-dependent tunneling. In the 3D case, such calculations were recently carried out for some specific materials and qualitatively new features were found. [@Mishra; @Sandu; @Wang]
2D tunnel structures are scantily studied to date. In particular, as far as we know, even the complex band structure of 2D electrons was not explored. Though the presence of evanescent states is obvious, only a few of works touched upon these modes. Usaj, Reynoso and Balseiro [@Usaj; @Reynoso] attracted evanescent states to study the electron scattering at the edges of 2D samples, but the total spectrum of evanescent states was not considered. The importance of evanescent modes in quasi-one-dimensional systems in the presence of the SOI was pointed out in a number of works. [@Governale; @Streda; @Lee; @Serra]
Khodas, Shekter, and Finkel’stein [@Khodas; @Shekhter] studied the electron beam propagation in 2D electron gas with spatially inhomogeneous SOI. They considered the transmission through a strip, in which the SOI strength differs from that in the rest of the 2D electron gas, to show that an initially unpolarized beam splits into two beams with different spin polarizations propagating in different directions. The consideration was restricted by propagating states since only the case of uniform potential landscape was studied. Spin-dependent reflection of electrons from a lateral barrier in 2D system was observed in InSb/InAlSb heterostuctures [@Chen] and described theoretically in Ref. . Silvestrov and Mishchenko [@Silvestrov] demonstrated the possibility to achieve spin-polarized currents in a 2D system with smooth potential barrier and spatially-uniform SOI by considering propagating modes within semiclassical approach.
In this paper we show that strong polarization effect appears in 2D structures when electrons pass through a lateral potential barrier, in which the SOI is stronger than in the outside 2D electron gas regions (reservoirs). The polarization arises even if the electric current is normal to the barrier. The highest polarization is attained when the electron energy is close to the conduction band bottom in the barrier. In this case the fact becomes important that the SOI effectively splits the barrier height so that some part of electrons passes through the barrier via propagating states while others do this via the evanescent modes. Since the spin and orbital degrees of freedom are coupled, rather strong spin filtering occurs.
We study the total spectrum of electron states in 2D bounded system with SOI to find that there are two type of evanescent states. The first-type states are characterized by an imaginary longitudinal wavevector. They fill in a gap in the propagating state spectrum. The states of the second type lie in the forbidden gap. They are described by a complex wavevector. The electron tunneling through a lateral barrier with SOI via these evanescent states exhibits unusual features, such as an oscillatory behavior of the transmission coefficient with the barrier width. But of most interest is the spin polarization of the electron current. The polarization efficiency is high enough even if the distribution function of incident electrons is symmetric with respect to the transverse momentum. We explore the polarization efficiency in a wide range of electron energy to find that most effective spin filtering occurs if the Fermi energy is close to the barrier height.
The paper is organized as follows. In Sec. \[ComplexStructure\] we describe the complex band structure and total spectrum of electron states. Sec. \[Tunneling\] is devoted to the tunneling through a barrier with SOI. In Sec. \[Polarization\] the electric and spin currents through the barrier with SOI are considered for a wide range of the Fermi energy in the reservoirs. The results obtained for the Rashba SOI are generalized to the Dresselhaus SOI in Sec. \[Dresselhaus\]. We summarize and conclude in Sec. \[Conclude\].
Complex band structure of a 2D electron system with SOI {#ComplexStructure}
=======================================================
We start by considering the total spectrum of propagating and evanescent electron states in 2D electron gas with the Rashba SOI. The Hamiltonian is [@Rashba] $$H_R=\dfrac{\hbar^{2}}{2m}(p_{x}^{2}+p_{y}^{2})+
\frac{\alpha}{\hbar}(p_{y}\sigma_{x}-p_{x}\sigma_{y})\,,
\label{HamiltR}$$ where $\alpha$ is the SOI constant, $\sigma_x,\sigma_y$ are Pauli matrices. The eigenfunctions are $$\psi_{\mathbf{k},s}=C e^{i(k_x x+k_y y)}
\binom{\chi_s(\mathbf{k})}{1}\,,
\label{psi}$$ where $s=+,-$ stands for spin states, $\mathbf{k}=(k_x,k_y)$, $C$ is a constant. The eigenenergy $\varepsilon_{\mathbf{k},s}$ and the spin function $\chi_s(\mathbf{k})$ are defined by following equations: $$\left\{
\begin{array}{rl}
(\zeta_{\mathbf{k},s}-k^2)\chi_s(\mathbf{k}) - 2a(k_y+ik_x)&=0 \\
- 2a(k_y-ik_x)\chi_s(\mathbf{k}) + (\zeta_{\mathbf{k},s}-k^2) &=0\;,
\end{array}
\right.
\label{zeta-chi}$$ where $a=m\alpha/\hbar^2$ is a characteristic wavevector of SOI, $k^2=k_x^2+k_y^2$ and $\zeta_{\mathbf{k},s}$ is the normalized eigenenergy: $$\zeta_{\mathbf{k},s}=\dfrac{2m\varepsilon_{\mathbf{k},s}}{\hbar^2}\,.$$ Using Eqs (\[zeta-chi\]) one obtains the dispersion equation $$(\zeta_{\mathbf{k},s}-k^2)^2-4a^2k^2=0
\label{zeta}$$ and the spin function $$\chi_s(\mathbf{k})=2a\dfrac{k_y+ik_x}{\zeta_{\mathbf{k},s}-k^2}\,.
\label{chi}$$
Let us analyze the dispersion equation. To be specific, assume that the system under consideration is infinite in $y$ direction and has a boundary in $x$ direction, i.e. the system is semi-infinite or finite in the $x$ direction. In this case, the $y$ component of the wavevector $k_y$ is real, while $k_x$ is generally complex, $k_x=k'_x+ik''_x$. Dividing the real and imaginary parts of Eq. (\[zeta\]) we obtain an equation set determining the energy $\zeta$ as a function of $k'_x,k''_x,k_y$ and trajectories in the ($k'_x,k''_x$) plane, along which $\zeta(k'_x,k''_x,k_y)$ is real: $$\begin{aligned}
\label{dis1}
&(\zeta-k'^2+k''^2)^2=4a^2(k'^2-k''^2)+4k'^2k''^2\,,\\
\label{dis2}
&(\zeta-k'^2+k''^2)k'k''=-2a^2k'k''\,,\end{aligned}$$ where $k'+ik''=\sqrt{(k'_x+ik''_x)^2+k_y^2}$.
![(color online). Total spectrum of 2D electron gas with SOI. Solid lines $1+$ and $1-$ are spin-splitted propagating modes of branch 1; lines $2+$ and $2-$ are evanescent states, branch 2; lines $3L$, $3R$ are evanescent states in the forbidden gap, branch 3. Dashed lines $1'$, $2'$ and $3L'$, $3R'$ are the trajectories corresponding to these branches on the complex plane ($k'_x,k''_x$).[]{data-label="f_spectrum"}](fig_1.eps){width="0.9\linewidth"}
Eq. (\[dis2\]) possesses a solution in the following cases: i) $k'=0$, ii) $k''=0$ and iii) $\zeta-k'^2+k''^2=-2a^2$ for $k',k''\neq 0$. The first case contradicts to Eq. (\[dis1\]) and hence cannot be realized. The second case generates two branches of the solution, which can be found after dividing the real and imaginary parts of Eq. (\[dis1\]):
1\. $k''_x=0$, $$\begin{split}
\zeta_{\mathbf{k},s}&=-a^2+\left(a\pm \sqrt{k_y^2+{k'_x}^2}\right)^2\,,\\
\chi_s(\mathbf{k})&=\pm\dfrac{k_y+ik'_x}{\sqrt{k_y^2+{k'_x}^2}}\,;
\end{split}
\label{branch1}$$
2\. $k'_x=0$, $|k''_x|<|k_y|$, $$\begin{split}
\zeta_{\mathbf{k},s}&=-a^2+\left(a\pm \sqrt{k_y^2-{k''_x}^2}\right)^2\,,\\
\chi_s(\mathbf{k})&=\pm\dfrac{k_y-k''_x}{\sqrt{k_y^2-{k''_x}^2}}\,.
\end{split}
\label{branch2}$$
The third case gives one further branch:
3\. This branch is defined for $k'_x,k''_x$ belonging to the folowing trajectory $${k'_x}^2{k''_x}^2+a^2(k_y^2+{k'_x}^2-{k''_x}^2)-a^4=0$$ in the complex plane ($k'_x,k''_x$). The eigenenergy and the spin function are $$\label{branch3}
\begin{split}
\zeta_{\mathbf{k},s}&=-a^2-\dfrac{{k'_x}^2{k''_x}^2}{a^2}\,,\\
\chi_s(\mathbf{k})&=-a\dfrac{k_y-k''_x+ik'_x}{a^2+ik'_xk''_x}\,.
\end{split}$$
The total spectrum is shown schematically in Fig. \[f\_spectrum\] where the energy $\varepsilon_{\mathbf{k},s}$ is presented as a function of $k'_x,k''_x$ for given transverse momentum $k_y$. The form of all three branches is different in the cases $|k_y|<a$ and $|k_y|>a$.
Branch 1 describes the propagating states with real $k_x$. This branch is splitted by spin. The energy gap, which opens at $k_x = 0$ for $k_y \neq 0$, depends on $|k_y|$.
Branch 2 describes purely decaying evanescent states defined on the imaginary $k_x$ axis. This branch connects the propagating state branches along the imaginary axis. The spectrum of branch 2 is also splitted by spin. The energy minimum $\varepsilon_m = -\hbar a^2/(2m)$ is common for branches 1 and 2. The minimum is attained on branch 1, if $|k_y|<a$, or on branch 2 in the opposed case.
Branch 3 corresponds to evanescent states in the forbidden gap, $\varepsilon_{\mathbf{k},s}<\varepsilon_m$. They are described by a complex longitudinal wavevector and therefore can be named “oscillating” evanescent states. The trajectories, along which these states are defined, obey the following equation: $$(a^2 - {k'_x}^2)(a^2 + {k''_x}^2) = a^2 k_y^2 \,.
\label{trajectories}$$ They are shown in Fig. \[f\_trajectories\]. There are trajectories of two types. If $|k_y|<a$, the trajectories (lines 1, 2) intersect the imaginary axis $k''_x$. In the vicinity of the intersection point, $|k''_x| \ll |k'_x|$ and hence the wavefunction oscillates with the distance faster than decreases. If $|k_y|>a$ the trajectories (lines 4-6) intersect the $k'_x$ axis.
![(color online). Real-energy trajectories along which the “oscillating” evanescent states are defined. $k_y$ is fixed for each line: lines *1-6* correspond to $k_y/a = 0.4, 0.8, 1.0, 2.0, 5.0, 10.0$.[]{data-label="f_trajectories"}](fig_2.eps){width="0.9\linewidth"}
It is seen that there are four states for any energy, in accordance with the number of the degrees of freedom of the system. At a given energy all the states are distinguished by the wavevector components $k'_x,k_x''$ and the spin function. The propagating states and the evanescent states of branch 2 have two subbranches divided by the energy, while the evanescent states of branch 3 are splitted in the complex momentum plane.
Though the wavefunctions of the third branch are complex, they do not carry the current. Using the Hamiltonian (\[HamiltR\]) one obtains the following expression for the particle current in the state $$\label{psi_spinor}
\psi_s=\binom{\psi_1}{\psi_2}\,,$$ $$\label{current}
\begin{split}
\mathbf{j}_s&= \dfrac{i\hbar}{2m}\left(\psi_1\nabla\psi^*_1-\psi^*_1\nabla\psi_1+\psi_2\nabla\psi^*_2-\psi^*_2\nabla\psi_2\right)\\ & -\dfrac{i\alpha}{\hbar}\left(\psi_1\psi^*_2-\psi^*_1\psi_2\right)\mathbf{e}_x +\dfrac{\alpha}{\hbar}\left(\psi^*_1\psi_2+\psi_1\psi^*_2\right)\mathbf{e}_y .
\end{split}$$ The wavefunctions of branch 3, calculated with using Eqs (\[psi\]) and (\[branch3\]), are easily seen to turn the $x$ component of the current (\[current\]) to zero while $y$ component is nonzero.
Tunneling currents through a barrier with SOI {#Tunneling}
=============================================
In this section we study the electron tunneling via the “oscillating” evanescent states of branch 3. Consider a 2D electron gas without SOI divided into two semiplanes (reservoirs) by a rectangular barrier, in which the SOI is present. The barrier height is $U$ and the width is $d$.
Let us calculate the transmission probability for electrons incident on the barrier from the left reservoir. We use the presentation of the wavefunctions in the reservoirs in the basis of eigenstates $|s\rangle$ of the $\sigma_z$ matrix. The basis functions are $|k_x,k_y,s\rangle$, where $k_x$ and $k_y$ are the wavevector components in the reservoirs, $s=\uparrow,\downarrow$.
In the left reservoir ($x<0$) the wavefunction is $$|\psi^{(L)}_{k_x,k_y,s}\rangle =| k_x,k_y,s\rangle + \sum_{s'}r_{s,s'}|-k_x,k_y,s'\rangle\;,
\label{Lwave}$$ where $|k_x,k_y,s\rangle$ is the state vector of incident electrons. The wavefunction of transmitted electrons ($x>d$) is $$|\psi^{(R)}_{k_x,k_y,s}\rangle = \sum_{s'}t_{s,s'}|k_x,k_y,s'\rangle\;.
\label{Rwave}$$ Here $r_{s,s'}$ and $t_{s,s'}$ are reflection and transmission matrices.
The wavefunction in the barrier is expanded in the eigenstates (\[psi\]) of the Hamiltonian (\[HamiltR\]) $$|\psi^{(B)}_{k_x,k_y,s}\rangle =\sum_{r,r'=+,-} b_{r,r'}^s|rK'_x,r'K''_x,k_y\rangle \,,
\label{Bwave}$$ where $K_x=K'_x+iK''_x$ denotes the complex wavevector in the barrier; $r,r'=\pm$ are indexes labeling all four evanescent states in the barrier. They are described by the spinors (\[psi\]), in which the real and imaginary parts of $K_x$ should be taken with different signs.
The matrices $r_{s,s'}$, $t_{s,s'}$ and $b_{r,r'}^s$ are determined by an equation set, which follows from the boundary conditions at the interfaces of the barrier and 2D electron reservoirs.
Boundary conditions for wavefunctions are obtained in a standard way by integrating the Schrödinger equation over the infinitesimal vicinity of the boundary. These conditions are well known for a boundary between regions with different strength of SOI [@Molenkamp; @Khodas]. In our case it is necessary to take into account that the lateral potential step at the boundary also contributes to the SOI. In the transition region, where the potential $U(x)$ varies with $x$, the following additional term should be added to the Hamiltonian (\[HamiltR\]) $$\label{boundarySOI}
H^b_{so}=\dfrac{\gamma}{\hbar}\dfrac{dU}{dx}\sigma_z p_y\,,$$ where $\gamma$ is SOI constant connected with $\alpha$ by $\alpha=-e\gamma F_z$, $F_z$ being the electric field perpendicular to the 2D layer. This term having been integrated over the transition region gives a finite contribution to the boundary conditions. Finally one obtains the following equations for the spinor (\[psi\_spinor\]) components: $$\begin{aligned}
\label{boundary_cond1}
&\psi_1|_{-0}^{+0} = \psi_2|_{-0}^{+0} =0 \,;\\
\label{boundary_cond2}
&\dfrac{1}{m(x)}\left[\dfrac{\partial \psi_1}{\partial x} \pm \beta(x) k_y\psi_1- a(x) \psi_2\right]_{-0}^{+0}=0\\
\label{boundary_cond3}
&\dfrac{1}{m(x)}\left[\dfrac{\partial \psi_2}{\partial x} \mp \beta(x)k_y \psi_2 + a(x) \psi_1\right]_{-0}^{+0}=0\,,\end{aligned}$$ where the parameter $$\beta= a \dfrac{2U}{eF_z}\,
\label{beta}$$ describes the Rashba SOI caused by the in-plane field. The upper and lower signs in Eqs (\[boundary\_cond2\]),(\[boundary\_cond3\]) correspond to the boundaries at which $U(x)$ increases or decreases with $x$.
Applying these boundary conditions to the system under consideration we put $a=0$ in the 2D electron reservoirs and keep $a\ne 0$ in the barrier. For simplicity, we ignore the difference in the effective masses of electrons in the barrier and reservoirs.
In addition, the wavevectors in the barrier and the reservoirs should be matched. The tangential components $k_y$ must be equal. The relation of the normal components is determined by equaling the energy $E$ of an incident electron to the electron energy in the barrier $$E(k_x,k_y) = U + \varepsilon(K_x,k_y,s)\;,$$ where $\varepsilon(K_x,k_y,s)$ is defined by Eqs (\[branch1\]),(\[branch2\]),(\[branch3\]) in accordance with the energy spectrum branch (1, 2 or 3), which is considered. In the present section we restrict ourselves by branch 3.
Finally one obtains two equation sets for the cases of the incident spin directed up and down. Each equation set contains eight equations. Dropping elementary calculations and combersome expressions for matrices $t_{s,s'}$, $r_{s,s'}$ and $b_{r,r'}^s$, we turn directly to main results.
First, note that the $t_{s,s'}$ matrix obeys the following symmetry relations: $$\label{symmetry}
\begin{split}
t_{\uparrow\uparrow}(K_x,k_y)=&t_{\downarrow\downarrow}(K_x,-k_y)\,,\\
t_{\uparrow\downarrow}(K_x,k_y)=&-t_{\downarrow\uparrow}(K_x,-k_y)\,.
\end{split}$$
The particle current through the barrier is $$\label{current_K}
j_{k_x,k_y,s}=\dfrac{\hbar k_x}{m}T_s(k_x,k_y)\;,$$ where $T_s$ is the transmission coefficient, $T_s=\sum_{s'}|t_{s,s'}|^2$, which has a meaning of the probability for an electron to tunnel with any spin in the final state. The peculiarity of the tunneling in the presence of the SOI consists in the involvement of four interfering states with different spin structure.
One of unusual consequences of this fact is an oscillatory dependence of the tunneling coefficient on the barrier width. This feature is demonstrated in Fig. \[tunnel\_oscillat\]. The oscillations exist when the electron energy is close to the top of the third branch: $U-E_{so}-E\ll E_{so}$, where $E_{so}=\hbar^2a^2/(2m)$ is a characteristic energy scale of SOI. The oscillations fade away as the energy decreases deep into the forbidden band, because $K''_x$ exceeds $K'_x$. The oscillations disappear also if the tangential momentum $k_y$ goes to zero.
![(color online). Tunneling coefficients for incident electrons with spin up and down as functions of the barrier width. $U=2E_{so}$, $E=0.95E_{so}$, $k_y=0.49 a$ and $\beta=0.1$.[]{data-label="tunnel_oscillat"}](fig_3.eps){width="0.9\linewidth"}
Fig. \[tunnel\_oscillat\] shows clearly that the barrier filters incident electrons by spin. This process depends evidently on the incident angle and the energy of electrons. Consider the spin polarization of transmitted electrons in more detail. Let an unpolarized electron flow with the wavevector $(k_x,k_y)$ is incident on the barrier. The flow consists of two components $|k_x,k_y,\uparrow\rangle$ and $|k_x,k_y,\downarrow\rangle$ with opposed spins. The transmitted flow acquires spin polarization. The spin density of transmitted electrons is $$\vec{\mu}_{k_x,k_y}=\dfrac{\hbar}{2}\sum_{s=\pm 1}\langle \psi_{k_x,k_y,s}^{(R)}|\hat{\vec{\sigma}}|\psi_{k_x,k_y,s}^{(R)}\rangle\;.
\label{mu_k-s}$$ With using Eq. (\[Rwave\]), the spin density takes the form: $$\vec{\mu}_{k_x,k_y}\! =\! \dfrac{\hbar}{2}|C|^2 \! \sum_{s,s'}\!\left[(\mathbf{e}_x\!-\! i \mathbf{e}_y s') t^*_{s,s'}t_{s,-s'}+ \mathbf{e}_z s'|t_{s,s'}|^2 \right],
\label{mu_k-s1}$$ where $|C|^2$ is a normalization constant.
The components of the spin polarization are shown in Fig. \[tunnel\_polariz\] as functions of the transverse momentum $k_y$ for a given energy of incident electrons. The main property of the acquired polarization is that the $x$ and $z$ components of the spin polarization are odd functions of $k_y$, while the $y$ component is an even function of $k_y$. This property is independent of the electron energy, the barrier height, the SOI strength, and thus is universal for the Rashba SOI. Two consequences follow from this fact. First, the unpolarized electron follow acquires spin polarization even if the distribution function of the incident electrons is symmetric with respect to the tangential momentum direction, the polarization being directed parallel to the barrier. Second, if the distribution function is not symmetric about $k_y$, the spin polarization arises also in the $x$ and $z$ directions.
![(color online). Spin polarization of transmitted electrons, $\tilde{\mu}=2\mu_{x,y,z}/\hbar |C|^2$. The curves are marked by letters corresponding to the polarization components. The incident electron energy $E=9.5~E_{so}$, the barrier height $U=11~E_{so}$, the barrier width $d=a$, the lateral SOI parameter $\beta=0.1$.[]{data-label="tunnel_polariz"}](fig_4.eps){width="0.9\linewidth"}
Detail analysis shows that the spin polarization is caused mainly by the SOI in the barrier, as it is described by the Hamiltonian (\[HamiltR\]). The SOI at the boundaries of the barrier (described by Eq. (\[boundarySOI\])) does not essentially affect the results if the parameter $\beta$ defined by Eq. (\[beta\]) is small. This case corresponds to realistic situation in experiments. Numerical estimations for InAs quantum well ($\alpha \sim 6\times 10^{-9}$ eVcm, $F_z\sim 10^5$ V/cm, $U\sim 20$ meV) give $\beta \sim 0.1$. If $\beta \gtrsim 1$, the boundary SOI changes the value of polarization, but main features (such as the symmetry relations, the oscillatory behavior of the tunneling coefficients) remain qualitatively similar.
Spin polarization of electrons by a barrier with SOI {#Polarization}
====================================================
In this section we turn from the separate electron states to the total spin polarization produced by all electron states contributing to the current through a barrier with SOI. In addition, we consider a wide range of the incident electron energy to include all three branches of the electron spectrum. This allows one to find out conditions under which the electron current is polarized most effectively.
Consider an electron current directed normally to the barrier. For definiteness, let the current be caused by a voltage $V$ applied across the 2D electron reservoirs, as it is shown in Fig. \[scheme\]. The incident electron states, which contribute to the current, occupy an energy layer near the Fermi level. They are located in a semi-ring in the $k_x,k_y$ plane, shown in the insertion. The particle and spin currents through the barrier are determined by summing over all these states. We carry out this calculation for various positions of the Fermi level $E_F$ relative to the barrier height $U$ to find the spin-polarization efficiency as a function of $E_F$.
![(color online). A schematic view of the barrier with SOI. Lines L and R show the electron dispersion relations in the left and right reservoirs. Lines 1,2,3 image schematically corresponding branches of the dispersion relation in the barrier. The insertion is the $k_x,k_y$ space, in which the shaded region shows the electron states contributing to the current.[]{data-label="scheme"}](fig_5.eps){width="0.9\linewidth"}
For simplicity suppose that the voltage is small compared to other energies, $eV\ll U, E_F, E_{so}$. This simplification allows one to restrict the summation by the integration in $\mathbf{k}$ space over the azimuthal angle at a given energy $E$. The integration is conveniently to carry out over $k_y$, but in doing this the fact should be taken into account that the set of branches, which must be used at given $k_y$ and $E$, can change with varying $k_y$. This happens because the gap $\Delta \varepsilon_{12}$ between spin-splitted subbranches of propagating modes (curves 1+ and 1- in Fig. \[f\_spectrum\]) increases with $k_y$ ($\Delta \varepsilon_{12}=2\hbar^2a|k_y|/m$) and furthermore the form of the dispersion curves describing the propagating (1+, 1-) and evanescent (2+, 2-) modes changes. The switches between the actual branches occur when any extremum of the dispersion curves (which is a function of $k_y$) coinsides with the energy $E$. Physically this means that electrons incident on the barrier at different angles and in different spin states feel different effective barrier height.
Fig. \[diagram\] presents a diagram showing which branches of the dispersion relation are accessible for electrons with given $E$ and $k_y$. Within each region bounded by thick lines, there are two different branches with two fundamental eigenfunctions on each ones or one branch with four solutions.
![The diagram of the spectrum branches accessible for electrons with energy $E$ and tangential momentum $k_y$. Thick lines divide the $(E,k_y)$ plane into 6 regions, in which corresponding branches are denoted by numbers 1,2,3 and chirality indexes $+,-$ in accordance with Fig. \[f\_spectrum\]. Line $\mathcal L$ is defined by equation $ k_y/a = (\sqrt{(E-U-E_{so})/E_{so} +1} \pm 1)^2$[]{data-label="diagram"}](fig_6.eps){width="1.0\linewidth"}
For each region of the diagram the wavefunctions are determined in the same way as in Sec. \[Tunneling\]. The only difference is that the eigenfunctions of the branches specified in the diagram are to be used in Eq. (\[Bwave\]) instead of the wavefunctions of branch 3. As a result of these calculations the transmission $t_{s,s'}(E,k_y)$ and reflection $r_{s,s'}(E,k_y)$ matrices are obtained for the whole $(E,k_y)$ plane. Using $t_{s,s'}(E,k_y)$ we find the particle and spin currents in the right electron reservoir.
The particle current is found by the summation of Eq. (\[current\_K\]) over all states of incident electrons: $$J=\dfrac{eV}{2\pi h}\int_{-k_F}^{k_F}\!\! dk_y \left(|t_{\uparrow \uparrow}|^2\!+\!|t_{\uparrow \downarrow}|^2\!+\!|t_{\downarrow \uparrow}|^2\!+\!|t_{\downarrow \downarrow}|^2\right)\,,$$ where the integration symbol implies also the summation over all regions of the $(E,k_y)$ plane which fall within the $(-k_F,k_F)$ interval, with $k_F$ being the Fermi wavevector in the 2D reservoirs.
The transmitted spin current in the left reservoir is defined in a standard way [@Rashba1; @Shi] using the following expression for the current in a state $|k_x,k_y\rangle$: $$J_{s,i}^j(k_x,k_y)= \dfrac{1}{2}\langle v_i \sigma_j + \sigma_j v_i \rangle\,,$$ where $i=\{x, y\}$ denotes the velocity components in the plane, $j=\{x, y, z\}$ denotes the spin components in 3D space. In the case under consideration, the $x$ component of the total spin current is $$\label{spin_current}
J^j_s=\dfrac{eV}{2\pi h}\int_{-k_F}^{k_F}\!\! dk_y \!\left(
\begin{array}{l}
\!2 \mathrm{Re} (t_{\uparrow \uparrow}t^*_{\uparrow \downarrow}\!+\!t_{\downarrow \downarrow}t^*_{\downarrow \uparrow})\\
\!2 \mathrm{Im} (-t_{\uparrow \uparrow}t^*_{\uparrow \downarrow}\!+\!t_{\downarrow \downarrow}t^*_{\downarrow \uparrow})\\
\!|t_{\uparrow \uparrow}|^2\!-\!|t_{\uparrow \downarrow}|^2\!+\!|t_{\downarrow \uparrow}|^2\!-\!|t_{\downarrow \downarrow}|^2
\end{array}
\!\!\right)\!,$$ where three lines in the RHS correspond to the $x,y,z$ components of the spin polarization for the spin current directed along $x$ axis.
The efficiency of spin polarization is characterized by the ratio of the spin current to the particle current $$\mathcal{P}_j = \dfrac{J^j_s}{J}\;.$$
Using the symmetry relations (\[symmetry\]) and Eq. (\[spin\_current\]) one finds that $x$ and $z$ components of the spin current are absent in the case of the Rashba SOI, and only the $y$ component is nonzero. Of course, this is a consequence of the symmetry of the distribution function with respect to the sign of $k_y$. If the current had not be perpendicular to the barrier, the polarization would appear also in the $x$ and $z$ directions. The polarization efficiency turns out to be sufficiently high. The dependencies of the polarization on the Fermi energy and the barrier width are nontrivial because they reflect a complex structure of the electron spectrum. Below two most significant results are considered.
The dependence of $\mathcal{P}_y$ on the barrier width is shown in Fig. \[polar-d\_3\] for energies below the top of the energy band where the evanescent states of branch 3 exist in the barrier, $E_F<U-E_{so}\equiv E_b$. The polarization efficiency is seen to oscillate with $d$ because of the interference of four oscillating evanescent modes. The effect is rather strong and becomes the stronger the closer is the energy to the band top. The oscillation period is of the order of $\pi/a$.
![Dependence of the polarization efficiency on the barrier width for the Fermi energy below $E_b$ for tunneling through branch 3. $U=5 E_{so}$, $E_F=3.5 E_{so}, \beta=0.1$.[]{data-label="polar-d_3"}](fig_7.eps){width="0.9\linewidth"}
The dependence of $\mathcal{P}_y$ on the Fermi energy is presented in Fig. \[polar-E\] for a wide energy range including the energy both below and above the barrier. It is seen that there is a critical energy $E=8 E_{so}$, which coincides with the top energy of the third branch band, $E_b=U-E_{so}$, in the barrier. At energies well below $E_b$, the polarization efficiency $\mathcal{P}_y$ increases with the energy. In the vicinity of the threshold $E_b$ an oscillatory behavior appears as a result of the interference of four slowly decaying waves. Above this energy the transmission process goes via the propagating states of branch 1 and the evanescent states of branch 2. At $E_F$ slightly higher than $E_b$ the polarization efficiency attains the highest value. With further increasing the energy the efficiency $\mathcal{P}_y$ decreases. It is worth noting that the highest efficiency of the spin polarization slightly depends on the SOI strength, but the barrier width, at which this high polarization is attained, varies inversely with the SOI constant.
Fig. \[polar-E\] demonstrates also how $\mathcal{P}_y$ changes with the barrier width. Increasing $d$ above $\sim \pi/a$ leads to more pronounced oscillatory dependence of $\mathcal{P}_y$ on the energy caused by the four-wave interference.
![(color online). Dependence of the polarization efficiency on the Fermi energy in 2D electron reservoirs. $U=9 E_{so}$ for $d=5/a$ (solid line 1) and $d=3/a$ (dashed line 2); $\beta=0.1$. Vertical dashed line marks the threshold energy $E_b=U-E_{so}$, which divides the evanescent states of branch 3 (on the left of $E_b$) from the states of branches 2 and 1.[]{data-label="polar-E"}](fig_8.eps){width="0.95\linewidth"}
The physical mechanism, owing to which the polarization of normally incident electron current appears, is connected with the splitting of electron waves in the barrier because of the SOI. Let us consider first a simplified case of semiinfinite barrier region with SOI. Let the energy is high enough so that electrons occupy propagating states (branches 1+,1-). The incident electron states can be represented in terms of two chiral modes: $$\chi^{(0)}_{\pm}=\binom{\pm \chi}{1}\;, \qquad \chi=\dfrac{k_y+ik_x}{\sqrt{k_y^2+k_x^2}}\,.$$ In the barrier region each electron beam splits into two beams, which propagate at different angles and have different chiralities: $$\label{chi_12}
\chi^{(1)}_{1,2}=\binom{\chi_{1,2}}{1}\;, \qquad \chi_{1,2}=\pm \dfrac{k_y+iK_{1,2}}{\sqrt{k_y^2+K_{1,2}^2}}\,,$$ where $K_1$ is $x$-component of the wavevector of the upper mode with positive chirality $\chi_1$ and $K_2$ corresponds to the lower mode with negative chirality $\chi_2$. It is essential that $K_1<K_2$. The incident and refracted beams as well as their spin polarizations are illustrated in Fig. \[birefringence\]. The transmitted beam amplitudes are $A_+$ and $B_+$ for the incident beam with positive chirality, and $A_-$ and $B_-$ for negative chirality. The $x$ and $y$ components of the spin polarization in the barrier are estimated as $$\begin{split}
S_x\propto (|A_+|^2\!+\!|A_-|^2)\mathrm{Re}\chi_1\!-\!(|B_+|^2\!+\!|B_-|^2)\mathrm{Re}\chi_2\!+ \dots ,\\
S_y\propto (|A_+|^2\!+\!|A_-|^2)\mathrm{Im}\chi_1\!-\!(|B_+|^2\!+\!|B_-|^2)\mathrm{Im}\chi_2\!+ \dots\,,
\end{split}$$ where the dots denote spatially dependent terms originating from the interference.
![(color online). Refraction and spin polarization of electron beams incident on the semiinfinite barrier with SOI. Solid arrows show the spin polarization in the case of positive chirality of the incident beam. Dashed arrows are the polarization for negative incident chirality. $A_{\pm}$ and $B_{\pm}$ are amplitudes for incident beam with $k_y>0$. The primed letters denote the same amplitudes for $k_y>0$.[]{data-label="birefringence"}](fig_9.eps){width="0.9\linewidth"}
The total spin polarization is determined by the sum over all incident angles. To estimate this sum, consider the dependence of the spin components on $k_y$. It is clear that $|A_{\pm}|^2$ and $|B_{\pm}|^2$ are even functions of $k_y$. Real and imaginary parts of $\chi_{1,2}$ are seen from Eq. (\[chi\_12\]) to be correspondingly odd and even functions of $k_y$. Therefore $S_x$ is an even function of $k_y$ and $S_y$ is an odd function, as it is illustrated in Fig. \[birefringence\]. Thus the total $S_x$ component vanishes while $S_y$ is nonzero.
It is easy to find the direction of the total spin for the energy close to the barrier height. If $k_y=0$, the amplitudes $|A_+|$ and $|B_-|$ are equal, $|A_-|=|B_+|=0$ and $\chi_1=\chi_2=i$. Hence, the total spin density is zero because the spins of opposed chiralities cancel each other. At small $k_y$, the spin polarization appears. Since the amplitudes are functions of $k_y^2$, they remain unchanged in the first approximation, so that the resulting spin is determined by the difference $(\chi_1-\chi_2)$ and $$S_y \propto \dfrac{K_1}{\sqrt{k_y^2+K_1^2}}-\dfrac{K_2}{\sqrt{k_y^2+K_2^2}}\,.$$ Since $K_1<K_2$, the $y$-component of the spin density is negative, i.e. the sign of the spin polarization is determined by the lower-energy branch of propagating mode.
If the barrier width is finite, there are four modes in the barrier, but the above property remains unchanged. It is also kept for other branches of the electron spectrum.
The case of Dresselhaus SOI {#Dresselhaus}
===========================
All the above results are generalized to the case of Dresselhaus SOI. For a 2D system oriented along \[001\] crystallographic direction, the SOI Hamiltonian is [@Winkler] $$H_D=\dfrac{\hbar^{2}}{2m}(p_{x}^{2}+p_{y}^{2})+
\frac{\alpha}{\hbar}(p_{x}\sigma_{x}-p_{y}\sigma_{y})\,.
\label{HamiltD}$$ It is well known that Dresselhaus and Rashba Hamiltonians are unitary equivalent. [@RashbaSheka] An unitary matrix $$\label{R-D_matrix}
U= \left( \begin{array}{cc}
0 & i\\ 1 & 0
\end{array}
\right)$$ transforms the Rashba Hamiltonian (\[HamiltR\]) to the Dresselhaus one: $\tilde{H}_R=U^+H_RU=H_D$. Therefore, the Dresselhaus SOI case does not require separate calculations. It is enough to carry out this transformation. Taking into account that matrix (\[R-D\_matrix\]) transforms the spin matrices as follows: $\tilde {\sigma}_x=-\sigma_y$, $\tilde {\sigma}_y=-\sigma_x$, and $\tilde {\sigma}_z=-\sigma_z$, we arrive at the conclusions: (i) the dispersion equation is the same as in the Rashba case (\[zeta\]);
\(ii) the spin functions differ from those defined by Eq. (\[chi\]) by a simple substitution $k_x\leftrightarrows k_y$;
\(iii) the spin components of transmitted electrons in the incident states $|k_x,k_y,\uparrow\rangle$ and $|k_x,k_y,\downarrow\rangle$ differ from those of Sec.\[Tunneling\] by replacements: $\mu_x\to-\mu_y$, $\mu_y\to-\mu_x$ and $\mu_z\to-\mu_z$;
\(iv) the spin polarization of the current transmitted through a barrier with SOI has only $x$ component, if the distribution function is even with respect to the transverse momentum. In particular, Figs \[polar-d\_3\], \[polar-E\] are valid for the spin polarization normal to the barrier, $\mathcal{P}_x$.
Of course, if the Rashba and Dresselhaus mechanisms act simultaneously the results change qualitatively.
Conclusion {#Conclude}
==========
We have found the total spectrum of electron states in a bounded 2D electron gas with SOI. It addition to well known propagating states it contains two branches of evanescent states. Their wavefunctions decay with the distance in the direction $x$ perpendicular to the boundary. One branch (“purely decaying” evanescent mode) is described by an imaginary wavevector. The energy of this state is splitted by spin, so that there are two subbranches. They fill in the gap, which opens in the propagating state spectrum at $k_y\ne0$. Other branch (“oscillating” evanescent mode) is characterized by a complex wavevector $K_x$. These states lie in the forbidden gap.
We have studied the electron transmission through a lateral potential barrier with the SOI. In the energy range, where electrons tunnel via the oscillating evanescent states, the tunneling reveals unusual features, such as an oscillatory dependence of the transmission coefficients on the barrier width and the energy. But of most importance is the spin polarization of the electron current. The value and direction of the polarization depend on the angle of incidence and the energy of incident electrons. The polarization appears even if the distribution function of incident electrons is symmetric with respect to the transverse momentum. In this case the polarization is directed parallel to the barrier (in the Rashba SOI case) or perpendicular to it (for Dresselhaus SOI). The highest efficiency of the spin polarization is attained when the Fermi energy is close to the barrier height. In this case, electrons pass through the barrier partially via the propagating states and partially via the purely decaying evanescent states. Under this condition the most effective spin filtering occurs. The maximal polarization efficiency depends on the barrier height and can exceed 0.5 if the barrier width is on the order of $\pi/a$.
This work was supported by Russian Foundation for Basic Research (project No. 05-02-16854), Russian Academy of Sciences (programs “Quantum Nanostructures”and “Strongly Correlated Electrons in Semiconductors, Metals, Superconductors, and Magnetic Materials”), RF Ministry of Education and Science.
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abstract: 'The phenomenon of quantum vacuum polarization in the presence of a gravitational field is well understood and is expected to have a physical reality, but studies of its back-reaction on the dynamics of spacetime are practically non-existent outside the specific context of homogeneous cosmologies. Building on previous results of quantum field theory in curved spacetimes, in this letter we first derive the semiclassical equations of stellar equilibrium in the $s$-wave Polyakov approximation. It is highlighted that incorporating the polarization of the quantum vacuum leads to a generalization of the classical Tolman-Oppenheimer-Volkoff equation. Despite the complexity of the resulting field equations, it is possible to find exact solutions. Aside from being the first known exact solutions that describe relativistic stars including the non-perturbative backreaction of semiclassical effects, these are identified as a non-trivial combination of the black star and gravastar proposals.'
author:
- 'Raúl Carballo-Rubio'
bibliography:
- 'black\_stars.bib'
title: Stellar equilibrium in semiclassical gravity
---
*Introduction.–*The recent detection of gravitational waves [@Abbott2016] has revived interest (e.g., [@Cardoso2016a; @Cardoso2016; @Abedi2016; @Barcelo2017; @Price2017; @Nakano2017; @Volkel2017; @Mark2017; @Maselli2017; @Krishnendu2017] and references therein) in theoretical scenarios in which black holes are replaced by horizonless ultra-compact configurations. Current electromagnetic and gravitational wave data definitively leave room for these alternatives to exist [@Cardoso2017n; @Cardoso2017]. This letter does not discuss the diverse motivations for these theoretical considerations or review different proposals available in the literature, for which we refer the reader to [@Cardoso2017; @Visser2009; @Barcelo2015u] for instance; it focuses rather on the following question: Is it possible to obtain horizonless ultra-compact configurations from known physics?
The theoretical status of horizonless ultra-compact objects is far from satisfactory at best, as their construction has been so far based on assuming large deviations from known physics, postulating geometries with no complete mathematical framework to justify the origin of these deviations, or to control that other predictions of general relativity are not affected. The lack of theoretical constraints leads to huge uncertainties on the properties of these hypothetical objects, even if considering the simpler problem of describing static configurations.
This letter aims to represent a first step in changing this situation, providing a framework in which the theoretical assumptions are minimal and under control, and everything else is obtained just by solving a consistent set of field equations. In particular, here we deal with the most conservative extension of general relativity, the semiclassical Einstein field equations [@Hartle1981; @Flanagan1996; @Hu2001] $$G_{\mu\nu}=8\pi G(T_{\mu\nu}+\hbar N\langle\hat{T}_{\mu\nu}\rangle)+\mathscr{H}_{\mu\nu},
\label{eq:sefq}$$ using a standard definition of the renormalized stress-energy tensor $\langle\hat{T}_{\mu\nu}\rangle$ describing the quantum vacuum polarization of $N\gg 1$ matter fields. $\mathscr{H}_{\mu\nu}$ stands for terms that, while being generally non-zero, are negligible for the situations studied in this letter. These can be further classified into two categories. The first category contains $\mathscr{O}(1/N)$ contributions with respect to the term proportional to $\hbar\langle\hat{T}_{\mu\nu}\rangle$ (i.e., subleading terms in the $1/N$ expansion). The second category includes contributions from possible higher-derivative curvature corrections to the Einstein-Hilbert Lagrangian which, on general grounds, can be written schematically as $\hbar^{n-1}G^{n-2}\mathscr{R}^n$ with $n\geq2$ and $\mathscr{R}^n$ a scalar polynomial of order $n$ on the Riemann curvature or its derived curvature tensors. For instance, the $n=2$ contributions are needed in order to renormalize $\langle\hat{T}_{\mu\nu}\rangle$ [@Birrell1982; @Visser2002] and are proportional to $N$. The semiclassical approximation is meaningful as long as curvature remains small enough. Hence, in practice we will study the field equations with $\mathscr{H}_{\mu\nu}=0$ identically, justifying later that this is indeed a good approximation for the solutions to be described.
We start presenting the new set of equations of stellar equilibrium that follows from these semiclassical equations, and continue with the description of a family of exact solutions. Let us remark that here we deal with static situations only. Our results do not imply by themselves that (perhaps some) astrophysical black holes are horizonless ultra-compact objects, as this conclusion could only follow from a detailed analysis of dynamical situations. However, that it is possible to obtain the static properties of such hypothetical configurations from a set of solid principles certainly makes this possibility more plausible from a theoretical perspective, and is therefore of clear relevance for dynamical studies.
*Setting.–*Our first goal is showing that the semiclassical Einstein field equations can be expressed in a closed and simple way under the two following mild assumptions: (i) spherical symmetry: this is common in the analytical study of relativistic stars in hydrostatic equilibrium. For instance, the Tolman-Oppenheimer-Volkoff (TOV) equation [@Tolman1930; @Oppenheimer1939], to be generalized here, is found in classical general relativity in this approximation; (ii) $s$-wave Polyakov approximation: this neglects quantum fluctuations that are not spherically symmetric, as well as the effects of backscattering, by means of a dimensional reduction (projection) to a 2-dimensional manifold. This is a well-known approximation that is routinely used in black hole physics and permits us to obtain closed analytical expressions for quantities of interest in the semiclassical theory (e.g., [@Birrell1982; @Nojiri1999; @Fabbri2005book]), preserving the qualitative features of all the relevant vacuum states [@Davies1976; @Visser1996].
Hence we will deal with spherically symmetric spacetimes, with line element $$\text{d}s^2=\text{d}s^2_{(2)}+r^2\text{d}\Omega^2=g_{ab}(y)\text{d}y^a\text{d}y^b+r^2(y)\text{d}\Omega^2(\theta,\varphi),\label{eq:sph4}$$ where $\text{d}\Omega^2(\theta,\varphi)$ is the angular line element on the 2-sphere. Greek indices such as $\mu$, $\nu$, take four values, while latin indices such as $a$, $b$, take just two values. For static situations, the metric in Eq. has a time-like Killing vector field $\xi$.
Regarding the classical source in Eq. , for simplicity we consider a perfect fluid in a spherically symmetric and static configuration. The stress-energy tensor is therefore given by $T_{\mu\nu}=(\rho+p)u_\mu u_\nu+pg_{\mu\nu}$, where the velocity field of the fluid $u$ is the vector field associated with the time-like Killing vector field $\xi$ of the spacetime geometry but properly normalized such that $u_\mu u^\mu=-1$. This guarantees that we are describing a static configuration.
To this classical source, we are adding the vacuum polarization as described by the (renormalized) vacuum expectation value of the stress-energy tensor of $N$ non-interacting scalar fields (following the usual practice [@Hartle1981; @Flanagan1996; @Hu2001]). This additional piece is given in the $s$-wave Polyakov approximation [@Fabbri2005book] by $N$ times $$\langle\hat{T}_{\mu\nu}\rangle=\frac{\delta^a_\mu\delta_\nu^b}{4\pi r^2}\langle\hat{T}_{ab}\rangle^{(2)},\label{eq:semisource}$$ where $\langle\hat{T}_{ab}\rangle^{(2)}=\langle0|\hat{T}_{ab}|0\rangle^{(2)}$ is evaluated in the 2-dimensional spacetime $\text{d}s^2_{(2)}=g_{ab}(y)\text{d}y^a\text{d}y^b$ [@Polyakov1981; @Maggiore2016]. In order to describe a static configuration, the expectation value in Eq. is taken in the Boulware state $|0\rangle$, namely the state associated with the time-like Killing vector field $\xi$.
*Quantum vacuum polarization in 2 dimensions.–*Let us recall how to calculate $\langle\hat{T}_{ab}\rangle^{(2)}$ in the Boulware state [@Davies1977; @Davies1977b]. For spherically symmetric spacetimes, the 2-dimensional line element in Eq. can be always put in the form $$\text{d}s^2_{(2)}=-C(r)\text{d}t^2+\frac{\text{d}r^2}{1-2Gm(r)/r}.\label{eq:sph2}$$ A closed expression for the renormalized stress-energy tensor $\langle\hat{T}_{ab}\rangle^{(2)}$ is most easily obtained in null coordinates, and then it is transformed back to the initial $(t,r)$ coordinates. Equivalently, the renormalized stress-energy tensor can be written in an explicitly tensorial way [@Barcelo2011] as $\langle\hat{T}_{ab}\rangle^{(2)}=\left(R^{(2)}g_{ab}+A_{ab}-\frac{1}{2}g_{ab}A\right)/48\pi$, where $R^{(2)}$ is the Ricci scalar of the 2-dimensional metric with line element and $A_{ab}=4|\xi|^{-1}\nabla_a\nabla_b|\xi|$ with $\xi=\partial_t$ for the Boulware state, so that $|\xi|=\sqrt{C(r)}$. The result of these computations are the following components of the renormalized stress-energy tensor in the Boulware state: $$\begin{aligned}
24\pi\langle\hat{T}_{rr}\rangle^{(2)}=&-\frac{1}{4}\left(\frac{C'}{C}\right)^2,\qquad \langle\hat{T}_{tr}\rangle^{(2)}=\langle\hat{T}_{rt}\rangle^{(2)}=0,\nonumber\\
24\pi\langle\hat{T}_{tt}\rangle^{(2)}=&\left(1-\frac{2Gm}{r}\right)C''-C'\left(\frac{Gm}{r}\right)'\nonumber\\
&-\frac{3}{4}\left(1-\frac{2Gm}{r}\right)\frac{{C'}^2}{C}.\label{eq:2renc}\end{aligned}$$ From now on, $f'=\text{d}f/\text{d}r$ for any function $f$.
*Field equations and semiclassical TOV equation.–*The semiclassical source in Eq. is identically conserved, as it can be checked explicitly using its components in Eq. . The Bianchi identities imply then the conservation of the classical source $T_{\mu\nu}$, which translates into the usual continuity equation $$p'=-\frac{1}{2}(\rho+p)\frac{C'}{C}.\label{eq:conteq}$$ Taking into account that the Einstein tensor $G_{\mu\nu}$ of the geometry and both classical and semiclassical sources are diagonal, this amounts in principle to five differential equations: the diagonal components $(t,t)$, $(r,r)$, $(\theta,\theta)$, and $(\varphi,\varphi)$ of the semiclassical field equations , plus the continuity equation above. However, as in general relativity, only three of these five equations are independent. We choose to work with the continuity equation and the $(t,t)$ and $(r,r)$ components of the semiclassical field equations . These two are given, respectively, by $$\begin{aligned}
\frac{2Gm'}{r^2}=\,&8\pi G \rho+\frac{\ell_{\rm P}^2}{r^2}\left[\left(1-\frac{2Gm}{r}\right)\frac{C''}{C}\right.\nonumber\\
&-\frac{C'}{C}\left(\frac{Gm}{r}\right)'-\left.\frac{3}{4}\left(1-\frac{2Gm}{r}\right)\left(\frac{C'}{C}\right)^2\right],\label{eq:semitemp}\end{aligned}$$ (where we have defined the “renormalized” Planck length as $\ell_{\rm P}^2=\hbar G N/12\pi$), and $$\frac{C'}{rC}-\frac{2Gm}{r^2(r-2Gm)}=\frac{8\pi Gp}{\displaystyle1-2Gm/r}-\frac{\ell_{\rm P}^2}{4}\left(\frac{C'}{rC}\right)^2.\label{eq:radeq}$$ It is illuminating to combine the latter equation with the continuity equation to obtain the semiclassical TOV equation $$\begin{aligned}
p'\left(1-\frac{\ell_{\rm P}^2}{2r}\frac{p'}{\rho+p}\right)=-\frac{Gm}{r^2}\rho\frac{(1+p/\rho)(1+4\pi r^3 p/m)}{1-2Gm/r}.\label{eq:semitov}\end{aligned}$$ The right-hand side of this equation is the same as the right-hand side of the well-known classical equation, and it is arranged in a way that relativistic modifications beyond Newtonian gravity are easily identified. The left-hand side contains new (dimensionless) contributions describing semiclassical modifications due to vacuum polarization. Eq. is then an extension of the TOV equation that describes the hydrostatic equilibrium of (spherically symmetric) relativistic stars including semiclassical effects. The equations above can be written more compactly in terms of the enthalpy $h(r)$, defined through the relation $h'/h=p'/(\rho+p)$.
*Exact solutions.–*The semiclassical TOV equation is a second-order polynomial equation for the gradient of the pressure $p'$, which has two roots. One of these two roots leads to the classical TOV equation in the formal limit $\hbar\rightarrow0$. The other root describes solutions for which the semiclassical modifications in Eq. are dominant, and are therefore non-perturbative (i.e., cannot be obtained by a perturbative expansion around classical solutions). The new exact solutions we have found are associated with this second root, $$p'=\frac{r(\rho+p)}{\ell_{\rm P}^2}\left(1+\sqrt{\displaystyle1+\frac{2G\ell_{\rm P}^2}{r^3}\frac{m+4\pi r^3 p}{1-2Gm/r}}\right).\label{eq:posroot}$$ As in the classical theory, an additional constraint (that usually takes the form of an equation of state relating directly $p$ and $\rho$) is needed in order to guarantee that there are the same number of differential equations than independent unknown functions. The constraint that the quantity inside the square root in the previous equation equals a constant $\lambda^2$, with $\lambda>1$, permits us to solve exactly the previous equation and obtain $$\begin{aligned}
&m(r)=\frac{r}{2G}\left[1+\frac{f(r)}{\lambda-1}\right]\frac{1}{1+\ell_{\rm P}^2/r^2(\lambda^2-1)},\nonumber\\
&\rho(r)=\frac{(1+\lambda)f(r)}{8\pi G\ell_{\rm P}^2}-\frac{f'(r)}{8\pi G r},\nonumber\\
&p(r)=-\frac{(1+\lambda)f(r)}{8\pi G\ell_{\rm P}^2}.\label{eq:soll1}\end{aligned}$$ The details of this integration are not needed for our discussion here, as the assertion that these expressions solve Eq. and satisfy the mentioned constraint can be confirmed by direct substitution. On the other hand, using Eq. , Eq. can be now integrated to yield $$C(r)=C(R)e^{(1+\lambda)(R^2-r^2)/\ell_{\rm P}^2}.\label{eq:soll4}$$ The value of the integration constant $C(R)$ will be determined below. The function $f(r)$ in Eq. satisfies $f(R)=0$ in order to ensure the standard condition that the pressure vanishes at the surface of the star, $p(R)=0$.
Eqs. and solve identically Eqs. and , and therefore Eqs. and . The expressions we have obtained display an unknown function $f(r)$. If these expressions are inserted into the only equation that remains to be solved, namely Eq. , it results in a differential equation for $f(r)$ that can be integrated. Hence, we have reduced the problem (at least for this family of solutions) to the integration of a first-order ordinary differential equation that has a unique solution satisfying the boundary condition $f(R)=0$ for each value of $\lambda$. It is straightforward to obtain this differential equation, as well as analytical expressions for $f(r)$ for different values of $\lambda$. It is more useful for our purposes here to note that its leading order in $(\ell_{\rm P}/r)^2$ is given by $$\begin{aligned}
f(r)=\frac{\ell_{\rm P}^2(1+\mathscr{O}[(\ell_{\rm P}/r)^2])}{(1+\lambda)r^2}\left[1-\frac{r^2}{R^2}\,e^{(1+\lambda)(r^2-R^2)/\ell_{\rm P}^2}\right].\label{eq:ssol5}\end{aligned}$$ Eq. permits to obtain simple expressions at leading order in $(\ell_{\rm P}/r)^2$ for all the physical quantities, e.g., pressure and density: $$\begin{aligned}
&p(r)=\frac{-1+\mathscr{O}[(\ell_{\rm P}/r)^2]}{8\pi G r^2R^2}\left[R^2-r^2e^{(1+\lambda)(r^2-R^2)/\ell_{\rm P}^2}\right],\nonumber\\
&\rho(r)=\frac{1+\mathscr{O}[(\ell_{\rm P}/r)^2]}{8\pi G r^2R^2}\left[R^2+r^2e^{(1+\lambda)(r^2-R^2)/\ell_{\rm P}^2}\right].\label{eq:presdens}\end{aligned}$$ The three point-wise energy conditions [@Poisson2004], null, weak, and dominant are satisfied as $\rho(r)+p(r)>0$, but the strong energy condition would be violated by the perfect fluid alone. As the strong energy condition must apply to the complete source of the field equations (it is ultimately a statement about the Ricci tensor), that it is violated by the classical source only does not have any physical significance, as this very configuration of the perfect fluid cannot exist by itself: the semiclassical source must be included. The three energy conditions that are satisfied have a clear physical meaning in terms of the local properties of the perfect fluid, and guarantee that its properties are that of standard matter (positive density and causal fluxes along time-like and null trajectories). Note that the perfect fluid reaches densities that are many orders of magnitude greater than the typical average density of a neutron star.
*Validity of the approximation.–*The renormalized stress-energy tensor on the right-hand side of Eq. is the leading order in the $1/N$ expansion of quantum matter fields [@Hartle1981; @Flanagan1996; @Hu2001]. For the solutions described above this semiclassical contribution is comparable to the classical contribution $T_{\mu\nu}$. It is therefore necessary to show that higher orders are negligible, namely that these solutions are consistent with the truncation $\mathscr{H}_{\mu\nu}=0$ of the field equations . In static situations (to which this letter is restricted) it is enough to realize that (i) terms in the first category of $\mathscr{H}_{\mu\nu}$ are suppressed at least as $1/N$, and (ii) curvature is well below Planckian values almost everywhere for these solutions, which makes non-suppressed higher-derivative curvature terms irrelevant. The first consideration is a corollary of the $1/N$ expansion, but the second one is non-trivial, as not all possible solutions of Eq. have to verify it necessarily. Using the expressions above for the solutions described, the Ricci scalar and other curvature invariants can be calculated and shown to be small (in units of $\ell_{\rm P}^{-2}$) for $r/\ell_{\rm P}\gg 1$. Curvature invariants become $\mathscr{O}(\ell_{\rm P}^{-2})$ only in the region in which $r\leq\mathscr{O}(\ell_{\rm P})$, which for typical values of $R$ and $N$ is many (more than 30) orders of magnitude smaller than the radius $R$ of the star; the semiclassical (as well as the classical) field equations are no longer reliable in this region, which is reasonable as quantum gravity is expected to be necessary in order to describe these small distances. The expressions above display a curvature singularity in the $r\rightarrow0$ limit, which is however inside this region in which the semiclassical approximation (and therefore these expressions) cannot be trusted. Pending a detailed analysis of the quantization of the geometry of the exact solutions described in this letter, this issue can be approached from an effective perspective as it is done in the study of regular black holes (e.g., [@Hayward2005]), for instance, replacing $r^2\rightarrow r^2+\ell_{\rm P}^2$ and $R^2\rightarrow R^2+\ell_{\rm P}^2$ in Eq. . This prescription leads to regular geometries and guarantees that all physical quantities remain bounded.
*Regarding stability.–* Potential alternatives to black holes should be stable (or decay over long enough timescales). A first non-trivial check of the solutions studied here can be motivated by analogy with the analysis of secular stability in the classical theory (see the Chap. 9 in [@Friedman2013book] and references therein). The density at the center of the structure is given by $\rho_{\rm c}=\left.m'(r)/4\pi r^2\right|_{r=0}$, while from Eq. it follows that $M=m(R)=R/2G[1+\ell_{\rm P}^2/R^2(\lambda^2-1)]$ is a monotonically increasing function with $R$. Calculating the derivative of these functions along the parameter of the family $\lambda$, it is straightforward to show that $\text{d}M/\text{d}\rho_{\rm c}=4\pi\ell_{\rm P}^4\{1+\mathscr{O}[(\ell_{\rm P}/R)^2]\}/3R(\lambda^2-1)^2>0$ for all the solutions in the family, so that there are no turning points. The occurrence of turning points would have pointed to the unstable nature of the bulk by itself. The word “bulk” is used here in order to emphasize that, as described just below, these exact solutions display non-trivial surface properties. Hence stability cannot be concluded without taking into account these additional features of the surface and their interplay with the bulk dynamics. This analysis, which is out of the scope of this letter, would follow similar conceptual steps as in the gravastar proposal [@Visser2003; @Pani2009]; note however that the junction conditions are different than in general relativity. An additional subtlety is associated with the regularization of the core with size roughly of $\ell_{\rm P}$, which implies that suitable boundary conditions must be imposed on perturbations at the boundary of the core.
*Junction conditions at the surface.–*As in general relativity, the exact interior solution describing the geometry inside the perfect fluid has to be glued with the exterior geometry outside the fluid at the surface in which the pressure vanishes, $p(R)=0$ (see Fig. \[fig:fig1\]). The surface is located at $r=R=2GM+\gamma\ell_{\rm P}^2/2GM>2GM$ with a minimum value $\gamma=\mathscr{O}(1)$. The numerical integration of the semiclassical field equations for the Boulware state outside the gravitational radius is well understood and has been carried out for instance in [@Fabbri2005; @Fabbri2005b; @Ho2017], resulting that the exterior geometry is approximated well by the Schwarzschild geometry but close to the Schwarzschild radius. Using these results permits us to fix the two unspecified constants as $C(R)=1-2GM/R+\mathscr{O}[(\ell_{\rm P}/R)^2]>0$ and $\lambda-1=\mathscr{O}(1)>0$, where the particular value of the latter is tied up to the value of $\gamma$.
The root of Eq. that must be used in order to obtain the correct classical limit far away from the relativistic star verifies $\lim_{r\rightarrow R^+}\left.C'/C\right|_{+}=-2 R(1-\lambda)/\ell_{\rm P}^2$, while for the interior solution we have from Eqs. and that $\lim_{r\rightarrow R^-}\left.C'/C\right|_{-}=-2R(1+\lambda)/\ell_{\rm P}^2$. This implies that the surface of the star has a non-zero surface tension; i.e., it behaves similarly to a soap bubble. Writing $C'/C=\left.C'/C\right|_{+}\theta(r-R)+\left.C'/C\right|_{-}\theta(R-r)$ with the usual convention $\theta(0)=1/2$, it is possible to show that the semiclassical field equations are well-defined in the distributional sense (following a similar analysis as in general relativity, e.g., [@Poisson2004]). The corresponding distributional source at $r=R$ has a stress-energy tensor $S_{\mu\nu}\delta(r-R)\sqrt{1-2M/R}$ [@Mansouri1996]. From Eq. it follows that the surface energy density is $S_{00}\sqrt{1-2M/R}=\Sigma=-\lambda\ell_{\rm P}^2/2\pi G R^3[(\lambda^2-1)+\ell_{\rm P}^2/R^2]$. This represents a negative but small surface density with respect to the value of the bulk density at the surface, $|\Sigma/R\rho(R)\sqrt{1-2M/R}|=\mathscr{O}\left(\ell_{\rm P}/R\right)\ll1$. On the other hand, the angular equations, that we have not discussed explicitly above (as these follow from the conservation equation), contain terms proportional to $C''/C$. The associated surface tension is $S_{\varphi\varphi}\sqrt{1-2M/R}=\sin^2\theta S_{\theta\theta}\sqrt{1-2M/R}=\sin^2\theta\,\Pi$, with $\Pi=\lambda/4\pi GR[(\lambda^2-1)+\ell_{\rm P}^2/R^2]$. The radial pressure cannot have a distributional part, as all the remaining quantities in Eq. are continuous. Hence an alternative way to obtain $\Pi$, and which needs only the independent equations used for the analysis in the bulk, is extending Eq. to include an anisotropic pressure $p_{\theta}=p+\Pi\delta(r-R)$ and density $\rho+\Sigma\delta(r-R)$, both containing a distributional part. This extension is given (see [@Lemaitre1933; @Cattoen2005] for instance) by $p'=-[\rho+\Sigma\delta(r-R)+p]C'/2C+\delta(r-R)2\Pi/r$. Let us stress that these surface properties arise naturally as a consequence of matching the interior geometry with the exterior geometry at the surface in which the pressure of the relativistic star vanishes.
*Summary.–*The effects of quantum vacuum polarization, which are widely expected to have a physical reality but to be negligible in most situations, are important enough to allow new static configurations of relativistic stars. We have shown this explicitly by constructing and solving a closed system of equations of stellar equilibrium (and in particular a generalized TOV equation) that includes semiclassical effects as described by quantum field theory in curved spacetimes and semiclassical gravity. The classical equations of stellar equilibrium are recovered at low density, but at high density, semiclassical effects cannot be disregarded. The analogy with the physics of neutron stars is appealing, with vacuum polarization providing a new kind of pressure of quantum-mechanical origin (analogous to the degeneracy pressure).
The exact solutions we have found describe relativistic stars with very specific properties. Although a detailed analysis will be presented elsewhere, let us give a brief summary here. Part of these properties correspond to a particular kind of horizonless ultra-compact object introduced [@Barcelo2007; @Barcelo2009] and studied [@Barcelo2010; @Barcelo2014e; @Barcelo2014; @Barcelo2015; @Barcelo2016] during the last decade: black stars. The three main proposed characteristics of black stars, all of them satisfied by the exact solutions found here, are [@Barcelo2007; @Barcelo2010] (i) an interior made of extremely dense matter supported by quantum vacuum polarization, (ii) a matter density profile proportional to the inverse of the square of the distance to the center, and (iii) a compactness $2Gm(r)/r$ arbitrarily close to (but slightly smaller than) unity for any value of $r$ (not only at the surface). On the other hand, other properties are strongly reminiscent of the gravastar proposal [@Mazur2004; @Mottola2010; @Mazur2015] (see also the generalizations in [@Visser2003; @Lobo2005; @Danielsson2017]). These are (iv) an equation of state of the form $p(r)\simeq-\rho(r)$ —see [@Gliner1966; @Sakharov1966] for arguments connecting this equation of state with hypothetical new states of matter at high densities— (let us note that in the original gravastar model there is no classical matter in the interior, which is therefore optically thin, leading to specific observational imprints due to the defocusing of light rays [@Mazur2015] that should not be present in the case analyzed here), and (v) non-zero surface tension and energy density. That all these properties (i)-(v) arise naturally and together is remarkable. These results provide also an important theoretical basis for the study of phenomenological implications of horizonless ultra-compact alternatives to black holes, such as gravitational wave echoes [@Cardoso2016a; @Cardoso2016; @Abedi2016; @Barcelo2017; @Price2017; @Nakano2017; @Volkel2017; @Mark2017; @Maselli2017].
![The complete geometry of static ultra-compact stars in semiclassical gravity. The exterior geometry (white) is a perturbative solution of the semiclassical Einstein field equations, while the interior geometry (dark gray) is a non-perturbative solution. Interior and exterior geometries are glued at the surface of the star, the position of which is determined by the condition that the pressure of the perfect fluid vanishes. The difference between $R$ and $2GM$ has been vastly exaggerated for illustrative purposes.[]{data-label="fig:fig1"}](black_star.png){width="40.00000%"}
Financial support was provided by the Claude Leon Foundation of South Africa. I would like to thank Carlos Barcel[ó]{}, Luis J. Garay, Vitor Cardoso and Paolo Pani for useful discussions and for their critical reading of a previous version of the manuscript. I also appreciate the positive criticism received from anonymous referees, which has facilitated improving the manuscript.
|
---
abstract: 'The relatively small family of ultra-compact X-ray binary systems is of great interest for many areas of astrophysics. We report on a detailed X-ray spectral study of the persistent neutron star low mass X-ray binary [1RXS J170854.4-321857]{}. We analysed two observations obtained in late 2004 and early 2005 when, in agreement with previous studies, the system displayed an X-ray luminosity (0.5–10 keV) of $\sim1\times10^{36}\lum$. The spectrum can be described by a Comptonized emission component with $\Gamma\sim$1.9 and a distribution of seed photons with a temperature of $\sim0.23$ keV. A prominent residual feature is present at soft energies, which is reproduced by the absorption model if over-abundances of Ne and Fe are allowed. We discuss how similar observables, that might be attributed to the peculiar (non-solar) composition of the plasma donated by the companion star, are a common feature in confirmed and candidate ultra-compact systems. Although this interpretation is still under debate, we conclude that the detection of these features along with the persistent nature of the source at such low luminosity and the intermediate-long burst that it displayed in the past confirms [1RXS J170854.4-321857]{}as a solid ultra-compact X-ray binary candidate.'
author:
- |
M. Armas Padilla,$^{1,2}$and E. López-Navas,$^{1,2,3}$\
$^{1}$Instituto de Astrofísica de Canarias (IAC), Vía Láctea s/n, La Laguna 38205, S/C de Tenerife, Spain\
$^{2}$Departamento de Astrofísica, Universidad de La Laguna, La Laguna, E-38205, S/C de Tenerife, Spain\
$^{3}$Instituto de Física y Astronom’ía, Facultad de Ciencias, Universidad de Valparaíso, Gran Bretana N 1111, Playa Ancha, Valparaíso, Chile
bibliography:
- '4U1705-32\_MAP.bib'
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: 'On the ultra-compact nature of the neutron star system [1RXS J170854.4-321857]{}: insights from X-ray spectroscopy'
---
\[firstpage\]
accretion, accretion discs – stars: individuals: [1RXS J170854.4-321857]{} – stars: neutron – X-rays: binaries
[ccccccc]{} ID & Date & Exposure &EPIC Camera & Net exposure & Net count rate\
&(yyyy-mm-dd) &\[ks\] & &\[ks\] &\[$\cnts$\]\
0206990201 &2004-10-01 & 13.4 &MOS1 &11.6 & 1.05$\pm$0.01\
& & &MOS2 &12. 1 & 0.98$\pm$0.01\
\
0206991101 &2005-02-19 & 12.9 &MOS1 &10.2 &1.5$\pm$0.01\
& & &MOS2 &10.7 &2.1$\pm$0.01\
& & &PN &7.6 &4.3$\pm$0.03\
\[tab:log\]
Introduction {#sec:intr}
============
The vast majority of the known galactic population of stellar-mass black holes (BHs) and a significant fraction of the neutron stars (NS) are found in low mass X-ray binaries (LMXBs). These stellar systems have sub–solar companion stars that transfer material to the compact object via Roche lobe overflow. Among LMXBs, the population of ultra-compact X-ray binaries (UCXBs), comprised by systems with orbital periods shorter than 80 min, is of particular interest. These short periods imply small Roche lobes, in which only degenerated (hydrogen poor) donor stars can fit. UCXBs are unique laboratories to study accretion processes in hydrogen deficient environments as well as some of the fundamental stages of binary evolution [e.g., common-envelope phase, @Nelemans2009; @Nelemans2010; @Tauris2018]. Last but not least, UCXBs will be primary sources for gravitational waves studies at low-frequencies by the forthcoming LISA mission [@Nelemans2018; @Tauris2018].
Although the predicted number of UCXBs in our galaxy is $\sim10^{6}$ [@Nelemans2009], at present only 14 systems have been confirmed (i.e. with reported orbital periods; see e.g., @Strohmayer2018; @Heinke2013, and references therein). One of the reasons of this scarcity is that measuring the orbital periods is not always straightforward. Standard techniques include the search for periodic eclipses/modulations for sources seen at high inclination or Doppler-delayed pulses in the case of pulsar acretors. In the optical regime, modulations due to accretion disc super-humps or triggered by irradiation of the companion star also enable to infer orbital solutions. When direct measurements are not feasible, we need to use indirect evidences to identify UCXB candidates.
@IntZand2007 proposed a method based on the disc instability model (DIM; see @Lasota2001 for a review) that identifies systems persistently accreting at very low luminosities as potential UCXBs. Indeed, a fraction of the now confirmed UCXB population were initially candidates proposed by this method. In addition, optical spectroscopic studies have been used to test the ultra-compact nature of several systems. This is done by investigating the absence/presence of emission/absorption features in the spectra, which provide hints on the chemical composition of the accreted material, and thus, on the nature of the companion star [@Nelemans2004; @Nelemans2006; @IntZand2008; @Santisteban2017]. Likewise, X-ray spectroscopy has been used to search for UCXB candidates. This is based on the presence of a common feature at $\sim$0.7–1 keV, attributed to an enhanced Ne/O ratio in the plasma donated by the (white dwarf) companion star [@Juett2001; @Juett2003b]. Despite this technique reported successfully confirmed candidates, it has to be taken with caution as the Ne/O ratio was observed to vary with luminosity in some sources [@Juett2005 see section \[sec:Disc\]]
[1RXS J170854.4-321857]{} is a NS LMXB that has been detected by several X-ray missions since the early 70’s [@IntZand2005 and references therein]. The source has shown a persistently low unabsorbed flux of a few times $10^{-11}~\flux$ [@IntZand2004; @IntZand2005], while only near-infrared upper-limits of $\gtrsim$20 mags (J, K and I bands; @Revnivtsev2013) have been reported. An intermediate-long type I X-ray burst ($\sim$10 min) showed mild photospheric radius expansion, from which – assuming either a pure hydrogen or helium atmosphere – a distance of 13$\pm$2 kpc was derived [@IntZand2005]. This translates into a persistent luminosity of (0.5–10 keV)$\sim1\times10^{36}~\lum$. This persistently low , together with the aforementioned detection of the 10-min burst, which occur predominantly on UCXBs (albeit not exclusively; see e.g., @Galloway2017 [@IntZand2019]) suggested that the system is an UCXB [@IntZand2005; @IntZand2007]. In this work, we present a detailed spectral study that provides additional support to the ultra-compact nature of [1RXS J170854.4-321857]{}.
Observations and data reduction {#sec:obs}
===============================
The observatory [@Jansen2001] pointed at [1RXS J170854.4-321857]{} on 2004 January 1 and 2005 February 19 for $\sim$12 ksec, respectively. Only the two Metal Oxide Semi-conductor cameras [MOS1 and MOS2 @Turner2001] of the European Photon Imaging Camera (EPIC) were active during the first observation, while the PN camera [@Struder2001] was also available for the second pointing (see Table \[tab:log\] for the observing log). The detectors were operated in imaging (full-frame window) mode with the medium optical blocking filter during both observations. We extract calibrated events and scientific products using the Science Analysis Software (<span style="font-variant:small-caps;">sas</span>, v.16.0.0). We excluded episodes of background flaring by removing data with high-energy count rates $>$ 0.3 and $>$ 0.5 $\cnts$ for MOS and PN cameras, respectively. We extracted the source events using a circular region of 75 arcsec radius excising a radius of 15 arcsec (MOS1 and MOS2 for the 2004 observation), 10 arcsec (MOS1 and PN for the 2005 observation) and 5 arcsec (MOS2 of the 2005 observation) in order to mitigate pile-up effects. Background events were extracted using a circular region of 100 arcsec radius placed in a source-free region of the CCDs. We selected events with pixel pattern below 5 and 13 for PN and MOS detectors, respectively. We extracted light curves and spectra and generated response matrix files (RMFs) and ancillary response files (ARFs) following the standard analysis threads[^1]. We rebinned the spectrum in order to include a minimum of 25 counts in every spectral channel and avoiding to oversample the full width at half-maximum of the energy resolution by a factor larger than 3. We used <span style="font-variant:small-caps;">xspec</span> (v.12.9.1; @Arnaud1996) to analyse the spectra.
Finally, we also inspected and extracted the data from the Reflection Grating Spectrometers [RGS; @Herder2001], but we did not include them in the in our analysis due to their low statistics.
Analysis and results {#sec:res}
====================
We investigated the light curves in order to search for features revealing the orbital period, such as eclipses or absorption dips [@White1995]. We created a 100s-bin light-curve using the PN data, from which we derive a upper limit on the mean aperiodic varibility of <5%, a constraint that it is limited by the quality of our data. A visual inspection to the (2$\times$3 h) light-curves do not show any clear high inclination feature. In order to test this further, we performed a flux vs hardness diagram (not shown; see e.g. fig 5 in @Kuulkers2013), which does not reveal any clear trend, as it would be expected for the case of variable absorption (e.g. dips). Thus, we conclude that either [1RXS J170854.4-321857]{} has an orbital inclination $i\lesssim65\degr$ [@White1985; @Cantrell2010] or that the orbital period is longer than 3.5 h. The latter scenario, given the X-ray characteristics of the source, seems unlikely.
$\phantom{!t}$
-- --
-- --
We started our spectral study by analysing the 2005–observation, which includes the three EPIC cameras and hence has better statistics. We simultaneously fitted the 0.5–10 keV PN, MOS1 and MOS2 spectra with the parameters tied between the tree detectors. In order to account for cross-calibration uncertainties between the cameras, we included a constant factor (CONSTANT) in our models. We fix it to 1 for the PN spectrum and leave it free to vary for the MOS spectra. In our first attempt, we used a single thermally Comptonized continuum model (NTHCOMP in <span style="font-variant:small-caps;">xspec</span>; @Zdziarski1996 [@Zycki1999]) modified by photoelectric absorption which we modelled using the Tuebingen-Boulder Interstellar Medium absorption model (TBABS in <span style="font-variant:small-caps;">xspec</span>) with cross-sections of @Verner1996 and abundances of @Wilms2000. The corona electron temperature parameter ($kT_{\rm e}$) typically adopts values above 10 keV in the low-hard state of LMXBs, which is beyond our 0.5–10 keV spectral coverage. Therefore, we fixed $kT_{\rm e}$ to 25 keV, which is the typical value displayed by NS systems at this low-luminous states [@Burke2017]. We note that variations of this value do not have an impact on the results (within errors). This simple absorbed Comptonization model was not able to reproduce our data. The fit residuals exhibited a strong feature around $\sim$1 keV (see left plot on Fig. \[fig:spectra\], bottom panel) with of 661 for 416 degrees of freedom (dof) (p-value[^2] of $2\times10^{-13}$). We added a thermal component (DISKBB or BBODYRAD) assuming that the soft residuals might be produced by emission from the accretion disc or NS surface/boundary layer, as commonly observed in NSs accreting a low luminosities [e.g. @ArmasPadilla2017; @ArmasPadilla2018]. However, the extra thermal component does not account for the soft residuals and an evident structure remains below 2 keV.
Soft excesses below $\sim$1 keV have been observed in spectra of highly obscured X-ray binaries obtained in EPIC-PN Timing mode (see XMM-SOC-CAL-TN-0083 [^3]). In some cases this feature has been attributed to residual uncertainties in the redistribution calibration [@Hiemstra2011]. However, it is unlikely that this is the cause of our residuals since (i) our spectra are taken in Imaging mode, (ii) the source is only mildly absorbed, and (iii) the structure is present in all three EPIC detectors. On the other hand, similar features have been observed in several (candidate) UCXBs [e.g. @Juett2001; @Juett2003b; @Farinelli2003; @IntZand2008]. These are proposed to result from over-abundances in the absorbing material, generally enhanced O, Ne and Fe, which is probably intrinsic to the sources. Following the same procedure reported in the literature, we replaced the TBABS model by the absorption model TBNEW, which allows the abundances to vary [@IntZand2008; @Madej2014; @VandenEijnden2018]. We started again with a simple absorbed Comptonization model (i.e., TBNEW\*NTHCOMP). We kept fixed all the TBnew parameters at their default values, except the equivalent hydrogen column density (). We tried several different fits by allowing both individual elements and combination of elements (e.g., C, O, Fe, Ne, etc) to vary. We found that soft residuals were present to some extend unless the abundances of Ne and Fe were allowed to vary freely. Indeed, this combination provided the best fit, which significantly improved previous attempts ($\Delta$=341 for 3 dof with respect to the same model with Solar abundances). Although we are aware that the fit is still poor from a purely statistical point of view (p-value is still $<$0.05), we can not decrease the by adding extra continuum components (e.g., DISKBB, BBODYRAD) to our model. Moreover, there is not any other feature in our residuals clearly suggesting that a component is missed. Therefore, we concluded that the impossibility of improvement might be caused by remaining uncertainties in the cross-calibration between the different detectors (@Kirsch2004; XMM-SOC-CAL-TN-0052). We obtained an of (3.4$\pm$0.4)$\times 10^{21}$(consistent with the reported value by @IntZand2005 using data) and over-abundances of Ne ($A_{\rm Ne}$=5.4$\pm$0.4) and Fe ($A_{\rm Fe}$=2.5$\pm$0.7). The Comptonization asymptotic power-law photon index ($\Gamma$) is $1.91\pm0.02$ and the temperature of the up-scattered seed photons ($kT_{\rm seed}$) assuming black body geometry is 0.23$\pm$0.06 keV (consistent values are obtained assuming a disc geometry). The inferred 0.5–10 keV unabsorbed flux of (6.3$\pm$0.2)$\times10^{-11}~\flux$ is implies a luminosity of $\sim1.7\times10^{36}\lum$ assuming a distance of 13 kpc [@IntZand2005]. We report best-fit results in Table \[tab:res\] (see also left panel in Fig.\[fig:spectra\]). Uncertainties are given at 90 per cent confidence level.
For the 2004 observation, that only includes MOS1 and MOS2 data, we fixed the constant (cross-calibration) factor to 1 for the former and leave it free to vary for MOS2. We fixed to the value obtained in the best fit of our higher signal–to–noise 2005 observation (=3.4$\times 10^{21}$). As for the 2005 data, a simple absorbed Comptonization model was not able to reproduce the data. This could not be solved by adding soft thermal components ($\cong$319 for 239 dof; p-value=$4\times10^{-4}$). Although it is less evident than for the 2005 observation, some residuals below 1 keV are also present (see bottom–right panel in Fig.\[fig:spectra\]). Thus, we repeated the fit using TBNEW\*NTHCOMP with free Ne and Fe abundances. This produced an acceptable fit, with = 275 for 243 dof (p-value$\cong$0.07). The fit and corresponding residuals are shown in Fig.\[fig:spectra\] (right plot). The resulting spectral values are consistent with those obtained from the 2005 observation (Table \[tab:res\]), albeit the system was slightly brighter in 2004.
Component 2004 2005
-------------------------------------- ---------------- ---------------
$C_{\rm PN}$ – 1 (fix)
$C_{\rm MOS1}$ 1 (fix) 1.04$\pm$0.02
$C_{\rm MOS2}$ 1.05$\pm$0.02 0.94$\pm$0.01
($\times 10^{22}$ ) 0.34 (fix) 0.34$\pm$0.4
$A_{\rm Ne}$ 4.1$\pm$1 5.4$\pm$0.4
$A_{\rm Fe}$ 3.5$\pm$1 2.5$\pm$0.7
$\Gamma$ 1.98$\pm$0.04 1.91$\pm$0.02
$kT_{\rm e}$ (keV) 25 (fix) 25 (fix)
$kT_{\rm seed}$ (keV) 0.21$\pm$0.02 0.23$\pm$0.03
$N_{\rm nthcomp}$ ($\times 10^{-2}$) 1.6$\pm$0.1 1.06$\pm$0.1
(dof) 283(249) 522 (414)
($\times10^{-11} \flux$) 8.5$\pm$0.2 6.3$\pm$0.2
$^{\rm a}$ ($\times10^{36}\lum$) 1.72 $\pm$0.04 1.27$\pm$0.04
: Fitting results for the 2004 and 2005 data using the TBNEW\*NTCHOMP model. Uncertainties are expressed at 90 per cent confidence level.
[X-ray luminosity assuming a distance of 13 kpc.]{}
\[tab:res\]
[ l c c c c c c c ]{} Source &Period & P/T & & Interm-long& & (Energy band)\
& (minutes) & & X-ray & Optical &burst &$\times10^{21}$& $\lum$ (keV)\
2S 0918-549 & 17.4 & P & Ne/O & He? & IB$^{a}$ & 3.0 &$\sim3.5\times10^{35}$ (2–10) $^{b}$\
4U 1543-624 &18.2 & P & Ne/O & C/O & – &3.5 &$\sim2-4\times10^{36}$ (2–10) $^{b}$\
4U 1850-087 & 20.6 & P & Ne/O &? & IB$^{a}$ & 3.9 &$\sim1\times10^{36}$ (0.5–10) $^{c,d}$\
IGR J17062-6143 & 37.97$^{e}$ & P & O$^{f}$ & lack $^{g}$ & IB$^{a}$ & 2.4$^{h}$ & $\sim2.9\times10^{35}$ (0.3–79) $^{h}$\
4U 1626-67 &42 & P & Ne/O, C, O, Ne & C/O & – &1.4 & $\sim1\times10^{36}$ (0.5–10) $^{i}$\
\
4U 0614+091 & 51? & P & Ne/O, O & C/O & IB$^{j}$ & 3.0 & $\sim1\times10^{36}$ (0.5–10) $^{c}$\
1A 1246-588 & ? & P & Ne$^{k}$ & Featureless$^{k}$ & IB$^{a}$ &2.5$^{k}$ & $\sim6\times10^{35}$ (0.3–10) $^{k}$\
**[1RXS J170854.4-321857]{}** & ? & P & Ne, Fe & ? & IB$^{l}$ & 3.4 & $\sim1\times10^{36}$ (0.5–10)\
[ Based on data presented in @Paradijs1994, @Nelemans2006, @VanHaaften2012, @Heinke2013.]{}
[References: $^{a}$@IntZand2019, $^{b}$@Juett2003b, $^{c}$@Juett2001, $^{d}$@Juett2005, $^{e}$@Strohmayer2018, $^{f}$@VandenEijnden2018, $^{g}$@Santisteban2017,$^{h}$@Degenaar2017, $^{i}$@Schulz2001, $^{j}$@Kuulkers2010, $^{k}$@IntZand2008, $^{l}$@IntZand2005, ]{}
\[tab:UCXB\]
Discussion {#sec:Disc}
==========
[1RXS J170854.4-321857]{} is a NS LMXB that has been detected without exception at $\sim$0.01 by several missions over the past decades. Within the framework of the disc instability model [@Lasota2001; @Coriat2012] this persistent nature at such low luminosity implies a short orbital period, possibly within the UCXB regime [@IntZand2005]. To investigate this further, we have carried out a detailed analysis of two archival observations. The 0.5–10 keV X-ray luminosity during these observations was $\sim1\times10^{36}\lum$, which is consistent with previous measurements reinforcing the persistent nature of the system [e.g., @Forman1978; @Markert1979; @Revnivtsev2004; @IntZand2005]. Our spectra are well described by a thermally Comptonized model with $\Gamma\cong1.9$ and a distribution of seed photons characterised by $kT_{\rm seed}\cong0.2$ keV arising from either the accretion disc or NS surface/boundary layer (possibly from both regions; see @ArmasPadilla2018). We do not detect direct emission from the accretion disc or NS surface as no softer thermal component was required in our analysis. NS systems accreting at luminosities $<10^{35}\lum$ usually show a soft thermal component, generally attributed to low-level accretion on to the neutron star surface [@Zampieri1995; @ArmasPadilla2013c; @Bahramian2014]. This typically contributes at the $\sim$30–50 per cent level [e.g., @Degenaar2013; @ArmasPadilla2013b; @Arnason2015; @ArmasPadilla2018]. However, at luminosities above $\sim 10^{35}\lum$ this contribution becomes lower or, in some cases, totally vanishes [@ArmasPadilla2013b; @Allen2015; @Wijnands2015]. The non-detection of the thermal component together with the relatively soft photons index ($\Gamma \sim 1.9$) concurs with the latter scenario.
Interestingly, the spectra showed a broad residual structure around $\sim$1 keV, which we were able to account for by allowing relative abundances to vary in the absorption model. We found =(3.4$\pm$0.4)$\times 10^{21}$ with enhanced Ne and Fe abundances. This is consistent with the value obtained by @IntZand2005, albeit their abundances are in agreement with the standard interstellar value. The reason for this difference might be related to the significantly worse statistics of their spectra (with only 58 dof). We obtain Ne and Fe relative abundances of $A_{\rm Ne}$=5.4$\pm$0.4 and $A_{\rm Fe}$=2.5$\pm$0.7, while a maximum value of $\sim$1.4 have been measured for the interstellar medium towards LMXBs [@Pinto2013]. Hence, the additional absorption required to fit both the 2004 and 2005 data is unlikely to have an interstellar origin, being probably intrinsic to the system. Although these over-abundances are rather high, similar values have been reported for some other UCXBs [e.g, @IntZand2008; @Madej2011]. Nevertheless, we note that due to the relatively low spectral resolution of our data and the systematics likely involved, to derive the actual abundances and species producing the soft X-ray feature presented here is beyond the scope of the paper.
A number of confirmed (and candidate) UCXBs have exhibited similar phenomenology to that reported here, which, in most cases, has been interpreted as due to over-abundances in the absorber. This generally involves excesses in the K-edges of neutral O and Ne and the L-edge of neutral Fe, which origin have been suggested to be intrinsic to the binaries. @Juett2001 first proposed the ultra-compact nature of several systems based on the detection of enhanced Ne/O ratios using X-ray spectra [see also, @Juett2003b]. This feature had been originally found in the confirmed UCXBs 4U 1626-67 and 4U 1850-087. In light of these findings, it was suggested that systems showing this peculiarity harbour C–O or O–Ne–Mg white dwarfs as companions. Also, a white dwarf donor was proposed for IGR J17062-6143, the last confirmed UCXB to date [@Strohmayer2018]. It showed a residual structure at $\sim$1 keV, that was suggested to arise from an over-abundance of O – likely related with circumbinary material – while Ne edge abundances were consistent with the interstellar medium [@VandenEijnden2018; @Degenaar2017]. The presence of strong emission lines of highly ionised species of Ne and O in the high resolution X-ray spectroscopy of 4U 1626-67 also suggested a C–O rich white dwarf donor [@Schulz2001; @Krauss2007]. Optical spectroscopic studies of some of these sources support the above conclusions. The detection of C and O emission lines evidences the presence of metal–rich material in the accretion disc, consistent with C–O white dwarf donors. [@Nelemans2004; @Werner2006; @Nelemans2006]. Additionally, the lack of strong H and He emission lines, typically the strongest features in the optical spectra of LMXBs [@Charles2006; @MataSanchez2018], supports the ultra-compact nature of some of the systems that have shown the peculiar X-ray features [e.g.; @Nelemans2006; @IntZand2008; @Santisteban2017]. In this context, it is important to emphasise that the Ne/O ratio was questioned as UCXB marker due to the hydrogen and helium lines found in the optical spectrum of the NS system 4U 1556-60 [@Nelemans2006; @Nelemans2010]. However, 4U 1556-60 has not shown any of the X-ray features classically associated with uncommon abundances [@Farinelli2003], and therefore, it was not (and has never been as far as we know) an UCXB candidate.
High resolution X-ray spectra of some UCXBs have shown, in addition to over-abundances in the absorber, the presence of a broad Ly$_{\alpha}$ emission line [e.g. @Madej2010; @Schulz2010; @Madej2011]. This, together with the Fe K$_{\alpha}$ line, is predicted to be the most prominent fluorescent line of the reflection spectrum if X-rays are reprocessed in a oxygen-rich accretion disc. Indeed, reflection models have been modified to account for the non-solar composition of the accretion disc in order to reproduce the reflection spectra of UCXBs [@Koliopanos2013; @Madej2014]. Moreover, @Madej2014 found that better results were obtained if the ionization structure of the illuminated disc was also taken in account. Likewise, it was suggested that ionization effects might be responsible for the epoch-to-epoch variations of the Ne/O ratio found in some systems [@Juett2003b; @Juett2005; @IntZand2008]. In this scenario, changes in the continuum spectral properties would modify the ionization state of the different (absorber related) species in the accretion disc. Thus, the measured Ne/O ratio would not reflect the donor composition. However, this does not explain why the $\sim$1 keV structure is so common in UCXBs as compared with the whole LMXB population. In Table \[tab:UCXB\] we list candidate and confirmed UCXBs that, to the best of our knowledge, have shown the soft X-ray features compatible with peculiar Ne and O abundances. Interestingly, these systems have a persistent nature, $<5\times10^{21}$and were accreting at $\lesssim$0.01 at the time they showed the soft feature.
Finally, numerous UCXBs have displayed intermediate-long type I X-ray burst, albeit these are not exclusively observed in UCXBs [e.g. @Falanga2008]. These thermonuclear explosions are thought to be associated with deep He ignitions from material accreted at very low rates ($\lesssim0.01$ ), suggesting He-rich donors (@IntZand2005b [@Cumming2006; @Falanga2008; @Kuulkers2010; @Galloway2017; @IntZand2019]). As a matter of fact, @IntZand2005b proposed that the anomalous Ne/O ratios could result from a reduction of the O abundance due to CNO processes in helium white dwarfs. This scenario might apply to [1RXS J170854.4-321857]{}. In this regard, the detection of a $\sim$10 min X-ray burst, the persistently low accretion rate ($\sim$0.01 ) and the apparent abundances anomalies that we have reported here agree with the above picture. Higher X-ray spectral resolution data at high signal-to-noise are required in order to further investigate the nature of [1RXS J170854.4-321857]{} in particular, and the physical processes behind the abundance-anomalies related features commonly observed in UCXBs.
Acknowledgements {#acknowledgements .unnumbered}
================
MAP acknowledge support by the Spanish MINECO under grant AYA2017-83216-P. MAP’s research is funded under the Juan de la Cierva Fellowship Programme (IJCI-2016-30867). ELN acknowledge the IAC Summer Fellowship, during which part of the research leading to this results were carried out. is an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA.
\[lastpage\]
[^1]: <https://www.cosmos.esa.int/web/xmm-newton/sas-threads>
[^2]: The probability value (p-value) represents the probability that the deviations between the data and the model are due to chance alone. In general, a model can be rejected when the p-value is smaller than 0.05.
[^3]: [ http://xmm.vilspa.esa.es/docs/documents/CAL-TN-0083.pdf]( http://xmm.vilspa.esa.es/docs/documents/CAL-TN-0083.pdf)
|
---
abstract: 'We report the measurement of $\nu$-$e$ elastic scattering from solar neutrinos with 3MeV energy threshold by the Borexino detector in Gran Sasso (Italy). The rate of solar neutrino-induced electron scattering events above this energy in Borexino is $0.217\pm 0.038 (stat)\pm 0.008 (syst)$ cpd/100t, which corresponds to $\Phi^{\rm ES}_{\rm ^8B}$ = [2.4 $\pm$ 0.4$\pm$ 0.1]{}$\times$10$^6$ cm$^{-2}$ s$^{-1}$, in good agreement with measurements from SNO and SuperKamiokaNDE. Assuming the neutrino flux predicted by the high metallicity Standard Solar Model, the average survival probability above 3 MeV is measured to be 0.29$\pm$0.10. The survival probabilities for and neutrinos as measured by Borexino differ by 1.9 $\sigma$. These results are consistent with the prediction of the MSW-LMA solution of a transition in the solar survival probability between the low energy vacuum-driven and the high-energy matter-enhanced solar neutrino oscillation regimes.'
author:
- 'G. Bellini'
- 'J. Benziger'
- 'S. Bonetti'
- 'M. Buizza Avanzini'
- 'B. Caccianiga'
- 'L. Cadonati'
- 'F. Calaprice'
- 'C. Carraro'
- 'A. Chavarria'
- 'A. Chepurnov'
- 'F. Dalnoki-Veress'
- 'D. D’Angelo'
- 'S. Davini'
- 'H. de Kerret'
- 'A. Derbin'
- 'A. Etenko'
- 'K. Fomenko'
- 'D. Franco'
- 'C. Galbiati'
- 'S. Gazzana'
- 'C. Ghiano'
- 'M. Giammarchi'
- 'M. Goeger-Neff'
- 'A. Goretti'
- 'E. Guardincerri'
- 'S. Hardy'
- Aldo Ianni
- Andrea Ianni
- 'M. Joyce'
- 'G. Korga'
- 'D. Kryn'
- 'M. Laubenstein'
- 'M. Leung'
- 'T. Lewke'
- 'E. Litvinovich'
- 'B. Loer'
- 'P. Lombardi'
- 'L. Ludhova'
- 'I. Machulin'
- 'S. Manecki'
- 'W. Maneschg'
- 'G. Manuzio'
- 'Q. Meindl'
- 'E. Meroni'
- 'L. Miramonti'
- 'M. Misiaszek'
- 'D. Montanari'
- 'V. Muratova'
- 'L. Oberauer'
- 'M. Obolensky'
- 'F. Ortica'
- 'M. Pallavicini'
- 'L. Papp'
- 'L. Perasso'
- 'S. Perasso'
- 'A. Pocar'
- 'R.S. Raghavan'
- 'G. Ranucci'
- 'A. Razeto'
- 'A. Re'
- 'P. Risso'
- 'A. Romani'
- 'D. Rountree'
- 'A. Sabelnikov'
- 'R. Saldanha'
- 'C. Salvo'
- 'S. Schönert'
- 'H. Simgen'
- 'M. Skorokhvatov'
- 'O. Smirnov'
- 'A. Sotnikov'
- 'S. Sukhotin'
- 'Y. Suvorov'
- 'R. Tartaglia'
- 'G. Testera'
- 'D. Vignaud'
- 'R.B. Vogelaar'
- 'F. von Feilitzsch'
- 'J. Winter'
- 'M. Wojcik'
- 'A. Wright'
- 'M. Wurm'
- 'J. Xu'
- 'O. Zaimidoroga'
- 'S. Zavatarelli'
- 'G. Zuzel'
title: Measurement of the solar neutrino rate with a liquid scintillator target and 3MeV energy threshold in the Borexino detector
---
Introduction {#sec:intro}
============
Solar -neutrino spectroscopy has been so far performed by the water [Čerenkov]{} detectors KamiokaNDE, SuperKamiokaNDE, and SNO [@Hir89; @SKII08; @SKI05; @SNO07]. The first two experiments used elastic $\nu$-$e$ scattering for the detection of neutrinos, whereas SNO also exploited nuclear reaction channels on deuterium with heavy water as target. These experiments provided robust spectral measurements with $\sim$5MeV threshold or higher for scattered electrons; a recent SNO analysis reached a 3.5MeV threshold [@SNO09].
We report the first observation of solar -neutrinos with a liquid scintillator detector, performed by the Borexino experiment [@BXD08; @BX09] via elastic $\nu$-$e$ scattering. Borexino is the first experiment to succeed in suppressing all major backgrounds, above the 2.614MeV $\gamma$ from the decay of , to a rate below that of electron scatterings from solar neutrinos. This allows to reduce the energy threshold for scattered electrons by solar neutrinos to 3MeV, the lowest ever reported for the electron scattering channel. To facilitate a comparison to the results of SuperKamiokaNDE [@SKI05] and SNO D$_2$O phase [@SNO07], we also report the measured neutrino interaction rate with 5MeV threshold.
Since Borexino also detected low energy solar neutrinos [@BX07; @BX08], this is the first experiment where both branches of the solar -cycle have been measured simultaneously in the same target. The large mixing angle solution (LMA) of the MSW effect [@MSW] predicts a transition in the $\nu_e$ survival probability from the vacuum oscillation regime at low energies to the matter dominated regime at high energies. Results on solar and neutrinos from Borexino, combined with prediction on the absolute neutrino fluxes from the Standard Solar Model [@BS07; @Pen08; @Ser09], confirm that our data are in agreement with the MSW-LMA prediction within 1$\sigma$.
Experimental Apparatus and Energy Threshold {#sec:detector}
===========================================
The Borexino detector is located at the underground Laboratori Nazionali del Gran Sasso (LNGS) in central Italy, at a depth of 3600m.w.e.. Solar neutrinos are detected in Borexino exclusively via elastic $\nu$-$e$ scattering in a liquid scintillator. The active target consists of 278t of pseudocumene (PC, 1,2,4-trimethylbenzene), doped with 1.5g/l of PPO (2,5-diphenyloxazole, a fluorescent dye). The scintillator is contained in a thin (125$\mu$m) nylon vessel of 4.25m nominal radius, and is shielded by two concentric PC buffers (323 and 567t) doped with 5.0g/l of a scintillation light quencher (dimethylphthalate). Scintillator and buffers are contained in a Stainless Steel Sphere (SSS) with a diameter of 13.7m and the scintillation light is detected via 2212 8” photomultiplier tubes (PMTs) uniformly distributed on the inner surface of the SSS. The two PC buffers are separated by a second thin nylon membrane to prevent diffusion of the radon emanated by the PMTs and by the stainless steel of the sphere into the scintillator. The SSS is enclosed in a 18.0 m diameter, 16.9 m high domed Water Tank (WT), containing 2100t of ultra-pure water as an additional shielding against external gamma- and neutron background. 208 8” PMTs in the WT detect the [Čerenkov]{} light produced by muons in the water shield, serving as a highly efficient muon veto. A complete description of the Borexino detector can be found in Ref. [@BXD08].
Scintillator detectors, with their high light yield, are sensitive to lower energy events than [Čerenkov]{} detectors. In this analysis, the 3MeV energy threshold is imposed mainly by the 2.614MeV $\gamma$-rays from the decay of ( chain, $Q$=5.001MeV) in the PMTs and in the SSS and by the finite energy resolution of the detector: a tail of 2.6 MeV $\gamma$ events leaks at higher energies, along with a very small percentage of combined $\gamma$’s from ( chain, $Q$=3.272MeV). The 3MeV energy threshold eliminates sample contamination from such events.
Potential background sources above 3MeV include the radioactive decays of residual and within the liquid scintillator, decays of cosmogenic isotopes (see Table \[tab:cosmogenic\] later), high energy $\gamma$-rays from neutron capture, and cosmic muons. No background from $\alpha$ decays is expected at these energies, since the light quenching of $\alpha$’s in organic liquid scintillators reduces their visible energy in the electron-equivalent scale below 1MeV. A measurement of neutrinos with a 3MeV energy threshold is contingent upon high radiopurity of the scintillator target. and concentrations in the Borexino scintillator have been measured at (1.6$\pm$0.1$)\times$10$^{-17}$g/g and (6.8$\pm$1.5)$\times$10$^{-18}$g/g, respectively, and record low levels of backgrounds in the energy range 0.2-5.0MeV have been reported [@BXD08; @BX08]. The dominant background in the energy range of interest for solar neutrinos originates from spallation processes of high energy cosmic muons. This paper demonstrates that, thanks to the LNGS depth and the Borexino muon veto system, cosmogenic background can be reduced below the rate of interaction of neutrinos, thus allowing the neutrino rate to be measured.
Energy and Position Response of the Detector and Associated Systematics {#sec:calibration}
=======================================================================
We tuned the response of energy and position reconstruction algorithms with a dedicated calibration campaign. We used an off-axis source insertion system designed to position radioactive and/or luminous sources at several locations throughout the detector active target. The source position can be measured with a set of stereoscopic cameras installed on the SSS with an uncertainty of $\pm$2cm in $x$, $y$, and $z$ [@Bac04] ($\pm$3.5cm in radius).
As explained above, Borexino has a unique sensitivity to electron scattering from low energy solar neutrinos, thanks to its unmatched record on background below the natural radioactivity barrier. To preserve this capability, the off-axis source insertion system was designed to respect stringent limits on leak tightness and cleanliness of the mechanics in contact with the liquid scintillator. A detailed description of the technique used for the calibration and for the position reconstruction in Borexino is in preparation.
Energy Scale {#sec:energy}
------------
Calibration of the energy scale allows to establish with high confidence the energy threshold for the neutrino analysis and the error in its determination, and to calibrate the energy scale to allow the energy spectrum of the electrons scattered by neutrinos to be determined. Distortions of the energy scale are due to physical effects (quenching), geometrical effects (light collection), and to the electronics, which was designed for optimal performance in the low-energy range of neutrinos, in a regime where a single photoelectron is expected for each PMT. At higher energies, the electronics response to multiple photoelectron hits on a single channel is not linear. For each triggered channel, the charge from photoelectrons in a 80ns gate is integrated and recorded, but photoelectrons in the following 65ns dead time window are lost. The resulting fraction of lost charge increases with energy and can reach $\sim$10% at $\sim$10 MeV. Moreover, the number of detected photoelectrons depends on the event position in the active volume, due to differences in PMT coverage. For an accurate determination of the energy scale, the dominant non-linearities have been reproduced with a Monte Carlo simulation and with calibration measurements.
To avoid contamination during calibration, $\beta$ sources were not put in direct contact with the scintillator. Instead, we calibrated the response of the detector with encapsulated $\gamma$ sources. The scintillation induced by $\gamma$-rays is due to the ionizing tracks of the secondary electrons, thus, the two energy scales are closely related.
Establishing a correlation between the $\beta$ and $\gamma$ energy scales required extensive simulations with the [[G4Bx]{}]{} Monte Carlo code. [[G4Bx]{}]{} is based on [Geant4]{} [@Ago03; @All06] and simulates in detail all of the detector component, and includes scintillation, [Čerenkov]{} photon production, absorption and scattering of light in the scintillator and in the buffer, as well as the PMT response. Each secondary electron in a $\gamma$-induced Compton electron cascade is affected by energy-dependent ionization quenching, which amplifies the distortion in the $\gamma$ energy scale. The quenching effect is modeled with the Birks formalism [@Bir51]. A second package, [BxElec]{}, simulates in detail the response of the electronics. Finally, Monte Carlo data are processed by the same reconstruction code used for real data. A detailed description of the Monte Carlo codes and of the energy reconstruction algorithm is in preparation.
To calibrate the detector energy response to neutrinos, we used an $^{241}$Am$^9$Be neutron source positioned at the center of the detector and at several positions at 3 m radius. Neutron capture on $^1$H and on $^{12}$C in the scintillator results in the emission of $\gamma$-rays from the 2.223MeV and 4.945MeV excited states, respectively. In addition, neutron capture on the stainless steel of the insertion system produces $\gamma$-rays from the 7.631MeV ($^{56}$Fe) and 9.298MeV ($^{54}$Fe) excited states. We validate the Monte Carlo code by simulating the four $\gamma$-rays in both the positions. In Figure \[fig:enecal\] we show the results of the calibration of the $\gamma$-equivalent energy scale in the detector center. Monte Carlo simulations reproduce $\gamma$ peak positions and resolutions at $\sigma_1$ = 1% precision in the detector center (as shown in Figure \[fig:enecal\]), and at $\sigma_2$ = 4% precision at 3m from the detector’s center. Assuming the same accuracy for the $\beta$-equivalent energy scale, we extrapolate it by simulating electrons uniformly distributed in the scintillator, and then selecting those with reconstructed position within the fiducial volume. The error on the energy scale is obtained with a linear interpolation from $\sigma_1$ in the detector center to $\sigma_2$ at 3 m, along the radius. The $\beta$-equivalent energy scale, in the energy region above 2 MeV, can be parametrized as:
$$N = a \cdot E + b,
\label{eq1}$$
where $N$ is the number of photoelectrons (p.e.) detected by the PMTs, $a$=459$\pm$11p.e./MeV and $b$=115$\pm$38p.e.. The non-zero intercept $b$ is related to the fact that this description is valid only in this energy range and that the overall relation between $N$ and $E$ is non linear.
The anticipated 3MeV (5MeV) energy threshold for the analysis corresponds to 1494p.e. (2413p.e.) within a 3m radial distance from the center of the detector. The uncertainty associated to the 3MeV (5MeV) energy threshold is obtained by propagating the errors of Eq. \[eq1\] and is equal to 51 p.e. (68 p.e.).
![ Black dots are the measured peak positions of $\gamma$ radiation induced by neutron captures in $^{1}$H (2.223 MeV), $^{12}$C (4.945 MeV), $^{56}$Fe (7.631 MeV) and $^{54}$Fe (9.298 MeV) in the detector center. Red line is the Monte Carlo prediction for $\gamma$ rays generated in the detector center.[]{data-label="fig:enecal"}](1_neutron_data_mc.eps){width="49.00000%"}
Vertex Reconstruction
---------------------
The positions of scintillation events are reconstructed with a photon time-of-flight method. We computed with [[G4Bx]{}]{} a probability density function (PDF) for the time of transit of photons from their emission point to their detection as photoelectron signals in the electronics chain. We refined the PDF with data collected in the calibration campaign. Event coordinates ($x_0$, $y_0$, $z_0$) and time ($t_0$) are obtained by minimizing:
$${\cal{L}}(x_0,y_0,z_0,t_0) = \prod_i {\rm PDF}\left( t_i - t_0 - \frac{d_{0,i} \cdot n_{\rm eff}}{c} \right)$$
where the index $i$ runs over the triggered PMTs, $t_i$ is the time of arrival of the photoelectron on the $i^{\rm th}$ electronic channel, and $d_{0,i}$ is the distance from the event position and the $i^{\rm th}$ PMT. $n_{\rm eff}$ is an empirically-determined effective index of refraction to account for any other effect that is not accounted for in the reconstruction algorithm but impacts the distribution of PMT hit times, both in the optics (e.g. Rayleigh scattering) and the electronics (e.g. multiple photoelectron occupancy).
The Borexino electronics records the time of each detected photoelectron introducing a dead time of 145 ns after each hit for each individual channel. Therefore, the timing distribution is biased at high energy, where multiple photoelectrons are detected by each channel, and the position reconstruction is energy dependent. To measure this effect, we deployed the $^{241}$Am$^9$Be neutron source at the six cardinal points of the sphere defining the fiducial volume, [*i.e.*]{} those points lying on axis through the center of the detector, with off-center coordinates from the set $x$=$\pm$3m, $y$=$\pm$3m, and $z$=$\pm$3m. The recoiled proton from neutron scattering allows us to study the reconstructed position as function of the collected charge up to $\sim$5000 p.e.. Figure \[fig:recon\] shows the ratio of measured versus nominal position of the $^{241}$Am$^9$Be source. This data was used to define the fiducial volume $R_{\rm nom}$$<$3m. The non-homogeneous distribution of live PMTs, in particular the large deficit of live PMTs in the bottom hemisphere [@BXD08], is responsible for the different spatial response at mirrored positions about the $x$-$y$ plane. Thus, as shown in Figure \[fig:recon\], two radial functions have been defined for positive and negative $z$ positions.
![Ratio of the reconstructed radial position of $\gamma$ events from the $^{241}$Am$^9$Be source in Borexino to the source radial position measured by the CCD camera system, as a function of the measured charge.[]{data-label="fig:recon"}](2_radcorr.eps){width="49.00000%"}
After all post-calibration improvements to the event reconstruction algorithm, typical resolution in the event position reconstruction is 13$\pm$2cm in $x$ and $y$, and 14$\pm$2cm in $z$ at the relatively high energies. The spatial resolution is expected to scale as $1/\sqrt{N}$ where $N$ is the number of triggered PMTs, and this was confirmed by determining the spatial resolution to be 41$\pm$6cm (1$\sigma$) at 140keV [@BXD08; @BX08]. Systematic deviations of reconstructed positions from the nominal source position are due to the 3.5cm accuracy of the CCD cameras in the determination of the calibration source position, and in 1.6 cm introduced by the energy dependency. The overall systematics are within 3.8cm throughout the 3m-radius fiducial volume.
-neutrino flux {#sec:b8nu}
==============
We report our results for the rate of electron scattering above 3MeV from neutrino interactions in the active target. We also report the result above the threshold of 5MeV, to facilitate the comparison with results reported by SNO [@SNO07] and SuperKamiokaNDE phase-I [@SKI05] at the same threshold. This energy range is unaffected by the scintillator intrinsic background, since the light quenching effect reduces the visible energy of ($Q$=5.001MeV) from $^{232}$Th contamination in the scintillator below the energy threshold of 5MeV.
The analysis in this paper is based on 488 live days of data acquisition, between July 15, 2007 and August 23, 2009, with a target mass of 100t, defined by a fiducial volume cut of radius 3m. The total exposure, after applying all the analysis cuts listed in the next section, is 345.3 days.
Muon Rejection
--------------
The cosmic muon rate at LNGS is 1.16$\pm$0.03m$^{-2}$hr$^{-1}$ with an average energy of 320$\pm4_{\rm stat}\pm11_{\rm sys}$GeV [@MAC99]. Each day, $\sim$4300 muons deposit energy in Borexino’s inner detector. Depending on deposited energy and track length, there is a small but non-zero chance that a cosmic muon induces a number of photoelectrons comparable to the multi-MeV electron scatterings of interest for this analysis, and is mistaken for a point-like scintillation event. A measurement of the neutrino interaction rate in Borexino requires high performance rejection of muon events and an accurate estimate of the muon tagging efficiency.
As mentioned earlier, the Borexino WT is instrumented with 208 PMTs to serve as a muon veto. If an Inner Detector (ID) event coincides in time with an Outer Detector (OD) trigger (i.e. more than 6 PMTs in the WT are hit within a 150ns window), the event is tagged as muon and rejected. However, the OD efficiency is not unity and depends on the direction of the incoming cosmic muon.
In addition, we perform pulse-shape discrimination on the hit time distribution of inner detector PMTs, since for track-like events, like muons, such distribution generally extends to longer times than for point-like events, like $\beta$-decays and $\nu-e$ scattering. We exclude muons from the event sample in the energy range of interest (3.0–16.3MeV, or 1413–6743p.e.) by imposing the following requirements (ID cuts):
The peak of the reconstructed hits time distribution, with respect to the first hit, is between 0ns and 30ns.
The mean value of the reconstructed hits time distribution, with respect to the first hit, is between 0ns and 100ns.
The efficiency of the selection cuts was evaluated on a sample of 2,170,207 events, identified by the OD as muons. Only 22 of these events, a fraction of (1.0$\pm$0.2)$\times10^{-5}$, survive the ID cuts in the energy and spatial region of interest, and are tagged as possible scintillation events.
We do not have an absolute value for the OD muon veto efficiency, but we estimate it to be larger than 99%, from [[G4Bx]{}]{} simulations. The residual muon rate, due to the combined inefficiency of the two tagging systems, taking into account the fact that the two detectors are independent, is muons/day/100t, or muons/day/100t above 5MeV.
Cosmogenic Background Rejection
-------------------------------
----------- -------- --------- ----------- ------------------- ---------- ---------------------------------- ---------------------------------- --
Isotopes $\tau$ $Q$ Decay Expected Rate Fraction Expected Rate $>3~MeV$ Measured Rate $>3~MeV$
\[MeV\] \[cpd/100t\] $>3~MeV$ \[cpd/100t\] \[cpd/100t\]
$^{12}$B 0.03s 13.4 $\beta^-$ 1.41 $\pm$ 0.04 0.886 1.25$\pm$ 0.03 1.48 $\pm$ 0.06
$^{8}$He 0.17s 10.6 $\beta^-$ 0.026 $\pm$ 0.012 0.898
$^{9}$C 0.19s 16.5 $\beta^+$ 0.096 $\pm$ 0.031 0.965 (1.8 $\pm$0.3 )$\times$10$^{-1}$ (1.7 $\pm$ 0.5)$\times$10$^{-1}$
$^{9}$Li 0.26s 13.6 $\beta^-$ 0.071 $\pm$ 0.005 0.932
$^{8}$B 1.11s 18.0 $\beta^+$ 0.273 $\pm$ 0.062 0.938
$^{6}$He 1.17s 3.5 $\beta^-$ NA 0.009 (6.0 $\pm$ 0.8)$\times$10$^{-1}$ (5.1 $\pm$ 0.7)$\times$10$^{-1}$
$^{8}$Li 1.21s 16.0 $\beta^-$ 0.40 $\pm$ 0.07 0.875
$^{10}$C 27.8s 3.6 $\beta^+$ 0.54 $\pm$ 0.04 0.012 (6.5$\pm$0.5)$\times$10$^{-3}$ (6.6$\pm$1.8) $\times$10$^{-3}$
$^{11}$Be 19.9s 11.5 $\beta^-$ 0.035 $\pm$ 0.006 0.902 (3.2 $\pm$ 0.5)$\times$10$^{-2}$ (3.6$\pm$3.5)$\times$10$^{-2}$
----------- -------- --------- ----------- ------------------- ---------- ---------------------------------- ---------------------------------- --
### Fast cosmogenic veto {#sec:fast}
Table \[tab:cosmogenic\] presents a list of expected cosmogenic isotopes produced by muons in Borexino. The short-lived cosmogenics ($\tau < 2s$), as well as the $\gamma$-ray capture on , are rejected by a 6.5s cut after each muon, with a 29.2% fractional dead time. Figure \[fig:TimeAfterMuon\] shows the time distribution of events following a muon. The data is well fit by three exponentials with characteristic times of 0.031$\pm$0.002s (), 0.25$\pm$0.21s ($^{8}$He, $^{9}$C, $^{9}$Li), 1.01$\pm$0.36s ($^{8}$B, $^{6}$He , $^{8}$Li), in good agreement with the lifetimes of the short-lived isotopes (see Table \[tab:cosmogenic\]). From the fit we estimate the production rates of these cosmogenic isotopes in Borexino. We conclude that rejection of events in a 6.5s window following every muon crossing the SSS reduces the residual contamination of the short lived isotopes to cpd/100t (cpd/100t above 5 MeV).
The expected rates (R) quoted in Table \[tab:cosmogenic\] are obtained by scaling the production rates (R$^0$) measured by KamLAND [@Abe09] with: $$R = R^0\left(\frac{E_{\mu}}{E_{\mu}^0}\right)^\alpha \frac{\Phi_{\mu}}{\Phi_{\mu}^0} ,
\label{eq:scaling}$$ where E$_{\mu}$ and $\Phi_{\mu}$ are the Borexino mean muon energy (320$\pm4_{\rm stat}\pm11_{\rm sys}$GeV) and flux (1.16$\pm$0.03m$^{-2}$ hr$^{-1}$), as measured by MACRO [@MAC99], and E$_{\mu}^0$ (260$\pm$4GeV) and $\Phi_{\mu}^0$ (5.37$\pm$0.41 m$^{-2}$ hr$^{-1}$) are the corresponding KamLAND values. $\alpha$ is a scaling parameter to relate cosmogenic production rate at different mean energies of the incoming muon flux; it is obtained in Ref. [@Abe09] by fitting the production yield of each isotope, simulated by FLUKA, as a function of muon beam energy. Overall, Borexino data results are in agreement with the values quoted in Table \[tab:cosmogenic\] within 15%.
### Neutron rejection {#sec:neutron}
The cosmogenic background in Borexino includes decays of radioactive isotopes due to spallation processes on the nuclei in the scintillator, as well as the $\gamma$-rays from the capture of neutrons that are common by-products of such processes. The capture time for neutrons in the Borexino scintillator has been measured to be 256.0$\pm$0.4$\mu$s, using a neutron calibration source, and the energy of the dominant $\gamma$-rays from neutron capture on at 2.223MeV is below the energy threshold of the present analysis. On the other hand, the 4.9MeV $\gamma$-rays from neutron captures on ${\mbox{$^{12}$C}}$ is a potential background for this analysis. The rate is estimated by scaling the cosmogenic neutron capture rate on $^1$H by the fraction of captures on ${\mbox{$^{12}$C}}$ with respect to the total, measured with the $^{241}$Am$^9$Be neutron source. The neutron capture rate on ${\mbox{$^{12}$C}}$ is 0.86$\pm$0.01cpd/100 t.
![ Cumulative distribution of events with energy $>$ 3 MeV within a 5 s window after a muon in Borexino. The time distribution has been fit to three decay exponentials. The ensuing exponential lifetimes are $\tau =$ 0.031$\pm$0.002s, 0.25$\pm$0.21s, 1.01$\pm$0.36s and corresponds to the contribution from , $^{8}$He $+$ $^{9}$C $+$ $^{9}$Li and $^{8}$B $+$ $^{6}$He $+$ $^{8}$Li, respectively. The expected and measured rates for these cosmogenic isotopes are summarized in Table \[tab:cosmogenic\].[]{data-label="fig:TimeAfterMuon"}](3_cosmogenic.eps){width="49.00000%"}
The fast cosmogenic veto, described in the [*Fast Cosmogenic Veto*]{} section, rejects neutrons produced in the scintillator or in the buffer by muon spallation with 99.99% efficiency. To reject neutrons produced in water, a second 2 ms veto is applied after each muon crossing the Water Tank only. The rejection efficiency for neutrons produced in water is 0.9996. The overall survival neutron rate in the energy range of interest and in the fiducial volume is cpd/100 t.
### identification and subtraction
A separate treatment is required for long-lived ($\tau$$>$2s) cosmogenic isotopes. Since ($\tau=76.9$d, $Q$–value$=$0.9MeV) and ($\tau=29.4$min, $Q$–value$=$2.0MeV) are below the energy threshold, we focus on and .
Taking into account the energy response of Borexino, the fraction of the energy spectrum above 3MeV is 1.2%. When is produced in association with a neutron, candidates are tagged by the three-fold coincidence with the parent muon and subsequent neutron capture in the scintillator [@Gal05]. The efficiency of the Borexino electronics in detecting at least one neutron soon after a muon has been estimated to be 94% by two parallel (1-channel and 8-channel) DAQ systems that digitize data for 2ms after every OD trigger at 500MHz. The rate of muons associated with at least 1 neutron, measured by the Borexino electronics, is $\sim$67cpd. Thus, to reject from the analysis we exclude all data within a 120s window after a $\mu$+$n$ coincidence and within a 0.8m distance from the neutron capture point. The efficiency of this cut is 0.74$\pm$0.11, for a 0.16% dead time. A time profile analysis of events tagged by this veto above 2.0MeV returns a characteristic time of 30$\pm$4s, consistent with the lifetime of , and a total rate of (0.50$\pm$0.13)cpd/100t, in production channels with neutron emission. Thus, the residual contamination from neutron-producing channels above 3MeV is cpd/100 t.
The dominant neutron-less production reaction is $^{12}$C($p$,$t$)$^{10}$C. We extrapolated its rate by scaling the $^{12}$C($p$,$d$)$^{11}$C production rate [@Gal05; @Gal052], by the ratio between the $^{12}$C($p$,$d$)$^{11}$C and $^{12}$C($p$,$t$)$^{10}$C cross sections measured in Ref. [@Yas77]. The $^{12}$C($p$,$t$)$^{10}$C rate is cpd/100 t. The residual background above 3MeV from is cpd/100 t.
The overall rate above 3 MeV, (6.6$\pm$1.8) $\times$10$^{-3}$cpd/100 t, agrees with the expected one, quoted in Table \[tab:cosmogenic\].
### estimation
Figure \[fig:be11\] shows the time profile of events within 240 s after a muon and within a 2 m distance from its track in the entire Borexino active volume (278 t). The efficiency of the distance cut is assumed to be the same as the one measured for cosmogenic (84%) by performing fits to the time distribution of events after a muon before and after the track cut. The measured rate above 3 MeV is cpd/100t, consistent with the cpd/100t rate extrapolated from the KamLAND measurements [@Abe09]. Since all measured rates deviated less than 18% from the extrapolated value, we adopt the latter as the residual rate of in our sample.
![Time profile of events with energy $>$ 3 MeV within 240 s after a muon and within 2 m from its track in the entire Borexino active volume (278 t). The distribution has been fit to the three decay exponentials as in Figure \[fig:TimeAfterMuon\], plus the $^{11}$Be component, with fixed mean-lives. []{data-label="fig:be11"}](4_be.eps){width="\columnwidth"}
Radioactive Background Rejection
--------------------------------
------------------------- -------------- --------------
Cut Counts Counts
3.0–16.3 MeV 5.0–16.3 MeV
All counts 1932181 1824858
*Muon and neutron cuts* 6552 2679
*FV cut* 1329 970
*Cosmogenic cut* 131 55
removal 128 55
removal 119 55
subtraction 90$\pm$13 55$\pm$7
$^{11}$Be subtraction 79$\pm$13 47$\pm$8
Residual subtraction 75$\pm$13 46$\pm$8
Final sample 75$\pm$13 46$\pm$8
BPS09(GS98) $\nu$ 86$\pm$10 43$\pm$6
BPS09(AGS05) $\nu$ 73$\pm$7 36$\pm$4
------------------------- -------------- --------------
: Effect of the sequence of cuts on the observed counts. The cosmogenic cut introduces a reduction of the detector live-time of 29.2%. The resulting effective live-time is 345.3d. The analysis is done with 100t fiducial mass target, after the FV cut. The expected counts are calculated from current best parameters for the MSW-LMA [@Fog08] and Standard Solar Models, BPS09(GS98) and BPS09(AGS05) [@BS07; @Pen08; @Ser09][]{data-label="tab:cuts"}
### External background
The 3MeV energy threshold is set by the 2.614MeV $\gamma$-rays from the $\beta$-decay of , due to radioactive contamination in the PMTs and in the SSS. Above 3MeV, the sources of radioactive background include the radioactive decays of residual ( chain, $Q$=3.272MeV) and ( chain, $Q$=5.001MeV) in the liquid scintillator. The fiducial volume cut is very effective against the and background due to and emanated from the nylon vessel, as well as residual external $\gamma$-ray background. In Figure \[fig:radfit\] the radial distribution of all scintillation events above 3MeV has been fit to a model which takes into account the three sources of backgrounds: a uniform distribution in the detector for internal events, a delta-function centered on the vessel radius for the point-like radioactive background in the nylon, and an exponential for external $\gamma$-ray background. All the three components are convoluted with the detector response function. From this radial analysis we conclude that within the fiducial volume there is a small contribution of events from surface contamination and the exterior of cpd/100 t (cpd/100 t) for events above 3MeV (5MeV).
![Fit of the radial distribution for events with E$>$3 MeV. The red line represents the uniformly distributed event component in the active mass, the green line the surface contamination, and the blue line is external background.[]{data-label="fig:radfit"}](5_radfit.eps){width="\columnwidth"}
### contamination
The suppression of from contamination in the FV relies on the delayed coincidence ($\tau$=237$\mu$s). We look for coincidences in time between 20$\mu$s and 1.4ms ($\epsilon$ = 0.91) with a spatial separation $<$1.5m ($\epsilon$ = 1) and Gatti parameter, a pulse shape discrimination estimator introduced in Ref. [@BXD08; @Bac08], larger than -0.008 ($\epsilon$ = 1) within the fiducial volume throughout the entire data set. The $\alpha$–decays are selected in the 0.3–1.2MeV ($\epsilon$ = 1) energy range. The remaining contribution of to the $\nu$-$e$ scattering sample is negligible (1.1$\pm$0.4)$\times$10$^{-4}$cpd/100t.
### contamination
Amongst the daughters of naturally present in the scintillator, decays are the only ones which contribute background above 3 MeV. The parent of is $\alpha$-decays into with a branching ratio of 36% and a lifetime of $\tau$ = 4.47 min. In the second channel with branching ratio 64%, $\beta$-decays into with a lifetime of $\tau=$ 431 ns. We estimate the rate from the fast coincidences. events are selected in a time window between 400 and 1300 ns, with an efficiency of 0.35, and requiring a maximum spatial distance between the two events of 1 m ($\epsilon$= 1). and are selected in \[20–1200\] p.e. ($\epsilon$ = 1) and \[420–580\] p.e. ($\epsilon=$ 0.93) energy regions, respectively. The –$\alpha$ quenched energy is estimated from the and peaks, optimal signatures for the $\alpha$-quenching calibration. We found 21 coincidences in the entire data set, within the FV. Accounting for the efficiency of the selection cuts and the branching ratios of the decays, this corresponds to a contamination in our neutrino sample of 29$\pm$7 events, or a (8.4$\pm$2.0)$\times$10$^{-2}$ cpd/100t rate.
A summary of the analysis sequence described above is shown in Table \[tab:cuts\]. The energy spectrum of the final sample, compared with simulated spectra of neutrinos and of each residual background component listed in Table \[tab:residual\], is shown in Figure \[fig:b8spectrum\].
----------------------- ----------------------------- ---------------
Background Rate \[10$^{-4}$cpd/100 t\]
$>$3 MeV $>$5 MeV
*Muons* 4.5$\pm$0.9 3.5$\pm$0.8
*Neutrons* 0.86$\pm$0.01 0
*External background* 64$\pm$2 0.03$\pm$0.11
*Fast cosmogenic* 17$\pm$2 13$\pm$2
22$\pm$2 0
1.1$\pm$0.4 0
840$\pm$20 0
320$\pm$60 233$\pm$44
*Total* 1270$\pm$63 250$\pm$44
----------------------- ----------------------------- ---------------
: Residual rates of background components after the data selection cuts above 3 and 5 MeV.[]{data-label="tab:residual"}
Neutrino interaction rates and electron scattering spectrum {#sec:neutrinos}
===========================================================
The mean value for neutrinos in the sample above 3 MeV (5 MeV) is 75$\pm$13 (46$\pm$8) counts.
The dominant sources of systematic errors are the determinations of the energy threshold and of the fiducial mass, both already discussed in the previous sections. The first introduces a systematic uncertainty of +3.6% -3.2% (+6.1% -4.8% above 5MeV). The second systematic source is responsible for a $\pm$3.8% uncertainty in the neutrino rate. A secondary source of systematics, related to the effect of the energy resolution on the threshold cuts, has been studied on a simulated neutrino spectrum and is responsible for a systematic uncertainty of +0.0% -2.5% (+0.0% -3.0% above 5MeV).
The total systematic errors are shown in Table \[tab:syst\].
The resulting count rate with E$>$3MeV is: $$0.217\pm 0.038 (stat)^{+0.008}_{-0.008}(syst)~cpd/100\,t$$ and with E$>$5MeV: $$0.134\pm 0.022(stat)^{+0.008}_{-0.007}(syst)~cpd/100\,t.$$ The final energy spectrum after all cuts and residual background is shown in Figure \[fig:b8fromModel\]. It is in agreement with the scenario which combines the high metallicity Standard Solar Model, called BPS09(GS98) [@Ser09], and the prediction of the MSW-LMA solution.
------------------- ------------ ------------ ------------ ------------
Source E$>$3 MeV E$>$5 MeV
$\sigma_+$ $\sigma_-$ $\sigma_+$ $\sigma_-$
Energy threshold 3.6% 3.2% 6.1% 4.8%
Fiducial mass 3.8% 3.8% 3.8% 3.8%
Energy resolution 0.0% 2.5% 0.0% 3.0%
Total 5.2% 5.6% 7.2% 6.8%
------------------- ------------ ------------ ------------ ------------
: Systematic errors.[]{data-label="tab:syst"}
![Comparison of the final spectrum after data selection (red dots) to Monte Carlo simulations (black line). The expected electron recoil spectrum from to oscillated $\nu$ interactions (filled blue histogram), $^{208}$Tl (green), $^{11}$Be (cyan) and external background (violet), are equal to the measured values in Table \[tab:residual\].[]{data-label="fig:b8spectrum"}](6_spectrum.eps){width="50.00000%"}
![Comparison of the final spectrum after data selection and background subtraction (red dots) to Monte Carlo simulations (blue) of oscillated $\nu$ interactions, with amplitude from the Standard Solar Models BPS09(GS98) (high metallicity) and BPS09(AGS05) (low metallicity), and from the MSW-LMA neutrino oscillation model.[]{data-label="fig:b8fromModel"}](7_compare.eps){width="50.00000%"}
Solar neutrino flux and neutrino oscillation parameters {#sec:implications}
========================================================
The equivalent unoscillated neutrino flux, derived from the electron scattering rate above 5 MeV (Table \[tab:comparison\]), is (2.7$\pm$0.4$_{\rm stat}$$\pm$0.2$_{\rm syst}$)$\times$10$^6$ cm$^{-2}$s$^{-1}$, in good agreement with the SuperKamiokaNDE-I and SNO D$_2$O measurements with the same threshold, as reported in Table \[tab:fluxes\]. The corresponding value above 3MeV, is (2.4$\pm$0.4$_{\rm stat}$$\pm$0.1$_{\rm syst}$)$\times$10$^6$ cm$^{-2}$s$^{-1}$. The expected value for the case of no neutrino oscillations, including the theoretical uncertainty on the flux from the Standard Solar Model [@BS07; @Pen08; @Ser09], is (5.88$\pm$0.65)$\times$10$^6$ cm$^{-2}$s$^{-1}$ and, therefore, solar $\nu_e$ disappearance is confirmed at 4.2$\sigma$.
To define the neutrino electron survival probability averaged in the energy range of interest, we define the measured recoiled electron rate $R$, through the convolution:
$$R = \int_{T_e>T_0} dT_e \int_{{ - \infty}}^{{ + \infty}} dt \left(\frac{dr}{dt}(t) \cdot g(t-T_e)\right)
\label{eq:meanpee}$$
between the detector energy response $g$, assumed gaussian, with a resolution depending on the energy, and differential rate:
$$\frac{dr}{dT_e}(T_e) = \int_{E_{\nu}>E_{0}} dE_{\nu} \left( \overline P_{ee}\cdot\Psi_{e} + (1 - \overline P_{ee}) \cdot \Psi_{\mu-\tau} \right)
\label{eq:meanpee}$$
with:
$$\Psi_{e} =\frac{d\sigma_e}{dT_e}(E_{\nu},T_e) \cdot N_e \cdot \frac{d\Phi_e}{dE_{\nu}} (E_{\nu}),
\label{eq:meanpee}$$
and:
$$\Psi_{\mu-\tau} = \frac{d\sigma_{\mbox{$\mu$-$\tau$}}}{dT_e}(E_{\nu},T_e)\cdot N_e \cdot \frac{d\Phi_e}{dE_{\nu}} (E_{\nu}).
\label{eq:meanpee}$$
where $T_e$ and $E_{\nu}$ are the electron and neutrino energies, and $\sigma_{x}$ ($x$=$e$,$\mu$-$\tau$) are the cross sections for elastic scattering for different flavors. $T_0$=3MeV is the energy threshold for scattered electrons, corresponding to a minimum neutrino energy of $E_0$=3.2MeV, $N_e$ is the number of target electrons, and $d\Phi_e/dE_{\nu}$ is the differential solar neutrino flux [@Bah96].
3.0–16.3MeV 5.0–16.3MeV
-------------------------------------------------------- ------------------------ ------------------------
Rate \[cpd/100 t\] 0.22$\pm$0.04$\pm$0.01 0.13$\pm$0.02$\pm$0.01
$\Phi^{\rm ES}_{\rm exp}$ \[10$^6$ cm$^{-2}$s$^{-1}$\] 2.4$\pm$0.4$\pm$0.1 2.7$\pm$0.4$\pm$0.2
*$\Phi^{\rm ES}_{\rm exp} / \Phi^{\rm ES}_{\rm th}$* 0.88$\pm$0.19 1.08$\pm$0.23
: Measured event rates in Borexino and comparison with the expected theoretical flux in the BPS09(GS98) MSW-LMA scenario [@MSW].[]{data-label="tab:comparison"}
Using the above equation, we obtain =0.29$\pm$0.10 at the mean energy of 8.9MeV for neutrinos. Borexino is the first experiment to detect in real time, and in the same target, neutrinos in the low energy, vacuum dominated- and in the high energy, matter enhanced-regions. Borexino already reported a survival probability of 0.56$\pm$0.10 for neutrinos, at the energy of 0.862MeV [@BX08]. The distance between the two survival probabilities is 1.9 $\sigma$.
![Electron neutrino survival probability as function of the neutrino energy, evaluated for the neutrino source assuming the BPS09(GS98) Standard Solar Model [@BS07; @Pen08; @Ser09] and the oscillation parameters from the MSW-LMA solution $\Delta$m$^2$=7.69$\times$10$^{-5}$ eV$^2$ and $\tan^2{\theta}$=0.45 [@Fog08]). Dots represent the Borexino results from and measurements. The error bars include also the theoretical uncertainty of the expected flux from the Standard Solar Model BPS09(GS98). []{data-label="fig:pee"}](8_PeeLMAB8data.eps){width="\columnwidth"}
Future precision measurements of and (and, possibly, ) neutrinos in Borexino could provide an even more stringent test of the difference in for low- and high-energy neutrinos predicted by the MSW-LMA theory: assuming other 4 years of data taking, and to reduce the overall uncertainty on neutrino rate at 5%, the distance between the two survival probabilities can be improved at 3 $\sigma$.
------------------------------ ----------- -------------------------------------------
Threshold $\Phi^{\rm ES}_{\rm ^8B}$
\[MeV\] \[10$^6$ cm$^{-2}$ s$^{-1}$\]
SuperKamiokaNDE I [@SKI05] 5.0 2.35$\pm$0.02$\pm$0.08
SuperKamiokaNDE II [@SKII08] 7.0 2.38$\pm$0.05$^{+0.16}_{-0.15}$
SNO D$_2$O [@SNO07] 5.0 2.39$^{+0.24}_{-0.23}$ $^{+0.12}_{-0.12}$
SNO Salt Phase [@SNO05] 5.5 2.35$\pm$0.22$\pm$0.15
SNO Prop. Counter [@SNO08] 6.0 1.77$^{+0.24}_{-0.21}$$^{+0.09}_{-0.10}$
Borexino 3.0 2.4$\pm$0.4$\pm$0.1
Borexino 5.0 2.7$\pm$0.4$\pm$0.2
------------------------------ ----------- -------------------------------------------
: Results on solar neutrino flux from elastic scattering, normalized under the assumption of the no-oscillation scenario reported by SuperKamiokaNDE, SNO, and Borexino.[]{data-label="tab:fluxes"}
We are grateful to F. Vissani and F. Villante for useful discussions and comments. The Borexino program was made possible by funding from INFN (Italy), NSF (U.S., PHY-0802646, PHY-0802114, PHY-0902140), BMBF, DFG and MPG (Germany), Rosnauka (Russia), MNiSW (Poland). We acknowledge the generous support of the Laboratori Nazionali del Gran Sasso (LNGS). This work was partially supported by PRIN 2007 protocol 2007JR4STW.
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---
abstract: 'We continue our study of operator algebras with contractive approximate identities (cais). In earlier papers we have introduced and studied a new notion of positivity in operator algebras, with an eye to extending certain $C^*$-algebraic results and theories to more general algebras. In the first part of this paper we do a more systematic development of this positivity and its associated ordering, proving many foundational facts. In much of this it is not necessary that the algebra have an approximate identity. In the second part we study interesting examples of operator algebras with cais, which in particular answer questions raised in previous papers in this series. Indeed the present work solves almost all of the open questions raised in these papers. Many of our results apply immediately to function algebras, but we will not take the time to point these out, although most of these applications seem new.'
address:
- 'Department of Mathematics, University of Houston, Houston, TX 77204-3008'
- 'Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England'
author:
- 'David P. Blecher'
- Charles John Read
title: 'Operator algebras with contractive approximate identities, III'
---
[^1]
Introduction
============
An [*operator algebra*]{} is a closed subalgebra of $B(H)$, for a Hilbert space $H$. We are mostly interested in operator algebras with contractive approximate identities (cai’s). We also call these [*approximately unital*]{} operator algebras.
In earlier papers [@BRI; @BRII; @Read] we introduced and studied a new notion of positivity in operator algebras, with an eye to extending certain $C^*$-algebraic results and theories to more general algebras. We are also simultaneously developing such extensions (see also e.g. [@Bnew; @BNI; @BNII]). With the same goal in mind, in the first part (Sections 2–6) of the present paper, we undertake a systematic study of foundational aspects of this positivity, and of the associated ordering. In particular, a central role is played by the set ${\mathfrak F}_A = \{ a \in A :
\Vert 1 - a \Vert \leq 1 \}$ (here $1$ is the identity of the unitization if $A$ is nonunital), and the cones $${\mathfrak c}_A =
\operatorname{{\mathbb R}}_+ \, {\mathfrak F}_A, \; \; \text{and} \; \; {\mathfrak r}_A = \overline{{\mathfrak c}_A} = \{ a \in A : a + a^* \geq 0 \}.$$ Elements of these sets, and their roots, play the role in many situations of positive elements in a $C^*$-algebra. In Section 2 we study general properties of these cones and the related real state space. Section 3 is a collection of results on positivity, some of which are used elsewhere in this paper, and some in forthcoming work. In Section 4 we study ‘strictly positive’ elements, a topic that is quite important for $C^*$-algebras. In Section 5 we solve a problem from [@BRI] concerning when a right ideal $xA$ is already closed, and use this to characterize algebraically finitely generated r-ideals in operator algebras. Section 6 presents versions of our previous Urysohn lemma for operator algebras (see e.g. [@BRI; @BNII]), but now insisting that the ‘interpolating element’ is ‘positive’ in our new sense, and is as close as one would wish to being positive in the usual sense. This solves the problems raised at the end of [@BNII]. See [@CGK] for a recent paper containing a kind of ‘Urysohn lemma with positivity’ for function algebras. Indeed many results in Sections 2–6 apply immediately to function algebras (uniform algebras), that is to uniformly closed subalgebras of $C(K)$, since these are special cases of operator algebras. We will not take the time to point these out, although some of these applications are new.
In Section 7, we construct an interesting new approximately unital operator algebra, and use it to solve questions arising in our earlier work, and which we now describe some background for. We recall that a semisimple Banach algebra $A$ is a modular annihilator algebra iff no element of $A$ has a nonzero limit point in its spectrum [@Pal Theorem 8.6.4]). If $A$ is also commutative then this is equivalent to the Gelfand spectrum of $A$ being discrete [@LN p. 400]. We write $M_{a,b} : A \to A : x \mapsto axb$, where $a, b \in A$. Recall that a Banach algebra is [*compact*]{} if the map $M_{a,a}$ is compact for all $a \in A$. We say that $A$ is [*weakly compact*]{} if $M_{a,a}$ is weakly compact for all $a \in A$. If $A$ is approximately unital and commutative then $A$ is weakly compact iff $A$ is an ideal in its bidual $A^{**}$. (We use the same symbol $*$ for the Banach dual and for the involution or adjoint operator, the reader will have to determine which is meant from the context.) In the noncommutative case $A$ is weakly compact iff $A$ is a [*hereditary subalgebra*]{} (or [*HSA*]{}, defined below) in its bidual. It is known [@Pal] that every compact semisimple Banach algebra is a modular annihilator algebra (and conversely every semisimple ‘annihilator algebra’, or more generally any Banach algebra with dense socle, is compact). Thus it is of interest to know if there are any connections for operator algebras between being a semisimple modular annihilator algebra, and being weakly compact. See the discussion after Proposition 5.6 in [@ABR], where some specific questions along these lines are raised. We solve these here; indeed we have solved almost all open questions posed in our previous papers [@BRI; @BRII; @ABR]. In particular we know that 1) a semisimple approximately unital operator algebra which is a modular annihilator algebra need not be weakly compact, 2) an approximately unital commutative weakly compact semisimple operator algebra $A$ need not have countable or scattered spectrum (in fact the spectrum of some of its elements can have nonempty interior), and 3) a radical operator algebra can have a semisimple bidual. We present the first and second of these examples here.
We now turn to notation and some background facts mostly needed for Sections 2–6. In this paper $H$ will always be a Hilbert space, usually the Hilbert space on which our operator algebra is acting, or is completely isometrically represented. We recall that by a theorem due to Ralf Meyer, every operator algebra $A$ has a unitization $A^1$ which is unique up to completely isometric homomorphism (see [@BLM Section 2.1]). Below $1$ always refers to the identity of $A^1$ if $A$ has no identity. We write oa$(x)$ for the operator algebra generated by $x$ in $A$, the smallest closed subalgebra containing $x$. A [*state*]{} of an approximately unital operator algebra $A$ is a functional with $\Vert \varphi \Vert = \lim_t \, \varphi(e_t) = 1$ for some (or any) cai $(e_t)$ for $A$. These extend to states of $A^1$. See [@BLM Section 2.1] for details. With this in mind, we define a state on any nonunital operator algebra $A$ to be a norm $1$ functional that extends to a state on $A^1$. We write $S(A)$ for the collection of such states; this is the [*state space*]{} of $A$. These extend further by the Hahn-Banach theorem to a state on any $C^*$-algebra generated by $A^1$, and therefore restrict to a positive functional on any $C^*$-algebra $B$ generated by $A$. The latter restriction is actually a state, since it has norm $1$ (even on $A$). Conversely, every state on $B$ extends to a state on $B^1$, and this restricts to a state on $A^1$. From these considerations it is easy to see that states on an operator algebra $A$ may equivalently be defined to be norm $1$ functionals that extend to a state on any $C^*$-algebra $B$ generated by $A$.
For us a [*projection*]{} is always an orthogonal projection, and an [*idempotent*]{} merely satisfies $x^2 = x$. If $X, Y$ are sets, then $XY$ denotes the closure of the span of products of the form $xy$ for $x \in X, y \in
Y$. We write $X_+$ for the positive operators (in the usual sense) that happen to belong to $X$. We write $M_n(X)$ for the space of $n \times n$ matrices over $X$, and of course $M_n = M_n(\operatorname{{\mathbb C}})$. The second dual $A^{**}$ is also an operator algebra with its (unique) Arens product, this is also the product inherited from the von Neumann algebra $B^{**}$ if $A$ is a subalgebra of a $C^*$-algebra $B$. Note that $A$ has a cai iff $A^{**}$ has an identity $1_{A^{**}}$ of norm $1$, and then $A^1$ is sometimes identified with $A + \operatorname{{\mathbb C}}1_{A^{**}}$. In this case the multiplier algebra $M(A)$ is identified with the idealizer of $A$ in $A^{**}$ (that is, the set of elements $\alpha\in A^{**}$ such that $\alpha A\subset A$ and $A
\alpha\subset A$). It can also be viewed as the idealizer of $A$ in $B(H)$, if the above representation on $H$ is nondegenerate.
For an operator algebra, not necessarily approximately unital, we recall that $\frac{1}{2} {\mathfrak F}_A = \{ a \in A : \Vert 1 - 2 a \Vert \leq 1 \}$. Here $1$ is the identity of the unitization $A^1$ if $A$ is nonunital. As we said, $A^1$ is uniquely defined, and can be viewed as $A + \operatorname{{\mathbb C}}I_H$ if $A$ is completely isometrically represented as a subalgebra of $B(H)$. Hence so is $A^1 + (A^1)^*$ uniquely defined, by e.g. 1.3.7 in [@BLM]. We define $A + A^*$ to be the obvious subspace of $A^1 + (A^1)^*$. This is well defined independently of the particular Hilbert space $H$ on which $A$ is represented, as shown at the start of Section 3 in [@BRII]. Thus a statement such as $a + b^* \geq 0$ makes sense whenever $a, b \in A$, and is independent of the particular $H$ on which $A$ is represented. Hence the set ${\mathfrak r}_A = \{ a \in A : a + a^* \geq 0 \}$ is independent of the particular representation too. Elements in ${\mathfrak r}_A$, that is elements in $A$ with ${\rm Re}(x) \geq 0$ will sometimes be called [*accretive*]{}. We recall that an [*r-ideal*]{} is a right ideal with a left cai, and an [*$\ell$-ideal*]{} is a left ideal with a right cai. We say that an operator algebra $D$ with cai, which is a subalgebra of another operator algebra $A$, is a HSA (hereditary subalgebra) in $A$, if $DAD \subset D$. See [@BHN] for the basic theory of HSA’s. HSA’s in $A$ are in an order preserving, bijective correspondence with the r-ideals in $A$, and with the $\ell$-ideals in $A$. Because of this symmetry we will usually restrict our results to the r-ideal case; the $\ell$-ideal case will be analogous. There is also a bijective correspondence with the [*open projections*]{} $p \in A^{**}$, by which we mean that there is a net $x_t \in A$ with $x_t = p x_t \to p$ weak\*, or equivalently with $x_t = p x_t p \to p$ weak\* (see [@BHN Theorem 2.4]). These are also the open projections $p$ in the sense of Akemann [@Ake2] in $B^{**}$, where $B$ is a $C^*$-algebra containing $A$, such that $p \in A^{\perp \perp}$. If $A$ is approximately unital then the complement $p^\perp = 1_{A^{**}} - p$ of an open projection for $A$ is called a [*closed projection*]{} for $A$. A closed projection $q$ for which there exists an $a \in {\rm Ball}(A)$ with $aq = qa = q$ is called [*compact*]{}. This is equivalent to $A$ being a closed projection with respect to $A^1$, if $A$ is approximately unital. See [@BNII; @BRII] for the theory of compact projections in operator algebras.
If $x \in {\mathfrak r}_A$ then it is shown in [@BRII Section 3] that the operator algebra oa$(x)$ generated by $x$ in $A$ has a cai, which can be taken to be a normalization of $(x^{\frac{1}{n}})$, and the weak\* limit of $(x^{\frac{1}{n}})$ is the support projection $s(x)$ for $x$. This is an open projection. We recall that if $x \in \frac{1}{2} \, {\mathfrak F}_A$ then the [*peak projection*]{} associated with $x$ is $u(x) = \lim_n \, x^n$ (weak\* limit). We have $u(x^{\frac{1}{n}}) = u(x)$, for $x \in \frac{1}{2} \, {\mathfrak F}_A$ (see [@BNII Corollary 3.3]). Compact projections in approximately unital algebras are precisely the infima (or decreasing weak\* limits) of collections of such peak projections [@BNII]. Note that $x \in {\mathfrak c}_A = \operatorname{{\mathbb R}}_+ {\mathfrak F}_A$ iff there is a positive constant $C$ with $x^* x \leq C(x+x^*)$.
In this paper we will sometimes use the word ‘cigar’ for the wedge-shaped region consisting of numbers $re^{i \theta}$ with argument $\theta$ such that $|\theta| < \rho$ (for some fixed small $\rho > 0$), which are also inside the circle $|z - \frac{1}{2}| \leq \frac{1}{2}$. If $\rho$ is small enough so that $|{\rm Im}(z)| < \epsilon/2$ for all $z$ in this region, then we will call this a ‘horizontal cigar of height $< \epsilon$ centered on the line segment $[0,1]$ in the $x$-axis’.
By [*numerical range*]{}, we will mean the one defined by states, while the literature we quote usually uses the one defined by vector states on $B(H)$. However since the former range is the closure of the latter, as is well known, this will cause no difficulties. For any operator $T \in B(H)$ whose numerical range does not include strictly negative numbers, and for any $\alpha \in [0,1]$, there is a well-defined ‘principal’ root $T^\alpha$, which obeys the usual law $T^\alpha T^\beta = T^{\alpha + \beta}$ if $\alpha + \beta \leq 1$ (see e.g. [@MP; @LRS]). If the numerical range is contained in a sector $S_\psi = \{ r e^{i \theta} : 0 \leq r , \, \text{and} \,
-\psi \leq \theta \leq \psi \}$ where $0 \leq \psi < \pi$, then things are better still. For fixed $\alpha \in (0,1]$ there is a constant $K > 0$ with $\Vert T^\alpha - S^\alpha \Vert \leq K \Vert T - S \Vert^\alpha$ for operators $S, T$ with numerical range in $S_\psi$ (see [@MP; @LRS]). Our operators $T$ will in fact be accretive (that is, $\psi \leq \frac{\pi}{2}$), and then these powers obey the usual laws such as $T^\alpha T^\beta = T^{\alpha + \beta}$ for all $\alpha, \beta > 0$, $(T^\alpha)^\beta
= T^{\alpha \beta}$ for $\alpha \in (0,1]$ and any $\beta > 0$, and $(T^*)^\alpha = (T^\alpha)^*$. We shall see in Lemma \[rootf\] that if $\psi < \frac{\pi}{2}$ then $T
\in {\mathfrak c}_{B(H)}$. The numerical range of $T^\alpha$ lies in $S_{\alpha \frac{\pi}{2}}$ for any $\alpha \in (0,1)$. Indeed if $n \in \operatorname{{\mathbb N}}$ then $T^{\frac{1}{n}}$ is the unique $n$th root of $T$ with numerical range in $S_{\frac{\pi}{2 n}}$. See e.g. [@NF Chapter IV, Section 5] and [@Haase] for all of these facts. Some of the following facts are no doubt also in the literature, since we do not know of a reference we sketch short proofs.
\[roots\] For an accretive operator $T \in B(H)$ we have:
- $(cT)^\alpha = c^\alpha T^\alpha$ for positive scalars $c$, and $\alpha \geq 0$.
- $\alpha \mapsto T^{\alpha}$ is continuous on $(0,\infty)$.
- $T^\alpha \in {\rm oa}(T)$, the operator algebra generated by $T$, if $\alpha > 0$.
\(1) This is obvious if $\alpha = \frac{1}{n}$ for $n \in \operatorname{{\mathbb N}}$ by the uniqueness of $n$th roots discussed above. In general it can be proved e.g. by a change of variable in the Balakrishnan representation for powers (see e.g. [@Haase]).
\(2) By a triangle inequality argument, and the inequality for $\Vert T^\alpha - S^\alpha \Vert$ above, we may assume that $T \in {\mathfrak c}_{B(H)}$. By (1) we may assume that $T \in \frac{1}{2} {\mathfrak F}_{B(H)}$. Define $$f(z) = ((1-z)/2)^{\alpha} - ((1-z)/2)^{\beta} \; , \qquad z \in \operatorname{{\mathbb C}}, |z| \leq 1.$$ Via the relation $T^\alpha T^\beta = T^{\alpha + \beta}$ above, we may assume that $\beta \in (0,1]$. Fix such $\beta$. By complex numbers one can show that $|f(z)| \leq g(|\alpha - \beta|)$ on the unit disk, for a function $g$ with $\lim_{t \to 0^+} \, g(t) = 0$. By von Neumann’s inequality, used as in [@BRII Proposition 2.3], we have $$\Vert T^\alpha - T^\beta \Vert= \Vert f(1 - 2T) \Vert \leq g(|\alpha - \beta|).$$ Now let $\alpha \to \beta$.
\(3) We proved this in the second paragraph of [@BRII Section 3] if $\alpha = \frac{1}{n}$ for $n \in \operatorname{{\mathbb N}}$. Hence for $m \in \operatorname{{\mathbb N}}$ we have by the paragraph above the lemma that $T^{\frac{m}{n}} = (T^{\frac{1}{n}})^m \in {\rm oa}(T)$. The general case for $\alpha > 0$ then follows by the continuity in (2).
In particular, ${\mathfrak r}_A$ is closed under taking roots for any operator algebra $A$.
Positivity in operator algebras
===============================
In earlier papers [@BRI; @BRII; @Read] we introduced and studied a new notion of positivity in operator algebras. In this section and the next several sections, we study foundational aspects of this positivity, and of the associated ordering, which we call the ${\mathfrak r}$-[*ordering*]{}.
Let $A$ be an operator algebra, not necessarily approximately unital for the present. Note that ${\mathfrak r}_A = \{ a \in A : a + a^* \geq 0 \}$ is a closed cone in $A$, hence is Archimedean, but it is not proper (hence is what is sometimes called a [*wedge*]{}). On the other hand ${\mathfrak c}_A = \operatorname{{\mathbb R}}_+ {\mathfrak F}_A$ is not closed in general, but it is a a proper cone (that is, ${\mathfrak c}_A \cap (-{\mathfrak c}_A) = (0)$). Indeed suppose $a \in {\mathfrak c}_A \cap (-{\mathfrak c}_A)$. Then $\Vert 1 - t a \Vert
\leq 1$ and $\Vert 1 + s a \Vert
\leq 1$ for some $s, t > 0$. By convexity we may assume $s = t$ (by replacing them by $\min \{ s , t \}$). It is well known that in any Banach algebra with an identity of norm $1$, the identity is an extreme point of the ball. Applying this in $A^1$ we deduce that $a = 0$ as desired.
The ${\mathfrak r}$-[*ordering*]{} is simply the order induced by the above closed cone; that is $b$ is ‘dominated’ by $a$ iff $a - b \in {\mathfrak r}_A$. If $A$ is a subalgebra of an operator algebra $B$, it is clear from a fact mentioned in the introduction (or at the start of [@BRII Section 3]) that the positivity of $a + a^*$ may be computed with reference to any containing $C^*$-algebra, that ${\mathfrak r}_A \subset {\mathfrak r}_B$. If $A, B$ are approximately unital subalgebras of $B(H)$ then it follows from [@BRII Corollary 4.3 (2)] that $A \subset B$ iff ${\mathfrak r}_A \subset {\mathfrak r}_B$. As in [@BRI Section 8], ${\mathfrak r}_A$ contains no idempotents which are not orthogonal projections, and no nonunitary isometries $u$ (since by the analogue of [@BRI Corollary 2.8] we would have $u u^* = s(u u^*) = s(u^* u) = I$). In [@BRII] it is shown that $\overline{{\mathfrak c}_A} = {\mathfrak r}_A$.
For any operator algebra $A$, $x \in {\mathfrak r}_A$ iff ${\rm Re}(\varphi(x)) \geq 0$ for all states $\varphi$ of $A^1$.
Such $\varphi$ extend to states on $C^*(A^1)$. So we may assume that $A$ is a unital $C^*$-algebra, in which case the result is well known ($x + x^* \geq 0$ iff $2 {\rm Re}(\varphi(x)) = \varphi(x+x^*) \geq 0$ for all states $\varphi$).
[**Remark.**]{} For an operator algebra which is not approximately unital, it is not true that $x \in {\mathfrak r}_A$ iff ${\rm Re}(\varphi(x)) \geq 0$ for all states $\varphi$ of $A$, with states defined as in the introduction. An example would be $\operatorname{{\mathbb C}}\oplus \operatorname{{\mathbb C}}$, with the second summand given the zero multiplication.
If $A$ is unital, then $A= {\mathfrak r}_A - {\mathfrak r}_A$; indeed any $a \in {\rm Ball}(A)$ may be written as $\frac{1}{2}(1 + a) - \frac{1}{2}(1 -a) \in
\frac{1}{2} {\mathfrak F}_A - \frac{1}{2} {\mathfrak F}_A$. Hence if $A$ is approximately unital, then $A^{**} = {\mathfrak r}_{A^{**}} - {\mathfrak r}_{A^{**}}$. Since the weak\* closure of ${\mathfrak r}_A$ is ${\mathfrak r}_{A^{**}}$ (see [@BRII Corollary 3.6]), it follows by some variant of the Hahn-Banach theorem that the norm closure of ${\mathfrak r}_A - {\mathfrak r}_A$ is $A$. We shall see next that the norm closure is unnecessary.
\[dualcor\] Let $A$ be an approximately unital operator algebra. Any $x \in A$ with $\Vert x \Vert < 1$ may be written as $x = a-b$ with $a, b \in {\mathfrak r}_A$ and $\Vert a \Vert < 1$ and $\Vert b \Vert < 1$. In fact one may choose such $a, b$ to also be in $\frac{1}{2} {\mathfrak F}_{A}$.
Assume that $\Vert x \Vert = 1$. By what we said above the theorem, $x = \eta - \xi$ for $\eta, \xi \in \frac{1}{2} {\mathfrak F}_{A^{**}}$. By [@BRI Lemma 8.1] we deduce that $x$ is in the weak closure of the convex set $\frac{1}{2} {\mathfrak F}_{A} - \frac{1}{2} {\mathfrak F}_{A}$. Therefore it is in the norm closure, so given $\epsilon > 0$ there exists $a_0, b_0 \in \frac{1}{2} {\mathfrak F}_{A}$ with $\Vert x - (a_0 - b_0) \Vert < \frac{\epsilon}{2}$. Similarly, there exists $a_1, b_1 \in \frac{1}{2} {\mathfrak F}_{A}$ with $\Vert x - (a_0 - b_0) - \frac{\epsilon}{2} (a_1 - b_1) \Vert < \frac{\epsilon}{2^2}$. Continuing in this manner, one produces sequences $(a_k), (b_k)$ in $\frac{1}{2} {\mathfrak F}_{A}$. Setting $a' = \sum_{k=1}^\infty \, \frac{1}{2^k} \, a_k$ and $b' = \sum_{k=1}^\infty \, \frac{1}{2^k} \, b_k$, which are in $\frac{1}{2} {\mathfrak F}_{A}$ since the latter is a closed convex set, we have $x = (a_0 - b_0) + \epsilon (a' - b')$. Let $a = a_0 + \epsilon a'$ and $b = b_0 + \epsilon b'$. By convexity $\frac{1}{1 + \epsilon} a \in \frac{1}{2} {\mathfrak F}_{A}$ and $\frac{1}{1 + \epsilon} b \in \frac{1}{2} {\mathfrak F}_{A}$.
If $\Vert x \Vert < 1$ choose $\epsilon > 0$ with $\Vert x \Vert (1 + \epsilon) < 1$. Then $x/\Vert x \Vert = a - b$ as above, so that $x = \Vert x \Vert \, a - \Vert x \Vert \, b$. We have $$\Vert x \Vert \, a
= (\Vert x \Vert (1 + \epsilon)) \cdot (\frac{1}{1 + \epsilon} a ) \in [0,1) \cdot \frac{1}{2} {\mathfrak F}_{A}
\subset \frac{1}{2} {\mathfrak F}_{A} ,$$ and similarly $\Vert x \Vert \, b \in \frac{1}{2} {\mathfrak F}_{A}$.
In the language of ordered Banach spaces, the above shows that ${\mathfrak r}_A$ and ${\mathfrak c}_A$ are [*generating*]{} cones (this is sometimes called [*positively generating*]{} or [*directed*]{} or [*co-normal*]{}).
[**Remarks.**]{} 1) Can every $x \in {\rm Ball}(A)$ be written as $x = a-b$ with $a,b
\in {\mathfrak r}_A \cap {\rm Ball}(A)$? As we said above, this is true if $A$ is unital. We can show that in general $x \in {\rm Ball}(A)$ cannot be written as $x = a-b$ with $a, b \in \frac{1}{2} {\mathfrak F}_{A}$. To see this let $A$ be the set of functions in the disk algebra vanishing at $-1$, an approximately unital function algebra. Let $W$ be the closed connected set obtained from the unit disk by removing the ‘slice’ consisting of all complex numbers with negative real part and argument in a small open interval containing $\pi$. By the Riemann mapping theorem it is easy to see that there is a conformal map $h$ of the disk onto $W$ taking $-1$ to $0$, so that $h \in {\rm Ball}(A)$. By way of contradiction suppose that $h = a-b$ with $a,b \in \frac{1}{2} {\mathfrak F}_{A}$. Then it is easy to see from the geometry of the circles $B(0,1)$ and $B(\frac{1}{2}, \frac{1}{2})$, that $a + b = 1$ on a nontrivial arc of the unit circle, and hence everywhere. However $a(-1) + b(-1) = 0$, which is the desired contradiction.
2\) Applying Theorem \[dualcor\] to $ix$ for $x \in A$, one gets a similar decomposition $x = a-b$ with the ‘imaginary parts’ of $a$ and $b$ positive. One might ask if, as is suggested by the $C^*$-algebra case, one may write for each $\epsilon$, any $x \in A$ with $\Vert x \Vert < 1$ as $a_1 - a_2 + i(a_3 - a_4)$ for $a_k$ with numerical range in a thin horizontal cigar of height $<
\epsilon$ centered on the line segment $[0,1]$ in the $x$-axis. In fact this is false, as one can see in the case that $A$ is the set of upper triangular $2 \times 2$ matrices with constant diagonal entries.
\[Ahasc\] An operator algebra $A$ has a cai iff $A = {\mathfrak c}_A - {\mathfrak c}_A$.
This follows from Theorem \[dualcor\] and [@BRII Corollary 4.3].
Most of the results in this section apply to approximately unital operator algebras. We offer a couple of results that are useful in applying the approximately unital case to algebras with no approximate identity. We will use the space $A_H$ studied in [@BRII Section 4], this is the largest approximately unital subalgebra of $A$; it is actually a HSA in $A$ (and will be an ideal if $A$ is commutative).
\[Ahasc2\] For any operator algebra $A$, $$A_H = {\mathfrak r}_A - {\mathfrak r}_A = {\mathfrak c}_A - {\mathfrak c}_A.$$ In particular these spaces are closed, and form a HSA of $A$.
In the language of [@BRII Section 4], and using [@BRII Corollary 4.3], ${\mathfrak r}_A = {\mathfrak r}_{A_H}$, and $$A_H = \overline{{\rm Span}}({\mathfrak r}_{A_H}) = {\mathfrak r}_{A_H} - {\mathfrak r}_{A_H}
= {\mathfrak r}_A - {\mathfrak r}_A,$$ by Theorem \[dualcor\]. A similar argument works with ${\mathfrak r}_{A_H}$ replaced by ${\mathfrak c}_{A_H}$ using Corollary \[Ahasc\] and facts from [@BRII Section 4] about ${\mathfrak F}_{A_H}$.
\[mnah\] Let $A$ be any operator algebra. Then for every $n \in \operatorname{{\mathbb N}}$, $$M_n(A_H) = M_n(A)_H \; , \; \; \; \; \; {\mathfrak r}_{M_n(A)} = {\mathfrak r}_{M_n(A_H)}
\; , \; \; \; \; \; {\mathfrak F}_{M_n(A)} = {\mathfrak F}_{M_n(A_H)}$$ (these are the matrix spaces).
Clearly $M_n(A_H)$ is an approximately unital subalgebra of $M_n(A)$. So $M_n(A_H)$ is contained in $M_n(A)_H$, since the latter is the largest approximately unital subalgebra of $M_n(A)$. To show that $M_n(A)_H \subset M_n(A_H)$ it suffices by Corollary \[Ahasc2\] to show that ${\mathfrak r}_{M_n(A)} \subset M_n(A_H)$. So suppose that $a = [a_{ij}] \in M_n(A)$ with $a + a^* \geq 0$. Then $a_{ii} + a_{ii}^* \geq 0$ for each $i$. We also have $\sum_{i,j} \, \bar{z_i} \, (a_{ij} + a_{ji}^*) \, z_j \geq 0$ for all scalars $z_1, \cdots , z_n$. So $\sum_{i,j} \, \bar{z_i} \, a_{ij} z_j \in {\mathfrak r}_A$. Fix an $i, j$, which we will assume to be $1, 2$ for simplicity. Set all $z_k = 0$ if $k \notin \{ i, j \} = \{ 1, 2 \}$, to deduce $$\bar{z_1} z_2 a_{12} + \bar{z_2} z_1 a_{21} = \sum_{i,j = 1}^2 \, \bar{z_i} \, a_{ij} z_j \, - \, (|z_1|^2 a_{11} +
|z_2|^2 a_{22}) \in {\mathfrak r}_A - {\mathfrak r}_A = A_H.$$ Choose $z_1 = 1$; if $z_2 = 1$ then $a_{12} + a_{21} \in A_H$, while if $z_2 = i$ then $i(a_{12} - a_{21}) \in A_H$. So $a_{12}, a_{21} \in A_H$. A similar argument shows that $a_{ij} \in A_H$ for all $i, j$. Thus $M_n(A_H) = M_n(A)_H$, from which we deduce by [@BRII Corollary 4.3 (1)] that $${\mathfrak r}_{M_n(A)} = {\mathfrak r}_{M_n(A)_H} = {\mathfrak r}_{M_n(A_H)}.$$ Similarly ${\mathfrak F}_{M_n(A)} = {\mathfrak F}_{M_n(A)_H}
= {\mathfrak F}_{M_n(A_H)}$.
The last result is used in [@BBS].
We write ${\mathfrak c}^{\operatorname{{\mathbb R}}}_{A^*}$ for the real dual cone of ${\mathfrak r}_A$, the set of continuous $\operatorname{{\mathbb R}}$-linear $\varphi : A \to \operatorname{{\mathbb R}}$ such that $\varphi({\mathfrak r}_A) \subset [0,\infty)$. Since $\overline{{\mathfrak c}_A} = {\mathfrak r}_A$ this is also the real dual cone of ${\mathfrak c}_A$.
We collect some facts about real states. Much of this parallels the development of ordinary states, but we include a brief sketch of the details to save others having to check these each time they are needed in the future. In the rest of this section $A$ is an approximately unital operator algebra. A [*real state*]{} on $A$ will be a contractive $\operatorname{{\mathbb R}}$-linear $\operatorname{{\mathbb R}}$-valued functional on $A$ such that $\varphi(e_t) \to 1$ for some cai $(e_t)$ of $A$. This is equivalent to $\varphi^{**}(1) = 1$, where $\varphi^{**}$ is the canonical $\operatorname{{\mathbb R}}$-linear extension to $A^{**}$, and $1$ is the identity of $A^{**}$ (here we are using the canonical identification between real second duals and complex second duals of a complex Banach space [@Li]). Hence $\varphi(e_t) \to 1$ for every cai $(e_t)$ of $A$.
Since we can identify $A^1$ with $A + \operatorname{{\mathbb C}}1_{A^{**}}$ if we like, by the last paragraph it follows that real states of $A$ extend to real states of $A^1$, hence by the Hahn-Banach theorem they extend to real states of $C^*(A^1)$. We claim that a real state $\psi$ on a $C^*$-algebra $B$ is positive on $B_+$, and is zero on $i B_+$. To see this, we may assume that $B$ is a von Neumann algebra (by extending the state to its second dual similarly to as in the last paragraph). For any projection $p \in B$, $C^*(1,p) \cong \ell^\infty_2$, and it is an easy exercise to see that real states on $\ell^\infty_2$ are positive on $(\ell^\infty_2)_+$ and are zero on $i (\ell^\infty_2)_+$. Thus $\psi(p) \geq 0$ and $\psi(ip) = 0$ for any projection $p$, hence $\psi$ is positive on $B_+$ and zero on $i B_+$ by the Krein-Milman theorem.
We deduce:
\[duals\] Real states on an approximately unital operator algebra $A$ are in ${\mathfrak c}^{\operatorname{{\mathbb R}}}_{A^*}$.
If $a + a^* \geq 0$, and $\tilde{\varphi}$ is the real state extension above to $B = C^*(A^1)$, then $$\varphi(a) = \frac{1}{2} \tilde{\varphi}(a + a^*) +
\frac{1}{2} \tilde{\varphi}(-i \cdot i (a - a^*)) =
\frac{1}{2} \tilde{\varphi}(a + a^*) \geq 0 ,$$ since $i(a-a^*) \in B_{\rm sa} = B_+ - B_+$, and $\tilde{\varphi}(i(B_+ - B_+)) = 0$, as we said above.
\[dualc\] Suppose that $A$ is an approximately unital operator algebra. The real dual cone ${\mathfrak c}^{\operatorname{{\mathbb R}}}_{A^*}$ equals $\{ t \, {\rm Re}(\psi) : \psi \in S(A) , \, t \in [0,\infty) \}$. It also equals the set of restrictions to $A$ of the real parts of positive functionals on any $C^*$-algebra containing (a copy of) $A$ as a closed subalgebra. Also, ${\mathfrak c}^{\operatorname{{\mathbb R}}}_{A^*}$ is a proper cone.
To see that ${\mathfrak c}^{\operatorname{{\mathbb R}}}_{A^*}$ is a proper cone, let $\rho, -\rho \in {\mathfrak c}^{\operatorname{{\mathbb R}}}_{A^*}$. Then $\rho(a) = 0$ for all $a \in {\mathfrak r}_A$, and hence $\rho = 0$ by the fact above that the norm closure of ${\mathfrak r}_A - {\mathfrak r}_A$ is $A$.
The restriction to $A$ of the real part of any positive functional on a $C^*$-algebra containing $A$, is easily seen to be in ${\mathfrak c}^{\operatorname{{\mathbb R}}}_{A^*}$. For the converse, if $\varphi \in {\mathfrak c}^{\operatorname{{\mathbb R}}}_{A^*}$ define $\tilde{\varphi}(a) = \varphi(a) - i \varphi(ia)$ for $a \in A$. As is checked in a basic functional analysis course, $\tilde{\varphi}
\in A^*$ (the complex dual space), and note that $\tilde{\varphi}({\mathfrak r}_A) \subset {\mathfrak r}_{\operatorname{{\mathbb C}}}$. This is also true at the matrix level, since in the notation of the proof of Lemma \[mnah\], $$\sum_{i,j} \overline{z_i} \varphi(a_{ij} + a_{ji}^*) z_j
= \varphi(\sum_{i,j} \overline{z_i} (a_{ij} + a_{ji}^*) z_j) \geq 0 .$$ That is, $\varphi$ is real completely positive in the sense of [@BBS Section 2], and so by that paper $\tilde{\varphi}$ extends to a positive functional $\psi$ on $C^*(A)$, and $\varphi = {\rm Re}(\psi)$.
The remaining statements follow from the well known facts that, first, states of an approximately unital operator algebra $A$ extend to states of any $C^*$-algebra generated by $A$, as we said in the introduction, and second, positive functionals on any $C^*$-algebra $B$ are just the nonnegative multiples of states on $B$.
Below we will use silently a couple of times the obvious fact that if $\varphi, \psi$ are two complex valued functionals on $A$ with ${\rm Re}(\varphi(a)) = {\rm Re}(\psi(a))$ for all $a \in A$, then $\varphi = \psi$ on $A$.
\[dualc2\] Suppose that $A$ is an approximately unital operator algebra.
- If $f \leq g \leq h$ in $B(A,\operatorname{{\mathbb R}})$ in the ${\mathfrak c}_{A^*}$-ordering, then $\Vert g \Vert \leq 2 \max \{ \Vert f \Vert , \Vert h \Vert \}$.
- The cone ${\mathfrak c}_{A^*}$ is [*additive*]{} (that is, the norm on $B(A,\operatorname{{\mathbb R}})$ is additive on ${\mathfrak c}^{\operatorname{{\mathbb R}}}_{A^*}$).
- If $(\varphi_t)$ is an increasing net in ${\mathfrak c}^{\operatorname{{\mathbb R}}}_{A^*}$ which is bounded in norm, then the net converges in norm, and its limit is the least upper bound of the net.
\(1) This may be deduced e.g.from Theorem \[dualcor\] by [@AE Theorem 1.5].
\(2) If $\psi$ is a positive map on $C^*(A)$ and $(e_t)$ is a cai for $A$ then then $$\Vert \psi \Vert = \lim_t \psi(e_t) = \lim_t \, {\rm Re} \, \psi(e_t)
\leq \Vert {\rm Re} \, \psi_{\vert A} \Vert \leq \Vert \psi_{\vert A} \Vert
\leq \Vert \psi \Vert.$$ Hence $\Vert \varphi \Vert = \langle 1 , \varphi \rangle$ for all $\varphi \in {\mathfrak c}^{\operatorname{{\mathbb R}}}_{A^*}$, and it follows that the norm on $B(A,\operatorname{{\mathbb R}})$ is additive on ${\mathfrak c}^{\operatorname{{\mathbb R}}}_{A^*}$.
\(3) Follows from [@AE Proposition 3.2].
[**Remarks.**]{} 1) In the language of ordered Banach spaces, the norm estimate in Lemma \[dualc2\] (1) is saying that ${\mathfrak c}_{A^*}$ is a (2-)[*normal cone*]{}. The best constant here is $2$; it is not hard to show there exist approximately unital algebras $A$ with ${\mathfrak c}^{\operatorname{{\mathbb R}}}_{A^*}$ not $C$-normal for any $C < 2$. Indeed if it was, then ${\mathfrak r}_{A}$ would be $C'$-directed for some $C' < 2$ by [@AE Theorem 1.5]. However if $A = A(\operatorname{{\mathbb D}})$, the disk algebra, it is easy to see that if $z = f-g$ for $f, g \in A$ having positive real part, then Re$(f(1)) \geq 1$. Hence $\Vert f \Vert \geq 1$. Similarly, $\Vert g \Vert \geq 1$, so that $\Vert f \Vert + \Vert g \Vert \geq 2$.
2\) It is probably never true for an approximately unital operator algebra $A$ that $B(A,\operatorname{{\mathbb R}}) = {\mathfrak c}^{\operatorname{{\mathbb R}}}_{A^*} - {\mathfrak c}^{\operatorname{{\mathbb R}}}_{A^*}$. Indeed, in the case $A = \operatorname{{\mathbb C}}$ the latter space has real dimension $1$. However the complex span of the (usual) states of an approximately unital operator algebra $A$ is $A^*$ (the complex dual space). Indeed by a result of Moore [@Moore; @AE2], the complex span of the states of any unital Banach algebra $A$ is $A^*$. In the approximately unital operator algebra case one can prove this same fact by a reduction to the unital case by noting that by the argument in the second last paragraph of [@BRII Section 4], the span of the states on $A$ is the span of the restrictions to $A$ of the states on $A^1$.
3\) Every element $x \in \frac{1}{2}
{\mathfrak F}_A$ need not achieve its norm at a state, even in $M_2$ (consider $x = (I + E_{12})/2$ for example).
\[rsaj\] The real states on an approximately unital operator algebra $A$ are just the real parts of ordinary states on $A$.
Certainly the real part of an ordinary state is a real state. If $\varphi$ is a real state on $A$, then by Lemma \[duals\] and Lemma \[dualc\] we have $\varphi = t {\rm Re} \, \psi$ for a state $\psi$ on $A$ which is the restriction of a state on $C^*(A)$. In the proof of Lemma \[dualc2\] we saw that $\Vert {\rm Re} \, \psi \Vert = 1$, so that $t = 1$.
\[uext\] Any real state on an approximately unital closed subalgebra $A$ of an approximately unital operator algebra $B$ extends to a real state on $B$. If $A$ is a HSA in $B$ then this extension is unique.
The first part f is as in [@Ped Proposition 3.1.6]. Suppose that $A$ is a HSA in $B$ and that $\varphi_1, \varphi_2$ are real states on $B$ extending a real state on $A$. By the above we may write $\varphi_i = {\rm Re} \, \psi_i$ for ordinary states on $B$. Since $\varphi_1 = \varphi_2$ on $A$ we have $\psi_1 = \psi_2$ on $A$. Hence $\psi_1 = \psi_2$ on $B$ by [@BHN Theorem 2.10]. So $\varphi_1 = \varphi_2$ on $B$.
\[dualco1\] Let $A$ be an approximately unital operator algebra. The second dual cone of ${\mathfrak r}_A$ (that is, the (real) dual cone of ${\mathfrak c}^{\operatorname{{\mathbb R}}}_{A^*}$) is ${\mathfrak r}_{A^{**}}$. The (real) predual cone of the dual cone ${\mathfrak c}^{\operatorname{{\mathbb R}}}_{A^*}$ is ${\mathfrak r}_A$.
We use Lemma \[dualc\]. Suppose that $a \in A$ with ${\rm Re} \, \varphi(a) \geq 0$ for all states $\varphi$ on $C^*(A)$. Then $\varphi(a + a^*) \geq 0$ for all states $\varphi$, so that $a + a^* \geq 0$.
A similar proof works if $a \in A^{**}$ to yield the first assertion. Indeed if Re$ \, a(\varphi) \geq 0$ for all states $\varphi$ on $A$, or equivalently if $(a + a^*)(\varphi) \geq 0$ for all states $\varphi$ on $C^*(A)$, then $a + a^* \geq 0$ in $C^*(A)^{**}$, so that $a \in {\mathfrak r}_{A^{**}}$.
It follows immediately from Theorem \[dualcor\] that any approximately unital operator algebra $A$ is a directed set. That is, if $x, y \in A$ then there exists $z \in A$ with $z-x, z-y \in {\mathfrak r}_A$. In fact more is true. We recall that the positive part of the open unit ball of a $C^*$-algebra is a directed set. The following generalizes this to operator algebras:
\[dirset\] If $A$ is an approximately unital operator algebra then the open unit ball of $A$ is a directed set with respect to the ${\mathfrak r}$-ordering. That is, if $x, y \in A$ with $\Vert x \Vert , \Vert y \Vert < 1$, then there exists $z \in A$ with $\Vert z\Vert < 1$ and $z-x, z-y \in {\mathfrak r}_A$.
This follows from Lemma \[dualc2\] and [@AE Corollary 3.6].
For a $C^*$-algebra $B$, a natural ordering on the open unit ball of $B$ turns the latter into a net which is a positive cai for $B$ (see e.g. [@Ped]). We do not know if something similar is true for general operator algebras via Corollary \[dirset\].
Some results on positivity in operator algebras {#morp}
===============================================
In this section we collect several useful facts concerning our positivity, some of which are used in this paper, and some in forthcoming work.
The ${\mathfrak F}$-transform
-----------------------------
In [@BRII] the sets $\frac{1}{2}{\mathfrak F}_{A}$ and ${\mathfrak r}_{A}$ were related by a certain transform. We now establish a few more basic properties of this transform. The Cayley transform $\kappa(x) = (x-I)(x+I)^{-1}$ of an accretive $x \in A$ exists since $-1 \notin {\rm Sp}(x)$, and is well known to be a contraction. Indeed it is well known (see e.g. [@NF]) that if $A$ is unital then the Cayley transform maps ${\mathfrak r}_A$ bijectively onto the set of contractions in $A$ whose spectrum does not contain $1$, and the inverse transform is $T \mapsto (I+T) (I - T)^{-1}$. The Cayley transform maps the accretive elements $x$ with ${\rm Re}(x) \geq \epsilon 1$ for some $\epsilon > 0$, onto the set of elements $T \in A$ with $\Vert T \Vert < 1$ (see e.g. 2.1.14 in [@BLM]). The ${\mathfrak F}$-transform ${\mathfrak F}(x) = 1 - (x+1)^{-1} = x (x+1)^{-1}$ may be written as ${\mathfrak F}(x) = \frac{1}{2} (1 + \kappa(x))$. Equivalently, $\kappa(x) = -(1 - 2 {\mathfrak F}(x))$.
For any operator algebra $A$, the ${\mathfrak F}$-transform maps ${\mathfrak r}_{A}$ bijectively onto the set of elements of $\frac{1}{2}{\mathfrak F}_{A}$ of norm $< 1$.
First assume that $A$ is unital. By the last equations ${\mathfrak F}({\mathfrak r}_A)$ is contained in the set of elements of $\frac{1}{2}{\mathfrak F}_{A}$ whose spectrum does not contain $1$. The inverse of the ${\mathfrak F}$-transform on this domain is $T (I-T)^{-1}$. To see for example that $T (I-T)^{-1} \in {\mathfrak r}_A$ if $T \in \frac{1}{2}{\mathfrak F}_{A}$ note that 2Re$(T (I-T)^{-1})$ equals $$(I-T^*)^{-1}(T^*(I-T) + (I-T^*) T) (I-T)^{-1} = (I-T^*)^{-1}(T + T^* - 2T^* T) (I-T)^{-1}$$ which is positive since $T^* T$ is dominated by Re$(T)$ if $T \in \frac{1}{2}{\mathfrak F}_{A}$. Hence for any (possibly nonunital) operator algebra $A$ the ${\mathfrak F}$-transform maps ${\mathfrak r}_{A^1}$ bijectively onto the set of elements of $\frac{1}{2}{\mathfrak F}_{A^1}$ whose spectrum does not contain $1$. However this equals the set of elements of $\frac{1}{2}{\mathfrak F}_{A^1}$ of norm $< 1$. Indeed if $\Vert {\mathfrak F}(x) \Vert = 1$ then $\Vert \frac{1}{2} (1 + \kappa(x)) \Vert = 1$, and so $1 - \kappa(x)$ is not invertible by [@ABS Proposition 3.7]. Hence $1 \in {\rm Sp}_{A^1}(\kappa(x))$ and $1 \in {\rm Sp}_A({\mathfrak F}(x))$. Since ${\mathfrak F}(x) \in A$ iff $x \in A$, we are done.
Thus in some sense we can identify ${\mathfrak r}_{A}$ with the strict contractions in $\frac{1}{2}{\mathfrak F}_{A}$. This for example induces an order on this set of strict contractions.
Roots of accretive elements
---------------------------
\[rootf\] Let $A$ be an operator algebra, and $x \in A$.
- If the numerical range of $x$ is contained in a sector $S_{\rho}$ for $\rho < \frac{\pi}{2}$ (see notation above Lemma [\[roots\]]{}), then $x/\Vert {\rm Re}(x) \Vert
\in \frac{\sec^2 \rho}{2} \, {\mathfrak F}_A$. So $x \in {\mathfrak c}_A$.
- If $x \in {\mathfrak r}_A$ then $x^\alpha \in {\mathfrak c}_A$ for any $\alpha \in (0,1)$.
\(1) Write $x = a + ib$, for positive $a$ and selfadjoint $b$ in a containing $B(H)$. By the argument in the proof of [@BRI Lemma 8.1], there exists a selfadjoint $c \in B(H)$ with $b = a^{\frac{1}{2}} c a^{\frac{1}{2}}$ and $\Vert c \Vert \leq \tan \rho$. Then $x = a^{\frac{1}{2}} (1 + ic) a^{\frac{1}{2}}$, and $$x^* x = a^{\frac{1}{2}} (1 + ic)^* a (1 + ic) a^{\frac{1}{2}} \leq
C a.$$ By the $C^*$-identity $\Vert (1 + ic)^* a (1 + ic) \Vert$ equals $$\Vert a^{\frac{1}{2}} (1 + ic) (1 + ic)^* a^{\frac{1}{2}} \Vert
\leq \Vert a \Vert (1 + \Vert c \Vert^2) \leq
\Vert a \Vert (1 + \tan^2 \rho) = \Vert a \Vert \sec^2 \rho.$$ So we can take $C = \Vert a \Vert \, \sec^2 \rho$. Saying that $x^* x \leq
C {\rm Re}(x)$ is the same as saying that $x \in \frac{C}{2} {\mathfrak F}_A$.
\(2) This follows from (1) since in this case the numerical range of $x^\alpha$ is contained in a sector $S_{\rho}$ with $\rho < \frac{\pi}{2}$.
[**Remark.**]{} The last result is related to the remark before [@BRI Lemma 8.1].
Of course $\Vert {\rm Im}(x^{\frac{1}{n}}) \Vert \to 0$ as $n \to \infty$, for $x \in {\mathfrak r}_A$ (as is clear e.g. from the above).
For $x \in {\mathfrak r}_A$, unlike the ${\mathfrak F}_A$ case, we do not have $||x^{1/m}|| \leq ||x||^{1/m}$. We are indebted to Christian Le Merdy for the example $$\left[ \begin{array}{ccl} 1 & i \\ i & 0 \end{array} \right].$$ Hence if $||x|| \leq 1$ we cannot say that $||x^{1/m}|| \leq 1$ always. However we have:
\[Bal\] If $||x|| \leq 1$ and $x \in {\mathfrak r}_A$, for an operator algebra $A$, then $||x^{1/m}||
\leq \frac{m^2}{(m-1) \pi} \sin(\frac{\pi}{m}) \leq \frac{m}{m-1}$. More generally, $||x^{\alpha}|| \leq \frac{\sin(\alpha \pi)}{\pi \alpha (1 - \alpha)}$ if $0 < \alpha < 1$. Hence for any $\alpha \in (0,1)$ there is a $\delta > 0$ with $\Vert x \Vert \leq \delta$ implying $||x^{\alpha} || \leq 1$ for all $x \in {\mathfrak r}_A$.
This follows from the well known A.V. Balakrishnan representation of powers $x^\alpha$ as $\frac{\sin(\alpha \pi)}{\pi} \int_0^\infty \, t^{\alpha - 1} \, (t + x)^{-1} x \, dt$ (see e.g. [@Haase]). If we use the simple fact that $\Vert (t + x)^{-1} \Vert \leq \frac{1}{t}$ for accretive operators $x$, and $$\Vert (t + x)^{-1} x \Vert = \Vert (1 + \frac{x}{t})^{-1} \frac{x}{t} \Vert
= \Vert {\mathfrak F}(\frac{x}{t}) \Vert \leq 1,$$ then the norm of $x^\alpha$ is dominated by $$\frac{\sin(\alpha \pi)}{\pi} (\int_0^1 t^{\alpha - 1} \, \cdot 1 dt + \int_1^\infty \, t^{\alpha - 1}
\frac{1}{t} \, dt ) = \frac{\sin(\alpha \pi)}{\pi \alpha (1 - \alpha)}.$$ The rest is clear from this.
\[strsq\] If $\alpha \in (0,1)$ then there exists a constant $K$ such that if $a, b \in {\mathfrak r}_{B(H)}$ for a Hilbert space $H$, and $ab = ba$, then $\Vert (a^{\alpha} - b^\alpha) \zeta \Vert
\leq K \Vert (a-b) \zeta \Vert^\alpha$, for $\zeta \in H$.
By the Balakrishnan representation in the last proof, if $\zeta \in {\rm Ball}(H)$ we have $$(a^{\alpha} - b^\alpha) \zeta =
\frac{\sin(\alpha \pi)}{\pi} \int_0^\infty \, t^{\alpha - 1} \, [(t + a)^{-1} a - (t + b)^{-1} b] \zeta \, dt.$$ By the inequality $\Vert (t + x)^{-1} \Vert \leq \frac{1}{t}$ for accretive operators $x$, we have $$\Vert [(t + a)^{-1} a - (t + b)^{-1} b] \zeta \Vert =
\Vert (t + a)^{-1} (t + b)^{-1} (a-b) t \zeta \Vert
\leq \frac{1}{t} \Vert (a-b) \zeta \Vert ,$$ and so as in the proof of Lemma \[Bal\], $\Vert \int_0^\infty \, t^{\alpha - 1} \, [(t + a)^{-1} a - (t + b)^{-1} b] \zeta \, dt \Vert$ is dominated by $$2 \int_0^\delta \, t^{\alpha - 1} \, dt + \int_{\delta}^\infty \, t^{\alpha - 2} \, dt \, \Vert (a-b) \zeta \Vert
= \frac{2}{\alpha} \delta^{\alpha} + \frac{\delta^{\alpha -1}}{1-\alpha} \, \Vert (a-b) \zeta \Vert$$ for any $\delta > 0$. We may now set $\delta = \Vert (a-b) \zeta \Vert$ to obtain our inequality.
[**Remark.**]{} The proof above uses a trick from [@MP Theorem 1], and perhaps can be extended to the class of operators considered there.
\[vpow\] If $a \in {\mathfrak r}_A$ for an operator algebra $A$, and $v$ is a partial isometry in any containing $C^*$-algebra $B$ with $v^* v = s(a)$, then $v a v^* \in {\mathfrak r}_B$ and $(v a v^*)^r = v a^r v^*$ if $r \in (0,1) \cup \operatorname{{\mathbb N}}$.
This is clear if $r = k \in \operatorname{{\mathbb N}}$. It is also clear that $v a v^* \in {\mathfrak r}_B$. We will use the Balakrishnan representation above to check that $(v a v^*)^r = v a^r v^*$ if $r \in (0,1)$ (it can also be deduced from the ${\mathfrak F}_A$ case in [@BNII]). Claim: $(t + v a v^*)^{-1} v a v^* = v (t + a)^{-1} a v^*$. Indeed since $v^* v a = a$ we have $$(t + v a v^*) v (t + a)^{-1} a v^*
= v (t+a) (t + a)^{-1} a v^* = v a v^*,$$ proving the Claim. Hence for any $\zeta, \eta \in H$ we have $$\langle (t + v a v^*)^{-1} v a v^* \zeta, \eta \rangle
= \langle v (t + a)^{-1} a v^* \zeta, \eta \rangle
= \langle (t + a)^{-1} a v^* \zeta, v^* \eta \rangle.$$ Hence by the Balakrishnan representation $\langle (v a v^*)^r \zeta, \eta \rangle$ equals $$\frac{\sin(r \pi)}{\pi} \int_0^\infty \, t^{r - 1} \, \langle
(t + vav^*)^{-1} v a v^* \zeta , \eta \rangle \, dt =
\frac{\sin(r \pi)}{\pi} \int_0^\infty \, t^{r - 1} \, \langle (t + a)^{-1} a v^* \zeta, v^* \eta \rangle \, dt,$$ which equals $\langle v a^r v^* \zeta, \eta \rangle$, as desired.
The last result generalizes [@BNI Lemma 1.4]. With the last few results in hand, particularly Lemma \[Bal\] and Lemma \[vpow\], it appears that all of the results in [@BNI] stated in terms of ${\mathfrak F}_A$ (or $\frac{1}{2} {\mathfrak F}_A$ or ${\mathfrak c}_A$), should generalize without problem to the ${\mathfrak r}_A$ case. We admit that we have not yet carefully checked every part of every result in [@BNI] for this though, but hope to in forthcoming work.
Commuting operators in ${\mathfrak r}_A$
----------------------------------------
If $S \subset {\mathfrak r}_A$, for an operator algebra $A$, and if $xy = yx$ for all $x, y \in S$, write oa$(S)$ for the smallest closed subalgebra of $A$ containing $S$.
\[commt\] If $S$ is a commuting subset of ${\mathfrak r}_A$ then ${\rm oa}(S)$ has a cai.
Let $C = {\rm oa}(S)$. Then $C$ contains oa$(x)$ for each $x \in S$, and hence $C^{\perp \perp}$ contains $s(x)$. Thus $C^{\perp \perp}$ contains $p = \vee_{x \in S} \, s(x))$ (by the comments on ‘meets’ and ‘joins’ on p. 190 of [@BRI]). Now it is easy to see that $p$ is an identity for $C^{\perp \perp}$, so that $C$ has a cai (by e.g. [@BLM Proposition 2.5.8]).
If $x, y$ are commuting elements of $\frac{1}{2} {\mathfrak F}_A$ or ${\mathfrak r}_A$, for an operator algebra $A$, then it is not necessarily true that $xy$ has numerical range excluding the negative real axis. (Indeed this can be false even in $M_2$, and even if the ‘imaginary parts’ of $x$ and $y$ have tiny norms compared to their ‘real parts’, so that their numerical range consists of numbers in the unit disk with argument very close to zero. Here $x$ and $y$ are very close to being positive). Thus one cannot define $(xy)^{\frac{1}{2}}$ as in e.g. [@LRS]; rather in such cases one should define $(xy)^{\frac{1}{2}}$ to be $x^{\frac{1}{2}} y^{\frac{1}{2}}$. It is proved in [@BBS] that $(xy)^{\frac{1}{2}}$ is again in $\frac{1}{2} {\mathfrak F}_A$. More generally, the product of the $n$th roots of $n$ mutually commuting members of $\frac{1}{2} {\mathfrak F}_A$ (resp.${\mathfrak r}_A$) is again in $\frac{1}{2} {\mathfrak F}_A$ (resp. in ${\mathfrak r}_A$). This fact has a nice application in Section \[pury\] below.
Concavity, monotonicity, and operator inequalities
--------------------------------------------------
The usual operator concavity/convexity results for $C^*$-algebras seem to fail for the ${\mathfrak r}$-ordering. That is, results of the type in [@Ped Proposition 1.3.11] and its proof fail. Indeed, functions like Re$(z^{\frac{1}{2}}),
{\rm Re}(z (1+z)^{-1}), {\rm Re}(z^{-1})$ are not operator concave or convex, even for operators $x, y \in \frac{1}{2} {\mathfrak F}_A$. In fact this fails even in the simplest case $A = \operatorname{{\mathbb C}}$, taking $x = \frac{1}{2}, y = \frac{1+i}{2}$. Similar remarks hold for ‘operator monotonicity’ with respect to the ${\mathfrak r}_A$-ordering for these functions.
For the ${\mathfrak r}$-ordering, a way one can often prove operator inequalities, or that something is increasing, is as follows. Suppose for example that $f, g$ are functions in the disk algebra, with Re$(f(z)) \leq {\rm Re}(g(z))$ for all $z \in \operatorname{{\mathbb C}}$ with $|z| \leq 1$. If $x \in \frac{1}{2} {\mathfrak F}_A$, for an operator algebra $A$, then we can deduce that Re$(f(1-2x)) \leq {\rm Re}(g(1-2x))$. Here e.g. $f(1-2x)$ is the ‘disk algebra functional calculus’, arising from von Neumann’s inequality for the contraction $1-2x$. The reason for this is that ${\rm Re}(g(z) - f(z)) \geq 0$ on the disk, and so by [@NF Proposition 3.1, Chapter IV] we have that ${\rm Re}(g(1-2x) - f(1-2x)) \geq 0.$
As an illustration of this principle, it follows easily by this idea that for any $x \in \frac{1}{2} {\mathfrak F}_A$, the sequence $({\rm Re}(x^{\frac{1}{n}}))$ is increasing (see [@BBS]). The last fact is another example of $\frac{1}{2} {\mathfrak F}_A$ behaving better than ${\mathfrak r}_A$: for contractions $x \in {\mathfrak r}_A$, we do not in general have $({\rm Re} (x^{1/m}))$ increasing with $m$. The matrix example just above Lemma \[Bal\] will demonstrate this. However one can show that for any $x \in {\mathfrak r}_A$ there exists a constant $c > 0$ such that $({\rm Re} ((x/c)^{1/m}))_{m \geq 2}$ is increasing with $m$. Indeed if $c = (2 \Vert {\rm Re}(x^{\frac{1}{2}}) \Vert)^2$, then by Lemma \[rootf\] (2) we have $(x/c)^{\frac{1}{2}} \in \frac{1}{2} {\mathfrak F}_A$. Thus ${\rm Re} ((x/c)^{t})$ increases as $t \searrow 0$ (see the proof of the [@BBS Proposition 3.4]), from which the desired assertion follows.
\[incai\] If $A$ is a separable approximately unital operator algebra, then $A$ has a commuting cai which is increasing in the ${\mathfrak r}$-ordering.
As in [@BRI Corollary 2.18], there is a $x \in \frac{1}{2} {\mathfrak F}_A$ such that $(x^{1/n})$ is a commuting cai. The result now follows by the fact in the last paragraph.
We do not know if this is true in the nonseparable case. See also the remarks after Corollary \[dirset\].
Finally, we clarify a few imprecisions in a couple of the positivity results in [@BRI; @BRII]. At the end of Section 4 of [@BRII], states on a nonunital algebra should probably also be assumed to have norm 1 (although the arguments there do not need this). In [@BRI Proposition 4.3] we should have explicitly stated the hypothesis that $A$ is approximately unital. There are some small typo’s in the proof of [@BRI Theorem 2.12] but the reader should have no problem correcting these.
Strictly real positive elements {#morp2}
===============================
An element in $A$ with ${\rm Re}(\varphi(x)) > 0$ for all states on $A^1$ whose restriction to $A$ is nonzero, will be called [*strictly real positive*]{}. Such $x$ are in $ {\mathfrak r}_A$. This includes the $x \in A$ with Re$(x)$ strictly positive in some $C^*$-algebra generated by $A$. If $A$ is approximately unital, then these conditions are in fact equivalent, as the next result shows. Thus the definition of strictly real positive here generalizes the definition given in [@BRI] for approximately unital operator algebras.
\[secto\] Let $A$ be an approximately unital operator algebra, which generates a $C^*$-algebra $C^*(A)$. An element $x \in A$ is strictly real positive in the sense above iff ${\rm Re}(x)$ is strictly positive in $C^*(A)$.
The one direction follows because any state on $A^1$ whose restriction to $A$ is nonzero, extends to a state on $C^*(A)^1$ which is nonzero on $C^*(A)$. The restriction to $C^*(A)$ of the latter state is a positive multiple of a state.
For the other direction recall that we showed in the introduction that any state on $C^*(A)$ gives rise to a state on $A^1$. Since any cai of $A$ is a cai of $C^*(A)$, the latter state cannot vanish on $A$.
[**Remark.**]{} Note that if ${\rm Re}(x) \geq \epsilon 1$ in $C^*(A)^1$, then there exists a constant $C > 0$ with ${\rm Re}(x) \geq \epsilon 1 \geq C x^* x$, and it follows that $x \in \operatorname{{\mathbb R}}_+ {\mathfrak F}_A$. Thus if $A$ is unital then every strictly real positive in $A$ is in $\operatorname{{\mathbb R}}_+ {\mathfrak F}_A$. However this is false if $A$ is approximately unital (it is even easily seen to be false in the $C^*$-algebra $A = c_0$). Conversely, note that if $A$ is an approximately unital operator algebra with no r-ideals and no identity, then every nonzero element of $\operatorname{{\mathbb R}}_+ {\mathfrak F}_A$ is strictly real positive by [@BRI] Theorem 4.1.
We also remark that it is tempting to define an element $x \in A$ to be strictly real positive if Re$(x)$ strictly positive in some $C^*$-algebra generated by $A$. However this definition can depend on the particular generated $C^*$-algebra, unless one only uses states on the latter that are not allowed to vanish on $A$ (in which case it is equivalent to other definition). As an example of this, consider the algebra of $2 \times 2$ matrices supported on the first row, and the various $C^*$-algebras it can generate.
We next discuss how results in [@BRI] generalize, particularly those related to strict real positivity if we use the definition at the start of the present section. We will need some of this in Section \[fgsect\] below. We recall that in [@BRI], many ‘positivity’ results were established for elements in ${\mathfrak F}_A$ or $\frac{1}{2} {\mathfrak F}_A$, and by extension for the proper cone ${\mathfrak c}_A =
\operatorname{{\mathbb R}}_+ {\mathfrak F}_A$. In [@BRII Section 3] we pointed out several of these facts that generalized to the larger cone ${\mathfrak r}_A$, and indicated that some of this would be discussed in more detail in [@BBS]. In [@BRII Section 4] we pointed out that the hypothesis in many of these results that $A$ be approximately unital could be simultaneously relaxed. In the next few paragraphs we give a few more details, that indicate the similarities and differences between these cones, particularly focusing on the results involving strictly real positive elements. The following list should be added to the list in [@BRII Section 3], and some complementary details are discussed in [@BBS].
In [@BRI Lemma 2.9] the ($\Leftarrow$) direction is correct for $x \in {\mathfrak r}_A$ with the same proof. Also one need not assume there that $A$ is approximately unital, as we said towards the end of Section 4 in [@BRII]. The other direction is not true in general (not even in $A = \ell^\infty_2$, see example in [@BBS]), but there is a partial result, Lemma \[sect\] below.
In [@BRI Lemma 2.10], (v) implies (iv) implies (iii) (or equivalently (i) or (ii)), with ${\mathfrak r}_A$ in place of ${\mathfrak F}_A$, using the ${\mathfrak r}_A$ version above of the ($\Leftarrow$) direction of [@BRI Lemma 2.9], and [@BRII Theorem 3.2] (which gives $s(x) = s({\mathfrak F}(x))$). However none of the other implications in that lemma are correct, even in $\ell^\infty_2$.
Proposition 2.11 and Theorem 2.19 of [@BRI] are correct in their ${\mathfrak r}_A$ variant, which should be phrased in terms of strictly real positive elements in ${\mathfrak r}_A$ as defined above at the start of the present section. Indeed this variant of Proposition 2.11 is true even for nonunital algebras if in the proof we replace $C^*(A)$ by $A^1$. Theorem 2.19 of [@BRI] may be seen using the parts of [@BRI Lemma 2.10] which are true for ${\mathfrak r}_A$ in place of ${\mathfrak F}_A$, and [@BRII Theorem 3.2] (which gives $s(x) = s({\mathfrak F}(x))$). Lemma 2.14 of [@BRI] is clearly false even in $\operatorname{{\mathbb C}}$, however it is true with essentially the same proof if the elements $x_k$ there are strictly real positive elements, or more generally if they are in ${\mathfrak r}_A$ and their numerical ranges in $A^1$ intersects the imaginary axis only possibly at $0$. Also, this does not effect the correctness of the important results that follow it in [@BRI Section 2]. Indeed as stated in [@BRII], all descriptions of r-ideals and $\ell$-ideals and HSA’s from [@BRI] are valid with ${\mathfrak r}_A$ in place of ${\mathfrak F}_A$, sometimes by using [@BRII Corollaries 3.4 and 3.5]). We remark that Proposition 2.22 of [@BRI] is clearly false with ${\mathfrak F}_A$ replaced by ${\mathfrak r}_A$, even in $\operatorname{{\mathbb C}}$.
Similarly, in [@BRI] Theorem 4.1, (c) implies (a) and (b) there with ${\mathfrak r}_A$ in place of ${\mathfrak F}_A$. However the Volterra algebra [@BRI Example 4.3] is an example where (a) in [@BRI] Theorem 4.1 holds but not (c) (note that the Volterra operator $V \in {\mathfrak r}_A$, but $V$ is not strictly real positive in $A$). The results in Section 3 of [@BRI] will be discussed later in Section \[fgsect\].
\[sect\] In an operator algebra $A$, suppose that $x \in {\mathfrak r}_A$ and either $x$ is strictly real positive, or the numerical range $w(x)$ of $x$ in $A^1$ is contained in a sector $S_\psi$ of angle $\psi < \pi/2$ (see notation above Lemma [\[roots\]]{}). If $\varphi$ is a state on $A$ or more generally on $A^1$, then $\varphi(s(x)) = 0$ iff $\varphi(x) = 0$.
The one direction is as in [@BRI Lemma 2.9] as mentioned above. The strictly real positive case of the other direction is obvious (but non-vacuous in the $A^1$ case). In the remaining case, write $\varphi = \langle \pi(\cdot) \xi , \xi \rangle$ for a unital $*$-representation $\pi$ of $C^*(A^1)$ on a Hilbert space $H$, and a unit vector $\xi \in H$. Then $w(\pi(x))$ is contained in a sector of the same angle. By Lemma 5.3 in Chapter IV of [@NF] we have $\Vert \pi(x) \xi \Vert^2
= \varphi(x^* x) = 0$. As e.g. in the proof of [@BRI Lemma 2.9] this gives $\varphi(s(x)) = 0$.
\[sx\] Let $x \in {\mathfrak r}_A$ for an operator algebra $A$. If $\varphi(x^{\frac{1}{n}}) = 0$ for some $n \in \operatorname{{\mathbb N}}, n \geq 2$, and state $\varphi$ on $A$, then $\varphi(s(x)) = 0$ and $\varphi(x^{\frac{1}{m}}) = 0$ for all $m \in \operatorname{{\mathbb N}}$. Thus if $\varphi(s(x)) \neq 0$ for a state $\varphi$ on $A$, then ${\rm Re}(\varphi(x^{\frac{1}{n}})) > 0$ for all $n \in \operatorname{{\mathbb N}}, n \geq 2$.
It is clear that $s(x) = s(x^{\frac{1}{m}})$ for all $m \in \operatorname{{\mathbb N}}$, by using for example the fact from [@BRII Section 3] that $x^{\frac{1}{n}} \to s(x)$ weak\*. Since the numerical range of $x^{\frac{1}{n}}$ in $A^1$ is contained in a sector centered on the positive real axis of angle $< \pi$, $\varphi(s(x)) = \varphi(s(x^{\frac{1}{n}})) = 0$ by Lemma \[sect\]. As we said above, this implies that $\varphi(x) = 0$, and the same argument applies with $x$ replaced by $x^{\frac{1}{m}}$ to give $\varphi(x^{\frac{1}{m}}) = 0$.
The last statement follows from this, since ${\rm Re}(\varphi(x^{\frac{1}{n}})) > 0$ is equivalent to $\varphi(x^{\frac{1}{n}}) \neq 0$ if $n \geq 2$.
[**Remark.**]{} Examining the proofs of the last three results show that they are valid if states on $A$ are replaced by nonzero functionals that extend to states on $A^1$, or equivalently extend to a $C^*$-algebra generated by $A^1$.
\[sx3\] In an operator algebra $A$, if $x \in {\mathfrak r}_A$ and $x$ is strictly real positive, then $x^{\frac{1}{n}}$ is strictly real positive for all $n \in \operatorname{{\mathbb N}}$.
If $x^{\frac{1}{n}}$ is not strictly real positive for some $n \geq 2$, then $\varphi(x^{\frac{1}{n}}) = 0$ for some state $\varphi$ of $A^1$ which is nonzero on $A$. Such a state extends to a state on $C^*(A^1)$. By the last Remark, $\varphi(x) = 0$ by Corollary \[sx\], a contradiction.
Principal $r$-ideals {#fgsect}
====================
Section 3 in our earlier paper [@BRI] was entitled “When $xA$ and $Ax$ are closed". We showed there that if $x \in {\mathfrak F}_A$ then both $xA$ and $Ax$ are closed iff $x$ is pseudo-invertible (that is, there exists $y \in A$ with $xy x = x$). We can improve this result as follows:
\[ws\] For an operator algebra $A$, if $x \in {\mathfrak r}_A$, then the following are equivalent:
- $s(x) \in A$,
- $xA$ is closed,
- $Ax$ is closed,
- $x$ is pseudo-invertible in $A$,
- $x$ is invertible in ${\rm oa}(x)$.
Moreover, all of Theorem [3.2]{} of [@BRI] is valid for $x \in {\mathfrak r}_A$.
We recall that $(x^{\frac{1}{m}})_{m \in \operatorname{{\mathbb N}}}$ is a bai for ${\rm oa}(x)$, by [@BRII Theorem 3.1], and it has weak\* limit $s(x) \in {\rm oa}(x)^{\perp \perp} \subset A^{**}$.
\(ii) $\Rightarrow$ (i) Suppose $xA$ is closed. Then $$x^{\frac{1}{2}} \in {\rm oa}(x) \subset \overline{x {\rm oa}(x)} \subset
\overline{xA} = xA,$$ so $x^{\frac{1}{2}} = xy$ for some $y \in A$. Thus if $z = x^{\frac{1}{2}} y \in A$ then $x = x^{\frac{1}{2}} x y = x z$, and so $a = a z$ for every $a \in {\rm oa}(x)$. Now $s(x) z = z$ since $x^{\frac{1}{2}} \in {\rm oa}(x)$ for example. On the other hand $s(x) z = s(x)$ since $s(x)$ is a weak\* limit of the bai in ${\rm oa}(x)$. Thus $s(x) = z \in A$.
\(i) $\Rightarrow$ (iv) Since $s(x)$ is the identity of ${\rm oa}(x)^{**}$, (i) is equivalent to ${\rm oa}(x)$ being unital. This implies by the Neumann lemma that $x$ is invertible in ${\rm oa}(x)$, hence that $x$ is pseudo-invertible in $A$.
\(iv) $\Rightarrow$ (ii) (iv) implies that $xA = xy A$ is closed since $xy$ is idempotent.
That (iii) is equivalent to the others follows from (ii) and the symmetry in (i) or (iv). That (v) is equivalent to (i) is as in [@BRI Theorem 3.2].
The rest follows almost identically to [@BRI Theorem 3.2], but using the next Proposition in place of [@BRI Theorem 2.12] (in the first line on page 204 one needs to replace the disk by the right half plane).
The next result, used in the last paragraph of the last proof, is an analogue of [@BRI Theorem 2.12]:
\[okins\] If $A$ is a subalgebra of $B(H)$, and $x \in {\mathfrak r}_A$, then $x$ is invertible in $B(H)$ iff $x$ is invertible in $A$, and iff ${\rm oa}(x)$ contains $I_H$; and in this case $s(x) = I_H$.
It is clear by the Neumann lemma that if ${\rm oa}(x)$ contains $I_H$ then $x$ is invertible in ${\rm oa}(x)$, and hence in $A$. Conversely, if $x$ is invertible in $B(H)$ then by the equivalences (i)–(iv) proved in the last theorem, with $A$ replaced by $B(H)$ we have $s(x) \in B(H)$, and this is the identity of ${\rm oa}(x)$. If $x y = I_H$ in $B(H)$, then $I_H = xy = s(x) x y = s(x)$. The latter implies that $I_H \in {\rm oa}(x) \subset
A$.
\[Aha3\] Let $A$ be an operator algebra. A closed right ideal $J$ of $A$ is of the form $x A$ for some $x \in {\mathfrak r}_A$ iff $J = eA$ for a projection $e \in A$.
If $xA$ is closed for a nonzero $x \in {\mathfrak r}_A$ then by the theorem $e = s(x) \in A$. Hence $xA = eA$ by [@BRII Corollary 3.5] . The other direction is trivial.
\[Aha2\] If a nonunital operator algebra $A$ contains a nonzero $x \in {\mathfrak r}_A$ with $xA$ closed, then $A$ contains a nontrivial projection. If also $A$ is approximately unital then this implies that $A$ has a nontrivial r-ideal.
By the above $xA = eA$ for a projection $e \in A$. Now $e \neq 0$ since $x \in \overline{x {\rm oa}(x)} \subset xA = eA$. If $eA = A$ then $e$ is a left identity for $A$, hence is a two-sided identity if $A$ is approximately unital (since $e e_t = e_t \to e$ for the cai $(e_t)$). This contradiction shows that $eA$ is a nontrivial r-ideal.
\[Aha\] If an operator algebra $A$ has a cai but no identity, then $x A \neq A$ for all $x \in {\mathfrak r}_A$.
If $xA = A$ then the last proof shows that $A$ is unital.
It follows from this, as in [@BRI], that if $x$ is a strictly real positive element (in new our sense above) in a nonunital approximately unital operator algebra $A$, then $xA$ is not closed.
\[Aha4\] Let $A$ be an operator algebra. A closed $r$-ideal $J$ in $A$ is algebraically finitely generated as a right module over $A$ iff $J = eA$ for a projection $e \in A$. This is also equivalent to $A$ being algebraically finitely generated as a right module over $A^1$.
Let $J$ be an r-ideal which is algebraically finitely generated over $A$ by elements $x_1, \cdots, x_n \in A$. By [@BRI Theorem 2.15], $J$ is the closure of $\cup_t \, J_t$ of an increasing net of r-ideals $J_t = \overline{a_t A}$. By [@ST Lemma 1], $J = \cup_t \, J_t$. It follows that for one of these $t$ we have $x_k \in J_t$ for all $k = 1, \cdots, n$, and so $J = J_t$. By [@ST Lemma 1] again, $J = a_t A$. By Corollary \[Aha3\], $J = eA$.
If $J$ is algebraically finitely generated over $A^1$ then by the above $J = eA^1$. Clearly $e \in A$, and so $J = \{ x \in A : e x = x \} = eA$.
[**Remark.**]{} One of our few remaining open questions concerning the ${\mathfrak r}$-ordering, is if $q({\mathfrak r}_A) = {\mathfrak r}_{A/I}$, where $q$ is the canonical quotient map from an approximately unital operator algebra $A$ onto its quotient by an approximately unital ideal $I$. Nor are we sure if the ‘strictly real positive’ variant of this is true if $A$ is approximately unital but nonunital. It is easy to see that this is all true if the support projection of $I$ is in $M(A)$. We showed in [@BRI Section 6] that $q({\mathfrak F}_A) = {\mathfrak F}_{A/J}$. These may be regarded as analogues of the ‘lifting of positive or selfadjoint elements in $C^*$-algebras. They may also be regarded as peak interpolation results (see [@Bnew] for a brief survey of peak interpolation). For example, that $q({\mathfrak F}_A) = {\mathfrak F}_{A/J}$ is saying in the case of a unital function algebra on a compact set $K$ that given a function $f \in A$ with $|1-2f(x)| \leq 1$ on a $p$-set $E$ in $K$, there exists a $g \in A$ with $|1-2g(x)| \leq 1$ on all of $K$ and $g = f$ on $E$.
Positivity in the Urysohn lemma {#pury}
===============================
In our previous work [@BHN; @BRI; @BNII; @BRII] we had two main settings for noncommutative Urysohn lemmata for a subalgebra $A$ of a $C^*$-algebra $B$. In both settings we have a compact projection $q \in A^{**}$, dominated by an open projection $u$ in $B^{**}$, and we seek to find $a \in {\rm Ball}(A)$ with $aq = q a = q$, and both $a \, u^{\perp}$ and $u^{\perp} \, a$ either small or zero. In the first setting $u \in A^{**}$ too, whereas this is not required in the second setting. We now ask if in both settings one may also have $a$ very close to a positive operator (in the usual sense), and $a$ ‘positive’ in our new sense (involving ${\mathfrak r}_A$ or sets related to ${\mathfrak F}_A$)? In the first setting, all works perfectly:
\[urysII\] Let $A$ be an operator algebra (not necessarily approximately unital), and let $q \in A^{**}$ be a compact projection, which is dominated by an open projection $u \in A^{**}$. Then there exists an $a \in \frac{1}{2} {\mathfrak F}_A$ with $a q = qa = q$, and $a u = ua = a$. Indeed for any $\epsilon > 0$ this can be done with in addition the numerical range (and spectrum) of $a$ within a horizontal cigar centered on the line segment $[0,1]$ in the $x$-axis, of height $< \epsilon$. Such $a$ is within distance $\epsilon$ of a positive operator in $A + A^*$.
The proofs of [@BNII Theorem 2.6] and [@BRII Theorem 6.6 (2)] show that this all can be done with $a \in \frac{1}{2} {\mathfrak F}_A$. Then $a^{\frac{1}{n}} q = q a^{\frac{1}{n}} = q$, as is clear for example using the power series form $a^{\frac{1}{n}} = \sum_{k=0}^\infty
\, { 1/n \choose k} \, (-1)^k (1-a)^k$ from [@BRI Section 2], where it is also shown that $a^{\frac{1}{n}} \in \frac{1}{2} {\mathfrak F}_A$. Similarly $a^{\frac{1}{n}} u = u a^{\frac{1}{n}} = a^{\frac{1}{n}}$, since $u$ is the identity multiplier on oa$(a)$ which contains these roots [@BRI Section 2]. That the numerical range of $a^{\frac{1}{n}}$ lies in the desired cigar is as in the proof of [@BRI Theorem 2.4]. The distance $\epsilon$ assertion is explained in [@BRI] after the just cited result, indeed the element is within distance $\epsilon$ of its real part, which is positive.
We now turn to the second setting (see e.g. [@BRII Theorem 6.6 (1)]), where the dominating open projection $u$ is not required to be in $A^{\perp \perp}$. Of course if $A$ has no identity or cai then one cannot expect the ‘interpolating’ element $a$ to be in $\frac{1}{2} {\mathfrak F}_A$ or ${\mathfrak r}_A$. This may be seen clearly in the case that $A$ is the functions in the disk algebra vanishing at $0$. Here $\frac{1}{2} {\mathfrak F}_A$ and ${\mathfrak r}_A$ are $(0)$. Indeed by the maximum modulus theorem for harmonic functions there are no nonconstant functions in this algebra which have nonnegative real part. The remaining question is the approximately unital case ‘with positivity’. We solve this next, also solving the questions posed at the end of [@BNII].
\[urys\] Let $A$ be an approximately unital subalgebra of a $C^*$-algebra $B$, and let $q \in A^{\perp \perp}$ be a compact projection.
- If $q$ is dominated by an open projection $u \in B^{**}$. For any $\epsilon > 0$, there exists an $a \in
\frac{1}{2} {\mathfrak F}_A$ with $a q = q$, and $\Vert a (1-u) \Vert < \epsilon$ and $\Vert (1-u) a \Vert < \epsilon$. Indeed this can be done with in addition the numerical range (and spectrum) of $a$ within a horizontal cigar centered on the line segment $[0,1]$ in the $x$-axis, of height $< \epsilon$. Again, such $a$ is within distance $\epsilon$ of a positive operator in $A + A^*$.
- $q$ is a weak\* limit of some net $(y_t) \in \frac{1}{2}
\, {\mathfrak F}_A$ with $y_t q = q y_t = q$.
\(2) First assume that $q = u(x)$ for some $x \in \frac{1}{2} {\mathfrak F}_A$. We may replace $A$ by the commutative algebra ${\rm oa}(x)$, and then $q$ is a minimal projection, since $q \, p(x) \in \operatorname{{\mathbb C}}q$ for any polynomial $p$. Now $q$ is closed and compact in $(A^{1})^{**}$, so the unital case of (2), which follows from [@BRI Theorem 2.24] and the closing remarks to [@BNII], there is a net $(z_t) \in \frac{1}{2}
\, {\mathfrak F}_{A^1}$ with $z_t q = q z_t = q$ and $z_t \to q$ weak\*. Let $y_t = z_t^{\frac{1}{2}} x^{\frac{1}{2}}$. By the paragraph after Lemma \[commt\] (namely by a result from [@BBS] referenced there), we have $y_t
\in \frac{1}{2} {\mathfrak F}_{A^1}
\cap A = \frac{1}{2} {\mathfrak F}_{A}$. Also, $x^{\frac{1}{2}} q = q x^{\frac{1}{2}} = q$ by considerations used in the last proof, and similarly $z_t^{\frac{1}{2}} q = q z_t^{\frac{1}{2}} = q$. Thus $y_t^{\frac{1}{2}} q = q y_t^{\frac{1}{2}} = q$. If $A$ is represented nondegenerately on a Hilbert space $H$, and we identify $1_{A^1}$ with $I_H$, then for any $\zeta \in H$ we have by Lemma \[strsq\] that $$\Vert (y_t - q) \zeta \Vert
= \Vert (z_t^{\frac{1}{2}} - q) x^{\frac{1}{2}} \zeta \Vert
\leq K \Vert (z_t - q) x^{\frac{1}{2}} \zeta \Vert^{\frac{1}{2}} \to 0 .$$ Thus $y_t \to q$ strongly and hence weak\*.
\(1) If $A$ is unital, the first assertion of (1) is [@BRI Theorem 2.24]. In the approximately unital case, by the ideas in the closing remarks to [@BNII], the first assertion of (1) should be equivalent to (2). Indeed, by feeding such a net $(y_t)$ into the proof of [@BNII Theorem 2.1] one obtains the first assertion of (1).
Next, for an arbitrary compact projection $q \in A^{\perp \perp}$, by [@BNII Theorem 3.4] there exists a net $x_s \in \frac{1}{2} {\mathfrak F}_A$ with $u(x_s) \searrow q$. By the last paragraph there exist nets $y_t^s \in \frac{1+\epsilon}{2} \, {\mathfrak F}_A$ with $y_t^s \, u(x_s) = u(x_s) \, y_t^s = u(x_s)$, and $y_t^s \to u(x_s)$ weak\*. Then $$y_t^s q = y_t^s \, u(x_s) \, q = u(x_s) \, q = q,$$ for each $t, s$. It is clear that the $y_t^s$ can be arranged into a net weak\* convergent to $q$. This proves the Claim.
Finally, we obtain the ‘cigar’ assertion. For $(y_t)$ as in our Claim, we feed the net $(y_t^{\frac{1}{m}})$ into the proof of [@BNII Theorem 2.1]. Here $m$ is a fixed integer so large that the numerical range of $y_t^{\frac{1}{m}}$ lies within the appropriate horizontal cigar, as in the proof of the previous theorem. As in that proof $y_t^{\frac{1}{m}} q = q y_t^{\frac{1}{m}} = q$ and $y_t^{\frac{1}{m}} \to q$ weak\*. The distance assertion also follows as in the previous proof.
[**Remarks.**]{} 1) The recent paper [@CGK] contains a special kind of ‘Urysohn lemma with positivity’ for function algebras. Some of the conditions of our Urysohn lemma are more general than theirs; also our interpolating elements have range in a cigar in the right half plane, as opposed to their Stolz region which contains $0$ as an interior point. Hopefully our results could be helpful in such applications.
2\) Part (2) of the theorem easily gives a characterization of compact projections $q$ in approximately unital operator algebras, as weak\* limits of nets $(y_t)$ in $A$, where $y_t q = q$ and $y_t \in \frac{1}{2} {\mathfrak F}_A$, and $y_t$ is ‘as close to being positive as one likes’.
A semisimple operator algebra which is a modular annihilator algebra but is not weakly compact {#notw}
==============================================================================================
In [@ABR p. 76] we asked if every approximately unital semisimple operator algebra which is a modular annihilator algebra, is weakly compact, or is nc-discrete. We recall that $A$ is [*nc-discrete*]{} if all the open projections in $A^{**}$ are also closed (or equivalently lie in the multiplier algebra $M(A)$). In this section we will construct an interesting operator algebra $A$ which answers these questions in the negative.
Let $(c_n)$ be an unbounded increasing sequence in $(0,\infty)$. For each $n \in \operatorname{{\mathbb N}}$ let $d_n$ be the diagonal matrix in $M_n$ with $c_n^k$ as the $k$th diagonal entry. If $M$ is the von Neumann algebra $\oplus_n^\infty \, (M_n \oplus
M_n)$, we let $N$ be its weak\*-closed unital subalgebra consisting of tuples $((x_n , d_n x_n d_n^{-1}))$, for all $(x_n) \in \oplus_n^\infty \, M_n$. We define $A_{00}$ to be the finitely supported tuples in $N$, and $A_0$ to be the closure of $A_{00}$. That is, $A_0$ is the intersection of the $c_0$-sum $C^*$-algebra $\oplus_n^{\circ} \, (M_n \oplus
M_n)$ with $N$. We sometimes simply write $(x_n)$ for the associated tuple in $N$.
\[lem1\] Let $A$ be any closed subalgebra of $N$ containing $A_0$. Then $A$ is semisimple.
For any nonzero $x = (x_n) \in A$, choose $m$ and $i$ with $z = x_m e_i \neq 0$, where $(e_i)$ is the usual basis of $\operatorname{{\mathbb C}}^m$. Choose $y_m \in M_m$ with $y_m z = e_i$, and otherwise set $y_n = 0$. Then $y = (y_n) \in A_0$, and the copy of $e_i$ is in the kernel of $I-yx$. Hence $I - yx$ is not invertible in $A^1$, and so $x$ is not in the Jacobson radical by a well known characterization of that radical. Thus $A$ is semisimple.
Endow $M_n$ with a norm $p_n(x) = \max \{ \Vert x \Vert , \Vert d_n x d_n^{-1} \Vert \}$. Then $N \cong \oplus^\infty_n \, (M_n, p_n(\cdot))$ isometrically, and we write $p(\cdot)$ for the norm on the latter space, so $p((x_n)) = \sup_n \, p_n(x_n)$. We sometimes view $p$ as the norm on $N$ via the above identification. Let $L_n$ be the left shift on $\operatorname{{\mathbb C}}^n$, so that in particular $L_n e_1 = 0$. Note that $d_n L_n d_n^{-1} = \frac{1}{c_n} \, L_n$, and that $p_n(L_n) = 1$ if $n \geq 2$. For $n, k \in \operatorname{{\mathbb N}}$ with $n \geq k$ define an ‘integer interval’ $E_{n,k} = \operatorname{{\mathbb N}}_0 \cap [\frac{n}{k} ,
\frac{2n}{k}]$. Set $\mu_{n,k} = |E_{n,k}|$ if $n \geq k$, with $\mu_{n,k} = 1$ if $n < k$. Then $\mu_{n,k}$ is strictly positive for all $n, k$. For $n \geq k$ define $u_{n,k} = \frac{1}{\mu_{n,k}} \, \sum_{i \in E_{n,k}} \, (L_n)^i \in M_n$. If $n < k$ set $u_{n,k} = I_n$. Define $u_k = (u_{n,k})_{n \in \operatorname{{\mathbb N}}}$. We have $$p_n(u_{n,k}) \leq \max_{i \in E_{n,k}} \, p((L_n)^i) \leq 1 , \qquad n \geq k,$$ and so $$p(u_k) \leq 1 , \qquad k \in \operatorname{{\mathbb N}}.$$
The operator algebra we are interested in is $$A = \{ a \in N : p(a u_k -a) + p(u_k a - a) \to 0 \} .$$ This will turn out to be the largest subalgebra of $N$ having $(u_k)$ as a cai. First, a preliminary estimate:
\[lem2\] Let $L \in {\rm Ball}(B)$ for a Banach algebra $B$. Suppose that $E_1$ is a set of $\mu_1$ integers from $[0, n]$, and $E_2$ is a set of $\mu_2$ consecutive nonnegative integers. If $u_i = \frac{1}{\mu_{i}} \, \sum_{i \in E_{i}} \, L^i$ then $$\Vert u_1 u_2 - u_2 \Vert \leq \frac{2 n}{\mu_2}.$$
If $n \geq \mu_2$ then $$\Vert u_1 u_2 - u_2 \Vert \leq \Vert u_1 \Vert \Vert u_2 \Vert + \Vert u_2 \Vert
\leq 2 \leq \frac{2n}{\mu_2}.$$ So we may assume that $n < \mu_2$. Let $m_0 = \min \, E_2$. Then $$u_1 u_2 = \frac{1}{\mu_1 \, \mu_2} \, \sum_{j \in E_1, k \in E_2} \,
L^{j+k}
= \sum_{m_0 \leq m < m_0 + n + \mu_2} \, \lambda_m L^m ,$$ where $\lambda_m$ is $\frac{1}{\mu_1 \, \mu_2}$ times the number of pairs in $E_1 \times E_2$ which sum to $m$. Since $$\mu_1 \leq n+ 1 \leq \mu_2,$$ and since the number of such pairs cannot exceed $\mu_1 = |E_1|$, we have $$0 \leq \lambda_m \leq \frac{1}{\mu_2}.$$ If $m \in [m_0 + n, m_0 + \mu_2)$ then $m - k \in E_2$ for any integer $k$ in $[0,n]$, and so $m - E_1 \subset E_2$. We deduce that $$\lambda_m = \frac{1}{\mu_2} , \qquad m \in [m_0 + n, m_0 + \mu_2
).$$ Since $u_2 = \frac{1}{\mu_2} \, \sum_{m_0 \leq m < m_0 + \mu_2} \, L^m$ we have $$u_1 u_2 - u_2 = \sum_{m_0 \leq m < m_0 + n} \,
(\lambda_m - \frac{1}{\mu_2}) \, L^m \, + \, \sum_{m_0 + \mu_2
\leq m < m_0 + n + \mu_2} \, \lambda_m L^m .$$ No coefficient in the last sum has modulus greater than $\frac{1}{\mu_2}$, and there are $2n$ nonzero coefficients, so $$\Vert u_1 u_2 - u_2 \Vert \leq \frac{2n}{\mu_2} \, \max_m \,
\Vert L^m \Vert = \frac{2n}{\mu_2} .$$ as desired.
\[isab\] Let $A = \{ a \in N : p(a u_k -a) + p(u_k a - a) \to 0 \}$. Then $A$ is a semisimple operator algebra with cai $(u_k)$, and $A_0$ is an ideal in $A$.
We first show $u_r \in A$ for all $r \in \operatorname{{\mathbb N}}$. Let $k \geq r$. If $n \geq k$ then $E_{n,k}$ is a subset of $[0, \frac{2n}{k}]$, and $\mu_{n,k}$ is either $\lfloor \frac{n}{k} \rfloor$ or $\lfloor \frac{n}{k}+1 \rfloor$. By Lemma \[lem2\], we have $$p_n(u_{n,k} u_{n,r} - u_{n,r}) \leq \frac{2 \, \lfloor \frac{2n}{k} \rfloor}{\lfloor \frac{n}{r} \rfloor} ,
r \geq n \geq k .$$ If $n < k$ then $p_n(u_{n,k} u_{n,r} - u_{n,r}) = 0$. If $k \geq 2 t r$ for an integer $t > 1$ then $$\frac{2 \, \lfloor \frac{2n}{k} \rfloor}{\lfloor \frac{n}{r} \rfloor} \leq
\frac{\lfloor\frac{n}{tr} \rfloor}{\lfloor \frac{n}{r} \rfloor} \leq \frac{1}{t} .$$ Thus $p_n(u_{n,k} u_{n,r} - u_{n,r}) \leq \frac{2}{t}$ for $k \geq 2 t r$, so $$p(u_{k} u_r - u_r) = \sup_n \, p_n(u_{n,k} u_{n,r} - u_{n,r}) \leq \frac{2}{t}
, \qquad k \geq 2 t r .$$ So $u_k u_r \to u_r$ with $k$, and so $u_r \in A$ for all $r \in \operatorname{{\mathbb N}}$.
It is now obvious that $A$, being a subalgebra of the operator algebra $N$, is an operator algebra with cai $(u_k)$. It is elementary that for any matrix $x$ in the copy $M_n'$ of $M_n$ in $A_0$ we have $x u_k \to x$ and $u_k x \to x$, since for example $u_k x = x$ for $k > n$. Hence $A_0 \subset A$, so that $A$ is semisimple by Lemma \[lem1\]. Since $M_n'$ is an ideal in $N$, so is $A_0$, giving the last statement.
In the following result, and elsewhere, $\Vert \cdot \Vert$ denotes the usual norm on $M_n$ or on $\oplus^\infty \, M_n$.
\[lem3\] For each $n \in \operatorname{{\mathbb N}}$ and $k \leq n$, we have $\Vert u_{n,k} \Vert \geq 1 - \frac{2}{k}$ and $\Vert u_{n,k}^3 \Vert \geq 1 - \frac{6}{k}$.
If $\eta$ is the unit vector $(\frac{1}{\sqrt{n}}, \cdots , \frac{1}{\sqrt{n}})$ in $\operatorname{{\mathbb C}}^n$, then it is easy to see that $$\langle (L_n)^k \eta, \eta
\rangle = 1 - \frac{k}{n} , \qquad 0 \leq k \leq n.$$ Since $u_{n,k}$ is an average of powers $(L_n)^j$ with $0 \leq j \leq \frac{2n}{k}$, we have $$\langle u_{n,k} \eta, \eta
\rangle \geq 1 - \frac{\frac{2n}{k}}{n} = 1 - \frac{2}{k} .$$ Similarly, $u_{n,k}^3$ is a weighted average of powers $(L_n)^j$ with $0 \leq j \leq \frac{6n}{k}$.
We note that the diagonal matrix units $e^n_{i,i}$ are projections, and are also minimal idempotents in $A$ (that is, have the property that $e A e = \operatorname{{\mathbb C}}e$).
\[nowk\] $A$ is not weakly compact, and is not separable.
Note that $A$ is an $\ell^\infty$-bimodule via the action $$(\alpha_n) \cdot (T_n) = (T_n) \cdot (\alpha_n)
= (\alpha_n T_n) , \qquad (\alpha_n) \in \ell^\infty,
(T_n) \in A .$$ We will use this to embed $\ell^\infty$ isomorphically in $xAx$, where $x = u_{r}$ for large enough $r$. Note that $$\ell^\infty \cdot x^3 = x (\ell^\infty \cdot x)x \subset xAx .$$ Choosing $r$ with $1 - \frac{6n}{r} \geq \frac{1}{2}$, we have that $\Vert u_{n,r}^3 \Vert \geq \frac{1}{2}$ for all $n \in \operatorname{{\mathbb N}}$ (recall $u_{n,r} = I$ if $n < r$). Thus for $\vec \alpha = (\alpha_n) \in \ell^\infty$ we have $$p(\vec \alpha \cdot x^3) \geq \Vert \vec \alpha \cdot x^3 \Vert = \Vert \vec \alpha \cdot u_{r}^3 \Vert = \sup_n \, |\alpha_n| \Vert u_{n,r}^3 \Vert \geq \frac{1}{2} \sup_n \, |\alpha_n|,$$ and so the map $\vec \alpha \mapsto \vec \alpha \cdot x^3$ is a bicontinuous injection of $\ell^\infty$ into $xAx$. Thus $A$ is not weakly compact, nor separable.
\[lem4\] If $T = (T_n) \in A$, then $\Vert d_n T_n d_n^{-1} \Vert \to 0$ as $n \to \infty$. Thus the spectral radius $r(T_n) \to 0$ as $n \to \infty$.
Given $\epsilon > 0$ there exists an $m \in \operatorname{{\mathbb N}}$ such that $$p_n(u_{n,m} T_n - T_n) + p_n(T _n u_{n,m} - T_n) < \frac{\epsilon}{2} p(u_m T - T) + p(T u_m - T) < \frac{\epsilon}{2} ,
n \in \operatorname{{\mathbb N}}.$$ We have noted that $d_n L_n d_n^{-1} = \frac{1}{c_n} L_n,$ and for $n \geq m$ the operator $u_{n,m}$ is an average of powers $L^j_n$, so for $n \geq m$ we have $$\Vert d_n u_{n,m} d_n^{-1} \Vert \leq \max_{j \in \operatorname{{\mathbb N}}} \, \Vert d_n L_n^j d_n^{-1} \Vert \leq \frac{1}{c_n} .$$ Thus $$\Vert d_n T_n u_{n,m} d_n^{-1} \Vert \leq \frac{1}{c_n} \Vert d_n T_n d_n^{-1} \Vert \leq \frac{1}{c_n} p(T) .$$ Consequently, for $n \geq m$ the quantity $\Vert d_n T_n d_n^{-1} \Vert$ is dominated by $$\Vert d_n (T_n u_{n,m} - T_n) d_n^{-1} \Vert + \Vert d_n T_n u_{n,m} d_n^{-1} \Vert \leq
p_n(T_n u_{n,m} - T_n) + \frac{1}{c_n} p(T) \leq \frac{\epsilon}{2} + \frac{1}{c_n} p(T).$$ The result is clear from this.
For a matrix $B$ write $\overline{\Delta}_U B$ for the upper triangular projection of $B$ (that is, we change $b_{ij}$ to $0$ if $i > j$). Similarly, write $\Delta_L B$ for the strictly lower triangular part of $B$. In the next results, as usual ${r \choose s} = 0$ if $0 \leq r < s$ are integers.
\[lem5\] If $0 \neq T = (T_n) \in A$, and $\epsilon > 0$ is given, there exist $k, m \in \operatorname{{\mathbb N}}$ such that for all $r \in \operatorname{{\mathbb N}}_0$ and $n \geq \max \{ k, m \}$, we have $$\Vert (\overline{\Delta}_U T_n)^r \Vert \leq \sum_{s=0}^{k-1} \, {r \choose s} \, (2 p(T))^r \, \epsilon^{r-s} .$$
The $i$-$j$ entry $T_{n,i,j}$ of $T_n$ equals $\langle T_n e_j , e_i \rangle = c_n^{j-i} \, \langle d_n T_n d_n^{-1} e_j , e_i \rangle$, and so $$|T_{n,i,j} | = c_n^{j-i} \, |\langle d_n T_n d_n^{-1} e_j , e_i \rangle| \leq c_n^{j-i} \, p_n(T_n) , \qquad T = (T_n) \in A.$$ It follows from this that $$\Vert \sum_{j = 1}^{n-r} \, T_{n,j+r,j} \, E_{j+r,j} \Vert = \max_{j \leq n-r} \, |T_{n,j+r,j} | \leq c_n^{-r} \, p_n(T_n) ,$$ if $r < n$. Since $\sum_{r =1}^{n-1} \, (\sum_{j = 1}^{n-r} \, T_{n,j+r,j} \, E_{j+r,j}) = \Delta_L T_n$, we deduce that $$\label{lse} \Vert \Delta_L T_n \Vert = \Vert T_n - \overline{\Delta}_U T_n
\Vert \leq \sum_{r =1}^{n-1} \, c_n^{-r} \, p_n(T_n) \leq \frac{p_n(T_n)}{c_n - 1} \leq
\frac{p(T)}{c_n - 1}.$$
Given $\epsilon > 0$ choose $k$ with $p(u_k T - T) < \epsilon p(T),$ and let $n \geq k$. Then $$\Vert u_{n,k} T_n - T_n \Vert \leq p_n(u_{n,k} T_n - T_n) < \epsilon p(T),$$ and so $$\Vert u_{n,k} \overline{\Delta}_U T_n - \overline{\Delta}_U T_n \Vert \leq \epsilon p(T) + \Vert (u_{n,k} - I) (T_n - \overline{\Delta}_U T_n) \Vert \leq p(T) (\epsilon + \frac{2}{c_n - 1}),$$ since $$u_{n,k} \overline{\Delta}_U T_n - \overline{\Delta}_U T_n = (I - u_{n,k}) (T_n - \overline{\Delta}_U T_n) + (u_{n,k} T_n - T_n) .$$ Let $S_1 = u_{n,k} \overline{\Delta}_U T_n$ and $S_2 = \overline{\Delta}_U T_n - S_1$, then $\Vert S_2 \Vert \leq p(T) (\epsilon + \frac{2}{c_n - 1}),$ by the last displayed equation. Also, $$\Vert S_1 \Vert \leq \Vert \overline{\Delta}_U T_n \Vert \leq p(T) + \Vert (I - \overline{\Delta}_U) T_n \Vert \leq p(T) + \frac{p(T)}{c_n - 1}
= p(T) \frac{c_n}{c_n - 1}$$ by (\[lse\]).
Now $\overline{\Delta}_U T_n = S_1 + S_2$, so $(\overline{\Delta}_U T_n)^r$ is a sum from $s = 0$ to $r$, of ${r \choose s}$ times terms which are a product of $r$ factors, $s$ of which are $S_1$ and $r-s$ of which are $S_2$. Note that any product of upper triangular $n \times n$ matrices that has $k$ or more factors which equal $S_1$, is zero. This is because multiplication of an upper triangular matrix $U$ by $u_{n,k}$ (and hence by $S_1$) decreases the number of nonzero ‘superdiagonals’ of $B$ by a number $\geq \frac{n}{k}$, so after $k$ such multiplications we are left with the zero matrix. Thus we can assume that $s < k$ above. Using the estimates at the end of the last paragraph, we deduce that $$\Vert (S_1 + S_2)^r \Vert \leq \sum_{s=0}^{k-1} \, {r \choose s} \Vert S_1 \Vert^s \Vert S_2 \Vert^{r-s} \leq
\sum_{s=0}^{k-1} \, {r \choose s} \, (p(T) \frac{c_n}{c_n - 1})^s \, (p(T) (\epsilon + \frac{2}{c_n - 1}))^{r-s}.$$ Since $c_n \to \infty$ we may choose $m$ such that $\frac{c_n}{c_n - 1} < 2$ and $\epsilon + \frac{2}{c_n - 1} < 2 \epsilon$ for all $n \geq m$. Thus for $n \geq \max \{ k, m \}$, we have $$\Vert (\overline{\Delta}_U T_n)^r \Vert = \Vert (S_1 + S_2)^r \Vert \leq \sum_{s=0}^{k-1} \, {r \choose s} \, (2 p(T))^r \epsilon^{r-s}$$ as desired.
For $k \in \operatorname{{\mathbb N}}$ and positive numbers $b, \epsilon$, define a quantity $K(k,b,\epsilon) = \frac{1}{2 b
(1- \epsilon) \, \epsilon^k}$.
\[lem6\] If $0 \neq T = (T_n) \in A$, and $\epsilon > 0$ is given, there exist $k, m \in \operatorname{{\mathbb N}}$ such that for all $\lambda \in \operatorname{{\mathbb C}}$ with $|\lambda| > 4 p(T) \epsilon$, and $n \geq \max \{ k, m \}$, we have $\lambda I - \overline{\Delta}_U T_n$ and $\lambda I - T_n$ invertible in $M_n$, and both $$\Vert (\lambda I - \overline{\Delta}_U T_n)^{-1} \Vert \leq K(k,
p(T), \epsilon)$$ and $$\Vert (\lambda I - T_n)^{-1} \Vert \leq 2 K(k,
p(T), \epsilon).$$
If $|\lambda| > 2 p(T) \epsilon$ then $$\sum_{r=0}^{\infty} \, \Vert \lambda^{-r-1} \, (\overline{\Delta}_U T_n)^r
\Vert \leq |\lambda|^{-1} \, \sum_{r=0}^{\infty} \, \sum_{s=0}^{k-1} \, {r \choose s} \, (\frac{2 p(T)}{|\lambda|})^r \, \epsilon^{r-s} ,$$ by Lemma \[lem5\], for $n \geq \max \{ k, m \}$, where $k, m$ are as in that lemma. However the latter quantity equals $$|\lambda|^{-1} \, \sum_{s=0}^{k-1} \, \sum_{r=0}^{\infty} \, {r \choose s} \, (\frac{2 p(T) \epsilon}{|\lambda|})^{r-s} \,
(\frac{2 p(T)}{|\lambda|})^{s} = |\lambda|^{-1} \, \sum_{s=0}^{k-1} \,
(\frac{2 p(T)}{|\lambda|})^{s} \, (1 - \frac{2 p(T) \epsilon}{|\lambda|})^{-s-1}$$ using the binomial formula. This is finite, so $\sum_{r=0}^{\infty} \, \lambda^{-r-1} \, (\overline{\Delta}_U T_n)^r$ converges, and this is clearly an inverse for $\lambda I - \overline{\Delta}_U T_n$. If $|\lambda| > 4 p(T) \epsilon$, then the sum in the last displayed equation is dominated by $$\frac{1}{4 p(T) \epsilon} \, \sum_{s=0}^{k-1} \,
(\frac{1}{2 \epsilon})^s \, 2^{s+1} =
\frac{1}{2 p(T) (1 - \epsilon)} \frac{1 - \epsilon^k}{\epsilon^k}
\leq K(k,p(T),\epsilon) .$$ We also obtain $$\label{deleq} \Vert (\lambda I - \overline{\Delta}_U T_n)^{-1} \Vert
\leq K(k,p(T), \epsilon).$$
By increasing $m$ if necessary, we can assume that $c_n - 1 > 2 \, p(T) \, K(k,
p(T), \epsilon)$. Then by (\[lse\]) we have $$\Vert T_n - \overline{\Delta}_U T_n \Vert \leq \frac{p(T)}{c_n - 1} < \frac{1}{2 K(k,
p(T), \epsilon)}.$$ A simple consequence of the Neumann lemma is that if $R$ is invertible and $\Vert H \Vert < \frac{1}{2 \Vert R^{-1} \Vert}$, then $R + H$ is invertible and $\Vert (R + H)^{-1} \Vert \leq 2 \Vert R^{-1} \Vert$. Setting $R = \lambda I - \overline{\Delta}_U T_n$ and $H = \overline{\Delta}_U T_n - T_n$, we have $$\Vert H \Vert < \frac{1}{2 K(k,
p(T), \epsilon)} < \frac{1}{2 \Vert R^{-1} \Vert}$$ by (\[deleq\]). Hence $R + H = \lambda I - T_n$ is invertible, and by (\[deleq\]) again the norm of its inverse is dominated by $2 \Vert R^{-1} \Vert \leq 2
K(k, p(T), \epsilon).$
The quantity $K(k,p(T), \epsilon)$ above is independent of $n$, which gives:
\[isnotma\] The spectrum of every element of $A$ is finite or a null sequence and zero. Hence $A$ is a modular annihilator algebra.
Let $0 \neq T = (T_n) \in A$. We will show that the spectrum of $T$ is finite or a null sequence and zero. It is sufficient to show that if $\epsilon > 0$ is given, there exists $m_0 \in \operatorname{{\mathbb N}}$ such that if $|\lambda| > 4 p(T) \epsilon$, and if $\lambda$ is not in the spectrum of $T_1, \cdots , T_{m_0}$, then $\lambda \notin {\rm Sp}_A(T)$. So assume these conditions, and let $m_0 = \max \{ k, m \}$ as in Lemma \[lem6\]. For $n \geq m_0$ we have by Lemma \[lem6\] that $\lambda I - T_n$ is invertible, and the usual matrix norm of its inverse is bounded independently of $n$. By assumption this is also true for $n < m_0$. By Lemma \[lem4\] there is a $q$ such that $\Vert d_n T_n d_n^{-1} \Vert < \epsilon$ for $n \geq q$. If $|\lambda| > \epsilon$ then $(\lambda I - T_n)^{-1} =
\sum_{r=0}^{\infty} \, \lambda^{-r-1} \, T_n^r$ and $$\Vert d_n (\lambda I - T_n)^{-1} d_n^{-1} \Vert =
\Vert \sum_{r=0}^{\infty} \, \lambda^{-r-1} \, d_n T_n^r d_n^{-1} \Vert \leq \sum_{r=0}^{\infty} \, |\lambda|^{-r-1} \, \epsilon^r
= |\lambda|^{-1} \, (1 - \frac{\epsilon}{|\lambda|})^{-1} .$$ Thus $(p_n((\lambda I - T_n)^{-1}))$ is bounded independently of $n$. Hence $((\lambda I - T_n)^{-1}) \in N$, and this is an inverse in $N$ for $\lambda I - T$. Thus the spectrum of $T$ in $N$ is finite or a null sequence and zero. The spectrum in $A$ might be bigger, but since the boundary of its spectrum cannot increase, Sp$_A(T)$ is also finite or a null sequence and zero.
The last statements follow from [@Pal Chapter 8].
We point out some more features of our example $A$, in hope that these may further its future use as a counterexample in the subject:
\[multis\] The multiplier algebra of $A$ may be taken to be $\{ x \in N : x A + Ax \subset A \}$. This is also valid with $N$ replaced by $M$.
Viewing $M = \oplus_n^\infty \, (M_n \oplus
M_n)$ as represented on $H =\oplus^2_n (\operatorname{{\mathbb C}}^n \oplus \operatorname{{\mathbb C}}^n)$, it is clear that $D_0$, and hence also $A$, acts nondegenerately on $H$. So the multiplier algebra $M(A)$ may be viewed as a subalgebra of $B(H)$. We also see that the weak\* continuous extension $\tilde{\pi} : A^{**} \to N$ of the ‘identity map’ on $A$, is a completely isometric homomorphism from the copy of $M(A)$ in $A^{**}$ onto the copy of $M(A)$ in $B(H)$, and in particular, the latter is contained in $N$. So the latter is $M(A) = \{ x \in N : x A + Ax \subset A \}$. A similar argument works with $M$ replaced by $N$.
We note that if $D_n$ is the commutative diagonal $C^*$-algebra in $M_n$, then there is a natural isometric copy $D$ of $\oplus_n^\infty \, D_n$ inside $N$, namely the tuples $((x_n , x_n))$ for a bounded sequence $x_n \in D_n$.
We assume henceforth that $c_n > 1$ for all $n$.
\[diaga\] The diagonal $\Delta(A) = A \cap A^*$ of $A$ equals the natural copy $D_0$ of the $c_0$-sum $C^*$-algebra $\oplus_n^\circ \, D_n$ inside $A$.
If $((x_n , d_n x_n d_n^{-1}))$ is selfadjoint, then $x_n$ is selfadjoint, and $d_n x_n d_n^{-1}$ is selfadjoint, which forces $d_n^2$ to commute with $x_n$. However this implies that $x_n$ is diagonal. Since $\Delta(N) = N \cap N^*$ is spanned by its selfadjoint elements it follows that $\Delta(N) = D$. Therefore $\Delta(A) = D \cap A$, and this contains $D_0$ since $D_0 \subset A_0 \subset A$ by Corollary \[isab\]. The reverse containment follows easily from Lemma \[lem4\], but we give a shorter proof. Let $(a_n) \in D \cap A$, with $a_n \in D_n$ for each $n$. If $\epsilon > 0$ is given, choose $k$ such that $p(u_k (a_n) - (a_n)) < \epsilon$. Choose $m$ with $u_{n,k}$ strictly upper triangular for all $n \geq m$. Then for $n \geq m$ we have $|a_n(i) |$, which is the modulus of the $i$-$i$ entry of $(u_k (a_n) - (a_n))$, is dominated by $$\Vert u_{n,k} \, a_n - a_n \Vert \leq p(u_k \, (a_n) - (a_n)) < \epsilon.$$ Thus $\Vert a_n \Vert < \epsilon$ for $n \geq m$, so that $(a_n) \in D_0$.
\[ispro\] Projections in $A^{**}$ which are both open and closed, or equivalently which are in $M(A)$, must be also in $D$. Thus they are diagonal matrices with $1$’s as the only permissible nonzero entries.
This follows from Proposition \[multis\] and the fact from the proof of Proposition \[diaga\] that $\Delta(N) = D$.
Note that the natural approximate identity for $\Delta(A) = D_0$ is not an approximate identity for $A$ (since $D_0 A \subset A_0 A \subset A_0 \neq A$). Thus $A$ is not $\Delta$-dual in the sense of [@ABS]. In [@ABR] we showed that any operator algebra which is weakly compact is nc-discrete, and we asked if every semisimple modular annihilator algebra was nc-discrete. To see that our example $A$ is not nc-discrete in the sense of [@ABS] note that $A_0$ is an r-ideal in $A$ (and an $\ell$-ideal), and its support projection $p$ in $A^{**}$, which is central in $A^{**}$, coincides with the support projection of $D_0$ in $A^{**}$, and this is an open projection in $A^{**}$ which we will show is not closed.
\[iscr\] The algebra $A$ above is not nc-discrete.
We saw that $p$ above was open. If $p$ also was closed in $A^{**}$, or equivalently in $M(A)$, then $\tilde{\pi}(1-p)$ would be a nonzero central projection in the copy of $M(A)$ in $M$. Also $\tilde{\pi}(1-p) e^n_{i,i}$ is nonzero for some $n$ and $i$, because the SOT sum of the $e^n_{i,i}$ in $M$ is $1$. On the other hand, since $e^n_{i,i}$ is in the ideal supported by $p$ we have $$\tilde{\pi}(p) e^n_{i,i} =
\tilde{\pi}(p \, e^n_{i,i}) = \tilde{\pi}(e^n_{i,i}) =e^n_{i,i},$$ and so $$\tilde{\pi}(1-p) \, e^n_{i,i} = \tilde{\pi}(1-p) \, \tilde{\pi}(p) \, e^n_{i,i} = 0 .$$ This contradiction shows that $A$ is not nc-discrete.
Indeed $A_0$ is a nice r- and $\ell$-ideal in $A$ which is supported by an open projection which is not one of the obvious projections, and is not any projection in $M(A)$. Note that $A$ is not a left or right annihilator algebra in the sense of e.g. [@Pal Chapter 8], since for example by [@Pal Chapter 8] this implies that $A$ is compact, whereas above we showed that $A$ is not even weakly compact. The spectrum of $A$ is discrete, and every left ideal of $A$ contains a minimal left ideal, by [@Pal Theorem 8.4.5 (h)]. Also every idempotent in $A$ belongs to the socle by [@Pal Theorem 8.6.6], hence to $A_{00}$ by the next result. From this it is clear what all the idempotents in $A$ are.
\[issoc\] The maximal modular right (resp. left) ideals in $A$ are exactly the ideals of the form $(1-e) A$ (resp. $A(1-e)$) for a minimal idempotent $e$ in $A$ which is the canonical copy in $A$ of a minimal idempotent in $M_n$ for some $n \in \operatorname{{\mathbb N}}$. The socle of $A$ is $A_{00}$, namely the set of $(a_n) \in A$ with $a_n = 0$ except for at most finitely many $n$.
Let $e = (e_n)$ be a (nonzero) minimal idempotent in $A$. Then $e_n$ is an idempotent in $M_n$ for each $n$. If $e^n_{i,i}$ is as above, then because the SOT sum of the $e^n_{i,i}$ in $M$ is $1$, we must have $e e^n_{i,i} e \neq 0$ for some $n$ and $i$. Since $e$ is minimal, for such $n$, $e$ is in the copy of $M_n$ in $A_0$. So this $n$ is unique, and $e$ is clearly a minimal idempotent in this copy of $M_n$ in $A_0$. Now it is easy to see the assertion about the socle of $A$. By [@Pal Proposition 8.4.3], it follows that the maximal modular left ideals in $A$ are the ideals $A(1-e)$ for an $e$ as above. We have also used the fact here that $A$ has no right annihilators in $A$. Similarly for right ideals.
\[oncom\] The only compact projections in $A^{**}$ for the algebra $A$ above are the obvious ‘main diagonal’ ones; that is the projections in $D_0 \cap A_{00}$.
Let $T = (T_n) \in A$, and $\epsilon \in (0,\frac{1}{4 p(T)})$ be given. As in the proof of Corollary \[isnotma\] there exists $m_0 \in \operatorname{{\mathbb N}}$ such that if $|\lambda| > 4 p(T) \epsilon$ then $\lambda I - T_n$ is invertible for $n \geq m_0$, and the usual matrix norm of its inverse is bounded independently of $n \geq m_0$. As in that proof, if $S_n = T_n$ for $n \geq m_0$, and $S_n = 0$ for $n < m_0$, then $\lambda I - S$ is invertible in $N$. Thus the spectral radius $r(S) \leq 4 p(T) \epsilon < 1$. Hence $\lim_{k \to \infty} \, S^k = 0$ in norm. Let $q$ be the central projection in $A$ corresponding to the identity of $\oplus_{n=1}^{m_0 -1} \, M_n$. If now also $T \in \frac{1}{2} {\mathfrak F}_A$, then $T^k$ converges weak\* to its peak projection $u(T)$ weak\* by [@BNII Lemma 3.1, Corollary 3.3], as $k \to \infty$. Thus $T^k q \to u(T) q$ and $T^k (1-q) = S^k \to u(T) (1-q)$ weak\*. Clearly it follows that $u(T) q$ is a projection in $A$, hence in $D_0 \cap A_{00}$ as we said above. On the other hand, since $S^k \to 0$ we have $u(T) (1-q) = 0$. Thus $u(T)$ is a projection in $D_0 \cap A_{00}$.
Finally we recall from [@BNII] that the compact projections in $A^{**}$ are decreasing limits of such $u(T)$. Thus any compact projection is in $D_0 \cap A_{00}$.
One may ask if there exists a [*commutative*]{} semisimple approximately unital operator algebra which is a modular annihilator algebra but is not weakly compact. After this paper was submitted we were able to check that the algebra constructed in [@BRIV] was such an algebra. However this example is quite a bit more complicated than the interesting noncommutative example above.
We end this paper by mentioning a complement to the example above. In [@ABR p. 76] we asked if for an approximately unital commutative operator algebra $A$, which is an ideal in its bidual (which means that multiplication by any fixed element of $A$ is weakly compact), is the spectrum of every element at most countable; and is the spectrum of $A$ scattered? In particular, is it a modular annihilator algebra (we recall that compact semisimple algebras are modular annihilator algebras [@Pal Chapter 8]). There is in fact a simple counterexample to these questions, which is quite well known in other contexts. The example may be described either in the operator theory language of weighted unilateral shifts, and the $H^p(\beta)$ spaces that occur there, or in the Banach algebra language of weighted convolution algebras $l^p(\operatorname{{\mathbb N}}_0,\beta)$. These are equivalent (in particular, $H^2(\beta) = l^2(\operatorname{{\mathbb N}}_0,\beta)$). For brevity we just mention the operator theory angle (a previous draft had a much fuller exposition). Let $T$ be a weighted unilateral shift which is one-to-one (that is, none of the weights are zero), and let $A$ be the algebra generated by $T$. Then $A$ is isomorphic to a Hilbert space if $T$ is [*strictly cyclic*]{} in Lambert’s sense [@Sh]. Central to the theory of weighted shifts is the convolution algebra $H^2(\beta) = l^2(\operatorname{{\mathbb N}}_0,\beta)$, and its space of ‘multipliers’ $H^\infty(\beta)$ . These spaces can canonically be viewed as spaces of converging (hence analytic) power series on a disk, via the map $(\alpha_n) \mapsto \sum_{n = 0}^\infty \, \alpha_n z^n$. By the well known theory in [@Sh], $T$ is unitarily equivalent to multiplication by $z$ on $H^2(\beta)$, the latter viewed as a space of power series on the disk of radius $r(T)$. In our strictly cyclic case, $A$, which equals its weak closure, is unitarily equivalent via the same unitary to $H^\infty(\beta)$. Since $H^\infty(\beta) H^2(\beta) \subset H^2(\beta)$ and the constant polynomial is in $H^2(\beta)$, it is clear that $H^\infty(\beta) \subset H^2(\beta)$. However, since the constant polynomial $1$ is a strictly cyclic vector, we in fact have $H^\infty(\beta) = H^\infty(\beta) 1 = H^2(\beta)$ (see p. 94 in that reference). On the same page of that reference we see that the closed disk $D$ of radius $r(T)$ is the maximal ideal space of $H^\infty(\beta)$, and the spectrum of any $f \in H^\infty(\beta)$ is $f(D)$. In particular, Sp$_A(T) =D$, and $A$ is semisimple.
[*Acknowledgements.*]{} We thank Garth Dales and Tomek Kania for discussions on algebraically finitely generated ideals in Banach algebras, and in particular drawing our attention to the results in [@ST]. The first author wishes to thank the departments at the Universities of Leeds and Lancaster, and in particular the second author and Garth Dales, for their warm hospitality during a visit in April–May 2013. We also gratefully acknowledge support from UK research council grant EP/K019546/1 for largely funding that visit. We thank Ilya Spitkovsky for some clarifications regarding roots of operators and results in [@MP], and Alex Bearden for spotting several typos in the manuscript.
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[^1]: The first author was supported by a grant from the NSF. The second author is grateful for support from UK research council grant EP/K019546/1
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---
abstract: 'We perform next-to-leading order calculations of the single-diffractive and non-diffractive cross sections for dijet production in proton-antiproton collisions at the Tevatron. By comparing their ratio to the data published by the CDF collaboration for two different center-of-mass energies, we deduce the rapidity-gap survival probability as a function of the momentum fraction of the parton in the antiproton. Assuming Regge factorization, this probability can be interpreted as a suppression factor for the diffractive structure function measured in deep-inelastic scattering at HERA. In contrast to the observations for photoproduction, the suppression factor in proton-antiproton collisions depends on the momentum fraction of the parton in the Pomeron even at next-to-leading order.'
author:
- Michael Klasen
- Gustav Kramer
title: 'Survival probability for diffractive dijet production in $p\bar{p}$ collisions from next-to-leading order calculations'
---
DESY 09-130\
LPSC 09-112\
Introduction
============
Diffractive events in high-energy $p\bar{p}$ or $ep$ collisions are characterized by the presence of a leading proton or antiproton, which remains intact, and/or by a rapidity gap, defined as a (pseudo-)rapidity region devoid of particles. Theoretically, diffractive interactions are described in the framework of Regge theory [@1] as the exchange of a trajectory with vacuum quantum numbers, the so-called Pomeron ($\p$) trajectory. Diffractive scattering involving hard processes (hard diffraction) such as the production of high-$E_T$ jets has been studied experimentally to investigate the parton content of the Pomeron (or additional lower-lying Regge poles). In this framework, $p\bar{p}$ hard diffraction can be expressed as a two-step process, $p+\bar{p} \to p+
\p+\bar{p}' \to 2~{\rm jets}+\bar{p}'+X$, and similarly diffractive deep-inelastic scattering (DDIS) as $\gamma^{*}+p \to \gamma^{*}+\p+p'
\to p'+X$. The subprocess $\gamma^{*}+\p \to X$ is interpreted as deep-inelastic scattering (DIS) on the Pomeron target for the case that the virtuality of the exchanged photon $Q^2$ is sufficiently large. In analogy to DIS on a proton target, $\gamma^{*}+p \to X$, the cross section for DIS on a Pomeron target is expressed as a convolution of partonic cross section and universal parton distribution functions (PDFs) of the Pomeron. The partonic cross sections are the same as for $\gamma^{*}p$ DIS. The Pomeron PDFs are multiplied with vertex functions for the vertex $p \to \p+p'$, yielding the diffractive PDFs (DPDFs). The additional vertex functions depend on the fractional momentum loss $\xi$ and the four-momentum transfer squared $t$ of the recoiling proton. The DDIS experiments measure the diffractive structure function of the proton $F_2^{\rm D}(\xi,\beta,Q^2)$ integrated over $t$, where $\beta=x/\xi$ is the momentum fraction of the parton in the Pomeron and $Q^2$ is the virtuality of the $\gamma^{*}$. The $Q^2$ evolution of the DPDFs is calculated with the usual DGLAP [@2] evolution equations known from $\gamma^{*}+p \to X$ DIS. Except for the $Q^2$ evolution, the DPDFs are not calculable in the framework of perturbative QCD and must be determined from experiment. Such DPDFs have been obtained from the HERA inclusive measurements of $F_2^{\rm D}$ [@3; @4]. The presence of a hard scale such as the squared photon virtuality $Q^2$ in DIS or a large transverse jet energy $E_T^{\rm jet}$ in hard diffractive processes, as for example in $p\p \to {\rm jets}+X$ or $\gamma \p \to {\rm jets}+X$, allows for the calculation of the corresponding partonic cross sections using perturbative QCD. The central issue is whether such hard diffractive processes obey QCD factorization, i.e. can be calculated in terms of parton-level cross sections convolved with universal DPDFs.
For DIS processes, QCD factorization has been proven to hold [@5], and DPDFs have been extracted at low and intermediate $Q^2$ [@3; @4] from high-precision inclusive measurements of the process $e+p \to e'+
p'+X$ using the usual DGLAP evolution equations. The proof of the factorization formula also appears to be valid for the direct part of photoproduction ($Q^2 \simeq 0$) of jets [@5]. However, factorization does not hold for hard processes in diffractive hadron-hadron scattering. The problem is that soft interactions between the ingoing hadrons and their remnants occur in both the initial and final states. This was also the result of experimental measurements by the CDF collaboration at the Tevatron [@6], where it was found that the single-diffractive dijet production cross section was suppressed by up to an order of magnitude as compared to the prediction based on DPDFs determined earlier by the H1 collaboration at HERA [@7]. In the CDF experiment [@6], the suppression factor was determined by comparing single-diffractive (SD) and non-diffractive (ND) events. SD events are triggered on a leading antiproton in the Roman pot spectrometer and at least one jet, while the ND trigger requires only a jet in the CDF calorimeters. The ratio $R(x,\xi,t)$ of SD to ND dijet production rates $N_{\rm JJ}$ is in a first approximation proportional to the ratio of the corresponding structure functions $F_{\rm JJ}$, i.e. R(x,,t) = . An approximation to the SD structure function $F_{\rm JJ}^{\rm SD}(x,Q^2,\xi,t)$, $\F_{\rm JJ}^{\rm D}(\beta)$, was obtained by multiplying the above ratio of rates by the known effective F\_[JJ]{}\^[ND]{}(x) = x\[g(x)+\_[i]{}q\_i(x)\] after integrating this ratio over $\xi$ and $t$ and changing variables from $x$ to $\beta$ using $x \to \beta \xi$. The result was then compared to the DPDFs from H1 [@7] using the same approximate formula, Eq. (2), relating the structure function to gluon and quark DPDFs as in the ND case. The above formula for the ratio $R(x,\xi,t)$ is certainly not sufficient for estimating the suppression factor for diffractive dijet production in $p\bar{p}$ collisions. It is based on a leading order (LO) calculation of the cross section in the numerator and in the denominator. Furthermore, it is assumed that the convolutions of the PDFs in the numerator and the denominator with the partonic cross sections are identical and drop out in the ratio together with the PDFs for the ingoing proton. These approximations are not valid in next-to-leading order (NLO), where, in particular, the cross sections in the numerator and denominator depend on the jet algorithm and on the kinematics of the SD and ND processes.
Since 2002, the two HERA collaborations have presented results for diffractive dijet photoproduction in order to establish a possible suppression factor. The factorization breaking was first investigated on the basis of NLO predictions by us in 2004 [@8; @9] by comparing to preliminary H1 data [@10]. Already in 2004 it became clear that in photoproduction the breaking could be shown only by comparing with NLO predictions, which produced by a factor of two larger cross sections than the LO predictions. Concerning factorization breaking, the conclusions were the same based on a preliminary ZEUS analysis [@11]. Both collaborations, H1 and ZEUS, have now published their final experimental data [@12; @13]. Whereas H1 confirm in [@12] their earlier findings based on the analysis of their preliminary data and preliminary DPDFs, the ZEUS collaboration [@13] reached somewhat different conclusions from their analysis. In particular, the H1 collaboration [@12] obtained a global suppression of their measured cross sections as compared to the NLO calculation of approximately $S = 0.5$. In addition they concluded that also the direct cross section together with the resolved one does not obey factorization. The ZEUS collaboration, however, concluded from their analysis [@13] that, within the large uncertainties of the NLO calculations and the differences in the DPDF input, their data are compatible with the NLO QCD calculation, i.e. a suppression could not be deduced from their data. In several recent reviews, we have shown, however, that the ZEUS data are compatible with the older H1 [@12] and with even more recent H1 data [@14], if one adjusts the ZEUS large rapidity-gap inclusive DIS diffractive data to the analogous H1 data, which are the basis of the recent H1 DPDFs [@4] and which are used to predict the diffractive dijet photoproduction cross sections. In these recent reviews [@15] we also investigated whether the NLO prediction with resolved suppression only, which would be more in line with the findings in [@5], will also describe the H1 and ZEUS data in a satisfactory way. The result is, that this is indeed possible, and the resolved suppression factor is of the order of $S \approx 0.3$. For the global suppression, i.e. direct and resolved component equally, the suppression factor is larger, and in addition, depends on $E_T^{\rm
jet}$, which is not the case for the resolved suppression only.
In this work we want to bring the theoretical analysis of diffractive dijet production in $p\bar{p}$ collisions to the same level as has been done for diffractive dijet photoproduction, i.e. to calculate the cross sections up to NLO and then compare with the CDF data, to establish the suppression factor in the Tevatron energy range. For this purpose we shall calculate the ratio $R(x,\xi,t)$. For this we need the NLO cross sections for SD and ND with the cuts as in the CDF measurements. The outline of the paper is as follows. In Section 2 we shall describe shortly the kinematic restrictions for the CDF analysis based on measurements at Run I for $\sqrt{s}=1800$ GeV [@6] and on measurements at $\sqrt{s}=630$ GeV and $\sqrt{s}=1800$ GeV [@16] obtained for comparison at two different center-of-mass energies. In this section we shall also specify the various inputs for our calculation. Our results and the comparison with the CDF data are presented in Sect. 3. The first $1800$ GeV data are compared with the calculations in Sect. 3.1. The comparative study of the 630 and the new 1800 GeV cross sections are presented in Sect. 3.2. An interpretation of the observed suppression factor is given in Sect. 3.3. Sect. 4 contains a summary and our conclusions.
Kinematic cuts and input for the calculations
=============================================
The data, which we want to compare our NLO calculations with, are published in Ref. [@6] and Ref. [@16]. In the first paper [@6], the CDF collaboration measured non-diffractive and single-diffractive dijet cross sections at a center-of-mass energy of $\sqrt{s}=1800$ GeV using Run IC (1995-1996) data. From an inclusive sample of single-diffraction (SD) events, $\bar{p}p \to \bar{p'}X$, triggering on a $\bar{p}$ detected in a forward Roman pot spectrometer, a diffractive dijet subsample with transverse energy $E_T^{\rm jet}>7$ GeV was selected. In addition to the two leading jets, these events contain other lower-$E_T$ jets. Similarly, a non-diffractive (ND) dijet sample was selected. From the $E_T$ and the rapidity $\eta$ of the jets, the fraction $x_{\bar{p}}$ of the momentum of the antiproton carried by the struck parton was calculated, where $x_{\bar{p}}$ is given by x\_[|[p]{}]{}= \_[i]{} E\_T\^i e\^[-\^i]{}. The jets were detected and their energy measured by calorimeters covering the pseudorapidity range $|\eta| < 4.2$. The $E_T^{\rm jet}$ was defined as the sum of the calorimeter $E_T$’s within an $\eta-\phi$ cone of radius 0.7. The jet energy correction included a subtraction of an average underlying event of $E_T$ of 0.54 (1.16) GeV for diffractive (non-diffractive) events. The recoil antiproton fractional momentum loss $\xi$ and four-momentum transfer squared $t$ were in the range $0.035<\xi
<0.095$ and $(-t) < 3$ GeV$^2$, respectively, which was in the final sample restricted to $(-t) < 1$ GeV$^2$. In the second paper [@16], the study of diffractive dijet events was extended to $\sqrt{s}=630$ GeV. These data were compared to new measurements at $\sqrt{s}=1800$ GeV in order to test Regge factorization. This study is similar to the previous diffractive dijet study in experimental setup and methodology. For the SD sample, the $\xi$-region is the same, $0.035<\xi<0.095$, but $(-t)<0.2$ GeV$^2$. Again in the SD sample events with at least two jets with $E_T^{\rm jet}>7$ GeV were selected, where again the $E_T^{\rm jet}$ was defined as the sum of the calorimeter $E_T$’s within a cone of 0.7 in $\eta-\phi$ space. The jet energy correction included a subtraction of an average underlying event of 0.5 (0.9) GeV for SD (ND) events.
The calculation of the cross sections for dijet production in non-diffractive and single-diffractive processes has been performed up to NLO. For the comparison we have calculated these cross sections also in LO. For our calculations, we rely on our work on dijet production in the reaction $\gamma+p \to {\rm jets}+X$ [@17], in which we have calculated the cross sections for inclusive one-jet and two-jet production up to NLO for both the direct and the resolved contribution. The version for the resolved contribution can be used immediately for two-jet production in $p\bar{p}$ collisions by substituting for the photon PDF the antiproton PDF (for ND) or the Pomeron PDF (for SD). For the (anti-)proton PDF we have chosen the version CTEQ6.6M [@18] for the NLO calculation with $N_f=5$ active flavors. The strong coupling constant $\alpha_s$ is calculated from the two-loop formula with $\Lambda^{(5)}_{\overline{\rm MS}}=226$ MeV. For the calculation in LO we have chosen CTEQ6L1 [@19] with $\alpha_s$ determined from the one-loop formula and $\Lambda^{(5)}=165$ MeV. The diffractive PDFs are taken from the recent H1 fits to the inclusive diffractive DIS data [@4]. They are only available at NLO and come in two versions, ’H1 2006 fit A’ and ’H1 2006 fit B’. These differ mostly in the gluon density, which is poorly constrained by the inclusive diffractive scattering data, since there is no direct coupling of the photon to gluons, so that the gluon density is constrained only through the evolution. The ’H1 2006 fit A’ has a much larger gluon for larger momentum fractions $\beta$ at the starting scale of $Q_0=\sqrt{8.5}$ GeV than ’fit B’, which leads to a larger gluon also for larger scales $Q$. The original fit on the data in [@4] is performed with $N_f=3$ massless flavors. The production of charm quarks was treated in the Fixed-Flavor Number Scheme (FFNS) in NLO with non-zero charm-quark mass yielding a diffractive $F_2^c$. This $F_2^c$ is contained in the ’H1 fit 2006 A, B’ parameterizations and is then converted into a charm PDF. The H1 collaboration constructed a third set of DPDFs, which is called the ’H1 2007 fit jets’ and which is obtained through a simultaneous fit to the diffractive DIS inclusive and dijet cross sections [@20]. It is performed under the assumption that there is no factorization breaking in the diffractive dijet cross sections. Including the diffractive DIS dijet cross section in the analysis leads to additional constraints, mostly on the diffractive gluon distribution. On average, the ’H1 2007 fit jets’ is similar to the ’H1 2006 fit B’, except for the gluon distribution at large momentum fraction and small factorization scale. The DPDFs of H1 contain as a factor the vertex function $f_{\p/p}(\xi,t)$, which describes the coupling of the Pomeron to the proton, i.e. the proton-proton-Pomeron vertex. This vertex function is parameterized by the Pomeron trajectory $\alpha_{\p}(t)$ and an additional exponential dependence on $t$. This function is used also for our calculations, as it has been determined by the H1 collaboration when fitting their data. The normalization factor $N$ of this function is included in the Pomeron PDFs. Therefore the H1 DPDFs are products of the Pomeron flux factors and the Pomeron PDFs. These H1 DPDFs include also low-mass proton dissociative processes with invariant mass $M_Y<1.6$ GeV, which increases the inclusive diffractive DIS cross section as compared to cross sections with a pure (anti-)proton final state. We have to keep this in mind, when we compare to the CDF data, which use a forward Roman pot spectrometer to trigger on the final antiproton and therefore have no antiproton dissociative contributions.
Results
=======
In this section we present our results and compare them to the experimental data obtained with $\sqrt{s}=1.8$ TeV in [@6] and to the more recent data with $\sqrt{s}=0.63$ TeV and $\sqrt{s}=1.8$ TeV published in [@16]. In this latter publication, the kinematic constraints differ in some points from the constraints used in [@6]. First we compare to the normalized differential cross sections $\d\sigma
/\d \overline{E_T}$ and $\d\sigma/\d\overline{\eta}$ for non-diffractive and diffractive dijet production. Second, the ratio $\R(x_{\bar{p}})$ of the number of SD dijet events to the number of ND dijets is compared to the CDF data. This function $\R(x_{\bar{p}})$, obtained by integrating the cross sections in the numerator of Eq. (1) over $\xi$ and $t$, is the main result, and from the theoretical and experimental distribution as a function of $x_{\bar{p}}$ the suppression factor $\R^{\rm exp}(x_{\bar{p}})/\R^{\rm (N)LO}(x_{\bar{p}})$ can be deduced and can be studied for the three H1 DPDFs, ’H1 2006 fit A’, ’H1 2006 fit B’ [@4] and ’H1 2007 fit jets’ [@20] for the NLO and the LO (which has been done only for the ’H1 2006 fit B’) calculations.
Comparison with 1800 GeV data
-----------------------------
First we have calculated the distribution $\frac{1}{\sigma}\frac{\d
\sigma}{\d\overline{E_T}}$ as a function of $\overline{E_T}=(E_T^{\rm
jet1}+E_T^{\rm jet2})/2$, where $E_T^{\rm jet1} (E_T^{\rm jet2})$ refers to the jet with the largest (second largest) $E_T$ for ND and SD dijet production with $\sqrt{s}=1800$ GeV center-of-mass energy, integrated over the rapidities of the jets in the range $|\eta|<4.2$. Jets are defined with the usual cone algorithm within a chosen $\eta-
\phi$ cone of radius $R=0.7$ and a partonic distance $R_{\rm sep}=1.3R$ to match the experimental analysis [@20a]. $\sigma$ is the integrated cross section with the cut $E_T^{\rm jet1(2)} > 7.0(6.5)$ GeV. The lower limit of the leading and subleading jet differ slightly in order to avoid infrared sensitivity in the computation of the NLO cross sections, when integrated over $\overline{E_T}$ [@21]. Unfortunately in the experimental analysis such an asymmetric choice of $E_T^{\rm jet1}$ and $E_T^{\rm jet2}$ has not been made, since both $E_T^{\rm jet1}$ and $E_T^{\rm jet2}$ are restricted by $E_T^{\rm jet1,2} > 7.0$ GeV, so that we do not know whether the choice $E_T^{\rm jet2} > 6.5$ GeV is in accord with the experimental analysis. Therefore we have also varied the $E_T^{\rm jet2}$ cut slightly to $E_T^{\rm jet2} > 6.6$ GeV. The results for $\frac{1}{\sigma}\frac{\d\sigma}{\d\overline{E_T}}$ are
![\[fig:1\]Normalized average transverse-energy distributions of the non-diffractive (left) and single-diffractive (right) dijet cross section at Run I of the Tevatron. The CDF data (points) are compared with our predictions at NLO (full) and LO (dotted) and also with a varied cut on the subleading jet $E_T$ (dot-dashed). The NLO scale uncertainty is shown as a shaded band (color online).](fig01a "fig:"){width="0.49\columnwidth"} ![\[fig:1\]Normalized average transverse-energy distributions of the non-diffractive (left) and single-diffractive (right) dijet cross section at Run I of the Tevatron. The CDF data (points) are compared with our predictions at NLO (full) and LO (dotted) and also with a varied cut on the subleading jet $E_T$ (dot-dashed). The NLO scale uncertainty is shown as a shaded band (color online).](fig01b "fig:"){width="0.49\columnwidth"}
shown in Fig. 1 (left) for $E_T^{\rm jet2} > 6.5$ GeV (full histogram) and $E_T^{\rm jet2} > 6.6$ GeV (dot-dashed histogram), respectively. Together with the NLO cross section, we also show the LO cross section (dotted histogram) and the scale variation of the NLO result (shaded band), where the renormalization and factorization scales are varied simultaneously by factors of 0.5 and 2.0 compared to the default scale, which is chosen equal to $E_T^{\rm jet1}$, i.e. the largest $E_T$ of both jets. As is often the case, the scale uncertainty is relatively small in the normalized distributions. In Fig. 1 (left) we have included also the measured cross section from [@6], which unfortunately is given without the experimental uncertainties. Besides the statistical errors, which should be similar to those in the single-diffractive distributions given the similar number of ND and SD events [@20a], there are also systematic errors, as for example those associated with the jet energy scale. These would be needed for a fair comparison. Second, the theoretical cross sections should be corrected for hadronization effects when comparing to data. These are not known to us, but could be calculated through Monte Carlo models. If we compare the calculations in Fig. 1 (left) with the data, we observe that the results with $E_T^{\rm jet2} > 6.5$ GeV agree reasonably well with the data in the large $\overline{E_T}$ range, $\overline{E_T} > 20$ GeV, but much less for the low and medium $\overline{E_T}$ range. Conversely, for $E_T^{\rm jet2} > 6.6$ GeV the small and medium $\overline{E_T}$ range agrees better and the large $\overline{E_T}$ range less. The experimental errors will be larger in the large $\overline{E_T}$ range. Therefore the cross section with the cut $E_T^{\rm jet2} > 6.6$ GeV would be preferred, in particular also because we have perfect agreement in the first, second and third $\overline{E_T}$ bin, which are the most important ones for the integrated cross section $\sigma$, which determines the normalization.
The equivalent comparison for the SD dijet $\overline{E_T}$-distribution is shown in Fig. 1 (right) for $E_T^{\rm jet1(2)}>7.0
(6.5)$ GeV (full) and $E_T^{\rm jet1(2)}>7.0(6.6)$ GeV (dot-dashed). Here we have chosen only the ’H1 2006 fit B’ as DPDF. The comparison of data versus theory in Fig. 1 (right) shows the same pattern as for the ND distributions in Fig. 1 (left). In general the agreement with the data is even somewhat better now, in particular for the $E_T^{\rm
jet1(2)}>7.0(6.6)$ GeV cut. As for the ND distribution, we present also the LO prediction (dotted). From the unnormalized distributions (not shown), we obtain ratios of NLO to the LO cross sections ($K$-factors), which increase from relatively small values of 0.5 (0.6) in the infrared-sensitive region close to the $E_T^{\rm jet1,2}$ cuts to unity at larger $\overline{E_T}$ for the ND (SD) cross sections, indicating good perturbative stability and no sensitivity to the cut on $E_T^{\rm jet2}$ there.
The ND $\bar{\eta}$-distribution $\frac{1}{\sigma}\frac{\d\sigma}
{\d\bar{\eta}}$, where $\bar{\eta}=(\eta^{\rm jet1}+\eta^{\rm
jet2})/2$, is plotted in Fig. 2 (left), again for the two
![\[fig:2\]Same as Fig. 1, but for the normalized average rapidity distributions (color online).](fig02a "fig:"){width="0.49\columnwidth"} ![\[fig:2\]Same as Fig. 1, but for the normalized average rapidity distributions (color online).](fig02b "fig:"){width="0.49\columnwidth"}
$E_T^{\rm jet2}$ cuts, and the SD $\bar{\eta}$-distribution in Fig. 2 (right). For ND and SD, the two choices for the $E_T^{\rm jet2}$ have little influence on the result. The cross sections are somewhat smaller (by about 20%) at the maximum as compared to the ND experimental data, again given without experimental errors, but are in agreement with the SD data. The theoretical diffractive $\bar{\eta}$-distribution is boosted towards positive $\bar{\eta}$, as is the experimental one. We conclude from these comparisons that there is satisfactory agreement between the measured $\overline{E_T}$ and $\overline{\eta}$ distributions and our theoretical predictions based on the ’H1 2006 fit B’ DPDF. This motivates us to move on to the calculation of the ratio $\R(x_{\bar{p}})$ of SD to ND cross sections. Actually, if the experimental cross sections above had been known to us in the unnormalized form, we would have been in the position to deduce suppression factors as a function of $\overline{E_T}$ and $\bar{\eta}$.
The ratio $\R(x_{\bar{p}})$ of the SD to ND cross sections is evaluated as a function of $x_{\bar{p}}$, the fraction of the momentum of the antiproton transferred to the struck parton. It is calculated from the $E_T^i$ and $\eta^i$ of the jets with the relation in Eq. (3), where the sum over $i$ is taken over the two leading jets plus the next highest $E_T$ jet with $E_T > 5$ GeV. The cross section in the numerator is integrated over $\xi$ in the range $0.035 < \xi < 0.095$ and over $(-t)$ in the range $(-t) < 1$ GeV$^2$ and over the $E_T^{\rm jet}$ of the highest and second highest $E_T^{\rm jet}$ with $E_T^{\rm jet1(2)} >
7.0(6.5)$ GeV for both the SD and ND jet sample. As already mentioned, the data samples have the constraint $E_T^{\rm jet1(2)} > 7.0(7.0)$ GeV. We have checked that the choice $E_T^{\rm jet2} > 6.6$ GeV for the ND and SD cross sections has negligible influence on $\R(x_{\bar{p}})$.
![\[fig:3\]Left: Ratio $\R$ of SD to ND dijet cross sections as a function of the momentum fraction of the parton in the antiproton, computed at NLO (with three different DPDFs) and at LO and compared to the Tevatron Run I data from the CDF collaboration. Right: Double ratio of experimental over theoretical values of $\R$, equivalent to the factorization-breaking suppression factor required for an accurate theoretical description of the data (color online).](fig03a "fig:"){width="0.49\columnwidth"} ![\[fig:3\]Left: Ratio $\R$ of SD to ND dijet cross sections as a function of the momentum fraction of the parton in the antiproton, computed at NLO (with three different DPDFs) and at LO and compared to the Tevatron Run I data from the CDF collaboration. Right: Double ratio of experimental over theoretical values of $\R$, equivalent to the factorization-breaking suppression factor required for an accurate theoretical description of the data (color online).](fig03b "fig:"){width="0.49\columnwidth"}
The results are plotted in Fig. 3 (left) as a function of $\log_{10}
(x_{\bar{p}})$ for three choices of the DPDFs, ’H1 2006 fit B’ (full), ’H1 2006 fit A’ (dashed) and ’H1 2007 fit jets’ (dot-dashed). All three are calculated in NLO. The NLO scale uncertainty for ’H1 2006 fit B’ (shaded band) cancels out to a large extent in this ratio of cross sections. The LO prediction for ’H1 2006 fit B’ is also given (dotted). The CDF data, which are plotted in Ref. [@6] in six $\xi$ bins of width $\Delta \xi = 0.01$, have been integrated to give $\R(x_{\bar{p}})$ in the range $0.035 < \xi <0.095$. They were available in numerical form with statistical errors [@22] and are also plotted in Fig. 3 (left). From these presentations it is obvious that the theoretical ratios $\R(x_{\bar{p}})$ are, depending on $x_{\bar{p}}$, by up to an order of magnitude larger than the measured $\R(x_{\bar{p}})$ in agreement with the result in [@6]. There are quite some differences for the different DPDF choices. In general ’fit B’ and ’fit jets’ lie closely together, whereas ’fit A’ deviates more or less from these two depending on the $x_{\bar{p}}$ range. For ’fit B’ we also show the scale error and the LO prediction. The hierarchy between the three DPDFs at large $x_{\bar{p}}$ is easily explained by the fact that the corresponding gluon DPDFs are at large $x_{\bar{p}}$ the largest for ’fit A’ and the smallest for ’fit jets’ [@4; @20]. The same pattern between the different DPDFs is seen even more clearly if we plot the ratio of the experimental $\R(x_{\bar{p}})$ and the theoretical $\R(x_{\bar{p}})$ as a function of $\log_{10}(x_{\bar{p}})$. The result for this (double) ratio $\R^{\rm exp}(x_{\bar{p}})/\R^{\rm (N)LO}
(x_{\bar{p}})$ is seen in Fig. 3 (right). As can be seen, this ratio varies in a rather similar way for the three DPDFs in NLO and for ’fit B’ in LO in the range $10^{-3} < x_{\bar{p}} < 10^{-1}$. The variation is strongest for the ’fit A’ DPDF, where this ratio varies by more than a factor of seven. For the other two DPDFs this variation is somewhat less, but still appreciable. Actually, we would expect that the ratio plotted in Fig. 3 (right), which gives us the suppression factor, should vary only moderately with $x_{\bar{p}}$. After presenting the $\sqrt{s}=630$ GeV and the more recent $\sqrt{s}=1800$ GeV data below, we shall discuss possible interpretations of this result. We also observe that the suppression factor for ’fit B’ in NLO and LO are different, in particular for the very small $x_{\bar{p}}$.
In Ref. [@6], the ratio $\R(x_{\bar{p}})$ was multiplied with an effective PDF governing the ND cross section to obtain the effective DPDF $\F_{\rm JJ}^{\rm D}(\beta)$ as a function of $\beta=x_{\bar{p}}/\xi$. This effective non-diffractive PDF $F_{\rm JJ}^{\rm ND}(x)$ is calculated from the formula in Eq. (2), where the gluon PDF $g(x)$ and the quark PDFs $q_i(x)$ are taken from the GRV98 LO parton density set [@23] and evaluated at the scale $Q^2=75$ GeV$^2$, corresponding to the average $E_T^{\rm jet}$ of the SD and ND jet cross sections. Then, the effective diffractive PDF $\F_{\rm JJ}^{\rm D}(\beta)$ of the antiproton is obtained from the equation \_[JJ]{}\^[D]{}() = (x=) \_[JJ]{}\^[ND]{}(x ). We use this relation for the experimental and theoretical values of $\R(x_{\bar{p}})$. However, both are integrated over $\xi$ and are not given as function of $\xi$. We consider them as only moderately $\xi$-dependent and evaluate $\R(x=\beta\xi)$ and $\F_{\rm JJ}^{\rm ND}(x\to
\beta \xi)$ at an average value of $\bar{\xi}=0.0631$. This works quite well over the $\beta$-range of interest, if we compare the $\F_{\rm JJ}^{\rm D}(\beta)$ values obtained in this way with the $\F_{\rm JJ}^{\rm D}(\beta)$ in the CDF publication [@6], as is seen in Fig. 4
![\[fig:4\]Left: Effective diffractive structure function $\F_{\rm JJ}^{\rm D}$ of the partons with momentum fraction $\beta$ in the Pomeron as measured in dijet production at the Tevatron and compared to our (N)LO calculations. Right: Ratio of experimental over theoretical values of $\F_{\rm JJ}^{\rm D}$, equivalent to the factorization-breaking suppression factor required for an accurate theoretical description of the data (color online).](fig04a "fig:"){width="0.49\columnwidth"} ![\[fig:4\]Left: Effective diffractive structure function $\F_{\rm JJ}^{\rm D}$ of the partons with momentum fraction $\beta$ in the Pomeron as measured in dijet production at the Tevatron and compared to our (N)LO calculations. Right: Ratio of experimental over theoretical values of $\F_{\rm JJ}^{\rm D}$, equivalent to the factorization-breaking suppression factor required for an accurate theoretical description of the data (color online).](fig04b "fig:"){width="0.49\columnwidth"}
(left), where the full published points from [@6], denoted CDF PRL 84, coincide rather well with the open points deduced from the above equation and the published values of $\R$, denoted CDF PRL 84 (KK). The ratio $\F_{\rm JJ}^{\rm D,exp}(\beta)/\F_{\rm JJ}^{\rm D,(N)LO}
(\beta)$, plotted in Fig. 4 (right) as a function of $\beta$ linearly, gives the suppression factor as a function of $\beta$ instead of $x_{\bar{p}}$ as in Fig. 3. For example, for the ’fit B’ DPDF it varies between 0.3 at $\beta= 0.05$ and 0.13 at $\beta=0.1$ to 0.07 at $\beta=0.9$. In the range $0.3<\beta<0.9$ the suppression factor varies only moderately with $\beta$, but increases strongly for $\beta<0.3$, independently of the chosen DPDF. Above $\beta=0.3$, ’fit B’ and ’fit jets’ show the most constant behavior. Here one should note that the result in Fig.4 (right) is independent of the assumptions inherent in Eq. (4), since $F_{\rm JJ}^{\rm ND}(x\to\beta\xi)$ cancels in the ratio. The information in this figure concerning the suppression factor is equivalent to Fig. 4 of the CDF publication [@6]. The main difference to the CDF plot is the fact that now the suppression factor is given by comparing to calculated NLO cross sections without using the approximate formula Eq. (4) above, which can be justified only in LO.
To obtain an idea how large the effect of our NLO dijet evaluation compared to a simple combination of LO parton densities in the Pomeron is, we have calculated the ratio $\F_{\rm JJ}^{\rm D,NLO}(\beta)/
\F_{\rm JJ}^{\rm D,LO}(\beta)$ for the three DPDFs. Here the numerator is the $\F_{\rm JJ}^{\rm D}$ from Eq. (4) with $\R$ evaluated in NLO, i.e.Fig. 4 (left), and the denominator is $\F_{\rm JJ}^{\rm D,LO}(\beta)$ calculated from the formula \_[JJ]{}\^[D,LO]{}() = t f\_[/|[p]{}]{}(,t) , where the Pomeron flux factor $f_{\p/\bar{p}}(\xi,t)$ and the gluon and quark PDFs in the Pomeron $g(\beta)$ and $q_i(\beta)$ are taken from the fits ’H1 fit A’, ’H1 fit B’ and ’H1 fit jets’ at the scale $Q^2=75$ GeV$^2$, respectively. At $\beta=0.1$, we obtain ratios of 0.95, 1.05 and 1.1 for these three fits, respectively, indicating that our more accurate NLO calculations lead to very similar suppression factors as the simple approximation in Eq. (5) for all three DPDFs. This ratio is more or less constant as a function of $\beta$ in the considered range, meaning that already in the CDF publication [@6] one has the strong variation of the suppression factor with $\beta$ as mentioned above. It is interesting to note that replacing the approximate Eq.(5) with the experimentally used Eq. (4) compensates the effect of the NLO corrections, as the ratio of SD to ND $K$-factors, or equivalently the ratio of the NLO over the LO value of $\R$, is approximately $1.35$ for the $1800$ GeV calculation discussed here and $1.6$ for the $630$ GeV calculation presented in the next subsection. To compute the effect of this approximation alone, i.e.the ratio of Eq. (4) at LO over Eq. (5), one must divide the values of 0.95, 1.05 and 1.1 by the ratio of $K$-factors, i.e. 1.35.
Comparison with 630 GeV and new 1800 GeV data
---------------------------------------------
In a second publication, the CDF collaboration presented data for diffractive and non-diffractive jet production at $\sqrt{s}=630$ GeV and compared them with a new measurement at $\sqrt{s}=1800$ GeV [@16]. From both measurements they deduced diffractive structure functions using the formula Eq. (4) with the expectation that $\F_{\rm JJ}^{\rm D}
(\beta)$ is larger at $\sqrt{s}=630$ GeV than at $\sqrt{s}=1800$ GeV. The experimental cuts are similar to the cuts in the first analysis [@6] with the exception that now $(-t) \leq 0.2$ GeV$^2$ and in addition to the $E_T^{\rm jet1,2} > 7.0$ GeV cut they require $\overline{E_T} > 10$ GeV. This second cut on $\overline{E_T}$ is very important for the comparison with the NLO predictions, since with this additional constraint the infrared sensitivity is not present anymore.
With these cuts and the integration over $\xi$ in the range $0.035 <
\xi < 0.095$, we have calculated the normalized cross sections $(1/\sigma)\d\sigma /\d\overline{E_T}$ and $(1/\sigma)\d\sigma/\d
\overline{\eta}$ as in the previous subsection, but now for $\sqrt{s}=630$ GeV. For ND (left) and SD (right) jet production, the results are shown in Fig. \[fig:6\] and compared to the data from Ref.
![\[fig:6\]Same as Fig. 1, but for a reduced center-of-mass energy of 630 GeV at the Tevatron (color online).](fig06a "fig:"){width="0.49\columnwidth"} ![\[fig:6\]Same as Fig. 1, but for a reduced center-of-mass energy of 630 GeV at the Tevatron (color online).](fig06b "fig:"){width="0.49\columnwidth"}
[@16]. Here, the ND data sample was larger by about two orders of magnitude compared to the SD data sample, so that the statistical errors, which were not given in Ref. [@16], should be smaller by about a factor of ten [@20a]. Again, no information about systematic errors was available. We find reasonably good agreement in the medium-$\overline{E_T}$ range. In these figures, we have also plotted the LO predictions (dotted). For the DPDF, we have chosen as before the ’H1 2006 fit B’ set. Due to the large experimental errors for $\overline{E_T} > 15$ GeV for the SD case, we also find good agreement in the large-$\overline{E_T}$ range.
The equivalent result and comparison with the data for the $\overline{\eta}$-distribution is shown in Fig. \[fig:7\], again for ND (left)
![\[fig:7\]Same as Fig. 2, but for a reduced center-of-mass energy of 630 GeV at the Tevatron (color online).](fig07a "fig:"){width="0.49\columnwidth"} ![\[fig:7\]Same as Fig. 2, but for a reduced center-of-mass energy of 630 GeV at the Tevatron (color online).](fig07b "fig:"){width="0.49\columnwidth"}
and SD (right) jet production for NLO (full) and LO (dotted) predictions. The agreement between the theoretical results and the CDF data is similar as in the previous subsection, where we compared to the $\sqrt{s}=1800$ GeV data. This justifies to go on with the calculation of the ratio $R(x_{\bar{p}})$ of SD to ND cross sections.
The momentum fraction $x_{\bar{p}}$ is calculated as before from Eq.(3), and then the cross sections $\d\sigma/\d x_{\bar{p}}$ can be calculated with the same restrictions on the number of included jets as before. The only difference is the different cut on $(-t)$. The results for $\R(x_{\bar{p}})$ at $\sqrt{s}=630$ GeV are presented in Fig. \[fig:8\]
![\[fig:8\]Ratios $\R$ of SD to ND dijet cross sections as a function of the momentum fraction of the parton in the antiproton, computed at NLO (with three different DPDFs) and at LO and compared to the Tevatron data at $\sqrt{s}=630$ (left) and 1800 GeV (right) from the CDF collaboration (color online).](fig08a "fig:"){width="0.49\columnwidth"} ![\[fig:8\]Ratios $\R$ of SD to ND dijet cross sections as a function of the momentum fraction of the parton in the antiproton, computed at NLO (with three different DPDFs) and at LO and compared to the Tevatron data at $\sqrt{s}=630$ (left) and 1800 GeV (right) from the CDF collaboration (color online).](fig08b "fig:"){width="0.49\columnwidth"}
(left) for the three choices of DPDFs, ’fit B’ (NLO and LO), ’fit A’ (NLO) and ’fit jets’ (NLO). In this figure, also the experimental data from Ref. [@16] are included. The range of $x_{\bar{p}}$ is now much smaller than for the $\sqrt{s}=
1800$ GeV case. It ranges from $x_{\bar{p}} = 0.025$ to $x_{\bar{p}}=
0.1$. From this plot, the suppression of the SD cross section is clearly visible. The suppression factor is of the same order of magnitude as in the previous subsection. The same plot for the new $\sqrt{s}=1800$ GeV data [@16] together with the predictions is given in Fig. \[fig:8\] (right).
From the two plots in Fig. \[fig:8\] we have calculated the corresponding suppression factors $\R^{\rm exp}(x_{\bar{p}})/\R^{\rm (N)LO}
(x_{\bar{p}})$, exhibited in Fig. \[fig:9\] (left: $\sqrt{s}=630$ GeV; right:
![\[fig:9\]Double ratios of experimental over theoretical values of $\R$, equivalent to the factorization-breaking suppression factor required for an accurate theoretical description of the data from the Tevatron at $\sqrt{s}=630$ (left) and 1800 GeV (right) (color online).](fig09a "fig:"){width="0.49\columnwidth"} ![\[fig:9\]Double ratios of experimental over theoretical values of $\R$, equivalent to the factorization-breaking suppression factor required for an accurate theoretical description of the data from the Tevatron at $\sqrt{s}=630$ (left) and 1800 GeV (right) (color online).](fig09b "fig:"){width="0.49\columnwidth"}
$\sqrt{s}=1800$ GeV). In both figures we observe that the LO and NLO results for the suppression factors differ significantly (LO only given for ’fit B’), but also the three different DPDFs give different suppression factors, although with smaller variation compared to the LO and NLO result. Due to the variation of this factor with $x_{\bar{p}}$ it is difficult to compare the suppression of the $\sqrt{s}=630$ GeV result (left) with the $\sqrt{s}=1800$ GeV result (right) in Fig. \[fig:9\]. On average, it seems that for larger $x_{\bar{p}}$ the two suppression factors are more or less equal and we cannot say that the suppression factor for $\sqrt{s}=630$ GeV is larger than for $\sqrt{s}=1800$ GeV, as we would expect it. In the region $x_{\bar{p}} \geq 0.02$, the suppression factors for both $\sqrt{s}$ are fairly constant ($\simeq 0.05$), in particular for the DPDF ’fit jets’. This is not the case for the analysis in the previous subsection, where, as we see in Fig. 4, the suppression factor varies already much more in this particular $x_{\bar{p}}$ range. From the results in Fig. \[fig:8\], we have calculated $\F_{\rm JJ}^{\rm D}
(\beta)$ by changing variables from $x_{\bar{p}}$ to $\beta$ with $\beta=x_{\bar{p}}/\bar{\xi}$ and $\bar{\xi} = 0.0631$ and multiplying with the effective PDF for ND jet production as in Eq. (4). The results, together with the corresponding experimental data from Ref.[@16] and those calculated with the chosen $\bar{\xi}$, which agree inside errors except for two points at small $\beta$, are shown in Fig. \[fig:10\] (left: $\sqrt{s}=630$ GeV, right: $\sqrt{s}=1800$ GeV).
![\[fig:10\]Effective diffractive structure function $\F_{\rm JJ}^{\rm D}$ of the partons with momentum fraction $\beta$ in the Pomeron as measured in dijet production at the Tevatron with $\sqrt{s}=630$ (left) and 1800 GeV (right) and compared to our (N)LO calculations (color online).](fig10a "fig:"){width="0.49\columnwidth"} ![\[fig:10\]Effective diffractive structure function $\F_{\rm JJ}^{\rm D}$ of the partons with momentum fraction $\beta$ in the Pomeron as measured in dijet production at the Tevatron with $\sqrt{s}=630$ (left) and 1800 GeV (right) and compared to our (N)LO calculations (color online).](fig10b "fig:"){width="0.49\columnwidth"}
From these results we have again calculated, as in the previous subsection, the suppression factor as a function of $\beta$ in the range $0<\beta<0.8$. The plots for the ratios $\F_{\rm JJ}^{\rm exp}/
\F_{\rm JJ}^{\rm (N)LO}$ are seen in Fig. \[fig:11\] for the lower (left) and
![\[fig:11\]Ratios of experimental over theoretical values of $\F_{\rm JJ}^{\rm D}$ for $\sqrt{s}=630$ (left) and 1800 GeV (right), equivalent to the factorization-breaking suppression factors required for an accurate theoretical description of the data (color online).](fig11a "fig:"){width="0.49\columnwidth"} ![\[fig:11\]Ratios of experimental over theoretical values of $\F_{\rm JJ}^{\rm D}$ for $\sqrt{s}=630$ (left) and 1800 GeV (right), equivalent to the factorization-breaking suppression factors required for an accurate theoretical description of the data (color online).](fig11b "fig:"){width="0.49\columnwidth"}
the higher center-of-mass energy data (right), again for the three DPDF fits in NLO and ’fit B’ also in LO. First we observe that the ratios in Fig. \[fig:11\] differ very little, except perhaps at very small $\beta$. This means that from these data there is no essential difference seen in the suppression at $\sqrt{s}=630$ GeV and $\sqrt{s}=1800$ GeV. Second, we notice that with the ’fit jets’ we have the most constant behavior of the suppression for $\beta > 0.2$. Furthermore, comparing Fig. \[fig:11\] (right) with Fig. 4 (right) we see some differences. While the general pattern is the same, the suppression factor for ’fit jets’ in particular is less constant and larger in Fig. 4 (right) than in Fig.\[fig:11\] (right), which is obviously correlated with the more restrictive cuts on $\overline{E_T}$ and $t$ in the latter.
For completeness we also compared our NLO dijet calculation to the approximate LO formula in Eq. (5). For $\sqrt{s}=1800$ GeV and $\beta=0.1$, we obtain the same values of 0.95, 1.05, and 1.1 for ’H1 2006 fit A’, ’H1 2006 fit B’, and ’H1 2007 fit jets’ as for the older CDF analysis. They depend again weakly on $\beta$. For $\sqrt{s}=630$ GeV and $\beta=0.1$ we obtain larger values of 1.15, 1.35, and 1.45, which is in line with the larger ratio of $K$-factors (1.6 instead of 1.35) for SD and ND events at this lower center-of-mass energy.
As stated above, the calculation of the effective diffractive structure function $\F^D_{\rm JJ}(\beta)$ from the ratio $\R(x_{\bar{p}})$ was based on the assumption that the latter was only weakly $\xi$-dependent, so that Eq. (4) could be evaluated at an average value of $\bar{\xi}=0.0631$. This weak $\xi$-dependence is indeed observed in the newer CDF data, published in the lower part of Fig. 4 of Ref. [@16] and reproduced in our Fig. \[fig:13\] (full circles). These data agree well with
![\[fig:13\]Effective diffractive structure function in Eq. (4) from (N)LO dijet cross sections for fixed $\beta$ as a function of the momentum fraction of the Pomeron in the antiproton $\xi$, compared to the Tevatron data at $\sqrt{s}=1800$ GeV from the CDF collaboration [@16] (color online).](fig13){width="0.98\columnwidth"}
the $\xi$-dependent values of $\R(x_{\bar{p}})$ published in Fig. 3 of Ref. [@6] when transformed into $\F^D_{\rm JJ}(\xi)$ using Eq. (4) and $\xi=x_{\bar{p}}/\beta$ with $\beta=0.1$ (open circles). The same weak $\xi$-dependence is also observed in our theoretical calculations when using the same procedure, except with different normalization, reflecting the $\xi$-dependence of the H1 fits to the Pomeron flux factors $f_{\p/\bar{p}}(\xi,t)\propto \xi^{-m}$ with $m\simeq 1.1$ (0.9 in the CDF fit to their data). At the considered value of $\beta=0.1$, the NLO suppression factors for ’fit A,B’ and ’fit jets’ are 0.15, 0.12 and 0.11, respectively, and are almost independent of $\xi$. At LO, the suppression factor for ’fit B’ is larger, i.e.0.15, which corresponds to the fact that the ratio of SD over ND $K$-factors is 1.35. Note that Fig. \[fig:13\] is based on the higher statistics CDF data without the stronger cuts on $\overline{E_T}$ and $t$ and should therefore be not compared to Fig. \[fig:11\] (right), but to Fig. 4 (right), where consistency of these numbers with the values shown at $\beta=0.1$ can be found.
The (small) difference of the theoretical (1.1) and experimental (0.9) values of $m$ can be explained by a subleading Reggeon contribution, which has not been included in our predictions. To study its importance, we have computed the ratio of the Reggeon over the Pomeron contribution to the LO single-diffractive cross section at $\sqrt{s}=1800$ GeV as a function of $\overline{E_T}$, $\bar{\eta}$, and $x_{\bar{p}}$. The Reggeon flux factor was obtained from ’H1 2006 fit B’ and convolved, as it was done in this fit, with the parton densities in the pion of Owens [@24]. Very similar results were obtained for the ’H1 2007 fit jets’ Reggeon flux. On average, the Reggeon adds a 5% contribution to the single-diffractive cross section, which is almost independent of $\overline{E_T}$, but is smaller in the proton direction at large $\bar{\eta}$ and small $x_{\bar{p}}=\xi\beta$ and $\xi$ (2.5%) than at large $x_{\bar{p}}$ and $\xi$ (8%). This corresponds to the graphs shown in Figs. 5 ($\xi=0.01$) and 6 ($\xi=0.03$) of the H1 publication [@4], e.g. at $Q^2=90$ GeV$^2$. While the Reggeon contribution thus increases the diffractive cross section and reduces the suppression factor at large $x_{\bar{p}}$ in Figs. 3, 8, and 9, making the latter more constant, the same is less true at small values of $x_{\bar{p}}$.
Interpretation of the observed suppression factor
-------------------------------------------------
Our main results are the plots for the suppression factors as a function of $\log_{10}(x_{\bar{p}})$ in Fig. 3 (right) deduced from the data of Ref.[@6] and in Fig. \[fig:9\] from the data of [@16] at $\sqrt{s}=630$ GeV (left) and $\sqrt{s}=1800$ GeV (right). The qualitative behavior of the suppression factor in these three figures is very similar. We observe an appreciable dependence of the suppression factor on the chosen DPDFs and a dependence on $x_{\bar{p}}$ with a minimum at $x_{\bar{p}} \simeq 0.032$ ($\log_{10}(x_{\bar{p}})\simeq -1.5)$ and a rise towards smaller $x_{\bar{p}}$ by up to a factor of five. The equivalent result as a function of $\beta$ is shown in Fig. 4 (right) for the data of Ref. [@6] and in Fig. \[fig:11\] (left) for the $\sqrt{s}=630$ GeV data of Ref. [@16] and in Fig.\[fig:11\] (right) for the $\sqrt{s}=1800$ GeV data of Ref. [@16]. Depending on the chosen DPDFs, the suppression factor as a function of $\beta$ is minimal with the value $\simeq 0.05$ at $\beta = 0.5$ and rises with decreasing $\beta$ to a value $\simeq 0.12$ in Fig. 4 (right) and Fig. \[fig:11\] (right) and to a value $\simeq 0.1$ in Fig. \[fig:11\] (left) at $\beta=0.1$ (considering ’fit B’ as an example). Of course, this rise of the suppression factor towards small $\beta$ is directly related to its rise as a function of $x_{\bar{p}}$ towards small $x_{\bar{p}}$.
A comparison of the H1 data [@4], which are used to obtain the DPDFs applied in our calculation, with a similar measurement, in which the leading proton is directly detected [@H1LP], yields a ratio of cross sections for $M_Y < 1.6$ GeV and $M_Y = m_p$ of $1.23\pm 0.03(stat.)\pm 0.16(syst.)$ [@4]. Since the CDF measurements are performed by triggering on the leading antiproton, these measurements must be multiplied by this ratio to normalize them to the $M_Y<1.6$ GeV constraint for the H1 DPDFs. Therefore, all suppression factors obtained so far must be multiplied by this ratio.
Any model calculation of the suppression factor, which is also sometimes called the rapidity gap survival factor, must try to explain two points, first the amount of suppression, which is $\simeq 0.1$ at $\beta=0.1$, and second its dependence on the variable $\beta$ (or $x_{\bar{p}}$). Such a calculation has been performed by Kaidalov et al. [@K]. In this calculation, which we call KKMR, the hard scattering cross section for the diffractive production of dijets was supplemented by screening or absorptive corrections on the basis of eikonal corrections in impact parameter ($b$) space. The parameters of the eikonal were obtained from a two-channel description of high-energy inelastic diffraction. The exponentiation of the eikonal stands for the exchange of multi-Pomeron contributions, which violate Regge and QCD factorization and modify the predictions based on single Pomeron and/or Regge exchange. The obtained suppression factor $S$ is not universal, but depends on the details of the hard subprocess as well as on the kinematic configurations. The first important observation in the KKMR analysis is that in the Tevatron dijet analysis the mass squared of the produced dijet system $M_{\rm JJ}^2=
x_p\beta \xi s$ as well as $\xi$ are almost constant, so that small $\beta$ implies large $x_p$. The second important ingredient in the KKMR model is the assumption that the absorption cross section of the valence and the sea components, where the latter includes the gluon, of the incoming proton are different, in particular, that the valence and sea components correspond to smaller and larger absorption. For large $x_p$ or small $\beta$, the valence quark contribution dominates, which produces smaller absorptive cross sections as compared to the sea quark and gluon contributions, which dominate at small $x_p$. Hence the survival probability (or suppression factor) increases as $x_p$ increases and $\beta$ decreases. In Ref. [@K], the convolution of the old H1 DPDFs [@H1] and the $\beta$-dependent absorption corrections produced a $F^D_{\rm JJ}(\beta)$-distribution corrected for the soft rescattering, which was in very good agreement with the corresponding experimental distribution in the CDF publication [@6] (see Fig. 4 in [@K]). We have no doubt that using our single-diffractive NLO cross sections based on the more recent DPDFs of H1 [@4] will lead to a very similar result.
An alternative model for the calculation of the suppression factor was developed by Gotsman et al. [@G]. However, these authors did not convolve their suppression mechanism with the hard scattering cross section. Therefore a direct comparison to the CDF data is not possible.
At variance with the above discussion of diffractive dijet production in hadron-hadron scattering, the survival probability in diffractive dijet photoproduction was found to be larger ($\simeq0.5$ for global suppression, $\simeq0.3$ for resolved photon suppression only) and fairly independent of $\beta$ (or $z_{\p}$) [@12; @15]. This can be explained by the fact that the HERA analyses are restricted to large values of $x_\gamma\geq 0.1$ (as opposed to small and intermediate values of $x_p=0.02$ ... $0.2$), where direct photons or their fluctuations into perturbative or vector meson-like valence quarks dominate. The larger suppression factor in photoproduction corresponds also to the smaller center-of-mass energy available at HERA.
Conclusions
===========
In conclusion, we have performed the first next-to-leading order calculation of single-diffractive and non-diffractive cross sections for dijet production in proton-antiproton collisions at the Tevatron, using recently obtained parton densities in the (anti-)proton from global fits and in the Pomeron from inclusive deep-inelastic scattering and DIS dijet production at HERA. The normalized distributions in the average transverse energy and rapidity of the two jets agreed well with those measured by the CDF collaboration at two different center-of-mass energies of $\sqrt{s}=1800$ and 630 GeV. However, the ratios of single-diffractive and non-diffractive cross sections had two be multiplied by factors of about 0.05 and up to 0.3, depending on the momentum fraction of the parton in the antiproton, the center-of-mass energy, the order of the calculation, and the DPDF. Assuming Regge factorization, the ratios of cross sections were interpreted as ratios of effective diffractive structure functions, exhibiting similar suppression factors.
We found that the ratios of SD over ND $K$-factors of 1.35 and 1.6 at $\sqrt{s}=
1800$ and 630 GeV, respectively, were partially compensated by the simplification inherent in the definition of the effective structure functions, but that the suppression factors were still smaller at NLO than at LO. They were also less dependent on the momentum fraction of the parton in the Pomeron at NLO than at LO, in particular at the lower center-of-mass energy and to a smaller extent also for the more restricted kinematics at the higher $\sqrt{s}$. The DPDF fit by the H1 collaboration using DIS dijet data to better constrain the gluon density in the Pomeron showed the most constant behavior.
We pointed out that all suppression factors obtained so far must be corrected by a factor of $1.23\pm 0.03(stat.)\pm 0.16(syst.)$ due to the fact that the DPDFs were obtained from H1 data that includes diffractive dissociation, while the CDF data were triggered on a leading antiproton. We also recalled that the remaining momentum-fraction dependence can be explained by a two-channel eikonal model that predicts different behaviors for the regions dominated by valence quarks and sea quarks and gluons in the proton. This is in contrast to the constant behavior observed in photoproduction, which is governed by direct photon or valence-like quark contributions. We finally confirmed that the single-diffractive data are dominated by a single Pomeron exchange, since its momentum fraction dependence in the antiproton is well described in shape by the Pomeron flux factors fitted to the H1 DIS data. An additional Reggeon exchange would add only two to eight percent to the single-diffractive cross section for smaller and larger values of the Reggeon momentum fraction.
We thank K. Hatakeyama for useful discussions concerning the CDF data analyses. This work has been supported by the Theory-LHC-France initiative of the CNRS/IN2P3.
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---
abstract: 'We report on a theoretical study of collective electronic excitations in single-layer antimony crystals (antimonene), a novel two-dimensional semiconductor with strong spin-orbit coupling. Based on a tight-binding model, we consider electron-doped antimonene and demonstrate that the combination of spin-orbit effects with external bias gives rise to peculiar plasmon excitations in the mid-infrared spectral range. These excitations are characterized by low losses and negative dispersion at frequencies effectively tunable by doping and bias voltage. The observed behavior is attributed to the spin-splitting of the conduction band, which induces interband resonances, affecting the collective excitations. Our findings open up the possibility to develop plasmonic and optoelectronic devices with high tunability, operating in a technologically relevant spectral range.'
author:
- 'D. A. Prishchenko'
- 'V. G. Mazurenko'
- 'M. I. Katsnelson'
- 'A. N. Rudenko'
bibliography:
- 'apstemplate.bib'
title: 'Gate-tunable infrared plasmons in electron-doped single-layer antimony'
---
The growing field of plasmonics continues to gather attention from the material science community. Collective oscillations of electron density provide a way to couple indecent electromagnetic radiation to matter, which enables one to confine and enhance local field inside the material, essentially turning optical signal to electrical. Their practical use is diverse and depends on the desirable frequency region. The related fields include biosensing, light harvesting, optical thermal heating, lasers, photodetection and others [@Maier; @Murray; @Boriskina; @kneipp1997; @mayer2011; @rodrigo2015; @sorger2012]. Mid-infrared (IR) wavelengths is especially attractive spectral range as it offers a large set of unique and technologically relevant applications [@Zhong].
Among the diversity of plasmonic materials two-dimensional (2D) structures stand out as especially appealing candidates for plasmonics [@Low2016; @adv2016; @Pfnur; @C8NR01395K]. For example, graphene, the most known 2D material, exhibits in many ways unique optoelectronic properties, showing high energy confinement and large tunability [@Geim; @koppens2011; @Fei; @chen2012; @Fei2012; @Grigorenko2012; @kim2012; @brar2013; @StauberReview; @Rodrigo2017; @yao2018]. Intensive research has also been focused on other two-dimensional materials. Among them are transition-metal dichalcogenides and black phosphorus [@Stauber; @wang2015; @mishra2016; @LowBP; @Serrano; @locplasm]. The former exhibits plasmon resonances in the visible and near ultraviolet ranges, while the latter demonstrates strongly anisotropic optical properties, which makes it suitable for hosting hyperbolic plasmons [@Nemilentsau]. On the other hand, emerging 2D materials with magnetic degrees of freedom [@Gong; @Huang] open up another exciting direction in the field of nanoplasmonics [@Armelles]. In this work, we study plasmon excitations in electron-doped single-layer antimony (SL-Sb), a recently fabricated 2D semiconductor with remarkable environmental stability [@Ares2016; @Wu2017], and presumably high carrier mobility [@antmob]. Electronic structure of SL-Sb is strongly influenced by the spin-orbit interaction (SOI) [@Zhao2015; @Rudenko], which presumes additional functionalities and control. We find that under application of the gate voltage, electron-doped SL-Sb demonstrates unusual low-loss plasmonic excitations in the mid-IR region. The observed excitations are characterized by negative dispersion at small wavevectors, and turn out to be highly tunable by either bias potential or charge doping. The effect mainly originates from the SOI-induced spin-splitting of the conduction band, resulting in the interband resonances, significantly affecting the dielectric response. Antimonene has a hexagonal A7-type crystal structure (space group $D^3_{3d}$) with the lattice parameter $a$=4.12 Å and two sublattices displaced vertically by $b$=1.65 Å [@Rudenko]. SL-Sb is predicted to be an indirect gap semiconductor with the gap in the near-IR range [@ant1]. The electronic structure of SL-Sb can be accurately described over a wide energy range using a tight-binding (TB) model proposed in Ref. . The model is defined in the basis of $p$ orbitals and explicitly takes into account SOI. In the presence of a vertical bias the corresponding Hamiltonian has the following form:
$$H=\sum_{ij\sigma \sigma'}t_{ij}^{\sigma \sigma'}c_{i\sigma}^{\dag}c_{j\sigma'}+\frac{V}{d}\sum_{i\sigma}z_{i}c_{i\sigma}^{\dag}c_{i\sigma},$$
where $c_{i\sigma}^{\dag}$ ($c_{j\sigma'}$) is the creation (annihilation) operator of electrons with spin $\sigma$ ($\sigma'$) at orbital $i$ ($j$), $z_i$ is the $z$-component of the position operator of the orbital $i$, $t^{\sigma \sigma'}_{ij}$ is the spin-dependent matrix of hopping integrals, $V$ is bias voltage applied to the upper and lower planes of the system, and $d$ is the vertical displacement between the sublattices \[Fig. \[band\](d)\].
Fig. \[band\](a) shows energy dispersion and density of states (DOS) of the conduction states of SL-Sb. The conduction band minimum corresponds to a low-symmetry $\Sigma$-point ($C_{2v}$ point group), which is located at 0.56 Å$^{-1}$ from the $\Gamma$-point along the $\Gamma$–M direction of the Brillouin zone. Low energy dispersion at the band edge can be described by two effective masses, $m^{\parallel}_{\Sigma}=0.43m_0$ and $m^{\perp}_{\Sigma}=0.13m_0$, corresponding to the direction along and perpendicular to $\Gamma$–M, respectively. This gives rise to six ellipsoidal valleys formed around the zone center. In the presence of a vertical bias (or perpendicular static electric field), the spin degeneracy is lifted as a result of inversion symmetry breaking [@Lugovskoi]. The resulting spin splitting is shown in Fig. \[band\](b), which reaches 0.1 eV at the bias voltage $V=1$ eV. In this situation, the effective masses enhance to $m^{\parallel}_{\Sigma}=0.47m_0$ and $m^{\perp}_{\Sigma}=0.17m_0$ for both bands. The corresponding Fermi energy contours are shown in Fig. \[band\](c), where projections on the opposite spin directions is shown by color. The splitting of electron states in SL-Sb is different from the Rashba splitting typical to narrow gap 2D electron gas, but rather resembles exchange splitting in the ferromagnets [@RashbaHamilt]. Indeed, the expectation value of the spin operator projected to the direction perpendicular to $\Gamma$–M, $\langle S^{\perp}_{\Gamma \mathrm{M}}({\bf k}) \rangle = \langle \psi^{\sigma}_i({\bf k})|S^{\perp \sigma \sigma'}_{\Gamma \mathrm{M}}| \psi^{\sigma'}_i({\bf k}) \rangle$, shows that the two states within each valley correspond to the opposite ($\pm \hbar/2$) spin projections \[see Fig. \[band\](c)\]. In contrast to ferromagnets, time reversal symmetry is preserved in biased SL-Sb, leading to zero net magnetization. The combination of a gate-controlled band splitting and finite DOS at the Fermi energy achievable by doping opens up the possibility to tune plasmonic resonances in SL-Sb, which is of interest for practical applications. Here, we restrict ourselves to the case of electron doping only, as it represents the most interesting case. We only note that the properties of hole-doped SL-Sb can be with high accuracy described by the well-studied Rashba model [@Rashba].
To investigate optical properties of SL-Sb, we first calculate frequency-dependent dielectric matrix $\epsilon_{ij}^{(\textbf{q})}(\omega)$. To this end, we use the random phase approximation assuming no dielectric background (free-standing sample):
$$\epsilon_{\sigma \sigma'}^{(\textbf{q})}(\omega)=\delta_{\sigma \sigma'}-\frac{2\pi e^2}{qS}\Pi_{\sigma \sigma'}^{(\textbf{q})}(\omega),$$
where $2\pi e^2/qS$ is the long-wavelength approximation of the bare Coulomb interaction density in 2D, and $\Pi_{\sigma \sigma'}^{({\bf q})}(\omega)$ is the polarizability matrix. For the purpose of our study it is sufficient to ignore local field effects related to the orbital degrees of freedom, while the effects of the spin subsystem turn out to be important. Using the definition given above, the spectrum of plasma excitations is determined by the equation $\mathrm{det}[\epsilon_{\sigma \sigma'}^{({\bf q})}(\omega_p)]=i\gamma^{({\bf q})}(\omega_p)$, where $\gamma^{({\bf q})}(\omega_p)$ is the damping factor, and $\omega_p=\omega_p^{({\bf q})}$ is the plasma frequency. In the spinor basis, the polarizability can be defined as [@graf_electromagnetic_1995]:
$$\begin{gathered}
\begin{split}
\hspace{-0.3cm}&\Pi_{\sigma \sigma'}^{({\bf q})}(\omega)=\sum_{ \substack{ij{\bf k} \\ mn} } ( f_{m}^{(\textbf{k})}-f_{n}^{(\textbf{k}')} ) \frac{C_{i\sigma m}^{(\textbf{k})}C^{*(\textbf{k}')}_{i\sigma n}C^{*(\textbf{k})}_{j\sigma' m}C^{(\textbf{k}')}_{j\sigma' n} }{E_{m}^{(\textbf{k})}-E_{n}^{(\textbf{k}')}+\omega+i\eta} \hspace{-0.45cm} %\\
%& \times C_{i\sigma m}^{(\textbf{k})}C^{*(\textbf{k}+\textbf{q})}_{i\sigma n}C^{*(\textbf{k})}_{j\sigma' m}C^{(\textbf{k}+\textbf{q})}_{j\sigma' n},
\end{split}
\label{Pi}\end{gathered}$$
where $C_{i\sigma m}^{(\textbf{k})}$ is the contribution of the $i$th orbital $w_{i\sigma}^{{\bf R}}({\bf r})$ with spin $\sigma$ to the Hamiltonian eigenstate $\psi_{m}^{{\bf k}}({\bf r})=\sum_{i\sigma {\bf R}} C_{i\sigma m}^{({\bf k})} e^{i{\bf k}\cdot {\bf R}}w_{i\sigma}^{{\bf R}}({\bf r})$ with energy $E_{m}^{({\bf k})}$, ${\bf k}'={\bf k}+{\bf q}$, $f_m^{({\bf k})}=(\mathrm{exp}[(E_m^{({\bf k})}-\mu)/T]+1)^{-1}$ is Fermi-Dirac occupation factor, $\mu$ is the chemical potential determined by the carrier concentration $n$, and $\eta$ is a broadening term. In our calculations, we used $T=300$ K, $\eta=5$ meV, and two representative values of electron doping, $n=10^{13}$ and $10^{14}$ cm$^{-2}$. Brillouin zone integration has been performed on a grid of $\sim$10$^6$ [**k**]{}-points.
![\[band\] Band structure and density of states of SL-Sb calculated in the absence (a), and in the presence (b) of a vertical bias with $V=1$ eV. In each case, black horizontal line marks the Fermi energy corresponding the electron doping with concentrations $n_1=10^{13}$ and $n_2=10^{14}$ cm$^{-2}$. (c) Fermi surface contour for the concentration $n_2$ and bias potential $V = 1$ eV, with colors corresponding to the expectation value (in units of $\hbar/2$) of the spin operator projected on the direction perpendicular to $\Gamma$–M, $S^{\perp}_{\Gamma \mathrm{M}}({\bf k})$. Blue corresponds to the clockwise direction, while red is for the anticlockwise direction. The direction of the total spin per valley is shown by the black arrows. (d) Schematic representation of SL-Sb embedded in electric field controlled by the gate voltage.](bands2.png){width="\columnwidth"}
![\[plasm2\] Plasmon loss function $L(\textbf{q},\omega)$ in SL-Sb calculated for electron doping concentrations $n=10^{13}$ and 10$^{14}$ cm$^{-2}$ (top and bottom panels), and bias voltages $V=0$ and 1 eV (left and right panels). White line represents boundaries of the particle-hole continuum determined by the poles of the polarization function \[Eq. \[Pi\]\], $\omega_0^{({\bf q})} = \mathrm{max}\{E_n^{({\bf k}+{\bf q})}-E_m^{({\bf k})}\}$.[]{data-label="loss"}](losses.jpg){width="\columnwidth"}
To understand the extent to which one can tune the optical properties of SL-Sb, we calculate the plasmon loss function $L(\textbf{q},\omega)=\mathrm{Im}(1/\mathrm{det}[\epsilon^{({\bf q})}_{\sigma \sigma'}(\omega)])$, and study its behavior with respect to the carrier doping and external potential strength. The results are presented in Fig. \[loss\], which also shows boundaries of the particle-hole continuum, $\omega_0^{(\bf q)}$. In the absence of external potential electron occupy the bottom of a single parabolic band with no interband transition allowed \[see Fig. \[band\](a)\]. In this situation, optical response is determined by the plasma oscillations of nonrelativistic 2D electron gas, for which one has $\omega_p^2 \approx an|{\bf q}| + bE_Fq^2$, where $a$ and $b$ are constants, and $E_F$ is the Fermi energy [@Stern]. The corresponding loss function for $n=10^{13}$ and $10^{14}$ cm$^{-2}$ is shown in Figs. \[loss\](a) and (c), from which one can see a “classical” $\sqrt{q}$ plasmon dispersion at low frequencies. At $\omega^{({\bf q})}_p < \omega_0^{({\bf q})}$ the plasmon dispersion enters single-particle excitation continuum and decays into electron-hole pairs. The energy scale of phonon excitations in SL-Sb lies in the far-IR region [@Lugovskoi], meaning the absence of phonon-plasmon resonances [@Low2014] in the relevant spectral range.
The plasmon spectrum changes drastically when we introduce external bias potential with magnitude 1 eV, see Figs. \[loss\](b) and (d). In this case, a second (“optical”) plasmon branch appears. The new branch has large spectral weight and lies in the mid-IR region, independently of the electron concentrations considered. These excitations have a peculiar parabolic-like negative dispersion at small $q$. Their origin is related to the SOI-mediated splitting of the conduction band \[Fig. \[band\](b)\] allowing for the interband transitions, reminiscent to that in bilayer graphene [@Low2014; @Gamayun]. The frequency of the excitations at small $q$ can be effectively tuned by gate voltage, as it is shown in Fig. \[disp\]. Depending on the level of electron doping (10$^{13}$ or 10$^{14}$ cm$^{-2}$), one can smoothly tune $\omega_p(q\rightarrow0)$ from 0 to 0.3 (or 0.5) eV by applying bias voltage in the range up to 1 eV. In all cases relevant excitations lie above the Landau damping region ($\omega_p > \omega_0$), indicating fully coherent plasmon modes. On the contrary, “classical” plasmon mode in biased SL-Sb falls inside the particle-hole continuum, and turns out to be essentially damped. As a consequence of the Kramers-Kronig sum rule [@KK-sum], the entire spectral weight at long wavelengths is transferred to the “optical” mode. We note that further tunability toward lower frequencies can be achieved by the dielectric substrate (not considered here).
![\[disp\] Solutions of the equation Re{$\mathrm{det}[\epsilon^{({\bf q})}_{\sigma \sigma'}(\omega_p)]\}=0$, determining lossless plasmon modes calculated for different bias potentials $V$ (in eV), and for two representative values of electron doping $n$ in SL-Sb. The corresponding region of Landau damping is shown in each case by the same color. For clarity, only the highest (“optical”) plasmon mode is shown at each gate voltage. ](disp.jpg){width="\columnwidth"}
To gain further insights into the origin of plasma excitations in SL-Sb, we analyze effective dielectric functions $\epsilon^{({\bf q})}_{\mathrm{eff}}(\omega)=\mathrm{det}[\epsilon_{\sigma \sigma'}^{({\bf q})}(\omega)]$, shown in Fig. \[det\] for $n=10^{13}$ cm$^{-2}$. Without bias potential \[Fig. \[det\](a)\] one has typical behavior $\epsilon_{\mathrm{eff}}(\omega) \approx 1-\omega_p^2/(\omega^2+i\omega\gamma)$ at large enough frequencies with $\omega_p^2\sim q$. At $\omega < \omega_p$ there is another solution of the equation $\mathrm{Re}[\epsilon^{({\bf q})}_{\mathrm{eff}}(\omega)]=0$ with $\omega \sim q$. This solution is known as the “acoustic” plasmon mode corresponding to out-of-phase charge density oscillations observed in 2D materials with finite thickness, including bilayer graphene [@Hwang], transition metal dichalcogenides [@andersen_plasmons_2013], and phosphorene [@BP-plasmons]. Similar to the other systems, this mode is strongly damped as it lies in the particle-hole continuum. If we introduce bias potential \[Fig. \[det\](b)\], $\epsilon_{\mathrm{eff}}(\omega)$ exhibits a discontinuity at $\omega=\omega_0(V)$, and has the characteristic shape typical to a conductor with resonant scatterers [@Marder], $$\epsilon_{\mathrm{eff}}(\omega) \approx 1 - \sum_l \frac{\omega_{p,l}^2}{\omega^2-\omega_{0,l}^2+i\omega\gamma_l},
\label{eps_eff2}$$ where $l$ denotes different scattering channels, which in our case can be associated with intraband and interband transitions. Eq. (\[eps\_eff2\]) allows for the existence of plasmons with negative dispersion if there is $l$ for which $\partial \omega^2_{0,l} / \partial q <0$. As can be seen from Figs. \[loss\](b) and (d), this condition is fulfilled in biased SL-Sb at $q \lesssim 0.02$ Å$^{-1}$, which coincides with the region of negative plasmon dispersion. Apart from this prominent solution, there is an overdamped plasmon mode at frequencies close to the interband resonance \[Fig. \[det\](b)\], while the “acoustic” branch turns out to be fully suppressed at large enough $V$.
![\[det\] Frequency dependence of $\mathrm{Re}[\epsilon_{\mathrm{eff}}(\omega,q)]$ in SL-Sb calculated for a series of [**q**]{}-points along the $\Gamma$–M path. (a) and (b) correspond to $V = 0$ and $V = 1$ eV, respectively. In both cases, electron doping of 10$^{13}$ cm$^{-2}$ is assumed. The inset shows a zoom-in of the region where $\mathrm{Re}[\epsilon_{\mathrm{eff}}(\omega,q)]=0$, determining the “optical” plasmon modes. Arrows point to the corresponding solutions for the given set of [**q**]{}-points.[]{data-label="det"}](fig4.eps){width="0.8\columnwidth"}
Plasmon excitations with negative dispersion is uncommon but not unique phenomenon. It was first appeared in the context of bulk Cs crystal [@Felde], but was further observed in other materials [@Schuster; @Wezel]. A recent study reports similar behavior in electron-doped monolayer MoS$_2$ [@MoS2]. Negative dispersion is associated with negative group velocity, indicating negative energy flux. This phenomenon gives rise to an intriguing subfield of nanoplasmonics with a number of exotic optical effects including negative refraction index [@Agranovich; @Feigenbaum; @Compaijen]. The fact that the corresponding frequencies in SL-Sb fall in the technologically relevant spectral range make this system prospective for further experimental studies. Plasmons in 2D systems can be accessed by a variety of methods, including electron energy-loss spectroscopy [@Geim], IR optical measurements [@Fei], and scanning probe microscopy [@Fei2012], performed earlier for graphene. High tunability of plasmon excitations in SL-Sb offered by the strong SOI is another appealing aspect to be explored in the context of nanoplasmonic applications. To experimentally observe the peculiar character of plasmons in SL-Sb, strong electric fields on the order of 0.1–0.5 eV/Å may be required. This can be achieved, for example, by the encapsulation of SL-Sb in polar semiconductors [@semicond], or by means of heavy alkali metal doping [@ScienceBP]. To conclude, we theoretically studied optoelectronic properties of SL-Sb at realistic electron concentrations by varying the applied gate voltage. In addition to the classical 2D plasmon, we find that SOI-induced spin splitting gives rise to a new lossless plasmon branch in the mid-IR region at frequencies highly sensitive to the bias voltage. Remarkably, the new excitations exhibit negative dispersion in a wide range of wavevectors. This behavior is attributed to the strong SOI and inversion symmetry breaking, as well as indicates an important role of the local field effects in the spin channel. Our findings suggest SL-Sb to be an appealing nanoplasmonic material with great gate-tunability, which paves the way for further experimental and theoretical studies in this field.
This work was supported by the Russian Science Foundation, Grant No. 17-72-20041. Part of the research was carried out using high performance computing resources at Moscow State University [@lomonosov].
|
---
abstract: |
Composite structures must endure a great variety of multi-axial stress states during their lifespan while guaranteeing their structural integrity and functional performance. Understanding the fatigue behavior of these materials, especially in the presence of notches that are ubiquitous in structural design, lies at the hearth of this study which presents a comprehensive investigation of the fracturing behavior of notched quasi-isotropic \[+45/90/$-$45/0\]$_{s}$ and cross-ply \[0/90\]$_{2s}$ laminates under multi-axial quasi-static and fatigue loading.
The investigation of the S-N curves and stiffness degradation, and the analysis of the damage mechanisms via micro-computed tomography clarified the effects of the multi-axiality ratio and the notch configuration. Furthermore, it allowed to conclude that damage progression under fatigue loading can be substantially different compared to the quasi-static case.
Future efforts in the formulation of efficient fatigue models will need to account for the transition in damaging behavior in the context of the type of applied load, the evolution of the local multi-axiality ratio, the structure size and geometry, and stacking sequence. By providing important data for model calibration and validation, this study represents a first step towards this important goal.
address: 'William E. Boeing Department of Aeronautics and Astronautics, University of Washington, Seattle, Washington 98195, USA'
author:
- Yao Qiao
- Antonio Alessandro Deleo
- Marco Salviato
title: 'A Study on the Multi-axial Fatigue Failure Behavior of Notched Composite Laminates'
---
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0.5in
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[**INTERNAL REPORT No. 19-07/03E**]{}\
Multiaxiality ,Fatigue ,Fracture ,Stiffness ,Strength
Introduction
============
The last few decades have seen a tremendous increase in the use of composite materials in aerospace [@rana; @baker], automotive[@Ahmed; @kaiser], and civil [@Kar03; @Brandt08; @Ceccato17; @Carloni16; @Feo13] applications as well as wind energy production [@sorensen; @chortis]. While, fostered by the need for lightweight and durable structures, the market of composites will continuously expand in future years, the widespread use of these materials is exposing the need for safe and reliable design rules, especially in the context of the complex multiaxial cyclic stress states during service.
Although far less attention has been devoted to fatigue compared to quasi-static loading, the characterization and modeling of the behavior of composite materials under uniaxial and multiaxial loading have been the subjects of countless advances since the pioneering works of Owen and co-authors [@owen1; @owen2; @owen3; @owen4; @owen5; @owen6]. Still today, these are considered among the most comprehensive experimental investigations of the multiaxial fatigue behavior of polymer composites. A formidable set of data was also provided by Fujii *et al*. [@Fuji1; @Fuji2; @Fuji3; @Fuji4], who investigated the fatigue behavior of woven Glass Fiber Reinforced Polymers (GFRPs) under a variety of multiaxial stress states and even in the presence of stress concentrations induced by notches [@Fuji5; @Fuji6; @Fuji7].
A very insightful discussion of fatigue damage in composites under on-axis tension was provided by Talreja in [@talreja1; @talreja2] who identified three distinct mechanisms: (i) fiber failure at high applied strains with nonprogressive failure, (ii) matrix fatigue cracking and progression by failing fibers or by debonding, and (iii) matrix cracking confined by fibers preventing its propagation.
The damage mechanisms under multiaxial fatigue were investigated, among others, by Wang *et al*. [@wang1; @wang2] who conducted cryogenic tests in tension/torsion on tubular specimens. They found that the dominating damage mechanisms included matrix cracking, fiber/matrix debonding, microbuckling of fiber bundles, and delamination. The relative contribution of each mechanism was found to be a function of the multiaxiality ratio.
By testing cruciform GFRP specimens, Smith and Pascoe [@pascoe] identified rectilinear cracking, shear degradation of the fiber/matrix interface, and delamination as the main fatigue mechanisms. Similar results were obtained on carbon/epoxy composites by Chen and Matthews [@matthews].
A step towards the quantification of microscale fatigue damage is represented by the work of Adden and Horst [@horst] who characterized the crack density evolution in tension/torsion loading in NCF glass/epoxy composites. In this context, a significant contribution was provided by Quaresimin and co-workers who investigated the damage evolution *in-situ* in various loading configurations and geometries [@quaresimin1; @quaresimin2; @quaresimin3; @quaresimin4; @quaresimin5; @quaresimin6].
On the modeling side, an early attempt of capturing the fatigue behavior of composites is due to Sims and Brogdon [@SimBro77] who tried to extend the Tsai-Hill [@tsaihill] static failure criterion to fatigue by substituting the strength parameters with suitable S-N curves.
A similar approach was pursued by Francis *et al* [@francis] while Fujii and Lin [@Fuji4] investigated the extension of the Tsai-Wu criterion [@tsaiwu] for the description of tension-torsion fatigue of glass/polyester tubes under different multiaxiality ratios. On similar grounds, several other authors proposed alternative extensions (see e.g. [@wafa; @kawai; @philippidis]).
An excellent discussion on polynomial criteria for multiaxial fatigue was provided by Quaresimin *et al*. [@quaresimin7] who verified their predictive capability through a comprehensive re-analysis of a large bulk of experimental data. They concluded that the accuracy of some multiaxial criteria was fair although, in few cases, largely unsafe predictions were obtained undermining the general validity of the models. They also stressed the importance of an insightful understanding of the local fatigue damage mechanisms and their incorporation in micromechanical models as an answer to the foregoing limitations.
In this context, a notable example is the Synergistic Damage Mechanics (SDM) approach proposed by Singh and Talreja [@talreja3] which combines micro-damage mechanics and continuum damage mechanics to predict the stiffness degradation due to presence of transverse cracks. Thanks to the microscale description of damage, the model can be used for various stacking sequences and loading conditions. In this direction is the outstanding work of McCartney and co-workers [@mccartney1; @mccartney2; @mccartney3; @mccartney4; @mccartney5; @mccartney6] and Lundmark and Varna [@lundmark1; @lundmark2] on the modeling of matrix microcracking in transverse plies.
Later, Carraro *et al*. [@carraro1; @carraro2] proposed an analytical model that can capture the microcracking in multidirectional laminates with any stacking sequence. Thanks to analytical framework, the model is extremely robust and inexpensive from the computational point of view while still extremely accurate.
The foregoing results show the significant progress in the understanding of the fatigue behavior of composites. However, while the uniaxial and multiaxial fatigue response in smooth specimens has been the subject of extensive research, the study of notched structures is yet elusive.
As a first step towards addressing this important problem, this work presents an experimental investigation of the fracturing behavior of notched quasi-isotropic \[+45/90/$-$45/0\]$_{s}$ and cross-ply \[0/90\]$_{2s}$ laminates under multiaxial quasi-static and fatigue loading. To have full control of both the multiaxiality ratio and the notch configuration relative to the fiber orientation and specimen geometry, the tests were conducted leveraging an Arcan rig which enables the application of combinations of nominal normal and shear stresses. Thanks to the synergistic use of stiffness degradation data, Digital Image Correlation (DIC), and X-ray Micro-computed tomography the study sheds more light on the effect of the multiaxial stress state, the layup and notch configuration on the fatigue behavior.
Before moving to the next sections, it is worth pointing out that all the results reported in this work are presented in terms of nominal stresses. Accordingly, the multiaxiality ratio refers to the “global" multiaxial state of stress rather than the “local", which may differ from lamina to lamina. Although it is well known that the damage mechanisms driving the fatigue behavior in composites are controlled by the *in-situ* stress field, the choice of using the nominal stress was made to provide objective results for the calibration and validation of multiaxial fatigue models. In fact, in the presence of a stress raiser, the local stress state is in continuous evolution due to the stress/strain re-distribution associated to the progressive damage occurring in the Fracture Process Zone (FPZ). The calculation of such a stress state depends on the ability of the damage model to capture the main damage mechanisms and their evolution.
Materials and Methods
=====================
Specimen Preparation
--------------------
The unidirectional system used for all the tests was a Glass Fiber Reinforced Polymer (GFRP) by Mitsubishi Composites [@Rock].
The investigated stacking sequences included quasi-isotropic \[+45/90/$-$45/0\]$_{s}$ and cross-ply \[0/90\]$_{2s}$ layups. To manufacture the laminates, the prepreg was hand laid-up and then vacuum-bagged by using a Vacmobile mobile vacuum system [@pump]. A Despatch LAC1-38A programmable oven was used to cure the panels by ramping up the temperature from room temperature to $275^{\circ}$F in one hour, soaking for one hour, and cooling down to room temperature. After curing, the panels were cut into specimen of about 200 $\times$ 44 mm by using a water-cooled circular saw with a diamond-coated blade. The specimen thickness was about 1.72 mm and the gauge length was about 25 mm.
The effect of an open hole or an intra-laminar central crack on composite materials under multi-axial quasi-static and fatigue loading was studied. To this end, three different geometries, illustrated in Figure \[fig:geometry\]b-d, were prepared: (1) specimen with a central circular hole of diameter $a_0 = 10$ mm, (2) specimen with a central crack of 10 mm, and (3) specimen with a central crack of 18 mm. The open-hole specimens featured only one stacking sequence of \[+45/90/$-$45/0\]$_{s}$ and a circular hole drilled by a tungsten carbide drill bit. In contrast, the cracked specimens featured both \[0/90\]$_{2s}$ and \[+45/90/$-$45/0\]$_{s}$ as stacking sequence. For the former layup, a crack $a_0=18$ mm was considered while, for the latter $a_0$ was equal to 10 mm. The crack was manufactured by firstly drilling a pre-notch using a 0.4 mm tungsten carbide drill bit and then completing the crack leveraging a 0.4 mm wide diamond-coated saw.
In addition to the forgoing notched specimens, unnotched specimens with different dimensions and layups were tested under quasi-static and fatigue loading conditions. The purpose of these tests was to complete the information on the quasi-static and fatigue behavior provided by the manufacturer. The dimensions of the specimens followed ASTM D3039/D3039M [@ASTM] and the details can be found in Table \[tab:mechanicalproperties\].
Test Method
-----------
A modified version of Arcan rig, illustrated in Figure \[fig:geometry\]a, was specifically designed to guarantee infinite life for multi-axial fatigue tests on composite materials. This modified Arcan rig comprises four identical plates (two fronts and two backs) which are used to clamp the specimens by friction. A 17-4 PH stainless steel was used to manufacture these four plates.
The specimens were clamped to the modified Arcan rig by using twelve M14 high-strength bolts, with each bolt torqued at $130 \mbox{ N m}$. This ensures that enough clamping pressure was applied on the specimen tabs to avoid slipping during the tests. The modified Arcan rig is capable of applying multi-axial loads by varying loading angle $\theta$, the angle between loading direction and the longitudinal direction of the specimen. As illustrated in Figure \[fig:geometry\]a, tension is applied when $\theta$ equals $0^{\circ}$ while pure shear is applied when $\theta$ equals $90^{\circ}$. A combination of tension and shear is achieved by an intermediate loading angle between $0^{\circ}$ and $90^{\circ}$. To describe the loading configuration, multiaxiality ratio can be defined as $\lambda=$ arctan$(\tau_{N}/\sigma_{N})$ where $\sigma_{N} = P$cos$\theta/[(w-a_0)t]$ is the nominal normal stress and $\tau_{N} = P$sin$\theta/[(w-a_0)t]$ is the nominal shear stress applied to the specimen. In the definitions of stress, $P$ is the instantaneous load, $\theta$ is the loading angle, $w$ is the specimen width, $a_0$ is the crack length or hole diameter and $t$ is the specimen thickness. It is worth mentioning here that the in-plane and out-of-plane bending are negligible in the foregoing tests. This was verified by using multiple strain gauges on the different locations of the specimen or Digital Image Correlation (DIC) on the displacement contours of the specimen under multi-axial loading condition as discussed by Tan *et al.* [@Tan].
The multi-axial, quasi-static, and fatigue tests were performed in a servo-hydraulic 8801 Instron machine with closed-loop control. The speckled specimens were analyzed by means of an open source Digital Image Correlation system programmed in MATLAB software developed at Georgia Tech [@Ncorrverify; @Blader].
Uniaxial tests on notch-free specimens {#sec:tensiletests}
--------------------------------------
Notch-free specimens were tested under quasi-static and fatigue loading conditions to characterize the uniaxial constitutive behavior and S-N curves. The quasi-static tests were performed under displacement control with a displacement rate of 0.01 mm/s whereas the fatigue tests were under load control with a stress ratio of $R=0.1$ and a low frequency of $f=5$ Hz. It is worth mentioning that both \[0/90\]$_{2s}$ and \[+45/90/$-$45/0\]$_{s}$ stacking sequence were investigated. These configurations were the same adopted for the multi-axial tests. The forgoing experimental campaign was performed to provide sufficient information on the uniaxial tensile behavior of the material before investigating the multi-axial behavior.
Multi-axial tests on notched specimens {#sec:multiaxial}
--------------------------------------
### Multi-axial quasi-static tests
To study the failure behavior of notched laminates under multi-axial quasi-static loading, five sets of multiaxiality ratios were investigated: $\lambda=$ 0, 0.262, 0.785, 1.309 and 1.571. Such multiaxiality ratios corresponded to $\theta$ = $0^{\circ}$, $15^{\circ}$, $45^{\circ}$, $75^{\circ}$ and $90^{\circ}$. At least three specimens were tested for each configuration. As for the tensile quasi-static tests on unnotched specimens, the load rate for the multi-axial quasi-static tests was 0.01 mm/s.
### Multi-axial fatigue tests
In the case of multi-axial fatigue tests, the same multiaxiality ratios as the quasi-static tests on notched specimens were investigated. At least three specimens were tested for each multiaxiality ratio in order to produce a statistically significant set of data. To study the fatigue failure behavior of notched laminates under multi-axial fatigue loading, three sets of loading cases were studied: 70% $P_{max}$, 55% $P_{max}$ and 40% $P_{max}$ where $P_{max}$ is the average peak force determined from quasi-static tests for each multiaxiality ratio. A stress ratio of $R=0.1$ and a low frequency of $f=5$ Hz were kept for all the forgoing loading cases.
Damage detection and analysis
-----------------------------
A NSI X5000 X-ray micro-tomography scanning system [@northstar] was used to observe the sub-critical fatigue damage mechanisms of the specimens with a X-ray tube setting of 90 kV voltage and 220 $\mu$A current. This non-destructive technique was significantly important to guarantee that no additional damage was created during the damage visualization process. In addition, a dye penetrant composed of zinc iodide powder (250 g), isopropyl alcohol (80 ml), Kodak photo-flow solution (1 ml) and distilled water (80 ml) was used in all the scans as a supplement to improve the visualization of the damage mechanisms [@dye; @dye1]. Prior to the scanning, the specimens were soaked in the dye penetrant mixture for approximately one day. The sub-critical damage in each ply and interface were identified by slicing through the reconstructed 3D images of the specimens leveraging ImageJ software [@imagej].
Experimental Results
====================
Uniaxial tests on notch-free specimens {#uniaxial-tests-on-notch-free-specimens}
--------------------------------------
The mechanical properties of the unnotched specimens under quasi-static loading conditions are tabulated in Table \[tab:mechanicalproperties\]. These properties were measured based on the stress and strain curves with the strain obtained from Digital Imaging Correlation (DIC) to exclude the compliance of the testing system. On the other hand, in terms of the fatigue behavior of unnotched specimens, the normalized S-N curves of unnotched \[+45/90/$-$45/0\]$_{s}$ and \[0/90\]$_{2s}$ specimens under tensile fatigue loading conditions are plotted in Figure \[fig:unnotched\]a. As can be noted from the figure, the slope of S-N curve for the cross-ply specimen is slightly larger than the one of the quasi-isotropic specimen. The larger slope of S-N curve reflects the specimen with better resistance to fatigue loading. This is consistent with the results on the normalized stiffness degradation curves, as shown in Figure \[fig:unnotched\]b, in which the structural stiffness of the cross-ply specimen deteriorates less significantly in comparison with the one of the quasi-isotropic specimen. The structural stiffness is defined as $K=(P_{peak}-P_{m})/(u_{peak}-u_{m})$ where $P_{peak}$ is the peak load, $P_{m}$ is the mean load, $u_{peak}$ is the displacement at the peak load for each cycle, and $u_{m}$ is the displacement at the mean load for each cycle. However, the endurance limit, generally considered as 2 million cycles, was achieved when the peak load in applied cyclic load reached roughly 30%-35% of the critical quasi-static load for both unnotched specimens.
Multi-axial quasi-static tests on notched specimens {#multiaxialquasistatictests}
---------------------------------------------------
The nominal stress and strain curves obtained from the multi-axial quasi-static tests are plotted in Figures \[fig:quasistaticcrossplynormal\]-\[fig:quasistaticquasishear\] for the following three specimen configurations: (1) \[0/90\]$_{2s}$ layup with a 18 mm central crack, (2) \[+45/90/$-$45/0\]$_{s}$ layup with a 10 mm hole, and (3) \[+45/90/$-$45/0\]$_{s}$ layup with a 10 mm central crack. In these figures, the normal and shear stresses are the nominal stresses in the net section, $\sigma_{N} = P$cos$\theta/[(w-a_0)t]$ and $\tau_{N} = P$sin$\theta/[(w-a_0)t]$, whereas the strain is calculated based on the difference in the displacements at two ends of the gauge area of the specimen obtained from Digital Imaging Correlation (DIC).
As can be noted from Figures \[fig:quasistaticcrossplynormal\]-\[fig:quasistaticquasishear\], the stress-strain curves under multi-axial quasi-static loading are characterized by a significant non-linear behavior when the specimens are subjected to a combination of tension and shear, regardless of the layup. This phenomenon becomes more and more significant with increasing multiaxiality ratios due to the emergence of diffused, sub-critical matrix micro-cracking, splitting and delamination. These damage mechanisms contribute to the dissipation of the strain energy stored in the specimen inducing a significant non-linearity before reaching the failure load. A similar mechanism was reported in uniaxial tests on unidirectional laminates and two dimensional textile composites [@waas; @sal1; @kirane]. The mechanical behavior after peak stress is characterized by catastrophic failure due to snap-back instability [@Baz4; @sal2] for all the investigated layups when the specimens are subjected to tension-dominated loading. On the other hand, the failure behavior becomes more and more stable with increasing the shear load component. Eventually, as shown in Figures \[fig:quasistaticcrossplynormal\]-\[fig:quasistaticquasishear\], the post-peak behavior becomes stable and strain softening becomes evident. This is an indication of more pronounced quasi-brittleness with increasing the shear load component, a phenomenon even more pronounced for the $[+45/90/-45/0]_{s}$ specimens. It is worth mentioning here that the snap-back instability in tension-dominated loads is a structural, not a material phenomenon. To explore the post-peak behavior, one could leverage the new device proposed by the authors [@patent]. However, this is beyond the scope of the present work.
It is interesting to investigate the relationship between the normal and shear strength (critical net cross-section stress) as a function of the multiaxiality ratio. The failure envelopes for the foregoing notched specimens under multi-axial quasi-static tests are plotted in Figure \[fig:failureenvelop\] and the details can be found in Table \[tab:notchedproperties\]. In this plot, the normal and shear strengths are defined as $\sigma_{N,max}=P_{max}$cos$\theta/[(w-a_0)t]$ and $\tau_{N,max}=P_{max}$sin$\theta/[(w-a_0)t]$ respectively where $P_{max}$ is the critical load. As can be noted from Figure \[fig:failureenvelop\], the failure envelope of quasi-isotropic \[+45/90/$-$45/0\]$_{s}$ layup with a 10 mm central crack encompasses the one with a 10 mm open hole hinting the fact that the hole affects the structural strength more severely than a crack. A similar phenomenon, associated to the greater splitting development in load-bearing plies for the specimens with an intra-laminar central crack compared to the ones with an open hole, has been reported by several authors on notched Carbon Fiber Reinforced Polymer (CFRP) laminates [@Tan; @Xu]. However, since the failure behavior of notched composites depends on the size of the non-linear Fracture Process Zone (FPZ) occurring in the presence of a large stress-free crack compared to the specimen width [@Baz1; @Sal; @Baz2; @Baz3; @Yao1; @Yao2; @Yao4; @Seung1; @Seung2; @Deleo], geometrically scaled specimens with an open hole or a central crack need to be investigated to better understand this phenomenon. The size effect tests are undergoing and will be presented in future publications. Another subject of future investigations is the evolution of the fracture toughness with the multiaxiality ratio which has been studied in nanocomposites [@Zap; @Samit].
Multi-axial fatigue tests on notched specimens
----------------------------------------------
### Evolution of structural stiffness vs. multiaxiality ratio {#sec:evolutionofstiffness}
Having discussed the failure behavior of notched laminates under multi-axial quasi-static loading, the fatigue failure behavior needs to be investigated. The evolution of the structural stiffness obtained from the multi-axial fatigue tests is represented in Figures \[fig:stiffnessdegradationcrossply70\]-\[fig:stiffnessdegradationisotropic55\] for all the studied notched specimens and two loading cases (70% and 55% of $P_{max}$). In these plots, the stiffness has the same meaning defined for the analysis of unnotched specimens under tensile fatigue loading condition.
As can be noted from Figures \[fig:stiffnessdegradationcrossply70\]-\[fig:stiffnessdegradationcrossply55\], for cross-ply \[0/90\]$_{2s}$ specimen with a central crack, the stiffness decreases of roughly 10% in the early stage of tension-dominated fatigue tests ($\lambda=0, 0.262$) while no significant degradation can be observed throughout fatigue life of specimens subjected to shear-dominated fatigue loading. This is mainly due to the growth of transverse matrix cracking during tension-dominated fatigue loading [@talreja1]. Eventually, as illustrated in Figure \[fig:degradationmultiaxialratio\]c, the stiffness decreases of about 20% before catastrophic failure for multiaxiality ratio $\lambda=0$ whereas only 5% degradation in stiffness was observed for multiaxiality ratio $\lambda=1.571$. This can be attributed to a significant reduction of the splitting in $90^{\circ}$ plies under shear-dominated loading compared to tension-dominated loading before catastrophic failure as shown in Figure \[fig:crossplydamage\] through the quantitative damage analysis by using X-ray micro-computed tomography ($\mu$-CT). Despite this reduction leading to a lower total crack volume for shear-dominated loading, the final failure is eventually triggered by the rapid growth of delamination in the late stage of fatigue tests as shown in Figure \[fig:crossdamageanalysis\].
However, this is not the case for quasi-isotropic \[+45/90/$-$45/0\]$_{s}$ specimen with an open hole or a central crack since in both cases a gradual stiffness degradation throughout the fatigue life can be noted for various multiaxiality ratios. In contrast to cross-ply \[0/90\]$_{2s}$ specimens, the stiffness of quasi-isotropic specimens with an open hole or a central crack deteriorates roughly 19% to 25% for all the investigated multiaxiality ratios as shown in Figures \[fig:degradationmultiaxialratio\]a-b. This consistent deterioration is mainly due to the significant development of splitting and delamination before catastrophic failure for all the multi-axial tests as shown in Figure \[fig:quasidamage\] exemplifying the typical damage in specimens featuring an open hole. Similar damage was observed in specimens featuring a central crack.
In addition to the foregoing observations, it can be noted from Figures \[fig:stiffnessdegradationisotropic70\]-\[fig:stiffnessdegradationisotropic55\] that the fatigue performance of quasi-isotropic \[+45/90/$-$45/0\]$_{s}$ specimens with a central crack is generally worse than the one of the same layup featuring an open hole. As shown in Figure \[fig:quasidamageanalysis\], the quantitative damage analysis by using $\mu$-CT technique confirms that the worse performance of the quasi-isotropic specimens weakened by a central crack compared to the ones weakened by a central hole is associated to a significant larger amount of crack volume. It is interesting to note that this is in contrast to the quasi-static results which pointed to the central hole case as being the most critical. This structural phenomenon is due to a different evolution of the Fracture Process Zone (FPZ) in quasi-static regime compared to fatigue. There is no doubt that the process of FPZ development is strongly affected by the size of the specimens (width, notch size, etc.). This size effect deserves further studies and it will be the subject of future publications by the authors.
### S-N curve vs. multiaxiality ratio
To gain a better understanding of the fatigue behavior of notched laminates under various multiaxiality ratios, S-N curves were constructed by leveraging the fatigue tests with three different loading cases (70%, 55% and 40% of $P_{max}$). As illustrated in Figure \[fig:SNcurves\], the peak nominal normal stress is plotted as a function of number of cycles in the semi-logarithmic coordinate for notched specimens. In these plots, the slopes of S-N curves decrease as multiaxiality ratio increases which indicates that the deterioration of the fatigue behavior happens severely when the loading condition transits from tension to shear. This phenomenon is consistent with the previous experimental investigations on both tabular and cruciform specimens featuring a circular hole [@quaresimin8; @francis; @jones; @Fuji5; @Fuji6; @Fuji7]. It is worth mentioning here that, for specimens under the fatigue loading of pure shear ($\lambda=1.571$), the slope of S-N curve is zero due to the definition of the multiaxiality ratio. Thus, additional S-N curves are plotted in Figure \[fig:SNcurves\] for these specific cases.
As far as the fatigue life is concerned, as illustrated in Figure \[fig:fatiguelifevsmultiaxialratio\] and tabulated in Table \[tab:fatiguetimelifes\], the number of cycles to failure decreases with increasing the multiaxiality ratio for both two fatigue loading conditions (70% and 55% of $P_{max}$). This cause the decrease of the slopes of S-N curves as mentioned in the forgoing discussion. Furthermore, it was shown in Figure \[fig:fatiguelifevsmultiaxialratio\]a that the endurance limit (2 million cycles) of notched cross-ply \[0/90\]$_{2s}$ specimens under the fatigue loading of pure tension can be achieved approximately when 55% of $P_{max}$ is applied as a peak load in the cyclic load. This is much higher than the condition for the forgoing unnotched specimens featuring the same layup under tensile fatigue loading condition. This is probably due to the reduced stress concentration of a central crack caused by splitting in $0^{\circ}$ plies occurring at the notch tip during the fatigue [@kortschot; @spearing; @morais]. However, the fatigue loading condition for the endurance limit of notched cross-ply \[0/90\]$_{2s}$ specimens at multiaxiality ratios except for pure tension is about 40% of $P_{max}$, the similar condition for notched quasi-isotropic \[+45/90/$-$45/0\]$_{s}$ specimens at all the investigated multiaxiality ratios in order to reach the endurance limit during the fatigue. This lower limit is close to the cases of investigated unnotched quasi-isotropic and cross-ply specimens under tensile fatigue loading condition which indicates that the effect of the splitting in those cases is mitigated and does not help to significantly reduce the stress concentration at the notch tip thus not improving the fatigue resistance. In fact, the splitting in $0^{\circ}$ plies occurring at the notch tip for those cases is not a dominant mechanism as shown in Figure \[fig:crossplydamage\]. The main damage mechanism transits from the significant splitting in $0^{\circ}$ plies to the delamination between $0^{\circ}$ and $90^{\circ}$ plies when the multiaxiality ratio increases. More detailed information on the fracturing morphology of notched quasi-isotropic and cross-ply laminates under multi-axial quasi-static and fatigue loading will be discussed in a future publication focusing on the damage mechanisms and the quantitative damage analysis by using X-ray micro-computed tomography ($\mu$-CT).
Conclusions
===========
This work investigated the failure behavior of notched quasi-isotropic and cross-ply laminates under both multi-axial quasi-static and fatigue loading. Based on the results obtained in this study, the following conclusions can be elaborated:
1\. for notched laminates under multi-axial quasi-static loading, the stress-strain behavior is characterized by a significant non-linearity. This phenomenon becomes more and more significant with increasing shear load components due to the emergence of diffused, sub-critical matrix micro-cracking and delamination. This conclusion is supported by micro-computed tomography analysis.
2\. The multiaxiality ratio influences significantly also the post-peak behavior. Catastrophic failure due to snap-back instability occurs when the specimens are subjected to tension-dominated loading whereas the post-peak behavior becomes stable and strain softening becomes evident in the case of shear-dominated loading;
3\. Leveraging the multi-axial data, failure envelops in terms of peak nominal normal and shear stresses were constructed. Interestingly, the quasi-isotropic laminates featuring an open hole of diameter $a_0$ exhibited a lower strength compared to specimens weakened by a central crack with length equal to $a_0$. This phenomenon depends on the evolution of the Fracture Process Zone (FPZ) as a function of the multiaxiality ratio and specimen size. Further size effect tests are undergoing to better clarify this phenomenon.
4\. the forgoing quasi-static data are in contrast to the results of the fatigue tests showing the central crack case being the most critical. This structural phenomenon may be due to a different evolution of the Fracture Process Zone (FPZ) in quasi-static regime compared to fatigue. This aspect is particularly important for structural design since it shows that failure under quasi-static loading and fatigue can occur by a completely different damage progression. The efficient fatigue design of composite structures demands the formulation of damage models that can capture such evolution in the context of the loading configuration, stacking sequence, and structure size and geometry.
5\. the evolution of the structural stiffness degradation in notched quasi-isotropic and cross-ply laminates is substantially different. While the structural stiffness exhibits a similar amount of degradation before catastrophic failure for quasi-isotropic laminates at different multiaxiality ratios, the stiffness of cross-ply laminates deteriorates about 20% before catastrophic failure for a multiaxiality ratio $\lambda=0$ but only 5% degradation for a multiaxiality ratio $\lambda=1.571$. This is mainly due to a significant reduction of transverse matrix cracking in specimens under shear-dominated loading compared to tension-dominated loading before catastrophic failure;
6\. the S-N curves clearly shows the detrimental effects of the shear load component on the fatigue life of all the investigated notched laminates since the slopes of S-N curves decrease with increasing the shear load component. On the other hand, the conditions to achieve the endurance limit are approximately the same (40% of $P_{max}$) for both notched and notch-free laminates at different multiaxiality ratios except for notched cross-ply laminates under tensile fatigue loading;
7\. in this case, the endurance limit is about 55% of $P_{max}$ which is much higher than the one of notch-free laminates featuring the same layup. This is probably due to the reduced stress concentration of a central crack caused by splitting occurring at the notch tip during the fatigue since the development of splitting is the main damage mechanism for this loading case;
8\. the foregoing results are of utmost importance for the structural design of polymer matrix composites under multi-axial loading condition but so far rarely investigated. The lack of experimental data on this topic in the literature hindered the development of more accurate models which can guarantee a safe design in particular when composite structural components subject to multi-axial fatigue loading. As a first step towards filling the forgoing knowledge gap, this study provides a comprehensive experimental data on this topic.
Acknowledgments {#acknowledgments .unnumbered}
===============
Marco Salviato acknowledges the financial support from the Haythornthwaite Foundation through the ASME Haythornthwaite Young Investigator Award and from the University of Washington Royalty Research Fund. This work was also partially supported by the Joint Center For Aerospace Technology Innovation through the grant titled “Design and Development of Non-Conventional, Damage Tolerant, and Recyclable Structures Based on Discontinuous Fiber Composites".
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Figures and Tables {#figures-and-tables .unnumbered}
==================
\[H\] ![(a) Test setup and Arcan rig used for the multi-axial tests. Geometry of notched specimens used for the multi-axial tests: (b) $[+45/90/-45/0]_s$ specimen with a 10 mm central crack, (c) $[+45/90/-45/0]_s$ specimen with a 10 mm hole, and (d) $[0/90]_{2s}$ specimen with a 18 mm central crack. Note that the gauge length is about 25 mm.[]{data-label="fig:geometry"}](geometry.pdf "fig:")
\[H\] ![(a) Normalized S-N curves for unnotched quasi-isotropic and cross-ply specimens under tensile fatigue loading conditions; (b) Normalized stiffness vs. number of cycles (60% $P_{max}$). Note that $N_{en}$ equals to 2 million cycles as endurance limit in this work and $\sigma_{N,max}$ represents the quasi-static uniaxial strength of the specimen.[]{data-label="fig:unnotched"}](unnotched.pdf "fig:")
\[H\] ![Nominal normal stress vs. nominal normal strain obtained from the multi-axial quasi-static tests on the $[0/90]_{2s}$ specimen weakened by a 18 mm central crack.[]{data-label="fig:quasistaticcrossplynormal"}](quasistaticcrossplynormal.pdf "fig:")
\[H\] ![Nominal shear stress vs. nominal shear strain obtained from the multi-axial quasi-static tests on the $[0/90]_{2s}$ specimen weakened by a 18 mm central crack.[]{data-label="fig:quasistaticcrossplyshear"}](quasistaticcrossplyshear.pdf "fig:")
\[H\] ![Nominal normal stress vs. nominal normal strain obtained from the multi-axial quasi-static tests on the $[+45/90/-45/0]_{s}$ specimens weakened by a 10 mm central crack and a 10 mm hole respectively. All the tests summarized in this figure exhibited snap-back instability.[]{data-label="fig:quasistaticquasiisonormal"}](quasistaticquasiisonormal.pdf "fig:")
\[H\] ![Nominal shear stress vs. nominal shear strain obtained from the multi-axial quasi-static tests on the $[+45/90/-45/0]_{s}$ specimens weakened by a 10 mm central crack and a 10 mm hole respectively.[]{data-label="fig:quasistaticquasishear"}](quasistaticquasiisoshear.pdf "fig:")
\[H\] ![Failure envelopes of notched $[0/90]_{2s}$ and $[+45/90/-45/0]_{s}$ specimens under multi-axial quasi-static loading conditions. Note that the nominal normal and shear strength are defined as $P_{max}cos\theta/[(w-a)t]$ and $P_{max}sin\theta/[(w-a)t]$ and the multiaxiality ratio is defined as $\lambda= arctan(\tau_{N}/\sigma_{N})$.[]{data-label="fig:failureenvelop"}](failureenvelope.pdf "fig:")
\[H\] ![Evolution of structural stiffness vs. number of cycles measured during the multi-axial fatigue tests (70% $P_{max}$) on the $[0/90]_{2s}$ specimen with a 10 mm central crack.[]{data-label="fig:stiffnessdegradationcrossply70"}](Stiffnessdegradationcrossply70.pdf "fig:")
\[H\] ![Evolution of structural stiffness vs. number of cycles measured during the multi-axial fatigue tests (55% $P_{max}$) on the $[0/90]_{2s}$ specimen with a 10 mm central crack.[]{data-label="fig:stiffnessdegradationcrossply55"}](Stiffnessdegradationcrossply55.pdf "fig:")
\[H\] ![Evolution of structural stiffness vs. number of cycles measured during the multi-axial fatigue tests (70% $P_{max}$) on the $[+45/90/-45/0]_{s}$ specimens with a 10 mm central crack and a 10 mm hole.[]{data-label="fig:stiffnessdegradationisotropic70"}](Stiffnessdegradationisotropic70.pdf "fig:")
\[H\] ![Evolution of structural stiffness vs. number of cycles measured during the multi-axial fatigue tests (55% $P_{max}$) on the $[+45/90/-45/0]_{s}$ specimens with a 10 mm central crack and a 10 mm hole.[]{data-label="fig:stiffnessdegradationisotropic55"}](Stiffnessdegradationisotropic55.pdf "fig:")
\[H\] ![The percentage of stiffness degradation right before catastrophic failure vs. multiaxiality ratio for notched quasi-isotropic and cross-ply specimens. Note that $K_{0}$ is the initial stiffness.[]{data-label="fig:degradationmultiaxialratio"}](degradationmultiaxialratio.pdf "fig:")
\[H\] ![Analysis of main fatigue damage mechanisms before final failure by micro-computed tomography. Comparison on fatigue damage in $[0/90]_{2s}$ specimens weakened by a 18 mm central crack for multiaxiality ratio $\lambda=0$ and $\lambda=1.571$. Note that the original colors of the images were inverted for a better visualization on the damage.[]{data-label="fig:crossplydamage"}](crossplydamage.pdf "fig:")
\[H\] ![The evolution of the total crack volume and delamination area for the $[0/90]_{2s}$ specimens weakened by a 18 mm central crack as a function of the percentage of fatigue life for three multiaxiality ratios. Results are based on micro-computed tomography.[]{data-label="fig:crossdamageanalysis"}](crossdamageanalysis.pdf "fig:")
\[H\] ![Analysis of main fatigue damage mechanisms before final failure by micro-computed tomography. Comparison on fatigue damage in \[+45/90/$-$45/0\]$_{s}$ specimens weakened by a 10 mm open hole for multiaxiality ratio $\lambda=0$ and $\lambda=1.571$. Note that the original colors of the images were inverted for a better visualization on the damage.[]{data-label="fig:quasidamage"}](quasidamage.pdf "fig:")
\[H\] ![The evolution of the total crack volume for notched \[+45/90/$-$45/0\]$_{s}$ specimens as a function of the percentage of fatigue life for three multiaxiality ratios. The graphs compare hole and crack results obtained from micro-computed tomography.[]{data-label="fig:quasidamageanalysis"}](quasidamageanalysis.pdf "fig:")
\[H\] ![S-N curves vs. multiaxiality ratio measured from the multi-axial fatigue tests on notched quasi-isotropic and cross-ply specimens.[]{data-label="fig:SNcurves"}](SNcurves.pdf "fig:")
\[H\] ![Number of cycles to failure vs. multiaxiality ratio for notched quasi-isotropic and cross-ply specimens.[]{data-label="fig:fatiguelifevsmultiaxialratio"}](fatiguelifevsmultiaxialratio.pdf "fig:")
|
---
abstract: 'Considering $CuO_{2}$ conducting layers present in high Tc superconductors as plasma sheets, we proposed a prescription for the $T_{c}$ pressure behavior taken into account the Casimir effect. The Casimir energy arises from these parallel plasma sheets (Cu-O planes) when it take placed in the regime of nanometer scale (small *d* distance). The charge reservoir layer supplies carries to the conducting $nCuO_{2}$ layers, which are a source of superconductivity. The pressure induced charge transfer model (PICTM) makes use of an intrinsic term, which description is still unclear. Considering Casimir energy describing the $T_{c}$ for the case of hight-$T_{c}$ superconductors, we propose an explicit expression to the intrinsic term. Realistic parameters used in the proposed expression have shown an agreement with experimental intrinsic term data observed in some high-$T_{c}$ compounds.'
author:
- 'H. Belich, M. T. D. Orlando, E. M. Santos, L. J. Alves, and J. M. Pires'
- 'T. Costa-Soares'
title: 'Pressure effect on high-$T_{c}$ superconductors and Casimir Effect in nanometer scale'
---
Introduction
============
The high Tc cuprates superconductors have been described since its discovery in 1986 [@Berd] as composed by two major constituents in their unit cell; $MBa_{2}O_{4-d}$ as a charge reservoir block (M=Cu,Tl,Hg,Bi,C) and a conducting $nCuO_{2}$ layers (n=2,3,4,5,6). The charge reservoir layer supplies carries to the conducting $nCuO{2}$ layers and these carriers in $%
CuO_{2}$ planes are a source of superconductivity.
In 1993, Putilin *et al.* [@putilin] have obtained a new family $%
HgBa_{2}Ca_{n-1}Cu_{n}O_{y}$ (n=1,2,3 ...), which has presented the highest $%
T_{c}$ (134K) for n=3. This Hg-cuprate system loss its superconducting properties due to $CO_{2}$ contamination, however this matter has been overcome by partial substitution of mercury (Hg) by rhenium (Re) [@Shi1; @Kis]. Taken into account the rhenium (Re) substitution, its was possible for other research groups to study the physical properties of this family without problems like sample degradation and with a precise oxygen content control.
Our group has investigated the $Hg_{0.8}Re_{0.2}Ba_{2}Ca_{2}Cu_{3}O_{8+%
\delta}$ in the ceramic form (polycristaline) since 1998 [@sin; @mtdo]. This compound can be described as three $CuO_{2}$ conducting planes separated by layers of essentially insulating material, which is a feature that high Tc cuprates have in common.
The mercury family $HgBa_{2}Ca_{n-1}Cu_{n}O_{y}$ (n=1,2,3 ...) has a number of $CuO_{2}$ conducting layers proportional to $n$. Taken into account the existence of these $nCuO_{2}$ conducting layers in cuprate superconductors, in 2003 it was indicated [@kempf] that Casimir effect [@casimir] should occur between the parallel superconducting layers in high Tc superconductors. For ideal conductors layers separated by vacuum the Casimir energy is described as:
$$E_{c}(d)=-\frac{\pi ^{2}\hbar cA}{720d^{3}}$$
where $A$ is the plate’s area, and is larger as compared with the distance *d*. This equation describes the Casimir energy for two parallel plasma sheets with larger separation *d*.
Our previous study about $Hg_{0.8}Re_{0.2}Ba_{2}Ca_{2}Cu_{3}O_{8+\delta}$ [@mtdo1], with optimally oxygen content ($\delta = 8.79$ and $T_{c}^{max}=133$K), has indicated $dT_{c}/dP$=$1.9(2)$K$GPa^{-1}$. Considering that the optimally oxygen content represent the optimally condition for carrier transport in the [*[Cu-O]{}*]{} cluster formed by $3-CuO_{2}$ layers, we attributed the $T_{c}$ increment with external pressure as a reduction of the *d* distance between the [*[Cu-O]{}*]{} clusters. In this scenario we propose a investigation of the nanometric distances correlation between [*[Cu-O]{}*]{} clusters and $T_{c}$ variation in the frame of Casimir energy.
Casimir effect in high-$T_{c}$ superconductors
==============================================
In high-$T_{c}$ superconductors the hole of the Casimir plates can be attributed to the $nCuO_{2}$ layers, which form a [*[Cu-O]{}*]{} non-superconducting charge carriers layers initially, and are able to form the superconductors layers below $T_{c}$. As these superconducting [*[Cu-O]{}*]{} cluster of layers are separated by two orders of magnitude smaller than the London penetration depth, the Casimir effect is reduced by several orders of magnitude. Taken into account a small *d* (nanometric scale), Bordag [@bordag] has proposed for a Transverse Magnetic (TM) mode, a modification on Casimir energy, as following:
$$E_{c}(a)=-5.10^{-3}\hbar cAd^{-5/2}\sqrt{\frac{nq^{2}}{2mc^{2}\epsilon _{0}}}$$
In the equation (2) $A$ is the sheet area, $d$ is a nanometric distance between the sheets, and $n$ represents the carrier density.
In the regime of small distances (nanometric scale) between the clusters of [*[Cu-O]{}*]{} layers, the Casimir effect becomes a van der Waals type effect dominated by contributions from TM surface plasmons propagating along the [*[ab]{}*]{} planes [@kempf]. Within the Kempf model, the superconducting condensation energy is the same order of magnitude as the Casimir energy. Taken into account the density of states in the case of a Fermi gas in two dimensions, the transition temperature $T_c$ was predicted [@kempf; @kempf1] as below:
$$T_{c}=\frac{2^{3/4}\pi^{1/2}\hbar^{3/2}e^{1/2}n^{1/4}} {10\eta
k_{B}m^{3/4}\epsilon_{0}^{1/4} d^{5/4}}$$
The equation (3) presents $m=2*\alpha*m_{e}$ as a carrier effective mass, $n$ as a carrier $CuO_{2}$ layer density, $\eta=1.76$ BSC parameter, $d$ as a distance between two $nCuO_{2}$ clusters of layers, $\epsilon_{0}$ as vacuum electrical permeability.
Our interpretation is that $\alpha$ in $m=2*\alpha*m_{e}$ represents a factor associated with effective mass of the conducting superconductor carrier, which came from the convolution of local symmetry of $CuO_{2}$ (Ex. Octahedral, pyramidal or plane) with the crystal symmetry. The main relation pointed out by the equation (3) is that $T_{c}$ is a function of $%
\alpha^{3/4}$, $n^{1/4}$, and $d^{-5/4}$. In order to verify the equation (3), it was built the Table I using realistic values found in our laboratory and in the literature [@mtdo1; @Armstrong; @Kleche; @Novikov; @Crisan; @Takahashi; @Zhi; @Cornelius; @Schirber].
Pressure effect on high-$T_{c}$
===============================
As a high-$T_{c}$ superconductor probe, it was investigated the effect of hydrostatic pressure under $Hg_{0.82}\-Re_{0.18}\-Ba_{2}\-Ca_{2}\-Cu_{3}\-O_{8+%
\delta}$, labeled here as (Hg,Re)-1223. First of all, to describe the effect of hydrostatic pressure, it was assumed that the volume compressibility of (Hg,Re)-1223 is the same one determined for $Hg_{1}\-Ba_{2}\-Ca_{2}\-Cu_{3}%
\-O_{8+\delta}$ compound (labeled as Hg-1223), which is close to 1%/GPa [@hunter]. For (Hg,Re)-1223, when the hydrostatic pressure is closer to 0.9GPa, the crystal unitary cell volume is reduced down to -0.8%. The variation of hydrostatic pressure up to 1.2 GPa on (Hg,Re)-1223, with different $\delta$ causes different $T_{c}$ changes [@mtdo1]. The reduction of the unitary cell, under hydrostatic pressure, leads to an variation of $T_{c}$ and it is associated to contraction of the [*[a, b and c]{}*]{}-axis. The different $T_{c}$ dependence, concerning external hydrostatic pressure, may be interpreted by the pressure induced charge transfer model (PICTM) modified by Almasan et al. [@alm]. The variation on $T_{c}$ can be described by Neumeier and Zimmermann [@neu] equation:
$$\frac{dT_{c}}{dP}=\frac{\partial T_{c}^{i}}{\partial P}+\frac{\partial T_{c}%
}{\partial n}\frac{\partial n}{\partial P}$$
where the first term is an intrinsic variation of $T_{c}$ with pressure and the second is related to changes in $T_{c}$ due to variation on the carrier concentration in $nCuO_{2}$ conducting layers, which are caused by the pressure’s change. For the case of external hydrostatic pressure effects on samples with optimally oxygen content, such as $%
Hg_{0.8}Re_{0.2}Ba_{2}Ca_{2}Cu_{3}O_{8+\delta}$ with($\delta=8.79$), the second term in equation (4) vanish. Then, under this condition we have:
$$\frac{dT_{c}}{dP}=\frac{\partial T_{c}^{i}}{\partial P}$$
So, for this case, the $T_{c}$ variation will be determined only by the intrinsic term. The non-negligible intrinsic term $\partial
T_{c}^{i}/\partial P$ suggests an effective contribution of the lattice to the mechanism of high-$T_{c}$ superconductivity against the role of carriers. The $Hg_{0.8}Re_{0.2}Ba_{2}Ca_{2}Cu_{3}O_{8.79}$ has shown $\partial T_{c}^{i}/\partial P = 1.9 K/GPa$ [@mtdo1] and the $YBa_{2}Cu_{3}O{7}$ compound has presented $\partial T_{c}^{i}/\partial P = 0.9 K/GPa$ [@neu].
The intrinsic term has been presented with a physical meaning, but without an exact description since its introduction in 1992 [@alm]. However, if the $T_{c}$ can be associated to the temperature from Casimir energy, as proposed by equation (3), the equation (4) can be rewriting as following:
$$\frac{dT_{c}}{dP}=\frac{\partial T_{c}}{\partial d}\frac{\partial d}{%
\partial P}+ \frac{\partial T_{c}}{\partial \alpha}\frac{\partial \alpha}{%
\partial P} + \frac{\partial T_{c}}{\partial n}\frac{\partial n}{\partial P}$$
For samples with optimally oxygen content the third term is vanish, as justified before. For this optimally conditions there is a direct correspondence between the intrinsic (5) term and the other two significant terms (6), as specified below:
$$\frac{\partial T_{c}^{i}}{\partial d}= \frac{\partial T_{c}}{\partial d}%
\frac{\partial d}{\partial P}+ \frac{\partial T_{c}}{\partial \alpha}\frac{%
\partial \alpha}{\partial P}$$
Substituting the equation (3) in (7) we have an explicit expression to the intrinsic term, as following:
$$\frac{\partial T_{c}^{i}}{\partial d} = \frac{-5}{4}T_{c}\frac{1}{d}\frac{%
\partial d}{\partial P} + \frac{-3}{4}T_{c}\frac{1}{\alpha}\frac{\partial
\alpha}{\partial P}$$
Discussion
==========
The signal of both terms in the equation (8) are negative, however the intrinsic term is positive, when the pressure is increase. This behavior can be justified by the negative signal of the both derivative terms,
$$\frac{\partial d}{\partial P}, \frac{\partial \alpha}{\partial P}
\rightarrow (P\uparrow) \rightarrow \frac{\partial d}{\partial P}<0 ,and ~
\frac{\partial \alpha}{\partial P} <0$$
The first derivative term represents the compression coefficient in *c* axis direction. The crystallography *c* axis is associated with the nanometric distance *d* between the two $nCuO_{2}$ clusters of layers, located each one in different adjacent crystals cells. As consequence one can write the following expression:
$$K_{c}= - \frac{1}{c}\frac{\partial c}{\partial P}= -\frac{1}{d}\frac{%
\partial d}{\partial P}$$
The second derivative term in (9) represents a variation of the effective carrier mass, which came from the change on the dispersion relation under pressure. X-ray diffraction and XANES analysis of the (Hg,Re)-1223, with optimally oxygen content has indicated a tendency of [*[O-Cu-O]{}*]{} bond angle being closest $180^{o}$ [@luis]. The effect of increase the external pressure is to change this [*[O-Cu-O]{}*]{} bond angle to $180^{o}$. In our point of view, $\alpha$ coefficient is related with the convolution of $CuO_{2}$ local symmetry and crystal symmetry. As consequence, $\alpha$ value is reduced as comparing with the initial value (ambient pressure), when the pressure is increase.
$$(P\uparrow) \rightarrow \frac{\partial \alpha}{\partial P}<0$$
Therefore, the final signal of $\partial T_{c}^{i}/\partial P$ is positive, and the intrinsic term will present a positive behavior under external pressure.
In order to verify the agreement of the equation (8), it was built the Table II using realistic values found in the literature [@hunter; @Armstrong; @Kleche; @Novikov; @Crisan; @Cornelius; @Schirber] and in our laboratory for compounds with $nCuO_{2}$ superconducting layers. Moreover, the recent discovery (April 2008) of a new superconductor family by Takahashi *et al.* [@Takahashi] with ($FeAs$) superconducting layers suggested that we included the $SmOFeAs$ [@Zhi] compound in the Table II also.
The values reduction of effective mass coefficient $\alpha$ suggested a relation with the total symmetry (local $CuO_{2}$ + crystal) configuration. Computing simulation are going on in order to verify the variation of dispersion relation in the reciprocal space in order to compare the values suggested in the Table II.
Conclusion
==========
The Casimir energy was related with the superconducting condensation energy [@kempf; @kempf1], taken into account the density of states in the case of a Fermi gas in two dimensions. As consequence, the transition temperature $T_c$ was predicted as function of $m^{3/4}$, $n^{1/4}$, and $d^{-5/4}$. Within this scenario, the $\alpha$ coefficient in $m=2*\alpha*m_{e}$ was interpreted as the effective carrier mass factor from the dispersion relation, taken into account the convolution between local symmetry of $CuO_{2}$ (Ex. Octahedral, pyramidal or plane) and the crystal symmetry. The values found by $T_{c}$ expression is in agreement with the experimental $T_{c}$ values found in the principal superconductors described in the literature and (Hg,Re)-1223 measured in our laboratory. The $T_{c}$’s behavior under external hydrostatic pressure (described by PICTM) shows an intrinsic term, which is identified here with the variation of Casimir energy. This intrinsic term’s pressure dependence presents an explicit expression proportional to the compressibility coefficient of *c* axis and the effective mass of carrier charge. For the best of our knowledge, the $\partial
T_{c}^{i}/\partial P$ has not presented an explicit expression before. Our propose describe the dependence of intrinsic term with pressure in agreement with the values found in the literature.
We would like to thank CNPq Grant CT-Energ 504578/2004-9, CNPq 471536/2004-0, and CAPES for financial supports. Thanks also to Companhia Siderúgica de Tubarão (ArcelorMittal). We gratefully acknowledge to National Laboratory of Light Synchrotron - LNLS, Brazil (XPD, XAS and DXAS projects), International Center for Condensed Matter (60 Years of Casimir Effect) Brasilia, Brazil, June 2008. H. Belich would like to expresses his gratitude to the High Energy Section of the Abdus Salam ICTP, for the kind hospitality during the period of this work was done.
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List of Tables
==============
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Compound d(nm) n($m^{-2}$) $\alpha$ $T_{c}^{Cas}$(K) $T_{c}^{ref}$ Cu-O Sym. Crystal Sym.
------------------------------------------ --------- ----------------- -------------- ---------------------- ------------------- ------------------------- ------------------
$La_{2}CuO_{4}$[@Schirber] $1.32$ $1$x$10^{18}$ $24$ $38$ $40$ tilted Octahe. Tetra.
$SmOFeAs$ [@Zhi] $0.85$ $0.6$x$10^{18}$ $24$ $58$ $55 \*tilted Pyram. (Fe-As) Tetra.
$
$YBa_{2}Cu_{3}O_{7}$ [@Cornelius] $1.16$ $2$x$10^{18}$ $% $92$ $92$ distorted Pyram. Ortho.
12$
$Hg-1201$ [@Kleche][@hunter] $0.95$ $1.2$x$10^{18}$ $% $96$ $98$ Octahe. Tetra.
13$
$Hg-1212$ [@Kleche][@hunter] $1.27$ $2.4$x$10^{18}$ $7$ $126$ $127$ Pyram. Tetra.
$Hg-1223$ [@Armstrong][@Kleche][@hunter] $1.58$ $% $5$ $135$ $134$ Pyram. Tetra.
3.1$x$10^{18}$
$(Hg,Re)-1223$ [@mtdo1] $1.56$ $3.2$x$10^{18}$ $5$ $% $133$ Pyram. Tetra.
134$
$Hg-1234$ [@Novikov] $1.89$ $4.4$x$10^{18}$ $4.5$ $125 $125$ Pyram. Tetra.
$
$Hg-1245$ [@Crisan] $2.21$ $5.5$x$10^{18}$ $4.5$ $108$ $108$ Pyram. Tetra.
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Critical temperature evaluated by Casimir energy $T_{c}^{Cas}$, $%
T^{ref}_{c}$ obtained in the references, and correlations[]{data-label="temperature"}
Tetra is Tetragonal, Orth is Orthorhombic, Pyram means Pyramidal, and Octahe is Octahedral. The [*[(Fe-As)]{}*]{} is a new system (2008).
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Compound $K_{c}$($10^{-3}GPa^{-1}$) $\partial T_{c}^{exp}/\partial P $\partial T_{c}^{i}/\partial P$ $\frac{-5}{4}T_{c}\frac{1}{d}\frac{% $\frac{-3}{4}T_{c}\frac{1}{\alpha}\frac{% $\frac{1}{\alpha}\frac{\partial \alpha}{%
$ \partial d}{\partial P}$ \partial \alpha}{\partial P}$ \partial P}$ ($10^{-3}GPa^{-1}$)
----------------------------------- ------------------------------ ------------------------------------ ------------------------------------ --------------------------------------- -------------------------------------------- --------------------------------------------
$Hg-1201$ [@hunter] $5.8$ $1.7$ $1.7$ $0.7$ $1.0$ $-13.8$
$Hg-1212$ [@hunter] $6.0$ $1.7$ $1.7$ $0.9$ $0.8$ $-8.4$
$Hg-1223$ [@Armstrong][@hunter] $5.6$ $1.7$ $1.7$ $0.9$ $% $-7.9$
0.8$
$(Hg,Re)-1223$[@mtdo1] $5.6$ $1.9$ $1.9$ $0.8$ $1.1$ $-11.0$
$Hg-1234$[@Novikov] $5.8$ $1.2$ $1.2$ $0.8$ $0.4$ $-4.3$
$YBa_{2}Cu_{3}O_{7}$ [@Cornelius] $4.2$ $0.9$ $0.9$ $0.3$ $0.6$ $-8.7$
$La_{2}CuO_{4}$ [@Schirber] $1.6$ $2.3$ $2.3$ $0.1$ $2.2$ $%
-73$
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Intrinsic term pressure dependence evaluated by Casimir energy[]{data-label="intrinsic"}
|
---
abstract: 'We consider a model of interacting neurons where the membrane potentials of the neurons are described by a multidimensional piecewise deterministic Markov process (PDMP) with values in $\R^N, $ where $ N$ is the number of neurons in the network. A deterministic drift attracts each neuron’s membrane potential to an equilibrium potential $m.$ When a neuron jumps, its membrane potential is reset to a resting potential, here $0,$ while the other neurons receive an additional amount of potential $\frac{1}{N}.$ We are interested in the estimation of the jump (or spiking) rate of a single neuron based on an observation of the membrane potentials of the $N$ neurons up to time $t.$ We study a Nadaraya-Watson type kernel estimator for the jump rate and establish its rate of convergence in $L^2 .$ This rate of convergence is shown to be optimal for a given Hölder class of jump rate functions. We also obtain a central limit theorem for the error of estimation. The main probabilistic tools are the uniform ergodicity of the process and a fine study of the invariant measure of a single neuron.'
address:
- 'P. Hodara: CNRS UMR 8088, Département de Mathématiques, Université de Cergy-Pontoise, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France.'
- 'N. Krell: Université de Rennes 1, Institut de Recherche mathématique de Rennes, CNRS-UMR 6625, Campus de Beaulieu. Bâtiment 22, 35042 Rennes Cedex, France.'
- 'E. Löcherbach: CNRS UMR 8088, Département de Mathématiques, Université de Cergy-Pontoise, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France.'
author:
- 'P. Hodara, N. Krell, E. Löcherbach'
date: 'September 22, 2016'
title: 'Non-parametric estimation of the spiking rate in systems of interacting neurons.'
---
Introduction
============
This paper is devoted to the statistical study of certain Piecewise Deterministic Markov Processes (PDMP) modeling the activity of a biological neural network. More precisely, we are interested in estimating the the underlying jump rate of the process, *i.e.* the spiking rate function of each single neuron.
Piecewise Deterministic Markov Processes (PDMP’s) have been introduced by Davis ([@Davis84] and [@Davis93]) as a family of càdlàg Markov processes following a deterministic drift with random jumps. PDMP’s are widely used in probabilistic modeling of *e.g.* biological or chemical phenomena (see *e.g.* [@CDMR] or [@PTW-10], see [@ABGKZ] for an overview). In the present paper, we study the particular case of PDMP’s which are systems of interacting neurons. Building a model for the activity of a neural network that can fit biological considerations is crucial in order to understand the mechanics of the brain. Many papers in the literature use Hawkes Processes in order to describe the spatio-temporal dependencies which are typical for huge systems of interacting neurons, see [@ae], [@hrbr] and [@HL] for example. Our model can be interpreted as Hawkes process with memory of variable length (see [@aenew]); it is close to the model presented in [@bresiliens]. It is of crucial interest for modern neuro-mathematics to be able to statistically identify the basic parameters defining the dynamics of a model for neural networks. The most relevant mechanisms to study are the way the neurons are connected to each other and the way that a neuron deals with the information it receives. In [@dglo] and in [@hrbr], the authors build an estimator for the interaction graph, in discrete or in continuous time. In the present work, we assume that we observe a subsystem of neurons which are all interconnected and behaving in a similar way. We then focus on the estimation of the firing rate of a neuron within this system. This rate depends on the membrane potential of the neuron, influenced by the activity of the other neurons.
More precisely, we consider a process $X_t=(X_t^1,...,X_t^N),$ where $N$ is the number of neurons in the network and where each variable $X_t^i$ represents the membrane potential of neuron $ i, $ for $ 1 \le i \le N.$ Each membrane potential $X_t^i $ takes values in a compact interval $ [0, K], $ where $0$ is interpreted as resting potential (corresponding to $ \sim - 90 mV$ in real neurons) and where $ K \sim 140 m V $ (see *e.g.* [@I-09]). This process has the following dynamic. A deterministic drift attracts the membrane potential of each neuron to an equilibrium potential $m \in \R_+$ with an exponential speed of parameter $\lambda \in \R_+ .$ Moreover, a neuron with membrane potential $x$ “fires" (*i.e.*, jumps) with intensity $f(x),$ where $f:\R_+ \to \R_+ $ is a given intensity function. When a neuron fires, its membrane potential is reset to $0,$ interpreted as resting potential, while the membrane potentials of the other neurons are increased by $\frac{1}{N}$ until they reach the maximal potential height $K.$
The goal of this paper is to explore the statistical complexity of the model described above in a non-parametric setting. We aim at giving precise statistical characteristics (such as optimal rates of convergence, estimation procedures) such that we are able to compare systems of interacting neurons to benchmark non-parametric models like density estimation or nonlinear regression. More precisely, given the continuous observation [^1] of the system of interacting neurons over a time interval $[0,t]$ (with asymptotics being taken as $t \rightarrow \infty$), we infer on the different parameters of the model which are: the equilibrium potential $m, $ the speed of attraction $\lambda $ and the spiking rate function $f$. Since in a continuous time setting, the coefficients $ \lambda $ and $m$ are known (they can be identified by any observation of the continuous trajectory of a neuron’s potential between two successive jumps), the [*typical*]{} problem is the estimation of the unknown spiking rate $f (\cdot).$
Therefore we restrict our attention to the estimation of the unknown spiking rate $f (\cdot) .$ We measure smoothness of the spiking rate by considering Hölder classes of possible shapes for the spiking rate and suppose that the spiking rate has smoothness of order $\beta $ in a Hölder sense. To estimate the jump rate $f$ in a position $a,$ we propose a Nadaraya-Watson type kernel estimator which is roughly speaking of the form $$\hat f_t(a)=\frac{\sharp \mbox{ spikes in positions in $ B_h(a)$ during $[0, t ] $} }{\mbox{ occupation time of $ B_h(a)$ during $[0, t ]$ } },$$ where $B_h(a)$ is a neighborhood of size $h$ of the position $a$ where we estimate the jump rate function $f.$ A rigorous definition of the estimator is given in terms of the jump measure and an occupation time measure of the process $X.$ The convergence of the estimator is implied by the fact that the compensator of the jump measure is the occupation time measure integrated against the jump rate function $f,$ together with uniform ergodicity of the process. Assuming that the jump rate function $f$ has smoothness of order $\beta $ in a Hölder sense, we obtain the classical rate of convergence of order $t^{-\frac{\beta}{2 \beta +1}}$ for the point-wise $L^2 -$error of the estimator. This rate is shown to be optimal. We also state two important probabilistic tools that are needed in order to obtain the statistical results. The first one is the uniform positive Harris recurrence of process. The second one is the existence of a regular density function of the invariant measure of a single neuron.
In the literature, non-parametric estimation for PDMP’s has already been studied, see for example [@ADG-P] and, more particularly concerning the estimation of the jump rate, [@AM-G]. On the contrary to these studies, the framework of the present work is more difficult for two reasons. The first reason is the fact that our process is multidimensional, presenting real interactions between the neurons. Of course, estimation problems for multidimensional PDMP’s have already been studied. However, in all cases we are aware of, a so-called “many-to-one formula" (see [@Nathalie], see also [@hoffmann-olivier]) allows to express the occupation time measure of the whole system in terms of a single “typical” particle. This is not the case in the present paper – and it is for this reason that we have to work under the relatively strong condition of uniform ergodicity which is implied by compact state space – a condition which is biologically meaningful. The second, more important, reason is the fact that the transition kernel associated to jumps is degenerate. This is why the construction of our estimator is different from other constructions in previous studies. The degeneracy of the transition kernel also leads to real difficulties in the study of the regularity of the invariant density of a single neuron, see [@evanew] and the discussions therein.
In Section \[sec:results\], we describe more precisely our model and state our main results. We first provide two probabilistic results necessary to prove the convergence of the estimator: firstly, the positive Harris recurrence of the process $X$ in Theorem \[theo:harrisok\] and secondly the properties of the invariant measure in Theorem \[theo:invmeasure\]. The speed of convergence of our estimator is established in Theorem \[theo:main\]. Finally, Theorem \[theo:lowerbound\] states that our speed of convergence is optimal for the point-wise $L^2-$error, uniformly in $f.$ The key tool to prove this optimality is to study the asymptotic properties of the likelihood process for a small perturbation of the function $f$ close to $a.$
The proofs of Theorems \[theo:harrisok\],\[theo:main\] and \[theo:lowerbound\] are respectively given in Sections \[sec:Harris\], \[sec:proofmain\] and \[sec:optimal\]. We refer the reader to [@evanew] for a proof of Theorem \[theo:invmeasure\].
The model {#sec:results}
=========
The dynamics
------------
Let $N > 1 $ be fixed and $(N^i(ds, dz))_{i=1,\dots,N}$ be a family of *i.i.d.* Poisson random measures on $\R_+ \times \R_+ $ having intensity measure $ds dz.$ We study the Markov process $X_t = (X^{ 1 }_t, \ldots , X^{ N}_t )$ taking values in $[0, K]^N$ and solving, for $i=1,\dots,N$, for $t\geq 0$, $$\begin{aligned}
\label{eq:dyn}
X^{ i}_t &= & X^{i}_0 - \lambda \int_0^t ( X^{ i}_s - m) ds - \int_0^t \int_0^\infty
X^{ i}_{s-} 1_{ \{ z \le f ( X^{ i}_{s-}) \}} N^i (ds, dz) \\
&&+ \sum_{ j \neq i } \int_0^t\int_0^\infty a_K( X_{s-}^i ) 1_{ \{ z \le f ( X^{j}_{s-}) \}} N^j (ds, dz).
\nonumber\end{aligned}$$ In the above equation, $\lambda > 0 $ is a positive number, $ m$ is the equilibrium potential value such that $0 < m < K.$ Moreover, we will always assume that $ K \geq \frac2N .$ Finally, the functions $a_K : [0, K ] \to [0, K ] $ and $f:\R_+\mapsto \R_+$ satisfy (at least) the following assumption.
\[ass:1\] \
1. $a_K : [0, K ] \to [0, \frac{1}{N}] $ is non-increasing and smooth, $ a_K ( x) = \frac1N, $ for all $x < K- \frac2N$ and $ a_K ( x) < K-x$ for all $x \geq K- \frac2N .$\
2. $f\in C^1(\mathbb R_+),$ $f$ is non-decreasing, $f(0) = 0,$ and there exists $f_{min}:\R_+\mapsto \R_+,$ non-decreasing, such that $f(x) \geq f_{min}(x) > 0 $ for all $x > 0$.
All membrane potentials take values in $ [0, K ], $ where $K$ is the maximal height of the membrane potential of a single neuron. $0$ is interpreted as resting potential (corresponding to $ \sim - 90 mV$ in real neurons) and $ K \sim 140 m V $ (see *e.g.* [@I-09]). In , $\lambda $ gives the speed of attraction of the potential value of each single neuron to an equilibrium value $m .$ The function $a_K$ denotes the increment of membrane potential received by a neuron when an other neuron fires. For neurons with membrane potential away from the bound $K,$ this increment is equal to $\frac{1}{N}.$ However, for neurons with membrane potential close to $K,$ this increment may bring their membrane potential above the bound $K.$ This is why we impose this dynamic close to the bound $K.$
In what follows, we are interested in the estimation of the intensity function $f,$ assuming that the parameters $K, f_{min}$ and $a_K$ are known and that the function $f$ belongs to a certain Hölder class of functions. The parameters of this class of functions are also supposed to be known. The assumption $f(0)=0$ comes from biological considerations and expresses the fact that a neuron, once it has fired, has a refractory period during which it is not likely to fire.
The generator of the process $X$ is given for any smooth test function $ \varphi : [0,K]^N \to \R $ and $x \in [0, K]^N$ by $$\label{eq:generator0}
L \varphi (x ) = \sum_{ i = 1 }^N f(x_i) \left[ \varphi ( \Delta_i ( x) ) - \varphi (x) \right]
- \lambda \sum_i \left( \frac{\partial \varphi}{\partial x_i} (x) \left[ x_i -m \right] \right) ,$$ where $$\label{eq:delta}
(\Delta_i (x))_j = \left\{
\begin{array}{ll}
x_j + a_K ( x_j ) & j \neq i \\
0 & j = i
\end{array}
\right\}
.$$
The existence of a process $X$ with such dynamics is ensured by an acceptance/rejection procedure that allows to construct solutions to (\[eq:dyn\]) explicitly. More precisely, since each neuron spikes at maximal intensity $f(K),$ we can work conditionally on the realization of a Poisson process $\bar N$ with intensity $Nf(K).$ We construct the process $X$ considering the jump times $\bar T_n$ of $\bar N$ as candidates for the jump times of $X$ and accepting them with probability $$\frac{\sum_{i=1}^Nf\left( X_{\bar T_n -}^i \right) }{Nf(K)}.$$ It is then possible to construct a solution to step by step, following the deterministic drift between the jump times of $\bar N,$ and jumping according to this acceptance/rejection procedure. We refer the reader to Theorem 9.1 in chapter IV of [@IW] for a proof of the existence of the process $(X_t)_t.$
We denote by $P_x$ the probability measure under which the solution $(X_t)_t$ of starts from $X_0 = x \in [0,K]^N . $ Moreover, $P_\nu = \int_{[0, K]^N } \nu (dx) P_x $ denotes the probability measure under which the process starts from $X_0 \sim \nu.$ Figure \[Fig. 1\] is an example of trajectory for $N=5$ neurons, choosing $f=Id, \; \lambda=1, \; m=1,$ and $K=2.$
![Trajectory of 5 neurons[]{data-label="Fig. 1"}](trajf1l1.pdf)
The aim of this work is to estimate the unknown firing rate function $f$ based on an observation of $ X$ continuously in time. Notice that for all $ 1 \le i \le N , $ $X^i$ reaches the value $0$ only through jumps. Therefore, the following definition gives the successive spike times of the $i-$th neuron, $1 \le i \le N.$ We put $$T^i_0 = 0 , T^i_n = \inf\{ t > T^i_{n-1} : X^i_{ t- } > 0 , X^i_t = 0 \} , n \geq 1 ,$$ and introduce the jump measures $$\mu^i ( ds, dy ) = \sum_{n \geq 1 } 1_{ \{ T^i_n < \infty \} } \delta_{ (T^i_n, X^i_{T^i_n - }) } (dt, dy), \quad \mu ( dt, dx) = \sum_{i=1}^N \mu^i ( ds, dx) .$$ By our assumptions, $ \mu^i$ is compensated by $ \hat \mu^i ( ds, dy ) = f( X^i_s) ds \delta_{ X^i_s} ( dy) ,$ and therefore the compensator $\hat \mu $ of $\mu $ is given by $$\hat \mu (dt, dy) = f(y) \eta(dt,dy) , \mbox{ where } \eta(A \times B) = \int_A \left( \sum_{i= 1}^N 1_B ( X^i_s) \right) ds$$ is the total occupation time measure of the process $X.$
We will also write $T_n , n \geq 0, $ for the successive jump times of the process $X,$ *i.e.*$$T_0 = 0 , T_n = \inf\{ T_k^i : T_k^i > T_{ n -1}, k \geq 1 , 1 \le i \le N \}, n \geq 1 .$$
For some kernel function $Q$ such that $$\label{eq:intkern}
Q \in C_{c} ( \R) , \int_{\R} Q(y) dy = 1 ,$$ we define the kernel estimator for the unknown function $f$ at a point $a$ with bandwidth $h$, based on observation of $X$ up to time $t$ by $$\label{eq:kernelest}
\hat f_{t,h} (a) =\frac{\int_0^t \int_{\R} Q_h(y-a) \mu(ds,dy) }{\int_0^t \int_{\R} Q_h(y-a) \eta(ds,dy)}, \mbox{ where } Q_h(y):=\frac{1}{h}Q\left(\frac{y}{h}\right) \mbox{ and } \frac{0}{0} := 0 .$$ For $h$ small, $ \hat f_{t,h} (a)$ is a natural estimator for $ f( a) .$ Indeed, this expression as a ratio follows the intuitive idea to count the number of jumps that occurred with a position close to $a$ and to divide by the occupation time of a neighborhood of $a,$ which is natural to estimate an intensity function depending on the position $a.$ More precisely, by the martingale convergence theorem, the numerator $\int_0^t \int_{\R} Q_h(y-a) \mu(ds,dy) $ should behave, for $ t$ large, as $ \int_0^t \int_{\R} Q_h(y-a) f (y) \eta (ds,dy) .$ But by the ergodic theorem, $$\frac{\int_0^t \int_{\R} Q_h(y-a) f (y) \eta (ds,dy)}{\int_0^t \int_{\R} Q_h(y-a) \eta(ds,dy)} \to \frac{\pi_1( Q_h (\cdot - a) f ) }{\pi_1 ( Q_h ( \cdot - a ) ) }$$ as $t \to \infty ,$ where $ \pi_1$ is the stationary measure of each neuron $X_t^i .$ Finally, if the invariant measure $\pi_1$ is sufficiently regular, then $$\frac{\pi_1( Q_h (\cdot - a) f ) }{\pi_1 ( Q_h ( \cdot - a ) ) } \to f(a)$$ as $h \to 0 . $
We restrict our study to fixed Hölder classes of rate functions $f.$ For that sake, we introduce the notation $ \beta = k + \alpha $ for $ k=\lfloor
\beta \rfloor \in \N $ and $ 0 \le \alpha < 1.$ We consider the following Hölder class for arbitrary constants $F, L > 0,$ and a function $f_{min}$ as in Assumption \[ass:1\]. $$\begin{gathered}
\label{eq:Hspace}
H( \beta,F, L ,f_{min}) = \{ f \in C^k ( \R_+) : | \frac{d^l}{d x^l } f(x) | \leq F , \mbox{ for all } 0 \le l \le k, x \in [0, K] , \; \\
f(x) \geq f_{min}(x) \mbox{ for all $x\in [0, K]$,} \; \ | f^{(k)} (x) - f^{(k) } (y) | \le L |x- y |^\alpha \mbox{ for all } x, y \in [0, K]\} .\end{gathered}$$
Probabilistic results
---------------------
In this Section, we collect important probabilistic results. We first establish that the process $(X_t)_{t \geq 0} $ is recurrent in the sense of Harris.
\[theo:harrisok\] Grant Assumption \[ass:1\]. Then the process $X$ is positive Harris recurrent having unique invariant probability measure $\pi ,$ *i.e.* for all $ B \in {\mathcal B} ( [0, K ]^N) , $ $$\label{eq:definrec}
\pi (B) > 0 \; \mbox{ implies } P_x \left( \int_0^\infty 1_B (X_s) ds = \infty \right) =1$$ for all $ x \in [0, K]^N .$ Moreover, there exist constants $C > 0 $ and $ \kappa > 1 $ which do only depend on the class $H( \beta , F, L, f_{min} ) , $ but not on $f,$ such that $$\label{eq:ergodic}
\sup_{ f \in H( \beta , F, L, f_{min} ) } \| P_t (x, \cdot ) - \pi \|_{TV} \le C \kappa^{ - t } .$$
It is well-known that the behavior of a kernel estimator such as the one introduced in depends heavily on the regularity properties of the invariant probability measure of the system. Our system is however very degenerate. Firstly, it is a piecewise deterministic Markov process (PDMP) in dimension $N,$ with interactions between particles. Hence, no Brownian noise is present to smoothen things. Moreover, the transition kernels associated to the jumps of system are highly degenerate (recall ). The transition kernel $$K ( x, dy) = {\mathcal L} ( X_{T_1} |X_{T_1- } = x ) (dy ) = \sum_{i=1}^N \frac{f ( x^i )}{\bar f (x) } \delta_{ \Delta^i (x) } ( dy )$$ with $\bar f(x):= \sum_{i=1}^N f(x^i)$ puts one particle (the one which is just spiking) to the level $0.$ As a consequence, the above transition does not create density – and it even destroys smoothness due to the reset to $0$ of the spiking neuron. Finally, the only way that “smoothness” is generated by the process is the smoothness which is present in the “noise of the jump times” (which are basically of exponential density). For this reason, we have to stay away from the point $x=m,$ where the drift of the flow vanishes. Moreover, the reset-to-$0$ of the spiking particles implies that we are not able to say anything about the behavior of the invariant density of a single particle in $0 $ (actually, near to $0$) neither. Finally, we also have to stay strictly below the upper bound of the state space $K.$ That is why we introduce the following open set $ S_{d , \beta }$ given by $$\label{eq:estimset}
S_{d , \beta } := \{ w \in [0, K] : \frac{\lfloor
\beta \rfloor}{N} < w < K- \frac{\lfloor
\beta \rfloor}{N} , |w-m| > d \} ,$$ where $\beta$ is the smoothness of the fixed class $ H( \beta , F, L, f_{min}) $ that we consider and where $ d $ is fixed such that $d > \frac{\lfloor
\beta \rfloor +2}{N}.$ Notice that $S_{d , \beta } $ also depends on $K, m $ and $N$ which are supposed to be known. We are able to obtain a control of the invariant measure only on this set $S_{d , \beta }.$ The dependence in $\beta$ is due to the fact that the regularity of $f$ is transmitted to the invariant measure by the means of successive integration by parts (see [@evanew] for more details).
We quote the following theorem from [@evanew].
\[theo:invmeasure\] (Theorem 5 of [@evanew])
Suppose that $f \in H( \beta , F, L, f_{min} ).$ Let $$\pi_1 := {\mathcal L}_\pi ( X_t^1)$$ be the invariant measure of a single neuron, *i.e.* $ \int g d \pi_1 = E_\pi ( g( X^1_t) ) .$ Then $\pi_1 $ possesses a bounded continuous Lebesgue density $\pi^1 $ on $S_{d , \beta } $ for any $d$ such that $d > (\lfloor
\beta \rfloor +2)/N ,$ which is bounded on $S_{d , \beta } ,$ uniformly in $f \in H( \beta , F, L, f_{min} ).$ Moreover, $\pi^1 \in C^k ( S_{d , \beta }) $ and $$\label{eq:imctrl}
\sup_{\ell \le \lfloor
\beta \rfloor , w \in S_{d , \beta } } | \pi_1^{(\ell )} ( w) | + \sup_{w \neq w' , w, w' \in S_{d , \beta }} \frac{\pi_1^{(\lfloor
\beta \rfloor )} (w) - \pi_1^{(\lfloor
\beta \rfloor )} (w') }{|w-w'|^\alpha } \le C_F,$$ where the constant $C_F $ depends on $d $ and on the smoothness class $ H( \beta , F, L, f_{min} ),$ but on nothing else.
Statistical results
-------------------
We can now state the main theorem of our paper which describes the quality of our estimator in the minimax theory. We assume that $ m$ and $\lambda $ are known and that $f$ is the only parameter of interest of our model. We shall always write $ P_x^f $ and $E_x^f$ in order to emphasize the dependence on the unknown $f.$ Fix some $r > 0 $ and some suitable point $a \in S_{d , \beta } .$ For any possible rate of convergence $(r_t )_{t \geq 0 } $ increasing to $\infty $ and for any process of ${\mathcal F}_t-$measurable estimators $\hat f_t $ we shall consider point-wise square risks of the type $$\sup_{ f \in H( \beta , F, L, f_{min} )} r_t^2 E_x^f \left[ | \hat f_t ( a) - f(a) |^2 | A_{t,r} \right] ,$$ where $$A_{t,r}:= \left\{\frac{1}{Nt} \int_0^t \int_{\R} Q_h(y-a) \eta(ds,dy) \geq r \right\}$$ is roughly the event ensuring that sufficiently many observations have been made near $a,$ during the time interval $[0, t ].$ We are able to choose $r$ small enough such that $$\label{eq:noloss}
\underset{t \to \infty }{\lim\inf} \inf_{f \in H( \beta , F, L, f_{min} ) } P^f_x (A_{t,r} ) = 1,$$ see Proposition \[prop:imlb\] below.
Recall that the kernel $Q$ is chosen to be of compact support. Let us write $R$ for the diameter of the support of $Q,$ therefore $Q(x) = 0 $ if $ |x| \geq R.$ For any fixed $a \in S_{d , \beta } , $ write $h_0 := h_0 ( a, R, \beta , d ) := \sup \{ h > 0 : B_{ h R} (a) \subset S_{d/2, \beta } \} .$ Here, $ B_{h R } (a) = \{ y \in \R_+ : |y - a | < h R \} .$
\[theo:main\] Let $f \in H( \beta , F, L, f_{min} ) $ and choose $Q \in C_{c} ( \R)$ such that $\int_{\R} Q(y) y^j dy = 0 $ for all $ 1 \le j \le \lfloor \beta \rfloor ,$ and $ \int_{\R} |y|^\beta Q(y) dy < \infty .$ Then there exists $r^*>0$ such that the following holds for any $ a \in S_{d , \beta } , r \le r^* $ and for any $ h_t \le h_0.$\
(i) For the kernel estimate with bandwidth $h_t = t^{ - \frac{1}{2 \beta + 1 } } ,$ for all $x \in [0,K],$ $$\underset{t \to \infty }{\lim\sup} \sup_{ f \in H( \beta , F, L, f_{min} ) }t^{\frac{2 \beta }{2 \beta +1} } E_x^f \left[ | \hat f_{t, h_t} (a) - f(a) |^2 | A_{t,r} \right] < \infty .$$
\(ii) Moreover, for $ h_t = o ( t^{ - 1 /(1 + 2 \beta ) } ) ,$ for every $f \in H( \beta , F, L, f_{min} ) $ and $ a \in S_{d , \beta } $ $$\sqrt{th_t} \left( \hat f_{t,h_t} ( a) - f(a) \right) \to {\mathcal N} ( 0, \Sigma ( a) )$$ weakly under $ P_x^f ,$ where $\Sigma ( a) = \frac{f(a)}{N \pi_1 (a) } \int Q^2 (y) dy .$
The next theorem shows that the rate of convergence achieved by the kernel estimate $ \hat f_{t, t^{- 1/(2 \beta + 1 ) } } $ is indeed optimal.
\[theo:lowerbound\] Let $ a \in S_{d , \beta } $ and $x \in [0, K ] $ be any starting point. Then we have $$\underset{t \to \infty }{\lim\inf} \inf_{ \hat f_t} \sup_{ f \in H( \beta , F, L, f_{min} ) } t^{ \frac{2 \beta }{1 + 2 \beta } } E_x^f [ | \hat f_t ( a) - f(a) |^2 ] > 0 ,$$ where the infimum is taken over the class of all possible estimators $\hat f_t (a) $ of $f(a) .$
The proofs of Theorems \[theo:main\] and \[theo:lowerbound\] are given in Sections \[sec:proofmain\] and \[sec:optimal\].
Simulation results
------------------
In this subsection, we present some results on simulations, for different jump rates $f.$ The other parameters are fixed: $N=100, \; \lambda=1, \; K=2$ and $m=1.$ The dynamics of the system are the same when $\lambda$ and $f$ have the same ratio. In other words, variations of $\lambda$ and $f$ keeping the same ratio between the two parameters lead to the same law for the process rescaled in time. This is why we fix $\lambda=1$ and propose different choices for $f.$ The kernel $Q$ used here is a truncated Gaussian kernel with standard deviation 1. We present for each choice of a jump rate function $f$ the associated estimated function $\hat f$ and the observed distribution of $X$ or more precisely of $\bar X=\frac{1}{N} \sum_{i=1}^N X^i.$ Figures 2, 3 and 4 correspond respectively to the following definitions of $f: \; f(x)=x, \; f(x)= \log(x+1)$ and $f(x)= \exp(x)-1.$
For Figures \[Fig. 2\], \[Fig. 3\] and \[Fig. 4\], we fixed the length of the time interval for observations respectively to $t=200,$ $300$ and $150.$ This allows us to obtain a similar number of jump for each simulation, respectively equal to $17324,$ $18579$ and $21214.$ These simulations are realized with the software R.
The optimal bandwidth $h_t=t^{-\frac{1}{2 \beta +1}}$ depends on the regularity of $f$ given by the parameter $\beta.$ Therefore, we propose a data-driven bandwidth chosen according to a Cross-validation procedure. For that sake, we define the sequence $\left( Z_k \right)_{k \in \N^*}$ by $Z_k^i=X_{T_k^i-}^i$ for all $1 \leq i \leq N.$ For each $a \in [0,K]$ and each sample $Z=(Z_1,...,Z_n),$ for $1\leq \ell\leq n$ we define the random variable $\hat \pi_1^{\ell, n,h} (a)$ by $$\hat \pi_1^{\ell, n,h} (a) = \frac{1}{(n-\ell)N} \sum_{k=\ell +1}^n \sum_{i=1}^N Q_h(Z_k^i-a).$$ $\hat \pi_1^{\ell, n,h} (a)$ can be seen as an estimator of the invariant measure $\pi^Z_1$ of the discrete Markov chain.
We propose an adaptive estimation procedure at least for this simulation part. We use a Smoothed Cross-validation (SCV) to choose the bandwidth (see for example the paper of Hall, Marron and Park [@HaMa]), following ideas which were first published by Bowmann [@Bo] and Rudemo [@Ru]. As the bandwidth is mainly important for the estimation of the invariant probability $\pi_1^{Z} $, we use a Cross validation procedure for this estimation. More precisely, we use a first part of the trajectory to estimate $\widehat \pi_1^{\ell, n,h}$ and then another part of the trajectory to minimize the Cross validation $SCV(h)$ in $h.$ In order to be closer to the stationary regime, we chose the two parts of the trajectory far from the starting time. Moreover we chose two parts of the trajectory sufficiently distant from each other. This is why we consider $m_1, m_2$ and $\ell$ such that $1<<m_1 \leq m_2 << \ell \leq n.$
We use the method of the least squares Cross validation and minimize $$SCV(h)=\int \left( \widehat \pi_1^{\ell , n,h} (x) \right)^2 dx-\frac{2}{N(m_2-m_1)} \sum_{k=m_1+1}^{m_2} \sum_{i=1}^N \widehat \pi_1^{\ell, n,h} (Z_k^i)$$ (where we have approximated the integral term by a Riemann approximation), giving rise to a minimizer $\hat h .$ We then calculate the estimator $\hat f$ the long of the trajectory. In the next figure, we use this method to find the reconstructed $ f $ with an adaptive choice of $h.$
![Estimation of the intensity function $f(x)=x$[]{data-label="Fig. 2"}](estimf1al2.pdf)
![Estimation of the intensity function $f(x)=\log (x+1)$[]{data-label="Fig. 3"}](estimflogal2.pdf)
![Estimation of the intensity function $f(x)=\exp (x) - 1$[]{data-label="Fig. 4"}](estimfexpal2.pdf)
As expected, we can see that the less observations we have, the worse is our estimator. Note that close to $0$ the observed density of $X$ explodes. This was also expectable due to the reset to $0$ of the jumping neurons. Moreover, the simulations show a lack of regularity of the observed density close to $m,$ which is consistent with our results, but this does not seem to affect the quality of the estimator.
Harris recurrence of $X$ and speed of convergence to equilibrium – Proof of Theorem \[theo:harrisok\] {#sec:Harris}
=====================================================================================================
In this section, we give the proof of Theorem \[theo:harrisok\] and show that the process $ (X_t)_{t \geq 0} $ is positive recurrent in the sense of Harris. We follow a classical approach and prove the existence of regeneration times. This is done in the next subsection and follows ideas given in Duarte and Ost [@bresiliens].
Regeneration
------------
The main idea of proving a regeneration property of the process is to find some uniform “noise” for the whole process on some “good subsets” of the state space. Since the transition kernel associated to the jumps of our process is not creating any density (and actually destroys it for the spiking neurons which are reset to $0$), the only source of noise is given by the random times of spiking. These random times are then transported through the deterministic flow $ \gamma_{s,t} ( v ) = ( \gamma_{s, t } ( v^1 ) , \ldots , \gamma_{s, t } ( v^N) ),$ which is given for any starting configuration $v \in [0, K]^N$ by $$\label{eq:flow}
\gamma_{s,t} ( v^i) = e^{ -\lambda (t-s) } v^i + ( 1 - e^{ - \lambda (t-s) } ) m , \; 0 \le s \le t , \; \gamma_t ( v^i ) := \gamma_{0, t } ( v^i ) .$$
The key idea of what follows – which is entirely borrowed from [@bresiliens] – is the following.
Write $I_n, n \geq 1, $ for the sequence giving the index of the spiking neuron at time $T_n,$ *i.e.* $I_n = i $ if and only if $ T_n = T_k^i $ for some $ k \geq 1 .$ It is clear that in order to produce an absolute continuous law with respect to the Lebesgue measure on $ [0, K ]^N, $ we need at least $N$ jumps of the process. On any event of the type $ \{ T_1 = t_1, I_1 = i_1 , \ldots , T_N = t_N , t_N < t < T_{N+1} , I_N = i_N \} ,$ it is possible to write the position of the process at time $t $ as a concatenation of the deterministic flows given by $$\label{eq:concatenation}
\Gamma_{ (t_1, \ldots, t_N, i_1, \ldots , i_N )} ( t, v) = \gamma_{t_{N}, t} ( \Delta_{i_N} ( \gamma_{t_{N-1} , t_N} ( \Delta_{i_{N-1} } ( \ldots \Delta_{i_1} ( \gamma_{0, t_1} ( v) ) ) ) ) ) .$$ Proving absolute continuity amounts to prove that the determinant of the Jacobian of the map $ (t_1, \ldots , t_N ) \to \Gamma_{ (t_1, \ldots, t_N, i_1, \ldots , i_N )} ( t, v) $ does not vanish. For general sequences of $ (i_1, \ldots , i_N) ,$ this will not be true (think *e.g.* of the sequence $ (i_1 = \ldots = i_N = 1)$).
The main idea is however to consider the sequence $ i_1 = 1, i_2 = 2, \ldots , i_N = N $ and to use the [*regeneration property of spiking*]{}, *i.e.* the fact that the neuron $k$ spiking at time $t_k$ is reset to zero at time $t_k.$ In this case, for all later times, its position does not depend on $ t_1, \ldots , t_{k-1} $ any more. In other words, the Jacobian of $\Gamma_{ (t_1, \ldots, t_N, 1, \ldots , N )} ( t, v) $ is a diagonal matrix, and all we have to do is to control that all diagonal elements do not vanish. The second idea is to linearize the flow, *i.e.* to consider the flow during very short time durations, and to use that, just after spiking, each diagonal element is basically of the form $$\frac{\partial \gamma_{s, t } (0) } { \partial s } \sim - \lambda m , \; \mbox{ as $t -s \to 0.$}$$ The important fact here is that the absolute value of the drift term of the deterministic flow of one neuron is strictly positive when starting from the initial value $0.$
In the following, this idea is made rigorous. Our proof follows the approach given in Section 4 of [@bresiliens]. We fix $ \ge > 0 $ and put $$A_\ge = \{ i \ge - \ge / 4 < T_i < i \ge , i = 1 , \ldots, N \}$$ and $$S = \{ I_1 =1, I_2 = 2 , \ldots , I_N = N \}$$ which is the event that all $N$ neurons have spiked in the fixed order given by their numbers, *i.e.* neuron $1$ spikes first, then neuron $2,$ then $3,$ and so on. We introduce $$u^* = \left( \frac{N-1}{N}, \frac{N-2}{N}, \ldots, \frac1N , 0 \right)$$ which would be the position of neurons after $N$ spikes and on the event $S,$ if $\lambda = 0 $ (here, we suppose w.l.o.g. that $ K > 1 + \frac1N $).
Now we fix any initial configuration $v \in [0, K]^N $ and introduce the sequence of configurations $ v (k) , 0 \le k \le N ,$ given by $v(0) = v,$ $v_k(k) = 0 $ and $$\label{eq:vk}
v_i(k) = \left\{
\begin{array}{ll}
\frac{k-i}{N} ,& i < k \\
\underbrace{a_K \circ \ldots \circ a_K}_{k \; \mbox{{\small times}}} ( v_i) , & i > k
\end{array}
\right\} .$$ Notice that $ \underbrace{a_K \circ \ldots \circ a_K}_{k \; \mbox{{\small times}}} ( v_i) = v_i + \frac{k}{N} $ if $ v_i < K - \frac{2+k}{N} .$ Notice also that $v(N) = u^* .$
We cite the following lemma from [@bresiliens].
\[lem:positionbresiliens\] If $ X_0 = u \in B_\delta ( v) , $ then on the event $ A_\ge \cap S , $ we have for all $ 1 \le k \le N , $\
(i) $X_i ( T_k) = v_i ( k) + \sum_{ r= i+1}^k \lambda (T_r - T_{r-1} ) d_i (r-1) + R_{ \delta \ge} ( T_1^k , u ) + R_{ \ge^2 }( T_1^k , u ) , $ if $i < k ,$\
(ii) $X_i ( T_k) = v_i ( k) + \sum_{ r= i}^k \lambda (T_r - T_{r-1} ) d_i (r-1) +R_{\delta } ( u) + R_{\delta \ge} ( T_1^k , u ) + R_{\ge^2 }(T_1^k , u ) , $ if $i > k ,$\
(iii) $ \bar X^N ( T_k ) = \bar v (k) + R_{\delta } (u) + R_{ \ge }( T_1^k , u) , $ if $ k < N ,$ and $ \bar X^N ( T_N ) = \bar u^* + R_{\ge }( T_1^N , u ) .$
Here, $ d_i ( r) = m - v_i ( r) $ and $ T_1^k = (T_1, \ldots, T_k).$ Moreover, the remainder functions are of order $$R_{ \delta \ge} ( T_1^k , u ) = O ( \delta \ge) , R_{ \ge^2 }( T_1^k , u ) = O ( \ge^2 ) , R_{\delta } ( u) = O ( \delta ) ,\ldots ,$$ and all partial derivatives are of order either $\delta $ or $\ge ,$ uniformly in $ v .$
Our model is slightly different from the model in [@bresiliens]: instead of an attraction to the empirical mean of the system, we have an attraction to a fixed equilibrium value $m.$ This leads to our definition of $ d_i ( r) $ which is slightly different from the one used in [@bresiliens].
Put $ t^* = N \ge .$ Then we have on $ A_\ge \cap S, $ $$\label{eq:14}
X_i ( t^*) = u_i^* + \lambda (t^* - T_N) d_i^* + \sum_{ r= i+1}^N \lambda (T_r - T_{r-1} ) d_i (r-1) + R_{ \delta \ge} ( T_1^N , u ) + R_{ \ge^2 }( T_1^N , u ) ,$$ where $ d_i^* = m - u^*_i .$
We put as in [@bresiliens] $\gamma^0 (t_1^N ) = ( \gamma^0_1 ( t_1^N) , \ldots , \gamma^0_N ( t_1^N) ) ,$ where $$\gamma^0_i ( t_1^N ) := u_i^* + \lambda (t^* - t_N) d_i^* + \sum_{ r= i+1}^N \lambda (t_r - t_{r-1} ) d_i (r-1) , 1 \le i \le N.$$ Hence $\gamma^0_i (t_1^N) $ models how the $N$ successive jump times $ t_1 < t_2 < \ldots < t_N$ are mapped, through the deterministic flow, into a final position at time $t^* -$ on the event $\{ T_1 = t_1, \ldots , T_N = t_N \} \cap A_\ge \cap S.$ In order to control how the law of the $ N$ successive jump times $ t_1 , \ldots , t_N $ is transported through this flow, we calculate the partial derivatives of $ \gamma^0 $ with respect to $ t_i, 1 \le i \le N.$ One sees immediately that $$\frac{\partial \gamma^0_i}{\partial t_k} = 0 , k < i , \quad \frac{\partial \gamma^0_i}{\partial t_i} = - \lambda m , 1 \le i \le N ,$$ whence
For each $ u \in B_\delta ( v) , $ the determinant of the Jacobian of the map $ \{ i \ge - \ge / 4 < t_i < i \ge , i = 1 , \ldots, N \} \ni t_1^N \mapsto \gamma^0 (t_1^N) + R_{ \delta \ge} ( t_1^N , u ) + R_{ \ge^2 }( t_1^N , u ) $ is given by $$\lambda^N m^N + R_\ge (t_1^N , u) + R_\delta ( t_1^N , u)$$ which is different from zero for $ \ge $ and $\delta $ small enough, for all $ u \in B_\delta ( v) .$
As in Proposition 4.1 of [@bresiliens], we now have two important conclusions from the above discussion.
There exists $ \delta^* > 0$ and $\ge > 0,$ such that for $ t^* = N \ge ,$ $$\label{eq:regen}
P_{t^* } ( x, \cdot ) \geq \eta_1 1_{ B_{\delta^*} ( u^*) } ( x) \nu ,$$ where $ \nu $ is a probability measure and $\eta_1 \in ]0, 1 [.$
The lower bound is a local Doeblin condition, and its proof is given in Proposition 4.1 of [@bresiliens]. We call $B_{\delta^*} ( u^*)$ a regeneration set: if the process visits this regeneration set, then after a time $t^*$ there is a probability $\eta_1$ that the law of the process is independent from its initial position $x \in B_{\delta^*} ( u^*).$
To be able to make use of the local Doeblin condition, we have to be sure that the process actually does visit the regeneration set $ B_{\delta^*} ( u^*).$ This is granted by the following result.
\[prop:control\] There exist $ \ge > 0 $ and $ \eta_2 > 0$ such that $$\inf_{f \in H( \beta , F, L, f_{min} )} \; \inf_{ v \in [0,K]^N } P_{t^* } ( v , B_{\delta^*} ( u^* ) ) \geq \eta_2 ,$$ for $ t^* = N \ge .$
By (\[eq:14\]), there exists $ \ge $ such that for all $ v \in [0,K]^N, $ we have that $X(t^* ) \in B_{\delta^* }( u^* )$ on $A_\ge \cap S ,$ when $ X( 0 ) = v.$ Hence $$P_{t^* } ( v , B_{\delta^*} ( u^* ) ) \geq P_v({A_\ge \cap S} ) .$$
Recalling (\[eq:flow\]), we then obtain $$\begin{gathered}
P_v({A_\ge \cap S} ) = \int_{ \ge - \ge / 4 }^{\ge} f( \gamma_{0, t_1} ( v^1) ) e^{ - \int_0^{t_1} \bar f( \gamma_{0,s} ( v) )ds } dt_1 \\
\int_{ 2 \ge - \ge / 4 }^{2 \ge}f( \gamma_{ t_1, t_2} ( v (1)^2 ) ) e^{ - \int_{t_1}^{t_2} \bar f( \gamma_{t_1,s}( v (1) ) )ds } dt_2 \ldots \\
\int_{ N \ge - \ge / 4 }^{N \ge}f( \gamma_{ t_{N-1}, t_N} ( v (N-1)^N ) ) e^{ - \int_{t_{N-1}}^{t_N} \bar f( \gamma_{t_{N-1},s} ( v (N-1)) ) )ds } dt_N ,\end{gathered}$$ where $ \bar f ( x) = \sum_{i=1}^N f( x^i) , $ where the sequence $ v(1) , \ldots , v( N-1) $ is given as in (\[eq:vk\]). Since by assumption $ v \in [0,K]^N ,$ it is immediate to see that $ \gamma_{s, t } ( v (k)^i ) \le C, $ for a constant $C,$ for all $ 0 \le s \le t \le t^* , $ for all $ k \le N$ and for all $i \le N.$ Moreover, $$\gamma_{0, t_1}^1 ( v^1) \geq (1- e^{ - \lambda \frac34 \ge } ) m > 0, \mbox{ on } t_1 \geq \frac34 \ge ,$$ and since $f$ is non decreasing, satisfying $ f (x) \geq f_{min}( x) $ and $ \| f \|_\infty \le F,$ this implies that $$f( \gamma_{0, t_1} ( v^1) ) e^{ - \int_0^{t_1} \bar f( \gamma_{0,s}( v) )ds } \geq f( (1- e^{ - \lambda \frac34 \ge } ) m) e^{ - N t_1 f( C ) } \geq f_{min} \left( (1- e^{ - \lambda \frac34 \ge } ) m \right) e^{ - N t_1 F },$$ on $ t_1 \geq \frac34 \ge .$ Similar arguments show that all consecutive terms are strictly lower bounded uniformly in $f \in H( \beta , F, L, f_{min} ) $ as well. As a consequence, $$P_v({A_\ge \cap S} ) \geq \left( \frac{\ge}{4} f_{min} \left( (1- e^{ - \frac34 \lambda \ge} ) m \right) e^{ - t^*N F } \right)^N > 0,$$ which concludes the proof.
\[rq:regenunif\] In the proof of Proposition 4.1 of [@bresiliens], the authors have no need to obtain (\[eq:regen\]) uniformly in $f \in H( \beta , F, L, f_{min} ).$ However, it is easy to see that we can rewrite their proof using the bounds for $ f \in H( \beta , F, L, f_{min} ) $ appearing in the proof of Proposition \[prop:control\] above. As a consequence, we obtain $$\label{eq:regenunif}
\inf_{f \in H( \beta , F, L, f_{min} )} P_{t^* } ( x, \cdot ) \geq \eta_1 1_{ B_{\delta^*} ( u^*) } ( x) \nu ,$$ for some $\eta_1 > 0.$
Once we dispose of the uniform local Doeblin condition and of the control given in Proposition \[prop:control\], it is classical, using regeneration arguments, to show that the process is recurrent in the sense of Harris.
Harris recurrence and invariant measure
---------------------------------------
Using the regeneration procedure, we can prove that the process $X$ is positive Harris recurrent. We denote by $ \| \cdot \|_{TV} $ the total variation distance, *i.e.* $\| \nu_1 - \nu_2 \|_{TV} = \sup_{ B \in {\mathcal B} ( [0, K ]^N ) } \left| \nu_1 ( B) - \nu_2 (B) \right| , $ for any two probability measures $\nu_1 , \nu_2 $ on $ ( [0, K ]^N , {\mathcal B} ( [0, K ]^N )) .$
We first show that the process is indeed Harris. For that sake, define the sequence of stopping times $(\tilde S_n )_{n \in \N}$ $$\tilde S_1:= \inf \{ t > 0 : X_t \in B_{\delta^*} ( u^*) \},$$ and for all $n \geq 1,$ $$\tilde S_{n+1}:= \inf \{ t > \tilde S_n + t^* : X_t \in B_{\delta^*} ( u^*) \}.$$
Let $(U_n )_{n \in \N}$ be a sequence of *i.i.d.* uniform random variables on $[0,1],$ which are independent of the process $X.$ Then, working conditionally on the realization of $(U_n )_{n \in \N},$ we define the sequence $ ( S_n )_{ n \in \N }$ and the sequence $(R_n )_{n \in \N}$ of regeneration times as follows.
$$S_1:= \inf \{ \tilde S_n : U_n < \eta_1 \}, R_1 := S_1 + t^* ,$$ and for all $n \geq 1,$ $$S_{n+1}:= \inf \{ \tilde S_k > S_n : U_k < \eta_1 \}, R_{n+1} := S_{n+1} + t^* ,$$ where $\eta_1 $ is given in .
\[rem:iidregen\] (\[eq:regenunif\]) allows us to construct the process $(X_t)_{t \geq 0} $ on a bigger probability space in such a way that for all $n, X_{R_n} \sim \nu$ and $\left(X_{R_n+t}\right)_{t \in \R^+}$ is independent from $ {\mathcal F}_{S_n-}. $ This construction is known as Nummelin splitting, we refer the interested reader to Chapter 6 of Löcherbach (2013) [@cours].
\[lem:FTregen\] For all $x \in [0, K]^N , E_x (R_1) < \infty$ and $E_x (R_2-R_1) < \infty.$
The proof of this lemma is postponed to the next subsection where we prove a stronger result. Now the following result implies that our process is actually positive Harris recurrent.
\[prop:HR\] $X$ is Harris recurrent with invariant probability measure $ \pi $ which is given by $$\pi (B):=\frac{1}{ E_x ( R_2 - R_1) } E_x \left( \int_{R_1}^{R_2} 1_B(X_s) ds \right) .$$
Fix $B \in {\mathcal B} ( [0, K]^N)$ and define the process $A_t$ by $$A_t:= \int_0^t 1_B(X_s)ds.$$ Assume that $\pi(B)>0,$ then, according to the definition of Harris recurrence, it is enough to show that for all $x, \lim_{t \to +\infty} \frac{A_t}{t}>0.$
We denote by $\tilde N_t$, $\tilde N^e_t$ and $\tilde N^o_t$ the counting processes respectively associated with the sequences of stopping times $(R_n)_{n \in \N^*}, (R_{2n})_{n \in \N^*}$ and $(R_{2n+1})_{n \in \N}:$
$$\tilde N_t := \sum_{n=1}^\infty 1_{R_n \leq t}, \; \; \tilde N_t^e := \sum_{n=1}^\infty 1_{R_{2n} \leq t} \; \mbox{and} \; \tilde N_t^o := \sum_{n=0}^\infty 1_{R_{2n+1} \leq t}.$$
For all $t$ we have $\tilde N_t=\tilde N^e_t + \tilde N^o_t$ and $$\begin{gathered}
\frac{A_t}{t}=\frac{1}{t} \left( \int_0^{R_1} 1_B(X_s)ds + \sum_{n=1}^{\tilde N_t} \int_{R_n}^{R_{n+1}} 1_B(X_s)ds - \int_{R_{\tilde N_t}}^t 1_B(X_s)ds \right)
\\
=\frac{1}{t} \left( \int_0^{R_1} 1_B(X_s)ds + \sum_{k=1}^{\tilde N^e_t} \int_{R_{2k}}^{R_{2k+1}} 1_B(X_s)ds + \sum_{k=1}^{\tilde N^o_t} \int_{R_{2k-1}}^{R_{2k}} 1_B(X_s)ds - \int_{R_{\tilde N_t}}^t 1_B(X_s)ds \right).\end{gathered}$$
When $t$ goes to $\infty,$ we obtain, using Lemma \[lem:FTregen\] to deal with the first and the last terms,
$$\begin{gathered}
\lim_{t \to +\infty} \frac{A_t}{t}
\\
= \lim_{t \to +\infty} \frac{\tilde N_t}{t} \frac{1}{\tilde N_t} \left( \tilde N^e_t \left( \frac{1}{\tilde N^e_t} \sum_{k=1}^{\tilde N^e_t} \int_{R_{2k}}^{R_{2k+1}} 1_B(X_s)ds \right) + \tilde N^o_t \left( \frac{1}{\tilde N^o_t} \sum_{k=1}^{\tilde N^o_t} \int_{R_{2k-1}}^{R_{2k}} 1_B(X_s)ds \right) \right).\end{gathered}$$
The decomposition between even and odd regeneration times is used here to be able to apply the strong law of large numbers, based on Remark \[rem:iidregen\]. In this way we obtain that $$\lim_{t \to +\infty} \frac{1}{\tilde N^e_t} \sum_{k=1}^{\tilde N^e_t} \int_{R_{2k}}^{R_{2k+1}} 1_B(X_s)ds= \lim_{t \to +\infty} \frac{1}{\tilde N^o_t} \sum_{k=1}^{\tilde N^o_t} \int_{R_{2k-1}}^{R_{2k}} 1_B(X_s)ds = \pi(B) > 0 \; \mbox{a.s.}$$ We can use the same decomposition to obtain that $$\lim_{t \to +\infty} \frac{\tilde N_t}{t} = \frac{1}{E_x(R_2-R_1)} \; \mbox{a.s.}$$ Putting all together we have $$\lim_{t \to +\infty} \frac{A_t}{t}=\frac{\pi(B)}{E_x(R_2-R_1)}$$ and we can conclude the proof using Lemma \[lem:FTregen\] once again.
Speed of convergence to equilibrium – Proof of (\[eq:ergodic\]) in Theorem \[theo:harrisok\] {#speed-of-convergence-to-equilibrium-proof-of-eqergodic-in-theorem-theoharrisok .unnumbered}
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We now show how to couple two processes $X$ and $Y$ following the same dynamics (\[eq:dyn\]) using Proposition \[prop:control\] and the lower bound (\[eq:regenunif\]) of Remark \[rq:regenunif\]. This coupling will give us a control of the distance in total variation between $P_x$ and $P_y,$ where $x$ and $y$ are the respective starting points of processes $X$ and $Y.$
The coupling procedure consists in using the same realization of uniform random variables $(U_n)_{n \in \N } $ for both processes, relying on (\[eq:regenunif\]), when both processes $X$ and $Y$ are in the regeneration set $B_{\delta^*} ( u^*)$ at the same time. More precisely, we let evolve $X$ and $Y$ independently up to the first time that they are both in the set $ B_{\delta^*} ( u^*).$ We introduce the sequence of stopping times $$\bar S_1 = \inf \{ t > 0 : (X_t , Y_t) \in B_{\delta^*} ( u^*) \times B_{\delta^*} ( u^*) \}$$ and $$\bar S_{n} = \inf \{ t > \bar S_{n-1} + t^* : (X_t , Y_t) \in B_{\delta^*} ( u^*) \times B_{\delta^*} ( u^*) \} ,\; n \geq 1.$$
Applying Proposition \[prop:control\] to two independent processes $X$ and $Y,$ we obtain
$$\label{eq:alignment}
\inf_{f \in H( \beta , F, L, f_{min} )} \; \inf_{ v_1 , v_2 \in [0,K]^N } P^{\otimes 2}_{t^* } ( ( v_1 , v_2) , B_{\delta^*} ( u^* )^2 ) \geq \eta_2^2.$$
As a consequence, $ \bar S_n < \infty $ almost surely for all $n,$ and $ P_{(v_1, v_2)} ( \bar S_1 > n t^* ) \le (1 - \eta_2^2 )^n ,$ *i.e.* $\bar S_1 $ and $ \bar S_{n+1} - \bar S_n$ possess exponential moments $$E_{(v_1, v_2) } [ e^{ \alpha \bar S_1 } ] < \infty$$ uniformly in the starting configuration $ (v_1, v_2) $ for all $ \alpha < \frac{- \ln ( 1 - \eta_2^2 ) }{t^*} .$
We are now able to couple the processes $X$ and $Y.$ We work conditionally on the realization of a sequence of *i.i.d.* uniform random variables $(U_n)_{n \in \N } $ and define the coupling time $\tau$ by $$\tau:= \inf \{ \bar S_n : U_n \le \eta_1 \} + t^* .$$ Using the regenerative construction described in the previous subsection based on (\[eq:regenunif\]), it is evident that $X$ and $Y$ can be constructed jointly in such a way that $ X_\tau = Y_\tau \sim \nu $ and such that $ X_t = Y_t $ for all $t \geq \tau .$ Since $\tau $ is constructed by sampling within the sequence $ (\bar S_n)_{n \in \N} $ at an independent geometrical time, it is immediate to see that there exists $\kappa > 1 $ such that $$\label{eq:ctmoment}
\sup_{ v_1, v_2 \in [0, K]^N } E_{(v^1,v^2)}(\kappa^{\tau}) < + \infty.$$
Notice that the regeneration time $R_1$ can be compared to $\tau $ and that $R_1 \le \tau .$ As a consequence, implies a proof of Lemma \[lem:FTregen\].
Since the two processes $X$ and $Y$ follow the same trajectory after time $\tau, $ we obtain the following classical upper bound on the total variation distance. $$\label{eq:couplingspeed}
\| P_t(x,\cdot ) -P_t(y,\cdot ) \| _{TV} \leq P_{(x,y)}(\tau >t) \leq \kappa^{-t} E_{(x,y)}( \kappa^{\tau}).$$
Now putting $C:=\sup_{x , y \in [0,K]^N} E_{(x, y )}( \kappa^{\tau}),$ the integration of (\[eq:couplingspeed\]) with respect to the invariant measure $\pi (dy)$ implies that $$\sup_{x \in [0,K]^N} \left\Vert P_t(x,\cdot ) - \pi \right\Vert_{TV} \leq C \kappa^{-t}.$$ This finishes the proof of Theorem \[theo:harrisok\]. $\qed$
Estimates on the invariant density of a single particle
-------------------------------------------------------
We start with some simple preliminary estimates. Recall that $$\mu ( ds, dx) = \sum_n \delta_{T_n } (ds) \delta_{X_{T_n-}} (dx)$$ denotes the jump measure of the system, with compensator $$\hat \mu ( ds, dx ) = \bar f (X_s ) ds \delta_{X_s} (dx), \mbox{ with } \bar f (x) = \sum_{i=1}^N f (x^i ).$$ Let $Z_k = X_{T_k- } , k \geq 1 , $ be the jump chain. Then the following holds.
\[prop:3\] $(Z_k)_k$ is Harris recurrent with invariant measure given by $$\pi^Z ( g) = \frac{1}{\pi ( \bar f ) } \pi ( \bar f g ) ,$$ for any $g : \R_+^N \to \R$ measurable and bounded.
Let $ g$ be a bounded test function. We have to prove that $$\frac{1}{n} \sum_{k=1}^n g ( Z_k) \to \pi^Z ( g)$$ as $n \to \infty, $ $P_x-$almost surely, for any fixed starting point $x \in [0, K]^N.$ But $$\frac{1}{n} \sum_{k=1}^n g ( Z_k) = \frac{1}{n} \sum_{k=1}^n g ( X_{T_k-}),$$ and, putting $ N_t = \sup \{ n : T_n \le t \},$ $$\lim_{ n \to \infty } \frac{1}{n} \sum_{k=1}^n g ( X_{T_k-}) = \lim_{t \to \infty}\frac{t}{N_t} \frac{1}{t} \sum_{k=1}^{N_t} g ( X_{T_k-})
= \lim_{t \to \infty}\frac{t}{N_t} \frac{1}{t} \int_0^t\int_{\R_+^N} g (x) \mu ( ds, dx) .$$ By the law of large numbers, $ N_t/t \to \int \bar f(x) \mu (dx ) = \mu ( \bar f ) ,$ and this convergence holds almost surely. Moreover, $$\label{eq:martingaleplusbiaisconv}
\frac{1}{t} \int_0^t\int_{\R_+^N} g (x) \mu ( ds, dx) = \frac{1}{t} M_t + \frac{1}{t} \int_0^t\int_{\R_+^N} g (x) \hat \mu ( ds, dx) ,$$ where $ M_t = \int_0^t \int g ( x) [\mu (ds, dx) - \hat \mu (ds, dx) ].$ Then $M_t$ is in ${\mathcal M}^{2,d}_{loc} ,$ the set of all locally square integrable purely discontinuous martingales, with predictable quadratic covariation process $$<M>_t\; =\int_0^t g^2 (X_s) \bar f (X_s )ds$$ where $$\frac{< M>_t}{t} \to \pi ( g^2 \bar f )$$ almost surely, as $t \to \infty.$ By the martingale convergence theorem, see *e.g.* Jacod-Shiryaev (2003) [@js], chapter VIII, Corollary 3.24 , $ t^{-1/2} M_t $ converges in law to a normal distribution. As a consequence, $M_t/t \to 0$ almost surely.
We now treat the second term in . By the ergodic theorem for integrable additive functionals, $$\frac{1}{t} \int_0^t\int_{\R_+^N} g (x) \hat \mu ( ds, dx) = \frac{1}{t} \int_0^t g (X_s) \bar f( X_s) ds \to \pi ( \bar f g ),$$ and this finishes the proof.
[**Exchangeability of the invariant measure**]{} We denote by $X^i : \R_+^N \to \R_+, ( x^1 , \ldots , x^N ) \mapsto x^i $ the $i-$th coordinate map.
For all $ 1 \le i \le N, $ ${\mathcal L}_\pi ( X^i ) = {\mathcal L}_\pi ( X^1).$
Fix an initial configuration $x = (x^1, x^1 , \ldots , x^1) \in [0, K]^N$ consisting of $N$ particles which are all in the same position. Let $ \check g : \R_+ \to \R $ be a bounded test function and introduce $g ( x) := \check g (x^1 ) ,$ *i.e.* $g$ depends only on the first coordinate. By the ergodic theorem, $$\frac1t \int_0^t g ( X_s) ds \to \int_\R \check g d {\mathcal L}_\pi ( X^1)$$ $P_x-$almost surely.
Now, introduce the system $ Y_t = (Y_t^1, \ldots , Y_t^N ) $ given by $ Y_t^k = X_t^k $ for all $ k \neq 1, i $ and $ Y_t^1 = X_t^i , $ $ Y_t^i = X_t^1 .$ Since the generator of $X$ is invariant under permutations, $ (Y_t)_{t \geq 0 } \stackrel{\mathcal L}{=} (X_t)_{t \geq 0}.$ In particular, $$\int_\R \check g d {\mathcal L}_\pi ( X^1) = \lim_{t \to \infty } \frac1t \int_0^t g ( X_s) ds = \lim_{ t \to \infty } \frac1t \int_0^t g ( Y_s) ds$$ On the other hand, $$\lim_{ t \to \infty } \frac1t \int_0^t g ( Y_s) ds = \lim_{ t \to \infty } \frac1t \int_0^t \check g ( X^i_s) ds = \int_\R \check g d {\mathcal L}_\pi ( X^i),$$ and this finishes the proof.
We are now going to study the support properties of the invariant measure of a single neuron. For that sake define for all $x \in [0,K], \; b(x):=\lambda (x-m)$ and recall that $\gamma_t (x^i) \in \R_+$ denotes the solution of $ d \gamma_t (x^i) =- b( \gamma_t (x^i)) dt ,$ given by $$\gamma_t (x^i) = e^{- \lambda t }x^i + ( 1 - e^{- \lambda t })m.$$ Moreover, for $ x = (x^1 , \ldots , x^N), $ $ \gamma_t ( x ) = ( \gamma_t (x^1 ), \ldots , \gamma_t ( x^N) ) .$ Finally, let $$K(x, dy) = \sum_{i=1}^N f( x^i ) \delta_{ \Delta^i ( x) } (dy) , \quad H_f^x (t) = e^{ - \int_0^t \bar f ( \gamma_s ( x) ) ds } ,$$ where $ \bar f ( \gamma_s ( x) ) = \sum_{ i = 1 }^N f ( \gamma_s ( x^i ) ) $ and where $ \Delta^i ( x) $ was defined in before.
We will use the change of variable, for a fixed value of $y,$ $$\label{eq:covz}
z = \gamma_t ( y^1) , dz = - b (z) dt = - \lambda ( z-m) dt$$ and denote by $ \kappa_y (z) $ the inverse function of $t \to \gamma_t ( y^1) .$
These definitions permit to obtain an expression of $\pi_1$.
For all $z \in S_{d,k},$ we have $$\label{eq:pi1dens}
\pi_1 (z) = \int \pi ( dx) \int K(x, dy ) H_f^y (\kappa_y ( z) ) \frac{1_{I_y^m}(z)}{|b ( z)| } .$$ Here the notation $I_y^m$ denotes either $]y,m[$ if $y<m$ or $]m,y[$ if $m<y.$
We have, by Proposition \[prop:3\], $$\pi ( G) = E_\pi ( G ( X_t )) = E_\pi ( \bar f (X_t) G ( X_t) \frac{1}{\bar f ( X_t) } )= \pi ( \bar f ) E_{\pi^Z} \left( G ( Z_n ) \frac{1}{\bar f (Z_n) }\right) .$$ We use that $ \pi^Z = {\mathcal L}(X_{T_2-} | X_{T_1-} \sim \pi^Z ) . $ Then we obtain $$\begin{aligned}
\label{eq:piG}
\pi (G) &=& \pi ( \bar f ) \pi^Z \left( \frac{G ( Z_n ) }{\bar f (Z_n) }\right) \nonumber \\
&=& \int_{\R_+^N} \bar f (x) \pi (dx) \sum_{i=1}^N \frac{f ( x^i ) }{\bar f (x) } \int_0^\infty \bar f ( \gamma_t ( \Delta_i (x) )) \nonumber \\
&&\quad e^{- \int_0^t \bar f ( \gamma_s ( \Delta_i (x) )) ds } G ( \gamma_t ( \Delta_i ( x) ) ) \frac{1 }{\bar f ( \gamma_t ( \Delta_i ( x) ))} dt \nonumber \\
&=& \int_{\R_+^N} \pi ( dx) \int K(x, dy) \int_0^\infty H_f^y (t) G( \gamma_t (y) ) dt . \end{aligned}$$ Now, let $g \in C^\infty_c (S_{d , \beta }) $ be a smooth test function having compact support in $ S_{d , \beta }.$ Using , we obtain $$\label{eq:pi1g}
\pi_1 ( g) =\int_{\R_+^N} \pi ( dx) \int K(x, dy) \int_0^\infty H_f^y (t) g( \gamma_t (y^1) ) dt .$$ Then, with the change of variable announced in , we can rewrite in the following way: $$\pi_1 (z) = \int \pi ( dx) \int K(x, dy ) H_f^y (\kappa_y ( z) ) \frac{1_{I_y^m}(z)}{|b ( z)| } .$$
Support of the invariant measure
--------------------------------
\[prop:suptrans\] For all $y \in ]0,K[,$ all $\delta > 0,$ we have $$\inf_{f \in H( \beta , F, L, f_{min} )} \; \pi_1 \left(B_\delta(y) \right)>0 .$$
Fix $y \in ]0,K[$ and let $k \in \N$ and $s \in [0,\frac{1}{N}[$ be such that $y=\frac{k}{N}+s.$
We define the time $t_s$ such that $m \left( 1-e^{-\lambda t_s} \right) = s$ and consider, for a fixed $\varepsilon > 0,$ the following events:
$$A_y = \left\{ \frac{\ge}{2} < T_1 < \ge ; t_s +(i-1)\ge < T_i < t_s + (i-1) \ge + \frac{\ge}{2} \; \forall i = 2 , \ldots, k +1 \right\}$$ and $$S_y= \{ I_1 =1, I_2 \neq 1 , \ldots , I_{k+1} \neq 1 \}.$$ The idea of the proof is that the event $A_y \cap S_y$ leads the neuron 1 to a position close to $y$ after a time $t_y:=t_s + k \ge:$
At time $T_1$ the neuron 1 jumps so that its position is reset to $0,$ the time $t_s$ is defined such that at time $T_{2-}$ the position of neuron 1 is close to $s,$ then in an interval of time short enough for the deterministic drift to be insignificant, we impose that the other neurons jump $k$ times so that at time $T_{k+1},$ the position of neuron 1 is indeed close to $y.$
In other words we can use similar arguments to the ones used in the proof of Lemma \[lem:positionbresiliens\] to obtain that, for all $x \in [0,K]^N,$ if $ X_0 = x,$ then on the event $ A_y \cap S_y , $ we have $X_{T_{k+1}}^1=y+O(\ge),$ and we can choose $\ge$ such that $X_{T_{k+1}}^1 \in B_\delta(y).$
Now we have to prove that $$\inf_{f \in H( \beta , F, L, f_{min} )} \; \inf_{ x \in [0,K]^N } P_x (A_y \cap S_y ) > 0 ,$$ which can be done as in the proof of proposition \[prop:control\].
Finally, integrating this result against the measure $\pi$ gives us the conclusion of the proof.
We can now obtain as corollary of the following Proposition.
\[prop:imlb\] We have that $$\label{eq:imlb}
r^* := \inf_{a \in S_{d , \beta }} \; \inf_{f \in H( \beta , F, L, f_{min} )} \; \pi_1 (a) > 0,$$ and for all $x \in [0,K]^N,$and for all $r \le r^*, $ $$\label{eq:imdlb}
\lim \inf_{t \to \infty } \inf_{f \in H( \beta , F, L, f_{min} ) } P^f_x (A_{t,r} ) = 1.$$
Recalling the construction of $ \pi_1$ in (\[eq:pi1dens\]), we have $$\begin{gathered}
\pi_1(a)= \int \pi ( dx) \int K(x, dy ) H_f^y (\kappa_y ( a) ) \frac{1_{I_y^m}(a)}{|b ( a)| } \\
= \int \pi ( dx) \int \left( \sum_{i=1}^N f(x^i) \delta_{\Delta^i(x)}(dy) \right) \exp \left( -\int_0^{\kappa_y(a)} \bar f (\gamma_u(x))du \right) \frac{1_{I_y^m}(a)}{|b ( a)| }.\end{gathered}$$ To obtain a lower bound uniform in $f$ of this expression we use again the bounds of the class of function $H( \beta , F, L, f_{min} ):$ $$\forall f \in H( \beta , F, L, f_{min} ), \; f_{min}(x) \leq f(x) \leq F.$$ Doing this, we will also need an upper bound for $\kappa_y(a).$ This is possible due to the term $1_{I_y^m}(a):$ since $y$ is such that $a \in I_y^m \cap S_{d , \beta },$ the flow starting from $y$ can reach $a$ in a finite time, even if we consider the worst cases where $y=0$ or $K.$
Thanks to Proposition \[prop:suptrans\], we have $\pi_1 \left( \left\{ y: a \in I_y^m \right\} \right) >0$ implying that the integration of $1_{I_y^m}(a)$ against the measure $\pi_1(dy)$ is not 0. Finally, due to the definition of $S_{d , \beta },$ we have no problem to obtain this lower bound uniformly in $a \in S_{d , \beta },$ and this finishes the proof of .
is obtained easily from thanks to the ergodic theorem: we have
$$\lim_{t \to + \infty} \frac{1}{Nt} \int_0^t \int_\R Q_h (y-a) \eta(ds,dy) = \pi_1(a),$$ (recall that $\int_\R Q(x)dx=1$), which concludes the proof.
proof of theorem \[theo:main\] {#sec:proofmain}
==============================
Convergence of the estimator
----------------------------
We now study the speed of convergence of our estimator. First we have the following classical kernel approximation:
\[prop:KernApprox\] For any Hölder function $g$ of order $\beta=k + \alpha$ satisfying $$\label{eq:holderg}
\sup_{w \neq w'} \big| g^{(k)} (w) - g^{(k)} (w') \big| \le C_g |w-w'|^\alpha$$ for some constant $C_g$ and for a kernel $Q$ as in Theorem \[theo:main\], we have: $$\left| \int_\R Q_h (y-a) g(y) dy - g(a) \right| \leq \frac{C_g \parallel Q \parallel_{L^1} R^\beta}{k!} h^{\beta},$$ where we recall that $R$ is the diameter of the support of $Q$ and where $\parallel Q \parallel_{L^1}$ denotes the $L^1$-norm of $Q.$
Using the property $\int_\R Q(x)dx =1$ and the change of variable $x=\frac{y-a}{h},$ we obtain $$\int_\R Q_h (y-a) g(y) dy - g(a) = \int_\R Q(x) \left( g(a+xh) - g(a) \right) dx.$$ Then, a Taylor-Lagrange expansion of the function $g$ gives us $$\int_\R Q_h (y-a) g(y) dy - g(a) = \int_\R Q(x) \left( \sum_{l=1}^k \frac{g^{(l)}(a)}{l!} (xh)^l + \frac{g^{(k)}(z)-g^{(k)}(a)}{k!}(xh)^k \right) dx,$$ for some $z \in ]a,a+xh[ \cup ]a + xh , a[ .$ By the assumptions of Theorem \[theo:main\], $\int_{\R} Q(y) y^j dy = 0 $ for all $ 1 \le j \le k.$ Then condition (\[eq:holderg\]) allows to conclude.
Fix $a \in S_{d , \beta } $ and define, for all $t \in \R^+, \; \tilde \mu = \mu - \hat \mu$ the centered jump measure.
\[prop:ernum\] Under the conditions of Theorem \[theo:main\], there exists a constant $C_1$ depending only on $\beta , F, L, N, f_{min}$ and $Q,$ such that for all $f \in H( \beta , F, L, f_{min} ), $ for all $x \in [0,K]$ and for a bandwidth of the form $h = h_t = t^{- \alpha } $ for some $ 0 < \alpha < 1,$ $$\label{eq:ernumx}
E_x \left[ \left( \frac{1}{Nt} \int_{ [0, t]} \int_\R Q_h (y-a) \tilde \mu (ds, dy) \right)^2 \right] \leq \frac{C_1}{ht}.$$
We start working under the invariant regime in the first part of the proof, *i.e.* we will work under $ E_\pi .$ In a second time we will use Theorem \[theo:harrisok\] to obtain the result for any starting point $x \in [0,K]^N.$
We use the properties of the compensator $\hat \mu_t$ and its explicit expression to write $$\begin{gathered}
E_{\pi} \left[ \left( \frac{1}{Nt} \int_{[0, t ]}\int_\R Q_h (y-a) \tilde \mu(ds, dy) \right)^2 \right]
= \frac{1}{(Nt)^2} E_{\pi} \left[ \int_0^t \int_\R \left( Q_h (y-a) \right)^2 \hat \mu(ds, dy) \right] \\
= \frac{1}{(Nt)^2} E_{\pi} \left[ \int_0^t \int_\R \left( Q_h (y-a) \right)^2 f(y) \eta(ds,dy) \right].\end{gathered}$$
Now, since we are in the invariant regime, we can use the density of the invariant measure of a single particle (recall Theorem \[theo:invmeasure\]) to obtain $$E_{\pi} \left[ \left( \frac{1}{Nt} \int_{[0, t ]} \int_\R Q_h (y-a) \tilde \mu(ds, dy) \right)^2 \right] = \frac{1}{Nt} \int_\R \left( Q_h (y-a) \right)^2 f(y) \pi_1 (y)dy.$$ Our aim is to obtain a control of $ \int_\R h f(y) ( Q_h(y -a))^2 \pi_1(y)dy$ independently of $h.$ To do this we use the change of variable $x=\frac{y-a}{h}$ and write $$E_{\pi} \left[ \left( \frac{1}{Nt} \int_{[0, t ]} \int_\R Q_h (y-a) \tilde \mu(ds, dy) \right)^2 \right] \\
= \frac{1}{Nht} \int_\R Q^2 (x) f(a+xh) \pi_1 (a+xh)dx .$$ This yields $$\label{eq:numinvreg}
E_{\pi} \left[ \left( \frac{1}{Nt} \int_{[0, t]}\int_\R Q_h (y-a) \tilde \mu(ds, dy) \right)^2 \right] \leq \frac{F}{Nht} \| Q \|_{L^2 }^2 \sup_{ x \in S_{d/2 , k }} \pi_1 ( x).$$ This result holds in stationary regime, but thanks to the exponential speed of convergence of Theorem \[theo:harrisok\], we can obtain it for any starting point $x \in [0,K]^N$ as we are going to show now. For that sake we fix the bandwidth $h$ in function of $t$ so that this speed of convergence depends only on $t.$ For the moment, we will assume that $h$ is of the form $$\label{eq:htmalpha}
h_t:=t^{-\alpha}$$ for some constant $\alpha \in ]0,1[.$ As in the beginning of the proof, we can write $$E_x \left[ \left( \frac{1}{Nt} \int_{[0, t ]}\int_\R Q_h (y-a) \tilde \mu(ds, dy) \right)^2 \right] = \frac{1}{(Nt)^2} E_x \left[ \int_0^t \int_\R \left( Q_h (y-a) \right)^2 f(y) \eta(ds,dy) \right].$$ Now, we have the following decomposition $$\begin{gathered}
E_x \left[ \left( \frac{1}{Nt} \int_{[0, t ]}\int_\R Q_h (y-a) \tilde \mu(ds, dy) \right)^2 \right] \\
= \frac{1}{(N t)^2} E_x \left[ \sum_{i=1}^N \int_0^t \left( Q_h (X_s^i-a) \right)^2 f(X_s^i) ds \right] - \frac{1}{(N t)^2} E_{\pi} \left[ \sum_{i=1}^N \int_0^t \left( Q_h (X_s^i-a) \right)^2 f(X_s^i) ds \right] \\
+ \frac{1}{(N t)^2} E_{\pi} \left[ \sum_{i=1}^N \int_0^t \left( Q_h (X_s^i-a) \right)^2 f(X_s^i) ds \right].\end{gathered}$$ The last term is controlled by . We will deal with the difference in the second line using Theorem \[theo:harrisok\] as follows: for all $p \in ]0,1-\alpha[,$ we have $$\begin{gathered}
\frac{1}{(N t)^2} E_x \left[ \sum_{i=1}^N \int_0^t \left( Q_h (X_s^i-a) \right)^2 f(X_s^i) ds \right] - \frac{1}{(N t)^2} E_{\pi} \left[ \sum_{i=1}^N \int_0^t \left( Q_h (X_s^i-a) \right)^2 f(X_s^i) ds \right] \\
= \frac{1}{(N t)^2} E_x \left[ \sum_{i=1}^N \int_0^{t^p} \left( Q_h (X_s^i-a) \right)^2 f(X_s^i) ds \right] - \frac{1}{(N t)^2} E_{\pi} \left[ \sum_{i=1}^N \int_0^{t^p} \left( Q_h (X_s^i-a) \right)^2 f(X_s^i) ds \right] \\
+ \frac{1}{(N t)^2} \sum_{i=1}^N \int_{t^p}^t \Big( E_x \left[ \left( Q_h (X_s^i-a) \right)^2 f(X_s^i) \right] - E_{\pi} \left[ \left( Q_h (X_s^i-a) \right)^2 f(X_s^i) \right] \Big) ds.\end{gathered}$$ To conclude, we use the upper bounds $\parallel Q \parallel_\infty$ and $F$ for $Q$ and $f$ to control the second line and we use Theorem \[theo:harrisok\] to control the last term. As a consequence, $$\begin{gathered}
\left| \frac{1}{(N t)^2} E_x \left[ \sum_{i=1}^N \int_0^t \left( Q_h (X_s^i-a) \right)^2 f(X_s^i) ds \right] - \frac{1}{(N t)^2} E_{\pi} \left[ \sum_{i=1}^N \int_0^t \left( Q_h (X_s^i-a) \right)^2 f(X_s^i) ds \right] \right| \\
\leq \frac{F \parallel Q \parallel_\infty^2}{Nh^2 t^2} \left( 2 t^p + C \int_{t^p}^t \kappa^{-s} ds \right) = \frac{F \parallel Q \parallel_\infty^2}{Nh^2 t^2} \left( 2 t^p + C \frac{\kappa^{-t^p}-\kappa^{-t}}{\ln (\kappa)} \right) = \frac{1}{ht} {\mathcal O} \left( \frac{t^p}{ht} \right) .\end{gathered}$$ Now recall that $h=h_t=t^{-\alpha}$ by (\[eq:htmalpha\]) and that $p \in ]0,1-\alpha[.$ Thus $$\frac{1}{(N t)^2} E_x \left[ \sum_{i=1}^N \int_0^t \left( Q_h (X_s^i-a) \right)^2 f(X_s^i) ds \right] - \frac{1}{(N t)^2} E_{\pi} \left[ \sum_{i=1}^N \int_0^t \left( Q_h (X_s^i-a) \right)^2 f(X_s^i) ds \right] = o \left( \frac{1}{ht} \right),$$ which allows to conclude.
Proposition \[prop:ernum\] will help us to control the numerator of our estimator. We want to establish the same kind of result for the denominator and this leads to the following proposition:
\[prop:erden\] For all $a \in S_{d , \beta },$ define $$\label{eq:defQtilde}
\tilde Q_{h,f}(y):=Q_h ( y - a ) \Big( f(y)-f(a) \Big) - \pi_1 \Big( Q_h ( \cdot - a ) \Big( f(\cdot)-f(a) \Big) \Big).$$ Under the conditions of Theorem \[theo:main\], there exists a constant $C_2$ depending only on $\beta , F, L, N, f_{min}$ and $Q,$ such that for all $f \in H( \beta , F, L, f_{min} ), $ for all $x \in [0,K]$ and for a bandwidth of the form $h = h_t = t^{- \alpha } $ for some $ 0 < \alpha < 1,$ $$\label{eq:erdenx}
E_x \left[ \left( \frac{1}{Nt} \int_0^t \int_\R \tilde Q_{h,f}(y) \eta(ds,dy) \right)^2 \right] \leq \frac{C_2}{t} h^{2 \left( 1\wedge \beta \right) -1}.$$
As in the preceding proof we start by working in the stationary regime, *i.e.* under $E_\pi .$ $$\begin{gathered}
\label{eq:errorstartden}
E_{\pi} \left[ \left( \frac{1}{Nt} \int_0^t \int_\R \tilde Q_{h,f}(y) \eta(ds,dy) \right)^2 \right]
\\
\leq \frac{2}{(Nt)^2} E_{\pi} \left[ \int_0^t \int_\R \Big| \tilde Q_{h,f}(x) \Big| \eta(ds,dx) \Big| E_{\pi} \left( \int_s^t \int_\R \tilde Q_{h,f}(y) \eta(du,dy) \Big| {\mathcal F}_s \right) \Big| \right].\end{gathered}$$ We deal with the conditional expectation using the Markov property and write $$\begin{gathered}
E_{\pi} \left( \int_s^t \int_\R \tilde Q_{h,f}(y) \eta(du,dy) \Big| {\mathcal F}_s \right)
\\
= E_{X_s} \left( \int_0^{t-s} \sum_{i=1}^N \tilde Q_{h,f}(X_u^i) du\right) = \int_0^{t-s} \sum_{i=1}^N E_{X_s} \left( \tilde Q_{h,f}(X_u^i) \right) du.\end{gathered}$$ Now going back to the definition of $\tilde Q_{h,f},$ we can use Theorem \[theo:harrisok\] and write $$\begin{gathered}
E_{X_s} \left( \tilde Q_{h,f}(X_u^i) \right) = E_{X_s} \Big( Q_h ( X_u^i - a ) \left( f(X_u^i) - f(a) \right) \Big) - \pi_1 \Big( Q_h ( \cdot - a ) \left( f(\cdot ) - f(a) \right) \Big) \\
\leq \frac{C}{h} (F\vee L)(Rh)^{1\wedge \beta} \parallel Q \parallel_\infty \kappa^{-u},\end{gathered}$$ due to the assumption (\[eq:Hspace\]) on the Hölder space containing $f.$ (Recall that $R$ is the diameter of the support of $Q.$) The integrability of the function $ u \to \kappa^{-u} $allows to deduce from this that $$\Big| E_{\pi} \left( \int_s^t \int_\R \tilde Q_h (y-a) \eta(du,dy) \Big| {\mathcal F}_s \right) \Big| \leq \frac{N \tilde C}{h} (F\vee L)(Rh)^{1\wedge \beta} \parallel Q \parallel_\infty$$ for some constant ${\tilde C}.$ Taking this result into account in (\[eq:errorstartden\]), we obtain $$E_{\pi} \left[ \left( \frac{1}{Nt} \int_0^t \int_\R \tilde Q_{h,f}(y) \eta(ds,dy) \right)^2 \right] \leq \frac{2{\tilde C}}{Nht^2} (F\vee L)(Rh)^{1\wedge \beta} \parallel Q \parallel_\infty E_{\pi} \left[\int_0^t \int_\R \Big| \tilde Q_{h,f}(x) \Big| \eta(ds,dx) \right].$$ The end of the proof is similar to the one of Proposition \[prop:ernum\]: the fact that we are in the invariant regime allows to use the density of the invariant measure of a single particle and its control given by Theorem \[theo:invmeasure\]. Then we use the same change of variable $x=\frac{y-a}{h}$ to obtain $$E_{\pi} \left[ \left( \frac{1}{Nt} \int_0^t \int_\R \tilde Q_{h,f}(y) \eta(ds,dy) \right)^2 \right] \leq \frac{4 {\tilde C} }{ht} (F\vee L)^2(Rh)^{2 \left( 1\wedge \beta \right) } \parallel Q \parallel_\infty \parallel Q \parallel_{L^1} \sup_{ x \in S_{d/2 , k }} \pi_1 ( x).$$ This result is established under the invariant regime, but we are able to extend it to any starting point $x \in [0,K]^N,$ using the same trick as the one in the proof of Proposition \[prop:ernum\]. This finishes the proof.
Proof of Theorem \[theo:main\], $(i)$
-------------------------------------
Introducing $$D^{t,h} = \frac{1}{Nt} \int_\R \frac1h Q\left( \frac{ y-a}{h} \right) \eta_t (dy ),$$ we have $$\begin{gathered}
D^{t,h} ( \hat f_{t,h} ( a) - f (a) ) = \frac{1}{Nt} \int_{[0, t ]}\int_\R Q_h (y-a) \mu( ds, dy) -f(a)D^{t,h} \\
= \frac{1}{Nt} \int_{[0, t ]}\int_\R Q_h (y-a) \tilde \mu( ds, dy) + \frac{1}{Nt} \int_0^t \int_\R \frac1h Q\left( \frac{ y-a}{h} \right) \left( f(y) - f(a) \right) \eta (ds, dy ).\end{gathered}$$ With the definition of $\tilde Q_{h,f}$ in (\[eq:defQtilde\]), we have the following decomposition: $$\begin{gathered}
\label{eq:erdecomp}
D^{t,h} ( \hat f_{t,h} ( a) - f (a) ) \\
= \frac{1}{Nt} \int_{[0, t ]}\int_\R Q_h (y-a) \tilde \mu(ds, dy) + \frac{1}{Nt} \int_0^t \int_\R \tilde Q_{h,f}(y) \eta (ds, dy ) + \pi_1 \Big( Q_h ( \cdot - a ) \left( f(\cdot ) - f(a) \right) \Big) .\end{gathered}$$ The first two terms of the previous sum are controlled respectively by Propositions \[prop:ernum\] and \[prop:erden\]. We deal with the third term using Proposition \[prop:KernApprox\] as follows:
$$\begin{gathered}
\pi_1 \Big( Q_h ( \cdot - a ) \left( f(\cdot ) - f(a) \right) \Big) = \int_\R Q_h ( y - a ) \left( f(y ) - f(a) \right) \pi_1 (y) dy \\
= \left( \int_\R Q_h ( y - a ) \left( f(y) \pi_1 (y) - f(a) \pi_1 (a) \right) dy \right) + f(a) \left( \int_\R Q_h ( y - a ) \left( \pi_1 (a) - \pi_1 (y) \right) dy \right).\end{gathered}$$
Both functions $\pi_1$ and $f \pi_1$ are Hölder of order $\beta$ (recall Theorem \[theo:invmeasure\]) and we can apply Proposition \[prop:KernApprox\] to each of the last two terms, using the upper bound $F$ for $f(a).$
Putting all together in (\[eq:erdecomp\]), we have $$\label{eq:ersum}
\parallel D^{t,h} \left( \hat f_{t,h} ( a) - f (a) \right) \parallel_{L^2(P^f_x)} \leq \sqrt{\frac{C_1}{ht}} + \sqrt{\frac{C_2}{ht}}h^{1\wedge \beta} + C_3 h^\beta ,$$ with constants $C_1 , C_2$ and $C_3$ depending only on $\beta , F, L, f_{min}$ and $Q.$ As in the proof of Proposition \[prop:erden\], we will fix the dependence in $t$ of $h$ putting $h_t:=t^{-\alpha}$ and choosing $\alpha \in ]0,1[$ to obtain an optimal speed of convergence.
This leads to the choice $\alpha := \frac{1}{2\beta + 1}$ and $h=h_t=t^{-\frac{1}{2\beta + 1}}$ which gives us $$\parallel D^{t,h_t} \left( \hat f_{t,h_t} ( a) - f (a) \right) \parallel_{L^2(P_x^f)} \leq C(\beta , F, L, f_{min}, Q) t^{-\frac{\beta}{2 \beta +1}}.$$ To finish the proof of Theorem \[theo:main\], we have to work conditionally on the event $A_{t,r}$, for $ r \le r^*, $ on which we have $D^{t,h} \geq r .$ $$\begin{gathered}
E_x \left[ \left( \hat f_{t,h_t} ( a) - f (a) \right)^2 \Big| A_{t,r} \right] = \frac{1}{P_x \left( A_{t,r} \right) } E_x \left[ \left( \hat f_{t,h_t} ( a) - f (a) \right)^2 1_{A_{t,r}} \right] \\
\leq \frac{1}{r^2 P_x \left( A_{t,r} \right) } \parallel D^{t,h_t} \left( \hat f_{t,h_t} ( a) - f (a) \right) \parallel_{L^2(P_x^f)}^2 \leq \frac{C(\beta , F, L,f_{min}, Q)^2 t^{-\frac{2 \beta}{2 \beta +1}}}{r^2 P_x \left( A_{t,r} \right) },\end{gathered}$$ and the conclusion follows thanks to (\[eq:imdlb\]). $\qed $
Proof of Theorem \[theo:main\] $(ii)$:
--------------------------------------
The proof relies on the martingale convergence theorem given in Corollary 3.24 of [@js] chapter VIII. We use the following decomposition $$\label{eq:decomptcl}
D^{t,h} ( \hat f_{t,h} ( a) - f (a) ) = \frac{1}{N\sqrt{th}} M^{t,h} + \frac{1}{Nt} \int_0^t \int_\R Q_h (y-a) \left( f(y)-f(a) \right) \eta(du,dy ),$$ where $$M^{t,h}:= \frac{1}{\sqrt{th}} \int_{[0, t ]}\int_\R Q \left(\frac{y-a}{h} \right) \tilde \mu(ds, dy).$$ We define for all $t \in \R_+$ $$(M^t)_s := \frac{1}{\sqrt{th}} \int_{[0, ts]} \int_\R Q \left(\frac{y-a}{h} \right) \tilde \mu(du, dy)$$ and show that the Assumption 3.23 of [@js] chapter VIII is satisfied for this sequence of processes. Therefore, we have to study, for all $\varepsilon>0$ and all $s \in \R_+,$ the limit of $$\frac{1}{th} \int_0^{ts} \sum_{i=1}^N f \left( X_u^i \right) Q^2 \left( \frac{X_u^i-a}{h} \right) 1_{ \{ \frac{1}{\sqrt{th}} Q \left( \frac{X_u^i-a}{h} \right)> \varepsilon \} } du$$
as $t$ goes to $+\infty.$ Since $Q$ is bounded and $\lim_{t \to +\infty} th_t = +\infty ,$ there exists $t_0$ such that for all $t>t_0,$ $1_{ \{ \frac{1}{\sqrt{th}} Q \left( \frac{X_u^i-a}{h} \right)> \varepsilon \} }=0.$ Consequently, the above limit is $0$ and Assumption 3.23 of [@js] chapter VIII is indeed satisfied.
Moreover, $$\left< M^t, M^t \right>_s = \frac{1}{th} \int_0^{ts} \int_\R Q^2 \left( \frac{y-a}{h} \right) f(y) \eta(du,dy).$$ Since our process is positive Harris recurrent, by the ergodic theorem, we have the following proposition.
$\left< M^t, M^t \right>_s$ converges in $P_x$-Probability as $t$ goes to $+\infty$ to $$Nsf(a) \pi_1(a) \int Q^2(x)dx \; {\mbox a.s.}$$
Since our process is positive Harris recurrent, $f$ being continuous and $Q$ with compact support, we have $$\lim_{t \to + \infty} E_x \left[ \left( \frac{1}{th} \int_0^{ts} \int_\R Q^2 \left( \frac{y-a}{h} \right) f(y) \eta(du,dy) - \frac{N}{th} \int_0^{ts} \int_\R Q^2 \left( \frac{y-a}{h} \right) f(y) \pi_1(y)dy \right)^2 \right] =0.$$ Then the result is obtained by continuity of $\pi_1$ and $f$ on $S_{d,k}.$
Consequently, Corollary 3.24 of [@js] chapter VIII with $s=1$ gives us the weak convergence of $M^{t,h}$ to ${\mathcal N} \left( 0, Nf(a) \pi_1(a) \int Q^2(x)dx \right).$
We deal with the second term of (\[eq:decomptcl\]) as in the previous subsection and obtain $$\left\Vert \frac{1}{Nt} \int_0^t \int_0^K Q_h (y-a) \left( f(y)-f(a) \right) \eta(du,dy ) \right\Vert_{L^2(P^f_x)} \leq \sqrt{\frac{C_2}{ht}}h^{1\wedge \beta} + C_3 h^\beta .$$ Therefore, when $t$ goes to $+ \infty,$ (\[eq:decomptcl\]) gives us the following weak convergence: $$\sqrt{th_t}D^{t,h_t} ( \hat f_{t,h_t} ( a) - f (a) ) \longrightarrow {\mathcal N} \left( 0, \frac{ f(a) \pi_1(a)}{N} \int Q^2(x)dx \right),$$ since $ h_t = o ( t^{ - 1 /(1 + 2 \beta ) } ).$
Finally, we deal with the additive functional $D^{t,h_t}$ using the ergodic theorem. Recall that $$D^{t,h} = \frac{1}{Nt} \int_0^t \int_\R \frac1h Q\left( \frac{ y-a}{h} \right) \eta(ds,dy ).$$ Thanks to , $\pi_1(a) >0,$ and the ergodic theorem gives us the almost sure convergence to $\pi_1(a)$ (since $\int Q(x)dx=1$), which allows us to conclude. $\qed $
Proof of Theorem \[theo:lowerbound\] {#sec:optimal}
====================================
The proof of Theorem \[theo:lowerbound\] follows closely the proof of Theorem 8 of Hoffmann and Olivier (2015) [@hoffmann-olivier], going back to similar ideas developed in [@hhl]. Let $ h_t = t^{ - \frac{1}{2 \beta + 1 } }$ and fix any test rate function $f_0 \in H( \beta, F - \delta , L- \delta, f_{min} ) ,$ for some fixed $ \delta \in ] 0, F \wedge L[ .$ Then, as in [@hoffmann-olivier], we define a perturbation $f_t $ of $f_0$ by $$f_t ( x ) = f_0 ( x) + b h_t^{\beta +1} \chi_{h_t} ( x - a ) ,$$ where $ b > 0 $ is a positive constant, $\chi \in C_c ( \R_+, \R_+) $ is a positive kernel function of compact support included in $[- 1, 1 ] $ such that $ \chi ( 0 ) = 1 , $ $ \chi ( x) \le 1 $ for all $x$ and $$\label{eq:chiht}
\chi_{h_t} ( x) = \frac{1}{h_t} \chi ( \frac{x}{h_t} ) .$$
Notice that the first $l$ derivatives of $ \chi_{h_t} $ are of order $ h_t^{ - (l+1)} ,$ therefore the factor $ h_t^{\beta +1}$ implies that $f_t \in H( \beta , F, L, f_{min} ) ,$ if we choose $b $ sufficiently small. An important point in the above choice of $f_t$ is that $$\label{eq:auchgut}
f_t ( a) - f_0 ( a) = b h_t^\beta = b t^{ - \frac{\beta }{2 \beta +1}} ,$$ since $\chi ( 0 ) = 1.$
In the following, we shall write shortly $ \P_0 := ( P_x^{f_0})_{| {\mathcal F_t}} $ and $ \P_t := (P_x^{f_t})_{| {\mathcal F_t}} $ for the associated probability measures in restriction to $ {\mathcal F}_t.$ The following lower bound is by now classical. For any fixed constant $C > 0,$ using Markov’s inequality and denoting by $L_t^{f_t/f_0} =\frac{ d \P_0}{d \P_t } $ the likelihood ratio of $ \P_0 $ with respect to $\P_t, $ on ${\mathcal F}_t,$ $$\begin{aligned}
&&\sup_{ f \in H( \beta , F, L, f_{min} ) } t^{ \frac{2\beta}{1 + 2 \beta } } E_x^f [ | \hat f_t ( a) - f(a) |^2 ] \\
&& \geq t^{ \frac{2\beta}{1 + 2 \beta } } \left[ \frac12 \E_0 [ | \hat f_t ( a) - f_0 ( a) |^2 ] + \frac12 \E_t [ | \hat f_t ( a) - f_t ( a) |^2 ] \right] \\
&&\geq \frac{C^2 }{2} \left[ \P_0 \left( t^{ \frac{\beta}{1 + 2 \beta } }| \hat f_t ( a) - f_0 ( a) | \geq C \right) + \P_t \left( t^{ \frac{\beta}{1 + 2 \beta } }| \hat f_t ( a) - f_t ( a) | \geq C \right) \right] \\
&& = \frac{C^2 }{2} \left[ \P_0 \left( t^{ \frac{\beta}{1 + 2 \beta } }| \hat f_t ( a) - f_0 ( a) | \geq C \right) + \E_0 \left( L_t^{f_t/f_0} 1_{\{ t^{ \frac{\beta}{1 + 2 \beta } }| \hat f_t ( a) - f_t ( a) | \geq C \}}\right) \right] .
$$ Now, $$t^{ \frac{\beta}{1 + 2 \beta } }[ | \hat f_t ( a) - f_0 ( a) | + | \hat f_t ( a) - f_t ( a) | ] \geq t^{ \frac{\beta}{1 + 2 \beta } } | f_0(a) - f_t ( a) | \geq b ,$$ which is due to . As a consequence, if we choose $
C = b /2 ,
$ then $$1_{\{ t^{ \frac{\beta}{1 + 2 \beta } }| \hat f_t ( a) - f_0 ( a) | \geq C \}} + 1_{\{ t^{ \frac{\beta}{1 + 2 \beta } }| \hat f_t ( a) - f_t ( a) | \geq C \}} \geq 1 ,$$ in particular, $$1_{\{ t^{ \frac{\beta}{1 + 2 \beta } }| \hat f_t ( a) - f_t ( a) | \geq C \}} \geq 1_{\{ t^{ \frac{\beta}{1 + 2 \beta } }| \hat f_t ( a) - f_0 ( a) | < C \}} .$$ We conclude that $$\begin{aligned}
&&\sup_{ f \in H( \beta , F, L, f_{min} ) } t^{ \frac{2\beta}{1 + 2 \beta } } E_x^f [ | \hat f_t ( a) - f(a) |^2 ]\\
&& \geq \frac{ b^2}{8}
\E_0 \left[ 1_{\{ t^{ \frac{\beta}{1 + 2 \beta } }| \hat f_t ( a) - f_0 ( a) | \geq \frac{b}{2} \}} + L_t^{ f_t/f_0} 1_{\{ t^{ \frac{\beta}{1 + 2 \beta } }| \hat f_t ( a) - f_0 ( a) | < \frac{b}{2} \}} \right]
\\
&&\geq \frac{ b^2}{8} e^{- s} \P_0 ( L_t^{f_t/ f_0 } \geq e^{-s} ) ,\end{aligned}$$ for any $s > 0 .$ Therefore, in order to achieve the proof of Theorem \[theo:lowerbound\], it suffices to show that $$\label{eq:sufficientcondlr}
\lim \sup_{t \to \infty } \E_0 [ | \log L_t^{f_t/f_0 } | ] < \infty .$$
Indeed, we can deduce from the following statements: $$\begin{gathered}
\exists M, \forall t, \E_0 \left( | \log L_t^{f_t/f_0 } | \right) \leq M, \\
\exists M, \forall t, \P_0 \left( \log L_t^{f_t/f_0 } < -2M \right) \leq \frac{1}{2}, \\
\exists s, \forall t, \P_0 \left( \log L_t^{f_t/f_0 } \geq -s \right) \geq \frac{1}{2}.\end{gathered}$$
Recall that by construction, $f_t \geq f_0.$ Moreover, since the support of $ \chi $ is included in $ [-1 , 1],$ $ f_t (y) \neq f_0 (y) $ implies $ y \in J_t := [ a - h_t, a+h_t ].$ Now, Theorem 3.5 of Löcherbach (2002) [@evaold], applied to the particular case without branching, shows that $\P_0$ and $\P_t $ are equivalent on $ {\mathcal F}_t, $ with density $$\label{eq:llr}
\log L_t^{f_t/f_0} = \int_0^t \int_{J_t} \log ( \frac{f_t}{f_0} (y)) \mu (ds, dy ) - \int_{J_t} \left ( \frac{f_t}{f_0 } - 1 \right)(y) f_0 (y) \eta_t (dy ) .$$ We now proceed exactly as in [@hhl], proof of Lemma 11. The $\P_0-$ martingale part within is given by $$\int_0^t \int_{J_t} \left ( \frac{f_t}{f_0 } - 1 \right) ( \mu - \hat \mu^{f_0} ) (ds, dy ) ,$$ where $ \hat \mu^{f_0} (ds , dy ) = \sum_{i=1}^N f_0 ( X^i_s ) \delta_{X^i_s} ( dy ) ds $ is the $ \P_0-$compensator of $ \mu .$ Its angle bracket is $$\begin{gathered}
b^2 h_t^{ 2 \beta +2} \int_{J_t} \left( \frac{ \chi^2_{h_t} ( y - a ) }{f_0 ( y ) } \right) \eta_t (dy ) \le \frac{ b^2 }{\inf_{y \in J_t} ( f_0 (y)) } t^{ - \frac{ 2 \beta}{ 2 \beta + 1 } } h_t^2 \int_{J_t} \chi^2_{h_t} ( y - a ) \eta_t (dy ) \\
\le \frac{ b^2 }{\inf_{y \in J_t} ( f_0 (y)) } t^{ - \frac{ 2 \beta}{ 2 \beta + 1 } } \eta_t ( J_t) = \frac{ b^2 }{\inf_{y \in J_t} ( f_0 (y)) } t^{ \frac{ 1}{ 2 \beta + 1 } } \frac{1}{t} \eta_t ( J_t),\end{gathered}$$ since $ \chi ( \cdot ) \le 1 ,$ by definition of $\chi_{h_t}$ (recall ). All other terms in are treated exactly as in [@hhl]. Therefore, it only remains to show that $$\label{eq:finalouf}
\lim \sup_{ t \to \infty } \E_0 \left( \frac{1}{t h_t} \eta_t (J_t) \right) < \infty .$$ We apply once more Theorem \[theo:harrisok\] and rewrite $$\begin{gathered}
\E_0 ( \eta_t ( J_t) ) = \int_0^t E_x^{f_0} ( \bar 1_{J_t} (X_s) ) ds =
\int_0^t E_x^{f_0} \left( \bar 1_{J_t} (X_s) - \pi^{f_0} ( \bar 1_{J_t} ) \right) ) ds + t \pi^{f_0} ( \bar 1_{J_t} )
\\
\le C N \int_0^t \kappa^{ - s} ds + t \pi^{f_0} ( \bar 1_{J_t} )
\le CN \frac{1}{\ln \kappa} + t \pi^{f_0} ( \bar 1_{J_t} ) \\
= CN \frac{1}{\ln \kappa} + N t \int_{J_t} \pi_1^{f_0} (y ) dy ,\end{gathered}$$ where $\bar 1_{J_t} (x):= \sum_{i=1}^N 1_{J_t}(x^i)$ for $x \in \R^N,$ and $\pi_1^{f_0} (y ) $ denotes the Lebesgue density of $ \pi_1^{f_0}, $ which exists on $J_t$ by choice of $a, $ for $t$ sufficiently large. Using the change of variables $ z = (y- a)/ h_t ,$ we obtain $$\E_0 ( \eta_t ( J_t) ) \le CN \frac{1}{\ln \kappa} + N t h_t \int_{-1}^1 \pi_1^{f_0} ( a + h_t z) dz \le CN \frac{1}{\ln \kappa} + 2 N t h_t \sup_{ x \in B_{ h_t ( a) } } \pi_1^{f_0} ( x) ,$$ which implies finally by Theorem \[theo:invmeasure\]. $\qed$
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank an anonymous referee for helpful comments and suggestions. This research has been conducted as part of the project Labex MME-DII (ANR11-LBX-0023-01), as part of the Agence Nationale de la Recherche PIECE 12-JS01-0006-01 and as part of the activities of FAPESP Research, Dissemination and Innovation Center for Neuromathematics (grant 2013/07699-0, S. Paulo Research Foundation).
[99]{}
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[^1]: A short remark concerning the continuous time observation scheme : Presumably, if we deal with discrete time samples, observed at sufficiently high frequency such that with huge probability at most one jump can take place during one sampling step, it would be possible to reconstruct the continuous trajectory of the process with hight probability and to perform our estimation procedure also in this frame.
|
---
abstract: |
We study actions of finitely generated groups on ${{\mathbb {R}}}$-trees under some stability hypotheses. We prove that either the group splits over some controlled subgroup (fixing an arc in particular), or the action can be obtained by gluing together actions of simple types: actions on simplicial trees, actions on lines, and actions coming from measured foliations on $2$-orbifolds. This extends results by Sela and Rips-Sela. However, their results are misstated, and we give a counterexample to their statements.
The proof relies on an extended version of Scott’s Lemma of independent interest. This statement claims that if a group $G$ is a direct limit of groups having suitably compatible splittings, then $G$ splits.
author:
- Vincent Guirardel
bibliography:
- 'published.bib'
- 'unpublished.bib'
title: 'Actions of finitely generated groups on ${{\mathbb {R}}}$-trees.'
---
Actions of groups on ${{\mathbb {R}}}$-trees are an important tool in geometric group theory. For instance, actions on ${{\mathbb {R}}}$-trees are used to compactify sets of hyperbolic structures ([@MS_valuationsI; @Pau_topologie]) or Culler-Vogtmann’s Outer space ([@CuMo; @BF_outer; @CoLu_very]). Actions on ${{\mathbb {R}}}$-trees are also a main ingredient in Sela’ approach to acylindrical accessibility [@Sela_acylindrical] (see [@Delzant_accessibilite] or [@KaWe_acylindrical] for alternative approaches) and in some studies of morphisms of a given group to a (relatively) hyperbolic group [@Sela_hopf; @BeleSzcz_endomorphisms; @DrutuSapir_tree-graded]. Limit groups and limits of (relatively) hyperbolic groups [@Sela_diophantine1; @Sela_diophantine7; @Gui_limit; @Alibegovic_MR; @Groves_limit_hypII], and Sela’s approach to Tarski’s problem ([@Sela_diophantine6]) are also studied using ${{\mathbb {R}}}$-trees as a basic tool.
To use ${{\mathbb {R}}}$-trees as a tool, one needs to understand the structure of a group acting on an ${{\mathbb {R}}}$-tree. The main breakthrough in this analysis is due to Rips. He proved that any finitely generated group acting freely on an ${{\mathbb {R}}}$-tree is a free product of surface groups and of free abelian groups (see [@GLP1; @BF_stable]).
More general results involve *stability* hypotheses. Roughly speaking, these hypotheses prohibit infinite sequences of nested arc stabilizers. A simple version of these stability hypotheses is the *ascending chain condition*: for any decreasing sequence of arcs $I_1\supset I_2\supset ...$ whose lengths converge to $0$, the sequence of their pointwise arc stabilizers $G(I_1) \subset G(I_2)\subset\dots$ stabilizes. The ascending chain condition implies BF-stability used below. See Definition \[dfn\_stable\] for other versions of stability and relations between them.
Another theorem by Rips claims that if $G$ is finitely presented and has a BF-stable action on an ${{\mathbb {R}}}$-tree $T$, then either $T$ has an invariant line or $G$ splits over an extension of a subgroup fixing an arc by a cyclic group (see [@BF_stable; @Gui_approximation]). Using methods in the spirit of [@GLP1], the author proved under the same hypotheses that one can *approximate* the action $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ by a sequence of actions on simplicial trees $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_k$ converging to $T$ ([@Gui_approximation]). Edge stabilizers of $T_k$ are extensions by finitely generated free abelian groups of subgroups fixing an arc in $T$. The convergence is in the length functions topology.\
Next results give a very precise structure of the action on the ${{\mathbb {R}}}$-tree. This is the basis for Sela’s shortening argument (see for instance [@RiSe_structure; @Sela_acylindrical]). These results say that the action on the ${{\mathbb {R}}}$-tree can be obtained by gluing together simple building blocks. The building blocks are actions on simplicial trees, actions on a line, and actions coming from a measured foliation on a $2$-orbifold. Two building blocks are glued together along one point, and globally, the combinatorics of the gluing is described by a simplicial tree (see section \[sec\_goa\] for more details). When $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ arises in this fashion, we say that $T$ splits as a *graph of actions on ${{\mathbb {R}}}$-trees*.
In [@RiSe_structure] and [@Sela_acylindrical], Rips-Sela and Sela give a structure theorem for actions of finitely presented groups and of finitely generated groups on ${{\mathbb {R}}}$-trees. The proof uses the following super-stability hypothesis: an arc $I\subset T$ is *unstable* if there exists $J\subset I$ with $G(I){\varsubsetneq}G(J)$; an action $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ is *super-stable* if the stabilizer of any unstable arc is trivial. In particular, super-stability implies that chains of arc stabilizers have length at most 2. Sela’s result is as follows: consider a minimal action $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ of a finitely generated group on an ${{\mathbb {R}}}$-tree, and assume that this action is super-stable, and has trivial tripod stabilizers. Then either $G$ splits as a free product, or $T$ can be obtained from simple building blocks as above ([@Sela_acylindrical Theorem 3.1]).
Actually, Sela’s result is stated under the ascending chain condition instead of super-stability. However, the proof really uses the stronger hypothesis of super-stability. In section \[sec\_example\], we give a counterexample to the more general statement. This does not affect the rest of Sela’s papers since super-stability is satisfied in the cases considered. Since the counterexample is an action of a finitely presented group, this also shows that one should include super-stability in the hypotheses of Rips-Sela’s statement [@RiSe_structure Theorem 10.8] (see also [@Sela_acylindrical Theorem 2.3]).\
Our main result generalizes Sela’s result in two ways. First, we don’t assume that tripod stabilizers are trivial. Moreover, we replace super-stability by the ascending chain condition together with weaker assumptions on stabilizers of unstable arcs.
Consider a non-trivial minimal action of a finitely generated group $G$ on an ${{\mathbb {R}}}$-tree $T$ by isometries. Assume that
$T$ satisfies the ascending chain condition;
for any unstable arc $J$,
$G(J)$ is finitely generated;
$G(J)$ is not a proper subgroup of any conjugate of itself [i. e. ]{}$\forall g\in G$, $G(J)^g\subset G(J)\Rightarrow G(J)^g= G(J)$.
Then either $G$ splits over the stabilizer of an unstable arc or over the stabilizer of an infinite tripod, or $T$ has a decomposition into a graph of actions where each vertex action is either
simplicial: $G_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$ is a simplicial action on a simplicial tree;
of Seifert type: the vertex action $G_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$ has kernel $N_v$, and the faithful action $G_v/N_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$ is dual to an arational measured foliation on a closed $2$-orbifold with boundary;
axial: $Y_v$ is a line, and the image of $G_v$ in ${\mathop{\mathrm{Isom}}}(Y_v)$ is a finitely generated group acting with dense orbits on $Y_v$.
An *infinite tripod* is the union of three semi-lines having a common origin $O$, and whose pairwise intersection is reduced to $\{O\}$.
If one assumes triviality of stabilizers of tripods and of unstable arcs (super-stability) in Main Theorem, one gets Sela’s result.
The non-simplicial building blocks are canonical. Indeed, non-simplicial building blocks are *indecomposable* which implies that they cannot be split further into a graph of actions (see Definition \[dfn\_indecomposability\] and Lemma \[lem\_indec\_component\]). If $T$ is not a line, simplicial building blocks can also be made canonical by imposing that each simplicial building block is an arc which intersects the set of branch points of $T$ exactly at its endpoints.
For simplicity, we did not state the optimal statement of Main Theorem, see Theorem \[thm\_main\] for more details. In particular, one can say a little more about the tripod stabilizer on which $G$ splits. This statement also includes a relative version: $G$ is only assumed to be finitely generated relative to a finite set of elliptic subgroups ${{\mathcal {H}}}=\{H_1,\dots,H_p\}$, and each $H_i$ is conjugate into a vertex group in the splittings of $G$ produced.\
When $T$ has a decomposition into a graph of actions as in the conclusion of Main Theorem, then $G$ splits over an extension of an arc stabilizer by a cyclic group except maybe if $T$ is a line. Thus we get:
[corBF]{}
Under the hypotheses of Main Theorem, either $T$ is a line, or $G$ splits over a subgroup $H$ which is an extension of a cyclic group by an arc stabilizer.
In the conclusion of the corollary, $H$ is an extension by a *full* arc stabilizer, and not of a subgroup fixing an arc. This contrasts with [@BF_stable Theorem 9.5].
Here is a simple setting where the main theorem applies. A group $H$ is *small* (resp. *slender*) if it contains no non-abelian free group (resp. if all its subgroups are finitely generated). The following corollary applies to the case where $G$ is hyperbolic relative to slender groups.
[cor\_petit]{}
Let $G$ be a finitely generated group for which any small subgroup is finitely generated. Assume that $G$ acts on an ${{\mathbb {R}}}$-tree $T$ with small arc stabilizers.
Then either $G$ splits over the stabilizer of an unstable arc or over a tripod stabilizer, or $T$ has a decomposition into a graph of actions as in Main Theorem. In particular, $G$ splits over a small subgroup.
In some situations, one can control arc stabilizers in terms of tripod stabilizers. For instance, we get:
[cor\_tripod]{}
Consider a finitely generated group $G$ acting by isometries on an ${{\mathbb {R}}}$-tree $T$. Assume that
arc stabilizers have a nilpotent subgroup of bounded index (maybe not finitely generated);
tripod stabilizers are finitely generated (and virtually nilpotent);
no group fixing a tripod is a proper subgroup of any conjugate of itself;
any chain $H_1\subset H_2\dots$ of subgroups fixing tripods stabilizes.
Then either $G$ splits over a subgroup having a finite index subgroup fixing a tripod, or $T$ has a decomposition as in the conclusion of Main Theorem.\
Our proof relies on a particular case of Sela’s Theorem, assuming triviality of arc stabilizers. For completeness, we give a proof of this result in appendix \[sec\_Sela\]. To reduce to this theorem, we prove that under the hypotheses of Main Theorem, the action is *piecewise stable*, meaning that any segment of $T$ is covered by finitely many stable arcs (see Definition \[dfn\_stable\]). Piecewise stability implies that $T$ splits into a graph of actions where each vertex action $G_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_v$ has trivial arc stabilizers up to some kernel (Theorem \[thm\_pw2triv\]). Because $G_v$ itself need not be finitely generated, we need to extend Sela’s result to finitely generated pairs (Proposition \[prop\_sela\_rel\]).
So the main step in the proof consists in proving piecewise stability (Theorem \[thm\_acc2pw\]). All studies of actions on ${{\mathbb {R}}}$-trees use resolutions by foliated $2$-complexes. Such foliated $2$-complexes have a dynamical decomposition into two kinds of pieces: *simplicial* pieces where each leaf is finite, and *minimal components* where every leaf is dense. These resolutions give sequence of actions $G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_k$ converging strongly to $T$, where $G$ is the inductive limit of $G_k$.
Maybe surprisingly, the main difficulty in the proof arises from the simplicial pieces. This is because minimal components give large stable subtrees (Lemma \[lem\_indec\_stab\]). In particular, a crucial argument is a result saying that if all the groups $G_k$ split in some nice compatible way, then so does $G$. In the case of free splittings, this is Scott’s Lemma. In our setting, we need an extended version of Scott’s Lemma, which is of independent interest.
[scott]{}
Let $G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}S_k$ be a sequence of non-trivial actions of finitely generated groups on simplicial trees, and $({\varphi}_k,f_k):G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}S_k{\rightarrow}G_{k+1}{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}S_{k+1}$ be epimorphisms. Assume that $({\varphi}_k,f_k)$ does not increase edge stabilizers in the following sense: $$\forall e\in E(S_k),\forall e'\in E(S_{k+1}),\quad e'\subset f_k(e)\Rightarrow G_{k+1}(e')={\varphi}_k(G_k(e))$$
Then the inductive limit $G=\displaystyle\lim_{{\rightarrow}} G_k$ has a non-trivial splitting over the image of an edge stabilizer of some $S_k$.
The paper is organized as follows. Section \[sec\_prelim\] is devoted to preliminaries, more or less well known, except for the notion of indecomposability which seems to be new. Section \[sec\_scott\] deals with Extended Scott’s Lemma. Section \[sec\_acc2pw\] proves piecewise stability. Section \[sec\_pw2triv\] shows how piecewise stability and Sela’s Theorem for actions with trivial arc stabilizers allow to conclude. Section \[sec\_proof\] gives the proof of the corollaries. Section \[sec\_example\] contains our counterexample to Sela’s misstated result. Appendix \[sec\_Sela\] gives a proof of the version of Sela’s result we need.
Preliminaries {#sec_prelim}
=============
Basic vocabulary
----------------
An ${{\mathbb {R}}}$-tree is a $0$-hyperbolic geodesic space. In all this section, we fix an isometric action of a group $G$ on an ${{\mathbb {R}}}$-tree $T$. A subgroup $H\subset G$ is *elliptic* if it fixes a point in $T$. The action $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ is *trivial* if $G$ is elliptic. The action $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ is *minimal* if $T$ has no proper $G$-invariant subtree. We consider the trivial action of $G$ on a point as minimal. When $G$ contains a hyperbolic element, then $T$ contains a unique minimal $G$-invariant subtree ${T_\mathrm{min}}$, and ${T_\mathrm{min}}$ is the union of axes of all hyperbolic elements.
An *arc* is a set homeomorphic to $[0,1]$. A subtree is *non-degenerate* if it contains an arc.
Say that an action on an ${{\mathbb {R}}}$-tree $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ is *simplicial* it can be obtained from an action on a combinatorial tree by assigning equivariantly a positive length for each edge. If $S$ is a simplicial tree, we denote by $V(S)$ its set of vertices, and $E(S)$ its set of oriented edges. A *morphism of ${{\mathbb {R}}}$-trees* $f:T{\rightarrow}T'$ is a $1$-Lipschitz map such that any arc of $T$ can be subdivided into a finite number of sub-arcs on which $f$ is isometric.
For basic facts about ${{\mathbb {R}}}$-trees, see [@Sh_dendrology; @Chi_book].
Stabilities
-----------
Stability hypotheses say how arc stabilizers are nested. By stabilizer, we always mean *pointwise* stabilizer. When we will talk about the global stabilizer, we will mention it explicitly. If $G$ acts on $T$, and $X\subset T$, we will denote by $G(X)$ the (pointwise) stabilizer of $X$.
\[dfn\_stable\] Consider an action of a group $G$ on an ${{\mathbb {R}}}$-tree $T$.
A non-degenerate subtree $Y\subset T$ is called *stable* if for every arc $I\subset Y$, $G(I)=G(Y)$.
$T$ is *BF-stable* (in the sense of Bestvina-Feighn [@BF_stable]) if every arc of $T$ contains a stable arc.
$T$ satisfies *the ascending chain condition* if for any sequence of arcs $I_1\supset I_2\supset \dots$ whose lengths converge to $0$, the sequence of stabilizers $G(I_1)\subset G(I_2)\subset\dots$ is eventually constant.
$T$ is *piecewise stable* if any arc of $T$ can covered by finitely many stable arcs. Equivalently, the action is piecewise stable if for all $a\neq b$, the arc $[a,b]$ contains a stable arc of the form $[a,c]$ (one common endpoint).
$T$ is *super-stable* if any arc with non-trivial stabilizer is stable.
The first definition of piecewise stability clearly implies the second one. Conversely, assume that any arc $[a,b]$ contains a stable arc of the form $[a,c]$. Let $I$ be any arc of $T$. Thus any point of $I$ has a neighbourhood in $I$ which is the union of at most two stable arcs. By compactness, $I$ is covered by finitely many stable arcs.
Clearly, super-stability or piecewise stability imply the ascending chain condition which implies BF-stability. If $T$ is a stable subtree of itself, then any arc stabilizer fixes $T$; in other words, if $N$ is the kernel of the action $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ ([i. e. ]{}$N$ is the set of elements acting as the identity), $G/N$ acts with trivial arc stabilizers on $T$.
Graphs of actions on ${{\mathbb {R}}}$-trees and transverse coverings {#sec_goa}
---------------------------------------------------------------------
Graphs of actions on ${{\mathbb {R}}}$-trees are a way of gluing equivariantly ${{\mathbb {R}}}$-trees (see [@Skora_combination] or [@Lev_graphs]). Here, we rather follow [@Gui_limit section 4].
A *graph of actions on ${{\mathbb {R}}}$-trees* ${{\mathcal {G}}}=(S,(Y_v)_{v\in V(S)},(p_e)_{e\in E(S)})$ consists of the following data:
a simplicial tree $S$ called the *skeleton* with a simplicial action without inversion of a group $G$;
for each vertex $v\in S$, an ${{\mathbb {R}}}$-tree $Y_v$ (called vertex tree or vertex action);
for each oriented edge $e$ of $S$ with terminal vertex v, an *attaching point* $p_e\in Y_v$.
All this data should be invariant under $G$: $G$ acts on the disjoint union of the vertex trees so that the projection $Y_v\mapsto v$ is equivariant; and for every $g\in G$, $p_{g.e}=g.p_e$.
Some vertex trees may be reduced to a point (but they are not allowed to be empty). The definition implies that $Y_v$ is $G(v)$-invariant and that $p_e$ is $G(e)$-invariant.
To a graph of actions ${{\mathcal {G}}}$ corresponds an ${{\mathbb {R}}}$-tree $T_{{\mathcal {G}}}$ with a natural action of $G$. Informally, $T_{{\mathcal {G}}}$ is obtained from the disjoint union of the vertex trees by identifying the two attaching points of each edge of $S$ (see [@Gui_limit] for a formal definition).
Alternatively, one can define a graph of actions as a graph of groups $\Gamma$ with an isomorphism $G\simeq\pi_1(\Gamma)$ together with an action of each vertex group $G_v$ on an ${{\mathbb {R}}}$-tree $Y_v$ and for each oriented edge $e$ with terminus vertex $v$, a point of $Y_v$ fixed by the image of $G_e$ in $G_v$. This is where the terminology comes from.
Say that $T$ splits as a graph of actions ${{\mathcal {G}}}$ if there is an equivariant isometry between $T$ and $T_{{\mathcal {G}}}$.
Transverse coverings are very convenient when working with graphs of actions.
A *transverse covering* of an ${{\mathbb {R}}}$-tree $T$ is a covering of $T$ by a family of subtrees ${{\mathcal {Y}}}=(Y_v)_{v\in V}$ such that
every $Y_v$ is a closed subtree of $T$
every arc of $T$ is covered by finitely many subtrees of ${{\mathcal {Y}}}$
for $v_1\neq v_2\in V$, $Y_{v_1}\cap Y_{v_2}$ contains at most one point
When $T$ has an action of a group $G$, we always require the family ${{\mathcal {Y}}}$ to be $G$-invariant.
In the definition above, if some subtrees $Y_v$ are reduced to a point, we may as well forget them in ${{\mathcal {Y}}}$. Moreover, given a covering by subtrees $Y_v$ which are not closed, but which satisfy the two other conditions for a transverse covering, then the family $({\overline{Y_v}})_{v\in V}$ of their closure is a transverse covering.
The relation between graphs of actions and transverse coverings is contained in the following result.
\[lem\_transverse\_cov\] Assume that $T$ splits as a graph of actions with vertex trees $(Y_v)_{v\in V(S)}$. Then the family ${{\mathcal {Y}}}=(Y_v)_{v\in V(S)}$ is a transverse covering of $T$.
Conversely, if $T$ has a transverse covering by a family ${{\mathcal {Y}}}=(Y_v)_{v\in V}$ of non-degenerate trees, then $T$ splits as a graph of actions whose non-degenerate vertex trees are the $Y_v$.
We recall for future use the definition of the graph of actions induced by a transverse covering ${{\mathcal {Y}}}$. We first define its skeleton $S$. Its vertex set $V(S)$ is $V_0(S)\cup V_1(S)$ where $V_1(S)$ is the set of subtrees $Y\in{{\mathcal {Y}}}$, and $V_0(S)$ is the set of points $x\in T$ lying in the intersection of two distinct subtrees in ${{\mathcal {Y}}}$. There is an edge $e=(x,Y)$ between $x\in V_0(S)$ and $Y\in V_1(S)$ if $x\in Y$. The vertex tree of $v\in V(S)$ is the corresponding subtree of $T$ (reduced to a point if and only if $v\in V_0(S)$). The two attaching points the edge $(x,Y)$ are the copies of $x$ in $\{x\}$ and $Y$ respectively.
Let $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ be a simplicial action. Then the family of its edges is a transverse covering of $T$. If $T$ has no terminal vertex, then the skeleton of this transverse covering is the barycentric subdivision of $T$.
Here is another general example which will be useful.
\[lem\_decompo\_minimal\] Assume that $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ contains a hyperbolic element and denote by ${T_\mathrm{min}}$ be the minimal subtree of $T$. Let ${{\mathcal {Y}}}_0$ be the set of closures of connected components of $T\setminus {\overline{{T_\mathrm{min}}}}$.
Then $\{{\overline{{T_\mathrm{min}}}}\}\cup {{\mathcal {Y}}}_0$ is a transverse covering of $T$.
If $Y_v\in{{\mathcal {Y}}}_0$, then its global stabilizer $G_v$ fixes the point $Y_v\cap {\overline{{T_\mathrm{min}}}}$. In particular, this lemma says that a finitely supported action (in the sense of Definition \[dfn\_span\]) can be decomposed into a graph of actions where each vertex action is either trivial or has a dense minimal subtree.
Any arc $I\subset T$ is covered by ${\overline{{T_\mathrm{min}}}}$ and at most two elements of ${{\mathcal {Y}}}_0$, containing the endpoints of $I$. The other properties of the transverse covering are clear.
Consider a graph of actions ${{\mathcal {G}}}=(S,(Y_v)_{v\in V(S)},(p_e)_{e\in E(S)})$, and $T_{{\mathcal {G}}}$ the corresponding ${{\mathbb {R}}}$-tree. Given a $G$-invariant subset $V'\subset V(S)$, we want to define an ${{\mathbb {R}}}$-tree by collapsing all the trees $(Y_v)_{v\in V'}$. For each $v\in V'$, replace $Y_v$ by a point, and for each edge $e$ incident on $v\in V'$, change $p_e$ accordingly. Let ${{\mathcal {G}}}'=(S,(Y'_v)_{v\in V(S)},(p'_e)_{e\in E(S)})$ the corresponding graph of actions, and $T_{{\mathcal {G}}}'$ the corresponding ${{\mathbb {R}}}$-tree.
\[dfn\_collapse\] We say that the tree $T_{{{\mathcal {G}}}'}$ is obtained from $T_{{\mathcal {G}}}$ by collapsing the trees $(Y_v)_{v\in V'}$.
Let $p_v:Y_v{\rightarrow}Y'_v$ the natural map (either the identity or the constant map), and $p:T_{{\mathcal {G}}}{\rightarrow}T_{{\mathcal {G}}}'$ the induced map.
We say that a map $f:T{\rightarrow}T'$ *preserves alignment* if the three following equivalent conditions hold:
- for all $x\in [y,z]$, $f(x)\in [f(y),f(z)]$.
- the preimage of a convex set is convex,
- the preimage of a point is convex.
See for instance [@Gui_coeur Lemma 1.1] for the equivalence.
\[lem\_collapse\] The map $p:T_{{\mathcal {G}}}{\rightarrow}T_{{\mathcal {G}}}'$ preserves alignment. In particular, if $T_{{\mathcal {G}}}$ is minimal, so is $T_{{\mathcal {G}}}'$.
Consider a point $x\in T_{{{\mathcal {G}}}'}$. Let $E'_v=\{x\}\cap Y'_v$. If $u,v$ are such that $E'_{u},E'_v\neq{\emptyset}$ for $i=1,2$, there is a path $u=v_0,\dots,v_n=v$ such that the attaching points of the edge $v_iv_{i+1}$ coincide with the point in $E'_{v_i}$ and $E'_{v_{i+1}}$ (in particular $E'_{v_i}\neq {\emptyset}$).
Consider $E_v=p_v{^{-1}}(E'_v)$. For each $v$, either $E_v=Y_v$ or $E_v$ consists of at most one point. The existence of the path above shows that the image of $E_v$ in $T_{{\mathcal {G}}}$ is connected. This proves that $p{^{-1}}(\{x\})$ is convex so $p$ preserves alignment.
Let $Y'\subset T_{{\mathcal {G}}}'$ be a non-empty $G$-invariant subtree. Then $p{^{-1}}(Y')$ is a non-empty $G$-invariant subtree, so $p{^{-1}}(Y')=T_{{\mathcal {G}}}$. Since $p$ is onto, $Y'=T_{{{\mathcal {G}}}'}$, and minimality follows.
Actions of pairs, finitely generated pairs
------------------------------------------
By a pair of groups $(G,{{\mathcal {H}}})$, we mean a group $G$ together with a finite family of subgroups ${{\mathcal {H}}}=\{H_1,\dots,H_p\}$. We also say that the groups $H_i$ are peripheral subgroups of $G$. In fact, each peripheral subgroup is usually only defined up to conjugacy so it would be more correct to define ${{\mathcal {H}}}$ as a finite set of conjugacy class of subgroups.
An action of a pair $(G,{{\mathcal {H}}})$ on a tree is an action of $G$ in which each subgroup $H_i$ is elliptic. When the tree is simplicial, we also say that this is a splitting of $(G,{{\mathcal {H}}})$ or a splitting of $G$ *relative* to ${{\mathcal {H}}}$.
\[dfn\_relfg\] A pair $(G,{{\mathcal {H}}})$ is *finitely generated* if there exists a finite set $F\subset G$ such that $F\cup H_1\cup\dots\cup H_p$ generates $G$. We also say in this case that $G$ is finitely generated relative to ${{\mathcal {H}}}$.
We say that $F$ is a generating set of the pair $(G,{{\mathcal {H}}})$.
If $(G,\{H_1,\dots,H_p\})$ is finitely generated, then for any $g_i\in G$, so is $(G,\{H_1^{g_1},\dots,H_p^{g_p}\})$: just add $g_i$ to $F$. Of course, if $G$ is finitely generated, then $(G,{{\mathcal {H}}})$ is finitely generated for any ${{\mathcal {H}}}$.
By Serre’s Lemma, if $(G,{{\mathcal {H}}}){\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ is a non-trivial action of a finitely generated pair, then $G$ contains a hyperbolic element ([@Serre_arbres Proposition 6.5.2]).
If $G_v$ is a vertex group in a finite graph of groups $\Gamma$, then it has a natural structure of pair $(G_v,{{\mathcal {H}}}_v)$ consisting of (representatives of) conjugacy classes of the images in $G_v$ of the incident edge groups. We call this pair the *peripheral structure* of $G_v$ in $\Gamma$.
\[lem\_relfg\] Consider a graph of groups $\Gamma$ and $G=\pi_1(\Gamma)$.
If $\Gamma$ is finite and $G$ is finitely generated, then the peripheral structure $(G_v,{{\mathcal {H}}}_v)$ of each vertex group $G_v$ is finitely generated.
We use the notations of [@Serre_arbres]. Choose a maximal subtree $\tau$ of $\Gamma$. For each vertex $v$, the vertex group $G_v$ is now identified with a subgroup of $\pi_1(\Gamma,\tau)$. Let $F$ be a finite generating set of $\pi_1(\Gamma,\tau)$. Since we can write each $g\in F$ as a product of edges of $\Gamma$ and of elements of the vertex groups, there is a generating set of $\pi_1(\Gamma,\tau)$ consisting of the edges of $\Gamma$ and of a finite set $F'$ of elements of the vertex groups of $\Gamma$.
For each vertex $v\in\Gamma$, let $H_v\subset G_v$ be the subgroup generated by the elements of $F'$ which lie in $G_v$, and by the image in $G_v$ of incident edge groups. Of course, $\pi_1(\Gamma,\tau)$ is generated by the groups $H_v$ and the edges of $\Gamma$.
Fix $v\in\Gamma$. We shall prove that $G_v=H_v$. Take $g\in G_v$, and write $g$ as $g=g_0e_1g_1e_2...e_kg_k$ where $v_0 e_1
v_1\dots e_k v_k$ is a loop based at $v$, and each $g_i$ is in $H_{v_i}$. If the word has length $0$, there is nothing to prove. If not, we shall find a shorter word of this form representing $g$, and the lemma will be proved by induction. If this word has not length $0$, it is not reduced as a word in the graph of groups. Therefore, there are two consecutive edges $e_i,e_{i+1}$ such that $e_{i+1}=\Bar{e_i}$, and $g_i\in j_{e_i}(G_{e_i})$ (where $j_e$ denotes the edge morphisms of $\Gamma$). Thus, we can write $h=g_0e_1\dots
e_{i-1}g_{i-1}g'_ig_{i+1}e_{i+2}\dots e_kg_k$ where $g'_i\in
j_{e_{i+1}}(G_{e_{i+1}})$. Note that $g_{i-1}g'_ig_{i+1}\in
H_{v_{i-1}}=H_{v_{i+1}}$ hence we found a shorter word of the required form representing $g$.
\[lem\_relfg\_rel\] Let $(G,{{\mathcal {H}}})$ be a finitely generated pair, and consider a (relative) splitting of $(G,{{\mathcal {H}}})$. Let $v$ be a vertex of the corresponding graph of groups.
Then $G_v$ is finitely generated relative to a family ${{\mathcal {H}}}_v$ consisting of the images in $G_v$ of the incident edge groups together with at most one conjugate of each $H_i$.
Let $G\simeq\pi_1(\Gamma)$ be a splitting of $G$ as a graph of groups. We can assume that $H_i\subset G_{v_i}$ for some vertex $v_i$ of $\Gamma$. For each index $i$, consider a finitely generated group $\Hat H_i$ containing $H_i$. Consider the graph of groups $\Hat \Gamma$ obtained from $\Gamma$ by adding for each $i$ a new edge $e_i$ edge carrying $H_i$ incident on a new vertex $\Hat v_i$ carrying $\Hat H_i$, and by gluing the other side of $e_i$ on $v_i$. Thus, $e_i$ carries the amalgam $G_{v_i}*_{H_i} \Hat H_i$. The fundamental group of $\Hat\Gamma$ is finitely generated. By Lemma \[lem\_relfg\], the peripheral structure of $G_v$ in $\Hat G_v$ is a finitely generated pair. The lemma follows.
Finitely supported actions
--------------------------
A *finite tree* in an ${{\mathbb {R}}}$-tree is the convex hull of a finite set.
\[dfn\_span\] A set $K\subset T$ *spans* $T$ if any arc $I\subset T$ is covered by finitely many translates of $K$. The tree $T$ is *finitely supported* if it is spanned by a finite tree $K$.
If $T$ is a simplicial tree, then $T$ is finitely supported if and only if $T/G$ is a finite graph. Any minimal action of a finitely generated group is finitely supported. More generally we have:
A minimal action of a finitely generated pair on an ${{\mathbb {R}}}$-tree is finitely supported.
Consider $(G,\{H_1,\dots,H_p\})$ a finitely generated pair acting minimally on $T$. Consider a finite generating set $\{f_1\dots,f_q\}$ of the pair $(G,\{H_1,\dots,H_p\})$. Let $x_i\in T$ be a point fixed by $H_i$ and $x\in T$ be any point. Let $K$ be the convex hull of $\{x,f_1.x,\dots,f_q.x,x_1,\dots,x_p\}$. Then $G.K$ is connected because for each generator $g\in \{f_1,\dots,f_q\}\cup H_1\cup\dots \cup H_p$, $g.K\cap K\neq{\emptyset}$. By minimality, $G.K=T$ and $T$ is finitely supported.
\[lem\_supp\_fini\] Consider an action of a finitely generated pair $(G,{{\mathcal {H}}}){\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$, ${{\mathcal {Y}}}$ a transverse covering of $T$, and $S$ the skeleton of ${{\mathcal {Y}}}$.
Then any $H\in{{\mathcal {H}}}$ is elliptic in $S$. Moreover, if $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ is minimal (resp. finitely supported) then so is $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}S$.
The statement about minimality is proved in [@Gui_limit Lemma 4.9].
We use the description of the skeleton given after Lemma \[lem\_transverse\_cov\]. We can assume that every subtree $Y\in{{\mathcal {Y}}}$ is non-degenerate. Let $K$ be a finite tree spanning $T$. It is covered by finitely many trees of ${{\mathcal {Y}}}$. It follows that ${{\mathcal {Y}}}/G=V_1(S)/G$ is finite. Choose a lift of each element of $V_1(S)/G$ in $V_1(S)$, and consider the convex hull $L\subset S$ of those vertices. Let $x_i\in T$ be a point fixed by $H_i$. Each $H_i\in {{\mathcal {H}}}$ fixes a vertex $v_i\in S$, lying in $V_0({{\mathcal {Y}}})$ or $V_1({{\mathcal {Y}}})$ according to whether $x_i$ lies in two distinct elements of ${{\mathcal {Y}}}$ or not.
Let $\{f_1,\dots,f_q\}$ be a finite generating set of $(G,{{\mathcal {H}}})$. Let $L_0$ be the convex hull of $L\cup f_1.L\cup\dots \cup f_q.L\cup\{v_1,\dots,v_p\}$. The set $S_0=G.L_0$ is a subtree of $S$ containing $V_1(S)$. It is finitely supported. If $S_0\neq S$, consider $x\in V_0(S)\setminus S_0$. Recall that the vertex $x\in V_0(S)$ corresponds to a point $x\in T$. Any edge of $S$ incident on $x$ corresponds to a subtree $Y\in V_1(S)$ such that $x\in Y$. Since $S_0$ is a subtree and contains $V_1(S)$, there is exactly one such edge. If follows that $x$ belongs to exactly one $Y\in{{\mathcal {Y}}}$, contradicting the definition of $V_0(S)$. Therefore, $S=S_0$ is finitely supported.
If $T$ splits as a graph of actions, vertex actions may fail to be minimal, even if $T$ is minimal. However, the property of being finitely supported is inherited by vertex actions:
\[lem\_support\] Assume that $T$ splits as a graph of actions and that $T$ is finitely supported. Then each vertex action is finitely supported.
Let $K$ be a finite tree spanning of $T$. Let ${{\mathcal {Y}}}$ be a transverse covering of $T$, and $Y_0\in{{\mathcal {Y}}}$ be a vertex tree. Consider a finite subset $F$ of ${{\mathcal {Y}}}$ which covers $K$, and let $\{g_1.Y_0,\dots,g_p.Y_0\}$ be the set of elements of $F$ lying in the orbit of $Y_0$. Consider the convex hull $K_0$ of the finite trees $Y_0\cap g_i{^{-1}}K$ for $i\in\{1,\dots, p\}$. It is easy to check that the finite tree $K_0$ spans $Y_0$.
Indecomposability
-----------------
Indecomposability is a slight modification of the *mixing* property introduced by Morgan in [@Mo_ergodic].
\[dfn\_indecomposability\] A non-degenerate subtree $Y\subset T$ is called *indecomposable* if for every pair of arcs $I,J\subset Y$, there is a finite sequence $g_1.I,\dots,g_n.I$ which covers $J$ and such that $g_i.I\cap g_{i+1}.I$ is non-degenerate (see figure \[fig\_indecomposability\]).
![Indecomposability\[fig\_indecomposability\].](indecompo.eps)
In the definition of indecomposability, one cannot assume in general that $g_i.I\cap J$ is non-degenerate for all $i$. This is indeed the case if $Y=T$ is dual to a measured foliation on a surface having a 4-pronged singularity, $J$ is an arc in $T$ represented by a small transverse segment containing this singularity and joining two opposite sectors of the singularity, and $I$ is represented by a transverse segment disjoint from the singularities.
The following property explains the choice of the terminology.
\[lem\_indec\_component\] If $Y\subset T$ is indecomposable, and if $T$ splits as a graph of actions, then $Y$ is contained in a vertex tree.
Consider an arc $I$ contained in $Y\cap Y_v$ for some vertex tree $Y_v$ of the decomposition. Consider an arc $J\subset Y$, and a finite sequence $g_1.I,\dots,g_n.I$ which covers $J$ and such that $g_i.I\cap g_{i+1}.I$ is non-degenerate. We have $g_1.I\subset g_1.Y_v$, but $g_1.Y_v\cap
g_2.Y_v$ is non-degenerate so $g_1.Y_v=g_2.Y_v$. By induction, we get that all the translates $g_i.I$ lie in the same vertex tree $g_1.Y_v$, so that $J$ lies in a vertex tree. Since this is true for every arc $J\subset Y$, the lemma follows.
\[lem\_indecompo\]
If $f:T{\rightarrow}T'$ is a morphism of ${{\mathbb {R}}}$-trees, and if $Y\subset
T$ is indecomposable, then so is $f(Y)$.
If $Y\subset T$ is indecomposable, then the orbit of any point $x\in Y$ meets every arc $I\subset Y$ in a dense subset.
If $T$ itself is indecomposable, then it is minimal (it has no non-trivial invariant subtree).
Assume that $(Y_v)_{v\in V}$ is a transverse covering of $T$. Let $Y_{v_0}$ be a vertex tree and $H$ its global stabilizer. If $Y_{v_0}$ is an indecomposable subtree of $T$, then $Y_{v_0}$ is indecomposable for the action of $H$.
Let $I=[f(a),f(b)]$ and $J=[f(x),f(y)]$ in $f(Y)$. Choose $I'\subset [a,b]$ so that $f(I')\subset I$, and let $J'=[x,y]$, so $f(J')$ contains $J$. Indecomposability of $Y$ now clearly implies indecomposability of $f(Y)$.
Statement 2 follows from the fact that for any arc $I'\subset I$, there exists $g_1$ such that $x\in g_1.I'$, so $g_1{^{-1}}.x\in I'$.
Let’s prove statement 3. It follows from statement 2 that $G$ contains a hyperbolic element. Consider ${T_\mathrm{min}}$ the minimal $G$-invariant subtree. By statement 2, every orbit meets ${T_\mathrm{min}}$; if follows that $T={T_\mathrm{min}}$.
For statement 4, consider $I,J\subset Y_{v_0}$, and $g_1,\dots,g_n\in G$ such that $J\subset g_1.I\cup \dots \cup g_n.I$ with $g_i.I\cap g_{i+1}.I$ non-degenerate. In particular, $g_i.Y_{v_0}\cap g_{i+1}.Y_{v_0}$ is non-degenerate, so $g_i.Y_{v_0}= g_{i+1}.Y_{v_0}$, and $g_{i+1}g_i{^{-1}}\in H$. Consider an index $i$ such that $g_i.I\cap J$ is non-degenerate. Since $J\subset Y_{v_0}$, $g_i.Y_{v_0}\cap Y_{v_0}$ is non-degenerate, so $g_i\in H$. Therefore, $g_1,\dots, g_n\in H$.
The conjunction of indecomposability and stability has a nice consequence:
\[lem\_indec\_stab\] If $T$ is BF-stable, and if $Y\subset T$ is indecomposable, then $Y$ is a stable subtree.
First, one easily checks that if $K_1,K_2$ are two stable subtrees of $T$ such that $K_1\cap K_2$ is non-degenerate, then $K_1\cup K_2$ is a stable subtree. Consider a stable arc $I\subset Y$, and any other arc $J\subset Y$. Since $J$ is contained in a union $g_1.I\cup\dots\cup g_n.I$ with $g_i.I\cap g_{i+1}.I$ non-degenerate, $J$ is stable. Since this holds for any arc $J$, $Y$ is a stable subtree of $T$.
Geometric actions and strong convergence
----------------------------------------
We review some material from [@LP] where more details can be found.
### Strong convergence {#strong-convergence .unnumbered}
Given two actions $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$, $G'{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T'$, we write $({\varphi},f):G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T{\rightarrow}G'{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T'$ when ${\varphi}:G{\rightarrow}G'$ is a morphism, and $f:T{\rightarrow}T'$ is a ${\varphi}$-equivariant map. We say that $({\varphi},f)$ is onto if both $f$ and ${\varphi}$ are. We say that $({\varphi},f)$ is morphism of ${{\mathbb {R}}}$-trees if $f$ is.
A *direct system* of actions on ${{\mathbb {R}}}$-trees is a sequence of actions of finitely generated groups $G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_k$ and an action $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$, with surjective morphisms of ${{\mathbb {R}}}$-trees $({\varphi}_k,f_k):G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_k{\twoheadrightarrow}G_{k+1}{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_{k+1}$ and $(\Phi_k,F_k):G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_k{\twoheadrightarrow}G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ such that the following diagram commutes: $$\xymatrix@1@R=0.5cm{T_k\ar[r]_{f_k} \ar@/^0.9cm/[rrr]|{F_k}
\ar@(dl,dr)[]
& T_{k+1} \ar@/^0.5cm/[rr]|{F_{k+1}} \ar@(dl,dr)[] & \cdots
& T \ar@(dl,dr)[] \\
G_k\ar[r]^{{\varphi}_k} \ar@/_0.7cm/[rrr]|{\Phi_k}
&G_{k+1} \ar@/_0.4cm/[rr]|{\Phi_{k+1}} &\cdots & G}$$
For convenience, we will use the notation $f_{kk'}=f_{k'-1}\circ\dots\circ f_k:T_k{\rightarrow}T_{k'}$ and ${\varphi}_{kk'}={\varphi}_{k'-1}\circ\dots\circ {\varphi}_k:G_k{\rightarrow}G_{k'}$.
A direct system of finitely supported actions of groups on ${{\mathbb {R}}}$-trees $G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_k$ *converges strongly* to $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ if
$G$ is the direct limit of the groups $G_k$
for every finite tree $K\subset T_k$, there exists $k'\geq k$ such that $F_{k'}$ restricts to an isometry on $f_{kk'}(K)$,
The action $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ is the *strong limit* of this direct system.
In practice, strong convergence allows to lift the action of a finite number of elements on a finite subtree of $T$ to $T_k$ for $k$ large enough.
As an application of the definition, we prove the following useful lemma.
\[lem\_strong\_minimal\] Assume that $G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_k$ converges strongly to $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$. Assume that $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ is minimal.
Then for $k$ large enough, $G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_k$ is minimal.
By definition of strong convergence, $T_0$ is finitely supported. Let $K_0\subset T_0$ be a finite tree spanning $T_0$. Let $K$ (resp. $K_k$) be the image of $K_0$ in $T$ (resp. in $T_k$). Choose some hyperbolic elements $g_1,\dots,g_p\in G$ whose axes cover $K$. Let $K'$ be the convex hull of $K\cup g_1.K\cup\dots\cup g_p.K$. Choose $ g_1^0,\dots g_p^0\in G_0$ some preimages of $g_1,\dots g_p$ and let $g_i^k$ be the image of $g_i^0$ in $G_k$. Let $K'_0\subset T_0$ be the convex hull of $K_0 \cup g_1^0.K_0\cup\dots\cup g_p^0.K_0$. Let $K'_k$ be the image of $K'_0$ in $T_k$. Take $k$ large enough so that $F_k$ induces an isometry between $K'_k$ and $K'$. Then the axes of $g_1^k,\dots,g_p^k$ cover $K_k$. Since $K_k$ spans $T_k$, $T_k$ is a union of axes, so $T_k$ is minimal.
### Geometric actions {#geometric-actions .unnumbered}
A *measured foliation* ${{\mathcal {F}}}$ on a $2$-complex $X$ consists of the choice, for each closed simplex $\sigma$ of $X$ of a non-constant affine map $f_\sigma:\sigma{\rightarrow}{{\mathbb {R}}}$ defined up to post-composition by an isometry of ${{\mathbb {R}}}$, in a way which is consistent under restriction to a face: if $\tau$ is a face of $\sigma$, then $f_\tau={\varphi}\circ(f_\sigma)_{|\tau}$ for some isometry of ${\varphi}$ of ${{\mathbb {R}}}$. Level sets of $f_\sigma$ define a foliation on each closed simplex. Leaves of the foliations on $X$ are defined as the equivalence classes of the equivalence relation generated by the relation *$x,y$ belong to a same closed simplex $\sigma$ and $f_\sigma(x)=f_\sigma(y)$*.
This also defines a transverse measure as follows: the transverse measure $\mu(\gamma)$ of a path $\gamma:[0,1]{\rightarrow}\sigma$ to the foliation is the length of the path $f_\sigma\circ\gamma$. The transverse measure of a path which is a finite concatenation of paths contained in simplices is simply the sum of the transverse measures of the pieces. The transverse measure is invariant under the holonomy along the leaves. We also view this transverse measure as a metric on each transverse edge.
For simplicity, we will say *foliated 2-complex* to mean a $2$-complex endowed with a measured foliation.
The pseudo-metric $$\delta(x,y)=\inf\{\mu(\gamma)\text{ for $\gamma$ joining $x$ to $y$}\}$$ is zero on each leaf of $X$. By definition, the *leaf space made Hausdorff* $X/{{\mathcal {F}}}$ of $X$ is the metric space obtained from $X$ by making $\delta$ Hausdorff, [i. e. ]{}by identifying points at pseudo-distance $0$. In nice situations, $\delta(x,y)=0$ if and only if $x$ and $y$ are on the same leaf. In this case, $X/{{\mathcal {F}}}$ coincides with the space of leaves of the foliation, and we say that the *leaf space is Hausdorff*.
\[thm\_LP\_leaves\] Let $(X,{{\mathcal {F}}})$ be a foliated $2$-complex. Assume that $\pi_1(X)$ is generated by free homotopy classes of curves contained in leaves.
Then $X/{{\mathcal {F}}}$ is an ${{\mathbb {R}}}$-tree.
\[dfn\_dual\] An action of a finitely generated group $G$ on an ${{\mathbb {R}}}$-tree $T$ is *geometric* if there exists a foliated $2$-complex $X$ endowed with a free, properly discontinuous, cocompact action of $G$, such that
each transverse edge of $X$ isometrically embeds into $X/{{\mathcal {F}}}$
$T$ and $X/{{\mathcal {F}}}$ are equivariantly isometric.
In this case, we say that $T$ is *dual* to $X$.
The definition in [@LP] is in terms of a compact foliated $2$-complex $\Sigma$ (here $\Sigma=X/G$) and of a Galois covering ${\overline{\Sigma}}$ with deck group $G$ (here ${\overline{\Sigma}}=X$). The two points of view are clearly equivalent.
This definition requires $G$ to be finitely generated. For finitely generated pairs, one should weaken the assumption of cocompactness, but we won’t enter into this kind of consideration.
### Decomposition of geometric actions {#decomposition-of-geometric-actions .unnumbered}
\[prop\_decompo\] Let $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ be a geometric action dual to a $2$-complex $X$ whose fundamental group is generated by free homotopy classes of curves contained in leaves.
Then $T$ has a decomposition into a graph of actions where each non-degenerate vertex action is either indecomposable, or is an arc containing no branch point of $T$ except possibly at its endpoints.
The arcs in the lemma above are called *edges*, and the indecomposable vertex actions are called *indecomposable components*.
The basis of the proof of this proposition is a version of a theorem by Imanishi giving a dynamical decomposition of a compact foliated $2$-complex $\Sigma$ ([@Imanishi]). A leaf of $\Sigma$ is *regular* if it contains no vertex of $\Sigma$. More generally, a leaf segment ([i. e. ]{}a path contained in a leaf) is *regular* if it contains no vertex of $\Sigma$. A leaf or leaf segment which is not regular is *singular*.
Let $\Sigma^*=\Sigma\setminus V(\Sigma)$ be the complement of the vertex set of $\Sigma$. It is endowed with the restriction of the foliation of $\Sigma$. Let $C^*\subset \Sigma^*$ be the union of leaves of $\Sigma^*$ which are closed but not compact.
\[dfn\_cut\] We call the set $C=C^*\cup V(\Sigma)$ *the cut locus* of $\Sigma$.
The cut locus is a finite union of leaf segments joining two vertices of $\Sigma$. In particular, $C$ is compact, and $\Sigma\setminus C$ consists of finitely many connected components. Each component of $\Sigma\setminus C$ is a union of leaves of $\Sigma^*$.
The dynamical decomposition of $\Sigma$ is as follows:
Let $U$ be a component of $\Sigma\setminus C$. Then either every leaf of $U$ is compact, or every leaf of $U$ is dense in $U$.
Let $X$ be a foliated $2$-complex such that $T$ is dual to $X$. Denote by $q:X{\rightarrow}T$ the natural map (recall that $T$ is the leaf space made Hausdorff of $X$). The quotient $\Sigma=X/G$ is a compact foliated $2$-complex and the quotient map $\pi:X{\rightarrow}\Sigma$ is a covering map. Let $C$ be the cut locus of $\Sigma$ and $\Tilde C$ its preimage in $X$.
Let $(U_v)_{v\in V}$ be the family of connected components of $X\setminus \Tilde C$. Let ${\overline{U}}_v$ be the closure of $U_v$ in $X$, and $Y_v=q({\overline{U}}_v)\subset T$. Note that ${\overline{U}}_v\setminus U_v\subset \Tilde C$ is contained in a union of singular leaves and that any leaf segment in $U_v$ is regular. We shall first prove that the family ${{\mathcal {Y}}}=(Y_v)_{v\in V}$ is a transverse covering of $T$. We will need the following result.
\[prop\_LP\_separation\] Assume that $\pi_1(X)$ is generated by free homotopy classes of curves contained in leaves.
Then there exists a countable union of leaves ${{\mathcal {S}}}$ such that for all $x,y\in X\setminus{{\mathcal {S}}}$, $q(x)=q(y)$ if and only if $x,y$ are in the same leaf.
Since there are finitely many orbits of singular leaves in $X$, we can choose such a set ${{\mathcal {S}}}$ containing every singular leaf.
Assume that $Y_v\cap Y_{w}$ is non-degenerate. There is an uncountable number of regular leaves of $X$ meeting both ${\overline{U}}_v\cap {\overline{U}}_{w}$. Since ${\overline{U}}_v\setminus U_v\subset \Tilde C$ is contained in a union of singular leaves, there is a regular leaf meeting both $U_v$ and $U_w$, so $U_v=U_w$, and $v=w$.
We denote by ${X^{(t)}}$ the subset of the $1$-skeleton of $X$ consisting of the union of all closed transverse edges. All the paths we consider are chosen as a concatenation of leaf segments and of arcs in ${X^{(t)}}$. Let $I$ be an arc in $T$. Let $a,b\in\Sigma$ be two preimages of the endpoints of $I$ in ${X^{(t)}}$. Choose a path $\gamma$ in $X$ joining $a$ to $b$. We view such a path both as a subset of $X$ and as a map $[0,1]{\rightarrow}X$. Since $T$ is an ${{\mathbb {R}}}$-tree, $q(\gamma)\supset I$. Since the cut locus $C$ is a finite union of leaf segments, $\gamma\cap\Tilde C$ consists of a finite number of points and of a finite number of leaf segments. In particular, $\gamma$ is a concatenation of a finite number of paths which are contained in $\Tilde C$ or in some ${\overline{U}}_v$. Since a path contained in $\Tilde C$ is mapped to a point in $T$, we get that $I$ is covered by finitely many elements of ${{\mathcal {Y}}}$.
To prove that the subtree $Y_v$ is closed in $T$, it is sufficient to check that given a semi-open interval $[a_0,b_0)$ contained in $Y_v$, $b_0$ is contained in $Y_v$. Consider $ a, b\in {X^{(t)}}$ some preimages of $a_0,b_0$, and a path $\gamma:[0,1]{\rightarrow}X$ joining $a$ to $b$. By restricting $\gamma$ to a smaller interval, we can assume that for all $t<1$, $q\circ \gamma(t)\neq b_0$. Since the image of each transverse edge of $X$ is an arc in $T$, $K=q(\gamma)$ is a finite tree, and $b_0$ is a terminal vertex of $K$. In particular, there exists ${\varepsilon}>0$ such that for $q([1-{\varepsilon},1))\subset [a_0,b_0) \subset Y_v$. We can also assume that $\gamma([1-{\varepsilon},1])$ is contained in a transverse edge of $X$. For all but countably many $t$’s, $\gamma(t)\notin {{\mathcal {S}}}$ and lies in a regular leaf. Since $q\circ\gamma(t)\in Y_v$, the leaf $l$ through $\gamma(t)$ meets ${\overline{U}}_v$. Since $\Tilde C$ consists of singular leaves, $l\subset U_v$. In particular, $\gamma(t)\in U_v$, so $\gamma(1)\in{\overline{U}}_v$ and $b_0\in Y_v$. We have proved that ${{\mathcal {Y}}}$ is a transverse covering of $T$.
Assume that $v$ is such that any leaf of $\pi(U_v)$ is compact. By definition of the cut locus, this means that every leaf of $U_v$ is regular. In particular, $U_v$ is a union of leaves of $X$ and $q(U_v)$ is open in $T$. Since every transverse edge of $X$ is embedded into $T$, the holonomy along any regular leaf is trivial. It follows that $U_v$ is a foliated product: is homeomorphic to $U_v\simeq J\times l$ foliated by $\{*\}\times l$, where $J$ is an open interval in a transverse edge. Therefore, $q(U_v)$ is an open interval isometric to $J$. Since $q(U_v)$ is open in $T$, $Y_v$ is an arc containing no branch point of $T$, except possibly at its endpoints.
Assume that leaves of $\pi(U_v)$ are dense. We shall prove that $Y_v$ is indecomposable. First, we claim that any point in $Y_v$ has a preimage in $U_v$. This will follow from the fact that for any $x\in{\overline{U}}_v\setminus U_v$, the leaf through $x$ intersects $U_v$. Assume on the contrary that there is a leaf $l$ of $X$ which meets ${\overline{U}}_v$ but does not intersect $U_v$. Consider $l_0$ a connected component of $l\cap{\overline{U}}_v$. This is a leaf of the foliated $2$-complex ${\overline{U}}_v$. Let $G_v$ be the global stabilizer of $U_v$, and consider the foliated $2$-complex ${\overline{U}}_v/G_v$. The natural map ${\overline{U}}_v/G_v {\rightarrow}\Sigma$ restricts to an isomorphism between $U_v/G_v$ and a connected component $\Sigma\setminus C$ (namely $\pi(U_v)$). Since $U_v/G_v$ consists of finitely many open cells, ${\overline{U}}_v/G_v$ is compact. Moreover ${\overline{U}}_v/G_v\setminus U_v/G_v$ consists of finitely many vertices and finitely many edges contained in leaves. The image $\lambda_0$ of $l_0$ in ${\overline{U}}_v/G_v$ is a leaf which does not intersect $U_v/G_v$. Therefore, this leaf is compact and so is any leaf of ${\overline{U}}_v/G_v$ close to $\lambda_0$. This contradicts the fact that every leaf of $\pi(U_v)$ is dense.
Let’s prove indecomposability. Let $I_0,J_0$ be two arcs in $Y_v$. By taking $I_0$ smaller, one can assume that $I_0=q(I)$ for some arc $I\subset U_v$ contained in a transverse edge. Consider $a,b$ some preimages of the endpoints of $J_0$ in $U_v\cap {X^{(t)}}$. Consider a path $\gamma\subset U_v$ joining $a$ to $b$, and write $\gamma$ as a concatenation $l_0.\tau_1.l_1.\tau_2\dots \tau_p.l_p$ where $\tau_i$ is contained in a transverse edge and $l_i$ is a leaf segment. Since every leaf is dense in $\pi(U_v)$, for all $x\in \tau_1\cup\dots \cup \tau_p$, there exists $g_x\in G$ and a leaf segment $l_x$ in $U_v$ joining $x$ to $g_x.\rond I$. Since any leaf segment in $U_v$ is regular, this is still valid with the same $g_x$ on a neighbourhood of $x$. By compactness, one can choose each $g_x$ in a finite set $F=\{g_1,\dots,g_n\}$. Clearly, $J_0$ is contained in $g_1.q(I)\cup\dots\cup g_n.q(I)$. Denote by $x$ and $y$ the endpoints of $l_i$. There is a (regular) leaf segment in $U_v$ joining $g_x.\rond I$ to $g_y.\rond I$, so $q(g_x.I)\cap q(g_y.I)$ is non-degenerate. Indecomposability follows, which completes the proof of Proposition \[prop\_decompo\].
\[rem\_decompo\_geom\] Each indecomposable vertex action $Y_v$ is geometric, and dual to the $2$-complex ${\overline{U}}_v$. Moreover, if $\pi_1(X)$ is generated by free homotopy classes of curves contained in leaves, then so is $\pi_1({\overline{U}}_v)$ (this will be useful in Appendix \[sec\_Sela\]). Indeed, consider a curve $\gamma\subset {\overline{U}}_v$, and a map $f$ from a planar surface $S$ to $X$ such that one boundary component is mapped to $\gamma$, and every other boundary component is mapped to a leaf. Let $S_0\subset S$ be the connected component of $f{^{-1}}({\overline{U}}_v)$ containing ${\overline{U}}_v$. Since every connected component of $\partial {\overline{U}}_v$ is contained in a leaf, each component of $\partial S_0$ is mapped into a leaf, which proves that $\pi_1({\overline{U}}_v)$ is generated by free homotopy classes of curves contained in leaves.
### Strong approximation by geometric actions {#strong-approximation-by-geometric-actions .unnumbered}
We now quote a result from [@LP].
\[thm\_strongCV\] Consider a minimal action of a finitely generated group $G$ on an ${{\mathbb {R}}}$-tree $T$.
Then $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ is a strong limit of a direct system of geometric actions $\{(\Phi_k,F_k):G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_k{\rightarrow}G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T\}$ such that
$\Phi_k$ is one-to-one in restriction to each arc stabilizer of $T_k$,
$T_k$ is dual to a $2$-complex $X$ whose fundamental group is generated by free homotopy classes of curves contained in leaves.
The second statement follows from the construction in [@LP Theorem 2.2].
Since $T_k$ is geometric, it has a decomposition into a graph of actions as described above. We show how to adapt this result to actions of finitely generated pairs.
\[prop\_strongCV\_rel\] Consider a minimal action $(G,{{\mathcal {H}}}){\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ of a finitely generated pair on an ${{\mathbb {R}}}$-tree. Then there exists a direct system of minimal actions $(G_k,{{\mathcal {H}}}_k){\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_k$ converging strongly to $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ and such that
${\varphi}_k$ and $\Phi_k$ are one-to-one in restriction to arc stabilizers of $T_k$
${\varphi}_k$ (resp. $\Phi_k$) restricts to an isomorphism between ${{\mathcal {H}}}_k$ and ${{\mathcal {H}}}_{k+1}$ (resp. ${{\mathcal {H}}}$).
$T_k$ splits as a graph of actions where each non-degenerate vertex action is indecomposable, or is an arc containing no branch point of $T_k$ except at its endpoints.
We follow Theorem 2.2 and 3.5 from [@LP]. Let $S\subset G$ be a finite set such that $S_\infty=S\cup H_1\dots\cup H_p$ generates $G$. Let $S_k$ be an exhaustion of $S_\infty$ by finite subsets. Let $K_k$ be an exhaustion of $T$ by finite trees. We assume that for each $i\in\{1,\dots,p\}$, $K_k$ contains a point $a_i$ fixed by $H_i$.
For each $s\in S_k$, consider $A_s=K_k\cap s{^{-1}}K_k$ and $B_s=s.A_s=K_k\cap s.K_k$. We obtain a foliated $2$-complex $\Sigma_k$ as follows. For each $s\in S_k$, consider a band $A_s\times [0,1]$ foliated by $\{*\}\times [0,1]$, glue $A_s\times \{0\}$ on $K$ using the identity map on $A_s$, and glue $A_s\times \{1\}$ using the restriction of $s:A_s{\rightarrow}K_k$. For each $s\in H_i\setminus S_k$, add an edge and glue its endpoints on $a_i$; this edge is contained in a leaf.
The fundamental group of $\Sigma_K$ is isomorphic to the free group $F(S_\infty)$. Let ${\varphi}:F(S_\infty){\rightarrow}G$ be the natural morphism. Let $N_k$ be the subgroup of $\ker{\varphi}$ generated by free homotopy classes of curves contained in leaves (in particular, any relation among the elements of some $H_i$ lies in $N_k$). This is a normal subgroup of $F(S_\infty)$, and let $G_k=F(S_\infty)/N_k$. We denote by ${\varphi}_k:G_k{\rightarrow}G_{k+1}$ and $\Phi_k:G_k{\rightarrow}G$ the natural morphisms. Clearly, $G$ is the direct limit of $G_k$. Let $H_i^{(k)}$ be the image in $G_k$ of the subgroup of $F(S_\infty)$ generated by the elements of $H_i$. Let ${{\mathcal {H}}}_k=\{H_1^{(k)},\dots,H_p^{(k)}\}$. The pair $(G_k,{{\mathcal {H}}}_k)$ is finitely generated, and ${\varphi}_k$ and $\Phi_k$ restrict to isomorphisms between $H_i^{(k)}$, $H_i^{(k+1)}$, and $H_i$.
Let $\Tilde \Sigma_k$ be the Galois covering of $\Sigma_k$ corresponding to $N_k$. By definition of $N_k$, $\pi_1(\Tilde \Sigma_k)$ is generated by free homotopy classes of curves contained in leaves. Let $T_k$ be the leaf space made Hausdorff of $\Tilde \Sigma_k$. This is an ${{\mathbb {R}}}$-tree by Theorem \[thm\_LP\_leaves\]. Each $H_i^{(k)}$ fixes a point in $T_k$. By Proposition \[prop\_LP\_separation\], each arc stabilizer of $T_k$ embeds into $G$. Finally, the argument of [@LP Th. 2.2] applies to prove that $G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_k$ converges strongly to $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$. Minimality follows from Lemma \[lem\_strong\_minimal\].
The $2$-complex $\Sigma_k$ is a finite foliated complex, with a finite set of infinite roses attached. In particular, $\Sigma_k$ has the same dynamical decomposition as a finite foliated $2$-complex so one can repeat the argument of Proposition \[prop\_decompo\].
Extended Scott’s Lemma {#sec_scott}
======================
Scott’s Lemma claims that if $G$ is a direct limit of groups $G_k$ having compatible decompositions into free products, then $G$ itself has such a decomposition (this follows from [@Scott_coherent Th 1.7]). Scott’s Lemma is usually proved using Scott’s complexity ([@Scott_coherent Th 1.7]). Delzant has defined a refinement of this complexity for morphisms which has many important applications ([@Swarup_Delzant]).
The main result of this section is an extension of Scott’s Lemma for more general splittings. This will be an essential tool to prove piecewise stability of $T$.
In the following statement, an *epimorphism* $({\varphi}_k,f_k)$ consists of an onto morphism ${\varphi}_k:G_k{\twoheadrightarrow}G_{k+1}$, and of a continuous map $f_k$ sending an edge to a (maybe degenerate) edge path.
[scott]{}
\[thm\_scott\] Let $G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}S_k$ be a sequence of non-trivial actions of finitely generated groups on simplicial trees, and $({\varphi}_k,f_k):G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}S_k{\rightarrow}G_{k+1}{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}S_{k+1}$ be epimorphisms. Assume that $({\varphi}_k,f_k)$ does not increase edge stabilizers in the following sense:
$$\forall e\in E(S_k),\forall e'\in E(S_{k+1}),\quad e'\subset f_k(e)\Rightarrow G_{k+1}(e')={\varphi}_k(G_k(e))\quad (*)$$
Then $\displaystyle\lim_{{\rightarrow}} G_k$ has a non-trivial splitting over the image of an edge stabilizer of some $S_k$.
When edge stabilizers of $S_k$ are trivial, we obtain Scott’s Lemma.
Decomposition into folds.
-------------------------
#### Collapses, folds, and group-folds.
We recall and adapt definitions of [@Dun_folding; @BF_complexity]. Consider a finitely generated group $G$ acting on a simplicial tree $S$, without inversion (no element of $G$ flips an edge). Given an edge $e$ of $S$, collapsing all the edges in the orbit of $e$ defines a new tree $S'$ with an action of $G$. We say that $S'$ is a *collapse* of $S$ and we call the natural map $S{\rightarrow}S'$ a *collapse*.
Consider two distinct oriented edges $e_1=uv_1$, $e_2=uv_2$ of $S$ having the same origin $u$; assume that $uv_1$ and $v_2u$ are not in the same orbit (as oriented edges). Identifying $g.e_1$ with $g.e_2$ for every $g\in G$ defines a new tree $S'$ on which $G$ acts (without inversion). We say that $S'$ is obtained by *folding $e_1$ on $e_2$* and we call the natural map $S{\rightarrow}S'$ a *fold*.
Consider a vertex $v\in S$, and $N_v$ a normal subgroup of $G(v)$. Let $N$ be the normal subgroup of $G$ generated by $N_v$ and let $G'=G/N$. The graph $S'=S/N$ is a tree (see fact \[fact\_elliptic\] below) on which $G'$ acts. We say that $S'$ is obtained from $S$ by a *group-fold* and we call the natural map $S{\rightarrow}S'$ a *group-fold*. This map is ${\varphi}$-equivariant where ${\varphi}:G{\rightarrow}G'$ is the quotient map.
#### Decomposition into folds.
The following result is a slight variation on a result by Dunwoody ([@Dun_folding Theorem 2.1]) . For previous results of this nature, see [@BF_complexity; @Stallings_topology].
\[prop\_fold\] Let $(\Phi,F):G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T{\rightarrow}G'{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T'$ be an epimorphism between finitely supported simplicial actions. Assume that $$\forall e\in E(T),\ \forall e'\in E(T'),\quad e'\subset F(e)\Rightarrow G'(e')=\Phi(G(e)) \quad(*)$$
Then we may subdivide $T$ and $T'$ so that there exists a finite sequence of simplicial actions $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T=G_0{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_0,\dots,G_n{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_n=G'{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T'$, some epimorphisms $({\varphi}_i,f_i):G_i{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_i{\rightarrow}G_{i+1}{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_{i+1}$ such that $\Phi={\varphi}_{n-1}\circ\dots\circ {\varphi}_0$, each $({\varphi}_i,f_i)$ satisfies $(*)$, and is either
a collapse
a group-fold
or a fold between two edges $uv,uv'$ of $T_i$ such that $G_i(u)$ injects into $G'$ under $\Phi_{i}$.
We don’t claim that $F=f_{n-1}\circ\dots\circ f_1$.
We may change $F$ so that it is linear in restriction to each edge of $T$. Then we can subdivide $T$ and $T'$ so that $F$ maps each vertex to a vertex and each edge to an edge or a vertex. The new map satisfies $(*)$. Set $T_0=T$, $G_0=G$.
We describe an iterative procedure. We assume that the construction has begun, and that we are given $(\Phi_i,F_i):G_i{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_i{\rightarrow}G'{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T'$ satisfying $(*)$.
Step 1. Assume that $F_i$ maps an edge $e$ of $T_i$ to a point. Let $T_{i+1}$ be the tree obtained by collapsing $e$ and let $G_{i+1}=G_i$, ${\varphi}_i={\mathrm{id}}$ and $\Phi_{i+1}=\Phi_i$. Define $f_i:T_i{\rightarrow}T_{i+1}$ as the collapse map, and $F_{i+1}:T_{i+1}{\rightarrow}T'$ as the map induced by $F_i$. Then return to step 1.
Clearly, $({\mathrm{id}},f_i)$ and $(\Phi_{i+1},F_{i+1})$ satisfy $(*)$. Since $T_{i+1}$ has fewer orbits of edges than $T_i$, we can repeat step 1 until $F_i$ does not collapse any edge of $T_i$. Step 2. Assume that there is a vertex $v\in T_i$ such that the kernel $N_v$ of $\Phi_i{}_{|G_i(v)}:G_i(v){\rightarrow}G'$ is non-trivial. Let $N$ the normal closure of $N_v$ in $G_i$, and consider $G_{i+1}=G_i/N{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_{i+1}=T_i/N$ the action obtained by the corresponding group-fold. Define $f_i:T_i{\rightarrow}T_{i+1}$ and ${\varphi}_i:G_i{\rightarrow}G_{i+1}$ as the quotient maps. Define $F_{i+1}:T_{i+1}{\rightarrow}T'$ and $\Phi_{i+1}:G_{i+1}{\rightarrow}G'$ as the maps induced by $F_i$ and $\Phi_i$. Then return to step 2.
The group fold $({\varphi}_i,f_i)$ automatically satisfies (\*). Moreover, $(\Phi_{i+1},F_{i+1})$ inherits property (\*) from $(\Phi_i,F_i)$. This step will repeated only a finite number of times since step 2 decreases the number of orbits of vertices with $N_v\neq\{1\}$.
Step 3. Assume that there exists two edges $e_1=uv_1$, $e_2=uv_2$ of $T_i$ such that $F_i(e_1)=F_i(e_2)$. Define $T_{i+1}$ as the tree obtained by folding $e_1$ on $e_2$, $f_i:T_i{\rightarrow}T_{i+1}$ as the folding map, and $F_{i+1}:T_{i+1}{\rightarrow}T'$ as the induced map. Define $G_{i+1}=G_i$, ${\varphi}_i={\mathrm{id}}$ and $\Phi_{i+1}=\Phi_i$. Then return to step 2.
Denote by ${\varepsilon}$ the common image of $e_1$ and $e_2$ in $T_{i+1}$. Since at the beginning of step 3, no group fold can be done, $\Phi_i{}_{|G_i(u)}$ is one-to-one. We claim that $e_1$ and $e_2$ cannot be in the same orbit. Indeed, if $e_1=g.e_2$ for some $g\in G$, then $g\in G_i(u)$ (because $T'$ has no inversion) so $\Phi_i(G_i(e_1)){\varsubsetneq}\Phi_i(\langle G_i(e_1),g\rangle)= \Phi_{i+1}(G_{i+1}({\varepsilon}))\subset G'(F_{i}(e_1))$, contradicting $(*)$. It follows that step 3 will be repeated only finitely many times because it decreases the number of orbits of edges, and step 2 does not change it. Since $F_i$ satisfies (\*) and $\Phi_i{}_{|G_i(u)}$ is one-to-one, we get $\Phi_i(G_i(e_1))=\Phi_i(G_i(e_2))$ so $G_i({\varepsilon})=G_i(e_1)$ and $f_i$ satisfies $(*)$. The fact that $F_{i+1}$ inherits $(*)$ is clear.
When step 3 cannot be repeated any more, $F_i$ is an isometry. If follows that any $g\in\ker \Phi_i$ fixes $T_i$ pointwise. Since $\Phi_i{}_{|G_i(u)}$ is one-to-one for every vertex, $\Phi_i$ is an isomorphism.
Proof of Extended Scott’s Lemma
-------------------------------
One can assume that each $S_k$ is minimal. By Proposition \[prop\_fold\], we may assume that each map $f_k$ is a collapse, a group-fold or a fold. Let ${\mathrm{Ell}}(S_k)$ be the subset of $G_k$ consisting of elements fixing a point in $S_k$. The first Betti number of the graph $S_k/G_k$ coincides with the rank of the free group $G_k/\langle{\mathrm{Ell}}(S_k)\rangle$. In particular, this Betti number is non-increasing and we can assume that it is constant.
We work at the level of quotient graph of groups ${\overline{S}}_k=S_k/G_k$ and we denote by $x\mapsto {\overline{x}}$ the quotient map. Consider an oriented edge ${\overline{e}}$ of ${\overline{S}}_k$ with terminal vertex ${\overline{v}}=t({\overline{e}})$. Say that ${\overline{e}}$ carries the symbol $=$ if the edge morphism $i_{{\overline{e}}}:G_{{\overline{e}}}{\rightarrow}G_{{\overline{v}}}$ is onto. Otherwise, we say that ${\overline{e}}$ carries $\neq$. Define $W_k$ as the set of vertices ${\overline{v}}\in{\overline{S}}_k$ such that there is an oriented edge ${\overline{e}}$ with $t({\overline{e}})={\overline{v}}$ carrying $\neq$. At the level of the tree, $W_k$ is the set of orbits of vertices $v\in S_k$ for which there exists an edge $e$ incident on $v$ with $G_k(e){\varsubsetneq}G_k(v)$. Note that $\#W_k$ is invariant under subdivision.
We claim that $\#W_k$ is non-increasing, and that folds and collapses which don’t decrease $\#W_k$ don’t change ${\mathrm{Ell}}(S_k)$.
Assume that $f_k:S_k{\rightarrow}S_{k+1}$ is induced by the collapse of an edge $e=uv$.
Let ${\Bar f}_k:{\overline{S}}_k{\rightarrow}{\overline{S}}_{k+1}$ be the induced map. The endpoints of ${\overline{e}}$ are distinct since otherwise, the first Betti number would decrease. If both ${\overline{u}}$ and ${\overline{v}}$ are outside $W_k$, then the stabilizer of $u$ coincides with that of $f_k(u)$; in particular, ${\Bar f}_k({\overline{u}})\notin W_k$ and $\#W_k$ is non-increasing. If ${\mathrm{Ell}}(S_k)$ increases under the collapse, then both orientations of ${\overline{e}}$ carry $\neq$, in which case both ${\overline{u}}$ and ${\overline{v}}$ belong to $W_k$, so $\#W_{k+1}< \# W_k$.
Now assume that $f_k:S_k{\rightarrow}S_{k+1}$ is the fold of $e_1=uv_1$ with $e_2=uv_2$ (in particular $G_k=G_{k+1}$). Denote by $v'$ (resp $e'$) the common image of $v_1,v_2$ (resp. $e_1,e_2$) in $S_{k+1}$, and $u'$ the image of $u$. Since $f_k$ satisfies $(*)$, $e_1$ and $e_2$ are in distinct orbits and $G_k(e')=G_k(e_1)=G_k(e_2)$. Since the first Betti number of ${\overline{S}}_k$ is constant, ${\overline{v}}_1\neq {\overline{v}}_2$. It may happen that ${\overline{u}}={\overline{v}}_i$ for some $i\in\{1,2\}$. By subdividing $e_1$ and $e_2$ and replacing the original fold by two consecutive folds, we can ignore this case. Thus we assume that ${\overline{u}},{\overline{v}}_1,{\overline{v}}_2$ are distinct. One has $G_k(v')=\langle G_k(v_1),G_k(v_2)\rangle$. It follows that if both ${\overline{v}}_1$ and ${\overline{v}}_2$ lie outside $W_k$, then ${\overline{v}}\notin W_{k+1}$. Since $G_k(u)=G_k(u')$, ${\overline{u}}\in W_k$ if and only if ${\overline{u}}'\in W_{k+1}$. It follows that $\#W_k$ is non-increasing. If ${\mathrm{Ell}}(S_k)$ increases, then $G_k(v)=\langle G_k(v_1),G_k(v_2)\rangle$ is distinct from both $G_k(v_1)$ and $G_k(v_2)$. Therefore, ${\overline{e}}_1$ can’t carry $=$ at ${\overline{v}}_1$, since this would imply $G_k(v_1)=G_k(e_1)=G_k(e_2)\subset G_k(v_2)$. Similarly, ${\overline{e}}_2$ carries $\neq$ at ${\overline{v}}_2$. Therefore, ${\overline{v}}_1,{\overline{v}}_2\in W_k$, and $\#W_{k+1}< \#W_k$. This proves the claim.\
Without loss of generality, we assume that $\#W_k$ is constant. For each $k$, either $f_k$ is a group-fold, or ${\mathrm{Ell}}(S_k)={\mathrm{Ell}}(S_{k+1})$. Let $N_k=\ker {\varphi}_{k-1}\circ\dots\circ{\varphi}_1$. We prove by induction on $k$ that $S_0/N_k$ is a tree and ${\mathrm{Ell}}(S_0/N_k)={\mathrm{Ell}}(S_k)\subset G_k$. We will use the following standard fact:
\[fact\_elliptic\] Let $T$ be a simplicial tree and $N$ a group of isometries. Then $T/N$ is a tree if and only if $N$ is generated by elliptic elements.
If the graph of groups $T/N$ is a tree, then its fundamental group is generated by its vertex groups. If $T/N$ is not a tree, killing its vertex groups gives a non-trivial free group.
Assume that $S_0/N_k$ is a tree. If $f_k$ is not a group-fold, then $G_k=G_{k+1}$, $N_k=N_{k+1}$ and ${\mathrm{Ell}}(S_k)={\mathrm{Ell}}(S_{k+1})$ so we are done. If $f_k$ is a group fold, then $G_{k+1}$ is the quotient of $G_k$ by a normal subgroup $K$ generated by subset of ${\mathrm{Ell}}(S_k)$ and $S_{k+1}=S_k/K$. It follows that $S_0/N_{k+1}=(S_0/N_k)/K$ is a tree. For $g\in G_{k}$, denote by ${\overline{g}}$ its image in $G_{k+1}=G_k/K$. We get: ${\overline{g}}\in {\mathrm{Ell}}(S_k/K)\Leftrightarrow \exists k\in K, gk\in {\mathrm{Ell}}(S_K)$ $\Leftrightarrow \exists k\in K, gk\in {\mathrm{Ell}}(S_0/N_k)$ $\Leftrightarrow {\overline{g}}\in {\mathrm{Ell}}(S_0/N_{k+1})$. The induction follows.
Since $S_0/N_k$ is a tree for all $k$, $N_k$ is generated by elliptic elements, and so is $N=\cup_k N_k$. This way, we get an action of $G=\displaystyle\lim_{{\rightarrow}} G_k=G_0/N$ on the tree $S_0/N$. Assume that this action has a global fix point and argue towards a contradiction. Let $\{g_1,\dots,g_p\}$ be a generating set of $G_0$. There exists $x\in S_0$ and $n_1,\dots,n_p\in N$ such that $g_in_i.x=x$. Choose $k$ large enough so that $n_1,\dots,n_p\in N_k$. Then $G_k$ has a global fix point in $S_0/N_k$. Since ${\mathrm{Ell}}(S_0/N_k)={\mathrm{Ell}}(S_k)$, $G_k$ fixes a point in $S_k$, a contradiction.
Finally, an edge stabilizer in $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}S_0/N$ is the image of an edge stabilizer in $G_0{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}S_0$, which concludes the proof of Extended Scott’s Lemma.
Relative version of Extended Scott’s Lemma
------------------------------------------
\[thm\_scott\_rel\] Consider $(G_k,\{H_1^k,\dots H_p^k\}){\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}S_k$ a sequence of non-trivial actions of finitely generated pairs on simplicial trees. Let $({\varphi}_k,f_k):G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}S_k{\rightarrow}G_{k+1}{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}S_{k+1}$ be epimorphisms mapping $H_i^k$ onto $H_i^{k+1}$. Consider $G=\displaystyle\lim_{{\rightarrow}} G_k$ the inductive limit and $\Phi_k:G_k{\rightarrow}G$ the natural map. Assume that
\[hyp1\] $\forall e\in E(S_k),\forall e'\in E(S_{k+1}),\quad e'\subset f_k(e)\Rightarrow G_{k+1}(e')={\varphi}_k(G_k(e))\quad (*)$
\[hyp2\] $\forall e\in E(S_k),\forall i$, $\Phi_k(H_i^k)\not\subset\Phi_k(e)$
Then the pair $(G,{{\mathcal {H}}})$ has a non-trivial splitting over the image of an edge stabilizer of some $S_k$. Moreover, any subgroup $H\subset G$ fixing a point in some $S_k$ fixes a point in the obtained splitting of $G$.
The additional assumption is necessary as shows the following example. Let $A$ be an unsplittable finitely generated group containing a finitely generated free group $F$. Let $\{a_1,a_2,\dots\}\subset F$ be an infinite basis of a free subgroup of $F$, and $F'=F(a'_1,a'_2,\dots)$ another free group with infinite basis. Consider $$G_k=\left\langle F*F'\ |\ a_1=a'_1, \dots, a_k=a'_k\right\rangle*_F A.$$ The group $G_k$ is finitely generated relative to $F'$. This sequence of splittings (over $F$) satisfies condition $(*)$ but not the additional assumption. The inductive limit of $G_k$ is the unsplittable $A$.
Since an action of a finitely generated pair is finitely supported, we can use the decomposition into folds of Proposition \[prop\_fold\]. The proof of Extended Scott’s Lemma does not use finite generation until the proof of the non-triviality of the obtained splitting. Recall the notations of the end of the proof of Theorem \[thm\_scott\]: $\Phi_k:G_k{\rightarrow}G$ is the natural morphism, $N=\ker \Phi_0:G_0{\rightarrow}G$, $N_k=\ker{\varphi}_{0k}:G_0{\rightarrow}G_k$ so that $N=\bigcup_k N_k$.
At this point, we know that we can forget finitely many terms in our sequence of actions so that $S_0/N$ is a tree, and that for all $k$, the action $G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}S_0/N_k$ is non-trivial. We assume that $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}G_0/N$ has a global fix point and argue towards a contradiction. Let $S^0\subset G_0$ be a finite set such that $S^0\cup H^0_1\cup\dots\cup H_p^0$ generates $G$. Consider $\{g_1,\dots,g_q\}$ such that $\{g_1,\dots,g_q\}\cup H^0_1\cup\dots\cup H_p^0$ generates $G_0$. There exists $a\in S_0$ and $n_1,\dots,n_q\in N$ such that $g_jn_j.a=a$ for all $j\in\{1,\dots,q\}$. Let $b_i\in S_0$ be a fix point of $H_i^0$. By the second hypothesis, $H_i^0$ fixes no edge in $S_0/N$ so it fixes a unique point in $S_0/N$. In particular, the images of $b_i$ and $a$ in $S_0/N$ coincide. Therefore, $b_i=n'_i.a$ for some $n'_i\in N$. Choose $k$ large enough so that $N_k$ contains all those elements $n_j,n'_i$. Then $G_k$ fixes the image of $a$ in $S_0/N_k$. This contradicts the non-triviality of $G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}S_0/N_k$.
Getting piecewise stability {#sec_acc2pw}
===========================
\[thm\_acc2pw\] Consider a minimal action of a finitely generated pair $(G,{{\mathcal {H}}})$ on an ${{\mathbb {R}}}$-tree $T$. Assume that
$T$ satisfies the ascending chain condition;
there exists a finite family of arcs $I_1,\dots,I_p$ such that $I_1\cup\dots\cup I_p$ spans $T$ and such that for any unstable arc $J$ contained in some $I_i$,
$G(J)$ is finitely generated;
$G(J)$ is not a proper subgroup of any conjugate of itself [i. e. ]{}$\forall g\in G$, $G(J)^g\subset G(J)\Rightarrow G(J)^g= G(J)$.
Then either $(G,{{\mathcal {H}}})$ splits over the stabilizer of an unstable arc contained in some $I_i$, or $T$ is piecewise-stable.
By enlarging some peripheral subgroups, we may assume that each $H\in{{\mathcal {H}}}$ is either finitely generated, or is not contained in a finitely generated elliptic subgroup of $G$.
By Proposition \[prop\_strongCV\_rel\], consider $(G_k,{{\mathcal {H}}}_k){\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_k$ a sequence of actions on ${{\mathbb {R}}}$-trees converging strongly to $T$, and such that $T_k$ splits as a graph of actions where each vertex action is either an indecomposable component, or an edge. Denote by ${\varphi}_k:G_k{\rightarrow}G_{k+1}$, $\Phi_k:G_k{\rightarrow}G$, $f_k:T_k{\rightarrow}T_{k+1}$, and $F_k:T_k{\rightarrow}T$ the morphisms of the corresponding direct system. Without loss of generality, we can assume that there exists $\Tilde I_1,\dots, \Tilde I_p\subset T_0$ which map isometrically to $I_1,\dots, I_p$. By subdividing the arcs $\Tilde I_i$ and the edges occurring in the decomposition of $T_k$, we may assume that each $\Tilde I_i$ is either an edge or an indecomposable arc of $T_0$. All the arcs of $T_k$ we are going to consider embed isometrically into $T$. Say that an arc $I\subset T_k$ is *pre-stable* if its image in $T$ (under $F_k$) is a stable arc. Otherwise, we say that $I$ is *pre-unstable*. If $I$ is contained in an indecomposable component, then it is pre-stable by Lemma \[lem\_indec\_stab\] and Assertion 1 of Lemma \[lem\_indecompo\].
To do some bookkeeping among pre-unstable edges, we shall construct inductively a combinatorial graph ${{\mathcal {T}}}={{\mathcal {T}}}(\Tilde I_1){\sqcup}\dots {\sqcup}{{\mathcal {T}}}(\Tilde I_p)$ as a disjoint union of rooted trees. We use standard terminology for rooted trees: the father of $v$ is the neighbour of $v$ closer to the root than $v$, a child of $v$ is a neighbour of $v$ which is not the father of $v$, an ancestor of $v$ if a vertex on the segment joining $v$ to the root. The level of a vertex is its distance to the root. We denote by ${{\mathcal {T}}}_k(\Tilde I_i)$ (resp. ${{\mathcal {T}}}_{\geq k}(\Tilde I_i)$) the set of vertices of ${{\mathcal {T}}}(\Tilde I_i)$ of level $k$ (resp. at least $k$), and ${{\mathcal {T}}}_k={{\mathcal {T}}}_k(\Tilde I_1)\cup\dots\cup{{\mathcal {T}}}_k(\Tilde I_p)$. (resp. ${{\mathcal {T}}}_{\geq k}={{\mathcal {T}}}_{\geq k}(\Tilde I_1)\cup\dots\cup{{\mathcal {T}}}_{\geq k}(\Tilde I_p)$).
Each vertex $v$ of level $k$ of ${{\mathcal {T}}}(\Tilde I_i)$ will be labeled by a pre-unstable edge $J_v$ contained in $f_{0k}(\Tilde I_i)\subset T_k$. We label the root of ${{\mathcal {T}}}(\Tilde I_i)$ by $\Tilde I_i$. Assume that ${{\mathcal {T}}}$ has been constructed up to the level $k$. Subdivide the edge structure of $T_{k+1}$ so that, for each $v\in{{\mathcal {T}}}_k$, $f_k(J_v)$ is a finite union of edges and of indecomposable arcs. We may also assume that the length of each edge is at most $1/2^k$. The indecomposable pieces of $f_k(J_v)$ are pre-stable, and we discard them. We also discard pre-stable edges of $f_k(J_v)$. For each pre-unstable edge $J'$ contained in $f_k(J_v)$, we add new child of $v$ labeled by $J'$.
If ${{\mathcal {T}}}$ is finite, then each $\Tilde I_i$ is contained in a finite union of stable arcs. Since $\Tilde I_1\cup\dots \Tilde I_p$ spans $T$, $T$ is piecewise stable. So we assume that ${{\mathcal {T}}}$ is infinite.
To each vertex $v\in{{\mathcal {T}}}_k$, we attach two subgroups of $G$: $A_v=\Phi_k(G_k(J_v))$, and $B_v=G(F_k(J_v))$. Clearly, $A_v\subset B_v$. Given $u,v\in{{\mathcal {T}}}$, write $B_u< B_{v}$ if $B_{u}$ is properly contained in some conjugate of $B_{v}$. Say that $v\in{{\mathcal {T}}}_k$ is *minimal* if there is no $u\in{{\mathcal {T}}}_k$ with $B_u < B_v$. Let ${{\mathcal {M}}}\subset{{\mathcal {T}}}$ be the set of minimal vertices, and ${{\mathcal {M}}}_k={{\mathcal {M}}}\cap{{\mathcal {T}}}_k$. Since no $B_v$ is a proper conjugate of itself, for each $v$, either $v\in{{\mathcal {M}}}_k$ or there exists $u\in{{\mathcal {M}}}_k$ with $B_u<B_v$.
\[lem\_immortel\] There exists $k_0$ such that for all $k\geq k_0$, the following hold:
\[it\_notmin\] if $v\in {{\mathcal {T}}}_k\setminus{{\mathcal {M}}}_k$, then no child of $v$ is minimal;
\[it\_min\] if $v\in {{\mathcal {M}}}_k$ then for any child $v'\in{{\mathcal {M}}}_{k+1}$ of $v$, $A_v=A_{v'}$;
Say that $v$ is a *clone* of $v'$ if $B_v=B_{v'}$ and $A_v=A_{v'}$. A *genealogical line* is a sequence $v_0,v_1,\dots$ of vertices of ${{\mathcal {T}}}$ where $v_{i+1}$ is a child of $v_i$. Say that $v\in {{\mathcal {T}}}$ is *immortal* if there is a genealogical line starting at $v$ and consisting of clones of $v$.
We claim that any genealogical line $v_0,v_1,\dots$ eventually consists of clones. By the ascending chain condition in $T$, there are at most finitely many indices $i$ such that $B_{v_i}{\varsubsetneq}B_{v_{i+1}}$. So we can assume that $B_{v_i}=B_{v_0}$ for all $i$. Since $J_{v_0}$ is pre-unstable, $B_{v_0}$ is finitely generated. The strong convergence implies that for $k$ large enough, every generator of $B_{v_0}$ has a preimage in $G_{k}(f_{k_0k}(J_{v_0}))$ (where $k_0$ is the level of $v_0$). Since for each $v_i$ of level at least $k$, $J_{v_i}\subset f_{k_0k}(J_{v_0})$, we get $B_{v_0}\subset A_{v_i}\subset B_{v_i}=B_{v_0}$. Therefore, all the vertices $v_i$ of sufficiently large level are clones of each other. This proves the claim. In particular, this also proves that if $v$ is immortal, then $A_v=B_v$.
Say that $v$ is *post-immortal* if it is immortal or if it has an immortal ancestor. We claim that all but finitely many vertices of ${{\mathcal {T}}}$ are post-immortal. Indeed, for each $i\in\{1,\dots,p\}$, the set of vertices of ${{\mathcal {T}}}(\Tilde I_i)$ which are not post immortal is a rooted subtree of ${{\mathcal {T}}}(\Tilde I_i)$. If it is infinite, it contains a genealogical line. Since this line contains many immortal vertices, this is a contradiction.
Let $k_0$ be such that ${{\mathcal {T}}}_{\geq k_0}$ consists of post-immortal vertices. We shall use several times the following fact: any $v\in{{\mathcal {M}}}_{\geq k_0}$ is the clone of an immortal vertex. Indeed, let $u'$ be an immortal ancestor of $v$, and let $u$ be a clone of $u'$ of same level as $v$; then $A_{u'}=B_{u'}\subset A_v\subset B_v$, and since $v$ is minimal, inclusions are equalities, and $u'$ is a clone of $v$. It follows that for all $v\in{{\mathcal {M}}}_{\geq k_0}$, $A_v=B_v$.
If $v\in {{\mathcal {T}}}_k\setminus{{\mathcal {M}}}_k$, there exists $u\in {{\mathcal {M}}}_k$ with $B_u<B_v$. Since $u$ is the clone of an immortal vertex, there exists $u'$ of level $k+1$ with $B_u=B_{u'}$ and no child of $v$ can be minimal. This proves assertion \[it\_notmin\].
If $v\in {{\mathcal {M}}}_k$, and $v'\in {{\mathcal {M}}}_{k+1}$ is a child of $v$, then $v$ has an immortal clone, so has an immortal clone $u'$ of level $k+1$. We have $B_{u'}=A_{u'}=A_v\subset A_{v'}\subset B_{v'}$ and since $v'$ is minimal, $v'$ is a clone of $v$. Assertion \[it\_min\] follows.
We always consider $k\geq k_0$. Let ${{\mathcal {E}}}_k$ be the set of edges of our decomposition of $T_k$ which are in the orbit of some $J_v$ for some $v\in{{\mathcal {M}}}_k$. In our decomposition of $T_k$ as a graph of actions, collapse all the vertex trees which are not in ${{\mathcal {E}}}_k$ (see Definition \[dfn\_collapse\]). The resulting tree $S_k$ is a graph of actions where each vertex action is an edge, so it is a simplicial tree. The action $G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}S_k$ is minimal by Lemma \[lem\_collapse\] and is therefore non-trivial. By assertion \[it\_notmin\] of Lemma \[lem\_immortel\], $f_k$ induces a natural map $S_k{\rightarrow}S_{k+1}$. Since any $H\in {{\mathcal {H}}}_k$ fixes a point in $T_k$, it also does in $S_k$, so the action on $S_k$ is an action of the pair $(G_k,{{\mathcal {H}}}_k)$.
Let’s check that the actions $G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}S_k$ satisfy the hypotheses of Extended Scott’s lemma. Consider an edge $e\in{{\mathcal {E}}}_k$. We may translate $e$ so that $e$ corresponds to an arc $J_v$, with $v\in {{\mathcal {M}}}_k$. Let $e'\in {{\mathcal {E}}}_{k+1}$ with $e'\subset f_k(e)$. Since $e'\in {{\mathcal {E}}}_{k+1}$, it is pre-unstable, so $e'=J_{v'}$ for some child $v'\in{{\mathcal {T}}}$ of $v$. Moreover, since $e'\in{{\mathcal {E}}}_{k+1}$, $e'$ is in the orbit of some $J_{w'}$ with $w'$ minimal, so $v'\in {{\mathcal {M}}}_{k+1}$. By assertion \[it\_min\] of Lemma \[lem\_immortel\], $\Phi_{k+1}(G_{k+1}(e'))=A_{v'}=A_v=\Phi_k(G_k(e))$. Since $\Phi_k$ and $\Phi_{k+1}$ are one-to-one in restriction to arc stabilizers, we get ${\varphi}_k(G_k(e))=G_{k+1}(e')$.
If ${{\mathcal {H}}}={\emptyset}$, this is enough to apply Extended Scott’s Lemma, so $G$ splits over some group $A_v$. Since $A_v=B_v$ is the stabilizer of an unstable arc in $T$, the theorem is proved in the non-relative case.
If ${{\mathcal {H}}}\neq{\emptyset}$, we need to modify slightly the argument to ensure that the second hypothesis of the relative version of Extended Scott’s Lemma holds. Let ${{\mathcal {H}}}'\subset {{\mathcal {H}}}$ be the subset consisting of subgroups which are not finitely generated and ${{\mathcal {H}}}'_k\subset {{\mathcal {H}}}_k$ the subset corresponding to ${{\mathcal {H}}}'$. The pairs $(G,{{\mathcal {H}}}')$ and $(G_k,{{\mathcal {H}}}'_k)$ are finitely generated. Recall that no $H\in {{\mathcal {H}}}'$ is contained in a finitely generated elliptic subgroup. In particular, for all $H\in{{\mathcal {H}}}'_k$, $\Phi_k(H)$ fixes no unstable arc of $T$. For each $e\in E(S_k)$, $\Phi_k(G_k(e))$ fixes an unstable arc in $T$ so Theorem \[thm\_scott\_rel\] applies. We thus get a non-trivial splitting $(G,{{\mathcal {H}}}'){\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}S$.
In view of the moreover part of Theorem \[thm\_scott\_rel\], since any $H\in{{\mathcal {H}}}_0$ fixes a point in $S_0$, any $H\in{{\mathcal {H}}}$ fixes a point in $S$. Thus, $S$ defines a non-trivial splitting of $(G,{{\mathcal {H}}})$.
Piecewise stable actions {#sec_pw2triv}
========================
The goal of this section is the following result:
\[thm\_pw2triv\] Let $(G,{{\mathcal {H}}})$ be a finitely generated pair having a piecewise stable action on an ${{\mathbb {R}}}$-tree $T$.
Then, either $(G,{{\mathcal {H}}})$ splits over the stabilizer $H$ of an infinite tripod (and the normalizer of $H$ contains a non-abelian free group generated by two hyperbolic elements whose axes don’t intersect), or $T$ has a decomposition into a graph of actions where each vertex action is either
simplicial: $G_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$ is a simplicial action on a simplicial tree;
of Seifert type: the vertex action $G_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$ has kernel $N_v$, and the faithful action $G_v/N_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$ is dual to an arational measured foliation on a closed $2$-orbifold with boundary;
axial: $Y_v$ is a line, and the image of $G_v$ in ${\mathop{\mathrm{Isom}}}(Y_v)$ is a finitely generated group acting with dense orbits on $Y_v$.
The proof relies on the following particular case of a result by Sela:
[thm\_sela]{}
Consider a minimal action of a finitely generated group $G$ on an ${{\mathbb {R}}}$-tree $T$ with trivial arc stabilizers.
Then, either $G$ is freely decomposable, or $T$ has a decomposition into a graph of actions where each vertex action is either
of surface type: the vertex action $G_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$ is dual to an arational measured foliation on a closed $2$-orbifold with boundary;
axial: $Y_v$ is a line, and the image of $G_v$ in ${\mathop{\mathrm{Isom}}}(Y_v)$ is a finitely generated group acting with dense orbits on $Y_v$.
We shall give a proof of this result in Appendix \[sec\_Sela\].
From piecewise stability to trivial arc stabilizers
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\[lem\_pw2goa\] Let $G$ be a group having a piecewise stable action on an ${{\mathbb {R}}}$-tree $T$.
Then $T$ has a decomposition into a graph of actions such that, denoting by $N_v$ the kernel of the vertex action $G_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$, then $G_v/N_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$ has trivial arc stabilizers.
The lemma doesn’t assume any finite generation of $G$.
Remember that a subtree $Y\subset T$ is stable if the stabilizer of any arc $J\subset Y$ fixes $Y$. Given a stable arc $I\subset T$, consider $Y_I$ the maximal stable subtree containing $I$. This is a well defined subtree because if two stable subtrees contain $I$, their union is still stable, and an increasing union of stable subtrees is stable. Moreover, $Y_I$ is closed in $T$ because the closure of a stable subtree is stable.
Let $(Y_v)_{v\in V}$ be the family of all maximal stable subtrees of $T$. By piecewise stability, any arc of $T$ is contained in a finite union of them. By maximality, if $Y_u\cap Y_v$ is non-degenerate, then $Y_u=Y_v$. Therefore, the family $(Y_v)_{v\in V}$ is a transverse covering of $T$. Denote by $G_v$ the global stabilizer of $Y_v$, and by $N_v$ its pointwise stabilizer. Since $Y_v$ is a stable subtree, $G_v/N_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$ has trivial arc stabilizers.
Relative version of Sela’s Theorem {#sec_relfg}
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Consider a decomposition of $T$ as a graph of actions as in Lemma \[lem\_pw2goa\]. Consider a vertex group $G_v$ of the corresponding graph of groups $\Gamma$. Even if $G$ is finitely generated, $G_v$ may fail to be finitely generated. However, in this case, the peripheral structure $(G_v,{{\mathcal {H}}}_v)$ of $G_v$ in $\Gamma$ is finitely generated (Lemma \[lem\_supp\_fini\] and Lemma \[lem\_relfg\]), and the action on $T_v$ is an action of the pair $(G_v,{{\mathcal {H}}}_v)$. If we started with a finitely generated pair $(G,{{\mathcal {H}}})$, $G_v$ is finitely generated with respect to a set ${{\mathcal {H}}}_v$ consisting of the peripheral structure of $G_v$ in $\Gamma$ together with some conjugates of elements of ${{\mathcal {H}}}$ (Lemma \[lem\_relfg\_rel\]). We will need a version of Sela’s result applying in this context.
\[prop\_sela\_rel\] Consider a minimal action of a finitely generated pair $(G,{{\mathcal {H}}})$ on an ${{\mathbb {R}}}$-tree $T$ with trivial arc stabilizers.
Then, either $(G,{{\mathcal {H}}})$ is freely decomposable, or $T$ has a decomposition into a graph of actions where each non-degenerate vertex action is either
of surface type: the vertex action $G_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$ is dual to an arational measured foliation on a closed $2$-orbifold with boundary;
axial: $Y_v$ is a line, and the image of $G_v$ in ${\mathop{\mathrm{Isom}}}(Y_v)$ is a finitely generated group acting with dense orbits on $Y_v$.
We shall embed $G$ into a finitely generated group $\Hat G$ and apply Sela’s non-relative result. Let ${{\mathcal {H}}}=\{H_1,\dots,H_p\}$. We may assume that each $H_i$ is non-trivial. For $i=1,\dots,p$, choose some finitely generated group $\Hat H_i$ containing $H_i$. We may assume that $\Hat H_i$ is freely indecomposable by changing $\Hat H_i$ to $\Hat H_i\times {{\mathbb {Z}}}/2{{\mathbb {Z}}}$. Consider the graph of groups $\Gamma$ below.
We denote by $u$ (resp. $u_i$) the vertex labeled by $G$ (resp. $\Hat H_i$). Clearly, the group $\Hat G=\pi_1(\Gamma)$ is finitely generated.
We shall add a structure of a graph of actions on ${{\mathbb {R}}}$-trees on $\Gamma$. Let $Y_u$ be a copy of $T$ endowed with its natural action of $G$, and let $Y_{u_i}$ be a point endowed with the trivial action of $\Hat H_i$. For each edge $uu_i$ of $\Gamma$, we define its attaching point in $Y_u$ as the (unique) point of $T$ fixed by $H_i$. Let $\Hat G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}\Hat T$ be the ${{\mathbb {R}}}$-tree dual to this graph of actions.
Since $T$ is a union of axes of elements of $G$, and since $\Hat T$ is covered by translates of $T$, $\Hat T$ is a union of axes of elements of $\Hat G$ so $\Hat T$ is minimal.
This action has trivial arc stabilizers because $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ does. Therefore, we can apply Sela’s Theorem \[thm\_Sela\] to $\Hat G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}\Hat T$. Assume first that $\Hat G$ is freely decomposable, [i. e. ]{}that $\Hat G$ acts non-trivially on a simplicial tree $R$ with trivial edge stabilizers. Then $\Hat H_i$ is elliptic in $R$ because it is freely indecomposable. If $G$ fixes a point $x\in R$, then each $H_i$ fixes $x$ and cannot fix any other point because edge stabilizers are trivial. It follows that each $\Hat H_i$ fixes $x$ and that $\Hat G$ fixes $x$, a contradiction. Thus, the action of $G$ on $R$ defines a non-trivial free decomposition of $G$ relative to $H_i$.
Assume now that $\Hat T$ has a decomposition into a graph of actions where each vertex action is axial or of surface type. In particular, each vertex action is indecomposable. Let $(Z_i)$ be the transverse covering of $\Hat T$ induced by this decomposition. Since $Z_i$ is indecomposable, if $Z_i\cap T$ is non-degenerate, then $Z_i\subset T$ (Lemma \[lem\_indec\_component\]). The family of subtrees $Z_i$ contained in $T$ is therefore a transverse covering of $T$. This gives a decomposition of $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ as a graph of actions where each vertex action is axial or of surface type.
Proof of Theorem \[thm\_pw2triv\]
---------------------------------
Consider a transverse covering ${{\mathcal {Y}}}=(Y_v)_{v\in V}$ of $T$ as in Lemma \[lem\_pw2goa\]. Denote by $G_v$ the global stabilizer of $Y_v$, and by $N_v{\vartriangleleft}G_v$ the kernel of the vertex action $G_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$. We denote by ${\overline{G}}_v=G_v/N_v$ the group acting with trivial arc stabilizers on $Y_v$.
We shall prove that either $(G,{{\mathcal {H}}})$ splits over a tripod stabilizer with the desired properties, or that each vertex action $G_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$ has a decomposition into a graph of actions of the right kind. The theorem will follow.
If $Y_v$ is a line, then $G_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$ is either simplicial or of axial type, so we can assume that $Y_v$ is not a line.
By Lemma \[lem\_decompo\_minimal\], we can first decompose $Y_v$ into a graph of actions where each vertex action has a global fixed point or a dense minimal subtree. Therefore, we can assume without loss of generality that for each $v$, ${\overline{G}}_v$ has either a global fixed point or a dense minimal subtree in $Y_v$.
If ${\overline{G}}_v$ fixes a point in $Y_v$, then $Y_v$ is a simplicial tree.
Using Lemma \[lem\_support\], consider a finite tree $K$ spanning $Y_v$. Let $x_0\in K$ be a ${\overline{G}}_v$-invariant point. Denote by $K_1,\dots,K_n$ the closure of the connected components of $K\setminus \{x_0\}$. If $K_i,K_j$ are such that there exists $g\in {\overline{G}}_v$ with $g.K_i\cap K_j$ non-degenerate, replace $K_i$ and $K_j$ by $g.K_i\cup K_j$. This way, one can assume that for all $i\neq j$ and all $g\in {\overline{G}}_v$, $g.K_i\cap K_j=\{x_0\}$. Since arc stabilizers are trivial, for all $g\in {\overline{G}}_v\setminus\{1\}$, $g.K_i\cap K_i=\{x_0\}$. Thus, $Y_v$ is the union of translates of $K_1,\dots,K_n$, all glued along $\{x_0\}$. Since each $K_i$ is a finite tree, $Y_v$ is a simplicial tree.
Now we consider the case where ${\overline{G}}_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$ has a dense minimal subtree. Since $Y_v$ is not a line, $Y_v$ contains an infinite tripod. This tripod is fixed by $N_v$. Moreover, consider ${\overline{g}},{\overline{h}}\in {\overline{G}}_v$ two hyperbolic elements having distinct axes. In particular $[{\overline{g}},{\overline{h}}]\neq 1$. Since arc stabilizers are trivial, the axes of ${\overline{g}}$ and ${\overline{h}}$ have compact intersection. Therefore, ${\overline{g}}$ and ${\overline{g}}'={\overline{g}}^{{\overline{h}}^k}$ have disjoint axes for $k$ large enough. Let $g,g'$ be some preimages of ${\overline{g}},{\overline{g}}'$ in $G$. Then $\langle g,g'\rangle$ is a free group generated by two elements having disjoint axes in $T$, and which normalizes $N_v$.
If $Y_v$ is not minimal, then $G$ splits over $N_v$.
Let $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}S$ be the skeleton $S$ of the transverse covering ${{\mathcal {Y}}}$. Since $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ is minimal, $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}S$ is minimal (Lemma \[lem\_supp\_fini\]). Therefore, we need only to prove that there is an edge of $S$ whose stabilizer is $N_v$.
Let $x\in Y_v\setminus \min(Y_v)$. Since $\min(Y_v)$ is dense in $Y_v$ and does not contain $x$, $Y_v\setminus x$ is connected. If $x$ does not lie in any other tree $Y\in{{\mathcal {Y}}}$, then $T\setminus \{x\}$ is convex, so $T\setminus G.x$ is a $G$-invariant subtree, contradicting the minimality of $T$.
Therefore, $x$ is a vertex of $S$ and $(x,Y_v)$ is an edge of $S$ (see section \[sec\_goa\]). Denote by $G(x,Y_v)=\{g\in G_v| g.x=x\}$ its stabilizer. Clearly, $N_v\subset G(x,Y_v)$. Conversely, consider $g\in G(x,Y_v)$ and $y\in Y_v\setminus\{x\}$. Since $y$ and $g.y$ both lie in the convex set $Y_v\setminus\{x\}$, $[x,y]\cap [x,g.y]$ is a non-degenerate arc fixed by $g$. Therefore, $g\in N_v$.
There remains to analyse the case where $Y_v$ is a not a line and $G_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$ is minimal without global fix point. Consider the graph of groups $\Gamma=S/G$. We identify $v$ with a vertex of $\Gamma$. By Lemma \[lem\_relfg\_rel\] (or \[lem\_relfg\] if $G$ is finitely generated), $G_v$ is finitely generated relative to the incident edge groups together with at most one conjugate of each $H\in{{\mathcal {H}}}$. When ${{\mathcal {H}}}\neq {\emptyset}$, we make sure that for any $H\in{{\mathcal {H}}}$ having a conjugate in $G_v$, ${{\mathcal {H}}}_v$ contains a conjugate of $H$. Let ${\overline{{{\mathcal {H}}}}}_v$ be the image of ${{\mathcal {H}}}_v$ in ${\overline{G}}_v=G_v/N_v$. Then we can apply the relative version of Sela’s Theorem (Proposition \[prop\_sela\_rel\]) to $({\overline{G}}_v,{\overline{{{\mathcal {H}}}}}_v){\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$.
Assume first that $({\overline{G}}_v,{\overline{{{\mathcal {H}}}}}_v)$ is freely decomposable. Therefore, $G_v$ splits over $N_v$ relative to ${{\mathcal {H}}}_v$. Thus, we can refine $\Gamma$ at $v$ using this splitting, so $G$ splits over $N_v$. This is really a splitting of $(G,{{\mathcal {H}}})$ because we made sure that for each $H\in{{\mathcal {H}}}$, either $H$ has a conjugate in ${{\mathcal {H}}}_v$, or $H$ is conjugate in some other vertex group of $\Gamma$.
Assume now that ${\overline{G}}_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$ has a decomposition into a graph of actions where each vertex action is either axial or of surface type. Then clearly, the action $G_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$ as a decomposition into a graph of actions where each vertex action is either axial or of Seifert type. This completes the proof of Theorem \[thm\_pw2triv\].
Proof of Main Theorem and corollaries {#sec_proof}
=====================================
Recall that an action of the pair $(G,{{\mathcal {H}}})$ on a tree is an action of $G$ where each $H\in{{\mathcal {H}}}$ fixes a point. In terms of graphs of groups, a splitting of the pair $(G,{{\mathcal {H}}})$ is an isomorphism of $G$ with a graph of groups such that each $H_i$ is contained in a conjugate of a vertex group (see section \[sec\_relfg\]).
\[thm\_main\] Consider a minimal action of finitely generated pair $(G,{{\mathcal {H}}})$ on an ${{\mathbb {R}}}$-tree $T$ by isometries. Assume that
$T$ satisfies the ascending chain condition;
there exists a finite family of arcs $I_1,\dots,I_p$ such that $I_1\cup\dots\cup I_p$ spans $T$ (see Definition \[dfn\_span\]) and such that for any unstable arc $J$ contained in some $I_i$,
$G(J)$ is finitely generated;
$G(J)$ is not a proper subgroup of any conjugate of itself [i. e. ]{}$\forall g\in G$, $G(J)^g\subset G(J)\Rightarrow G(J)^g= G(J)$.
Then either $(G,{{\mathcal {H}}})$ splits over the stabilizer of an unstable arc, or over the stabilizer of an infinite tripod (whose normalizer contains a non-abelian free group generated by two elements having disjoint axes), or $T$ has a decomposition into a graph of actions where each vertex action is either
simplicial: $G_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$ is a simplicial action;
of Seifert type: the vertex action $G_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$ has kernel $N_v$, and the faithful action $G_v/N_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$ is dual to an arational measured foliation on a closed $2$-orbifold with boundary;
axial: $Y_v$ is a line, and the image of $G_v$ in ${\mathop{\mathrm{Isom}}}(Y_v)$ is a finitely generated group acting with dense orbits on $Y_v$.
The group over which $G$ splits ([i. e. ]{}the stabilizer of an unstable arc or of a tripod) is really its full pointwise stabilizer. This contrasts with [@BF_stable Theorem 9.5].
Let $(G,{{\mathcal {H}}}){\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ be as in Main Theorem. By Theorem \[thm\_acc2pw\], if $(G,{{\mathcal {H}}})$ does not split over the stabilizer of an unstable arc, $T$ is piecewise stable. By Theorem \[thm\_pw2triv\], either $(G,{{\mathcal {H}}})$ splits over the stabilizer of an infinite tripod with the required properties, or $T$ has a decomposition into a graph of actions of the desired type.
[corBF]{}
\[cor\_BF\] Under the hypotheses of Main Theorem, either $T$ is a line or $(G,{{\mathcal {H}}})$ splits over a subgroup $H$ which is an extension of a cyclic group (maybe finite or trivial) by an arc stabilizer.
We can assume that $T$ splits as a graph of actions ${{\mathcal {G}}}$ where each vertex action $G_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$ is either simplicial, of Seifert type, or axial. Let $(Y_v)_{v\in V}$ be the family of non-degenerate vertex trees. Let $S$ be the skeleton of this transverse covering (see Lemma \[lem\_transverse\_cov\]).
First, consider the case where $S$ is reduced to a point $v$. This means that $T=Y_v$, and $G=G_v$, and that $T$ is itself simplicial or of Seifert type ($T$ is not a line so cannot be of axial type). If $T$ is simplicial, the result is clear, so assume that $T$ is of Seifert type. Let $N_v{\vartriangleleft}G_v$ be the kernel of this action and let $\Sigma$ be a $2$-orbifold with boundary, with cone singularities, such that $G_v/N_v=\pi_1(\Sigma)$ and holding an arational measured foliation to which $G_v/N_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$ is dual. Consider a splitting of $G_v/N_v$ corresponding to an essential simple closed curve ([i. e. ]{}a curve which cannot be homotoped to a point, a cone point, or to the boundary). Such a curve exists because $\Sigma$ holds an arational measured foliation. This defines a splitting of $\pi_1(\Sigma)$ over a cyclic group, and a splitting of $G=G_v$ over an extension of a cyclic group by $N_v$. Any subgroup $H\subset G$ elliptic in $T$ corresponds to the fundamental group of a cone point or of a boundary component of $\Sigma$. Thus $H$ is elliptic in this splitting and we get a splitting of the pair $(G,{{\mathcal {H}}})$.
Now assume that $S$ is not reduced to a point. By Lemma \[lem\_supp\_fini\], $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}S$ is minimal and any $H\in{{\mathcal {H}}}$ is elliptic in $S$. In particular, given any edge $e$ of $S$, the corresponding splitting of $G$ over $G(e)$ is non-trivial. We shall prove that for some edge $e$ of $S$, $G(e)$ is an extension of an arc stabilizer by a cyclic group, and the corollary will follow.
Assume that some action $G_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v\simeq{{\mathbb {R}}}$ is axial. Consider an edge $e=(x,Y_v)$ of $S$ incident on $v$ (see Lemma \[lem\_transverse\_cov\]). Its stabilizer $G(e)$ is the stabilizer of $x$ in $G_v$. Since the stabilizer of $x$ in $G_v/N_v$ is either trivial or ${{\mathbb {Z}}}/2{{\mathbb {Z}}}$, $G(e)$ is a extension of a cyclic group by $N_v$.
Assume that some action $G_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$ is of Seifert type. Consider an edge $e=(x,Y_v)$ incident on $v$ in $S$. Since $G(e)$ is the stabilizer of $x$ in $G_v$, $G(e)$ is an extension by $N_V$ of the stabilizer of a point in $\pi_1(\Sigma){\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$ which is cyclic. The only remaining case is when each $G_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$ is simplicial. In this case $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ is simplicial, and the result is clear.
[cor\_petit]{}
\[cor\_small\] Let $G$ be a finitely generated group for which any small subgroup is finitely generated. Assume that $G$ acts on an ${{\mathbb {R}}}$-tree $T$ with small arc stabilizers.
Then $G$ splits over the stabilizer of an unstable arc, or over a tripod stabilizer, or $T$ has a decomposition into a graph of actions as in Main Theorem. In particular, $G$ splits over a small subgroup.
We prove that all hypotheses of the Main Theorem are satisfied. The set of small subgroups is closed under increasing union. Since small subgroups are finitely generated, any ascending chain of small subgroups is finite. The ascending chain condition follows. Similarly, if $H{\varsubsetneq}H^g$ for some small subgroup $H\subset G$, then the set of small subgroup $H^{g^n}$ is an infinite increasing chain, a contradiction. Then Main Theorem and corollary \[cor\_BF\] apply.
In the following situation, stabilizers of unstable arcs can be controlled by tripod stabilizers.
[cor\_tripod]{}
\[cor\_tripod\] Consider a finitely generated group $G$ acting by isometries on an ${{\mathbb {R}}}$-tree $T$. Assume that
arc stabilizers have a nilpotent subgroup of bounded index (maybe not finitely generated);
tripod stabilizers are finitely generated (and virtually nilpotent);
no group fixing a tripod is a proper subgroup of any conjugate of itself;
any chain $H_1\subset H_2\dots$ of tripods stabilizers stabilizes.
Then either $G$ splits over a subgroup having a finite index subgroup fixing a tripod, or $T$ has a decomposition as in the conclusion of Main Theorem.\
Let $k$ be a bound on the index of a nilpotent subgroup in an arc stabilizer. We first claim that the stabilizer of an unstable arc has a subgroup of bounded index fixing a tripod. Indeed, let $A$ be the stabilizer of an unstable arc $I$, and let $B\supsetneq A$ be the stabilizer of a sub-arc $J$. If $A$ has index at most $k$ in $B$, then $A$ contains a normal subgroup $A'$ of index at most $k!$. Since for $g\in B\setminus A$, $g.I$ is an arc distinct from $I$ and containing $J$, $B.I$ contains a tripod. This tripod is fixed by $A'$. Suppose that the index of $A$ in $B$ is larger than $k$. Let $N_B$ be a nilpotent subgroup of index at most $k$ in $B$ and $N_A=A\cap N_B$. Because of indices, $N_A{\varsubsetneq}N_B$. Let $N$ be the normalizer of $N_A$ in $N_B$. It is an easy exercise to check that in a nilpotent group, no proper subgroup is its own normalizer so $N_A{\varsubsetneq}N$. Therefore, $N.I$ contains a tripod, and this tripod is fixed by $N_A$.
It follows that the stabilizer of an unstable arc is finitely generated. Since each tripod stabilizer is slender, any ascending chain of subgroups fixing tripods stabilizes. It follows that the stabilizer of an unstable arc cannot be properly conjugated into itself. Consider $A_1{\varsubsetneq}\dots{\varsubsetneq}A_n{\varsubsetneq}$ an ascending chain of arc stabilizers. Let $N_n$ be a subgroup of bounded index of $A_n$ fixing a tripod, and let $N_{i,n}$ be its intersection with $A_i$ for $i\leq n$. Since $N_{i,n}$ has bounded index in $A_i$, it takes finitely many values for each $i$. By a diagonal argument, we get an ascending chain of subgroups fixing tripods, a contradiction. Thus, Main Theorem applies, and the corollary is proved.
An example {#sec_example}
==========
The goal of this section is to give an example of an action with trivial tripod stabilizers providing a counter-example to [@RiSe_structure Theorem 10.8] and [@Sela_acylindrical Theorem 2.3 and 3.1]. Note however that the proof of Theorem 3.1 in [@Sela_acylindrical] is valid under the stronger assumption that $T$ is *super-stable*, which is a natural hypothesis since it is satisfied in applications in [@RiSe_structure; @Sela_acylindrical; @Sela_diophantine1].
There exists a non-trivial minimal action of a group $G$ on an ${{\mathbb {R}}}$-tree $T$ such that
$G$ is finitely presented (and even word hyperbolic) and freely indecomposable
tripod stabilizers are trivial
$T$ satisfies the ascending chain condition
$T$ has no decomposition into a graph of actions as in Main Theorem.
Let $A$ be a freely indecomposable hyperbolic group containing a free malnormal subgroup $M_1=\langle a,b_1\rangle$. For instance, one can take for $A$ a surface group, and for $M_1$, the fundamental group of a punctured torus contained in this surface. Let $A'$ be another copy of the group $A$, $C=\langle a\rangle$, and let $G=A*_C A'$. By Bestvina-Feighn’s Combination Theorem, $G$ is hyperbolic [@BF_combination].
For $i>1$, define inductively $M_i\subset M_{i-1}$ by $M_i=\langle a,b_i\rangle$ where $b_i=b_{i-1}ab_{i-1}^2$. One can easily check that $M_{i}$ is malnormal in $M_{i-1}$ (one can also use the software Magnus to do so [@Magnus_software]). Moreover, one also easily checks that $\cap_{i\geq 1} M_i=C$ since any reduced word $w$ on $\{a,b_i\}$ defines a reduced word on $\{a,b_{i-1}\}$ by the obvious substitution, and its length is strictly larger if $w$ is not a power of $a$.
Let $\Gamma_k$ be the graph of groups and $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_k$ be the corresponding Bass-Serre tree as shown in Figure \[fig\_exple\].
![The graph of groups $\Gamma_k$ and the tree $T_k$\[fig\_exple\]](exemple.eps)
Let $a_k,a'_k$ be the vertices of $T_k$ fixed by $A$ and $A'$ respectively. Denote by $e^k_i\subset[a_k,a'_k]$ the edge of $T_k$ such that $G(e^k_i)=M_i$, and by ${\varepsilon}^k\subset [a_k,a'_k]$ the edge such that $G({\varepsilon}^k)=C$. Figure \[fig\_exple\] shows a neighbourhood of $[a_k,a_k']$ in $T_k$, and the action of the vertex stabilizers. The main feature is that from each vertex $v$ with stabilizer $M_i$, the neighbourhood of $v$ consists of one orbit of edges $M_i.e_{i+1}^k$, and one single edge $e_i^k$.
Assign length $1/2^i$ to the edge $e_i^k$, and length $1/2^k$ to ${\varepsilon}^k$. This way, $d(a_k,a'_k)=1$ for all $k$. There is a natural morphism of ${{\mathbb {R}}}$-trees $f_k:T_k{\rightarrow}T_{k+1}$ sending $e^k_i$ to $e^{k+1}_i$ and ${\varepsilon}^k$ to $e_{k+1}^{k+1}\cup{\varepsilon}^{k+1}$.
The length function of $T_{k+1}$ is not larger than the length function of $T_{k}$, so $T_k$ converges in the length function topology to an action on an ${{\mathbb {R}}}$-tree $T$. This action is non-trivial since for any $g\in A\setminus C$ and $g'\in A'\setminus C$, the translation length of $gg'$ in every $T_k$ is $2$.
We shall prove that $T_k$ converges strongly to $T$. Consider $\alpha,\beta$ two distinct edges of $T_0$ sharing a vertex $v$ ($T_0$ is the tree dual to the amalgam $G=A*_C A'$). If $v$ is in the orbit of $a'_0$, then the image of $\alpha$ and $\beta$ in $T_k$ share only one point. Now assume without loss of generality that $v=a$, and $\alpha={\varepsilon}^0$. Let $g\in A\setminus C$ be such that $g.\alpha=\beta$. Consider $k_0$ the smallest integer such that, $g\notin M_{k_0}$. Then for $k\geq k_0$, the image of $\alpha$ and $\beta$ in $T_k$ share exactly an initial segment of length $1-1/2^{k_0}$. In particular, $f_k$ is one-to-one in restriction to the union of the images of $\alpha$ and $\beta$. Now if $K$ is a finite subtree of $T_0$, it follows that for $k$ large enough, $f_k$ is one-to-one in restriction to the image of $K$ in $T_k$. This proves the strong convergence of $T_k$.
Say that an arc $I\subset T_k$ is *immersed in $\Gamma_k$* if the restriction to $I$ of the quotient map $T_k{\rightarrow}\Gamma_k$ is an immersion. We claim that for any arc $I\subset T_k$ which is not immersed in $\Gamma_k$, $G(I)$ is trivial. It is sufficient to prove it for $I$ of the form $I=\alpha\cup \beta$ where $\alpha,\beta\in E(T_k)$ are incident on a common vertex and are in the same orbit. The triviality of $G(I)=G(\alpha)\cap G(\beta)$ follows easily from the malnormality of $M_1$ in $A$, of $M_{i+1}$ in $M_i$, and of $C$ in $M_k$ and $A'$. Since a tripod cannot be immersed, it follows that tripod stabilizers of $T_k$ are trivial. Going to the limit, tripod stabilizers of $T$ are trivial.
Now we study arc stabilizers of $T$. Denote by $F_k:T_k{\rightarrow}T $ the natural map, and by $a=F_k(a_k)$ and $a'=F_k(a'_k)$. Let $I=[u,v]\subset T$ be a non-degenerate arc with non-trivial stabilizer. By strong convergence, if $g\in G(I)$ then there exists a lift $I_k\subset T_k$ in restriction to which $F_k$ is isometric, and which is fixed by $g$. The argument above shows that the image of $I_k$ in $\Gamma_k$ is immersed. In particular, we can assume that $I_k\subset [a_k,a'_k]$. If $I$ contains $a'$, then since $F_k{^{-1}}(a')=\{a'_k\}$, $I_k$ intersects ${\varepsilon}_k$ in a non-degenerate segment, so $G(I_k)=C$ and $G(I)=C$. If $I$ does not contain $a'$, for $k$ large enough, $I_k$ does not intersect ${\varepsilon}_k$ so $G(I_k)=M_i$ for some $i$ independent of $k$. If follows that $G(I)=M_i$. The ascending chain condition for $T$ follows immediately.
Since the restriction of $F_k$ to the ball of radius $1-1/2^k$ around $a_k$ is an isometry, the segment $[a,a']\subset T$ contains an infinite number of branch points. In particular, $T$ is not a simplicial tree.
We claim that no subtree $Y$ of $T$ is indecomposable. Since $[a,a']$ spans $T$, we may assume that $Y\cap [a,a']$ contains a non-degenerate arc $I$. By indecomposability, the orbit of any point of $I$ is dense in $I$, a contradiction.
Finally, assume that $T$ has a decomposition as in Main Theorem. Since there are no indecomposable subtrees in $T$, $T$ is simplicial, a contradiction.
Sela’s Theorem {#sec_Sela}
==============
[thm\_sela]{}
\[thm\_Sela\] Consider a minimal action of a finitely generated group $G$ on an ${{\mathbb {R}}}$-tree $T$ with trivial arc stabilizers.
Then, either $G$ is freely decomposable, or $T$ has a decomposition into a graph of actions where each non-degenerate vertex action is either
of surface type: the vertex action $G_v{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_v$ is dual to an arational measured foliation on a closed $2$-orbifold with boundary;
axial: $Y_v$ is a line, and the image of $G_v$ in ${\mathop{\mathrm{Isom}}}(Y_v)$ is a finitely generated group acting with dense orbits on $Y_v$.
Preliminaries {#preliminaries}
-------------
The following lemma is essentially contained in [@LP].
\[lem\_cv2\] Let $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ be a minimal action with trivial arc stabilizers.
There exists a sequence of ${{\mathbb {R}}}$-trees $G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_k$ converging strongly to $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ such that $T_k$ is geometric, dual to a foliated 2-complex $G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}X_k$ whose leaf space is Hausdorff, and such that $\Phi_k:G_k{\rightarrow}G$ is one-to-one in restriction to each point stabilizer.
Consider a sequence of actions $G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_k$ dual to foliated $2$-complexes $X_k$ as in Theorem \[thm\_strongCV\]. Let $N_k$ be the kernel of $\Phi_k:G_k{\rightarrow}G$. Let $L_k$ be the subgroup of $N_k$ generated by elements which preserve a leaf of $X_k$. This is a normal subgroup of $G_k$. Let $X'_k=X_k/L_k$ be the quotient foliated $2$-complex, endowed with the natural action of $G'_k=G_k/L_k$. By construction, for each leaf $l$ of $X'_k$, the global stabilizer of $l$ embeds into $G$ under the induced morphism $\Phi'_k:G'_k{\rightarrow}G$.
The natural map $X_k{\rightarrow}X'_k$ is a covering. Since $\pi_1(X_k)$ is generated by free homotopy classes of curves contained in leaves, and since $L_k$ is generated by elements preserving a leaf, $\pi_1(X'_k)$ generated by free homotopy classes of curves contained in leaves. Let $T'_k$ be the leaf space made Hausdorff of $X'_k$. It is an ${{\mathbb {R}}}$-tree by Theorem \[thm\_LP\_leaves\]. Since $L_k\subset N_k$, the map $f_k:T_k{\rightarrow}T$ factors through the natural map $T_k{\rightarrow}T'_k$. It follows that $T'_k$ is geometric, dual to $X'_k$, and that $G'_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T'_k$ converge strongly to $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$.
By Proposition \[prop\_LP\_separation\], there exists a countable union of leaves ${{\mathcal {S}}}$ on the complement of which two points of $X'_k$ are identified in $T'_k$ if and only if they lie on the same leaf. It follows that any element $g\in G'_k$ fixing an arc in $T'_k$ preserves a leaf of $X'_k$. Since arc stabilizers of $T$ are trivial, the image of $g$ in $G$ is trivial, so $g=1$. Thus, arc stabilizers of $G'_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T'_k$ are trivial.
By proposition \[prop\_hausdorff\] below, the leaf space of $X'_k$ is Hausdorff. In particular, a point stabilizer of $T'_k$ coincides with a leaf stabilizer of $X'_k$, which embeds into $G$. The lemma follows.
We will say that $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ is *nice* if
$T$ is geometric, dual to a foliated $2$-complex $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}X$ such that $\pi_1(X)$ is generated by free homotopy classes of curves contained in leaves;
arc stabilizers of $T$ are trivial.
The following result follows from the concatenation of Lemma 3.5 and 3.4 in [@LP]:
\[prop\_hausdorff\] Assume that $T$ is nice.
Then the leaf space of $X$ is Hausdorff: any two points of $X$ identified in $T$ are in the same leaf.
Assume that $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ is nice. Let $Y$ be an indecomposable component of $T$ (as in Proposition \[prop\_decompo\]), and $H$ its global stabilizer.
Then $H{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y$ is nice.
Let $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}X$ be a foliated $2$-complex such that $T_k$ is dual to $X$. Let $C\subset X/G$ be its cut locus (see Definition \[dfn\_cut\]), and $\Tilde C$ its preimage in $X$. The tree $Y$ is dual to the closure ${\overline{U}}$ of a connected component $U$ of $X\setminus \Tilde C$. By Remark \[rem\_decompo\_geom\], $\pi_1({\overline{U}})$ is generated by free homotopy classes of curves contained in leaves. This means that $Y$ is nice.
\[prop\_3types\] Assume that $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ is nice and let $H{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y$ be an indecomposable component. Then one of the following holds:
Axial type: $Y$ is a line, and the image of $H$ in ${\mathop{\mathrm{Isom}}}(Y)$ is a finitely generated group acting with dense orbits on $Y$;
Surface type: $H$ is the fundamental group of a $2$-orbifold with boundary $\Sigma$ holding an arational measured foliation and $Y$ is dual to $\Tilde \Sigma$;
Exotic type: $H$ has a non-trivial free decomposition $H{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}S$ in which any subgroup of $H$ fixing a point in $Y$ fixes a point in $S$.
This proposition is essentially well known: it is a way of describing the output of the Rips machine. This would follow from [@BF_stable] if we knew that $H$ is finitely presented (because of the finiteness hypothesis in [@BF_stable Definition 5.1]). But we can apply some arguments of [@Gui_approximation] where finite presentation is not assumed.
We recall some vocabulary from [@GLP1] or [@Gui_approximation]. A closed multi-interval $D$ is a finite union of compact intervals. A *partial isometry* of $D$ is an isometry between closed sub-intervals of $D$. A *system of isometries* ${{\mathcal {S}}}$ on $D$ is a finite set of partial isometries of $D$. Its *suspension* is the foliated $2$-complex obtained from $D$ by gluing for each partial isometry ${\varphi}:I{\rightarrow}J\in{{\mathcal {S}}}$ by a foliated band $I\times [0,1]$ on $D$ joining $I$ to $J$ whose holonomy is given by ${\varphi}$. A *singleton* is a partial isometry ${\varphi}:I{\rightarrow}J$ where $I$ is reduced to a point. We also call singleton the band corresponding to a singleton in the suspension of a system of isometries. The $\rond{{\mathcal {S}}}$-orbits are the equivalence classes for the equivalence relation generated by $x\sim y$ if $y={\varphi}(x)$ for some non-singleton ${\varphi}:I{\rightarrow}J$ such that $x\in \rond I$. A system of isometries ${{\mathcal {S}}}$ is *minimal* if ${{\mathcal {S}}}$ has no singleton and every $\rond{{\mathcal {S}}}$-orbit of any point in $D\setminus \partial D$ is dense. The system ${{\mathcal {S}}}$ is *simplicial* if every ${{\mathcal {S}}}$-orbit is finite.
The proposition holds if $H{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y$ is dual to a minimal system of isometries: this follows from [@Gui_approximation] section 5 (axial case), proposition 7.2 (exotic case), and section 8 (surface case).
Now $H{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y$ is dual to some foliated $2$-complex $H{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}X$, occurring as a Galois covering of the suspension of some system of isometries ${{\mathcal {S}}}$. Following [@GLP1], [@BF_stable] or [@Gui_approximation Prop 3.1], one can perform a sequence of Rips moves on ${{\mathcal {S}}}$ so that ${{\mathcal {S}}}$ becomes a disjoint union finitely many systems of isometries which are either minimal or simplicial, together with a finite set of singletons joining them. This decomposition of induces a decomposition of $Y$ as a graph of actions as in Proposition \[prop\_decompo\]. Since $Y$ is indecomposable, this decomposition is trivial, and $Y$ is dual to the suspension of a minimal component of ${{\mathcal {S}}}$.
Proof of the theorem
--------------------
Let $G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_k$ be a sequence of geometric actions converging strongly to $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ as in lemma \[lem\_cv2\]. Arc stabilizers of $G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_k$ are trivial. By Proposition \[prop\_decompo\], $T_k$ splits as a graph of actions ${{\mathcal {G}}}$, where each non-degenerate vertex action is either simplicial or indecomposable.
Assume that for all $k$, the simplicial part of $T_k$ is non-empty. We shall prove that $G$ is freely decomposable. Collapse the indecomposable components of $T_k$ as in Definition \[dfn\_collapse\]. Let $S_k$ be the obtained tree. Clearly, $S_k$ is a simplicial tree, and edge stabilizers are trivial because arc stabilizer of $T_k$ are trivial. The action $G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}S_k$ is minimal by Lemma \[lem\_collapse\] and therefore non-trivial. The map $f_k:T_k{\rightarrow}T_{k+1}$ maps an indecomposable component of $T_k$ into an indecomposable component of $T_{k+1}$ so $f_k$ induces a map $S_k{\rightarrow}S_{k+1}$. By Scott’s Lemma, $G$ is freely decomposable.
Now, we assume that for all $k$, $T_k$ splits as a graph of indecomposable actions. Let $S_k$ be the skeleton of the corresponding transverse covering of $T_k$. Recall that its vertex set $V(S)$ is $V_0(S)\cup V_1(S)$ where $V_1(S)={{\mathcal {Y}}}$, and $V_0(S)$ is the set of points $x\in T$ lying in the intersection of two distinct trees of ${{\mathcal {Y}}}$. Since $f_k$ maps an indecomposable tree into an indecomposable tree, $f_k$ induces a map $V_1(S_k){\rightarrow}V_1(S_{k+1})$. Moreover, $f_k$ induces a map $V_0(S_k){\rightarrow}V_0(S_{k+1})\cup V_1(S_{k+1})$: for $x\in V_0(S_k)$, if $f_k(x)$ belongs to two distinct indecomposable components, we map $x$ to $f_k(x)\in V_0(S_{k+1})$, otherwise, we map $x$ to the only indecomposable component containing $f_k(x)$. This map extends to a map $g_k:S_k{\rightarrow}S_{k+1}$ sending an edge to an edge or a vertex. The number of orbits of edges of $S_k$ is non-increasing so for $k$ large enough, no edge of $S_k$ is collapsed by $g_k$, and any pair of edges $e_1,e_2$ identified by $g_k(e_1)$ are in the same orbit. Moreover, using Scott’s Lemma, we can assume that for $k$ large enough, no edge of $S_k$ has trivial stabilizer.
The following lemma will be proved in next sections.
\[lem\_inj\] For $k$ large enough, the following holds. Consider an indecomposable component $H{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y$ of $T_k$, and let $H'{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y'$ be the indecomposable component of $T_{k+1}$ containing $f_k(Y)$. Then $f_k{}_{|Y}$ is an isometry from $Y$ onto $Y'$. Moreover, if $Y$ is not a line, then ${\varphi}_k(H)=H'$.
Using this lemma, we shall prove that for $k$ large enough, $f_k:T_k{\rightarrow}T_{k+1}$ is an isometry. If this is not the case, then there exists two arcs $J_1,J_2\subset T_k$ with $J_1\cap J_2=\{x\}$ and $f_k(J_1)=f_k(J_2)$. By shortening them, we may assume that $J_1$ and $J_2$ lie in some indecomposable components $Y_1$, $Y_2$. By Lemma \[lem\_inj\], $Y_1\neq Y_2$. Therefore, the edges $(x,Y_1)$ and $(x,Y_2)$ of $S_k$ are identified under $g_k$. By the assumption above, they lie in the same orbit. Consider $g\in G_k(x)$ such that $g.Y_1=Y_2$. Let $Y'=g_k(Y_1)=g_k(Y_2)$.
First, assume that $Y_1$ is not a line. Since ${\varphi}_k(g)$ preserves $Y'$, there exists $\Tilde g\in G_k(Y_1)$ with ${\varphi}_k(\Tilde g)={\varphi}_k(g)$ by Lemma \[lem\_inj\]. Since $f_k$ is isometric in restriction to $Y_1$ and since ${\varphi}_k(\Tilde g)={\varphi}_k(g)$ fixes $f_k(x)$, $\Tilde g$ fixes $x$. In particular, $g{^{-1}}\Tilde g$ fixes $x$ and lies in the kernel of ${\varphi}_k$. By Lemma \[lem\_cv2\], ${\varphi}_k$ is one-to-one in restriction to point stabilizers, so $g=\Tilde g$. This is a contradiction because $g \notin G_k(Y_1)$.
Therefore, $Y_1$ is a line. Since ${\varphi}_k$ is one-to-one in restriction to $G_k(x)$, ${\varphi}_k(g)\neq 1$. Since ${\varphi}_k(g)$ preserves the line $Y'$, and since arc stabilizers are trivial, ${\varphi}_k(g)$ acts on $Y'$ as the reflection fixing $f_k(x)$. By the assumption above, edge stabilizers of $S_{k}$ are non-trivial so consider $h\neq 1$ in the stabilizer of the edge $(x,Y_1)$. Since $h$ fixes a point in $T_k$ (namely, $x$), ${\varphi}_k(h)\neq 1$. So ${\varphi}_k(h)$ acts as the same reflection as ${\varphi}_k(g)$ on $Y'$. Since arc stabilizers are trivial, ${\varphi}_k(g)={\varphi}_k(h)$. Since $g,h$ are contained in a point stabilizer, we get $g=h$. This is a contradiction since $g\notin G_k(Y_1)$.
This proves that for $k$ large enough, $f_k$ is an isometry. It follows that ${\varphi}_k$ is an isomorphism because any element of $\ker {\varphi}_k$ must fix $T_k$ pointwise.
Thus, Proposition \[prop\_3types\] gives us a decomposition of $T$. If there is an exotic type vertex action $H{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y$, the free splitting of $H$ given in Proposition \[prop\_3types\] can be used to refine the decomposition of $G$ induced by the skeleton $S_k$, thus giving a decomposition of $G$ as a free product.
Finally, there remains prove the finite generation claimed in the axial case. Let $H{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y$ be an indecomposable component such that $Y$ is a line. In particular, $H$ occurs as a vertex group in the graph of groups given by the decomposition of $T$. By Lemma \[lem\_relfg\], $H$ is finitely generated relative to the stabilizers of incident edges $H_1,\dots, H_p$. Denote by $\psi:H{\rightarrow}{\mathop{\mathrm{Isom}}}(Y)$ the map induced by the action. Since each $H_i$ fixes a point in $Y$, $\psi( H_i)$ is cyclic of order at most 2. Since $\psi(H)$ is finitely generated relative to $\psi(H_1),\dots, \psi( H_p)$, $\psi(H)$ is finitely generated.
Standard form for indecomposable components
-------------------------------------------
We set up the material needed for the proof of Lemma \[lem\_inj\]. Consider $H{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y$ a nice and indecomposable action. Given an arc $I\subset Y$ and a finite generating set $S\subset H$, we construct a foliated $2$-complex $X(I,S)$ as a kind of thickened Cayley graph. Alternatively, one can view $X(I,S)$ as the covering space with deck group $H$ of the suspension of a system of isometries on $I$. The point here is to start with an arbitrary arc $I$ and to allow $S$ to be large in order to ensure that $Y$ is dual to $X(I,S)$.
We first define $X(I,S)$ for any arc $I\subset Y$, and any finite subset $S\subset H$. Start with $H\times I$, endowed with the action of $H$ given by $g.(h,x)=(gh,x)$. For each $s\in S$, consider $K_s=(I\cap s{^{-1}}I)$, and for each $g\in H$, add a foliated band $K_s\times [0,1]$ joining $\{g\}\times K_s$ to $\{gs\}\times s.K_s$ whose holonomy is given by the restriction of $s$ to $K_s$. The map sending $(g,x)$ to $g.x$ extends uniquely to a map $p:X(I,S){\rightarrow}Y$ which is constant on each leaf.
\[lem\_dual\] Consider $H{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y$ a nice indecomposable action. For every arc $I\subset Y$, there exists a finite generating set $S$ of $H$ such that
the leaf space of $X(I,S)$ is Hausdorff;
$Y$ is dual to $X(I,S)$.
Before proving the lemma, we introduce some tools. A *holonomy band* in a foliated $2$-complex $\Sigma$ is a continuous map $b:I\times [0,1]{\rightarrow}\Sigma$ such that $b_{|I\times \{0\}}$ (resp. $b_{|I\times \{1\}}$) is an isometric map to a subset of a transverse edge of $\Sigma$, and for each $x\in I$, $b(\{x\}\times [0,1])$ is contained in a leaf segment, and this leaf segment is regular if $x\notin \partial I$. The following property is a restatement of Theorem 2.3 in [@GLP2].
Let $\Sigma$ be a compact foliated $2$-complex. Consider two arcs $I,J\subset \Sigma^{(1)}$ and an isometry ${\varphi}:I{\rightarrow}J$ such that for all but countably many $x\in I$, $x$ and ${\varphi}(x)$ are in the same leaf.
Then there exist a subdivision $I=I_1\cup \dots\cup I_p$, and for each $i\in\{1,\dots,p\}$ a holonomy band $b_i$ joining $I_i$ to ${\varphi}(I_i)$ whose holonomy is the restriction of ${\varphi}$.
\[cor\_bands\] Assume that $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ is nice, dual to a foliated $2$-complex $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}X$.
Consider two arcs $I,J$ contained in transverse edges of $X$ and having the same image in $T$. Then there exist subdivisions $I=I_1\cup \dots\cup I_p$, $J=J_1\cup \dots\cup J_p$, such that for each $i\in\{1,\dots,p\}$ there exists a holonomy band joining $I_i$ to $J_i$.
Let ${\varphi}:I{\rightarrow}J$ be an isometry such that for all $x\in I$, $x$ and ${\varphi}(x)$ map to the same point in $T$. By Proposition \[prop\_LP\_separation\], for all but countably many $x\in I$, $x$ lies in the same leaf as ${\varphi}(x)$. Apply segment closed property to the images of $I,J$ in the compact foliated $2$-complex $X/G$. By lifting the obtained holonomy bands to $X$, we obtain a subdivision $I=I_1\cup \dots\cup I_p$ and for each $i\in\{1,\dots,p\}$ a holonomy band $b_i$ joining $I_i$ to $g_i.{\varphi}(I_i)\subset g_i.J$ for some $g_i\in G$. Since $x$ and ${\varphi}(x)$ map to the same point in $T$, $g_i$ fixes the arc $I_i$, so $g_i=1$.
Consider a foliated $2$-complex $H{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}X$ such that $Y$ is dual to $X$. Let $\pi:X{\rightarrow}X/H$ be the covering map. Consider the cut locus $C\subset X/H$, $\Tilde C$ its preimage in $X$. The set $U=X\setminus \Tilde C$ is connected: otherwise, $Y$ would split as a graph of actions with at least two non-degenerate vertex trees by Proposition \[prop\_decompo\]; this is impossible because $Y$ is indecomposable (Lemma \[lem\_indec\_component\]). Let $q:X{\rightarrow}Y$ the quotient map ($Y$ is the leaf space of $X$).
Recall that ${X^{(t)}}\subset X^{(1)}$ denotes the union of all closed transverse edges of $X$. We aim to construct a collection of partial isometries $\Phi$ from ${X^{(t)}}$ to $H\times I$, and use $\Phi$ to transport the holonomy of the triangles of $X$ to bands on $H\times I$.
Consider a finite set of arcs $J_1,\dots, J_p$ contained in edges of ${X^{(t)}}$ such that $H.(J_1\cup\dots\cup J_p)={X^{(t)}}$ and $q(J_1)\cup\dots \cup q(J_p)\supset I$. By subdividing each $J_i$, one may assume that either $q(J_i)\subset I$, or $q(J_i)\cap I$ is degenerate. Since $Y$ is indecomposable, $I$ spans $Y$, so up to further subdivision of each $J_i$, one may assume that for each $i\in\{1,\dots,p\}$ there exists $g_i\in H$ such that $q(g_i.J_i)\subset I$. By replacing $J_i$ by $g_i.J_i$ for all $i$ such that $q(J_i)\cap I$ is degenerate, we get $q(J_1)\cup\dots \cup q(J_p)=I$.
Let ${\varphi}_i:J_i{\rightarrow}q(J_i)$ be the partial isometry defined as the restriction of $q$ to $J_i$. We view ${\varphi}_i$ as a partial isometry whose domain of definition is ${\mathop{\mathrm{dom}}}{\varphi}_i=J_i$ and whose range is ${\mathop{\mathrm{Im}}}{\varphi}_i=\{1\}\times q(J_i)\subset X(I,S)$. For $g\in H$, we consider the $g$-conjugate of ${\varphi}_i$ defined by $${\varphi}_i^g=g \circ {\varphi}_i\circ g{^{-1}}:g.J_i{\rightarrow}\{g\}\times I.$$ Let $\Phi=\{{\varphi}_i^g|g\in H,i=1,\dots,p\}$. By construction, $\bigcup_{{\varphi}\in\Phi}{\mathop{\mathrm{dom}}}\Phi={X^{(t)}}$, and $\bigcup_{{\varphi}\in\Phi}{\mathop{\mathrm{Im}}}\Phi=H\times I$.
There is a kind of commutative diagram $$\xymatrix@1@R=0.5cm{
{X^{(t)}}\ar[rr]^-{\Phi}\ar[rd]_q&& H\times I\ar[dl]^p\\
&Y&
}$$ meaning that for each $x\in {X^{(t)}}$, and any ${\varphi}\in\Phi$ defined on $x$, $q(x)=p({\varphi}(x))$.
We now build a foliated complex $X'$ by gluing foliated bands on $H\times I$. For each pair of arcs $g_1.J_{i_1}, g_2.J_{i_2}\subset {X^{(t)}}$ whose intersection $K$ is non-empty, we add a foliated band joining ${\varphi}_{i_1}^{g_1}(K)$ to ${\varphi}_{i_2}^{g_2}(K)$ with holonomy ${\varphi}_{i_2}^{g_2}\circ ({\varphi}_{i_1}^{g_1}){^{-1}}$. For each triangle $\tau$ of $X$, and for each pair of arcs $g_1.J_{i_1}, g_2.J_{i_2}\subset \partial \tau$ such that the holonomy along the leaves of $\tau$ defines a partial isometry $\psi:K_1\subset g_1.I_1{\rightarrow}K_2\subset g_2.I_2$, we add a foliated band joining ${\varphi}_{i_1}^{g_1}(K_1)$ to ${\varphi}_{i_2}^{g_2}(K_2)$, with holonomy ${\varphi}_{i_2}^{g_2}\circ \psi\circ ({\varphi}_{i_1}^{g_1}){^{-1}}$.
The important property is that $\Phi$ maps leaves to leaves in the following sense: if $x,y\in {X^{(t)}}$ are in the same leaf, and if ${\varphi},\psi\in \Phi$ are defined on $x$ and $y$ respectively, then ${\varphi}(x)$ and $\psi(y)$ are in the same leaf of $X'$.
Since $X$ is locally finite, so is $X'$. Therefore, the natural free, properly discontinuous action of $H$ on $X'$ is cocompact. Let $S\subset H$ be the finite set of elements $s\in H$ such that there is a band connecting $(1,I)$ to $(s,I)$. Then $X'$ is naturally contained in $X(I,S)$. And $\Phi$, viewed as a collection of maps ${X^{(t)}}{\rightarrow}X(I,S)$, still maps leaves to leaves.
Let $T(I,S)$ be the leaf space made Hausdorff of $X(I,S)$ and $\Tilde p:X(I,S){\rightarrow}T(I,S)$ be the quotient map. The map $p:H\times I{\rightarrow}Y$ being constant on any leaf of $X(I,S)$ (or rather on its intersection with $H\times I$), and isometric in restriction to any connected component of $H\times I$, it induces a natural distance decreasing map $f:T(I,S){\rightarrow}Y$. $$\xymatrix@1@R=0.8cm{
{X^{(t)}}\ar[r]^-{\Phi}\ar[d]_q& H\times I\ar[dl]_p \ar[d]^-{\Tilde p} \\
Y\ar@/_3mm/@{.>}[r]_-g &T(I,S)\ar[l]_f
}$$
Since $\Phi$ maps leaves to leaves, and since $Y$ is the set of leaves of ${X^{(t)}}$ (it is Hausdorff by Proposition \[prop\_hausdorff\]), $\Phi$ induces a map $g:Y{\rightarrow}T(I,S)$. This map is distance decreasing because any path in $X$ written as a concatenation of leaf segments and of transverse arcs in ${X^{(t)}}$ defines a path having the same transverse measure in $X(I,S)$ as follows: subdivide its transverse pieces so that they can be mapped to $X(I,S)$ using $\Phi$, and join the obtained paths by leaf segments of $X(I,S)$. Since $f$ and $g$ are distance decreasing, both are isometries. In particular $Y$ is dual to $X(I,S)$.
We prove that the leaf space of $X(I,S)$ is Hausdorff. Assume that $x,y\in H\times I$ have the image under $p$. Consider $\Tilde x,\Tilde y\in X$ and ${\varphi},\psi\in\Phi$ such that ${\varphi}(\Tilde x)=x$ and $\psi(\Tilde y)=y$. Then $q(\Tilde x)=q(\Tilde y)$, so $\Tilde x$ and $\Tilde y$ are in the same leaf because the leaf space of $X$ is Hausdorff. Therefore, $x$ and $y$ are in the same leaf of $X(I,S)$.
If the lemma holds for some $S\subset H$, it also holds for any $S'$ containing $S$.
Pseudo-groups
-------------
Up to now we considered partial isometries between closed intervals. We now need to consider partial isometries between *open* intervals. We use notations like $\rond D,\rond {\varphi}$ to emphasize this point. An open *multi-interval* $\rond D$ is a finite union of copies of bounded open intervals of ${{\mathbb {R}}}$. The *pseudo-group of isometries* generated by some partial isometries $\rond {\varphi}_1,\dots,\rond {\varphi}_n$ between open intervals of $D$, is the set of partial isometries $\rond{\varphi}:\rond I{\rightarrow}\rond J$ such that for any $x\in \rond I$, there exists a composition $\rond{\varphi}_{i_1}^{\pm 1}\circ\dots\circ \rond{\varphi}_{i_k}^{\pm 1}$ which is defined on a neighbourhood of $x$, and which coincides with $\rond{\varphi}$ on this neighbourhood. Two points $x,y\in\rond{D}$ are in the same orbit if there exists $\rond{\varphi}\in\Lambda$ such that $y=\rond{\varphi}(x)$. A pseudo-group is *minimal* if its orbits are dense in $\rond D$.
Consider an arc $I\subset Y$, where $H{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y$ is an indecomposable nice action. Let $\rond I=I\setminus \partial I$ and $\rond{\varphi}_s$ be the restriction of $s$ to the interior of $s{^{-1}}.\rond I\cap \rond I$. Let $\Lambda_0$ be the pseudo-group of isometries on $\rond{I}$ generated $\{\rond{\varphi}_s|s\in S\}$. Let $\rond D\subset \rond{I}$ be the set of points whose orbit under $\Lambda_0$ is infinite.
Let $\Lambda(I,S)$ be the restriction of $\Lambda_0$ to $\rond D$. We claim that $\rond I\setminus \rond D$ is finite (in particular, $\rond D$ is an open multi-interval) and that $\Lambda(I,S)$ is minimal. Let $C$ be the cut locus of $\Sigma=X(I,S)/H$. Since $Y$ is indecomposable, $\Sigma\setminus C$ consists of only one minimal component, so every leaf $l$ of $\Sigma\setminus C$ is dense in $\Sigma\setminus C$. Since for any such leaf $l$, $l\cap \rond I$ is contained in a $\Lambda_0$-orbit, the claim follows.
A pseudo-group of isometries $\Lambda$ is *homogeneous* if for each $\rond{\varphi}\in\Lambda$, any partial isometry $\rond\psi$ extending $\rond{\varphi}$ lies in $\Lambda$. When $\Lambda(I,S)$ is homogeneous, $Y$ is a line (see for instance [@Gui_approximation section 5]).
\[thm\_minimax\] Let $\Lambda$ be a minimal finitely generated pseudo-group of isometries of an open multi-interval $\rond D$. Then the set of non-homogeneous pseudo-groups of isometries containing $\Lambda$ is finite.
In [@Lev_pseudo], the result is proved for orientable pseudogroups of isometries of the circle. Theorem \[thm\_minimax\] is an easy generalisation, proved in [@Gui_these], but is not published. The proof follows step by step the proof in [@Lev_pseudo], using Gusmão’s extension of Levitt’s results ([@Gus_feuilletages]). If $\Lambda$ is non-orientable, one can deduce Theorem \[thm\_minimax\] from the orientable case by a straightforward 2-fold covering argument. The only remaining unpublished case is when $\Lambda$ is orientable, and we allow larger pseudo-groups to be non-orientable, but this case is not needed in our argument.
Stabilisation of indecomposable components
------------------------------------------
We are now ready to prove the stabilisation of indecomposable components.
Let $G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_k$ be a sequence of geometric actions converging strongly to $G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ as in lemma \[lem\_cv2\]. Recall that $({\varphi}_k,f_k):G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_k{\rightarrow}G_{k+1}{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_{k+1}$ and $(\Phi_k,F_k):G_k{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T_k{\rightarrow}G{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}T$ denote the maps of the corresponding direct system.
Let $H_{k_0}{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y_{k_0}$ be an indecomposable component of $T_{k_0}$. Consider $k\geq k_0$. Since $f_{{k_0}k}(Y_{k_0})$ is indecomposable, it is contained in an indecomposable component $Y_k$ of $T_k$ (see Lemma \[lem\_indec\_component\] and \[lem\_indecompo\]). Let $I_{k_0}\subset Y_{k_0}$ be an arc which embeds into $T$ under $F_{k_0}$, and let $I_k=f_{{k_0}k}(I_{k_0})$.
By Lemma \[lem\_dual\], $Y_k$ is dual to $X(I_k,S_k)$ for some finite set $S_k\subset H_k$. By enlarging each $S_k$, we can assume that for all $k$, ${\varphi}_k(S_k)\subset S_{k+1}$. Let $\Lambda(I_k,S_k)$ be the corresponding minimal pseudo-group of isometries on $\rond D_k\subset \rond I_k$. Under the natural identification between $I_k$ and $I_{k+1}$, we get $\rond D_{k}\subset \rond D_{k+1}$, and $\Lambda(I_{k},S_{k})\subset \Lambda(I_{k+1},S_{k+1})$. Since $I_k\setminus \rond D_k$ is finite, for $k$ large enough, $\rond D_{k+1}=\rond D_k$.
By Theorem \[thm\_minimax\] above, for $k$ large enough, either $\Lambda(I_k,S_k)$ is homogeneous, or $\Lambda(I_k,S_k)=\Lambda(I_{k+1},S_{k+1})$. It is an exercise to show that if $\Lambda$ is a minimal homogeneous pseudo-group of isometries, then any pseudo-group of isometries containing $\Lambda$ is homogeneous. Thus, in the first case, $\Lambda(I_{k+1},S_{k+1})$ is homogeneous. It follows that $Y_k$ and $Y_{k+1}$ are lines. Since $H_k$ acts with dense orbits on $Y_k$, the morphism of ${{\mathbb {R}}}$-trees $f_{k}{}_{|Y_k}:Y_k{\rightarrow}Y_{k+1}$ is necessarily one-to-one: there exists an arc of $Y_k$ which is embedded under $f_{k}$ and using the action of $H_k$, we see that $f_k$ is locally isometric. It follows that $f_k(Y_k)=Y_{k+1}$.
In the second case, the following result concludes the proof.
Assume that $({\varphi},f):H{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y{\rightarrow}H'{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y'$ is a morphism between nice indecomposable actions. Consider $I\subset Y$ such that $f$ is isometric in restriction to $I$ and let $I'=f(I)$. Consider finite subsets $S\subset H$ and $S'\subset H'$ such that ${\varphi}(S)\subset S'$.
Assume that $Y$ is dual to $X(I,S)$, that $Y'$ is dual to $X(I',S')$, and that $\Lambda(I,S)=\Lambda(I',S')$.
Then $f:Y{\rightarrow}Y'$ is a surjective isometry and ${\varphi}:H{\rightarrow}H'$ is an isomorphism.
![The pseudo-group determines the action[]{data-label="fig_121"}](pseudogroupe.eps)
Assume that $f$ is not one-to-one. There exists two arcs $[x,a_1],[x,a_2]\subset Y$ such that $[x,a_1]\cap [x,a_2]=\{x\}$ and $f([x,a_1])=f([x,a_2])$. Since $I$ spans $Y$, one may shorten these arcs and assume without loss of generality that $[x,a_1]\subset I$ and $[x,a_2]\subset h.I$ for some $h\in H$.
Consider $a'=f(a_1)=f(a_2)$, $x'=f(x)$ and $h'={\varphi}(h)$ (see Figure \[fig\_121\]). The two points $\Tilde a'=h'{}{^{-1}}a'$ and $\Tilde x'=h'{}{^{-1}}x'$ lie in $I'$. The subsets $\{1\}\times [x',a']$ and $\{h'\}\times \Tilde [\tilde x',\Tilde a']$ of $X(I',S')$ map to the same arc in $Y'$. By Corollary \[cor\_bands\], there exist arcs $[x',b']\subset [x',a']$, $[\Tilde x',\Tilde b']\subset [\Tilde x',\Tilde a']$, and a holonomy band in $X(I',S')$ joining $\{1\}\times [x',b']$ to $\{h'\}\times [\Tilde x',\Tilde b']$. This holonomy band defines a partial isometry $\rond\psi{}'\in\Lambda(I',S')$ such that $\rond\psi{}'((x',b'))=(\Tilde x',\Tilde b')$. By hypothesis, $\rond\psi{}'$ corresponds to an element of $\rond\psi\in\Lambda(I,S)$, sending $(x,b)$ to $(\Tilde x,\Tilde b)$ where $b,\Tilde x,\Tilde b$ are the points of $I$ corresponding to $b',\Tilde x',\Tilde b'$ under the natural identification.
We now apply segment closed property to the partial isometry induced by $\rond\psi$ in $X(I,S)/H$. We get the existence of arcs $[x,c]\subset [x,b]$ and $[\Tilde x,\Tilde c]\subset [\Tilde x,\Tilde b]$ and of a holonomy band in $X(I,S)/H$ whose holonomy coincides with $\rond\psi_{|(x,c)}$. Lifting this holonomy band to $X(I,S)$, we get a holonomy band joining $\{1\}\times [x,c]$ to $\{\Tilde h\}\times [\Tilde x,\Tilde c]$ for some $\Tilde h\in H$. Since $S'\supset {\varphi}(S)$, this holonomy band defines a holonomy band in $X(I',S')$ joining $\{1\}\times [x',c']$ to $\{{\varphi}(\Tilde h)\}\times [\Tilde x',\Tilde c']$ where $\Tilde c'=f(\Tilde c)$. In particular, ${\varphi}(\Tilde hh{^{-1}})$ fixes the arc $[\Tilde x',\Tilde c']$, so ${\varphi}(\Tilde hh{^{-1}})=1$. But since $\{\Tilde h\}\times\{ \Tilde x\}$ is in the same leaf as $\{h\}\times \{\Tilde x\}$, $\Tilde hh{^{-1}}$ fixes the point $x$ in $Y$. Since ${\varphi}$ is one-to-one in restriction to point stabilizers of $Y$, $h=\Tilde h$. Since $\{h\}\times [\Tilde x,\Tilde c]$ maps into $[x,a_2]$ in $Y$, we get that $[x,a_2]\cap [x,a_1]$ contains $[x,c]$, a contradiction.
This proves that $f$ is one-to-one. It follows that ${\varphi}$ is one-to one since an element of its kernel must fix $Y$ pointwise.
Let’s prove that ${\varphi}(H)\supset H'$, the other inclusion being trivial. The fact that $f(Y)=Y'$ will follow because $H'{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y'$ is minimal. Since $H'{\,\raisebox{1.8ex}[0pt][0pt]{\begin{turn}{-90}\ensuremath{\circlearrowright}\end{turn}}\,}Y$ is indecomposable, $H'$ is generated by the set of elements $g\in H'$ such that $g.I'\cap I'$ is non-degenerate. Let $h'\in H'$ be such that $[x',a']=h'.I'\cap I'$ is non-degenerate. Repeating the argument above, there exists an arc $[x',b']\subset [x',a']$ and $\rond\psi{}'\in\Lambda(I',S')$, $\rond\psi{}':(x',b'){\rightarrow}(\Tilde x',\Tilde b')$ where $\Tilde x'=h{^{-1}}x'$, $\Tilde b'=h{^{-1}}b'$. Let $\rond\psi\in\Lambda(I,S)$ be the corresponding partial isometry. By segment closed property, there exist arcs $[x,c]\subset [x,b]$ and $[\Tilde x,\Tilde c]\subset[\Tilde x,\Tilde b]$ and a holonomy band in $X(I,S)/H$ joining them in $X(I,S)/H$, whose holonomy coincides with $\rond\psi$. This band lifts in $X(I,S)$, to a band joining $\{1\}\times [x,c]$ to $\{\Tilde h\}\times [\Tilde x,\Tilde c]$ for some $\Tilde h\in H$. In $X(I',S')$, this bands joins the corresponding arcs $\{1\}\times [x,c]$ to $\{\Tilde h\}\times [\Tilde x,\Tilde c]$. It follows that in $Y$, the actions of ${\varphi}(\Tilde h)$ and $h$ coincide on $[x',c']$. Since arc stabilizers are trivial, $h={\varphi}(\Tilde h)$. We conclude that ${\varphi}(H)=H'$.
Vincent Guirardel\
Institut de Mathématiques de Toulouse, UMR 5219\
Université Paul Sabatier\
31062 Toulouse cedex 9.\
France.\
*e-mail:*`guirardel@math.ups-tlse.fr`\
|
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abstract: 'We study evolutionary games in a spatial diluted grid environment in which agents strategically interact locally but can also opportunistically move to other positions within a given migration radius. Using the imitation of the best rule for strategy revision, it is shown that cooperation may evolve and be stable in the Prisoner’s Dilemma game space for several migration distances but only for small game interaction radius while the Stag Hunt class of games become fully cooperative. We also show that only a few trials are needed for cooperation to evolve, i.e. searching costs are not an issue. When the stochastic Fermi strategy update protocol is used cooperation cannot evolve in the Prisoner’s Dilemma if the selection intensity is high in spite of opportunistic migration. However, when imitation becomes more random, fully or partially cooperative states are reached in all games for all migration distances tested and for short to intermediate interaction radii.'
author:
- Pierre Buesser
- Marco Tomassini
- Alberto Antonioni
title: Opportunistic Migration in Spatial Evolutionary Games
---
=1
Introduction {#intro}
============
Spatially embedded systems are very important in biological and social sciences since most interactions among living beings or artificial actors take place in physical two- or three-dimensional space [@Geometry]. Along these lines, game-theoretical interactions among spatially embedded agents distributed according to a fixed structure in the plane have been studied in detail, starting from the pioneering works of Axelrod [@axe84] and Nowak and May [@nowakmay92]. The related literature is very large; see, for instance, the review article by Nowak and Sigmund [@nowak-sig-00] and references therein for a synthesis. Most of this work was based on populations of agents arranged according to planar regular grids for mathematical simplicity and ease of numerical simulation. Recently, some extensions to more general spatial networks have been discussed in [@buesser-tom-space]. The study of strategic behavior on fixed spatial structures is necessary in order to understand the basic mechanisms that may lead to socially efficient global outcomes such as cooperation and coordination. However, in the majority of real situations both in biology and in human societies, actors have the possibility to move around in space. Many examples can be found in biological and ecological sciences, in human populations, and in engineered systems such as ad hoc networks of mobile communicating devices or robot teams. Mobility may have positive or negative effects on cooperation, depending on several factors. An early investigation was carried out by Enquist and Leimar [@Enquist] who concluded that mobility may seriously restrict the evolution of cooperation. In the last decade there have been several new studies of the influence of mobility on the behavior of various games in spatial environments representing essentially two strands of research: one in which the movement of agents is seen as a random walk, and a second one in which movement may contain random elements but it is purposeful, or strategy-driven.
Random diffusion of mobile agents through space, either in continuous space or, more commonly, on diluted grids has been investigated in [@Meloni; @Arenzon1; @Arenzon2]. In the present study we focus on situations where, instead of randomly diffusing, agents possess some basic cognitive abilities and they actively seek to improve their situation by moving in space represented as a discrete grid in which part of the available sites are empty and can thus be the target of the displacement. This approach has been followed, for example, in [@helbingPNAS; @Helb-Mobil; @adaptive-mig; @droz-09; @chen-perc; @reput-migr; @SwarmPD; @aktipis]. The mechanisms invoked range from success-driven migration [@helbingPNAS], adaptive migration [@adaptive-mig], reputation-based migration [@reput-migr], risk-based migration [@chen-perc], flocking behavior [@SwarmPD], and cooperators walking away from defectors [@aktipis]. In spite of the difference among the proposed models, the general qualitative message of this work is that purposeful contingent movement may lead to highly cooperating stable or quasi-stable population states if some conditions are satisfied.
Our approach is based on numerical simulation and is inspired by the work of Helbing and Yu [@Helb-Mobil; @helbingPNAS] which they call “success-driven migration” and which has been shown to be able to produce highly cooperative states. In this model, locally interacting agents playing either defection or cooperation in a two-person Prisoner’s Dilemma are initially randomly distributed on a grid such that there are empty grid points. Agents update their strategies according to their own payoff and the payoff earned by their first neighbours but they can also “explore” an extended square neighborhood by testing all the empty positions up to a given distance. If the player finds that it would be more profitable to move to one of these positions then she does it, choosing the best one among those tested, otherwise she stays at her current place. Helbing and Yu find that robust cooperation states may be reached by this mechanism, even in the presence of random noise in the form of random strategy mutations and random agent relocation. Our study builds upon this work in several ways. In the first place, whilst Helbing and Yu had a single game neighborhood and migration neighborhood, we systematically investigate these two parameters showing that only some combination do foster cooperation using success-driven migration. Secondly, cost issues are not taken into account in [@helbingPNAS]. However, it is clear that moving around to test the ground is a costly activity. In a biological setting, this could mean using up energy coming from metabolic activity, and this energy could be in short supply. In a human society setting, it is the search time that could be limited in a way or another. Additionally to physical energy, cognitive abilities could also limit the search. We present results for a whole game phase space including the Hawk-Dove class of games, and the Stag Hunt coordination class. Helbing’s and Yu’s agents based their strategy change on the imitation of the most successful neighbour in terms of accumulated payoff. We kept this rule but also added the Fermi strategy-updating rule, a choice that allows us to introduce a parametrized amount of imitation noise. With the imitiation of the best policy we find that cooperation prevails in the Stag Hunt and may evolve in the Prisoner’s Dilemma for small interaction radius. With the Fermi rule fully cooperative states are reached for the standard neighborhoods independently of the migration distances when the rate of random strategy imitation is high enough.
Methods
=======
The Games Studied
-----------------
We investigate three classical two-person, two-strategy, symmetric games classes, namely the Prisoner’s Dilemma (PD), the Hawk-Dove Game (HD), and the Stag Hunt (SH). These three games are simple metaphors for different kinds of dilemmas that arise when individual and social interests collide. The Harmony game (H) is included for completeness but it doesn’t originate any conflict. The main features of these games are well known; more detailed accounts can be found elsewhere e.g. [@Hofbauer1998; @vega-redondo-03; @weibull95]. The games have the generic payoff matrix $M$ (equation \[eq:payoff\]) which refers to the payoffs of the row player. The payoff matrix for the column player is simply the transpose $M^\top$ since the game is symmetric.
$$\bordermatrix{\text{}&C& D\cr
C&R&S\cr
D&T&P\cr
}
\label{eq:payoff}$$ The set of strategies is $\Lambda=\{C,D\}$, where $C$ stands for “cooperation” and $D$ means “defection”. In the payoff matrix $R$ stands for the *reward* the two players receive if they both cooperate, $P$ is the *punishment* if they both defect, and $T$ is the *temptation*, i.e. the payoff that a player receives if he defects while the other cooperates getting the *sucker’s payoff* $S$. For the PD, the payoff values are ordered such that $T > R > P > S$. Defection is always the best rational individual choice, so that $(D,D)$ is the unique Nash Equilibrium (NE). In the HD game the payoff ordering is $T > R > S > P$. Thus, when both players defect they each get the lowest payoff. $(C,D)$ and $(D,C)$ are NE of the game in pure strategies. There is a third equilibrium in mixed strategies which is the only dynamically stable equilibrium [@weibull95; @Hofbauer1998]. In the SH game, the ordering is $R > T > P > S$, which means that mutual cooperation $(C,C)$ is the best outcome and a NE. The second NE, where both players defect is less efficient but also less risky. The third NE is in mixed strategies but it is evolutionarily unstable [@weibull95; @Hofbauer1998]. Finally, in the H game $R>S>T>P$ or $R>T>S>P $. In this case $C$ strongly dominates $D$ and the trivial unique NE is $(C,C)$. The game is non-conflictual by definition; it is mentioned to complete the quadrants of the parameter space.
There is an infinite number of games of each type since any positive affine transformation of the payoff matrix leaves the NE set invariant [@weibull95]. Here we study the customary standard parameter space [@santos-pach-06; @anxo1], by fixing the payoff values in the following way: $R=1$, $P=0$, $-1 \leq S \leq 1$, and $0 \leq T \leq 2$. Therefore, in the $TS$ plane each game class corresponds to a different quadrant depending on the above ordering of the payoffs as depicted in Fig. \[PhaseSpace\], left image. The right image depicts the well mixed replicator dynamics stable states for future comparison.
[cccccccc]{} {width="6cm"} & {width="6cm"} &
Population Structure {#pop-str}
--------------------
The Euclidean two-dimensional space is modeled by a discrete square lattice of side $L$ with toroidal borders. Each vertex of the lattice can be occupied by one player or be empty. The *density* is $\rho=N/L^2$, where $N \le L^2$ is the number of players. Players can interact with $k$ neighbours which lie at an Euclidean distance smaller or equal than a given constant $R_g$. Players can also migrate to empty grid points at a distance smaller than $R_m$. We use three neighborhood sizes with radius $1.5$, $3$, and $5$; they contain, respectively, $8$, $28$, and $80$ neighbours around the central player.
Payoff Calculation and Strategy Update Rules {#revision-protocols}
--------------------------------------------
Each agent $i$ interacts locally with a set of neighbours $V_i$ lying closer than $R_g$. Let $\sigma_i(t)$ be a vector giving the strategy profile at time $t$ with $C= (1, 0)$ and $D = (0, 1)$ and let $M$ be the payoff matrix of the game (equation \[eq:payoff\]). The quantity $$\Pi_i(t) = \sum _{j \in V_i} \sigma_i(t)\; M\; \sigma_{j}^\top(t)
\label{payoffs}$$ is the cumulated payoff collected by player $i$ at time step $t$.
We use two imitative strategy update protocols. The first is the Fermi rule in which the focal player $i$ is given the opportunity to imitate a randomly chosen neighbour $j$ with probability: $$p(\sigma_i \rightarrow \sigma_j) = \frac{1}{ 1+exp(-\beta(\Pi_j - \Pi_i))}
\label{fermi}$$ where $\Pi_j -\Pi_i$ is the difference of the payoffs earned by $j$ and $i$ respectively and $\beta$ is a constant corresponding to the inverse temperature of the system. When $\beta \to 0$ the probability of imitating $j$ tends to a constant value $0.5$ and when $\beta \to \infty$ the rule becomes deterministic: $i$ imitates $j$ if $(\Pi_j - \Pi_i)>0$, otherwise it doesn’t. In between these two extreme cases the probability of imitating neighbour $j$ is an increasing function of $\Pi_j - \Pi_i$.
The second imitative strategy update protocol is to switch to the strategy of the neighbour that has scored best in the last time step. In contrast with the previous one, this rule is deterministic. This *imitation of the best* (IB) policy can be described in the following way: the strategy $\sigma_i(t)$ of individual $i$ at time step $t$ will be $$\sigma_i(t) = \sigma_j(t-1),
\label{ib}$$ where $$j \in \{V_i \cup i\} \;s.t.\; \Pi_j = \max \{\Pi_k(t-1)\}, \; \forall k \in \{V_i \cup i\}.
\label{ib2}$$ That is, individual $i$ will adopt the strategy of the player with the highest payoff among its neighbours including itself. If there is a tie, the winner individual is chosen uniformly at random.
Population Dynamics and Opportunistic Migration {#migration}
-----------------------------------------------
We use an asynchronous scheme for strategy update and migration, i.e. players are updated one by one by choosing a random player in each step with uniform probability and with replacement. Then the player migrates with probability $1/2$, otherwise it updates its strategy. If the pseudo-random number drawn dictates that $i$ should migrate, then it considers $N_{test}$ randomly chosen positions in the disc of radius $R_m$ around itself. The quantity $N_{test}$ could be seen as a kind of “energy” available to a player for moving around and doing its search. $N_{test}$ being fixed for a given run, it follows that an agent will be able to make a more complete exploration of its local environment the smaller the $R_m$. For each trial position the player computes the payoff that it would obtain in that place with its current strategy. The positions already occupied are just discarded from the possible choices. Then player $i$ stays at its current position if it obtains there the highest payoff, or migrates to the most profitable position among those explored during the test phase. If several positions, including its current one, share the highest payoff then it chooses one at random. We call this migration *opportunistic* or *fitness-based*. The protocol described in Helbing and Yu [@helbingPNAS] is slightly different: the chosen player chooses the strategy of the best neighbour including itself with probability $1-r$, and with probability $r$, with $r \ll 1-r$, its strategy is randomly reset. Before this imitation step $i$ deterministically chooses the highest payoff free position in a square neighborhood of size $(2M+1) \times (2M+1)$ cells surrounding the current player and including itself, where $M$ can take the values $0,1,2,5$. If several positions provide the same payoff, the one that is closer is selected.
Simulation Parameters {#Simulation Parameters}
---------------------
The $TS$ plane has been sampled with a grid step of $0.1$ and each value in the phase space reported in the figures is the average of $50$ independent runs. The evolution proceeds by first initializing the population by distributing $N=1000$ players with uniform probability among the available cells. Then the players’ strategies are initialized uniformly at random such that each strategy has a fraction of approximately $1/2$. To avoid transient states, we let the system evolve for a period of $\tau=1000$ time steps and, in each time step, $N$ players are chosen for update. At this point almost always the system reaches a steady state in which the frequency of cooperators is stable except for small statistical fluctuations. We then let the system evolve for $50$ further steps and take the average cooperation value in this interval. We repeat the whole process $50$ times for each grid point and, finally, we report the average cooperation values over those $50$ repetitions.
Results
=======
Imitation of the Best and Opportunistic Migration {#IBAM}
-------------------------------------------------
[cccccccc]{} {width="6cm"} & {width="6cm"} &
In this section we study cooperation with the IB rule and fitness-based opportunistic migration, and we explore the influence of different radii $R_m$ and $R_g$ and other parameters such as the density $\rho$ and the number of trials $N_{test}$. The left image of Fig. \[IB1\] displays the TS plane with the IB rule, a density $\rho=0.5$, and $N_{test}= 20$. For small $R_g=1.5$ full cooperation is achieved in the SH quadrant for all $R_m$. The average levels of cooperation in the PD games are $0.33, 0.31,0.30$ for $R_m = 1.5, 3, 5$ and $R_g = 1.5$ respectively. It is remarkable that cooperation emerges in contrast to the well mixed population case (Fig. \[PhaseSpace\], right image), and also that better results are obtained with respect to a fully populated grid in which agents cannot move [@anxo1]. The HD doesn’t benefit in the same way and the cooperation levels are almost the same in the average. Cooperation remains nearly constant as a function of $R_m$ for a given $R_g$ value but increasing $R_g$ has a negative effect. For higher game radius, $R_g\in\{3, 5\}$ cooperation is progressively lost in the PD games while there is little variation in the HD quadrant among the different cases due to the dimorphic structure of these populations. In the SH quadrant there is a large improvement compared to the well mixed case but the gain tends to decrease with increasing $R_g$. In the PD with high $R_g$, cooperators cannot increase their payoff by clustering, since the neighborhood of defectors covers adjacent small clusters of cooperators, the payoff of defectors becomes higher and they can invade cooperators clusters. Figure \[grids\] illustrates in an idealized manner what happens to a small cooperators cluster when the game radius $R_g$ increases using a full grid for simplicity.
[cccccccc]{} {width="7cm"}
For $R_g=1.5$ (left image) the cooperator cluster is stable as long as $8R > 3T$ since the central cooperator gets a payoff of $8R$, while the best payoff among the defectors is obtained by the individual marked $D$ (and by the symmetrically placed defectors) and is equal to $3T$ since $P=0$. Under this condition all the cooperators will thus imitate the central one. On the other hand, the defector will turn into a $C$ as long as $5R+3S > 3T$, thus provoking cooperator cluster expansion for parameter values in this range. On the contrary, for $R_g=3$ (right image) the central cooperator gets $8R+20S$ whilst the central defector at the border has a payoff of $7T$. Thus the cooperator imitates the defector if $7T > 8R + 20S$, i.e. $7T > 8 + 20S$ since $R=1$. This qualitative argument helps to explain the observed cooperation losses for increasing $R_g$.
This inequality is satisfied almost everywhere in the PD quadrant except in a very small area in its upper left corner.
Helbing and Yu [@helbingPNAS] found very encouraging cooperation results in their analysis but they only had a small game radius corresponding to the Von Neumann neighborhood which is constituted, in a full lattice, by the central individual and the four neighbours at distance one situated north, east, south, and west. We also find similar results for our smallest neighborhood having $R_g=1.5$, which corresponds to the eight-points Moore neighborhood but, as $R_g$ gets larger, we have just seen that a sizable portion of the cooperation gains are lost. We think that this is an important point since there are certainly situations in which those more extended neighborhoods are the natural choice in a spatially extended population.
[cccccccc]{} {width="6cm"} & {width="6cm"} &
The number of trials $N_{test}$ could also be a critical parameter in the model. The right image of Fig. \[IB1\] refers to the same case as the left one, i.e. the IB update rule with opportunistic migration and $\rho=0.5$, except for the number of trials which is one instead of $20$. We observe that practically the same cooperation levels are reached at steady state in both cases for $R_m = 5$ and $R_m=3$, while there is a small increase of the average cooperation in the PD games for $R_g = 1.5$ which goes from $0.33$, $0.31$, and $0.30$ for $N_{test}=20$ to $0.41$, $0.36$, and $0.33$ for $N_{test}=1$, for $R_m=1.5,3,5$ respectively. On the whole, it is apparent that $N_{test}$ does not seem to have a strong influence. However, one might ask whether the times to convergence are shorter when more tests are used, a fact that could compensate for the extra work spent in searching. But Figs. \[ConvergenceTimes\] show that convergence times are not very different and decrease very quickly with the number of essays $N_{test}$. This is shown for two particular games, one in the middle of the SH quadrant (left image), and the other near the upper left corner of the PD space (right image). Thus, a shorter time does not compensate for the wasted trials. Since moving around to find a better place is a costly activity in any real situation, this result is encouraging because it says that searching more intensively doesn’t change the time to convergence for more than four tests. Thus, quite high levels of cooperation can be achieved by opportunistic migration at low search cost, a conclusion that interestingly extends the results presented in [@helbingPNAS].
In diluted grids, density is another parameter that influences the evolution of cooperation [@Arenzon1; @Arenzon2], also in the presence of intelligent migration [@helbingPNAS; @adaptive-mig]. Too high densities should be detrimental because clusters of cooperators are surrounded by a dense population of defectors, while low densities allow cooperator clusters to have less defectors in their neighborhood once they are formed. We have performed numerical simulations for two other values of the density besides $0.5$, $\rho=0.2$ and $\rho=0.8$. We do not show the figures to save space but the main remark is that there is a monotone decrease of cooperation going from low to higher densities in the low $S$ region that influences mainly the PD and, to a smaller extent, the SH games.
Opportunistic Migration and Noisy Imitation
-------------------------------------------
In this section we use the more flexible strategy update protocol called the Fermi rule which was described in Sect. \[revision-protocols\] and in which the probability to imitate a random neighbour’s strategy depends on the parameter $\beta$. We have seen that using the IB rule with adaptive migration leads to full cooperation in the SH quadrant and improves cooperation in a part of the PD quadrant (Fig. \[IB1\]). This result does not hold with the Fermi rule with $\beta\ge1$, and we are back to full defection in the PD and almost $50\%$ cooperation as in the well mixed case in the SH; this behavior can be appreciated in the leftmost image of Fig. \[Fermi0p5\].
[cccccccc]{} {width="3.7cm"} & {width="3.7cm"} & {width="3.7cm"} & {width="3.7cm"} &
An interesting new phenomenon appears when $\beta$ becomes small, of the order of $10^{-2}$. In this case, the levels of cooperation increase in all games for $R_g$ values up to $3$ and cooperation raises to almost $100\%$ in all game phase space for $R_g=1.5$, for all migration radii, see the third image of Fig. \[Fermi0p5\]. The positive trend continues with decreasing $\beta$ (see rightmost image) and cooperation prevails almost everywhere. As we said above, the Fermi rule with $\beta=0.01$ or less implies that the decision to imitate a random neighbour becomes almost random itself. Thus, the spectacular gains in cooperation must depend in some way from opportunistic migration for the most part. Figure \[FermiRand\] illustrates the dynamical behavior of a particular case in the PD space. Here $T=1.5$, $S=-0.5$, $R=1$, $P=0$; that is, the game is in the middle of the PD quadrant. The other parameters are: $\beta=0.01$, $R_g=1.5$, and $R_m=3$. This particular game would lead to full defection in almost all cases but here we can see that it leads to full cooperation instead.
{width="14cm"}
This is a surprising phenomenon that needs an explanation. At the beginning, due to opportunistic migration, cooperators will be likely to form small clusters between themselves more than defectors, as the latter tend to follow cooperators instead of clustering between themselves since the $(D,D)$ payoff is equal to $0$. The low $\beta$ value will make strategy change close to random and thus strategy update will have a neutral effect. Indeed, as soon as cooperator clusters form due to migration, defectors that enter a cooperator cluster thanks to random imitation cannot invade them. The situation there is akin to a full grid and the number of defectors inside the cluster will fluctuate. Meanwhile, defectors at the border of a cooperator cluster will steadily turn into cooperators thus extending the cluster. This is due to the fact that lone defectors at the border will tend to imitate cooperators since defectors are less connected, and strategy imitation is almost random. Finally, the defectors inside the clusters will reach the border and turn into cooperators as well. The phenomenon is robust with respect to the migration radius $R_m$, as can be seen in the lower part of the third and fourth images of Fig. \[Fermi0p5\]. Cooperation prevails even when $P$ becomes positive which increases the payoff for defectors to aggregate. We have simulated the whole phase space for $P=0.2$ and $P=-0.2$. The results are similar to those with $P=0$ except that cooperation decreases slightly with increasing $P$. On the same images it can be seen that the game radius $R_g$ has a large influence and cooperation tends to be lost for radii larger than $1.5$. The reasons for this are very similar to those advocated in Sect. \[IBAM\] where Fig. \[grids\] schematically illustrates the fact that increasing $R_g$ makes the situation more similar to a well mixed population. In these conditions, the payoff-driven strategy imitation process becomes more important and may counter the benefits of opportunistic migration. However, since we believe that system possessing locality are important in practice, the findings of this section seem very encouraging for mobile agents that are better at finding more profitable positions and moving to them rather than at strategic reasoning.
Discussion and Conclusions
==========================
In this work we have explored some possibilities that arise when agents playing simple two-person, two-strategy evolutionary games may also move around in a certain region seeking better positions for themselves. The games examined are the standard ones, like the Prisoner’s Dilemma, the Hawk-Dove, or the Stag Hunt. In this context, the ability to move around in space is extremely common in animal as well as human societies and therefore its effect on global population behavior is an interesting research question. As already pointed out by other researchers [@helbingPNAS; @Helb-Mobil; @adaptive-mig; @reput-migr; @SwarmPD; @aktipis], adding a form of contingent mobility may result in better capabilities for the population to reach socially valuable results. Among the existing models, we have started from a slightly modified form of the interesting Helbing’s and Yu’s model [@helbingPNAS] and have explored some further avenues that were left untouched in the latter work. In the model agents live and move in a discrete two-dimensional grid space in which part of the cells are unoccupied. Using a strategy update rule that leads an agent to imitate her most successful neighbour as in [@helbingPNAS], and having the possibility to explore a certain number of free positions around oneself to find a better one, the gains in cooperative behavior are appreciable in the Prisoner’s Dilemma, in qualitative agreement with [@helbingPNAS]. In the Hawk-Dove games the gains in cooperation are small but, in addition, we find that cooperation is fully promoted in the class of Stag Hunt games which were not considered in [@helbingPNAS]. In Helbing and Yu the exploration of the available cells in search of a better one was fixed and deterministic. The question of the amount of effort needed to improve the agent’s situation was left therefore open, although this is clearly an important point, given that in the real world more exploration usually entails an increasing cost be it in terms of energy, time, or money. By using a similar search strategy but to random positions within a given radius, and by varying the number of searches available to the agent, we have seen that the convergence times to reach a given average level of cooperation do not degrade significantly by using fewer trials. This is a reassuring finding, given the above remarks related to the search cost. Helbing and Yu explored migration effects under a number of sizes of the square neighborhood around a given agent. However, they only had a single neighborhood for the game interactions, the standard five-cells Von Neumann neighborhood. We have explored this aspect more deeply and presented results for several combinations of game radius $R_g$ and migration radius $R_m$. In fact, it turns out that increasing the interaction radius has an adverse effect on cooperation to the point that, at $R_g=5$, cooperation levels are similar to those of a well mixed population, in spite of fitness-based migration. Thus, positive results are only obtained when agents interact locally in a relatively small neighborhood which, fortunately, seems to be a quite common condition in actual spatial systems.
Most importantly, we have explored another important commonly used strategy update rule, the Fermi rule. This rule is also imitative but allows to control the intensity of selection by varying a single parameter $\beta$. When $\beta$ is high, i.e. larger than one, almost all the cooperation gains observed with the imitation of the best rule are lost and we are back to a scenario of defection in the Prisoner’s Dilemma space and the Stag Hunt games are also influenced negatively. Migration does not help in this case. However, when $\beta$ is low, of the order of $0.01$, a very interesting phenomenon emerges: cooperation prevails everywhere in the game space for small game radius and for all migration radii, including in the PD space, which is notoriously the most problematic class of games. With $\beta=0.01$ or lower the strategy update is close to random; however, fitness-based migration is active and thus we see that migration, and not strategy update, is the main force driving the population towards cooperation and we have hypothesized a qualitative mechanism that could explain this striking result. Cooperation is robust with respect to the migration radius $R_m$ but increasing $R_g$ affects the results negatively for $R_g \ge 3$. The effect is mitigated the more random the strategy update, i.e. by further decreasing $\beta$.
Acknowledgments
---------------
The authors thank the Swiss National Foundation for their financial support under contracts 200021-14661611 and 200020-143224.
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abstract: 'Following Radford’s proof of Lagrange’s theorem for pointed Hopf algebras, we prove Lagrange’s theorem for Hopf monoids in the category of connected species. As a corollary, we obtain necessary conditions for a given subspecies ${\mathbf k}$ of a Hopf monoid ${\mathbf h}$ to be a Hopf submonoid: the quotient of any one of the generating series of ${\mathbf h}$ by the corresponding generating series of ${\mathbf k}$ must have nonnegative coefficients. Other corollaries include a necessary condition for a sequence of nonnegative integers to be the dimension sequence of a Hopf monoid in the form of certain polynomial inequalities, and of a set-theoretic Hopf monoid in the form of certain linear inequalities. The latter express that the binomial transform of the sequence must be nonnegative.'
address:
- |
Department of Mathematics\
Texas A&M University\
College Station, TX 77843
- |
Department of Mathematics and Statistics\
Loyola University Chicago\
Chicago, IL 60660
author:
- Marcelo Aguiar
- Aaron Lauve
bibliography:
- 'bibl.bib'
date: 18 August 2011
title: '[L]{}agrange’s Theorem for [H]{}opf Monoids in Species'
---
Introduction {#introduction .unnumbered}
============
Lagrange’s theorem states that for any subgroup $K$ of a group $H$, $H\cong K\times Q$ as (left) $K$-sets, where $Q=H/K$. In particular, if $H$ is finite, then $|K|$ divides $|H|$. Passing to group algebras over a field $\field$, we have that $\field H \cong \field K \otimes \field Q$ as (left) $\field K$-modules, or that $\field H$ is free as a $\field K$-module. Kaplansky [@Kap:1975] conjectured that the same statement holds for Hopf algebras—group algebras being principal examples. It turns out that the result does not hold in general, as shown by Oberst and Schneider [@ObeSch:1974 Proposition 10] and [@Mon:1993 Example 3.5.2]. On the other hand, the result does hold for certain large classes of Hopf algebras, including the finite dimensional ones by a theorem of Nichols and Zoeller [@NicZoe:1989], and the pointed ones by a theorem of Radford [@Rad:1977]. Further (and finer) results of this nature were developed by Schneider [@Sch:1990; @Sch:1992]. Additional work on the conjecture includes that of Masuoka [@Mas:1992] and Takeuchi [@Tak:1979]; more information can be found in Sommerhäuser’s survey [@Som:2000].
The main result of this paper (Theorem \[t:main\]) is a version of Lagrange’s theorem for Hopf monoids in the category of connected species: if ${\mathbf h}$ is a connected Hopf monoid and ${\mathbf k}$ is a Hopf submonoid, there exists a species ${\mathbf q}$ such that ${\mathbf h}={\mathbf k}{\bm\cdot}{\mathbf q}$. An immediate application is a test for Hopf submonoids (Corollary \[c:Sp series\]): if any one of the generating series for a species ${\mathbf k}$ does not divide in ${\mathbb{Q}}_{\geq 0}\llb x\rrb$ the corresponding generating series for the Hopf monoid ${\mathbf h}$ (in the sense that the quotient has at least one negative coefficient), then ${\mathbf k}$ is not a Hopf submonoid of ${\mathbf h}$. A similar test also holds for connected graded Hopf algebras (Corollary \[c:cgVec series\]). The proof of Theorem \[t:main\] for Hopf monoids in species parallels Radford’s proof for Hopf algebras. (Hopf algebras are Hopf monoids in the category of vector spaces).
The paper is organized as follows. In Section \[s:Hopf algebras\], we recall Lagrange’s theorem for Hopf algebras, focusing on the case of connected graded Hopf algebras. In Section \[s:Hopf monoids\], we recall the basics of Hopf monoids in species and prove Lagrange’s theorem in this setting. Examples and applications are given in Section \[s:applications\]. Among these, we derive certain polynomial inequalities that a sequence of nonnegative integers must satisfy in order to be the dimension sequence of a connected Hopf monoid in species. In the case of a set-theoretic Hopf monoid structure, we obtain additional necessary conditions in the form of linear inequalities which express that the binomial transform of the enumerating sequence must be nonnegative. In Section \[s:dimensions\] we provide information on the growth and support of the dimension sequence of a connected Hopf monoid. The latter must be an additive submonoid of the natural numbers. We conclude in Section \[s:kernels\] with information on the species ${\mathbf q}$ entering in Lagrange’s theorem. In the dual setting, ${\mathbf q}$ is the Hopf kernel of a morphism, and for cocommutative Hopf monoids it can be described in terms of Lie kernels and primitive elements via the Poincaré-Birkhoff-Witt theorem.
All vector spaces are over a fixed field $\field$ of characteristic $0$, except in Section \[s:dimensions\], where the characteristic is arbitrary.
Lagrange’s theorem for Hopf algebras {#s:Hopf algebras}
====================================
We begin by recalling a couple of versions of this theorem.
Let $H$ be a finite dimensional Hopf algebra over a field $\field$. If $K\subseteq H$ is any Hopf subalgebra, then $H$ is a free left (and right) $K$-module.
This is the Nichols-Zoeller theorem [@NicZoe:1989]; see also [@Mon:1993 Theorem 3.1.5]. We will not make direct use of this result, but instead of the related results discussed below.
A Hopf algebra $H$ is if all its simple subcoalgebras are $1$-dimensional. Equivalently, the group-like elements of $H$ linearly span the coradical of $H$.
Given a subspace $K$ of $H$, set $$K_+:=K\cap\ker(\epsilon),$$ where $\epsilon:H\to\field$ is the counit of $H$. Let $K_+H$ denote the right $H$-ideal generated by $K_+$.
\[t:pointed\] Let $H$ be a pointed Hopf algebra. If $K\subseteq H$ is any Hopf subalgebra, then $H$ is a free left (and right) $K$-module. Moreover, $$H\cong K\otimes (H/K_+H)$$ as left $K$-modules.
The first statement is due to Radford [@Rad:1977 Section 4] and the second (stronger) statement to Schneider [@Sch:1990 Remark 4.14], [@Sch:1992 Corollary 4.3]. Various generalizations can be found in these references as well as in Masuoka [@Mas:1992] and Takeuchi [@Tak:1979]; see also Sommerhaüser [@Som:2000]. We are interested in the particular variant given in Theorem \[t:connected\] below.
A Hopf algebra $H$ is if there is given a decomposition $$H=\bigoplus_{n\geq0} H_n$$ into linear subspaces that is preserved by all operations. It is if in addition $H_0$ is linearly spanned by the unit element.
\[t:connected\] Let $H$ be a connected graded Hopf algebra. If $K\subseteq H$ is a graded Hopf subalgebra, then $H$ is a free left (and right) $K$-module. Moreover, $$H\cong K\otimes (H/K_+H)$$ as left $K$-modules and as graded vector spaces.
Since $H$ is connected graded, its coradical is $H_0 = \field$, so $H$ is pointed and Theorem \[t:pointed\] applies. Radford’s proof shows that there exists a graded vector space $Q$ such that $$H \cong K \otimes Q$$ as left $K$-modules and as graded vector spaces. (The argument we give in the parallel setting of Theorem \[t:main\] makes this clear.) Note that $K_{+\!}=\bigoplus_{n\geq1} K_n$, hence $K_+H$ and $H/K_+H$ inherit the grading of $H$. To complete the proof, it suffices to show that $Q\cong H/K_+H$ as graded vector spaces.
Let $\varphi:K\otimes Q\to H$ be an isomorphism of left $K$-modules and of graded vector spaces. We claim that $$\varphi(K_+\otimes Q)=K_+H.$$ In fact, since $\varphi$ is a morphism of left $K$-modules, $$\varphi(K_+\otimes Q) = K_+\varphi(1\otimes Q)\subseteq K_+H.$$ Conversely, if $k\in K_+$ and $h\in H$, writing $h=\sum_i \varphi(k_i\otimes q_i)$ with $k_i\in K$ and $q_i\in Q$, we obtain $$kh=\sum_i \varphi(kk_i\otimes q_i)\in \varphi(K_+\otimes Q),$$ since $K_+$ is an ideal of $K$.
Now, since $K=K_0\oplus K_+$, we have $$K\otimes Q =(K_0\otimes Q)\oplus (K_+\otimes Q)$$ and therefore $$H/K_+H = \varphi(K\otimes Q)/\varphi(K_+\otimes Q) \cong \varphi (K_0\otimes Q)\cong Q$$ as graded vector spaces.
Given a graded Hopf algebra $H$, let ${\mathcal O_{H}(x)}\in {\mathbb{N}}\llb x\rrb$ denote its —the ordinary generating function for the sequence of dimensions of its graded components, $$\begin{gathered}
{\mathcal O_{H}(x)} := \sum_{n\geq0} \dim H_n \, x^n .\end{gathered}$$ Suppose $H$ is connected graded and $K$ is a graded Hopf subalgebra. In this case, their Poincaré series are of the form $$1+a_1x+a_2x^2+\cdots$$ with $a_i\in{\mathbb{N}}$ and the quotient ${\mathcal O_{H}(x)}/{\mathcal O_{K}(x)}$ is a well-defined power series in ${\mathbb{Z}}\llb x\rrb$.
\[c:cgVec series\] Let $H$ be a connected graded Hopf algebra. If $K\subseteq H$ is any graded Hopf subalgebra, then the quotient ${\mathcal O_{H}(x)} / {\mathcal O_{K}(x)}$ of Poincaré series is nonnegative, i.e., belongs to ${\mathbb{N}}\llb x\rrb$.
By Theorem \[t:connected\], $H \cong K \otimes Q$ as graded vector spaces, where $Q=H/K_+H$. Hence ${\mathcal O_{H}(x)} = {\mathcal O_{K}(x)} \, {\mathcal O_{Q}(x)}$ and the result follows.
Consider the Hopf algebra ${\textsl{QSym}}$ of quasisymmetric functions in countably many variables, and the Hopf subalgebra ${\textsl{Sym}}$ of symmetric functions. They are connected graded, so by Theorem \[t:connected\], ${\textsl{QSym}}$ is a free module over ${\textsl{Sym}}$. Garsia and Wallach prove this same fact for the algebras ${\textsl{QSym}}_n$ and ${\textsl{Sym}}_n$ of (quasi) symmetric functions in $n$ variables (where $n$ is a finite number) [@GarWal:2003]. While ${\textsl{QSym}}_n$ and ${\textsl{Sym}}_n$ are quotient algebras of ${\textsl{QSym}}$ and ${\textsl{Sym}}$, they are not quotient coalgebras. Since a Hopf algebra structure is needed in order to apply Theorem \[t:connected\], we cannot derive the result of Garsia and Wallach in this manner. The papers [@GarWal:2003] and [@LauMas:2011] provide information on the space $Q_n$ entering in the decomposition ${\textsl{QSym}}_n\cong{\textsl{Sym}}_n\otimes Q_n$.
Lagrange’s theorem for Hopf monoids in species {#s:Hopf monoids}
==============================================
We first review the notion of Hopf monoid in the category of species, following [@AguMah:2010], and then prove Lagrange’s theorem in this setting. We restrict attention to the case of connected Hopf monoids.
Hopf monoids in species {#ss:species}
-----------------------
The notion of species was introduced by Joyal [@Joy:1981]. It formalizes the notion of combinatorial structure and provides a framework for studying the generating functions which enumerate these structures. The book [@BerLabLer:1998] by Bergeron, Labelle and Leroux expounds the theory of set species.
Joyal’s work indicates that species may also be regarded as algebraic objects; this is the point of view adopted in [@AguMah:2010] and in this work. To this end, it is convenient to work with vector species.
A is a functor ${\mathbf q}$ from [finite sets]{} and bijections to [vector spaces]{} and linear maps. Specifically, it is a family of vector spaces ${\mathbf q}[I]$, one for each finite set $I$, together with linear maps ${\mathbf q}[\sigma]: {\mathbf q}[I] \to {\mathbf q}[J]$, one for each bijection $\sigma:I\to J$, satisfying $${\mathbf q}[\id_I] = \id_{{\mathbf q}[I]} \quad\hbox{and}\quad
{\mathbf q}[\sigma\circ\tau] = {\mathbf q}[\sigma]\circ {\mathbf q}[\tau]$$ whenever $\sigma$ and $\tau$ are composable bijections.
The notation ${\mathbf q}[a,b,c,\ldots]$ is shorthand for ${\mathbf q}[\{a,b,c,\ldots\}]$ and ${\mathbf q}[n]$ is shorthand for ${\mathbf q}[1,2,\ldots,n]$. The space ${\mathbf q}[n]$ is an $S_n$-module via $\sigma\cdot v = {\mathbf q}[\sigma](v)$ for $v\in{\mathbf q}[n]$ and $\sigma\in S_n$.
A species ${\mathbf q}$ is if each vector space ${\mathbf q}[I]$ is finite dimensional. In this paper, all species are finite dimensional.
A morphism of species is a natural transformation of functors. Let ${\sf Sp}$ denote the category of (finite dimensional) species.
We give two elementary examples that will be useful later.
\[ex: example species\] Let ${\mathbf E}$ be the , defined by ${\mathbf E}[I] = \field\{\ast_I\}$ for all $I$. The symbol $\ast_I$ denotes an element canonically associated to the set $I$ (for definiteness, we may take $\ast_I=I$). Thus, ${\mathbf E}[I]$ is a $1$-dimensional space with a distinguished basis element. A richer example is provided by the species ${\mathbf L}$ of , defined by ${\mathbf L}[I] = \field\{\hbox{linear orders on }I\}$ for all $I$ (a space of dimension $n!$ when $|I|=n$).
We use ${\bm\cdot}$ to denote the of two species. Specifically, $$\bigl({\mathbf p}{\bm\cdot}{\mathbf q}\bigr)[I] := \bigoplus_{S{\sqcup}T = I} {\mathbf p}[S] \otimes {\mathbf q}[T]
\quad\hbox{for all finite sets }I.$$ The notation $S{\sqcup}T = I$ indicates that $S\cup T = I$ and $S \cap T = \emptyset$. The sum runs over all such of $I$, or equivalently over all subsets $S$ of $I$: there is one term for $S{\sqcup}T$ and another for $T{\sqcup}S$. The Cauchy product turns ${\sf Sp}$ into a symmetric monoidal category. The braiding simply switches the tensor factors. The unit object is the species ${\mathbf{1}}$ defined by $${\mathbf{1}}[I] := \begin{cases}
\field & \text{if $I$ is empty,} \\
0 & \text{otherwise.}
\end{cases}$$
A in the category $({\sf Sp},{\bm\cdot})$ is a species ${\mathbf m}$ together with a morphism of species $\mu: {\mathbf m}{\bm\cdot}{\mathbf m}\to {\mathbf m}$, i.e., a family of maps $$\mu_{S,T} : {\mathbf m}[S] \otimes {\mathbf m}[T] \to {\mathbf m}[I],$$ one for each ordered decomposition $I = S{\sqcup}T$, satisfying appropriate associativity and unital conditions, and naturality with respect to bijections. Briefly, to each ${\mathbf m}$-structure on $S$ and ${\mathbf m}$-structure on $T$, there is assigned an ${\mathbf m}$-structure on $S{\sqcup}T$. The analogous object in the category of graded vector spaces is a graded algebra.
For the species ${\mathbf E}$, a monoid structure is defined by sending the basis element $\ast_S \otimes \ast_T$ to the basis element $\ast_I$. For ${\mathbf L}$, a monoid structure is provided by concatenation of linear orders: $\mu_{S,T}(\ell_1 \otimes \ell_2) = (\ell_1, \ell_2)$.
A in the category $({\sf Sp},{\bm\cdot})$ is a species ${\mathbf c}$ together with a morphism of species $\Delta:{\mathbf c}\to {\mathbf c}{\bm\cdot}{\mathbf c}$, i.e., a family of maps $$\Delta_{S,T} : {\mathbf c}[I] \to {\mathbf c}[S] \otimes {\mathbf c}[T],$$ one for each ordered decomposition $I=S{\sqcup}T$, which are natural, coassociative and counital.
For the species ${\mathbf E}$, a comonoid structure is defined by sending the basis vector $\ast_I$ to the basis vector $\ast_S \otimes \ast_T$. For ${\mathbf L}$, a comonoid structure is provided by restricting a total order $\ell$ on $I$ to total orders on $S$ and $T$: $\Delta_{S,T}(\ell) = \ell\vert_S\otimes \ell\vert_T$.
We assume that our species ${\mathbf q}$ are , i.e., ${\mathbf q}[\emptyset]=\field$. In this case, the (co)unital conditions for a (co)monoid force the maps $\mu_{S,T}$ ($\Delta_{S,T}$) to be the canonical identifications if either $S$ or $T$ is empty. Thus, in defining a connected (co)monoid structure one only needs to specify the maps $\mu_{S,T}$ ($\Delta_{S,T}$) when both $S$ and $T$ are nonempty.
A in the category $({\sf Sp}, {\bm\cdot})$ is a monoid and comonoid whose two structures are compatible in an appropriate sense, and which carries an antipode. In this paper we only consider connected Hopf monoids. For such Hopf monoids, the existence of the antipode is guaranteed. The species ${\mathbf E}$ and ${\mathbf L}$, with the structures outlined above, are two important examples of connected Hopf monoids.
For further details on Hopf monoids in species, see Chapter 8 of [@AguMah:2010]. The theory of Hopf monoids in species is developed in Part II of this reference; several examples are discussed in Chapters 12 and 13.
Lagrange’s theorem for connected Hopf monoids
---------------------------------------------
Given a connected Hopf monoid ${\mathbf k}$ in species, we let ${\mathbf k}_+$ denote the species defined by $${\mathbf k}_+[I] = \begin{cases}
{\mathbf k}[I] & \text{ if }I\neq\emptyset, \\
0 & \text{ if }I=\emptyset.
\end{cases}$$ If ${\mathbf k}$ is a submonoid of a monoid ${\mathbf h}$, then ${\mathbf k}_+{\mathbf h}$ denotes the right ideal of ${\mathbf h}$ generated by ${\mathbf k}_+$. In other words, $$({\mathbf k}_+{\mathbf h})[I]=\sum_{\substack{S\sqcup T=I\\ S\neq\emptyset}}\mu_{S,T}\bigl({\mathbf k}[S]\otimes{\mathbf h}[T]\bigr).$$
\[t:main\] Let ${\mathbf h}$ be a connected Hopf monoid in the category of species. If ${\mathbf k}$ is a Hopf submonoid of ${\mathbf h}$, then ${\mathbf h}$ is a free left ${\mathbf k}$-module. Moreover, $${\mathbf h}\cong {\mathbf k}{\bm\cdot}({\mathbf h}/{\mathbf k}_+{\mathbf h})$$ as left ${\mathbf k}$-modules (and as species).
The proof is given after a series of preparatory results. Our argument parallels Radford’s proof of the first statement in Theorem \[t:pointed\] [@Rad:1977 Section 4]. The main ingredient is a result of Larson and Sweedler [@LarSwe:1969] known as the fundamental theorem of Hopf modules [@Mon:1993 Theorem 1.9.4]. It states that if $(M,\rho)$ is a left Hopf module over $K$, then $M$ is free as a left $K$-module and in fact is isomorphic to the Hopf module $K\otimes Q$, where $Q$ is the space of for the coaction $\rho \colon M\to K\otimes M$. Takeuchi extends this result to the context of Hopf monoids in a braided monoidal category with equalizers [@Tak:1999 Theorem 3.4]; a similar result (in a more restrictive setting) is given by Lyubashenko [@Lyu:1995 Theorem 1.1]. As a special case of Takeuchi’s result, we have the following.
\[p:FTHM\] Let ${\mathbf m}$ be a left Hopf module over a connected Hopf monoid ${\mathbf k}$ in species. There is an isomorphism ${\mathbf m}\cong {\mathbf k}{\bm\cdot}{\mathbf q}$ of left Hopf modules, where $${\mathbf q}[I] := \bigl\{m\in {\mathbf m}[I] \mid \text{$\rho_{S,T}(m) = 0$ for $S\sqcup T=I$, $T\neq I$} \bigr\}.$$ In particular, ${\mathbf m}$ is free as a left ${\mathbf k}$-module.
Here $\rho:{\mathbf m}\to{\mathbf k}{\bm\cdot}{\mathbf m}$ denotes the comodule structure, which consists of maps $$\rho_{S,T}: {\mathbf m}[I] \to {\mathbf k}[S]\otimes{\mathbf m}[T],$$ one for each ordered decomposition $I=S\sqcup T$.
Given a comonoid ${\mathbf h}$ and two subspecies ${\mathbf u},{\mathbf v}\subseteq {\mathbf h}$, the of ${\mathbf u}$ and ${\mathbf v}$ is the subspecies ${\mathbf u}\wedge{\mathbf v}$ of ${\mathbf h}$ defined by $${\mathbf u}\wedge{\mathbf v}:= \Delta^{-1}({\mathbf u}{\bm\cdot}{\mathbf h}+ {\mathbf h}{\bm\cdot}{\mathbf v}).$$
The remaining ingredients needed for the proof are supplied by the following lemmas.
\[l:lemma 1\] Let ${\mathbf h}$ be a comonoid in species. If ${\mathbf u}$ and ${\mathbf v}$ are subcomonoids of ${\mathbf h}$, then:
- ${\mathbf u}\wedge{\mathbf v}$ is a subcomonoid of ${\mathbf h}$ and ${\mathbf u}+{\mathbf v}\subseteq {\mathbf u}\wedge{\mathbf v}$;
- ${\mathbf u}\wedge{\mathbf v}= \Delta^{-1}\bigl({\mathbf u}{\bm\cdot}({\mathbf u}\wedge{\mathbf v}) + ({\mathbf u}\wedge{\mathbf v}){\bm\cdot}{\mathbf v}\bigr)$.
\(i) The proofs of the analogous statements for coalgebras given in [@Abe:1980 Section 3.3] extend to this setting.
\(ii) From the definition, $\Delta^{-1}\bigl({\mathbf u}{\bm\cdot}({\mathbf u}\wedge{\mathbf v}) + ({\mathbf u}\wedge{\mathbf v}){\bm\cdot}{\mathbf v}\bigr)\subseteq {\mathbf u}\wedge{\mathbf v}$. Now, since ${\mathbf u}\wedge{\mathbf v}$ is a subcomonoid, $$\Delta({\mathbf u}\wedge{\mathbf v})\subseteq \bigr(({\mathbf u}\wedge{\mathbf v}){\bm\cdot}({\mathbf u}\wedge{\mathbf v})\bigr)\cap
({\mathbf u}{\bm\cdot}{\mathbf h}+ {\mathbf h}{\bm\cdot}{\mathbf v}) = {\mathbf u}{\bm\cdot}({\mathbf u}\wedge{\mathbf v}) + ({\mathbf u}\wedge{\mathbf v}){\bm\cdot}{\mathbf v},$$ since ${\mathbf u},{\mathbf v}\subseteq {\mathbf u}\wedge{\mathbf v}$. This proves the converse inclusion.
\[l:lemma 1.5\] Let ${\mathbf h}$ be a Hopf monoid in species and ${\mathbf k}$ be a submonoid. Let ${\mathbf u},{\mathbf v}\subseteq{\mathbf h}$ be subspecies which are left ${\mathbf k}$-submodules of ${\mathbf h}$. Then ${\mathbf u}\wedge {\mathbf v}$ is a left ${\mathbf k}$-submodule of ${\mathbf h}$.
Since ${\mathbf h}$ is a Hopf monoid, the coproduct $\Delta:{\mathbf h}\to{\mathbf h}{\bm\cdot}{\mathbf h}$ is a morphism of left ${\mathbf h}$-modules, where ${\mathbf h}$ acts on ${\mathbf h}{\bm\cdot}{\mathbf h}$ via $\Delta$. Hence it is also a morphism of left ${\mathbf k}$-modules. By hypothesis, ${\mathbf u}{\bm\cdot}{\mathbf h}+{\mathbf h}{\bm\cdot}{\mathbf v}$ is a left ${\mathbf k}$-submodule of ${\mathbf h}{\bm\cdot}{\mathbf h}$. Hence, ${\mathbf u}\wedge{\mathbf v}=\Delta^{-1}({\mathbf u}{\bm\cdot}{\mathbf h}+ {\mathbf h}{\bm\cdot}{\mathbf v})$ is a left ${\mathbf k}$-submodule of ${\mathbf h}$.
\[l:lemma 2\] Let ${\mathbf h}$ be a Hopf monoid in species and ${\mathbf k}$ a Hopf submonoid. Let ${\mathbf c}$ be a subcomonoid of ${\mathbf h}$ and a left ${\mathbf k}$-submodule of ${\mathbf h}$. Then $({\mathbf k}\wedge {\mathbf c})/{\mathbf c}$ is a left ${\mathbf k}$-Hopf module.
By Lemma \[l:lemma 1.5\], ${\mathbf k}\wedge{\mathbf c}$ is a left ${\mathbf k}$-submodule of ${\mathbf h}$. Therefore, the quotient $({\mathbf k}\wedge{\mathbf c})/{\mathbf c}$ by the left ${\mathbf k}$-submodule ${\mathbf c}$ is a left ${\mathbf k}$-module.
We next argue that $({\mathbf k}\wedge{\mathbf c})/{\mathbf c}$ is a ${\mathbf k}$-comodule. Consider the composite $${\mathbf k}\wedge{\mathbf c}\map{\Delta} {\mathbf k}{\bm\cdot}({\mathbf k}\wedge{\mathbf c}) + ({\mathbf k}\wedge{\mathbf c}){\bm\cdot}{\mathbf c}\onto {\mathbf k}{\bm\cdot}\bigl({\mathbf k}\wedge{\mathbf c})/{\mathbf c},$$ where the first map is granted by Lemma \[l:lemma 1\] and the second is the projection modulo ${\mathbf c}$ on the second coordinate. Since ${\mathbf c}$ is a subcomonoid, the composite factors through ${\mathbf c}$ and induces $$({\mathbf k}\wedge{\mathbf c})/{\mathbf c}\to {\mathbf k}{\bm\cdot}\bigl({\mathbf k}\wedge{\mathbf c})/{\mathbf c}.$$ This defines a left ${\mathbf k}$-comodule structure on $({\mathbf k}\wedge{\mathbf c})/{\mathbf c}$.
Finally, the compatibility between the module and comodule structures on $({\mathbf k}\wedge{\mathbf c})/{\mathbf c}$ is inherited from the compatibility between the product and coproduct of ${\mathbf h}$.
We are nearly ready for the proof of the main result. First, recall the of a connected comonoid in species [@AguMah:2010 Section 8.10]. Given a connected comonoid ${\mathbf c}$, define subspecies ${\mathbf c}_{(n)}$ by $${\mathbf c}_{(0)}={\mathbf{1}}{\quad\text{and}\quad}{\mathbf c}_{(n)} = {\mathbf c}_{(0)} \wedge {\mathbf c}_{(n-1)} \text{ \ for all $n\geq 1$.}$$ We then have $${\mathbf c}_{(0)} \subseteq {\mathbf c}_{(1)} \subseteq \dotsb\subseteq {\mathbf c}_{(n)}\subseteq \dotsb {\mathbf c}{\quad\text{and}\quad}{\mathbf c}= \bigcup_{n\geq 0} {\mathbf c}_{(n)}.$$
We show that there is a species ${\mathbf q}$ such that ${\mathbf h}\cong {\mathbf k}{\bm\cdot}{\mathbf q}$ as left ${\mathbf k}$-modules. As in the proof of Theorem \[t:connected\], it then follows that ${\mathbf q}\cong {\mathbf h}/{\mathbf k}_+{\mathbf h}$.
Define a sequence ${\mathbf k}^{(n)}$ of subspecies of ${\mathbf h}$ by $${\mathbf k}^{(0)}={\mathbf k}{\quad\text{and}\quad}{\mathbf k}^{(n)} = {\mathbf k}\wedge {\mathbf k}^{(n-1)} \text{ \ for all $n\geq 1$.}$$
Each ${\mathbf k}^{(n)}$ is a subcomonoid and a left ${\mathbf k}$-submodule of ${\mathbf h}$. This follows from Lemmas \[l:lemma 1\] and \[l:lemma 1.5\] by induction on $n$. Then Lemma \[l:lemma 2\] provides a left ${\mathbf k}$-Hopf module structure on the quotient species ${\mathbf k}^{(n)}/{\mathbf k}^{(n-1)}$ for all $n\geq1$. Hence ${\mathbf k}^{(n)}/{\mathbf k}^{(n-1)}$ is a free left ${\mathbf k}$-module, by Proposition \[p:FTHM\].
We claim that there exists a sequence of species ${\mathbf q}_n$ such that $${\mathbf k}^{(n)} \cong {\mathbf k}{\bm\cdot}{\mathbf q}_n$$ as left ${\mathbf k}$-modules for all $n\geq 0$; that is, each ${\mathbf k}^{(n)}$ is a free left ${\mathbf k}$-module. Moreover, we claim that the ${\mathbf q}_n$ can be chosen so that $${\mathbf q}_0\subseteq {\mathbf q}_1\ \subseteq \dotsb\subseteq {\mathbf q}_n \subseteq\dotsb$$ and the above isomorphisms are compatible with the inclusions ${\mathbf q}_{n-1}\subseteq{\mathbf q}_n$ and ${\mathbf k}^{(n-1)}\subseteq{\mathbf k}^{(n)}$.
We prove the claims by induction on $n\geq 0$. We start by letting ${\mathbf q}_0={\mathbf{1}}$. For $n\geq 1$, we have $${\mathbf k}^{(n-1)}\cong {\mathbf k}{\bm\cdot}{\mathbf q}_{n-1}
{\quad\text{and}\quad}{\mathbf k}^{(n)}/{\mathbf k}^{(n-1)}\cong {\mathbf k}{\bm\cdot}{\mathbf q}'_n$$ for some species $ {\mathbf q}'_n$ (the former by induction hypothesis and the latter by the above argument). Let $${\mathbf q}_{n} = {\mathbf q}_{n-1}\oplus {\mathbf q}'_n.$$ By choosing an arbitrary ${\mathbf k}$-module section of the map ${\mathbf k}^{(n)}\onto{\mathbf k}^{(n)}/{\mathbf k}^{(n-1)}\cong {\mathbf k}{\bm\cdot}{\mathbf q}'_n$ (possible by freeness), we obtain an isomorphism $${\mathbf k}^{(n)}\cong {\mathbf k}{\bm\cdot}{\mathbf q}_{n}$$ extending the isomorphism ${\mathbf k}^{(n-1)}\cong {\mathbf k}{\bm\cdot}{\mathbf q}_{n-1}$. This proves the claims.
We utilize the coradical filtration of ${\mathbf h}$ to finish the proof. Since ${\mathbf h}$ is connected, ${\mathbf h}_{(0)}={\mathbf{1}}\subseteq {\mathbf k}= {\mathbf k}^{(0)}$, and by induction, $${\mathbf h}_{(n)} \subseteq {\mathbf k}^{(n)} \quad\hbox{for all }n\geq0.$$ Hence, $${\mathbf h}= \bigcup_{n\geq 0} {\mathbf h}_{(n)}
=\bigcup_{n\geq 0} {\mathbf k}^{(n)}
\cong\bigcup_{n\geq 0} {\mathbf k}{\bm\cdot}{\mathbf q}_{n}
\cong{\mathbf k}{\bm\cdot}{\mathbf q}\text{ \ where \ }
{\mathbf q}=\bigcup_{n\geq 0} {\mathbf q}_n.$$ Thus ${\mathbf h}$ is free as a left ${\mathbf k}$-module.
Let $\pi:{\mathbf h}\to{\mathbf k}$ be a morphism of Hopf monoids. The of $\pi$ is the species defined by $$\label{e:hker}
{\mathrm{Hker}}(\pi)=\ker\bigl({\mathbf h}\map{\Delta} {\mathbf h}{\bm\cdot}{\mathbf h}\map{\pi_{+}{\bm\cdot}\,\id} {\mathbf k}_{+\!}{\bm\cdot}{\mathbf h}\bigr),$$ where $\pi_+:{\mathbf h}\to{\mathbf k}_+$ is $\pi$ followed by the canonical projection ${\mathbf k}\onto{\mathbf k}_+$.
For the following result, we employ duality for Hopf monoids [@AguMah:2010 Section 8.6.2]. (We assume all species are finite dimensional.)
\[t:maindual\] Let ${\mathbf h}$ be a connected Hopf monoid in the category of species and ${\mathbf k}$ a quotient Hopf monoid via a morphism $\pi:{\mathbf h}\onto{\mathbf k}$. Then ${\mathbf h}$ is a cofree left ${\mathbf k}$-comodule. Moreover, $${\mathbf h}\cong {\mathbf k}{\bm\cdot}{\mathrm{Hker}}(\pi)$$ as left ${\mathbf k}$-comodules (and as species).
By duality, ${\mathbf k}^*$ is a Hopf submonoid of ${\mathbf h}^*$, so ${\mathbf h}^*\cong{\mathbf k}^*{\bm\cdot}({\mathbf h}^*/{\mathbf k}^*_+{\mathbf h}^*)$ by Theorem \[t:main\]. Dualizing again we obtain the result, since $${\mathbf h}^*/{\mathbf k}^*_+{\mathbf h}^*={\mathrm{coker}}\bigl({\mathbf k}^*_{+\!}{\bm\cdot}{\mathbf h}^* \map{\pi^*_+{\bm\cdot}\,\id}
{\mathbf h}^{*\!}{\bm\cdot}{\mathbf h}^* \map{\Delta^*}
{\mathbf h}^* \bigr). \qedhere$$
Applications and examples {#s:applications}
=========================
A test for Hopf submonoids
--------------------------
Two invariants associated to a (finite dimensional) species ${\mathbf q}$ are the ${\mathcal E_{{\mathbf q}}(x)}$ and the ${\mathcal T_{{\mathbf q}}(x)}$. They are given by $${\mathcal E_{{\mathbf q}}(x)} = \sum_{n\geq 0} \dim {\mathbf q}[n]\, \frac{x^n}{n!}
{\quad\text{and}\quad}{\mathcal T_{{\mathbf q}}(x)} = \sum_{n\geq 0} \dim {\mathbf q}[n]_{S_n}\, x^n,$$ where $${\mathbf q}[n]_{S_n} ={\mathbf q}[n]/\field\{v-\sigma\cdot v \mid v\in {\mathbf q}[n],\ \sigma\in S_n\}.$$ Both are specializations of the ${\mathcal Z_{{\mathbf q}}(x_1,x_2,\dotsc)}$; see [@BerLabLer:1998 Section 1.2] for the definition. Specifically, $$\begin{gathered}
{\mathcal E_{{\mathbf q}}(x)} = {\mathcal Z_{{\mathbf q}}(x,0,0,\dotsc)}
{\quad\text{and}\quad}{\mathcal T_{{\mathbf q}}(x)} = {\mathcal Z_{{\mathbf q}}(x,x^2,x^3,\dotsc)} .\end{gathered}$$
The cycle index series is multiplicative under Cauchy product [@BerLabLer:1998 Section 1.3]: if ${\mathbf h}= {\mathbf k}{\bm\cdot}{\mathbf q}$, then ${\mathcal Z_{{\mathbf h}}(x_1,x_2,\dotsc)} = {\mathcal Z_{{\mathbf k}}(x_1,x_2,\dotsc)} \, {\mathcal Z_{{\mathbf q}}(x_1,x_2,\dotsc)}$. By specialization, the same is true for the exponential and type generating series.
Let ${\mathbb{Q}}_{\geq 0}$ denote the nonnegative rational numbers. An immediate consequence of Theorems \[t:main\] and \[t:maindual\] is the following.
\[c:Sp series\] Let ${\mathbf h}$ and ${\mathbf k}$ be connected Hopf monoids in species. Suppose ${\mathbf k}$ is either a Hopf submonoid or a quotient Hopf monoid of ${\mathbf h}$. Then the quotient of cycle index series ${\mathcal Z_{{\mathbf h}}(x_1,x_2,\dotsc)} / {\mathcal Z_{{\mathbf k}}(x_1,x_2,\dotsc)}$ is nonnegative, i.e., belongs to ${\mathbb{Q}}_{\geq 0}\llb x_1,x_2,\dotsc \rrb$. In particular, the quotients ${\mathcal E_{{\mathbf h}}(x)}/{\mathcal E_{{\mathbf k}}(x)}$ and ${\mathcal T_{{\mathbf h}}(x)}/{\mathcal T_{{\mathbf k}}(x)}$ are also nonnegative.
Given a connected Hopf monoid ${\mathbf h}$ in species, we may use Corollary \[c:Sp series\] to determine if a given species ${\mathbf k}$ may be a Hopf submonoid (or a quotient Hopf monoid).
\[ex: submonoid\] A of a set $I$ is an unordered collection of disjoint nonempty subsets of $I$ whose union is $I$. The notation $ab\pmrg c$ is shorthand for the partition $\bigl\{\{a,b\},\,\{c\}\bigr\}$ of $\{a,b,c\}$.
Let $\bm\Pi$ be the species of set partitions, i.e., $\bm\Pi[I]$ is the vector space with basis the set of all partitions of $I$. Let $\bm\Pi'$ denote the subspecies linearly spanned by set partitions with distinct block sizes. For example, $$\bm\Pi[a,b,c] = \field\bigl\{ abc, a\pmrg bc, ab\pmrg c, a\pmrg bc, a\pmrg b\pmrg c \bigr\}
{\quad\text{and}\quad}\bm\Pi'[a,b,c] = \field\bigl\{ abc, a\pmrg bc, ab\pmrg c, a\pmrg bc\bigr\}.$$ The sequences $(\dim\bm\Pi[n])_{n\geq 0}$ and $(\dim\bm\Pi'[n])_{n\geq 0}$ appear in [@Slo:oeis] as A000110 and A007837, respectively. We have $${\mathcal E_{\bm\Pi}(x)} = \exp\bigl(\exp(x)-1\bigr) = 1+x+x^2+\frac{5}{6}x^3+\frac{5}{8}x^4+\dotsb$$ and $${\mathcal E_{\bm\Pi'}(x)} = \prod_{n\geq 1}\bigl(1+\frac{x^n}{n!}\bigr) = 1+x+\frac{1}{2}x^2+\frac{2}{3}x^3+\frac{5}{24}x^4+\dotsb \,.$$
A Hopf monoid structure on $\bm\Pi$ is defined in [@AguMah:2010 Section 12.6]. There are many linear bases of $\bm\Pi$ indexed by set partitions, and many ways to embed $\bm\Pi'$ as a subspecies of $\bm\Pi$. Is it possible to embed $\bm\Pi'$ as a Hopf submonoid of $\bm\Pi$? We have $${\mathcal E_{\bm\Pi}(x)} \big/ {\mathcal E_{\bm\Pi'}(x)}
=
1+\frac12 x^2 - \frac1{3} x^3 + \frac{1}{2} x^4 - \frac{11}{30} x^5 + \dotsb \,,$$ so the answer is negative by Corollary \[c:Sp series\]. In fact, it is not possible to embed $\bm\Pi'$ as a Hopf submonoid of $\bm\Pi$ for any potential Hopf monoid structure on $\bm\Pi$.
We remark that the type generating series quotient for the pair of species in Example \[ex: submonoid\] is nonnegative: $$\begin{aligned}
{\mathcal T_{{\bm\Pi}}(x)} \ &= \ 1+x+2x^2+3x^3+5x^4+7x^5+11x^6+15x^7 + \dotsb \,, \\
{\mathcal T_{{\bm\Pi'}}(x)} \ &= \ 1 + x+x^2+2x^3+2x^4+3x^5+4x^6+5x^7 + \dotsb \,, \\
{\mathcal T_{{\bm\Pi}}(x)} \big/ {\mathcal T_{{\bm\Pi'}}(x)} \ &= \ 1+x^2+2x^4+3x^6+5x^8+7x^{10} + \dotsb \,. \end{aligned}$$ This can be understood by appealing to the Hopf algebra ${\textsl{Sym}}$ of symmetric functions. A basis for its homogenous component of degree $n$ is indexed by integer partitions of $n$, so ${\mathcal O_{{\textsl{Sym}}}(x)} = {\mathcal T_{{\bm\Pi}}(x)}$. Moreover, ${\mathcal T_{{\bm\Pi'}}(x)}$ enumerates the integer partitions with odd part sizes and ${\textsl{Sym}}$ does indeed contain a Hopf subalgebra with this Poincaré series. It is the algebra of Schur $Q$-functions. See [@Mac:1995 III.8]. Thus ${\mathcal T_{{\bm\Pi}}(x)} \big/ {\mathcal T_{{\bm\Pi'}}(x)}$ is nonnegative by Corollary \[c:cgVec series\].
Tests for Hopf monoids {#ss:ord-exp}
----------------------
Let $(a_n)_{n\geq 0}$ be a sequence of nonnegative integers with $a_0=1$. Does there exist a connected Hopf monoid ${\mathbf h}$ with $\dim {\mathbf h}[n]=a_n$ for all $n$? The next result provides conditions that the sequence $(a_n)_{n\geq 0}$ must satisfy in order for this to be the case. The proof makes use of the of Hopf monoids [@AguMah:2010 Sections 8.1 and 8.13]. If ${\mathbf h}$ and ${\mathbf k}$ are Hopf monoids, so is ${\mathbf h}\times{\mathbf k}$, with $({\mathbf h}\times{\mathbf k})[I] = {\mathbf h}[I]\otimes {\mathbf k}[I]$ for each finite set $I$. The exponential species ${\mathbf E}$ is the unit element for the Hadamard product.
\[c:ord-exp\] For any connected Hopf monoid in species ${\mathbf h}$, $$\Bigl(\sum_{n\geq 0} \dim {\mathbf h}[n]\, x^n\Bigr) \Big/ \Bigl(\sum_{n\geq 0} \frac{\dim {\mathbf h}[n]}{n!} \, x^n\Bigr) \in {\mathbb{Q}}_{\geq 0}\llb x\rrb.$$
Consider the canonical morphism of Hopf monoids ${\mathbf L}\onto {\mathbf E}$ [@AguMah:2010 Section 8.5]; it maps any linear order $\ell\in{\mathbf L}[I]$ to the basis element $\ast_I\in{\mathbf E}[I]$. The Hadamard product then yields a morphism of Hopf monoids $${\mathbf L}\times{\mathbf h}\onto {\mathbf E}\times{\mathbf h}\cong{\mathbf h}.$$ By Corollary \[c:Sp series\], ${\mathcal E_{{\mathbf L}\times{\mathbf h}}(x)}/{\mathcal E_{{\mathbf h}}(x)}\in
{\mathbb{Q}}_{\geq 0}\llb x\rrb$. Since ${\mathcal E_{{\mathbf L}\times{\mathbf h}}(x)}=\sum_{n\geq 0} \dim {\mathbf h}[n]\, x^n$, the result follows.
Let $a_n=\dim {\mathbf h}[n]$. Corollary \[c:ord-exp\] states that the ratio of the ordinary to the exponential generating function of the sequence $(a_n)_{n\geq 0}$ must be nonnegative. This translates into a sequence of polynomial inequalities, the first of which are as follows: $$\label{e:ord-exp}
5a_3\geq 3a_2a_1, \quad
23a_4+12a_2a_1^2 \geq 20a_3a_1+6a_2^2.$$ In particular, not every nonnegative sequence arises as the dimension sequence of a Hopf monoid.
The following test is of a similar nature, but involves the type instead of the exponential generating function. The conditions then depend not just on the dimension sequence of ${\mathbf h}$, but also on its species structure.
\[c:full-bosonic\] For any connected Hopf monoid in species ${\mathbf h}$, $$\Bigl(\sum_{n\geq 0} \dim {\mathbf h}[n]\, x^n\Bigr) \Big/ \Bigl(\sum_{n\geq 0} \dim {\mathbf h}[n]_{S_n}\, x^n\Bigr) \in {\mathbb{N}}\llb x\rrb.$$
We argue as in the proof of Corollary \[c:ord-exp\], using type generating functions instead. Since we have ${\mathcal T_{{\mathbf L}\times{\mathbf h}}(x)}= \sum_{n\geq 0} \dim {\mathbf h}[n]\, x^n$, the result follows.
The previous result may also be derived as follows. According to [@AguMah:2010 Chapter 15], associated to the Hopf monoid ${\mathbf h}$ there are two graded Hopf algebras ${\mathcal K}({\mathbf h})$ and ${\overline{{\mathcal K}}}({\mathbf h})$, as well as a surjective morphism $${\mathcal K}({\mathbf h}) \onto {\overline{{\mathcal K}}}({\mathbf h}).$$ Moreover, the Poincaré series for these Hopf algebras are $${\mathcal O_{{\mathcal K}({\mathbf h})}(x)} = \sum_{n\geq 0} \dim {\mathbf h}[n]\, x^n
{\quad\text{and}\quad}{\mathcal O_{{\overline{{\mathcal K}}}({\mathbf h})}(x)} = \sum_{n\geq 0} \dim {\mathbf h}[n]_{S_n}\, x^n.$$ Corollary \[c:full-bosonic\] now follows from (the dual form of) Corollary \[c:cgVec series\].
Additional tests for Hopf monoids {#ss:addtests}
---------------------------------
The method of Section \[ss:ord-exp\] can be applied in multiple situations in order to deduce additional inequalities that the dimension sequence of a connected Hopf monoid must satisfy. We illustrate this next.
Let $k$ be a fixed nonnegative integer. Let ${\mathbf E}^{{\bm\cdot}k}$ denote the $k$-th Cauchy power of the exponential species ${\mathbf E}$. The space ${\mathbf E}^{{\bm\cdot}k}[I]$ has a basis consisting of functions $f:I\to [k]$. The species ${\mathbf E}^{{\bm\cdot}k}$ carries a Hopf monoid structure [@AguMah:2010 Examples 8.17 and 8.18] and any fixed inclusion $[k]\hookrightarrow[k{+}1]$ gives rise to an injective morphism of Hopf monoids ${\mathbf E}^{{\bm\cdot}k}\hookrightarrow{\mathbf E}^{{\bm\cdot}(k+1)}$. Employing the Hadamard product as in Section \[ss:ord-exp\], we obtain an injective morphism of Hopf monoids $${\mathbf E}^{{\bm\cdot}k}\times{\mathbf h}\hookrightarrow {\mathbf E}^{{\bm\cdot}(k+1)}\times{\mathbf h}$$ where ${\mathbf h}$ is an arbitrary connected Hopf monoid. From the nonnegativity of the first coefficients of ${\mathcal E_{{\mathbf E}^{{\bm\cdot}(k+1)}\times{\mathbf h}}(x)} \big/ {\mathcal E_{{\mathbf E}^{{\bm\cdot}k}\times{\mathbf h}}(x)}$ we obtain $$(2k+1)a_2 \geq 2k a_1^2
{\quad\text{and}\quad}(3k^2+3k+1)a_3 \geq 3(3k^2+k)a_2a_1 - 6k^2a_1^3.$$ These inequalities hold for every $k\in{\mathbb{N}}$. Letting $k\to\infty$ we deduce $$\label{e:addcond}
a_2\geq a_1^2
{\quad\text{and}\quad}a_3\geq 3a_2a_1-2a_1^3.$$
Consider the species ${\mathbf{e}}$ . The set $I$ is a basis of the space ${\mathbf{e}}[I]$, so the dimension sequence of ${\mathbf{e}}$ is $a_n=n$. This sequence does not satisfy the second inequality in . Therefore, the species ${\mathbf{e}}$ does not carry any Hopf monoid structure.
A test for Hopf monoids over ${\mathbf E}$
------------------------------------------
Our next result is a necessary condition for a Hopf monoid in species to contain or surject onto the exponential species ${\mathbf E}$.
Given a sequence $(a_n)_{n\geq0}$, its $(b_n)_{n\geq0}$ is defined by $$b_n := \sum_{i=0}^n \binom{n}{i}(-1)^{i}\,a_{n-i}.$$
\[c:e-test\] Suppose ${\mathbf h}$ is a connected Hopf monoid that either contains the species ${\mathbf E}$ or surjects onto ${\mathbf E}$ (in both cases as a Hopf monoid). Let $a_n=\dim {\mathbf h}[n]$ and $\overline{a}_n=\dim {\mathbf h}[n]_{S_n}$. Then the binomial transform of $(a_n)_{n\geq0}$ must be nonnegative and $(\overline{a}_n)_{n\geq0}$ must be nondecreasing.
More plainly, in this setting, we must have the following inequalities: $$a_1\geq a_0, \quad
a_2\geq 2a_1-a_0, \quad
a_3\geq 3a_2-3a_1+a_0, \ \ \dotsc$$ and $\overline{a}_n\geq \overline{a}_{n-1}$ for all $n\geq 1$.
By Corollary \[c:Sp series\], the quotient ${\mathcal E_{{\mathbf h}}(x)} / {\mathcal E_{{\mathbf E}}(x)}$ is nonnegative. But ${\mathcal E_{{\mathbf E}}(x)} = \exp(x)$, so the quotient is given by $$b_0 + b_1 x + b_2 \frac{x^2}{2} + b_3 \frac{x^3}{3!} + \dotsb \,,$$ where $(b_n)_{n\geq0}$ is the binomial transform of $(a_n)_{n\geq0}$. Replacing exponential for type generating functions yields the result for $(\overline{a}_n)_{n\geq0}$, since ${\mathcal T_{{\mathbf E}}(x)} = \frac{1}{1-x}$.
Myhill’s theory of [@Dek:1990; @Myh:1958] provides necessary and sufficient conditions that a sequence $(a_n)_{n\geq0}$ must satisfy in order for its binomial transform to be nonnegative: the sequence must arise from a particular type of operator defined on finite sets. Work of Menni [@Men:2009] expands on this from a categorical perspective. It would be interesting to relate these ideas to the ones of this paper.
We make a further remark regarding connected Hopf monoids. These are Hopf monoids of a set theoretic nature. See [@AguMah:2010 Section 8.7] for details. Briefly, there are set maps $\mu_{S,T}:{\mathrm{H}}[S]\times{\mathrm{H}}[T]\to{\mathrm{H}}[I]$ and $\Delta_{S,T}:{\mathrm{H}}[I]\to{\mathrm{H}}[S]\times{\mathrm{H}}[T]$ which produce a Hopf monoid in (vector) species when the set species ${\mathrm{H}}$ is linearized. It follows that if ${\mathbf h}$ is a linearized Hopf monoid other than the trivial Hopf monoid ${\mathbf{1}}$, then there is a morphism of Hopf monoids from ${\mathbf h}$ onto ${\mathbf E}$. Thus, Corollary \[c:e-test\] provides a test for existence of a linearized Hopf monoid structure on ${\mathbf h}$.
We return to the species $\bm\Pi'$ of set partitions into distinct block sizes. We might ask if this can be made into a linearized Hopf monoid in some way (after Example \[ex: submonoid\], this would *not* be as a Hopf submonoid of $\bm\Pi$). With $a_n$ and $b_n$ as above, we have: $$\begin{aligned}
(a_n)_{n\geq0} \ &= \ 1, \ 1, \ 1, \ 4, \ \ 5, \ 16, \ \ 82, \ \ 169, \ 541, \dotsc \,, \\[.5ex]
(b_n)_{n\geq0} \ &= \ 1, \ 0, \ 0, \ 3, \, -8,\ 25, \, -9, \, -119, \ 736, \dotsc \,.\end{aligned}$$ Thus $\bm\Pi'$ fails the ${\mathbf E}$-test and the answer to the above question is negative.
A test for Hopf monoids over ${\mathbf L}$
------------------------------------------
Let ${\mathbf h}$ be a connected Hopf monoid in species. Let $a_n=\dim {\mathbf h}[n]$ and $\overline{a}_n=\dim {\mathbf h}[n]_{S_n}$. Note that the analogous integers for the species ${\mathbf L}$ of linear orders are $b_n=n!$ and $\overline{b}_n=1$. Now suppose that ${\mathbf h}$ contains ${\mathbf L}$ or surjects onto ${\mathbf L}$ as a Hopf monoid. An obvious necessary condition for this situation is that $a_n\geq n!$ and $\overline{a}_n\geq 1$. Our next result provides stronger conditions.
\[c:l-test\] Suppose ${\mathbf h}$ is a connected Hopf monoid that either contains the species ${\mathbf L}$ or surjects onto ${\mathbf L}$ (in both cases as a Hopf monoid). If $a_n=\dim {\mathbf h}[n]$ and $\overline{a}_n=\dim {\mathbf h}[n]_{S_n}$, then $$a_n\geq n a_{n-1} {\quad\text{and}\quad}\overline{a}_n\geq \overline{a}_{n-1} \quad(\forall\,n\geq1).$$
It follows from Corollary \[c:Sp series\] that both ${\mathcal E_{{\mathbf h}}(x)} / {\mathcal E_{{\mathbf L}}(x)}$ and ${\mathcal T_{{\mathbf h}}(x)} / {\mathcal T_{{\mathbf L}}(x)}$ are nonnegative. These yield the first and second set of inequalities, respectively.
Before giving an application of the corollary, we introduce a new Hopf monoid in species. A of a set $I$ is an ordered collection of disjoint nonempty subsets of $I$ whose union is $I$. The notation $ab\cmrg c$ is shorthand for the composition $\bigl(\{a,b\},\,\{c\}\bigr)$ of $\{a,b,c\}$.
Let $\mathbf{Pal}$ denote the species of set compositions whose sequence of block sizes is palindromic. So, for instance, $$\begin{gathered}
\mathbf{Pal}[a,b] = \field\bigl\{ ab, \ a\cmrg b, \ b\cmrg a\bigr\}\\
\intertext{and}
\mathbf{Pal}[a,b,c,d,e] = \field\bigl\{abcde, \ a\cmrg bcd\cmrg e, \ ab\cmrg c\cmrg de, \ a\cmrg b\cmrg c\cmrg d\cmrg e, \ \dotsc \bigr\}.\end{gathered}$$ The latter space has dimension $171=1 + 5\binom{4}{3} + \binom{5}{2}3 + 5!$ while $\dim \mathbf{Pal}[5]_{S_5}=4$ (accounting for the four types of palindromic set compositions shown above).
Given a palindromic set composition $F=F_1\cmrg \dotsb\cmrg F_r$, we view it as a triple $F=(F^-,F^0,F^+)$, where $F^-$ is the initial sequence of blocks, $F^0$ is the central block if this exists (if the number of blocks is odd) and otherwise it is the empty set, and $F^+$ is the final sequence of blocks. That is, $$F^-=F_1\cmrg \dotsb \cmrg F_{\lfloor r/2\rfloor},
\qquad\quad
F^0 = \begin{cases} F_{\lfloor r/2\rfloor+1} & \hbox{if $r$ is odd,}\\
\emptyset & \hbox{if $r$ is even,} \end{cases}
\qquad\quad
F^+ = F_{\lceil r/2+1\rceil}\cmrg \dotsb \cmrg F_{r}.$$
Given $S\subseteq I$, let $$F\vert_S := F_1\cap S\,\cmrg\, F_2\cap S\,\cmrg\, \dotsb \,\cmrg\, F_r\cap S\,,$$ where empty intersections are deleted. Then $F\vert_S$ is a composition of $S$. Let us say that $S$ is for $F$ if $$\#\bigl( F_i\cap S\bigr) = \# \bigl(F_{r+1-i}\cap S\bigr) \ \text{ \ for all $i=1,\ldots,r$.}$$ In this case, both $F\vert_S$ and $F\vert_{I\setminus S}$ are palindromic.
We employ the above notation to define product and coproduct operations on $\mathbf{Pal}$. Fix a decomposition $I = S{\sqcup}T$.\
[*Product. *]{} Given palindromic set compositions $F\in\mathbf{Pal}[S]$ and $G\in\mathbf{Pal}[T]$, we put $$\mu_{S,T}(F \otimes G) := \bigl(F^-\cmrg G^-, F^0\cup G^0, G^+\cmrg F^+\bigr).$$ In other words, we concatenate the initial sequences of blocks of $F$ and $G$ in that order, merge their central blocks, and concatenate their final sequences in the opposite order. The result is a palindromic composition of $I$. For example, with $S=\{a,b\}$ and $T=\{c,d,e,f\}$, $$(a\cmrg b) \otimes (c\cmrg de\cmrg f) \mapsto a\cmrg c\cmrg de\cmrg f\cmrg b.$$ [*Coproduct. *]{} Given a palindromic set composition $F\in\mathbf{Pal}[I]$, we put $$\Delta_{S,T}(F) := \begin{cases}
F\vert_S \otimes F\vert_T & \hbox{if $S$ is admissible for $F$,} \\
0 & \hbox{otherwise.}
\end{cases}$$ For example, with $S$ and $T$ as above, $$ad\cmrg b\cmrg e\cmrg cf \mapsto 0
{\quad\text{and}\quad}e\cmrg abcd\cmrg f \mapsto (ab) \otimes (e\cmrg cd\cmrg f).$$ These operations endow $\mathbf{Pal}$ with the structure of Hopf monoid, as may be easily checked.
A linear order may be seen as a palindromic set composition (with singleton blocks). Both Hopf monoids $\mathbf{Pal}$ and ${\mathbf L}$ are cocommutative and not commutative. We may then ask if $\mathbf{Pal}$ contains (or surjects onto) ${\mathbf L}$ as a Hopf monoid.
Writing $a_n=\dim\mathbf{Pal}[n]$, we have: $$\begin{gathered}
(a_n)_{n\geq0} \ = \ 1, \ 1, \ 3, \ 7, \ 43, \ 171, \ 1581, \ 8793, \ 108347, \dotsc \,.\end{gathered}$$ However, $$\begin{gathered}
(a_n - na_{n-1})_{n\geq1} \ = \ 0, \ 1, \ -2, \ 15, \ -44, \ 555, \ -2274, \ \ 38003, \dotsc \,,\end{gathered}$$ so $\mathbf{Pal}$ fails the ${\mathbf L}$-test and the answer to the above question is negative.
Examples of nonnegative quotients
---------------------------------
We comment on a few examples where the quotient power series ${\mathcal E_{{\mathbf h}}(x)}/{\mathcal E_{{\mathbf k}}(x)}$ is not only nonnegative but is known to have a combinatorial interpretation as a generating function.
Consider the Hopf monoid $\bm\Pi$ of set partitions. It contains ${\mathbf E}$ as a Hopf submonoid via the map that sends $\ast_I$ to the partition of $I$ into singletons. We have $${\mathcal E_{\bm\Pi}(x)}/{\mathcal E_{{\mathbf E}}(x)} = \exp\bigl(\exp(x)-x-1\bigr),$$ which is the exponential generating function for the number of set partitions into blocks of size strictly bigger than $1$. This fact may also be understood with the aid of Theorem \[t:main\], as follows.
Given $I=S\sqcup T$, the product of a partition $\pi\in\bm\Pi[S]$ and a partition $\rho\in\bm\Pi[T]$ is the partition $\pi\cdot\rho\in\bm\Pi[I]$ each of whose blocks is either a block of $\pi$ or a block of $\rho$. (In the notation of [@AguMah:2010 Section 12.6], we are employing the $h$-basis of $\bm\Pi$.) Now, the $I$-component of the right ideal ${\mathbf E}_+\bm\Pi$ is linearly spanned by elements of the form $\ast_S\cdot\pi$ where $I=S\sqcup T$ and $\pi$ is a partition of $T$. Then, since $\ast_S=\ast_{\{i\}}\cdot\ast_{S\setminus\{i\}}$ (for any $i\in S$), we have that ${\mathbf E}_+\bm\Pi[I]$ is linearly spanned by elements of the form $\ast_{\{i\}}\cdot\pi$ where $i\in I$ and $\pi$ is a partition of $I\setminus\{i\}$. But the above description of the product shows that these are precisely the partitions with at least one singleton block.
\[eg:derangement\] Consider the Hopf monoids ${\mathbf L}$ and ${\mathbf E}$ and the surjective morphism $\pi:{\mathbf L}\onto{\mathbf E}$ (as in the proof of Corollary \[c:ord-exp\]). We have $${\mathcal E_{{\mathbf L}}(x)}=\frac{1}{1-x}
{\quad\text{and}\quad}{\mathcal E_{{\mathbf E}}(x)}=\exp(x).$$ It is well-known [@BerLabLer:1998 Example 1.3.9] that $$\frac{\exp(-x)}{1-x} = \sum_{n\geq 0} \frac{d_n}{n!}\,x^n$$ where $d_n$ is the number of of $[n]$. Together with Theorem \[t:maindual\], this suggests the existence of a basis for the Hopf kernel of $\pi$ indexed by derangements. We construct such a basis and expand on this discussion in Section \[ss:derangement\].
Let $\bm{\Sigma}$ be the Hopf monoid of set compositions defined in [@AguMah:2010 Section 12.4]. It contains ${\mathbf L}$ as a Hopf submonoid via the map that views a linear order as a composition into singletons. (In the notation of [@AguMah:2010 Section 12.4], we are employing the $H$-basis of $\bm{\Sigma}$.) This and other morphisms relating ${\mathbf E}$, ${\mathbf L}$, $\bm{\Pi}$ and $\bm{\Sigma}$, as well as other Hopf monoids, are discussed in [@AguMah:2010 Section 12.8].
The sequence $(\dim \bm{\Sigma}[n])_{n\geq 0}$ is A000670 in [@Slo:oeis]. We have $${\mathcal E_{\bm{\Sigma}}(x)}=\frac{1}{2-\exp(x)}.$$ Moreover, it is known from [@Sta:1999 Exercise 5.4.(a)] that $$\frac{1-x}{2-\exp(x)} = \sum_{n\geq 0} \frac{s_n}{n!}\,x^n$$ where $s_n$ is the number of graphs with vertex set $[n]$ and no isolated vertices. Together with Theorem \[t:main\], this suggests the existence of a basis for $ \bm{\Sigma}/{\mathbf L}_+ \bm{\Sigma}$ indexed by such graphs. We do not pursue this possibility in this paper.
The dimension sequence of a connected Hopf monoid {#s:dimensions}
=================================================
Let ${\mathbf h}$ be a connected Hopf monoid and $a_n=\dim{\mathbf h}[n]$ for $n\in{\mathbb{N}}$. The results of Sections \[ss:ord-exp\] and \[ss:addtests\], derived from Theorem \[t:main\], impose restrictions on the sequence $a_n$ in the form of polynomial inequalities. The results of this section are neither weaker nor stronger than those of Section \[s:applications\], but provide supplementary information on the dimension sequence $a_n$. They do not make use of Theorem \[t:main\]. In this section, the base field $\field$ is of arbitrary characteristic.
\[p:aiaj\] For any $n,i$ and $j$ such that $n=i+j$, $$\label{e:aiaj}
a_n\geq a_ia_j.$$
Since ${\mathbf h}$ is connected, the compatibility axiom for Hopf monoids (diagram (8.18) in [@AguMah:2010 Section 8.3.1]) implies that the composite $${\mathbf h}[S]\otimes{\mathbf h}[T] \map{\mu_{S,T}} {\mathbf h}[I] \map{\Delta_{S,T}} {\mathbf h}[S]\otimes{\mathbf h}[T]$$ is the identity. The result follows by choosing any decomposition $I=S\sqcup T$ with ${\lvertI\rvert}=n$, ${\lvertS\rvert}=i$, and ${\lvertT\rvert}=j$.
The second inequality in may be combined with to obtain $$a_3-a_2a_1 \geq 2a_1(a_2-a_1^2)\geq 0.$$ Considerations of this type show that neither set of inequalities , or follows from the others.
As a first consequence of Proposition \[p:aiaj\], we derive a result on the growth of the dimension sequence.
\[c:expgrowth\] If $a_1\geq 1$, then the sequence $a_n$ is weakly increasing. If moreover there exists $k\geq 1$ such that $$a_k\geq 2 {\quad\text{and}\quad}a_i\geq 1\ \forall\, i=0,\ldots,k-1,$$ then $a_n=O(2^{n/k})$.
The first statement follows from $a_n\geq a_1a_{n-1}$. Now fix $k$ as in the second statement. Given $n\geq k$, write $n=qk+r$ with $q\in{\mathbb{N}}$ and $0\leq r\leq k-1$. From we obtain $$a_n\geq a_k^q a_r\geq 2^q = 2^{-r/k}2^{n/k}.$$ Thus $a_n=O(2^{n/k})$.
Define the of ${\mathbf h}$ to be the support of its dimension sequence; namely, $$\operatorname{supp}({\mathbf h})=\{n\in{\mathbb{N}}\mid a_n\neq 0\}.$$ We turn to consequences of Proposition \[p:aiaj\] on the support.
\[c:aiaj\] The set $\operatorname{supp}({\mathbf h})$ is a submonoid of $({\mathbb{N}},+)$.
By (co)unitality of ${\mathbf h}$, $0\in \operatorname{supp}({\mathbf h})$. (In fact, $a_0=1$ by connectedness.) By Proposition \[p:aiaj\], the set $\operatorname{supp}({\mathbf h})$ is closed under addition.
We mention that, conversely, given any submonoid $S$ of $({\mathbb{N}},+)$, there exists a connected Hopf monoid ${\mathbf h}$ such that $\operatorname{supp}({\mathbf h})=S$. Indeed, let $\bm\Pi_S[I]$ be the space spanned by the set of partitions of $I$ whose block sizes belong to $S\setminus\{0\}$. Then $\bm\Pi_S$ is a quotient Hopf monoid of $\bm\Pi$ and $\operatorname{supp}(\bm\Pi_S)=S$. (The former follows from the formulas in [@AguMah:2010 Section 12.6.2]; we employ the $h$-basis of $\bm\Pi$.)
Consider the special case of the previous paragraph in which $S$ is the submonoid of even numbers. Then $\Pi_S$ is the species of set partitions into blocks of even size. In particular, $a_n=0$ for all odd $n$, so the dimension sequence is neither increasing nor of exponential growth. This example shows that the hypotheses of Corollary \[c:expgrowth\] cannot be removed.
\[c:numerical\] The set $\operatorname{supp}({\mathbf h})$ is either $\{0\}$ or infinite. The set ${\mathbb{N}}\setminus\operatorname{supp}({\mathbf h})$ is finite if and only if $\gcd\bigl(\operatorname{supp}({\mathbf h})\bigr)=1$.
These statements hold for all submonoids of ${\mathbb{N}}$, hence for $\operatorname{supp}({\mathbf h})$ by Corollary \[c:aiaj\]. For the second statement, see [@RosGar:2009 Lemma 2.1].
We comment on counterparts for connected graded Hopf algebras of the results of this section.
Consider the polynomial Hopf algebra $H=\field[x_1,\ldots,x_k]$, in which the generators $x_i$ are primitive and of degree $1$. The dimension sequence is $a_n=\binom{n+k-1}{k-1}$. In contrast to Corollary \[c:expgrowth\], this sequence is polynomial even if $a_1>1$. It follows that Proposition \[p:aiaj\] has no counterpart for connected graded Hopf algebras $H$, and that the multiplication map $H_i\otimes H_j \to H_{i+j}$ is not injective in general.
Corollaries \[c:aiaj\] and \[c:numerical\] fail for connected Hopf algebras over a field of positive characteristic. In characteristic $p$, a counterexample is provided by $H=\field[x]/(x^p)$ with $x$ primitive and of degree $1$.
On the other hand, if the field characteristic is $0$, then the set $S=\{n\in{\mathbb{N}}\mid H_n\neq 0\}$ is a submonoid of $({\mathbb{N}},+)$. This follows from the fact that in this case any connected Hopf algebra is a domain. We expand on this point in the Appendix.
Hopf kernels for cocommutative Hopf monoids {#s:kernels}
===========================================
Hopf kernels enter in the decomposition of Theorem \[t:maindual\] (and in dual form, in Theorem \[t:main\]). For cocommutative Hopf monoids, Hopf kernels and Lie kernels are closely related, as discussed in this section. We provide a simple result that allows us to describe the Hopf kernel in certain situations and we illustrate it with the case of the canonical morphism ${\mathbf L}\onto{\mathbf E}$.
Hopf and Lie kernels
--------------------
The species ${\mathcal{P}}({\mathbf h})$ of of a connected Hopf monoid ${\mathbf h}$ is defined by $${\mathcal{P}}({\mathbf h})[I] =
\{ x \in {\mathbf h}[I] \mid \Delta(x) = 1 \otimes x + x \otimes 1\}$$ for each nonempty finite set $I$, and ${\mathcal{P}}({\mathbf h})[\emptyset]=0$. Equivalently, $${\mathcal{P}}({\mathbf h})[I] = \bigcap_{\substack{S\sqcup T=I \\ S,T\neq\emptyset}} \ker\bigl(\Delta_{S,T}:{\mathbf h}[I]\to {\mathbf h}[S]\otimes{\mathbf h}[T]\bigr).$$ It is a Lie submonoid of ${\mathbf h}$ under the commutator bracket. See [@AguMah:2010 Sections 8.10 and 11.9] for more information on primitive elements.
Let $\pi:{\mathbf h}\to{\mathbf k}$ be a morphism of connected Hopf monoids. It restricts to a morphism of Lie monoids ${\mathcal{P}}({\mathbf h})\to{\mathcal{P}}({\mathbf k})$, which we still denote by $\pi$. We define the of $\pi$ as the species $${\mathrm{Lker}}(\pi)=\ker\bigl(\pi:{\mathcal{P}}({\mathbf h})\to{\mathcal{P}}({\mathbf k})\bigr).$$ It is a Lie ideal of ${\mathcal{P}}({\mathbf h})$. The Hopf kernel ${\mathrm{Hker}}(\pi)$ is defined in .
\[l:lker\] Let $\pi:{\mathbf h}\to{\mathbf k}$ be a morphism of connected Hopf monoids. Then $${\mathrm{Lker}}(\pi)\subseteq{\mathrm{Hker}}(\pi).$$
Let $x\in{\mathrm{Lker}}(\pi)$. Then $$(\pi_{+\!}{\bm\cdot}\id)\Delta(x)=(\pi_{+\!}{\bm\cdot}\id)(1\otimes x + x\otimes 1)=0,$$ since $\pi_+(1)=0$ and $\pi(x)=0$. Thus $x\in{\mathrm{Hker}}(\pi)$.
\[l:hker\] Let $\pi:{\mathbf h}\to{\mathbf k}$ be a morphism of connected Hopf monoids. Then ${\mathrm{Hker}}(\pi)$ is a submonoid of ${\mathbf h}$.
By definition, $${\mathrm{Hker}}(\pi)=\Delta^{-1}\bigl({\mathrm{Eq}}(\pi{\bm\cdot}\id,\, \iota\epsilon{\bm\cdot}\id)\bigr),$$ where $\iota:{\mathbf{1}}\to{\mathbf k}$ is the unit of ${\mathbf k}$, $\epsilon:{\mathbf h}\to{\mathbf{1}}$ is the counit of ${\mathbf h}$, and ${\mathrm{Eq}}$ denotes the equalizer of two maps. Since $\pi$ and $\iota\epsilon$ are morphisms of monoids ${\mathbf h}\to{\mathbf k}$, the above equalizer is a submonoid of ${\mathbf h}{\bm\cdot}{\mathbf h}$. Since $\Delta$ is a morphism of monoids, ${\mathrm{Hker}}(\pi)$ is a submonoid of ${\mathbf h}$.
The following result provides the announced connection between Lie and Hopf kernels for cocommutative Hopf monoids. It makes use of the Poincaré-Birkhoff-Witt and Cartier-Milnor-Moore theorems for species, which are discussed in [@AguMah:2010 Section 11.9.3].
\[p:kernels\] Let $\pi:{\mathbf h}\to{\mathbf k}$ be a surjective morphism of connected cocommutative Hopf monoids. Then ${\mathrm{Hker}}(\pi)$ is the submonoid of ${\mathbf h}$ generated by ${\mathrm{Lker}}(\pi)$.
Lemmas \[l:lker\] and \[l:hker\] imply one inclusion. To conclude the equality, it suffices to check that the dimensions agree, or equivalently, that the exponential generating series are the same. (We are assuming finite dimensionality throughout.)
First of all, from Theorem \[t:maindual\], we have $${\mathcal E_{{\mathrm{Hker}}(\pi)}(x)}={\mathcal E_{{\mathbf h}}(x)}/{\mathcal E_{{\mathbf k}}(x)}.$$ Now, since ${\mathbf h}$ is cocommutative, we have $${\mathbf h}\cong {\mathcal{U}}\bigl({\mathcal{P}}({\mathbf h})\bigr) \cong {\mathcal{S}}\bigl({\mathcal{P}}({\mathbf h})\bigr) = {\mathbf E}\circ {\mathcal{P}}({\mathbf h}).$$ The first is an isomorphism of Hopf monoids (the Cartier-Milnor-Moore theorem), the second is an isomorphism of comonoids (the Poincaré-Birkhoff-Witt theorem), and the third is the definition of the species underlying ${\mathcal{S}}\bigl({\mathcal{P}}({\mathbf h})\bigr)$ [@AguMah:2010 Section 11.3]. It follows that $${\mathcal E_{{\mathbf h}}(x)} = \exp\bigl({\mathcal E_{{\mathcal{P}}({\mathbf h})}(x)}\bigr).$$ For the same reason, $${\mathcal E_{{\mathbf k}}(x)} = \exp\bigl({\mathcal E_{{\mathcal{P}}({\mathbf k})}(x)}\bigr),$$ and therefore $${\mathcal E_{{\mathrm{Hker}}(\pi)}(x)}=\exp\bigl({\mathcal E_{{\mathcal{P}}({\mathbf h})}(x)} - {\mathcal E_{{\mathcal{P}}({\mathbf k})}(x)}\bigr).$$
On the other hand, since the functors ${\mathcal{U}}$ and ${\mathcal{P}}$ define an adjoint equivalence, they preserve surjectivity of maps. Thus, the induced map $\pi:{\mathcal{P}}({\mathbf h})\to{\mathcal{P}}({\mathbf k})$ is surjective, and we have an exact sequence $$0\to {\mathrm{Lker}}(\pi) \to {\mathcal{P}}({\mathbf h}) \to {\mathcal{P}}({\mathbf k}) \to 0.$$ Hence, $${\mathcal E_{{\mathrm{Lker}}(\pi)}(x)}={\mathcal E_{{\mathcal{P}}({\mathbf h})}(x)} - {\mathcal E_{{\mathcal{P}}({\mathbf k})}(x)}.$$ Since ${\mathrm{Lker}}(\pi)$ is a Lie submonoid of ${\mathcal{P}}({\mathbf h})$, the submonoid of ${\mathbf h}$ generated by ${\mathrm{Lker}}(\pi)$ identifies with ${\mathcal{U}}\bigl({\mathrm{Lker}}(\pi)\bigr)$. Therefore, as above, the generating series for the latter submonoid is $$\exp\bigl({\mathcal E_{{\mathrm{Lker}}(\pi)}(x)}\bigr)=
\exp\bigl({\mathcal E_{{\mathcal{P}}({\mathbf h})}(x)} - {\mathcal E_{{\mathcal{P}}({\mathbf k})}(x)}\bigr)={\mathcal E_{{\mathrm{Hker}}(\pi)}(x)},$$ which is the desired equality.
The results of this section hold also for connected (not necessarily graded or finite dimensional) Hopf algebras. See [@BCM:1986 Example 4.20] for a proof of Proposition \[p:kernels\] in this setting. The proof above used finite dimensionality of the (components of the) species, but this hypothesis is not necessary. The proof in [@BCM:1986] may be adapted to yield the result for arbitrary species.
The Lie kernel of $\pi:{\mathbf L}\onto{\mathbf E}$ {#ss:cycle}
---------------------------------------------------
We return to the discussion in Example \[eg:derangement\]. The primitive elements of the Hopf monoids ${\mathbf E}$ and ${\mathbf L}$ are described in [@AguMah:2010 Example 11.44]. We have that $${\mathcal{P}}({\mathbf E})={\mathbf{X}}{\quad\text{and}\quad}{\mathcal{P}}({\mathbf L})={\mathbf{Lie}}$$ where ${\mathbf{X}}$ is the species of , $${\mathbf{X}}[I] =
\begin{cases}
\field & \text{ if ${\lvertI\rvert}=1$, } \\
0 & \text{ otherwise,}
\end{cases}$$ and ${\mathbf{Lie}}$ is the species underlying the . It follows that the Lie kernel of the canonical morphism $\pi:{\mathbf L}\onto{\mathbf E}$ is given by $$\label{e:lker}
{\mathrm{Lker}}(\pi)[I]=
\begin{cases}
{\mathbf{Lie}}[I] & \text{ if ${\lvertI\rvert}\geq 2$, } \\
0 & \text{ otherwise.}
\end{cases}$$
Before moving on to the Hopf kernel of $\pi$, we provide some more information on the species ${\mathbf{Lie}}$.
Let $I$ be a finite nonempty set and $n={\lvertI\rvert}$. It is known that the space ${\mathbf{Lie}}[I]$ is of dimension $(n-1)!$. We proceed to describe a linear basis indexed by on $I$. A cyclic order on $I$ is an equivalence class of linear orders on $I$ modulo the action $$i_1\cmrg \cdots \cmrg i_{n-1} \cmrg i_n \ \mapsto \ i_n\cmrg i_1\cmrg \cdots \cmrg i_{n-1}$$ of the cyclic group of order $n$. Each class has $n$ elements so there are $(n-1)!$ cyclic orders on $I$. We use $(b,a,c)$ to denote the equivalence class of the linear order $b\cmrg a\cmrg c$.
We fix a finite nonempty set $I$ and choose a linear order $\ell_0$ on $I$, say $$\ell_0 = i_1\cmrg i_2 \cmrg \cdots \cmrg i_n.$$ The basis of ${\mathbf{Lie}}[I]$ will depend on this choice. Given a cyclic order $\gamma$ on $I$, let $S$ be the subset of $I$ consisting of the elements encountered when traversing the cycle from $i_1$ to $i_2$ clockwise, including $i_1$ but excluding $i_2$ (these are the first and second elements in $\ell_0$, respectively). Let $T$ consist of the remaining elements (from $i_2$ to $i_1$). Note that $i_1\in S$ and $i_2\in T$, so both $S$ and $T$ are nonempty. The cyclic order $\gamma$ on $I$ induces cyclic orders on $S$ and $T$. We denote them by $\gamma|_S$ and $\gamma|_T$. An element $p_\gamma\in {\mathbf L}[I]$ is defined recursively by $$p_\gamma := [p_{\gamma|_S}, p_{\gamma|_T}]= p_{\gamma|_S}\cdot p_{\gamma|_T}-
p_{\gamma|_T}\cdot p_{\gamma|_S}.$$ The elements $p_{\gamma|_S}\in{\mathbf L}[S]$ and $p_{\gamma|_T}\in{\mathbf L}[T]$ are themselves defined with respect to the induced linear orders $(\ell_0)|_S$ and $(\ell_0)|_T$. The recursion starts with the case when $I$ is a singleton $\{a\}$. In this case, we set $$p_{(a)}:=a\in {\mathbf L}[a]$$ (the unique linear order).
Clearly $a\in{\mathbf L}[a]$ is a primitive element. Since the primitive elements are closed under commutators, we have $p_\gamma\in {\mathbf{Lie}}[I]$. Moreover, we have the following.
\[p:liebase\] For fixed $I$ and $\ell_0$ as above, the set $$\bigl\{p_\gamma \mid \text{$\gamma$ is a cyclic order on $I$}\bigr\}$$ is a linear basis of ${\mathbf{Lie}}[I]$.
The construction of the elements $p_\gamma$ is a reformulation of the familiar construction of the *Lyndon* basis of a free Lie algebra [@Lot:1997; @Reu:1993; @Reu:2003]. Reading the elements of the cyclic order $\gamma$ clockwise starting at the minimum of $\ell_0$ gives rise to a Lyndon word on $I$ (without repeated letters). The cyclic orders $\gamma|_S$ and $\gamma|_T$ give rise to the Lyndon words in the canonical factorization of this Lyndon word.
For example, suppose that $I=\{a,b,c,d\}$, $\ell_0=a\cmrg b\cmrg c\cmrg d$ and $\gamma=(b,a,c,d)$. Then $$\begin{aligned}
p_{(b,a,c,d)} &= [p_{(a,c,d)}, p_{(b)}]=\bigl[ [p_{(a)}, p_{(c,d)}], p_{(b)}\bigr]
=\Bigl[ \bigl[ p_{(a)}, [p_{(c)}, p_{(d)}] \bigr], p_{(b)} \Bigr]
=\Bigl[ \bigl[ a, [c, d] \bigr], b \Bigr] \\
&=
a\cmrg c\cmrg d\cmrg b - a\cmrg d\cmrg c\cmrg b - c\cmrg d\cmrg a\cmrg b
+d\cmrg c\cmrg a\cmrg b
-b\cmrg a\cmrg c\cmrg d +b\cmrg a\cmrg d\cmrg c +b\cmrg c\cmrg d\cmrg a
-b\cmrg d\cmrg c\cmrg a.\end{aligned}$$
The vector species ${\mathbf{Lie}}$ is *not* the linearization of the set species of cycles. Note also that, for a general bijection $\sigma:I\to J$, the $p$-basis of ${\mathbf{Lie}}[I]$ will not map to the $p$-basis of ${\mathbf{Lie}}[J]$ under ${\mathbf L}[\sigma]$.
The Hopf kernel of $\pi:{\mathbf L}\onto{\mathbf E}$ {#ss:derangement}
----------------------------------------------------
The above description of the Lie kernel of $\pi:{\mathbf L}\onto{\mathbf E}$ together with Proposition \[p:kernels\] imply that the Hopf kernel of $\pi$ is given by $${\mathrm{Hker}}(\pi)[I]=\sum_{k\geq 1}\sum_{\substack{S_1\sqcup\cdots\sqcup S_k=I \\ {\lvertS_r\rvert}\geq 2\,\,\forall r}}
{\mathbf{Lie}}[S_1]\cdots{\mathbf{Lie}}[S_k].$$ An element in ${\mathbf{Lie}}[S_1]\cdots{\mathbf{Lie}}[S_k]$ is a $k$-fold product of primitive elements $x_r\in{\mathbf{Lie}}[S_r]$; each $S_r$ must have at least $2$ elements. We proceed to describe a linear basis for ${\mathrm{Hker}}(\pi)[I]$.
As in Section \[ss:cycle\], we fix a linear order $\ell_0=i_1\cmrg i_2 \cmrg \cdots \cmrg i_n$ on $I$. The basis will be indexed by of $\ell_0$. A derangement of $\ell_0$ is a linear order $\ell=j_1\cmrg j_2 \cmrg \cdots \cmrg j_n$ on $I$ such that $i_r\neq j_r$ for all $r=1,\ldots,n$.
View linear orders as bijections $[n]\to I$ and define $\sigma:=\ell\circ\ell_0^{-1}$. Then $\sigma$ is a permutation of $I$ and $\ell$ is a derangement of $\ell_0$ precisely when $\sigma$ has no fixed points.
Let $\ell$ be a derangement of $\ell_0$ and $\sigma$ the associated permutation. Let $S_1,\ldots,S_k$ be the orbits of $\sigma$ on $I$ labeled so that $$\min S_1<\cdots<\min S_k
\text{ \ according to $\ell_0$,}$$ and let $\gamma_r$ be the cyclic order on $S_r$ induced by $\sigma$. In other words, $\sigma=\gamma_1\cdots\gamma_k$ is the factorization of $\sigma$ into cycles, ordered in this specific manner.
Employing the $p$-basis of ${\mathbf{Lie}}$ from Section \[ss:cycle\] (defined with respect to $\ell_0$ and the orders induced by $\ell_0$ on subsets of $I$), we define an element $p_{\ell}\in{\mathbf L}[I]$ by $$p_{\ell} :=p_{\gamma_1}\cdots p_{\gamma_k}.$$ By assumption, ${\lvertS_r\rvert}\geq 2$ for all $r$. Hence $p_{\gamma_r}\in{\mathrm{Lker}}(\pi)[S_r]$ and $p_{\ell}\in{\mathrm{Hker}}(\pi)[I]$.
For example, let $I=\{e,i,m,s,t\}$, $\ell_0 =s\cmrg m \cmrg i \cmrg t \cmrg e$ and $\ell = i \cmrg t \cmrg e \cmrg m \cmrg s$. Then $$\sigma = (s,i,e)(m,t), \quad S_1 = \{i,e,s\},\quad S_2 = \{m,t\},$$ and $$p_\ell = p_{(s,i,e)}p_{(m,t)} = \left[p_{(s)},p_{(i,e)}\right]p_{(m,t)} = \bigl[ s,[i,e]\bigr][m,t].$$
\[p:hopfbase\] For fixed $I$ and $\ell_0$ as above, the set $$\bigl\{p_\ell \mid \text{$\ell$ is a derangement of $\ell_0$}\bigr\}$$ is a linear basis of ${\mathrm{Hker}}(\pi)[I]$.
This follows from Proposition \[p:liebase\] and the Poincaré-Birkhoff-Witt theorem.
We describe the $p$-basis of ${\mathrm{Hker}}(\pi)[I]$ in low cardinalities. Throughout, we choose $$\ell_0=a\cmrg b \cmrg c \cmrg\cdots\,.$$ The space ${\mathrm{Hker}}(\pi)[a,b]$ is $1$-dimensional, linearly spanned by $$p_{b\cmrg a} = p_{(a,b)} =[a,b].$$ The space ${\mathrm{Hker}}(\pi)[a,b,c]$ is $2$-dimensional, linearly spanned by $$\begin{aligned}
p_{b\cmrg c\cmrg a} = p_{(a,b,c)} = [ p_{(a)},p_{(b,c)}] = \bigl[ a,[b,c]\bigr],\\
p_{c\cmrg a\cmrg b} = p_{(a,c,b)} = [p_{(a,c)},p_{(b)}] = \bigl[ [a,c], b\bigr].\end{aligned}$$ The space ${\mathrm{Hker}}(\pi)[a,b,c,d]$ is $9$-dimensional. There are $6$ basis elements corresponding to $4$-cycles, such as $$p_{c\cmrg a\cmrg d\cmrg b} = p_{(a,c,d,b)} = \Bigl[ \bigl[ a, [c, d] \bigr], b \Bigr],$$ and $3$ basis elements corresponding to products of two $2$-cycles, such as $$p_{b\cmrg a\cmrg d\cmrg c} = p_{(a,b)}p_{(c,d)} = [a,b]\cdot [c,d].$$
Appendix {#appendix .unnumbered}
========
The following fact was referred to in the last remark in Section \[s:dimensions\].
\[p:domain\] Let $H$ be a connected (not necessarily graded) Hopf algebra over a field of characteristic $0$. Then $H$ is a domain.
This result is proven in [@WZZ:2011 Lemma 1.8(a)], where it is attributed to Le Bruyn. We provide a different proof here.
Let $K$ denote the associated graded Hopf algebra with respect to the coradical filtration of $H$. Since $H$ is connected, $K$ is commutative [@AguSot:2005 Remark 1.7]. Now by [@Rad:1979 Proposition 1.2.3], $K$ embeds in a shuffle Hopf algebra. The latter is a free commutative algebra [@Rad:1979 Corollary 3.1.2], hence a domain. It follows that $K$ and hence also $H$ are domains.
Over a field of positive characteristic, the restricted universal enveloping algebra $\mathfrak u(\mathfrak g)$ of a finite dimensional nonzero Lie algebra $\mathfrak g$ is a connected Hopf algebra that is not a domain. Indeed, in this case $u(\mathfrak g)$ is finite dimensional [@Jac:1962 Theorem V.12] and so has a nontrivial idempotent, being a Hopf algebra with integrals [@Mon:1993 Theorem 2.1.3].
|
---
abstract: |
This is the first part of the article we promised at the end of [@foootech Section 1]. We discuss the foundation of the virtual fundamental chain and cycle technique, especially its version appeared in [@FO] and also in [@fooobook2 Section A1, Section 7.5], [@fooo09 Section 12], [@fooo091]. In Part 1, we focus on the construction of the virtual fundamental chain on a single space with Kuranishi structure. We mainly discuss the de Rham version and so work over $\R$-coefficients, but we also include a self-contained account of the way how to work over $\Q$-coefficients in case the dimension of the space with Kuranishi structure is $\le 1$.
Part 1 of this document is independent of our earlier writing [@foootech]. We also do not assume the reader have any knowledge on the pseudo-holomorphic curve, in Part 1.
Part 2 (resp. Part 3), which will appear in the near future, discusses the case of a system of Kuranishi structures and its simultaneous perturbations (resp. the way to implement the abstract story in the study of moduli spaces of pseudo-holomorphic curves).
address:
- 'Simons Center for Geometry and Physics, State University of New York, Stony Brook, NY 11794-3636 U.S.A. & Center for Geometry and Physics, Institute for Basic Sciences (IBS), Pohang, Korea'
- 'Center for Geometry and Physics, Institute for Basic Sciences (IBS), Pohang, Korea & Department of Mathematics, POSTECH, Pohang, Korea'
- 'Graduate School of Mathematics, Nagoya University, Nagoya, Japan'
- 'Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan'
author:
- 'Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono'
date: 'Feb 7th, 2015'
title: 'Kuranishi structure, Pseudo-holomorphic curve, and Virtual fundamental chain: Part 1'
---
[^1]
Introduction
============
The purpose of this article is to present the virtual fundamental chain technique and its application to the moduli space of pseudo-holomorphic curves, in as much detail as possible. The technique of virtual fundamental cycles and chains was invented in the year 1996 by groups of mathematicians [@FO], [@LiTi98], [@LiuTi98], [@Rua99], [@Siebert] and applied to many purposes subsequently. Several references such as [@fooo06], [@ChenTian], [@ChenLieWang], [@LuT] provide various versions and details of this technique.
In this article (besides various improvement of the detail of the proof, presentation etc.) we add the following points to our earlier writing [@foootech] of similar nature.
1. We present the detail of the chain level argument, especially in the de-Rham version.
2. We provide a package of the statements which arise from the virtual fundamental chain and cycle technique, in the way one can directly quote and use without referring to their proofs.
3. We give constructions of algebraic structures from systems of spaces with Kuranishi structures. We separate the topological and algebraic issues from analytic ones of the story so that this part of the construction can be rigorously stated and proved without referring to the construction of such system.[^2]
4. We discuss in detail the construction of a system of spaces with Kuranishi structures that satisfies the axioms we mentioned in Item (3). Compared to [@foootech], we put more emphasis on the consistency of Kuranishi structures we obtain. Namely our emphasis lies in the construction of such a system of Kuranishi structures rather than that of a single Kuranishi structure.
5. We also provide details of the proof of certain fundamental properties of Gromov-Witten invariants. In this article we clarify the fact that we can work out this proof without using continuous family of perturbations. In other words, we need only the technique that already appeared in [@FO] but not those developed in the later literature.
6. We also explain a method of working with the $\Q$-coefficient without using a triangulation of the perturbed moduli space. This is possible if we only need to study moduli space of virtual dimension 0,1 and negative. Note that these cases handle all the applications appearing in [@FO].
This article is divided into three parts.
Part 1 studies space with Kuranishi structure. Here we consider only single space with Kuranishi structure and not its system. We also write Part 1 (and Part 2) so that it can be read without any knowledge of pseudo-holomorphic curves. Especially readers’ knowledge on analytic details on pseudo-holomorphic curves is not required. Occasionally we mention the moduli space of pseudo-holomorphic curves in Parts 1 and 2, but none of the proof relies on them. This is because we do so only at the place where we think some motivation of the construction or the definition etc. would be useful. We believe that for the reader whose background is more on topology and/or algebra but not on analysis, this way of writing is useful. We also believe that our writing clarifies the fact that problem of transversality of the moduli space of pseudo-holomorphic curves lies in the realm of differential topology rather than one in functional or nonlinear analysis.
Item (1) mentioned above comprises Part 1.
In Part 2 we study systems of Kuranishi structures and provide a systematic way of building algebraic structures on certain chain complex arising from such a system. We discuss the case of Floer homology (of periodic Hamiltonian system) and of $A_{\infty}$ algebra appearing in Lagrangian Floer theory. Item (3) above comprises Part 2.
In Part 3 we present in detail the construction of the moduli spaces of pseudo-holomorphic curves and its Kuranishi structure which satisfies the axioms we formulate in Part 2. We discuss them for the case of Floer homology (of periodic Hamiltonian system) and of $A_{\infty}$ algebra appearing in Lagrangian Floer theory. We also discuss the case of Gromov-Witten invariants. (Item (5) is in Part 3.)
We remark Parts 1 and 2 of this article are self-contained. It does not use the results of [@foootech]. In other words, we reprove certain parts of [@foootech] in this article. (We quote some general topology issue from [@foooshrink].) In Part 3, we quote [@foootech] in several places.
Introduction to each part follows.
Now we upload Part 1. In the text below we quote results, sections etc. of Parts 2 and 3. They will appear later on. Note we quote Parts 2 and 3 in order to provide a view how the results of Part 1 are used or to provide perspective to related results. As far as the logic of the proof concerns, the discussion of Part 1 is complete without referring Parts 2 and 3.
\[sec:introduction\]
Introduction to Part 1 {#subsec:intro1}
----------------------
In Part 1 we discuss the space with Kuranishi structure and define its virtual fundamental chain.
Kuranishi structure is a ‘generalization of the notion of manifold which allows singularity’. We entirely work in $C^{\infty}$-category. In general, singularity in $C^{\infty}$-category can be much wilder than one in analytic category. Our aim is not developing a general theory of ‘$C^{\infty}$ analogue of scheme’ but a more restricted one, which is, to define virtual fundamental chains and cycles. Because of this, our definitions and constructions are designed so that they are useful for that particular purpose.
Now an outline of the main results proved in Part 1 is in order. Since most basic and conceptual ideas behind the definitions of Kuranishi structures and virtual fundamental chains have been explained in various other previous writings of ours (see [@fooo010] [@Fusuugaku] [@fooobook Section A1] [@fooo09 Section 12] [@FO] for example), here we discuss more technical aspect of the story, that is, the way how we rigorously realize those basic concepts. In fact the aim of this article is [*not*]{} to explain the basic concepts of the theory or to present the main ideas of the construction of virtual fundamental chains (which, according to the opinion of the authors of this article, is definitely more important than the details provided here) but to present its [*technical*]{} details with as much details as possible
Roughly speaking Kuranishi structure of a space $X$ represents $X$ locally as a zero set $s^{-1}(0)$ where $s$ is a section of vector bundle $E \to U$ over an orbifold $U$. More precisely a Kuranishi chart is a quadruple $\mathcal U = (U,E,s,\psi)$ where $s$ is a section of a vector bundle $E \to U$ over an orbifold $U$ and $\psi : s^{-1}(0) \to X$ is a homeomorphism to its image. (See Definition \[kuranishineighborhooddef\].) We call $s$ a *Kuranishi map*. In fact this description in the situation of various applications appears as a Kuranishi model of the local description of the moduli problem. We also need appropriate notion of coordinate changes among them.
The main difference of the coordinate change between Kuranishi charts from the coordinate change between manifold or orbifold charts lies in the fact that the coordinate change of Kuranishi charts may not be a local ‘isomorphism’. This is because the dimension of the space with Kuranishi structure is, by definition, $\dim U - {\rm rank} E$ and so the orbifolds $U_i$ of two Kuranishi charts $(U_i,E_i,s_i,\psi_i)$ of the same space $X$ may have different dimension. (Namely $\dim U_1 \ne \dim U_2$ in general.) Recall that in the description of the manifold or orbifold structure one uses a pseudo-group or a groupoid. This description certainly is [*not*]{} useful for the Kuranishi structure since the most important axiom of groupoid (the invertibility of the morphisms) is exactly the one we give up. One may use a version of category of fractions to invert the arrows appearing in the definition of coordinate change of Kuranishi charts to adapt the groupoid language. This is the route taken, for example, by D. Joyce in [@joyce2], which however is not the one we take.
There are two slightly different versions of the definition of ‘spaces with Kuranishi type structures’ in [@FO]. One is what we call ‘a Kuranishi structure’ the other ‘a good coordinate system’.
In the manifold theory (or in the orbifold theory) we consider a maximal set of the coordinate charts compatible with one another, and call this set a structure of $C^{\infty}$ manifold. Equivalently for a manifold $M$ we assign a (germ of a) coordinate neighborhood to each point $p$ of $M$. This way of defining the manifold structure is somehow more canonical. On the other hand to perform various operations using the given manifold structure, we usually take a locally finite cover consisting of coordinate charts in the given manifold structure. The latter is especially necessary when we use a partition of unity as in the case of defining integration of differential form on a manifold $M$. Certain amount of general topology is needed to be worked out in the manifold course to prove existence of an appropriate locally finite cover extracted from the set of infinitely many charts defining the manifold structure. This technicality is not entirely trivial but is never a conceptual heart of the theory. [^3]
The relationship between the Kuranishi structure and the good coordinate system is similar to the one between these two ways of describing a $C^{\infty}$ structure of manifolds. A Kuranishi structure of $X$ assigns a Kuranishi chart $\mathcal U_p$ to each point $p \in X$. So in particular it contains uncountably many charts. On the other hand, a good coordinate system of $X$ consists of locally finite (finite if $X$ is compact) Kuranishi charts $\mathcal U_{\frak p}$. ($\frak p \in \frak P$. Here $\frak P$ is certain index set.)
Note coordinate changes between Kuranishi charts are not necessarily a local isomorphism. In general we have coordinate change only in one direction. In the case of Kuranishi structure the coordinate change from $\mathcal U_{q}$ to $\mathcal U_{p}$ is defined only when $q \in U_{p}$. (Here $U_p$ is a neighborhood of $p$.) We remark that $q \in U_p$ does not imply $p \in U_q$.
In the case of good coordinate system the set of Kuranishi charts is parameterized by a certain partially ordered set $\frak P$ and the coordinate change from $\mathcal U_{\frak q}$ to $\mathcal U_{\frak p}$ is defined only when $\frak q \le \frak p$.
(Chen-Tian’s approach) In the paper [@ChenTian] [^4] (and several other references which follow Chen-Tian’s approach) the set of charts are parameterized by a subset of certain finite set $I$. (Namely $\frak P \subseteq 2^I$.) The partial order in Chen-Tian’s approach is the usual inclusion of subsets. (See [@ChenTian Definition 2.1].) If we examine the proof of the construction of Kuranishi structure in [@FO], we find that the partially ordered set which appears there is actually parameterized by a subset of certain subset of the moduli space. (Namely $I$ is a finite subset of $X$.) Thus our approach is not so far away from Chen-Tian’s approach in applications. We, however, think our way of including arbitrary partially ordered set is more flexible than Chen-Tian’s. (Their approach certainly works at least for their purpose.)
For example the approach using $\frak P \subseteq 2^I$ does not seem to work literally in the case when we try to construct a system of Kuranishi structures on moduli spaces of pseudo-holomorphic curves (without boundary) so that it is consistent with the fiber product description of the stratum. (See Part 3.)[^5] One can certainly modify or use its variant to include that case. However modifying the definition when new applications appear is not a good way to develop general theory, especially such a lengthy one we are building. Because then one need either to repeat all the proofs again or to say ‘We can prove it in the same way’. In this article we want to avoid saying this sentence as much as possible though it might be impossible to completely avoid it.[^6] The structure of partially ordered set is certainly the structure which remains in all the expected generalizations and is also the very structure we need for the inductive argument we will use to work on good coordinate system.
The main reason why the notion of Kuranishi structure was introduced is to work out the transversality issue appearing in the theory of pseudo-holomorphic curves and etc. For this purpose it is important to perturb the Kuranishi map, denoted by $s$, so that the resulting perturbed map becomes transversal to $0$. The original proof of transversality theorem due to R. Thom [@thom] constructed perturbation by using induction on charts. We follow this proof. We use partial order on the index set $\frak P$ of Kuranishi charts for this purpose. Note in order to perform this inductive construction it is essential to use good coordinate system rather than Kuranishi structure since it is hard to work on uncountably many coordinate charts consisting Kuranishi structure. Thus construction of good coordinate system out of Kuranishi structure is an important step in the construction of an appropriate perturbation and resolving transversality issue.
On the other hand, the fact that the Kuranishi structure is more canonical than the good coordinate system can be seen from the following point. We can define the product of two spaces with Kuranishi structure in a canonical way. However the definition of product of two spaces with good coordinate system is more complicated and is less natural. For our application we need to construct virtual fundamental chains in the way that is compatible with fiber product or with direct product. For this purpose working with Kuranishi structure is more appropriate.
Thus during the various constructions, we need to go from a Kuranishi structure to a good coordinate system and back several times.
Concerning this transition from one to the other, there is one difference between the current story and the manifold theory. In the manifold theory, one starts with a $C^{\infty}$ structure (which consists of all the compatible coordinate charts) and then pick up an appropriate locally finite cover. It is a fact that one can always recover original $C^{\infty}$ structure from the charts consisting of the chosen locally finite cover. In the current story of Kuranishi structures, we start with a Kuranishi structure $\widehat{\mathcal U}$ and construct a compatible good coordinate system $\widetriangle{\mathcal U}$. We can use the resulting good coordinate system $\widetriangle{\mathcal U}$ to construct another Kuranishi structure $\widehat{\mathcal U^+}$. Unfortunately $\widehat{\mathcal U^+}$ is different from $\widehat{\mathcal U}$ in general. In other words we lose certain information while we go from a Kuranishi structure to its associated good coordinate system. Some portion of the discussion in Part 1 is devoted to showing that this loss does not affect the construction of virtual fundamental chain.
This technical trouble seems to reflect the fact that the notion of Kuranishi structure is conceptually less canonical than that of manifold. For example we can construct a Kuranishi structure on (practically all the) moduli spaces of pseudo-holomorphic curves. However the Kuranishi structure we obtain is not unique but depends on some choices. (We compensate this shortcoming by using an appropriate notion of cobordism.) In case the moduli space happens to be a manifold (and the equation defining the moduli space is Fredholm regular) the $C^{\infty}$ structure on the moduli space is certainly canonical.[^7] Another sign that the notion of Kuranishi structure is not canonical enough is that we do not know how to define a morphism between two spaces with Kuranishi structure.
It seems that the route taken by Joyce [@joyce2], [@joyce4] resolves these two issues. See Remark \[joycerem\] for the reason why we nevertheless use our definitions of Kuranishi structure and good coordinate system.
The history of the 20 century mathematics tells us that in the realm of algebro-geometric or complex analytic category, going to the ultimatum in making all the constructions, definitions and etc. as canonical as possible is often the correct point of view[^8] even if it looks cumbersome at the beginning. It also tells us that in the realm of differential geometry or in that of $C^{\infty}$ functions, an attempt to realize a canonical construction in the ultimate level frequently fails [^9] and so we are forced to find the suitable place to compromise and to be content ourselves with being able to achieve the particular purpose we pursue.
The whole exposition of Part 1 is designed in the way that most of the statements and proofs find their analogs in the corresponding statements and proofs of the standard theory of manifolds. Once the right statements are given, the proofs are fairly obvious most of the times,[^10] although it is not entirely so because there are also some differences between the Kuranishi structure and the manifold structure in certain technical points. These differences result in cumbersome and technically heavy proofs in those cases. However we emphasize that these are rather technical points and do not comprise the conceptional heart of the theory. Especially for the researchers who have in their mind the construction of virtual fundamental chains in various concrete geometric problems, thorough understanding of cumbersome details of these technicality should not be a part of everyone’s required background. [^11]
It seems to us that the whole theory now becomes nontrivial only because its presentation is heavy and lengthy. The method of making the proof ‘locally trivial’, which we are taking here, has been used in various branches of mathematics as an established method for building the foundation of a new theory, especially when the theory is conceptually simple but meets certain complicated technicality in its rigorous details. We choose the ‘Bourbaki style’ way of writing this article. Especially, we do our best effort to make explicit the assumptions we work with and the conclusions we obtain. As its consequence, there appear so many definitions and statements in the text, which we acknowledge is annoying but may be inevitable.
We hope that in the near future many users of virtual techniques via the Kuranishi structure appreciate that they do not need to know anything more than a small number of basic definitions and theorems together with the fact that the story of Kuranishi structure is mostly similar to that of smooth manifolds. Then one can safely dispose the details such as those we provide here and use the conclusions as a ‘black box’. It is the present authors’ opinion that advent of such an enlightening and agreement in the area will be important for the development of symplectic geometry and related fields. In fact, one main obstacle to its smooth development has been the nuisance of working out similar, but not precisely the same, details each time when one tries to use the moduli spaces of pseudo-holomorphic curves of various kinds in various situations.
Now we provide description of the main results of Part 1 sectionwise. In Section \[sec:skuraterm\], we give the definitions of Kuranishi structures (Definition \[kstructuredefn\]) and good coordinate systems (Definition \[gcsystem\]). These definitions are based on a version of the definitions of an orbifold, its embedding and a vector bundle on it, which we describe in Section \[sec:ofd\]. Of course the basic concepts and the mathematical contents of orbifold were established long time ago [@satake]. However there are a few different ways of defining an orbifold in the technical point of view. More significantly, the notion of morphisms between orbifolds is rather delicate to define. We refer readers to the discussion of the book [@ofdruan] especially its section 1 for these points. In this article, we restrict ourselves to the world of effective orbifolds and use only embeddings as maps between them. Then those delicate points and troubles disappear.
In Section \[sec:fiber\], we define the fiber and the direct product of spaces with Kuranishi structure. We remark that in the category theory the notion of fiber product is defined in purely abstract language of objects and morphisms. The definition of fiber product we give in this section, however, is [*not*]{} the one given in the category theory.[^12] In fact, we never define the notion of morphisms between spaces equipped with Kuranishi structure in general. Therefore we define a fiber product of spaces with Kuranishi structure directly in Section \[sec:fiber\]. Here we consider only the fiber product over a manifold and require an appropriate transversality to define a fiber product. Although the definition we provide is not in the sense of category theory, they are so natural and canonical that the basic properties expected for the fiber product are fairly manifest. For example, associativity of the fiber product follows rather immediately from its definition. We do not define a fiber or a direct product of spaces with good coordinate systems here since its definition necessarily becomes more technical and complicated.
In this article we assume all the Kuranishi structures and good coordinate systems are oriented (Definitions \[kuraorient\] and \[gcsystem\]) unless otherwise mentioned.
In Section \[sec:thick\] we discuss the process going from a Kuranishi structure to a good coordinate system and back. As we mentioned before, we start from a Kuranishi structure $\widehat{\mathcal U}$ and construct a good coordinate system $\widetriangle{\mathcal U}$ as follows.
[**Theorem \[Them71restate\].**]{} [*For any Kuranishi structure $\widehat{\mathcal U}$ of $Z \subseteq X$ there exist a good coordinate system ${\widetriangle{\mathcal U}}$ of $Z \subseteq X$ and a KG-embedding $\widehat{\mathcal U} \to {\widetriangle{\mathcal U}}$.*]{}
See Definition \[defn32020202\] for KG-embedding. (The good coordinate system ${\widetriangle{\mathcal U}}$ above is said to be [*compatible*]{} with the Kuranishi structure $\widehat{\mathcal U}$.) We also start from a good coordinate system $\widetriangle{\mathcal U}$ and construct a Kuranishi structure $\widehat{\mathcal U^{+}}$. (Proposition \[lemappgcstoKu\].) When $\widehat{\mathcal U^{+}}$ is obtained from $\widehat{\mathcal U}$ by combining these two processes, the Kuranishi structure $\widehat{\mathcal U^{+}}$ is in general different from $\widehat{\mathcal U}$ but $\widehat{\mathcal U^{+}}$ is related to $\widehat{\mathcal U}$ in a way that $\widehat{\mathcal U^{+}}$ is a thickening of $\widehat{\mathcal U}$. We define this notion of thickening and study its properties in Section \[sec:thick\].
To formulate the notion of compatibility between a Kuranishi structure and a good coordinate system, and compatibility among Kuranishi structures or among good coordinate systems, we use the notion of embeddings. We define 4 possible versions of such embeddings and their compositions (there are 8 possible versions of compositions) in Section \[sec:thick\]. See Tables 5.1 and 5.2. In Section \[sec:contgoodcoordinate\], the proof of basic results of existence of good coordinate system compatible with given Kuranishi structure (Theorem \[Them71restate\]) is given.
In Section \[sec:multisection\], we define the notion of multisections and multi-valued perturbations and their transversality. The basic result is the following existence theorem of transversal perturbation, which is proved in Section \[sec:constrsec\].�� Here we leave the precise definitions of the terminology used in the statement to the later text.
[**Theorem \[prop621\].**]{}
*Let ${\widetriangle{\mathcal U}}$ be a good coordinate system of $Z \subseteq X$ and $\mathcal K$ its support system.*
1. There exists a multivalued perturbation $\widetriangle{\frak s} = \{\frak s^n_{\frak p}\}$ of $({\widetriangle{\mathcal U}},\mathcal K)$ such that each branch of $\frak s^n_{\frak p}$ is transversal to $0$.
2. Suppose $\widetriangle f : (X,Z;{\widetriangle{\mathcal U}}) \to N$ is strongly smooth and is transversal to $g : M \to N$, where $g$ is a map from a manifold $M$. Then we may choose $\widetriangle{\frak s}$ such that $\widetriangle f$ is strongly transversal to $g$ with respect to $\widetriangle{\frak s}$.
A technical but nontrivial result we prove in Section \[sec:multisection\] is compactness of the zero set of a multi-valued perturbation, which is a part of Corollary \[cor69\]. Corollary \[cor69\] also claims the fact that the zero set of multi-valued perturbation converges to the zero set of the Kuranishi map in Hausdorff topology as our perturbation converges to the Kuranishi map. (We remark that the zero set of the Kuranishi map is nothing but the space $X$ itself on which we define our good coordinate system.) These and other related results play an important role to work out the technical detail of the proof of well-defined-ness of the virtual fundamental chain (Propositions \[indepofukuracont\] and \[prop14777\]).
We remark that in Section \[sec:multisection\] we work on a space with good coordinate system and not on a space with Kuranishi structure. As we mentioned before, we use an induction over the charts in the construction of a transversal multisection. This induction works with the good coordinate system but not with the Kuranishi structure itself.
In Section \[sec:contfamily\], we define the notion of continuous family perturbation (abbreviated as CF-perturbation) and study their properties in relation to the good coordinate systems and the integration over the fiber. We use this notion to define the integration of differential forms on the space with good coordinate system. (Moreover we also define the integration along the fiber.)
The framework with which we define the integration along the fiber of differential form is as follows. (See also Situation \[smoothcorr\].)
\[situ13\] We consider $X$, $M$, $\widetriangle{\mathcal U}$, $\widetriangle{f}$ with the following properties.
1. We are given a good coordinate system $\widetriangle{\mathcal U}
= \{(U_{\frak p},E_{\frak p},s_{\frak p},\psi_{\frak p})\}$ of a space $X$.
2. We are given submersions $f_{\frak p} : U_{\frak p} \to M$ to a manifold $M$ such that they are compatible with the coordinate change in an appropriate sense. (Namely we assume that there is a weakly submersive map $\widetriangle{f} = \{f_{\frak p} \mid \frak p \in \frak P\}
: (X,\widetriangle{\mathcal U}) \to M$.) (Definition \[mapkura\].)$\blacksquare$
We next define the notion of continuous family perturbation (CF-perturbation). We do so in 2 steps. We first define such notion on a single chart $(U_{\frak p},E_{\frak p},s_{\frak p},\psi_{\frak p})$. We then discuss its compatibility with coordinate change and use it to define the notion of CF-perturbation of a good coordinate system. We define the notion of differential form on the space with good coordinate system. (It assigns a differential form on $U_{\frak p}$ to each $\frak p \in \frak P$ which are compatible with coordinate changes. See Definition \[defndiffformgcs\].) We use them to define integration along the fiber $$\label{intalongfiber}
\widetriangle{h} \mapsto \widetriangle f!(\widetriangle{h};\widetriangle{\frak S^{\epsilon}}) \in \Omega^d(M),$$ for any sufficiently small $\epsilon > 0$. Here the degree $d$ is $\deg \widetriangle{h} + \dim M - \dim (X,\widetriangle{\mathcal U})$ and $\widetriangle{\frak S}$ is a CF-perturbation, which satisfies an appropriate transversality assumption. (More precisely we assume that $\widetriangle{f}$ is strongly submersive with respect to $\widetriangle{\frak S}$. See Definition \[smoothfunctiononvertK\].)
In the case when $(X,\widetriangle{\mathcal U})$ is a manifold or an orbifold (that is, when all the obstruction bundles $E_{\frak p}$ are trivial), the operation (\[intalongfiber\]) reduces to the standard integration along the fiber of a differential form. Note $\widetriangle{\frak S}$ is a one-parameter family of perturbations parameterized by $\epsilon >0$. The integration along the fiber [*does*]{} depend on $\epsilon$ as well as CF-perturbation. We also remark that typically (\[intalongfiber\]) diverges as $\epsilon$ goes to $0$. We firmly believe there is [*no*]{} way of defining the integration along the fiber in the way independent of the choice of CF-perturbation. This is related to the following most basic point of the whole story of virtual fundamental chains:
In the case we need to construct a virtual fundamental *chain* but *not a cycle*, that is, as in the case when our (moduli) space has a boundary or a corner, the virtual fundamental chain depends on the choice of perturbations. However we can make (many) choices in a consistent way so that the resulting algebraic system is independent of such choices modulo certain homotopy equivalence.
We will discuss this point further in Part 2.
On the other hand, the integration along the fiber (\[intalongfiber\]) is independent of various other choices involved. Especially it is independent of the choice of partition of unity we use to define the integration. We define the partition of unity in Definition \[pounity\] in the current context and prove its existence in Proposition \[pounitexi\]. The above mentioned independence is proved as Proposition \[indepofukuracont\].
In Section \[sec:stokes\] we prove Stokes’ formula for the integration along the fiber (\[intalongfiber\]). We begin with discussing the boundary of a space with Kuranishi structure or with good coordinate system. We can define the notion of a boundary $\partial M$ of a manifold with corners $M$, which we call the normalized boundary. $\partial M$ is again a manifold with corner and there is a map $\partial M \to M$ which is generically one to one and is a surjection to the boundary of $M$. Set theoretically $\partial M$ is not a subset of $M$. For example, codimension 2 corner points of $M$ appear twice in $\partial M$. These issues are not deep and are basically well-known. Because a systematic discussion of these issues is not easy to find and also because they are needed for a systematic study of Kuranishi structure with corner, which is important for the chain level argument (especially those appearing in Part 2), we include a systematic discussion of these issues here. Once these points are understood for the case of manifolds, it is straightforward to generalize them to the Kuranishi structure or to the good coordinate system with boundary and corners.
Stokes’ formula in this context is (\[stokesinintro\]) given below. In Situation \[situ13\], suppose $(X,\widetriangle{\mathcal U})$ has normalized boundary, $\partial(X,\widetriangle{\mathcal U}) = (\partial X,\partial \widetriangle{\mathcal U})$. We assume the restriction $\widetriangle{f_{\partial}}$ of $\widetriangle{f}$ to $(\partial X,\partial \widetriangle{\mathcal U})$ is still weakly submersive.[^13] Suppose also we are given a CF-perturbation $\widetriangle{\frak S}$ of $(X,\widetriangle{\mathcal U})$. We assume $\widetriangle{f}$ is strongly submersive with respect to $\widetriangle{\frak S}$. Then $\widetriangle{\frak S}$ induces a CF-perturbation $\widetriangle{\frak S_{\partial}}$ on the boundary with respect to which the restriction $\widetriangle{f_{\partial}}$ of $\widetriangle{f}$ is strongly submersive. Stokes’ formula for a good coordinate system now is stated as Theorem \[Stokes\]: For any sufficiently small $\epsilon > 0$ we have $$\label{stokesinintro}
d\left(\widetriangle f!(\widetriangle h;\widetriangle{{\frak S}^{\epsilon}})\right)
=
\widetriangle f!(d\widetriangle h;\widetriangle{{\frak S}^{\epsilon}})
+
\widetriangle f_{\partial}!(\widetriangle {h_{\partial}};\widetriangle{{\frak S}_{\partial}^{\epsilon}}).$$ This is proved in Subsection \[subsec:Stokesgcs\]. We note that in case $M$ (the target of $\widetriangle f$) is a point, the integration along the fiber (\[intalongfiber\]) is nothing but the integration of differential form and is a real number. In that case we write $$\int_{(X,\widetriangle{\mathcal U},\widetriangle{\frak S^{\epsilon}})}h
= \widetriangle f!(\widetriangle h;\widetriangle{{\frak S}^{\epsilon}}).$$ Then (\[stokesinintro\]) becomes $$\label{formstakessimple}
\int_{(X,\widetriangle{\mathcal U},\widetriangle{\frak S^{\epsilon}})}dh
=
\int_{\partial(X,\widetriangle{\mathcal U},\widetriangle{\frak S^{\epsilon}})}h.$$ If all the obstruction bundles are trivial, (\[formstakessimple\]) is nothing but the usual Stokes’ formula.
In Section \[sec:stokes\], we also include one easy application of Stokes’ formula. Namely we prove that if $(X,\widetriangle{\mathcal U})$ is equipped with a good coordinate system [*without boundary*]{} and $\widetriangle{f}$ is as in Situation \[situ13\], then push-forward of $1$ (which is a differential $0$-form of $(X,\widetriangle{\mathcal U})$) gives rise to a smooth differential form on M $$\widetriangle f!(1;\widetriangle{{\frak S}^{\epsilon}}) \in \Omega^{\dim M - \dim (X,\widetriangle{\mathcal U})}(M).$$ Then (\[stokesinintro\]) implies that this form is closed. Moreover its de Rham cohomology class is independent of the choices (of $\widetriangle{{\frak S}}$ and $\epsilon$ also). This is a consequence of Propositions \[relextendgood\], \[cobordisminvsmoothcor\] below. To state them we introduce the notion of [*smooth correspondence*]{} under the following situation. Note that the assumptions on the weak submersivity as above (also see Situation \[situ71\] below) are satisfied, for example, for the case of Gromov-Witten invariant. So we can use this result to prove well-defined-ness of Gromov-Witten invariant.
\[situ71\] \[Situation \[smoothcorr\]\] Let $X$ be a compact metrizable space, and $\widehat{\mathcal U}$ a Kuranishi structure of $X$ (with or without boundaries or corners). Let $M_s$ and $M_t$ be $C^{\infty}$ manifolds. We assume $\widehat{\mathcal U}$, $M_s$ and $M_t$ are oriented[^14].
Let $\widehat f_s : (X;\widehat{\mathcal U}) \to M_s$ be a strongly smooth map and $\widehat f_t : (X;\widehat{\mathcal U}) \to M_t$ a weakly submersive strongly smooth map. We call $\frak X = ((X;\widehat{\mathcal U});\widehat f_s,\widehat f_t)$ a [*smooth correspondence*]{} from $M_s$ to $M_t$. $\blacksquare$
By Theorem \[Them71restate\] for $Z=X$, we have a good coordinate system $(X, \widetriangle{\mathcal U})$ compatible with $(X,\widehat{\mathcal U})$. Moreover we have $\widetriangle f_s : (X;{\widetriangle{\mathcal U}})
\to M_s$ and $\widetriangle f_t : (X;{\widetriangle{\mathcal U}})
\to M_t$ such that $\widehat f_s$, $\widehat f_t$ are pullbacks of $\widetriangle f_s$, $\widetriangle f_t$ respectively, and $\widetriangle f_t$ is weakly submersive (Proposition \[le614\] (2)). Thus we have the correspondence denoted by $${\frak X} = ((X;\widetriangle{\mathcal U});\widetriangle f_s,
\widetriangle f_t).$$
We take a CF-perturbation $\widetriangle{\frak S}$ of $(X;{\widetriangle{\mathcal U}})$ such that $\widetriangle f_t$ is strongly submersive with respect to ${\widetriangle{\mathcal U}}$ (Theorem \[existperturbcont\] (2)). Then we define a map $$\label{Into:corr}
{\rm Corr}_{(\frak X,\widetriangle{{\frak S}^{\epsilon}})}
:
\Omega^k(M_s) \ni h
\to
(\widetriangle{f_t})!
(\widetriangle{f_s^*}h;\widetriangle{\frak S^{\epsilon}})
\in \Omega^{\ell+k}(M_t)$$ (Definition \[defn748\]), which we call the [*smooth correspondence map associated to $\frak X = ((X;\widetriangle{\mathcal U});\widetriangle f_s,\widetriangle f_t)$*]{}. Here $\ell = \dim M_t -\dim (X;\widetriangle{\mathcal U})$. Then Stokes formula yields that for any sufficiently small $\epsilon >0$ we have $$d \circ {\rm Corr}_{(\frak X,\widetriangle{{\frak S}^{\epsilon}})}
-
{\rm Corr}_{(\frak X,\widetriangle{{\frak S}^{\epsilon}})}
\circ d
=
{\rm Corr}_{(\partial\frak X ,\widetriangle{\frak S^{\partial,\epsilon}})}$$ (Corollary \[Stokescorollary\]). Using this formula, we show the following propositions.
[**Proposition \[relextendgood\].**]{}
*Consider Situation \[situ71\] and assume that our Kuranishi structure on $X$ has no boundary. Then the map $
{\rm Corr}_{(\frak X,\widetriangle{{\frak S}^{\epsilon}})}
:
\Omega^k(M_s) \to \Omega^{\ell+k}(M_t)
$ defined above is a chain map.*
Moreover, provided $\epsilon$ is sufficiently small, the map ${\rm Corr}_{(\frak X,\widetriangle{{\frak S}^{\epsilon}})}$ is independent of the choices of our good coordinate system ${\widetriangle{\mathcal U}}$ and CF-perturbation $\widetriangle{\frak S}$ and of $\epsilon>0$, up to chain homotopy.
Thus in the situation of Proposition \[relextendgood\], the correspondence map ${\rm Corr}_{(\frak X,\widetriangle{{\frak S}^{\epsilon}})}$ on differential forms descends to a map on [*cohomology*]{} which is independent of the choices of $\widetriangle{\mathcal U}$ and $\widetriangle{{\frak S}^{\epsilon}}$. We write the cohomology class as $[{\rm Corr}_{\frak X}(h)] \in H(M_t)$ for any closed differential form $h$ on $M_s$ by removing $\widetriangle{{\frak S}^{\epsilon}}$ from the notation.
[**Proposition \[cobordisminvsmoothcor\].**]{} [*Let $\frak X_i = ((X_i,\widehat{\mathcal U^i}),\widehat f_s^i,\widehat f_t^i)$ be smooth correspondence from $M_s$ to $M_t$ such that $\partial X_i = \emptyset$. Here $i=1,2$ and $M_s$, $M_t$ are independent of $i$. We assume that there exists a smooth correspondence $\frak Y = ((Y,\widehat{\mathcal U}),\widehat f_s,\widehat f_t)$ from $M_s$ to $M_t$ with boundary (but without corner) such that $$\partial \frak Y = \frak X_1 \cup -\frak X_2.$$ Here $-\frak X_2$ is the smooth correspondence $\frak X_2$ with opposite orientation. Then we have $$\label{chomotopyrelation22}
[{\rm Corr}_{\frak X_1}(h)] = [{\rm Corr}_{\frak X_2}(h)]
\in H(M_t),$$ where $h$ is a closed differential form on $M_s$.*]{}
Summing up these propositions, if the Kuranishi structure of $X$ has [*no boundary*]{}, the virtual fundamental cycle of $X$ and the smooth correspondence map are well-defined on [*cohomology level*]{}. These are proved in Section \[sec:stokes\].
Besides Stokes’ formula, an important property we use for the integration along the fiber is the composition formula. To formulate a composition formula we need to study the fiber product of CF-perturbations. Since a fiber product is to be better defined for the Kuranishi structure than for the good coordinate system, we rewrite the story of CF-perturbation and integration along the fiber with respect to the good coordinate system into the one with respect to the Kuranishi structure. This is the content of Section \[sec:kuraandgood\]. The results addressed in Section \[sec:kuraandgood\] are really necessary for the [*chain level argument*]{}, in particular in the later Part of this manuscript.
We can define the notion of CF-perturbation of a Kuranishi structure in the same way as that of a good coordinate system. Namely it assigns a CF-perturbation to each chart $\mathcal U_p$ of $p \in X$ so that they are compatible with coordinate changes. (See Definition \[defn81\].) Note however it is very difficult to construct a CF-perturbation with appropriate transversality property on a given Kuranishi structure $\widehat{\mathcal U}$ of $X$, since the proof should be by induction on charts as we mentioned several times already. So we first construct a good coordinate system $\widetriangle{\mathcal U}$ compatible to a given Kuranishi chart $\widehat{\mathcal U}$ and a CF-perturbation $\widetriangle{\frak S}$ on the good coordinate system $\widetriangle{\mathcal U}$ we obtained. Then we construct another Kuranishi structure $\widehat{\mathcal U^{+}}$ in such a way that $\widetriangle{\frak S}$ induces a CF-perturbation $\widehat{\frak S^+}$ of $(X,\widehat{\mathcal U^{+}})$. (Lemma \[lemappgcstoKucont\].) Thus in place of constructing a CF-perturbation of given $(X,\widehat{\mathcal U})$, we construct one of its thickening $(X,\widehat{\mathcal U^{+}})$.
In this way we have arrived in the situation where we are given a Kuranishi structure equipped with a CF-perturbation satisfying appropriate transversality property needed. We formulate this situation as follows.
\[situ155\] We consider $X$, $M$, $\widehat{\mathcal U}$, $\widehat{f}$, $\widehat{\frak S}$ with the following properties.
1. We are given a Kuranishi structure $\widehat{\mathcal U} = \{(U_{p},E_{p},s_{p},\psi_{p})\}$ of a space $X$.
2. We are given submersions $f_{p} : U_{p} \to M$ to a manifold $M$ such that they are compatible with coordinate change in an appropriate sense. (Namely, we assume that there is a weakly submersive map $\widehat{f} = \{f_{p} \mid p \in X\}
: (X,\widehat{\mathcal U}) \to M$.) (Definition \[mapkura\].)
3. $\widehat{\frak S}$ is a CF-perturbation such that $\widehat f$ is strongly transversal to $0$ with respect to $\widehat{\frak S}$. $\blacksquare$
We can define the notion of differential forms on the space with Kuranishi structure in the same way as on the space equipped with a good coordinate system. Now we define the integration along the fiber $$\label{intalongfiber2}
\widehat{h} \mapsto \widehat f!(\widehat{h};\widehat{\frak S^{\epsilon}}) \in \Omega^d(M)$$ as follows. We take a good coordinate system $\widetriangle{\mathcal U}$ on which $\widehat{\frak S^{\epsilon}}$, $\widehat{h}$, $\widehat{f}$ induce corresponding objects $\widetriangle{\frak S^{\epsilon}}$, $\widetriangle{h}$, $\widetriangle{f}$. Then by definition $$\label{formula1.5}
\widehat f!(\widehat{h};\widehat{\frak S^{\epsilon}})
=
\widetriangle f!(\widetriangle{h};\widetriangle{\frak S^{\epsilon}}).$$ The main result of Section \[sec:kuraandgood\] claims that the right hand side of (\[formula1.5\]) is independent of the choice of $\widetriangle{\mathcal U}$, $\widetriangle{\frak S^{\epsilon}}$, $\widetriangle{h}$, $\widetriangle{f}$ but depend only on $\widehat{\mathcal U}$, $\widehat{\frak S^{\epsilon}}$, $\widehat{h}$, $\widehat{f}$. This is Theorem \[theorem915\]. (We note that integration along the fiber is hard to define directly with Kuranishi structure since we use a partition of unity to define the integration.)
Stokes’ formula with respect to the good coordinate system is easily translated to one for the Kuranishi structure. Namely we have (Proposition \[Stokeskura\]) $$\label{stokesinintro2}
d\left(\widehat f!(\widehat h;\widehat{{\frak S}^{\epsilon}})\right)
=
\widehat f!(d\widehat h;\widehat{{\frak S}^{\epsilon}})
+
\widehat f_{\partial}!(\widehat {h_{\partial}};\widehat{{\frak S}_{\partial}^{\epsilon}}).$$
In Section \[sec:composition\], we state and prove a composition formula whose outline is in order. We begin with $X,M_s,M_t,\widehat{\mathcal U},\widehat f_s, \widehat f_t, \widehat{\frak S}$ satisfying the following properties. (Here the indices $s$ and $t$ stand for the source and the target, respectively.)
\[situ1665\]
1. $X,M_t,\widehat{\mathcal U},\widehat{f_t}$ play the role of $X$, $M$, $\widehat{\mathcal U}$, $\widehat{f}$ in Situation \[situ155\].
2. $X,M_s,\widehat{\mathcal U},\widehat{f_s}$ play the role of $X$, $M$, $\widehat{\mathcal U}$, $\widehat{f}$ of Situation \[situ155\] except we do not assume weak submersivity of $\widehat f_s$ but assume only strong smoothness.
3. $\widehat{\frak S}$ is a CF-perturbation such that $X,M_t,\widehat{\mathcal U},\widehat{f_t}$ together with $\widehat{\frak S}$ satisfy the transversality assumption required to define integration along the fiber by $\widehat{f_t}$. (Namely we assume $\widehat f_t$ is strongly submersive with respect to $\widehat{\frak S}$. See Definition \[defn929292\].)$\blacksquare$
We call $(X,M_s,M_t,\widehat{\mathcal U},\widehat{f_s}, \widehat{f_t})$ a *smooth correspondence* (See Situation \[smoothcorr\]) and $(X,M_s,M_t,\widehat{\mathcal U},\widehat{\frak S},\widehat{f_s}, \widehat{f_t})$ a *perturbed smooth correspondence* (Definition \[defn839\]). To each such correspondence $\frak X = (X,M_s,M_t,\widehat{\mathcal U},\widehat{f_s}, \widehat{f_t},
\widehat{\frak S})$ and sufficiently small $\epsilon > 0$, we associate a linear map $${\rm Corr}^{\epsilon}_{\frak X} : \Omega^*(M_s) \to \Omega^{*+d}(M_t)$$ by $$\label{correspondintro}
{\rm Corr}^{\epsilon}_{\frak X}(h)
= \widehat{f_t}!(\widehat f_s^*\widehat{h};\widehat{\frak S^{\epsilon}}).$$ (See Definition \[def92111\].) Here $\widehat{f_s^*}$ is the pull-back operation which assigns a differential form on $(X,\widehat{\mathcal U})$ to a differential form on $M_s$. The pull-back is defined for an arbitrary strongly smooth map $\widehat{f_{s}}$. (See Definition \[defn75555\]. Here we do not need to assume weak or strong submersivity.) We define an integer $d$ to be $$d = \dim M_t - \dim (X,\widehat{\mathcal U}).$$ Our composition formula claims that the assignment $\frak X \mapsto {\rm Corr}^{\epsilon}_{\frak X}$ is compatible with compositions.
Suppose that both $$\begin{aligned}
\frak X_{21} & = & (X_{21},M_1,M_2,\widehat{\mathcal U_{21}},
\widehat{\frak S_{21}},\widehat{f_{1,21}}, \widehat{f_{2,21}})\\
\frak X_{32} & = & (X_{32},M_2,M_3,\widehat{\mathcal U_{32}},
\widehat{\frak S_{32}},
\widehat{f_{2,32}}, \widehat{f_{3,32}})\end{aligned}$$ satisfy Situation \[situ1665\]. We define their composition $\frak X_{31} = \frak X_{32} \circ \frak X_{21}$ as follows.
First we consider the fiber product of Kuranishi structures $$(X_{31},\widehat{\mathcal U_{31}}) = (X_{21},\widehat{\mathcal U_{21}}) \times_{M_2} (X_{32},\widehat{\mathcal U_{32}})$$ to define the space with Kuranishi structure $(X_{31},\widehat{\mathcal U_{31}})$ on which the CF-perturbations $\widehat{\frak S_{21}}$ and $\widehat{\frak S_{32}}$ induce a CF-perturbation. (Definition \[defn837\].) $\widehat{f_{1,21}}$ induces $\widehat f_{1,31} : (X_{31},\widehat{\mathcal U_{31}}) \to M_1$ and $\widehat{f_{3,32}}$ induces $\widehat f_{3,31} : (X_{31},\widehat{\mathcal U_{31}}) \to M_3$. We then put $$\frak X_{31} = \frak X_{32} \circ \frak X_{21}
=
(X_{31},M_1,M_3,\widehat{\mathcal U_{31}},
\widehat{\frak S_{31}},
\widehat{f_{1,31}}, \widehat{f_{3,31}}).$$ It satisfies Situation \[situ1665\]. (Lemma \[lem838\] etc.) Now the composition formula is stated as
[**Theorem \[compformulaprof\].**]{} [*Suppose that $\tilde{\frak X}_{i+1 i} =
(X_{i+1 i},\widehat{\mathcal U_{i+1 i}},\widehat{\frak S_{i+1 i}},
\widehat{f_{i,i+1 i}},
\widehat{f_{i+1,i+1 i}})$ are perturbed smooth correspondences for $i=1,2$. Then $$\label{formula814}
{\rm Corr}^{\epsilon}_{\tilde{\frak X}_{32}\circ\tilde{\frak X}_{21}}
=
{\rm Corr}^{\epsilon}_{\tilde{\frak X}_{32}} \circ {\rm Corr}^{\epsilon}_{\tilde{\frak X}_{21}}$$ for each sufficiently small $\epsilon >0$.*]{}
Integration along the fiber of differential form on spaces with Kuranishi structure is written in [@fooo09 Section 12]. Especially Stokes’ formula and the Composition formula was given in [@fooo09 Lemma 12.13] and [@fooo09 Lemma 12.15] respectively. Here we present them in greater details. In [@fooo09], the process going from a Kuranishi structure to a good coordinate system and back was not written explicitly. (One reason is because the main focus of [@fooo09] lies in its application to the Lagrangian Floer theory of torus orbits of toric manifolds but not in the foundation of the general theory.) Here we provide thorough detail. This theory is actually very similar to the manifold theory.
In Section \[sec:contgoodcoordinate\], we prove the existence of a good coordinate system that is compatible with the given Kuranishi structure (Theorem \[Them71restate\]). We also prove its several variations. The proof we give there is basically the same as those presented in [@FO] which itself is more detailed in [@foootech]. We simplify the proof in several places as well as provide details of several points whose proofs were rather sketchy in the previous writings. We separate the discussion on general topology issue from other parts and put it in a separate paper [@foooshrink].
Section \[sec:contfamilyconstr\] is devoted to the proof of existence of a CF-perturbation satisfying appropriate transversality properties. This proof is split into 3 parts.
In the first part (Subsection \[subsec:confapersingle\]) we prove such an existence result for a single Kuranishi chart. For this, we use the language of sheaf. In particular we prove that the assignment $$U \mapsto \{\text{all CF-perturbations on $U$}\}$$ for each open subset $U \subset U_{\frak p}$ defines a sheaf. We also consider the sub-sheaf consisting of CF-perturbations satisfying appropriate transversality properties. The main result is stated as a softness of these sheaves. (Proposition \[prop123123\].)
The second part of the proof (Subsection \[subsec:extembandcfp\]) discusses the case where we have a coordinate change of Kuranishi chart from $\mathcal U_1$ to $\mathcal U_2$. Assuming we are given a CF-perturbation on $\mathcal U_1$ with various transversality properties, we show that we can find a CF-perturbation of $\mathcal U_2$ with certain transversality properties such that these two CF-perturbations are compatible with the coordinate change (Proposition \[prop1221\]). (Actually Proposition \[prop1221\] includes a relative version of this statement, which we also need for our exposition.)
In the third part (Subsection \[subsec:cfpgoodcsys\]) we complete the proof of existence theorem of a CF-perturbation (Theorem \[existperturbcont\] and its variants) combining the results of the earlier two subsections.
As far as the de Rham version is concerned, the results up to Section \[sec:contfamilyconstr\] provide a package we need for the case we work on a single space with Kuranishi structure. The contents of Sections \[sec:constrsec\] and \[sec:onezerodim\] will be used in Parts 2 and 3 to verify that most part of the story works when the ground field is $\Q$. The results of Sections \[sec:constrsec\] and \[sec:onezerodim\] are not necessary over the ground field $\R$ or $\C$. Actually de Rham version of the story is in various sense easier to work out than proving the corresponding results in the singular homology version. (Maybe the reason is similar to the reason why teaching an under-graduate homology theory of manifold based on de Rham theory is easier than teaching on based on the singular homology theory. This is especially so when one teaches cup product.)
In Section \[sec:constrsec\], we prove an existence theorem of multi-valued perturbation (See Theorem \[prop621\]). The proof is similar to the proof of Theorem \[existperturbcont\]. There are two differences:
The first difference is as follows. In the case we work with the de Rham complex, we can construct a CF-perturbation of $(X,\widetriangle{\mathcal U})$ so that a given weakly submersive map $\widetriangle f : (X,\widetriangle{\mathcal U}) \to M$ becomes strongly submersive and use it. (This means that the restriction of $f_{\frak p}$ to the zero set of perturbed section is a submersion.) This makes it possible to define the integration along the fiber. On the other hand we cannot expect to find multi-valued perturbation such that restriction of $f_{\frak p}$ to its zero set is submersive. (We can find a multi-valued perturbation that is transversal to $0$.) This difference makes it a bit harder to use multi-valued perturbation to work out chain level argument. We can still do it but we need to break symmetry more. This point is explained, for example, in [@fooobook2 Subsection 7.2.2], [@fooo091] and etc.. In this article we use multi-valued perturbation only in the case when the dimension of our good coordinate system is 1,0 or negative because of this issue.
The second difference is rather technical and is explained in Subsection \[subsec:nastyreason\].
In Sections \[sec:onezerodim\], we discuss the virtual fundamental chain (over $\Q$) through a multi-valued perturbation. The de Rham version of this section is Sections \[sec:contfamily\] and \[sec:stokes\]. In this section, we study only the case where the (virtual) dimension of our space $(X,\widetriangle{\mathcal U})$ with good coordinate system is negative, $0$ or $1$ and prove the following.
1. In case $\dim (X,\widetriangle{\mathcal U}) < 0$ the zero set of multisection which is transversal to $0$ is an empty set.
2. In case $\dim (X,\widetriangle{\mathcal U}) = 0$ the zero set of multisection which is transversal to $0$ consists of finitely many points and is away from the boundary of $X$. By defining appropriate weight to each point of this zero set and taking weighted sum, we can define a rational number which is the virtual fundamental chain. (This number in general depends on the choice of multisection.)
3. Suppose $\dim (X,\widetriangle{\mathcal U}) = 1$. Then $\dim \partial(X,\widetriangle{\mathcal U}) = 0$. By Item (2) we can define a virtual fundamental chain $[\partial(X,\widetriangle{\mathcal U},\widetriangle{\frak s^{n}})]$ of $\partial(X,\widetriangle{\mathcal U})$. (Here $\widetriangle{\frak s^{n}}$ is the multi-valued perturbation we use to define it.) Then we have: $$[\partial(X,\widetriangle{\mathcal U},\widetriangle{\frak s^{n}})] = 0.$$
Item (1) is Lemma \[lem1311111\] (1). Item (2) is Lemma \[lem1311111\] (2), Lemma \[lem134\] and Definition \[defn1355\]. Item (3) is Theorem \[prop13777\], which corresponds to Stokes’ formula in our situation and is the main result of Section \[sec:onezerodim\].
[**Theorem \[prop13777\].**]{} [*Let $({\widetriangle{\mathcal U}},\widetriangle{\frak s})$ be a good coordinate system with multivalued perturbation of $X$ (see Definition \[gcswithperturb\]) and assume $\dim (X,{\widetriangle{\mathcal U}}) = 1$. We consider its normalized boundary $\partial(X,{\widetriangle{\mathcal U}}) =
(\partial X,\partial{\widetriangle{\mathcal U}})$ where $\widetriangle{\frak s}$ induces a multivalued perturbation $\widetriangle{\frak s_{\partial}}$ thereof and $(\partial{\widetriangle{\mathcal U}},\widetriangle{\frak s_{\partial}^{n}})$ is a good coordinate system with multivalued perturbation of $\partial X$ with $\dim (\partial X, \partial \widetriangle{\mathcal U})=0$. Then the following formula holds. $$[(\partial X,\partial{\widetriangle{\mathcal U}},\widetriangle{\frak s_{\partial}^{n}})]
=0.$$*]{}
Furthermore we also show
[**Corollary \[cobordisminvsmoothcormulti\].**]{} [*Let $\frak X_i = (X_i,\widehat{\mathcal U^i})$, $i=1,2$ be spaces with Kuranishi structure without boundary of dimension $0$. Suppose that there exists a space with Kuranishi structure $\frak Y = (Y,\widehat{\mathcal U})$ (but without corner) such that $$\partial \frak Y = \frak X_1 \cup -\frak X_2.$$ Here $-\frak X_2$ is the smooth correspondence $\frak X_2$ with opposite orientation. Then we have $$\label{chomotopyrelation2}
[(X_1,\widehat{\mathcal U^1})]
=
[(X_2,\widehat{\mathcal U^2})].$$*]{}
The proof of Theorem \[prop13777\], which was given in [@FO], goes as follows. We take a transversal multisection of $(X,\widetriangle{\mathcal U})$ extending the given one, $\widetriangle{\frak s^{n}}$, on the boundary. We may take it so that its zero set has a triangulation and it, when equipped with an appropriate weight, defines a chain. (In our situation it is a singular chain of a space which consists of a single point.) The boundary of this chain is the degree $0$ chain which is the virtual fundamental chain $[\partial(X,\widetriangle{\mathcal U},
\widetriangle{\frak s^{n}})]$. Therefore it is zero. (Degree zero singular chain of a point is zero if it is homologous to zero.)
This proof is correct as it is. The only nontrivial issue (besides the existence of transversal multisection) to be clarified is the triangulability of the zero set. Nontrivial point of the argument is the following. If we take each branch of multisection $\frak s_i^{n}$, its zero set is a one dimensional manifold. We need to check however union of various $(\frak s_i^{n})^{-1}(0)$ (for different branches) has triangulation. In case the intersection of $(\frak s_i^{n})^{-1}(0)$ with $(\frak s_j^{n})^{-1}(0)$ is wild this triangulability fails. Other point is a discussion of the behavior of the zero set at the locus where the number of branches changes. As we mentioned in [@FO] we can resolve these points by taking an appropriate generic choice of perturbations. (See [@FO page 946].) In our situation where our space (with good coordinate system) is 1 dimensional it is not so difficult to work it out.[^15]
We will discuss triangulation of the zero set of multisection elsewhere in more detail. In this article, we provide an alternative proof of Theorem \[prop13777\] which may be simpler and more transparent. In this proof we take a function $f : (X,\widetriangle{\mathcal U}) \to \R_{\ge 0}$ such that $f^{-1}(0) = \partial X$ and its gradient vector field of $f$ along $\partial X$ points inward. For generic $s > 0$, the level set $f^{-1}(s) = X_s$ carries a good coordinate system induced from $\widetriangle{\mathcal U}$ which we write $\widetriangle{\mathcal U_s}$. The dimension of $(X_s,\widetriangle{\mathcal U_s})$ is zero. Using compactness of $X$ we find that $X_s$ is an empty set for sufficiently large $s$. Since $(X_0,\widetriangle{\mathcal U_0}) = \partial(X,\widetriangle{\mathcal U})$, to prove Theorem \[prop13777\], it suffices to prove that the virtual fundamental chain $[(X_s,\widetriangle{\mathcal U_s},\widetriangle{\frak s^{n}}\vert_{\widetriangle{\mathcal U_s}})] \in \Q$ is independent of $s$. Note the zero set of $(\widetriangle{\frak s^{n}})^{-1}(0) \cap f^{-1}(0)$ is zero dimensional, which is a finite set. So the required independence follows by locally studying the zero set of $\widetriangle{\frak s^{n}}$.
\[joycerem\] We would like to again mention a relationship with [@joyce2], [@joyce4]. In [@joyce4], Joyce gave an alternative version of ‘space with Kuranishi structure’. In his version he relaxes the condition of compatibility of coordinate changes so that it is required only at the zero set of Kuranishi map. He requires compatibility including the derivative up to the first order. In that way Joyce succeeds in inverting the arrows of the coordinate change of Kuranishi charts so that the trouble coming from noninvertibility of coordinate change disappears. His way has an advantage that one can define the notion of morphisms between ‘spaces with Kuranishi structures’.
We however use our version of Kuranishi structure and good coordinate system in this article. The reason is as follows. Our goal is to define a system of operators from a system of smooth correspondences (which is the object such as Situation \[situ1665\] (1)(2)). We need to choose a chain model on which we realize our operations forming the algebraic structure. Our choice in this article is the de Rham complex. (In some other occasion we also use the singular chain complex.) Note the space $X$ which has Kuranishi structure may have pathological topology in general. So singular homology does not behave nicely for $X$. In the case of de Rham model, the situation gets even worse. Namely it seems impossible to define the notion of differential forms on $X$. [^16]
Thus, we need to take a union of charts of $X$ which has a positive size to work with de Rham or singular homology. Namely we need a system of spaces $U_{\frak p}$ which is a manifold or orbifold and containing $X$. Both singular homology and de Rham cohomology of such spaces behave nicely. We remark that to define the notion of differential forms of a good coordinate system or of Kuranishi structure, we need compatibility of coordinate change in our sense, that is stronger than Joyce assumed in [@joyce4]. Namely, we also need to assume compatibility at some points outside $X$ (i.e., outside the zero set of Kuranishi map $s$). By this reason, it seems that we can use neither de Rham cohomology nor singular homology directly if we use the definition of Kuranishi structure in the sense of [@joyce4].
As far as we understand, Joyce’s plan is to use a version of Kuranishi homology ([@joyce]) as the cohomology theory which makes sense in his version of Kuranishi structure. It seems likely that this approach works. One potential trouble however is the Poincaré duality. Joyce in [@joyce] provides a chain level intersection paring. However the intersection ‘number’ in his chain level intersection paring is not a number but is an element of some huge complex (whose cohomology group is $\Q$). Though this construction provides the same amount of information in the homology level there is a trouble using it for the chain level argument. While we work on the chain level argument, sometimes we need to convert some input variables (of algebraic operation we will obtain) to output variables. We use the (chain level) Poincaré duality for this purpose. Note the pairing $$(u,v) \mapsto \int_M u \wedge v \in \R$$ works in the chain level in de Rham theory. This identifies an element of de Rham complex with an element in its dual. Although the dual space of the space of differential forms is the set of distributions and is different from the set of differential forms, the difference between the spaces of differential forms and of distributions is relatively small so that we can still use the chain level Poincaré duality to convert certain input variables to output variables. It seems that to realize this Poincaé duality in the situation of Kuranishi homology, one needs to work more on the side of homological algebra. The amount of algebraic work to be done for this purpose might be very heavy, although it is plausible.
As we mentioned in the beginning, we previously wrote an article [@foootech] which provides a detailed explanation of similar nature as that of the present article. We refer readers to [@foootech Part 6] for some documentation of certain activities in which we were involved concerning the foundation of the virtual fundamental chain techniques, around the time when [@foootech] was written. After we posted [@foootech] in arXiv in 2013 September, we have continued our effort of accommodating the demand for more details of this technique which came from some part of symplectic geometry community.
The first named author, together with other mathematicians, organized a semester-long program in the Simons Center for Geometry and Physics to discuss the foundation of the virtual fundamental chain techniques. Two one-week long conferences were held on the subject as well as a series of eleven lectures are presented by the first named author which are closely related to the content of this article. (The video of the conferences and the first named author’s lectures are available in the web page of the Simons Center for Geometry and Physics.) In addition, in our attempt to clarify the ‘Hausdorffness issue’ raised by D. McDuff and K. Werheim in their joint lectures given in the Institute for Advanced Study in early 2012, we separately wrote a paper [@foooshrink].
While we have been writing this article and during these activities occurred (that is, between September 2013 and March 2015), the present authors have not been aware of any explicit mathematical questions unanswered on the foundation of virtual fundamental chain or cycle technique or on its application to the moduli space of pseudo-holomorphic curves. [^17]
Notations {#sec:notations}
=========
1. ${\rm Int}\, A$, $\ring A$: Interior of a subset $A$ of a topological space.
2. $\overline A$: Closure of a subset $A$ of a topological space.
3. ${\rm Perm}(k)$: The permutation group of order $k!$.
4. ${\rm Supp}(h)$, ${\rm Supp}(f)$: The support of a differential form $h$, a function $f$, etc..
5. $\varphi^{\star}\mathscr F$: Pullback of a sheaf $\mathscr F$ by a map $\varphi$.
6. $X$: A paracompact metrizable space. (Part I).
7. $Z$: A compact subspace of $X$. (Part I).
8. $\mathcal U =(U,\mathcal E,\psi,s)$: A Kuranishi chart, Definition \[defnKchart\].
9. $\mathcal U\vert_{U_0} =(U_0,\mathcal E\vert_{U_0},\psi\vert_{U_0\cap s^{-1}(0)},s\vert_{U_0})$: open subchart of $\mathcal U =(U,\mathcal E,\psi ,s)$, Definition \[defnKchart\].
10. $\Phi = (\varphi,\widehat\varphi)$: Embedding of Kuranishi charts, Definition \[defKchart\].
11. $o_p, o_p(q)$: Points in a Kuranishi neighborhood $U_p$ of $p$. Definition \[kuranishineighborhooddef\].
12. $\Phi_{21} = (U_{21},\varphi_{21},\widehat\varphi_{21})$: Coordinate change of Kuranishi charts from $\mathcal U_1$ to $\mathcal U_2$, Definition \[coordinatechangedef\].
13. $\widehat{\mathcal U} = (\{\mathcal U_p\},\{\Phi_{pq}\})$: Kuranishi structure, Definition \[kstructuredefn\].
14. $(X,\widehat{\mathcal U})$, $(X,Z;\widehat{\mathcal U})$: K-space, relative K-space, Definition \[Kspacedef\].
15. ${\widetriangle{\mathcal U}}
= (({\frak P},\le), \{\mathcal U_{\frak p}\},
\{\Phi_{\frak p\frak q}\})$: Good coordinate system, Definition \[gcsystem\].
16. $\vert{\widetriangle{\mathcal U}}\vert$: Definition \[defofveruver\].
17. $\widehat\Phi : \widehat{\mathcal U} \to \widehat{\mathcal U'}$: KK-embedding. An embedding of Kuranishi structures, Definition \[defn311\].
18. $\widetriangle\Phi : \widetriangle{\mathcal U} \to \widetriangle{\mathcal U'}$ : GG-embedding. An embedding of good coordinate systems, Definition \[defn31222\].
19. $\widehat\Phi : \widehat{\mathcal U} \to \widetriangle{\mathcal U}$: KG-embedding, An embedding of a Kuranishi structure to a good coordinate system, Definition \[defn32020202\].
20. $\widehat\Phi : \widetriangle{\mathcal U} \to\widehat{\mathcal U}$: GK-embedding. An embedding of good coordinate system to a Kuranishi structure, Definition \[embgoodtokura\].
21. $\widehat f : (X,Z;\widehat{\mathcal U}) \to Y$ and $\widetriangle f : (X,Z;\widetriangle{\mathcal U}) \to Y$ : Strongly continuous map, Definitions \[mapkura\] and \[definition32727\].
22. $(X,Z;\widehat{\mathcal U}) \times_{N} M$, $(X_1,Z_1;\widehat{\mathcal U}_1) \times_{M}
(X_2,Z_2;\widehat{\mathcal U}_2)$: Fiber product of Kuranishi structures, Definition \[firberproddukuda\].
23. $S_k(X,Z;\widehat{\mathcal U})$, $S_k(X,Z;\widehat{\mathcal U})$: Corner structure stratification, Definition \[dimstratifidef\].
24. $\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U})$, $\mathcal S_{\frak d}(X,Z;\widetriangle{\mathcal U})$: Dimension stratification, Definition \[stratadim\]
25. $\widehat{\mathcal U} < \widehat{\mathcal U^+}$: $\widehat{\mathcal U^+}$ is a thickening of $\widehat{\mathcal U}$. Definition \[thickening\].
26. $\mathcal S_{\frak p}(X,Z;{\widetriangle {\mathcal U}};\mathcal K)$: Definition \[situ61\] (4).
27. $
\mathcal K = \{\mathcal K_{\frak p}\mid {\frak p
\in \frak P}\}$: A support system. Definition \[situ61\] (2).
28. $(\mathcal K^1,\mathcal K^2)$ or $(\mathcal K^-,\mathcal K^+)$: A support pair, Definition \[situ61\] (3).
29. $\mathcal K^1 < \mathcal K^2$: Definition \[situ61\].
30. $\vert\mathcal K\vert$: Definition \[situ61\].
31. $B_{\delta}(A)$: Metric open ball, (\[defmetricball\]).
32. $\mathcal S_x = (W_x,\omega_x,\{{\frak s}_x^{\epsilon}\})$: CF-perturbation (=continuous family perturbation) on one orbifold chart. Definition \[defn73ss\].
33. $\mathcal S_x^{\epsilon} = (W_x,\omega_x,{\frak s}_x^{\epsilon})$ for each $\epsilon >0$: Definition \[C0convconti\].
34. $\frak S = \{(\frak V_{\frak r},\mathcal S_{\frak r})\mid{\frak r\in
\frak R}\}$: Representative of a CF-perturbation on Kuranishi chart $\mathcal U$. Definition \[semiglobalocntpert\].
Here $\frak V_{\frak r}=(V_{\frak r},E_{\frak r},\Gamma_{\frak r},\phi_{\frak r},\widehat{\phi}_{\frak r})$ is an orbifold chart of $(U,\mathcal E)$ and $\mathcal S_{\frak r} = (W_{\frak r} ,\omega_{\frak r}, \{{\frak s}_{\frak r} ^{\epsilon}\})$ is a CF-perturbation of $\mathcal U$ on $\frak V_{\frak r}$.
35. $\frak S^{\epsilon} = \{(\frak V_{\frak r},\mathcal S_{\frak r}^{\epsilon})\mid{\frak r\in
\frak R}\}$ for each $\epsilon >0$. Definition \[semiglobalocntpert\].
36. $\widetriangle{\frak S} = \{\frak S_{\frak p} \mid \frak p \in \frak P\}$: CF-perturbation of good coordinate system. Definition \[defn7732\].
37. $\widehat{\frak S}$: CF-perturbation of Kuranishi structure. Definition \[defn81\].
38. $\mathscr{S}$: Sheaf of CF-perturbations. Proposition \[prop721\].
39. $
\mathscr S_{\pitchfork 0}
$, $
\mathscr S_{f \pitchfork}
$, $
\mathscr S_{f \pitchfork g}
$: Subsheaves of $\mathscr S$. Definition \[strosubsemiloc\].
40. ${\widetriangle f}!(\widetriangle h;\widetriangle{{\frak S}^{\epsilon}})$: Pushout or integration along the fiber of $\widetriangle{h}$ by $(\widetriangle{f},\widetriangle{{\frak S}^{\epsilon}})$ on good coordinate system. Definition \[pushforwardKuranishi\]
41. ${\widehat f}!(\widehat h;\widehat{{\frak S}^{\epsilon}})$: Pushout or integration along the fiber of $\widehat{h}$ by $(\widehat{f},\widehat{{\frak S}^{\epsilon}})$ on Kuranishi structure. Definition \[deflemgg\].
42. ${\rm Corr}_{(\frak X,\widetriangle{{\frak S}^{\epsilon}})}$: Smooth correspondence associated to good coordinate system. Definition \[defn748\].
43. ${\rm Corr}_{(\frak X,\widehat{\frak S^{\epsilon}})}$: Smooth correspondence of Kuranishi structure Definition \[def92111\].
44. $(\frak s_{\frak p}^{n})^{-1}(0)$: The zero set of multisection.
45. $\Pi((\mathfrak S^{\epsilon})^{-1}(0))$: Support set of a CF-perturbation $\mathfrak S^{\epsilon}$. Definition \[defn767\].
46. $(V,\Gamma,\phi)$: Orbifold chart, Definitions \[2661\], \[defn26550\].
47. $(V,E,\Gamma,\phi,\widehat\phi)$: Orbifold chart of a vector bundle, Definitions \[defn2613\], \[defn2655\].
48. $(X,\mathcal E)$: Orbibundle, Definition \[defn2820\].
[**Convention on the way to use several notations.**]{}
1. We use ‘hat’ such as $\widehat{\mathcal U}$, $\widehat{f}$, $\widehat{\frak S}$, $\widehat{h}$ of an object defined on a Kuranishi structure $\widehat{\mathcal U}$. We use ‘triangle’ such as $\widetriangle{\mathcal U}$, $\widetriangle{f}$, $\widetriangle{\frak S}$, $\widetriangle{h}$ of an object defined on a good coordinate system $\widetriangle{\mathcal U}$.
2. For a Kuranishi structure $\widehat{\mathcal U}$ on $Z\subseteq X$ we write $\mathcal U_p$ for its Kuranishi chart, where $p \in Z$. (We use an italic letter $p$.) For a good coordinate system $\widetriangle{\mathcal U}$ on $Z\subseteq X$ we write $\mathcal U_{\frak p}$ for its Kuranishi chart, where $\frak p \in \frak P$. (We use a German character $\frak p$.) Here $\frak P$ is a partial ordered set.
3. The mark $\blacksquare$ indicates the end of Situation. See Situation \[opensuborbifoldchart\], for example.
Kuranishi structure and good coordinate system {#sec:skuraterm}
==============================================
Kuranish structure {#subsec:kuranishi}
------------------
The notion of Kuranishi structure in this document is the same as one in [@fooo09 Section A1] and [@foootech], except we include the existence of tangent bundle in the definition of Kuranishi structure. The notion of good coordinate system in this document is the same as one in [@foootech]. We introduce some more notations which are useful to shorten the account of this article. We refer Section \[sec:ofd\], for the definition of (effective) orbifold, vector bundle on an oribifold, and their embeddings. Our orbifold is always assumed to be [*effective*]{} unless otherwise mentioned explicitly.
Throughout Part 1, $X$ is always a separable metrizable space.
\[defnKchart\] A [*Kuranishi chart*]{} of $X$ is $\mathcal U =(U,\mathcal E,\psi,s)$ with the following properties.
1. $U$ is an orbifold.
2. $\mathcal E$ is a vector bundle on $U$.
3. $s$ is a smooth section of $\mathcal E$.
4. $\psi : s^{-1}(0) \to X$ is a homeomorphism onto an open set.
We call $U$ a [*Kuranishi neighborhood*]{}, $\mathcal E$ an [*obstruction bundle*]{}, $s$ a [*Kuranishi map*]{} and $\psi$ a [*parametrization*]{}.
If $U'$ is an open subset of $U$, then by restricting $\mathcal E$, $\psi$ and $s$ to $U'$, we obtain a Kuranishi chart which we write $\mathcal U\vert_{U'}$ and call it an [*open subchart*]{}.
The [*dimension*]{} $\mathcal U =(U,\mathcal E,\psi,s)$ is by definition $$\dim \mathcal U = \dim U - {\rm rank} \mathcal E.$$ Here ${\rm rank} \mathcal E$ is the dimension of the fiber $\mathcal E \to U$.
We say that $\mathcal U =(U,\mathcal E,\psi,s)$ is [*orientable*]{} if $U$ and $E$ are orientable. An [*orientation*]{} of $\mathcal U =(U,\mathcal E,\psi,s)$ is a pair of orientations of $U$ and of $\mathcal E$. An open subchart of an oriented Kuranishi chart is oriented.
\[defKchart\] Let $\mathcal U = (U,\mathcal E,\psi,s)$, $\mathcal U'
= (U',\mathcal E',\psi',s')$ be Kuranishi charts of $X$. An [*embedding*]{} of Kuranishi charts $: \mathcal U\to \mathcal U'$ is a pair $\Phi = (\varphi,\widehat\varphi)$ with the following properties.
1. $\varphi : U \to U'$ is an embedding of orbifolds. (See Definition \[def262220\].)
2. $\widehat\varphi : \mathcal E \to \mathcal E'$ is an embedding of vector bundles over $\varphi$. (See Definition \[defn2820\].)
3. $\widehat\varphi \circ s = s' \circ \varphi$.
4. $\psi' \circ \varphi = \psi$ holds on $s^{-1}(0)$.
5. For each $x \in U$ with $s(x) = 0$, the (covariant) derivative $
D_{\varphi(x)}s'
$ induces an isomorphism $$\label{form3.1111}
\frac{T_{\varphi(x)}U'}{(D_x\varphi)(T_xU)}
\cong
\frac{\mathcal E'_{\varphi(x)}}{\widehat\varphi(\mathcal E_x)}.$$
In other words, the map (\[form3.1111\]) is the right vertical arrow of the next commutative diagram. $$\label{diagrampart1333}
\begin{CD}
T_xU
@ > {D_x\varphi} >>
T_{\varphi(x)}U'
@ >>>
\frac{T_{\varphi(x)}U'}{(D_x\varphi)(T_xU)}
\\
@ V{D_xs}VV @ VV{D_{\varphi(x)}s'}V @VVV\\
\widehat\varphi(\mathcal E_x) @>{\widehat{\varphi}}>>\mathcal E'_{\varphi(x)}
@>>>\frac{\mathcal E'_{\varphi(x)}}{\widehat\varphi(\mathcal E_x)}
\end{CD}$$
If $\dim U = \dim U'$ in addition, we call $(\varphi,\widehat\varphi)$ an [*open embedding*]{}.
In the situation of Definition \[defKchart\], suppose $\mathcal U$ and $\mathcal U'$ are oriented. Then the orientations induce trivialization of ${\rm Det} TU' \otimes {\rm Det}{\mathcal E}'$ and of ${\rm Det} TU \otimes {\rm Det} {\mathcal E}$. (Here ${\rm Det} {\mathcal E}$ is a real line bundle which is the determinant line bundle of $\mathcal E$. ${\rm Det} TU'$ etc. are defined in the same way.)
We say $\Phi = (\varphi,\widehat\varphi)$ is [*orientation preserving*]{} if the isomorphism $${\rm Det}T_{\varphi(x)}U' \otimes ({\rm Det}T_{x}U)^*
\cong
{\rm Det}\mathcal E'_{\varphi(x)} \otimes ({\rm Det}\mathcal E_x)^*$$ induced by (\[form3.1111\]) is compatible with these trivializations.
The composition of embeddings of Kuranishi charts is again an embedding of Kuranishi charts. There is an obvious embedding of Kuranishi charts from $\mathcal U$ to itself, that is, the identity. We can define the notion of [*isomorphism*]{} of Kuranishi charts by using the above two facts in an obvious way.
\[kuranishineighborhooddef\] For $A \subseteq X$, a [*Kuranishi neighborhood*]{} of $A$ is a Kuranishi chart such that ${\rm Im}(\psi)$ contains $A$. In case $A=\{p\}$ we call it a Kuranishi neighborhood of $p$ or a Kuranishi chart at $p$.
When $\mathcal U_p = (U_p,\mathcal E_p,\psi_p,s_p)$ is a Kuranishi neighborhood of $p$, we denote by $o_p \in U_p$ the point such that $s_p(o_p)= 0$ and $\psi_p(o_p) = p$. If $q \in {\rm Im}(\psi_p)$ we denote by $o_p(q) \in U_p$ the point such that $s_p(o_p(q))= 0$ and $\psi_p(o_p(q)) = q$. Note such $o_p$ and $o_p(q)$ are unique.
\[coordinatechangedef\] Let $\mathcal U_1 = (U_1,\mathcal E_1,\psi_1,s_1)$, $\mathcal U_2
= (U_2,\mathcal E_2,\psi_2,s_2)$ be Kuranishi charts of $X$. A [*coordinate change in weak sense*]{} (resp. [*in strong sense*]{}) from $\mathcal U_1$ to $\mathcal U_2$ is $\Phi_{21} = (U_{21},\varphi_{21},\widehat\varphi_{21})$ with the following properties (1) and (2) (resp. (1), (2) and (3)):
1. $U_{21}$ is an open subset of $U_1$.
2. $(\varphi_{21},\widehat\varphi_{21})$ is an embedding of Kuranishi charts $: \mathcal U_1\vert_{U_{21}}
\to \mathcal U_2$.
3. $\psi_1(s_1^{-1}(0) \cap U_{21}) = {\rm Im}(\psi_1) \cap {\rm Im}(\psi_2)$.
In case $\mathcal U_1$ and $\mathcal U_2$ are oriented $\Phi_{21}$ is said to be [*orientation preserving*]{} if it is so as an embedding.
We use coordinate changes in weak sense for Kuranishi structures (Definition \[kstructuredefn\]), while we use coordinate changes in strong sense for good coordinate systems (Definition \[gcsystem\]). From now on, [*coordinate changes appearing in Kuranishi structures are in weak sense*]{} and [*coordinate changes appearing in good coordinate systems are in strong sense.*]{}
Hereafter in Part 1, $Z$ is assumed to be a compact subset of $X$, unless otherwise specified.
\[kstructuredefn\] A [*Kuranishi structure*]{} $\widehat{\mathcal U}$ of $Z \subseteq X$ assigns a Kuranishi neighborhood $\mathcal U_p
= (U_p,\mathcal E_p,\psi_p,s_p)$ of $p$ to each $p \in Z$ and a coordinate change in weak sense $\Phi_{pq} = (U_{pq},\varphi_{pq},\widehat\varphi_{pq}) : \mathcal U_q \to \mathcal U_p$ to each $p$, $q \in {\rm Im}(\psi_p) \cap Z$ such that $q \in \psi_q(s_q^{-1}(0) \cap U_{pq})$ and the following holds for each $p$, $q \in {\rm Im}(\psi_p) \cap Z$, $r \in \psi_q(s_q^{-1}(0) \cap U_{pq}) \cap Z$.
We put $U_{pqr} = \varphi_{qr}^{-1}(U_{pq}) \cap U_{pr}$. Then we have $$\label{form3333}
\Phi_{pr}\vert_{U_{pqr}} = \Phi_{pq}\circ
\Phi_{qr}\vert_{U_{pqr}}.$$
We call $Z$ the [*support set* ]{} of our Kuranishi structure.
We also require that the dimension of $\mathcal U_p$ is independent of $p$ and call it the [*dimension*]{} of $\widehat{\mathcal U}$.
\[rem3737\] We require that the equality (\[form3333\]) holds on the domain where both sides are defined. This is always the case of this kinds of equality when we require such an equality between the maps whose domain is a Kuranishi chart that is a member of a Kuranishi structure.
On the other hand, in case when we study maps whose domain is a Kuranishi chart that is a member of a good coordinate system defined in Definition \[gcsystem\], we sometimes require other conditions such as the condition that the domains of the two maps coincide. (See Definition \[defn31222\] (1) for example.) We mention explicitly those conditions when we require it.
\[kuraorient\] We say the Kuranishi structure $(\{\mathcal U_p\},\{\Phi_{pq}\})$ is [*orientable*]{} if we can choose orientation of $\mathcal U_p$ such that all $\Phi_{pq}$ are orientation preserving.
The notion of orientation of an orientable Kuranishi structure and of oriented Kuranishi structure is defined in an obvious way.
\[Kspacedef\] A [*K-space*]{} is a pair $(X,\widehat{\mathcal U})$ of a paracompact metrizable space $X$ and a Kuranishi structure $\widehat{\mathcal U}$ of $X$.
A [*relative K-space*]{} is a triple $(X,Z;\widehat{\mathcal U})$, where $Z \subseteq X$ is a compact subspace and $\widehat{\mathcal U}$ is a Kuranishi structure of $Z \subseteq X$.
In [@FO; @fooobook2; @foootech] we assumed that the orbifold appearing in Kuranishi structure is a global quotient. Namely we assumed $U = V/\Gamma$ where $V$ is a manifold and $\Gamma$ is a finite group acting on $V$ effectively and smoothly. There is no practical difference of the definition since we can always replace $U_p$ by a smaller open subset so that it becomes of the form $U_p = V_p/\Gamma_p$.
In [@joyce] etc. a space with Kuranishi structure is called Kuranishi space. However the name ‘Kuranishi space’ has been used for a long time for the deformation space of complex structure, which Kuranishi discovered in his celebrated work. The Kuranishi structure in our sense is much inspired by Kuranishi’s work, but a space with Kuranishi structure is different from the deformation space of complex structure (Kuranishi space). So we call it K-space in this document.
We also insist to call $s$ a Kuranishi map. This is the main notion discovered by Kuranishi.
From now on when we write Kuranishi neighborhood of $p$ as $\mathcal U_p$, $\mathcal U'_p$ etc. we use the notation $V_p,U_p$ etc. by $
\mathcal U_p = (U_p,\mathcal E_p,\psi_p,s_p)
$.
Good coordinate system {#subsec:goodcoordinate}
----------------------
\[gcsystem\] A [*good coordinate system*]{} of $Z \subseteq X$ is $${\widetriangle{\mathcal U}}
= (({\frak P},\le), \{\mathcal U_{\frak p}\mid \frak p \in \frak P\},
\{\Phi_{\frak p\frak q}
\mid \frak q \le \frak p\})$$ such that:
1. $({\frak P},\le)$ is a partially ordered set. We assume $\# \frak P$ is finite.
2. $\mathcal U_{\frak p}$ is a Kuranishi chart of $X$.
3. $
\bigcup_{\frak p \in \frak P} U_{\frak p} \supseteq Z.
$
4. Let $\frak q \le \frak p$. Then $\Phi_{\frak p \frak q} = (U_{\frak p \frak q},\varphi_{\frak p \frak q},\widehat\varphi_{\frak p \frak q})$ is a coordinate change in strong sense: $\mathcal U_{\frak q} \to \mathcal U_{\frak p}$ in the sense of Definition \[coordinatechangedef\].
5. If $\frak r \le \frak q \le \frak p$, then by putting $U_{\frak p\frak q\frak r} = \varphi_{\frak q\frak r}^{-1}(U_{\frak p\frak q}) \cap U_{\frak p\frak r}$ we have $$\Phi_{\frak p \frak r}\vert_{U_{\frak p \frak q \frak r}} = \Phi_{\frak p \frak q}\circ
\Phi_{\frak q \frak r}\vert_{U_{\frak p \frak q \frak r}}.$$
6. If ${\rm Im}(\psi_{\frak p}) \cap {\rm Im}(\psi_{\frak q})
\ne \emptyset$, then either $\frak p \le \frak q$ or $\frak q \le \frak p$ holds.
7. We define a relation $\sim $ on the disjoint union $\coprod_{\frak p \in \frak P}U_{\frak p}$ as follows. Let $x \in U_{\frak p}, y \in U_{\frak q}$. We define $x\sim y$ if and only if one of the following holds:
1. $\frak p = \frak q$ and $x=y$.
2. $\frak p \le \frak q$ and $y = \varphi_{\frak q\frak p}(x)$.
3. $\frak q\le \frak p$ and $x = \varphi_{\frak p\frak q}(y)$.
Then $\sim$ is an equivalence relation.
8. The quotient of $\coprod_{\frak p \in \frak P}U_{\frak p}/\sim$ by this equivalence relation is Hausdorff with respect to the quotient topology.
In case $Z = X$ we call it a good coordinate system of $X$.
In case ${\widetriangle{\mathcal U}}$ satisfies only (1)-(6), we call it [*a good coordinate system in the weak sense*]{}.
We call $Z$ the [*support set*]{} of our good coordinate system.
We also require that the dimension of $\mathcal U_{\frak p}$ is independent of $\frak p$ and call it the [*dimension*]{} of $\widetriangle{\mathcal U}$.
We say a good coordinate system structure ${\widetriangle{\mathcal U}}
= (({\frak P},\le), \{\mathcal U_{\frak p}\mid \frak p \in \frak P\},
\{\Phi_{\frak p\frak q}
\mid \frak q \le \frak p\}) $ is [*orientable*]{} if we can choose orientation of $\mathcal U_{\frak p}$ such that all $\Phi_{\frak p\frak q}$ are orientation preserving. The notion of orientation of orientable good coordinate system and of oriented good coordinate system is defined in an obvious way.
\[defofveruver\] We write $\vert{\widetriangle{\mathcal U}}\vert$ the quotient set of the equivalence relation in Definition \[gcsystem\] (7). (See [@foootech Remark 5.20] for its topology.)
1. Condition (7) above was not included in the definition of good coordinate system in [@FO], [@fooobook2]. This condition is due to Joyce [@joyce].
2. The fact that Condition (8) makes the argument used in our construction of the perturbations more transparent became clearer during the discussion at the google group Kuranishi. This condition was not included in the definition of good coordinate system in [@FO], [@fooobook2]. The authors thank the members of the google group Kuranishi who helped us much to polish the discussion here.
3. However, we note that, for each ‘good coordinate system’ in the sense of [@FO], [@fooobook2], for which (7), (8) are not necessarily satisfied, we can [*always*]{} shrink $U_{\frak p}$ so that (7), (8) are satisfied. The detailed proof of this fact is given in [@foooshrink]. Therefore, the statements in [@fooobook2] based on the definition of ‘good coordinate system’ in the sense of [@fooobook2], is correct [*as stated there without change*]{}[^18].
4. Throughout this document, we denote by $\mathcal U_{\frak p}$ etc. (where the index $\frak p$ is a German character) a Kuranishi chart which is a member of good coordinate system, and by $\mathcal U_{p}$ etc. (where the index $p$ is an italic letter) a Kuranishi chart which is a member of Kuranishi structure.
From now on, when we write coordinate change $\mathcal U_{\frak q} \to \mathcal U_{\frak p}$ as $\Phi_{\frak p\frak q}$ we use the notations $U_{\frak p\frak q}$, $\varphi_{\frak p \frak q}$ etc. where $\Phi_{\frak p\frak q} = (U_{\frak p \frak q},\varphi_{\frak p \frak q},\widehat\varphi_{\frak p \frak q})$. The same remark applies to $\Phi_{pq}$.
\[sumchart\] Let $\mathcal U_i
= (U_i,\mathcal E_i,\psi_i,s_i)$, $(i=1,2)$ be Kuranishi charts and $\Phi_{21} : \mathcal U_1 \to \mathcal U_2$ a coordinate change. We assume that $\dim U_1 = \dim U_2$ and the map $$U_{21} \to U_1 \times U_2$$ defined by $x \mapsto (\varphi_{21}(x),x)$ is proper. Then there exists a Kuranishi chart $\mathcal U_3
= (U_3,\mathcal E_3,\psi_3,s_3)$ and open KK-embeddings $\Phi_{3i} = (\varphi_{3i},\widehat\varphi_{3i})
: \mathcal U_i \to \mathcal U_3$ $(i=1,2)$ such that
1. $
\Phi_{32}\circ \Phi_{21}
= \Phi_{31}\vert_{U_{21}}.
$
2. $U_3 = {\rm Im}(\varphi_{31}) \cup {\rm Im}(\varphi_{32})$.
We can glue two orbifolds (of the same dimension) $U_1$ and $U_2$ by the diffeomorphism $U_{21} \to U_2$ to its image (that is an open set). Here $U_{21} \subset U_1$ is also an open set. By the properness we assumed the glued space is Hausdorff. Therefore, by gluing we obtain an orbifold, which we denote by $U_3$. We can glue $\mathcal E_1$ and $\mathcal E_2$ to obtain $\mathcal E_3$. The rest of the proof is obvious.
We note that $\mathcal U_3$ is unique up to isomorphism that is compatible with $\Phi_{3i}$. (The proof of this uniqueness is easy and is left to the reader.) We call the chart $\mathcal U_3$ the [*sum chart*]{}.
\[lem310\] Let $\mathcal U_i
= (U_i,\mathcal E_i,\psi_i,s_i)$ $(i=1,2,3)$ and $\Phi_{ij}$ ($1\le i\le j\le 3$) be as in Lemma \[sumchart\]. Let $\mathcal U_0= (U_0,\mathcal E_0,\psi_0,s_0)$ be another Kuranishi chart and $
\Phi_{0i} : \mathcal U_i \to \mathcal U_0
$ (resp. $
\Phi_{i0} : \mathcal U_0 \to \mathcal U_i
$) embeddings of Kuranishi charts for $i=1,2$. We assume $$\aligned
&\Phi_{02} \circ \Phi_{21}
= \Phi_{01}\vert_{U_{21}},\\
&\text{(resp.
$
U_{10} \cap U_{20} = \varphi_{10}^{-1}(U_{21})
$
and
$ \Phi_{21} \circ \Phi_{10}\vert_{U_{10} \cap U_{20} }
= \Phi_{20}\vert_{U_{10} \cap U_{20} }$.)}
\endaligned$$ Then there exists a unique embedding of Kuranishi charts $
\Phi_{03} : \mathcal U_3 \to \mathcal U_0
$ (resp. $
\Phi_{30} : \mathcal U_0 \to \mathcal U_3
$) such that $$\Phi_{03} \circ \Phi_{3i}
=
\Phi_{0i}
\qquad \text{(resp. $\Phi_{3i} \circ \Phi_{i0}
=
\Phi_{30}$).}$$
We note that our orbifold is always assumed to be effective and we only consider embeddings as maps between them. As its consequence, two such maps coincide if they coincide set-theoretically. Moreover, if we are given smooth maps (embedding) on open subsets of our orbifolds so that they coincide on the intersection of the domain, then we can glue them to obtain a smooth map (embedding). Lemma \[lem310\] is an immediate consequence of these facts.
It seems that Lemma \[lem310\] will become false if we include noneffective orbifold.
Embedding of Kuranishi structures I {#subsec:embedding1}
-----------------------------------
\[defn311\] Let $\widehat{\mathcal U} =(\{\mathcal U_p\},\{\Phi_{pq}\})$, $\widehat{\mathcal U'} =(\{\mathcal U'_p\},\{\Phi'_{pq}\})$ be Kuranishi structures of $Z \subseteq X$. A [*strict KK-embedding*]{} $\widehat\Phi = \{\Phi_{p}\}$ from $\widehat{\mathcal U}$ to $\widehat{\mathcal U'}$ assigns, to each $p \in Z$, an embedding of Kuranishi charts $\Phi_p = (\varphi_p,\widehat{\varphi}_p) : \mathcal U_p \to \mathcal U'_p$ such that for each $q \in {\rm Im}(\psi_p) \cap Z$ we have the following:
1. $
\Phi_p \circ \Phi_{pq}\vert_{U_{pq} \cap \varphi_q^{-1}(U'_{pq})}
=
\Phi'_{pq}\circ \Phi_q\vert_{U_{pq} \cap \varphi_q^{-1}(U'_{pq})}.
$
(See Remark \[rem3737\].)
We say that it is an [*open KK-embedding*]{} if $\dim U_p = \dim U'_p$ for each $p$.
We say that $\widehat{\mathcal U}$ is an [*open substructure*]{} of $\widehat{\mathcal U'}$ if there exists an open KK-embedding $\widehat{\mathcal U} \to \widehat{\mathcal U'}$.
A [*KK-embedding*]{} from $\widehat{\mathcal U}$ to $\widehat{\mathcal U^+}$ is a strict KK-embedding $\widehat{\mathcal U_0} \to \widehat{\mathcal U^+}$ from an open substructure $\widehat{\mathcal U_0}$ of $\widehat{\mathcal U}$.
In the situation of Definition \[defn311\] we assume that $\widehat{\mathcal U}$ and $\widehat{\mathcal U'}$ are oriented. We say that $\widehat\Phi = \{\Phi_{p}\}$ is [*orientation preserving*]{} if each of $\Phi_{p}$ is orientation preserving.
The notion orientation preserving embedding can be defined for other types of embeddings (there are 4 types of them see Table 5.1.) in an obvious way.
\[conv323\] Hereafter in this article we assume all the Kuranishi charts, Kuranish structures and good coordinate systems are oriented unless otherwise mentioned explicitly. We also assume all the coordinate change and embedding among Kuranishi charts, Kuranish structures and good coordinate systems are orientation preserving unless otherwise mentioned explicitly.
\[defn31222\] Let ${\widetriangle{\mathcal U}} =(\frak P,\{\mathcal U_{\frak p}\},\{\Phi_{\frak p\frak q}\})$, ${\widetriangle{\mathcal U'}} =(\frak P',\{\mathcal U'_{\frak p'}\},\{\Phi'_{\frak p'\frak q'}\})$ be good coordinate systems of $Z \subseteq X$. A [*GG-embedding*]{} $\widetriangle{\Phi} = \{\Phi_{\frak p}\}$ from ${\widetriangle{\mathcal U}}$ to ${\widetriangle{\mathcal U'}}$ assigns an order preserving map $\frak i : \frak P \to \frak P'$ and, to each ${\frak p}\in \frak P$, an embedding of Kuranishi charts $\Phi_{\frak p} = (\varphi_{\frak p},
\widehat{\varphi}_{\frak p}) : \mathcal U_{\frak p} \to \mathcal U'_{\frak i({\frak p})}$ such that for each $\frak q \le {\frak p}$ we have the following:
1. $
U_{\frak p\frak q} = \varphi_{
\frak q}^{-1}(U'_{\frak i(\frak p)\frak i(\frak q)}).
$
2. $
\Phi_{\frak p} \circ \Phi_{\frak p\frak q}
=
\Phi'_{\frak i(\frak p)\frak i(\frak q)}\circ \Phi_{\frak q}\vert_{U_{\frak p\frak q}}.
$
$$\label{diag33--}
\begin{CD}
\mathcal U_{\frak q}\vert_{U_{\frak p\frak q}
} @ > {\Phi_{\frak q}} >>
{\mathcal U}'_{\frak i(\frak q)}\vert_{{U}'_{\frak i(\frak p)\frak i(\frak q)}} \\
@ V{\Phi_{\frak p\frak q}}VV @ VV{\Phi'_{\frak i(\frak p)\frak i(\frak q)}}V\\
\mathcal U_{\frak p} @ > {\Phi_{\frak p}} >>{\mathcal U}'_{\frak i(\frak p)}
\end{CD}$$
We say that $\widetriangle\Phi$ is a [*weakly open GG-embedding*]{} if $\dim U_{\frak p} = \dim U'_{\frak i(\frak p)}$ for each $\frak p$. We say it is an [*open GG-embedding*]{} if $\frak P = \frak P'$ in addition. We say it is a [*strongly open GG-embedding*]{} if $$\psi_{\frak p}(U_{\frak p} \cap s_{\frak p}^{-1}(0)) = \psi_{\frak p}(U'_{\frak p} \cap (s'_{\frak p})^{-1}(0))$$ holds in addition.
We say that ${\widetriangle{\mathcal U}}$ is an [*open substructure*]{} (resp. [*weakly open substructure*]{}, [*strongly open substructure*]{}) of ${\widetriangle{\mathcal U'}}$ if there exists an open (resp. weakly open, strongly open) GG-embedding ${\widetriangle{\mathcal U}} \to {\widetriangle{\mathcal U'}}$.
We say a GG-embedding $\widetriangle{\Phi}$ is an [*isomorphism*]{} if the map $\frak i$ is a bijection and $\widehat{\varphi}_{\frak p}$ is an isomorphism for each $\frak p$.
Definition \[defn31222\], especially Item (1), implies that a GG-embedding ${\widetriangle{\mathcal U}} \to {\widetriangle{\mathcal U'}}$ induces an injective continuous map $\vert {\widetriangle{\mathcal U}}\vert
\to \vert {\widetriangle{\mathcal U'}}\vert$.
\[lem320\] Let $\widetriangle{\mathcal U} = (\frak P,\{\mathcal U_{\frak p}\},\{\Phi_{\frak p\frak q}\})$ be a good coordinate system of $Z \subseteq X$ and let $U^0_{\frak p} \subseteq U_{\frak p}$ be given open subsets such that $
Z \subset \bigcup_{\frak p \in \frak P} \psi_{\frak p}(s_{\frak p}^{-1}(0) \cap U^0_{\frak p}).
$ Then there exists a unique coordinate change $\Phi_{\frak p\frak q}^0$ such that $(\frak P,\{\mathcal U_{\frak p}\vert_{U^0_{\frak p}}\},\{\Phi^0_{\frak p\frak q}\})$ is an open substructure of $\widetriangle{\mathcal U}$.
Let $\Phi_{\frak p\frak q} = (U_{\frak p\frak q},\varphi_{\frak p\frak q},
\widehat\varphi_{\frak p\frak q})$. We put $$\label{eqform3636}
U^0_{\frak p\frak q}
=
U_{\frak p\frak q} \cap U^0_{\frak q} \cap \varphi_{\frak p\frak q}^{-1}(U^0_{\frak p})$$ and $\Phi^0_{\frak p\frak q} = (U^0_{\frak p\frak q},\varphi_{\frak p\frak q}\vert_{U^0_{\frak p\frak q}},
\widehat\varphi_{\frak p\frak q}\vert_{U^0_{\frak p\frak q}})$. It is easy to see that $(\frak P,\{\mathcal U_{\frak p}\vert_{U^0_{\frak p}}\},\{\Phi^0_{\frak p\frak q}\})$ is an open substructure of $\widetriangle{\mathcal U}$.
On the other hand, if $(\frak P,\{\mathcal U_{\frak p}\vert_{U^0_{\frak p}}\},\{\Phi^0_{\frak p\frak q}\})$ is an open substructure of $\widetriangle{\mathcal U}$, then Definition \[defn31222\] (1) implies that the domain $U^0_{\frak p\frak q}$ of $\Phi_{\frak p\frak q}$ must be as in (\[eqform3636\]).
\[lem321321\] Let $\widehat{\mathcal U} = (\{\mathcal U_{p}\},\{\Phi_{pq}\})$ be a Kuranishi structure of $Z \subseteq X$ and $U^0_{p} \subseteq U_{p}$ open subsets containing $p$. Then there exists $\Phi_{pq}^0$ such that $(\{\mathcal U_{p}\vert_{U^0_{p}}\},\{\Phi^0_{pq}\})$ is an open substructure of $\widehat{\mathcal U}$.
The uniqueness dose not hold in Lemma \[lem321321\] since there is no condition similar to Definition \[defn31222\] (1) for Kuranishi structure. They have, however, a common open substructure.
We put $$\label{eqform3636revrev}
U^0_{pq}
=
U_{pq} \cap U^0_{q} \cap \varphi_{pq}^{-1}(U^0_{p})$$ and $\Phi^0_{pq} = (U^0_{pq},\varphi_{pq}\vert_{U^0_{pq}},
\widehat\varphi_{pq}\vert_{U^0_{pq}})$. It is easy to see that $(\{\mathcal U_{p}\vert_{U^0_{p}}\},\{\Phi^0_{pq}\})$ is a Kuranishi structure.
\[defn32020202\] Let $\widehat{\mathcal U}$ be a Kuranishi structure and ${\widetriangle{\mathcal U}}$ a good coordinate system of $Z \subseteq X$. A [*strict KG-embedding*]{} of $\widehat{\mathcal U}$ to ${\widetriangle{\mathcal U}}$ assigns, for each $p\in Z$, $\frak p\in \frak P$ with $p \in {\rm Im}(\psi_{\frak p})$, an embedding of Kuranishi charts $
\Phi_{\frak p p} = (\varphi_{\frak p p},\widehat\varphi_{\frak p p}) :
\mathcal U_p \to \mathcal U_{\frak p}$ with the following properties.
If $\frak p,\frak q\in \frak P$, $\frak q \le \frak p$, $p \in {\rm Im}(\psi_{\frak p}) \cap Z$, $q \in {\rm Im}(\psi_{p}) \cap \psi_{\frak q}(U_{\frak p\frak q}) \cap Z$, then the following diagram commutes. $$\label{diag33}
\begin{CD}
\mathcal U_{q}\vert_{U_{pq}
\cap \varphi_{\frak q q}^{-1}(U_{\frak p \frak q})} @ > {\Phi_{\frak q q}} >>
{\mathcal U}_{\frak q}\vert_{{U}_{\frak p\frak q}} \\
@ V{\Phi_{pq}}VV @ VV{\Phi_{\frak p \frak q}}V\\
\mathcal U_{p} @ > {\Phi_{\frak p p}} >>{\mathcal U}_{\frak p}
\end{CD}$$
A [*KG-embedding*]{} of $\widehat{\mathcal U}$ to ${\widetriangle{\mathcal U}}$ is by definition a strict KG-embedding of an open substructure of $\widehat{\mathcal U}$ to ${\widetriangle{\mathcal U}}$.
If $\widehat{\mathcal U_0} \to \mathcal{\widehat{\mathcal U}}$ is a KK-embedding, ${\widetriangle{\mathcal U}}
\to \widetriangle{{\mathcal U}^+}$ is a GG-embedding, and $\widehat{\mathcal U} \to {\widetriangle{\mathcal U}}$ is a strict KG-embedding (resp. KG-embedding), then the [*composition*]{} $$\widehat{\mathcal U}_0 \to \widehat{\mathcal U}
\to {\widetriangle{\mathcal U}} \to \widetriangle{{\mathcal U}^+}$$ is defined as a strict KG-embedding (resp. a KG-embedding). (See Definition \[definition516161\] (3).)
The next result is the same as [@FO Lemma 6.3], [@foootech Theorem 7.1]. We will reproduce its proof together with various addenda in Section \[sec:contgoodcoordinate\]. In particular, Theorem \[Them71restate\] is proved in Subsection \[subsec:constgcsabs\].
\[Them71restate\] For any Kuranishi structure $\widehat{\mathcal U}$ of $Z \subseteq X$ there exist a good coordinate system ${\widetriangle{\mathcal U}}$ of $Z \subseteq X$ and a KG-embedding $\widehat{\mathcal U} \to {\widetriangle{\mathcal U}}$.
According to Convention \[conv323\], Theorem \[Them71restate\] contains the statement that ${\widetriangle{\mathcal U}}$ and the KG-embedding $\widehat{\mathcal U}\to {\widetriangle{\mathcal U}}$ are oriented (provided $\widehat{\mathcal U}$ is oriented). We do not repeat this kinds of remarks later on.
\[gcsystemcompa\] A good coordinate system $\widetriangle{\mathcal U}$ is said to be [*compatible*]{} with a Kuranishi structure ${\widehat{\mathcal U}}$ if there exists a KG-embedding $\widehat{\mathcal U} \to {\widetriangle{\mathcal U}}$.
1. The next terminology is due to Joyce [@joyce].
A good coordinate system is said to be [*excellent*]{} if $\frak P \subset \Z_{\ge 0}$, $\le$ is the standard inequality on $\Z_{\ge 0}$ and $\dim U_{\frak p} = \frak p$.
Starting with an arbitrary good coordinate system $\widetriangle{\mathcal U} = (\{U_{\frak p}\},\{\Phi_{\frak p\frak q}\})$, we can construct an excellent good coordinate system $\widetriangle{\mathcal U'}$ as follows. Suppose $\dim U_{\frak p_1} = \dim U_{\frak p_2}$. If ${\rm Im}(\psi_{\frak p_1})$ is disjoint from ${\rm Im}(\psi_{\frak p_2})$, we take its disjoint union as a new chart and remove these two charts. Suppose ${\rm Im}(\psi_{\frak p_1}) \cap {\rm Im}(\psi_{\frak p_2}) \ne \emptyset$. Then we may assume $\frak p_1 < \frak p_2$. Since an embedding between two orbifolds of the same dimension is necessarily a diffeomorphism, the coordinate change $\Phi_{\frak p_2\frak p_1}$ is an isomorphism. We can use this observation and Lemma \[sumchart\] to construct the sum chart of $\mathcal U_{\frak p_1}$ and $\mathcal U_{\frak p_2}$. We take it as a new chart and remove $\mathcal U_{\frak p_1}$ and $\mathcal U_{\frak p_2}$. The coordinate change between sum charts and other charts can be defined by using Lemma \[lem310\].
We can continue this process finitely many times until we get an excellent coordinate system. Note there is a weakly open GG-embedding $\widetriangle{\mathcal U} \to \widetriangle{\mathcal U'}$.
2. We note that in [@foootech Section 7] we introduced the notion of mixed neighborhood. It is basically equivalent to the notion of a pair of an excellent good coordinate system ${\widetriangle{\mathcal U}}$ and a KG-embedding $\widehat{\mathcal U} \to {\widetriangle{\mathcal U}}$. (The only difference is that the conclusion of [@foootech Lemma 7.32] is not assumed in [@foootech Definition 7.15]. This difference is not essential at all because of [@foootech Lemma 7.32].) Therefore Theorem \[Them71restate\] is equivalent to [@foootech Theorem 7.1].
\[mapkura\] Let $\widehat{\mathcal U}$ be a Kuranishi structure of $Z \subseteq X$.
1. A [*strongly continuous map $\widehat f$*]{} from $(X,Z;\widehat{\mathcal U})$ to a topological space $Y$ assigns a continuous map $f_{p}$ from $U_{p}$ to $Y$ for each $p\in X$ such that $f_p \circ \varphi_{pq} = f_q$ holds on $U_{pq}$.
2. In the situation of (1), the map $f:Z \to Y$ defined by $f(p) = f_p(p)$ is a continuous map from $Z$ to $Y$. We call $f : Z \to Y$ the [*underlying continuous map*]{} of $\widehat f$.
3. We require that the underlying continuous map $f : Z \to Y$ is extended to a continuous map $f : X \to Y$ and include it to the data defining a strong continuous map.
4. When $Y$ is a smooth manifold, we say $\widehat f$ is [*strongly smooth*]{} if each of $f_p$ is smooth.
5. A strongly smooth map is said to be [*weakly submersive*]{} if each of $f_p$ is a submersion.
We sometimes say $f$ is a strongly continuous map (resp. a strongly smooth map, a weakly submersive map) in place of $\widehat f$ is a strongly continuous map (resp. a strongly smooth map, weakly submersive), by an abuse of notation.
\[remrem327\] The continuity claimed in (2) follows from the next diagram $$\begin{CD}
s_p^{-1}(0) @ > {\psi_{p}} >>
Z \\
@ V{f_p}VV @ VV{f}V\\
Y @ = Y
\end{CD}$$ whose comutativity is a consequence of $f_p \circ \varphi_{pq} = f_q$.
In [@joyce], Joyce used the terminology ‘strong submersion’ instead of ‘weak submersion’ which we have been used since [@FO]. We insist on using the terminology ‘weakly submersive’ by the following two reasons.
1. Let $V_1 \subset V_2$ be a submanifold, $f_2$ a map from $V_2$ and $f_1$ a restriction of $f_2$ to $V_1$. The condition that $f_2$ is smooth is stronger than the condition that $f_1$ is smooth. So we used the word strongly smooth. On the other hand, in case $f_2$ is smooth, the condition that $f_2$ is a submersion at each point of $V_1$ is weaker than the condition that $f_1$ is a submersion. So we used the word weak submersion.
2. Later we will use the terminology ‘strongly submersive’ for a different notion. See Definitions \[transofdvect\] and \[transkurakuravect\].
\[definition32727\] Let ${\widetriangle{\mathcal U}}$ be a good coordinate system of $Z \subseteq X$.
1. A [*strongly continuous map $\widetriangle f$*]{} from $(X,Z;{\widetriangle{\mathcal U}})$ to a topological space $Y$ assigns a continuous map $f_{\frak p}$ from $U_{\frak p}$ to $Y$ to each $\frak p\in \frak P$ such that $f_{\frak p} \circ \varphi_{{\frak p}{\frak q}} = f_{\frak q}$ holds on $U_{{\frak p}{\frak q}}$.
2. In the situation of (1), the map $f: Z \to Y$ defined by $f(p) = f_{\frak p}(o_{\frak p}(p))$ (for $p \in {\rm Im}(\psi_{\frak p})\cap Z$) is a continuous map from $Z$ to $Y$. [^19] We call $f : Z \to Y$ the [*underlying continuous map*]{} of $\widetriangle f$.
3. We require that the underlying continuous map $f : Z \to Y$ is extended to a continuous map $f : X \to Y$ and include it to the data defining strong continuous map.
4. When $Y$ is a smooth manifold, we say $\widetriangle f$ is [*strongly smooth*]{} if each of $f_{\frak p}$ is smooth.
5. A strongly smooth map is said to be [*weakly submersive*]{} if each of $f_{\frak p}$ is a submersion.
We sometimes say $f$ is a strongly continuous map (resp. a strongly smooth map, weakly submersive) in place of $\widetriangle f$ is a strongly continuous map (resp. a strongly smooth map, weakly submersive), by an abuse of notation.
\[defn320\]
1. If $\widehat f : (X,Z;\widehat{\mathcal U'}) \to Y$ is a strongly continuous map and $\widehat\Phi = \{\Phi_{p}\} : \widehat{\mathcal U} \to \widehat{\mathcal U'}$ is a KK-embedding, then $f_p \circ \varphi_p : U_p \to Y$ defines a strongly continuous map, which we call the [*pullback*]{} and write $\widehat f \circ \widehat\Phi$.
2. Let $\widehat{\Phi}$ be a KK-embedding. If $\widehat f$ is strongly smooth, then so is $\widehat f \circ \widehat\Phi$. If $\widehat{\Phi}$ is an open embedding and $\widehat f$ is weakly submersive, then $\widehat f \circ \widehat\Phi$ is also weakly submersive.
3. The good coordinate system version of pullback of maps can be defined in the same way as (1). A similar statement as (2) holds as well.
4. If $\widetriangle f : (X,Z;{\widetriangle{\mathcal U}}) \to Y$ is a strongly continuous map from a good coordinate system and $\widehat\Phi : \widehat{\mathcal U} \to {\widetriangle{\mathcal U}}$ is a KG-embedding, then the pullback $\widetriangle f\circ \widehat\Phi$ can be defined in the same way as (1). A similar statement as (2) holds as well.
Fiber product of Kuranishi structures {#sec:fiber}
=====================================
Fiber product {#subsec:fibprod}
-------------
Before studying fiber product we consider direct product. Let $X_i$, $i=1,2$ be separable metrizable spaces, $Z_i \subseteq X_i$ compact subsets, and $\widehat{\mathcal U}_i$ Kuranishi structures of $Z_i \subseteq X_i$. We will define a Kuranishi structure of the direct product $Z_1 \times Z_2 \subseteq X_1 \times X_2$.
\[directproduct\] For $p_i \in Z_i$ let $\mathcal U_{p_i} = (U_{p_i},\mathcal E_{p_i},\psi_{p_i},s_{p_i})$ be their Kuranishi neighborhoods. Then the Kuranishi neighborhood of $p = (p_1,p_2) \in Z_1 \times Z_2$ is $\mathcal U_{p} = \mathcal U_{p_1} \times \mathcal U_{p_2} = (U_p,\mathcal E_p,\psi_p,s_p)$ where $$(U_p,\mathcal E_p,\psi_p,s_p) =
(U_{p_1}\times U_{p_2},\mathcal E_{p_1}\times \mathcal E_{p_2},\psi_{p_1}\times \psi_{p_2},s_{p_1}\times
s_{p_2}) .$$ This system satisfies the condition of Kuranishi neighborhood (Definition \[kuranishineighborhooddef\].)
Suppose $q_i \in Z_i$ and $q = (q_1,q_2) \in Z_1 \times Z_2$. If $q \in \psi_p(s^{-1}_p(0))$, then it is easy to see that $q_i \in \psi_{p_i}(s^{-1}_{p_i}(0))$ for $i=1,2$. Therefore there exist coordinate changes $\Phi_{p_iq_i} = ({\varphi_{p_iq_i}},\widehat{\varphi}_{p_iq_i},h_{p_iq_i})$ from $\mathcal U_{q_i}$ to $\mathcal U_{p_i}$. We define $$\aligned
\Phi_{pq} &=
\Phi_{p_1q_1} \times \Phi_{p_2q_2}=
(U_{pq},{\varphi}_{pq},\widehat{\varphi}_{pq})
\\
&=
(U_{p_1q_1} \times U_{p_2q_2},{\varphi}_{p_1q_1}\times {\varphi}_{p_2q_2},\widehat{\varphi}_{p_1q_1}\times \widehat{\varphi}_{p_2q_2}).
\endaligned$$ This satisfies the condition of coordinate change of Kuranishi charts (Definition \[coordinatechangedef\]).
Then it is also easy to show that $(\{\mathcal U_{p_1} \times \mathcal U_{p_2}\},\{\Phi_{p_1q_1} \times \Phi_{p_2q_2}\})$ defines a Kuranishi structure of $Z_1 \times Z_2 \subseteq X_1 \times X_2$ in the sense of Definition \[kstructuredefn\]. (We note that effectivity of an orbifold is preserved by the direct product.) We call this Kuranishi structure the [*direct product Kuranishi structure*]{}.
We can easily prove that the direct product of oriented Kuranishi structures ([@foootech Definition 4.5]) is also oriented.
Next we study fiber product. Let $(X,Z;\widehat{\mathcal U})$ be a relative K-space and $\widehat f = \{f_p\}: (X,Z;\widehat{\mathcal U}) \to N$ a strongly smooth map, where $N$ is a smooth manifold of finite dimension. Let $f' : M \to N$ be a smooth map between smooth manifolds. We assume $M$ is compact. In this section we define a Kuranishi structure on the pair of topological spaces $$\label{formula4141}
\aligned
Z \times_N M
&= \{(p,q) \in Z \times M \mid f(p) = f'(q)\},
\\
X \times_N M
&= \{(p,q) \in X \times M \mid f(p) = f'(q)\},
\endaligned$$ that is the fiber product. The assumption we need to require is certain transversality, which we define below.
\[transverse1\] We say $\widehat f$ is [*weakly transversal*]{} to $f'$ on $Z \subseteq X$ if the following holds. Let $(p,q) \in Z \times_N M$ and $\mathcal U_p = (U_p,E_p,s_p,\psi_p)$ be a Kuranishi neighborhood of $p$. We then require that for each $(x,y) \in U_p\times M$ with $f_p(x) = f'(y)$ we have $$\label{transformula}
(d_xf_p)(T_xU_p) + (d_yf')(T_yM) = T_{f(x)}N.$$ Let us explain the meaning of (\[transformula\]). We take an orbifold chart $(V_p(x),\Gamma_p(x),\phi_p(x))$ of $U_p$ at $x$. (Definition \[defn26550\].) The composition $$V_p(x) \overset{\phi_p(x)}\longrightarrow U_p \overset{f_p}\longrightarrow N$$ is by assumption a smooth map which we write $f_{p,x}$. (\[transformula\]) means $$(d_{o(x)}f_{p.x})(T_{o_p(x)}V_p(x)) + (d_yf')(T_yM) = T_{f(x)}N$$ where $o(x) \in V_p(x)$ is the base point which satisfies $\phi_p(x)(o(x)) = x$.
\[fiberexa\]
1. If $\widehat f : (X,Z;\widehat{\mathcal U}) \to N$ is weakly submersive in the sense of Definition \[mapkura\], then for any $f' : M \to N$, $f$ is weakly transversal to $f'$.
2. If $f' : M \to N$ is a submersion, then any strongly smooth map $\widehat f : (X,Z;\widehat{\mathcal U}) \to N$ is weakly transversal to $f'$.
3. The pullback of map in Definition \[defn320\] by an open embedding preserves the weak transversality.
4. Suppose $(X_i,Z_i)$ $(i=1,2)$ have Kuranishi structures $\widehat{\mathcal U_i}$ and the maps $\widehat f_i : (X_i,Z_i;\widehat{\mathcal U_i}) \to N$ are strongly smooth. We put: $$Z_1 \times_N Z_2
=
\{(p,q) \in Z_1 \times Z_2 \mid f_1(p) = f_2(q) \}.$$ Let $(p,q) \in Z_1 \times_N Z_2$. We denote by $\mathcal U_p$, $\mathcal U_q$ the Kuranishi neighborhoods of $p, \, q$ respectively and assume the condition $$\label{2transverse}
(d_x(f_1)_p)(T_xU_p) + (d_y(f_2)_q)(T_yU_q) = T_{(f_1)_p(x)}N$$ for each $(x,y) \in U_p\times U_q$ with $(f_1)_p(x) = (f_2)_q(y)$. (The precise meaning of (\[2transverse\]) can be defined in a similar way as the case of (\[transformula\]).) It is easy to see that (\[2transverse\]) is equivalent to the next condition. We consider the map $$f = (f_1,f_2) : X_1 \times X_2 \to N\times N.$$ We use the direct product Kuranishi structure (Definition \[directproduct\]) on $Z_1\times Z_2 \subseteq X_1\times X_2$. Then (\[2transverse\]) holds if and only if $f$ is transversal to the diagonal embedding $N \to N \times N$ in the sense of Definition \[transverse1\].
5. We can generalize the situation of (4) to the case when three or more factors are involved. In fact, in the study of the moduli space of pseudo-holomorphic curves, we will encounter the situation where we consider the fiber product of various factors which are organized by a tree or a graph. See Parts 2 and 3 or [@fooobook2 Subsection 7.1.1].
In the situation of Example \[fiberexa\] (4) we say $\widehat f_1$ is [*weakly transversal*]{} to $\widehat f_2$ if (\[2transverse\]) is satisfied.
Now we assume that $\widehat f : (X,Z;\widehat{\mathcal U}) \to N$ is weakly transversal to $f' : M \to N$ in the sense of Definition \[transverse1\] and define a Kuranishi structure on the fiber product (\[formula4141\]). Recalling that a Kuranishi neighborhood $U_p$ of $p \in Z$ is assumed to be an effective orbifold, we find the following.
\[fibereffec\] For each $(p,\frak x) \in Z \times_N M$ the fiber product $U_p\times_N M$ is again an effective orbifold.
Let $(p, {\mathfrak x}) \in Z \times_N M$. Pick an orbifold chart $(V_p,
\Gamma_p)$ at $p$. Denote by ${\tilde o}_p \in V_p$ the point, which is mapped to $o_p$ under the quotient map $V_p \to U_p$ by $\Gamma_p$ and by $\widetilde{f}_p$ the lift of $f_p:U_p
\to N$. Since $\widetilde{f}_p$ is $\Gamma_p$-invariant, we find that $K_{{\tilde o}_p}={\rm Ker~} d\widetilde{f}_p $ at ${\tilde o}_p$ is transversal to the tangent space $T_{{\tilde o}_p} V_p^{\Gamma_p}$ of the fixed point set $V_p^{\Gamma_p}$ by $\Gamma_p$-action. Hence the tangent space $T_{[{\tilde o}_p, x]} V_p \times_N M$ contains $K_{{\tilde o}_p}$. Since $\Gamma_p$ acts trivially on $V_p^{\Gamma_p}$ and $\Gamma_p$ acts effectively on $V_p$, $\Gamma_p$ acts effectively on $K_{{\tilde o}_p}$. It implies that the fiber product $U_p \times_N M$ is an effective orbifold.
Let $\mathcal U_p$ be the given Kuranishi neighborhood of $p$ and $(p,\frak x) \in Z \times_N M$. We define $
U_{(p,\frak x)} = U_p \times_N M.
$ Note $U_{(p,\frak x)}$ is a smooth orbifold by Definition \[transverse1\]. The bundle $\mathcal E_{(p,\frak x)}$ is the pullback of $\mathcal E_p$ by the map $U_{(p,\frak x)} \to U_p$ that is the projection to the first factor. The section $s_p$ induces $s_{(p,\frak x)}$ of $\mathcal E_{(p,\frak x)}$ in an obvious way. Note $
s_{(p,\frak x)}^{-1}(0) = s_p^{-1}(0) \times_N M.
$ Therefore $\psi_p : s_p^{-1}(0) \to X$ induces $$\psi_{(p,\frak x)} : s_{(p,\frak x)}^{-1}(0)
= s_p^{-1}(0) \times_N M
\to X \times_N M.$$ It is easy to see that $\psi_{(p,\frak x)}$ is a homeomorphism onto a neighborhood of $(p,\frak x)$.
In sum we have the following:
\[lem25\] $\mathcal U_{(p,\frak x)} = (U_{(p,\frak x)},\mathcal E_{(p,\frak x)},s_{(p,\frak x)},\psi_{(p,\frak x)})$ is a Kuranishi neighborhood of $(p,\frak x) \in X\times_N M$.
We next consider the coordinate change. Let $(p,\frak x), (q,\frak y) \in Z\times_N M$. We assume $
(q,\frak y) = \psi_{(p,\frak x)}(x,y)
$ where $(x,y) \in V_{(p,\frak x)}$. By definition we have $
q = \psi_p(x).
$ Therefore by Definition \[kstructuredefn\] there exists a coordinate change $\Phi_{pq} = (U_{pq},\varphi_{pq},\widehat{\varphi}_{pq})$ from $\mathcal U_q$ to $\mathcal U_p$ in the sense of Definition \[coordinatechangedef\].
Now we put
1. $
U_{(p,\frak x),(q,\frak y)} = U_{pq} \times_N M
$.
2. $\varphi_{(p,\frak x),(q,\frak y)} = \varphi_{pq} \times_N {\rm id}
: U_{pq} \times_N M \to U_p \times_N M.$
3. $\widehat\varphi_{(p,\frak x),(q,\frak y)} = \widehat\varphi_{pq} \times_N {\rm id} :
\mathcal E_q\vert_{U_{pq}} \times_N M \to \mathcal E_p\times_N M $.
\[lem27\] $\Phi_{(p,\frak x),(q,\frak y)} = (U_{(p,\frak x),(q,\frak y)}
,\varphi_{(p,\frak x),(q,\frak y)},\widehat\varphi_{(p,\frak x),(q,\frak y)})$ defines a coordinate change from $\mathcal U_{(q,\frak y)}$ to $\mathcal U_{(p,\frak x)}$.
The proof is immediate from the definition.
\[fiberkura22\] Suppose $Z \subseteq X$ has a Kuranishi structure $\widehat{\mathcal U}$ and $\widehat f : (X,Z;\widehat{\mathcal U}) \to N$ is weakly transversal to $f' : M \to N$. Then the Kuranishi neighborhoods in Lemma \[lem25\] together with coordinate changes in Lemma \[lem27\] define a Kuranishi structure of the fiber product of $Z \times_N M \subseteq X \times_N M$.
The proof is again immediate from the definition.
\[firberproddukuda\]
1. Suppose $\widehat f : (X,Z;\widehat{\mathcal U}) \to N$ is weakly transversal to $f' : M \to N$. We call the Kuranishi structure obtained in Lemma \[fiberkura22\], the [*fiber product Kuranishi structure*]{} and write the resulting relative K-space by $$(X,Z;\widehat{\mathcal U}) \,{}_{f}\times_{f'} M
\quad
\text{or}
\quad
(X,Z;\widehat{\mathcal U}) \times_{N} M.$$
2. Suppose $\widehat f_i : (X_i,Z_i;\widehat{\mathcal U}_i) \to N$ are strongly smooth maps. We assume $\widehat f_1$ and $\widehat f_2$ are weakly transversal in the sense of Example \[fiberexa\] (4). Then we define the [*fiber product*]{} $$(X_1,Z_1;\widehat{\mathcal U}_1) \,{}_{f_1}\times_{f_2}
(X_2,Z_2;\widehat{\mathcal U}_2)
\quad
\text{or}
\quad
(X_1,Z_1;\widehat{\mathcal U}_1) \times_{M}
(X_2,Z_2;\widehat{\mathcal U}_2)$$ as the fiber product $$\left(
(X_1,Z_1;\widehat{\mathcal U}_1) \times
(X_2,Z_2;\widehat{\mathcal U}_2)\right)
\,\,{}_{f_1 \times f_2}\times_{i} \Delta_M.$$ Here $i : \Delta_M \to M\times M$ is the embedding of the diagonal.
For the purpose of reference we also include another obvious statements.
We consider the situation of Lemma \[fiberkura22\].
1. If $\widehat g : (X,Z;\widehat{\mathcal U}) \to M'$ is also a strongly continuous map, then it induces a strongly continuous map $Z \times_N M \to M'$. It is weakly submersive if $(\widehat f,\widehat g) : (X,Z;\widehat{\mathcal U}) \to N \times M'$ is weakly submersive.
2. If $\widehat f$ is weakly submersive, then the projection $Z \times_N M \to M$ is weakly submersive .
Let $\widehat{\mathcal U}$, $\widehat{\mathcal U^+}$ be Kuranishi structures of $Z \subseteq X$ and $\widehat{\Phi} : \widehat{\mathcal U} \to \widehat{\mathcal U^+}$ a KK-embedding. Let $\widehat f : (X,Z;\widehat{\mathcal U^+}) \to N$ be a strongly smooth map and $g : M \to N$ a smooth map between manifolds.
1. If $\widehat f\circ \widehat{\Phi} : (X,Z;\widehat{\mathcal U}) \to N$ is weakly transversal to $g$, then $\widehat f : (X,Z;\widehat{\mathcal U^+}) \to N$ is weakly transversal to $g$.
2. In the situation of (1), $\widehat{\Phi}$ induces a KK-embedding $$\widehat{\Phi} \times_N M :
\widehat{\mathcal U} \times_N M
\to
\widehat{\mathcal U^+} \times_N M.$$
The same conclusions hold if we replace $g : M \to N$ by a strongly smooth map from a relative K-space $g : (X',Z';\widehat{\mathcal U'}) \to N$.
Boundary and corner I {#subsec:bdrycorn1}
---------------------
So far we study the case when our Kuranishi structures do not have boundary or corner. Its generalization to the case when our Kuranishi structure and/or the manifold $M$ has boundary or corner is straightforward. We however state them for the completeness’ sake. Later we need to and will study boundary and corner more systematically. (See Subsections \[subsec:kuranishiemb2\], \[subsection:normbdry2\], and Part 2.)
An orbifold with corner is a space locally homeomorphic to $V/\Gamma$ where $V$ is a smooth manifold with corner and $\Gamma$ is a finite group acting smoothly and effectively on $V$. We assume the smoothness of the coordinate change as usual.
See Definition \[orbifolddefn\] for more precise and detailed definition.
\[defn4111\] Let $M$ be an orbifold with corner. It has the following canonical stratification $\{S_k(M)\}$. The stratum $S_k(M)$ is the closure of the set of the points whose neighborhoods are diffeomorphic to open neighborhoods of $0$ of the space $([0,1)^k \times \R^{n-k})/\Gamma$. We call this stratification the [*corner structure stratification*]{} of $M$.
It is easy to see that $\overset{\circ}S_k(M) = S_k(M) \setminus S_{k+1}(M)$ carries a structure of a smooth orbifold of dimension $n-k$ without boundary or corner. However this orbifold may not be effective. In this document, we [*assume*]{} the next condition in addition as a part of the definition of orbifold with corners.
\[effectivitycorner\](Corner effectivity hypothesis) When we say $M$ is an orbifold with corners, we assume the orbifold $\overset{\circ}S_k(M)$ is always an [*effective*]{} orbifold in this document.
\[dimstratifidef\] For a relative K-space $(X,Z;\widehat{\mathcal U})$, we put $$\aligned
S_k(X,Z;\widehat{\mathcal U})
&=
\{p \in Z \mid o_p \in S_k(U_p)\},
\\
\overset{\circ}S_k(X,Z;\widehat{\mathcal U})
&=
S_k(X,Z;\widehat{\mathcal U})
\setminus
\bigcup_{k' > k}S_{k'}(X,Z;\widehat{\mathcal U}),
\endaligned$$ where $\mathcal U_p = (U_p,E_p,s_p,\psi_p)$ is the Kuranishi neighborhood of $p$. We call this stratification the [*corner structure stratification*]{} of $\widehat{\mathcal U}$. We can define corner structure stratification $\{S_k(X,Z;\widetriangle{\mathcal U})\}$ of a good coordinate system $\widetriangle{\mathcal U}$ in the same way.
For any compact subset $K$ of $\overset{\circ}S_k(X,Z;\widehat{\mathcal U}) \subseteq X$, the Kuranishi structure $\widehat{\mathcal U}$ induces a Kuranishi structure without boundary of dimension $\dim (X,Z;\widehat{\mathcal U}) - k$ on $K \subseteq \overset{\circ}S_k(X,Z;\widehat{\mathcal U})$.
The same conclusion holds for good coordinate system.
We put $$\overset{\circ}{\mathcal S}_k (\mathcal U_p) =
\left(\overset{\circ}S_k (\mathcal U_p), E_p\vert_{\overset{\circ}S_k (\mathcal U_p)},
s_p\vert_{\overset{\circ}S_k (\mathcal U_p)}, \psi_p\vert_{\overset{\circ}S_k (\mathcal U_p)}\right).
$$ Then we define a Kuranishi neighborhood of $K \subseteq \overset{\circ}S_k(X,Z;\widehat{\mathcal U})$ at $p$ by $
\overset{\circ}{\mathcal S}_k (\mathcal U_p).
$ Suppose $q = o_p(q) \in \psi_p(s_p^{-1}(0)) \cap Z$. Then $q \in S_{k'}(X,Z;\widehat{\mathcal U})$ if and only if $o_p(q) \in S_{k'}(U_p)$. Using this fact, we can restrict coordinate changes to $
\overset{\circ}{\mathcal S}_k (\mathcal U_p)
$ to obtain desired coordinate changes. The compatibility conditions follow from ones of $\widehat{\mathcal U}$.
1. In general the above Kuranishi structure on $K \subseteq \overset{\circ}S_k(X,Z;\widehat{\mathcal U})$ may not be orientable even if $\widehat{\mathcal U}$ is orientable.
2. In case $k=1$ the above Kuranishi structure of $K \subseteq \overset{\circ}S_1(X,Z;\widehat{\mathcal U})$ is orientable if $\widehat{\mathcal U}$ is orientable.
3. The Kuranishi structure induced to the normalized corner of $(X,Z;\widehat{\mathcal U})$ (see Part 2) is orientable if $\widehat{\mathcal U}$ is orientable.
\[defn417\] Let $M_1$ and $M_2$ be smooth orbifolds with corner, $N$ a smooth orbifold without boundary or corner and $f_i : M_i \to N$ smooth maps. We say that $f_1$ is [*transversal*]{} to $f_2$ if for each $k_1, k_2$ the restriction $f_1 : \overset{\circ}S_{k_1}(M_1) \to N$ is transversal to $f_2 : \overset{\circ}S_{k_2}(M_2) \to N$.
We can define the case of strongly continuous maps from relative K-spaces with corners to a manifold in the same way. The case of good coordinate system is the same.
\[fiberkurabdry\]
1. Suppose that $Z \subseteq X$ has a Kuranishi structure with boundary and/or corner and $\widehat f : (X,Z;\widehat{\mathcal U}) \to N$ is weakly transversal to $f' : M \to N$. Then the fiber product $Z {}\times_{N} M \subseteq X {}\times_{N} M$ has a Kuranishi structure with corner.
2. If $Z_i \subseteq X_i$ has a Kuranishi structure with boundary and/or corner and $\widehat f_i : (X_i,Z_i;\widehat{\mathcal U_i}) \to N$ a strongly smooth map to a manifold. Suppose they are weakly transversal to each other. Then the fiber product $Z_1 \times_N Z_2 \subseteq X_1 \times_N X_2$ has a Kuranishi structure with corners.
The proof is immediate from definition.
We call the Kuranishi structure obtained in Lemma \[fiberkurabdry\], the [*fiber product Kuranishi structure*]{}.
Basic property of fiber product {#subsec:fibbasic}
-------------------------------
One important property of fiber product is its associativity, which we state below [^20]. We consider the following situation. Suppose $(X_i,Z_i)$ have Kuranishi structures for $i=1,2,3$ and let $\widehat f_1 : (X_1,Z_1;\widehat{\mathcal U_1}) \to M_1$, $\widehat f_2 = (\widehat f_{2,1},\widehat f_{2,2}) : (X_2,Z_2;\widehat{\mathcal U_2}) \to M_1 \times M_2$, $\widehat f_3 : (X_3,Z_3;\widehat{\mathcal U_3}) \to M_2$ be maps which are weakly smooth. We assume $\widehat f_1$ is transversal to $\widehat f_{2,1}$ and $\widehat f_{2,2}$ is transversal to $\widehat f_{3}$.
\[lemassoc\] In the above situation, the following three conditions are equivalent.
1. The map $\widehat f_{3} : (X_3,Z_3;\widehat{\mathcal U_3}) \to M_2$ is transversal to the map $\widehat f'_{2,2} : (X_1,Z_1;\widehat{\mathcal U_1})\times_{M_1} (X_2,Z_2;\widehat{\mathcal U_2}) \to M_2$, which is induced by $\widehat f_{2,2}$.
2. The map $\widehat f_{1} : (X_1,Z_1;\widehat{\mathcal U_1}) \to M_1$ is weakly transversal to the map $\widehat f'_{2,1} : (X_2,Z_2;\widehat{\mathcal U_2})\times_{M_2} (X_3,Z_3;\widehat{\mathcal U_3}) \to M_1$, which is induced by $\widehat f_{2,1}$.
3. The map $$\label{form4545}
(\widehat f_1,\widehat f_2,\widehat f_3) :
(X_1,Z_1;\widehat{\mathcal U_1}) \times (X_2,Z_2;\widehat{\mathcal U_2}) \times (X_3,Z_3;\widehat{\mathcal U_3}) \to M_1^2 \times M_2^2$$ is weakly transversal to $$\Delta = \{(x_1,x_2,y_1,y_2) \in M_1 \times M_1 \times M_2 \times M_2 \mid x_1=x_2,\,\, y_1=y_2\}.$$ Here we use the direct product Kuranishi structure in the left hand side of (\[form4545\]).
In case those three equivalent conditions are satisfied, we have $$\label{associa}
\aligned
&\left((X_1,Z_1;\widehat{\mathcal U_1}) \times_{M_1} (X_2,Z_2;\widehat{\mathcal U_2})\right) \times_{M_2} (X_3,Z_3;\widehat{\mathcal U_3})
\\
&\cong
(X_1,Z_1;\widehat{\mathcal U_1}) \times_{M_1} \left((X_2,Z_2;\widehat{\mathcal U_2}) \times_{M_2} (X_3,Z_3;\widehat{\mathcal U_3})\right).
\endaligned$$
Here the isomorphism $\cong$ in (\[associa\]) is defined as follows.
\[defniso\] Suppose $(X_1,Z_1;\widehat{\mathcal U_1})$ and $(X_2,Z_2;\widehat{\mathcal U_2})$ are relative K-spaces. Let $f : (X_1,Z_1) \to (X_2,Z_2)$ be a homeomorphism. An [*isomorphism*]{} of relative K-spaces between $(X_1,Z_1;\widehat{\mathcal U_1})$ and $(X_2,Z_2;\widehat{\mathcal U_2})$ assigns the maps $f_p, \hat f_p$ to each $p \in X_1$ such that the following holds. Let $\mathcal U_p^1, \mathcal U_{f(p)}^2$ be the Kuranishi charts of $p, f(p)$ in $X_1, X_2$, respectively.
1. $
f_p : U^1_p \to U^2_{f(p)}
$ is a diffeomorphism of orbifolds.
2. $\hat f_p : \mathcal E^1_p \to \mathcal E^2_{f(p)}$ is a bundle isomorphism over $f_p$.
3. $s^p_2 \circ f_p = \hat f_p \circ s^p_1$.
4. $\psi_{f(p)}^2 \circ f_p = \psi_{p}^1$ on $s_{p}^{-1}(0)$.
5. $f_p(o^1_p) = o^2_p$.
This definition of isomorphism is too restrictive to be a natural notion of isomorphism between Kuranishi structures. To find a correct notion of morphisms between K-spaces and of isomorphism between them is interesting and is a highly nontrivial problem. We do not study it here since it is not necessary for our purpose. A slightly better notion is an equivalence as germs of Kuranishi structures. See [@Fu1].
The proof of Lemma \[lemassoc\] is easy and is omitted.
In the previous literature such as [@fooobook2 Section A1.2] we defined a fiber product using the notion of good coordinate system. There is one difficulty in defining the fiber product with a space equipped with good coordinate system, which we explain below.
For $i=1,2$, suppose that $X_i$ have good coordinate systems that are defined by $\frak P_i$, $\mathcal U_{p_i}
= (U^i_{\frak p},\mathcal E^i_{\frak p},\psi^i_{\frak p},s^i_{\frak p})$, and $\Phi_{\frak p_i\frak q_i}
= (U^i_{\frak p\frak q},\widehat\varphi^i_{\frak p\frak q},\varphi^i_{\frak p\frak q})$. Let $\widehat{f}_i = \{(f_i)_{\frak p}\} : (X_i,\mathcal U_{p_i}) \to Y$ be strongly smooth maps. We assume that $f_{1,\frak p_1} : U^1_{\frak p_1} \to Y$ is transversal to $f_{2,\frak p_2} : U^2_{\frak p_2} \to Y$ for each $\frak p_1 \in \frak P_1$ and $\frak p_2 \in \frak P_2$. Then we define $$U_{(\frak p_1,\frak p_2)} = U^1_{\frak p_1}\times_Y U^1_{\frak p_2}$$ and define other objects $\mathcal E_{(\frak p_1,\frak p_2)},\psi_{(\frak p_1,\frak p_2)},s_{(\frak p_1,\frak p_2)}$ by taking fiber product in a similar way and to define a good coordinate system.
This is written in [@fooobook2 Section A1.2]. A point to take care of in this construction is as follows. (This point is mentioned in [@fooo010 Remark 10 page 165] and is discussed in detail by Joyce in [@joyce].)
Let $\frak p_i, \frak q_i \in \frak P_i$ such that $\frak q_i < \frak p_i$. We assume that the fiber product $$(U^1_{\frak p_1\frak q_1} \cap (\frak s^1_{\frak q_1})^{-1}(0)) \times_Y (U^2_{\frak p_2\frak q_2} \cap (\frak s^2_{\frak q_2})^{-1}(0))$$ is nonempty. Then we have $$\psi_{(\frak q_1,\frak p_2)}(\frak s^{-1}_{(\frak q_1,\frak p_2)}(0)) \cap
\psi_{(\frak p_1,\frak q_2)}(\frak s^{-1}_{(\frak p_1,\frak q_2)}(0))
\ne \emptyset.$$ On the other hand, neither $(\frak q_1,\frak p_2) \le (\frak q_2,\frak p_1)$ nor $(\frak q_2,\frak p_1) \le (\frak q_1,\frak p_2)$. In fact it may happen that $
\dim U_{\frak q_i} < \dim U_{\frak p_i}
$. In such a case there is no way to define $\varphi_{(\frak q_1,\frak p_2),(\frak p_1,\frak q_2)}$ or $\varphi_{(\frak p_1,\frak q_2),(\frak q_1,\frak p_2)}$.
Note the same problem already occurs while we study the direct product.
We can resolve this problem by shrinking $U_{(\frak p_1,\frak p_2)}$ appropriately. Joyce [@joyce] gave a beautiful canonical way to perform this shrinking process so that the resulting fiber product is associative. (See also [@fooo010 Figure 14].)
In case we have a multisection (multivalued perturbation) on the Kuranishi structures on $X_i$ so that the fiber product over $Y$ is transversal on its zero set, we use the fiber product of this multisection. This is especially important when we work in the chain level. Joyce [@joyce] did not seem to discuss this point since, for his purpose in [@joyce], it is unnecessary. We have no doubt that we can incorporate the construction of multisection to Joyce’s fiber product so that we can perturb the Kuranishi structure in a way consistent with the fiber product and is also associativity of fiber product holds together with perturbation.
However, in this article we take a slightly different way. We define the fiber product among the spaces with Kuranishi structures themselves not those with good coordinate systems. Then the above mentioned problem does not occur. In other words, Kuranishi chart (of the fiber product) is defined as the fiber product of Kuranishi charts [*without shrinking it*]{}. (Lemmata \[lem25\], \[lem27\], \[fiberkura22\].) Moreover associativity holds obviously. (Lemma \[fiberkurabdry\]).
On the other hand, the compatibility of the multisection with fiber product still needs to be taken care of. In fact, to find a multisection with appropriate transversality properties, we used a good coordinate system. So we need to perform certain process to move from a good coordinate system back to a Kuranishi structure together with multisections on it. We will discuss this point in Sections \[sec:thick\] - \[sec:contfamilyconstr\].
Thickening of a Kuranishi structure {#sec:thick}
===================================
Background of introducing the notion of thickening {#subsec:thickback}
--------------------------------------------------
Let $X$ be a paracompact metrizable space, and let $\widehat{\mathcal U}
= (\{\mathcal U_p\},\{\Phi_{pq}\})$ be a Kuranishi structure on it. We consider a system of multisections[^21] $\{ \frak s_p \}$ of the vector bundle $\mathcal E_p \to U_p$ for each $p$ with the following property:
1. For each $p$ and $q \in {\rm Im} (\psi_{p})$ the pullback of $\frak s_p$ to $U_{pq}$ that is a multisection of $\varphi_{pq}^*\mathcal E_p$ is the image of the multisection $\frak s_q$ by the bundle embedding $\widehat{\varphi}_{pq}$.
This is a kind of obvious condition of multisection (multivalued perturbation) that is compatible with the Kuranishi structure $\widehat{\mathcal U}$. (We define such a notion precisely later in Definition \[compapertKuranishi\].)
However, we need to note the following: Let us take a good coordinate system ${\widetriangle{\mathcal U}}=
((\frak P,\le) ,
\{\mathcal U_{\frak p} \mid \frak p\in \frak P\},
\{\Phi_{\frak p\frak q} \mid \frak p,\frak q\in \frak P, \frak q \le \frak p\})$ such that ${\widetriangle{\mathcal U}}$ is compatible with $\widehat{\mathcal U}$ in the sense of Definition \[gcsystemcompa\] and use it to define a system of multisections $\frak s_{\frak p}$ on $U_{\frak p}$. Then it is usually impossible to use $\frak s_{\frak p}$ to obtain a system of multisections $\frak s_p$ of $\widehat{\mathcal U}$ that has property ($\bigstar$).
The reason is as follows. Let $p\in X$. We take $\frak p \in \frak P$ such that $p \in {\rm Im}(\psi_{\frak p})$. By definition of the compatibility of good coordinate system and Kuranishi structure, there exists an embedding $\Phi_{\frak p p} : \mathcal U_p \to
\mathcal U_{\frak p}$. We put $\Phi_{\frak p p} = (\varphi_{\frak p p},\widehat{\varphi}_{\frak p p})$. Then we consider $\varphi_{\frak p p}(o_p) \in U_{\frak p}$. (Here $\psi_p(o_p) = p$.) By definition (See Definition \[defn62\].) $$\frak s_{\frak p}(o_p) \in (\mathcal E_{\frak p}\vert_{o_p})^l.$$ On the other hand, $\widehat{\varphi}_{\frak p p}$ restricts to a linear embedding $
\mathcal E_p\vert_{o_p} \to \mathcal E_{\frak p}\vert_{\psi_{\frak p p}(o_p)}.
$ It induces $$\label{32formula}
(\mathcal E_p\vert_{o_p})^l \to (\mathcal E_{\frak p}\vert_{\psi_{\frak p p}(o_p)})^l.$$ By inspecting the construction of the multisection $\frak s_{\frak p}$ given in [@FO p 955], we find that $$\label{33formula}
\frak s_{\frak p}(\varphi_{\frak p p}(o_p)) \notin
{\rm Im} (\ref{32formula})$$ in general. So $\frak s_{\frak p}$ cannot be pulled back to a multisection of $\mathcal E_p$ on $U_{\frak p p}$.
To explain the reason why (\[33formula\]) occurs, we introduce some notations.
\[stratadim\] For a Kuranishi structure $\widehat{\mathcal U}$ of $Z \subseteq X$, we define the [*dimension stratification*]{} of $Z$ by $$\label{defstratifi}
\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U})
=
\{ p \in Z \mid \dim U_p \ge \frak d\}.$$ Here $\frak d \in \Z_{\ge 0}$.
When ${\widetriangle{\mathcal U}}$ is a good coordinate system of $Z \subseteq X$, we define the dimension stratification of $Z$ by $$\label{defstratifi2}
\mathcal S_{\frak d}(X,Z;{\widetriangle{\mathcal U}})
=
\{ p \in Z \mid \exists \,\frak p, \, \dim U_{\frak p} \ge \frak d,\,
p \in {\rm Im} \psi_{\frak p}\}.$$
1. $\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U})$, is a closed subset of $Z$. $\mathcal S_{\frak d}(X,Z;{\widetriangle{\mathcal U}})$ is an open subset of $Z$. Moreover if $\frak d' < \frak d$, then $$\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U}) \subseteq
\mathcal S_{\frak d'}(X,Z;\widehat{\mathcal U}),
\mathcal \qquad
\mathcal S_{\frak d}(X,Z;\widetriangle{\mathcal U}) \subseteq
\mathcal S_{\frak d'}(X,Z;\widetriangle{\mathcal U}).$$
2. If $\widehat{\mathcal U}$ is embedded into $\widehat{\mathcal U'}$ then $$\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U}) \subseteq
\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U'}).$$ The equality holds if and only if the embedding from $\widehat{\mathcal U}$ to $\widehat{\mathcal U'}$ is an open embedding.
The same holds for GG-embeddings. (The equality in $
\mathcal S_{\frak d}(X,Z;\widetriangle{\mathcal U}) \subseteq
\mathcal S_{\frak d}(X,Z;\widetriangle{\mathcal U'})
$ holds if the embedding is a strongly open embedding.)
If the good coordinate system ${\widetriangle{\mathcal U}}$ is compatible with $\widehat{\mathcal U}$, then $$\label{dimandUUU}
\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U}) \subseteq
\mathcal S_{\frak d}(X,Z;{\widetriangle{\mathcal U}}).$$
The proof is immediate from definition. However, we note that the equality almost never holds in (\[dimandUUU\]). Namely $\mathcal S_{\frak d}(X,Z;\widetriangle{\mathcal U})$ contains an open neighborhood of $\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U})$. This is the reason why (\[33formula\]) occurs.
Definition of thickening {#subsec:thickening}
------------------------
To go around this problem we introduce the notion of thickening.
\[thickening\] Let $\widehat{\mathcal U}$ be a Kuranishi structures of $Z \subseteq X$. We say that $(\widehat{\mathcal U^+},\widehat\Phi)$ is a [*thickening*]{} of $\widehat{\mathcal U}$ if the following condition is satisfied.
1. $\widehat{\mathcal U^+}$ is a Kuranishi structure of $Z \subseteq X$ and $\widehat\Phi : \widehat{\mathcal U} \to \widehat{\mathcal U^+}$ is a KK-embedding.
2. For each $p \in Z$ there exists a neighborhood $O_p$ of $p$ in $\psi_p((s_p)^{-1}(0))
\cap \psi^+_p((s^+_p)^{-1}(0)) \subset X$ with the following properties.
For each $q \in O_p \cap Z$ there exists a neighborhood $W_p(q)$ of $o_{p}(q)$ in $U_{p}$ such that:
1. $\varphi_p(W_p(q)) \subseteq \varphi^+_{pq}(U_{pq}^+)$.
2. For any $x \in W_p(q)$, $y\in U_{pq}^+$ with $ \varphi_p(x) = \varphi^+_{pq}(y)$, we have $$\widehat\varphi_p(E_p\vert_x)
\subseteq
\widehat\varphi^+_{pq}(E^+_q\vert_y).$$
We sometimes say $\widehat{\mathcal U^+}$ is a thickening of $\widehat{\mathcal U}$ by an abuse of notation.
We write $\widehat{\mathcal U} < \widehat{\mathcal U^+}$ if $\widehat{\mathcal U^+}$ is a thickening of $\widehat{\mathcal U}$.
$$\xymatrix{
U^+_{pq}\ar[rr]^{\varphi^+_{pq}} && U^+_p \\
W_p(q) \ar@{.>}[u] \ar@{^{(}-{>}}[rr] &&
U_p \ar[u]_{\varphi_p}
\\
q \ar@{|->}[d] & &s_p^{-1}(0) \ar@{^{(}-{>}}[u] \ar[d]_{\psi_p} \\
O_p\ar@{^{(}-{>}}[rr] && X
}$$
Condition (2) above implies that
1. $S_{\frak d}(X,Z;\widehat{\mathcal U^+})$ is a neighborhood of $S_{\frak d}(X,Z;\widehat{\mathcal U})$.
In fact if $q \in O_p \cap Z$, then $$\dim U^{+}_q \ge \dim W_q(p) = \dim U_p$$ by Condition (2)(a). In particular, $\widehat{\mathcal U}$ is almost never a thickening of itself.
In the case $\dim U^{+}_p$ is strictly greater than $\dim U_p$ and $\dim U^+_q$, Condition () may not imply Condition (2)(a).
If $(\widehat{\mathcal U^+},\widehat\Phi)$ is a thickening of $\widehat{\mathcal U}$ and $(\widehat{\mathcal U^{++}},\widehat{\Phi^+})$ is a thickening of $\widehat{\mathcal U^+}$, then $(\widehat{\mathcal U^{++}},\widehat{\Phi^+}\circ \widehat\Phi)$ is a thickening of $\widehat{\mathcal U}$.
Let $O_p$, $W_p(q)$ be as in Definition \[thickening\] (2) (a) for $\widehat\Phi : \widehat{\mathcal U} \to \widehat{\mathcal U^+}$ and let $O^+_p$, $W^+_p(q)$ be one for $\widehat{\Phi^+} : \widehat{\mathcal U^+} \to \widehat{\mathcal U^{++}}$. We put $O_p^{++} = O_p \cap O^+_p$ and $W_p^{++}(q) = W_p(q) \cap \varphi_p^{-1}(W^+_p(q))$. Suppose $q \in O_p^{++}$. Then we have $$(\varphi_p^+\circ\varphi_p)(W_p^{++}(q))=
\varphi_p^+(\varphi_p(W_p^{++}(q)))
\subseteq \varphi_p^+(W_p^+(q))
\subset \varphi_{p}^{++}(U_{pq}^{++}).$$ Thus we have checked Definition \[thickening\] (2) (a). The proof of Definition \[thickening\] (2) (b) is similar.
Existence of thickening {#subsec:thickexi}
-----------------------
We next prove the existence of thickening. We need some notation.
\[situ61\] Let ${\widetriangle {\mathcal U}}$ be a good coordinate system of $Z \subseteq X$.
1. A [*support system*]{} of ${\widetriangle {\mathcal U}}$ is $
\mathcal K = \{\mathcal K_{\frak p}\mid {\frak p
\in \frak P}\}$ where $\mathcal K_{\frak p} \subset U_{\frak p}$ is a compact subset for each $\frak p \in \frak P$ such that it is a closure of an open subset $\overset{\circ}{{\mathcal K}_{\frak p}}$ of $ U_{\frak p}$, and $$\label{form5.65.6}
\bigcup_{\frak p \in \frak P}
\psi_{\frak p}(\overset{\circ}{\mathcal K}_{\frak p} \cap s_{\frak p}^{-1}(0))
\supseteq Z.$$
2. A [*support pair*]{} $(\mathcal K^1,\mathcal K^2)$ is a pair of support systems $(\mathcal K^i_{\frak p})_{\frak p
\in \frak P}$ $i=1,2$, such that $${\mathcal K}_{\frak p}^1 \subset \overset{\circ}{{\mathcal K}_{\frak p}^2}.$$
We write $\mathcal K^1 < \mathcal K^2$ if $(\mathcal K^1,\mathcal K^2)$ is a support pair.
3. When $\mathcal K$ is a support system, we define $$\vert \mathcal K\vert = \left(\coprod_{\frak p \in \frak P} \mathcal K_{\frak p}\right) / \sim.$$ Here, for $x \in \mathcal K_{\frak p}$, $y \in \mathcal K_{\frak q}$, the relation $x \sim y$ is defined by: $x = \varphi_{\frak p \frak q}(y)$ or $y = \varphi_{\frak q \frak p}(x)$. On $\vert \mathcal K\vert$, we put the induced topology from $\vert \widehat{\mathcal U}\vert$. Then it follows from the definition and [@foootech Proposition 5.17] or [@foooshrink Proposition 5.1], that the space $\vert \mathcal K\vert$ is metrizable.
4. When $\mathcal K$ is a support system we define $$\aligned
\mathcal S_{\frak p}(X,Z;{\widetriangle {\mathcal U}};\mathcal K)
&=
\bigcup_{\frak q \ge \frak p}\psi_{\frak q}(s_{\frak q}^{-1}(0)\cap \mathcal K_{\frak q})\cap Z, \\
\overset{\circ}{\mathcal S}_{\frak p}(X,Z;{\widetriangle {\mathcal U}};\mathcal K)
&=
\mathcal S_{\frak p}(X,Z;{\widetriangle {\mathcal U}};\mathcal K)
\setminus
\bigcup_{\frak q > \frak p}\mathcal S_{\frak q}(X;{\widetriangle{\mathcal U}};\mathcal K).
\endaligned$$
\[exithicken\] For any Kuranishi structure $\widehat{\mathcal U}$ there exists its thickening.
By Theorem \[Them71restate\] we have a good coordinate system ${\widetriangle{\mathcal U}}
= (\frak P,\{\mathcal U_{\frak p}\},\{\Phi_{\frak p \frak q}\})$ compatible with $\widehat{\mathcal U}$ and its support pair $(\mathcal K^-,\mathcal K^+)$. By compatibility, there exists a KG-embedding $\widehat{\mathcal U} \to {\widetriangle{\mathcal U}}$ which we denote by $\widehat\Phi = \{\Phi_{\frak p p}\mid p\in X,\frak p \in \frak P, p\in \psi_{\frak p}(s^{-1}_{\frak p}(0))\}$.
The first step is to define $\widehat{\mathcal U^+}$.
Let $p \in Z$. There exists unique $\frak p \in \frak P$ such that $
p \in
\overset{\circ}{\mathcal S}_{\frak p}(X,Z;{\widetriangle {\mathcal U}};\mathcal K^-).
$
By definition $\overset{\circ}{\mathcal S}_{\frak p}(X,Z;{\widetriangle {\mathcal U}};\mathcal K^-)$ are disjoint from one another for different $\frak p$’s. By (\[form5.65.6\]) they cover $Z$. This finishes the proof.
We take an open neighborhood $U^+_p$ of $o_{\frak p}(p)
\in \mathcal K^-_{\frak p}$ in $\mathcal K^+_{\frak p}$ such that $$\label{eq5959}
\psi_{\frak p}(s^{-1}_{\frak p}(0) \cap U^+_{p})
\cap \psi_{\frak q}(s_{\frak q}^{-1}(0) \cap \mathcal K^-_{\frak q}) \ne \emptyset
\,\,\Rightarrow\,\,
\frak q \le \frak p.$$ Such a neighborhood exists by Condition (6) of Definition \[gcsystem\]. We define $$\mathcal U^+_p =
\mathcal U_{\frak p}\vert_{U^+_p}.$$ We next define a coordinate change. Let $q \in \psi_{\frak p}(s^{-1}_{\frak p}(0) \cap U^+_p)$. Take the unique $\frak q \in \frak P$ such that $
q \in
\overset{\circ}{\mathcal S}_{\frak q}(X,Z;{\widetriangle {\mathcal U}};\mathcal K^-)
$. Since $q \in \psi_{\frak p}(s^{-1}_{\frak p}(0) \cap U^+_{p})
\cap \psi_{\frak q}(s_{\frak q}^{-1}(0) \cap \mathcal K^-_{\frak q})$, (\[eq5959\]) implies $\frak q \le \frak p$. We put $$\label{U+pq}
U^+_{pq} = U_{q}^+ \cap U_{\frak p\frak q}
\cap \varphi_{\frak p\frak q}^{-1}(U_p^+).$$ This is a subset of $U_q^+$ $\subset U_{\frak q}$ and contains $o^+_q =
o_{\frak q}(q)$. We define $$\Phi^+_{pq} = \Phi_{\frak p\frak q}\vert_{U^+_{pq}}.$$ Clearly $\Phi^+_{pq}$ is a coordinate change from $U_{q}^+$ to $U_{p}^+$. Using Definition \[gcsystem\] applied to ${\widetriangle{\mathcal U}}$, we can easily show that $\mathcal U^+_p$ and $\Phi^+_{pq}$ define a Kuranishi structure. We denote it by $\widehat{\mathcal U^+}$.
We next define an open substructure $\widehat{\mathcal U_0}$ of $\widehat{\mathcal U}$ and a strict embedding $\widehat{\mathcal U_0} \to \widehat{\mathcal U^+}$. Let $p \in Z$ and we take $\frak p$ such that $
p \in
\overset{\circ}{\mathcal S}_{\frak p}(X,Z;{\widetriangle{\mathcal U}};\mathcal K^-)
$. We put $$U^0_p = \varphi_{\frak p p}^{-1}(U^+_p), \qquad
\mathcal U^0_p = \mathcal U_p\vert_{U^0_p}.$$ By restricting the coordinate change $\Phi_{pq}$ of $\widehat{\mathcal U}$ to $\varphi_{pq}^{-1}(U^0_p) \cap U_{qp} \cap U_q^0$, we obtain $\widehat{\mathcal U_0}$ that is an open substructure of $\widehat{\mathcal U}$. Then $
\Phi_{p} = \Phi_{\frak p p}\vert_{U^0_p}
$ is defined.
Definition \[defn311\] $\circledast$ follows from the fact that $\widehat{\Phi} : \widehat{\mathcal U} \to {\widetriangle{\mathcal U}}$ is a KG-embedding.
We finally prove that $\widehat{\mathcal U^{+}}$ is a thickening. Let $
p \in
\overset{\circ}{\mathcal S}_{\frak p}(X,Z;{\widetriangle {\mathcal U}};\mathcal K^-)
$. We choose $O_p$, a neighborhood of $p$ in $X$ so that the following condition ($\divideontimes$) is satisfied.
1. If $O_p \cap \psi_{\frak q}(s_{\frak q}^{-1}(0) \cap \mathcal K_{\frak q}^-) \ne \emptyset$, then $p \in \psi_{\frak q}(s_{\frak q}^{-1}(0) \cap
{\rm Int}\,\mathcal K_{\frak q}^+)$.
Let $q \in O_p \cap Z$. We take $\frak q$ such that $
q \in
\overset{\circ}{\mathcal S}_{\frak q}(X,Z;{\widetriangle {\mathcal U}};\mathcal K^-)
$. By Condition ($\divideontimes$) we have $p \in \psi_{\frak q}(s_{\frak q}^{-1}(0) \cap
{\rm Int}\,\mathcal K_{\frak q}^+)$. Therefore there exists an embedding $
\Phi_{\frak q p} : \mathcal U_p \to \mathcal U_{\frak q}.
$
Recall from that $q \in U^+_{\frak p \frak q} \subset U_{\frak p \frak q}$. We put $$W_{p}(q) = \varphi_{\frak q p}^{-1}(U_q^+) \cap U^0_p \cap \varphi_{\frak p \frak q}^{-1}(U_{\frak p \frak q}).$$ This is an open subset of $U_p^0$ and contains $o_p(q)$. Now we have $$\varphi_{\frak q p}(W_p(q)) \subset U^+_q \cap \varphi_{\frak q p}( \varphi_{\frak p p}^{-1}(U^+_p)) \cap U_{\frak p \frak q}.$$ Since $$\varphi_{\frak p \frak q} \bigl( \varphi_{\frak q p}( \varphi_{\frak p p}^{-1}(U^+_p)) \cap U_{\frak p \frak q} \bigr) \subset U^+_p,$$ we have $$\varphi_{\frak p p}(W_p(q)) = \varphi_{\frak p \frak q}(\varphi_{\frak q p}(W_p(q))
\subset \varphi_{\frak p \frak q}(U_{p q}^+)
= \varphi_{pq}(U_{p q}^+).$$ Thus we have proved Definition \[thickening\] (2)(a).
Using $\widehat \varphi_{\frak q p}$ we can prove Definition \[thickening\] (2)(b) in the same way. The proof of Proposition \[exithicken\] is complete.
Embedding of Kuranishi structures II {#subsec:kuranishiemb2}
------------------------------------
\[embgoodtokura\] Let ${\widetriangle{\mathcal U}} = (\frak P,\{\mathcal U_{\frak p}\},\{\Phi_{\frak p\frak q}\})$ be a good coordinate system of $Z \subseteq X$ and $\widehat{\mathcal U^+}$ a Kuranishi structure of $Z \subseteq X$. A [*GK-embedding*]{} $\widehat\Phi : {\widetriangle{\mathcal U}} \to \widehat{\mathcal U^+}$ is a collection $\{(U_{\frak p}(p),\Phi_{p\frak p})\}$ with the following properties.
1. $(U_{\frak p}(p),\Phi_{p \frak p})$ is defined when $\frak p \in \frak P$ and $p \in \psi_{\frak p}(s_{\frak p}^{-1}(0)) \cap Z$.
2. $U_{\frak p}(p)$ is an open neighborhood of $o_{\frak p}(p)$ in $U_{\frak p}$ where $\psi_{\frak p}(o_{\frak p}(p)) = p$.
3. $\Phi_{p \frak p} : \mathcal U_{\frak p}\vert_{U_{\frak p}(p)} \to {\mathcal U}^+_p$ is an embedding of Kuranishi charts.
4. If $\frak q \in \frak P$, $\frak q \le \frak p$, $q \in \psi_{\frak p}(U_{\frak p}(p) \cap s_{\frak p}^{-1}(0))$ and $q \in \psi_{\frak q}(s_{\frak q}^{-1}(0)) \cap Z$, then $q \in \psi_p^+(U_p^+ \cap (s_{p}^+)^{-1}(0))$ and the following diagram commutes.
$$\label{diagram58}
\begin{CD}
\mathcal U_{\frak q}\vert_{\varphi_{\frak p\frak q}^{-1}(U_{\frak p}(p))\cap (\varphi_{q\frak q}^+)^{-1}({U}_{pq}^+)} @ > {\Phi_{q\frak q}} >>
{\mathcal U}^+_{q}\vert_{{U}_{pq}^+} \\
@ V{\Phi{\frak p\frak q}}VV @ VV{\Phi_{pq}^+}V\\
\mathcal U_{\frak p}\vert_{U_{\frak p}(p)} @ > {\Phi_{p\frak p}} >>{\mathcal U}_{p}^+
\end{CD}$$
1. The case $p=q$ (but $\frak p \ne \frak q$) is included in Definition \[embgoodtokura\] (4). The case $\frak p = \frak q$ (but $p \ne q$) is also included.
2. Note $q \in \psi_p^+(U_p^+ \cap (s_{\frak p}^+)^{-1}(0))$ (in Definition \[embgoodtokura\] (4)) follows from the assumptions ($\frak q \in \frak P$ and $q \in \psi_{\frak q}(s_{\frak q}^{-1}(0))$, $\frak q \le \frak p$) and Definition \[embgoodtokura\] (3).
3. We include $U_{\frak p}(p)$ as a part of the data to define an embedding. We sometimes need to shrink it. Such a process is included in the discussion below.
We can pullback a strongly continuous (resp. strongly smooth) map from Kuranishi structure to one from a good coordinate system by a GK-embedding. Weak submersivity is preserved by an open embedding.
The proof is immediate from the definition.
Below we define compositions of embeddings of various types. See the table below. In the tables below, KS = Kuranishi structure, GCS = good coordinate system.
source target symbol Definition name comment
--------- --------- ------------------------------------------------------------- ------------------------------ -------------- ---------
[KS]{} [KS]{} $\widehat{\mathcal U} \to \widehat{\mathcal U^+}$ Definition \[defn311\] KK-embedding (1)
[KS]{} [GCS]{} $\widehat{\mathcal U} \to \widetriangle{\mathcal U}$ Definition \[defn32020202\] KG-embedding (2)
[GCS]{} [KS]{} $\widetriangle{\mathcal U} \to \widehat{\mathcal U^+}$ Definition \[embgoodtokura\] GK-embedding (3)
[GCS]{} [GCS]{} $\widetriangle{\mathcal U} \to \widetriangle{\mathcal U^+}$ Definition \[defn31222\] GG-embedding (4)
$${\mbox {Table 5.1}: \text{Definition of embedding}}$$ Comments: (1) Strict version and non-strict version exist. (2) Strict version and non-strict version exist. (3) None. (4) None.
1st structure 2nd structure 3rd structure definition comment
--------------- --------------- --------------- --------------------------------------- ---------
[KS]{} [KS]{} [KS]{} Definition \[definition516161\] (1)
[KS]{} [KS]{} [GCS]{} Definition \[defn32020202\] (2)
[KS]{} [GCS]{} [KS]{} Lemma \[lem513\] (3)
[KS]{} [GCS]{} [GCS]{} Definition \[defn32020202\] (4)
[GCS]{} [KS]{} [KS]{} Definition-Lemma \[kuragoodkuracomp\] (5)
[GCS]{} [KS]{} [GCS]{} Definition-Lemma \[henacomp\] (6)
[GCS]{} [GCS]{} [KS]{} Definition-Lemma \[kuragoodkuracomp\] (7)
[GCS]{} [GCS]{} [GCS]{} Definition \[defn31222\] (8)
$${\mbox {Table 5.2}: \text{Composition of embeddings}
\atop
\text{1st structure} \to \text{2nd structure} \to \text{3rd structure} }$$
Comments: (1) Strict version and non-strict version exist. Composition of non-strict version is well-defined only up to equivalence. (2) Strict version and non-strict version exist. Composition of non-strict version is well-defined only up to equivalence. (Definition \[definition516161\] (3)) (3) Need to restrict to an open substructure. The composition becomes a thickening. (4) Strict version and non-strict version exist. (5) None. (6) Need to restrict to a weakly open substructure. (7) None. (8) None.
1. Let $\widehat\Phi = \{(U_{\frak p}(p),\Phi_{\frak p p})\} : {\widetriangle{\mathcal U}} \to \widehat{\mathcal U^+}$ be a GK-embedding. Suppose for each $\frak p \in \frak P$ and $p \in \psi_{\frak p}(s_{\frak p}^{-1}(0)) \cap Z$ we are given an open subset $U'_{\frak p}(p)$ of $U_{\frak p}(p)$ such that $o_{\frak p}(p) \in U'_{\frak p}(p)$. Then $\{(U'_{\frak p}(p),\Phi_{\frak p p}\vert_{U'_{\frak p}(p)})\}$ is also an embedding ${\widetriangle{\mathcal U}} \to \widehat{\mathcal U^+}$. We call it an [*open restriction of the embedding*]{} $\widehat\Phi$.
2. Two embeddings ${\widetriangle{\mathcal U}} \to \widehat{\mathcal U^+}$ are said to be [*equivalent*]{} if they have a common open restriction. This is obviously an equivalence relation.
\[kuragoodkuracomp\] Let $\widehat\Phi : {\widetriangle{\mathcal U}} \to \widehat{\mathcal U^+}$ be a GK-embedding, $\widetriangle\Phi_1 : {\widetriangle{\mathcal U'}} \to {\widetriangle{\mathcal U}}$ a GG-embedding and $\widehat\Phi_2 : \widehat{\mathcal U^+} \to \widehat{\mathcal U^{++}}$ a strict KK-embedding. Then we can define the composition $${\widetriangle{\mathcal U'}} \longrightarrow {\widetriangle{\mathcal U}} \longrightarrow
\widehat{\mathcal U^+} \longrightarrow \widehat{\mathcal U^{++}}$$ which is a GK-embedding.
The proof is easy and left to the reader.
\[lem513\] If $\widehat{\Phi^1} : {\widehat{\mathcal U}} \to {\widetriangle{\mathcal U}}$ is a KG-embedding and $\widehat{\Phi^2} : {\widetriangle{\mathcal U}} \to \widehat{\mathcal U^+}$ is a GK-embedding, then we can find an open sub-structure ${\widehat{\mathcal U_0}}$ of ${\widehat{\mathcal U}}$ such that the composition of $
{\widehat{\mathcal U_0}} \to {\widehat{\mathcal U}}
\to {\widetriangle{\mathcal U}}$ and ${\widetriangle{\mathcal U}} \to \widehat{\mathcal U^+}
$ is defined as a strict KK-embedding $:
{\widehat{\mathcal U_0}} \to \widehat{\mathcal U^+}
$.
$\widehat{\mathcal U^+}$ is a thickening of ${\widehat{\mathcal U_0}}$.
Replacing $\widehat{\mathcal U}$ by its open substructure, we may assume that $\widehat{\Phi^1}$ is a strict KG-embedding. Let $p \in Z$. We define $U^0_{p} \subset U_p$ by $$U^0_{p} = \bigcap_{\frak p : p \in \psi_{\frak p}(s_{\frak p}^{-1}(0))}
(\varphi_{\frak p p}^1)^{-1}(U_{\frak p}(p)).$$ Here we define $U_{\frak p}(p)$ by $\Phi^2 = \{(U_{\frak p}(p),\Phi^2_{p \frak p})\}$. We use them to define our open substructure ${\widehat{\mathcal U_0}} = {\widehat{\mathcal U}}\vert_{\{U_p^0\}}$. Then $$\Phi^2_{p\frak p}\circ\Phi^1_{\frak p p}\vert_{U^0_{p}} : \mathcal U_p\vert_{U^0_{p}} \to \mathcal U^+_p$$ is well-defined and defines the required embedding.
We can prove that $\widehat{\mathcal U^+}$ is a thickening of ${\widehat{\mathcal U_0}}$ in the same way as the proof of Proposition \[exithicken\].
Lemma \[lem513\] implies that there exists a KK-embedding ${\widehat{\mathcal U}}
\to {\widehat{\mathcal U^+}}$ in the situation of Lemma \[lem513\]. This embedding is well-defined in the following sense.
\[embedkuraequiv\] Let ${\widehat{\mathcal U}}$ and ${\widehat{\mathcal U^+}}$ be Kuranishi structures. Suppose ${\widehat{\mathcal U_{0,i}}}$ $(i=1,2)$ are open substructures of ${\widehat{\mathcal U}}$ and $\widehat\Phi^{0,i} : {\widehat{\mathcal U_{0,i}}} \to {\widehat{\mathcal U^+}}$ are strict KK-embeddings. We say they are [*equivalent*]{} if there exists an open neighborhood $(U^{00})_{p}$ of $o_p$ in $U_p$ such that $$U^{00}_{p} \subset U^{0,1}_{p} \cap U^{0,2}_{p},
\qquad
\Phi^{0,1}_p\vert_{U^{00}_{p}} = \Phi^{0,2}_p\vert_{U^{00}_{p}}.$$ Here $U^{0,i}_{p}$ is the Kuranishi neighborhood of $p$ assigned by ${\widehat{\mathcal U_{0,i}}}$.
We can define an equivalence between two KG-embeddings $\widehat{\mathcal U}
\to \widetriangle{\mathcal U}$ in the same way.
\[definition516161\]
1. The [*composition*]{} of two strict KK-embeddings $\widehat{\mathcal U} \to \widehat{\mathcal U'}$, $\widehat{\mathcal U'} \to \widehat{\mathcal U''}$ is defined in an obvious way and it is a strict KK-embedding $\widehat{\mathcal U} \to \widehat{\mathcal U''}$.
2. We compose two KK-embeddings $\widehat{\mathcal U} \to \widehat{\mathcal U'}$, $\widehat{\mathcal U'} \to \widehat{\mathcal U''}$ and obtain a KK-embedding $\widehat{\mathcal U} \to \widehat{\mathcal U''}$. The composition is well-defined up to equivalence defined in Definition \[embedkuraequiv\].
3. The composition of a KK-embedding $\widehat{\mathcal U} \to \widehat{\mathcal U'}$ and a KG-embedding $\widehat{\mathcal U'} \to \widetriangle{\mathcal U}$ in Definition \[defn32020202\] is well-defined up to equivalence.
\[deflem517\] Let $\widehat\Phi : {\widetriangle{\mathcal U}} \to \widehat{\mathcal U^+}$ be a GK-embedding and $\widehat{\mathcal U^+_0}$ an open substructure of $\widehat{\mathcal U^+}$. Then there exists a GK-embedding $\widehat{\Phi^0} : {\widetriangle{\mathcal U}} \to \widehat{\mathcal U^+_0}$ such that the composition ${\widetriangle{\mathcal U}} \to \widehat{\mathcal U^+_0} \to \widehat{\mathcal U^+}$ is an open restriction of $\widehat\Phi$.
Let $\widehat\Phi = \{(U_{\frak p}(p),\Phi_{p \frak p})\}$. We put $
U^0_{\frak p}(p) = \varphi_{p \frak p}^{-1}(U_p^0) \subset U_{\frak p}(p).
$ We define $ \{(U^0_{\frak p}(p),\Phi_{p \frak p}\vert_{U^0_{\frak p}(p)})\}$ and obtain the required embedding.
The composition of embeddings in another case is slightly nontrivial.
\[henacomp\] [*Let $\widehat{\Phi} : {\widetriangle{\mathcal U}} \to \widehat{\mathcal U}$ be a GK-embedding, and $\widehat{\Phi^+} :{\widehat{\mathcal U}} \to {\widetriangle{\mathcal U^+}}$ a KG-embedding. Then there exists a weakly open substructure ${\widetriangle{\mathcal U_0}}$ of ${\widetriangle{\mathcal U}}$ such that $
{\widetriangle{\mathcal U_0}} \longrightarrow {\widetriangle{\mathcal U}} \overset{\widehat{\Phi}}{\longrightarrow}
\widehat{\mathcal U}
$ and $
\widehat{\mathcal U} \overset{\widehat{\Phi^+}}{\longrightarrow} {\widetriangle{\mathcal U^+}}
$ can be composed to a GG-embedding $
{\widetriangle{\mathcal U_0}} \longrightarrow {\widetriangle{\mathcal U^+}}.
$* ]{}
We will prove Definition-Lemma \[henacomp\] in Subsection \[subsec:proofofwdofcomp\], where we use it.
We may introduce the notion of germ of Kuranishi structures and use it to describe these situations. See Section [@Fu1].
\[defn517\] Let $\widehat{\mathcal U}$ be a Kuranishi structure on $X$ and $\widehat{\mathcal U^+}$ be its thickening. We say a good coordinate system ${\widetriangle{\mathcal U}}$ is [*in between $\widehat{\mathcal U}$ and $\widehat{\mathcal U^+}$*]{} and write $$\widehat{\mathcal U} < {\widetriangle{\mathcal U}} < \widehat{\mathcal U^+},$$ if the following holds.
1. There exist embeddings $\widehat{\mathcal U} \to {\widetriangle{\mathcal U}}$ and ${\widetriangle{\mathcal U}} \to \widehat{\mathcal U^+}$.
2. The composition $\widehat{\mathcal U} \to {\widetriangle{\mathcal U}}
\to \widehat{\mathcal U^+}$ is equivalent to the given embedding $\widehat{\mathcal U} \to \widehat{\mathcal U^+}$ in the sense of Definition \[embedkuraequiv\].
\[prop518\] Let $\widehat{\mathcal U}$ be a Kuranishi structure of $Z \subseteq X$ and $\widehat{\mathcal U^+}$ its thickening. Then there exists a good coordinate system ${\widetriangle{\mathcal U}}$ in between $\widehat{\mathcal U}$ and $\widehat{\mathcal U^+}$.
We also use the following version thereof.
\[prop519\] Let $\widehat{\mathcal U}$ be a Kuranishi structure of $Z \subseteq X$ and $\widehat{\mathcal U^+_a}$ $(a=1,2)$ thickenings of $\widehat{\mathcal U}$. Then there exists a good coordinate system ${\widetriangle{\mathcal U}}$ and embeddings $$\widehat{\mathcal U} \to {\widetriangle{\mathcal U}},
\qquad
{\widetriangle{\mathcal U}} \to \widehat{\mathcal U^+_a}
\quad
(a=1,2)$$ such that their compositions $\widehat{\mathcal U} \to {\widetriangle{\mathcal U}} \to
\widehat{\mathcal U^+_a}
$ are equivalent to the given embedding $\widehat{\mathcal U} \to \widehat{\mathcal U^+_a}$.
$$\xymatrix{
&&& \widehat{\mathcal U^+_1} \\
\widehat{\mathcal U} \ar[rrru] \ar@{.>}[rr] \ar@{>}[rrrd] && \widetriangle{\mathcal U} \ar@{.>}[ru] \ar@{.>}[rd] \\
&&& \widehat{\mathcal U^+_2}
}$$
The proofs are almost the same as the proof of Theorem \[Them71restate\]. We will review and prove them later in Subsection \[subsec:constgcsrel\].
Multivalued perturbation {#sec:multisection}
========================
Multisection {#subsec multi}
------------
### Multisection on an orbifold {#subsubsec multiofd}
We next define the notion of multivalued perturbations associated to a given good coordinate system. We will slightly modify the previously given definition to make explicit certain properties which we used to study its zero set (in [@foootech Section 2.6] for example.)[^22]
We begin with a review of multisections. We first introduce certain notations on vector bundles on orbifolds. See Section \[sec:ofd\] for detail.
\[defn61\] Let $U$ be an orbifold and $\mathcal E$ a vector bundle on it.
1. (Definitions \[2661\] (1) and \[defn26550\] (1)(3)) Let $x \in U$. We call $(V_x,\Gamma_x,\phi_x)$ an [*orbifold chart*]{} of $U$ at $x$ if the following holds.
1. $V_x$ is a smooth manifold on which a finite group $\Gamma_x$ acts effectively and smoothly.
2. $\phi_x : V_x
\to U$ is a $\Gamma_x$-invariant map which induces a diffeomorphism $\overline\phi_x : V_x/\Gamma_x
\to U$ [^23] onto an open neighborhood $U_x$ of $x$.
3. We require that there exists a unique point $o_x \in V_x$ such that $o_x$ is a fixed point of all elements of $\Gamma_x$ and $\phi_x([o_x]) = x$.
2. (Definition \[defn2655\] (3)) A [*trivialization of our obstruction bundle $\mathcal E = \widetilde{\mathcal E}/\Gamma_x$ on $U_x$*]{} is by definition $(E_x,\widehat\phi_x)$ such that
1. $E_x$ is a vector space on which $\Gamma_x$ acts linearly.
2. $\widehat\phi_x
: V_x \times E_x \to \widetilde{\mathcal E}$ is a $\Gamma_x$-invariant smooth map which induces an isomorphism of vector bundles $(V_x \times E_x)/\Gamma_x \to \mathcal E\vert_{V_x/\Gamma_x}$.
3. (Definition \[defn2655\] (2)(4)) We call $\frak V_x = (V_x,\Gamma_x,E_x,\phi_x,\widehat\phi_x)$ an [*orbifold chart*]{} of $(U,\mathcal E)$.
\[defn62\] Let $\frak V_x = (V_x,\Gamma_x,E_x,\phi_x,\widehat\phi_x)$ be an orbifold chart of $(U,\mathcal E)$.
1. A [*smooth $\ell$-multisection of $\mathcal E$*]{} on an orbifold chart $\frak V_x$ is $\frak s = (\frak s_1,\dots,\frak s_{\ell})$ with the following properties.
1. $\frak s$ is a smooth map $V_x \to E_x^{\ell}$.
2. For each $y \in V_x$ and $\gamma\in \Gamma_x$ there exists $\sigma \in {\rm Perm}(\ell)$ such that $$\frak s_{\sigma(k)}(y) = \gamma \frak s_{k}(y).$$
Hereafter we simply say [*$\ell$-multisection*]{} in place of smooth $\ell$-multisection.
2. Two $\ell$-multisections $(\frak s_1,\dots,\frak s_{\ell})$ and $(\frak s'_1,\dots,\frak s'_{\ell})$ are said to be [*equivalent as $\ell$-multisections on $\frak V_x$*]{} if for each $y$ there exists a permutation $\sigma \in {\rm Perm}(\ell)$ such that $
\frak s'_i(y) = \frak s_{\sigma(i)}(y)
$.
3. The [*$\ell'$-iteration*]{} of $\ell$-multisection $\frak s$ is the $\ell'\ell$-multisection $\frak s'$ such that $\frak s'_k = \frak s_m$ for $k\equiv m \mod \ell$. We denote the $\ell'$-iteration of $\frak s$ by $\frak s^{\times \ell'}$.
4. Let $\frak s_{(1)}$ be an $\ell_1$-multisection and $\frak s_{(2)}$ an $\ell_2$-multisection. We say $\frak s_{(1)}$ is [*equivalent to $\frak s_{(2)}$ as multisections*]{} if $\frak s_{(1)}^{\times \ell_{2}}$ is equivalent to $\frak s_{(2)}^{\times \ell_{1}}$ as $\ell_{1}\ell_{2}$-multisections.
5. It is easy to see that Item (4) defines an equivalence relation. We say its equivalence class a [*multisection*]{} on our orbifold chart.
\[opensuborbifoldchart\] Let $\frak V_x = (V_x,\Gamma_x,E_x,\phi_x,\widehat\phi_x)$ and $\frak V'_{x'} = (V'_{x'},\Gamma'_{x'},E'_{x'},\phi'_{x'},\widehat\phi'_{x'})$ be two orbifold charts of a vector bundle $(U,\mathcal E)$. We assume that $\phi'_{x'}(V'_{x'}) \subset
\phi_{x}(V_{x})$ and that there exist $\tilde\varphi_{xx'} : V'_{x'} \to V_{x}$, $h_{xx'} : \Gamma'_{x'} \to \Gamma_{x}$ such that $h_{xx'}$ is an injective group homomorphism and $\tilde\varphi_{xx'}$ is an $h_{xx'}$ equivariant smooth open embedding such that they induce the composition map $$(\overline{\phi_x})^{-1}\circ \overline{\phi'_{x'}} : V'_{x'}/\Gamma'_{x'} \to V_{x}/\Gamma_{x},$$ where $\overline{\phi_x}$ (resp. $\overline{\phi'_{x'}}$) is induced by $\phi_x$ (resp. $\phi'_{x'}$). In other words, $$\phi'_{x'}(y) \equiv \phi_{x}(\tilde\varphi_{xx'}(y)) \mod \Gamma_x.$$ Moreover we assume that the composition $$\left(\overline{\widehat\phi_x}\right)^{-1}\circ \overline{\widehat\phi'_{x'}} : (V'_{x'}\times E'_{x'})/\Gamma'_{x'} \to (V_{x}\times E_{x})/\Gamma_{x}$$ is induced by a smooth map $\breve\varphi_{xx'} : V'_{x'} \times E'_{x'} \to E_x$ that is linear in $E'_{x'}$ factor. In other words $$\widehat\phi'_{x'}(y,v) \equiv \widehat\phi_x(\tilde\varphi_{xx'}(y),\breve\varphi_{xx'}(y,v))
\mod \Gamma_x.$$ We put $
\Phi_{xx'} = (h_{xx'},\tilde\varphi_{xx'},\breve\varphi_{xx'}).
$$\blacksquare$
We call $\Phi_{xx'}$ a [*coordinate change*]{} from an orbifold chart $\frak V_{x'}$ to $\frak V_{x}$.
We put $
\tilde{\hat{\varphi}}_{xx'}(y,v) = (y,\breve\varphi_{xx'}(y,v))
$. Then $$(h_{xx'},\tilde\varphi_{xx'},\tilde{\hat{\varphi}}_{xx'})$$ is a [*local representative of an embedding of vector bundles*]{}, ${\rm id} : \mathcal E\vert_{{\rm Im}\overline{\widehat{\phi^{\prime}}}_{x'}} \to \mathcal E\vert_{{\rm Im}\overline{\widehat\phi}_{x}}$ in the sense of Definition \[def28262826\]. Moreover it is a fiberwise isomorphism.
\[multilocalrest\] Let $\frak s_x$ be an $\ell$-multisection on $\frak V_x$ and $\Phi_{xx'} : \frak V_{x'} \to \frak V_x$ a coordinate change. We define the [*restriction*]{} $\Phi_{xx'}^*\frak s$ by $$(\Phi_{xx'}^*\frak s)_k(y)
=
g_y^{-1}\frak s_k(\tilde\varphi_{xx'}(y))$$ where $
g_y : E_{x'} \to E_x
$ is defined by $
g_y(v) = \breve\varphi_{xx'}(y,v).
$
If $\frak s_x$ is equivalent to $\frak s'_x$ as multisections then $\Phi_{xx'}^*\frak s$ is equivalent to $\Phi_{xx'}^*\frak s'$ as multisections.
We omit the proof. (See the proof of a similar lemma, Lemma \[lem77\].)
Here is a notational remark.
\[xkararhe\] So far we have written $\frak V_x = (V_x,\Gamma_x,E_x,\phi_x,\widehat\phi_x)$. The point $x$ plays no particular role except we assume the existence of $o_x \in V_x$ such that $\phi_x(o_x) = x$ and $o_x$ is fixed by all the elements of $\Gamma_x$. If we change the choice of such $x$, the constructions so far do not change at all. So, from now on, we do not specify $x$ in our notation of local orbifold chart but only assume an existence of such $x$. We will write for example $\frak V_{\frak r} = (V_{\frak r},\Gamma_{\frak r},E_{\frak r},\phi_{\frak r},\widehat\phi_{\frak r})$ instead of $\frak V_{x} = (V_{x},\Gamma_{x},E_{x},\phi_{x},\widehat\phi_{x})$.
\[lem606969\]
\[defn6868\] Let $U$ be an orbifold and $\mathcal E$ a vector bundle on it.
1. A [*representative of a multisection of $\mathcal E$ on $U$*]{} is $(\{\frak V_{\frak r} \mid \frak r \in \frak R\},\{\frak s_{\frak r}
\mid \frak r \in \frak R\})$ with the following properties.
1. $\frak V_{\frak r}$ is an orbifold chart of a vector bundle $(U,\mathcal E)$ such that $\bigcup_{\frak r \in \frak R}{U_{\frak r}} = U$.
2. $\frak s_{\frak r}$ is a multisection of $\frak V_{\frak r}$.
3. For any $y \in \frak V_{\frak r_1} \cap \frak V_{\frak r_2}$, there exist an orbifold chart $\frak V_y$ and coordinate changes $\Phi_{\frak r_i y} : \frak V_y \to \frak V_{\frak r_i}$ such that $\Phi_{\frak r_1 y}^*\frak s_{\frak r_1}$ is equivalent to $\Phi_{\frak r_2 y}^*\frak s_{\frak r_2}$.
2. Let $(\{\frak V^{(i)}_{\frak r^{(i)}} \mid \frak r^{(i)} \in \frak R^{(i)}\},\{\frak s^{(i)}_{\frak r^{(i)}}
\mid \frak r^{(i)} \in \frak R^{(i)}\})$ be representatives of multisections of $\mathcal E$ on $U$ for $i=1,2$. We say they are [*equivalent*]{} if the following holds.
For any $x \in \frak V^{(1)}_{\frak r^{(1)}_1} \cap \frak V^{(2)}_{\frak r^{(2)}_2}$, there exist an orbifold chart $\frak V_x$ and coordinate changes $\Phi_{\frak r_i x} : \frak V_x \to \frak V^{(i)}_{\frak r_i^{(i)}}$ ($i=1,2$) such that $\Phi_{\frak r^{(1)}_1 x}^*\frak s_{\frak r^{(1)}_1}^{(1)}$ is equivalent to $\Phi_{\frak r^{(2)}_2 x}^*\frak s_{\frak r^{(2)}_2}^{(2)}$.
An equivalence class of this equivalence relation is called a [*multisection of $(U,\mathcal E)$*]{}.
3. (See [@FO Definition 3.10].) Let $\frak s^{n}$ be a sequence of multisections of $(U,\mathcal E)$. We say that [*it converges to a multisection $\frak s$ in $C^{k}$-topology*]{} ($k$ is any of $0,1,\dots,\infty$) if there exists a representative $(\{\frak V_{\frak r} \mid \frak r \in \frak R\},\{\frak s^{n}_{\frak r}
\mid \frak r \in \frak R\})$ of $\frak s^{n}$ for sufficiently large $n$ and $(\{\frak V_{\frak r} \mid \frak r \in \frak R\},\{\frak s_{\frak r}
\mid \frak r \in \frak R\})$ of $\frak s$ such that $\frak s^{n}_{\frak r}$ converges to $\frak s_{\frak r}$ in compact $C^k$-topology for each $\frak r$. We note that we assume $\frak V_{\frak r}$ and $\frak R$ are independent of $n$.
\[definition610\] Let $\frak s$ be a multisection of a vector bundle $(U,\mathcal E)$ on orbifold $U$ and $x \in U$. We put $\frak s = [(\{\frak V_{\frak r} \mid \frak r \in \frak R\},\{\frak s_{\frak r}
\mid \frak r \in \frak R\})]$. We take an orbifold chart $\frak V_x$ at $x$. A map germ $[s]$, where $s : O_{x} \to E_x$, is said to be a [*branch of $\frak s$ at $x$*]{} if the following holds.
1. $O_x$ is a neighborhood of $o_x$ in $V_x$.
2. Let $\frak r \in \frak R$ such that $x \in U_{\frak r}$. Then there exists $k$ such that $$\widehat\phi_{\frak r}(\tilde\varphi_{\frak r x}(y),\frak s_{\frak r,k}(\tilde\varphi_{\frak r x}(y))) = \widehat\phi_{x}(y,s(y))$$ if $y$ is on a neighborhood of $x$ in $O_x$. Here $\tilde\varphi_{\frak r x}$ is a part of a coordinate change $\frak V_x \to \frak V_{\frak r}$.
### Multisection on a good coordinate system {#subsubsec multigcs}
Let ${\widetriangle{\mathcal U}}=(\frak P, \{ \mathcal U_{\frak p}\}, \{\Phi_{\frak p \frak q}\})$ be a good coordinate system of $Z \subseteq X$.
\[defn612\] Let $\mathcal K = \{\mathcal K_{\frak p}\}$ be a support system of a good coordinate system ${\widetriangle {\mathcal U}}$ and $\frak s_{\frak p}^{n}$ multisections of $E_{\frak p}$ on a neighborhood of $\mathcal K_{\frak p}$ for each $n \in \Z_{\ge 0}$ and $\frak p$. We say that $\widetriangle{\frak s}
= \{\widetriangle{\frak s^n} \mid n \in \Z_{\ge 0}\} = \{\frak s_{\frak p}^{n} \mid n \in \Z_{\ge 0},
\frak p \in \frak P\}$ is a [*multivalued perturbation of $({\widetriangle U},\mathcal K)$*]{} if the following conditions are satisfied.
1. $
\frak s_{\frak p}^{n} \circ \varphi_{\frak p\frak q}
=
\widehat\varphi_{\frak p\frak q}\circ \frak s_{\frak q}^{n}
$ on a neighborhood of $\mathcal K_{\frak q}
\cap \varphi_{\frak p,\frak q}^{-1}(\mathcal K_{\frak p})$.
2. $
\lim_{n\to \infty} \frak s_{\frak p}^{n} = s_{\frak p}
$ in $C^1$-topology on a neighborhood of $\mathcal K_{\frak p}$.
A [*multivalued perturbation of $\widetriangle{\mathcal U}$*]{} is a collection $\{\frak s_{\frak p}^{n}\}$ such that (1)(2) hold for some support system $\mathcal K$.
Note that the Kuranishi map $s_{\frak p}$, which is a single valued section of $\mathcal E_{\frak p}$, can be regarded as a multisection by Lemma \[lem2626\]. The $C^1$-convergence in Definition \[defn612\] (2) therefore is defined in Definition \[defn6868\] (3).
Below we will elaborate on the equality in Definition \[defn612\] (1) further. Let $x \in \mathcal K_{\frak q}
\cap \varphi_{\frak p\frak q}^{-1}(\mathcal K_{\frak p})$ and $x' = \varphi_{\frak p\frak q}(x) \in \mathcal K_{\frak p}$. We can take orbifold charts $\frak V_{x}$ of $(U_{\frak q},\mathcal E_{\frak q})$, $\frak V_{x'}$ of $(U_{\frak p},\mathcal E_{\frak p})$ such that $(\varphi_{\frak p\frak q},\widehat\varphi_{\frak p\frak q})$ has a local representative $(h_{\frak p\frak q;x'x},\tilde\varphi_{\frak p\frak q;x'x},
\tilde{\hat{\varphi}}_{\frak p\frak q;x'x})$ with respect to these orbifold charts. (Lemma \[lem2622\].) We define $\breve\varphi_{\frak p\frak q;x'x} :
V_x \times E_x \to E_{x'}$ by the relation $$\tilde{\hat{\varphi}}_{\frak p\frak q;x'x}(y,v)
=
(\tilde\varphi_{\frak p\frak q;x'x}(y),\breve\varphi_{\frak p\frak q;x'x}(y,v)).$$
We may choose $\frak V_{x}$ and $\frak V_{x'}$ so small that $\frak s_{\frak p}^{n}$ and $\frak s_{\frak q}^{n}$ have representatives on the charts. Let $\frak s^n_{\frak p,x'}$ and $\frak s^n_{\frak q,x}$ be the representatives, which are $\ell_1$ and $\ell_2$ multisections, respectively. By taking an appropriate iteration we may assume $\ell_1 = \ell_2 = \ell$. We then require $$\label{form6161}
\frak s^n_{\frak p,x';k}(\tilde\varphi_{\frak p\frak q;x'x}(y)) =
\breve\varphi_{\frak p\frak q;x'x}(y,\frak s^n_{\frak q,x;\rho_y(k)}(y))$$ for $y \in V_x$, $k=1,\dots,\ell$, where $\rho_y \in {\rm Perm}(\ell)$. (\[form6161\]) is the precise form of Definition \[defn612\] (1).
We will use the $C^1$-convergence to prove certain important properties which we use in the next subsection. To state this property we need to prepare some notations.
We denote the normal bundle of our embedding $\varphi_{\frak p\frak q}: U_{\frak p\frak q} \to U_{\frak p}$ by $$N(\varphi_{\frak p\frak q};U_{\frak p})
: = \frac{\varphi_{\frak p\frak q}^*TU_{\frak p}}{TU_{\frak q}\vert_{U_{\frak p\frak q}}}$$ It defines a vector bundle over $U_{\frak p\frak q}$. For a compact subset $K \subset U_{\frak q}$ we denote by $N_K(\varphi_{\frak p\frak q};U_{\frak p})$ the restriction of this vector bundle to $K \cap U_{\frak p\frak q}$.
\[shit612712\] We fix a Riamannian metric on $U_{\frak p}$. It induces a metric on $N(\varphi_{\frak p\frak q};U_{\frak p})$. We denote by $N^\delta(\varphi_{\frak p\frak q};U_{\frak q})$ the $\delta$-disc bundle thereof for $\delta > 0$. Using the normal exponential map of the embedding $\varphi_{\frak p\frak q}$, we have a diffeomorphism: $${\rm Exp} : N_{K}^\delta(\varphi_{\frak p\frak q};U_{\frak p}) \to U_{\frak p\frak q} \times U_{\frak p}$$ which is given by ${\rm Exp}(x,v) = (x, \exp_x v)$ where $\exp_x$ is the exponential map of the metric given on $U_{\frak p}$. (See [@fooooverZ Lemma 6.5].) Here $\delta$ is a positive number depending on a compact subset $K$ of $U_{\frak q}$ and the embedding $\varphi_{\frak p\frak q}$. We put $$BN_{\delta'}(K;U_{\frak p})
= \bigcup_{x \in U_{\frak p\frak q} \cap K}
{\rm exp}_x(N_K^{\delta'}(\varphi_{\frak p\frak q};U_{\frak p})) \subset U_{\frak p}$$ for $\delta' \le \delta$. We denote by $\pi_{\delta'}: BN_{\delta'}(K;U_{\frak p}) \to U_{\frak p\frak q} \cap K$ the composition of ${\rm Ext}^{-1}$ with the projection $N(\varphi_{\frak p\frak q};U_{\frak p}) \to U_{\frak q}$ of the vector bundle. Note that on the image of $\varphi_{\frak p\frak q}$, the obstruction bundle $\mathcal E_{\frak p}$ has a subbundle $\widetilde\varphi_{\frak p\frak q}(\mathcal E_{\frak q})$. We consider a sub-bundle $$\mathcal E_{\frak q;\frak p} \subset \mathcal E_{\frak p}
= \pi_{\delta}^*(\widetilde\varphi_{\frak p\frak q}(\mathcal E_{\frak q}))$$ on $B_{\delta}(K;U_{\frak p})$ which restricts to the bundle $\widetilde\varphi_{\frak p\frak q}(\mathcal E_{\frak q})$ on $\varphi_{\frak p\frak q}(K) \subset U_{\frak p}$. We take the quotient bundle $
\mathcal E_{\frak p}/\mathcal E_{\frak q;\frak p}
$ and consider $$\overline{s_{\frak p}} \equiv s_{\frak p}
\mod \mathcal E_{\frak q;\frak p}$$ that is a section of $\mathcal E_{\frak p}/\mathcal E_{\frak q;\frak p}$. In a similar way for each branch $\frak s^{n}_{\frak p;k}$ of $\frak s^{n}_{\frak p}$ we obtain $$\label{coderivativeoffras}
\overline{\frak s^{n}_{\frak p;k}}(y) \in (\mathcal E_{\frak p}/\mathcal E_{\frak q;\frak p})_y.$$ $\blacksquare$
We denote by $E: BN_{\delta}(K;U_{\frak p}) \to N_{K}^\delta(\varphi_{\frak p\frak q};U_{\frak q})$ the inverse of ${\rm Exp}$ on its image. Namely $$\label{invexpmap}
E(y) = (x,v)
\qquad
\Leftrightarrow
\qquad
x = \pi_\delta(y), \, v = \exp_x^{-1}(y).$$
\[lem611\] Suppose we are in Situation \[shit612712\]. There exist $c>0$, $\delta_0>0$ and $n_0 \in \Z_{\ge 0}$ such that for $y \in BN_{\delta_0}(K;U_{\frak p})$ $$\label{normailityestimate}
\vert \overline{\frak s^{n}_{\frak p;k}}(y) \vert \ge c \vert E(y)\vert$$ and $$\label{normailityestimate22}
\vert\overline{s_{\frak p}}(y) \vert\ge c \vert E(y)\vert$$ hold for any branch $\overline{\frak s^{n}_{\frak p;k}}$, if $n > n_0$ and $d(\pi_{\delta}(y),s_{\frak q}^{-1}(0)) < \delta_0$.
We choose and fix a connection of the quotient bundle $\mathcal E_{\frak p}/\mathcal E_{\frak q;\frak p}$. For $y = {\rm Exp}(x,0) \in \varphi_{\frak p,\frak q}(K)$ we consider the covariant derivative $$\label{normalonimage}
V \mapsto D_V \overline{s_{\frak p}} : (N_K(\varphi_{\frak p\frak q};U_{\frak p}))_x
\to (\mathcal E_{\frak p}/\mathcal E_{\frak q;\frak p})_x.$$ By Definition \[defKchart\] (5), the map (\[normalonimage\]) is an isomorphism if $x \in \frak s_q^{-1}(0)$ in addition. Therefore we may choose $\delta_0$ so that the map (\[normalonimage\]) is an isomorphism if $d(x,s_{\frak q}^{-1}(0)) < \delta_0$. Then the existence of $\delta_0$, $c$ satisfying (\[normailityestimate22\]) is an immediate consequence of the fact $s_{\frak p}$ is smooth. The inequality (\[normailityestimate\]) then is a consequence of $C^1$-convergence.
\[rem612612\] Note the set theoretical fiber of a vector bundle over an orbifold is a quotient of a vector space by a finite group. The value $\overline{s_{\frak p}}(y)$ is well-defined as an element of vector space if we fix a local trivialization. When we do not specify the local trivialization, the value $\overline{s_{\frak p}}(y)$ is defined as an element of a quotient of a vector space by a finite group. The left hand side of (\[normailityestimate22\]) so makes sense.
In case of multisection $\overline{\frak s^{n}_{\frak p;k}}(y)$, this is well-defined as an element of vector space if we fix a local trivialization. When we change the local trivialization it changes by the permutation of $k$ and a finite group action. Therefore the validity of (\[normailityestimate\]) for all branches $k$ is independent of the choice of trivialization.
\[rem614\] Note that in Definition \[defn62\] (2) we allow the permutation $\sigma$ to [*depend*]{} on $y$ which lies in a neighborhood of $x$. By this reason the notion of branch of multisection should be studied rather carefully. Here is an example: We define $$e_{\epsilon_1\epsilon_2}(t)
=
\begin{cases}
\epsilon_1 e^{ -1/\vert t\vert} & t\le 0 \\
\epsilon_2 e^{ -1/\vert t\vert} & t\ge 0.
\end{cases}$$ Here $\epsilon_i$ is either plus or minus. Four functions $e_{++}$, $e_{--}$, $e_{-+}$, $e_{+-}$ are all smooth functions on $\R$. We define 2-multisections $\frak s$ and $\frak s'$ as follows. $$\frak s = (e_{++},e_{--}),
\qquad
\frak s' = (e_{+-},e_{-+}).$$ They are 2-multisections on $\R$ of a trivial line bundle. It is easy to see that $\frak s$ is equivalent to $\frak s'$ in the sense of Definition \[defn62\] (2). (This definition coincides with [@FO].) However, it is impossible to choose $\sigma$ appeared in Definition \[defn62\] (2) in a way independent of $y$. See Subsection \[subsec:nastyreason\] for more discussion about this point.
It is convenient to remove the assumption $d(\pi_{\delta}(y),s_{\frak q}^{-1}(0)) < \delta_0$ in Lemma \[lem611\]. Lemmata \[lem6767\] and \[lem6768\] below say that we can always do so by replacing our good coordinate system by a strongly open GG-embedding.
\[conds6.17\] Let $\widetriangle{\mathcal U}$ be a good coordinate system of $(X,Z)$ and $\mathcal K$ its support system. We consider the following condition for them.
For each $\frak p > \frak q$, there exist Riemannian metrics on $U_{\frak p}$ and the sub-bundle $\mathcal E_{\frak q;\frak p}$ as in Situation \[shit612712\]. (Here the compact set $K$ appearing in Situation \[shit612712\] is $\varphi_{\frak p\frak q}(\mathcal K_{\frak q})$.) and we have a connection on $\mathcal E_{\frak p}/\mathcal E_{\frak q;\frak p}$. They satisfy the following.
If $x \in \varphi_{\frak p\frak q}(\mathcal K_{\frak q})$, $V \in (N_K(\varphi_{\frak p\frak q};U_{\frak p}))_x$, $V \ne 0$ then $$\label{nmlderi0}
D_V \overline{s_{\frak p}} \ne 0.$$
Note the left hand side of (\[nmlderi0\]) is a covariant derivative which depends on the choice of the connection in general. It is independent of the choice of the connection when $x \in s_{\frak p}^{-1}(0))$.
\[lem6767\] Suppose $\widetriangle{\mathcal U}$ and $\mathcal K$ satisfy Condition \[conds6.17\]. Then there exist $c>0$ and $n_0 \in \Z_{\ge 0}$ $$\label{normailityestimate3}
\vert \overline{s_{\frak p}}(y) \vert \ge c \vert E(y)\vert$$ and $$\label{normailityestimate42}
\vert \overline{\frak s^{n}_{\frak p;k}}(y) \vert \ge c \vert E(y) \vert$$ hold for all branch $\overline{\frak s^{n}_{\frak p;k}}$, if $n > n_0$, $x \in \mathcal K_q$ and $y \in BN_{\delta_0}(K;U_{\frak p})$.
Here the map $E$ is as in (\[invexpmap\]). The proof is the same as the (second half of the) proof of Lemma \[lem611\].
We take a metric $d$ on $\vert \mathcal K^+\vert$ in the next lemma and $B_{\delta}(A) = \{x \in \vert \mathcal K^+\vert \mid d(x,A) < \delta\}$. We also regard $Z \subset \vert\mathcal K^+\vert$ by $\psi$.
\[lem6768\] Let $\widetriangle{\mathcal U}$ be a good coordinate system of $(X,Z)$ and $(\mathcal K^-,\mathcal K^+)$ be its support pair. Then there exists a support system $\mathcal K^{-\prime}$ and $\delta_0 > 0$ with the following properties.
1. $(\mathcal K^{-\prime},\mathcal K^+)$ is a support pair.
2. $\widetriangle{\mathcal U}$, $\mathcal K^{-\prime}$ satisfy Condition \[conds6.17\].
3. $$\label{neweq612}
B_{\delta}(\mathcal K^-_{\frak q} \cap Z) \cap Z \subseteq
B_{\delta}(\mathcal K^{-\prime}_{\frak q} \cap Z),$$
for any $\frak q \in \frak P$.
Take a support system $\mathcal K^{-+}$ such that $\mathcal K^- < \mathcal K^{-+} < \mathcal K^{+}$. Apply (the first half of the proof of) Lemma \[lem611\] to $K = \mathcal K_{\frak q}^{-+}$ and obtain $\delta_{\frak q}$ such that $$\label{newform6132}
D_V \overline{s_{\frak p}} \ne 0$$ for $V \in (N_K(\varphi_{\frak p\frak q};U_{\frak p}))_x$, $x \in B_{\delta_{\frak q}}(\mathcal K_{\frak q}^{-+} \cap s_{\frak q}^{-1}(0)) \cap U_{\frak q}$.
Put $\delta_1 = \min\{\delta_{\frak q} \mid \frak q \in \frak P\}$. Take $\delta_2>0$ such that $$\label{newform613}
B_{2\delta_2}(\mathcal K^-_{\frak q}) \cap U_{\frak q} \subset \overset{\circ}{{\mathcal K}_{\frak q}^{-+}},
\qquad
B_{2\delta_2}(\mathcal K^{-+}_{\frak q}) \cap U_{\frak q} \subset \overset{\circ}{{\mathcal K}_{\frak q}^{+}}$$ for all $\frak q$. We take $$\label{newform614}
\mathcal K^{-\prime}_{\frak q} = {\rm Close}(B_{\delta_2}(\mathcal K^-_{\frak q} \cap Z) \cap U_{\frak q}).$$ We take $\delta_0$ smaller than $\min\{\delta_1,\delta_2\}$.
Item (1) follows from (\[newform613\]). Item (2) follows from (\[newform6132\]) and (\[newform613\]).
We prove hat Item (3) holds if we replace $\delta_0$ by a smaller positive number if necessary. We remark that for any $\delta>0$ there exists $\delta' > 0$ such that $$\label{form616}
B_{\delta'}(\mathcal K^-_{\frak q} \cap Z) \cap U_{\frak p}
\subset
N_{{\rm Close}(B_{\delta}(\mathcal K^-_{\frak q} \cap Z) \cap U_{\frak q})}^{\delta}(\varphi_{\frak p\frak q};U_{\frak p}).$$ (\[form616\]) and Lemma \[lem611\] implies $$\label{newform617}
B_{\delta'}(\mathcal K^-_{\frak q} \cap Z) \cap U_{\frak p}
\subset U_{\frak q}$$ for sufficiently small $\delta'$ and $\frak q < \frak p$. (\[neweq612\]) follows from (\[newform613\]), (\[newform614\]) and (\[newform617\]).
\[rem61261222\] Note the way taken in [@foootech] or in the earlier literatures such as [@FO],[@fooobook2] is different from that in this document and proceed as follows. We fix the extension of the subbundle $\mathcal E_{\frak q;\frak p}$ and fix the choice of the splitting $
\mathcal E_{\frak q}
\equiv
\mathcal E_{\frak q;\frak p} \oplus \frac{\mathcal E_{\frak q}}{\mathcal E_{\frak q;\frak p}}.
$ We then assumed the equality $$\label{formula611}
\Pi_{\frac{\mathcal E_{\frak q}}{\mathcal E_{\frak q;\frak p}}}(\frak s_{\frak p,k}^n) =
\Pi_{\frac{\mathcal E_{\frak q}}{\mathcal E_{\frak q;\frak p}}}(s_{\frak p})$$ for any branch $\frak s_{\frak p,k}^n$ of our multisection $\frak s_{\frak p}^n$. (See [@FO (6.4.4)] for example.)[^24] We did [*not*]{} assume $\frak s_{\frak p,k}^n$ converge to $s_{\frak p}$ in $C^1$-topology but assumed only $C^0$-convergence.
However (\[formula611\]) together with the fact $$\vert\Pi_{\frac{\mathcal E_{\frak q}}{\mathcal E_{\frak q;\frak p}}}(s_{\frak p})\vert > c\vert v\vert$$ (which follows from the proof of Lemma \[lem6767\] (\[normailityestimate3\])) is enough to prove (\[normailityestimate42\]).
We have slightly modified the definition here, since by assuming $C^1$-convergence as in Definition \[defn612\] (2) we can prove Lemma \[lem6767\], which we will use instead of the assumption (\[formula611\]) made in the earlier literature. In fact, (\[normailityestimate42\]) is the property we need. (See the proof of Sublemma \[subsub67\] Case 4.)
We however emphasize that the results using the definition in the earlier literature is [*literally correct without change*]{} by the proof given there. We here are improving the presentation of the proof of the earlier literatures but are [*not*]{} correcting the proof therein.
In Part 2 of this document, we need to study a family of multivalued perturbations. The following notion is useful for the study of a family of multivalued perturbations.
\[uniformmulivalupert\] A $\sigma$ parameterized family of multivalued perturbations $\{ \{{\frak s}^n_{\sigma}\} \mid \sigma \in \mathscr A\}$ of $(\widetriangle{\mathcal U},\mathcal K)$ is said to be a [*uniform family*]{} if the convergence in Definition \[defn612\] is uniform. More precisely, we require the following.
For each $\epsilon$ there exists $n(\epsilon)$ such that if $n > n(\epsilon)$ then $$\vert s(y) - s_{\frak p}(y) \vert < \epsilon,
\qquad
\vert (Ds)(y) - (Ds_{\frak p})(y) \vert < \epsilon$$ hold for any branch $s$ of ${\frak s}^n_{\sigma}$ at any point $y \in \mathcal K_{\frak p}$ for any $\frak p \in \frak P$, $\sigma \in \mathscr A$.
Note that we assume that $(\widetriangle{\mathcal U},\mathcal K)$ is independent of $\sigma$.
Inspecting the proof of Lemma \[lem6768\], we have the following.
\[lem618\] If $\{ \{{\frak s}^n_{\sigma}\} \mid \sigma \in \mathscr A\}$ is a uniform family, then the constants $n_0$, $c$ and $\delta_0$ in Lemma \[lem6768\] can be taken independent of $\sigma$.
\[rmCkconverge\] Suppose $\mathscr A$ consists of one point. Then the condition assumed in Lemma \[lem618\] is slightly weaker than $C^1$ convergence in the sense of Definition \[defn6868\] (3). In fact, in Definition \[defn6868\] (3), we assumed, for example, that we may choose that the number of branches of $\frak s^n$ is independent of $n$.
If we define $C^k$ convergence of multisections to a [*multisection*]{} using branch in the same way as above, it seems rather complicated. Note we discuss here the case of $C^1$ convergence of multisections to a [*single-valued*]{} section, the Kuranishi map.
Support system and the zero set of multisection {#subset:supportzeromulti}
-----------------------------------------------
We use Lemma \[lem6767\] to prove Propositions \[splem2\] and \[lem715\] below. In this subsection we use a metric on subsets of $
\vert \widetriangle{\mathcal U} \vert = \bigcup_{\frak p \in \frak P} U_{\frak p} / \sim
$, which we choose as follows. We start with support systems $\mathcal K^i$, $i=1,2,3$ with $\mathcal K^1 <\mathcal K^2 <\mathcal K^3$. (See Definition \[situ61\] (2) for this notation.) The union of the images of $\mathcal K_{\frak p}^3$ in $\vert \widetriangle{\mathcal U}\vert$ is denoted by $\vert \mathcal K^3 \vert$. The quotient topology on $\vert \mathcal K^3 \vert$ is metrizable. (See [@foooshrink Proposition 5.1].) We use this topology or its induced topology. The space $X$ can be regarded as a subspace of $\vert \mathcal K^3 \vert$ and of $\vert \mathcal K^2 \vert$ or $\vert \mathcal K^1 \vert$. We take a metric on a compact neighborhood of $\vert \mathcal K^3\vert$ in $\vert \widetriangle{\mathcal U}\vert$ and use the induced metric on various spaces appearing below. Note that all the spaces $\mathcal K^i_{\frak p}$ etc. are contained in the compact neighborhood of $\vert \mathcal K^3\vert$ so have this metric.
For a subset $A \subset \vert \mathcal K^3 \vert$ we put $$\label{defmetricball}
B_{\delta}(A) = \{ x \in \vert \mathcal K^3\vert \mid d(x,A) < \delta\}.$$ For a point $p \in \vert \mathcal K^3 \vert$, we define $B_{\delta}(p) := B_{\delta}(\{p\})$.
Sometimes we identify a subset $\mathcal K^i_{\frak p}$ with its image in $\vert \mathcal K^3\vert$. Then for example, for a subset $A \subset \mathcal K^3_{\frak q} \cap U_{\frak p\frak q}$, we identify $A$ with $\varphi_{\frak p \frak q}(A)$.
\[splem2\] Let $\mathcal K^- < \mathcal K^+ < \mathcal K^2 < \mathcal K^{3}$ and let $\{\frak s_{\frak p}^{n}\}$ be a multivalued perturbation of $(\widetriangle{\mathcal U},\mathcal K^3)$. Then there exist $\delta >0$, $n_0 \in \Z_{\ge 0}$ such that for any $\frak q \in \frak P$ and $n >n_0$ $$B_{\delta}(\mathcal K_{\frak q}^- \cap Z) \cap
\bigcup_{\frak p} ((\frak s_{\frak p}^{n})^{-1}(0)
\cap \mathcal K^2_{\frak p})
\subset
\mathcal K_{\frak q}^+.$$
Here $(\frak s_{\frak p}^{n})^{-1}(0)$ is the set of the points in $\mathcal K_{\frak p}^3$ such that at least one of the branches of $\frak s_{\frak p}^{n}$ vanishes.
We first remark that in view of Lemma \[lem6768\] it suffices to prove the proposition when $\mathcal K^-$ satisfies Condition \[conds6.17\] in addition. In fact we apply Lemma \[lem6768\] to obtain $\mathcal K^{-\prime}$ satisfying Condition \[conds6.17\] in addition. By using Lemma \[lem6768\], we have $$B_{\delta}(\mathcal K_{\frak q}^- \cap Z) \cap
\bigcup_{\frak p} ((\frak s_{\frak p}^{n})^{-1}(0)
\cap \mathcal K^2_{\frak p})
\subset
B_{\delta}(\mathcal K_{\frak q}^{-\prime} \cap Z) \cap
\bigcup_{\frak p} ((\frak s_{\frak p}^{n})^{-1}(0)
\cap \mathcal K^2_{\frak p})$$ Therefore (\[splem2\]) with $\mathcal K^-$ replaced by $\mathcal K^{-\prime}$ implies (\[splem2\]).
Hereafter we assume the condition.
Let $x \in \mathcal K_{\frak q}^- \cap Z$. We first show the following lemma.
\[sublem66\] There exist $\delta_{x,\frak q} > 0$ and $n_{x,\frak q} > 0$ such that for $n > n_{x,\frak q}$ $$B_{\delta_{x,\frak q}}(x) \cap Z \subset \mathcal K_{\frak q}^+,
\quad B_{\delta_{x,\frak q}}(x) \cap
\bigcup_{\frak p} (\frak s_{\frak p}^{n})^{-1}(0)
\subset \mathcal K_{\frak q}^+.$$
During the proof of Lemma \[sublem66\] we fix $x$ and $\frak q$. The constants in Sublemmata \[sublem618\], \[subsub67\] depend on $x$ and $\frak q$.
\[sublem618\] There exists $\delta_1>0$ with the following properties.
1. If $x \notin \mathcal K_{\frak p}^2$, then $B_{\delta_1}(x) \cap \mathcal K_{\frak p}^2 = \emptyset$.
2. If $x \in \mathcal K_{\frak p}^-$, then $B_{\delta_1}(x) \cap
{\mathcal K_{\frak p}^2} = B_{\delta_1}(x) \cap {\mathcal K_{\frak p}^+}$.
3. If $x \in \mathcal K_{\frak p}^2$, $\frak q \le \frak p$, then $
B_{\delta_1}(x) \cap \mathcal K_{\frak q}^2 \subset \mathcal K_{\frak p}^{3} \cap \mathcal K_{\frak q}^2.
$
4. If $x \in \mathcal K_{\frak p}^2$, $\frak q \ge \frak p$, then $
B_{\delta_1}(x) \cap \mathcal K_{\frak p}^2 \subset \mathcal K_{\frak q}^+.
$
Statement (1) follows from compactness of $\mathcal K_{\frak p}^2$, (2) from $\mathcal K_{\frak p}^- \subset {\rm Int}\,
\mathcal K_{\frak p}^+$, (3) from $\mathcal K^2_{\frak p} \subset {\rm Int}\, {\mathcal K}_{\frak p}^{3}$, and (4) from $x \in \mathcal K_{\frak q}^- \cap Z$ and $\mathcal K_{\frak q}^- \subset {\rm Int}\,
\mathcal K_{\frak q}^+$, respectively.
\[subsub67\] There exists $\delta_{2,\frak p}$ for each $\frak p \in \frak P$ and $n_{1,\frak p} \in \Z_{\ge 0}$ such that $$\aligned
B_{\delta_{2,\frak p}}(x) \cap (s_{\frak p})^{-1}(0)
\cap \mathcal K_{\frak p}^2 \subset \mathcal K_{\frak q}^+,
\quad
B_{\delta_{2,\frak p}}(x) \cap (\frak s_{\frak p}^{n})^{-1}(0)
\cap \mathcal K_{\frak p}^2 \subset \mathcal K_{\frak q}^+
\endaligned$$ hold for $n > n_{1,\frak p}$.
We discuss 4 cases separately. In the first 3 cases we will prove $$\label{form6969}
B_{\delta_{2,\frak p}}(x) \cap \mathcal K^2_{\frak p}
\subset \mathcal K_{\frak q}^+.$$ (\[form6969\]) obviously implies the required inclusion in those cases.
(Case 1) Neither $\frak p \le \frak q$ nor $\frak q \le \frak p$. In this case we may choose $\delta_{2,\frak p} = \delta_1$ since the left hand side of (\[form6969\]) is an empty set by Sublemma \[sublem618\] (1).
(Case 2) $\frak p = \frak q$. We take $\delta_{2,\frak p} = \delta_1$. Then (\[form6969\]) follows from Sublemma \[sublem618\] (2).
(Case 3) $\frak p < \frak q$. We take $\delta_{2,\frak p} = \delta_1$. Then (\[form6969\]) follows from Sublemma \[sublem618\] (4).
(Case 4) $\frak q < \frak p$. This is the most important case. Let $d_{\frak p}$ be a metric function induced by a Riemannian metric $g_{\frak p}$ of $U_{\frak p}$. Note the metric $d$ which we used to define the metric ball in Sublemma \[sublem618\] is different from $d_{\frak p}$. However they define the same topology. We next prove that there exist, $\delta_3>0$ and $c>0$ such that $$\label{form67}
\vert s_{\frak p}(y)\vert \ge c d_{\frak p}(y,\mathcal K^2_{\frak q})$$ holds for $y \in B_{\delta_{3}}(x) \cap \mathcal K^2_{\frak p}$. To prove this we show the next subsublemma.
\[sublem628\] There exists $\delta_3 > 0$ such that if $y \in B_{\delta_3}(x)\cap \mathcal K_{\frak p}^2$ then there exists a minimal $g_{\frak p}$-geodesic $\ell : [0,d] \to U_{\frak p}$ of length $d$ such that $\ell(0) \in \varphi_{\frak p\frak q}(U_{\frak p\frak q})\cap \mathcal K_{\frak p}^{3} \cap \mathcal K_{\frak q}^2$ and $d = d_{\frak p}(y,\mathcal K^2_{\frak q})$.
Since ${\mathcal K}_{\frak q}^{2}$ is a relatively compact subset of ${\mathcal K}_{\frak q}^{3}$ and $x \in {\mathcal K}_{\frak q}^{2}$, there exists $\delta_3 > 0$ such that $d_{\frak p}(x,y) < \delta_3$ implies that there exists $z \in \mathcal K^3_{\frak q}$ with the property that the minimal geodesic joining $z$ and $y$ is perpendicular to $\mathcal K^3_{\frak q}$ and that $d_{\frak p}(z,y) \le d_{\frak p}(x,y)$. We can now use Sublemma \[sublem618\] (3) to show that we may choose $\delta_3$ such that $z \in \varphi_{\frak p\frak q}(U_{\frak p\frak q})\cap \mathcal K_{\frak p}^{3} \cap \mathcal K_{\frak q}^2$. The subsublemma follows.
The inequality (\[form67\]) follows from Subsublemma \[sublem628\] and Lemma \[lem6767\] (\[normailityestimate3\]).
Now (\[form67\]) implies that $$B_{\delta_{3}}(x) \cap (s_{\frak p})^{-1}(0) \subset \mathcal K_{\frak q}^2.$$ It implies $B_{\delta_{3}}(x) \cap (s_{\frak p})^{-1}(0) \subset \mathcal K_{\frak q}^+$ by Sublemma \[sublem618\] (2) applied to $\frak p = \frak q$.
We use Lemma \[lem6767\] (\[normailityestimate42\]) and Subsublemma \[sublem628\] in the same way as the proof of (\[form67\]) to show: $$\label{estimateofsepsilon}
\vert \frak s_{\frak p}^{n}(y)\vert \ge c d_{\frak p}(y,U_{\frak q})$$ for all $y \in B_{\delta_{4}}(x) \cap \mathcal K^2_{\frak p}$ and sufficiently large $n$. (Note that this inequality holds for any branch of $\frak s^{n}_{\frak q}$.) Using (\[estimateofsepsilon\]) in place of (\[form67\]) we prove $B_{\delta_{4}}(x) \cap (\frak s_{\frak p}^{n})^{-1}(0) \subset \mathcal K_{\frak q}^2$ in the same way as above. Then $B_{\delta_{4}}(x) \cap (\frak s_{\frak p}^{n})^{-1}(0) \subset \mathcal K_{\frak q}^+$ follows from Sublemma \[sublem618\] (2). Thus $\delta_{2,\frak p} = \min\{ \delta_1,\delta_3,\delta_4\}$ has the required properties. The proof of Sublemma \[subsub67\] is complete.
We put $\delta_{x,\frak q} = \min\{\delta_{2,\frak p}\mid \frak p \in \frak P\}$ and $n_{x,\frak q} = \max \{n_{1,\frak p} \mid \frak p \in \frak P\}$. Then Lemma \[sublem66\] follows from Sublemma \[subsub67\].
Now we take finitely many points $x_i \in \mathcal K_{\frak q}^- \cap Z$, $i =1,\dots, N_{\frak q}$ such that $$\bigcup_{i=1}^{N_{\frak q}} B_{\delta_{x,\frak q} }(x_i) \supset \mathcal K_{\frak q}^- \cap Z.$$ We put $\frak U_{\frak q} = \bigcup_{i=1}^{N_{\frak q}} B_{\delta_{x_i,\frak q}}(x_i)$. Then for any $n \ge \max \{ n_{x_i, \frak q} \mid i = 1, \dots ,N_{\frak q} \}$ we have $$\frak U_{\frak q} \supset \mathcal K_{\frak q}^- \cap Z, \qquad
\frak U_{\frak q} \cap \bigcup_{\frak p} ((\frak s_{\frak p}^{n})^{-1}(0)) \subset \mathcal K_{\frak q}^+.$$ Since $\frak U_{\frak q}$ is open and $\mathcal K_{\frak q}^- \cap Z$ is compact, there exists $\delta_{\frak q} > 0$ such that $
B_{\delta_{\frak q}}(\mathcal K_{\frak q}^- \cap Z) \subset \frak U_{\frak q}.
$ It is easy to see that $
\delta = \min \{ \delta_{\frak q} \mid \frak q \in \frak P\}
$ and $
n_0 = \max \{ n_{x_i,\frak q} \mid i=1,\dots, N_{\frak q},\,
\frak q \in \frak P\}
$ have the required properties. The proof of Proposition \[splem2\] is complete.
\[lem715\] Let $\mathcal K^1,\mathcal K^2,\mathcal K^3$ be a triple of support systems of a good coordinate system ${\widetriangle {\mathcal U}}$ of $Z \subseteq X$ with $\mathcal K^1<\mathcal K^2<\mathcal K^3$ and $\widetriangle{\frak s} = \{\frak s_{\frak p}^{n}\}$ a multivalued perturbation of $({\widetriangle {\mathcal U}},\mathcal K^3)$. Then there exists a neighborhood $\frak U(Z)$ of $Z$ in $\vert \mathcal K^2 \vert$ and $n_0\in \Z_{\ge 0}$ such that the following holds for any $n > n_0$. $$\left(\bigcup_{\frak p}((\frak s_{\frak p}^{n})^{-1}(0)
\cap \mathcal K^1_{\frak p})\right)
\cap \frak U(Z)
=
\left(\bigcup_{\frak p}((\frak s_{\frak p}^{n})^{-1}(0)
\cap \mathcal K^{2}_{\frak p})\right)
\cap \frak U(Z).$$
The inclusion $\subseteq$ is obvious for any $\frak U(Z)$. We will prove the inclusion of the opposite direction.
\[splem1\] For each $\delta >0$, there exists $\delta' > 0$ such that for every $\frak p\in \frak P$ $$B_{\delta'}(Z) \cap \mathcal K_{\frak p}^2 \subset B_{\delta}(Z \cap \mathcal K_{\frak p}^2).$$ Here we put $
B_{\delta}(A) = \{ x\in \vert \mathcal K^2 \vert \mid d(x,A) < \delta\}.
$
The proof is by contradiction. If the lemma does not hold, there exist $\frak p \in \frak P$, $\delta > 0$, a sequence $\delta_i \to 0$, and points $x_i \in B_{\delta_i}(Z) \cap \mathcal K_{\frak p}^2$ such that $x_i \notin B_{\delta}(Z \cap \mathcal K_{\frak p}^2)$.
Since $\mathcal K_{\frak p}^2$ is compact, we may assume that the sequence $x_i$ converges to a point $x \in \mathcal K_{\frak p}^2$. Then $x \in \mathcal K_{\frak p}^2 \cap Z$. Therefore $x _i \in B_{\delta}(Z \cap \mathcal K_{\frak p}^2)$ for sufficiently large $i$. This is a contradiction.
By Lemma \[splem1\] $$\label{form615454}
\aligned
\left(\bigcup_{\frak q}((\frak s_{\frak q}^{n})^{-1}(0)
\cap \mathcal K^{2}_{\frak q})\right)
\cap B_{\delta'}(Z)
&=
\bigcup_{\frak q}\left((\frak s_{\frak q}^{n})^{-1}(0)
\cap \mathcal K^{2}_{\frak q}
\cap B_{\delta'}(Z)\right) \\
&\subseteq
\bigcup_{\frak q}
\left((\frak s_{\frak q}^{n})^{-1}(0)
\cap B_{\delta}(Z \cap \mathcal K^{2}_{\frak q}) \cap \mathcal K^2_{\frak q}\right).
\endaligned$$ for sufficiently large $n$. We take a support system $\mathcal K^0 = (\mathcal K_{\frak p}^0)_{\frak p \in \frak P}$ such that $\mathcal K^0 < \mathcal K^1$. Since $\bigcup_{\frak q} Z \cap \mathcal K^{0}_{\frak q} = Z$, we have $$\label{+next66611}
\bigcup_{\frak q}
\left((\frak s_{\frak q}^{n})^{-1}(0)
\cap B_{\delta}(Z \cap \mathcal K^{2}_{\frak q}) \cap \mathcal K^2_{\frak q}\right)
\subseteq
\bigcup_{\frak p,\frak q}
\left((\frak s_{\frak p}^{n})^{-1}(0)
\cap B_{\delta}(Z \cap \mathcal K^{0}_{\frak q}) \cap \mathcal K^2_{\frak p}\right).$$ In fact $$\aligned
\bigcup_{\frak q}
\left((\frak s_{\frak p}^{n})^{-1}(0)
\cap B_{\delta}(Z \cap \mathcal K^{0}_{\frak q}) \cap \mathcal K^2_{\frak p}\right)
&=
(\frak s_{\frak p}^{n})^{-1}(0)
\cap B_{\delta}(Z)\cap \mathcal K^2_{\frak p}
\\
&\supseteq
(\frak s_{\frak p}^{n})^{-1}(0)
\cap B_{\delta}(Z \cap \mathcal K^{2}_{\frak p})
\cap \mathcal K^2_{\frak p}.
\endaligned$$ We apply Proposition \[splem2\] to $(\mathcal K^-,\mathcal K^+) = (\mathcal K^0,\mathcal K^1)$ and obtain $$\label{form617617}
\bigcup_{\frak p}
\left((\frak s_{\frak p}^{n})^{-1}(0)
\cap B_{\delta}(Z \cap \mathcal K^{0}_{\frak q})
\cap \mathcal K^{2}_{\frak p}\right)
\subset
\mathcal K^{1}_{\frak q}$$ for sufficiently large $n$. Note $$\bigcup_{\frak p}
\left((\frak s_{\frak p}^{n})^{-1}(0)
\cap \mathcal K^{1}_{\frak q}\right)
=
(\frak s_{\frak q}^{n})^{-1}(0)
\cap \mathcal K^{1}_{\frak q}.$$ Hence (\[form617617\]) implies $$\bigcup_{\frak p,\frak q}
\left((\frak s_{\frak p}^{n})^{-1}(0)
\cap B_{\delta}(Z \cap \mathcal K^{0}_{\frak q})
\cap \mathcal K^{2}_{\frak p}\right)
\subseteq
\bigcup_{\frak q}
\left((\frak s_{\frak q}^{n})^{-1}(0)
\cap \mathcal K^{1}_{\frak q}\right).$$ Combined with (\[form615454\]) and (\[+next66611\]), we have $$\left(\bigcup_{\frak q}(\frak s_{\frak q}^{n})^{-1}(0)
\cap \mathcal K^{2}_{\frak q})\right)
\cap B_{\delta'}(Z)
\subseteq
\bigcup_{\frak q}\left(
(\frak s_{\frak q}^{n})^{-1}(0)
\cap \mathcal K^{1}_{\frak q}\right).$$ We take $\frak U(Z) = B_{\delta'}(Z)$. The proof of Proposition \[lem715\] is then complete.
Proposition \[lem715\] corresponds to [@foootech Lemma 6.6]. The proof we gave above is based on the same idea.
\[cor69\] There exist a neighborhood $\frak U(Z)$ of $Z$ in $\vert\mathcal K^2\vert$ and $n_0 \in \Z_{\ge 0}$ such that the space $\left(\bigcup_{\frak p}((\frak s_{\frak p}^{n})^{-1}(0)
\cap \overset{\circ}{\mathcal K^2_{\frak p}})\right)
\cap \frak U(Z)$ is compact if $n > n_0$. Moreover, $$\label{hdfconv}
\lim_{n\to \infty}
\left(\bigcup_{\frak p}((\frak s_{\frak p}^{n})^{-1}(0)
\cap \overset{\circ}{\mathcal K^2_{\frak p}})\right)
\cap \frak U(Z)
\subseteq X.$$ Here the limit is taken in Hausdorff topology.
The first claim corresponds to [@foootech Lemma 6.11] and the second claim corresponds to [@foootech Lemma 6.12]. Using Proposition \[lem715\] the proof is also the same as those lemmata. We reproduce them here for reader’s convenience.
Proposition \[lem715\] implies that $$\label{form619619}
\left(\bigcup_{\frak p}((\frak s_{\frak p}^{n})^{-1}(0)
\cap \overset{\circ}{\mathcal K^2_{\frak p}})\right)
\cap \frak U(Z)
=
\left(\bigcup_{\frak p}((\frak s_{\frak p}^{n})^{-1}(0)
\cap {\mathcal K^1_{\frak p}})\right)
\cap \frak U(Z).$$ We may assume that $\frak U(Z)$ is compact. Then $
\left(\bigcup_{\frak p}((\frak s_{\frak p}^{n})^{-1}(0)
\cap {\mathcal K^1_{\frak p}})\right)
\cap \frak U(Z)
$ is compact. The compactness of $\left(\bigcup_{\frak p}((\frak s_{\frak p}^{n})^{-1}(0)
\cap \overset{\circ}{\mathcal K^2_{\frak p}})\right)
\cap \frak U(Z)$ follows from (\[form619619\]).
We next prove (\[hdfconv\]). Suppose (\[hdfconv\]) does not hold for any $\frak U(Z)$. Then there exist $\frak p \in \frak P$, $\delta > 0$, $n_i \to \infty$, and $x_i$ such that $x_i\in (\frak s_{\frak p}^{n_i})^{-1}(0)
\cap {\mathcal K^1_{\frak p}}
\cap \frak U(Z)$ and $d(x_i,X) \ge \delta$ for all $i$. We may assume that $x_i$ converges to $x$. (Note we may assume that $\frak U(Z)$ is compact.) Then $x\in (s_{\frak p})^{-1}(0)
\cap {\mathcal K^1_{\frak p}}
\cap \frak U(Z)$. Therefore $x \in X$. This contradicts to $d(x,X) \ge \delta > 0$.
To derive the estimate we used Definition \[defn6868\] (3) for the notion of $C^1$-convergence of the multivalued perturbations $\{ \frak s_{\frak p}^n \}$. However, we note that can be also obtained by using a slightly weaker notion of $C^1$-convergence defined by Definition \[uniformmulivalupert\]. See also Remark \[rmCkconverge\]. Therefore Propositions \[splem2\], \[lem715\] also hold even if we use this weaker notion of $C^1$-convergence of multivalued perturbations. Indeed, the next proposition, which concerns uniformity of the constants appearing in Propositions \[splem2\], \[lem715\] and Corollary \[cor69\], is also obtained under the weaker notion of $C^1$-convergence.
\[lem627\] Let $\{ \{{\frak s}^n_{\sigma}\mid n \in \Z_{\ge 0}\} \mid \sigma \in \mathscr A\}$ be a uniform family of multivalued perturbations of $(\widetriangle{\mathcal U},\mathcal K^3)$.
1. In Proposition \[splem2\] the constants $\delta$ and $n_0$ can be taken independent of $\sigma$.
2. In Proposition \[lem715\] the set $\frak U(Z)$ and the constant $n_0$ can be taken independent of $\sigma$.
3. In Corollary \[cor69\] the set $\frak U(Z)$ and the constant $n_0$ can be taken independent of $\sigma$. Moreover $$\lim_{n\to \infty}\sup\left\{ d_H\left(X,\left(\bigcup_{\frak p}((\frak s_{\sigma,\frak p}^{n})^{-1}(0)
\cap \overset{\circ}{\mathcal K^2_{\frak p}})\right)
\cap \frak U(Z)\right) \mid \sigma \in \mathscr A\right\}
= 0.$$
This is a consequence of Lemma \[lem618\] and the proofs of Propositions \[splem2\], \[lem715\] and Corollary \[cor69\].
\[transofdvect\] (Orbifold case)
1. Let $\frak s$ be a multisection of a vector bundle $\mathcal E$ on an orbifold $U$. We say it is [*transversal to $0$*]{} on $K\subset U$ if for each $x \in K$ and any branch $\frak s_k$ of $\frak s$ at $x$ such that $\frak s_k(x) = 0$, $\frak s_k$ is transversal to $0$. (Note $\frak s_k : V_x \to E_x$ is a smooth map and $(V_x,\Gamma_x,E_x,\phi_x,\widehat\phi_x)$ is an orbifold chart of $(U,\mathcal E)$.)
2. In the situation of (1), let $f : U \to N$ be a smooth map to a manifold. We say $f$ is [*strongly submersive*]{} with respect to $\frak s$, if for any branch $\frak s_k$ of $\frak s$ at $x \in K$ such that $\frak s_k(x) = 0$, the composition $$\label{form618618}
\frak s_k^{-1}(0) \hookrightarrow V \overset{\!\!\!\!\!\phi_x}{\longrightarrow U}
\overset{\!\! f}\longrightarrow N$$ is a submersion.
3. In the situation of (2), let $g : M \to N$ be a smooth map between manifolds. Suppose the multisection $\frak s$ is transversal to $0$ on $K$. We say $(\frak s,f)$ is [*strongly transversal to $g$*]{} if $\frak s$ is transversal to $0$ and, for any branch $\frak s_k$ of $\frak s$ at $x \in K$ such that $\frak s_k(x) = 0$, the composition (\[form618618\]) is transversal to $g$.
\[transkurakuravect\] (Good coordinate system case)
1. Let $\widetriangle{\frak s}$ be a multisection of $(\widetriangle{\mathcal U},\mathcal K)$, where $\widetriangle{\mathcal U}$ is a good coordinate system of $Z\subseteq X$ and $\mathcal K$ its support system. We say it is [*transversal to $0$*]{} if for each $\frak p \in \frak P$, $\frak s_{\frak p}$ is transversal to $0$ on $\mathcal K_{\frak p}$.
2. In the situation of (1), let $\widetriangle f : (X,Z;\widehat{\mathcal U}) \to N$ be a smooth map to a manifold. We say $f$ is [*strongly submersive*]{} with respect to $\widetriangle{\frak s}$, if for each $\frak p \in \frak P$, $f_{\frak p}$ is strongly submersive with respect to $\frak s_{\frak p}$.
3. In the situation of (2), let $g : M \to N$ be a smooth map between manifolds. Suppose the multisection $\widetriangle{\frak s}$ is transversal to $0$ on $K$. We say $(\widetriangle{\frak s},f)$ is [*strongly transversal to $g$*]{} if for each $\frak p \in \frak P$, $f_{\frak p}$ is strongly transversal to $g$ with respect to $\frak s_{\frak p}$.
4. A multivalued perturbation of a good coordinate system $\widetriangle{\frak s} = \{\widetriangle{\frak s^n}\}$ is said to be [*transversal to $0$*]{} if $\widetriangle{\frak s^n}$ is transversal to $0$ for sufficiently large $n$.
5. Strong submersivity of maps on good coordinate system with respect to a multivalued perturbation is defined in the same way. The definition of [*strong submersivity of a map $\widetriangle f : (X,Z;\widehat{\mathcal U}) \to N$ on good coordinate system to $g : M \to N$ with respect to a multivalued perturbation*]{} is defined in the same way.
\[prop621\] Let ${\widetriangle{\mathcal U}}$ be a good coordinate system of $Z \subseteq X$ and $\mathcal K$ its support system.
1. There exists a multivalued perturbation $\widetriangle{\frak s} = \{\frak s^n_{\frak p}\}$ of $({\widetriangle{\mathcal U}},\mathcal K)$ such that each branch of $\frak s^n_{\frak p}$ is transversal to $0$.
2. Suppose $\widetriangle f : (X,Z;{\widetriangle{\mathcal U}}) \to N$ is strongly smooth and is transversal to $g : M \to N$, where $g$ is a map from a manifold $M$. Then we may choose $\widetriangle{\frak s}$ such that $\widetriangle f$ is strongly transversal to $g$ with respect to $\widetriangle{\frak s}$.
This is actually proved during the proof of [@foootech Proposition 6.3]. We will prove it in Section \[sec:constrsec\].
Embedding of Kuranishi structure and multisection {#subset:embmultikura}
-------------------------------------------------
\[compapertKuranishi\] Let $\widehat{\mathcal U}$ be a Kuranishi structure of $Z \subseteq X$. A [*strictly compatible multivalued perturbation of $\widehat{\mathcal U}$*]{} is a collection $\widehat{\frak s} =
\{\widehat{\frak s^n}\} = \{ \frak s^{n}_p\}_{p \in Z}$ such that $\frak s^{n}_p$ is a multisection of $E_{p}$ on $U_p$ for each $p
\in X$ and $n \in \Z_{\ge 0}$, which have the following properties.
1. $
\frak s_{p}^{n} \circ \varphi_{pq}
=
\widehat\varphi_{pq}\circ \frak s_{q}^{n}
$ on $U_{pq} $.
2. $
\lim_{n\to \infty} \frak s_{p}^{n} = s_{p}
$ in $C^1$ sense on $U_{p}$.
The precise meaning of (1), (2) above is the same as in the case of Definition \[defn612\].
We use the terminology, strictly compatible multivalued perturbations, in Definition \[compapertKuranishi\]. The phrase ‘strictly compatible’ indicates that this is rather a strong condition and is hard to realize. For example, we may not expect such a perturbation exists for a given Kuranishi structure. Namely we need to replace the given Kuranishi structure to its appropriate thickening to obtain strictly compatible multivalued perturbation. (See Proposition \[lemappgcstoKu\].) Nevertheless we usually omit the phrase ‘strictly compatible’ and simply say multivalued perturbation.
\[defn692\]
1. Let $\widehat{\Phi}
= \{\Phi_{p}\}: \widehat{\mathcal U} \to {\widehat{\mathcal U'}}$ be a strict KK-embedding of Kuranishi structures. Let $\{\frak s^{n}_{p}\}$ and $\{\frak s^{\prime n}_{p}\}$ be multivalued perturbations of ${\widehat{\mathcal U}}$ and $\widehat{\mathcal U'}$, respectively. We say $\{\frak s^{n}_{p}\}$ and $\{\frak s^{\prime n}_{p}\}$ are [*compatible*]{} with $\widehat{\Phi}$ if $
\frak s_{p}^{\prime n} \circ \varphi_{p}
=
\widehat\varphi_{p}\circ \frak s_{p}^{n}.
$
2. Let $\widehat{\mathcal U_0}$ be an open substructure of a Kuranishi structure $\widehat{\mathcal U}$. Let $\{\frak s^{n}_{p}\}$ be a multivalued perturbation of ${\widehat{\mathcal U}}$. Then $\{\frak s^{n}_{p}\vert_{U_p^0}\}$ is a multivalued perturbation of $\widehat{\mathcal U_0}$. We call it the [*restriction*]{} of $\{\frak s^{n}_{p}\}$ and write $\{\frak s^{n}_{p}\}\vert_{\widehat{\mathcal U}^0}$.
3. Let $\widehat{\Phi}
= \{\Phi_{p}\}: \widehat{\mathcal U} \to {\widehat{\mathcal U'}}$ be a (not necessary strict) KK-embedding of Kuranishi structures. Let $\{\frak s^{n}_{p}\}$ and $\{\frak s^{\prime n}_{p}\}$ be multivalued perturbations of ${\widehat{\mathcal U}}$ and $\widehat{\mathcal U'}$, respectively. We say $\{\frak s^{n}_{p}\}$ and $\{\frak s^{\prime n}_{p}\}$ are [*compatible*]{} with $\widehat{\Phi}$ if a restriction $\{\frak s^{n}_{p}\}\vert_{\widehat{\mathcal U_0}}$ is compatible to $\{\frak s^{\prime n}_{p}\}$ with respect to a strict embedding $\widehat{\mathcal U_0} \to \widehat{\mathcal U'}$. Here $\widehat{\mathcal U_0}$ is an open substructure of $\widehat{\mathcal U}$
\[defn69233\] Let ${\widetriangle{\Phi} }
= \{\Phi_{p}\}: {\widetriangle{\mathcal U}} \to {{\widetriangle{\mathcal U'}}}$ be a GG-embedding, and $\widetriangle{\frak s} = \{\frak s^{n}_{\frak p}\}$ and $
\widetriangle{\frak s'} = \{\frak s^{\prime n}_{\frak p}\}$ multivalued perturbations of ${{\widetriangle{\mathcal U}}}$ and ${\widetriangle{\mathcal U'}}$, respectively. We say $\{\frak s^{n}_{\frak p}\}$ and $\{\frak s^{\prime n}_{\frak p}\}$ are [*compatible*]{} with ${\widetriangle{\Phi}}$ if $
\frak s_{\frak p}^{\prime n} \circ \varphi_{\frak p}
=
\widehat\varphi_{\frak p}\circ \frak s_{\frak p}^{n}.
$
\[defn69\] Let $\widehat{\Phi}
= (\{U_{\frak p}(p)\},\{\Phi_{p\frak p}\}): {\widetriangle{\mathcal U}} \to \widehat{\mathcal U^+}$ be a GK-embedding. Let $\widetriangle{\frak s} = \{\frak s^{n}_{\frak p}\}$ and $
\widehat{\frak s} = \{\frak s^{n}_{p}\}$ be multivalued perturbations of ${\widetriangle{\mathcal U}}$ and $\widehat{\mathcal U^+}$, respectively. We say $\{\frak s^{n}_{\frak p}\}$ and $\{\frak s^{n}_{p}\}$ are [*compatible*]{} with $\widehat{\Phi}$ if $
\frak s_{p}^{n} \circ \varphi_{p\frak p}
=
\widehat\varphi_{p\frak p}\circ \frak s_{\frak p}^{n}
$ holds on $U_{\frak p}(p)$.
There are various obvious statements about the composition of embeddings and its compatibilities with the multivalued perturbations. We leave to the interested readers to state and prove them.
\[situ610\] Let ${\widetriangle{\mathcal U}}$, ${\widetriangle{\mathcal U_0}}$ be good coordinate systems of $Z \subseteq X$. An open GG-embedding $\widetriangle\Phi : {\widetriangle{\mathcal U_0}}
\to {\widetriangle{\mathcal U}}$ is said to be [*relatively compact*]{} if, for each $\frak p$, the subset $\varphi_{\frak p}(U^0_{\frak p})$ is relatively compact in $U_{\frak p}$.
\[lemappgcstoKu\] Let ${\widetriangle{\mathcal U_0}}
\to {\widetriangle{\mathcal U}}$ be a relatively compact open GG-embedding of good coordinate systems of $Z \subseteq X$. Then there exist a Kuranishi structure $\widehat{\mathcal U}$ and a GK-embedding ${\widetriangle{\mathcal U_0}} \to \widehat{\mathcal U}$ with the following properties.
1. Let $\mathcal K$ be a support system of ${\widetriangle{\mathcal U}}$ and ${\widetriangle{\frak s}} = \{\frak s^{n}_{\frak p}\}$ a multivalued perturbation of $({\widetriangle{\mathcal U}},\mathcal K)$. We assume $
\overline{\varphi_{\frak p}(U^0_{\frak p})} \subset
{\rm Int} \,\, \mathcal K_{\frak p}
$.
Then there exists a multivalued perturbation $\widehat{\frak s} = \{\frak s^{n}_p\}$ of ${\widehat{\mathcal U}}$ such that $\widetriangle{\frak s}\vert_{{\widetriangle{\mathcal U}}_0}$ and ${\widehat{\frak s}}$ are compatible with the embedding ${\widetriangle{\mathcal U_0}}
\to {\widehat{\mathcal U}}$.
2. If ${\widetriangle f} : (X,Z;{\widetriangle{\mathcal U}}) \to Y$ is a strongly continuous (resp. strongly smooth) map, then there exists ${\widehat f} : (X,Z;{\widehat{\mathcal U}}) \to Y$ such that ${\widetriangle f}\vert_{{\widetriangle{\mathcal U_0}}}$ is a pullback of ${\widehat f}$ by the embedding ${\widetriangle{\mathcal U_0}}
\to {\widehat{\mathcal U}}$.
We will prove Proposition \[lemappgcstoKu\] together with the following relative version.
\[prop634\] In the situation of Proposition \[lemappgcstoKu\] (1) we assume the following in addition.
1. $\widehat{\mathcal U^+}$ is a Kuranishi structure of $Z
\subseteq X$ and ${\widetriangle{\mathcal U}} \to \widehat{\mathcal U^+}$ is a GK-embedding.
Then we may choose ${\widehat{\mathcal U}}$ and ${\widetriangle{\mathcal U_0}}
\to {\widehat{\mathcal U}}$ in Proposition \[lemappgcstoKu\] such that the following holds in addition.
There exists a KK-embedding ${\widehat{\mathcal U}} \to \widehat{\mathcal U^+}$ such that:
1. The composition of ${\widetriangle{\mathcal U_0}}
\to {\widehat{\mathcal U}}$ and ${\widehat{\mathcal U}} \to \widehat{\mathcal U^+}$ is equivalent to the composition of the embeddings ${\widetriangle{\mathcal U_0}} \to {\widetriangle{\mathcal U}}$, ${\widetriangle{\mathcal U}} \to \widehat{\mathcal U^+}$.
2. $\widehat{\mathcal U^+}$ is a thickening of ${\widehat{\mathcal U}}$.
3. In the situation of Proposition \[lemappgcstoKu\] (2), we assume that $\widetriangle f : (X,Z; {\widetriangle{\mathcal U}}) \to Y$ is a pullback of $\widehat{f^+} : (X,Z; {\widehat{\mathcal U^+}}) \to Y$, in addition. Then we may take $\widehat f$ such that it is a pullback of $\widehat{f^+}$.
$$\xymatrix{
& & & Y \\
\widetriangle {\mathcal U} \ar[urrr]^{\widetriangle f}
\ar[r] & \widehat{\mathcal U^+} \ar[urr]_{\widehat f^+} && \\
\widetriangle {\mathcal U^0} \ar@{^{(}-{>}}[u]\ar@{.>}[r] &\widehat {\mathcal U}
\ar@{.>}[u]\ar@{.>}[uurr]_{\widehat f} &&
}$$
\[Proof of Propositions \[lemappgcstoKu\] and \[prop634\]\] We put $\mathcal K^0_{\frak p} = \overline{\varphi_{\frak p}(U^0_{\frak p})}$ and $\mathcal K^0 = \{\mathcal K^0_{\frak p}\}$. Then $(\mathcal K^0,\mathcal K)$ is a support pair. Let $p \in Z$. We take $\frak p_p$ with the following properties.
\[property619\]
1. $p \in \psi_{\frak p_p}(s_{\frak p_p}^{-1}(0) \cap \mathcal K^0_{\frak p_p})$.
2. If $p \in \psi_{\frak q}(s_{\frak q}^{-1}(0) \cap \mathcal K^0_{\frak q})$ then $\frak q \le \frak p_p$.
Existence of such $\frak p_p$ follows from Definition \[gcsystem\] (6). We take an open neighborhood $U_p$ of $p$ in $\mathcal K_{\frak p_p}$ with the following properties.
\[property620\]
1. $U_p$ is relatively compact in ${\rm Int}\,\mathcal K_{\frak p_p}$.
2. If $\psi_{p}(\overline{U_p}\cap s_p^{-1}(0)) \cap
\psi_{\frak q}(\mathcal K^0_{\frak q} \cap s_{\frak q}^{-1}(0)) \cap Z \ne \emptyset$, then $\frak q \le \frak p_p$.
3. In the situation of Proposition \[prop634\] we require $U_p \subset U_{\frak p_p}(p)$ in addition. Here $U_{\frak p_p}(p)$ is an open neighborhood of $o_p$ in $U_{\frak p}$ that appears in the definition of the GK-embedding ${\widetriangle{\mathcal U}}
\to \widehat{\mathcal U^+}$. (Definition \[defn69\].)
We define a Kuranishi chart $\mathcal U_p$ by $\mathcal U_{\frak p_p}\vert_{U_p}$.
We next define a coordinate change among them. Let $q \in \psi_{\frak p_p}(s_{\frak p_p}^{-1}(0) \cap U_p)$. Property \[property619\] and Property \[property620\] (2) imply that $\frak p_q \le \frak p_p$. Therefore there exists a coordinate change $\Phi_{\frak p_p\frak p_q}$ of the good coordinate system ${\widetriangle{\mathcal U}}$. The coordinate change from $\mathcal U_q$ to $\mathcal U_p$ is by definition the restriction of $\Phi_{\frak p_p\frak p_q}$ to $U_q \cap \varphi_{\frak p_p\frak p_q}^{-1}(U_q)$. Compatibility of the coordinate changes follows from the compatibility of the coordinate changes of ${\widetriangle{\mathcal U}}$ and the commutativity of Diagram (\[diag33\]). We thus obtain the required Kuranishi structure $\widehat{\mathcal U}$.
Firstly we prove Proposition \[lemappgcstoKu\] (1). We define the GK-embedding ${\widetriangle{\mathcal U_0}}
\to \widehat{\mathcal U}$. Let $p \in U_{\frak p}^0 \cap X$. Since $\mathcal K_{\frak p}^0$ is the closure of $U_{\frak p}^0$, Property \[property619\] (2) implies $\frak p \le \frak p_p$. Therefore there exists a coordinate change $\Phi_{\frak p_p\frak p} : \mathcal U_{\frak p} \to \mathcal U_{\frak p_p}$. We put $$U^0_{\frak p}(p) = \varphi_{\frak p_p\frak p}^{-1}(U_p)\cap U_{\frak p}^0,
\qquad
\Phi^0_{p\frak p} = \Phi_{\frak p_p\frak p}\vert_{U^0_{\frak p}(p)}.$$ It is easy to see that they define the required GK-embedding ${\widetriangle{\mathcal U_0}}
\to \widehat{\mathcal U}$.
Secondly we prove Proposition \[lemappgcstoKu\] (2) (3). We define $\frak s^{n}_p = \frak s^{n}_{\frak p_p}\vert_{U_p}$. Its compatibility with coordinate change follows from one of $\frak s^{n}_{\frak p}$. Thus we obtain a multivalued perturbation $\{\frak s_{p}^{n}\}$. The strong compatibility of it with the GK-embedding $ {\widetriangle{\mathcal U_0}}
\to \widehat{\mathcal U}$ follows from the strong compatibility of $\frak s^{n}_{\frak p}$ with coordinate change.
If ${\widetriangle f} = \{f_{\frak p}\}$, then we define $
f_{p} = f_{\frak p_p}\vert_{U_p}$. It is easy to see that it has required properties.
Thirdly we prove Proposition \[prop634\] (i). We define a KK-embedding $: \widehat{\mathcal U} \to \widehat{\mathcal U^+}$. Let $p \in X$. We consider $\varphi_{p\frak p_p} : \mathcal U_{\frak p_p}\vert_{U_{\frak p}(p)} \to \mathcal U^+_{p}$ that is a part of the data defining the GK-embedding ${\widetriangle{\mathcal U}}
\to \widehat{\mathcal U^+}$ (Definition \[defn69\].) By Property \[property620\] (3) we can restrict it to $U_p$. It is easy to see that they are compatible with coordinate changes and define the required KK-embedding $: \widehat{\mathcal U} \to \widehat{\mathcal U^+}$.
Commutativity of Diagram \[diagram58\] implies that the composition ${\widetriangle{\mathcal U_0}}
\to \widehat{\mathcal U} \to \widehat{\mathcal U}^{+}$ is the given embedding ${\widetriangle{\mathcal U_0}} \to \widehat{\mathcal U}^{+}$.
We finally prove Proposition \[prop634\] (ii)(iii), that is, $\widehat{\mathcal U^+}$ is a thickening of $\widehat{\mathcal U}$. Let $p \in X$. We put $$O_p
=
\psi_{\frak p_p}
(
s_{\frak p_p}^{-1}(0) \cap U_{p} \cap {\rm Int}\, \mathcal K_{\frak p_p}
).$$ Note $U_{p} \subset U_{\frak p_p}(p) \subset U_{\frak p_p}$.
Let $q \in O_p$. Then $$q \in \psi_{\frak p_p}(s_{\frak p_p}^{-1}(0) \cap
{\rm Int}\, \mathcal K_{\frak p_p})
\cap \psi_p^+((s_{p}^+)^{-1}(0)).$$ Therefore there exist $O_{\frak p_p}(q) \subset U_{\frak p_p}$ and $\varphi_{q \frak p_p} : O_{\frak p_p}(q)
\to U_q^+$. We put $$W_{p}(q) = U_p \cap \varphi^{-1}_{q \frak p_p}(U_{pq}^+).$$ Then $$\varphi_p(W_p(q))
=
\varphi_{p \frak p_p}(W_p(q))
\subset
\varphi^+_{pq}(\varphi_{q \frak p_p}(W_p(q)))
\subset
\varphi^+_{pq}(U_{pq}^+).$$ We have thus checked Definition \[thickening\] (2)(a). Definition \[thickening\] (2)(b) can be checked in the same way by using $\widehat\varphi_{q \frak p_p}$. We have thus proved (ii).
\(iii) is a consequence of the fact that $U^+_{p}$ is an open subset of $U_{\frak p_p}$ and $f^+_{p}$ is a restriction of $f_{\frak p_p}$.
The proof of Propositions \[lemappgcstoKu\] and \[prop634\] is now complete.
Propositions \[lemappgcstoKu\] and \[prop634\] provide a way to transfer a multivalued perturbation of a good coordinate system to that of a Kuranishi structure. The next results describe the way of transferring them in the opposite direction. For this, we need one more definition.
\[defn6928\] Let $\widehat{\Phi}
= \{\Phi_{\frak p p}\} : \widehat{\mathcal U} \to {\widetriangle{\mathcal U}}$ be a KG-embedding. Let $\widehat{\frak s} = \{\frak s^{n}_{p}\}$ and $
\widetriangle{\frak s} = \{\frak s^{n}_{\frak p}\}$ be multivalued perturbations of ${\widehat{\mathcal U}}$ and ${\widetriangle{\mathcal U}}$, respectively. We say $\{\frak s^{n}_{p}\}$ and $\{\frak s^{n}_{\frak p}\}$ are [*compatible*]{} with $\widehat{\Phi}$ if they satisfy $$\varphi_{\frak p p} \circ \frak s_{p}^{n}
=
\frak s_{\frak p}^{n}\circ \widehat\varphi_{\frak p p}$$ on $U_{p}$.
\[le614\] Let $\widehat{\mathcal U}$ be a Kuranishi structure on $Z \subseteq X$ and $\widehat{\frak s} = \{\frak s^{n}_{p}\}$ a multivalued perturbation of ${\widehat{\mathcal U}}$. Then we can take a good coordinate system $\widetriangle{\mathcal U}$ and the strict KG-embedding $\widehat{\Phi} : \widehat{\mathcal U_0} \to \widetriangle{\mathcal U}$ in Theorem \[Them71restate\] so that the following holds in addition.
1. There exists a multivalued perturbation ${\widetriangle{\frak s}} = \{\frak s^{n}_{\frak p}\}$ of ${\widetriangle{\mathcal U}}$ such that ${\widehat{\frak s}}\vert_{\widehat{\mathcal U_0}}$ and ${\widetriangle{\frak s}}$ are compatible with the embedding $\widehat{\Phi}$.
2. If ${\widehat f} : (X,Z;{\widehat{\mathcal U}})\to Y$ is a strongly continuous map, then there exists ${\widetriangle f} : (X,Z;{\widetriangle{\mathcal U}})
\to Y$ such that ${\widetriangle f}\circ \widehat\Phi$ is a pullback of ${\widehat f}$. If ${\widehat f}$ is strongly smooth (resp. weakly submersive) then so is ${\widetriangle f}$. The transversality to $M \to Y$ is also preserved.
\[pro616\] Suppose we are in the situation of Propositions \[prop518\] (resp. \[prop519\]) and \[le614\]. Then we can take the GK-embedding $\widehat{\Phi^+} : \widetriangle{\mathcal U} \to
\widehat{\mathcal U^+}$ in Proposition \[prop518\] (resp. the GK-embeddings $\widehat{\Phi^+_a} : \widetriangle{\mathcal U} \to
\widehat{\mathcal U^+_a}$ in Proposition \[prop519\] ($a=1,2$)) so that the following holds.
1. If ${\widehat{\frak s^+}}$ is a multivalued perturbation of ${\widehat{\mathcal U^+}}$ such that ${\widehat{\frak s^+}}$, ${\widehat{\frak s}}$ are strongly compatible with the KK-embedding ${\widehat{\mathcal U}} \to \widehat{\mathcal U^+}$, then we may choose $\widetriangle{\frak s}$ such that $\widetriangle{\frak s}$, $\widehat{\frak s^+}$ are compatible with the embedding $\widehat{\Phi^+}$. (resp. If ${\widehat{\frak s^+_a}}$ ($a=1,2$) is a multivalued perturbation of ${\widehat{\mathcal U^+_a}}$ such that ${\widehat{\frak s^+_a}}$, ${\widehat{\frak s}}$ are strongly compatible with the embedding ${\widehat{\mathcal U}}
\to \widehat{\mathcal U^+}$, then we may choose $\widetriangle{\frak s}$ such that $\widetriangle{\frak s}$, $\widehat{\frak s^+_a}$ are both compatible with the embedding $\widehat{\Phi^+_a}$.)
2. If ${\widehat{f^+}} : (X,Z;\widehat{\mathcal U^+}) \to Y$ is a strongly continuous map so that the pull back of ${\widehat{f^+}}$ is ${\widehat f} : (X,Z;{\widehat{\mathcal U}})\to Y$, then we may choose $\widetriangle{f} : (X,Z;\widetriangle{\mathcal U}) \to Y$ such that ${\widehat{f^+}} \circ \widehat{\Phi^+} = \widetriangle{f}$. (resp. If ${\widehat{f^+_a}} : (X,Z;\widehat{\mathcal U_a^+}) \to Y$ are strongly continuous maps such that the pull back of both ${\widehat{f^+_a}}$ ($a=1,2$) are ${\widehat f} : (X,Z;{\widehat{\mathcal U}})\to Y$, then we may choose $\widetriangle{f} : (X,Z;\widetriangle{\mathcal U}) \to Y$ such that ${\widehat{f^+_a}} \circ \widehat{\Phi^+_a} = \widetriangle{f}$.)
We will prove Propositions \[le614\] and \[pro616\] in Subsection \[subsec:movingmulsectionetc\].
General strategy of construction of virtual fundamental chain {#bigremarkinsec6}
-------------------------------------------------------------
In this subsection we summarize a general strategy we will take and show how the results of this section will be used in the strategy.
Let us start with a Kuranishi structure $\widehat{\mathcal U}$ of $X$.
1. We find a good coordinate system ${\widetriangle{\mathcal U}}$ such that $\widehat{\mathcal U} < {\widetriangle{\mathcal U}}$, which means that $\widehat{\mathcal U}$ is embedded in ${\widetriangle{\mathcal U}}$. (Theorem \[Them71restate\].)
2. We find a multivalued perturbation ${\widetriangle{\frak s}}$ of ${\widetriangle{\mathcal U}}$ that has various transversality properties. (Theorem \[prop621\].)
3. We obtain a virtual fundamental chain associated to the perturbations of ${\widetriangle{\mathcal U}}$.
4. We next apply Proposition \[lemappgcstoKu\] to obtain Kuranishi structure $\widehat{\mathcal U^+}$ such that $$\widehat{\mathcal U} < {\widetriangle{\mathcal U}}< {\widehat{\mathcal U^+}}$$ and a multivalued perturbation ${\widehat{\frak s^+}}$ of it.
5. We apply Proposition \[prop518\] to $\widehat{\mathcal U^{+}}$ and obtain a good coordinate system ${\widetriangle{\mathcal U^{+}}}$ such that $$\widehat{\mathcal U} < {\widetriangle{\mathcal U}}< {\widehat{\mathcal U^+}} <{\widetriangle{\mathcal U^{+}}}.$$ Moreover, the multivalued perturbation ${\widehat{\frak s^+}}$ induces a multivalued perturbation ${\widetriangle{\frak s^+}}$ of ${\widetriangle{\mathcal U^{+}}}$.
6. The transversality of ${\widetriangle{\frak s}}$ implies one of ${\widehat{\frak s^+}}$ and then one of ${\widetriangle{\frak s^+}}$.
7. We obtain a virtual fundamental chain associated to ${\widetriangle{\frak s^+}}$.
Now an important statement is that the virtual fundamental chain obtained in Step 3 is the same as the virtual fundamental [*chain*]{} obtained in Step 7 for any sufficiently large $n$. (Roughly speaking, this is a consequence of Proposition \[lem715\].) Its de Rham version is Proposition \[integralinvembprop\].
This statement can be used as follows. Note the constructions in Step 5 is not unique. Namely for each given ${\widehat{\mathcal U^+}}$ there are many possible choices of ${\widetriangle{\mathcal U^{+}}}$. However the virtual fundamental chain associated to it is independent of such choices. Moreover it coincides with the virtual fundamental chain obtained in Step 3.
In other words, we can recover the virtual fundamental chain of ${\widetriangle{\frak s}}$ (that is defined on ${\widetriangle{\mathcal U}}$ in Step 3) from ${\widehat{\frak s^+}}$ that is defined on ${\widehat{\mathcal U^+}}$.
By this reason, we can forget the good coordinate system ${\widetriangle{\mathcal U}}$ and remember only the Kuranishi structure ${\widehat{\mathcal U^+}}$ and ${\widehat{\frak s^+}}$ on it. Since Kuranishi structure behaves better with fiber product than good coordinate system, we can use this fact to make the whole construction compatible with the fiber product description of the boundaries.
In Sections \[sec:contfamily\]-\[sec:contfamilyconstr\], we will work out this process in the de Rham model in great detail, where we will use [*CF-perturbations*]{}, which is an abbreviation of [*continuous family perturbations*]{}, instead of multivalued perturbations.
In Part 2 of this document, we will discuss in detail the way how we make the whole construction compatible when we start with the system of Kuranishi structures.
CF-perturbation and integration along the fiber (pushout) {#sec:contfamily}
=========================================================
Introduction to Sections \[sec:contfamily\]-\[sec:composition\] {#subsec:seccontintro}
---------------------------------------------------------------
As we mentioned in Introduction, we study systems of Kuranishi structures so that the boundary of each of its member is described by the fiber product of the other members. We will obtain an algebraic structure on certain chain complexes which realize the homology groups of certain spaces. They are the spaces over which we take fiber product between members of the system of the Kuranishi structures. To work out this process we need to make a choice of the homology theory we use. The choices are de Rham cohomology, singular homology, Čech cohomology, Morse homology or Kuranishi homology (see [@joyce]), and etc.. In [@fooobook] we took the most standard choice, that is, the singular homology. In this article, we mainly use de Rham cohomology. There are three advantages in using de Rham cohomology. One is that it seems shortest to write a detailed and rigorous proof. The second is that it is easiest to keep as much symmetry as possible. The third is, by using de Rham cohomology, we might clarify some direct relation to quantum field theory, (especially in the case of perturbation of the constant maps). There are certain disadvantage in using de Rham cohomology. The most serious disadvantage is that we can work only over real or complex numbers as a ground field. Certain technical points which appear when we use singular homology will be explained elsewhere. The way to use Morse homology is discussed in [@FOOO08III]. Actually Morse homology is one the authors of the present document had used around 20 years ago in [@Fuk97III],[@Oh96II], etc. See [@fooobook Remark 1.32].
The situation we work with is as follows.
\[smoothcorr\](See [@fooo09 Section 12]) Let $X$ be a compact metrizable space, and $\widehat{\mathcal U}$ a Kuranishi structure of $X$ (with or without boundaries or corners). Let $M_s$ and $M_t$ be $C^{\infty}$ manifolds. We assume $\widehat{\mathcal U}$, $M_s$ and $M_t$ are oriented[^25].
Let $\widehat f_s : (X;\widehat{\mathcal U}) \to M_s$ be a strongly smooth map and $\widehat f_t : (X;\widehat{\mathcal U}) \to M_t$ a weakly submersive strongly smooth map. We call $\frak X = ((X;\widehat{\mathcal U});\widehat f_s,\widehat f_t)$ a [*smooth correspondence*]{} from $M_s$ to $M_t$.$\blacksquare$
Our goal in Sections \[sec:contfamily\]-\[sec:composition\] is to associate a linear map $$\label{smoothcorr}
{\rm Corr}_{\frak X} : \Omega^k(M_s) \to \Omega^{k+ \ell}(M_t)$$ to a smooth correspondence $\frak X$ and study its properties. Here $\Omega^k(M_s)$ is the set of smooth $k$ forms on $M_s$ and $$\ell = \dim M_t - \dim (X,\widehat{\mathcal U}).$$
If $(X;\widehat{\mathcal U})$ is a smooth orbifold, namely if the obstruction bundles are all $0$, then we can define (\[smoothcorr\]) by $${\rm Corr}_{\frak X}(h) = f_t! (f_s^*(h)).$$ Here $f^*_s : \Omega^{k}(M_s) \to \Omega^{k}(X)$ is the pullback of the differential form and $f_t! : \Omega^{k}(X) \to \Omega^{k+\ell}(M_t)$ is [*the integration along the fiber*]{}, or [*pushout*]{}, which is characterized by $$\int_{M_t} f_t!(v)\wedge \rho = \int_X v \wedge f_t^*\rho.$$ (Note $\ell \le 0$ in this case.) Existence of such $f_t!$ is a consequence of the fact that $f_t$ is a proper submersion. (In our situation where $(X;\widehat{\mathcal U})$ is an orbifold this is a consequence of the weak submersivity of $\widehat f_t$.)
When the obstruction bundle is nontrivial, we need to perturb the space $X$ so that integration along the fiber is well-defined. However, taking a [*multivalued perturbation*]{} of $\widehat{\mathcal U}$ discussed in Section \[sec:multisection\] is not good enough for our purpose unless $M_t$ is a point. Let us elaborate on this point below.
Suppose $\widetriangle{\frak s} = \{\frak s_{\frak p}^{n}\}$ is a multivalued perturbation of ${\widetriangle{\mathcal U}}$ where ${\widetriangle{\mathcal U}}$ is a good coordinate system compatible with ${\widehat{\mathcal U}}$. If we assume that $\frak s_{\frak p}^{n}$ is transversal to $0$, then in case $\dim{\widehat{\mathcal U}} = \deg h$, we can define the number $$\int_{\bigcup_{\frak p}(\frak s_{\frak p}^{n})^{-1}(0)} f_s^*(h) \in \R.$$ However we can not expect the map $$f_t\vert_{\bigcup_{\frak p}(\frak s_{\frak p}^{n})^{-1}(0)} :
\bigcup_{\frak p}(\frak s_{\frak p}^{n})^{-1}(0) \to M_t$$ is a ‘submersion’ in any reasonable sense. In fact, there may happen the case when $\dim{\widehat{\mathcal U}}$ is strictly smaller than $\dim M_t$. Therefore the integration along the fiber $f_t\vert_{\bigcup_{\frak p}(\frak s_{\frak p}^{\epsilon})^{-1}(0)}!$ sends a differential form to a distributional form which may not be smooth. Thus we need to find an appropriate way to smooth it to define (\[smoothcorr\]). The way we take here is to use [*CF-perturbation*]{}, which is an abbreviation of [*continuous family perturbations*]{}. We had discussed this construction in [@fooobook2 Section 7.5], [@fooo010 Section 12], [@fooo091 Section 4], [@fooo09 Section 12].
Now the outline of Sections \[sec:contfamily\]-\[sec:composition\] is as follows. We review and describe the CF-perturbation and the integration along the fiber in greater detail and then combine them with the process to transfer various objects from a Kuranishi structure to a good coordinate system and back. More specifically, we first introduce the notion of a CF-perturbation of a single Kuranishi chart $\mathcal U$ in Subsection \[subsec:conper1chart\], where we find in Proposition \[prop721\] that the set of CF-perturbations of $\mathcal U$ turns out to be a sheaf $\mathscr S$. We introduce several subsheaves of $\mathscr S$ which satisfy various transversality conditions. Using these subsheaves we define the pushout of differential forms. Next in Subsection \[subsec:conpergcs\], we generalize these results to the case of CF-perturbations of a good coordinate system. Then we can formulate the pushout of differential forms and smooth correspondences in a good coordinate system. We also formulate and prove Stokes’ formula for good coordinate system in Section \[sec:stokes\]. So far, everything here are discussed based on good coordinate system. However, as mentioned at the end of Section \[sec:fiber\], it is more convenient and natural to use Kuranishi structure itself rather than good coordinate system, when we study the fiber product of $K$-spaces. For this purpose, we start from a Kuranishi structure and use certain embeddings into a good coordinate system and/or another Kuranishi structure introduced in Section \[sec:thick\] to translate the above results based on the good coordinate system into ones based on the Kuranishi structure and study their relationship. As a result, we show in Theorem \[theorem915\] that the pushout of differential forms for Kuranishi structure is indeed independent of choice of good coordinate system. After these foundational results on the pushout of differential forms are prepared, we prove a basic result about smooth correspondence, which is called [*composition formula of smooth correspondence*]{}, in Theorem \[compformulaprof\]. The proof of the existence of a CF-perturbation for any $K$-space is postponed till Section \[sec:contfamilyconstr\], where we also prove in Proposition \[prop123123\] that the sheaf $\mathscr S$ of CF-perturbations, together with the several subsheaves mentioned above, is soft.
CF-perturbation on a single Kuranishi chart {#subsec:conper1chart}
-------------------------------------------
We first consider the situation where we have only one Kuranishi chart as follows. After that we will introduce a CF-perturbation of good coordinate system in Subsection \[subsec:conpergcs\]. A CF-perturbation of Kuranishi structure will be defined in Subsection \[subsec:contfamiKura\].
\[smoothcorrsingle\] Let $\mathcal U = (U,\mathcal E,s,\psi)$ be a Kuranishi chart of $X$, and $f : U \to M$ a smooth submersion to a smooth manifold $M$, and $h$ a differential form on $U$ which has compact support. Assume that $U, \mathcal E$ and $M$ are oriented.$\blacksquare$
### CF-perturbation on one orbifold chart {#cftoneorbifoldchart}
Under Situation \[smoothcorrsingle\] let $\frak V_x = (V_x,\Gamma_x,E_x,\phi_x,\widehat\phi_x)$ be an orbifold chart of $(U,\mathcal E)$. (Definition \[defn2613\].) We assume $(V_x,\Gamma_x,\phi_x)$ is an oriented orbifold chart. (Definition \[defn281010\] (5).) Since $f$ is a submersion, the composition $
f\circ \phi_x : V_x \to U \to M
$ is a smooth submersion, which is denoted by $f_x$.
\[defn73ss\] A [*CF-perturbation (=continuous family perturbation)*]{} of $\mathcal U$ on our orbifold chart $\frak V_x = (V_x,\Gamma_x,E_x,\phi_x,\widehat\phi_x)$ consists of $\mathcal S_x = (W_x,\omega_x,\{{\frak s}_x^{\epsilon}\})$, $0 < \epsilon \le 1$, with the following properties:
1. $W_x$ is an open neighborhood of $0$ of a finite dimensional vector space $\widehat W_x$ on which $\Gamma_x$ acts linearly. $W_x$ is $\Gamma_x$ invariant.
2. $
{\frak s}_x^{\epsilon} : V_x \times W_x \to E_x
$ is a $\Gamma_x$-equivariant smooth map for each $0 < \epsilon \le 1$.
3. For $y \in V_x$ , $\xi \in W_x$ we have $$\label{C0convconti}
\lim_{\epsilon\to 0} {\frak s}_x^{\epsilon}(y,\xi)
= s_x(y)$$ in compact $C^1$-topology on $V_x \times W_x$.
4. $\omega_x$ is a smooth differential form on $W_x$ of degree $\dim W_x$ that is $\Gamma_x$ invariant, of compact support and $$\int_{W_x} \omega_x = 1.$$ We assume $\omega_x = \vert\omega_x\vert{\rm vol}_x$ here ${\rm vol}_x$ is a volume form of the oriented manifold $W_x$ and $\vert\omega_x\vert$ is a non-negative function.
For each $0 < \epsilon \le 1$, we denote the restriction of $\mathcal S_x$ at $\epsilon$, by $\mathcal S_x^{\epsilon} = (W_x,\omega_x,{\frak s}_x^{\epsilon})$.
1. In Definition \[defn73ss\] (3) we regard $s_x : V_x \to E_x$ as a $\Gamma_{x}$ equivariant map that is a local representative of the Kuranishi map in the sense of Definition \[defnlocex\].
2. In our earlier writings, we used a family of [*multi*]{}-sections parameterized by $W_x$. Here we use a family of sections parameterized by $W_x$ on $V_x$ such that it is $\Gamma_x$ equivariant as a map from $V_x \times W_x$. We also allow $W_x$ to have a nontrivial $\Gamma_x$ action. This formulation seems simpler.
3. We may regard $ {\frak s}_x^{\epsilon}$ as a local representative of a section of the vector bundle $(V_x \times W_x \times E_x)/\Gamma_x \to (V_x \times W_x)/\Gamma_x$. (Lemma \[lem2627\].)
\[conmultiequiv11\] Let $\mathcal S^i_x = (W^i_x,\omega^i_x,\{{\frak s}_{x}^{\epsilon,i}\})$ $(i=1,2)$ be two CF-perturbations of $\mathcal U$ on $\frak V_x$.
1. We say $\mathcal S^1_x$ is a [*projection*]{} of $\mathcal S^2_x$, if there exist a map $\Pi :
\widehat W^2_x \to \widehat W^1_x$ with the following properties.
1. $\Pi$ is a $\Gamma_x$ equivalent linear projection which sends $W^2_x$ to $W^1_x$ and satisfies $
\Pi!(\omega^2_x) = \omega^1_x.
$
2. For each $y \in V_x$ and $\xi \in W_x^1$ we have $${\frak s}_{x}^{\epsilon,1}(y,\Pi(\xi)) = {\frak s}_{x}^{\epsilon,2}(y,\xi).$$
2. We say $\mathcal S^1_x$ is [*equivalent*]{} to $\mathcal S^2_x$ on $\frak V_x$ if there exist $N$ and $\mathcal S^{(i)}_x$ for $i=0,\dots,2N$ with the following properties.
1. $\mathcal S^{(i)}_x$ is a CF-perturbation of $\mathcal U$ on $\frak V_x$.
2. $\mathcal S^{(0)}_x = \mathcal S^1_x$, $\mathcal S^{(2N)}_x = \mathcal S^2_x$.
3. $\mathcal S^{(2k-1)}_x$ and $\mathcal S^{(2k+1)}_x$ are both projections of $\mathcal S^{(2k)}_x$.
It is easy to see that the relations defined in Definition \[conmultiequiv11\] (2) is an equivalence relation.
$$\xymatrix{
& \mathcal S_x^{(1)} \ar[ld]\ar[rd] && \cdots\ar[ld]\ar[rd] &&
\mathcal S_x^{(2N-1)} \ar[ld]\ar[rd] \\
\mathcal S_x^{(0)} && \mathcal S_x^{(2)} & \cdots&
\mathcal S_x^{(2N-2)} && \mathcal S_x^{(2N)}
}
\nonumber$$
\[contipertlocalrest\] Let $
\Phi_{xx'} = (h_{xx'},\widetilde\varphi_{xx'},\breve\varphi_{xx'})
$ be a coordinate change from $\frak V_{x'}$ to $\frak V_{x}$. (See Situation \[opensuborbifoldchart\].) Let $\mathcal S_x = (W_x,\omega_x,{\frak s}_{x}^{\epsilon})$ be a CF-perturbation of $\mathcal U$ on $\frak V_x$. We define its [*restriction*]{} $\Phi_{xx'}^*\mathcal S_x$ by $$\Phi_{xx'}^*\mathcal S_x
=
(W_x,\omega_x,\{{\frak s}^{\epsilon \prime}_{x'}\})$$ where ${\frak s}^{\epsilon\prime}_{x'}$ is defined as follows. We associate a linear isomorphism $
g_y : E_{x'} \to E_x
$ to each $y \in V'_{x'}$ by $
\breve\varphi_{xx'}(y,v) = g_y(v).
$ Then we put $$\label{pullbacklocaldefcont}
{\frak s}^{\epsilon \prime}_{x'}(y,\xi)
= g_y^{-1}({\frak s}^{\epsilon}_{x}(\tilde\varphi_{xx'}(y),\xi)).$$
\[lem77\]
1. If $\mathcal S^1_x$ is equivalent to $\mathcal S^2_x$, then $
\Phi_{xx'}^*\mathcal S^1_x
$ is equivalent to $
\Phi_{xx'}^*\mathcal S^2_x
$.
2. The restriction $\Phi_{xx'}^*\mathcal S_x$ may depend on the choice of $\Phi_{xx'} = (h_{xx'},\tilde\varphi_{xx'},\breve\varphi_{xx'})$. However the equivalence class of $\Phi_{xx'}^*\mathcal S_x$ is independent of such a choice.
\(1) is obvious as far as we use the same $(h_{xx'},\tilde\varphi_{xx'},\breve\varphi_{xx'})$. We will prove (2). Let $(h^{i}_{xx'},\tilde\varphi^{i}_{xx'},\breve\varphi^{i}_{xx'})$ ($i=1,2$) be two choices and ${\frak s}^{\epsilon i \prime}_{x'}(y,\xi)$ the restrictions obtained by these two choices for $i=1,2$, respectively. Then by Lemma \[lem2715\] there exists $\gamma \in \Gamma_x$ such that $$h^{2}_{xx'}(\mu) = \gamma h^{1}_{xx'}(\mu) \gamma^{-1},
\quad
\tilde\varphi^{2}_{xx'} = \gamma \tilde\varphi^{1}_{xx'},
\quad
\breve\varphi^{2}_{xx'} = \gamma \breve\varphi^{1}_{xx'}.$$ We put $
\breve\varphi^{i}_{xx'}(y,v) = g^{i}_y(v).
$ Then $
g^2_y(v) = \gamma g^1_y(v).
$ Therefore (\[pullbacklocaldefcont\]) implies $$\label{lem78profcal}
\aligned
{\frak s}^{\epsilon 2 \prime}_{x'}(y,\xi)
&= (g_y^2)^{-1}({\frak s}^{\epsilon \prime}_{x}(\tilde\varphi^2_{xx'}(y),\xi))
\\
&=(g_y^1)^{-1}\gamma^{-1}({\frak s}^{\epsilon\prime}_{x}(\gamma\tilde\varphi^1_{xx'}(y),\xi))\\
&=(g_y^1)^{-1}({\frak s}^{\epsilon 1\prime}_{x}(\tilde\varphi^1_{xx'}(y),\gamma^{-1}\xi))
\\
&={\frak s}^{\epsilon 1 \prime}_{x'}(y,\gamma^{-1}\xi).
\endaligned$$ We note that $\gamma^{-1}$ induces a $\Gamma_{x'}$ linear isomorphism from $(W_x,h^2_{xx'})$ to $(W_x,h^1_{xx'})$. (Here the $\Gamma_{x'}$ action on $W_x$ is induced from the $\Gamma_{x}$ action by $h^{i}_{xx'}$ in case of $(W_x,h^i_{xx'})$, $i=1,2$. Then the map $\xi \mapsto \gamma^{-1}\xi$ is $\Gamma_x$ equivariant as a map $W_x \to W_x$, where $\Gamma_x$ acts in a different way on the source and the target.) Therefore ${\frak s}^{\epsilon 2 \prime}_{x'}(y,\xi)$ is equivalent to ${\frak s}^{\epsilon 1 \prime}_{x'}(y,\xi)$.
We next define the pushout of a differential form by using a CF-perturbation.
\[submersivepertconlocloc\] In Situation \[smoothcorrsingle\], let $\mathcal S_x = (W_x,\omega_x,\{{\frak s}_{x}^{\epsilon}\})$ be a CF-perturbation of $\mathcal U$ on $\frak V_x$.
1. We say $\mathcal S_x$ is [*transversal to $0$*]{} if there exists $\epsilon_0 > 0$ such that the map ${\frak s}_{x}^{\epsilon}$ is transversal to $0$ on a neighborhood of the support of $\omega_x$ for all $0 < \epsilon < \epsilon_0$. In particular $$({\frak s}_{x}^{\epsilon})^{-1}(0)
=
\{(y,\xi) \in V_x \times W_x\mid {\frak s}_{x}^{\epsilon}(y,\xi) = 0\}$$ is a smooth submanifold of $V_x \times W_x$ on a neighborhood of the support of $\omega_x$.
2. We say $f_x = f \circ \phi_x$ is [*strongly submersive*]{} with respect to $(\frak V_x,\mathcal S_x)$ if $\mathcal S_x$ is transversal to $0$ and there exists $\epsilon_0 > 0$ such that the map $$\label{form7575}
f_x \circ \pi_1\vert_{({\frak s}_{x}^{\epsilon})^{-1}(0)} :
({\frak s}_{x,k}^{\epsilon})^{-1}(0) \to M$$ is a submersion on a neighborhood of the support of $\omega_x$, for all $0 < \epsilon < \epsilon_0$. Here $\pi_1 : V_x \times W_x \to V_x$ is the projection.
3. Let $g : N \to M$ be a smooth map between manifolds. We say [*$f_x$ is strongly transversal to $g$*]{} with respect to $(\frak V_x,\mathcal S_x)$ if $\mathcal S_x$ is transversal to $0$ and there exists $\epsilon_0 > 0$ such that the map (\[form7575\]) is transversal to $g$, for all $0 < \epsilon < \epsilon_0$. Here $\pi_1 : V_x \times W_x \to V_x$ is the projection.
\[newlem79\] Suppose $\mathcal S^1_x$ is equivalent to $\mathcal S^2_x$.
1. $\mathcal S^1_x$ is transversal to $0$ if and only if $\mathcal S^2_x$ is transversal to $0$.
2. $f_x$ is strongly submersive with respect to $(\frak V_x,\mathcal S^1_x)$ if and only if $f_x$ is strongly submersive with respect to $(\frak V_x,\mathcal S^2_x)$.
3. $f_x$ is strongly transversal to $g : N \to M$ with respect to $(\frak V_x,\mathcal S^1_x)$ if and only if $f_x$ is strongly transversal to $g$ with respect to $(\frak V_x,\mathcal S^2_x)$.
It suffices to prove the lemma for the case when $\mathcal S^2_x$ is a projection of $\mathcal S^1_x$. This case follows from the fact that $\omega_x = \vert\omega_x\vert{\rm vol}_x$ where ${\rm vol}_x$ is a volume form of $W_x$ and $\vert\omega_x\vert$ is a non negative function, which is a part of Definition \[defn73ss\] (4).
We recall that a smooth differential form $h$ on $V_x/\Gamma_x$ is identified with a $\Gamma_x$ invariant smooth differential form $\tilde h$ on $V_x$. (Definition \[defn281010\] (2).) The [*support*]{} of $h$ is the quotient of the support of $\tilde h$ by $\Gamma_x$ and is a closed subset of $V_x/\Gamma_x
\cong U_x$. We denote it by ${\rm Supp}(h)$.
In Situation \[smoothcorrsingle\], let $\mathcal S_x = (W_x,\omega_x,\{{\frak s}_{x}^{\epsilon}\})$ be a CF-perturbation of $\mathcal U$ on $\frak V_x$. Let $h$ be a smooth differential form on $U_x$ that has compact support. Then we define a smooth differential form $f_x!(h;\mathcal S^{\epsilon}_x)$ on $M$ for each $\epsilon > 0$ by the equation (\[form72\]) below. We call it the [*pushout*]{} of $h$ with respect to $f_x$, $\mathcal S_x$.
Let $\rho$ be a smooth differential form on $M$. Then we require $$\label{form72}
\# \Gamma_x
\int_{M} f_x!(h;\mathcal S^{\epsilon}_x) \wedge \rho
=
\int_{({\frak s}_{x}^{\epsilon})^{-1}(0)}
\pi_1^*(\tilde h) \wedge \pi_1^*(f_x^*\rho) \wedge \pi_2^*\omega_x.$$ Here $\pi_1$ (resp. $\pi_2$) is the projection of $V_x \times W_x$ to the first (resp. second) factor.
Unique existence of such $f_x!(h;\mathcal S^{\epsilon}_x)$ is an immediate consequence of the existence of pushforward of a smooth form by a proper submersion.
In the left hand side of (\[form72\]) we crucially use the fact that $\Gamma_x$ is a finite group. It seems that this is the [*only*]{} place we use the finiteness of $\Gamma_x$ when we use de Rham theory to realize virtual fundamental chain. We might try to use a CF-perturbation and de Rham version together with an appropriate model of equivariant cohomology to study virtual fundamental chain in case the isotropy group can be a continuous compact group of positive dimension, such as the case of gauge theory or pseudo-holomorphic curves in a symplectic manifold acted by a compact Lie group.
\[pushequivlocal\] If $\mathcal S_x^1$ is equivalent to $\mathcal S_x^2$, then $$f_x!(h;\mathcal S^{1,\epsilon}_x)
=
f_x!(h;\mathcal S^{2,\epsilon}_x).$$
It suffices to prove the equality in the case when $\mathcal S_x^1$ is a projection of $\mathcal S_x^2$. This is immediate from definition.
\[resandtrans\] Suppose we are in the situation of Definition \[contipertlocalrest\] and Situation \[smoothcorrsingle\].
1. If $(\frak V_x,\mathcal S_x)$ is transversal to $0$, then $(\frak V_{x'},\Phi_{xx'}^*\mathcal S_x)$ is transversal to $0$.
2. If $f_x$ is strongly submersive with respect to $(\frak V_x,\mathcal S_x)$, then $f_{x'}$ is strongly submersive with respect to $(\frak V_{x'},\Phi_{xx'}^*\mathcal S_x)$.
3. If $f_x$ is strongly transversal to $g : N \to M$ with respect to $(\frak V_x,\mathcal S_x)$, then $f_{x'}$ is strongly transversal to $g : N \to M$ with respect to $(\frak V_{x'},\Phi_{xx'}^*\mathcal S_x)$.
The proof is immediate from definition.
### CF-perturbation on a single Kuranishi chart {#cftonekurachart}
In Subsubsection \[cftoneorbifoldchart\], we studied locally on a single chart $U_x = V_x/\Gamma_x$. We next work globally on an orbifold $U$. We apply Remark \[xkararhe\] hereafter.
In this subsubsection we consider a Kuranish chart $\mathcal U = (U,\mathcal E,s,\psi)$. However the parametrization $\psi$ does not play any role in this subsubsection.
\[semiglobalocntpert\] Let $\mathcal U =(U,\mathcal E,s,\psi)$ be a Kuranishi chart. A [*representative of a CF-perturbation of $\mathcal U$*]{} is the following object $\frak S = \{(\frak V_{\frak r},\mathcal S_{\frak r})\mid{\frak r\in
\frak R}\}$.
1. $\{U_{\frak r} \mid \frak r \in \frak R\}$ is a family of open subsets of $U$ such that $
U = \bigcup_{\frak r \in \frak R}U_{\frak r}.
$
2. $\frak V_{\frak r} = (V_{\frak r},\Gamma_{\frak r},E_{\frak r},\phi_{\frak r},\widehat\phi_{\frak r})$ is an orbifold chart of $(U,\mathcal E)$ such that $\phi_{\frak r}(V_{\frak r}) = U_{\frak r}$.
3. $\mathcal S_{\frak r} = (W_{\frak r} ,\omega_{\frak r}, \{{\frak s}_{\frak r} ^{\epsilon}\})$ is a CF-perturbation of $\mathcal U$ on $\frak V_{\frak r}$.
4. For each $x \in U_{\frak r_1} \cap U_{\frak r_2}$, there exists an orbifold chart $\frak V_{\frak r}$ with the following properties:
1. $
x \in U_{\frak r} \subset U_{\frak r_1} \cap U_{\frak r_2}.
$
2. The restriction of $\mathcal S_{\frak r_1}$ to $\frak V_{\frak r}$ is equivalent to the restriction of $\mathcal S_{\frak r_2}$ to $\frak V_{\frak r}$.
For each $\epsilon >0$, we write $$\frak S^{\epsilon}
= \{(\frak V_{\frak r},\mathcal S_{\frak r}^{\epsilon})\mid{\frak r\in
\frak R}\}.$$ See Definition \[defn73ss\] for the notation $\mathcal S^{\epsilon}_{\frak r}.$
\[projectcontfamilocal\] Let $\mathcal U = (U,\mathcal E,s,\psi)$ be as in Definition \[semiglobalocntpert\] and $\frak S^i = \{(\frak V_{\frak r}^i,\mathcal S_{\frak r}^i)\mid{\frak r\in
\frak R^i}\}$ ($i=1,2$) representatives of CF-perturbations of $\mathcal U$. We say that $\frak S^2$ is [*equivalent*]{} to $\frak S^1$ if, for each $x \in U_{\frak r_1} \cap U_{\frak r_2}$, there exists an orbifold chart $\frak V_{\frak r}$ with the following properties:
1. $x \in U_{\frak r} \subset U_{\frak r_1} \cap U_{\frak r_2}.$
2. The restriction of $\mathcal S^1_{\frak r_1}$ to $\frak V_{\frak r}$ is equivalent to the restriction of $\mathcal S^2_{\frak r_2}$ to $\frak V_{\frak r}$.
\[defn71717\] Suppose we are in Situation \[smoothcorrsingle\]. Let $\frak S = \{\frak S_{\frak r}\}$ be a representative of a CF-perturbation of $\mathcal U$. Let $U' \subseteq U$ be an open subset.
Let $\frak S_{\frak r} = (\frak V_{\frak r},\mathcal S_{\frak r})$, $\frak V_{\frak r} = (V_{\frak r},\Gamma_{\frak r},E_{\frak r},\phi_{\frak r},\widehat\phi_{\frak r})$. If ${\rm Im}(\phi_{\frak r}) \cap U' = \emptyset$, then we remove $\frak r$ from $\frak R$. Let $\frak R_0$ be obtained by removing all such $\frak r$ from $\frak R$. If ${\rm Im}(\phi_{\frak r}) \cap U' \ne \emptyset$, then $\frak V_{\frak r}\vert_{{\rm Im}\phi_{\frak r} \cap U'}$ is an orbifold chart of $(U,\mathcal E)$, which we write $\frak V_{\frak r}\vert_{U'} = (V'_{\frak r},\Gamma'_{\frak r},E_{\frak r},\phi'_{\frak r},\widehat\phi'_{\frak r})$. Let $\mathcal S_{\frak r} = (W_{\frak r} ,\omega_{\frak r} ,\{{\frak s}^{\epsilon}_{\frak r}\})$. We define $$\label{formform7676}
\mathcal S_{\frak r}\vert_{U'}
=
(W_{\frak r} ,\omega_{\frak r} ,\{{\frak s}^{\epsilon}_{\frak r}\vert_{V'_{\frak r} \times W_{\frak r}}\}).$$ Now we put: $$\frak S\vert_{U'}
=
\{(\frak V_{\frak r}\vert_{U'},\mathcal S_{\frak r}\vert_{U'}) \mid \frak r \in \frak R_0 \}.$$ It is a representative of a CF-perturbation of $\mathcal U\vert_{U'}$, which we call the [*restriction*]{} of $\frak S$ to $U'$.
\[lem718718\] If $\frak S^1$ is equivalent to $\frak S^2$, then $\frak S^1\vert_{\Omega}$ is equivalent to $\frak S^2\vert_{\Omega}$.
The proof is immediate from definition.
\[rem718\] Suppose we are in Situation \[smoothcorrsingle\].
1. A [*CF-perturbation*]{} on $\mathcal U$ is an equivalence class of a representative of a CF-perturbation with respect to the equivalence relation in Definition \[projectcontfamilocal\].
2. Let $\Omega$ be an open subset of $U$. We denote by $\mathscr{S}(\Omega)$ the set of all CF-perturbations on $\mathcal U\vert_{\Omega}$.
3. Let $\Omega_1 \subset \Omega_2 \subset U$. Then using Lemma \[lem718718\], the restriction defined in Definition \[defn71717\] induces a map $$\frak i_{\Omega_1\Omega_2} : \mathscr{S}(\Omega_2) \to \mathscr{S}(\Omega_1).$$ We call this map [*restriction map*]{}.
4. $\Omega \mapsto \mathscr{S}(\Omega)$ together with $\frak i_{\Omega_1 \Omega_2}$ defines a presheaf. (In fact, $\frak i_{\Omega_1\Omega_2}\circ \frak i_{\Omega_2\Omega_3} = \frak i_{\Omega_1\Omega_3}$ holds obviously.) The next proposition says that it is indeed a sheaf. We call it the [*sheaf of CF-perturbations*]{} on $\mathcal U$ .
Hereafter, by a slight abuse of notation, we also use the symbol $\frak S$ for a CF-perturbation. Namely we use this symbol not only for a representative of continuous family perturbation but also for its equivalence class, by an abuse of notation.
\[prop721\] The presheaf $\mathscr{S}$ is a sheaf.
This is mostly a tautology. We provide a proof for completeness’ sake.
Let $
\bigcup_{a\in A} \Omega_{a} = \Omega
$ be an open cover of $\Omega$. Suppose $\frak S_a \in \mathscr{S}(\Omega_a)$ and $\{(\frak V_{a,\frak r},\mathcal S_{a,\frak r})\mid{\frak r\in
\frak R_a}\}$ is a representative of $\frak S_a$. We put $\Omega_{ab} = \Omega_a \cap \Omega_{b}$. We assume $$\label{form7878}
\frak i_{\Omega_{ab}\Omega_a}(\frak S_{a}) = \frak i_{\Omega_{ab}\Omega_b}(\frak S_{b}).$$ To prove Proposition \[prop721\] it suffices to show that there exists a unique $\frak S \in \mathscr{S}(\Omega)$ such that $$\label{form7979}
\frak i_{\Omega_{a}\Omega}(\frak S) = \frak S_{a}.$$
Suppose that $\frak S, \frak S' \in \mathscr{S}(\Omega)$ both satisfy (\[form7979\]). Let $\{(\frak V_{\frak r},\mathcal S_{\frak r})\mid{\frak r\in
\frak R}\}$ and $\{(\frak V'_{\frak r'},\mathcal S'_{\frak r'})\mid{\frak r'\in
\frak R'}\}$ be representatives of $\frak S$ and $\frak S'$, respectively. We will prove that they are equivalent.
Let $x \in \Omega$. There exist $\frak r \in \frak R$, $\frak r' \in \frak R'$ such that $x \in U_{\frak r} \cap U'_{\frak r'}$, where $U_{\frak r} = {\rm Im}(\phi_{\frak r})$, $U'_{\frak r'} = {\rm Im}(\phi'_{\frak r'})$. We take $a$ such that $x \in \Omega_a$. By (\[form7979\]), $\mathcal S_{\frak r}\vert_{U_{\frak r} \cap \Omega_a}$ is equivalent to $\mathcal S'_{\frak r'}\vert_{U'_{\frak r'} \cap \Omega_a}$. Therefore there exists an orbifold chart $\frak V_x$ such that $U_x \subset U_{\frak r} \cap U'_{\frak r'}$ and the restriction of $\mathcal S_{\frak r}\vert_{U_{\frak r} \cap \Omega_a}$ to $\frak V_x$ is equivalent to the restriction of $\mathcal S'_{\frak r'}\vert_{U'_{\frak r'} \cap \Omega_a}$ to $\frak V_x$. Thus the restriction of $\mathcal S_{\frak r}$ to $\frak V_x$ is equivalent to the restriction of $\mathcal S'_{\frak r'}$ to $\frak V_x$.
Since this holds for any $x \in \Omega$, $\{(\frak V_{\frak r},\mathcal S_{\frak r})\mid{\frak r\in
\frak R}\}$ is equivalent to $\{(\frak V'_{\frak r'},\mathcal S'_{\frak r'})\mid{\frak r'\in
\frak R'}\}$ by definition.
Let $\frak V_{\frak r,a} = (V_{\frak r,a},\Gamma_{\frak r,a},E_{\frak r,a},\phi_{\frak r,a},\widehat\phi_{\frak r,a})$ and $U_{\frak r,a} = \phi_{\frak r,a}(V_{\frak r,a})$. Then $\{U_{\frak r,a} \mid a \in A,\,\, \frak r \in \frak R_a\}$ is an open cover of $\Omega$. We put $$\frak S = \coprod_{a \in A} \{(\frak V_{a,\frak r},\mathcal S_{a,\frak r})\mid{\frak r\in
\frak R_a}\}.$$ To show $\frak S \in \mathscr{S}(\Omega)$ it suffices to check Definition \[semiglobalocntpert\] (4). This is a consequence of (\[form7878\]) and the definitions of the equivalence of representatives of CF-perturbations and of the restriction. We can check (\[form7979\]) also by (\[form7878\]) in the same way as in the proof of uniqueness.
The proof of Proposition \[prop721\] is now complete.
We recall that the [*stalk*]{} $\mathscr S_x$ of the sheaf $\mathscr S$ at $x \in U$ is by definition $$\label{form72000}
\mathscr S_x = \varinjlim_{\Omega \ni x}\mathscr S(\Omega).$$
The stalk $\mathscr S_x$ is identified with the set of the equivalence classes of the equivalence relation defined in Item (2) on the set which is defined in Item (1).
1. We consider the set $\widetilde{\mathscr S}_x$ of pairs $(\frak V_{\frak r},\frak S_{\frak r})$ where $\frak V_{\frak r}$ is an orbifold chart of $(U,\mathcal E)$ at $x$ and $\frak S_{\frak r}$ is a CF-perturbation on $\frak V_{\frak r}$.
2. Let $(\frak V_{\frak r},\frak S_{\frak r}), (\frak V_{\frak r'},\frak S_{\frak r'}) \in \widetilde{\mathscr S}_x$. We say that they are equivalent if there exists an orbifold chart $\frak V_{x}$ at $x$ such that $U_x \subset U_{\frak r} \cap U_{\frak r'}$ and the restriction of $\frak S_{\frak r}$ to $\frak V_{x}$ equivalent to the restriction of $\frak S_{\frak r'}$ to $\frak V_{x}$.
The proof of obvious. The next lemma is standard in sheaf theory.
\[lem723\] The set $\mathscr S(\Omega)$ is identified with the following object:
1. For each $x \in \Omega$ it associates $\frak S_x \in \mathscr S_x$.
2. For each $x \in \Omega$, there exists a representative $(\frak V_{\frak r},\frak S_{\frak r})$ of $\frak S_x$, such that for each $y \in \phi_{\frak r}(V_{\frak r})$ the germ $\frak S_y$ is represented by $(\frak V_{\frak r},\frak S_{\frak r})$.
Let $K \subseteq U$ be a closed subset. A [*CF-perturbation of $K \subseteq U$*]{} is an element of the inductive limit: $
\varinjlim_{\Omega \supset K}\mathscr S(\Omega)
$. We denote the set of all CF-perturbations of $K \subseteq U$ by $\mathscr S(K)$. Namely $$\label{sheafoverclosedset}
\mathscr S(K) = \varinjlim_{U\supset \Omega \supset K;~ \Omega \text{ open}}\mathscr S(\Omega).$$
Integration along the fiber (pushout) on a single Kuranishi chart {#subsec:intonechart}
-----------------------------------------------------------------
\[strosubsemiloc\] Suppose we are in Situation \[smoothcorrsingle\]. Let $\Omega \subset U$ be an open subset and let $\frak S \in \mathscr S(\Omega)$ be a CF-perturbation. We consider its representative $\{(\frak V_{\frak r},\mathcal S_{\frak r})\mid{\frak r\in
\frak R}\}$.
1. We say that $(\mathcal U,\frak S)$ is [*transversal to $0$*]{} if, for each $\frak r$, $(\frak V_{\frak r},\mathcal S_{\frak r})$ is transversal to $0$. This is independent of the choice of representative.
2. We say that $f$ is [*strongly submersive with respect to $(\mathcal U,\frak S)$*]{} if, for each $\frak r$, the map $f$ is strongly submersive with respect to $(\frak V_{\frak r},\mathcal S_{\frak r})$. This is independent of the choice of representative.
3. Let $g : N \to M$ be a smooth map between manifolds. We say that the map $f$ is [*strongly transversal to $g$ with respect to $(\mathcal U,\frak S)$*]{} if, for each $\frak r$, $f$ is strongly transversal to $g$ with respect to $(\frak V_{\frak r},\mathcal S_{\frak r})$. This is independent of the choice of representative.
4. We denote by $
\mathscr S_{\pitchfork 0}(\Omega)
$ the set of all $\frak S \in \mathscr S(\Omega)$ transversal to $0$, $
\mathscr S_{f \pitchfork}(\Omega)
$ the set of all $\frak S \in \mathscr S(\Omega)$ such that $f$ is strongly submersive with respect to $(\mathcal U,\frak S)$, and by $
\mathscr S_{f \pitchfork g}(\Omega)
$ the set of all $\frak S \in \mathscr S(\Omega)$ such that $f$ is strongly transversal to $g$ with respect to $(\mathcal U,\frak S)$. They are subsheaves of $\mathscr S$.
5. For a closed subset $K \subseteq U$ we define $
\mathscr S_{\pitchfork 0}(K)
$, $
\mathscr S_{f \pitchfork}(K)
$ and $
\mathscr S_{f \pitchfork g}(K)
$ in the same way as (\[sheafoverclosedset\]).
The statements (1), (2), and (3) follow from Lemma \[newlem79\]. (4) is a consequence of the definition.
Next we introduce the notion of [*member of a CF-perturbation*]{}, which is an analogue of the notion of branch of multisection.
\[memberonechart\] Let $\frak V_{\frak r}$ be an orbifold chart of $(U,\mathcal E)$ and $\mathcal S_{\frak r}$ a CF-perturbation of $(\mathcal U,\frak V_{\frak r})$. Let $x \in V_{\frak r}$. We take $\epsilon \in (0,1]$.
Consider the germ of a map $y \mapsto \frak s(y)$, $O_x \to E_{\frak r}$ (where $O_x$ is a neighborhood of $x \in V_{\frak r}$). We say $\frak s$ is a [*member of $\mathcal S^{\epsilon}_{\frak r}$ at $x$*]{} if there exists $\xi \in W_x$ such that the germ of $y \mapsto \frak s_{\frak r}^{\epsilon}(y,\xi)$ at $x$ is $\frak s$ and $\xi \in {\rm supp}\,\omega_x$. (See Definition \[defn73ss\] for the notation $\mathcal S^{\epsilon}_{\frak r}$.)
The member of $\mathcal S^{\epsilon}_{\frak r}$ at $x$ depends on $\epsilon$. In other words, we define the notion of the member of $\mathcal S^{\epsilon}$ for each $\epsilon \in (0,1]$.
\[lem721\] In the situation of Definition \[memberonechart\], let $\mathcal S'_{\frak r}$ be a CF-perturbation of $(\mathcal U,\frak V_{\frak r})$ that it is equivalent to $\mathcal S_{\frak r}$. Then $\frak s$ is a member of $\mathcal S^{\epsilon}_{\frak r}$ if and only if $\frak s$ is a member of $\mathcal S^{\prime\epsilon}_{\frak r}$.
It suffices to consider only the case when $\mathcal S'_{\frak r}$ is a projection of $\mathcal S_{\frak r}$. This also follows from the fact that $\omega_x = \vert\omega_x\vert{\rm vol}_x$ where ${\rm vol}_x$ is a volume form of $W_x$ and $\vert\omega_x\vert$ is a non negative function, which is a part of Definition \[defn73ss\] (4).
Therefore we can define an element of the stalk $\mathscr S_x$ at $x$ of the sheaf $\mathscr S$ of CF-perturbations.
We now recall from Definition \[defn73ss\] that we denoted by $
\mathcal S_x^\epsilon = (W_x, \omega_x,\frak s_x^\epsilon)
$ the restriction of the CF-perturbation $\mathcal S_x = (W_x, \omega_x,\{\frak s_x^\epsilon\})$ at $\epsilon$. Recall by definition that $\{\frak s_x^\epsilon\}$ is an $\epsilon$-dependent family of parameterized section, i.e., a map $\frak s_x^\epsilon: V_x \times W_x \to E_x$.
Let $\frak S \in \mathscr S(\Omega)$ and $x \in \Omega$. A [*member of $\frak S^{\epsilon}$ at $x$*]{} is a member of the germ of $\frak S^{\epsilon}$ at $x$.
Let $U$ be an orbifold and $K\subset U$ a compact subset and $\{U_{\frak r}\}$ a set of finitely many open subsets such that $\cup_{\frak r} U_{\frak r} \supset K$. A [*partition of unity*]{} subordinate to $\{U_{\frak r}\}$ on $K$ assigns a smooth function $\chi_{\frak r}$ on $U$ to each $\frak r$ such that:
1. ${\rm supp} \chi_{\frak r}\subset U_{\frak r}$.
2. $
\sum_{\frak r\in\frak R} \chi_{\frak r}
\equiv 1$ on a neighborhood of $K$.
It is standard and easy to prove that a partition of unity always exists on an orbifold.
Suppose we are in Situation \[smoothcorrsingle\]. Let $h$ be a smooth differential form of compact support in $U$. Let $\frak S \in \mathscr S_{f \pitchfork}({\rm Supp}(h))$. Let $\{\chi_{\frak r}\}$ be a smooth partition of unity subordinate to the covering $\{U_{\frak r}\}$ on ${\rm Supp}(h)$. We define the [*pushout of $h$ by $f$ with respect to $\frak S^{\epsilon}$*]{} by $$f!(h;\frak S^{\epsilon})
=
\sum_{\frak r\in\frak R}
f!(\chi_{\frak r}h;\mathcal S^{\epsilon}_{\frak r})$$ for each $\epsilon >0$. It is a smooth form on $M$ of degree $$\deg f!(h,\mathcal S^{\epsilon})
= \deg h + \dim M - \dim \mathcal U,$$ where $
\dim \mathcal U = \dim U - {\operatorname{rank}}\mathcal E.
$ The well-defined-ness follows from Lemma \[lem721\] (3).
We also call pushout [*integration along the fiber.*]{}
In general $f!(h;\frak S^{\epsilon})$ [*depends*]{} on $\epsilon$. Moreover $\lim_{\epsilon\to 0}f!(h;\frak S^{\epsilon})$ typically diverges.
\[lem721\]
1. $
f!(h_1+h_2;\frak S^{\epsilon})
=
f!(h_1;\frak S^{\epsilon})
+
f!(h_2;\frak S^{\epsilon})
$ and $
f!(ch;\frak S^{\epsilon}) = cf!(h;\frak S^{\epsilon})
$ for $c \in \R$.
2. Pushout of $h$ is independent of the choice of partition of unity.
3. If $\frak S^1$ is equivalent to $\frak S^2$ then $$\label{icchiprojectint}
f!(h;\frak S^{1,\epsilon})
=
f!(h;\frak S^{2,\epsilon}).$$
\(1) is obvious as far as we use the same partition of unity on both sides. We will prove (2) and (3) at the same time. We take a partition of unity $\{\chi^i_{\frak r} \mid \frak r \in \frak R_i\}$ subordinate to $\frak S^i$ for $i=1,2$ and will prove (\[icchiprojectint\]). Here we use those partitions of unity to define the left and right hand sides of (\[icchiprojectint\]), respectively. The case $\frak S^1 = \frak S^2$ will be (2).
We put $h_0 = \chi_{\frak r_0}\widehat h$. In view of (1) it suffices to prove $$\label{foru735}
f!(h_0;\mathcal S^{1,\epsilon}_{\frak r_0})
=
\sum_{\frak r \in \frak R_2}
f!(\chi^2_{\frak r}h_0;\mathcal S^{2,\epsilon}_{\frak r}).$$ To prove (\[foru735\]), it suffices to show the next equality for each $\frak r \in \frak R_2$. $$\label{foru74}
f!(\chi^2_{\frak r}h_0;\mathcal S^{1,\epsilon}_{\frak r_0})
=
f!(\chi^2_{\frak r}h_0;\mathcal S^{2,\epsilon}_{\frak r}).$$ We put $h_1 = \chi^2_{\frak r}h_0$. Let $K = {\rm Supp}(h_1)$. For each $x \in K$ there exists $\frak V_x$ such that the restriction of $\mathcal S^{1,\epsilon}_{\frak r_0}$ to $U_{x}$ is equivalent to the restriction of $\mathcal S^{2,\epsilon}_{\frak r}$ to $U_x$. We cover $K$ by a finitely many such $U_{x_i}$, $i=1,\dots,N$. Let $\{\chi'_i \mid i=1,\dots,N\}$ be a partition of unity on $K$ subordinate to the covering $\{U_{x_i}\}$. Then we obtain: $$\aligned
f!(\chi^2_{\frak r}h_0;\mathcal S^{1,\epsilon}_{\frak r_0})
&=
\sum_{i=1}^N
f!(\chi'_{i}h_1;\mathcal S^{1,\epsilon}_{\frak r_0}\vert_{U_{x_i}})\\
&=
\sum_{i=1}^N
f\vert_{U_{x_i}}!(\chi'_{i}h_1;\mathcal S^{2,\epsilon}_{\frak r}\vert_{U_{x_i}})
=
f!(\chi^2_{\frak r}h_0 ;\mathcal S^{2,\epsilon}_{\frak r}).
\endaligned$$ The proof of Lemma \[lem721\] is now complete.
CF-perturbations of good coordinate system {#subsec:conpergcs}
------------------------------------------
To consider a CF-perturbation of good coordinate system, we first study the pullback of a CF-perturbation by an embedding of Kuranishi charts in this subsubsection. Using the pullback, we then define the notion of a CF-perturbation of good coordinate system.
### Embedding of Kuranishi charts and CF-perturbations {#subsub:embcfp}
To highlight the dependence on the Kuranishi structure, we write $\mathscr S^{\mathcal U^1}$, $\mathscr S^{\mathcal U^2}$ etc. in place of $\mathscr S$. Namely $\mathscr S^{\mathcal U^2}(\Omega)$ is the set of all continuous family perturbations of $\mathcal U^2\vert_{\Omega}$.
\[contfamipullbacksitu\] Let $\mathcal U^i = (U^i,\mathcal E^i,s^i,\psi^i)$ $(i=1,2)$ be Kuranishi charts and $\Phi_{21} = (\varphi_{21},\widehat\varphi_{21}): \mathcal U^1 \to \mathcal U^2$ an embedding of Kuranishi charts. Let $\frak S^2 \in \mathscr{S}^{\mathcal U^2}(U^2)$ and let $\{\frak S^2_{\frak r}\} = \{ (\frak V^2_{\frak r},\mathcal S^2_{\frak r})\}$ be its representative.$\blacksquare$
We will define the pullback $\Phi_{21}^*\frak S^2 \in \mathscr{S}
^{\mathcal U^1}(U_1)$. We need certain conditions for this pullback to be defined.
\[exitpullbackcont1\] Suppose we are in Situation \[contfamipullbacksitu\]. We require that there exists an orbifold chart $\frak V^1_{\frak r} = (V^1_{\frak r},\Gamma^1_{\frak r},E^1_{\frak r},\phi^1_{\frak r},\widehat\phi^1_{\frak r})$ such that $\phi^1_{\frak r}(V^1_{\frak r}) = \varphi_{21}^{-1}(U_{\frak r}^2)$.
(Recall $U_{\frak r}^2 = \phi_{\frak r}^2(V_{\frak r}^2)$ and $\frak V^2_{\frak r} = (V^2_{\frak r},\Gamma^2_{\frak r},E^2_{\frak r},\phi^2_{\frak r},\widehat\phi^2_{\frak r})$.)
For any given $\{ (\frak V^2_{\frak r},\mathcal S^2_{\frak r})\}$ there exists $\{ (\frak V^{2 \prime}_{\frak r},\mathcal S^{2 \prime}_{\frak r})\}$ which is equivalent to $\{ (\frak V^2_{\frak r},\mathcal S^2_{\frak r})\}$ and satisfies Condition \[exitpullbackcont1\].
For each $x \in U_2$ there exists $\frak V_x$ such that $\varphi_{21}^{-1}(U_{x}^2)$ has an orbifold chart and $\frak V_x \subset \frak V_{\frak r}$ for some $\frak r$. We cover $U_2$ by such $\frak V_x$ to obtain the required $\{ (\frak V^{2 \prime}_{\frak r},\mathcal S^{2 \prime}_{\frak r})\}$.
Therefore we may assume Condition \[exitpullbackcont1\]. Then we can represent the orbifold embedding $(\varphi_{21},\widehat\varphi_{21}) : (U^1,\mathcal E^1) \to (U^2,\mathcal E^2)$ in terms of the orbifold charts $\frak V^1_{\frak r}$, $\frak V^2_{\frak r}$ by $(h^{\frak r}_{21},\tilde\varphi^{\frak r}_{21},\breve\varphi^{\frak r}_{21})$ that have the following properties.
1. $h^{\frak r}_{21} : \Gamma_1^{\frak r} \to \Gamma_2^{\frak r}$ is an injective group homomorphism.
2. $\tilde\varphi^{\frak r}_{21} : V_1^{\frak r} \to V_2^{\frak r}$ is an $h^{\frak r}_{21}$-equivariant smooth embedding of manifolds.
3. $h^{\frak r}_{21}$ and $\tilde\varphi^{\frak r}_{21}$ induce an orbifold embedding $$\left(\overline{\phi_2^{\frak r}}\right)^{-1} \circ \varphi_{21} \circ \overline{\phi^{\frak r}_{1}} : V^1_{\frak r}/\Gamma^1_{\frak r}
\to V^2_{\frak r}/\Gamma^2_{\frak r}.$$
4. $\breve\varphi^{\frak r}_{21} : V_1^{\frak r} \times E_1^{\frak r} \to E_2^{\frak r}$ is an $h^{\frak r}_{21}$-equivariant smooth map such that for each $y\in V_1^{\frak r}$ the map $v \mapsto \breve\varphi^{\frak r}_{21}(y,v)$ is a linear embedding $E_1^{\frak r} \to E_2^{\frak r}$.
5. $\breve\varphi^{\frak r}_{21}$ induces a smooth embedding of vector bundles: $$\left(\overline{\widehat\phi^{\frak r}_{2}}\right)^{-1} \circ \widehat\varphi_{21} \circ \overline{\widehat\phi^{\frak r}_{1}} : (V^1_{\frak r} \times E^1_{\frak r})/\Gamma^1_{\frak r}
\to(V^2_{\frak r} \times E^2_{\frak r})/\Gamma^2_{\frak r}.$$ In other words for each $(y,v) \in V^1_{\frak r} \times E^1_{\frak r}$ we have $$\widehat\varphi_{21}(\widehat\phi^{\frak r}_{1}(y,v))
= \widehat\phi^{\frak r}_{2}(\tilde\varphi^{\frak r}_{21}(y),\breve\varphi^{\frak r}_{21}(y,v)).$$
See Lemma \[lem2622\].
The map $(h^{\frak r}_{21},\tilde\varphi^{\frak r}_{21},\breve\varphi^{\frak r}_{21})$ satisfying (1)-(5) above is not unique.
\[exitpullbackcont2\] We consider Situation \[contfamipullbacksitu\] and assume Condition \[exitpullbackcont1\]. We take $(h^{\frak r}_{21},\tilde\varphi^{\frak r}_{21},\breve\varphi^{\frak r}_{21})$ which satisfies Property \[proper728\]. Let $\frak S^2_{\frak r} = (W^2_{\frak r},\omega^2_{\frak r},\{\frak s^{2,\epsilon}_{\frak r}\})$. Then for each $y \in V^1_{\frak r}$, $\xi \in W^2_{\frak r}$, we require $$\label{s2haimage}
{\frak s}^{2,\epsilon}_{\frak r}(y,\xi) \in {\rm Im}(g_y)$$ where $g_y : E^1_{\frak r} \to E^2_{\frak r}$ is defined by $$\label{defofgy}
\breve\varphi^{\frak r}_{21}(y,v) = g_y(v).$$
In Situation \[contfamipullbacksitu\] we say that $\{ (\frak V^2_{\frak r},\mathcal S^2_{\frak r})\}$ [*can be pulled back to $\mathcal U^1$*]{} by $\Phi_{21}$ if and only if Conditions \[exitpullbackcont1\] and \[exitpullbackcont2\] are satisfied.
The [*pullback*]{} $\Phi_{21}^*\{ (\frak V^2_{\frak r},\mathcal S^2_{\frak r})\}$ of $\{ (\frak V^2_{\frak r},\mathcal S^2_{\frak r})\}$ is by definition $\{ (\frak V^1_{\frak r},\mathcal S^1_{\frak r}) \mid \frak r \in \frak R_0\}$ which we define below.
1. $\frak R_0$ is the set of all $\frak r$ such that $U_{\frak r}^2 \cap \varphi_{21}(U_{21}) \ne \emptyset$.
2. $\frak V^1_{\frak r}$ then is given by Condition \[exitpullbackcont1\].
3. $\mathcal S^1_{\frak r} = (W^2_{\frak r},\omega^2_{\frak r},\{s^{1,\epsilon}_{\frak r}\})$, where $\frak s^{1,\epsilon}_{\frak r}$ is defined by $$\label{formula718}
\frak s^{1,\epsilon}_{\frak r}(y,\xi) = g_y^{-1}(\frak s^{2,\epsilon}_{\frak r}(y,\xi)).$$ Here $g_y$ is as in (\[defofgy\]). The right hand side exists because of (\[s2haimage\]).
It is easy to see that $\{ (\frak V^1_{\frak r},\mathcal S^1_{\frak r}) \mid
\frak r \in \frak R_0\}$ is a representative of a continuous family perturbation of $\mathcal U^1$.
\[lem7414\]
1. If $\{ (\frak V^2_{\frak r},\mathcal S^2_{\frak r})\}$ can be pulled back to $\mathcal U^1$ by $\Phi_{21}$ and $\{ (\frak V^{\prime 2}_{\frak r'},\mathcal S^{\prime 2}_{\frak r'})\}$ is equivalent to $\{ (\frak V^2_{\frak r},\mathcal S^2_{\frak r})\}$, then $\{ (\frak V^{\prime 2}_{\frak r'},\mathcal S^{\prime 2}_{\frak r'})\}$ can be pulled back to $\mathcal U^1$ by $\Phi_{21}$. Moreover $\Phi^*_{21}\{ (\frak V^2_{\frak r},\mathcal S^2_{\frak r})\}$ is equivalent to $\Phi^*_{21}\{ (\frak V^{\prime 2}_{\frak r'},\mathcal S^{\prime 2}_{\frak r'})\}$.
2. The pullback $\Phi^*_{21}\{ (\frak V^2_{\frak r},\mathcal S^2_{\frak r})\}$ is independent of the choice of $(h^{\frak r}_{21},\varphi^{\frak r}_{21},\widehat\varphi^{\frak r}_{21})$ up to equivalence.
To prove (1) it suffices to consider only the case when $\mathcal S'_{\frak r}$ is a projection of $\mathcal S_{\frak r}$, which follows again from the fact that $\omega_x = \vert\omega_x\vert{\rm vol}_x$ where ${\rm vol}_x$ is a volume form of $W_x$ and $\vert\omega_x\vert$ is a non-negative function, which is a part of Definition \[defn73ss\] (4). The assertion (2) follows from Lemma \[lem2715\].
\[deflem743\] Suppose we are in Situation \[contfamipullbacksitu\].
1. We denote by $\mathscr S^{\mathcal U^1\triangleright \mathcal U^2}(U^2)$ the set of all elements of $\mathscr S^{\mathcal U^2}(U^2)$ whose representative can be pulled back to $\mathcal U^1$. This is well-defined by Lemma \[lem7414\] (1).
2. The pullback $$\Phi^*_{21} : \mathscr S^{\mathcal U^1\triangleright \mathcal U^2}(U^2) \to
\mathscr S^{\mathcal U^1}(U^1)$$ is defined by Lemma \[lem7414\] (2).
3. $\Omega \mapsto \mathscr S^{\mathcal U^1\triangleright \mathcal U^2}(\Omega)$ is a subsheaf of $\mathscr S^{\mathcal U^2}$.
4. The restriction map $\Phi^*_{21}$ is induced by a sheaf morphism: $$\Phi^{*}_{21} : \varphi^{\star}_{21}\mathscr S^{\mathcal U^1\triangleright \mathcal U^2} \to \mathscr S^{\mathcal U^1}.$$ Here the left hand side is the pullback sheaf. (In this document, we use $\star$ to denote the pullback sheaf to distinguish it from pullback map.)
In other words, the following diagram commutes for any open sets $\Omega, \Omega'$. $$\begin{CD}
\mathscr S^{\mathcal U^1\triangleright \mathcal U^2}(\Omega) @ > {\frak i_{\Omega'\Omega}} >>
\mathscr S^{\mathcal U^1\triangleright \mathcal U^2}(\Omega') \\
@ V{\Phi^*_{21}}VV
@ VV{\Phi^*_{21}}V\\
\mathscr S^{\mathcal U^1}(U^1 \cap \Omega) @ >>{\frak i_{\Omega'\cap U^1 \Omega\cap U^1 }}>
\mathscr S^{\mathcal U^1}(U^1 \cap \Omega')
\end{CD}$$
We can check the assertion directly. So we omit the proof.
Let $\Phi_{i+1 i} : \mathcal U^i \to \mathcal U^{i+1}$ ($i=1,2$) be embeddings of Kuranishi charts. We put $\Phi_{31} = \Phi_{32}\circ \Phi_{21}$.
1. We have $$(\Phi^{*}_{32})^{-1}(\mathscr S^{\mathcal U^1\triangleright \mathcal U^2})
\cap \mathscr S^{\mathcal U^2\triangleright \mathcal U^3}
\subseteq
\mathscr S^{\mathcal U^1\triangleright \mathcal U^3}$$ as subsheaves of $\mathscr S^{\mathcal U^3}$.
2. The next diagram commutes. $$\begin{CD}
\Phi^{\star}_{31}((\Phi^{*}_{32})^{-1}(\mathscr S^{\mathcal U^1\triangleright \mathcal U^2})
\cap \mathscr S^{\mathcal U^2\triangleright \mathcal U^3}) @ > {\Phi^*_{32}} >>
\Phi^{\star}_{21}(\mathscr S^{\mathcal U^1\triangleright \mathcal U^2}) \\
@ V{}VV
@ VV{\Phi^*_{21}}V\\
\Phi^{\star}_{31}\mathscr S^{\mathcal U^1\triangleright \mathcal U^3} @ >>{\Phi^*_{31}}>
\mathscr S^{\mathcal U^1}
\end{CD}$$ In other words, the next diagram commutes for each $\Omega \subset U^3$. $$\begin{CD}
(\Phi^{*}_{32})^{-1}\mathscr S^{\mathcal U^1\triangleright \mathcal U^2}(\Omega \cap U^2) \cap
\mathscr S^{\mathcal U^2\triangleright \mathcal U^3}(\Omega)
@ > {\Phi^*_{32}} >>
\mathscr S^{\mathcal U^1\triangleright \mathcal U^2}(\Omega \cap U^2) \\
@ V{}VV
@ VV{\Phi^*_{21}}V\\
\mathscr S^{\mathcal U^1\triangleright \mathcal U^3}
(\Omega) @ >>{\Phi^*_{31}}>
\mathscr S^{\mathcal U^1}(\Omega \cap U^1)
\end{CD}$$
This is a consequence of (\[formula718\]).
Next we generalize Lemma \[lem6767\] to the case of continuous families.
\[lem6767cont\] Suppose $\widetriangle{\mathcal U}$ and $\mathcal K$ satisfy Condition \[conds6.17\]. Then there exists $c>0$, $\delta_0>0$ and $\epsilon_0 >0$ such that for each $\epsilon < \epsilon_0$ and member $\frak s^{\epsilon}$ of $\frak S^{\epsilon}_{\frak p}$ at $y = {\rm Exp}(x,v) \in BN_{\delta_0}(K;U_{\frak p})$, $x \in \mathcal K_q$, we have $$\label{normailityestimate4}
\vert \frak s^{\epsilon}(y) \vert \ge c \vert v\vert$$
The proof is the same as the proof of Lemma \[lem6767\].
The constants $\epsilon_0$, $c$, $\delta_0$ can be taken to be independent of the choice of representative of CF-perturbations. In fact, the notion of member is independent of the choice of representatives of CF-perturbation.
### CF-perturbations on good coordinate system {#subsub:goodcsyscfp}
\[defn7732\] Let ${\widetriangle{\mathcal U}} = \{\mathcal U_{\frak p} \mid \frak p \in \frak P\}$ be a good coordinate system of $Z \subseteq X$. A [*CF-perturbation of $({\widetriangle{\mathcal U}},\mathcal K)$*]{} is by definition $\widetriangle{\frak S} = \{\frak S_{\frak p} \mid \frak p \in \frak P\}$ with the following properties.
1. $\frak S_{\frak p}
\in \mathscr S^{\mathcal U_{\frak p}}(\mathcal K_{\frak p})$.
2. If $\frak q \le \frak p$ then $\frak S_{\frak p} \in
\mathscr S^{\mathcal U^{\frak q}\triangleright \mathcal U^{\frak p}}(\mathcal K_{\frak p})$.
3. The pull back $\Phi_{\frak p\frak q}^*(\frak S_{\frak p})$ is equivalent to $\frak S_{\frak q}$ as an element of $\mathscr S^{\mathcal U_{\frak q}}(\varphi_{\frak p\frak q}^{-1}(\mathcal K_{\frak p}) \cap \mathcal K_{\frak q})$.
\[smoothfunctiononvertK\] Let $\widetriangle{\frak S}= \{\frak S_{\frak p} \mid \frak p \in \frak P\}$ be a CF-perturbation of a good coordinate system ${\widetriangle{\mathcal U}}$ and $\mathcal K$ its support system.
1. We say $\widetriangle{\frak S}$ is [*transversal to 0*]{} if each of $\frak S_{\frak p}$ is transversal to $0$.
2. Let $\widetriangle f : (X,Z;{\widetriangle{\mathcal U}}) \to M$ be a strongly smooth map that is weakly submersive. We say that $\widetriangle f$ is [*strongly submersive with respect to $\widetriangle{\frak S}$ on $\mathcal K$*]{} if for each $\frak p \in \frak P$ the map $f_{\frak p}$ is strongly submersive with respect to $\frak S_{\frak p}$ on $\mathcal K_{\frak p}$ in the sense of Definition \[submersivepertconlocloc\].
3. Let $g : N \to M$ be a smooth map between smooth manifolds. We say that $\widetriangle f$ is [*strongly transversal to $g$ with respect to $\widetriangle{\frak S}$ on $\mathcal K$*]{} if for each $\frak p \in \frak P$ the map $f_{\frak p}$ is strongly transversal to $g$ with respect to $\frak S_{\frak p}$ on $\mathcal K_{\frak p}$.
\[existperturbcont\] Let ${\widetriangle{\mathcal U}}$ be a good coordinate system of $Z \subseteq X$ and $\mathcal K$ its support system.
1. There exists a CF-perturbation $\widetriangle{\frak S}$ of $({\widetriangle{\mathcal U}},\mathcal K)$ transversal to $0$.
2. If $\widetriangle f : (X,Z;{\widetriangle{\mathcal U}}) \to M$ is a weakly submersive strongly smooth map, then we may take $\widetriangle{\frak S}$ with respect to which $\widetriangle f$ is strongly submersive.
3. If $\widetriangle f : (X,Z;{\widetriangle{\mathcal U}}) \to M$ is a strongly smooth map which is weakly transversal to $g : N \to M$, then we may take $\widetriangle{\frak S}$ with respect to which $\widetriangle f$ is strongly transversal to $g$.
The proof of Theorem \[existperturbcont\] is given in Subsection \[subsec:cfpgoodcsys\].
For a later use we include its relative version, Proposition \[prop7582752\] and Lemma \[lem753753\]. To state this relative version we need some digression.
\[defn735f\] Let $X$ be a separable metrizable space and $Z_1, Z_2 \subseteq X$ compact subsets. We assume $Z_1 \subset \ring{Z_2}$.
1. For each $i=1,2$, let ${\widetriangle{\mathcal U^i}}
= (\frak P_i,\{\mathcal U^i_{\frak p}\},\{\Phi^i_{\frak p\frak q}\})$ be a good coordinate system of $Z_i \subseteq X$. We say $\widetriangle{\mathcal U^2}$ [*strictly extends*]{} $\widetriangle{\mathcal U^1}$ if the following holds.
1. $
\frak P_1 = \{ \frak p \in \frak P_2 \mid
\psi_{\frak p}((s_{\frak p})^{-1}(0)) \cap Z_1 \ne \emptyset
\}.
$ The partial order of $\frak P_1$ is the restriction of one of $\frak P_2$.
2. If $\frak p \in \frak P_1$, then $\mathcal U^1_{\frak p}$ is an open subchart of $\mathcal U^2_{\frak p}$. Moreover ${\rm Im} (\psi^1_{\frak p}) \cap Z_1 = {\rm Im} (\psi^2_{\frak p}) \cap Z_1$.
3. If $\frak p, \frak q \in \frak P_1$, then $\Phi_{\frak p\frak q}^1$ is a restriction of $\Phi^2_{\frak p\frak q}$.
Note the case $Z_2 = X$ is included in this definition.
2. In the situation of (a), we say ${\widetriangle{\mathcal U}^2}$ [*extends*]{} ${\widetriangle{\mathcal U^1}}$ (resp. [*weakly extends*]{}) if it strictly extends an open substructure (resp. weakly open substructure) of ${\widetriangle{\mathcal U^1}}$.
3. Let $\widehat{\mathcal U^2}
= (\{\mathcal U_p^2 \mid p \in Z_2\},\{\Phi_{pq}^2 \mid q \in {\rm Im}(\psi_{p}), \, p,q \in Z_2\})$ be a Kuranish structure of $Z_2 \subset X$. Then $(\{\mathcal U_p^2 \mid p \in Z_1\},\{\Phi_{pq}^2 \mid q \in {\rm Im}(\psi_{p}), \, p,q \in Z_1\})$ is a Kuranishi structure of $Z_1 \subseteq X$. We call it the [*restriction*]{} of $\widehat{\mathcal U^2}$ and write $\widehat{\mathcal U^2}\vert_{Z_1}$.
For each $i=1,2$ let ${\widetriangle{\mathcal U^i}}
= (\frak P_i,\{\mathcal U^i_{\frak p}\},\{\Phi^i_{\frak p\frak q}\})$ be a good coordinate system of $Z_i \subseteq X$, and $\widehat{\mathcal U^2}$ a Kuranishi structure of $Z_2 \subseteq X$ such that $\widetriangle{\mathcal U^2}$ is compatible with $\widehat{\mathcal U^2}$. Suppose that $Z_1 \subset \ring{Z_2}$ and $\widetriangle{\mathcal U^2}$ strictly extends $\widetriangle{\mathcal U^1}$.
Then there exists a KG-embedding $\widehat{\mathcal U^2}\vert_{Z_1} \to \widetriangle{\mathcal U^1}$ with the following property. Let $p \in Z_1$ and $p \in {\rm Im}(\psi^1_{\frak p})$. Let $\Phi_{\frak p p}^1 : \mathcal U^2_p\vert_{U^1_{p}}
\to \mathcal U^1_{\frak p}$ be a part of the KG-embedding $\widehat{\mathcal U^2}\vert_{Z_1} \to \widetriangle{\mathcal U^1}$. (Here $\mathcal U^2_p\vert_{U^1_{p}}$ is an open subchart of $\mathcal U^2_p$.) Let $\Phi_{\frak p p}^2 : \mathcal U^2_p \to \mathcal U^2_{\frak p}$ be a part of the KG-embedding $: \widehat{\mathcal U^2} \to \widetriangle{\mathcal U^2}$. The next diagram commutes: $$\label{form726}
\begin{CD}
\mathcal U^2_p\vert_{U^1_{p}}
@ >>>
\mathcal U^2_p \\
@ V{\Phi_{\frak p p}^1}VV
@ VV{\Phi_{\frak p p}^2}V\\
\mathcal U^1_{\frak p}
@ >>>
\mathcal U^2_{\frak p}
\end{CD}$$ Note that the horizontal arrows are embeddings as open substructures.
Moreover, the KG embedding $\widehat{\mathcal U^2}\vert_{Z_1} \to \widetriangle{\mathcal U^1}$ satisfying this property is unique up to equivalence in the sense of Definition \[embedkuraequiv\].
The same conclusion holds for extension or weak extension.
Let $p \in Z_1$ and $p \in {\rm Im}(\psi^1_{\frak p})$. By Definition \[defn735f\] (1)(b), $\mathcal U^1_{\frak p}$ is an open subchart of $\mathcal U^2_{\frak p}$. In particular, $U^1_{\frak p}$ is an open subset of $U^2_{\frak p}$. Let $\Phi_p^2 : \mathcal U^2_p \to \mathcal U^2_{\frak p}$ be a part of the KG embedding$: \widehat{\mathcal U^2} \to \widetriangle{\mathcal U^2}$. We put $$U^1_{p} = (\varphi^2_p)^{-1}(U^1_{\frak p}) \subseteq U^2_{p}.$$ By Lemma \[lem321321\], there exists a Kuranishi structure of $Z_1 \subseteq X$ whose Kuranishi chart is $\mathcal U^2_p\vert_{U^1_{p}}$. We can define $\Phi_p^1$ by restricting $\Phi_p^2$. The commutativity of Diagram (\[form726\]) is obvious.
We recall Definition \[gcsystemcompa\] for the compatibility used in the next proposition.
\[prop7582752\] Let $X$ be a separable metrizable space, $Z_1, Z_2 \subseteq X$ be compact subsets. We assume $Z_1 \subset \ring{Z_2}$. Let $\widehat{\mathcal U^2}$ be a Kuranishi structure of $Z_2 \subseteq X$ and $\widetriangle{\mathcal U^1}$ a good coordinate system of $Z_1 \subseteq X$ which is compatible with $\widehat{\mathcal U^2}\vert_{Z_1}$. Then there exists a good coordinate system $\widetriangle{\mathcal U^2}$ of $Z_2 \subseteq X$ such that
1. $\widetriangle{\mathcal U^2}$ extends $\widetriangle{\mathcal U^1}$.
2. $\widetriangle{\mathcal U^2}$ is compatible with $\widehat{\mathcal U^2}$.
3. Diagram (\[form726\]) commutes.
The proof is given in Subsection \[subsec:moreversionegcs2\].
\[lem753753\] Suppose we are in the situation of Proposition \[prop7582752\]. We may choose $\widetriangle{\mathcal U^2}$ such that the following holds.
Let $\widetriangle{\mathcal U^1_0}$ be an open substructure of $\widetriangle{\mathcal U^1}$ strictly extended to $\widetriangle{\mathcal U^2}$. Let $\widetriangle {f^1} = \{f^1_{\frak p}\} : (X,Z_1;\widetriangle{\mathcal U^1_0}) \to M$ and $\widehat {f^2} = \{f^2_{p}\} : (X,Z_2;\widehat{\mathcal U^2}) \to M$ be strongly continuous maps. Assume that the equality $
f^1_{\frak p} \circ \varphi^1_{\frak p p} = f^2_{p}
$ holds on $U^1_{p} \subset U^2_p$ for each $p \in {\rm Im}\,(\psi_{\frak p}) \cap Z_1$. Then the following hold:
1. There exists a strongly continuous map $\widetriangle {f^2} = \{f^2_{\frak p}\} : (X,Z_2;\widetriangle{\mathcal U^2}) \to M$ such that:
1. $
f^2_{\frak p} \circ \varphi^2_{\frak p p} = f^2_{p}
$ for $p \in {\rm Im}\,(\psi_{\frak p}) \cap Z_2$.
2. $f^2_{\frak p} = f^1_{0,\frak p}$ on $U^1_{\frak p} \subset U^2_{\frak p}$.
$$\xymatrix{
& & & M \\
\widetriangle {\mathcal U^1_0} \ar[urrr]^{\widetriangle f^1}
\ar[r] & \widetriangle {\mathcal U^2} \ar@{.>}[urr]|{\widetriangle f^2} && \\
\widehat {\mathcal U^2}\vert_{Z_1} \ar[u]\ar[r] &\widehat {\mathcal U^2}
\ar[u]^{\{f^2_{\frak p p}\}} \ar[uurr]_{\widehat f^2} &&
}$$
2. If $\widetriangle {f^1}$ and $\widehat {f^2}$ are strongly smooth (resp. weakly submersive) then so is $\widetriangle {f^2}$.
3. If $\widetriangle {f^1}$ and $\widehat {f^2}$ are strongly transversal to $g : N \to M$ then so is $\widetriangle {f^2}$.
The proof is given in Subsection \[subsec:moreversionegcs2\].
\[defn738\] Let ${\widetriangle{\mathcal U^i}}
= (\frak P_i,\{\mathcal U^i_{\frak p}\},\{\Phi^i_{\frak p\frak q}\})$ be good coordinate systems of $Z_i \subseteq X$, $i=1,2$ and $\mathcal K^i$ support systems of ${\widetriangle{\mathcal U^i}}$ for $i=1,2$.
1. Suppose $\widetriangle{\mathcal U^2}$ strictly extends $\widetriangle{\mathcal U^1}$. We say that $\mathcal K^1$ is [*compatible with*]{} $\mathcal K^2$ if for each $\frak p \in \frak P_1$ we have $$\mathcal K^1_{\frak p} \subset \ring{\mathcal K^2_{\frak p}}.$$ (We note that $U^1_{\frak p} \subset U^2_{\frak p}$ by Definition \[defn735f\] (1)(c).) In this situation we say $(\widetriangle{\mathcal U^2},\mathcal K^2)$ [*strictly extends*]{} $(\widetriangle{\mathcal U^1},\mathcal K^1)$.
2. Suppose $\widetriangle{\mathcal U^2}$ extends $\widetriangle{\mathcal U^1}$. We say that $\mathcal K^1$ is [*compatible with*]{} $\mathcal K^2$ if the following holds.
By definition there exists an open substructure $\widetriangle{\mathcal U^1_0} = \{\mathcal U^1_{\frak p,0}\}$ of $\widetriangle{\mathcal U^1}$ such that $\widetriangle{\mathcal U^2}$ strictly extends $\widetriangle{\mathcal U^1_0}$. For each $\frak p \in \frak P_1$ we require $$\mathcal K^1_{\frak p} \subset U^1_{\frak p,0} \cap \ring{\mathcal K^2_{\frak p}}.$$ In this situation we say $(\widetriangle{\mathcal U^2},\mathcal K^2)$ [*extends*]{} $(\widetriangle{\mathcal U^1},\mathcal K^1)$.
\[defn738pert\] For each $i=1,2$ let ${\widetriangle{\mathcal U^i}}
= (\frak P_i,\{\mathcal U^i_{\frak p}\},\{\Phi^i_{\frak p\frak q}\})$ be a good coordinate system of $Z_i \subseteq X$, $\mathcal K^i$ a support system of ${\widetriangle{\mathcal U^i}}$, and $\widetriangle{\frak S^i}$ a CF-perturbation of $({\widetriangle{\mathcal U^i}},\mathcal K^i)$.
1. Suppose $(\widetriangle{\mathcal U^2},\mathcal K^2)$ strictly extends $(\widetriangle{\mathcal U^1},\mathcal K^1)$. We say $\widetriangle{\frak S^2}$ [*strictly extends*]{} $\widetriangle{\frak S^1}$ if the restriction of ${\frak S^2_{\frak p}}$ to $\mathcal K^1_{\frak p}$ is ${\frak S^1_{\frak p}}$ for each $\frak p \in \frak P_1$.
2. Suppose $(\widetriangle{\mathcal U^2},\mathcal K^2)$ extends $(\widetriangle{\mathcal U^1},\mathcal K^1)$. We say $\widetriangle{\frak S^2}$ [*extends*]{} $\widetriangle{\frak S^1}$ if the restriction of ${\frak S^2_{\frak p}}$ to $\mathcal K^1_{\frak p}$ is $\frak S^1_{\frak p}$ for each $\frak p \in \frak P_1$.
In Definition \[defn738pert\] (2) we note that $\mathcal K_{\frak p}^1$ can be regarded as a support system of an open substructure of $\widetriangle{\mathcal U^1}$ by Definition \[defn738\] (2).
\[existperturbcontrel\] Suppose we are in the situation of Proposition \[prop7582752\]. We may choose $\widetriangle{\mathcal U^2}$ such that the following holds.
Let $\widetriangle{\mathcal U^1_0}$ be an open substructure of $\widetriangle{\mathcal U^1}$ strictly extended to $\widetriangle{\mathcal U^2}$. Let $\mathcal K^1$, $\mathcal K^2$ be support systems of ${\widetriangle{\mathcal U^1}}$, ${\widetriangle{\mathcal U^2}}$ respectively such that $(\widetriangle{\mathcal U^2},\mathcal K^2)$ extends $(\widetriangle{\mathcal U^1},\mathcal K^1)$. Let $\widetriangle{\frak S^1}$ be a CF-perturbation of $({\widetriangle{\mathcal U^1}},\mathcal K^1)$. Then there exists a CF-perturbation $\widetriangle{\frak S^2}$ of $({\widetriangle{\mathcal U^2}},\mathcal K^2)$ which extends $\widetriangle{\frak S^1}$. Moreover the following holds.
1. If $\widetriangle{\frak S^1}$ is transversal to $0$, so is $\widetriangle{\frak S^2}$.
2. Suppose we are in the situation of Lemma \[lem753753\] (1) in addition. We assume $\widetriangle {f^1}$ is strongly submersive with respect to $\widetriangle{\frak S^1}$ and $\widetriangle {f^2}$ is weakly submersive. Then $\widetriangle {f^2}$ is strongly submersive with respect to $\widetriangle{\frak S^2}$.
3. Suppose we are in the situation of Lemma \[lem753753\] (1) in addition. We assume $\widetriangle {f^1}$ is strongly transversal to $g : N \to M$ with respect to $\widetriangle{\frak S^1}$ and $\widetriangle {f^2}$ is weakly transversal to $g$. Then $\widetriangle {f^2}$ is strongly transversal to $g : N \to M$ with respect to $\widetriangle{\frak S^2}$.
The proof of Proposition \[existperturbcontrel\] is given in Subsection \[subsec:cfpgoodcsys\].
Before we end this subsection, we consider a family of CF-perturbations. The next definition is a CF-perturbation version of Definition \[uniformmulivalupert\].
\[uniformcongpert\] A $\sigma$-parameterized family of CF-perturbations $\{ \widetriangle{\frak S_{\sigma}} \mid \sigma \in \mathscr A\}$ ($\widetriangle{\frak S_{\sigma}}
= \{\mathcal S^{\epsilon}_{\sigma,\frak p}\}$) of $(\widetriangle{\mathcal U},\mathcal K)$ is called a [*uniform family*]{} if the convergence in Definition \[defn73ss\] (3) is uniform. More precisely, we require the following.
For each $\frak o>0$ there exists $\epsilon_0(\frak o)>0$ such that if $0< \epsilon < \epsilon_0(\frak o)$, then $$\vert \frak s(y) - s_p(y) \vert < \frak o,
\qquad
\vert (D\frak s)(y) - (Ds_p)(y) \vert < \frak o$$ hold for any $\frak s$ which is a member of $\mathcal S^{\epsilon}_{\sigma,\frak p}$ at any point $y \in \mathcal K_{\frak p}$ for any $\frak p \in \frak P$ and $\sigma \in \mathscr A$.
\[lemma748\] If $\{ \widetriangle{\frak S_{\sigma}} \mid \sigma \in \mathscr A\}$ is a uniform family of CF-perturbations of $(\widetriangle{\mathcal U},\mathcal K)$, then the constant $\epsilon_0$ in Lemma \[lem6767cont\] can be taken independent of $\sigma$.
In the same way as Lemma \[lem618\] follows from the proof of Lemma \[lem611\], this follows from the proof of Lemma \[lem6767cont\], which is the same as the proof of Lemma \[lem6767\].
Partition of unity associated to a good coordinate system {#subsec:partitionunigcs}
---------------------------------------------------------
We next define the notion of partition of unity on spaces with good coordinate system.
\[situ5757\] Let $\widetriangle{\mathcal U}=(\frak P, \{\mathcal U_{\frak p} \},\{ \Phi_{\frak p \frak q}\})$ be a good coordinate system of $Z \subseteq X$ and $\mathcal K$ its support system.$\blacksquare$
\[def758\] In Situation \[situ5757\], let $\Omega$ be an open subset of $\vert{\mathcal K}\vert$ and $f : \Omega \to \R$ a continuous function on it. We say $f$ is [*strongly smooth*]{} if the restriction of $f$ to $\mathcal K_{\frak p} \cap \Omega$ is smooth for any $\frak p \in \frak P$.
We remark that the strongly smooth function $f : \Omega \to \R$ is nothing but a strongly smooth map to $\R$ regarded as a manifold in the sense of Definition \[definition32727\].
We take a support system $\mathcal K^+$ such that $(\mathcal K ,\mathcal K^+)$ is a support pair. We take a metric $d$ on $\vert{\mathcal K}^+ \vert$ and use it in the next definition.
\[def761\] In Situation \[situ5757\], let $\delta > 0$ be a positive number. We put $$\label{formula7120}
\mathcal K_{\frak p}(2\delta)
=
\{
x \in U_{\frak p} \mid d(x,\mathcal K_{\frak p})
\le 2\delta
\}.$$ $2\delta$-neighborhood of $\mathcal K_{\frak p}$. We assume $\mathcal K_{\frak p}(2\delta)$ is compact.
We put $\mathcal K(2\delta) = \{\mathcal K_{\frak p}(2\delta)\}_{\frak p \in \frak P}$, which is a support system of $\widetriangle{\mathcal U}$. We put $$\label{formula712}
\Omega_{\frak p}(\mathcal K,\delta)
=
B_{\delta}(\mathcal K_{\frak p})
=
\{x \in \vert \mathcal K(2\delta)\vert
\mid d(x,\mathcal K_{\frak p}) < \delta\}.$$
**Figure 7.1**
\[rem74444\] If $\frak q < \frak p$ and $\delta>0$ is sufficiently small, then $$\Omega_{\frak p}(\mathcal K,\delta) \cap \mathcal K_{\frak q}(2\delta)
\subset \mathcal K_{\frak p}(2\delta).$$ Moreover, $\Omega_{\frak p}(\mathcal K,\delta) \cap \mathcal K_{\frak p}(2\delta)
\subset \rm{Int}\,\mathcal K_{\frak p}(2\delta)$ and $\Omega_{\frak p}(\mathcal K,\delta) \cap \mathcal K_{\frak p}(2\delta)$ is an orbifold.
The first claim is a consequence of (\[formula712\]). The second claim follows from the fact that ${\rm Int}\,\mathcal K_{\frak p}(2\delta)$ is open in $\bigcup_{\frak q \le \frak p} \mathcal K_{\frak q}(2\delta)$.
\[pounity\] We say $\{\chi_{\frak p}\}$ is a [*strongly smooth partition of unity* ]{} of the quintuple $(X,Z,{\widetriangle{\mathcal U}},\mathcal K,\delta)$ if the following holds.
1. $\chi_{\frak p} : \vert \mathcal K(2\delta)\vert \to [0,1]$ is a strongly smooth function.
2. ${\rm supp} \, \chi_{\frak p} \subseteq \Omega_{\frak p}(\mathcal K,\delta)$.
3. There exists an open neighborhood $\frak U$ of $Z$ in $\vert \mathcal K(2\delta)\vert$ such that for each point $x \in \frak U$, we have $$\label{pounitymainequ}
\sum_{\frak p} \chi_{\frak p}(x) = 1.$$
In our earlier writings such as [@fooo09 Section 12], we defined a partition of unity in a slightly different way. Namely we require $\chi_{\frak p}$ to be defined on $U_{\frak p}$ and we required $$\label{pounitymainequ2}
\chi_{\frak p}(x)
+
\sum_{\frak q > \frak p, x \in U_{\frak q\frak p}}\chi_{\frak q}
(\varphi_{\frak q\frak p}(x))
+
\sum_{\frak p > \frak q, x \in {\rm Im}\varphi_{\frak p\frak q}}
\chi^{\delta}(\rho(x;U_{\frak p\frak q}))
\chi_{\frak q}(\pi(x))= 1$$ instead of (\[pounitymainequ\]). Here $\pi$ is the projection of a tubular neighborhood of $\varphi_{\frak p\frak q}(U_{\frak p\frak q})$ in $U_{\frak p}$, $\rho$ is a tubular distance function of this tubular neighborhood [^26] and $\chi^{\delta} : [0,\infty) \to [0,1]$ is a smooth function such that it is 1 in a neighborhood of $0$ and is $0$ on $(\delta,\infty)$. We required (\[pounitymainequ2\]) for all $x \in \mathcal K^1_{\frak p}$. Formula (\[pounitymainequ2\]) depends on the choice of the tubular neighborhood and we need to take certain compatible system of tubular neighborhoods (see [@Math73]) to define it.
Note the covering $
\vert {\widetriangle{\mathcal U}} \vert
= \bigcup_{\frak p \in \frak P} U_{\frak p}
$ is [*not*]{} an open covering. In fact $U_{\frak p}$ is not an open subset of $\vert {\widetriangle{\mathcal U}} \vert$ unless $\frak p$ is maximal. So a partition of unity in the above sense is different from one in the usual sense. Definition \[pounity\] seems to be simpler than the one in the previous literature. Also it is more elementary in the sense that we do not use any compatible system of tubular neighborhoods. However, by using the third term of (\[pounitymainequ2\]) we can extend the function $\chi_{\frak q}$ to its neighborhood in $\vert\mathcal K\vert$ and the compatibility of tubular neighborhoods implies that it becomes a strongly smooth function. So the present definition is not very different from the earlier one.
In the rest of this subsection we prove the existence of strongly smooth partition of unity. The proof is very similar to the corresponding one in manifold theory. However we prove it for completeness’ sake. We begin with the following lemma.
\[bumpfunctionlemma\] For any open set $W$ of $\vert \mathcal K(2\delta)\vert$ containing a compact subset $K$ of $Z$ there exists a strongly smooth function $g : \vert \mathcal K(2\delta)\vert
\to \R$ that has a compact support in $W$ and is $1$ on a neighborhood of $K$.
Let $\mathcal K^+$ be a support system such that $\mathcal K(2\delta) < \mathcal K^+$. We take a metric $d$ on $\vert \mathcal K^+\vert$. (See [@foooshrink Proposition 5.1].) For each $x \in K$ we consider $$\epsilon_x = \inf\{ d(x,\mathcal K_\frak q(2\delta)) \mid
\frak q \in \frak P, \,\,
x \notin \mathcal K_\frak q(2\delta)\}.$$ Let $\frak p_x \in \frak P$ be the element which is maximum among elements $\frak p \in \frak P$ such that $x \in \mathcal K_\frak p(2\delta)$. Let $W^+_x$ be an open neighborhood of $x$ in $
\mathcal K_{\frak p_x}^+ \cap B_{\epsilon_x/2}(x,\vert\mathcal K^+\vert).
$ We may choose it so small that if $x \in \mathcal K_{\frak q}(2\delta)$ then $W^+_x \cap \mathcal K_{\frak q}(2\delta)$ is open. (Here we use the fact that $x \in \mathcal K_{\frak q}(2\delta)$ implies $\frak q \le \frak p_x$.) Therefore $W_x = W^+_x \cap \ring{\mathcal K}_{\frak p_x}(2\delta)$ is open in $\vert\mathcal K(2\delta)\vert$. We may choose $W^+_x$ small so that $W_x \subset W$.
We can take a smooth function $f_{x} : W^+_x \to [0,1]$ that has a compact support and is 1 in a neighborhood $Q^+_x$ of $x$. (This is because $W^+_x$ is an orbifold.) We restrict $f_x$ to $W_x$ and extend it by $0$ to $\vert\mathcal K\vert$, which we denote by the same symbol. Then $f_x$ is a strongly smooth function with a compact support in $W_x$ and equal to $1$ on an open neighborhood $Q_x= Q^+_x \cap \vert\mathcal K(2\delta)\vert$ of $x$.
We find finitely many points $x_1,\dots,x_N$ of $K$ so that $$\bigcup_{i=1}^N Q_{x_i} \supset K.$$ We put $$f = \sum_{i=1}^N f_{x_i}.$$ Then the function $f$ is a strongly smooth function on $\vert\mathcal K(2\delta)\vert$ with a compact support in $W$ and satisfies $
f(x) \ge 1
$ if $x \in Q = \bigcup_{i=1}^N Q_{x_i}$. Let $\rho : \R \to \R$ be a nondecreasing smooth map such that $\rho(s) = 0$ if $s$ is in a neighborhood of $0$ and $\rho(s) = 1$ if $s \ge 1$. It is easy to see that $g = \rho\circ f$ has the required property.
\[pounitexi\] If $\delta>0$ is sufficiently small, then a strongly smooth partition of unity of $(X,Z,{\widetriangle{\mathcal U}},\mathcal K,\delta)$ exists.
We put $$\mathcal K_{\frak p}(-\delta)
=
\{x \in \mathcal K_{\frak p} \mid d(x,U_{\frak p} \setminus \mathcal K_{\frak p}) \ge \delta\}.$$ It is easy to see that $$\bigcup_{\frak p} \mathcal K_{\frak p}(-\delta) \supset Z.$$ for sufficiently small $\delta>0$. We apply Lemma \[bumpfunctionlemma\] to $(K,W) = ({\mathcal K_{\frak p}(-\delta)},\Omega_{\frak p}(\mathcal K,\delta))$ to obtain $f_{\frak p}$. Then there exists a neighborhood $W'$ of $Z$ such that $$\sum_{\frak p} f_{\frak p}(x) \ge 1/2$$ for $x\in W'$. We apply Lemma \[bumpfunctionlemma\] to $(K,W) = (Z,W')$ to obtain $f : \vert\mathcal K(2\delta)\vert \to [0,1]$. Now we define $$\chi_{\frak p}(x) =
\begin{cases}
\displaystyle
\frac{f(x)f_{\frak p}(x)}{\sum_{\frak p} f_{\frak p}(x)}
&\text{if $x \in W'$}\\
0
&\text{if $x \notin W'$}.
\end{cases}$$ Then it is easy to see that this is a strongly smooth partition of unity.
Differential form on good coordinate system and Kuranishi structure {#subsec:differentialforms}
-------------------------------------------------------------------
In this subsection we define differential forms of good coordinate system and Kuranishi structure, and give several basic definitions for differential forms.
\[defndiffformgcs\] Let ${\widetriangle{\mathcal U}}=(\frak P, \{\mathcal U_{\frak p}\}, \{ \Psi_{\frak p \frak q}\})$ be a good coordinate system and $\mathcal K$ its support system. A [*smooth differential $k$ form $\widetriangle h$ of $({\widetriangle{\mathcal U}},\mathcal K)$*]{} by definition assigns a smooth $k$ form $h_{\frak p}$ on a neighborhood of $\mathcal K_{\frak p}$ for each $\frak p \in \frak P$ such that the next equality holds on $\varphi_{\frak p\frak q}^{-1}(\mathcal K_{\frak p})\cap
\mathcal K_{\frak q}$. $$\label{compatidefform}
\varphi_{\frak p\frak q}^* h_{\frak p} = h_{\frak q}.$$ A [*smooth differential $k$ form $\widetriangle h$ of ${\widetriangle{\mathcal U}}$*]{} by definition assigns a smooth differential $k$ form $h_{\frak p}$ on $U_{\frak p}$ for each $\frak p$ such that (\[compatidefform\]) is satisfied on $U_{\frak p\frak q}$.
A [*differential $k$ form $\widehat h$ of a Kuranishi structure*]{} $\widehat{\mathcal U}$ of $Z\subseteq X$ assigns a differential $k$-form $h_p$ on $U_p$ for each $p \in Z$ such that $\varphi_{pq}^*h_p = h_q$.
\[defn75555\]
1. Let $\widetriangle f = \{f_{\frak p}\} :
(X,Z;{\widetriangle{\mathcal U}}) \to M$ be a strongly smooth map and $h$ a smooth differential $k$ form on $M_s$. Then $\widetriangle f^*h = \{f_{\frak p}^*h\}$ is a smooth differential $k$ form on ${\widetriangle{\mathcal U}}$, which we call the [*pullback of $h$ by $\widetriangle f = \{f_{\frak p}\}$*]{} and denote by $\widetriangle f^* h$.
2. A smooth differential $0$ form is nothing but a strongly smooth function in the sense of Definition \[def758\].
3. If $\widetriangle {h^i} = \{h^i_{\frak p}\}$ are smooth differential $k_i$ forms on ${\widetriangle{\mathcal U}}$ for $i = 1,2$, then $\{h^1_{\frak p} \wedge h^2_{\frak p}\}$ is a smooth differential $k_1 + k_2$ form on ${\widetriangle{\mathcal U}}$. We call it the [*wedge product*]{} and denote it by $\widetriangle {h^1} \wedge \widetriangle {h^2}$.
4. In particular, we can define a product of a smooth differential form and a strongly smooth function.
5. The sum of smooth differential forms of the same degree is defined by taking the sum for each $\frak p \in \frak P$.
6. If $\widetriangle h = \{h_{\frak p}\}$ is a smooth differential $k$ form on ${\widetriangle{\mathcal U}}$, then $\{dh_{\frak p}\}$ is a smooth differential $k+1$ form on ${\widetriangle{\mathcal U}}$, which we call the [*exterior derivative*]{} of $\widetriangle h$ and denote by $d\widetriangle h$.
7. The [*support*]{} ${\rm Supp}(\widetriangle h)$ of $\widetriangle h$ is the union of the supports of $h_{\frak p}$, $\frak p \in \frak P$, which is a subset of $\vert \mathcal K\vert$.
(1)-(6) have obvious versions in the case of differential forms of a Kuranishi structures. (7) is modified as follows.
1. If $\widehat h = \{h_{p} \mid p \in Z\}$ is a smooth differential $k$ form on a Kuranishi structure ${\widehat{\mathcal U}}$ of $Z \subseteq X$, its [*support*]{} ${\rm Supp}(\widehat h)$ is the set of the points $p \in Z$ such that $\widehat h_p$ is nonzero on any neighborhood of $o_p$ in $U_p$.
Note that ${\rm Supp}(\widehat h)$ is a subset of $Z$ in this case.
Let ${\widetriangle{\mathcal U}}$ be a good coordinate system of $Z \subseteq X$ and $\widetriangle h = \{h_{\frak p}\}$ a differential form on it. We say that $\widetriangle h$ has a [*compact support in $\ring{Z}$*]{} if $${\rm Supp}(\widetriangle h) \cap X
\subset \ring Z.$$ Here the intersection in the left hand side is taken on $\vert{\widetriangle{\mathcal U}}\vert$.
Integration along the fiber (pushout) on a good coordinate system {#subsec:integrationgcs}
-----------------------------------------------------------------
To define the pushout (integration along the fiber) of a differential form using a CF-perturbation, we need a CF-perturbation version of Propositions \[splem2\], \[lem715\]. To state them we introduce the notation of [*support set*]{} of a CF-perturbation.
\[defn767\]
1. Let $\mathcal U$ be a Kuranishi chart, $\mathfrak S$ a CF-perturbation of $\mathcal U$ and $\{(\frak V_{\frak r},\mathcal S_{\frak r}) \mid \frak r\in
\frak R\}$ its representative. For each $\epsilon >0$ we define the [*support set*]{} $\Pi((\mathfrak S^{\epsilon})^{-1}(0)) $ of $\frak S$ as the set of all $x \in U$ with the following property:
There exist $\frak r \in \frak R$ and $y \in V_{\frak r}$, $\xi \in W_{\frak r}$ such that $$\phi_{\frak r}([y]) = x, \qquad
s^{\epsilon}_{\frak r}(y,\xi) = 0,
\qquad \xi \in {\rm Supp}(\omega_x).$$ This definition is independent of the choice of representative because of Definition \[defn73ss\] (4).
2. Let ${\widetriangle{\mathcal U}}$ be a good coordinate system, $\mathcal K$ its support system and $\widetriangle{\mathfrak S}
= \{\mathfrak S_{\frak p}\}$ a CF-perturbation of $({\widetriangle{\mathcal U}},\mathcal K)$. The [*support set $\Pi((\widetriangle{\mathfrak S^{\epsilon}})^{-1}(0))$ of $\widetriangle{\mathfrak S}$*]{} is defined by $$\Pi((\widetriangle {{\mathfrak S}^{\epsilon}})^{-1}(0))
= \bigcup_{\frak p \in \frak P}
\left(\mathcal K_{\frak p} \cap \Pi(({\mathfrak S}_{\frak p}^{\epsilon})^{-1}(0))\right)$$ which is a subset of $\vert \mathcal K\vert$.
The CF-perturbation version of Propositions \[splem2\] is as follows
\[lem739\] Let ${\widetriangle{\mathcal U}}$ a good coordinate system of $Z \subseteq X$, $\mathcal K$ its support system and $\widetriangle{\mathfrak S}
= \{\mathfrak S_{\frak p} \mid \frak p \in \frak P\}$ a CF-perturbation of $({\widetriangle{\mathcal U}},\mathcal K)$. Let $\mathcal K^- < \mathcal K^+ < \mathcal K' < \mathcal K$. Then there exist positive numbers $\delta_0$ and $\epsilon_0$ such that $$B_{\delta}(\mathcal K_{\frak q}^{-} \cap Z) \cap
\bigcup_{\frak p \in \frak P}
(\mathcal K'_{\frak p} \cap \Pi((\widetriangle{{\mathfrak S}_{\frak p}^{\epsilon}})^{-1}(0)))
\subset
\mathcal K_{\frak q}^{+}$$ for $\delta < \delta_0, 0 < \epsilon < \epsilon_0$.
Using Lemma \[lem6767cont\], the proof is the same as the proof of Proposition \[splem2\].
The next lemma is the CF-perturbation version of Proposition \[lem715\].
\[lem740\] Let $\mathcal K^1,\mathcal K^2, \mathcal K^3$ be support systems of a good coordinate system ${\widetriangle{\mathcal U}}$ of $Z \subseteq X$ with $\mathcal K^1 < \mathcal K^2 < \mathcal K^3$ and $\widetriangle{\mathfrak S}$ a CF-perturbation of $({\widetriangle{\mathcal U}},\mathcal K^3)$. Then there exists a neighborhood $\frak U(Z)$ of $Z$ in $\vert \mathcal K^2\vert$ such that for $0 < \epsilon < \epsilon_0$ $$\frak U(Z) \cap
\bigcup_{\frak p}(\Pi((\widetriangle{{\mathfrak S}_{\frak p}^{\epsilon}})^{-1}(0)
)\cap \mathcal K^1_{\frak p})
=
\frak U(Z)\cap
\bigcup_{\frak p}(\Pi((\widetriangle{{\mathfrak S}_{\frak p}^{\epsilon}})^{-1}(0)
)\cap \mathcal K^2_{\frak p}).$$
The proof is the same as the proof of Proposition \[lem715\].
The next lemma is the CF-perturbation version of Corollary \[cor69\].
\[lem743\] In the situation of Lemma \[lem740\], $\left(\bigcup_{\frak p}((\frak s_{\frak p}^{\epsilon})^{-1}(0))
\cap \ring{\mathcal K}^1_{\frak p})\right)
\cap \frak U(Z)$ is compact for a sufficiently small neighborhood $\frak U(Z)$ of $Z$ in $\vert\mathcal K^2\vert$. Moreover, we have $$\lim_{\epsilon\to 0}
\left(\bigcup_{\frak p}(\Pi((\widetriangle{{\mathfrak S}_{\frak p}^{\epsilon}})^{-1}(0)
\cap \ring{\mathcal K}^1_{\frak p})\right)
\cap \frak U(Z)
\subseteq X$$ in Hausdorff topology.
The proof is the same as Corollary \[cor69\].
Now to define the pushout of a differential form we consider the following situation.
\[situ774\] Let $\widetriangle{\mathcal U}=(\frak P, \{\mathcal U_{\frak p} \},\{ \Phi_{\frak p \frak q}\})$ be a good coordinate system, $\mathcal K$ its support system, $\widetriangle h = \{h_{\frak p}\}$ a differential form on ${\widetriangle{\mathcal U}}$, $\widetriangle f
: (X,Z;\widetriangle{\mathcal U}) \to M$ a strongly smooth map, and $\widetriangle{\frak S}$ a CF-perturbation of $({\widetriangle{\mathcal U}}, \mathcal K)$. We assume that
1. ${\widetriangle f}$ is strongly submersive with respect to $\widetriangle{\frak S}$.
2. $\widetriangle h$ has a compact support in $\ring Z$.$\blacksquare$
\[cho776\] In Situation \[situ774\] we make the following choices.
1. A triple of support systems $\mathcal K^1,\mathcal K^2, \mathcal K^3$ with $\mathcal K^1 < \mathcal K^2< \mathcal K^3 < \mathcal K$.
2. A constant $\delta>0$ such that:
1. $\mathcal K^1(2\delta)$ is compact. (Definition \[def761\].)
2. There exists a strongly smooth partition of unity $\{\chi_{\frak p}\}$ of $(X,Z,{\widetriangle{\mathcal U}},\mathcal K^1,\delta)$. (Proposition \[pounitexi\].)
3. $\mathcal K^1(2\delta) < \mathcal K^2$.
4. $\delta$ satisfies the conclusion of Lemma \[lem739\] for $\mathcal K^- = \mathcal K^1$, $\mathcal K^+ = \mathcal K^2$, $\mathcal K' = \mathcal K^3$.
5. We put $$\delta_0 = \inf \{d(\mathcal K^2_{\frak p},\mathcal K^2_{\frak q}) \mid \text{neither $\frak p\le\frak q$ nor
$\frak q \le \frak p$}\},$$ where we use the metric $d$ of $\vert\mathcal K\vert$. Then $\delta < \delta_0/2$.
3. We take a strongly smooth partition of unity $\{\chi_{\frak p}\}$ of $(X,Z,{\widetriangle{\mathcal U}},\mathcal K^1,\delta)$.
4. We take an open neighborhood $\frak U(X)$ of $Z$ in $\vert\mathcal K\vert$ such that the conclusion of Lemma \[lem740\] holds.
\[pushforwardKuranishi\] In Situation \[situ774\], we make Choice \[cho776\]. We define a smooth differential form $
{\widetriangle f}!(\widetriangle h;\widetriangle{{\frak S}^{\epsilon}})$ on the manifold $M$ by (\[formula714\]). We call it the [*pushout*]{} or [*integration along the fiber*]{} of $\widetriangle h$ by $(\widetriangle f,\widetriangle{{\frak S}^{\epsilon}})$.
$$\label{formula714}
{\widetriangle f}!(\widetriangle h;\widetriangle{{\frak S}^{\epsilon}})
=
\sum_{\frak p\in \frak P}
f_{\frak p}!(\chi_{\frak p}h_{\frak p};{\frak S}^{\epsilon}_{\frak p}
\vert_{\frak U(Z) \cap \mathcal K^1_{\frak p}(2\delta)}).$$
We note that the restriction of $\chi_{\frak p}h_{\frak p}$ to $\mathcal K^1_{\frak p}(2\delta)$ has compact support in ${\rm Int}\,\mathcal K^1_{\frak p}$. Therefore the right hand side of (\[formula714\]) makes sense.
The degree is given by $$\deg {\widetriangle f}!(\widetriangle h;\widetriangle{{\frak S}^{\epsilon}})
=
\deg \widehat h + \dim M - \dim \widetriangle{\mathcal U}.$$
\[defnspadesuit\] Let $F_a : (0,\epsilon_a) \to \mathscr X$ be a family of maps parameterized by $a \in \mathscr B$. We say that $F_a$ is [*independent of the choice of $a$ in the sense of $\spadesuit$*]{} if the following holds.
1. For $a_1,a_2 \in \mathscr B$ there exists $0 < \epsilon_0
< \min\{\epsilon_{a_1},\epsilon_{a_2}\}$ such that $F_{a_1}(\epsilon) = F_{a_2}(\epsilon)$ for all $\epsilon < \epsilon_0$.
\[indepofukuracont\] In Situation \[situ774\] the pushout ${\widetriangle f}!(\widetriangle h;\widetriangle{{\frak S}^{\epsilon}})$ is independent of Choice \[cho776\] in the sense of $\spadesuit$.
However ${\widetriangle f}!(\widetriangle h;\widetriangle{{\frak S}^{\epsilon}})$ depends on the choices of $\epsilon$ and $\widetriangle{\frak S}$.
We first show the independence of $\frak U(Z)$. Lemma \[lem743\] implies that if $\frak U'(Z) \subset \frak U(Z)$ is another open neighborhood of $Z$ then, for sufficiently small $\epsilon$, the value of the right hand side of (\[formula714\]) does not change if we replace $\frak U(Z)$ by $\frak U'(Z)$. (We use Situation \[situ774\] (2) also here.) This implies independence of $\frak U(Z)$. Moreover it implies that we can always replace $\frak U(Z)$ by a smaller open neighborhood of $Z$.
We next show independence of $\mathcal K^2$, $\mathcal K^3$. Let $\mathcal K^{2 \prime}$, $\mathcal K^{3 \prime}$ be alternative choices of $\mathcal K^2$, $\mathcal K^3$. We take $\mathcal K^{2 \prime\prime}_{\frak p} =\mathcal K^2_{\frak p} \cup \mathcal K^{2 \prime}_{\frak p}$, $\mathcal K^{3 \prime\prime}_{\frak p} =\mathcal K^3_{\frak p} \cup \mathcal K^{3 \prime}_{\frak p}$. Then $\mathcal K^{2 \prime\prime}$, $\mathcal K^{3 \prime\prime}$ are support systems. Note in Definition \[pushforwardKuranishi\], the support system $\mathcal K^2$, $\mathcal K^3$ appears only when we apply Lemma \[lem740\] to obtain $\frak U(Z)$ and Lemma \[lem739\] to obtain $\delta$. Since we can always replace $\frak U(Z)$ by a smaller neighborhood of $Z$ (as far as $\epsilon>0$ is sufficiently small) and since we do not need to change $\delta$ in Lemma \[lem739\] when we replace $\mathcal K^2$, $\mathcal K^3$ by $\mathcal K^{2 \prime\prime} \supset \mathcal K^2$, $\mathcal K^{3 \prime\prime} \supset \mathcal K^3$, it follows that we obtain the same number in (\[formula714\]) if we replace $\mathcal K^2$ or $\mathcal K^{2 \prime}$ by $\mathcal K^{2 \prime\prime}$, as far as $\epsilon>0$ is sufficiently small. This implies independence of $\mathcal K^2$, $\mathcal K^3$.
It remains to prove the independence of $\mathcal K^1$ and of $\{\chi_{\frak p}\}$, $\delta$. We will prove the independence for this case below.
Let $\mathcal K^1_{\frak p}, \chi_{\frak p}, \delta$ and $\mathcal K^{1 \prime}_{\frak p},
\chi'_{\frak p}, \delta'$ be two such choices. We take $\mathcal K^{1 \prime\prime}_{\frak p}
= \mathcal K^1_{\frak p} \cup \mathcal K^{1 \prime}_{\frak p}$. Then $\mathcal K^{1 \prime\prime} < \mathcal K^2 <\mathcal K^3$. We can also take $\delta'' > 0$ and a strongly smooth partition of unity $\{\chi''_{\frak p}\}$ of $(X,Z,{\widetriangle{\mathcal U}},\mathcal K^{1 \prime\prime},\delta'')$. So it suffices to prove that the pushout defined by $\{\mathcal K^1_{\frak p}\}, \{\chi_{\frak p}\}, \delta$ coincides with one defined by $\{\mathcal K^{1 \prime\prime}_{\frak p}\},\{\chi''_{\frak p}\}, \delta''$ and that the pushout defined by $\{\mathcal K^{1 \prime}_{\frak p}\}, \{\chi'_{\frak p}\}
, \delta'$ coincides with one defined by $\{\mathcal K^{1 \prime\prime}_{\frak p}\},\{\chi''_{\frak p}\},
\delta''$. In other words, we may assume $\mathcal K^1_{\frak p}
\subset \mathcal K^{1 \prime}_{\frak p}$. We will prove the independence in this case.
We observe that $$\label{form715}
\widetriangle f!(\widetriangle h_1+\widetriangle h_2;\widetriangle{{\frak S}^{\epsilon}})
=
\widetriangle f!(\widetriangle h_1;\widetriangle{{\frak S}^{\epsilon}})
+
\widetriangle f!(\widetriangle h_2;\widetriangle{{\frak S}^{\epsilon}})$$ as far as we use the same strongly smooth partition of unity in all these three terms. (This is a consequence of Lemma \[lem721\] (2).) We take $\frak p_0 \in \frak P$ and put $$\label{form738373}
\widetriangle h_0 = \chi'_{\frak p_0} \widetriangle h.$$ In view of (\[form715\]) we find that to prove Proposition \[indepofukuracont\] it suffices to show the next formula. $$\label{form716}
f_{\frak p_0}!((\widetriangle h_0)_{\frak p_0};{\frak S}^{\epsilon}_{\frak p_0}
\vert_{\frak U(Z) \cap \mathcal K^{1\prime}_{\frak p_0}(2\delta')})
=
\sum_{\frak p}f_{\frak p}!((\chi_{\frak p}\widetriangle h_0)_{\frak p};{\frak S}^{\epsilon}_{\frak p}
\vert_{\frak U(Z) \cap \mathcal K^1_{\frak p}(2\delta)}).$$ By taking $\epsilon>0$ small, we may assume $\sum \chi_{\frak p} = 1$ on the intersection of $\frak U(Z)$ and the support set $\Pi((\widetriangle{{\mathfrak S}_{\frak p_0}^{\epsilon}})^{-1}(0))$. (This is a consequence of Lemma \[lem743\], Definition \[pounity\] (3) and Situation \[situ774\] (2).) Therefore, to prove (\[form716\]) it suffices to prove the next formula for each $\frak p$. $$\label{form717}
f_{\frak p_0}!((\chi_{\frak p}\widetriangle h_0)_{\frak p_0};{\frak S}^{\epsilon}_{\frak p_0}
\vert_{\frak U(X) \cap \mathcal K^{1\prime}_{\frak p_0}(2\delta')})
=
f_{\frak p}!((\chi_{\frak p}\widetriangle h_0)_{\frak p};{\frak S}^{\epsilon}_{\frak p}
\vert_{\frak U(X) \cap \mathcal K^1_{\frak p}(2\delta)})$$ whose proof is now in order.
In case $\frak p_0 = \frak p$, (\[form717\]) follows from $${\rm Supp} (\chi_{\frak p} h_{0,\frak p_0})
= {\rm Supp} (\chi_{\frak p_0} h_{0,\frak p_0})
\subseteq
\mathcal K^1_{\frak p_0}(2\delta) \cap \mathcal K^{1\prime}_{\frak p_0}(2\delta').$$ If neither $\frak p \le \frak p_0$ nor $\frak p_0 \le \frak p$, then both sides of (\[form717\]) are zero because $$\label{form743}
{\rm Supp}\,(\chi_{\frak p} h_{0,\frak p_0}) \subseteq
\Omega_{\frak p_0}(\mathcal K^{1\prime},\delta') \cap \Omega_{\frak p}(\mathcal K^{1},\delta)
= \emptyset.$$ Note the second equality of (\[form743\]) is a consequence of Choice \[cho776\] (2)(e).
We will discuss the other two cases below.
(Case 1): $\frak p > \frak p_0$.
By definition $\Omega_{\frak p_0}(\mathcal K^{1 \prime},\delta')
\subset B_{\delta'}(\mathcal K^{1\prime}_{\frak p_0})$. Therefore by (\[form738373\]) the support of $\widetriangle h_0$ is in $B_{\delta'}(\mathcal K^{1\prime}_{\frak p_0})$. By taking $\epsilon>0$ small, Lemma \[lem739\] implies $${\rm Supp}(\widetriangle h_0) \cap \Pi((\widetriangle{{\frak S}^{\epsilon}})^{-1}(0))
\subset
\mathcal K^{2\prime}_{\frak p_0} \cap B_{\delta'}(\mathcal K^{1\prime}_{\frak p_0})
\subset
\mathcal K^{1\prime}_{\frak p_0}(2\delta').$$ Therefore $$\label{form741741}
{\rm Supp}(\chi_{\frak p}\widetriangle h_0) \cap \Pi((\widetriangle{{\frak S}^{\epsilon}})^{-1}(0))
\cap \frak U(Z)
\subseteq
\mathcal K^{1\prime}_{\frak p_0}(2\delta')
\cap
\mathcal K^{1}_{\frak p}(2\delta)
\cap \frak U(Z).$$ Then (\[form717\]) follows from Definition \[defn7732\] (2)(3).
(Case 2): $\frak p_0 > \frak p$.
By definition $\Omega_{\frak p}(\mathcal K^1,\delta)
\subset B_{\delta}(\mathcal K^1_{\frak p})$. Therefore the support of $\chi_{\frak p}\widetriangle h_0$ is in $B_{\delta}(\mathcal K^1_{\frak p})$. Therefore by taking $\epsilon>0$ small, Lemma \[lem739\] implies $${\rm Supp}(\chi_{\frak p}\widetriangle h_0) \cap \Pi((\widehat{{\frak S}^{\epsilon}})^{-1}(0))
\subset \mathcal K^2_{\frak p} \cap B_{\delta}(\mathcal K^1_{\frak p_0})
\subset \mathcal K^1_{\frak p}(2\delta).$$ It implies (\[form741741\]). Then (\[form717\]) follows from Definition \[defn7732\] (2)(3).
The proof of Proposition \[indepofukuracont\] is complete.
The pushforward (\[formula714\]) is independent of the choice of the support system $\mathcal K$ appearing in Situation \[situ774\], as far as $\widetriangle{\frak S}$ and $\widetriangle h$ are defined on it. In fact $\mathcal K$ does not appear in the definition.
\[lem782\] Let $h,h_1,h_2$ be differential forms on $(X,Z;\widetriangle{\mathcal U})$ and $c_1, c_2 \in \R$.
1. $
\widetriangle f!(c_1\widetriangle h_1 + c_1\widetriangle h_2;\widetriangle{{\frak S}^{\epsilon}})
=
c_1\widetriangle f!(\widetriangle h_1;\widetriangle{{\frak S}^{\epsilon}})
+
c_2\widetriangle f!(\widetriangle h_2;\widetriangle{{\frak S}^{\epsilon}}).
$
2. If $\rho \in \Omega^*(M)$, then $
\widetriangle f!(\widetriangle h \wedge \widetriangle f^*(\rho);\widetriangle{{\frak S}^{\epsilon}})
=
\widetriangle f!(\widetriangle h;\widetriangle{{\frak S}^{\epsilon}}) \wedge \rho.
$
Formula (1) follows from Lemma \[lem721\] (1). Formula (2) is immediate from definition.
We now define the smooth correspondence.
\[cordefjyunbi\] Suppose we are in Situation \[smoothcorr\]. We construct objects as in Situation \[situ774\] as follows.
We put $Z=X$. We take a good coordinate system ${\widetriangle{\mathcal U}}$ compatible with ${\widehat{\mathcal U}}$ such that $\widehat f_s$ and $\widehat f_t$ are pullbacks of $\widetriangle f_s : (X;{\widetriangle{\mathcal U}})
\to M_s$ and $\widetriangle f_t : (X;{\widetriangle{\mathcal U}})
\to M_t$, respectively. Moreover we may take $\widetriangle f_t$ so that it is weakly submersive. (Proposition \[le614\] (2).)
We take a CF-perturbation $\widetriangle{\frak S}$ of $(X;{\widetriangle{\mathcal U}})$ such that $\widetriangle f_t$ is strongly submersive with respect to ${\widetriangle{\mathcal U}}$. (Theorem \[existperturbcont\] (2).)
Let ${\mathcal K}$ be a support system of ${\widetriangle{\mathcal U}}$. We consider the differential form $\widetriangle f_s^*h$ on $(X,{\widetriangle{\mathcal U}})$.
We denote the correspondence by $${\frak X} = ((X;\widetriangle{\mathcal U});\widetriangle f_s,
\widetriangle f_t).$$
\[defn748\] Using Construction \[cordefjyunbi\], we define $$\label{scordef}
{\rm Corr}_{(\frak X,\widetriangle{{\frak S}^{\epsilon}})}(h)
=
({\widetriangle f}_t)!(\widetriangle f_s^*h;\widetriangle{{\frak S}^{\epsilon}}).$$ We call the linear map $${\rm Corr}_{(\frak X,\widetriangle{{\frak S}^{\epsilon}})} :
\Omega^*(M_s)
\to
\Omega^{*+ \dim M_t - \dim \widetriangle{\mathcal U}}(M_t)$$ the [*smooth correspondence map associated to $\widetriangle{\frak X} = ((X;\widetriangle{\mathcal U});\widetriangle f_s,
\widetriangle f_t)$*]{}.
\[rem:785\]
1. Proposition \[indepofukuracont\] implies that the right hand side of (\[scordef\]) is independent of various choices appearing in Definition \[pushforwardKuranishi\] if $\epsilon>0$ is sufficiently small. However, it [*does*]{} depend on $\widetriangle{\frak S}$ and $\epsilon > 0$. So we keep the symbol $\epsilon$ in the notation $\widetriangle{{\frak S}^{\epsilon}}$ of the left hand side of (\[scordef\]).
2. There seems to be no way to define a smooth correspondence in the way that it becomes independent of the choices [*in the chain level*]{}. This is related to an important point of the story, that is, the construction of various structures from system of moduli spaces are well-defined only up to homotopy equivalence and only as a whole. (We however note that the method of [@joyce] seems to be a way to minimize this dependence.) This is the fundamental issue which appears in [*any*]{} approach. For example, it should remain to exist in the infinite dimensional approach to construct virtual fundamental chain, such as those by Li-Tian [@LiTi98], Liu-Tian [@LiuTi98], Siebert [@Siebert], Chen-Tian [@ChenTian], Chen-Li-Wang [@ChenLieWang] or Hofer-Wyscoski-Zehnder [@hwze].
3. Dependence of the good coordinate system ${\widetriangle{\mathcal U}}$ and the other choices made in Construction \[cordefjyunbi\] will be discussed in Section \[sec:kuraandgood\].
In Proposition \[indepofukuracont\], we have proved independence of the pushout of various choices for sufficiently small $\epsilon>0$. On the other hand, how $\epsilon$ must be small depends on our good coordinate system and CF-perturbation on it. In certain situations appearing in applications, we need to estimate this required smallness of $\epsilon$ uniformly from below when our CF-perturbations vary in a certain family. The next proposition can be used for such a purpose.
\[intheclubsuit\] Let $F_{\sigma,a} : (0,\epsilon_a) \to \mathscr X$ be a family of maps parameterized by $a \in \mathscr B$ and $\sigma \in \mathscr A$. We say that $F_{\sigma,a}$ is [*uniformly independent of the choice of $a$ in the sense of $\clubsuit$*]{} if the following holds.
1. For $a_1,a_2 \in \mathscr B$ there exists $0 < \epsilon_0
< \min\{\epsilon_{a_1},\epsilon_{a_2}\}$ independent of $\sigma$ such that $F_{\sigma,a_1}(\epsilon) = F_{\sigma,a_2}(\epsilon)$ for $0<\epsilon < \epsilon_0$ and any $\sigma \in \mathscr A$.
\[lem761\] We assume $\{ \widetriangle{\frak S_{\sigma}} \mid \sigma \in \mathscr A\}$ is a uniform family of CF-perturbations parameterized by $\sigma \in \mathscr A$ in the sense of Definition \[uniformcongpert\]. Then we can make Choice \[cho776\] in a way independent of $\sigma$.
Moreover the pushout ${\widetriangle f}!(\widetriangle h;\widetriangle{{\frak S}_{\sigma}^{\epsilon}})$ of Proposition \[indepofukuracont\] is uniformly independent of Choice \[cho776\] in the sense of $\clubsuit$.
From the proof of Proposition \[indepofukuracont\], the proof of Proposition \[lem761\] follows from the next lemma.
\[lem6878787\] Let $\{ \widetriangle{\frak S_{\sigma}} \mid \sigma \in \mathscr A\}$ be a uniform family of CF-perturbations. Then the following holds.
1. In Lemma \[lem739\] the constants $\delta_0$ and $\epsilon_0$ can be taken in dependent of $\sigma$.
2. In Lemma \[lem740\] the set $\frak U(Z)$ and the constant $\epsilon_0$ can be taken independent of $\sigma$.
3. In Lemma \[lem743\] the set $\frak U(Z)$ and the constant $\epsilon_0$ can be taken independent of $\sigma$. Moreover $$\lim_{n\to \infty}\sup\left\{ d_H\left(X, \bigcup_{\frak p}(\Pi((\widetriangle{{\mathfrak S}_{\sigma,\frak p}^{\epsilon}})^{-1}(0)
\cap \overset{\circ}{\mathcal K^2_{\frak p}}
\cap \frak U(Z))\right) \mid \sigma \in \mathscr A\right\}
= 0.$$
Using Lemma \[lemma748\], the proof of Lemma \[lem6878787\] is the same as that of Proposition \[lem627\].
Stokes’ formula {#sec:stokes}
===============
Boundary and corner II {#subsection:normbdry2}
----------------------
In this section, we state and prove Stokes’ formula. We first discuss the notion of boundary or corner of an orbifold and of a Kuranishi structure in more detail. (The discussion below is a detailed version of [@fooobook2 the last paragraph of page 762]. See also [@joyce page 11]. [@joyce3] gives a systematic account on this issue.
Let $U$ be an orbifold with boundary and corner. We defined its corner structure stratification $S_k(U)$ and $\overset{\circ}S_k(U)$ in Definition \[defn4111\]. Note $\overset{\circ}S_k(U)$ is an orbifold of dimension $\dim U -k$ without boundary. However we also note that we can [*not*]{} find a structure of orbifold with corner on $S_k(U)$ such that $\overset{\circ}S_0(S_k(U)) = \overset{\circ}S_k(U)$.
Let $U = \R_{\ge 0}^2$. Then $S_1(U)$ is homeomorphic to $\R$ and $S_2(U)$ is one point identified with $0 \in \R = S_1(U)$.
To obtain an orbifold with corner from $S_1(U)$ we need to modify it at its boundaries and corners. Let us first consider the case of manifolds.
\[lemma749\] Suppose $U$ is a manifold with corner. Then there exists a manifold with corner, denoted by $\partial U$, and a map $\pi : \partial U \to S_1(U)$ with the following properties.
1. For each $k$, $\pi$ induces a map $$\label{maponkstratum}
\overset{\circ}S_k(\partial U) \to \overset{\circ}S_{k+1}(U).$$
2. The map (\[maponkstratum\]) is a $(k+1)$-fold covering map.
3. $\pi$ is a smooth map $
\partial U \to U.
$
The smoothness claimed in Lemma \[lemma749\] (3) is defined as follows. Let $U$ be any smooth manifold with corner. We can find a smooth manifold without boundary or corner $U^+$ of the same dimension as $U$ and an embedding $U \to U^+$, such that for each point $p\in U$ there exists a coordinate of $U^+$ at $p$ by which $U$ is identified with an open subset of $[0,1)^{\dim U}$ by a diffeomorphism from $U^+$ onto an open subset of $\R^{\dim U}$. Then a map $F : U_1 \to U_2$ between two manifolds with corners is said to be smooth if $F$ extends to $F^+$ that is a smooth map from a neighborhood of $U_1$ in $U_1^+$ to $U_2^+$.
We fix a Riemannian metric on $U$ so that each $\epsilon$-ball $B_{\epsilon}(p)$ is convex. Let $p \in \overset{\circ}S_k(U)$. We consider $\alpha \in \pi_0(B_{\epsilon}(p) \cap \overset{\circ}S_{1}(U))$. The set of all such pairs $(p,\alpha)$ with $k\ge 1$ is our $\partial U$. The map $(p,\alpha) \to p$ is the map $\pi$.
By identifying $U$ locally with $[0,\infty)^n$, it is easy to construct the structure of manifold with corner on $\partial U$ and prove that they have the required properties.
\[defbdrofd\] Let $U$ be an orbifold. We call $\partial U$ a [*normalized boundary*]{} of $U$ and $\pi : \partial U \to S_1(U)$ the .
\[lem750\]
1. Let $U$ and $U'$ be as in Lemma \[lemma749\] and $F : U \to U'$ a diffeomorphism. Then $F$ uniquely induces a diffeomorphism $$F^{\partial} : \partial U \to \partial U'$$ such that $\pi \circ F^{\partial} = F \circ \pi$.
2. Suppose a finite group $\Gamma$ acts on $U$, where $U$ is as in Lemma \[lemma749\]. Suppose also that each connected component of $\overset{\circ}S_k(U)/\Gamma$ is an effective orbifold for each $k$. Then $\Gamma$ acts on $\partial U$ so that $\pi$ is $\Gamma$ equivariant and each connected component of $\overset{\circ}S_k(\partial U)/\Gamma$ is an effective orbifold for each $k$.
3. Let $U$ be as in Lemma \[lemma749\] and $U'$ its open subset. Then there exits an open embedding $\partial U' \to \partial U$ which commutes with $\pi$.
\(1) is immediate from the construction. Then the uniqueness implies (2) and (3).
Now we consider the case of an orbifold $U$. We cover $U$ by orbifold charts $\{ (V_i,\Gamma_i,\phi_i)\}$. We apply Lemma \[lemma749\] to $V_i$ and obtain $\partial V_i$. Then $\Gamma_i$ action on $V_i$ induces one on $\partial V_i$ by Lemma \[lem750\] (2). We thus obtain orbifolds $\partial V_i/\Gamma_i$. Using Lemma \[lem750\] (1) and (3) we can glue $\partial V_i/\Gamma_i$ for various $i$ and obtain $\partial U$. We obtain also a map $\pi : \partial U \to S_1(U)$. It induces a map $\overset{\circ}S_k(\partial U) \to \overset{\circ}S_{k+1}(U)$.
\[rem8ten6\]
1. We note that the map $\overset{\circ}S_k(\partial U)
\to \overset{\circ}S_{k+1}(U)$ is a $(k+1)$-fold orbifold covering of orbifolds in the sense we will define in Part 2.
2. In particular, $\overset{\circ}S_0(\partial U) \to \overset{\circ}S_{1}(U)$ is a diffeomorphism of orbifolds.
3. We also note that $\overset{\circ}S_k(\partial U) \to \overset{\circ}S_{k+1}(U)$ is not necessarily a $k+1$ to $1$ map set-theoretically. The following is a counter example. Let $$U = (\R_{\ge 0}^2 \times \R)/\Z_2$$ where the action is $(a,b,c) \mapsto (b,a,-c)$. Then $\partial U \cong \R_{\ge 0} \times \R$, $S_1(\partial U) \cong \R$, $S_2(U) \cong \R/\Z_2$ and the map $\pi$ is canonical projection $\R \to \R/\Z_2$ on $S_1(\partial U)$. So it is generically 2 to 1 but is 1 to 1 at $0$.
Next we consider the case of Kuranishi structure. We recall the following notation from Definition \[dimstratifidef\]. $$S_k(X,Z;\widehat{\mathcal U}) = \{ p \in Z \mid o_p \in S_k(U_p)\},
\qquad
\overset{\circ}{S_k}(X,Z;\widehat{\mathcal U}) = \{ p \in Z \mid o_p \in \overset{\circ}{S_k}(U_p)\},$$ where $\widehat{\mathcal U}$ is a Kuranishi structure of $Z \subseteq X$ and $$\aligned
S_k(X,Z;{\widetriangle{\mathcal U}}) &= \{ p \in Z \mid \exists \frak p \exists x \in S_k(U_{\frak p}), {\rm st}\,\,
s_{\frak p}(x) = 0, \psi_{\frak p}(x) = p\},
\\
\overset{\circ}{S_k}(X,Z;{\widetriangle{\mathcal U}}) &= \{ p \in Z \mid \exists \frak p \exists x \in \overset{\circ}{S_k}(U_{\frak p}), {\rm st}\,\,
s_{\frak p}(x) = 0, \psi_{\frak p}(x) = p\},
\endaligned$$ where $\widetriangle{\mathcal U}$ is a good coordinate system of $Z \subseteq X$.
We can rewrite the set $S_k(X,Z;{\widetriangle{\mathcal U}})$ as $$S_k(X,Z;{\widetriangle{\mathcal U}}) = \{ p \in X \mid \forall \frak p \forall x \in U_{\frak p},
s_{\frak p}(x) = 0, \psi_{\frak p}(x) = p \,\,\Rightarrow\,\, x \in {S_k}(U_{\frak p})
\}.$$ A similar remark applies to $\overset{\circ}{S_k}(X,Z;{\widetriangle{\mathcal U}})$.
\[lem754\]
1. Any compact subset of the space $\overset{\circ}{S_k}(X,Z;\widehat{\mathcal U})$ (resp. $\overset{\circ}{S_k}(X,Z;{\widetriangle{\mathcal U}})$) has Kuranishi structure without boundary (resp. good coordinate system without boundary) and of dimension $\dim (X,Z;\widehat{\mathcal U}) - k$ (resp. $\dim (X,Z;{\widetriangle{\mathcal U}}) - k$).
2. There exist a relative K-space with corner $\partial(X,Z;\widehat{\mathcal U})$ (resp. $\partial(X,Z;{\widetriangle{\mathcal U}})$) whose underlying topological spaces are $(\partial X,\partial Z)$ and a continuous map between their underlying topological spaces $\pi : \partial Z \to S_1(X,Z;\widehat{\mathcal U})$ (resp. $\pi : \partial Z \to S_1(X,Z;\widetriangle{\mathcal U})$) such that the following holds. We call $\partial(X,Z;\widehat{\mathcal U})$, $\partial(X,Z;{\widetriangle{\mathcal U}})$ the [normalized boundary]{} of $(X,Z;\widehat{\mathcal U})$, $(X,Z;{\widetriangle{\mathcal U}})$, respectively.
1. If $\pi(\tilde p) = p$, $\tilde p \in \partial Z$, $p \in Z$, then the Kuranishi neighborhood of $\tilde p$ is obtained by restricting $\mathcal U_p$ to $\partial U_p$, which is as in Definition \[defbdrofd\].
2. The coordinates of $\partial(X,Z;{\widetriangle{\mathcal U}})$ are obtained by restricting $\mathcal U_{\frak p}$ to $\partial U_{\frak p}$.
3. The coordinate change of $\partial(X,Z;\widehat{\mathcal U})$ (resp. $\partial(X,Z;{\widetriangle{\mathcal U}})$) is obtained by restricting one of $\partial \mathcal U_{p}$ (resp. $\partial \mathcal U_{\frak p}$).
4. The restriction of $\pi$ induces a map $$\overset{\circ}{S_0}(\partial(X,Z;\widehat{\mathcal U}))
\to \overset{\circ}{S_1}(X,Z;\widehat{\mathcal U})$$ that is an isomorphism of Kuranishi structures.
5. The restriction of $\pi$ induces a map $$\overset{\circ}{S_0}(\partial(X,Z;{\widetriangle{\mathcal U}}))
\to \overset{\circ}{S_1}(X,Z;{\widetriangle{\mathcal U}})$$ that is an isomorphism of good coordinate systems.
6. In the case of Kuranishi structure and $Z \ne X$, we need to replace ${\widehat{\mathcal U}}$ by its open substructure.
3. Various kinds of embeddings among Kuranishi structures and/or good coordinate systems induce embeddings of their normalized boundaries.
We first prove (2). Let $\mathcal U = (U,\mathcal E,s,\psi)$ be a Kuranishi chart. We restrict $\mathcal E$ and $s$ to $\partial U$ and obtain $\partial U,\mathcal E^{\partial}, s^{\partial}$. We will define underlying topological spaces $\partial X$, $\partial Z$, parametrization $\psi^{\partial}$ and the coordinate change.
Let $\Phi_{21}=(U_{21}, \varphi_{21},\widehat{\varphi}_{21}) : \mathcal U_1 \to \mathcal U_2$ be a coordinate change of Kuranishi charts. We note that we required the following condition for an embedding of orbifolds $\varphi_{21} : U_1 \to U_2$. $$\label{cornerorbemb}
\varphi_{21}(S_k(U_1)) \subset S_k(U_2),
\qquad
\overset{\circ}{S_k}(U_1) = \varphi_{21}^{-1}(\overset{\circ}{S_k}(U_2)).$$ We can generalize Lemma \[lem750\] (3) to the case when $U_1, U_2$ are orbifolds. Moreover by the condition (\[cornerorbemb\]) we can generalize Lemma \[lem750\] (1) to the embedding of orbifolds with corners. Thus $\varphi_{21}$ induces $\varphi^{\partial}_{21} : \partial U_1 \to \partial U_2$. In the same way $\widehat\varphi_{21} : \mathcal E_1 \to \mathcal E_2$ induces $\widehat\varphi_{21}^{\partial} : \mathcal E_1^{\partial}
\to \mathcal E_2^{\partial}$.
Thus the data consisting of the coordinate change of the Kuranishi charts given as in (2) (a),(b) are defined by (2) (c), except the underlying topological space $\partial X$, $\partial Z$ and parametrization $\psi$.
Below we will construct $\partial X$, $\partial Z$ and $\psi$.
We first consider the case of good coordinate system and $X = Z$. Let $\widetriangle{\mathcal U} =(\frak P,\{\mathcal U_{\frak p}\},\{\Phi_{\frak p\frak q}\})$. We consider $\partial U_{\frak p}$ and $\varphi_{\frak p\frak q}^{\partial}$, defined as above. We glue the spaces $\partial U_{\frak p}$ by $\varphi_{\frak p\frak q}^{\partial}$ and obtain a topological space $\vert\partial \widetriangle{\mathcal U}\vert$. The zero sets of the Kuranishi maps $s_{\frak p}^{\partial}$ on $\partial U_{\frak p}$ are glued to define a subspace of $\vert\partial \widetriangle{\mathcal U}\vert$, which we define to be $\partial X$. $\partial X$ is Hausdorff and metrizable.[^27] We define $\psi_{\frak p}^{\partial} : (s_{\frak p}^{\partial})^{-1}(0)
\to \partial X$ by mapping a point of $(s_{\frak p}^{\partial})^{-1}(0)$ to its equivalence class. Then $\mathcal U^{\partial}_{\frak p} = (\partial U_{\frak p},\mathcal E_{\frak p}^{\partial},s_{\frak p}^{\partial},
\psi_{\frak p}^{\partial})$ is a Kuranishi chart of $\partial X$. We put $\partial U_{\frak p\frak q} = U_{\frak p\frak q} \cap
\partial U_{\frak q}$. Then $\Phi_{\frak p\frak q}^{\partial} =
(\partial U_{\frak p\frak q},\varphi_{21}^{\partial},\widehat\varphi_{21}^{\partial})
$ is a coordinate change $\mathcal U^{\partial}_{\frak p}
\to \mathcal U^{\partial}_{\frak q}$. Thus we obtain a good coordinate system $\partial(X,Z;{\widetriangle{\mathcal U}})$ in case $Z = X$.
Next we consider the case of good coordinate system but $X \ne Z$. We glue the zero sets of Kuranishi map on $\partial U_{\frak p}$ in the same way as above to obtain a topological space $\partial X$. (See Remark \[rem8989\].) We define the subset $\partial Z \subset \partial X$ by $$\partial Z =
\bigcup_{\frak p \in \frak P}\{ x \in \partial U_{\frak p} \mid s_{\frak p}(x) = 0,
\psi_{\frak p}(\pi(x)) \in Z\}.$$ Here we identify $\partial U_{\frak p}$ with its image in $\vert\partial \widetriangle{\mathcal U}\vert$ and the union is taken in $\vert\partial \widetriangle{\mathcal U}\vert$. The rest of the proof is the same as the case of $X = Z$.
Finally we consider the case of Kuranishi structure. We take a good coordinate system $\widetriangle{\mathcal U}$ compatible to $\widehat{\mathcal U}$ and use the case of good coordinate system to define $\partial X$ and $\partial Z$.
It now remains to define the parameterization $\psi^{\partial}_p :
(s^{\partial}_p)^{-1}(0) \to \partial X$. Let $\mathcal U_p$ be a Kuranishi neighborhood of $p$ which is a part of the data of $\widehat{\mathcal U}$. In case the embedding $\widehat{\mathcal U} \to \widetriangle{\mathcal U}$ is strict, $(s^{\partial}_p)^{-1}(0) \subset U_{\frak p}$ for some $\frak p \in \frak P$. Therefore we obtain $\psi^{\partial}_p$ by restricting the parametrization map $\psi_{\frak p}$ of the good coordinate system $\partial\widetriangle{\mathcal U}$.
In the case $Z \ne X$ we replace $\widehat{\mathcal U}$ by its open substructure $\widehat{\mathcal U_0}$ such that there exists a strict embedding $\widehat{\mathcal U_0} \to \widetriangle{\mathcal U}$.
Suppose $Z = X$. We define $\psi^{\partial}_p : (s^{\partial}_p)^{-1}(0) \to \partial X$ (without taking open substructure) as follows. Let $x \in (s^{\partial}_p)^{-1}(0) \subset s_p^{-1}(0)$ and $q = \psi_p(x) \in X$. We have $o_q \in U_q$ such that $\varphi_{pq}(o_q) = x$. (\[cornerorbemb\]) implies $o_q \in \partial U_{q}$. Moreover $o_q \in \partial U_{0,q}$. (Here $U_{0,q}$ is the Kuranishi neighborhood of the open substructure $\widehat{\mathcal U_0}$.) Therefore $o_q$ may be regarded as an element of $\partial U_{\frak p}$ for some $\frak p \in \frak P$. We define $\psi_p(x)$ to be the equivalence class of $o_q \in \partial U_{\frak p}$, which is an element of $\partial X$.
Therefore the proof of the statement (2) is complete.
The statement (1) can be proved in the same way and the statement (3) is obvious from definition.
\[rem8989\] Here is a technical remark about the way to define underlying topological space $\partial X$ in Lemma-Definition \[lem754\].
1. Let $\widehat{\mathcal U}$ be a Kuranishi structure of $Z \subseteq X$. Then the parametrization $\psi_p : s_p^{-1}(0) \to X$ has $X$ as a target space. So $\widehat{\mathcal U}$ is [*not*]{} a Kuranishi structure of $Z$ itself.
2. The data consisting of $\widehat{\mathcal U}$ contain enough information to determine which points of $Z$ lie in the boundary. However the data do not contain such information for the points of $X \setminus Z$ which are far away from $Z$.
The space $\partial X$ that we defined in the proof of Lemma-Definition \[lem754\] consists of points which correspond to the ‘boundary points’ of $X$ that is sufficiently close to $Z$.
Since the image of $\psi_p$, $p\in Z$ lies in a neighborhood of $Z$, we need only a neighborhood of $Z$ in $X$ to define Kuranishi structure of $Z \subseteq X$. This is the reason why it suffices to define $\partial X$ in a neighborhood of $Z$.
3. On the other hand, as a consequence of (2), the topological space $\partial X$ is not canonically determined from $(X,Z;\widehat{\mathcal U})$. For example, the following phenomenon happens. Let $\widehat {\mathcal U}$ be a Kuranishi structure of $Z_2 \subseteq X$ and $Z_1 \subset \ring Z_2$. We restrict ${\mathcal U}$ to $Z_1$ to obtain ${\mathcal U}\vert_{Z_1}$. We consider $
\partial(X,Z_1;\widehat{\mathcal U}\vert_{Z_1})
$ and $
\partial(X,Z_2;\widehat{\mathcal U})
$. Let $(\partial_1X,\partial Z_1)$ and $(\partial_2X,\partial Z_2)$ be their underlying topological spaces. Then $\partial_1X$ may not be the same as $\partial_2X$.
There is no such an issue in the absolute case $Z=X$. In the applications the case $Z \ne X$ appears only together with a means of defining ${\partial}X$ given.
We note that all the arguments of Section \[sec:contfamily\] work for Kuranishi structure or good coordinate system with boundaries or corners.
The next lemma describes the way how to restrict CF-perturbations and etc. to the normalized boundary.
\[lemma755\] Let $\widetriangle{\mathcal U}$ be a good coordinate system of $Z \subseteq X$, ${\mathcal K}^1,{\mathcal K}^2,{\mathcal K}^3$ a triple of support systems of $\widetriangle{\mathcal U}$ with ${\mathcal K}^1 <{\mathcal K}^2<{\mathcal K}^3$, and $\widetriangle{\frak S^{\epsilon}}$ a CF-perturbation of $({\widetriangle{\mathcal U}},\mathcal K^3)$.
1. $\{\partial U_{\frak p} \cap \mathcal K^i_{\frak p}\}$ is a support system of $\partial (X,Z;{\widetriangle{\mathcal U}})$, which we denote by $\mathcal K^i_{\partial}$. ${\mathcal K}_{\partial}^1,{\mathcal K}_{\partial}^2,{\mathcal K}_{\partial}^3$ are support systems with ${\mathcal K}_{\partial}^1 < {\mathcal K}_{\partial}^2 < {\mathcal K}_{\partial}^3$. Here $\partial U_{\frak p} \cap \mathcal K^i_{\frak p}
= \pi^{-1}(\mathcal K^i_{\frak p}) \subset \partial U_{\frak p}$.
2. 1. For each $\frak p$ there exists ${\frak S}^{\partial}_{\frak p}$ that is a CF-perturbation of $\mathcal K^3_{\partial,\frak p} \subset \partial U_{\frak p}$.
2. The restriction of ${\frak S}^{\partial}_{\frak p}$ to $\overset{\circ}{S_0}(\partial U_{\frak p})$ is identified with the restriction of ${\frak S}_{\frak p}$ to $\overset{\circ}{S_1}(U_{\frak p})$ by the diffeomorphism in Lemma \[lem754\] (2)(e).
3. The collection $\{{\frak S}^{\partial}_{\frak p}\}$ is a CF-perturbation of $(\partial(X,Z;{\widetriangle{\mathcal U}}),\mathcal K^3_{\partial})$, which we denote by $\widetriangle{\frak S^{\partial}}$.
4. If $\widetriangle{\frak S}$ varies in a uniform family (in the sense of Definition \[uniformcongpert\]) then $\widetriangle{\frak S^{\partial}}$ varies in a uniform family.
3. 1. A strongly continuous map $\widetriangle f :
(X,{\widetriangle{\mathcal U}})\to M$ induces a strongly continuous map $\widetriangle{f_{\partial}}
: \partial(X,Z;{\widetriangle{\mathcal U}})\to M$.
2. The restriction of $\widetriangle{f_{\partial}}$ to $\overset{\circ}{S_0}(\partial(X,{\widetriangle{\mathcal U}}))$ coincides with the restriction of $\widetriangle f$ to $\overset{\circ}{S_1}(X,{\widetriangle{\mathcal U}})$.
3. If $\widetriangle f$ is strongly smooth (resp. weakly submersive), so is $\widetriangle{f_{\partial}}$.
4. 1. If $\widetriangle{\frak S}$ is transversal to $0$, so is $\widetriangle{\frak S^{\partial}}$.
2. If $\widetriangle f$ is strongly submersive with respect to $\widetriangle{\frak S}$ then $\widetriangle{f_{\partial}}$ is strongly submersive with respect to $\widetriangle{\frak S^{\partial}}$.
3. Let $g : N \to M$ be a smooth map between smooth manifolds. If $\widetriangle f$ is strongly smooth and weakly transversal to $g$ then so is $\widetriangle{f_{\partial}}$.
5. 1. A differential form $\widetriangle h$ on $(X,Z;{\widetriangle{\mathcal U}})$ induces a differential form on $\partial(X,Z;{\widetriangle{\mathcal U}})$, which we write $\widetriangle{h_{\partial}}$.
2. The restriction of $\widetriangle{h_{\partial}}$ to $\overset{\circ}{S_0}(\partial(X,Z;{\widetriangle{\mathcal U}}))$ coincides with the restriction of $\widetriangle h$ to $\overset{\circ}{S_1}(X,Z;{\widetriangle{\mathcal U}})$.
3. In particular, a strongly continuous function on $(X,Z,{\widetriangle{\mathcal U}})$ induces one on $\partial(X,Z;{\widetriangle{\mathcal U}})$, such that (b) above applies.
6. If $\{\chi_{\frak p}\}$ is a strongly smooth partition of unity of $(X,Z,{\widetriangle{\mathcal U}},\mathcal K^2,\delta)$ then $\{(\chi_{\frak p})_{\partial}\}$ is a strongly smooth partition of unity of $(\partial X,\partial Z,\partial{\widetriangle{\mathcal U}},\mathcal K_{\partial}^2,\delta)$. Here $(\chi_{\frak p})_{\partial}$ is one induced from $\{\chi_{\frak p}\}$ as in (5) (c) above.
In the case when $U$ is an orbifold with corners, various transversality or submersivity are defined by requiring the conditions not only to $ \overset{\circ}{S_0}(U)$ (the interior point) but also to all $\overset{\circ}{S_k}(U)$. Once we observe this point all the statements are obvious from the definition.
Stokes’ formula for a good coordinate system {#subsec:Stokesgcs}
--------------------------------------------
Now we are ready to state and prove Stokes’ formula.
[(Stokes’ formula, [@fooo09 Lemma 12.13])]{}\[Stokes\] Assume that we are in the situation of Lemma \[lemma755\] (1), (2), (3) (a)(b)(c), (4) (a)(b), (5) (a)(b). Then, for each sufficiently small $\epsilon>0$, we have $$d\left(\widetriangle f!(\widetriangle h;\widetriangle{{\frak S}^{\epsilon}})\right)
=
\widetriangle f!(d\widetriangle h;\widetriangle{{\frak S}^{\epsilon}})
+
\widetriangle f_{\partial}!(\widetriangle {h_{\partial}};\widetriangle{{\frak S}_{\partial}^{\epsilon}}).$$
Let $\{\chi_{\frak p}\}$ be a strongly smooth partition of unity of $(X,Z,{\widetriangle{\mathcal U}},\mathcal K^2,\delta)$. Let $(\chi_{\frak p})_{\partial}$ and ${\mathcal K}_{\partial}^1$ be defined by Lemma \[lemma755\]. We put $h_0 = \chi_{\frak p} h_{\frak p}$. It suffices to show $$\label{eq722}
\aligned
&d\left(f_{\frak p}!(h_0;{\frak S}_{\frak p}^{\epsilon})\vert_{\frak U(Z) \cap \mathcal K_{\frak p}^1(2\delta)}\right)
\\
&=
f_{\frak p}!(dh_0;{\frak S}_{\frak p}^{\epsilon}
\vert_{\frak U(Z) \cap \mathcal K_{\frak p}^1(2\delta)})
+
f^{\partial}_{\frak p}!(h_0;{\frak S}^{\partial,\epsilon}_{\frak p}
\vert_{\frak U(Z) \cap \mathcal K_{\partial,\frak p}^1(2\delta)}),
\endaligned$$ where $
\widetriangle{{\frak S}^{\partial,\epsilon}}
= \{{\frak S}^{\partial,\epsilon}_{\frak p} \mid \frak p \in \frak P\}.
$ Let ${\frak S}_{\frak p} = \{(\frak V_{\frak r},\mathcal S_{\frak r}^{\frak p}) \mid \frak r \in \frak R\}$ and $\{\chi_{\frak r}\}$ a partition of unity subordinate to $\{U_{\frak r}\}$. We put $h_1 = \chi_{\frak r}h_0$ and $f_{\frak r}
= f_{\frak p}\vert_{U_{\frak r}}$. To prove (\[eq722\]), it suffices to prove: $$\label{eq723}
\aligned
&d\left(f_{\frak r}!(h_1;{\mathcal S}_{\frak r}^{\epsilon})\vert_{\frak U(Z) \cap U_{\frak r}}\right)
\\
&=
f_{t,\frak r}!(dh_1;{\mathcal S}_{\frak r}^{\epsilon})\vert_{\frak U(Z) \cap U_{\frak r}})
+
f_{\frak r}^{\partial}!(h_1;{\mathcal S}^{\partial,\epsilon}_{\frak r}\vert_{\frak U(Z) \cap \partial U_{\frak r}}),
\endaligned$$ where $
\widetriangle{{\frak S}^{\epsilon}_{\partial,\frak p}}
=
\{(\partial \frak V_{\frak r},\mathcal S^{\partial,\epsilon}_{\frak r})\}
$. (\[eq723\]) follows from the next lemma.
\[lem759sss\] Let $\Omega$ be an open neighborhood of $0$ in $[0,1)^m \times \R^{n-m}$ and $f : \Omega \to M$ a smooth map. Let $h$ be a smooth differential $k$ form on $\Omega$ with compact support and $\rho$ a differential $(n-k-1)$-form on $M$. Then we have $$(-1)^k\int_{\Omega} h \wedge f^*d\rho
=
\int_{\Omega \cap \partial ([0,1)^m \times \R^{n-m})}h
\wedge f^*\rho
+
\int_{\Omega} dh \wedge \rho.$$
Lemma \[lem759sss\] is an immediate consequence of the usual Stokes’ formula. Thus the proof of Theorem \[Stokes\] is complete.
Using Stokes’ formula we can immediately prove the following basic properties of smooth correspondence.
\[Stokescorollary\] In Situation \[smoothcorr\] we apply Construction \[cordefjyunbi\]. Let $
{\rm Corr}_{((X,\widetriangle{\mathcal U}),\widetriangle{{\frak S}^{\epsilon}})}
:
\Omega^k(M_s) \to \Omega^{\ell+k}(M_t)
$ be the map obtained by Definition \[defn748\]. (Here $\ell = \dim M_t - \dim (X,\widehat{\mathcal U})$.) We define the boundary by $$\partial (X,\widetriangle{\mathcal U}) = (\partial(X,\widetriangle{\mathcal U}),
\widetriangle f_s\vert_{\partial(X,\widetriangle{\mathcal U})},
\widetriangle f_t\vert_{\partial(X,\widetriangle{\mathcal U})}).$$ $\widetriangle{{\frak S}^{\epsilon}}$ induces a CF-perturbation $\widetriangle{{\frak S}^{{\partial},\epsilon}}$ of it as in Lemma \[lemma755\] (4). $\partial (X,\widetriangle{\mathcal U})$ and $\widetriangle{{\frak S}^{{\partial},\epsilon}}$ define a map $
{\rm Corr}_{(\partial(X,\widetriangle{\mathcal U}), \widetriangle{\frak S^{{\partial},\epsilon}})}
:
\Omega^k(M_s) \to \Omega^{\ell+k+1}(M_t).
$ Then for any sufficiently small $\epsilon >0$, we have $$d \circ {\rm Corr}_{((X,\widetriangle{\mathcal U}),\widetriangle{{\frak S}^{\epsilon}})}
-
{\rm Corr}_{((X,\widetriangle{\mathcal U}),\widetriangle{{\frak S}^{\epsilon}})}
\circ d
=
{\rm Corr}_{(\partial(X,\widetriangle{\mathcal U}) ,\widetriangle{\frak S^{\partial,\epsilon}})}.$$ In particular, ${\rm Corr}_{((X,\widetriangle{\mathcal U}),\widetriangle{{\frak S}^{\epsilon}})}$ is a chain map if $\widehat{\mathcal U}$ is a Kuranishi structure without boundary.
This is immediate from Theorem \[Stokes\].
\[lem761rev\] We assume $\widetriangle{\frak S_{\sigma}}$ is a uniform family in the sense of Definition \[uniformcongpert\]. Then the positive number $\epsilon$ in Theorem \[Stokes\] and Corollary \[Stokescorollary\] can be taken independent of $\sigma$.
The proof is the same as that of Lemma \[lem761\].
Well-defined-ness of virtual fundamental cycle {#subsec:JFCstokes}
----------------------------------------------
We use Corollary \[Stokescorollary\] to prove well-defined-ness of virtual cohomology class, and well-defined-ness of the smooth correspondence [*in the cohomology level*]{}, when Kuranishi structure has no boundary.
\[relextendgood\] Consider Situation \[smoothcorr\] and assume that our Kuranishi structure on $X$ has no boundary. Then the map $
{\rm Corr}_{(\frak X,\widetriangle{{\frak S}^{\epsilon}})}
:
\Omega^k(M_s) \to \Omega^{\ell+k}(M_t)
$ defined in Definition \[defn748\] is a chain map.
Moreover, provided $\epsilon$ is sufficiently small, the map ${\rm Corr}_{(\frak X,\widetriangle{{\frak S}^{\epsilon}})}$ is independent of the choices of our good coordinate system ${\widetriangle{\mathcal U}}$ and CF-perturbation $\widetriangle{\frak S}$ and of $\epsilon>0$, up to chain homotopy.
The first half is repetition of Corollary \[Stokescorollary\]. We will prove the independence of the definition up to chain homotopy below. Let ${\widetriangle{\mathcal U}}$, ${\widetriangle{\mathcal U'}}$ be two choices of good coordinate system and $\widetriangle{\frak S}$, $\widetriangle{{\frak S}^{\prime}}$ CF-perturbations of $(X;{\widetriangle{\mathcal U}})$, $(X;{\widetriangle{\mathcal U'}})$ respectively. (During the proof of Proposition \[relextendgood\], we do not need to discuss the choice of support system, since the correspondence map is independent of it.)
We put direct product Kuranishi structure on $X\times [0,1]$. During the proof of Proposition \[relextendgood\], we do not need to make a specific choice of support system because Proposition \[indepofukuracont\] (see also Remark \[rem:785\]) shows the map ${\rm Corr}_{(\frak X,\widetriangle{{\frak S}^{\epsilon}})}$ is independent thereof. We identify $X = X \times \{0\}$. Then the good coordinate system ${\widetriangle{\mathcal U}}$ induces ${\widetriangle{\mathcal U}}\times [0,1/3)$ on $X \times [0,1/3)$ such that $\partial
(X \times [0,1/3),{\widetriangle{\mathcal U}}\times [0,1/3))
$ is isomorphic to $
(X;{\widetriangle{\mathcal U}})
$. Similarly we have a good coordinate system ${\widetriangle{\mathcal U'}}\times (2/3,1]$ on $X \times (2/3,1]$ such that $\partial(X \times (2/3,1];{\widetriangle{\mathcal U'}}\times (2/3,1])$ is isomorphic to $
(X;{\widetriangle{\mathcal U'}})
$ with opposite orientation. Here the notion of isomorphism of good coordinate system is defined in Definition \[defn31222\]. Then, by Proposition \[prop7582752\], there exists a good coordinate system ${\widetriangle{\mathcal U'}}\times [0,1]$ such that $$\label{boundarycobor77}
\partial
(X\times [0,1];{\widetriangle{\mathcal U'}}\times [0,1])
=
(X;{\widetriangle{\mathcal U}})
\cup
-(X,{\widetriangle{\mathcal U'}}).$$
We next consider two choices of CF-perturbations, which we denote by $\widetriangle{\frak S}$ and $\widetriangle{{\frak S}^{\prime}}$. We assume that $\widetriangle{f_t}$ is strongly submersive with respect to both of them. We define $\widetriangle{\frak S}\times [0,1/3)$ and $\widetriangle{{\frak S}'} \times (2/3,1]$ as follows.
We consider Situation \[smoothcorrsingle\]. Let $\frak V_x = (V_x,\Gamma_x,E_x,\phi_x,\widehat\phi_x)$ be an orbifold chart of $(U,\mathcal E)$ and $\mathcal S_x = (W_x,\omega_x,{\frak s}_{x}^{\epsilon})$ a CF-perturbation on it. (Definition \[defn73ss\].) Suppose $(f_t)_x$ is strongly submersive with respect to $\mathcal S_x$. We take $\frak V_x
\times [0,1/3) = (V_x\times [0,1/3) ,\Gamma_x,E_x\times [0,1/3) ,\phi_x\times
{\rm id},\widehat\phi_x\times {\rm id})$ that is an orbifold chart of $(U\times [0,1/3),\mathcal E\times [0,1/3))$. In an obvious way $\mathcal S_x$ induces a CF-perturbation of it, with respect to which $(f_t)_x \circ \pi$ is strongly submersive. Here $\pi : X \times [0,1/3) \to X$ is the projection. We denote it by $\mathcal S_x\times [0,1/3)$. (See Definition \[defn1022\] for detail.)
We perform this construction of multiplying $[0,1/3)$ for each chart (once for each orbifold chart and once for each Kuranishi chart) then it is fairly obvious that they are compatible with various coordinate changes. Thus we obtain $\widetriangle{\frak S} \times [0,1/3)$ that is a CF-perturbation of $X \times [0,1/3)$. We obtain $\widetriangle{{\frak S}'} \times (2/3,1]$ in the same way.
Now we use Proposition \[existperturbcontrel\] with $Z_1 = X \times \{0,1\}$, $Z_2 = X \times [0,1]$. Then we obtain a CF-perturbation $\widetriangle{{\frak S}^{[0,1]}}$ of $X \times [0,1]$ such that its restriction to $X \times \{0\}$ and $X \times \{1\}$ are $\widetriangle{\frak S}$ and $\widetriangle{{\frak S}'}$, respectively.
Now we use Corollary \[Stokescorollary\] and (\[boundarycobor77\]) to show: $$\label{chomotopyrelation}
\aligned
&d\circ {\rm Corr}_{(\frak X\times [0,1],
\widetriangle{{{\frak S}^{[0,1]}}^{\epsilon}})}
+
{\rm Corr}_{(\frak X\times [0,1],\widetriangle{{{\frak S}^{[0,1]}}^{\epsilon}})}\circ d \\
&=
{\rm Corr}_{(\frak X,\widetriangle{{\frak S}^{\epsilon}})}
-
{\rm Corr}_{(\frak X,\widetriangle{{\frak S}^{ \prime \epsilon}})}.
\endaligned$$ The independence of sufficiently small $\epsilon > 0$ follows from the following facts: For each $c > 0$ the family $\epsilon \mapsto \widetriangle{{\frak S}^{c\epsilon}}$ is also a CF-perturbation.
The proof of Proposition \[relextendgood\] is complete.
Therefore in the situation of Proposition \[relextendgood\], the correspondence map ${\rm Corr}_{(\frak X,\widetriangle{{\frak S}^{\epsilon}})}$ on differential forms descends to a map on cohomology which is independent of the choices of $\widetriangle{\mathcal U}$ and $\widetriangle{{\frak S}^{\epsilon}}$. We write the cohomology class as $[{\rm Corr}_{\frak X}(h)] \in H(M_t)$ for any closed differential form $h$ on $M_s$ by removing $\widetriangle{{\frak S}^{\epsilon}}$ from the notation.
In Proposition \[relextendgood\] we fixed our Kuranishi structure $\widehat{\mathcal U}$ on $X$. In fact, we can prove the same conclusion under milder assumption.
\[cobordisminvsmoothcor\] Let $\frak X_i = ((X_i,\widehat{\mathcal U^i}),\widehat f_s^i,\widehat f_t^i)$ be smooth correspondence from $M_s$ to $M_t$ such that $\partial X_i = \emptyset$. Here $i=1,2$ and $M_s$, $M_t$ are independent of $i$. We assume that there exists a smooth correspondence $\frak Y = ((Y,\widehat{\mathcal U}),\widehat f_s,\widehat f_t)$ from $M_s$ to $M_t$ with boundary (but without corner) such that $$\partial \frak Y = \frak X_1 \cup -\frak X_2.$$ Here $-\frak X_2$ is the smooth correspondence $\frak X_2$ with opposite orientation. Then we have $$\label{chomotopyrelation22}
[{\rm Corr}_{\frak X_1}(h)] = [{\rm Corr}_{\frak X_2}(h)]
\in H(M_t),$$ where $h$ is a closed differential form on $M_s$.
We take a good coordinate system $\widetriangle{\mathcal U}$ of $Y$ and a KG embedding $(Y,\widehat{\mathcal U}) \to (Y,\widetriangle{\mathcal U})$. (Theorem \[Them71restate\].) $\widehat f_t$ is pulled back from $\widetriangle f_t : (Y,\widetriangle{\mathcal U}) \to M_t$. $\widehat f_s$ is also pulled back from $\widetriangle f_s$ (Proposition \[le614\] (2).) We also obtain a CF perturbation $\widetriangle{\frak S}$ of $\widetriangle{\mathcal U}$ with respect to which $\widetriangle f$ is strongly submersive (Theorem \[existperturbcont\] (2)). They restrict to $(X_i,\widetriangle{\mathcal U_i})$, $\widetriangle{\frak S_i}$ and $\widetriangle f_t^i : (Y,\widetriangle{\mathcal U}) \to M_t$, $\widetriangle f_s^i$.
We remark that $\widetriangle f_t^i$ is strongly transversal to $\widetriangle{\frak S_i}$. (This is the consequence of the definition of strong transversality. Namely we required the transversality on each of the strata of corner structure stratification (Definition \[defn417\]).)
By Proposition \[relextendgood\] we can use $(X_i,\widetriangle{\mathcal U_i})$, $\widetriangle{\frak S_i}$, $\widetriangle f_t^i$, $\widetriangle f_s^i$ to define smooth correspondence ${\rm Corr}_{\frak X_i}$ (in the cohomology level.)
The proposition now follows from Corollary \[Stokescorollary\] and (\[boundarycobor77\]) applied to $(Y,\widetriangle{\mathcal U})$, $\widetriangle{\frak S}$, $\widetriangle f_t$, $\widetriangle f_t$. Namely we can calculate in the same way as (\[chomotopyrelation\]).
The proofs of Propositions \[relextendgood\], \[cobordisminvsmoothcor\] (Formulae (\[chomotopyrelation\]), (\[chomotopyrelation22\])) are a prototype of the proofs of various similar equalities which appear in our construction of structures and proof of its independence. We will apply a similar method in a slightly complicated situation in Part 2 systematically.
From good coordinate system to Kuranishi structure and back with CF-perturbations {#sec:kuraandgood}
=================================================================================
As we explained at the end of Section \[sec:fiber\], it is more canonical to define the notion of fiber product of spaces with Kuranishi structure than to define that of fiber product of spaces with good coordinate system. On the other hand, in Section \[sec:contfamily\], we gave the definition of CF-perturbation and of the pushout of differential forms by using good coordinate system. In this section, we describe the way how we go from a good coordinate system to a Kuranishi structure and back together with CF-perturbations on them, and prove Theorem \[theorem915\] that we can define the pushout by using Kuranishi structure itself in the way that the outcome is independent of auxiliary choice of good coordinate system.
CF-perturbation and embedding of Kuranish structure {#subsec:contfamiKura}
---------------------------------------------------
\[defn81\] Let $\widehat{\mathcal U}$ be a Kuranishi structure on $Z \subseteq X$. A [*CF-perturbation $\widehat{\frak S}$ of $\widehat{\mathcal U}$*]{} assigns $\frak S_p$ for each $p \in Z$ with the following properties.
1. $\frak S_p$ is a CF-perturbation of $\mathcal U_p$.
2. If $q \in {\rm Im}(\psi_{p}) \cap Z$, then $\frak S_p$ can be pulled back by $\Phi_{pq}$. Namely $$\frak S_p \in \mathscr S^{\mathcal U_q \triangleright \mathcal U_p}(U_p).$$
3. If $q \in {\rm Im}(\psi_{p}) \cap Z$, then $\frak S_p$, $\frak S_{q}$ are compatible with $\Phi_{pq}$. Namely $$\label{form91}
\Phi_{pq}^*(\frak S_p) = \frak S_q\vert_{U_{pq}} \in \mathscr S^{\mathcal U_q}(U_{pq}).$$
\[defn929292\] Suppose we are in the situation of Definition \[defn81\]. Let $\widehat f :
(X,Z;\widehat{\mathcal U}) \to M$ be a strongly smooth map. Here $M$ is a smooth manifold.
1. We say $\widehat{\frak S}$ is strictly [*transversal to $0$*]{} if each $\frak S_p$ is transversal to $0$. We say $\widehat{\frak S}$ is [*transversal to $0$*]{} if its restriction to an open substructure is strictly so.
2. We say $\widehat f$ is strictly [*strongly submersive with respect to $\widehat{\frak S}$*]{} if each of $f_p$ is strongly submersive with respect to $\frak S_p$. We say $\widehat f$ is [*strongly submersive with respect to $\widehat{\frak S}$*]{} if its restriction to an open substructure is strictly so.
3. We say $\widehat f$ is strictly [*strongly transversal to $g: N \to M$ with respect to $\widehat{\frak S}$*]{} if each of $f_p$ is strongly transversal to $g: N \to M$ with respect to $\frak S_p$. We say $\widehat f$ is [*strongly transversal to $g: N \to M$ with respect to $\widehat{\frak S}$*]{} if its restriction to an open substructure is strictly so.
We next define compatibility of CF-perturbations with various embeddings of Kuranishi structures and/or good coordinate systems and prove versions of several lemmata in Section \[sec:multisection\] corresponding to the current context of CF-perturbations.
\[defn83k\] Let $\widehat{\mathcal U}$ and $\widehat{\mathcal U^+}$ be Kuranishi structures of $Z \subseteq X$, ${\widetriangle{\mathcal U}}$ and ${\widetriangle{\mathcal U^+}}$ good coordinate systems of $Z \subseteq X$. Let $\mathcal K$ and $\mathcal K^+$ be support systems of ${\widetriangle{\mathcal U}}$ and ${\widetriangle{\mathcal U^+}}$, respectively. Let $\widehat{\frak S}$, $\widehat{\frak S^+}$, ${\widetriangle{\frak S}}$, ${\widetriangle{\frak S^+}}$ be CF-perturbations of $\widehat{\mathcal U}$, $\widehat{\mathcal U^+}$, $({\widetriangle{\mathcal U}},\mathcal K)$, $({\widetriangle{\mathcal U^+}},\mathcal K^+)$, respectively.
1. Let $\widehat\Phi : \widehat{\mathcal U} \to \widehat{\mathcal U^+}$ be a strict KK-embedding. We say $\widehat{\frak S}$, $\widehat{\frak S^+}$ are [*compatible*]{} with $\widehat\Phi$ if the following holds for each $p$.
1. $\frak S^+_p \in \mathscr S^{\mathcal U_p \triangleright \mathcal U^+_p}(U_{p}).$ Here we use the embedding $\Phi_{p}$ to define the subsheaf $\mathscr S^{\mathcal U_p \triangleright \mathcal U^+_p}$.
2. $
\Phi_{p}^*(\frak S^+_p) = \frak S_p \in \mathscr S^{\mathcal U_p}(U_{p})$.
2. Let $\widehat\Phi : \widehat{\mathcal U} \to \widehat{\mathcal U^+}$ be a KK-embedding. We say $\widehat{\frak S}$, $\widehat{\frak S^+}$ are [*compatible*]{} with $\widehat\Phi$ if there exist an open substructure $\widehat{\mathcal U_0}$, a CF-perturbation $\widehat{\frak S_0}$ of $\widehat{\mathcal U_0}$ and a strict KK-embedding $\widehat\Phi_0 : \widehat{\mathcal U_0} \to \widehat{\mathcal U^+}$ such that $\widehat{\frak S_0}$, $\widehat{\frak S^+}$ are compatible with $\widehat\Phi_0$ and $\widehat{\frak S_0}$, $\widehat{\frak S}$ are compatible with the open embedding $\widehat{\mathcal U_0} \to \widehat{\mathcal U}$.
3. Let $\widetriangle\Phi = (\{\Phi_{\frak p}\},\frak i) : {\widetriangle{\mathcal U}} \to {\widetriangle{\mathcal U^+}}$ be a GG-embedding. We say that $\mathcal K, \mathcal K^+$ is [*compatible*]{} with $\widehat\Phi$ if $
\varphi_{\frak p}(\mathcal K_{\frak p}) \subset \ring\mathcal K^+_{\frak i(\frak p)}
$ for each $\frak p \in \frak P$.
4. In the situation of (3), we say ${\widetriangle{\frak S}}$, ${\widetriangle{\frak S^+}}$ are [*compatible*]{} with $\widehat\Phi$ if the following holds for each $\frak p \in \frak P$.
1. ${\frak S}^+_{\frak i(\frak p)} \in \mathscr S^{\mathcal U_{\frak p} \triangleright \mathcal U^+_{\frak i(\frak p)}}(\mathcal K_{{\frak i(\frak p)}}).$ Here we use the embedding $\Phi_{\frak p} : \mathcal U_{\frak p} \to \mathcal U^+_{\frak i(\frak p)}$ to define the subsheaf $\mathscr S^{\mathcal U_{\frak p} \triangleright \mathcal U^+_{\frak i(\frak p)}}$.
2. $
\Phi_{\frak p}^*(\frak S^+_{\frak i(\frak p)}) = \frak S_{\frak p} \in \mathscr S^{\mathcal U_{\frak p}}(\mathcal K_{\frak p})$.
5. Let $\widehat\Phi : {\widehat{\mathcal U}} \to {\widetriangle{\mathcal U}}$ be a strict KG-embedding. We say ${\widehat{\frak S}}$, ${\widetriangle{\frak S}}$ are [*compatible*]{} with $\widehat\Phi$ if the following holds for each $\frak p$ and $p \in \psi_{\frak p}(\mathcal K_{\frak p}\cap s_{\frak p}^{-1}(0)) \cap Z$.
1. ${\frak S}_{\frak p} \in \mathscr S^{\mathcal U_{p} \triangleright \mathcal U_{\frak p}}(\mathcal K_{\frak p}).$ Here we use the embedding $\Phi_{\frak p p} : \mathcal U_p \to \mathcal U_{\frak p}$ to define the subsheaf $ \mathscr S^{\mathcal U_{p} \triangleright \mathcal U_{\frak p}}$.
2. $\Phi_{p}^*({\frak S}_{\frak p}) = {\frak S}_{p} \in \mathscr S^{\mathcal U_{p}}(U_{p})$.
6. In case $\widehat\Phi : {\widehat{\mathcal U}} \to {\widetriangle{\mathcal U}}$ is a KG-embedding, we can define compatibility of ${\widehat{\frak S}}$, ${\widetriangle{\frak S}}$ with $\widehat\Phi$ in the same way as Item (2) (using Items (1) and (5)).
7. Let $\widehat\Phi
= (\{U_{\frak p}(p)\},\{
\Phi_{p \frak p}\}) : {\widetriangle{\mathcal U}} \to \widehat{\mathcal U}$ be a GK-embedding. We say ${\widetriangle{\frak S}}$, $\widehat{\frak S}$ are [*compatible*]{} with $\widehat\Phi$ if the following holds for each $\frak p$ and $p \in \psi_{\frak p}(\mathcal K_{\frak p} \cap s_{\frak p}^{-1}(0)) \cap Z$.
1. ${\frak S}_{p} \in \mathscr S^{\mathcal U_{\frak p} \triangleright \mathcal U_{p}}(U_{p}).$ Here we use the embedding $\Phi_{p \frak p} : \mathcal U_{\frak p}\vert_{U_{\frak p}(p)}
\to \mathcal U_p$ to define the subsheaf $\mathcal U_{\frak p} \triangleright \mathcal U_{p}$.
2. $\Phi_{p\frak p}^*{\frak S}_p = \widehat{\frak S}_{\frak p}\vert_{U_{\frak p}(p)}
\in \mathscr S^{\mathcal U_{p}}(U_{\frak p}(p))$.
With these definitions of compatibility, we now prove the compatibilities relevant to various embeddings.
\[lem9494\] Let $\widetriangle\Phi = (\{\Phi_{\frak p}\},\frak i) : \widetriangle{\mathcal U} \to \widetriangle{\mathcal U^+}$ be a GG-embedding.
1. If $\mathcal K$ is a support system of $\widetriangle{\mathcal U}$, then there exist a support system $\mathcal K^+$ of $\widetriangle{\mathcal U^+}$ such that $\mathcal K$, $\mathcal K^+$ are compatible with $\widetriangle\Phi$.
2. If $\mathcal K_i$ ($i=1,\dots,m$) are support systems of $\widetriangle{\mathcal U}$ with $\mathcal K_i < \mathcal K_{i+1}$ then there exist support systems $\mathcal K^+_i$ ($i=1,\dots,m$) of $\widetriangle{\mathcal U^+}$ such that $\mathcal K_i$, $\mathcal K_i^+$ are compatible with $\widetriangle\Phi$ and $\mathcal K^+_i < \mathcal K^+_{i+1}$.
\(1) Let $\mathcal K = (\mathcal K_{\frak p})$. Let $
\mathcal K_{0,\frak p_+}$ be a closure of a sufficiently small neighborhood of $\bigcup_{\frak p \in \frak P \atop
\frak i(\frak p) = \frak p_+} \varphi_{\frak p}(\mathcal K_{\frak p})
$ for $\frak p_+ \in \frak P_+$. It is easy to see that $\mathcal K_{0}^+ = (\mathcal K_{0,\frak p_+})$ is a support system of $\widetriangle{\mathcal U^+}$. Any $\mathcal K^+ > \mathcal K_{0}^+$ has required properties. The proof of (2) is similar by using upward induction on $i$.
It seems possible to prove the following. For each $\mathcal K^+$ there exists $\mathcal K$ such that $\mathcal K$, $\mathcal K^+$ are compatible with $\widetriangle\Phi$. Its proof seems to be more complicated than that of Lemma \[lem9494\]. We do not try to prove it here since we do not use it.
\[compatiwithwkemb\] Let $\widetriangle{\Phi} : {\widetriangle{\mathcal U}} \to {\widetriangle{\mathcal U^+}}$ be a weakly open GG-embedding and $\mathcal K$, $\mathcal K^+$ support systems of ${\widetriangle{\mathcal U}}$, ${\widetriangle{\mathcal U^+}}$, respectively, which are compatible with $\widetriangle\Phi$. Then for any CF-perturbation $\widetriangle{\frak S^+}$ of $({\widetriangle{\mathcal U^+}},\mathcal K^+)$, there exists a unique CF-perturbation $\widetriangle{\frak S}$ of $({\widetriangle{\mathcal U}},\mathcal K)$ such that $\widetriangle{\frak S^+}$ and $\widetriangle{\frak S}$ are compatible with $\widetriangle{\Phi}$.
For any $\frak p \in \frak P$ we restrict ${\frak S_{\frak i(\frak p)}^+}$ to ${\mathcal U}_{\frak p}$ to obtain ${\frak S}_{\frak p}$. We thus obtain $\widetriangle{\frak S}$ . Since normal bundles are trivial in the case of weakly open embedding, the compatibility is automatic.
\[lem9797\] Various transversality or submersivity of the target of an open KK-embedding imply those of the source. The same holds for a weakly open GG-embedding.
This is an easy consequence of the definition.
The notion of compatibility of CF-perturbations to embeddings is preserved by the composition of embeddings of various kinds.
The proof is obvious.
The next lemma is a CF-perturbation version of Proposition \[lemappgcstoKu\].
\[lemappgcstoKucont\] In the situation of Proposition \[lemappgcstoKu\], let $\mathcal K_0$, $\mathcal K$ be support systems of $\widetriangle{\mathcal U_0}$, ${\widetriangle{\mathcal U}}$, respectively, which are compatible with the open embedding $\widetriangle{\mathcal U_0}\to {\widetriangle{\mathcal U}}$. Let $\widetriangle{\frak S}$ be a CF-perturbation of $({\widetriangle{\mathcal U}},\mathcal K)$, which restricts to a CF-perturbation ${\widetriangle{\frak S_0}}$ of $({\widetriangle{\mathcal U_0}},\mathcal K_0)$. Then the following holds.
1. There exists a CF-perturbation $\widehat{\frak S}$ of $\widehat{\mathcal U}$ such that $\widetriangle{\frak S_0}$ and $\widehat{\frak S}$ are compatible with the GK-embedding ${\widetriangle{\mathcal U_0}} \to \widehat{\mathcal U}$.
2. In the situation of Proposition \[lemappgcstoKu\] (2), if ${\widetriangle f}$ is strongly submersive with respect to ${\widetriangle{\frak S}}$, then ${\widehat f}$ is strongly submersive with respect to $\widehat{\frak S}$. The transversality to $M \to Y$ is also preserved.
The proof of Lemma \[lemappgcstoKucont\] is the same as that of Proposition \[lemappgcstoKu\]. In fact, the Kuranishi chart of $\widehat{\mathcal U}$ is a restriction of a Kuranishi chart of ${\widetriangle{\mathcal U_0}}$. Since $\mathcal U_{0,\frak p} \subset \mathcal K_{\frak p}$ (see Proposition \[lemappgcstoKu\] (1)), we can restrict ${\widetriangle{\frak S}}$ to the Kuranishi charts of $\widehat{\mathcal U}$.
We next state CF-perturbation versions of Propositions \[le614\] and \[pro616\]. (In Lemmas \[le714\] and \[le7155\] we do not specify support system for the CF-perturbations of good coordinate system. We take one but do not mention them.)
\[le714\] Let $\widehat{\mathcal U}$ be a Kuranishi structure on $Z \subseteq X$. Then we can take a good coordinate system $\widetriangle{\mathcal U}$ and the KG-embedding $\widehat{\Phi} : \widehat{\mathcal U_0} \to \widetriangle{\mathcal U}$ in Theorem \[Them71restate\] so that the following holds in addition.
1. If $\widehat h$ is a differential form of $\widehat{\mathcal U}$, then there exists a differential form ${\widetriangle h}$ on ${{\widetriangle {\mathcal U}}}$ such that $\widehat{\Phi}^*({\widetriangle h}) = \widehat h\vert_{\widehat{\mathcal U_0}}$. If $\widehat h$ has a compact support in $\ring Z$, then ${\widetriangle h}$ has a compact support in $\vert\widetriangle {\mathcal U}\vert$ and ${\rm Supp}({\widetriangle h}) \cap Z \subset \overset{\circ} Z$.
2. If $\widehat{\frak S}$ a CF-perturbation of ${\widehat{\mathcal U}}$, then there exists a CF-perturbation ${\widetriangle{\frak S}}$ of ${\widetriangle{\mathcal U}}$ such that ${\widehat{\frak S}}\vert_{\widehat{\mathcal U_0}}$ and ${\widetriangle{\frak S}}$ are compatible with the KG-embedding $\widehat{\Phi}$.
3. In the situation of (2) the following holds.
1. If $\widehat{\frak S}$ is transversal to $0$ then so is ${\widetriangle{\frak S}}$.
2. If ${\widehat f}$ is strongly submersive with respect to $\widehat{\frak S}$, then ${\widetriangle f}$ is strongly submersive with respect to ${\widetriangle{\frak S}}$.
3. If ${\widehat f}$ is strongly transversal to $g : M \to Y$ with respect to ${\widehat{\frak S}}$ then ${\widetriangle f}$ is strongly transversal to $g$ with respect to ${\widetriangle{\frak S}}$.
\[le7155\] Suppose we are in the situation of Propositions \[prop518\] (resp. Proposition \[prop519\]) and \[le614\]. Then we can take the GK-embedding $\widehat{\Phi^+} : \widetriangle{\mathcal U} \to
\widehat{\mathcal U^+}$ in Proposition \[prop518\] (resp. the GK-embeddings $\widehat{\Phi^+_a} : \widetriangle{\mathcal U} \to
\widehat{\mathcal U^+_a}$ in Proposition \[prop519\] ($a=1,2$)) so that the following holds.
1. If ${\widehat{\frak S^+}}$ is a CF-perturbation of ${\widehat{\mathcal U^+}}$ such that ${\widehat{\frak S^+}}$, ${\widehat{\frak S}}$ are strongly compatible with the embedding ${\widehat{\mathcal U}} \to \widehat{\mathcal U^+}$, then we may choose $\widetriangle{\frak S}$ such that $\widetriangle{\frak S}$, $\widehat{\frak S^+}$ are compatible with the embedding $\widehat{\Phi^+}$. (resp. If ${\widehat{\frak S^+_a}}$ ($a=1,2$) is a CF-perturbation of ${\widehat{\mathcal U^+_a}}$ such that ${\widehat{\frak S^+_a}}$, ${\widehat{\frak S}}$ are strongly compatible with the embedding ${\widehat{\mathcal U}} \to \widehat{\mathcal U^+}$, then we may choose $\widetriangle{\frak S}$ such that $\widetriangle{\frak S}$, $\widehat{\frak S^+_a}$ are both compatible with the embedding $\widehat{\Phi^+_a}$.)
2. If $\widehat{\frak S}$ is transversal to $0$, so is $\widetriangle{\frak S}$.
3. In the situation of Proposition \[pro616\] suppose $Y$ is a manifold $M$. Then if ${\widehat f}$ is strongly submersive with respect to ${\widehat{\frak S}}$, then ${\widetriangle{f}}$ is strongly submersive with respect to $\widetriangle{\frak S}$.
4. In the situation of Proposition \[pro616\] suppose $M$ is a manifold. Then if ${\widehat f}$ is strongly transversal to $g : N \to M$ with respect to ${\widehat{\frak S}}$, then ${\widetriangle{f}}$ is strongly transversal to $g : N \to M$ with respect to $\widetriangle{\frak S}$.
The proofs of Lemmata \[le714\] and \[le7155\] are given in Subsection \[subsec:movingmulsectionetc\].
Integration along the fiber (pushout) for Kuranishi structure {#subsec:intKurast}
-------------------------------------------------------------
\[sitsu8main\] Let $\widehat{\mathcal U}$ be a Kuranishi structure on $X$ and ${\widehat{\frak S}}$ a CF-perturbation of $(X,Z;\widehat{\mathcal U})$. Let $\widehat f : (X,Z;\widehat{\mathcal U}) \to M$ be a strongly smooth map that is strongly submersive with respect to ${\widehat{\frak S}}$. Let $\widehat h$ be a differential form on $\widehat{\mathcal U}$. By Lemma \[le714\], we obtain ${\widetriangle{\mathcal U}}$, $\widehat{\Phi}$, ${\widetriangle{\frak S}}$, ${\widetriangle f}$, ${\widetriangle{ h}}$.$\blacksquare$
\[deflemgg\] In Situation \[sitsu8main\], we define the [*pushout*]{}, or the [*integration along the fiber*]{} $\widehat f !(\widehat{ h};\widehat{{\frak S}^{\epsilon}})$ by $$\label{form9393}
\widehat f !(\widehat{ h};\widehat{{\frak S}^{\epsilon}})
=
{\widetriangle f} !\left({\widetriangle{h}};{\widetriangle{\frak S^{\epsilon}}}\right).$$ Here the right hand side is defined in Definition \[pushforwardKuranishi\]. Hereafter we mostly use the terminology ‘pushout’ in this document.
\[theorem915\] The right hand side of (\[form9393\]) is independent of choices of ${\widetriangle{\mathcal U}}$, $\widehat{\Phi}$, ${\widetriangle{\frak S}}$, ${\widetriangle f}$, ${\widetriangle{ h}}$ in the sense of $\spadesuit$ of Definition \[defnspadesuit\], but depends only on ${\widehat{\mathcal U}}$, ${\widehat{\frak S}}$, ${\widehat f}$, ${\widehat{ h}}$ and $\epsilon$.
The proof uses Proposition \[integralinvembprop\]. To state it we consider the following situation.
\[situ9016\] Let ${\widetriangle{\mathcal U}}$, ${\widetriangle{\mathcal U^+}}$ be good coordinate systems of $Z \subseteq X$, $\widetriangle\Phi :
{\widetriangle{\mathcal U}} \to {\widetriangle{\mathcal U^+}}$ a GG-embedding, and $\mathcal K$, $\mathcal K^+$ the respective support systems of ${\widetriangle{\mathcal U}}$, ${\widetriangle{\mathcal U^+}}$ compatible with $\widetriangle\Phi$. Let ${\widetriangle{\frak S}}$, ${\widetriangle{\frak S^+}}$ be CF-perturbations of $(\widetriangle{\mathcal U},\mathcal K)$, $(\widetriangle{\mathcal U^+},\mathcal K^+)$, respectively. Let ${\widetriangle{h^+}}$ be a differential form on ${\widetriangle{\frak U^+}}$ which has a compact support in $\ring Z$ and $\widetriangle {f^+} : (X,Z;{\widetriangle{\mathcal U^+}}) \to M$ a strongly smooth map. We put $\widetriangle h = \widetriangle\Phi^* \widetriangle{h^+}$ and $\widetriangle f = \widetriangle {f^+} \circ \widetriangle\Phi : (X,Z;{\widetriangle{\mathcal U}}) \to M$.
We assume that $\widetriangle f$ is strongly submersive with respect to $\widetriangle{\frak S}$ and $\widetriangle{f^+}$ is strongly submersive with respect to $\widetriangle{{\frak S}^+}$. $\blacksquare$
\[integralinvembprop\] In Situation \[situ9016\] we have $${\widetriangle f} !\left( {\widetriangle{h}};{\widetriangle{\frak S^{\epsilon}}}
\right)
=
{\widetriangle{f^+}} !\left({\widetriangle{h^+}};{\widetriangle{\frak S^{+\epsilon}}}
\right)$$ for each sufficiently small $\epsilon >0$.
We use Definition-Lemma \[henacomp\] also in this proof. $$\xymatrix{
& \widetriangle{\mathcal U_1} \ar[rr] && \widehat{\mathcal U_1^+} \ar[rr] &&\widetriangle{\mathcal U_1^{+}} \\
\widehat{\mathcal U} \ar[ru] \ar[rr]\ar[rd] && \widetriangle{\mathcal U_3} \ar[ru]\ar[rd] \\
& \widetriangle{\mathcal U_2} \ar[rr] && \widehat{\mathcal U_2^+} \ar[rr] &&\widetriangle{\mathcal U_2^{+}}
}$$ Let ${\widetriangle{\mathcal U_i}}$, $\widehat{\Phi_i}$, ${\widetriangle{\frak S_i}}$, ${\widetriangle f_i}$, ${\widetriangle{h_i}}$, $i=1,2$ be two choices. By Lemma \[le7155\] we have ${\widehat{\mathcal U_i^+}}$, ${\widehat{\frak S_i^+}}$, ${\widehat f_i^+}$, ${\widehat{h_i^+}}$, $i=1,2$ and GK-embeddings $\widehat{\Phi^+_i} : {\widetriangle{\mathcal U_i}} \to {\widehat{\mathcal U_i^+}}$ to which various objects are compatible.
By Lemma \[le7155\] we obtain ${\widetriangle{\mathcal U_3}}$, $\widehat{\Phi_3}$, ${\widetriangle{\frak S_3}}$, ${\widetriangle f_3}$, ${\widetriangle{h_3}}$ and GK-embeddings $\widehat{\Phi^{+-}_i} : {\widetriangle{\mathcal U_3}} \to {\widehat{\mathcal U_i^+}}$ to which various objects are compatible.
By Lemma \[le714\], we obtain ${\widetriangle{\mathcal U_i^{+}}}$, $\widehat{\Phi_i^{+}}$, ${\widetriangle{\frak S_i^{+}}}$, ${\widetriangle f_i^{+}}$, ${\widetriangle{ h_i^{+}}}$ and KG-embeddings $\widehat{\Phi^{+}_i} : {\widehat{\mathcal U^+_i}} \to {\widetriangle{\mathcal U_i^{+}}}$ to which various objects are compatible.
Now we claim $$\label{formula9696}
{\widetriangle{f_1}} !\left({\widetriangle{h_1}};{\widetriangle{\frak S_1^{\epsilon}}}
\right)
=
{\widetriangle{f_1^{+}}} !\left({\widetriangle{h_1^{+}}};{\widetriangle{\frak S_1^{+\epsilon}}}
\right).$$ In fact by Definition-Lemma \[henacomp\] there exists a weakly open substructure ${\widetriangle{\mathcal U_{0,1}}}$ of ${\widetriangle{\mathcal U_{1}}}$ and a GG-embedding ${\widetriangle{\mathcal U_{0,1}}} \to {\widetriangle{\mathcal U_1^{+}}}$. By Lemma \[compatiwithwkemb\] we can restrict ${\widetriangle{\frak S_1}}$ to ${\widetriangle{\frak S_{0,1}}}$, as well as other objects. Strong submersivity is preserved by Lemma \[lem9797\]. Therefore by Proposition \[integralinvembprop\] we find $${\widetriangle{f_1}} !\left({\widetriangle{h_1}};{\widetriangle{\frak S_1^{\epsilon}}}
\right)
=
{\widetriangle{f_{0,1}}} !\left({\widetriangle{h_{0,1}}};{\widetriangle{\frak S_{0,1}^{\epsilon}}}
\right)
=
{\widetriangle{f_1^{+}}} !\left({\widetriangle{h_1^{+}}};{\widetriangle{\frak S_1^{+\epsilon}}}
\right).$$ Here $\widetriangle{h_{0,1}}$ is the pull back of $\widetriangle{h_{1}}$ to ${\widetriangle{\mathcal U_{0,1}}}$.
We have thus proved (\[formula9696\]). Using the same argument three more times, we obtain $$\aligned
{\widetriangle{f_1}} !\left({\widetriangle{h_1}};{\widetriangle{\frak S_1^{\epsilon}}}
\right)
&=
{\widetriangle{f_1^{+}}} !\left({\widetriangle{h_1^{+}}};{\widetriangle{\frak S_1^{+\epsilon}}}
\right)=
{\widetriangle{f_3}} !\left({\widetriangle{h_3}};{\widetriangle{\frak S_3^{\epsilon}}}\right) \\
&=
{\widetriangle{f_2^{+}}} !\left({\widetriangle{h_2^{+}}};{\widetriangle{\frak S_2^{+\epsilon}}}
\right) =
{\widetriangle{f_2}} !\left({\widetriangle{h_2}};{\widetriangle{\frak S_2^{\epsilon}}}\right).
\endaligned$$ We have thus proved the required independence of $\widehat{\Phi}$, ${\widetriangle{\frak S}}$, ${\widetriangle f}$, ${\widetriangle{ h}}$.
Proof of Definition-Lemma \[henacomp\] {#subsec:proofofwdofcomp}
--------------------------------------
Recalling the notation for GK-embedding in Definition \[embgoodtokura\], we put $$\widehat{\Phi} = \{(U_{\frak p}(p),\Phi_{p\frak p})\} ~:~
\widetriangle{\mathcal U} \to \widehat{\mathcal U}$$ and a KG-embedding $\widehat{\Phi^+} :
\widehat{\mathcal U} \to {\widetriangle{\mathcal U^+}}$. We take a support system $\mathcal K$ of ${\widetriangle{\mathcal U}}$ and $\mathcal K^{+}$ of ${\widetriangle{\mathcal U^+}}$, respectively. For $\frak p \in \frak P$, $\frak q \in \frak P^+$ we define $$\label{defXpq}
Z_{\frak p\frak q} = (\mathcal K_{\frak p} \cap Z)
\cap (\mathcal K^{+}_{\frak q} \cap Z).$$ Here and hereafter the set theoretical symbols such as equality and the intersection in (\[defXpq\]) etc.. are regarded as those among the subsets of $\vert{\widetriangle {\mathcal U^+}}\vert$.
We will use Lemma \[lem816\] to obtain the partial ordered set $\frak P_0$ which is a part of weakly open substructure $\widetriangle{\mathcal U^0}$ of $\widetriangle{\mathcal U}$. (In this subsection we write $\widetriangle{\mathcal U^0}$ in place of $\widetriangle{\mathcal U_0}$.)
\[lem816\] There exist a finite subset $A_{\frak p\frak q}$ of $Z_{\frak p\frak q}$ for each $\frak p \in \frak P$, $\frak q \in \frak P^+$ and a subset $U_{(\frak p,p)}$ of $U_{\frak p}$ for each $\frak p$ and $p \in A_{\frak p\frak q}$ such that they have the following properties.
1. $p \in U_{(\frak p,p)}$. $U_{(\frak p,p)}$ is an open subset of $U_{\frak p}$.
2. $U_{(\frak p,p)} \subset U_{\frak p}(p)$.
3. If $p \in A_{\frak p\frak q}$, $p' \in A_{\frak p'\frak q'}$, $\frak p \le \frak p'$ and $$\varphi_{\frak p'\frak p}^{-1}(U_{(\frak p',p')}) \cap U_{(\frak p,p)}
\ne \emptyset,$$ then $\frak q\le \frak q'$.
4. For each $\frak p_0 \in \frak P$, $\frak q_0 \in \frak P^+$ we have $$\bigcup_{\frak p, \frak q :
\frak p_0\le \frak p, \frak q_0 \le \frak q,
\atop p\in A_{\frak p\frak q}}
\left(
U_{(\frak p,p)}\cap Z
\right)
\supseteq Z_{\frak p_0\frak q_0}.$$
5. If $(\frak p,\frak q) \ne (\frak p',\frak q')$ then $A_{\frak p\frak q} \cap A_{\frak p'\frak q'} = \emptyset$.
We recall:
Let $(\frak P,\le)$ be a partially ordered set. A subset $\frak I \subseteq \frak P$ is said to be an [*ideal*]{} if $\frak p \in \frak I$, $\frak p' \ge \frak p$ implies $\frak p' \in \frak I$.
We define a partial order on $\frak P \times \frak P^+$ such that $(\frak p,\frak q) \le (\frak p',\frak q') $ if and only if ‘$\frak p\le \frak p'$’ $\wedge$ ‘$\frak q\le \frak q'$’. (Note if $\frak p < \frak p'$ and $\frak q >\frak q'$, neither $(\frak p,\frak q) \le (\frak p',\frak q') $ nor $(\frak p,\frak q) \ge (\frak p',\frak q') $ hold.) Let $\frak I \subset \frak P \times \frak P^+$ be an ideal. We will prove the following by induction on $\#\frak I$.
\[setsekigomasubme\] For each $(\frak p,\frak q) \in \frak I$ there exist a finite subset $A_{\frak p\frak q}$ of $Z_{\frak p\frak q}$ and a subset $U_{(\frak p,p)}$ of $U_{\frak p}$ for each $p \in A_{\frak p\frak q}$ such that they satisfy (1)(2)(5) of Lemma \[lem816\] and the following conditions (3)’ and (4)’.
1. 1. If $(\frak p,\frak q), (\frak p',\frak q') \in \frak I$, then Lemma \[lem816\] (3) holds.
2. If $(\frak p,\frak q) \in \frak I$, $p \in A_{\frak p\frak q}$ and $(\frak p',\frak q') \in \frak P\times \frak P^+$ satisfies $$\overline{U_{(\frak p,p)}} \cap Z_{\frak p'\frak q'} \ne \emptyset,$$ then $(\frak p,\frak q) \ge (\frak p',\frak q')$.
2. For each $(\frak p_0,\frak q_0) \in \frak I$ we have $$\bigcup_{\frak p, \frak q :
\frak p_0\le \frak p, \frak q_0 \le \frak q,
\atop p\in A_{\frak p\frak q}}
\left(
U_{(\frak p,p)}\cap Z
\right)
\supseteq Z_{\frak p_0\frak q_0}.$$
Note that we do not assume $(\frak p',\frak q') \in \frak I$ in Sublemma \[setsekigomasubme\] (3)’ (b).
The case $\frak I = \emptyset$ is trivial. Suppose Sublemma \[setsekigomasubme\] is proved for all $\frak I'$ with $\#\frak I' < \#\frak I$. We will prove the case of $\frak I$.
Let $(\frak p_1,\frak q_1)$ be a minimal element of $\frak I$. Then $\frak I_- =\frak I \setminus \{(\frak p_1,\frak q_1)\}$ is an ideal of $\frak P \times \frak P^+$. By induction hypothesis, we obtain $A_{\frak p\frak q}$ for $(\frak p,\frak q) \in \frak I_-$ and $U_{(\frak p,p)}$ for each $p \in A_{\frak p\frak q}$, $(\frak p,\frak q) \in \frak I_-$. By induction hypothesis, Sublemma \[setsekigomasubme\] (4)’, the set $$O
= \bigcup_{(\frak p, \frak q) \in \frak I :
(\frak p_1,\frak q_1) < (\frak p,\frak q)
\atop p\in A_{\frak p\frak q}}
\left(
U_{(\frak p,p)}\cap Z_{\frak p_1\frak q_1}
\right)$$ is an open neighborhood of $$L = \left(\bigcup_{(\frak p, \frak q) \in \frak I :
(\frak p_1,\frak q_1) < (\frak p,\frak q)
}
Z_{\frak p\frak q}\right) \cap Z_{\frak p_1\frak q_1}$$ in $Z_{\frak p_1\frak q_1}$.
\[sublem8199\] If $x \in Z_{\frak p_1\frak q_1} \setminus O$ and $x \in Z_{\frak p \frak q}$, then $(\frak p,\frak q) \le (\frak p_1,\frak q_1)$.
Note that we do [*not*]{} assume $(\frak p,\frak q) \in \frak I$.
Since $$x \in \mathcal K_{\frak p} \cap \mathcal K_{\frak p_1}
\cap
\mathcal K^+_{\frak q} \cap \mathcal K^+_{\frak q_1} \cap Z,$$ Definition \[gcsystem\] (5) implies that ‘$\frak p \le \frak p_1$ or $\frak p \ge \frak p_1$’ holds and ‘$\frak q \le \frak q_1$ or $\frak q \ge \frak q_1$’ holds also.
Suppose $\frak p >\frak p_1$. Then we claim $(\frak p,\frak q)
> (\frak p_1,\frak q_1)$ can not occur. In fact, if $(\frak p,\frak q)
> (\frak p_1,\frak q_1)$, then $(\frak p,\frak q) \in \frak I_-$ because $\frak I$ is an ideal. This contradicts to $x \in Z_{\frak p_1\frak q_1} \setminus O$. (We use the induction hypothesis Sublemma \[setsekigomasubme\] (4)’ here.) Therefore $\frak q < \frak q_1$ must hold. Then $x \in \mathcal K_{\frak p} \cap \mathcal K^+_{\frak q_1} \cap Z$ and $(\frak p,\frak q_1) > (\frak p_1,\frak q_1) $. This contradicts $x \notin O$. We can find a contradiction from $\frak q > \frak q_1$ in a similar way. Therefore we obtain $(\frak p,\frak q) \le (\frak p_1,\frak q_1)$.
\[propety818\] For each $x \in Z_{\frak p_1\frak q_1} \setminus O$, there exists its neighborhood $W_x$ in $U_{\frak p_1}$ with the following properties.
1. $x \in W_x$ and $W_x$ is open in $U_{\frak p_1}$.
2. $W_x \subset U_{\frak p_1}(x)$.
3. If $W_x \cap Z_{\frak p\frak q} \ne \emptyset$ then $(\frak p,\frak q) \le (\frak p_1,\frak q_1)$.
4. If $\frak p \ge \frak p_1$, $p \in A_{\frak p\frak q}$, $(\frak p,\frak q) \in \frak I_-$ and $W_x \cap \varphi_{\frak p \frak p_1}^{-1}({U_{(\frak p,p)}}) \ne \emptyset$, then $\frak q \ge \frak q_1$.
Since $Z_{\frak p \frak q}$ is a closed set, Subsublemma \[sublem8199\] implies that (3) holds for a sufficiently small neighborhood $W_x$ of $x$.
We next prove that (4) holds for a sufficiently small neighborhood $W_x$ of $x$. Suppose $\frak p \ge \frak p_1$, $p \in A_{\frak p\frak q}$, $(\frak p,\frak q) \in \frak I_-$ and $x \in \overline{U_{(\frak p,p)}}$. Then $
x \in \overline{U_{(\frak p,p)}} \cap Z_{\frak p_1\frak q_1}.
$ We apply the induction hypothesis Sublemma \[setsekigomasubme\] (3)’ (b) to $\frak I_-$ and find $(\frak p,\frak q) \ge (\frak p_1,\frak q_1)$. In particular, $\frak q \ge \frak q_1$. Then we can take a sufficiently small neighborhood $W_x$ so that $$\frak p \ge \frak p_1, p \in A_{\frak p\frak q}, (\frak p,\frak q) \in \frak I_-,
W_x \cap \overline{\varphi_{\frak p p}^{-1}(U_{(\frak p,p)})}
\ne \emptyset
\,\,\Rightarrow\,\,
\frak q \ge \frak q_1.$$ Since $W_x$ is open, the condition $W_x \cap \varphi_{\frak p \frak p_1}^{-1}(\overline{U_{(\frak p,q)}}) \ne \emptyset$ is equivalent to the condition $W_x \cap \varphi_{\frak p \frak p_1}^{-1}({U_{(\frak p,q)}}) \ne \emptyset$. Thus we have proved (4).
We take an open neighborhood $W^0_x$ of $x$ such that $\overline{W^0_x} \subset W_x$. We take a finite subset $A_{\frak p_1\frak q_1} \subset Z_{\frak p_1\frak q_1} \setminus O$ such that $$\label{eq8333}
Z_{\frak p_1\frak q_1} \setminus O \subset
\bigcup_{x \in A_{\frak p_1\frak q_1}} W^0_x.$$ Lemma \[lem816\] (5) is obvious from definition.
For $x \in A_{\frak p_1\frak q_1}$, we put $$\label{eq09999}
U_{(\frak p_1,x)} = W^0_x.$$
\[cond819\] There exists an open neighborhood $U'_{(\frak p,p)}$ of $p$ for $(\frak p,\frak q) \in \frak I_-$ and $p \in A_{\frak p\frak q}$ such that the following holds.
1. $U'_{(\frak p,p)} \subset U_{(\frak p,p)}$.
2. Sublemma \[setsekigomasubme\] (1)(2)(4)’ hold for $U'_{(\frak p,p)}$.
3. If $p \in A_{\frak p\frak q}$, $(\frak p,\frak q) \in \frak I_-$, $\frak p_1\ge \frak p$, $x \in A_{\frak p_1\frak q_1}$, then $$\varphi_{\frak p_1\frak p}^{-1}(U_{(\frak p_1,x)}) \cap U'_{(\frak p,p)}
= \emptyset.$$
We take $$\label{form99}
U'_{(\frak p,p)} = U_{(\frak p,p)} \setminus \bigcup_{x \in A_{\frak p_1\frak q_1}}\overline{U_{(\frak p_1,x)}}.$$ Here we regard $U_{(\frak p,p)}$ and $\overline{U_{(\frak p_1,x)}}$ as subsets of $\vert {\widetriangle {\mathcal U^+}}\vert$. (1) (3) are immediate. We will prove (2). By Subsublemma \[propety818\] (3) and (\[eq09999\]), we have $$\label{form1000}
Z_{\frak p\frak q} \cap \overline{U_{(\frak p_1,x)}}
= \emptyset,$$ for each $(\frak p,\frak q) \in \frak I_-$, $x \in A_{\frak p_1\frak q_1}$. Therefore $p \in A_{\frak p\frak q}$, $(\frak p,\frak q) \in \frak I_-$ imply $p \notin \overline{U_{(\frak p_1,x)}}$. Hence $p \in U'_{(\frak p,p)} \subset U_{(\frak p,p)}$. This implies that Sublemma \[setsekigomasubme\] (1)(2) hold for $U'_{(\frak p,p)}$. Sublemma \[setsekigomasubme\] (4)’ is a consequence of (\[form1000\]) and (\[form99\]).
Hereafter we write $U_{(\frak p,p)}$ in place of $U'_{(\frak p,p)}$.
We will prove that they have the properties claimed in Sublemma \[setsekigomasubme\]. Sublemma \[setsekigomasubme\] (1),(2) follow from Subsublemma \[propety818\] (1),(2) and the induction hypothesis. Sublemma \[setsekigomasubme\] (4)’ follows from (\[eq8333\]) and induction hypothesis (which is claimed as Subsublemma \[cond819\] (2)).
Suppose $(\frak p,\frak q), (\frak p',\frak q') \in \frak I$, $p \in A_{\frak p\frak q}$, $p' \in A_{\frak p'\frak q'}$, $\frak p \le \frak p'$ and $
\varphi_{\frak p'\frak p}^{-1}(U_{(\frak p',p'))}) \cap U_{(\frak p,p)}
\ne \emptyset.
$ We will prove $\frak q \le \frak q'$.
The case $(\frak p,\frak q), (\frak p',\frak q') \in \frak I_-$ follows from the induction hypothesis.
Suppose $(\frak p',\frak q') = (\frak p_1,\frak q_1)$. Then $
\varphi_{\frak p_1\frak p}^{-1}(U_{(\frak p_1,p'))}) \cap U_{(\frak p,p)}
\ne \emptyset.
$ Subsublemma \[cond819\] (3) implies that $(\frak p,\frak q) \notin \frak I_-$. Therefore $(\frak p,\frak q) = (\frak p_1,\frak q_1)$. Hence $\frak q \le \frak q'$ as required.
We next assume $(\frak p_1,\frak q_1) = (\frak p,\frak q)$. Then Subsublemma \[propety818\] (4) implies $\frak q_1 \le \frak q'$ as required.
Suppose $(\frak p,\frak q) \in \frak I$, $p \in A_{\frak p\frak q}$ and $
\overline{U_{(\frak p,p)}} \cap Z_{\frak p'\frak q'} \ne \emptyset
$. We will prove $(\frak p,\frak q) \ge (\frak p',\frak q')$.
The case $(\frak p,\frak q) \in \frak I_-$ follows from the induction hypothesis. Suppose $(\frak p,\frak q) = (\frak p_1,\frak q_1)$ . Then $
\overline{U_{(\frak p_1,p)}} \cap Z_{\frak p'\frak q'} \ne \emptyset
$. Note $
\overline U_{(\frak p_1,x)} \subset \overline W^0_x \subset W_x.
$ Therefore Subsublemma \[propety818\] (3) implies $(\frak p',\frak q') \le (\frak p_1,\frak q_1)$, as required.
Therefore the proof of Sublemma \[setsekigomasubme\] is now complete.
Lemma \[lem816\] is the case $\frak I = \frak P \times \frak P^+$ of Sublemma \[setsekigomasubme\].
Now we put $$\frak P_0 = \bigcup_{(\frak p,\frak q) \in \frak P \times
\frak P^+} A_{\frak p\frak q} \times \{(\frak p,\frak q)\}.$$ We choose any linear order on $A_{\frak p\frak q}$ and define a partial order on $\frak P_0$ by the following: $$(x,(\frak p,\frak q))
\le (x',(\frak p',\frak q'))
\quad
\text{if and only if}
\quad
\begin{cases}
(\frak p,\frak q) < (\frak p',\frak q'))\\
\text{or $(\frak p,\frak q) = (\frak p',\frak q')$, $x\le x'$}.
\end{cases}$$ We define $$U^0_{(x,(\frak p,\frak q))} = U_{(\frak p,x)},
\quad
\mathcal U^0_{(x,(\frak p,\frak q))}
= \mathcal U_{\frak p}\vert_{ U^0_{(x,(\frak p,\frak q))} }.$$ We define coordinate changes among them by restricting those of ${\widetriangle{\mathcal U}}$. We thus obtain a good coordinate system ${\widetriangle{\mathcal U^0}}$. (Note we use Lemma \[lem816\] (3) to check Definition \[gcsystem\] (5).)
We will define a weakly open embedding ${\widetriangle{\mathcal U^0}} \to {\widetriangle{\mathcal U}}$. We first define a map $\frak P_0 \to \frak P$ by sending $(x,(\frak p,\frak q)) \mapsto \frak p$. This is order preserving. We also have an open embedding of Kuranishi charts $\mathcal U^0_{(x,(\frak p,\frak q))} = \mathcal U_{\frak p}\vert_{U^0_{(x,(\frak p,\frak q))}}
\to \mathcal U_{\frak p}$. They obviously commute with coordinate change.
We next define the embedding $
{\widetriangle{\mathcal U^0}} \to {\widetriangle{\mathcal U^+}}
$ that will be the composition of ${\widetriangle{\mathcal U^0}}
\to {\widetriangle{\mathcal U}}
\to {\widehat{\mathcal U}}
$ and $ {\widehat{\mathcal U}} \to {\widetriangle{\mathcal U^+}}$. We define a map $\frak P_0 \to \frak P^+$ by sending $(x,(\frak p,\frak q)) \mapsto \frak q$. This is an order preserving map. We next define a map $\mathcal U^0_{(x,(\frak p,\frak q))}
\to \mathcal U^+_{\frak q}$ as the composition of $$\mathcal U^0_{(x,(\frak p,\frak q))}
\to
\mathcal U_{\frak p}\vert_{U_{\frak p}(x)}
\to \mathcal U_x
\to \mathcal U^+_{\frak q}.$$ Here the first map $\mathcal U^0_{(x,(\frak p,\frak q))}
\to
\mathcal U_{\frak p}\vert_{U_{\frak p}(x)}$ is an open embedding that exists by Lemma \[lem816\] (2). The second map $\mathcal U_{\frak p}\vert_{U_{\frak p}(x)}
\to \mathcal U_x$ is a part of the GK-embedding ${\widetriangle{\mathcal U}}
\to {\widehat{\mathcal U}}$. The third map $ \mathcal U_x
\to \mathcal U^+_{\frak q}$ is a part of the KG-embedding ${\widehat{\mathcal U}}
\to {\widetriangle{\mathcal U^+}}$. The proof of Definition-Lemma \[henacomp\] is now complete.
Proof of Proposition \[integralinvembprop\] {#subsec:intwdonKura}
-------------------------------------------
Let $(\mathcal K^+_1,\mathcal K^+_2)$ (resp. $(\mathcal K_1,\mathcal K_2)$) be a support pair of ${\widetriangle{\mathcal U^+}}$ (resp. ${\widetriangle{\mathcal U}}$). We may choose them so that $
\varphi_{\frak p}(\mathcal K^i_{\frak p}) \subseteq \mathcal K^{i+}_{\frak i(\frak p)}
$. Let $\mathcal K_2^+ < \mathcal K_3^+$ and $\mathcal K_2 < \mathcal K_3$.
We will choose $\delta_+$, $\delta$ and $\frak U(Z)$, later. Let $\{\chi^+_{\frak p^+}\}$ (resp. $\{\chi_{\frak p}\}$) be a strongly smooth partition of unity of $(X,Z,{\widetriangle{\mathcal U^+}},{\widetriangle{\frak S^+}},\delta_+)$ (resp. $(X,Z,{\widetriangle{\mathcal U}},{\widetriangle{\frak S}},\delta)$). By inspecting the proof of Proposition \[pounitexi\], we can take $\chi_{\frak p}$ so that it is not only a strongly smooth function on $\vert \mathcal K_2\vert$ but also one on $\vert \mathcal K^+_2\vert$.
We take $\frak p^+_0 \in \frak P^+$ and set $h_0 = \chi^+_{\frak p^+_0}h_{\frak p^+_0}$. To prove Proposition \[integralinvembprop\] it suffices to show $$\label{formula84}
f_{\frak p^+_0}!\left( h_0; \frak S^{+\epsilon}_{\frak p^+_0} \vert_{\frak U(Z)\cap
\mathcal K^{+1}_{\frak p_0^+}(2\delta_+)}\right)
=
\sum_{\frak p \in \frak P}
f_{\frak p}!( (\chi_{\frak p}h_0)_{\frak p}; \frak S^{\epsilon}_{\frak p} \vert_{\frak U(Z)\cap
\mathcal K^{1}_{\frak p}(2\delta)}).$$ By taking $\epsilon >0$ sufficiently small, we may assume $\sum \chi_{\frak p}= 1$ on $\frak U(Z) \cap \Pi((\frak S^{+ \epsilon}_{\frak p^+_0})^{-1}(0))$. (This is a consequence of Lemma \[lem739\] and Definition \[pounity\] (3). Note the differential form $(\chi_{\frak p}h_0)_{\frak p_0}$ is defined since the function $\chi_{\frak p}$ is strongly smooth on $\vert\mathcal K_2^+\vert$.) Therefore to prove (\[formula84\]) it suffices to show $$\label{formula85}
f_{\frak p^+_0}!\left( (\chi_{\frak p}h_0)_{\frak p_0^+}; \frak S^{+\epsilon}_{\frak p^+_0}
\vert_{\frak U(Z)\cap
\mathcal K^{+1}_{\frak p_0^+}(2\delta_+)}\right)
=
f_{\frak p}!( (\chi_{\frak p}h_0)_{\frak p}; \frak S^{\epsilon}_{\frak p} \vert_{\frak U(Z)\cap
\mathcal K^{1}_{\frak p}(2\delta)})$$ for each $\frak p$. We will prove it below. There are three cases.
(Case 1) Neither $\frak i(\frak p) \le \frak p^+_0$ nor $\frak i(\frak p) \ge \frak p^+_0$: In this case we have $$\mathcal K^{1+}_{\frak p^+_0} \cap \mathcal K^{1}_{\frak p}
\subset
\mathcal K^{1+}_{\frak p^+_0} \cap \mathcal K^{1 +}_{\frak i(\frak p)}
= \emptyset.$$ Therefore in the same way as the proof of (\[form743\]), we can choose $\delta$,$\delta_+$ small so that $$\Omega^{1+}_{\frak p^+_0}(\mathcal K^{1+},\delta_+) \cap
\Omega^{1}_{\frak p}(\mathcal K^{1},\delta) = \emptyset.$$ Then both sides of (\[formula85\]) are zero.
(Case 2) $\frak i(\frak p) \le \frak p^+_0$: We consider the embedding $$\mathcal U_{\frak p} \,\,\overset{\Phi_{\frak p}}\longrightarrow \,\, \mathcal U^+_{\frak i(\frak p)}
\,\,\overset{\Phi_{\frak p^+_0 \frak i(\frak p)}}\longrightarrow \,\, \mathcal U^+_{\frak p_0}.$$ In the same way as the proof of (\[form741741\]) we can choose $\delta$, $\delta_+$, $\frak U(Z)$ small so that $$\label{form914914}
{\rm Supp}(\chi_{\frak p}\widetriangle h_0) \cap \Pi((\widetriangle{\frak S^{+\epsilon}})^{-1}(0))
\cap \frak U(Z)
\subset \mathcal K^{1 +}_{\frak p^+_0}(2\delta^+)
\cap
\mathcal K^{1}_{\frak p}(2\delta)
\cap \frak U(Z).$$ Then (\[formula85\]) follows.
(Case 3) $\frak i(\frak p) \ge \frak p^+_0$: In the same way as the proof of Proposition \[indepofukuracont\] Case 2, we can choose $\delta$, $\delta_+$, $\frak U(Z)$ small so that (\[form914914\]) holds. (\[formula85\]) is its consequence.
Thus the proof of Proposition \[integralinvembprop\] is complete.
CF-perturbations of correspondences {#subsec:confamicor}
-----------------------------------
\[def92111\] We consider Situation \[smoothcorr\]. Let $\widehat{\frak S}$ be a CF-perturbation of $\widehat{\mathcal U}$ such that $\widehat {f_t}$ is strongly submersive with respect to $\widehat{\frak S}$.
We call such $\widehat{\frak S}$ a [*CF-perturbation of Kuranishi correspondence*]{} $\frak X$. We then define $$\label{defn924}
{\rm Corr}_{(\frak X,\widehat{\frak S^{\epsilon}})}
: \Omega^k(M_s) \to \Omega^{k+\ell}(M_t)$$ by $${\rm Corr}_{(\frak X,\widehat{\frak S^{\epsilon}})}(h)
=
\widehat{f_t} !((\widehat{f_s})^*h;{\widehat{\frak S}}^{\epsilon}).$$ This is well-defined by Definition-Lemma \[deflemgg\]. We call the linear map ${\rm Corr}_{(\frak X,\widehat{\frak S^{\epsilon}})}$ a [*smooth correspondence map*]{} of Kuranishi structure and $\ell$ the [*degree*]{} of smooth correspondence $\frak X$ and write it $\deg \frak X$.
The next lemma says that in Situation \[smoothcorr\] we can always thicken our Kuranishi structure so that the assumptions of Definition \[def92111\] is satisfied.
For each smooth correspondence $(X,\widehat{\mathcal U},\widehat{f_s},\widehat{f_t})$ as in Situation \[smoothcorr\] there exist $\widehat{\mathcal U^+},\widehat{\frak S^+},\widehat{f^+_s},\widehat{f^+_t}$ with the following properties.
1. $(X,\widehat{\mathcal U^+},\widehat{f^+_s},\widehat{f^+_t})$ is a Kuranishi correspondence and $\widehat{\frak S^+}$ a CF-perturbation of Kuranishi correspondence.
2. $\widehat{\mathcal U^+}$ is a thickening of $\widehat{\mathcal U}$.
3. Let $\widehat\Phi : \widehat{\mathcal U} \to \widehat{\mathcal U^+}$ be the KK-embedding. Then $\widehat{f^+_s}$ and $\widehat{f^+_t}$ induce $\widehat{f_s}$ and $\widehat{f_t}$ by $
\widehat\Phi
$.
This is an immediate consequence of Lemmata \[lemappgcstoKucont\] and \[le714\].
Stokes’ formula for Kuranishi structure {#subsec:StokesKura}
---------------------------------------
We have Stokes’ formula in Theorem \[Stokes\], which is the formula for good coordinate system. In this subsection we translate it to one for Kuranishi structure.
\[stokeskurashitu\] Let $\widehat{\mathcal U}$ be a Kuranishi structure of $Z \subseteq X$, $\widehat{\frak S}$ its CF-perturbation, $\widehat f : (X,Z;\widehat{\mathcal U}) \to M$ a strongly submersive map with respect to $\widehat{\frak S}$, and $\widehat h$ a differential form on $ (X,Z;\widehat{\mathcal U})$.
Let $\partial(X,Z,\widehat{\mathcal U},\widehat{\frak S})
= (\partial X,\partial Z,\widehat{\mathcal U_{\partial}},\widehat{\frak S_{\partial}})$, where $(\partial X,\partial Z;\widehat{\mathcal U_{\partial}})$ is the normalized boundary of $(X,Z;\widehat{\mathcal U})$ on which $\widehat{\frak S}$ induces a CF-perturbation $\widehat{\frak S_{\partial}}$ by Lemma \[lemma755\] (2). Since $\widehat f$ induces a map $\widehat{f_{\partial}} : (\partial X,
\partial Z;\widehat{\mathcal U_{\partial}})
\to M$, which is strongly submersive with respect to $\widehat{\frak S_{\partial}}$ if $\widehat f$ is strongly submersive with respect to $\widehat{\frak S}$ (Lemma \[lemma755\] (4)).
Let $\widehat{h_{\partial}}$ be the restriction of $\widehat h$ to $(\partial X,\partial Z;\widehat{\mathcal U_{\partial}})$.$\blacksquare$
[(Stokes’ formula for Kuranishi structure.)]{}\[Stokeskura\] In Situation \[stokeskurashitu\] we have the next formula for each sufficiently small $\epsilon>0$: $$d\left(\widehat f !(\widehat h;\widehat{\frak S^{\epsilon}})\right)
=
\widehat f !(d\widehat h;\widehat{\frak S^{\epsilon}})
+
\widehat f_{\partial}!(\widehat {h_{\partial}};\widehat{\frak S_{\partial}^{\epsilon}}).$$
By Lemmata \[lemappgcstoKucont\] and \[le714\], there exist a good coordinate system ${\widetriangle{\mathcal U}}$ and an KG-embedding ${\widehat{\mathcal U}} \to {\widetriangle{\mathcal U}}$. Moreover there exists a CF-perturbation ${\widetriangle{\frak S}}$ of ${\widetriangle{\mathcal U}}$ such that ${\widehat{\frak S}}$, ${\widetriangle{\frak S}}$ are compatible with the KG-embedding ${\widehat{\mathcal U}} \to {\widetriangle{\mathcal U}}$. Furthermore there exist a strongly smooth map ${\widetriangle f} : (X,Z;{\widetriangle{\mathcal U}})
\to M$ and a differential form $\widetriangle h$, which are pulled back to $\widehat f$ and $\widehat h$, by the KG-embedding ${\widehat{\mathcal U}} \to {\widetriangle{\mathcal U}}$. Then ${\widetriangle{\mathcal U}}$, ${\widetriangle{\frak S}}$, ${\widetriangle f}$ and $\widetriangle h$, induce ${\widetriangle{\mathcal U_{\partial}}}$, ${\widetriangle{\frak S_{\partial}}}$, $\widetriangle{f_{\partial}}$ and $\widetriangle{h_{\partial}}$ on the boundary, respectively, which are compatible with corresponding objects on $(\partial X,\partial Z;\widehat{\mathcal U_{\partial}})$. Thus Proposition \[Stokeskura\] follows by applying Theorem \[Stokes\] to ${\widetriangle{\mathcal U}}$, ${\widetriangle{\frak S}}$, ${\widetriangle f}$, $\widetriangle h$, and ${\widetriangle{\mathcal U_{\partial}}}$, ${\widetriangle{\frak S_{\partial}}}$, $\widetriangle{f_{\partial}}$, $\widetriangle{h_{\partial}}$.
The next corollary is an immediate consequence of Proposition \[Stokeskura\].
In the situation of Definition \[def92111\] we have the next formula for each sufficiently small $\epsilon >0$: $$d \circ {\rm Corr}_{(\frak X,\widehat{\frak S^{\epsilon}})}
= {\rm Corr}_{(\frak X,\widehat{\frak S^{\epsilon}})} \circ d
+ {\rm Corr}_{\partial(\frak X,\widehat{\frak S^{\epsilon}})}.$$
Uniformity of CF-perturbations on Kuranishi structure {#subsec:unfcfpKura}
-----------------------------------------------------
In this subsection, we collect various facts which we use to show the existence of uniform bound of the constants $\epsilon$ that appear in Theorem \[theorem915\] etc..
Let $\widehat{\mathcal U}$ be a Kuranishi structure on $Z \subseteq X$ and $\widehat{\frak S_{\sigma}}$ be a $\sigma \in \mathscr A$ parameterized family of CF-perturbations. We say that $\widehat{\frak S_{\sigma}}$ is a [*uniform family*]{} if the convergence in Definition \[defn73ss\] is uniform. More precisely, we require the following.
For each $\frak o$ there exists $\epsilon_0 >0$ such that if $0<\epsilon < \epsilon_0$, $p \in Z$ then $$\vert s(y) - s_p(y) \vert < \frak o,
\qquad
\vert (Ds)(y) - (Ds_p)(y) \vert < \frak o,$$ hold for any $s$ which is any member of ${\frak S}^{\epsilon,p}_{\sigma}$ at any point $y \in U_{p}$ and $\sigma \in \mathscr A$.
\[lem92929\]
1. In the situation of Lemma \[lemappgcstoKucont\] if $\widetriangle{\frak S}$ varies in a uniform family then $\widehat{\frak S}$ varies in a uniform family.
2. In the situation of Lemma \[le714\] (2), if ${\widehat{\frak S}}$ varies in a uniform family then $\widetriangle{\frak S}$ varies in a uniform family.
3. In the situation of Lemma \[le7155\], if ${\widehat{\frak S^+}}$, ${\widehat{\frak S}}$ vary in a uniform family (resp. ${\widehat{\frak S^+_a}}$ ($a=1,2$) , ${\widehat{\frak S}}$ vary in a uniform family) then $\widetriangle{\frak S}$ varies in a uniform family.
The proof will be given at the end of Subsection \[subsec:movingmulsectionetc\].
In the situation of Theorem \[theorem915\] suppose ${\widetriangle{\frak S_{\sigma}}}$ varies in a uniform family. (We require that ${\widetriangle{\mathcal U}}$, $\widetriangle{\Phi}$ are independent of the parameter $\sigma$.) Then the pushout $\widehat f !(\widehat{ h};{\widehat{\frak S_{\sigma}}})$ is uniformly independent of the choices in the sense of $\clubsuit$ in Definition \[intheclubsuit\]. We may choose the constant $\epsilon$ in Proposition \[Stokeskura\] independent of $\sigma$ also.
Using Lemma \[lem92929\] the proof goes in the same way as the proof of Proposition \[lem761\].
We can choose $\epsilon_0$ independent of $\widehat f$ and $\widehat h$.
Composition formula of smooth correspondences {#sec:composition}
=============================================
The purpose of this section is to provide thorough technical detail of the proof of [@fooo09 Lemma 12.15] = Theorem \[compformulaprof\], where fiber product of Kuranishi structures is used as a way to define composition of smooth correspondences. For this purpose we work out the plan described in Subsection \[bigremarkinsec6\] in the de Rham model.
Direct product and CF-perturbation {#subsec:dprocontper}
----------------------------------
Firstly, we begin with defining direct product of CF-perturbations.
\[sit824\] For each $i=1,2$, $\mathcal U_i = (U_i,\mathcal E_i,s_i,\psi_i)$ is a Kuranishi chart of $X$, $x_i \in U_i$, $\frak V_{x_i}^i = (V^i_{x_i},\Gamma^i_{x_i},E^i_{x_i},\psi^i_{x_i},\widehat\psi^i_{x_i})$ is an orbifold chart of $(U_i,\mathcal E_i)$ as in Definition \[defn73ss\]. Let $\mathcal S_{x_i}^i = (W_{x_i}^i,\omega_{x_i}^i,\frak s_{x_i}^{i\epsilon})$ be a CF-perturbation $\mathcal U_i$ on $\frak V_{x_i}^i$.$\blacksquare$
\[defn1022\] In Situation \[sit824\], we define the [*direct product of $\mathcal S_{x_1}^1$ and $\mathcal S_{x_2}^2$*]{} by $$\mathcal S_{x_1}^1 \times \mathcal S_{x_2}^2
=
(W_{x_1}^1 \times W_{x_2}^2, \omega_{x_1}^1 \times \omega_{x_2}^2, \frak s_{x_1}^{1\epsilon} \times
\frak s_{x_2}^{2\epsilon}),$$ where $$( \frak s_{x_1}^{1\epsilon} \times
\frak s_{x_2}^{2\epsilon})(y_1,y_2,\xi_1,\xi_2)
=
( \frak s_{x_1}^{1\epsilon}(y_1,\xi_1),\frak s_{x_2}^{2\epsilon}(y_2,\xi_2))$$ for $y_i \in V^i_{x_i}$ $\xi_i \in W_{x_i}^i$.
\[lem826\]
1. $\mathcal S_{x_1}^1 \times \mathcal S_{x_2}^2$ is a CF-perturbation of $\mathcal U_1 \times \mathcal U_2$.
2. If $\mathcal S_{x_i}^i$ are equivalent to $\mathcal S_{x_i}^{i\prime}$ for $i=1,2$, then $\mathcal S_{x_1}^1 \times \mathcal S_{x_2}^2$ is equivalent to $\mathcal S_{x_1}^{1\prime} \times \mathcal S_{x_2}^{2\prime}$.
3. Let $\Phi^i : \frak V_{x'_i}^{i\prime} \to \frak V_{x_i}^i$ be an embedding of orbifold chart. Then $$(\Phi^1)^*\mathcal S_{x_1}^1 \times (\Phi^2)^*\mathcal S_{x_2}^2$$ is equivalent to $$(\Phi^1 \times \Phi^2)^*(\mathcal S_{x_1}^1 \times \mathcal S_{x_2}^2).$$
This is a direct consequence of definitions.
Suppose we are in Situation \[sit824\].
1. Let $
\frak S^i = \{(\frak V^i_{\frak r_i},\mathcal S^i_{\frak r_i}) \mid \frak r_i \in \frak R_i\}
$ be representatives of CF-perturbations of $\mathcal U^i$ for $i=1,2$. Then $$\{ (\frak V^1_{\frak r_1} \times \frak V^2_{\frak r_2},\mathcal S^1_{\frak r_1} \times \mathcal S^2_{\frak r_2})
\mid (\frak r_1,\frak r_2) \in \frak R_1 \times \frak R_2\}$$ is a representative of a CF-perturbation of $\mathcal U^1 \times \mathcal U^2$. We call it the [direct product]{} and write $\frak S^1 \times \frak S^2$.
2. If $\frak S^i$ is equivalent to $\frak S^{i\prime}$, then $\frak S^1 \times \frak S^2$ is equivalent to $\frak S^{1\prime} \times \frak S^{2\prime}$.
3. Therefore we can define direct product of CF-perturbations.
4. Direct product defines a sheaf morphism $$\label{form101}
\pi_1^{\star} \mathscr S^{\mathcal U_1} \times
\pi_2^{\star} \mathscr S^{\mathcal U_2}
\to
\mathscr S^{\mathcal U_1 \times \mathcal U_2},$$ where $\pi_i : U_1 \times U_2 \to U_i$ are projections.
This is an immediate consequence of Lemma \[lem826\].
\[lem828\] Let $\Phi^i : \mathcal U^{i} \to \mathcal U^{i +}$ be embeddings of Kuranishi charts and $\frak S^i$, $\frak S^{i + }$ CF-perturbations of $\mathcal U^{i}$, $ \mathcal U^{i +}$, for $i=1,2$, respectively.
1. If $\frak S^{i +}$ can be pulled back by $\Phi^i$ for $i=1,2$, then $\frak S^{1 +} \times \frak S^{2 +}$ can be pulled back by $\Phi^1 \times \Phi^2$.
2. If $\frak S^{i +}$, $\frak S^{i}$ are compatible with $\Phi^i$ for $i=1,2$, then $\frak S^{1 +} \times \frak S^{2 +}$ and $\frak S^{1} \times \frak S^{2}$ are compatible with $\Phi^1 \times \Phi^2$.
3. The next diagram commutes: $$\label{diagrampart1010}
\begin{CD}
(\Phi^1)^{\star}\pi_1^{\star} \mathscr S^{\mathcal U^1\triangleright \mathcal U^{1+}} \times
(\Phi^2)^{\star}\pi_2^{\star} \mathscr S^{\mathcal U^2\triangleright \mathcal U^{2+}}
@ > {(\ref{form101})} >>
(\Phi^1\times \Phi^2)^{\star} \mathscr S^{(\mathcal U_1
\times \mathcal U^2) \triangleright (\mathcal U^{1+} \times
\mathcal U^{2+})}
\\
@ VV{(\Phi^1)^* \times (\Phi^2)^*}V @VV{(\Phi^1\times \Phi^2)^*}V\\
\pi_1^{\star} \mathscr S^{\mathcal U^1} \times
\pi_2^{\star} \mathscr S^{\mathcal U^2}
@>> {(\ref{form101})} > \mathscr S^{\mathcal U^1 \times \mathcal U^2}
\end{CD}
\nonumber$$
This is a direct consequence of the definitions.
Let $\widehat{\mathcal U^i}= (\{\mathcal U^i_{p_i}\},\{\Phi^i_{p_iq_i}\})$ be Kuranishi structures of $Z_i \subseteq X_i$ for $i=1,2$, and $\widehat{\mathcal U^1} \times \widehat{\mathcal U^2}$ the direct product Kuranishi structure on $Z_1 \times Z_2 \subseteq X_1 \times X_2$. Let $\widehat{\frak S^i} = \{\frak S^i_{p_i}\}$ be CF-perturbations of $\mathcal U^i_{p_i}$. Then $ \{\frak S^1_{p_1} \times \frak S^2_{p_2}\}$ defines a CF-perturbation of $\widehat{\mathcal U^1} \times \widehat{\mathcal U^2}$. We call it the [direct product]{} of CF perturbations and denote it by $\widehat{\frak S^1} \times \widehat{\frak S^2}$.
This is an immediate consequence of Lemma \[lem828\].
We have thus defined the direct product of CF-perturbations.
We have defined the notion of direct product of CF-perturbations of Kuranishi structures, but [*not*]{} one of good coordinate systems. The reason is explained at the end of Section \[sec:fiber\].
Fiber product and CF-perturbation {#subsec:fprocontper}
---------------------------------
We next discuss the case of fiber product.
Let $\mathcal U = (U,\mathcal E,s,\psi)$ be a Kuranishi chart of $X$. For $x \in U$ let $\frak V_x$ be an orbifold chart of $(U,\mathcal E)$ and $\mathcal S_x = (W_x,\omega_x,\{\frak s^{\epsilon}_x\})$ a CF-perturbation of $\mathcal U$ on $\frak V_x$. Let $f : U \to M$ be a smooth map to a manifold $M$ and $g : N \to M$ a smooth map from a manifold $N$. Suppose $f$ is strongly transversal to $g$ with respect to $\mathcal S_x$ in the sense of Definition \[submersivepertconlocloc\] (3). Then we take the fiber product $X {}_f\times_g N$, fiber product Kuranishi chart $(\frak V_x) {}_f\times_g N$ and $(\mathcal S_x){}_f\times_g N= (W_x,\omega_x,\{(\frak s^{\epsilon}_x)\,\,{}_f\times_g N\})$. Here $$((\frak s^{\epsilon}_x) {}_f\times_g N) : ((V_x) {}_{f}\times_g N) \times W_x
\to E_x$$ is defined by $$((\frak s^{\epsilon}_x) {}_f\times_g N)((y,z),\xi) = \frak s^{\epsilon}_x(y,\xi).$$ We call $(\mathcal S_x){}_f\times_g N$ the [*fiber product CF-perturbation*]{}. It is a CF-perturbation of $\mathcal U {}_f\times_g N$.
\[lem826222\]
1. If $\mathcal S_{x}$ is equivalent to $\mathcal S'_{x}$ and $f$ is strongly transversal to $g$ with respect to $\mathcal S_x$, $f$ is strongly transversal to $g$ with respect to $\mathcal S'_x$. Moreover $(\mathcal S_x){}_f\times_g N$ is equivalent to $(\mathcal S'_x){}_f\times_g N$.
2. Let $\Phi : \frak V_{x'}^{\prime} \to \frak V_{x}$ be an embedding of orbifold charts. If $f \circ \Phi$ is strongly transversal to $g$ with respect to $\Phi^*\frak S_x$, $f$ is strongly transversal to $g$ with respect to $\frak S_x$. Moreover $$\Phi^*((\mathcal S_{x}) {}_f \times_g N)$$ is equivalent to $$(\Phi^*(\mathcal S_{x})) {}_f \times_g N.$$
This is a direct consequence of the definitions.
\[lemdef834\] Let $\mathcal U = (U,\mathcal E,s,\psi)$ be a Kuranishi chart of $X$, and $\frak S = \{(\frak V_{\frak r},\mathcal S_{\frak r})
\mid \frak r \in \frak R\}$ a representative of a CF-perturbation of $\mathcal U$. Let $f : U \to M$ be a smooth map to a manifold $M$ and $g : N \to M$ a smooth map from a manifold $N$.
1. If $f$ is strongly transversal to $g$ with respect to $\frak S$ in the sense of Definition-Lemma \[strosubsemiloc\] (3), then $$\frak S {}_f\times_g N= \{((\frak V_{\frak r}) {}_f\times_g N,(\mathcal S_{\frak r}) {}_f\times_g N) \mid \frak r \in \frak R\}$$ is a CF-perturbation of $\mathcal U{}_f\times_g N$.
2. If $\frak S$ is equivalent to $\frak S'$, then $\frak S {}_f\times_g N$ is equivalent to $\frak S' {}_f\times_g N$.
3. Therefore we can define a [fiber product of CF-perturbations]{} with a map $g : N \to M$ when $\widehat f$ is strongly transversal to $g$.
This is an immediate consequence of Lemma \[lem826222\].
\[lem828222\] Let $\Phi : \mathcal U \to \mathcal U^+$ be an embedding of Kuranishi charts and $\frak S$, $\frak S^{+ }$ CF-perturbations of $\mathcal U$, $ \mathcal U^+$, respectively. Suppose $\frak S$, $\frak S^{+ }$ are compatible with $\Phi$. Let $f^+ : \mathcal U^+ \to M$ be a strongly smooth map to a manifold $M$, $f = f^+ \circ \varphi : \mathcal U \to M$, and $g : N \to M$ a smooth map from a manifold $N$. We assume $f, f^+$ are strongly transversal to $g$ with respect to $\frak S$, $\frak S^{+ }$, respectively.
Then $\frak S^+ \,{}_{f^+}\times_g N$, $\frak S \,{}_f\times_g N$ are compatible to $\Phi \times {\rm id} :
\mathcal U \,{}_f\times_g N \to \mathcal U^+ \,{}_{f^+}\times_g N$.
This is a direct consequence of the definitions.
\[lemdef835\] Let $\widehat{\mathcal U}= (\{\mathcal U_{p}\},\{\Phi_{pq}\})$ be a Kuranishi structure of $Z \subseteq X$ and $\widehat{\frak S} = \{\frak S_{p}\}$ a CF-perturbation of $\widehat{\mathcal U}$. Suppose that a strongly smooth map $\widehat f : (X,\widehat{\mathcal U}) \to M$ is strongly transversal to a smooth map $g : N \to M$ with respect to $\widehat{\frak S}$ in the sense of Definition \[smoothfunctiononvertK\] (3).
Then $\{(\frak S_{p}) {}_f\times_g N\}$ is a CF-perturbation. We call it [a fiber product CF-perturbations]{} and write $(\frak S_{p}) {}_f\times_g N$.
Lemma-Definition \[lemdef835\] is a consequence of Lemma \[lem828222\].
\[defn837\] Let $\widehat{\mathcal U^i}= (\{\mathcal U^i_{p_i}\},\{\Phi^i_{p_iq_i}\})$ be Kuranishi structures of $Z_i \subseteq X_i$ and $\widehat{\mathcal U^1} \times \widehat{\mathcal U^2}$ the direct product Kuranishi structure on $Z_1 \times Z_2 \subseteq X_1 \times X_2$. Let $\widehat{\frak S^i} = \{\frak S^i_{p_i}\}$ be CF-perturbations of $\mathcal U^i_{p_i}$ and $\widehat{\frak S^1} \times \widehat{\frak S^2}$ their direct product. Let $\widehat{f^i} : (X_i,\widehat{\mathcal U^i}) \to M$ be strongly smooth maps to a manifold $M$.
1. We say that $\widehat{f^1}$ is [*strongly transversal to $\widehat{f^2}$ with respect to $\widehat{\frak S^1}$, $\widehat{\frak S^2}$*]{}, if and only if $$(\widehat{f^1},\widehat{f^2}) : (X_1\times X_2,\widehat{\mathcal U^1} \times \widehat{\mathcal U^2})
\to M \times M$$ is strongly transversal to the diagonal $\Delta_M =\{(x,x) \mid x \in M\}$, with respect to the direct product $\widehat{\frak S^1}\times \widehat{\frak S^2}$ in the sense of Definition \[smoothfunctiononvertK\] (3).
2. In the situation of (1), we define the [*fiber product of CF-perturbations*]{} by $$(\widehat{\frak S^1}) {}_{\widehat{f^1}} \times_{\widehat{f^2}} (\widehat{\frak S^2})
=
(\widehat{\frak S^1}\times\widehat{\frak S^2}) \,{}_{(\widehat{f^1},\widehat{f^2})}\times_{M\times M} \Delta_M.$$ Here the right hand side is defined by Lemma-Definition \[lemdef835\].
\[lem838\]
1. Suppose we are in the situation of Definition \[defn837\]. If $\widehat{f^1}$ is strongly submersive with respect to $\widehat{\frak S^1}$, then $\widehat{f^1}$ is strongly transversal to any $\widehat{f^2}$ with respect to $\widehat{\frak S^1}$ and $\widehat{\frak S^2}$, provided $\widehat{\frak S^2}$ is transversal to $0$.
2. In the situation of Definition \[defn837\], we assume $\widehat{f^1}$ is strongly submersive with respect to $\widehat{\frak S^1}$. Let $\widehat{f^3} : (X_2,\widehat{\mathcal U^2}) \to N$ be another strongly smooth map such that $\widehat{f^3}$ is strongly submersive with respect to $\widehat{\frak S^2}$.
Then $$\widetilde{\widehat{f^3}}
: (X_1\times X_2,\widehat{\mathcal U^1} \times \widehat{\mathcal U^2})
\to N$$ is strongly submersive with respect to $(\widehat{\frak S}^1) {}_{\widehat{f^1}} \times_{\widehat{f^2}} (\widehat{\frak S}^2)$. Here $
\widetilde{\widehat {f^3}}$ is the map induced from $\widehat{f^3}$.
3. The fiber product of uniform family of CF-perturbations is uniform.
It suffices to prove the corresponding statement on a single orbifold chart. Namely for each $i=1,2$ we consider $\frak V_{{\frak r}_i}^i$ an orbifold chart of a Kuranishi neighborhood of $\widehat{\mathcal U}^i$, a CF-perturbation $\mathcal S^i_{{\frak r}_i}$ of it, and maps $f^i_{{\frak r}_i} : U^i_{{\frak r}_i} \to M$, $f^3_{{\frak r}_2} : U^2_{{\frak r}_2} \to N$. We will prove this case below.
[*Proof of (1)*]{} : By assumption $$f^1_{{\frak r}_1}\vert_{(\mathcal S^{1\epsilon}_{{\frak r}_1})^{-1}(0)}
: (\mathcal S^{1\epsilon}_{{\frak r}_1})^{-1}(0) \to M$$ is a submersion. Therefore it is transversal to $$f^2_{{\frak r}_2}\vert_{(\mathcal S^{2\epsilon}_{{\frak r}_2})^{-1}(0)}
: (\mathcal S^{2\epsilon}_{{\frak r}_2})^{-1}(0) \to M$$ as required.
[*Proof of (2)*]{} : Let $
(y_i,\xi_i) \in (\mathcal S^{i\epsilon}_{{\frak r}_i})^{-1}(0).
$ Here $y_i \in V^i_{{\frak r}_i}$, $\xi_i \in W^i_{{\frak r}_i}$ where $\frak V_{x_i}^i = (V^i_{{\frak r}_i},\Gamma^i_{x_i},E^i_{{\frak r}_i},\psi^i_{{\frak r}_i},\widehat{\psi}^i_{{\frak r}_i})$, $\mathcal S^{i\epsilon}_{{\frak r}_i} = (W^i_{{\frak r}_i},\omega^i_{{\frak r}_i},\frak s^{i\epsilon}_{{\frak r}_i})$.
Suppose $f^1_{{\frak r}_1}(y_1) = f^2_{{\frak r}_2}(y_2) = z$ and $f^3_{{\frak r}_2}(y_2) = w$. We consider $$\aligned
(d_{y_1}f^1_{{\frak r}_1}
\oplus
d_{y_2}f^2_{{\frak r}_2})
\oplus
d_{y_2}f^3_{{\frak r}_2}
:
T_{(y_1,\xi_1)}(\mathcal S^{1\epsilon}_{{\frak r}_1})^{-1}(0)
&\oplus
T_{(y_2,\xi_2)}(\mathcal S^{2\epsilon}_{{\frak r}_2})^{-1}(0)
\\
&\to
T_zM \oplus T_z M \oplus T_wN.
\endaligned$$ Let $\frak v \in T_wN$. Then there exists $\tilde{\frak v_2}\in
T_{(y_2,\xi_2)}(\mathcal S^{2\epsilon}_{{\frak r}_2})^{-1}(0)$ such that $$\label{formula810}
(d_{y_2}f^3_{{\frak r}_2})(\tilde{\frak v_2}) = \frak v.$$ Then there exists $\tilde{\frak v_1}\in
T_{(y_1,\xi_1)}(\mathcal S^{1\epsilon}_{{\frak r}_1})^{-1}(0)$ such that $$\label{formula811}
(d_{y_1}f^1_{{\frak r}_1})(\tilde{\frak v_1})
=
(d_{y_2}f^2_{{\frak r}_2})(\tilde{\frak v_2}).$$ (\[formula811\]) implies that $$(\tilde{\frak v_1},\tilde{\frak v_2})
\in
T_{((y_1),(y_2))}
(
(\mathcal S^{1\epsilon}_{{\frak r}_1})^{-1}(0) \,{}_{f^1}\times_{f^2}
(\mathcal S^{2\epsilon}_{{\frak r}_1})^{-1}(0)
)$$ and (\[formula810\]) implies that $$(d_{((y_1),(y_2))} \overline{f^3})
(\tilde{\frak v_1},\tilde{\frak v_2})
= \frak v.$$ Here $\overline{f^3} :
(\mathcal S^{1\epsilon}_{{\frak r}_1})^{-1}(0) \,{}_{f^1}\times_{f^2}
(\mathcal S^{2\epsilon}_{{\frak r}_1})^{-1}(0)\to N$ is a local representative of $\widetilde{\widehat{f^3}}$. We have thus proved the required submersivity.
The proof of (3) is obvious from the definition.
Composition of smooth correspondences {#subsec:compsmcor}
-------------------------------------
In this subsection we define composition of smooth correspondences and its perturbation. Let us consider the following situation.
\[compositu\] Let $(X_{21},\widehat{\mathcal U_{21}})$, $(X_{32},\widehat{\mathcal U_{32}})$ be K-spaces and $M_{i}$ ($i=1,2,3$) smooth manifolds. Let $$\widehat{f_{i,ji}} : (X_{ji},\widehat{\mathcal U_{ji}}) \to M_i,
\qquad
\widehat{f_{j,ji}} : (X_{ji},\widehat{\mathcal U_{ji}}) \to M_j$$ be strongly smooth maps for $(i,j) = (1,2)$ or $(2,3)$. We assume $\widehat{f_{2,21}}$ and $\widehat{f_{3,32}}$ are weakly submersive. These facts imply that $$\frak X_{i+1 i} = ((X_{i+1 i},\widehat{\mathcal U_{i+1 i}}),\widehat{f_{i,i+1 i}},
\widehat{f_{i+1,i+1 i}})$$ is a smooth correspondence from $M_i$ to $M_{i+1}$ for $i=1,2$. (Lemma \[lem838\] (2).) Let $\widehat{\frak S_{i+1 i}}$ be a CF- perturbation of $(X_{i+1 i},\widehat{\mathcal U_{i+1 i}})$ for each $i=1,2$. We assume that $\widehat{f_{i+1,i+1 i}}$ is strongly submersive with respect to $\widehat{\frak S_{i+1 i}}$ for each $i=1,2$.$\blacksquare$
In Situation \[compositu\], we put $$X_{31} = X_{21} \times_{M_2} X_{32}
=
\{(x_{21},x_{32}) \in X_{21} \times X_{32}\mid f_{2,21}(x_{21}) = f_{2,32}(x_{32})\}.$$ We put the fiber product Kuranishi structure $$\label{compkurafiber}
\widehat{\mathcal U_{31}}
=
\widehat{\mathcal U_{21}} \times_{M_2} \widehat{\mathcal U_{32}}$$ on $X_{31}$ and define $$\label{mapf131}
\widehat{f_{1,31}} : (X_{31},\widehat{\mathcal U_{31}}) \to M_1,
\qquad
\widehat{f_{3,31}} : (X_{31},\widehat{\mathcal U_{31}}) \to M_3$$ as the compositions $$(X_{31},\widehat{\mathcal U_{31}}) \to (X_{21},\widehat{\mathcal U_{21}}) \to M_1,
\qquad
(X_{31},\widehat{\mathcal U_{31}}) \to (X_{32},\widehat{\mathcal U_{32}}) \to M_3,$$ where the first arrows are obvious projections. We write $$\frak X_{21} \times_{M_2} \frak X_{32}
=
((X_{31},\widehat{\mathcal U_{31}}),\widehat{f_{1,31}},\widehat{f_{3,31}})$$ and call it the [*composition of smooth correspondences*]{} $\frak X_{21}$ and $ \frak X_{32}$. We also denote it by $\frak X_{32} \circ \frak X_{21}$. $$\label{diagram1010}
\xymatrix{
&& \frak X_{31} \ar[ld]\ar[rd] \\
& \frak X_{21} \ar[ld]\ar[rd] && \frak X_{32}\ar[ld]\ar[rd] \\
M_1 && M_2 && M_3}$$
Note that we did not define the ‘maps’ $\frak X_{31} \to \frak X_{21}$, $\frak X_{31} \to \frak X_{32}$ in Diagram (\[diagram1010\]). This is because we never defined the notion of morphism between K-spaces in this document. However, the maps $\frak X_{31} \to M_1$ and $\frak X_{31} \to M_3$ are defined by composing the map $f_{21,p}$ or $f_{32,q}$ and the projection on each chart.
1. The fiber product (\[compkurafiber\]) is well-defined.
2. The map $(X_{31},\widehat{\mathcal U_{31}}) \to M_3$ is weakly submersive.
\(1) By assumption, $\widehat{f_{2,21}}$ is weakly submersive. This implies well-defined-ness of (\[compkurafiber\]).
\(2) Let $(p,q) \in X_{31}$, i.e., $p\in X_{21}$, $q \in X_{32}$ $f_{2,21}(p) = f_{2,32}(q)$. We put $x = f_{2,21}(p) = f_{3;32}(q)$ and $y \in f_{3,32}(q)$. By assumption $$d_{o_p}(f_{2,21})_p : T_{o_p}U_p \to T_xM_2,
\quad
d_{o_q}(f_{3,32})_q : T_{o_q}U_q \to T_yM_3$$ are surjective. Let $v_3 \in T_yM_3$. There exists $\tilde v_3 \in T_{o_q}U_q$ such that $(d_{o_q}(f_{3,32})_q)(\tilde v_3) = v_3$. There exists $\tilde v_2 \in T_{o_p}U_p$ such that $(d_{o_p}(f_{2,21})_p)(\tilde v_2) = (d_{o_q}(f_{2,32})_q)(\tilde v_3)$. Then $(\tilde v_2,\tilde v_3) \in T_{o_{(p,q)}}U_{(p,q)}$ and $(d_{o_{(p,q)}}(f_{3,31})_{(p,q)})(\tilde v_2,\tilde v_3) = v_3$ as required.
\[defn839\]
1. Let $\frak X = ((X,\widehat{\mathcal U}),\widehat{f_s},\widehat{f_t})$ be a smooth correspondence and $\widehat{\frak S}$ a CF-perturbation of $(X,\widehat{\mathcal U})$. We say that $(X,\widehat{\mathcal U},\widehat{\frak S},\widehat{f_s},
\widehat{f_t})$ is a [*perturbed smooth correspondence*]{} if $\widehat{f_t}$ is strongly submersive with respect to $\widehat{\frak S}$.
2. A perturbed smooth correspondence $\tilde{\frak X} = (\frak X,\widehat{\frak S}) =
(X,\widehat{\mathcal U},\widehat{\frak S},\widehat{f_s},\widehat{f_t})$ from $M_s$ to $M_t$ induces a linear map $
\Omega^*(M_s) \to \Omega^{* + \deg\frak X} (M_t)
$ by (\[defn924\]). We write it as ${\rm Corr}^{\epsilon}_{\tilde{\frak X}}$.
3. In Situation \[compositu\], let $\widehat{{\frak S}_{i+1 i}}$ be a CF-perturbation of $(X_{i+1 i},\widehat{{\mathcal U}_{i+1 i}})$ for each $i=1,2$. Suppose $\tilde{\frak X}_{i+1 i} = (X_{i+1 i},\widehat{\mathcal U_{i+1 i}},\widehat{\frak S_{i+1 i}},\widehat{f_{i,i+1 i}},\widehat{f_{i+1, i+1 i}})$ is a perturbed smooth correspondence for each $i=1,2$. Then by Lemma \[lem838\] $$\label{fppersmcorr}
\aligned
(X_{21}\,{}_{f_{2,21}}\times_{ f_{2,32}} X_{32},\,
&\widehat{\mathcal U_{21}}\,{}_{\widehat{f_{2,21}}}\times _{\widehat{f_{2,32}}},\widehat{\mathcal U_{32}},\\
&(\widehat{{\frak S}_{21}}) {}_{\widehat {f_{2,21}}} \times_{\widehat{f_{2,32}}} (\widehat{\frak S_{32}}),\widehat{f_{1,31}},\widehat{f_{3,31}})
\endaligned$$ is a perturbed smooth correspondence from $M_1$ to $M_3$. We call (\[fppersmcorr\]) the [*composition*]{} of $\tilde{\frak X}_{21}$ and $\tilde{\frak X}_{32}$ and write $\tilde{\frak X}_{32} \circ \tilde{\frak X}_{21}$. Here $$\aligned
& \widehat{f_{1,31}} : (X_{21}\,{}_{f_{2,21}}\times_{ f_{2,32}} X_{32},\,
\widehat{\mathcal U_{21}}\,{}_{\widehat{f_{2,21}}}\times _{\widehat{f_{2,32}}},\widehat{\mathcal U_{32}}) \to M_1, \\
& \widehat{f_{3,31}} : (X_{21}\,{}_{f_{2,21}}\times_{f_{2,32}} X_{32},\,
\widehat{\mathcal U_{21}}\,{}_{\widehat{f_{2,21}}}\times _{\widehat{f_{2,32}}},\widehat{\mathcal U_{32}}) \to M_3
\endaligned$$ are maps as in .
Composition formula {#subsec:compformulaub}
-------------------
The main result of this section is the following.
\[compformulaprof\][(Composition formula, [@fooo09 Lemma 12.15])]{} Suppose that $\tilde{\frak X}_{i+1 i} =
(X_{i+1 i},\widehat{\mathcal U_{i+1 i}},\widehat{\frak S_{i+1 i}},
\widehat{f_{i,i+1 i}},
\widehat{f_{i+1,i+1 i}})$ are perturbed smooth correspondences for $i=1,2$. Then $$\label{formula814}
{\rm Corr}^{\epsilon}_{\tilde{\frak X}_{32}\circ\tilde{\frak X}_{21}}
=
{\rm Corr}^{\epsilon}_{\tilde{\frak X}_{32}} \circ {\rm Corr}^{\epsilon}_{\tilde{\frak X}_{21}}$$ for each sufficiently small $\epsilon >0$.
Note that ${\rm Corr}^{\epsilon}_{\tilde{\frak X}_{**}}$ depends on the positive number $\epsilon$.
Let $h_1$ and $h_3$ be differential forms on $M_1$ and $M_3$, respectively. It suffices to show the next formula. $$\label{fpm109109}
\int_{M_3}
{\rm Corr}^{\epsilon}_{\tilde{\frak X}_{32}\circ\tilde{\frak X}_{21}}(h_1) \wedge h_3
=
\int_{M_3}
{\rm Corr}^{\epsilon}_{\tilde{\frak X}_{32}}({\rm Corr}^{\epsilon}_{\tilde{\frak X}_{21}}(h_1)) \wedge h_3.$$ We use the following notation.
In Situation \[sitsu8main\], we consider the case when $M$ is a point and put: $$\int_{(X,Z,\widehat{\mathcal U},{\widehat{\frak S^{\epsilon}}})}\widehat{ h}
=
\widehat f !(\widehat{h};{\widehat{\frak S^{\epsilon}}}).$$ We call it the [*integration of $\widehat h$ over $(X,Z,\widehat{\mathcal U},{\widehat{\frak S^{\epsilon}}})$.*]{} It is a real number depending on $(X,Z,\widehat{\mathcal U},\widehat{\frak S}), \epsilon$ and $\widehat h$. We also define $$\int_{(X,{\mathcal U},{\frak S}^{\epsilon})} h
=
f !({h};{{\frak S}}^{\epsilon}).$$ Here ${\mathcal U}$ is a Kuranishi chart of $X$, $h$ is a differential form of compact support on $U$, ${\frak S}^{\epsilon}$ is a CF-perturbation of $\mathcal U$ on the support of $h$ and $f : U \to $ a point is a trivial map, such that $f$ is strongly submersive with respect to ${\frak S}^{\epsilon}$. (Note the strong submersivity in the case when the map is trivial is nothing but transversality of the CF-perturbation to $0$.) We call it the [*integration of $h$ over $(\widehat{\mathcal U},\widehat{\frak S^{\epsilon}})$.*]{}
In the notation above we omit $Z$ if $Z=X$.
Using this notation the right hand side of (\[fpm109109\]) is $$\aligned
&\int_{(X_{32},\widehat{\mathcal U_{32}},\widehat{\frak S_{32}^{\epsilon}})} (\widehat{f_{2,32}})^*({\rm Corr}_{\tilde{\frak X}_{21}}(\widehat{h_1})) \wedge
(\widehat{f_{3,32}})^* \widehat{h_3}\\
&=
\int_{(X_{32},\widehat{\mathcal U_{32}},\widehat{\frak S_{32}^{\epsilon}})} (\widehat{f_{2,32}})^*(\widehat f_{1,21}!(\widehat{f_{1,21}})^*(\widehat{h_1});
\widehat{{\frak S}_{21}^{\epsilon}})) \wedge
(\widehat{f_{3,32}})^* \widehat{h_3}.
\endaligned$$ On the other hand the left hand side of (\[fpm109109\]) is $$\int_{(X_{31},\widehat{\mathcal U_{31}},{\widehat{\frak S^{\epsilon}_{31}}})}
(\widehat{f_{1,31}})^*(\widehat{h_1}) \wedge (\widehat{f_{3,31}})^*(\widehat{h_3}).$$ Therefore (\[fpm109109\]) follows from the next proposition.
\[fubiniprop\] For $i=1,2$, let $\widehat{\mathcal U_i}$ be Kuranishi structures of $Z_i \subseteq X_i$, $\widehat{\frak S_i}$ their CF-perturbations, $\widehat h_i$ smooth differential forms on $(X_i,\widehat{\mathcal U_i})$ which have compact support in $\ring Z_i$, and $\widehat f_i : (X_i,Z_i;\widehat{\mathcal U_i}) \to M$ strongly smooth maps. Supposed that $\widehat f_1$ is strongly submersive with respect to $\widehat{\frak S_1}$ and $\widehat{\frak S_2}$ is transversal to $0$ and denote by $(X,Z,\widehat{\mathcal U},\widehat{\frak S})$ the fiber product $$(X_1,Z_1,\widehat{\mathcal U_1},\widehat{\frak S_1})
\,{}_{\widehat{f_1}}
\times_{\widehat{f_2}}\,\,
(X_2,Z_2,\widehat{\mathcal U_2},\widehat{\frak S_2})$$ over $M$. Then $$\label{fubini}
\int_{(X,Z,\widehat{\mathcal U},\widehat{\frak S^{\epsilon}})} \widehat{h_1}\wedge
\widehat{h_2}
=
\int_{(X_2,Z_2,\widehat{\mathcal U}_2,\widehat{\frak S_2^{\epsilon}})}
(\widehat{f_2})^*(\widehat{f_1}!(\widehat{h_1};\widehat{\frak S_1^{\epsilon}}))
\wedge \widehat{h_2}.$$
We may regard Formula (\[fubini\]) as a version of Fubini formula.
$$\xymatrix{
& (X,Z,\widehat{\mathcal U},\widehat{\frak S}) \ar[ld]\ar[rd]\\
(X_1,Z_1,\widehat{\mathcal U_1},\widehat{\frak S_1}) \ar[rd]^{\widehat{f_1}} && (X_2,Z_2,\widehat{\mathcal U_2},\widehat{\frak S_2}) \ar[ld]_{\widehat{f_2}} \\
& M
}$$
For $i=1,2$ let ${\widetriangle{\mathcal U_i}}$ be good coordinate systems of $X_i$ and $\widehat{\mathcal U_i} \to {\widetriangle{\mathcal U_i}}$ KG-embeddings. We may assume that there exist CF-perturbations ${\widetriangle{\frak S_i}}$ of $(X_i,Z_i;{\widetriangle{\mathcal U_i}})$ such that ${\widetriangle{\frak S_i}}$, ${\widehat{\frak S_i}}$ are compatible with the KG-embeddings and $\widehat{h_i}$, $\widehat{f_i}$ are pull back of differential forms on ${\widetriangle{\mathcal U_i}}$ and of strongly smooth maps on ${\widetriangle{\mathcal U_i}}$, which we also denote by $\widetriangle{h_i}$, $\widetriangle{f_i}$, respectively. (Theorem \[Them71restate\], Proposition \[le614\] (2), Theorem \[existperturbcont\].)
Let $\{\chi^i_{\frak p_i}\}$ be strongly smooth partitions of unity of $(X_i,{\widetriangle{\mathcal U_i}})$. The functions $\chi^i_{\frak p_i}$ can be regarded as strongly smooth maps $(X_i,Z_i;{\widetriangle{\mathcal U_i}}) \to \R$. Therefore they induce strongly smooth maps $(X_i,Z_i;{\widehat{\mathcal U_i}}) \to \R$, which we also denote by $\chi^i_{\frak p_i}$. Then they induce strongly smooth functions on the fiber product $(X,Z;\widehat{\mathcal U})$.
To prove (\[fubini\]) it suffices to prove the next formula for each $\frak p_1$, $\frak p_2$ and $\epsilon$. $$\label{fubinionechart1}
\aligned
&\int_{(X,Z,\widehat{\mathcal U},\widehat{\frak S^{\epsilon}})}
\chi^1_{\frak p_1}\widehat{h_1}\wedge
\chi^2_{\frak p_2}\widehat{h_2} \\
&=
\int_{({\mathcal U}_{2,\frak p_2},{\frak S}^{\epsilon}_{2,\frak p_2})}
f_{2,\frak p_2}^*(f_{1,\frak p_1}!
(\chi^1_{\frak p_1}h_{1,\frak p_1};{\frak S}_{1,\frak p_1}^{\epsilon}))
\wedge \chi^2_{\frak p_2} h_{2,\frak p_2}.
\endaligned$$ We will use the following result.
\[lemma8845\] Let $Z_{(1)}$ and $Z_{(2)}$ be compact subsets of $X$ such that $Z_{(1)} \subset \ring Z_{(2)}$. Let ${\widehat{\mathcal U}}$ be a Kuranishi structure of $Z_{(2)} \subseteq X$ and $\widehat h$ a differential form on ${\widehat{\mathcal U}}$. Let $\widehat{f_{(2)}} : (X,Z_{(2)};{\widehat{\mathcal U}}) \to M$ be a strongly smooth map and $\widehat{\frak S_{(2)}}$ a CF-perturbation of ${\widehat{\mathcal U}}$. Denote by $\widehat{\frak S_{(1)}}$ the restriction of $\widehat{\frak S_{(2)}}$ to ${\widehat{\mathcal U}\vert_{Z_{(1)}}}$ (Definition \[defn71717\]). Suppose
1. $\widehat{f_{(2)}}$ is strongly submersive with respect to $\widehat{\frak S_{(2)}}$.
2. $\widehat h$ has compact support in $\ring Z_{(1)}$.
We denote by $\widehat h\vert_{Z_{(1)}}$ the differential form on ${\widehat{\mathcal U}\vert_{Z_{(1)}}}$ induced by $\widehat h$ via the condition (2). Then $$\widehat{f_{(2)}}!(\widehat h;\widehat{\frak S_{(2)}^{\epsilon}})
=
\widehat{f_{(1)}}!(\widehat h\vert_{Z_{(1)}};\widehat{\frak S_{(1)}^{\epsilon}}).$$ where $\widehat{f_{(1)}}$ is the restriction of $\widehat{f_{(2)}}$ to $(X,Z_{(1)};{\widehat{\mathcal U}\vert_{Z_{(1)}}})$.
By using differential forms $\rho$ on $M$ it suffices to consider the case $\deg h = \dim \widehat{\mathcal U}$ and prove the next formula (see Lemma \[lem782\] (2)) $$\label{form1014}
\int_{(X,Z_{(2)},\widehat{\mathcal U},\widehat{\frak S_{(2)}^{\epsilon}})} \widehat h
=
\int_{(X,Z_{(1)},\widehat{\mathcal U}\vert_{Z_{(1)}},\widehat{\frak S_{(1)}^{\epsilon}})} \widehat h.$$ We take good coordinate systems $\widetriangle{\mathcal U_{(i)}}$ of $Z_{(i)} \subseteq X$ such that:
1. $\widetriangle{\mathcal U_{(1)}}$ is compatible with $\widehat{\mathcal U}\vert_{Z_{(1)}}$ and $\widetriangle{\mathcal U_{(2)}}$ is compatible with $\widehat{\mathcal U}$.
2. $\widetriangle{\mathcal U_{(2)}}$ strictly extends $\widetriangle{\mathcal U_{(1)}}$.
3. There are CF-perturbations $\widetriangle{\frak S_{(i)}}$, differential forms $\widetriangle{h_{(i)}}$ on $(X,Z_{(i)};\widetriangle{\mathcal U_{(i)}})$ which are compatible each other and are compatible with corresponding objects on $\widetriangle{\mathcal U_{(1)}}$ and on $\widehat{\mathcal U}\vert_{Z_{(1)}}$.
4. $\widetriangle{\frak S_i}$ are transversal to $0$.
Existence of such objects is a consequence of Theorem \[Them71restate\], Propositions \[prop7582752\] and \[existperturbcontrel\].
Let $\widetriangle{\mathcal U_{(i)}} = (\frak P_{(i)},\{\mathcal U_{{(i)},\frak p}\},
\{\Phi_{{(i)},\frak p\frak q}\})$. By Definition \[defn735f\] (1)(a), $
\frak P_{(1)} = \{\frak p \in \frak P_{(2)} \mid {\rm Im}(\psi_{{(2)},\frak p})
\cap Z_{(1)} \ne \emptyset\}
$ and $\mathcal U_{{(1)},\frak p}$ is an open subchart of $\mathcal U_{{(2)},\frak p}$ for $\frak p \in \frak P_{(1)} \subset \frak P_{(2)}$. We choose support systems $\mathcal K^{(i)}$ and a neighborhood $\frak U_{1}(Z_{(1)})$ of $Z_{(1)}$ in $\vert\widetriangle{\mathcal U_{(2)}} \vert$ with the following properties.
1. If $\frak p \in \frak P_{(1)}\subset \frak P_{(2)}$ then $\mathcal K^{(1)}_{\frak p} \cap \frak U_{(1)}(Z_{(1)})= \mathcal K^{(2)}_{\frak p}
\cap \frak U_{1}(Z_{(1)})$.
2. If $\frak p \in \frak P_{(2)} \setminus \frak P_{(1)}$ then $\frak U_{1}(Z_{(1)}) \cap \mathcal K_{\frak p}^{(2)}= \emptyset$.
We take support pairs $(\mathcal K^{1,{(i)}},\mathcal K^{2,{(i)}})$ of $\widetriangle{\mathcal U^{(i)}}$ such that $\mathcal K^{2,{(i)}} < \mathcal K^{{(i)}}$. By (a), we may assume that there exists a neighborhood $\frak U_2(Z_{(1)})$ of $Z_{(1)}$ in $\vert\widetriangle{\mathcal U_{(2)}} \vert$ such that
1. If $\frak p \in \frak P_{(1)} \subset \frak P_{(2)}$ then $\mathcal K^{j,(1)}_{\frak p} \cap \frak U_2(Z_{(1)})= \mathcal K^{j,(2)}_{\frak p}
\cap \frak U_2(Z_{(1)})$ for $j=1,2$.
We can take $\delta_{(i)}$ and partition of unities $\{\chi^{(i)}_{\frak p}\}$ of $(X,Z_{(i)},{\widetriangle{\mathcal U_{(i)}}},\mathcal K^{1,{(i)}},\delta_{(i)})$ and a neighborhood $\frak U_3(Z_{(1)})$ of $Z_{(1)}$ in $\vert\widetriangle{\mathcal U_{(2)}} \vert$ for $i=1,2$, such that:
1. If $\frak p \in \frak P_{(1)} \subset \frak P_{(2)}$ then $\chi^{(1)}_{\frak p} = \chi^{(2)}_{\frak p}$ on $\frak U_3(Z_{(1)})$.
By definition we have $$\label{form1015}
\int_{(X,Z_{(i)},\widehat{\mathcal U_{(i)}},\widehat{\frak S_{(i)}^{\epsilon}})}
\widehat h
=
\sum_{\frak p \in \frak P_{(i)}}
\int_{\mathcal K^{1,{(i)}}_{\frak p}(2\delta_{(i)}) \cap \frak U(Z_{(i)})} \chi^{(i)}_{\frak p} h_{\frak p}$$ for sufficiently small neighborhoods $\frak U(Z_{(i)})$ of $Z_{(i)}$ in $\vert\widetriangle{\mathcal U_{(i)}}\vert$. By (a)(b)(c)(d) above and using the fact that $\widehat h$ has a compact support in $\ring Z_{(1)}$, (\[form1015\]) implies (\[form1014\]). In fact, if $\frak p \in \frak P_{(2)} \setminus \frak P_{(1)}$ then (b) implies that $$\int_{\mathcal K^{1,{(2)}}_{\frak p}(2\delta_{(2)}) \cap \frak U(Z_{(2)})} \chi^{(2)}_{\frak p} h_{\frak p} = 0$$ and if $\frak p \in \frak P_{(1)}$ then (c)(d) imply $$\int_{\mathcal K^{1,{(1)}}_{\frak p}(2\delta_{(1)}) \cap \frak U(Z_{(1)})} \chi^{(1)}_{\frak p} h_{\frak p} =
\int_{\mathcal K^{1,{(2)}}_{\frak p}(2\delta_{(2)}) \cap \frak U(Z_{(2)})} \chi^{(2)}_{\frak p} h_{\frak p}.$$ Thus the proof of Proposition \[lemma8845\] is complete.
We continue the proof of Proposition \[fubiniprop\]. We consider the fiber product Kuranishi chart $\mathcal U_{1,\frak p_1} \,{}_{f_{1,\frak p_1}}\times_{f_{2,\frak p_2}}
\, \mathcal U_{2,\frak p_2}$ and the fiber product CF-perturbation $
\frak S_{1,\frak p_1}\,{}_{f_{1,\frak p_1}}\times_{f_{2,\frak p_2}}
\, \frak S_{2,\frak p_2}
$ of it.
We apply Lemma \[lemma8845\] to $\widehat h = \chi^1_{\frak p_1}\widehat h_1\wedge
\chi^2_{\frak p_2}\widehat h_2$, $Z_{(2)} = Z$ and $$Z_{(1)} =
(\psi_{1,\frak p_1}(U_{1,\frak p_1} \cap s_{1,\frak p_1}^{-1}(0)))\,{}_{f_{1}}\times_{f_{2}}
\, (\psi_{2,\frak p_2}(U_{2,\frak p_2} \cap s_{2,\frak p_2}^{-1}(0))).$$ Note there exists a good coordinate system consisting of a single Kuranishi chart $\mathcal U_{1,\frak p_1} \,{}_{f_{1,\frak p_1}}\times_{f_{2,\frak p_2}}
\, \mathcal U_{2,\frak p_2}$ of $Z_{(1)} \subseteq X_1 \times_M X_2$. Therefore the left hand side of (\[fubinionechart1\]) is equal to $$\int_{(\mathcal U_{1,\frak p_1} \,{}_{f_{1,\frak p_1}}\times_{f_{2,\frak p_2}}
\, \mathcal U_{2,\frak p_2},\frak S^{\epsilon}_{1,\frak p_1} \,{}_{f_{1,\frak p_1}}\times_{f_{2,\frak p_2}}
\, \frak S^{\epsilon}_{2,\frak p_2})}
\chi^1_{\frak p_1}\widehat h_1\wedge
\chi^2_{\frak p_2}\widehat h_2.$$ Thus to prove (\[fubinionechart1\]) it suffices to prove the next lemma.
\[lem846\] Let $\mathcal U_i$ be Kuranishi charts of $X_i$, $f_i : U_i \to M$ smooth maps, $h_i$ differential forms on $U_i$ and $\frak S_i$ CF-perturbations of $\mathcal U_i$, for $i=1,2$. We assume that $f_1$ is strongly submersive with respect to $\frak S_1$ and $\frak S_2$ is transversal to $0$. Then $$\label{formula819}
\int_{(\mathcal U_1 \,{}_{f_1}\times_{f_2}\, \mathcal U_2,\frak S^{\epsilon}_1 \,{}_{f_1}\times_{f_2}\, \frak S_2^{\epsilon})}
h_1\wedge h_2
=
\int_{(\mathcal U_2,\frak S^{\epsilon}_2)} f_2^*
(
f_1!(h_1;\frak S_1^{\epsilon})
)
\wedge h_2.$$
Let $\frak S_i = (\{\frak V^i_{\frak r_i}\},\{\mathcal S^i_{\frak r_i}\})$ and $\{\chi_{\frak r_i}^i\}$ be a smooth partition of unities of orbifolds $U_i$ subordinate to its open cover $\{U^i_{\frak r_i}\}$. Then $\{\chi^1_{\frak r_1}\chi^2_{\frak r_2} \mid \frak r_1 \in \frak R_1,
\frak r_2 \in \frak R_2\}$ is a partition of unity subordinate to the covering $\{U_1\,{}_{f_1}\times_{f_2} U_2 \mid \frak r_1 \in \frak R_1,
\frak r_2 \in \frak R_2\}$. Therefore, to prove (\[formula819\]), it suffices to prove $$\label{formula820}
\aligned
&\int_{(\mathcal U^1_{\frak r_1} \,{}_{f_1}\times_{f_2}\, \mathcal U^2_{\frak r_2},\mathcal S^{1\epsilon}_{\frak r_1} \,{}_{f_1}\times_{f_2}\, \mathcal S^{2\epsilon}_{\frak r_2})}
\chi_{\frak r_1}^1h_1\wedge \chi_{\frak r_2}^2 h_2\\
&=
\int_{(\mathcal U^2_{\frak r_2},\mathcal S^{2\epsilon}_{\frak r_2})}
\chi_{\frak r_1}^1\chi_{\frak r_2}^2 f_2^*
(
f_1!(h_1;\mathcal S^{1\epsilon}_{\frak r_1})
)\wedge h_2.
\endaligned$$ (\[formula820\]) is an immediate consequence of the next lemma.
\[lem847\] For $i=1,2$ let $N_i$ be smooth manifolds and $f_i : N_i \to M$ smooth maps and $h_i$ smooth differential forms on $N_i$ of compact support. Suppose that $f_1$ is a submersion. Then we have $$\label{form10821}
\int_{N_1\,{}_{f_1}\times_{f_2} N_2} h_1 \wedge h_2
=
\int_{N_2} f_2^* (f_1!(h_1)) \wedge h_2.$$
By using a partition of unity, it suffices to prove the lemma in the following special case: $N_1 = \R^a \times M$, $f_1 : \R^a \times M \to M$ is the projection to the second factor, $N_2 = M \times \R^b$, $f_2 : M \times \R^b \to M$ is the projection to the first factor, and $M,N_1,N_2$ are open subsets of Euclidean spaces. We prove this case below.
In this case $N_1 \,{}_{f_1}\times_{f_2} N_2 \cong \R^a \times M \times \R^b$. Let $(x_1,\dots,x_m)$ be a coordinate of $M$, $(y_1,\dots,y_a)$ a coordinate of $\R^a$ and $(z_1,\dots,z_b)$ a coordinate of $\R^b$. Then $(y_1,\dots,y_a,x_1,\dots,x_m)$ is a coordinate of $N_1$, $(x_1,\dots,x_m,z_1,\dots,z_b)$ is a coordinate of $N_2$, and $(y_1,\dots,y_a,x_1,\dots,x_m,z_1,\dots,z_b)$ is a coordinate of $N_1\,{}_{f_1}\times_{f_2} N_2$.
We may write $$\aligned
h_1 &= \sum_{I} g_{1,I}(y_1,\dots,y_a,x_1,\dots,x_m)
dy_1 \wedge \dots \wedge dy_a \wedge dx_{I}
\\
h_2 &= \sum_{J} g_{2,J}(x_1,\dots,x_m,z_1,\dots,z_b)
dx_J \wedge dz_1 \wedge \dots \wedge dz_b
\endaligned$$ for certain smooth functions $g_{1,I}, g_{2,J}$. We write them as $g_{1,I}(y,x), g_{2,J}(x,z)$ for simplicity. Here $I,J \subset \{ 1, \dots ,m\}$ and $dx_I = dx_{i_1} \wedge \dots \wedge dx_{i_{\vert I \vert}}$ for $I=\{ i_1, \dots , i_{\vert I \vert} \}$ and $dx_{J}$ is defined in a similar way. We may assume $I \cup J =\{ 1,\dots , m\}$. Then the left hand side of (\[form10821\]) is given by $$\sum _{I,J}\int_{N_1\,{}_{f_1}\times_{f_2} N_2}
g_{1,I}(y,x) g_{2,J}(x,z)dydx_I dx_J dz.$$ On the other hand, the right hand side of (\[form10821\]) is given by $$\sum_{I,J}\int_{N_2}
\left(\int_{y\in \R^a}g_{1,I}(y,f_2(x,z))dy \right)g_{2,J}(x,z) dx_I dx_J dz.$$ Therefore Lemma \[lem847\] is an immediate consequence of Fubini’s theorem.
This completes the proof of Lemma \[lem846\].
This also completes the proof of Proposition \[fubiniprop\].
Therefore the proof of Theorem \[compformulaprof\] is now complete.
We finally remark the following.
When CF-perturbations $\tilde{\frak X}_{32}$, $\tilde{\frak X}_{21}$ vary in a uniform family, we can choose $\epsilon$ in Theorem \[compformulaprof\] in the way independent of the CF-perturbations in that family.
The proof goes in the same way as the proof of Proposition \[lem761\]. So we omit it.
Construction of good coordinate system {#sec:contgoodcoordinate}
======================================
In this section we prove Theorem \[Them71restate\] together with various addenda and variants.
Construction of good coordinate systems: the absolute case {#subsec:constgcsabs}
----------------------------------------------------------
This subsection will be occupied by the proof of Theorem \[Them71restate\].
[^28] Let $\widehat{\mathcal U}$ be a Kuranishi structure of $Z \subseteq X$. We use the dimension stratification $\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U})$ as in Definition \[stratadim\], for the inductive construction of a good coordinate system compatible to the given Kuranishi structure in the sense of Definition \[gcsystem\]. The main part of the proof is the proof of Proposition \[inductiveprop\] below. We first describe the situation under which Proposition \[inductiveprop\] is stated.
\[situation101\] Let $\frak d \in \Z_{\ge 0}$, $Z_0$ a compact subset of $$\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U})
\setminus
\bigcup_{\frak d' > \frak d}\mathcal S_{\frak d'}(X,Z;\widehat{\mathcal U}),$$ and $Z_1$ a compact subset of $\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U})$. We assume that $Z_1$ contains an open neighborhood of $$\label{biggedZZZ}
\bigcup_{\frak d' > \frak d}\mathcal S_{\frak d'}(X,Z;\widehat{\mathcal U})
\nonumber$$ in $\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U})$.
We also assume that we have a good coordinate system ${\widetriangle{\mathcal U}}
= (\frak P,\{\mathcal U_{\frak p}\},\{\Phi_{\frak p\frak q}\})$ of $Z_1^+ \subseteq X$, where $Z^+_1$ is a compact neighborhood of $Z_1$ in $X$, and a strict KG-embedding $$\widehat{\Phi^1}
= \{\Phi^1_{\frak p p} \mid
p \in {\rm Im}(\psi_{\frak p}) \cap Z^+_1\} : \widehat{\mathcal U}\vert_{Z^+_1} \to {\widetriangle{\mathcal U}},$$ where $\widehat{\mathcal U}\vert_{Z^+_1}$ is the restriction of $\widehat{\mathcal U}$, which is defined in Definition \[defn735f\] (3).
Let $\mathcal U_{\frak p_0} = (U_{\frak p_0},E_{\frak p_0},s_{\frak p_0},\psi_{\frak p_0})$ be a Kuranishi neighborhood of $Z^+_0$ such that $\dim U_{\frak p_0} = \frak d$. Here $Z^+_0$ is a compact neighborhood of $Z_0$ in $X$. We regard $\mathcal U_{\frak p_0}$ as a good coordinate system $\widetriangle{\mathcal U_{\frak p_0}}$ that consists of a single Kuranishi chart and suppose that we are given a strict KG-embedding $$\widehat{\Phi^0}
= \{\Phi^0_{\frak p_0 p} \mid
p \in {\rm Im}(\psi_{\frak p_0}) \cap Z^+_0\} : \widehat{\mathcal U}\vert_{Z^+_0} \to \widetriangle{\mathcal U_{\frak p_0}}.$$ We put $$\label{form111}
Z_+ = Z_1 \cup Z_0.$$ $\blacksquare$
We remark $Z_+ \subseteq \mathcal S_{\frak d}(X,Z;\widehat{\mathcal U})$. However in general $Z^+_0$, $Z^+_1$ are not subsets of $\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U})$.
\[inductiveprop\] In Situation \[situation101\], there exists a good coordinate system ${\widetriangle{\mathcal U^+}}
= (\frak P^+,\{\mathcal U^+_{\frak p}\},\{\Phi^+_{\frak p\frak q}\})$ of $Z^+_+ \subseteq X$ with the following properties. Here $Z_+^+$ is a compact neighborhood of $Z_+$ in $X$.
1. $\frak P^+ = \frak P \cup \{\frak p_0\}$. The partial order on $\frak P^+$ is the same as one on $\frak P$ among the elements of $\frak P$ and $\frak p > \frak p_0$ for any $\frak p \in \frak P$.
2. If $\frak p \in \frak P$ then $\mathcal U^+_{\frak p}$ is an open subchart $\mathcal U_{\frak p}\vert_{U^+_{\frak p}}$ where $U^+_{\frak p}$ is an open subset of $U_{\frak p}$. We have $$\bigcup_{\frak p \in \frak P}
\psi_{\frak p}^+ ((s_{\frak p}^+)^{-1}(0))
\supset Z_1.$$
3. $\mathcal U^+_{\frak p_0}$ is a restriction of $\mathcal U_{\frak p_0}$ to an open subset $U^+_{\frak p_0}$ of $U_{\frak p_0}$. We have $$\psi_{\frak p_0}^+ ((s_{\frak p_0}^+)^{-1}(0))
\supset
Z_0.$$
4. The coordinate change $\Phi^+_{\frak p\frak q}$ is the restriction of $\Phi_{\frak p\frak q}$ to $
U^+_{\frak q} \cap \varphi_{\frak p\frak q}^{-1}(U^+_{\frak p}),
$ if $\frak p,\frak q \in \frak P$.
5. There exists an open substructure $\widehat{\mathcal U^0}$ of $\widehat{\mathcal U}$, a strict KG-embedding $\widehat{\Phi^+}
= \{\Phi^+_{\frak p p} \mid
p \in {\rm Im}(\psi^+_{\frak p}) \cap Z^+_+\} : \widehat{\mathcal U^0} \to {\widetriangle{\mathcal U^+}}$ with the following properties.
1. If $\frak p \in \frak P$ and $p \in {\rm Im}(\psi^+_{\frak p}) \cap Z_+^+$, then we have the commutative diagram: $$\label{diagram103}
\begin{CD}
\mathcal U_{p}^0 @ > {\Phi_{\frak p p}^+} >>
{\mathcal U}_{\frak p}^+ \\
@ VVV @ VVV\\
\mathcal U_{p} @ > {\Phi_{\frak p p}^1} >>{\mathcal U}_{\frak p}
\end{CD}$$ where the vertical arrows are embeddings as open subcharts. (The commutativity of diagram means that the maps coincide when both sides are defined.)
2. If $p\in {\rm Im}(\psi^+_{\frak p_0}) \cap Z^+_+$, then we have the commutative diagram: $$\label{diagram104}
\begin{CD}
\mathcal U_{p}^0 @ > {\Phi_{\frak p_0 p}^+} >>
{\mathcal U}_{\frak p_0}^+ \\
@ VVV @ VVV\\
\mathcal U_{p} @ > {\Phi_{\frak p_0 p}^0} >>{\mathcal U}_{\frak p_0}
\end{CD}$$ where the vertical arrows are embeddings as open subcharts. (The commutativity of diagram means that the maps coincide when both sides are defined.)
Note the good coordinate system ${\widetriangle{\mathcal U^+}}$ has one more Kuranishi chart than ${\widetriangle{\mathcal U}}$. Moreover ${\widetriangle{\mathcal U^+}}$ is a good coordinate system of a neighborhood of $Z_{+}$ which contains $Z_1$. ${\widetriangle{\mathcal U}}$ is a good coordinate system of a neighborhood of $Z_{1}$. This is the reason why we use the symbol $+$ in ${\widetriangle{\mathcal U^+}}$.
On the other hand, each Kuranishi chart of ${\widetriangle{\mathcal U^+}}$ is either an open subchart of ${\widetriangle{\mathcal U}}$ or of ${\widetriangle{\mathcal U_{\frak p_0}}}$. In other words each Kuranishi chart of ${\widetriangle{\mathcal U^+}}$ is [*smaller*]{} than that of ${\widetriangle{\mathcal U}}$ or of ${\widetriangle{\mathcal U_{\frak p_0}}}$.
We take a support system $\mathcal K$ of ${\widetriangle{\mathcal U}}$. Note by definition we have: $$\bigcup_{\frak p} \psi_{\frak p}(\overset{\circ}{\mathcal K_{\frak p}} \cap
s_{\frak p}^{-1}(0))
\supset
Z^+_1.$$ We take a compact subset $\mathcal K_{\frak p_0}$ of $U_{\frak p_0}$ such that $$\psi_{\frak p_0}(\overset{\circ}{\mathcal K_{\frak p_0}} \cap
s_{\frak p_0}^{-1}(0))
\supset
Z^+_0.$$ Since $
\widehat{\Phi^1}
: \widehat{\mathcal U}\vert_{Z^+_1} \to {\widetriangle{\mathcal U}}
$ and $
\widehat{\Phi^0} : \widehat{\mathcal U}\vert_{Z^+_0} \to \widetriangle{\mathcal U_{\frak p_0}}
$ are [*strict*]{} KG-embeddings, they have the following properties.
\[proper113\]
1. If $q \in
\psi_{\frak p}(\mathcal K_{\frak p} \cap
s_{\frak p}^{-1}(0))$, then $\Phi^1_{\frak p q} = ( \varphi^1_{\frak p q},
\widehat{\varphi^1_{\frak p q}})$ is defined and is an embedding $\Phi^1_{\frak p q} : \mathcal U_q \to \mathcal U_{\frak p}$. Note the domain of $\varphi^1_{\frak p q}$ is $U_q$. We put $$U^1_q = U^1_{\frak p q} = U_q.$$
2. If $q \in \psi_{\frak p_0}(\mathcal K_{\frak p_0} \cap
s_{\frak p_0}^{-1}(0))$, then $\Phi^0_{\frak p_0 q} = (\varphi^0_{\frak p_0 q},
\widehat{\varphi^0_{\frak p_0 q}})$ is defined and is an embedding $\Phi^0_{\frak p_0 q} : \mathcal U_q \to \mathcal U_{\frak p_0}$. Note the domain of $\varphi^0_{\frak p_0 q}$ is $U_q$. We put $$U^0_q = U^0_{\frak p_0 q} = U_q.$$
For each $\frak p$ we use compactness to find a finite number of points $q^{\frak p}_1,\dots,q^{\frak p}_{N(\frak p)} \in
\psi_{\frak p_0}(\mathcal K_{\frak p_0} \cap
s_{\frak p_0}^{-1}(0)) \cap
\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U})
\cap \psi_{\frak p}(\mathcal K_{\frak p} \cap
s_{\frak p}^{-1}(0))$ such that $$\bigcup_{i=1}^{N(\frak p)}
\psi_{q_i^{\frak p}}(s_{q^{\frak p}_i}^{-1}(0) \cap U^1_{q^{\frak p}_i})
\supset
\psi_{\frak p_0}(\mathcal K_{\frak p_0} \cap
s_{\frak p_0}^{-1}(0))
\cap
\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U})
\cap \psi_{\frak p}(\mathcal K_{\frak p} \cap
s_{\frak p}^{-1}(0)).
\nonumber$$ We take relatively compact open subsets $U^2_{q_i^{\frak p}}$ of $U^1_{q_i^{\frak p}}$ such that $$\label{qcoversuru}
\bigcup_{i=1}^{N(\frak p)}
\psi_{q^{\frak p}_i}(s_{q^{\frak p}_i}^{-1}(0) \cap U^2_{q^{\frak p}_i})
\supset
\psi_{\frak p_0}(\mathcal K_{\frak p_0} \cap
s_{\frak p_0}^{-1}(0))
\cap
\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U})
\cap
\psi_{\frak p}(\mathcal K_{\frak p} \cap
s_{\frak p}^{-1}(0)).$$ We consider the following diagram: $$\xymatrix{
\mathcal U^1_{\frak p_0} \ar@{.>}[rr] && \mathcal U^1_{\frak p}\\
& \mathcal U^2_{{q_i^{\frak p}}}\ar[lu]^{\Phi^0_{\frak p_0 q_i^{\frak p}}}
\ar[ru]_{\Phi^1_{\frak p q_i^{\frak p}}}
}$$ Here $\mathcal U^2_{{q_i^{\frak p}}}$ is the restriction of $\mathcal U_{{q_i^{\frak p}}}$ to $U^2_{{q_i^{\frak p}}}$.
Note $\Phi^0_{\frak p_0 q_i^{\frak p}}$ is locally invertible since $\varphi^0_{\frak p_0 q_i^{\frak p}}$ is an orbifold embedding between orbifolds of the same dimension. So we can find an embedding (from an appropriate open subchart) written in dotted arrow in the diagram. Those maps for various $q_i^{\frak p}$ however may not coincide on the overlapped part. We use the next lemma to shrink the domains so that those maps are glued to define a coordinate change from an open subchart of $\mathcal U^1_{\frak p_0}$ to an open subchart of $\mathcal U^1_{\frak p}$, which we need to find to prove Proposition \[inductiveprop\].
Note that Property \[proper113\] also holds when we replace $q$ by $r$.
[(Compare [@foootech Lemma 7.8])]{}\[lemma114\] For each $r \in \psi_{\frak p_0}(\mathcal K_{\frak p_0} \cap
s_{\frak p_0}^{-1}(0))
\cap
\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U})
\cap \bigcup_{\frak p} \psi_{\frak p}(\mathcal K_{\frak p} \cap
s_{\frak p}^{-1}(0))$ there exists an open neighborhood $U^3_{r}$ of $o_r$ in $U_r = U^1_{\frak p r}$ with the following properties.
1. If $r \in \psi_{\frak p}(s_{\frak p}^{-1}(0) \cap \mathcal K_{\frak p})$ and $
\varphi^1_{\frak p r}(U^3_r) \cap \overline{\varphi^1_{\frak p q^{\frak p}_i}(U^2_{q_i^{\frak p}}})
\ne \emptyset
$, then $$\label{115555}
U^3_{r} \subset U_{q^{\frak p}_i r} \cap \varphi^{-1}_{q^{\frak p}_i r}
(U^1_{\frak p q^{\frak p}_i}).$$
2. If $
\varphi^0_{\frak p_0 r}(U^3_r) \cap \overline{\varphi^0_{\frak p_0 q^{\frak p}_i}(U^2_{q^{\frak p}_i}})
\ne \emptyset
$, then $$U^3_{r} \subset U_{q^{\frak p}_i r} \cap \varphi^{-1}_{q^{\frak p}_i r}(U^0_{\frak p_0 q^{\frak p}_i}).$$
We observe the following 3 points.
1. (1)(2) above require a finite number of conditions for each $U_r^3$.
2. If a choice of $U_r^3$ satisfies one of those (finitely many) conditions, then any smaller $U_r^3$ satisfies the same condition.
3. For any one of those (finitely many) conditions, there exists $U_r^3$ which satisfies that condition.
In fact, (c) is proved in the case of Lemma \[lemma114\] (1), for example, as follows. Suppose $r \in \psi_{\frak p_0}(\mathcal K_{\frak p_0} \cap
s_{\frak p_0}^{-1}(0))
\cap
\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U})
\cap \bigcup_{\frak p} \psi_{\frak p}(\mathcal K_{\frak p} \cap
s_{\frak p}^{-1}(0))$. Let $\overline{U^2_{q^{\frak p}_i}}$ be the closure of ${U^2_{q^{\frak p}_i}}$ in ${U^1_{q^{\frak p}_i}}$. This is a compact set.
Suppose $o_r \notin U_{q^{\frak p}_i r}
\cap \varphi_{q^{\frak p}_i r}^{-1}(\overline{U^2_{q^{\frak p}_i}})$. Since $o_r \in U_{q^{\frak p}_i r}$, we have $\varphi_{q^{\frak p}_i r}(o_r) \notin \overline{U^2_{q^{\frak p}_i}}$. Therefore we can find $U^3_{r}$ such that $\varphi^1_{\frak p r}(U^3_r) \cap \overline{\varphi^1_{\frak p q^{\frak p}_i}(U^2_{q_i^{\frak p}}}) = \emptyset$. Thus Lemma \[lemma114\] (1) is satisfied in this case.
Suppose $o_r \in U_{q^{\frak p}_i r}
\cap \varphi_{q^{\frak p}_i r}^{-1}(\overline{U^2_{q^{\frak p}_i}})$. Since $\overline{U^2_{q^{\frak p}_i}} \subset {U^1_{q^{\frak p}_i}}$, we can find $U^3_{r}$ satisfying (\[115555\]).
The lemma follows immediately from (a)(b)(c).
For each $\frak p \in \frak P$ we choose $J(\frak p)$ points $$r^{\frak p}_{1},\dots,r^{\frak p}_{J(\frak p)}
\in
\psi_{\frak p_0}(s_{\frak p_0}^{-1}(0) \cap \mathcal K_{\frak p_0})
\cap \psi_{\frak p}(s_{\frak p}^{-1}(0) \cap \mathcal K_{\frak p})
\cap
\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U})$$ such that $$\label{rtachicoversuru}
\bigcup_{j=1}^{J(\frak p)}
\psi_{r^{\frak p}_j}(U^3_{r^{\frak p}_j}\cap s^{-1}_{r^{\frak p}_j}(0))
\supset
\psi_{\frak p_0}(s_{\frak p_0}^{-1}(0) \cap \mathcal K_{\frak p_0}) \cap \psi_{\frak p}(s_{\frak p}^{-1}(0) \cap \mathcal K_{\frak p})
\cap
\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U}).$$ We define $$\label{118888}
U^1_{\frak p \frak p_0}
=
\bigcup_{i=1}^{N(\frak p)} \bigcup_{j=1}^{J(\frak p)}
\left(
\varphi^0_{\frak p_0 r_j^{\frak p}}(U^3_{r_j^{\frak p}})
\cap
\varphi^0_{\frak p_0 q^{\frak p}_i}(U^2_{q^{\frak p}_i}))
\right)
\subset
U_{\frak p_0}.$$ We put $$\aligned
U^0_{q^{\frak p}_i r_j^{\frak p}}
&=
(\varphi^0_{\frak p_0 r_j^{\frak p}})^{-1}
\left(
\varphi^0_{\frak p_0 r_j^{\frak p}}(U^3_{r_j^{\frak p}})
\cap
\varphi^0_{\frak p_0 q^{\frak p}_i}(U^2_{q_i^{\frak p}}))
\right)
\subset U^3_{r_j^{\frak p}},
\\
U^0_{r_j^{\frak p}q^{\frak p}_i}
&=
(\varphi^0_{\frak p_0 q^{\frak p}_i})^{-1}
\left(
\varphi^0_{\frak p_0 r_j^{\frak p}}(U^3_{r_j^{\frak p}})
\cap
\varphi^0_{\frak p_0 q^{\frak p}_i}(U^2_{q_i^{\frak p}}))
\right)
\subset U^2_{q_i^{\frak p}}.
\endaligned$$
\[lem115555\] $$\psi_{\frak p_0}(s_{\frak p_0}^{-1}(0) \cap U^1_{\frak p \frak p_0} )
\supset
\psi_{\frak p}(\mathcal K_{\frak p} \cap s^{-1}_{\frak p}(0))
\cap \psi_{\frak p_0}(\mathcal K_{\frak p_0} \cap s_{\frak p_0}^{-1}(0))
\cap
\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U}).$$
This is a consequence of (\[qcoversuru\]) and (\[rtachicoversuru\]).
\[lemma116\] There exists an embedding $\Phi^+_{\frak p \frak p_0} : \mathcal U\vert_{U^1_{\frak p \frak p_0}}
\to \mathcal U_{\frak p}$ such that: $$\label{formform1177}
\Phi^+_{\frak p \frak p_0}
\circ
\Phi^0_{\frak p_0 q^{\frak p}_i}\vert_{U^2_{q^{\frak p}_i}
\cap (\varphi^1_{\frak p q_i^{\frak p}})^{-1}(U^1_{\frak p\frak p_0})}
=
\Phi^1_{\frak p q^{\frak p}_i}\vert_{U^2_{q^{\frak p}_i}
\cap (\varphi^1_{\frak p q_i^{\frak p}})^{-1}(U^1_{\frak p\frak p_0})}.$$
$$\xymatrix{
\mathcal U^1_{\frak p_0} \ar@{.>}[rrrr] &&&& \mathcal U^1_{\frak p}\\
&
\mathcal U^2_{q_i^{\frak p}}\ar[lu]
\ar[rrru]
&&
\mathcal U^2_{q_{i'}^{\frak p}} \ar[ru]\ar[lllu]
\\
&&
\mathcal U^3_{r_j^{\frak p}} \ar[lu] \ar[ru]
}$$
Recalling the definition of $U^1_{\frak p\frak p_0}$ in (\[118888\]), we define a map $\varphi^+_{\frak p \frak p_0} : U^1_{\frak p\frak p_0} \to
U_{\frak p}$ by $$\label{labelfixedqidefcoochange}
\varphi^+_{\frak p \frak p_0}(x)
=
\varphi^1_{\frak p q^{\frak p}_i}(y_i)$$ for $x = \varphi^0_{\frak p_0 q^{\frak p}_i}(y_i) \in \varphi^0_{\frak p_0 q^{\frak p}_i}(U^2_{q^{\frak p}_i}) \cap U^1_{\frak p \frak p_0} $. We will prove that (\[labelfixedqidefcoochange\]) is well-defined below.
Suppose $x = \varphi^0_{\frak p_0 q^{\frak p}_i}(y_i) = \varphi^0_{\frak p_0 q^{\frak p}_{i'}}(y_{i'})$. (Here $y_i \in U^2_{q^{\frak p}_i}$, $y_{i'} \in U^2_{q^{\frak p}_{i'}}$.) There exist $r^{\frak p}_j$ and $z_j \in U^3_{r_j^{\frak p}}$ such that $x = \varphi^0_{\frak p_0 r_j^{\frak p}}(z_j)$. (This is a consequence of (\[118888\]).)
By Lemma \[lemma114\] (2), $z_j \in U_{q^{\frak p}_i r^{\frak p}_j} \cap U_{q^{\frak p}_{i'} r^{\frak p}_j}$ and $\varphi_{q^{\frak p}_i r^{\frak p}_j}(z_j) \in U^0_{\frak p_0 q^{\frak p}_i}$. We have $$\varphi^0_{\frak p_0 q^{\frak p}_i}(\varphi_{q^{\frak p}_i r^{\frak p}_j}(z_j))
= \varphi^0_{\frak p_0 r_j^{\frak p}}(z_j) = x = \varphi^0_{\frak p_0 q^{\frak p}_i}(y_i).$$ Here the first equality is a consequence of the fact that $\widehat{\Phi^0}$ is a strict KG-embedding.
Since $\varphi^0_{\frak p_0 q^{\frak p}_i}$ is injective, we have $y_i = \varphi_{q^{\frak p}_i r^{\frak p}_j}(z_j)$. In the same way we have $y_{i'} = \varphi_{q^{\frak p}_{i'} r^{\frak p}_j}(z_j)$. Therefore $$\varphi^1_{\frak p q^{\frak p}_i}(y_i)
=
\varphi^1_{\frak p q^{\frak p}_i}(\varphi_{q^{\frak p}_i r^{\frak p}_j}(z_j))
=
\varphi^1_{\frak p r^{\frak p}_j}(z_j).$$ This is a consequence of the fact that $\widehat{\Phi^1}$ is a strict KG-embedding.
We also have $$\varphi^1_{\frak p q^{\frak p}_{i'}}(y_{i'})
=
\varphi^1_{\frak p q^{\frak p}_{i'}}(\varphi_{q^{\frak p}_{i'} r^{\frak p}_j}(z_j))
=
\varphi^1_{\frak p r^{\frak p}_j}(z_j).$$ Thus $\varphi^1_{\frak p q^{\frak p}_i}(y_i) = \varphi^1_{\frak p q^{\frak p}_{i'}}(y_{i'})$ as required.
Thus we have defined a set theoretical map $\varphi^+_{\frak p \frak p_0} : U^1_{\frak p \frak p_0} \to U_{\frak p}$.
We note that it is locally written as $\varphi^1_{\frak p q^{\frak p}_i} \circ (\varphi_{\frak p_0 q^{\frak p}_i}^0)^{-1}$. Furthermore $\varphi^0_{\frak p_0 q^{\frak p}_i}$ is an embedding between orbifolds of the same dimension. Therefore $\varphi^0_{\frak p_0 q^{\frak p}_i}$ is locally a diffeomorphism. Hence $\varphi_{\frak p \frak p_0}^+$ is locally a composition of embedding and diffeomorphism and so is an embedding (of orbifolds). We can define the bundle map $\widehat{\varphi^+_{\frak p \frak p_0}}$ in the same way.
We note that the condition for $(\varphi^+_{\frak p \frak p_0},\widehat{\varphi^+_{\frak p \frak p_0}})$ to be an embedding of Kuranishi charts can be checked locally. Namely it suffices to check it at a neighborhood of each point. On the other hand, $\Phi^0_{* q^{\frak p}_i}$ is locally an isomorphism and $(\varphi^+_{\frak p \frak p_0},\widehat{\varphi^+_{\frak p \frak p_0}})$ is locally $\Phi_{\frak p q^{\frak p}_i} \circ (\Phi^0_{* q^{\frak p}_i})^{-1}$. Hence $(\varphi^+_{\frak p \frak p_0},\widehat{\varphi^+_{\frak p \frak p_0}})$ is an embedding of Kuranishi charts, as required. Then (\[formform1177\]) is immediate from definition.
We take a support system $\mathcal K^-$ of $(X,Z^+_1;\widetriangle{\mathcal U})$ such that $(\mathcal K^-,\mathcal K)$ is a support pair of $(X,Z^+_1;\widetriangle{\mathcal U})$. In particular, we have $$\bigcup_{\frak p \in \frak P} \psi_{\frak p}({\rm Int}~ \mathcal K^-_{\frak p}
\cap
s_{\frak p}^{-1}(0))
\supset
Z^+_1.$$ We also take a compact subset $\mathcal K_{\frak p_0}^-$ of $U_{\frak p_0}$ such that $${\mathcal K_{\frak p_0}^-}
\subset
{\rm Int}~ \mathcal K_{\frak p_0},
\qquad
Z^+_0 \subset \psi_{\frak p_0}
(s_{\frak p_0}^{-1}(0) \cap {\rm Int}~ \mathcal K_{\frak p_0}^-).$$ Put $U'_{\frak p} = {\rm Int}~ \mathcal K^-_{\frak p}$ and $U'_{\frak p_0} = {\rm Int}~ \mathcal K^-_{\frak p_0}$.
\[lem119\] There exist open subsets $U''_{\frak p} \subseteq U'_{\frak p}$, $U''_{\frak p_0} \subseteq U'_{\frak p_0}$ with the following properties:
1. We put $U''_{\frak p \frak p_0}
=
U^1_{\frak p\frak p_0}
\cap U''_{\frak p_0} \cap
(\varphi_{\frak p\frak p_0}^+)^{-1}(U''_{\frak p})$. Then $$\psi_{\frak p}(U''_{\frak p} \cap s_{\frak p}^{-1}(0))
\cap \psi_{\frak p_0}(U''_{\frak p_0} \cap s_{\frak p_0}^{-1}(0))
=
\psi_{\frak p_0}(U''_{\frak p\frak p_0} \cap s_{\frak p_0}^{-1}(0)).$$
2. $
\bigcup_{\frak p} \psi_{\frak p}(U''_{\frak p} \cap s_{\frak p}^{-1}(0))
\supseteq Z^+_1.
$
3. $
\psi_{\frak p_0}(U''_{\frak p_0} \cap s_{\frak p_0}^{-1}(0))
\supseteq Z^+_0.
$
This lemma implies that $\Phi_{\frak p \frak p_0}^+\vert_{U''_{\frak p\frak p_0}}$ is a coordinate change from $\mathcal U_{\frak p_0}\vert_{U''_{\frak p_0}}$ to $\mathcal U_{\frak p}\vert_{U''_{\frak p}}$ in the strong sense. In particular, Definition \[coordinatechangedef\] (3) is satisfied.
We take compact subsets $Z_{\frak p} \subset X$ such that
1. $
Z_{\frak p}\cap Z^+_{0}
\subseteq
\psi_{\frak p_0}(U^1_{\frak p\frak p_0} \cap s_{\frak p_0}^{-1}(0)).
$
2. $\psi_{\frak p}(U'_{\frak p}\cap s_{\frak p}^{-1}(0)) \supset Z_{\frak p}$.
3. $\bigcup Z_{\frak p} \supset Z^+_1$.
Existence of such $Z_{\frak p}$ is a consequence of Lemma \[lem115555\].
We next take open subsets $W_{\frak p}, W_{\frak p_0} \subset X$ such that
1. $
W_{\frak p}\cap W_{\frak p_0}
\subseteq
\psi_{\frak p_0}(U^1_{\frak p\frak p_0} \cap s_{\frak p_0}^{-1}(0)).
$
2. $\psi_{\frak p}(U'_{\frak p}\cap s_{\frak p}^{-1}(0)) \supset W_{\frak p} \supset Z_{\frak p}$.
3. $\psi_{\frak p_0}(U'_{\frak p_0}\cap s_{\frak p_0}^{-1}(0)) \supset W_{\frak p_0} \supset Z^+_{0}$.
Existence of such $W_{\frak p}, W_{\frak p_0}$ is a consequence of (i)(ii)(iii) above.
We then take open subsets $U''_{\frak p} \subset U'_{\frak p}$, $U''_{\frak p_0} \subset U'_{\frak p_0}$ such that
1. $W_{\frak p}
\supset \psi_{\frak p}(\overline{U''_{\frak p}}\cap s_{\frak p}^{-1}(0))
\supset \psi_{\frak p}({U''_{\frak p}}\cap s_{\frak p}^{-1}(0))
\supset Z_{\frak p}$.
2. $W_{\frak p_0}
\supset \psi_{\frak p_0}(\overline{U''_{\frak p_0}}\cap s_{\frak p_0}^{-1}(0))
\supset
\psi_{\frak p_0}({U''_{\frak p_0}}\cap s_{\frak p_0}^{-1}(0))
\supset Z^+_{0}$.
3. $\overline{U''_{\frak p}} \subset U'_{\frak p}$, $\overline{U''_{\frak p_0}} \subset U'_{\frak p_0}$ are compact.
Existence of such $U''_{\frak p}$, $U''_{\frak p_0}$ is a consequence of (b)(c).
Lemma \[lem119\] (3) is nothing but the last inclusion of (B). Lemma \[lem119\] (2) follows from (iii)(A).
We finally prove Lemma \[lem119\] (1).
\[sublem11111\] $$\psi_{\frak p}(U''_{\frak p} \cap s_{\frak p}^{-1}(0))
\cap
\psi_{\frak p_0}(U''_{\frak p_0} \cap s_{\frak p_0}^{-1}(0))
\subseteq
\psi_{\frak p_0}
(U^1_{\frak p\frak p_0} \cap s_{\frak p_0}^{-1}(0)).$$
This is a consequence of (a)(A)(B).
Note $U''_{\frak p \frak p_0}
=
U^1_{\frak p\frak p_0}
\cap U''_{\frak p_0} \cap
(\varphi^+_{\frak p\frak p_0})^{-1}(U''_{\frak p})$ by definition.
$$\psi_{\frak p}(U''_{\frak p} \cap s_{\frak p}^{-1}(0))
\cap
\psi_{\frak p_0}(U''_{\frak p_0} \cap s_{\frak p_0}^{-1}(0))
\subseteq
\psi_{\frak p_0}
(U''_{\frak p\frak p_0} \cap s_{\frak p_0}^{-1}(0)).$$
Let $\psi_{\frak p}(x) = \psi_{\frak p_0}(y)$ be in the left hand side. Since $\psi_{\frak p_0}$ is injective, Sublemma \[sublem11111\] implies $y \in U^1_{\frak p\frak p_0} \cap U''_{\frak p_0} \cap s_{\frak p_0}^{-1}(0)$. Since $\psi_{\frak p}(\varphi_{\frak p\frak p_0}^+(y)) =
\psi_{\frak p}(x)$, the injectivity of $\psi_{\frak p}$ implies $\varphi_{\frak p\frak p_0}^+(y) \in U''_{\frak p}$. Therefore $y \in U''_{\frak p \frak p_0}$ as required.
The opposite inclusion $$\psi_{\frak p}(U''_{\frak p} \cap s_{\frak p}^{-1}(0))
\cap
\psi_{\frak p_0}(U''_{\frak p_0} \cap s_{\frak p_0}^{-1}(0))
\supseteq
\psi_{\frak p_0}
(U''_{\frak p\frak p_0} \cap s_{\frak p_0}^{-1}(0))$$ is obvious. We have thus proved Lemma \[lem119\] (1). The proof of Lemma \[lem119\] is complete.
Thus $U''_{\frak p}$, $U''_{\frak p_0}$ together with the restriction of $\Phi^1_{\frak p \frak q}$, $\Phi_{\frak p \frak p_0}^+\vert_{U''_{\frak p
\frak p_0}}$ satisfy the conditions of Definition \[gcsystem\] except (7) and (8). Then we use Shrinking Lemma [@foooshrink Theorem 2.7] to shrink $U''_{\frak p}$, $U''_{\frak p_0}$ and $U''_{\frak p \frak p_0}$ so that Definition \[gcsystem\] (7) and (8) also hold.
We now change the notations $U''_{\frak p}$, $U''_{\frak p_0}$ to $U_{\frak p}$, $U_{\frak p_0}$ so that they denote the Kuranishi neighborhoods of the good coordinate system we have obtained above.
To complete the proof of Proposition \[inductiveprop\] it remains to prove Proposition \[inductiveprop\] (5). For this purpose, we will shrink $U_{\frak p}$, $U_{\frak p_0}$ further as follows.
We first take open subsets $U^{(1)}_{\frak p} \subset U_{\frak p}$ and $U^{(1)}_{\frak p_0} \subset U_{\frak p_0}$ with the following properties:
1. $\bigcup_{\frak p \in \frak P} \psi_{\frak p}(U^{(1)}_{\frak p}
\cap s_{\frak p}^{-1}(0)) \supset Z_1$.
2. $\bigcup_{\frak p \in \frak P} \psi_{\frak p}(U^{(1)}_{\frak p}
\cap s_{\frak p}^{-1}(0))
\subset {\rm Int}\, Z^+_1$.
3. $\psi_{\frak p_0}(U^{(1)}_{\frak p_0}
\cap s_{\frak p_0}^{-1}(0)) \supset Z_0$.
4. $\psi_{\frak p_0}(U^{(1)}_{\frak p_0}
\cap s_{\frak p_0}^{-1}(0))
\subset {\rm Int}\, Z^+_0$.
By Lemma \[lem320\] we have a good coordinate system of $Z_+^{++} \subset X$ whose Kuranishi charts are $\mathcal U_{\frak p}\vert_{U^{(1)}_{\frak p}}$ and $\mathcal U_{\frak p_0}\vert_{U^{(1)}_{\frak p_0}}$. (Here $Z_+^{++}$ is a compact neighborhood of $Z_+$. The compact neighborhood $Z^+_+$ we will obtain is a subset of $Z_+^{++}$.) We denote this good coordinate system by $\widetriangle{\mathcal U^{(1)}}$.
By the assumptions put in Situation \[situation101\] the following holds after replacing $\widehat{\mathcal U}$ by its open substructure if necessary.
1. If $p \in \psi_{\frak p_0}(U_{\frak p_0}^{(1)} \cap s_{\frak p_0}^{-1}(0)) \cap Z_+^{++}$, then there exists an embedding of Kuranishi charts $
\Phi^{(1)}_{\frak p_0 p} : \mathcal U_p \to \mathcal U_{\frak p_0}^{(1)}$. Here $\Phi^{(1)}_{\frak p_0 p}$ is an open restriction of $\Phi^{0}_{\frak p_0 p}$
2. If $p \in\psi_{\frak p}(U_{\frak p}^{(1)} \cap s_{\frak p}^{-1}(0)) \cap Z_+^{++}$and $\frak p \in \frak P$, then there exists an embedding of Kuranishi charts $\Phi^{(1)}_{\frak p p} :\mathcal U_p \to \mathcal U_{\frak p}$. Here $\Phi^{(1)}_{\frak p p}$ is an open restriction of $\Phi^{1}_{\frak p p}$
3. If $p \in {\rm Im}(\psi_{\frak p}) \cap Z_+^{++}$, $q \in {\rm Im}(\psi_{\frak q}) \cap Z_+^{++}$, $q \in {\rm Im}(\psi_{ p})$ and $\frak q \le \frak p$ then the following diagram commutes. $$\label{diag33repeat}
\begin{CD}
\mathcal U_{q}\vert_{U_{pq}
\cap \varphi_{\frak q q}^{-1}(U^{(1)}_{\frak p \frak q})} @ > {\Phi^{(1)}_{\frak q q}} >>
{\mathcal U}^{(1)}_{\frak q}\vert_{{U}^{(1)}_{\frak p\frak q}} \\
@ V{\Phi_{pq}}VV @ VV{\Phi^{(1)}_{\frak p \frak q}}V\\
\mathcal U_{p} @ > {\Phi^{(1)}_{\frak p p}} >>{\mathcal U}_{\frak p}
\end{CD}$$
Note we use Proposition \[inductiveprop\] (2)(4), when we take $U^{(1)}_{\frak p_0}$, $U^{(1)}_{\frak p}$ to show the properties (A)(B) above.
We also note that, to define a KG-embedding $\widehat{\mathcal U} \to {\widetriangle{\mathcal U^+}}$, we will use $\Phi^{(1)}_{\frak p p}$, $\Phi^{(1)}_{\frak p}$ or their open restrictions. Therefore the only remaining nontrivial part of the proof of Proposition \[inductiveprop\] (5) is the commutativity of the next diagram $$\label{diag33new}
\begin{CD}
\mathcal U^0_{q}\vert_{U^0_{pq}
\cap \varphi_{\frak p_0 q}^{-1}(U_{\frak p \frak p_0})} @ > {\Phi_{\frak p_0 q}} >>
{\mathcal U}^+_{\frak p_0}\vert_{{U}_{\frak p\frak p_0}
} \\
@ V{\Phi_{pq}\vert_{U^0_{pq}\cap \varphi_{\frak p_0 q}^{-1}(U_{\frak p \frak p_0})}}VV @ VV{\Phi^+_{\frak p \frak p_0}}V\\
\mathcal U^0_{p} @ > {\Phi_{\frak p p}} >>{\mathcal U}^+_{\frak p}
\end{CD}$$ which we need to prove under the assumption $p \in {\rm Im}(\psi_{\frak p}) \cap Z_+^+$, $q \in {\rm Im}(\psi_{\frak p_0}) \cap Z_+^+$, $q \in {\rm Im}(\psi_{p})$. Here $\widehat{\mathcal U^0}$ is an open substructure of $\widehat{\mathcal U}$ and $\mathcal U^0_{p}$ is its Kuranishi chart. $\Phi_{\frak p p}$ (resp. $\Phi_{\frak p_0 q}$) is an open restriction of $\Phi^{(1)}_{\frak p p}$ (resp. $\Phi^{(1)}_{\frak p_0 q}$). ${\mathcal U}^+_{\frak p}$ (resp. ${\mathcal U}^+_{\frak p_0}$) is an open subchart of $\mathcal U^{(1)}_{\frak p}$ (resp. $\mathcal U^{(1)}_{\frak p_0}$) and $\Phi^+_{\frak p \frak p_0}$ is an open restriction of $\Phi^{(1)}_{\frak p \frak p_0}$.
We will use the next lemma to show the commutativity of Diagram (\[diag33new\]). We take a support system $\{\mathcal K_{\frak p}^+
\mid \frak p \in \frak P\} \cup \{\mathcal K^+_{\frak p_0}\}$ of the good coordinate system on $Z_+$ that we have already obtained. Note that $\mathcal K^+_{\frak p_0} \subset \overset{\circ}{\mathcal K_{\frak p_0}}$, $\mathcal K^+_{\frak p} \subset \overset{\circ}{\mathcal K_{\frak p}}$.
\[119lem\] There exist open subsets $U_{\frak p}^+ \subset U_{\frak p}$, $U_{\frak p_0}^+ \subset U_{\frak p_0}$ with the following properties.
1. $
\psi_{\frak p_0}(\mathcal K_{\frak p_0}^+ \cap
s_{\frak p_0}^{-1}(0) \cap U_{\frak p_0}^+)
\cup
\bigcup_{\frak p} \psi_{\frak p}(\mathcal K^+_{\frak p} \cap
s_{\frak p}^{-1}(0) \cap U_{\frak p}^+)
\supset
Z_+.
$
2. If $
p \in \psi_{\frak p_0}(\mathcal K_{\frak p_0}^+ \cap
s_{\frak p_0}^{-1}(0) \cap U_{\frak p_0}^+)
\cap \psi_{\frak p}(\mathcal K^+_{\frak p} \cap
s_{\frak p}^{-1}(0) \cap U_{\frak p}^+),
$ then there exist $q^{\frak p}_i$ ($i=1,\dots,N(\frak p)$) such that $$p \in \psi_{q^{\frak p}_i}(s_{q^{\frak p}_i}^{-1}(0) \cap U_{q^{\frak p}_i}^2).$$
Recall $Z_+ \subset \mathcal S_{\frak d}(X,Z;\widehat{\mathcal U})$. (See (\[form111\]).) We also note that we do [*not*]{} assume $p \in \mathcal S_{\frak d}(X,Z;\widehat{\mathcal U})$ in Item (2) above.
We choose a decreasing sequence of relatively compact open subsets $U_{\frak p_0}^{[a]} \subset U$, $U_{\frak p}^{[a]}\subset U_{\frak p}$, ($a=1,2,\dots$) such that $$\overline{U_{\frak p}^{[a+1]}} \subset U_{\frak p}^{[a]},
\quad
\overline{U_{\frak p_0}^{[a+1]}} \subset U_{\frak p_0}^{[a]}$$ and $$\label{new1118}
\bigcap_{a=1}^{\infty} U_{\frak p_0}^{[a]}
= \psi_{\frak p_0}^{-1}(Z_0) \cap \mathcal K_{\frak p_0}^+,
\quad
\bigcap_{a=1}^{\infty} U_{\frak p}^{[a]}
= \psi_{\frak p}^{-1}(Z_1) \cap \mathcal K^+_{\frak p}.$$ Then, for each $a$, the choice $U_{\frak p_0}^+ = U_{\frak p_0}^{[a]}$ and $U_{\frak p}^+ = U_{\frak p}^{[a]}$ satisfies (1) above.
On the other hand, we can use (\[qcoversuru\]) to prove that (2) is satisfied for sufficiently large $a$ as follows. Suppose that (2) is not satisfied for $a_n \to \infty$. Then we have $$\label{phakokoniiru}
p_n \in \psi_{\frak p_0}(\mathcal K_{\frak p_0}^+ \cap
s^{-1}(0) \cap \overline{U_{\frak p_0}^{[a_n]}})
\cap \psi_{\frak p}(\mathcal K^+_{\frak p} \cap
s_{\frak p}^{-1}(0) \cap \overline{U_{\frak p}^{[a_n]}})$$ and $$\label{phaittenai}
p_n \notin
\bigcup_{i=1}^{N(\frak p)}
\psi_{q^{\frak p}_i}(s_{q^{\frak p}_i}^{-1}(0) \cap U_{q^{\frak p}_i}^{2}).$$ Note that we may assume $\frak p$ is independent of $n$ by taking a subsequence if necessary, since $\frak p \in \frak P$ and $\frak P$ is a finite set.
Since the right hand side of (\[phakokoniiru\]) is contained in $\psi_{\frak p_0}(\mathcal K_{\frak p_0}^+ \cap
s^{-1}(0) \cap \overline{U_{\frak p_0}^{[a_1]}})
\cap \psi_{\frak p}(\mathcal K^+_{\frak p} \cap
s_{\frak p}^{-1}(0) \cap \overline{U_{\frak p}^{[a_1]}})$ which is compact and independent of $n$, we may take a subsequence and may assume that $p_n$ converges. Then (\[new1118\]) and (\[phakokoniiru\]) imply $$\lim_{n\to \infty} p_n
\in Z_0 \cap \psi_{\frak p_0}(\mathcal K_{\frak p_0}^+
\cap s_{\frak p_0}^{-1}(0))
\cap
Z_1 \cap \psi_{\frak p}(\mathcal K^+_{\frak p}
\cap s_{\frak p}^{-1}(0)).$$ This contradicts to (\[phaittenai\]) since the right hand side of (\[phaittenai\]) is a neighborhood of $Z_0 \cap \psi_{\frak p_0}(\mathcal K_{\frak p_0}^+
\cap s_{\frak p_0}^{-1}(0))
\cap
Z_1 \cap \psi_{\frak p}(\mathcal K^+_{\frak p}
\cap s_{\frak p}^{-1}(0))$.
We consider ${\mathcal U}^+_{\frak p} =
{\mathcal U}_{\frak p}\vert_{U_{\frak p}^+}$, ${\mathcal U}^+_{\frak p_0} =
{\mathcal U}\vert_{U_{\frak p_0}^+}$. Put $$\frak s_{\frak p}^+ = \frak s_{\frak p}\vert_{{U}^+_{\frak p}},
\quad
\frak s_{\frak p_0}^+ = \frak s_{\frak p_0}\vert_{{U}^+_{\frak p_0}}$$ and denote by $\psi^+_{\frak p}$ and $\psi^+_{\frak p_0}$ the restrictions of the parametrization $\psi_{\frak p}$ and $\psi_{\frak p_0}$ to $(\frak s_{\frak p}^+)^{-1}(0)$ and $(\frak s_{\frak p_0}^+)^{-1}(0)$, respectively.
Next we note the following lemmata.
\[lem111441\] There exists an open neighborhood $U^0_p$ of $o_p$ in $U_p$ for each $p$ with the following properties.
1. If $p \in \psi_{q^{\frak p}_i}(s_{q^{\frak p}_i}^{-1}(0) \cap U_{q^{\frak p}_i}^2)$, then $$\label{eq1115}
U^0_p \subset U_{q^{\frak p}_i p}
\cap
\varphi_{q^{\frak p}_i p}^{-1}( U^2_{q^{\frak p}_i}).$$
2. If $p \in {\rm Im}(\psi_{q_i^{\frak p}})$ and $q \in \psi_{p}(U_{p}^0 \cap s_p^{-1}(0))$, then $q \in {\rm Im}(\psi_{q_i^{\frak p}})$.
We remark that, for each $p$, there exists only a finite number of $q^{\frak p}_i$ satisfying the assumptions of (1) or (2). The lemma immediately follows from this remark.
\[111515\] For each $p,q$ with $q \in \psi_{p}(U_{p}^0 \cap s_p^{-1}(0))$ there exists an open neighborhood $U_{pq}^0$ of $o_q$ in $U_{pq} \cap U^0_q$ with the following properties.
1. $\varphi_{pq}(U_{pq}^0) \subset U_p^0$.
2. If $p \in {\rm Im}(\psi_{q_i^{\frak p}})$ then $\varphi_{pq}(U_{pq}^0) \subset U_{q_i^{\frak p}p}$.
Since $\varphi_{pq}(o_q) \in U^0_p$ for $q \in \psi_{p}(U_{p}^0 \cap s_p^{-1}(0))$, (1) holds for a sufficiently small neighborhood $U^0_{pq}$ of $o_q$. Lemma \[lem111441\](2) implies (2) for a sufficiently small neighborhood $U^0_{pq}$ of $o_q$.
The pair $\mathcal U_p\vert_{U_p^0}$, $\Phi_{pq}\vert_{U_{pq}^0}$ define an open substructure of $\widehat{\mathcal U}$ by Lemma \[111515\] (1).
Now we will prove the commutativity of Diagram (\[diag33new\]). We take $Z_+^+$ so that it is contained in the left hand side of Lemma \[119lem\] (1). Suppose $p \in {\rm Im}(\psi^+_{\frak p}) \cap Z_+^+$, $q \in {\rm Im}(\psi^+_{\frak p_0}) \cap Z_+^+$, $q \in \psi_{p}(U_p^0\cap s_p^{-1}(0))$. Let $$y \in U_{pq}^0 \cap \varphi_{\frak p_0 q}^{-1}(U_{\frak p\frak p_0}).$$ There exists $i$ such that $
p \in \psi_{q^{\frak p}_i}(s_{q^{\frak p}_i}^{-1}(0) \cap U_{q^{\frak p}_i}^2)
$ by Lemma \[119lem\] (2). Then $
q \in \psi_{q^{\frak p}_i}(s_{q^{\frak p}_i}^{-1}(0) \cap U_{q^{\frak p}_i}^2)
$ by Lemma \[lem111441\] (2).
Now we have $$\aligned
(\varphi^+_{\frak p \frak p_0}\circ \varphi^0_{\frak p_0 q})(y)
&=
\varphi^+_{\frak p \frak p_0}(\varphi^0_{\frak p_0 q^{\frak p}_i}(\varphi_{q^{\frak p}_i q}(y)))
\\
&= \varphi^1_{\frak p q^{\frak p}_i}(\varphi_{q^{\frak p}_i q}(y))
\\
&= \varphi^1_{\frak p q^{\frak p}_i}(\varphi_{q^{\frak p}_i p}(\varphi_{p q}(y)))
\\
&=
\varphi^1_{\frak p p}(\varphi_{p q}(y)).
\endaligned$$ Here the equality in the first line follows from (\[eq1115\]) (with $p$ replaced by $q$) and the fact that $\widehat{\Phi^0}$ is a KG-embedding. The equality of the second line is the definition of $\varphi^+_{\frak p \frak p_0}$, that is, (\[labelfixedqidefcoochange\]). The equality of the third line is the consequence of the compatibility of coordinate change of Kuranishi structure and Lemma \[111515\] (2). The equality of the fourth line is the consequence of the fact that $\widehat{\Phi^1}$ is a strict KG-embedding.
We have proved the commutativity of Diagram (\[diag33new\]) for the maps between base orbifolds. The proof of the commutativity of the maps between bundles is the same. The proof of Proposition \[inductiveprop\] is now complete.
We use Proposition \[inductiveprop\] to prove Theorem \[Them71restate\] as follows.
We will construct a good coordinate system of a closed subset $\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U})$ of $Z$ by a downward induction on $\frak d$.
Let ${\widetriangle{\mathcal U_{\frak d+1}}}
= (\frak P,\{\mathcal U_{\frak p}\},\{\Phi_{\frak p\frak q}\})$ be a good coordinate system of a compact neighborhood of $\mathcal S_{\frak d+1}(X,Z;\widehat{\mathcal U})$. We will construct one of $\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U})$.
We put $$\frak P(\frak d) =
\{
\frak p \in \frak P \mid \dim U_{\frak p} > \frak d
\}.$$ We take a support system $\mathcal K_{\frak p}$ of $\widetriangle{\mathcal U_{\frak d+1}}$ and put $$B
=
\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U})
\setminus
\bigcup_{\frak p \in \frak P(\frak d)}
\psi_{\frak p}(s_{\frak p}^{-1}(0) \cap {\rm Int}\, \mathcal K_{\frak p}).$$ Then $B$ is a compact subset of $
\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U})
\setminus
\bigcup_{\frak d' > \frak d}
\mathcal S_{\frak d'}(X,Z;\widehat{\mathcal U})
$. We take a finite number of points $p_1,\dots,p_N
\in B$ such that $$\bigcup_{i=1}^N \psi_{p_i}(s_{p_i}^{-1}(0))
\supset B.$$ We take compact subsets $K_{p_i} \subset U_{p_i}$ such that $$\bigcup_{i=1}^N \psi_{p_i}(s_{p_i}^{-1}(0) \cap K_{p_i})
\supset B.$$ We put $Z_i = \psi_{p_i}(s_{p_i}^{-1}(0))$ and $$Z(0) =
\bigcup_{\frak p \in \frak P(\frak d)}
\psi_{\frak p}(s_{\frak p}^{-1}(0) \cap \mathcal K_{\frak p}) \cap Z.$$
Now using Proposition \[inductiveprop\] we can construct a good coordinate system on a compact neighborhood of $$Z(0) \cup \bigcup_{i=1}^n Z_i$$ by an induction over $n$. We thus obtain a good coordinate system on $\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U})$.
Now the downward induction over $\frak d$ is complete and we obtain a good coordinate system on $\mathcal S_{0}(X,Z;\widehat{\mathcal U}) = Z$. The proof of Theorem \[Them71restate\] is complete.
1. The heart of the proof of Theorem \[Them71restate\] is Lemma \[lemma116\]. There (and in several other places) we crucially use the fact that equality between embeddings of orbifolds is a local property. (It implies that the equality between embeddings of Kuranishi charts is also a local property). Thanks to this fact, we can check various equalities by looking finer and finer charts. For this main idea of the proof to work, we assumed our orbifold to be effective and the maps between them to be embeddings. Without this restriction the argument will become more cumbersome and lengthy. (We however believe that one can prove somewhat similar results without assuming effectivity of orbifold. Joyce may have done it in [@joyce2].)
2. The proof of Theorem \[Them71restate\] in [@foootech] is basically the same as one we gave above. However the argument of the proof of Lemma \[lemma116\] appeared twice in [@foootech] once during the proof of [@foootech Lemma 7.5] and once during the proof of [@foootech Lemma 7.24]. We reorganize the proof so that we need to use it only once. Also the argument used to prove the main theorem of [@foooshrink] appeared twice in [@foootech]. Besides these simplifications, we defined the notion of embedding of Kuranishi charts and use it systematically in this document. For example the statement of [@foootech Lemma 7.24] is nothing but the existence of an appropriate embedding of Kuranishi charts.
These changes make the presentation of the proof simpler and make the proof shorter, in this document, although its mathematical contents are the same as those in [@foootech].
We also prove compatibility of the good coordinate system to the Kuranishi structure (that is the existence of the embedding of the latter to the former) explicitly. (Namely we proved Proposition \[inductiveprop\] (5), explicitly.) In [@foootech], the proof of this part was said similar and was omitted. The proof we gave in this article is indeed similar to the other part of the proof. We repeated the argument here for completeness’ sake.
In Subsections \[subsec:constgcsrel\]-\[subsec:moreversionegcs2\], we will prove several variants of Theorem \[Them71restate\].
Construction of good coordinate systems: the relative case 1 {#subsec:constgcsrel}
------------------------------------------------------------
This section will be occupied by the proofs of Propositions \[prop518\] and \[prop519\].
We will prove Proposition \[prop519\] in this subsection in detail. Proposition \[prop518\] then follows by putting $\widehat{\mathcal U^+}
= \widehat{\mathcal U_1^+} = \widehat{\mathcal U_2^+}$.
We will use the next proposition in addition to Proposition \[inductiveprop\] in our inductive construction of good coordinate system.
\[prop1111\] Under Situation \[situation101\], we assume in addition that [for each $a=1,2$ there exists a Kuranishi structure $\widehat{\mathcal U^{+}_a}$ of $Z \subseteq X$]{} with the following properties.
1. There exists a strict KK-embedding $\widehat{\Phi^2_a} : \widehat{\mathcal U}
\to \widehat{\mathcal U^{+}_a}$.
2. There exists a GK-embedding $\widehat{\Phi^3_a} : \widetriangle{\mathcal U}
\to \widehat{\mathcal U^{+}_a}\vert_{Z_1^+}$ such that the composition $\widehat{\Phi^3_a}\circ\widehat{\Phi^1} :
\widehat{\mathcal U}\vert_{Z_1^+} \to {\widetriangle{\mathcal U}}
\to {\widehat{\mathcal U^{+}_a}}\vert_{Z_1^+}$ is an open restriction of $\widehat{\Phi^2_a}\vert_{Z_1^+}$.
3. There exists a GK-embedding $\widehat{\Phi^4_a} : \widetriangle{{\mathcal U}_{\frak p_0}}
\to {\widehat{\mathcal U^{+}_a}}\vert_{Z_0^+}$ such that the composition $\widehat{\Phi^4_a}\circ\widehat{\Phi^0} :
\widehat{\mathcal U}\vert_{Z_{0}^+}
\to \widetriangle{\mathcal U_{\frak p_0}} \to {\widehat{\mathcal U^{+}_a}}
\vert_{Z_0^+}$ is an open restriction of $\widehat{\Phi^2_a}\vert_{Z_0^+}$.
Then there exists ${\widetriangle{\mathcal U^+}}$ such as in the conclusion of Proposition \[inductiveprop\] that the following holds in addition.
1. There exists a GK-embedding $\widehat{\Phi^5_a} : {\widetriangle{\mathcal U^{+}}}
\to {\widehat{\mathcal U^{+}_a}}$.
2. The composition $\widehat{\Phi^5_a} \circ \widehat{\Phi^{+}} :
\widehat{\mathcal U}\vert_{Z_+^+} \to {\widetriangle{\mathcal U^+}}
\to {\widehat{\mathcal U^{+}_a}}$ is an open restriction of $\widehat{\Phi^2_a}\vert_{Z_+^+}$.
3. The restriction of $\widehat{\Phi^5_a}\vert_{Z_1}$ to a strictly open substructure of ${\widetriangle{\mathcal U^+}}\vert_{Z_1}$ coincides with $\widehat{\Phi^3_a}\vert_{Z_1}$.
4. The restriction of $\widehat{\Phi^5_a}\vert_{Z_0}$ to a strictly open substructure of ${\widetriangle{\mathcal U^+}}\vert_{Z_0}$ coincides with $\widehat{\Phi^4_a}\vert_{Z_0}$.
Let $\widetriangle{\mathcal U^+}$ be the good coordinate system constructed in the proof of Proposition \[inductiveprop\]. During the proof of Proposition \[inductiveprop\] we took a support system $\mathcal K$ of $\widetriangle{\mathcal U}$ and a compact subset $\mathcal K_{\frak p_0}$ of $U_{\frak p_0}$. We put $\mathcal K^+ = \mathcal K\cup \{\mathcal K_{\frak p_0}\}
= \{\mathcal K^+_{\frak p} \mid \frak p \in \frak P^+\}$. Then $\mathcal K^+$ is a support system of $\widetriangle{\mathcal U^+}$. We denote by $U^3_{a;\frak p}(q_i^{\frak p})$ and $U^4_{a;\frak p_0}(q_i^{\frak p})$ the domains of $\varphi^3_{a;q_i^{\frak p} \frak p}$ and $\varphi^4_{a;q_i^{\frak p}\frak p_0}$, which are part of data carried by $\widehat{\Phi^3_a}$ and $\widehat{\Phi^4_a}$. During this proof we choose $U^1_{q_i^{\frak p}}$ so that the following properties are also satisfied: $$\label{formula1125}
\varphi^1_{\frak p q^{\frak p}_i}(U^1_{q^{\frak p}_i})
\subset U^3_{a;\frak p}(q_i^{\frak p}), \quad
\varphi^0_{\frak p_0 q^{\frak p}_i}(U^1_{q^{\frak p}_i})
\subset U^4_{a;\frak p_0}(q_i^{\frak p})
\qquad \text{for $a=1,2$}.$$ Here $U^3_{a;\frak p}(q_i^{\frak p})$ and $U^4_{a;\frak p_0}(q_i^{\frak p})$ are domains of $\varphi^3_{a;q_i^{\frak p} \frak p}$ and $\varphi^4_{a;q_i^{\frak p}\frak p_0}$, which are parts of $\widehat{\Phi^3_a}$ and $\widehat{\Phi^4_a}$. Note that we required Property \[proper113\] to define $U^1_{q}$. We can certainly require $$\varphi^1_{\frak p q}(U^1_{q}) \subset U^3_{a;\frak p}(q),\quad
\varphi^0_{\frak p_0 q}(U^1_{q}) \subset U^4_{a;\frak p_0}(q)
\qquad \text{for $a=1,2$}$$ in addition, by replacing $\widehat{\mathcal U}$ by its open substructure if necessary. Then (\[formula1125\]) is satisfied.
We will further shrink $U^+_{\frak p}$ to $U^{+\prime}_{\frak p}$ so that it satisfies the conclusion of Proposition \[prop1111\]. (Here $U_{\frak p}^+$ is a Kuranishi neighborhood which is a part of $\widetriangle{\mathcal U^+}$.)
Let $p \in \psi^+_{\frak p}(U_{\frak p}^+ \cap (s^+_{\frak p})^{-1}(0))$. We will construct a neighborhood $U^5_{\frak p}(p) \subset U_{\frak p}^+$ of $p$ and an embedding of Kuranishi charts $\Phi^5_{a;p\frak p} : \mathcal U_{\frak p}^+\vert_{U^5_{\frak p}(p)}
\to \mathcal U^{+}_{a,p}$ for $a=1,2$. Here $\mathcal U^{+}_{a,p}$ is the Kuranishi neighborhood which ${\widehat{\mathcal U^{+}_a}}$ assigns to $p\in Z$.
(Case 1) $\frak p \in \frak P$.
By assumption (b) there exists a neighborhood $U^3_{a;\frak p}(p) \subset U_{\frak p}$ of $o_{\frak p}(p)$ and an embedding $\Phi^3_{a;p\frak p} : \mathcal U_{\frak p}\vert_{U^3_{a;\frak p}(p)}
\to \mathcal U^{+}_{a,p}$. We take $$U^{5\prime}_{\frak p}(p) = U^+_{\frak p} \cap \bigcap_{a=1}^2U^3_{a;\frak p}(p)$$ and $\Phi^{5\prime}_{a;p\frak p}: \mathcal U_{\frak p}\vert_{U^{5\prime}_{\frak p}(p)}
\to \mathcal U^{+}_{a,p}$ to be the restriction of $\Phi^3_{a;p\frak p}$.
(Case 2) $\frak p = \frak p_0$.
By assumption (c) there exists a neighborhood $U^4_{a;\frak p_0}(p) \subset U_{\frak p_0}$ of $o_{\frak p_0}(p)$ and an embedding $\Phi^4_{a;p\frak p_0} : \mathcal U_{\frak p_0}\vert_{U^4_{a;\frak p_0}(p)}
\to \mathcal U^{+}_{a;p}$. We take $$U^{5\prime}_{\frak p_0}(p) = U^+_{\frak p_0}\cap \bigcap_{a=1}^2U^4_{a;\frak p_0}(p)$$ and $\Phi^{5\prime}_{a;p\frak p_0}: \mathcal U_{\frak p_0}\vert_{U^{5\prime}_{\frak p_0}(p)}
\to \mathcal U^{+}_{a;p}$ to be the restriction of $\Phi^4_{a;p\frak p_0}$.
Most of the properties required for $(\{\Phi^{5\prime}_{a;p\frak p}\},\{U^{5\prime}_{\frak p}(p)\})$ to be a GK-embedding is a direct consequence of Assumptions (a), (b), (c). The nontrivial part to check is the commutativity of Diagram (\[diagram58\]) in the following case: Here we will further shrink $U^+_{\frak p}$, $U^+_{\frak p_0}$, $U^{5\prime}_{\frak p}(p)$, $U^{5\prime}_{\frak p_0}(q)$ to $U^{+-}_{\frak p}$, $U^{+-}_{\frak p_0}$, $U^{5}_{\frak p}(p)$, $U^{5}_{\frak p_0}(q)$, respectively, by Lemmas \[lem11166\] and \[1112label\] below. After these shrinking, we will prove the commutativity of Diagram (\[diagram58sec11\]), where $$p \in \psi^+_{\frak p}((s^+_{\frak p})^{-1}(0) \cap U^{+ -}_{\frak p}), \quad
q \in \psi^+_{\frak p_0}((s^+_{\frak p_0})^{-1}(0) \cap U^{+ -}_{\frak p_0})
\cap \psi^+_{\frak p}((s^+_{\frak p})^{-1}(0) \cap U^{5}_{\frak p}(p)).$$ $$\label{diagram58sec11}
\begin{CD}
\mathcal U_{\frak p_0}\vert_{
(U^+_{\frak p\frak p_0} \cap (\varphi_{\frak p\frak p_0}^+)^{-1}(U^{5}_{\frak p}(p)))
\cap ( U^{5}_{\frak p_0}(q)
\cap(\varphi_{q\frak p_0}^4)^{-1}({U}_{a;pq}^{+}))} @ > {\Phi_{a;q\frak p_0}^4} >>
{\mathcal U}^+_{a;q}\vert_{{U}_{a;pq}^+} \\
@ V{\Phi^+_{\frak p\frak p_0}}VV @ VV{\Phi_{a;pq}^{+}}V\\
\mathcal U_{\frak p}\vert_{U^{5}_{\frak p}(p)} @ > {\Phi_{a;p\frak p}^3} >>{\mathcal U}_{a;p}^{+}
\end{CD}$$ Here $\Phi_{a;pq}^{+}$ is the coordinate change of the Kuranishi structure $\widehat{\mathcal U^+_a}$ whose domain is ${U}_{a;pq}^+$, and $\Phi^+_{\frak p\frak p_0}$ is the coordinate change of the good coordinate system $\widetriangle{\mathcal U^+}$ whose domain is $U^+_{\frak p\frak p_0}$. Thus Diagram (\[diagram58sec11\]) is nothing but Diagram (\[diagram58\]) in the current context. During the proof below, we use the notations used in the proof of Proposition \[inductiveprop\].
Now we begin with describing the shrinkings. For each $i$, let $U^{2 \prime}_{q_i^{\frak p}}$ be a relatively compact open neighborhood of $q_i^{\frak p}$ in $U^{2}_{q_i^{\frak p}}$ such that $$\label{qcoversuru22}
\aligned
&\bigcup_{i=1}^{N(\frak p)}
\psi^+_{q^{\frak p}_i}((s_{q^{\frak p}_i}^+)^{-1}(0)
\cap U^{2\prime}_{q^{\frak p}_i})
\\
&\supset
\psi^+_{\frak p_0}(\mathcal K_{\frak p_0} \cap
(s^+_{\frak p_0})^{-1}(0))
\cap
\mathcal S_{\frak d}(X,Z;\widehat{\mathcal U})
\cap
\psi^+_{\frak p}(\mathcal K_{\frak p} \cap
(s^+_{\frak p})^{-1}(0)).
\endaligned$$ Such a choice is possible because of (\[qcoversuru\]). (Recall $U^2_{q_i^{\frak p}}$ is defined so that (\[qcoversuru\]) holds.)
\[lem11166\] We can take relatively compact opens subsets $U^{+-}_{\frak p} \subset U_{\frak p}^+$ and $U^{+-}_{\frak p_0} \subset U_{\frak p_0}^+$ with the following properties.
1. $
Z_1 \subset
\bigcup_{\frak p}\psi^+_{\frak p}(U^{+-}_{\frak p}
\cap (s^+_{\frak p})^{-1}(0))
$.
2. $
Z_0 \subset
\psi^+_{\frak p_0}(U^{+-}_{\frak p_0}
\cap (s^+_{\frak p_0})^{-1}(0))
$.
3. We put $U^{+-}_{\frak p\frak p_0} = U_{\frak p\frak p_0} \cap U^{+-}_{\frak p_0}
\cap (\varphi^+_{\frak p\frak p_0})^{-1}(U^{+-}_{\frak p})$. Then $$\overline{U^{+-}_{\frak p\frak p_0}}
\subset
\bigcup_{i=1}^{N(\frak p)}
\varphi^0_{\frak p_0 q^{\frak p}_i}(U^{2\prime}_{q^{\frak p}_i}).$$
We note that the good coordinate system $\widetriangle{\mathcal U^+}$ defines a Hausdorff space $\vert\widetriangle{\mathcal U^+}\vert$ and $U_{\frak p}^+$, $U_{\frak p_0}^+$, $Z_0$, $Z_1$ can be identified as subspaces of $\vert\widetriangle{\mathcal U^+}\vert$. Therefore the existence of such $U^{+-}_{\frak p} \subset U_{\frak p}^+$ and $U^{+-}_{\frak p_0} \subset U_{\frak p_0}^+$ is an easy consequence of (\[qcoversuru22\]).
By Lemma \[lem11166\] (1)(2), we can apply Lemma \[lem320\] to $\widetriangle{\mathcal U^+}$ and $U^{+-}_{\frak p}$, $U^{+-}_{\frak p_0}$ and obtain a good coordinate system whose Kuranishi neighborhoods are $U^{+-}_{\frak p}$, $U^{+-}_{\frak p_0}$. We denote this good coordinate system by $\widetriangle{\mathcal U^{+-}}$. It still satisfies the conclusion of Proposition \[inductiveprop\].
\[1112label\] We can take relatively compact neighborhoods $U^{5}_{\frak p}(p) \subseteq U^{5\prime}_{\frak p}(p)$ and $U^{5}_{\frak p_0}(q) \subseteq U^{5\prime}_{\frak p_0}(q)$ of $o_{\frak p}(p)$ and $o_{\frak p_0}(q)$ in $U^+_{\frak p}$ and $U^+_{\frak p_0}$, respectively, so that if $$\aligned
p &\in \psi^+_{\frak p}((s^+_{\frak p})^{-1}(0) \cap U^{+-}_{\frak p})\\
q &\in \psi^+_{\frak p_0}(U^{+-}_{\frak p_0} \cap (s^+_{\frak p_0})^{-1}(0))
\cap
\psi^+_{\frak p}(U^{5}_{\frak p}(p) \cap (s^+_{\frak p})^{-1}(0))
\endaligned$$ then there exists $i \in \{1,\dots, N(\frak p)\}$ such that:
1. $
(\varphi^1_{\frak p q^{\frak p}_i})^{-1}(U^{5}_{\frak p}(p)) \subset
U^{2\prime}_{q^{\frak p}_i}
$.
2. $
U^{5}_{\frak p_0}(q) \subset
\varphi^0_{\frak p_0 q^{\frak p}_i}(U^{2\prime}_{q^{\frak p}_i}).
$
3. $
U^{5}_{\frak p}(p) \subset
(\varphi^3_{a;p \frak p})^{-1}
(U^+_{a;q_i^{\frak p} p})
\cap U^3_{a;\frak p}(q_i^{\frak p}).
$
4. $U^{5}_{\frak p_0}(q)
\subset U^4_{a;\frak p_0}(q_i^{\frak p})$.
5. $\varphi^4_{a;q\frak p_0}(U^{5}_{\frak p_0}(q))
\subset
U^+_{{a;q_i^{\frak p}q}}
\cap (U^+_{a;pq} \cap
(\varphi^+_{a;pq})^{-1}(U^+_{a;q_i^{\frak p}p}))$.
We first observe that Lemma \[lem11166\] (3) and Definition \[coordinatechangedef\] (3) imply $$\label{112626}
\aligned
&{\rm Close}\left(
\psi^+_{\frak p}((s^+_{\frak p})^{-1}(0) \cap U^{+-}_{\frak p})
\cap
\psi^+_{\frak p_0}(U^{+-}_{\frak p_0} \cap (s^+_{\frak p_0})^{-1}(0))\right) \\
&\subset
\bigcup_{i=1}^{N(\frak p)}
\psi^+_{\frak p_0}(\varphi^0_{\frak p_0 q^{\frak p}_i}(U^{2\prime}_{q^{\frak p}_i})
\cap (s^+_{\frak p_0})^{-1}(0)).
\endaligned$$ Here ${\rm Close}$ denotes the closure. Since $$\psi_{\frak p_0}\circ \varphi^0_{\frak p_0 q^{\frak p}_i}
=
\psi_{q^{\frak p}_i}
=
\psi_{\frak p} \circ \varphi^1_{\frak p q^{\frak p}_i}$$ on $U^{2\prime}_{q^{\frak p}_i}
\cap s_{q^{\frak p}_i}^{-1}(0)$, (\[112626\]) implies $$\label{11262622}
\aligned
&{\rm Close}\left(\psi^+_{\frak p}((s^+_{\frak p})^{-1}(0) \cap U^{+-}_{\frak p})
\cap
\psi^+_{\frak p_0}(U^{+-}_{\frak p_0} \cap (s^+_{\frak p_0})^{-1}(0))
\right)\\
&\subset
\bigcup_{i=1}^{N(\frak p)}
\psi^+_{\frak p}(\varphi^1_{\frak p q^{\frak p}_i}(U^{2\prime}_{q^{\frak p}_i})
\cap (s^+_{\frak p})^{-1}(0)).
\endaligned$$ Moreover $$\label{11262622+}
\psi^+_{\frak p_0}(\varphi^0_{\frak p_0 q^{\frak p}_i}(U^{2\prime}_{q^{\frak p}_i})
\cap (s^+_{\frak p_0})^{-1}(0))
=
\psi_{\frak p}^+(\varphi^1_{\frak p q^{\frak p}_i}(U^{2\prime}_{q^{\frak p}_i})
\cap (s^+_{\frak p})^{-1}(0)).$$ By (\[11262622\]) we can find compact subsets $C_{q^{\frak p}_i}$ of $U^{2\prime}_{q^{\frak p}_i}$ such that $$\label{112626222}
\aligned
&\psi^+_{\frak p}((s^+_{\frak p})^{-1}(0) \cap U^{+-}_{\frak p})
\cap
\psi^+_{\frak p_0}(U^{+-}_{\frak p_0} \cap (s^+_{\frak p_0})^{-1}(0)) \\
&\subset
\bigcup_{i=1}^{N(\frak p)}
\psi^+_{\frak p}(\varphi^1_{\frak p q^{\frak p}_i}(C_{q^{\frak p}_i})
\cap (s^+_{\frak p})^{-1}(0)).
\endaligned$$
We may choose $U^{5}_{\frak p}(p)$ to be a sufficiently small neighborhood of $o_{\frak p}(p)$ so that the following holds.
Let $p \in \psi^+_{\frak p}((s^+_{\frak p})^{-1}(0) \cap U^{+-}_{\frak p})$. Then there exists $i \in \{1,\dots, N(\frak p)\}$ such that Lemma \[1112label\] (1) and (3) hold and $$\label{11262622333}
(\psi^+_{\frak p_0})^{-1}\left(
\psi^+_{\frak p_0}(U^{+-}_{\frak p_0} \cap (s^+_{\frak p_0})^{-1}(0))
\cap
\psi^+_{\frak p}(U^{5}_{\frak p}(p) \cap (s^+_{\frak p})^{-1}(0))
\right)
\subset
\varphi^0_{\frak p_0 q^{\frak p}_i}(U^{2\prime}_{q^{\frak p}_i}).$$
Let $p \in \psi^+_{\frak p}((s^+_{\frak p})^{-1}(0) \cap U^{+-}_{\frak p})$. We may choose $U^{5}_{\frak p}(p)$ sufficiently small so that the following holds.
1. If $$\psi^+_{\frak p}(U^{5}_{\frak p}(p) \cap (s^+_{\frak p})^{-1}(0))
\cap
\psi^+_{\frak p}(\varphi^1_{\frak p q^{\frak p}_i}(C_{q^{\frak p}_i})
\cap (s^+_{\frak p})^{-1}(0))\ne
\emptyset,$$ then $$p \in
\psi^+_{\frak p}(\varphi^1_{\frak p q^{\frak p}_i}(U^{2\prime}_{q^{\frak p}_i})
\cap (s^+_{\frak p})^{-1}(0)).$$
In fact we can prove the contraposition of (\*) by using the compactness of $C_{q^{\frak p}_i}$.
Let $q \in \psi^+_{\frak p_0}(U^{+-}_{\frak p_0} \cap (s^+_{\frak p_0})^{-1}(0))
\cap
\psi^+_{\frak p}(U^{5}_{\frak p}(p) \cap (s^+_{\frak p})^{-1}(0))$. By (\[112626222\]), there exists $i \in \{1,\dots, N(\frak p)\}$ such that $q \in \psi^+_{\frak p}(\varphi^1_{\frak p q^{\frak p}_i}(C_{q^{\frak p}_i})
\cap (s^+_{\frak p})^{-1}(0))$. For any such $i$, (\*) implies $
p \in
\psi_{\frak p}^+(\varphi^1_{\frak p q^{\frak p}_i}(U^{2\prime}_{q^{\frak p}_i})
\cap (s^+_{\frak p})^{-1}(0)).
$ We can further shrink $U^{5}_{\frak p}(p)$, if necessary, so that Lemma \[1112label\] (1) is satisfied.
Therefore (\[formula1125\]) and $U^{2\prime}_{q^{\frak p}_i} \subset U^1_{q^{\frak p}_i}$ imply $
o_{\frak p}(p) \in U^3_{a;\frak p}(q_i^{\frak p}).
$ We remark that $
p \in
\psi^+_{\frak p}(\varphi^1_{\frak p q^{\frak p}_i}(U^{2\prime}_{q^{\frak p}_i})
\cap (s^+_{\frak p})^{-1}(0))
$ implies $o_{\frak p}(p) \in U^3_{a;p\frak p}$. Moreover $$\varphi^3_{a;p\frak p}(o_{\frak p}(p)) = o_{p}(p)
\in U^+_{a;q^{\frak p}_i p}.$$ (In fact $U^+_{a;q^{\frak p}_i p}$ is nonempty since $
p \in
\psi_{q^{\frak p}_i}(s_{q^{\frak p}_i}^{-1}(0) \cap
U_{q^{\frak p}_i})
$ follows from Lemma \[1112label\] (1).) Hence $
o_{\frak p}(p) \in
(\varphi^3_{a;p \frak p})^{-1}
(U^+_{q_i^{\frak p} p})
\cap U^3_{a;\frak p}(q_i^{\frak p})
$. Therefore we can shrink $U^{5}_{\frak p}(p)$ so that Lemma \[1112label\] (3) is satisfied.
To prove (\[11262622333\]) it suffices to show $$\label{form1125}
\psi^+_{\frak p}(U^{5}_{\frak p}(p) \cap (s^+_{\frak p})^{-1}(0))
\subset
\psi^+_{\frak p_0}( \varphi^0_{\frak p_0 q^{\frak p}_i}(U^{2\prime}_{q^{\frak p}_i})).$$ The right hand side is $\psi^+_{\frak p}(\varphi^1_{\frak p q^{\frak p}_i}(U^{2\prime}_{q^{\frak p}_i})
\cap (s^+_{\frak p})^{-1}(0))$ by (\[11262622+\]). Therefore (\[form1125\]) follows from Lemma \[1112label\] (1).
By (\[11262622333\]) we find that $o_{\frak p_0}(q) \in \varphi^0_{\frak p_0 q^{\frak p}_i}(U^{2\prime}_{q^{\frak p}_i})$. Since $\varphi^0_{\frak p_0 q^{\frak p}_i}$ an open mapping, we can choose $U^{5}_{\frak p_0}(q)$ so that Lemma \[1112label\] (2) holds.
We will next shrink $U^{5}_{\frak p_0}(q)$ so that Lemma \[1112label\] (4)(5) are satisfied. Lemma \[1112label\] (2) implies $o_{\frak p_0}(q) \in \varphi^0_{\frak p_0 q^{\frak p}_i}(U^{2\prime}_{q^{\frak p}_i})
\subseteq \varphi^0_{\frak p_0 q^{\frak p}_i}(U^{1}_{q^{\frak p}_i})$. Therefore by (\[formula1125\]) $
o_{\frak p_0}(q) \in U^4_{a;\frak p_0}(q_i^{\frak p}).
$ Hence we can find a neighborhood $U^{5}_{\frak p_0}(q)$ of $o_{\frak p_0}(q)$ such that Lemma \[1112label\] (4) is satisfied.
Note Lemma \[1112label\] (1)(2) implies $$\label{1126262233343}
q \in \psi_{q_i^{\frak p}}(U_{q_i^{\frak p}}
\cap s_{q_i^{\frak p}}^{-1}(0)),
\qquad
p \in \psi_{q_i^{\frak p}}(U_{q_i^{\frak p}}
\cap s_{q_i^{\frak p}}^{-1}(0)).$$ In fact, the first half is a consequence of Lemma \[1112label\] (2) and $
q \in \psi^+_{\frak p_0}(U^{5}_{\frak p_0}(q)
\cap (s^+_{\frak p_0})^{-1}(0))
$. The second half follows from Lemma \[1112label\] (1) as we mentioned already.
Using the existence of a [*strict*]{} KK-embedding $\widehat{\Phi^2_a} : \widehat{\mathcal U} \to \widehat{\mathcal U^+_a}$ the second formula of (\[1126262233343\]) implies $p \in \psi_{a;q_i^{\frak p}}(U^+_{a;q_i^{\frak p}}
\cap s_{q_i^{\frak p}}^{-1}(0))$. Therefore the coordinate change $\varphi^+_{a;q_i^{\frak p}p}$ is defined by Definition \[kstructuredefn\].
The first formula of (\[1126262233343\]) and the existence of a strict KK-embedding $\widehat{\Phi^2_a} : \widehat{\mathcal U} \to \widehat{\mathcal U^+_a}$ imply $$q \in \psi_{a;q_i^{\frak p}}(U^+_{a;q_i^{\frak p}}
\cap s_{a;q_i^{\frak p}}^{-1}(0)).$$ Since Lemma \[1112label\] (1) implies $q \in \psi^+_{\frak p}((s^+_{\frak p})^{-1}(0) \cap U^{5}_{\frak p}(p))
\subseteq \psi_{a;p}(s_p^{-1}(0) \cap U^+_{a;p})$, the coordinate change $\varphi_{a;pq}^+$ is defined.
We may choose $U^5_{\frak p}(p)$ such that $\varphi_{a;p \frak p}^3(U^5_{\frak p}(p)) \subset U^+_{a;q_i^{\frak p} p}$ and so $$o_q \in U^+_{{a;q_i^{\frak p}q}}
\cap (U^+_{a;pq} \cap
(\varphi^+_{a;pq})^{-1}(U^+_{a;q_i^{\frak p}p})).$$ Therefore we can find a neighborhood $U^{5}_{\frak p_0}(q)$ of $o_{\frak p_0}(q)$ so that Lemma \[1112label\] (5) is satisfied. The proof of Lemma \[1112label\] is now complete.
Now we are ready to prove the commutativity of Diagram (\[diagram58sec11\]). Our assumption is $p \in \psi^+_{\frak p}(U^{+-}_{\frak p} \cap (s^+_{\frak p})^{-1}(0))$ and $q \in \psi^+_{\frak p_0}(U^{+-}_{\frak p_0} \cap (s^+_{\frak p_0})^{-1}(0))
\cap
\psi^+_{\frak p}(U^{5}_{\frak p}(p) \cap (s^+_{\frak p})^{-1}(0)).
$ We choose $i$ as in Lemma \[1112label\]. Let $$y \in (U^+_{\frak p\frak p_0} \cap (\varphi_{\frak p\frak p_0}^+)^{-1}(U^{5}_{\frak p}(p)))
\cap ( U^{5}_{\frak p_0}(q)
\cap(\varphi_{a;q\frak p_0}^4)^{-1}({U}_{a;pq}^{+})).$$ Then, since $y\in U^{5}_{\frak p_0}(q)$, Lemma \[1112label\] (2) implies $y = \varphi^0_{\frak p_0 q^{\frak p}_i}(\tilde y)$ with $\tilde y \in U^{2 \prime}_{q^{\frak p}_i}$.
Now we calculate $$\aligned
(\varphi^+_{a;q_i^{\frak p}p}
\circ \varphi^{+}_{a;pq}\circ\varphi^4_{a;q \frak p_0})(y)
&=
\varphi^{+}_{a;q_i^{\frak p}q}(\varphi^4_{a;q \frak p_0}(y)) \\
&=
\varphi^4_{a;q^{\frak p}_i \frak p_0}(y)\\
&=
\varphi^4_{a;q^{\frak p}_i \frak p_0}( \varphi^0_{\frak p_0 q^{\frak p}_i}(\tilde y))
\\
&=
\varphi^2_{a; q^{\frak p}_i}(\tilde y).
\endaligned$$ The first equality is the compatibility condition of the coordinate change of $\widehat{\mathcal U^+_a}$. (Here we use Lemma \[1112label\] (5) to apply the compatibility condition.) The second equality is the consequence of the fact that $\varphi^4_{a;q \frak p_0}$ is a part of the object consisting GK-embedding. (We use Lemma \[1112label\] (4)(5) to apply the compatibility condition.) The third line is the definition of $\tilde y$. The fourth equality follows from (\[formula1125\]) and Assumption (c).
On the other hand, we have $$\aligned
(\varphi^+_{a;q_i^{\frak p}p}
\circ\varphi^3_{a;p \frak p}\circ \varphi^+_{\frak p\frak p_0})(y)
&=
\varphi^+_{a;q_i^{\frak p}p}
(\varphi^3_{a;p \frak p}(\varphi^+_{\frak p\frak p_0}(y))) \\
&=
\varphi^3_{a;q_i^{\frak p} \frak p}(\varphi^+_{\frak p\frak p_0}(y)) \\
&=
\varphi^3_{a;q_i^{\frak p} \frak p}(\varphi^+_{\frak p\frak p_0}( \varphi^0_{\frak p_0 q^{\frak p}_i}(\tilde y))) \\
&=
\varphi^3_{a;q_i^{\frak p} \frak p}(\varphi^1_{\frak p q^{\frak p}_i}(\tilde y)) \\
&=
\varphi^2_{a; q^{\frak p}_i}(\tilde y).
\endaligned$$ The first equality is obvious. The second equality follows from the fact that $\widehat{\Phi^3}$ is a GK-embedding and Lemma \[1112label\] (3). (Note $\varphi^+_{\frak p\frak p_0}(y) \in U^{5}_{\frak p}(p) \subset U^{3}_{\frak p}(p)$.) The third equality is the definition of $\tilde y$. The fourth equality is the definition of $\varphi^+_{\frak p\frak p_0}$, that is (\[formform1177\]). The fifth equality follows from (\[formula1125\]) and Assumption (b).
Therefore we obtain $$(\varphi^+_{a;q_i^{\frak p}p}
\circ \varphi^{+}_{a;pq}\circ\varphi^4_{a;q \frak p_0})(y)
=
(\varphi^+_{a;q_i^{\frak p}p}
\circ\varphi^3_{a;p \frak p}\circ \varphi^+_{\frak p\frak p_0})(y).$$ Since $\varphi^+_{a;q_i^{\frak p}p}$ is injective, we have proved the commutativity of the Diagram (\[diagram58sec11\]).
The proof of Proposition \[prop1111\] is complete.
The rest of the proof of Proposition \[prop519\] is mostly the same as the proof of Theorem \[Them71restate\], using Proposition \[prop1111\] in addition to Proposition \[inductiveprop\]. We use the same notation as in the last part of the proof of Theorem \[Them71restate\]. We construct a good coordinate system of a compact neighborhood of $$Z \cup \bigcup_{i=1}^n Z_i$$ together with its GK-embedding to $\widehat{\mathcal U^{+}_a}$ by induction. Suppose we have a good coordinate system of a compact neighborhood of $
Z \cup \bigcup_{i=1}^{n-1} Z_i
$ together with its embedding to $\widehat{\mathcal U^{+}_a}$. To apply Propositions \[inductiveprop\], \[prop1111\], we need to find a good coordinate system on $Z_n$ together with the GK-embedding to $\widehat{\mathcal U^{+}_a}$. We use the following lemma for this purpose.
\[charthuyasilemma\] Let $\widehat{\mathcal U}$ be a Kuranishi structure and $\widehat{\mathcal U^{+}_a}$ its thickenings for $a=1,2$. Then for each $p
\in \mathcal S_{\frak d}(X,Z;\widehat{\mathcal U})$ there exists a good coordinate system $\widetriangle{\mathcal U_{\frak p_0}}$ of $\{p\} \subseteq X$ with the following properties.
1. $\widetriangle{\mathcal U_{\frak p_0}}$ consists of a single Kuranishi chart $\mathcal U_{\frak p_0}$ such that $\dim U_{\frak p_0} = \frak d$ and $\mathcal U_{\frak p_0}$ is a restriction to an open set of a Kuranishi chart of a point $p$ of $\widehat{\mathcal U}$.
2. For each $a=1,2$, there exists a GK-embedding $\widetriangle{\mathcal U_{\frak p_0}}
\to \widehat{\mathcal U_a^{+}}\vert_{Z_0}$, where $Z_0$ is a compact neighborhood of $p$ in $Z$.
Let $\mathcal U_p$ (resp. $\mathcal U^{+}_{a;p}$) be the Kuranishi chart of $p$ induced by the Kuranishi structure $\widehat{\mathcal U}$ (resp. $\widehat{\mathcal U^{+}_{a}}$). Let $O_{a;p}$ be the neighborhood of $p$ in $X$ as in Definition \[thickening\]. (Here we use thickness of $\widehat{\mathcal U^{+}_a}$.) We put $O_p = O_{1;p} \cap O_{2;p}$. We take an open neighborhood $U_{\frak p_0}$ of $o_p$ in $U_p$ such that $$\psi_{p}(U_{\frak p_0} \cap s_p^{-1}(0)) \subset O_p,$$ and set $\mathcal U_{\frak p_0} = \mathcal U_p\vert_{U_{\frak p_0}}$. It is obvious that $\mathcal U_{\frak p_0}$ satisfies (1). Let us prove (2). The subset $Z_0$ is any compact neighborhood of $p$ contained in $O_{p}$.
Let $q \in \psi(U_{\frak p_0} \cap s_{\frak p_0}^{-1}(0)) \cap Z_0$. We take $W_{a;p}(q) \subset U_p$ as in Definition \[thickening\] and put $U(q) = W_{1;p}(q) \cap W_{2;p}(q)$. Then by Definition \[thickening\] (2) (a) we have $$\varphi_{a;p}(U(q)) \subset \varphi^{+}_{a;pq}(U^{+}_{a;pq}).$$ Since $\varphi^{+}_{a;pq}$ is injective, we have a set theoretical map $\varphi_{a;q\frak p_0} : U(q) \to U^{+}_{a;pq}$ such that $\varphi^{+}_{a;pq} \circ \varphi_{a;q\frak p_0}
= \varphi_{a;p}$. Since $\varphi^{+}_{a;pq}$ and $\varphi_{a;p}$ are embeddings between orbifolds, the map $\varphi_{a;q \frak p_0}$ is an embedding of orbifolds. We can use Definition \[thickening\] (2) (b) in the same way to obtain an embedding of vector bundles $\widehat\varphi_{a;q\frak p_0} :
E_{\frak p_0} \to E^+_{a;q}$ such that it covers $\varphi_{a;q\frak p_0}$ and satisfies $\widehat\varphi^{+}_{a;pq}\circ \widehat\varphi_{a;q\frak p_0}
= \widehat\varphi_{a;p}$. We thus obtain $\Phi_{a;q\frak p_0} = (U(q),\widehat\varphi_{a;q\frak p_0})$ for each $q \in \psi(U_{\frak p_0} \cap s_{\frak p_0}^{-1}(0))$. It is easy to see that $\Phi_{a;q\frak p_0}$ defines an embedding of a good coordinate system to a Kuranishi structure. The proof of Lemma \[charthuyasilemma\] is complete.
The proof of this lemma is the place where we use the assumption that $\widehat{\mathcal U^{+}_a}$ is a thickening of $\widehat{\mathcal U}$.
Since ${\mathcal U}_{\frak p_0}$ is an open subchart of a Kuranishi chart of $\widehat{\mathcal U}$ there exists a KG-embedding $\widehat{\mathcal U}
\to \widetriangle{\mathcal U_{\frak p_0}}$. Moreover the composition $\widehat{\mathcal U}
\to\widetriangle{\mathcal U_{\frak p_0}} \to \widehat{\mathcal U^{+}_a}$ coincides with the given KK-embedding on a neighborhood of $p$. In other words Assumption (c) of Proposition \[prop1111\] is satisfied. Therefore we apply Proposition \[prop1111\] inductively to complete the proof of Proposition \[prop519\].
KG-embeddings and compatible perturbations {#subsec:movingmulsectionetc}
------------------------------------------
The present section will be occupied by the proofs of Propositions \[le614\], \[pro616\] and Lemmata \[le714\], \[le7155\], \[lem92929\].
We use the same induction scheme as in the proof of Theorem \[Them71restate\].
\[lem11115\] We consider the situation of Proposition \[inductiveprop\] and Situation \[situation101\].
1. Suppose there exists a system of multivalued perturbations $\widetriangle{\frak s} = \{\frak s^{n}_{\frak p}\}$ of ${\widetriangle{\mathcal U}}$, $\widetriangle{\frak s_{\frak p_0}} = \{\frak s^{n}_{\frak p_0}\}$ of $\widetriangle{\mathcal U_{\frak p_0}}$, and $\widehat{\frak s} = \{\frak s^{n}_{p}\}$ of ${\widehat{\mathcal U}}$. We assume that they are compatible with the KG-embeddings $\widehat{\Phi^0} : {\widehat{\mathcal U}}\vert_{Z_0^+} \to \widetriangle{\mathcal U_{\frak p_0}}$ and $\widehat{\Phi^1} : {\widehat{\mathcal U}}\vert_{Z_1^+} \to {\widetriangle{\mathcal U}}$.
Then there exists a system of multivalued perturbations $\widetriangle{\frak s^+} = \{\frak s^{n +}_{\frak p}\}$ of ${\widetriangle{\mathcal U^+}}$ with the following properties.
1. $\widetriangle{\frak s^+}$, $\widehat{\frak s}$ are compatible with the strict KG-embedding $\widehat{\Phi^+} :
{\widehat{\mathcal U_0}}\vert_{Z_+^+} \to {\widetriangle{\mathcal U^+}}$, where ${\widehat{\mathcal U_0}}$ is an open substructure of $\widehat{\mathcal U}$.
2. If $\frak p \in \frak P$, then $\frak s^{n +}_{\frak p}$ is the restriction of $\frak s^{n}_{\frak p}$ to $\mathcal U_{\frak p}^+$.
3. In case of $\frak p_0$, $\frak s^{n +}_{\frak p_0}$ is the restriction of $\frak s^{n}_{\frak p_0}$ to $\mathcal U_{\frak p_0}^+$.
4. If $\widetriangle{\frak s}$, $\widehat{\frak s}$ are transversal to $0$, then so is $\widetriangle{\frak s^+}$.
2. Suppose there exists a CF-perturbation $\widetriangle{
\frak S} = \{\frak S^{\epsilon}_{\frak p}\}$ of ${\widetriangle{\mathcal U}}$, $\widetriangle{\frak S_{\frak p_0}} = \{\frak S_{\frak p_0}^{\epsilon}\}$ of $\widetriangle{\mathcal U_{\frak p_0}}$, and $\widehat{
\frak S} = \{\frak S^{\epsilon}_{p}\}$ of $\widehat{{\mathcal U}}$. We assume that they are compatible with the KG-embeddings $\widehat{\Phi^0} : {\widehat{\mathcal U}}\vert_{Z_0^+} \to \widetriangle{\mathcal U_{\frak p_0}}$ and $\widehat{\Phi^1} : {\widehat{\mathcal U}}\vert_{Z_1^+} \to {\widetriangle{\mathcal U}}$.
Then there exists a CF-perturbation $\widetriangle{
\frak S^+} =\{\frak S^{\epsilon +}_{\frak p}\}$ of ${\widetriangle{\mathcal U^+}}$ with the following properties.
1. $\widehat{
\frak S}$, $\widetriangle{
\frak S^+}$ are compatible with the strict KG-embedding $\widehat{\Phi^+} :
{\widehat{\mathcal U}}_0 \to {\widetriangle{\mathcal U^+}}$, where ${\widehat{\mathcal U_0}}$ is an open substructure of $\widehat{\mathcal U}$.
2. If $\frak p \in \frak P$, then $\frak S^{\epsilon +}_{\frak p}$ is the restriction of $\frak S^{\epsilon}_{\frak p}$ to $\mathcal U_{\frak p}^+$.
3. In case of $\frak p_0$, $\frak S^{\epsilon +}_{\frak p_0}$ is the restriction of $\frak S^{\epsilon}$ to $\mathcal U_{\frak p_0}^+$.
4. If $\widetriangle{\frak S}$, $\widehat{\frak S}$ are transversal to $0$, then so is $\widetriangle{\frak S^+}$.
3. Suppose there exists a differential form (resp. strongly continuous map to a manifold $M$) $\widetriangle h = \{h_{\frak p}\}$ (resp. $
\widetriangle f = \{f_{\frak p}\}$) on ${\widetriangle{\mathcal U}}$, $\widetriangle{h_{\frak p_0}}$ (resp. $\widetriangle{f_{\frak p_0}}$) on $\widetriangle{\mathcal U_{\frak p_0}}$, and $\widehat{h} = \{h_{p}\}$ (resp. $\widehat{f} = \{f_{p}\}$) of ${\widehat{\mathcal U}}$. We assume $(\widehat{\Phi^{0}})^*(\widetriangle{h_{\frak p_0}}) =
\widehat{h}\vert_{Z_0^+}$ (resp. $\widetriangle{f_{\frak p_0}} \circ \widehat{\Phi^{0}}=
\widehat{f}\vert_{Z_0^+}$) and $(\widehat{\Phi^{1}})^*(\widetriangle h) = \widehat{h}\vert_{Z_1^+}$ (resp. $\widetriangle f \circ \widehat{\Phi^{1}} = \widehat{f}\vert_{Z_1^+}$).
Then there exists a differential form (resp. strongly continuous map to $M$) $\widetriangle{h^+} = \{h^{+}_{\frak p}\}$ (resp.$\widetriangle{f^+}
= \{f^{+}_{\frak p}\}$ ) of ${\widetriangle{\mathcal U^+}}$ with the following properties.
1. $(\widehat{\Phi^{+}})^*(\widetriangle{h^+}) = \widehat{h}\vert_{\widehat{\mathcal U_0}}$ (resp. $\widetriangle{f^+} \circ \widehat{\Phi^{+}} = \widehat{f}\vert_{\widehat{\mathcal U_0}}$) holds. Here $\widehat{\mathcal U_0}$ is an open substructure of $\widehat{\mathcal U}$ and $\widehat{\Phi^+} : {\widehat{\mathcal U}}_0 \to {\widetriangle{\mathcal U^+}}$ is a strict KG-embedding.
2. If $\frak p \in \frak P$, then $h^+_{\frak p}$ (resp. $f^+_{\frak p}$) is a restriction of $h_{\frak p}$ (resp.$f_{\frak p}$ ) to $\mathcal U_{\frak p}^+$.
3. In case of $\frak p_0$, $h^+_{\frak p_0}$ (resp. $f^+_{\frak p_0}$) is a restriction of $h_{\frak p_0}$ (resp. $f_{\frak p_0}$) to $\mathcal U_{\frak p_0}^+$.
4. Suppose we are in the situation of (1). Let $\widehat f$, $\widetriangle f$ be as in (3).
1. If $\widehat f$, $\widetriangle f$ are strongly submersive with respect to $\widehat{\frak s}$ and $\widetriangle{\frak s}$, respectively, then $\widetriangle{f^{+}}$ is strongly submersive with respect to $\widetriangle{\frak s^+}$.
2. Let $g : N\to M$ be a smooth map such that $\widehat f$, $\widetriangle f$ are strongly transversal to $g$ with respect to $\widehat{\frak s}$ and $\widetriangle{\frak s}$, respectively. Then $\widetriangle{f^{+}}$ is strongly transversal to $g$ with respect to $\widetriangle{\frak s^+}$.
5. Suppose we are in the situation of (2). Let $\widehat f$, $\widetriangle f$ be as in (3).
1. If $\widehat f$, $\widetriangle f$ are strongly submersive with respect to $\widehat{\frak S}$ and $\widetriangle{\frak S}$, respectively, then $\widetriangle{f^{+}}$ is strongly submersive with respect to $\widetriangle{\frak S^+}$.
2. Let $g : N\to M$ be a smooth map such that $\widehat f$, $\widetriangle f$ are strongly transversal to $g$ with respect to $\widehat{\frak S}$ and $\widetriangle{\frak S}$, respectively. Then $\widetriangle{f^{+}}$ is strongly transversal to $g$ with respect to $\widetriangle{\frak S^+}$.
We will prove (1),(4). The proofs of (2),(3) and (5) are entirely similar.
We note that (1) (b) and (c) uniquely determine $\{\frak s^{n +}_{\frak p}\}$. Its compatibility with the coordinate change $\Phi^+_{\frak p\frak q}$ for $\frak p,\frak q\in \frak P$ follows from the assumption, that is, the compatibility of $\{\frak s^{n}_{\frak p}\}$ with $\Phi_{\frak p\frak q}$. Therefore to complete the proof, it suffices to check the following three points.
1. $\frak s^{n +}_{\frak p}$, $\frak s^{n +}_{\frak p_0}$ are compatible with $\Phi^+_{\frak p \frak p_0}$.
2. Statement (1)(a) holds.
3. Statements (1)(d) and (4) hold.
Let $y \in U^+_{\frak p\frak p_0}$. Since $U^+_{\frak p\frak p_0} \subset U^1_{\frak p\frak p_0}$, (\[118888\]) implies that there exist $q_i^{\frak p}$ and $\tilde y \in U^2_{q_i^{\frak p}}$ such that $\varphi^0_{\frak p_0 q_i^{\frak p}}(\tilde y) = y$. We can take a representative $(\frak s^{n}_{\frak p_0,k})_{k=1,\dots,\ell}$ of $\frak s^{n}_{\frak p_0}$ (resp. $({\frak s}^{n}_{q_i^{\frak p},k})_{k=1,\dots,\ell}$ of $\frak s^{n}_{q_i^{\frak p}}$) on a neighborhood $U_y$ of $y$ (resp. $U_{\tilde y}$ of $\tilde y$) such that $$\label{112555}
\frak s^{n}_{\frak p_0,k}(\varphi^0_{\frak p_0 q_i^{\frak p}}(\tilde z))
=
\widehat{\varphi^0_{\frak p_0 q_i^{\frak p}}}(\frak s^{n}_{q_i^{\frak p},k}(\tilde z))$$ holds for any $\tilde z \in U_{\tilde y}$. This is a consequence of the compatibility of $\{\frak s_{\frak p_0}^{n}\}$ and $\{\frak s_{\frak p}^{n}\}$ with $\widehat{\Phi^0}$.
We put $\overline y = \varphi^+_{\frak p\frak p_0}(y) \in U^+_{\frak p}$. Then we have $\overline y = \varphi^1_{\frak pq_i^{\frak p}}(\tilde y)$. (This is the definition of $\varphi^+_{\frak p\frak p_0}$.) We can take a representative $(\frak s^{n}_{\frak p,k})_{k=1,\dots,\ell}$ of $\frak s^{n}_{\frak p}$ such that $$\label{112666}
\frak s^{n}_{\frak p,k}(\varphi^1_{\frak p q_i^{\frak p}}(\tilde z))
=
\widehat{\varphi^1_{\frak p q_i^{\frak p}}}(\frak s^{n}_{q_i^{\frak p},k}(\tilde z))
=
(\widehat{\varphi^+_{\frak p \frak p_0}}\circ
\widehat{\varphi^0_{\frak p_0 q_i^{\frak p}}})
(\frak s^{n}_{q_i^{\frak p},k}(\tilde z))$$ holds for any $\tilde z \in U_{\tilde y}$. Here the first equality is a consequence of the compatibility of $\{\frak s^{n}_{\frak p}\}$ and $\{\frak s^{n}_{p}\}$ with $\widehat{\Phi^1}$ and the second equality is the definition of $\widehat{\varphi^+_{\frak p \frak p_0}}$.
We also have $$\label{112777}
\varphi^1_{\frak p q_i^{\frak p}}(\tilde z)
=
\varphi^+_{\frak p \frak p_0}(\varphi^0_{\frak p_0 q_i^{\frak p}}(\tilde z))$$ by definition of $\varphi^+_{\frak p \frak p_0}$, which is (\[formform1177\]). Then (\[112555\]), (\[112666\]), (\[112777\]) imply $$\frak s^{n}_{\frak p,k}(\varphi^+_{\frak p \frak p_0}(z))
=
\widehat{\varphi^+_{\frak p \frak p_0}}(\frak s^{n}_{{\frak p}_0,k}(z))$$ for all $z \in \varphi^0_{\frak p_0 q_i^{\frak p}}(U^2_{q_i^{\frak p}})$. Since $\varphi^0_{\frak p_0 q_i^{\frak p}}(U^2_{q_i^{\frak p}})$ is a neighborhood of $y$, this implies (I).
Suppose $p \in {\rm Im}(\psi^+_{\frak p}) \cap Z_1^+$, $\frak p \in \frak P$. We need to prove $(\Phi^{+}_{\frak p p})^*(\frak s^{n +}_{\frak p})
= \frak s^{n}_p$, where $\Phi_{\frak p p}^+ : \mathcal U_{0,p} \to \mathcal U_{\frak p}$. This is a consequence of the fact that $\{\frak s^{n}_{\frak p}\}$, $\{ \frak s^{n}_p\}$ are compatible with the embedding $
\widehat{\Phi^1} : \widehat{\mathcal U}\vert_{Z_1^+}
\to {\widetriangle{\mathcal U}}$. We can prove the case of $\frak p = \frak p_0$ in the same way.
This is an immediate consequence of the fact that transversality to $0$, strong submersivity, and transversality to a map, is preserved under the restriction to open subsets.
The proof of Lemma \[lem11115\] is complete.
Using Lemma \[lem11115\], we can prove Proposition \[le614\] and Lemma \[le714\], by inspecting the proof of Theorem \[Them71restate\].
\[Proof of Proposition \[pro616\] and Lemmata \[le7155\], \[lem92929\]\] We use the next lemma for the proof.
\[lemma1116\] Suppose we are in the situation of Proposition \[prop1111\].
1. We suppose that the assumptions of Lemma \[lem11115\] (1) are satisfied, in addition. Moreover we assume that there exists a multivalued perturbation $\widehat{\frak s^+_a} = \{\frak s^{n +}_{a;p}\}$ of $\widehat{\mathcal U^{+}_a}$ such that:
1. $\widehat{\frak s^+_a}$, $\widehat{\frak s}$ are compatible with the KK-embedding $\widehat{\Phi^2_a} : \widehat{\mathcal U} \to \widehat{\mathcal U^{+}_a}\vert_{Z_1^+}$.
2. $\widehat{\frak s^+_a}$, $\widetriangle{\frak s}$ are compatible with the GK-embedding $\widehat{\Phi^3_a} : {\widetriangle{\mathcal U}} \to \widehat{\mathcal U^{+}_a}\vert_{Z_0^+}$.
3. $\widehat{\frak s^+_a}$, $\widetriangle{\frak s_{\frak p_0}}$ are compatible with the GK-embedding $\widehat{\Phi^4_a} : \widetriangle{\mathcal U_{\frak p_0}} \to \widehat{\mathcal U^{+}_a}$.
Then we can take the multivalued perturbation $\widetriangle{\frak s^+} = \{\frak s^{n +}_p\}$ of $\widetriangle{\mathcal U^+}$ as in Lemma \[lem11115\] (1) such that $\widehat{\frak s^+_a}$, $\widetriangle{\frak s^+}$ are compatible with the GK-embedding $\widehat{\Phi^5_a} : \widetriangle{\mathcal U^+} \to \widehat{\mathcal U^{+}_a}$.
2. A statement similar to (1) for the CF-perturbations holds.
3. A statements similar to (1) for the differential forms and for strongly continuous maps hold.
We omit the precise statement for (2)(3) above. We believe that it is not difficult to find it for the reader.
We prove (1). The proofs of (2) (3) are entirely similar. Let $p \in \psi^+_{\frak p}((s^+_{\frak p})^{-1}(0)) \cap Z$. It suffices to show that $\widehat{\frak s^+_a}$, $\widetriangle{\frak s^+}$ are compatible with the embedding $\Phi^5_{a;p\frak p} : \mathcal U^+_{\frak p}\vert_{U^+_{\frak p}(p)}
\to \mathcal U_{a;p}^{+}$. In case $\frak p \in \frak P$ this is a consequence of Lemma \[lem11115\] (1)(b) and (ii). In case $\frak p = \frak p_0$ this is a consequence of Lemma \[lem11115\] (1)(c) and (iii).
Using Lemma \[lemma1116\], we can prove Proposition \[pro616\] and Lemma \[le7155\] by inspecting the proof of Proposition \[prop519\].
Lemma \[lem92929\] is immediate from construction.
Extension of good coordinate systems: the relative case 2 {#subsec:moreversionegcs2}
---------------------------------------------------------
The present section will be occupied by the proofs of Proposition \[prop7582752\] and Lemma \[lem753753\].
In the statement of Proposition \[prop7582752\] we used the symbols $Z_1, Z_2$ for compact subsets of $X$. In the proof below we use the symbols $\mathcal Z_{(1)}, \mathcal Z_{(2)}$ for the compact subsets $Z_1, Z_2$ in Proposition \[prop7582752\] to distinguish them from $Z_0, Z_1$ that appear in Proposition \[inductiveprop\].
To prove Proposition \[prop7582752\] we use the same induction scheme as the proof of Theorem \[Them71restate\]. We will modify the statement of Proposition \[inductiveprop\] to Lemma \[lem1118\] below. We begin with modifying Situation \[situation101\].
In Proposition \[prop7582752\] we considered ${\widetriangle{\mathcal U^{1}}}$. We write ${\widetriangle{\mathcal U^{(1)}}}$ hereafter in this subsection in place of ${\widetriangle{\mathcal U^{1}}}$. (We also write ${\widetriangle{\mathcal U^{(2)}}}$ hereafter in this subsection in place of ${\widetriangle{\mathcal U^{2}}}$.) Let ${\widetriangle{\mathcal U^{(1)}}}
= (\frak P(\mathcal Z_{(1)}),\{\mathcal U^{(1)}_{\frak p'}\},
\{\Phi^{(1)}_{\frak p'\frak q'}\})$. (We denote elements of $\frak P(\mathcal Z_{(1)})$ by $\frak p'$, $\frak q'$, that is, by small German characters with prime.)
Let $(\mathcal K^{(1)},\mathcal K^{(1) +})$ be a support pair of ${\widetriangle{\mathcal U^{(1)}}}$. We put $$Z_{\frak p'} = \psi^{(1)}_{\frak p'}((s_{\frak p'}^{(1)})^{-1}(0) \cap \mathcal K^{(1)}_{\frak p'}).$$ Let $
\frak U(Z_{\frak p'})
$ be a relatively compact open neighborhood of $Z_{\frak p'}$ in $
\psi_{\frak p'}^{(1)}((s^{(1)}_{\frak p'})^{-1}(0) \cap {\rm Int}\,
\mathcal K^{(1) +}_{\frak p'})
$. In Proposition \[prop7582752\] a Kuranishi structure $\widehat{\mathcal U^2}$ is given as a part of the assumption. During the proof, we will write $\widehat{\mathcal U}$ in place of $\widehat{\mathcal U^2}$.
\[situation117\] Let $\frak d \in \Z_{\ge 0}$, and let $Z_0$ be a compact subset of $$\mathcal S_{\frak d}(X,\mathcal Z_{(2)};\widehat{\mathcal U})
\setminus
\bigcup_{\frak d' > \frak d}\mathcal S_{\frak d'}(X,\mathcal Z_{(2)};\widehat{\mathcal U})
\setminus
\bigcup_{\frak p' \in \frak P(\mathcal Z_{(1)})} Z_{\frak p'},$$ and $Z_1$ a compact subset of $$\mathcal S_{\frak d}(X,\mathcal Z_{(2)};\widehat{\mathcal U})
\cup
\bigcup_{\frak p' \in \frak P(\mathcal Z_{(1)})} \frak U(Z_{\frak p'}).$$ We assume that $Z_1$ contains an open neighborhood of $\mathcal Z_{(1)} \cup \bigcup_{\frak d' > \frak d}\mathcal S_{\frak d'}(X,
\mathcal Z_{(2)};\widehat{\mathcal U})$ in $\mathcal S_{\frak d}(X,\mathcal Z_{(2)};\widehat{\mathcal U})$. We put $$Z_+ = Z_0 \cup Z_1.$$
Let ${\widetriangle{\mathcal U}}
= (\frak P,\{\mathcal U_{\frak p}\},\{\Phi_{\frak p\frak q}\})$ be a good coordinate system on a compact neighborhood $Z_1^+$ of $Z_1$. We assume that $\frak P$ is written as $$\frak P = \frak P({\mathcal Z}_{(1)}) \cup \frak P_0$$ (disjoint union) and the inclusions $\frak P({\mathcal Z}_{(1)})
\to \frak P$, $\frak P_0
\to \frak P$ preserve the partial order. Moreover we assume that, for $\frak p' \in \frak P({\mathcal Z}_{(1)})$, the Kuranishi chart $\mathcal U_{\frak p'}$ of ${\widetriangle{\mathcal U}}$ is an open subchart of the Kuranishi chart $\mathcal U^{(1)}_{\frak p'}$ of ${\widetriangle{\mathcal U^{(1)}}}$ and $$U_{\frak p'} \cap \mathcal Z_{(1)} = U_{\frak p'}^{(1)}
\cap \mathcal Z_{(1)} .
\footnote{Compare Definition \ref{defn735f} (1)(b).}$$ Furthermore we assume $\dim U_{\frak p} \ge \frak d$ for $\frak p \in \frak P_0$.
Let $\widehat{\Phi^1}
= \{\Phi^1_{\frak p p} \mid
p \in {\rm Im}(\psi_{\frak p}) \cap Z^+_1\} : \widehat{\mathcal U}\vert_{Z^+_1} \to {\widetriangle{\mathcal U}}$ be a strict KG-embedding such that, for $\frak p' \in \frak P({\mathcal Z}_{(1)})$, the embedding $\Phi^1_{\frak p' p}$ is an open restriction of one that is a part of the given KG-embedding $\widehat{\mathcal U}\vert_{\mathcal Z_{(1)}}
\to {\widetriangle{\mathcal U
^{(1)}}}$. Let $Z_0^+$ be a compact neighborhood of $Z_0$ in $\mathcal S_{\frak d}(X,
\mathcal Z_{(2)};\widehat{\mathcal U})$ and $\mathcal U_{\frak p_0} = (U_{\frak p_0},E_{\frak p_0},s_{\frak p_0},\psi_{\frak p_0})$ a Kuranishi neighborhood of $Z_0^+$ such that $\dim U_{\frak p_0} = \frak d$. We regard $\mathcal U_{\frak p_0}$ as a good coordinate system $\widetriangle{\mathcal U_{\frak p_0}}$ that consists of a single Kuranishi chart and suppose that we are given a strict KG-embedding $\Phi^0
= \{\Phi^0_{\frak p_0} \mid
p \in {\rm Im}(\psi) \cap Z_0\} : \widehat{\mathcal U}\vert_{Z^+_0} \to {\widetriangle{\mathcal U_{\frak p_0}}}$.
We put $\frak P^+ = \frak P \cup \{\frak p_0\}$, so $\frak P^+ \supset \frak P(\mathcal Z_{(1)})$. $\blacksquare$
\[lem1118\] In Situation \[situation117\] there exists a good coordinate system ${\widetriangle{\mathcal U
^{+}}}$ of $Z_+^+$ satisfying conclusions (1)-(5) of Proposition \[inductiveprop\]. Here $Z_+^+$ is a compact neighborhood of $Z_+$ in $X$.
Moreover the following holds.
1. If $\frak p' \in \frak P(\mathcal Z_{(1)}) \subset \frak P^+$, then the Kuranishi chart $\mathcal U^+_{\frak p'}$ of ${\widetriangle{\mathcal U^+}}$ is an open subchart of the Kuranishi chart $\mathcal U^{(1)}_{\frak p'}$ of ${\widetriangle{\mathcal U^{(1)}}}$ and $$U^+_{\frak p'} \cap \mathcal Z_{(1)} = U^{(1)}_{\frak p'}
\cap \mathcal Z_{(1)}.$$
We put $$\frak P_{\ge \frak d}
=
\frak P \setminus \{\frak p' \in \frak P(\mathcal Z_{(1)})
\mid \dim U_{\frak p'} < \frak d\}.$$ Then ${\widetriangle{\mathcal U_{\ge \frak d}}}
= (\frak P_{\ge \frak d},\{\mathcal U_{\frak p}
\mid \frak p \in \frak P_{\ge \frak d}\},\{\Phi_{\frak p\frak q}
\mid \frak p,\frak q \in \frak P_{\ge \frak d}, \frak p \ge \frak q\})$ is a good coordinate system of any compact subset of $\mathcal Z_{(2)} \cap \bigcup_{\frak p' \in \frak P_{\ge \frak d}}
{\rm Im}(\psi_{\frak p'})$.
We take $\frak U'(Z_{\frak p'})$ which is an open neighborhood of $Z_{\frak p'}$ and is relatively compact in $\frak U(Z_{\frak p'})
$. We put $$Z'_1 = Z_1 \setminus \bigcup_{\frak p' \in \frak P(\mathcal Z_{(1)}),
\dim U_{\frak p'} < \frak d} \frak U'(Z_{\frak p'}).$$ We observe that we are then in Situation \[situation101\], where ${\widetriangle{\mathcal U_{\ge \frak d}}}$ (resp. $Z'_1$) plays the role of ${\widetriangle{\mathcal U}}$ (resp. $Z_1$) in Situation \[situation101\]. We apply Proposition \[inductiveprop\] to our situation and obtain ${\widetriangle{\mathcal U
^{+ \prime}}}$. Note that the union of the sets of Kuranishi charts of ${\widetriangle{\mathcal U
^{+ \prime}}}$ and $\{\mathcal U^{(1)}_{\frak p'} \mid \frak p' \in
\frak P(\mathcal Z_{(1)}),
\dim U_{\frak p'} < \frak d\}$ has most of the properties we need to prove. The only point to take care of is that, for $\frak p' \in
\frak P(\mathcal Z_{(1)})$ with $\dim U_{\frak p'} < \frak d$, neither the coordinate change $\Phi^+_{\frak p_0\frak p'}$ nor $\Phi^+_{\frak p'\frak p_0}$ is defined.
Let $\frak P^{+\prime}$ be the partial ordered set appearing in $\widetriangle{\mathcal U^{+\prime}}$. Then $\frak p_0 \in \frak P^{+\prime}$ and $\mathcal U^{+\prime}
_{\frak p_0}$ is a Kuranishi chart that is an open subchart of $\mathcal U_{\frak p_0}$. To take care of the point mentioned above we shrink $\mathcal U^{+\prime}
_{\frak p_0}$ to $\mathcal U^{+}
_{\frak p_0}$ so that these two coordinates will not intersect, as follows.
\[sublem119\] There exists an open subset $U^{+}
_{\frak p_0}$ of $U^{+ \prime}
_{\frak p_0}$ such that the following holds.
1. If $\frak p' \in
\frak P(\mathcal Z_{(1)}),
\dim U_{\frak p'} < \frak d$ then $$\psi^+_{\frak p_0}(s_{\frak p_0}^{-1}(0) \cap U^{+}
_{\frak p_0})
\cap \frak U'(Z_{\frak p'})
= \emptyset
.$$
2. $
\psi_{\frak p_0}(s_{\frak p_0}^{-1}(0) \cap U^{+ \prime}
_{\frak p_0})
\cap \mathcal S_{\frak d}(X,\mathcal Z_{(1)}; \widehat{\mathcal U})
=
\psi_{\frak p_0}(s_{\frak p_0}^{-1}(0) \cap U^{+}
_{\frak p_0})
\cap \mathcal S_{\frak d}(X,\mathcal Z_{(1)};\widehat{\mathcal U}).
$
By definition, we have $Z_{\frak p'}
\cap
\mathcal S_{\frak d}(X,\mathcal Z_{(1)};\widehat{\mathcal U}) = \emptyset
$ for $\frak p' \in
\frak P(\mathcal Z_{(1)}),
\dim U_{\frak p'} < \frak d$. Therefore we may choose $\frak U(Z_{\frak p'})$ so that $$\frak U(Z_{\frak p'})
\cap
\mathcal S_{\frak d}(X,\mathcal Z_{(1)};\widehat{\mathcal U}) = \emptyset$$ for such $\frak p'$. In fact, $\mathcal S_{\frak d}(X,\mathcal Z_{(1)};\widehat{\mathcal U})$ is a closed set. Since $\frak U'(Z_{\frak p'})$ is relatively compact in $\frak U(Z_{\frak p'})$, we have $$\overline{\frak U'(Z_{\frak p'})}
\cap
\mathcal S_{\frak d}(X,\mathcal Z_{(1)};\widehat{\mathcal U})
= \emptyset.$$ Sublemma \[sublem119\] is an immediate consequence of this fact.
We now put $
\mathcal U^{+}_{\frak p_0}
=
\mathcal U^{+\prime}_{\frak p_0}\vert_{U^{+}
_{\frak p_0}}
$. For $\frak p' \in
\frak P(\mathcal Z_{(1)})$ with $\dim U_{\frak p'} < \frak d$, we take an open subset $
U^{+}_{\frak p'} \subset U^{(1)}_{\frak p'}
$ such that $$\frak U'(Z_{\frak p'})
=
\psi_{\frak p'}^{(1)}((s_{\frak p'}^{(1)})^{-1}(0)
\cap U^{+}
_{\frak p'}).$$ Then we put $
\mathcal U^{+}_{\frak p'}
=
\mathcal U^{+\prime}_{\frak p'}\vert_{U^{+}
_{\frak p'}}
$. For $\frak p \in \frak P^{+ \prime} \setminus \{\frak p_0\}$ we put $
\mathcal U^{+}_{\frak p}
=
\mathcal U^{+\prime}_{\frak p}$. We define a partial order $\le$ on $\frak P_+
= \frak P^{+\prime} \cup \frak P(\mathcal Z_{(1)})$ such that $\le$ coincides with the partial orders on $\frak P^{+\prime}$ and on $\frak P(\mathcal Z_{(1)})$. Moreover we define $\le$ so that for $\frak p' \in \frak P^{+\prime}$ with $\dim U_{\frak p'} < \frak d$, neither $\frak p'\le \frak p_0$ nor $\frak p'\ge \frak p_0$.
We can define coordinate change among them by restricting of the coordinate change of either $\widetriangle{{\mathcal U
^{+ \prime}}}$ or of ${\widetriangle{\mathcal U}}$. Sublemma \[sublem119\] (1) implies that these two cases exhaust the cases we need to define coordinate change. The proof of Lemma \[lem1118\] is complete.
Using Lemma \[lem1118\] we discuss in the same way as the last step of the proof of Theorem \[Them71restate\] to complete the proof of Proposition \[prop7582752\].
Using Lemma \[lem1118\], we can prove it in the same way as in Subsection \[subsec:movingmulsectionetc\].
Construction of CF-perturbations {#sec:contfamilyconstr}
================================
In this section, we give a thorough detail of the proof of existence of CF-perturbations with respect to which a given weakly submersive map becomes strongly submersive. We also prove its relative version.
Construction of CF-perturbations on a single chart {#subsec:confapersingle}
--------------------------------------------------
We first study the case of a single Kuranishi chart.
\[situ121\] $\mathcal U = (U,E,s,\psi)$ is a Kuranishi chart of $X$ and $f : U \to M$ is a smooth map. $\blacksquare$
\[situ122\] In Situation \[situ121\], we assume that $g : N \to M$ is a smooth map between manifolds and that $f$ is transversal to $g$. $\blacksquare$
The main result of Subsection \[situ121\] is Proposition \[prop123123\] below. We recall the following well-known definition.
A sheaf (of sets) $\mathscr F$ on a topological space $V$ is said to be [*soft*]{} if the restriction map $$\mathscr F(V) \to \mathscr F(K)$$ is surjective for any closed subset $K$ of $V$. (We note $\mathscr F(K) = \varinjlim_{W \supset K, \text{open}} \mathscr F(W)$.)
\[prop123123\] Suppose we are in Situation \[situ121\].
1. The sheaf $\mathscr S$ in Proposition \[prop721\] is soft.
2. The sheaf $\mathscr S_{\pitchfork 0}$ in Lemma-Definition \[strosubsemiloc\] is soft.
3. Suppose $f$ is a submersion. Then, the sheaf $\mathscr S_{f \pitchfork}$ in Lemma-Definition \[strosubsemiloc\] is soft.
4. In Situation \[situ122\] the sheaf $\mathscr S_{f \pitchfork g}$ in Lemma-Definition \[strosubsemiloc\] is soft.
The rest of this subsection will be occupied by the proof of this proposition.
We first prove (1). We use partition of unity to glue sections of $\mathscr S$. Note our sheaf $\mathscr S$ is a sheaf of sets. Nevertheless we can apply partition of unity, as we will discuss below.
\[situ12999\] Let $A$ be a subset of $U$ and let $\{U_{\frak r} \mid \frak r \in \frak R\}$ be a locally finite open cover of a subset $A$ in $U$ and $\{\chi_{\frak r}\}$ a smooth partition of unity subordinate to this covering. In other words, $\chi_{\frak r} : U \to [0,1]$ is a smooth function of $U$ which has compact support in $U_{\frak r}$, and $$\sum_{\frak r \in \frak R} \chi_{\frak r}(x) =1$$ for $x \in A$.
We assume that an element $\frak S_{\frak r} \in \mathscr S(A)$ is given for each $\frak r\in \frak R$. $\blacksquare$
Below we will define the sum $$\sum_{\frak r} \chi_{\frak r} \frak S_{\frak r} \in \mathscr S
(A).$$ For $x \in A$, let $\frak V_x = (V_x,\Gamma_x,E_x,\phi_x,\widehat\phi_x)$ be an orbifold chart of $(U,\mathcal E)$ at $x$. We may assume that, for each $\frak r$ with $x \in U_{\frak r}$, we are given a representative $\mathcal S_{\frak r}$ of $\frak S_{\frak r}$ on a neighborhood of $x$. It consists of $\frak V_{\frak r} = (V_{\frak r},\Gamma_{\frak r},E_{\frak r},\psi_{\frak r},
\widehat{\psi}_{\frak r})$ and $(W_{\frak r},\omega_{\frak r},\{{\frak s}_{\frak r}^{\epsilon} \mid \epsilon\})$ where $\frak V_{\frak r} = (V_{\frak r},\Gamma_{\frak r},E_{\frak r},\psi_{\frak r},
\widehat{\psi}_{\frak r})$ is an orbifold chart of $(U,\mathcal E)$ at $x$ and $(W_{\frak r},\omega_{\frak r},\{{\frak s}_{\frak r}^{\epsilon} \mid \epsilon\})$ is as in Definition \[defn73ss\].
We put $$\frak R(x) = \{\frak r \in \frak R \mid x \in {\rm Supp}(\chi_{\frak r})\}.$$
By shrinking $V_x$ if necessary we may assume $\psi_x(V_x) \subset U_{\frak r}$ for each $\frak r \in \frak R(x)$ and $\chi_{\frak r} \equiv 0$ on $\psi_x(V_x)$ for each $\frak r \notin \frak R(x)$.
Furthermore we may choose $U_{x}$ so that there exist $$\label{coorchange124}
\aligned
&h_{\frak r x} : \Gamma_{x} \to \Gamma_{\frak r}, \\
&\widetilde{\varphi}_{\frak r x} : V_{x} \to V_{\frak r}, \\
&\breve\varphi_{\frak r x} : V_{x} \times E_{x} \to E_{\frak r}
\endaligned$$ as in Situation \[opensuborbifoldchart\], for each $\frak r \in \frak R(x)$. (See Lemma \[lem2622\].)
\[defn123\] We put $$W_{x} = \prod_{\frak r \in \frak R(x)} W_{\frak r},
\quad
\omega_{x} = \prod_{\frak r \in \frak R(x)} \omega_{\frak r}.$$ We define $\frak s^{\epsilon}_{x}
: V_{x} \times W_{x} \to E_{x}$ by the following formula: $$\label{formula12555}
\frak s^{\epsilon}_{x}(y,(\xi_{\frak r})_{\frak r \in \frak R(x)})
=
s_{x}(y) + \sum_{\frak r \in \frak R(x)}
\chi_{\frak r}(\psi_x(y))
g_{\frak r,y}^{-1}(\frak s_{\frak r}^{\epsilon}(\widetilde{\varphi}_{\frak r x}(y),\xi_{\frak r})
- s_{\frak r}(\widetilde{\varphi}_{\frak r x}(y)).$$ Here $s_{x} : V_{x} \to E_{x}$ and $s_{\frak r} : V_{\frak r} \to E_{\frak r}$ are the local expressions of the Kuranishi map (Definition \[defnlocex\].) and $g_{\frak r,y} : E_{x} \to E_{\frak r}$ is defined by $
\breve{\varphi}_{\frak r x}(y,\xi) = g_{\frak r,y}(\xi).
$
We put $\mathcal S_x = (W_{x},\omega_{x},\{\frak s^{\epsilon}_{x}\})$.
\[lem1241\]
1. $\mathcal S_{x}$ is a CF-perturbation of $\mathcal U$ on $\frak V_{x}$.
2. The germ $[\mathcal S_{x}] \in \mathscr S_x$ represented by $\mathcal S_{x}$ depends only on $\{\chi_{\frak r}\}$, $\{\mathcal S_{\frak r}\}$, $x$ and is independent of the choices of $\frak V_x$, the coordinate changes (\[coorchange124\]), and the representatives of $\{\mathcal S_{\frak r}\}$.
3. $x \mapsto [\mathcal S_{x}] \in \mathscr S_x$ defines a (global) section of the sheaf $\mathscr S$.
Statement (1) is an immediate consequence of the construction.
We prove Statement (2). We first prove independence of the coordinate changes (\[coorchange124\]). Let $(h'_{\frak r x},\widetilde{\varphi}'_{\frak r x},\breve\varphi'_{\frak r x})$ be an alternative choice. Then there exists $\gamma_{\frak r} \in \Gamma_{\frak r}$ such that $h'_{\frak r x}
=\gamma_{\frak r}h_{\frak r x}\gamma_{\frak r}^{-1}
$, $
\widetilde{\varphi}'_{\frak r x}
= \gamma_{\frak r}\widetilde{\varphi}_{\frak r x}
$, $
\breve\varphi'_{\frak r x}
= \gamma_{\frak r}\breve\varphi_{\frak r x}.
$ The third equality implies $g'_{\frak r,y} = \gamma_{\frak r} g_{\frak r,y}$ by Lemma \[lem2715\]. Let $\frak s^{\epsilon \prime}_{x}$ be obtained from this alternative choice. Then we have $$\label{form126126}
\aligned
&\frak s^{\epsilon \prime}_{x}(y,\xi) \\
&=
s_{x}(y) + \sum_{\frak r \in \frak R(x)}
\chi_{\frak r}(\psi_x(y)) (g'_{\frak r,y})^{-1}
(\frak s_{\frak r}^{\epsilon}(\widetilde{\varphi}'_{\frak r x}(y),\xi_{\frak r})
- s_{\frak r}(\widetilde{\varphi}'_{\frak r x}(y))) \\
&=
s_{x}(y) + \sum_{\frak r \in \frak R(x)}
\chi_{\frak r}(\psi_x(y)) g_{\frak r,y}^{-1}
(\frak s_{\frak r}^{\epsilon}(\widetilde{\varphi}_{\frak r x}(y),\gamma_{\frak r}^{-1}\xi_{\frak r})
- s_{\frak r}(\widetilde{\varphi}_{\frak r x}(y))).
\endaligned$$ Here we use $\Gamma_{\frak r}$ equivariance of $\frak s_{\frak r}^{\epsilon}$ and of $s_{\frak r}$.
We define a $\Gamma_{x}$ action on $W_{\frak r}$ by $\mu\cdot\xi
= h_{\frak r x}(\mu)\xi$. We write $W_{\frak r}$ with this action by $W_{\frak r}^{h_{\frak r x}}$. The notation $W_{\frak r}^{h'_{\frak r x}}$ is defined in a similar way. Its product in Definition \[defn123\] is denoted by $W_x^h$ and $W_x^{h'}$, respectively.
Then $\xi_{\frak r} \mapsto \gamma_{\frak r}^{-1}\xi_{\frak r}$ (resp. $(\xi_{\frak r}) \mapsto (\gamma_{\frak r}^{-1}\xi_{\frak r})$) is a $\Gamma_{x}$ equivariant linear map $: W_{\frak r}^{h_{\frak r x}}
\to W_{\frak r}^{h'_{\frak r x}}$ (resp. $W_x^h \to W_x^{h'}$). Therefore (\[form126126\]) implies that the equivalence class $[\mathcal S_{x}]$ is independent of the choices of the coordinate changes (\[coorchange124\]).
Secondly we prove independence of the representative of $\mathcal S_{\frak r}$. We consider one of $\frak r_0 \in \frak R(x)$ and take an alternative choice $\mathcal S'_{\frak r_0}$ of $\mathcal S_{\frak r_0}$. It suffices to consider the case when other $\mathcal S_{\frak r}$’s for $\frak r \ne \frak r_0$ are the same for both. We may also assume that $\mathcal S'_{\frak r_0}$ is also a projection of $\mathcal S_{\frak r_0}$. Then it is immediate from definition that $\mathcal S'_x$ obtained by using $\mathcal S'_{\frak r_0}$ is a projection of $\mathcal S_x$ which is obtained by using $\mathcal S_{\frak r_0}$. We have thus proved the independence of the representative of $\mathcal S_{\frak r}$.
Thirdly we prove independence of the orbifold chart $\frak V_x = (V_x,\Gamma_x,E_x,\phi_x,\widehat\phi_x)$. Let $\frak V'_x = (V'_x,\Gamma'_x,E'_x,\phi'_x,\widehat\phi'_x)$ and suppose we obtain $\mathcal S'_x$ when we use $\frak V'_x$.
By shrinking $V'_x$ if necessary we may assume that there exists a coordinate change $(h_x,\widetilde{\varphi}_x,\breve\varphi_x)$ from the chart $\frak V'_x$ to $\frak V_x$. Let $(h_{\frak r x},\widetilde{\varphi}_{\frak r x},\breve\varphi_{\frak r x})$ be the coordinate change as in (\[coorchange124\]). Then by putting $$h'_{\frak r x} = h_{\frak r x}\circ h_x,
\quad
\widetilde{\varphi}'_{\frak r x} = \widetilde{\varphi}_{\frak r x}\circ \widetilde{\varphi}_x,
\quad
\breve\varphi'_{\frak r x} = \breve\varphi_{\frak r x}\circ
\breve\varphi_x,$$ $(h'_{\frak r x},\widetilde{\varphi}'_{\frak r x},\breve\varphi'_{\frak r x})$ becomes a coordinate change from $\frak V'_x$ to $\frak V_{\frak r}$ as in (\[coorchange124\]).
Then $$\label{form12612622}
\aligned
&\frak s^{\epsilon \prime}_{x}(y,\xi)\\
&=
s_{x}(y) + \sum_{\frak r \in \frak R(x)}
\chi_{\frak r}(\psi'_x(y)) (g'_{\frak r,y})^{-1}
(\frak s_{\frak r}^{\epsilon}(\widetilde{\varphi}'_{\frak r x}(y),\xi_{\frak r})
- s_{\frak r}(\widetilde{\varphi}'_{\frak r x}(y))) \\
&=
s_{x}(y) + \sum_{\frak r \in \frak R(x)}
\chi_{\frak r}(\psi_x(\widetilde{\varphi}_x(y))) g_{\frak r,y}^{-1}
(\frak s_{\frak r}^{\epsilon}(\widetilde{\varphi}_{\frak r x}(\widetilde{\varphi}_x(y)),\gamma_{\frak r}^{-1}\xi_{\frak r})
- s_{\frak r}(\widetilde{\varphi}_{\frak r x}(\widetilde{\varphi}_x(y))))
\\
&= \frak s^{\epsilon}_{x}(\widetilde{\varphi}_x(y),\xi).
\endaligned
\nonumber$$ This implies the required independence of the coordinate $\frak V_x$. The proof of Statement (2) is complete.
We now prove Statement (3). Let $\mathcal S_x = (W_{x},\omega_{x},\{\frak s^{\epsilon}_{x}\})$ as above. Suppose $y \in s_{x}^{-1}(0) \cap U_x$ and $y = \phi_x(\tilde y)$. We denote $\Gamma_{\tilde y} = \{\gamma \in \Gamma_x \mid
\gamma \tilde y = \tilde y\}$ and take a $\Gamma_{\tilde y}$ invariant neighborhood $V_y$ of $\tilde y$. Then $\frak V_y = (V_y,\Gamma_y,E_x,\phi_x\vert_{V_y},\widehat\phi_x\vert_{V_y})$ is an orbifold chart of $(U,E)$ at $y$. It is easy to see that $\mathcal S_x\vert_{V_y} = (W_{x},\omega_{x},\{\frak s^{\epsilon}_{x}\vert_{V_y \times E_x}\})$ is a CF-perturbation on $\frak V_y$.
\[sublem128\] $\mathcal S_x\vert_{V_y}$ is equivalent to $\mathcal S_y$ in the sense of Definition \[conmultiequiv11\].
We consider $\frak R(y) = \{\frak r \in \frak R \mid
y \in {\rm Supp}(\chi_{\frak r})\}$. Since we chose $U_x \subseteq U_{\frak r}$ such that $U_x \cap \text{\rm Supp} (\chi_{\frak r}) = \emptyset$ for ${\frak r} \notin {\frak R}(x)$, we have $\frak R(y)
\subseteq \frak R(x)$. Therefore there exists an obvious projection $$\pi : \widehat W_x = \prod_{\frak r\in \frak R(x)} W_{\frak r}
\to \widehat W_y = \prod_{\frak r\in \frak R(y)} W_{\frak r}.$$ It is easy to see that $\pi!(\omega_x) = \omega_y$.
We may choose $V_y$ so small that for $z \in V_y$ and $\frak r \in \frak R(x)
\setminus \frak R(y)$ we have $\chi_{\frak r}(z) = 0$. Therefore by definition $$\frak s^{\epsilon}_{x}(\tilde\varphi_x(z),\xi)
=
\frak s^{\epsilon}_{y}(\tilde\varphi_y(z),\pi(\xi))$$ for $z \in V_y$. Thus $\mathcal S_y$ is a projection of $\mathcal S_x\vert_{V_y}$.
Statement (3) follows from Sublemma \[sublem128\] and Lemma \[lem723\].
\[defn1213\] We denote by $$\sum_{\frak r} \chi_{\frak r} \frak S_{\frak r}$$ the element $x \mapsto [\mathcal S_{x}] \in \mathscr S_x$ of $\mathscr S(A)$ obtained by Lemma \[lem1241\].
Suppose $\frak S \in \mathscr S(U)$ and $\{U_{\frak r}\mid \frak r \in \frak R\}$ is a locally finite cover of $U$. We can define an element of $\mathscr S(U)$ by $$\sum_{\frak r} \chi_{\frak r} \frak S\vert_{U_{\frak r}}$$ as above. (Here $\frak S\vert_{U_{\frak r}} \in \mathscr S(U_{\frak r})$ is the restriction of $\frak S$.) However this section is in general [*different*]{} from the originally given $\frak S \in \mathscr S(U)$.
Proposition \[prop123123\] (1) follows easily from Definition \[defn1213\] and the results we proved above. (See also the end of this subsection where the proof of Proposition \[prop123123\] (2)(3)(4) are completed.)
We next prove Proposition \[prop123123\] (2)(3)(4). We begin with the next definition.
\[strongtransvers\] Suppose we are in Situation \[situ121\]. Let $\frak S_x \in \mathscr S_x$ be a germ of the sheaf $\mathscr S$ at $x \in U$.
We say $\frak S_x$ is [*strongly transversal*]{} if its representative $(W_x,\omega_x,\{\frak s^{\epsilon}_x\})$ (which is defined on the orbifold chart $\frak V_x = (V_x,\Gamma_x,E_x,\phi_x,\widehat\phi_x)$ (Definition \[defn61\](3))) has the following properties.
1. For all sufficiently small $\epsilon > 0$, the map $\frak s^{\epsilon}_x : V_x \times W_x \to E_x$ is transversal to $c \in E_x$ on a neighborhood of $\{o_x\} \times {\rm Supp}(\omega_x)$ for any $c \in E_x$. (Here $o_x \in V_x$ is the point such that $\phi_x(o_x) = x$.)
2. For $\xi \in {\rm Supp}(\omega_x)$ and $c= \frak s^{\epsilon}_x(o_x,\xi)$ the projection $$T_{(o_x,\xi)} (\frak s^{\epsilon}_x)^{-1}(c) \to T_{o_x} V_x$$ is surjective.
We write $(\mathscr S_{\pitchfork\pitchfork 0})_x$ the set of all germs of the sheaf $\mathscr S$ at $x \in U$ that is strongly transversal.
It is easy to see that there exists a subsheaf of $\mathscr S$ whose stalk at $x$ is $(\mathscr S_{\pitchfork\pitchfork 0})_x$. We denote this sheaf by $\mathscr S_{\pitchfork\pitchfork 0}$.
It is easy to see that the above properties (1)(2) are independent of the choice of the representative $(W_x,\omega_x,\{\frak s^{\epsilon}_x\})$ and of the orbifold chart $(V_x,\Gamma_x,E_x,\phi_x,\widehat\phi_x)$ but depend only on $\frak S_x$.
\[lemma12770\] In Situation \[situ121\], the set $(\mathscr S_{\pitchfork\pitchfork 0})_x$ is nonempty.
Let $\frak V_x = (V_x,\Gamma_x,E_x,\phi_x,\widehat\phi_x)$ be an orbifold chart of $(U,\mathcal E)$ at $x$. We put $\widehat W_x = E_x$ and $W_x$ is a sufficiently small $\Gamma_x$ invariant neighborhood of $0$ in $\widehat W_x$ and $\omega_x$ is a $\Gamma_x$ invariant differential form of compact support on $W_x$ of degree $\dim W_x$ with $\int \omega_x =1$. We define $$\frak s^{\epsilon}(x,\xi) = s(x) + \epsilon \xi.$$ It is easy to see that $(W_x,\omega_x,\{{\frak s}_x^{\epsilon} \mid \epsilon\})$ is a CF-perturbation on $\frak V_x$. Moreover it is easy to show that the projection $
(\frak s^{\epsilon})^{-1}(c) \to V_x
$ is a submersion for any $c$. Lemma \[lemma12770\] follows.
\[lem1244440\] Suppose we are in Situation \[situ121\].
1. $(\mathscr S_{\pitchfork\pitchfork 0})_x \subseteq (\mathscr S_{\pitchfork})_x$.
2. If $f$ is a submersion at $x$, then, $(\mathscr S_{\pitchfork\pitchfork 0})_x \subseteq (\mathscr S_{f \pitchfork})_x$.
3. In Situation \[situ122\], we have $(\mathscr S_{\pitchfork\pitchfork 0})_x \subseteq (\mathscr S_{f \pitchfork g})_x$.
This is immediate from the definition.
\[lem124444\] Suppose we are in Situation \[situ121\].
1. The stalk $(\mathscr S_{\pitchfork 0})_x$ is nonempty for any $x \in U$.
2. Suppose $f$ is a submersion at $x$. Then, the stalk $(\mathscr S_{f \pitchfork})_x$ is nonempty.
3. In Situation \[situ122\], the stalk $(\mathscr S_{f \pitchfork g})_x$ is nonempty.
This is an immediate consequence of Lemmata \[lemma12770\] and \[lem1244440\].
To prove Proposition \[prop123123\] (2)(3)(4), we need one more result (Proposition \[lem1211515\] below.)
\[situ121515\] Suppose we are in Situation \[situ12999\]. We put $\frak R = \{\frak r_0\} \cup \frak R'$ and assume that for $\frak r \in \frak R'$ the section $\frak S_{\frak r} \in \mathscr S(U_{\frak r})$ is strongly transversal. $\blacksquare$
\[lem1211515\] In Situation \[situ121515\], the following holds for $i=1,2,3,4$.
If $\frak S_{\frak r_0}$ has Property $(i)$ below and $\frak S_{\frak r} \in \mathscr S_{\pitchfork\pitchfork 0}(U_{\frak r})$ for $\frak r \in \frak R'$, then the sum $$\frak S = \sum_{\frak r\in \frak R} \chi_{\frak r} \frak S_{\frak r}$$ has the same property $(i)$.
1. $\frak S_{\frak r_0} \in \mathscr S_{\pitchfork\pitchfork 0}(U_{\frak r_0})$.
2. $\frak S_{\frak r_0} \in \mathscr S_{\pitchfork 0}(U_{\frak r_0})$.
3. $\frak S_{\frak r_0} \in \mathscr S_{f \pitchfork}(U_{\frak r_0})$.
4. We are in Situation \[situ122\] and $\frak S_{\frak r_0} \in \mathscr S_{f \pitchfork g}(U_{\frak r_0})$.
To prove Proposition \[lem1211515\] we rewrite the strong transversality as follows. Let $\mathcal S_x = (W_x,\omega_x,\{\frak s^{\epsilon}_x\})$ be a representative of a germ $\mathscr S_x$ which is defined on an orbifold chart $\frak V_x = (V_x,\Gamma_x,E_x,\phi_x,\widehat\phi_x)$ of $(U,\mathcal E)$.
\[lem1217\] $\mathcal S_x$ is strongly transversal if and only if the derivative $$\nabla^W_{(x,\xi)} \frak s^{\epsilon}_x : T_{\xi}W_x \to T_cE_x$$ in $W_x$ direction is surjective for all $\xi $ in the support of $\omega_x$. Here $c = \frak s^{\epsilon}_x(o_x,\xi)$.
We consider the following commutative diagram where all the horizontal and vertical lines are exact. $$\begin{CD}
&&
&&
0
&&
0
\\
&&
&& @VVV @VVV
\\
&&
&&
T_{o_x,\xi} (\frak s_x^{\epsilon})^{-1}(c)
@>>>
T_{o_x}V_x
\\
&&
&& @ VVV @VVV
\\
0 @>>>T_{\xi}W_x
@>>>T_{(o_x,\xi)}(V_x \times W_x)
@>>>
T_{o_x}V_x
@>>> 0
\\
&& @ VVV @VVV @VVV\\
0 @>>>
T_c E_x @>>>T_c E_x
@>>>0
\\
&& && @VVV
\\
&&&& 0
\end{CD}$$ The required strong transversality is nothing but the surjectivity of the second horizontal map $: T_{o_x,\xi} (\frak s_x^{\epsilon})^{-1}(c) \to T_{o_x}V_x$ and the map $
\nabla^W_{(x,\xi)} \frak s^{\epsilon}_x : T_{\xi}W_x \to T_cE_x
$ is the second vertical map. The equivalence of the surjectivities of them is a consequence of simple diagram chase.
The next lemma is a half of the proof of Proposition \[lem1211515\].
\[lem121818\] Suppose we are in Situation \[situ121515\] and $\chi_{\frak r_1}(x) \ne 0$ for some $\frak r_1 \in \frak R'$. Then the germ $\frak S_x$ of $\frak S$ at $x$ is strongly transversal.
A representative of $\frak S_x$ is $(W_x,\omega_x,\frak s_x^{\epsilon})$ where $$\frak s^{\epsilon}_{x}(y,(\xi_{\frak r})_{\frak r \in \frak R(x)})
=
s_{x}(y) + \sum_{\frak r \in \frak R(x)}
\chi_{\frak r}(\psi_x(y))
g_{\frak r,y}^{-1}(\frak s_{\frak r}^{\epsilon}(\tilde{\varphi}_{\frak r x}(y),\xi_{\frak r})
- s_{\frak r}(\tilde{\varphi}_{\frak r x}(y)).$$ Here $\frak R(x) \subseteq \frak R$ and $\frak r_1 \in \frak R(x)$. The derivative of $\frak s^{\epsilon}_{x}$ in $W_{\frak r_1}$ direction is $$\label{form12118}
\chi_{\frak r_1}(\psi_x(y))
g_{\frak r_1,y}^{-1}(\nabla^{W_{\frak r_1}}\frak s_{\frak r_1}^{\epsilon}\vert_{(\tilde{\varphi}_{\frak r x}(y),\xi_{\frak r_1})}).$$ By Lemma \[lem1217\] the derivative $\nabla^{W_{\frak r_1}}\frak s_{\frak r_1}^{\epsilon}$ is surjective (to $T_cE_{x}$). Therefore (\[form12118\]) is surjective. Therefore by Lemma \[lem1217\] $\frak S_x$ is strongly transversal.
Now we are ready to complete the proof of Proposition \[lem1211515\]. By Lemma \[lem121818\] the germ $\frak S_x$ has the property claimed in Proposition \[lem1211515\] unless $\chi_{\frak r}(x) = 0$ for all $\frak r \in \frak R'$. We may also assume that $\frak s^{\epsilon}(x,(\xi_{\frak r})_{\frak r \in \{\frak r_0\} \cup \frak R(x)}) = 0$. We consider such a point $x$. A representative of $\frak S_x$ is $(W_x,\omega_x,\frak s_x^{\epsilon})$ where $$\aligned
\frak s^{\epsilon}_{x}(y,&(\xi_{\frak r})_{\frak r \in \{\frak r_0\} \cup \frak R(x)}) \\
=
s_{x}(y) & +
\chi_{\frak r_0}(\psi_x(y))
g_{\frak r_0,y}^{-1}(\frak s_{\frak r_0}^{\epsilon}(\tilde{\varphi}_{\frak r_0 x}(y),\xi_{\frak r})
- s_{\frak r}(\tilde{\varphi}_{\frak r x}(y))\\
&+ \sum_{\frak r \in \frak R(x)}
\chi_{\frak r}(\psi_x(y))
g_{\frak r,y}^{-1}(\frak s_{\frak r}^{\epsilon}(\tilde{\varphi}_{\frak r x}(y),\xi_{\frak r})
- s_{\frak r}(\tilde{\varphi}_{\frak r x}(y)).
\endaligned$$ Here $\frak R(x) \subseteq \frak R'$. We remark that $\chi_{\frak r_0}(x) = 1$ and takes maximum there. (Note $\chi_{\frak r}$ is a smooth map to $[0,1]$.) Therefore the first derivative at $x$ of $\chi_{\frak r_0}$ is zero. In a similar way we can show that the first derivatives at $x$ of $\chi_{\frak r}$ are all zero.
We also remark that $\frak s^{\epsilon}_{\frak r_0}(x,\xi_{\frak r_0}) = 0$. Therefore $$\label{form121010}
T_{(x,\xi_{\frak r_0})}(\frak s^{\epsilon}_{\frak r_0})^{-1}(0)
\times \prod_{\frak r \in \frak R(x)} T_{\xi_{\frak r}}W_{\frak r}
\subseteq
T_{(x,(\xi_{\frak r})_{\frak r \in \{\frak r_0\} \cup \frak R(x)})}
(\frak s^{\epsilon}_{x})^{-1}(0).$$ (\[form121010\]) implies that if $\frak S_{\frak r_0}$ has Property $(i)$ at $x$ then $
\frak S = \sum_{\frak r} \chi_{\frak r} \frak S_{\frak r}
$ has the same property $(i)$ at $x$, where $\chi_{\frak r_0}(x) = 1$.
This fact together with Lemmata \[lem124444\] and \[lem121818\] imply Proposition \[lem1211515\] .
We are now in the position to complete the proof of Proposition \[prop123123\] (2)(3)(4). Let $K \subset U$ be a closed subset and $\frak S_K \in \mathscr S(K)$. By definition there exists an open neighborhood $U_{\frak r_0}$ of $K$ such that $\frak S_K$ is a restriction of $\frak S_{\frak r_0}
\in \mathscr S(U_{\frak r_0})$. We take an index set $\frak R'$ and an open covering $$\label{form121111}
U = U_{\frak r_0} \cup \bigcup_{\frak r \in \frak R'} U_{\frak r}$$ with the following properties.
1. The covering (\[form121111\]) is locally finite.
2. $\mathscr S_{\pitchfork\pitchfork 0}(U_{\frak r}) \ne \emptyset$ for $\frak r \in \frak R'$.
3. $K \cap U_{\frak r} = \emptyset$ for $\frak r \in \frak R'$.
Existence of such covering is a consequence of paracompactness of $U$ and Lemma \[lemma12770\].
Let $\frak S_{\frak r} \in \mathscr S_{\pitchfork\pitchfork 0}(U_{\frak r})$ and $\chi_{\frak r}$ a partition of unity subordinate to the covering (\[form121111\]). We put $$\frak S = \sum_{\frak r \in \{\frak r_0\} \cup \frak R'} \chi_{\frak r}\frak S_{\frak r}.$$ Property (c) implies that $\frak S$ restricts to $\frak S_K$. We have thus proved the softness of $\mathscr S$.
To prove the softness of $\mathscr S_{\pitchfork 0}$, we may assume $\frak S_{\frak r_0} \in
\mathscr S_{\pitchfork 0}(U_{\frak r_0})$. Then by Proposition \[lem1211515\], $\frak S \in \mathscr S_{\pitchfork 0}(U)$.
The proof of softness of $\mathscr S_{f \pitchfork }$ and of $\mathscr S_{f \pitchfork g}$ is the same. The proof of Proposition \[prop123123\] is complete.
The same argument also proves softness of $\mathscr S_{\pitchfork\pitchfork 0}$.
Embedding of Kuranishi charts and extension of CF-perturbations {#subsec:extembandcfp}
---------------------------------------------------------------
In this subsection, we study the case of good coordinate system and in the next we prove Theorem \[existperturbcont\], and its relative version Proposition \[existperturbcontrel\].
For that purpose we need to study other kinds of extension. Namely we will study extension of a CF-perturbation defined on an embedded orbifold to its neighborhood. (Then, by Proposition \[prop123123\], we can extend a CF-perturbation defined on this neighborhood.) We consider the following Situation \[sit121111\].
\[sit121111\] Let $\mathcal U_i = (U_i,\mathcal E_i,s_i,\psi_i)$ $(i=1,2)$ be Kuranishi charts of $X$, $\Phi_{21} = (\varphi_{21},\widehat{\varphi}_{21}) : \mathcal U_1 \to \mathcal U_2$ an embedding of Kuranishi charts and $K$ a closed subset of $X$ such that $K \subset \psi_1(s_1^{-1}(0)) \subset \psi_2(s_2^{-1}(0))$.
Hereafter in this subsection, we regard $K$ also as a subset of $U_1$ or $U_2$ via the parameterizations $\psi_1$, $\psi_2$ respectively.
Let $f_2 : U_2 \to M$ be a strongly smooth map and we put $f_1 = f_2 \circ \varphi_{21}$. $\blacksquare$
We now consider the following commutative diagram. $$\begin{CD}
\varphi_{21}^{\star}\mathscr S_{\sharp}^{\mathcal U^1\triangleright \mathcal U^2}(U_1)
@>{\frak i_{KU_1}}>>
\varphi_{21}^{\star}\mathscr S_{\sharp}^{\mathcal U^1\triangleright \mathcal U^2}(K)
\\
@V{\Phi^*_{21}}VV @V{\Phi^*_{21}}VV
\\
\mathscr S_{\sharp}^{\mathcal U^1}(U_1)
@>{\frak i_{KU_1}}>>
\mathscr S_{\sharp}^{\mathcal U^1}(K)
\end{CD}$$ Here $\sharp$ stands for any of $\pitchfork 0$, $f \pitchfork$, $f \pitchfork g$, or $\pitchfork\pitchfork 0$. Recall $\Phi^*_{21}$ denotes the restriction map and $\varphi_{21}^{\star}$ stands for pullback sheaf. (See Definition \[deflem743\] (4).)
\[prop1221\] The following holds for ${\sharp}=\pitchfork 0$, $f \pitchfork$, $f \pitchfork g$, or $\pitchfork\pitchfork 0$.
Let $\frak S^1 \in \mathscr S_{\sharp}^{\mathcal U^1}(U_1)$, $\frak S^{2,K} \in \varphi_{21}^{\star}\mathscr S_{\sharp}^{\mathcal U^1\triangleright \mathcal U^2}(K)$ such that $$\frak i_{KU_1}(\frak S^1) = \Phi^*_{21}(\frak S^{2,K}).$$ Then for any compact subset $Z \subset U_1$ containing $K$, there exists $\frak S^{2} \in \varphi_{21}^{\star}\mathscr S_{\sharp}^{\mathcal U^1\triangleright \mathcal U^2}(Z)$ such that $$\label{form1214}
\frak i_{KZ}(\frak S^2) = \frak S^{2,K},
\qquad
\Phi^*_{21}(\frak S^{2}) = \frak S^{1}.$$
The proof occupies the rest of this subsection.
For the proof, we will use the notion of bundle extension data. To give its definition, we first introduce the definition of a tubular neighborhood of an orbifold embedding.
\[lem123000\] Let $X \to Y$ be an embedding of orbifolds, $Z\subset X$ a compact subset and $U$ be an open neighborhood of $K$ in $Y$. We say that $\pi : U \to X$ is [*diffeomorphic to the projection of normal bundle*]{} if the following holds.
Let ${\rm pr} : N_XY \to X$ be the normal bundle. Then there exists a neighborhood $U'$ of $Z \subset X \subset N_XY$ and a diffeomorphism of orbifolds $h : U' \to U$ such that $\pi\circ h = {\rm pr}$. We also require that $h(x) = x$ for $x$ in a neighborhood of $Z$ in $X$.
Using this, we give the definition of a bundle extension datum.
\[defn1230\] Let $\mathcal U^i$ $(i=1,2)$ be Kuranishi charts and $\Phi_{21} : \mathcal U^1 \to \mathcal U^2$ be an embedding of Kuranishi charts. Let $Z \subset U_1$ be a compact subset.
A quadruple $(\pi_{12},\tilde\varphi_{21},\Omega_{12}, \Omega_1)$ is called a [*bundle extension data*]{} of $(\Phi_{21},Z)$ if
1. $\pi_{12} : \Omega_{12} \to \Omega_1$ is a continuous map, where $\Omega_{12}$ is a neighborhood of $\varphi_{21}(Z)$ in $U_2$ and $\Omega_{1}$ is a neighborhood of $Z$ in $U_1$.
2. $\pi_{12}$ is diffeomorphic to the projection of the normal bundle in a neighborhood of $Z$ in the sense of Definition \[lem123000\].
3. $\tilde\varphi_{21} : \pi_{12}^*\mathcal E_1 \to \mathcal E_2$ is an embedding of vector bundle. (See Definition \[def26222\].)
4. The map $$\varphi_{21}^{*} \pi^*_{12} \mathcal E_{1} \to \varphi_{21}^*\mathcal E_{1}$$ that is induced from $\tilde\varphi_{21}$ and $\varphi_{21}$ coincides with the bundle map $$\widehat{\varphi}_{21} : \mathcal E_{1} \to \mathcal E_{2}$$ which covers $\varphi_{21}$.
Here $\pi^*_{12}\mathcal E_1$ and $\varphi_{12}^*\mathcal E_1$ denote the pullback bundles.
Note here $\pi_{12}$ is a map between orbifolds but is not an embedding. So it violates our thesis that we consider only an embedding as a morphism of orbifolds. This map however is identified with a restriction of the projection of vector bundle so is also a map discussed in Section \[sec:ofd\]. We use this fact to define the pullback $\pi_{12}^*\mathcal E$ of the vector bundle $\mathcal E$. (See Definition-Lemma \[pullbackbyproj\].)
\[lem1224\] Suppose we are in Situation \[sit121111\]. Then for any compact subset $Z$ of $U_1$ there exists a bundle extension datum $(\pi_{12},\tilde\varphi_{21},\Omega_{12}, \Omega_1)$.
Let $\Omega_1$ be a relatively compact open neighborhood of $Z$. Let $\pi : N_{U_1}U_2 \to U_1$ be the normal bundle. Then by taking a Riemannian metric and exponential map, we can find an open neighborhood $\tilde\Omega_{12}$ of the zero section of $N_{U_1}U_2\vert_{\Omega_1}$ and a diffeomorphism $\tilde\Omega_{12} \to \Omega_{12}$ onto an open neighborhood $\Omega_{12}$ of $\varphi_{21}(\Omega_1)$. (See for example [@fooooverZ Lemma 6.5].) Therefore we find $\pi_{12} : \Omega_{12} \to U_1$ as the composition of the diffeomorphism $I _{12}: \Omega_{12} \cong \tilde\Omega_{12}$ and the projection $\pi : N_{U_1}U_2 \to U_1$.
The diffeomorphism $I _{12}$ is homotopic to the composition of $\pi_{12} : \Omega_{12} \to \Omega_1$ and $\varphi'_{21} : \Omega_1 \to \tilde\Omega_{12}$. (Here $\varphi'_{21}$ is the embedding as the zero section of vector bundle.) Therefore $\mathcal E_{2}\vert_{\Omega_{12}}$ is isomorphic to the pullback of $\mathcal E_2$ by $I _{12}^{-1}\circ \varphi'_{21}\circ \pi_{12}$. (See Proposition \[homotopicpulback\].) Note $I _{12}^{-1}\circ \varphi'_{21}\circ \pi_{12} = \varphi_{21}\circ \pi_{12}$. Thus $({\varphi}_{21}\circ \pi_{12})^* \mathcal E_2 \cong \mathcal E_2$.
$\widehat{\varphi}_{21} : \mathcal E_1 \to \mathcal E_2$ induces a bundle inclusion $
\pi_{12}^*\mathcal E_1 \to ({\varphi}_{21}\circ \pi_{12})^* \mathcal E_2.
$ Therefore by using isomorphism $({\varphi}_{21}\circ \pi_{12})^* \mathcal E_2 \cong \mathcal E_2$ we obtain the required embedding: $
\tilde\varphi_{21} : (\pi_{12}^*\mathcal E_1)\vert_{\Omega_{12}} \to (\mathcal E_2)\vert_{\Omega_{12}}.
$
We can take the homotopy $I _{12} \sim \varphi'_{21}\circ \pi_{12}$ so that its composition with the inclusion $\varphi_{21}\vert_{\Omega_1} : \Omega_1 \to \Omega_{12} \subset U_2$ is the trivial homotopy between ${\varphi}_{21} : \Omega_1 \to \tilde\Omega_{12}$ and ${\varphi}'_{21}\circ \pi \circ {\varphi}_{21} = {\varphi}_{21}$. We can use this fact to check Definition \[defn1230\] (4). It is easy to check Definition \[defn1230\] (1)(2)(3).
Let $(\pi_{12},\tilde\varphi_{21},\Omega_{12}, \Omega_1)$ be a bundle extension datum as in Lemma \[lem1224\] and $\frak S^1 \in \mathscr S(\Omega_1)$. We will define $\frak S^2 \in \mathscr S(\Omega_{12})$ which is compatible with $\frak S^1$ below.
Let $x_2 \in \Omega_{12}$. We put $x_1 = \pi_{12}(x_2)$. Since $\pi_{12}$ is diffeomorphic to a restriction of a projection of a vector bundle we can find orbifold charts $(V_i,\Gamma_i,\phi_i)$ of $x_i$ in $U_i$ such that $(V_i,\Gamma_i,\phi_i)$, $\pi_{12}$ have the following properties: (See Definition \[defn2613\] (1).)
\[proper1227\]
1. $\Gamma_1 = \Gamma_2$.
2. $V_2$ is identified with an open neighborhood of $V_1 \times \{0\}$ in $V_1 \times F$ where $F$ is a vector space, which is the fiber of the normal bundle $N_{U_1}U_2$.
3. $\Gamma_1 = \Gamma_2$ acts on $V_1$ and has a linear action on $F$.
4. The diagram $$\label{diagram1215}
\xymatrix{
V_1 \times F \ar@{{<}-^{)}}[r]\ar[d] & V_2\ar[r]^{\phi_{2}}\ar[d] & \Omega_{12}\ar^{\pi_{12}}[d] \\
V_1 \ar@{=}[r] & V_1\ar^{\phi_1}[r] & \Omega_1
}$$ commutes, where the first vertical arrow is the projection to the first factor.
For each given representative $(W_{1},\omega_{1},\{\frak s_1^{\epsilon}\})$ of $\frak S^1$, we define $\frak s_2^{\epsilon} : V_2 \times W_1 \to E_2$ by $$\label{defextSSSSS}
\frak s_2^{\epsilon}(y,\xi)
=
s_2(y)
+
\tilde\varphi_{21}
\left(
\frak s_1^{\epsilon}(\pi(y),\xi)
- s_1(\pi(y))
\right).$$
Define $(W_2,\omega_2,\{\frak s_2^{\epsilon}\}) = (W_1,\omega_1,
\{\frak s_2^{\epsilon}\})$. Then it is a CF-perturbation of $\mathcal U_2$.
The conditions (1),(2) and (4) of Definition \[defn73ss\] obviously hold by definition.
The $C^1$ convergence, $\lim_{\epsilon \to 0}\frak s_2^{\epsilon} = s_2$, is a consequence of that of $\lim_{\epsilon \to 0}\frak s_1^{\epsilon} = s_1$. If $y \in V_1 \times \{0\}$ then $s_2(y) = s_1(y)$, the second term of the right hand side of (\[defextSSSSS\]) is $\frak s_1^{\epsilon}(y,\xi) -s_1(y)$. Therefore the right hand side of (\[defextSSSSS\]) coincides with $\frak s_1^{\epsilon}(y,\xi)$ on $V_1 \times \{0\}$. Definition \[defn73ss\] (3) holds.
\[lem1225new\] The equivalence class of $(W_2,\omega_2,\{\frak s_2^{\epsilon}\})$ is independent of the choice of the orbifold charts $(V_i,\Gamma_i,\phi_i)$ of $x_i$ in $U_i$ satisfying (1)(2)(3)(4) above.
The proof is easy from definition and Lemma \[lem2622\].
We remark that the equivalence class of $(W_2,\omega_2,
\{\frak s^{\epsilon}_2\})$ depends on the choice of bundle extension datum.
\[lemma1217\] Suppose $(W_1,\omega_1,\{\frak s^{\epsilon}_1\})$ is equivalent to $(W'_1,\omega'_1,\{\frak s^{\epsilon \prime}_1\})$ in the sense of Definition \[conmultiequiv11\]. We take and fix a bundle extension datum $(\pi_{12},\tilde\varphi_{21},\Omega_{12}, \Omega_1)$. We use it to define $\frak s^{\epsilon}_2$ (resp. $\frak s^{\epsilon \prime}_2$) by formula (\[defextSSSSS\]) for $(W_1,\omega_1,\{\frak s^{\epsilon}_1\})$ (resp. for $(W'_1,\omega'_1,\{\frak s^{\epsilon \prime}_1\})$) and obtain $(W_1,\omega_1,\{\frak s^{\epsilon}_2\})$ (resp. $(W'_1,\omega'_1,\{\frak s^{\epsilon \prime}_2\})$).
Then $(W_1,\omega_1,\{\frak s^{\epsilon}_2\})$ is equivalent to $(W'_1,\omega'_1,\{\frak s^{\epsilon \prime}_2\})$.
It suffices to show the lemma in the case when $(W_1,\omega_1,\{\frak s^{\epsilon}_1\})$ is a projection of $(W'_1,\omega'_1,\{\frak s^{\epsilon \prime}_1\})$. Let $\Pi : W'_1 \to W_1$ be the projection such that $\Pi!(\omega'_1) = \omega_1$ and $\frak s_1^{\epsilon}(y,\Pi(\xi)) = \frak s_1^{\epsilon \prime}(y,\xi)$. Then (\[defextSSSSS\]) implies $\frak s_2^{\epsilon}(y,\Pi(\xi)) = \frak s_2^{\epsilon \prime}(y,\xi)$. This implies the lemma.
We use these lemmata to prove Proposition \[prop1221\] as follows.
Take a bundle extension datum $(\pi_{12},\tilde\varphi_{21},\Omega_{12}, \Omega_1)$ as in Lemma \[lem1224\].
We put $\mathcal O^1 = \Omega_1$, and denote $\mathcal O^2 = \Omega_{12}$ which is an open neighborhood of $K$ in $U_2$ where $\frak S^{2,K}$ is defined on. We take open sets $O^i_1,O^i_2 \subset U_i$ ($i=1,2$) such that $$K \subset O^i_1 \subset \overline{O^i_1} \subset O^i_2 \subset \overline{O^i_2} \subset \mathcal O^i$$ and $\varphi_{21}^{-1}(O^2_j) = O^1_j$. Put $C = \overline{O^1_2} \setminus O^1_1$.
Then we take a smooth function $\chi : U_1 \to [0,1]$ such that $$\chi(p) =
\begin{cases}
1 &\text{on a neighborhood of $\overline{O^1_1}$}, \\
0 &\text{on a neighborhood of $U_1 \setminus {O^1_2}$}.
\end{cases}$$
Now we are ready to define an extension $\mathfrak S^2$ of $\mathfrak S^1$. Let $\frak z \in \mathcal O^1$. We will define $(\mathfrak S^2)_{\frak z}
\in \mathscr S_{\varphi_{21}(\frak z)}$ in the following three cases separately.
(Case 1): $\frak z \in \overline{O^1_1}$.
In this case, we set $(\mathfrak S^2)_{\frak z}$ to be a germ of $\frak S^{2,K}$.
(Case 2): $\frak z \in \mathcal O^1 \setminus {O^1_2}$.
In this case, we use $(\pi_{12},\tilde\varphi_{21},\Omega_{12}, \Omega_1)$ to extend $\frak S^1$ to $\mathcal O^2$ by Formula (\[defextSSSSS\]). We then obtain $(\mathfrak S^2)_{\frak z}$.
(Case 3): $\frak z \in C$.
We take a representative of the germ of $\frak S^{1}$ at ${\frak z}$, which we denote by $(\frak V^1_{{\frak z}},\mathcal S^{1}_{{\frak z}})$. We put $\mathcal S^{1}_{{\frak z}}
= (W,\omega,\{\frak s^{\epsilon}_1\})$ and $\frak V^1_{{\frak z}}
= (V_1,E_1,\Gamma_1,\phi_1,\widehat{\phi}_1)$. Then, by shrinking $O^2$ if necessary, we may assume that the germ $\frak S^{2,K}_{{\frak z}}
\in (\mathscr S^{\mathcal U^1\triangleright \mathcal U^2})_{\varphi_{21}(\frak z)}$ is represented by $(\frak V^2_{{\frak z}},\mathcal S^{2,K}_{{\frak z}})$ such that
1. $\frak V^2_{{\frak z}} = (V_2,E_2,\Gamma_2,\phi_2,\widehat{\phi}_2)$, where $(V_i,\Gamma_i,\phi_i)$, $\pi_{12}$ have Property \[proper1227\].
2. $\mathcal S^{2,K}_{{\frak z}}
= (W,\omega,\{\frak s^{\epsilon}_{2,K}\})$ where $W$ and $\omega$ are the same as those appearing in $\frak S^{1}$.
The map $\tilde \varphi_{21}$, which is a part of bundle extension data, rise to a map $$\breve\varphi_{21,\frak z} :
V_2 \times E_1 \to E_2$$ which is $\Gamma_2$ equivariant.
Now we define $(\mathfrak S^2)_{\frak z}
\in \mathscr S_{\varphi_{21}(\frak z)}$ as $(W,\omega,\{\frak s^{\epsilon}_{2}\})$ where $$\label{defext3030}
\aligned
\frak s^{\epsilon}_{2}(y,\xi)
=
s_{2}(y)
&+
\chi([y]) (\frak s^{\epsilon}_{2,K}(y,\xi) - s_{2}(y)) \\
&+
(1-\chi([y]))
\breve \varphi_{21,\frak z}
\left(y,
\frak s^{\epsilon}_1(\pi(y),\xi))
- s_{1}(\pi(y))
\right).
\endaligned$$ Here $\pi : V_2 \to V_1$ is a restriction of the projection $V_1 \times F \to V_1$, which represents the map $\pi_{12}$, which is a part of bundle extension data. We denote by $[y] \in V_2/\Gamma_2$ the equivalence class of $y$ which we identify with an element of $U_2$ by an abuse of notation.
1. $(\mathfrak S^2)_{\frak z}$ defined above is independent of the choices made in the definition and depends only on $(\pi,\tilde \varphi_{21})$, $\chi$ and $\frak S^{2,K}$, $\frak S^{1}$.
2. Moreover $(\mathfrak S^2)_{\frak z}$ for various $\frak z$ defines a section of $\mathfrak S^2$ of $\varphi_{21}^{\star}\mathscr S$.
3. $\mathfrak S^2$ is a section of $\varphi_{21}^{\star}\mathscr S^{\mathcal U^1\triangleright \mathcal U^2}$ and $\Phi_{21}^*(\mathfrak S^2) = \mathfrak S^1$.
In Case 1, the well-defined-ness is obvious. In Case 2, the well-defined-ness follows from Lemmata \[lem1225new\] and \[lemma1217\]. To prove the well-defined-ness in Case 3, it suffices to consider the case of projection, which follows immediately from (\[defext3030\]). We have thus proved Statement (1).
To prove Statement (2), it suffices to show the next two facts (a)(b).
1. If $\frak z \in \overline{O^1_1} \cap C$ then $(\mathfrak S^2)_{\frak z}$ obtained by applying Case 1 is equivalent to $(\mathfrak S^2)_{\frak z}$ obtained by applying Case 3.
2. If $\frak z \in (U'_1 \setminus {O^1_2}) \cap C$ then $(\mathfrak S^2)_{\frak z}$ obtained by applying Case 2 is equivalent to $(\mathfrak S^2)_{\frak z}$ obtained by applying Case 3.
To prove (a) we remark that $\chi = 1$ on a neighborhood of $\frak z$. Therefore (\[defext3030\]) becomes $$\frak s^{\epsilon}_{2}(y,\xi)
=
s_{2}(y)
+
(\frak s^{\epsilon}_{2,K}(y,\xi) - s_{2}(y))
=
\frak s^{\epsilon}_{2,K}(y,\xi),$$ as required.
To prove (b) we remark that $\chi = 0$ on a neighborhood of $\frak z$. Therefore (\[defext3030\]) becomes $$\frak s^{2,\epsilon}_{\frak z}(y,(\xi^1,\xi^2))
=
s_2(y) +
\breve \varphi_{21,\frak z}
\left(y,
\frak s^{\epsilon}_1(\pi(y),\xi))
- s_{1}(\pi(y))
\right).$$ The right hand side coincides with (\[defextSSSSS\]), as required.
By the way how we defined $(\mathfrak S^2)_{\frak z}$ in Case 1, it is easy to see that the restriction of $\frak S^2\vert_{U_2''}$ to $U_2'' \cap O^2_1$ is equivalent to the restriction of $\frak S^{2,K}$ on $U_2'' \cap O^2_1$. This implies the first formula of (\[form1214\])
The second formula of (\[form1214\]) follows from the way how we defined $(\mathfrak S^2)_{\frak z}$ in Cases 2 and 3. In fact if $y = \varphi_{21}(x)$ then (\[defext3030\]) becomes $$\aligned
\frak s^{\epsilon}_{2}(y,\xi)
=
s_{2}(y)
&+
\chi([x]) (\frak s^{\epsilon}_{2,K}(\varphi_{21}(x),\xi) - s_{2}(y)) \\
&+
(1-\chi([x]))
\breve \varphi_{21,\frak z}
\left(\varphi_{21}(x),
\frak s^{\epsilon}_1(x,\xi))
- s_{1}(x)
\right) \\
=
s_{1}(x)&+
\chi([x]) (\frak s^{\epsilon}_{1}(x,\xi) - s_{1}(x)) \\
&+
(1-\chi([x]))
(\breve \varphi_{21,\frak z}
\left(
\frak s^{\epsilon}_1(x,\xi))
- s_{1}(x)
\right) \\
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!= \frak s^{\epsilon}_{1}(x,\xi).
\endaligned$$
To complete the proof of Proposition \[prop1221\], it remains to prove that if $\frak S^1 \in \mathscr S_{\star}^{\mathcal U^1}(U_1)$, $\frak S^{2,K} \in (\varphi_{21}^{\star}\mathscr S_{\sharp}^{\mathcal U^1\triangleright \mathcal U^2})(K)$ then $\frak S^2 \in (\varphi_{21}^{\star}\mathscr S_{\sharp}^{\mathcal U^1\triangleright \mathcal U^2})(Z)$. This is actually an immediate consequence of the fact that the condition for a section of $\mathscr S^{\mathcal U^2}$ to be a section of $\mathscr S_{\sharp}^{\mathcal U^2}$ is an open condition.
Construction of CF-perturbations on good coordinate system {#subsec:cfpgoodcsys}
----------------------------------------------------------
In this subsection we complete the proof of Theorem \[existperturbcont\].
We first discuss the absolute case. We will construct a CF-perturbation on the Kuranishi charts $\mathcal U_{\frak p}$ by the [*upward*]{} induction on $\frak p \in \frak P$ with respect to the partial order of $\frak P$. (We remark that during the construction of good coordinate system we used downward induction on the partial order of $\frak P$. So our construction here goes the opposite direction from that of Section \[sec:contgoodcoordinate\].)
We say a subset $\frak F \subseteq \frak P$ a [*filter*]{} if $\frak p,\frak q \in \frak P$, $\frak p \ge \frak q$, $\frak p \in \frak F$ imply $\frak q \in \frak F$. In this article we regard $\emptyset$ as a filter. (This may be different from the usual convention.) Let $\frak F \subseteq \frak P$ be a filter. The main ingredient of the proof of Theorem \[existperturbcont\] is Proposition \[existontiiindc1t\] below, which we prove by an upward induction on $\#\frak F$. To state Proposition \[existontiiindc1t\] we need several notations. For an arbitrary subset $\frak F \subseteq \frak P$ we put $$\label{form1210}
\frak T(\frak F,\mathcal K)
=
\bigcup_{\frak p \in \frak F}
\psi_{\frak p}(s_{\frak p}^{-1}(0) \cap \mathcal K_{\frak p}).$$ This is a compact subset of $X$. We next define a good coordinate system on a neighborhood of $\frak T(\frak F,\mathcal K)$. We take $\mathcal K^+$ so that $(\mathcal K,\mathcal K^+)$ is a support pair of ${\widetriangle{\mathcal U}}$ and put $$\label{form1211}
\frak Y(\frak F)
=
\bigcup_{\frak p \in \frak F}\psi_{\frak p}(s_{\frak p}^{-1}(0) \cap
\mathcal {\rm Int}\,K^+_{\frak p}).$$ We consider the set of Kuranishi charts $\{ \mathcal U_{\frak p}\vert_{ {\rm Int}\,K^+_{\frak p}}
\mid \frak p \in \frak F\}$. Together with the restrictions of the coordinate changes of ${\widetriangle{\mathcal U}}$ it defines a good coordinate system of $\frak Y(\frak F)$ on $\frak T(\frak F,\mathcal K)$. We denote this good coordinate system by ${\widetriangle{\mathcal U}}(\frak F,\mathcal K^+)$. Note $\{\mathcal K_{\frak p} \mid \frak p \in \frak F\}$ is a support system of ${\widetriangle{\mathcal U}}(\frak F,\mathcal K^+)$. We denote it by $\mathcal K(\frak F)$.
The main difference between ${\widetriangle{\mathcal U}}(\frak F,\mathcal K^+)$ and ${\widetriangle{\mathcal U}}$ lies on the fact that for ${\widetriangle{\mathcal U}}(\frak F,\mathcal K^+)$ we take the Kuranishi charts $\mathcal U_{\frak p}$ with $\frak p \in \frak F$ only.
We now prove the following proposition by an induction. This inductive proof is the same as one we had written in [@FO page 955 line 17-24] in a similar situation of multivalued perturbation. Here we provide much more detail.
\[existontiiindc1t\] For any filter $\frak F \subseteq \frak P$, there exists a CF-perturbation $\widetriangle{\frak S^{\frak F}}$ of $({\widetriangle{\mathcal U}}(\frak F,\mathcal K^+),\mathcal K(\frak F))$ on $\frak T(\frak F,\mathcal K)$. It satisfies the following properties.
1. $\widetriangle{\frak S^{\frak F}}$ is transversal to $0$.
2. If $\widetriangle{f} : (X,Z;\widetriangle{\mathcal U}) \to M$ is a weakly submersive strongly smooth map, then $\widetriangle{\frak S^{\frak F}}$ can be taken so that $\widetriangle{f}$ is strongly submersive with respect to $\widetriangle{\frak S^{\frak F}}$.
3. If $\widetriangle{f} : (X,Z;\widetriangle{\mathcal U}) \to M$ is a strongly smooth map which is weakly transversal to $g : N \to M$, then $\widetriangle{\frak S^{\frak F}}$ can be taken so that $\widetriangle{f}$ is strongly transversal to $g$ with respect to $\widetriangle{\frak S^{\frak F}}$.
The proof is by induction of $\# \frak F$. If $\frak F = \emptyset$, there is nothing to show.
Suppose Proposition \[existontiiindc1t\] is proved for all $\frak F'$ with $\# \frak F' < \# \frak F$. We will prove the case of $\frak F$.
Let $\frak p_0$ be a maximal element of $\frak F$. Then $\frak F_- =\frak F \setminus \{ \frak p_0\}$ is a filter. By the standing induction hypothesis, there exists a CF-perturbation $\widetriangle{\frak S^{\frak F_-}}$ of $({\widetriangle{\mathcal U}}(\frak F_-,\mathcal K^+),\mathcal K(\frak F_-))$ on $\frak T(\frak F_-,\mathcal K)$.
We will extend this CF-perturbation to a CF-perturbation $\widetriangle{\frak S^{\frak F}}$ in two steps. In the first step we use Proposition \[prop1221\] to extend it to a CF-perturbation on $({\widetriangle{\mathcal U}}(\frak F_,\mathcal K^+),\mathcal K(\frak F_-))$ of $\frak T(\frak F_-,\mathcal K)$. We then obtain $\widetriangle{\frak S^{\frak F}}$ in the second step.
Note the difference between ${\widetriangle{\mathcal U}}(\frak F_-,\mathcal K^+)$ and ${\widetriangle{\mathcal U}}(\frak F,\mathcal K^+)$ is that an open subchart of $\mathcal U_{\frak p_0}$ is included in the latter. So, the main integrant of the first step is defining a CF-perturbation on $\mathcal U_{\frak p_0}$ in a neighborhood of $\frak T(\frak F_-,\mathcal K)$. The second step uses Proposition \[prop123123\] to extend it to a Kuranishi neighborhood of $\frak T(\frak F,\mathcal K)$.
We remark that by definition there exists a GG-embedding $$\widetriangle{\Phi_{\frak F\frak F_-}} :
{\widetriangle{\mathcal U}}(\frak F_-,\mathcal K^+)
\to {\widetriangle{\mathcal U}}(\frak F,\mathcal K^+).$$ (We recall $\frak F = \frak F_- \cup \{\frak p_0\}$.) By induction hypothesis, we have a CF-perturbation $\widetriangle{{\frak S}^{\frak F_-}}$ of $({\widetriangle{\mathcal U}}(\frak F_-,\mathcal K^+),\mathcal K(\frak F_-))$ on $\frak T(\frak F_-,\mathcal K)$.
We consider an ideal $\frak I \subseteq \frak F_-$. Namely $\frak I$ is a subset such that $\frak p \in \frak I$, $\frak q \in \frak F_-$ and $
\frak p \le \frak q$ imply $\frak q \in \frak I$.
In the next lemma, we take a neighborhood $U_{\frak p_0}(\frak I)$ of $\psi_{\frak p_0}^{-1}(\frak T(\frak I,\mathcal K)) \cap {\mathcal K}_{\frak p_0}$ in $\mathcal K_{\frak p_0}^+$. Then we replace the Kuranishi neighborhood $\mathcal U_{\frak p_0}$, which is a member of the good coordinate system $\widetriangle{\mathcal U^{\frak F}}$, by $\mathcal U_{\frak p_0}\vert_{U_{\frak p_0}(\frak I)}$. We denote the resulting good coordinate system by $\widetriangle{\mathcal U}(\frak F,\frak I;\mathcal K^+)$. It is a good coordinate system of $\frak T(\frak F_-,\mathcal K)$. (The difference between $\widetriangle{\mathcal U}(\frak F,\frak I;\mathcal K^+)$ and $\widetriangle{\mathcal U^{\frak F_-}}$ is that $\widetriangle{\mathcal U}(\frak F,\frak I;\mathcal K^+)$ has one more Kuranishi chart $\mathcal U_{\frak p_0}\vert_{U_{\frak p_0}(\frak I)}$ than $\widetriangle{\mathcal U^{\frak F_-}}$.)
We can define a GG embedding $\widetriangle{\mathcal U^{\frak F_-}}
\to \widetriangle{\mathcal U}(\frak F,\frak I;\mathcal K^+)$ so that the map induced on the index set of the Kuranishi charts is an obvious embedding $\frak F_- \to \frak F$, which we denote by $\widehat\Phi_{\frak F_-\frak F;\frak I}$.
\[122222\] For any ideal $\frak I \subseteq \frak F_-$, there exist $U_{\frak p_0}(\frak I)$ and a CF-perturbation $\widetriangle{\frak S^{\frak F}}(\frak I)$ of $(\widetriangle{\mathcal U}(\frak F,\frak I;\mathcal K^+),
\frak T(\frak F_-,\mathcal K))$ with the following properties.
1. $\widetriangle{\frak S^{\frak F}}(\frak I)$, $\widetriangle{\frak S^{\frak F_-}}$ are compatible with the embedding $\widehat\Phi_{\frak F_-\frak F;\frak I}$.
2. 1. If $\widetriangle{{\frak S}^{\frak F_-}}$ is transversal to $0$ so is $\widetriangle{\frak S^{\frak F}}(\frak I)$.
2. If $\widetriangle f$ is strongly submersive with respect to $\widetriangle{{\frak S}^{\frak F_-}}$ and $\widetriangle f$ is weakly submersive then $\widetriangle f$ is strongly submersive with respect to $\widetriangle{\frak S^{\frak F}}(\frak I)$.
3. If $\widetriangle f$ is strongly transversal to $g : N \to M$ with respect to $\widetriangle{{\frak S}^{\frak F_-}}$ and $\widetriangle f$ is weakly transversal to $g : N \to M$ then $\widetriangle f$ is strongly transversal to $g : N \to M$ with respect to $\widetriangle{\frak S^{\frak F}}(\frak I)$.
Recall $\frak I \subseteq \frak F_- = \frak F \setminus \{\frak p_0\}$. The proof is by an upward induction over $\#\frak I$. For $\frak I = \emptyset$, we put $U_{\frak p_0}(\emptyset) = \emptyset$. Then $\widetriangle{\mathcal U}(\frak F,\emptyset;\mathcal K^+)
= \widetriangle{\mathcal U}(\frak F_-;\mathcal K^+)
$. We put $\widetriangle{\frak S^{\frak F}}(\emptyset)
= \widetriangle{\frak S^{\frak F_-}}$. It is easy to see that it has the required properties.
We assume that the lemma is proved for $\frak I'$ with $\#\frak I' <\#\frak I$ and will prove the case of $\frak I$.
Let $\frak p_1$ be a minimal element of $\frak I$ and denote $\frak I_- = \frak I \setminus \{\frak p_1\}$. Then $\frak I_-$ is an ideal. We use the induction hypothesis to obtain $\widetriangle{\frak S^{\frak F}}(\frak I_-)$ where $\widetriangle{\frak S^{\frak F}}(\frak I_-)$ is a CF-perturbation of $(\widetriangle{\mathcal U}(\frak F,\frak I_-;\mathcal K^+),
\frak T(\frak F_-,\mathcal K))$.
We apply Proposition \[prop1221\] by putting: $$\label{form121212}
\aligned
&\frak S^{2,K} = (\widetriangle{\frak S^{\frak F}}(\frak J_-))_{\frak p_0},
\qquad
\frak S^1 = (\widetriangle{\frak S^{\frak F_-}})_{\frak p_1}, \\
&
\mathcal U_1 = \mathcal U_{\frak p_1},\quad
\mathcal U_2 = \mathcal U_{\frak p_0},
\qquad
K = \frak T(\frak I_-,\mathcal K).
\endaligned$$ We denote by $(\widetriangle{\frak S^{\frak F}}(\frak J_-))_{\frak p_0}$ a CF-perturbation induced by $\widetriangle{\frak S^{\frak F}}(\frak J_-)$ on an open subchart $\mathcal U_{\frak p_0}\vert_{U_{\frak p_0}(\frak I_-)}$ of $\mathcal U_{\frak p_0}$. The definition of $(\widetriangle{\frak S^{\frak F_-}})_{\frak p_1}$ is similar.
We now apply Proposition \[prop1221\] to obtain $U_{\frak p_0}(\frak I)$ and a CF-perturbation $(\widetriangle{\frak S^{\frak F}}(\frak J))_{\frak p_0}$ of an open subchart $\mathcal U_{\frak p_0}\vert_{U_{\frak p_0}(\frak I)}$ of $\mathcal U_{\frak p_0}$.
Among the CF-perturbations on Kuranishi charts consisting $\widetriangle{\frak S^{\frak F}}(\frak J)$, we replace $(\widetriangle{\frak S^{\frak F}}(\frak J_-))_{\frak p_0}$ by $(\widetriangle{\frak S^{\frak F}}(\frak J))_{\frak p_0}$ and obtain a required CF-perturbation $\widetriangle{\frak S^{\frak F}}(\frak I)$.
We have thus completed Step 1 of the proof of Proposition \[existontiiindc1t\].
We take the case $\frak I = \frak F_-$ of Lemma \[122222\]. Then we have $\widetriangle{\frak S^{\frak F}}(\frak F_-)$ which include $(\widetriangle{\frak S^{\frak F}}(\frak F_-))_{\frak p_0}$ such that:
1. $(\widetriangle{\frak S^{\frak F}}(\frak F_-))_{\frak p_0}$ is a CF-perturbation of $\mathcal U_{\frak p_0}$ on $\frak T(\frak F_-,\mathcal K)$.
2. $\widetriangle{\frak S^{\frak F}}(\frak F_-)$, $\widetriangle{\frak S^{\frak F_-}}$ are compatible with the embedding $\widehat{\Phi_{\frak F\frak F_-}}$ on a neighborhood of $\frak T(\frak F_-,\mathcal K)$.
We now apply Proposition \[prop123123\] to $\mathcal U_{\frak p_0}$ and find that $$\label{1213formula}
\mathscr F_{\sharp}(V) \to \mathscr F_{\sharp}(K)$$ is surjective. Here $\sharp$ is one of ${\pitchfork 0}$, ${f \pitchfork}$, ${f \pitchfork g}$. We put $$\label{12132formula}
K = \frak T(\frak F_-,\mathcal K),
\qquad
V = \text{an open neighborhood of $\mathcal K_{\frak p_0}$}.$$ Therefore we can extend $(\widetriangle{\frak S^{\frak F}}(\frak F_-))_{\frak p_0}$ to a CF perturbation $(\widetriangle{\frak S^{\frak F}}(\frak F))_{\frak p_0}$ on a Kuranishi neighborhood of $\frak T(\frak F,\mathcal K)$ that is an open subchart of $\mathcal U_{\frak p_0}$.
Replacing $(\widetriangle{\frak S^{\frak F}}(\frak F_-))_{\frak p_0}$ in $\widetriangle{\frak S^{\frak F}}(\frak F_-)$ by this extension we obtain $\widetriangle{\frak S^{\frak F}}$.
The proof of Proposition \[existontiiindc1t\] is complete.
Theorem \[existperturbcont\] is the case of Proposition \[existontiiindc1t\] when $\frak F = \frak P$.
Proposition \[existperturbcontrel\] is a relative version of Theorem \[existperturbcont\] and the proof is mostly the same. We use the symbol $\mathcal Z_{(i)}$ in place of $Z_i$ during the proof of Proposition \[existperturbcontrel\] (since we used $Z$ for other objects in this section already).
We replace (\[form1210\]) by $\frak T(\frak F,\mathcal K;\mathcal Z_{(1)}) = \frak T(\frak F,\mathcal K)
\cup \mathcal Z_{(1)}$ and (\[form1211\]) by $\frak Y(\frak F) \cup \mathcal Z_{(1)} =\frak Y(\frak F;\mathcal Z_{(1)})$. We have a good coordinate system ${\widetriangle{\mathcal U}}(\frak F,\mathcal K^+) \cup {\widetriangle{\mathcal U^{\mathcal Z_{(1)}}}}$ on $\frak Y(\frak F;\mathcal Z_{(1)})$, which we denote by ${\widetriangle{\mathcal U}}(\frak F,\mathcal K^+;\mathcal Z_{(1)})$. We take a support system $\mathcal K^{\mathcal Z_{(1)}}$ of ${\widetriangle{\mathcal U^{\mathcal Z_{(1)}}}}$. Together with $\mathcal K(\frak F)$ it gives a support system of $\frak T(\frak F,\mathcal K;\mathcal Z_{(1)})$, which we denote by $\mathcal K(\frak F;\mathcal Z_{(1)})$.
Proposition \[existontiiindc1t\] is replaced by the following:
1. There exists a CF-perturbation $\widetriangle{\frak S^{\frak F,\mathcal Z_{(1)}}}$ of $({\widetriangle{\mathcal U}}(\frak F,\mathcal K^+;\mathcal Z_{(1)}),
\mathcal K(\frak F;\mathcal Z_{(1)}))$ on $\frak T(\frak F,\mathcal K;\mathcal Z_{(1)})$. The same properties as (1)(2)(3) of Proposition \[existontiiindc1t\] are satisfied.
Moreover the CF-perturbation $\widetriangle{\frak S^{\frak F,\mathcal Z_{(1)}}}$ coincides with a restriction of $\widetriangle{\frak S^1}$ (given as a part of assumption) on the charts of $\widetriangle{\mathcal U^{(1)}}$.
We prove (\*) by the same induction as the proof of Proposition \[existontiiindc1t\]. Namely we prove:
1. There exists a CF-perturbation $\widetriangle{\frak S^{\frak F,\mathcal Z_{(1)}}}(\frak I)$ of $({\widetriangle{\mathcal U}}(\frak F,\frak I;\mathcal K^+;\mathcal Z_{(1)}),\mathcal K(\frak F;\mathcal Z_{(1)}))$ on $\frak T(\frak I,\mathcal K;\mathcal Z_{(1)})$ with the following properties.
1. $\widetriangle{\frak S^{\frak F,\mathcal Z_{(1)}}}(\frak I)$, $\widetriangle{{\frak S}^{\frak F_-,\mathcal Z_{(1)}}}$ are compatible with the embedding $\widehat{\Phi_{\frak F\frak F_-;\frak I;\mathcal Z_{(1)}}}$ on a neighborhood of $\frak T(\frak I,\mathcal K;\mathcal Z_{(1)})$.
2. The same properties as (1)(2)(3) of Proposition \[existontiiindc1t\] are satisfied.
Here we take $U_{\frak p_0}(\frak I)$ and define ${\widetriangle{\mathcal U}}(\frak F,\frak I;\mathcal K^+;\mathcal Z_{(1)})$ in the same way as ${\widetriangle{\mathcal U}}(\frak F,\frak I;\mathcal K^+)$. The embedding $\widehat{\Phi_{\frak F\frak F_-;\frak I;\mathcal Z_{(1)}}}$ is obtained from $\widehat{\Phi_{\frak F\frak F_-}}$ by using the identity embedding for the charts in ${\widetriangle{\mathcal U^{\mathcal Z_{(1)}}}}$.
The proof of (\*\*) is by the same induction as the proof of Lemma \[122222\], where we replace (\[form121212\]) by $$\aligned
&
\frak S^{2,K} = (\widetriangle{\frak S^{\frak F,\mathcal Z_{(1)}}}(\frak J_-))_{\frak p_0},
\qquad
\frak S_1 = (\widetriangle{\frak S^{\frak F_-,\mathcal Z_{(1)}}})_{\frak p_1},
\\
&\mathcal U_1 = \mathcal U_{\frak p_1},
\qquad\qquad\quad\,\,
\mathcal U_2 = \mathcal U_{\frak p_0},
\quad\qquad\qquad
K = \frak T(\frak I_-,\mathcal K;\mathcal Z_{(1)}).
\endaligned
\nonumber$$ Using (\*\*) we can prove (\*) in the same way as the last step in the proof of Theorem \[existperturbcont\]. The proof of Proposition \[existperturbcontrel\] is complete.
Construction of multisections {#sec:constrsec}
=============================
In Sections \[sec:constrsec\] we discuss the multivalued perturbation, especially its existence result, Theorem \[prop621\]. This result will be used in Section \[sec:onezerodim\]. One of the advantages using multivalued perturbations is it enables us to work with $\Q$ coefficients. In the construction based on de Rham theory we can work only with $\R$ or $\C$. For many applications, it is enough to work with coefficients $\R$ or $\C$. For these cases, we do not need to use the results of Sections \[sec:constrsec\] and \[sec:onezerodim\].
Construction of multisection on a single chart {#subsec:musecexsingle}
----------------------------------------------
The proof of Theorem \[prop621\] is similar to the proof of Theorem \[existperturbcont\].
We begin with proving a version of Proposition \[prop123123\] (2)(4). (We remark that the multisection version of Proposition \[prop123123\] (3) does not seem to exist.)
\[prop127777ver\] In Situation \[situ121\], let $K \subset U$ be a compact subset and $\widetriangle{\frak s_K} = \{\frak s_{K}^n\}$ a multivalued perturbation of $\mathcal U$ on a neighborhood of $K$ transversal to $0$.
Let $\Omega \subset U$ be an relatively compact open subset such that $K \subseteq \Omega$.
1. There exists a multivalued perturbation $\widetriangle{\frak s} = \{\widetriangle{\frak s^n}\}$ of $\mathcal U$ on $\Omega$ such that $\widetriangle{\frak s}$ is transversal to $0$ and is equal to $\widetriangle{\frak s_K}$ on a neighborhood of $K$.
2. Suppose we are in Situation \[situ122\]. Then we may choose $\widetriangle{\frak s}$ so that $f$ is strongly transversal to $g$ with respect to $\widetriangle{\frak s^n}$ for sufficiently large $n$. (See Definition \[transofdvect\].)
We use Lemma \[lemma12770\] to obtain $\{\frak V_{\frak r} \vert \frak r \in \frak R\}$ and $\{\mathcal S_{\frak r} \vert
\frak r \in \frak R \}$ with the following properties. Let $U_0$ be an open neighborhood of $K$ which we will fix later.
1. $\frak V_{\frak r} = (V_{\frak r},\Gamma_{\frak r},E_{\frak r},\phi_{\frak r},\widehat\phi_{\frak r})$ is an orbifold chart of $U$. We put $U_{\frak r} = {\rm Im}(\phi_{\frak r})$ and assume $U_{\frak r} \cap K = \emptyset$.
2. $$U_0 \cup \bigcup_{\frak r \in \frak R} U_{\frak r}$$
is an open covering of $\overline{\Omega}$.
3. $\mathcal S_{\frak r} = (W_{\frak r},\omega_{\frak r},\{\frak s_{\frak r}^{\epsilon}\})$ is a CF-perturbation of $\mathcal U$ on $\frak V_{\frak r}$. (Definition \[contipertlocalrest\].)
4. $\mathcal S_{\frak r}$ is strongly transversal. (Definition \[strongtransvers\].)
We may assume that the given $\Gamma_{\frak r}$ action on $W_{\frak r}$ is effective, by replacing $W_{\frak r}$ with the product $W_{\frak r} \times W'$ if necessary, where $W'$ is a faithful representation of the finite group $\Gamma_{\frak r}$. Then we define $\frak s_{\frak r}^{\epsilon}$ to be the pull-back of a multisection defined on $W_{\frak r}$ by the projection $W_{\frak r} \times W' \to W_{\frak r}$ and define our $\omega_{\frak r}$ on $W_{\frak r} \times W'$ to be the product form of a smooth $\Gamma_{\frak r}$ invariant top degree form of compact support on $W'$ and the given smooth top degree form on $W_{\frak r}$.
For each $\frak r$ we take an open subset $W_{\frak r}^0$ of $W_{\frak r}$ such that $$\label{form132222}
\gamma W_{\frak r}^0 \cap W_{\frak r}^0 = \emptyset,$$ if $\gamma \in \Gamma_{\frak r} \setminus \{1\}$ (we use effectivity of $\Gamma_{\frak r}$ action here) and put $$\label{form1217777}
W_0 = \prod_{\frak r \in \frak R}W_{\frak r}^0.$$
For each $x \in K$ we take a representative of $\frak s^n_K$ in an orbifold chart $\frak V_x$ at $x$ and denote it by $(\frak s^n_{x,1},\dots,\frak s^n_{x,\ell(x)})$. We take a finite number of points $x_1,\dots,x_k$ of $K$ so that $\bigcup_{a=1}^k U_{{x_a}} \supset K$ and the sections $\frak s^n_{x,k}$ are transversal to $0$ on $\bigcup_{a=1}^k U_{{x_a}}$ which is possible by the openness of transversality condition. We put $\frak V_a = \frak V_{x_a}$, $U_a = U_{x_a}$, $U^0_a = U^0_{x_a}$, and $(\frak s_{1}^{a,n},\dots,\frak s_{\ell_a}^{a,n})
= (\frak s^n_{x_a,1},\dots,\frak s^n_{x_a,\ell({x_a})})$. Then we define $$\frak U(K) = \bigcup_{a=1}^k U_{a}.$$ We also take a relatively compact subset $U_{a}^0$ of $U_{a}$ for each $a$ such that $$\frak U_0(K) = \bigcup_{a=1}^k U^0_{a} \supset K.$$ We then fix an open neighborhood $U_0$ of $K$ to be $U_0 = \frak U_0(K)$ .
Let $\{\chi_0\} \cup \{\chi_{\frak r} \mid \frak r \in \frak R\}$ be a set of strongly smooth functions $\chi_* : \Omega^+ \to [0,1]$ satisfying the following properties.
1. The support of $\chi_0$ is contained in $U_0$. The support of $\chi_{\frak r}$ is contained in $U_{\frak r}$.
2. $
\chi_0 + \sum_{\frak r \in \frak R} \chi_{\frak r} \equiv 1
$ on $\overline{\Omega}$.
For each $x \in \overline{\Omega}$ we take $U_x$ with the following properties.
\[prop12888rev\]
1. If $x \in U_0 = \frak U_0(K)$ then $U_x \subset \frak U_0(K)$.
2. If $x \in U_{\frak r}$ for $\frak r\in \frak R$, then $U_x \subset U_{\frak r}$. Moreover there exists $(h_{\frak r x},\tilde\varphi_{\frak r x},\breve\varphi_{\frak r x})$ as in Property \[proper728\].
3. If $x \notin U_{\frak r}$ then $\overline U_x \cap {\rm Supp}(\chi_{\frak r}) =
\emptyset$.
4. If $x \in \frak U(K)$ then $U_x \subset U_{a}$ for some $a \in \{1,\dots,k\}$. Moreover there exists $(h_{ax},\tilde\varphi_{ax},\breve\varphi_{ax}) : \frak V_x \to \frak V_a$ as in Property \[proper728\].
For each $
\vec\xi = (\xi_{\frak r})_{\frak r\in \frak R} \in W_0
$ and $n \in \Z_{\ge 0}$ we will define a multivalued perturbation $\{\frak s^{n,\vec\xi}_{x}\}$ on $\frak V_x$ as follows.
(Case 1) $x \in U_0 = \frak U_0(K)$ then $U_x \subset \frak U_0(K)$.
Then $U_x \subset \frak U_0(K)$. We take $a \in \{1,\dots,k\}$ such that $x \in U^{0}_{a}$. By Property \[prop12888rev\] (4) we can pullback $(\frak s_{1}^{a,n},\dots,\frak s_{\ell_a}^{a,n})$ to $\frak V_{x}$. This pullback is our $\frak s_{x}^{n,\vec\xi}$. (This is independent of $\vec\xi$.)
(Case 2) $x \in \frak U(K) \setminus \frak U_0(K)$.
We take $a \in \{1,\dots k\}$ such that $U_x \subset U_{a}$. By Property \[prop12888rev\] (4) we have $(h_{ax},\tilde\varphi_{ax},\breve\varphi_{ax})$. We put $$\label{formula123rev}
\frak R(x) = \{\frak r \in \frak R
\mid x \in {\rm Supp}(\chi_{\frak r})\}.$$ We take $(h_{\frak rx},\tilde\varphi_{\frak rx},\breve\varphi_{\frak r x})$ as in Property \[prop12888rev\] (3) for each $\frak r \in \frak R(x)$. We put $$I = \{1,\dots,\ell_a\}\times \prod_{\frak r \in \frak R(x)}\Gamma_{\frak r}.$$ We define $\frak s^{n,\vec\xi}_{x,i}
: V_{x} \to E_{x}$ for each $i \in I$ by Formula (\[furmula124rev\]) below. We take a sequence $\epsilon_n > 0$ with $\lim_{n\to\infty}\epsilon_n = 0$ and fix it throughout the proof. (For example we may take $\epsilon_n = 1/n$.) We put $i = (j,(\gamma_{\frak r}))$. $$\label{furmula124rev}
\aligned
\frak s^{n,\vec\xi}_{x,i}(y)
=
s_{x}(y) &+
\chi_{0}([y]) g_{a,y}^{-1}(\frak s^{n}_{a,j}(\tilde\varphi_{ax}(y)) - s_{a}(\tilde\varphi_{ax}(y)))
\\
&+\sum_{\frak r \in \frak R(x)}
\chi_{\frak r}([y])
g_{\frak r,y}^{-1}(
\frak s_{\frak r}^{\epsilon_n}(\tilde\varphi_{\frak r x}(y),\gamma_{\frak r}^{-1}\xi_{\frak r})
- s_{\frak r}(\tilde\varphi_{\frak r x}(y))
\endaligned$$ Explanation of the notations in Formula (\[furmula124rev\]) is in order. $s_{a} : V_{a} \to E_{a}$ is the representative of the Kuranishi map, $g_{a,y} : E_{a} \to E_{x}$ is defined by $
\breve{\varphi}_{ax}(y,\eta) = g_{a,y}(\eta),
$ $s_{\frak r} : V_{\frak r} \to E_{\frak r}$ is the representative of the Kuranishi map, $g_{\frak r,y} : E_{\frak r} \to E_{x}$ is defined by $
\breve{\varphi}_{\frak r x}(y,\eta) = g_{\frak r,y}(\eta).
$ $[y] \in V_a/\Gamma_a$ is the equivalence class of $y$, which we regard as an element of $U$.
(Case 3) $x \in U \setminus \frak U(K)$.
We define $\frak R(x)$ by (\[formula123rev\]). We take $(h_{\frak rx},\tilde\varphi_{\frak rx},\breve\varphi_{\frak rx})$ as in Property \[prop12888rev\] (3) for each $x\in \frak R(x)$. We put $$I = \prod_{\frak r \in \frak R(z)}\Gamma_{\frak r}.$$ We define $\frak s^{n,\vec\xi}_{x,i}
: V_{x} \to E_{x}$ for each $i = (\gamma_{\tau})_{\tau \in \frak R(x)}\in I$ by the following formula. $$\label{furmula124rev22}
\frak s^{n,\vec\xi}_{x,i}(y)
=
s_{x}(y) +
\sum_{\frak r \in \frak R(x)}
\chi_{\frak r}([y])
g_{\frak r,y}^{-1}(
\frak s_{\frak r}^{\epsilon_n}(\tilde\varphi_{\frak r x}(y),\gamma_{\frak r}^{-1}\xi_{\frak r})
- s_{\frak r}(\tilde\varphi_{\frak r x}(y)).$$ Here the notations are the same as (\[furmula124rev\]).
\[lem1227\] $(\frak s^{n,\vec\xi}_{x,i})_{i\in I}$ defines a multisection on $\frak V_{x}$. Moreover it satisfies the following properties.
1. In Case 1, it is independent of the choice of $a$ and $(h_{ax},\tilde\varphi_{ax},\breve\varphi_{ax})$.
2. In Case 2, it is independent of the choice of $(h_{\frak r\frak z},\tilde\varphi_{\frak rx},\breve\varphi_{\frak rx})$, and $a$, $(h_{ax},\tilde\varphi_{ax},\breve\varphi_{ax})$.
3. In Case 3, it is independent of the choice of $(h_{\frak rx},\tilde\varphi_{\frak rx},\breve\varphi_{\frak rx})$.
4. The restriction of $(\frak s^{n,\vec\xi}_{x,i})_{i\in I}$ to $U_{x} \cap U_{x'}$ is equivalent to restriction of $(\frak s^{n,\vec\xi}_{x',i})_{i\in I}$ to $U_{x} \cap U_{x'}$.
In Cases 2 and 3 we will show that $(\frak s^{\epsilon,\vec\xi}_{x,i}( y))_{i\in I}$ is a permutation of $(\gamma\frak s^{n,\vec\xi}_{x,i}(\gamma^{-1}y))_{i\in I}$ for $\gamma \in \Gamma_{x}$. We calculate $$\label{form1222new}
\gamma g_{\frak r,\gamma^{-1}y}^{-1}(\xi_{\frak r})
=
\gamma \tilde\varphi_{\frak r x}(\gamma^{-1}y,\xi)
=
\tilde\varphi_{\frak r x}(y,\gamma\xi)
= g_{\frak r,y}^{-1}(\gamma\xi_{\frak r}).$$ This implies that the third term of (\[furmula124rev\]) and the second term of (\[furmula124rev22\]) is invariant under $\gamma$ action modulo permutation of the indices in $I$. The second term of (\[furmula124rev\]) is invariant under $\gamma$ action modulo permutation of $1,\dots,a$ since $(\frak s_{a,1}^n,\dots,\frak s_{a,\ell_a}^n)$ is a multisection. We have thus proved that $(\frak s^{n,\vec\xi}_{x,i})_{i\in I}$ is a multisection. (In Case 1 this fact is obvious.)
Statement (1) is a consequence of the definition of multivalued perturbation, that is the well-defined-ness of $\frak s_K$.
To prove Statement (2), we observe that different choices of $(h_{\frak rx},\tilde\varphi_{\frak rx},\breve\varphi_{\frak rx})$ are related one another by the action of $\gamma\in \Gamma_{\frak r}$. (Lemma \[lem2715\].) Then using (\[form1222new\]) we can show that the third term of (\[furmula124rev\]) changes by the permutation of $\gamma_{\frak r}$. The first and the second terms of (\[furmula124rev\]) do not change.
By changing $(h_{ax},\tilde\varphi_{ax},\breve\varphi_{ax})$ the second term of (\[furmula124rev\]) changes by the permutation of $j$ and the third term of (\[furmula124rev\]) does not change.
The proof of (3) is easier.
To prove (4) it suffices to consider the case $U_{x'} \subset U_{x}$. If Case 1 is applied to both $x$ and $x'$ it is a consequence of the well-defined-ness of the restriction of multisection.
If Case 3 is applied to both, then $\frak R(x') \subset \frak R(x)$. In this case the right hand side of (\[furmula124rev22\]) is independent of $\Gamma_{\frak r}$ factor for $\frak r \notin \frak R(x')$ on $U_{\frak z'}$ . Therefore the restriction of $(\frak s^{n,\vec\xi}_{x,i})_{i\in I}$ to $U_{x} \cap U_{x'}$ is a permutation of the $\prod_{\frak r \in \frak R(\frak z') \setminus \frak R(\frak z)} \# \Gamma_{\frak r}$ iteration of the restriction of $(\frak s^{\epsilon,\vec\xi}_{x',i})_{i\in I}$ to $U_{\frak z} \cap U_{x'}$.
When Case 2 is applied to both we can prove (4) by combining the above those two cases.
What remains to prove is the case where the Case 2 is applied to $x$ and Case 1 or Case 3 is applied to $x'$. If Case 1 is applied to $x'$ then $\chi_0$ becomes 1 on $U_{x'}$. Therefore the required equivalence follows from the well-defined-ness of the restriction of the multisection. If Case 3 is applied to $x'$ then $\chi_0$ becomes 0 on $U_{x'}$. Therefore the second term of (\[furmula124rev\]) vanishes. So the restriction of $(\frak s^{n,\vec\xi}_{x,i})_{i\in I}$ to $U_{x'}$ is a permutation of the $\ell_a\prod_{\frak r \in \frak R(x') \setminus \frak R(x)} \# \Gamma_{\frak r}$ iteration of the restriction of $(\frak s^{n,\vec\xi}_{x',i})_{i\in I}$ to $U_{x'}$.
\[lem1228\] For each sufficiently large $n$, the set of $\vec\xi$ such that $(\frak s^{n,\vec\xi}_{x,i})_{i\in I}$ is transversal to $0$ for all $x \in \overline{\Omega}$ is open and dense in $W_0$.
If we are in Situation \[situ122\] in addition, then the set of $\vec\xi$ such that $f$ is strongly transversal to $g$ with respect to $(\frak s^{n,\vec\xi}_{x,i})_{i\in I}$ is dense in $W_0$.
It suffices to show the conclusion on a neighborhood $U_x$ of each fixed $x \in \overline{\Omega}$, since we can cover $\overline{\Omega}$ by countably many such $U_{x}$’s.
In Case 1 this follows from the assumption that $\frak s_K$ is transversal to $0$.
In Case 2 the set $$\{(y,\vec \xi) \mid \frak s^{n,\vec\xi}_{x,i}(y) = 0\}$$ is a smooth submanifold of $V_{x} \times W_{0}$. Therefore we can prove the lemma by applying Sard’s theorem to its projection to $W_{0}$. (We use (\[form132222\]) here.)
We consider Case 3. Suppose $\chi_0([x]) \ne 1$. Then we can shrink the domain $U_{x}$ if necessary and assume $\chi_{\frak r}([y]) \ne 0$ on $U_x$. Then by the same argument as Case 2 we can show the required transversality for the dense of of $\xi_{\frak r}$.
Suppose $\chi_0([x]) = 1$. Then all the functions $\chi_{\frak r}$ together with its first derivative is small in a neighborhood of $x$. Moreover the first derivative of $\chi_0$ is small in a neighborhood of $x$. We also remark that $\frak s_K$ is transversal to $0$ at $x$. Therefore by using the openness of transversality, we can shrink the neighborhood $U_x$ so that $(\frak s^{n,\vec\xi}_{x,i})_{\in I}$ reminds to be transversal to $0$ on $U_x$ for any $\xi$ contained in a compact subset of $W_0$.
The proof of the second half is similar. In Case 2 above we consider the set $$\{(y,\vec \xi,z) \in V_x \times W \times N \mid \frak s^{n,\vec\xi}_{x,i}(y) = 0, \,\, f(y) = g(z)\}.$$ This is a smooth submanifold of $V_x \times W \times N$. Therefore we have the required transversality result in this case by applying Sard’s theorem to the projection to $W$ from this manifold. The rest of the proof is entirely the same.
The proof of Proposition \[prop127777ver\] is complete.
Compatible system of bundle extension data {#subsection:bdlextcompa}
------------------------------------------
There is one nasty point in proving a multisection version of Proposition \[prop127777ver\]. (See Subsection \[subsec:nastyreason\].) To go around it, we will take the bundle extension data for each coordinate change of the good coordinate system so that they are compatible to one another. This is closely related to the idea of compatible system of tubular neighborhoods by Mather. However in our case its construction is rather easy.
In this section, when we consider a bundle extension datum $(\pi_{12},\tilde{\varphi}_{12},\Omega_{12},\Omega_1)$ of $(\Phi,\mathcal K)$, we sometimes shrink $\Omega_{12}, \Omega_1$ and restrict $(\pi_{12},\tilde\varphi_{21})$ thereto respectively. It will be a bundle extension datum if $\Omega_{12}$, $\Omega_1$ still remain to be neighborhoods of $\varphi_{21}(K)$, $K$ respectively. So we sometimes say $(\pi_{12},\tilde\varphi_{21})$ is a bundle extension data without specifying $\Omega_{12}, \Omega_1$.
\[defn1232\] Let ${\widetriangle{\mathcal U}}$ be a good coordinate system and $\mathcal K$ a support system.
We call $(
\{\pi_{\frak q\frak p}\},\{\tilde\varphi_{\frak p\frak q}\},\{\Omega_{\frak q\frak p}\},\{\Omega_{\frak p}\})$ a [*system of bundle extension data*]{} of $({\widetriangle{\mathcal U}},\mathcal K)$ if they have the following properties. (Here $\frak r \le \frak q \le \frak p$ are elements of $\frak P$.)
1. $(\pi_{\frak q\frak p},\tilde\varphi_{\frak p\frak q},
\Omega_{\frak q\frak p},\Omega_{\frak p})$ is a bundle extension datum of $(\Phi_{\frak p\frak q},\mathcal K_{\frak p\frak q})$, where $\mathcal K_{\frak p\frak q}
= \varphi_{\frak p\frak q}^{-1}(\mathcal K_{\frak p}) \cap \mathcal K_{\frak q}$.
2. $\pi_{\frak r\frak q} \circ \pi_{\frak q\frak p} = \pi_{\frak r\frak p}$ on a neighborhood of $\varphi^{-1}_{\frak q\frak r}(\varphi^{-1}_{\frak p\frak q}(\mathcal K_{\frak p}))
\cap \varphi^{-1}_{\frak q\frak r}(\mathcal K_{\frak q})
\cap \varphi^{-1}_{\frak p\frak r}(\mathcal K_{\frak p})
\cap \mathcal K_{\frak r}$.
3. $\tilde\varphi_{\frak p\frak q} \circ \tilde\varphi_{\frak q\frak r} = \tilde\varphi_{\frak p\frak r}$ on a neighborhood of $\varphi^{-1}_{\frak q\frak r}(\varphi^{-1}_{\frak p\frak q}(\mathcal K_{\frak p}))
\cap \varphi^{-1}_{\frak q\frak r}(\mathcal K_{\frak q})
\cap \varphi^{-1}_{\frak p\frak r}(\mathcal K_{\frak p})
\cap \mathcal K_{\frak r}$.
The precise meaning of equality in Condition (3) is as follows. We pull back $\tilde\varphi_{\frak q\frak r} : \pi_{\frak r\frak q}^*\mathcal E_{\frak r} \to \mathcal E_{\frak q}$ by $\pi_{\frak q\frak p}$ and obtain $\pi_{\frak q\frak p}^*\tilde\varphi_{\frak q\frak r} : \pi_{\frak r\frak p}^*\mathcal E_{\frak r} \to \pi_{\frak q\frak p}^*\mathcal E_{\frak q}$. We compose it with $\tilde\varphi_{\frak p\frak q}: \pi_{\frak q\frak p}^*\mathcal E_{\frak q} \to \mathcal E_{\frak p}$ and obtain a map $: \pi_{\frak r\frak p}^*\mathcal E_{\frak r} \to \mathcal E_{\frak p}$. We denote it by $\tilde\varphi_{\frak p\frak q} \circ \tilde\varphi_{\frak q\frak r}$. Condition (3) requires that this map coincides with $\tilde\varphi_{\frak p\frak r}$.
During the discussion of this section, we shrink $,\{\Omega_{\frak q\frak p}\},\{\Omega_{\frak p}\}$ several times and restrict $\pi_{\frak q\frak p},\tilde\varphi_{\frak p\frak q}$ thereto. We call $(
\{\pi_{\frak q\frak p}\},\{\tilde\varphi_{\frak p\frak q}\})$ a compatible system of bundle extension data sometimes, in case we do not need to specify the domain.
If $\pi_{\frak r\frak q}$, $\pi_{\frak q\frak p}$ are diffeomorphic to the projections of normal bundles, then the composition $\pi_{\frak r\frak q}\circ \pi_{\frak q\frak p}$ is diffeomorphic to the projections of normal bundle.
The proof is easy and is omitted.
\[prop12333\] For any pair $({\widetriangle{\mathcal U}},\mathcal K)$ there exists a system of bundle extension data associated thereto.
We first construct $\{\pi_{\frak q \frak p}\}$.
\[lem1226\] Let $\mathcal K$ be a support system of ${\widetriangle{\mathcal U}}$. Then there exists $\Omega_{\frak q\frak p}$ for $\frak p > \frak q$, $\Omega_{\frak p}$ and $\pi_{\frak q\frak p}$ with the following properties.
1. $\Omega_{\frak q\frak p}$ is a neighborhood of $\varphi_{\frak p\frak q}
(\mathcal K_{\frak q} \cap U_{\frak p\frak q}) \cap \mathcal K_{\frak p}$ in $U_{\frak p}$.
2. $\Omega_{\frak p}$ is a neighborhood of $\mathcal K_{\frak p}$ in $U_{\frak p}$.
3. $\pi_{\frak q\frak p} : \Omega_{\frak q\frak p} \to \Omega_{\frak q}$ is a continuous map which is diffeomorphic to the restriction of a projection of a vector bundle to a neighborhood of $0$ section. (See Definition \[lem123000\].)
4. $\pi_{\frak q\frak p} \circ \varphi_{\frak p\frak q} = {\rm id}$ on a neighborhood of $\varphi^{-1}_{\frak p\frak q}(\mathcal K_{\frak p})
\cap \mathcal K_{\frak q}$.
5. $\pi_{\frak r\frak q} \circ \pi_{\frak q\frak p} = \pi_{\frak r\frak p}$ on a neighborhood of $\varphi^{-1}_{\frak q\frak r}(\varphi^{-1}_{\frak p\frak q}(\mathcal K_{\frak p}))
\cap \varphi^{-1}_{\frak q\frak r}(\mathcal K_{\frak q})
\cap \varphi^{-1}_{\frak p\frak r}(\mathcal K_{\frak p})
\cap \mathcal K_{\frak r}$.
The proof is by induction on the number ${\rm differ}(\frak p,\frak q)$, which we define now.
For $\frak p, \frak q \in \frak P$ with $\frak q < \frak p$ we put $${\rm differ}(\frak p,\frak q)
=
\max \{ n \mid \exists \frak r_1,\dots,\frak r_n \in \frak P,
\,\, \frak q = \frak r_1 < \dots < \frak r_n = \frak p \}.$$
We will prove the following statement by induction on $n$.
1. The conclusion of Lemma \[lem1226\] holds for $\frak p,\frak q$ with ${\rm differ}(\frak p,\frak q) \le n$.
Note we will shrink $\Omega_{\frak q\frak p}$, $\Omega_{\frak p}$ several times during the proof and restrict $\pi_{\frak q\frak p}$ to the shrinked domain. We however use the same symbol for the shrinked open sets and retraction on it.
If $n=0$ there is nothing to prove. Next assuming (\*) holds for all $\frak p, \, \frak q$ with ${\rm differ}(\frak p,\frak q) < n$, we will prove the case of pair $(\frak p,\frak q)$ with ${\rm differ}(\frak p,\frak q)=n$. Let $\frak p,\frak q \in \frak P$ with $\frak p > \frak q$ and ${\rm differ}(\frak p,\frak q) = n$.
For $\frak r \in \frak P$ with $\frak p > \frak r > \frak q$ we take a neighborhood $
\Omega_{\frak q\frak r\frak p}
$ of $$\varphi^{-1}_{\frak r\frak q}(\varphi^{-1}_{\frak p\frak r}(\mathcal K_{\frak p}))
\cap \varphi^{-1}_{\frak r\frak q}(\mathcal K_{\frak r})
\cap \varphi^{-1}_{\frak p\frak q}(\mathcal K_{\frak p})
\cap \mathcal K_{\frak q}.$$ The composition $\pi_{\frak q\frak r} \circ \pi_{\frak r\frak p}$ is defined there.
We may take $\Omega_{\frak q\frak r\frak p}$ such that $$\pi_{\frak q\frak r} \circ \pi_{\frak r\frak p}
=
\pi_{\frak q\frak r'} \circ \pi_{\frak r'\frak p}$$ holds on $\Omega_{\frak q\frak r\frak p} \cap \Omega_{\frak q\frak r'\frak p}$.
By Definition \[gcsystem\] (5), we may shrink $\Omega_{\frak q\frak r\frak p}$ so that $\Omega_{\frak q\frak r\frak p} \cap \Omega_{\frak q\frak r'\frak p}
\ne \emptyset$ implies either $\frak r < \frak r'$ or $\frak r' > \frak r$. We may assume $\frak r < \frak r'$ without loss of generality. Then, by the induction hypothesis, we have $$\pi_{\frak q\frak r} \circ \pi_{\frak r\frak p}
=
\pi_{\frak q\frak r'} \circ \pi_{\frak r'\frak r} \circ \pi_{\frak r\frak p}
=
\pi_{\frak q\frak r'} \circ \pi_{\frak r'\frak p}$$ on a neighborhood of $$\aligned
&\varphi^{-1}_{\frak r\frak q}(\varphi^{-1}_{\frak p\frak r}(\mathcal K_{\frak p}))
\cap \varphi^{-1}_{\frak r\frak q}(\mathcal K_{\frak r})
\cap \varphi^{-1}_{\frak p\frak q}(\mathcal K_{\frak p})
\cap \mathcal K_{\frak q}\\
&\cap
\varphi^{-1}_{\frak r'\frak q}(\varphi^{-1}_{\frak p\frak r'}(\mathcal K_{\frak p}))
\cap \varphi^{-1}_{\frak r'\frak q}(\mathcal K_{\frak r'})
\cap \varphi^{-1}_{\frak p\frak q}(\mathcal K_{\frak p})
\cap \mathcal K_{\frak q}.
\endaligned$$ because $${\rm differ}(\frak r,\frak q), \, {\rm differ}(\frak p,\frak r), \, {\rm differ}(\frak p,\frak r'),\,
{\rm differ}(\frak r',\frak r), \, {\rm differ}(\frak r',\frak q),\, {\rm differ}(\frak p,\frak r') < n.$$ The sublemma follows easily.
Thus we can define $\pi_{\frak q \frak p}$ to be $\pi_{\frak q\frak r} \circ \pi_{\frak r\frak p}$ on a neighborhood of $$\label{form1212}
\varphi^{-1}_{\frak p\frak q}(\mathcal K_{\frak p})
\cap \mathcal K_{\frak q}
\cap
\bigcup_{\frak r}
\varphi^{-1}_{\frak r\frak q}(\varphi^{-1}_{\frak p\frak r}(\mathcal K_{\frak p}))
\cap \varphi^{-1}_{\frak r\frak q}(\mathcal K_{\frak r})$$ which will then satisfy the required properties. Then by shrinking the neighborhood of (\[form1212\]) a bit, we can use Proposition \[prop2949\] to extend it to a neighborhood of $\varphi^{-1}_{\frak p\frak q}(\mathcal K_{\frak p})
\cap \mathcal K_{\frak q}$. Thus Lemma \[lem1226\] is proved by induction on $n$.
The proof of this lemma is easier than the proof of existence of system of normal bundles of Mather since in our case stratification is locally linear ordered by Definition \[gcsystem\] (5).
\[lem12343\] Let $(\{\Omega_{\frak q\frak p}\},\{\Omega_{\frak p}\},\{\pi_{\frak q\frak p}\})$ be as in Lemma \[lem1226\]. Then by shrinking $\Omega_{\frak q\frak p}$ and $\Omega_{\frak p}$ if necessary, there exists $\tilde\varphi_{\frak p\frak q}$ such that $(
\{\pi_{\frak q\frak p}\},\{\tilde\varphi_{\frak p\frak q}\},\{\Omega_{\frak q\frak p}\},\{\Omega_{\frak p}\})$ becomes a system of bundle extension data of $({\widetriangle{\mathcal U}},\mathcal K)$.
We can prove Lemma \[lem12343\] by the same induction as Lemma \[lem1226\].
The proof of Proposition \[prop12333\] is complete.
\[1219\] In the situation of Definition \[defn1230\], let $\frak s_1$ (resp. $\frak s_2$) be mutlisections of $\mathcal U_1$ (resp. $\mathcal U_2$) defined on a neighborhood of $K$ (resp. $\varphi_{21}(K)$).
We say that $\frak s_1$ and $\frak s_2$ are [*compatible with the bundle extension data* ]{} $(\pi_{12},\tilde\varphi_{21})$ if $\frak s_1$ and $\frak s_2$ are represented both by $\ell$ multisection and there exists a permutation $\sigma : \{1,\dots,\ell\} \to \{1,\dots,\ell\}$ such that $$\label{form1226}
(y,\frak s_{2,i}(y))
=
\tilde\varphi_{21}(y,\frak s_{1,\sigma(i)}(\pi_{12}(y))),$$ holds if $y$ is in a neighborhood of $\varphi_{21}(K)$ in $U_2$. Here $\sigma$ depends on $\pi_{12}(y)$.
The equality (\[form1226\]) implies that $\frak s_1$ and $\frak s_2$ are compatible with the embedding $\Phi_{21}$ automatically.
In the situation of Definition \[defn1232\], let $\frak s =\{\frak s_{\frak p}^{n}\}$ be a multivalued perturbation of $({\widetriangle{\mathcal U}},\mathcal K)$.
We say that $\frak s$ is compatible with the compatible system of bundle extension data $(
\{\pi_{\frak q\frak p}\},\{\tilde\varphi_{\frak p\frak q}\})$ if for each $\frak p > \frak q$ and $n \in \Z_{\ge 0}$, the pair $\frak s_{\frak p}^{n}$, $\frak s_{\frak q}^{n}$ is compatible with $(\pi_{\frak q\frak p},\tilde\varphi_{\frak p\frak q})$ in the sense of Definition \[1219\].
Embedding of Kuranishi charts and extension of multisections {#subsec:extmultisec}
------------------------------------------------------------
We now consider the following situation.
\[situ1219rev\] In the situation of Definition \[1219\], let $K \subset X$ be a compact subset contained in a relatively compact open subset $W \subset X$, $\{\frak s_{K,2}^{n}\}$ a mutivalued perturbations of $\mathcal U_2$ on a neighborhood of $K$, and $\{\frak s_1^{n}\}$ a mutivalued perturbations of $\mathcal U_1$ on a neighborhood of $\overline W$.
We assume that $\{\frak s_{K,2}^n\}$ and a restriction of $\{\frak s^{n}_1\}$ to a neighborhood of $K$ are compatible with $\Phi_{21}$ and also compatible with the restriction of the bundle extension data $(\pi_{12},\tilde\varphi_{21})$ to a neighborhood of $K$. $\blacksquare$
\[prop1220rev\] In Situation \[situ1219rev\] there exists a multivalued perturbation $\{\frak s_2^{n}\}$ of $\mathcal U_2$ on a neighborhood of $\overline W$ such that
1. The restriction of $\{\frak s_2^{n}\}$ to a neighborhood of $K$ coincides with $\{\frak s_{K,2}^n\}$ .
2. $\{\frak s_{K,2}^n\}$ , $\{\frak s_1^{n}\}$ are compatible with $\Phi_{21}$ on a neighborhood of $K$. Moreover they are compatible with the bundle extension data $(\pi_{12},\tilde\varphi_{21})$ in a neighborhood of $\overline W$.
3. If $\{\frak s_n^{K,2}\}$, $\{\frak s_1^{n}\}$ are transversal to $0$ for sufficiently large $n$ in addition then $\frak s_2^{n}$ can be chosen to be transversal to $0$ for sufficiently large $n$.
4. Suppose that $g : N \to M$ is a smooth map between manifolds and $f_1$ is strongly transversal to $g$ with respect to $\frak s_1^{n}$ for sufficiently large $n$. Then we may choose $\frak s_2^{n}$ such that $f_2$ is strongly transversal to $g$ with respect to $\frak s_2^{n}$ for sufficiently large $n$.
We define $\frak s_{2,i}(\pi_{12}(y))$ by $$\label{form1226rev}
(y,\frak s_{2,i}(y))
=
\tilde\varphi_{21}(y,\frak s_{1,i}(\pi_{12}(y))),$$ (Note this is Formula (\[form1226\]) except we take $\sigma$ to be the identity.) Statements (1) and (2) are obvious. We can shrink the neighborhood of $\overline W$ so that statements (3) and (4) hold.
The proof of Proposition \[prop1220rev\] is much simpler than that of Proposition \[prop1221\]. In fact in the proof of Proposition \[prop1221\] we use a bump function to glue two CF-perturbations. Here Property (2) is automatic without using bump function since we assumed $\{\frak s_{K,2}^n\}$ is compatible with bundle extension data.
We need a few more definitions:
\[bundleextembedding\] Let $\widehat{\Phi} = (\{\Phi_{\frak p}\},\frak i) : {\widetriangle{\mathcal U}} \to {\widetriangle{\mathcal U^+}}$ be an embedding of good coordinate systems $\mathcal K$, $\mathcal K^+$ be their support systems such that $\varphi_{\frak p}(\mathcal K_{\frak p}) \subset \mathcal K^+_{\frak i(\frak p)}$. Let $\Xi =
(\{\pi_{\frak q\frak p}\},\{\tilde\varphi_{\frak p\frak q}\},\{\Omega_{\frak q\frak p}\},\{\Omega_{\frak p}\})$ and $\Xi^+ = (
\{\pi^+_{\frak q\frak p}\},\{\tilde\varphi^+_{\frak p\frak q}\},\{\Omega^+_{\frak q\frak p}\},\{\Omega^+_{\frak p}\})$ be systems of bundle extension data of $({\widetriangle{\mathcal U}},\mathcal K)$ and $({\widetriangle{\mathcal U^+}},\mathcal K^+)$ respectively.
A bundle extension datum of $(\widehat{\Phi},\mathcal K,\Xi,\Xi^+)$ consists of the objects $\{(\tilde\varphi_{\frak p},\pi_{\frak p})\}$ that satisfy the following properties:
1. For each $\frak p \in \frak P$ we have a bundle extension datum $(\tilde\varphi_{\frak p},\pi_{\frak p})$ of the embedding $\Phi_{\frak p}$ on $\mathcal K_{\frak p}$.
2. If $\frak q < \frak p$, then
1. $\pi_{\frak q\frak p} \circ \pi_{\frak i(\frak p)}
=
\pi_{\frak q} \circ \pi^+_{\frak i(\frak q)\frak i(\frak p)}$ on a neighborhood of $(\varphi^+_{\frak i(\frak p)\frak i(\frak q)}\circ \varphi_{\frak q})(\varphi^{-1}_{\frak p\frak q}(\mathcal K_{\frak p})
\cap \mathcal K_{\frak q})$.
2. $\tilde\varphi_{\frak p} \circ \tilde\varphi_{\frak p\frak q} = \tilde\varphi^+_{\frak i(\frak p)\frak i(\frak q)} \circ \tilde\varphi_{\frak q}$ on a neighborhood of $\varphi^{-1}_{\frak p\frak q}(\mathcal K_{\frak p})
\cap \mathcal K_{\frak q}$.
Here $(\pi_{\frak q\frak p},\tilde\varphi_{\frak p\frak q})$ is a part of the bundle extension datum of $({\widetriangle{\mathcal U}},\mathcal K)$ and $(\pi^+_{\frak i(\frak q)\frak i(\frak p)}, \tilde\varphi^+_{\frak i(\frak p)\frak i(\frak q)})$ is a part of the bundle extension datum of $({\widetriangle{\mathcal U^+}},\mathcal K^+)$. (See Diagram \[diag33–\].)
\[def1325\] In the situation of Definition \[bundleextembedding\], let $\widetriangle{\frak s}$ (resp. $\widetriangle{\frak s^+}$) be a multivalued perturbation of $({\widetriangle{\mathcal U}},\mathcal K)$ (resp. $({\widetriangle{\mathcal U^+}},\mathcal K^+)$), such that $\widetriangle{\frak s}$ (resp. $\widetriangle{\frak s^+}$) is compatible with the bundle extension data $\Xi$ (resp. $\Xi^+$).
We say that $\widetriangle{\frak s}$, $\widetriangle{\frak s^+}$ are compatible with $\widehat{\Phi},
\{(\tilde\varphi_{\frak p},\pi_{\frak p})\}$ if for each $\frak p$, the pair $\frak s_{\frak p}$, $\frak s^+_{\frak i(p)}$ are compatible with $(\tilde\varphi_{\frak p},\pi_{\frak p})$ in the sense of Definition \[1219\].
We will work out the induction scheme of the proof of Proposition \[existperturbcont\] for a multivalued perturbation compatible with the system of bundle extension data as produced in Proposition \[prop12333\]. The detail is now in order.
We write the bundle extension data we use by $\Xi$. We use the notations in the proof of Proposition \[existperturbcont\]. We replace Proposition \[existontiiindc1t\] by the following.
\[existontiiindc1trev\] There exists a multivalued of perturbations $\widetriangle{{\frak s}^{\frak F}}$ of $({\widetriangle{\mathcal U}}(\frak F,\mathcal K^+),\mathcal K(\frak F))$ on $\frak T(\frak F,\mathcal K)$ with the following properties.
1. $\widetriangle{{\frak s}^{\frak F}}$ is compatible with the bundle extension data which is a restriction of $\Xi$ to $({\widetriangle{\mathcal U}}(\frak F,\mathcal K^+),\mathcal K(\frak F))$.
2. $\widetriangle{{\frak s}^{\frak F}}$ is transversal to $0$.
3. If $\widetriangle{f} : (X,Z;\widetriangle{\mathcal U}) \to M$ is weakly transversal to $g : N \to M$, then we may take $\widetriangle{{\frak s}^{\frak F}}$ so that the restriction of $\widetriangle f$ is strongly transversal to $g$ with respect to $\widetriangle{{\frak s}^{\frak F}}$.
To prove Proposition \[existontiiindc1trev\] we replace Lemma \[122222\] by the following. (We use the notation of Lemma \[122222\].)
\[122222rev\] For any ideal $\frak I \subseteq \frak F_-$, there exist an open neighborhood $U_{\frak p_0}(\frak I)$ of $\psi_{\frak p_0}^{-1}(\frak T(\frak I,\mathcal K)) \cap {\mathcal K}_{\frak p_0}$ in $U_{\frak p_0}$ and a multivalued perturbation $\widetriangle{\frak s^{\frak F}}(\frak I)$ of $({\widetriangle{\mathcal U}}(\frak F,\mathcal K^+),\mathcal K(\frak F))$ on $\frak T(\frak I,\mathcal K)$ with the following properties.
1. $\widetriangle{\frak s^{\frak F}}(\frak I)$ is compatible with the system of the bundle extension data obtained by restricting $\Xi$.
2. $\widetriangle{\frak s^{\frak F}}(\frak I)$, $\widetriangle{{\frak s}^{\frak F_-}}$ are compatible with the embedding $\widehat\Phi_{\frak F\frak F_-;\frak I}$ and its bundle extension data in the sense of Definition \[def1325\].
3. $\widetriangle{\frak s^{\frak F}}(\frak I)$ is transversal to $0$.
4. If $\widetriangle f : (X,\widetriangle{\mathcal U}) \to M$ is weakly transversal to $g : N\to M$ then we can choose $\widetriangle{\frak s^{\frak F}}(\frak I)$ such that the restriction of $\widetriangle f$ is strongly transversal to $g$ with respect to it.
We remark that the embedding $\widehat\Phi_{\frak F\frak F_-;\frak I}$ is obtained by restricting the coordinate change of ${\widetriangle{\mathcal U}}$. Therefore $\Xi$ induces a bundle extension datum of each embedding of the Kuranishi charts which makes up $\widehat\Phi_{\frak F\frak F_-;\frak I}$. The compatibility condition for $\widehat\Phi_{\frak F\frak F_-;\frak I}$, Definition \[bundleextembedding\] (2), is a consequence of the compatibility condition for $\Xi$, Definition \[defn1232\]. We thus obtain a bundle extension data of embedding $\widehat\Phi_{\frak F\frak F_-;\frak I}$ that we mentioned in Item (2).
The proof is the same as the proof of Proposition \[122222\]. Namely we replace Proposition \[prop1221\] by Proposition \[prop1220rev\].
We are now ready to complete the proof of Proposition \[existontiiindc1trev\]. In the proof of Proposition \[existontiiindc1t\] we replace Lemma \[122222\] by Lemma \[122222rev\]. We also replace Proposition \[prop123123\] by Proposition \[prop127777ver\]. This proves Proposition \[existontiiindc1trev\].
Theorem \[prop621\] follows from Proposition \[existontiiindc1trev\].
Relative version of the existence of multisection {#subsec:relexmulti}
-------------------------------------------------
We next prove a relative version of Theorem \[prop621\]. We need a relative version of Proposition \[prop12333\].
\[sotu1248\]
1. Let ${\widetriangle{\mathcal U^{\mathcal Z_{(1)}}}}$ be a good coordinate system of $X$ and $\mathcal Z_{(1)} \subset X$ is a compact subset. Let $\mathcal K^{\mathcal Z_{(1)}}$ be a support system of ${\widetriangle{\mathcal U^{\mathcal Z_{(1)}}}}$ in the sense of Definition \[defn738\] (1).
2. Let ${\widetriangle{\mathcal U}}$ be a good coordinate system of $\mathcal Z_{(2)} \subset X$ with $\mathcal Z_{(1)} \subset {\rm Int}\,\mathcal Z_{(2)}$ and suppose ${\widetriangle{\mathcal U^{\mathcal Z_{(1)}}}}$ strictly extends to ${\widetriangle{\mathcal U}}$ in the sense of Definition \[defn735f\] (4). Let $\mathcal K$ be a support system of ${\widetriangle{\mathcal U}}$ which extends $\mathcal K^{\mathcal Z_{(1)}}$ in the sense of Definition \[defn738\] (2).$\blacksquare$
In Situation \[sotu1248\] (1), we call $(
\{\pi^{\mathcal Z_{(1)}}_{\frak q\frak p}\},\{\tilde\varphi^{\mathcal Z_{(1)}}_{\frak p\frak q}\},\{\Omega^{\mathcal Z_{(1)}}_{\frak q\frak p}\},\{\Omega^{\mathcal Z_{(1)}}_{\frak p}\})$ a [*system of bundle extension data*]{} of $({\widetriangle{\mathcal U^{\mathcal Z_{(1)}}}},\mathcal K^{\mathcal Z_{(1)}})$ if they have the following properties.
1. $(\pi^{\mathcal Z_{(1)}}_{\frak q\frak p},\tilde\varphi^{\mathcal Z_{(1)}}_{\frak p\frak q},
\Omega^{\mathcal Z_{(1)}}_{\frak q\frak p},\Omega^{\mathcal Z_{(1)}}_{\frak p})$ is a bundle extension data of $(\Phi^{\mathcal Z_{(1)}}_{\frak p\frak q},\mathcal K^{\mathcal Z_{(1)}}_{\frak p\frak q})$, where $\mathcal K^{\mathcal Z}_{\frak p\frak q}
= (\varphi^{\mathcal Z_{(1)}}_{21})^{-1}(\mathcal K^{\mathcal Z_{(1)}}_{\frak p}) \cap \mathcal K^{\mathcal Z_{(1)}}_{\frak q}$.
2. $\pi^{\mathcal Z_{(1)}}_{\frak r\frak q} \circ \pi^{\mathcal Z_{(1)}}_{\frak q\frak p} = \pi^{\mathcal Z_{(1)}}_{\frak r\frak p}$ on a neighborhood of $(\varphi^{\mathcal Z_{(1)}}_{\frak q\frak r})^{-1}(\varphi^{\mathcal Z_{(1)}}_{\frak p\frak q})^{-1}(\mathcal K^{\mathcal Z_{(1)}}_{\frak p}))
\cap (\varphi^{\mathcal Z_{(1)}}_{\frak q\frak r})^{-1}(\mathcal K^{\mathcal Z_{(1)}}_{\frak q})
\cap (\varphi^{\mathcal Z_{(1)}}_{\frak p\frak r})^{-1}(\mathcal K^{\mathcal Z_{(1)}}_{\frak p})
\cap \mathcal K^{\mathcal Z_{(1)}}_{\frak r}$.
3. $\tilde\varphi^{\mathcal Z_{(1)}}_{\frak p\frak q} \circ \tilde\varphi^{\mathcal Z_{(1)}}_{\frak q\frak r} = \tilde\varphi^{\mathcal Z_{(1)}}_{\frak p\frak r}$ on a neighborhood of $(\varphi^{\mathcal Z_{(1)}}_{\frak q\frak r})^{-1}((\varphi^{\mathcal Z_{(1)}}_{\frak p\frak q})^{-1}(\mathcal K^{\mathcal Z_{(1)}}_{\frak p}))
\cap (\varphi^{\mathcal Z_{(1)}}_{\frak q\frak r})^{-1}(\mathcal K^{\mathcal Z_{(1)}}_{\frak q})
\cap (\varphi^{\mathcal Z_{(1)}}_{\frak p\frak r})^{-1}(\mathcal K^{\mathcal Z_{(1)}}_{\frak p})
\cap \mathcal K^{\mathcal Z_{(1)}}_{\frak r}$.
\[prop1250\] In Situation \[sotu1248\] (1)+(2), let $\Xi^{\mathcal Z_{(1)}}$ be a bundle extension data of $({\widetriangle{\mathcal U^{\mathcal Z_{(1)}}}},\mathcal K^{\mathcal Z_{(1)}})$. Then there exists a bundle extension data $\Xi$ of $({\widetriangle{\mathcal U}},\mathcal Z_{(2)};\mathcal K)$ which coincides with $\Xi^{\mathcal Z_{(1)}}$ in a neighborhood of $\mathcal Z_{(1)}$.
The proof is the same as the proof of Proposition \[existontiiindc1trev\].
\[1252situ\] In Situation \[sotu1248\] (1) + (2), suppose we are given systems of bundle extension data $\Xi^{\mathcal Z_{(1)}}$ and $\Xi$ respectively as in Proposition \[prop1250\].
Let $\widetriangle{\frak s^{\mathcal Z_{(1)}}}$ is a multivalued perturbation of $({\widetriangle{\mathcal U^{\mathcal Z_{(1)}}}},\mathcal K^{\mathcal Z_{(1)}})$. We assume that $\widetriangle{\frak s^{\mathcal Z_{(1)}}}$ is transversal to $0$. $\blacksquare$
\[existperturbmultires\]
In Situation \[1252situ\], there exists multivalued perturbation $\widetriangle{\frak s}$ of $({\widetriangle{\mathcal U}},\mathcal K)$ such that
1. $\widetriangle{\frak s}$ is transversal to $0$.
2. $\widetriangle{\frak s}$ coincides with $\frak s^{\mathcal Z_{(1)}}$ in a neighborhood of $\mathcal Z_{(1)}$.
3. $\widetriangle{\frak s}$ is compatible with $\Xi$.
4. If $\widetriangle{f} : (X,\mathcal Z_{(2)};\widetriangle{\mathcal U}) \to M$ is weakly transversal to $g : N \to M$ and the restriction of $\widetriangle{f}$ is strongly transversal to $g$ with respect to $\widetriangle{\frak s}$ then we may choose $\widetriangle{\frak s}$ such that $\widetriangle{f}$ is strongly transversal to $g$ with respect to $\widetriangle{\frak s}$.
We can modify the proof of Proposition \[existperturbcontrel\] in exactly the same way as we modified the proof of Proposition \[existperturbcont\] to the proof of Theorem \[prop621\]. It proves Proposition \[existperturbmultires\].
Remark on the number of branches of extension of multisection {#subsec:nastyreason}
-------------------------------------------------------------
We now explain a certain delicate point we encounter when we try to extend a multisection or multivalued perturbation. (This point does not appear while we extend a CF-perturbation.)
Note that during the proof of Proposition \[prop1221\] we used the same parameter space while we extend our CF-perturbation on a subset of $U_1$ to its neighborhood in $U_2$. To extend a multivalued perturbation it is important that we do not change the number of branches. This point is mentioned in [@FO page 955 line 20-24] and [@foootech Remark 6.4]. We elaborate it below.
Let $U_1 \subset U_2$ be an embedded submanifold. Suppose $U_1$ is expressed as the union $U_1 = U_{1,1} \cup U_{1,2}$ of two open subsets $U_{1,1}, \, U_{1,2}$ and we are given multisection $\frak s$ on $U_1$. We also assume that $\frak s\vert_{U_{1,j}}$ is extended to its neighborhood in $U_2$ and we denote this extension by $\tilde{\frak s}_{j}$. (We assume that the number of branches of $\tilde{\frak s}_{j}$ is the same as one of $\tilde{\frak s}$ around each point of $U_{1,j}$.) We try to glue $\tilde{\frak s}_{1}$ and $\tilde{\frak s}_{2}$ to obtain an extension of $\frak s$ to a neighborhood of $U_1$ in $U_2$.
Let $p \in U_{1,1} \cap U_{1,2}$. We represent $\frak s$ in a neighborhood of $p$ as $(s_1,\dots,s_{\ell})$, where $s_i$ are branches of $\frak s$ in a neighborhood of $p$. We might say that extension $\tilde{\frak s}_{j}$ gives $(s_{j,1},\dots,s_{j,\ell})$ and we might try to glue them as $$\label{1226form}
s_i(p) =
\chi(\pi(p)) s_{1,i}(p) + (1-\chi(\pi(p))) s_{2,i}(p)$$ where $\chi$ is a function on $U_1$ such that $\chi$ has a support in $U_{1,1}$ and $1-\chi$ has a support in $U_{1,2}$. $\pi$ is the projection of the tubular neighborhood of $U_1$ in $U_2$.
However, we can not define $s_i$ by (\[1226form\]), because of the following problem. As we explained in Remark \[rem614\], the way we take a representative $(s_1,\dots,s_{\ell})$ of $\frak s$ is not unique. (The uniqueness modulo permutation also fails.) Therefore, when we extend $\frak s$ to $\tilde{\frak s}_{j}$, we may take a representative different from $(s_1,\dots,s_{\ell})$ in a neighborhood of $p$. So to add $s_{1,i}(p)$ and $s_{2,i}(p)$ may not make sense.
Note the representative $(s_{j,1}(p),\dots,s_{j,\ell}(p))$ does make sense at $p$ (modulo permutation). Namely it makes sense point-wise.
We remark that Definition \[1219\] works point-wise. Namely the right hand side of (\[form1226\]) makes sense point-wise and we do not need to take the representative of $\frak s_i^1$ [*locally*]{} but we only need to take its representative [*at*]{} $y = \pi(p)$. This is because we fix a bundle extension datum which is well-defined globally (and is compatible with coordinate changes etc. in the sense of Definition \[defn1232\]). In fact this is [*the*]{} reason why we use the system of bundle extension data (Definition \[defn1232\]) to extend multivalued perturbations.
We note that, in Formula (\[defext3030\]), we do not use system of bundle extension data to extend [*CF-perturbations*]{}.
The definition of CF-perturbation is similar to that of multivalued perturbation where we use an open set of vector space in the former and a finite set in the latter. The definition of equivalence of CF-perturbation in Definition \[conmultiequiv11\] is different from that of multivalued perturbation in Definition \[defn62\]. In fact the former is a local condition and the latter is a point-wise condition.
We can slightly modify the definition of multivalued perturbation by imitating the way taken in the case of CF-perturbation. Then we do not need to use system of bundle extension data.
In this document we use system of bundle extension data since we want the definition of the multisection to be exactly the same as one in [@FO]. We want to do so since the definition of [@FO] had been used by various people including ourselves.
Zero and one dimensional cases via multisection {#sec:onezerodim}
===============================================
In Sections \[sec:contfamily\] - \[sec:composition\], we discussed smooth correspondence and defined virtual fundamental chain based on de Rham theory and CF-perturbations. In this section, we discuss another method based on multivalued perturbation. Here we restrict ourselves to the case when the dimension of K-spaces of our interest is $1$, $0$ or negative, and define virtual fundamental chain over $\Q$ in the $0$ dimensional case. In spite of this restriction, the argument of this section is enough for the purpose, for example, to prove all the results stated in [@FO]. We recall that in [@FO] we originally used a triangulation of the zero set of multisection to define a virtual fundamental chain. In this section we present a different way from [@FO]. Namely, we use Morse theory in place of triangulation. This change will make the relevant argument simpler and shorter for this restricted case. We will explain the thorough detail about the triangulation of the zero set of multisection elsewhere.
Statements of the results {#subsec:zero1state}
-------------------------
We start with the following :
\[lem1311111\] Let ${\widetriangle{\mathcal U}}$ be a good coordinate system (which may or may not have boundary or corner), $\mathcal K$ its support system, and $\widetriangle{\frak s} = \{\frak s^{n}_{\frak p} \mid \frak p \in \frak P\}$ a multivalued perturbation of $({\widetriangle{\mathcal U}},\mathcal K)$ (Definition \[defn612\].) We assume that $\widetriangle{\frak s}$ is transversal to $0$. Then there exists a natural number $n_0$ with the following properties.
1. If the dimension of $(X,{\widetriangle{\mathcal U}})$ is negative, then $
(\frak s_{\frak p}^{n})^{-1}(0) \cap \vert \mathcal K\vert= \emptyset
$ for $n \ge n_0$.
2. If the dimension of $(X,{\widetriangle{\mathcal U}})$ is $0$, then $
(\frak s_{\frak p}^{n})^{-1}(0) \cap
\vert\partial{\widetriangle{\mathcal U}}\vert\cap \vert \mathcal K\vert
= \emptyset
$ for $n \ge n_0$. Moreover there exists a neighborhood $\frak U(X)$ of $X$ in $\vert{\widetriangle{\mathcal U}}\vert \cap \vert \mathcal K\vert$ such that the intersection $(\frak s_{\frak p}^{n})^{-1}(0) \cap \frak U(X)$ is a finite set for any $n \ge n_0$.
\(1) is obvious. Using the fact that the dimension of the boundary of $(X,{\widetriangle{\mathcal U}})$ is negative, we have $
(\frak s_{\frak p}^{\epsilon})^{-1}(0) \cap \vert\partial{\widetriangle{\mathcal U}}\vert
\cap \vert \mathcal K\vert
= \emptyset
$ in (2) also. The finiteness of the order of the set $(\frak s_{\frak p}^{\epsilon})^{-1}(0) \cap \frak U(X)
\cap \vert \mathcal K\vert$ is a consequence of its compactness, Corollary \[cor69\].
We now consider:
\[Situation132\] Let ${\widetriangle{\mathcal U}}$ be a good coordinate system (which may or may not have boundary or corner), and assume a support system $\mathcal K$ thereof is given. Let $\widetriangle{\frak s} = \{\frak s^{\epsilon}_{\frak p} \mid \frak p \in \frak P\}$ be a multivalued perturbation of $({\widetriangle{\mathcal U}},\mathcal K)$. We assume:
1. $(X,{\widetriangle{\mathcal U}})$ is oriented,
2. $\widetriangle{\frak s}$ is transversal to $0$.
Consider another support system $\mathcal K'$ with $\mathcal K' < \mathcal K$. We take a neighborhood $\frak U(X)$ of $X$ as in Corollary \[cor69\] for $\mathcal K_2 = \mathcal K'$ and $\mathcal K_3 = \mathcal K$. $\blacksquare$
\[gcswithperturb\] In Situation \[Situation132\] we call $(\widetriangle{\mathcal U}, \widetriangle{\frak s})$ a [*good coordinate system with multivalued perturbation*]{} of $X$.
\[defn13223\] In Situation \[Situation132\], we assume $\dim (X,{\widetriangle{\mathcal U}}) = 0$. We consider $p \in U_{\frak p} \cap \frak U(X) \cap \vert \mathcal K\vert$ such that $\frak s_{\frak p}^{n}(p) = 0$. (This means that there is a branch of $\frak s_{\frak p}^{n}$ that vanishes at $p$.) Let $\frak V = (V,\Gamma,E,\psi,\hat\psi)$ be an orbifold chart of $(U_{\frak p},E_{\frak p})$ at $p$. We take a representative $(\frak s_{\frak p,1}^{n},\dots,
\frak s_{\frak p,\ell}^{n})$ of $\frak s_{\frak p}^{n}$ on $\frak V$. Let $\tilde p \in V$ such that $[\tilde p] = p$.
1. For $i=1,\dots,\ell$ we put: $$\epsilon_{p,i} =
\begin{cases}
0 & \text{if $\frak s_{\frak p,i}^{n}(\tilde p) \ne 0$.} \\
+1 & \text{if $\frak s_{\frak p,i}^{n}(\tilde p) = 0$
and (\ref{iso1311111}) below is orientation preserving.}\\
-1 & \text{if $\frak s_{\frak p,i}^{n}(\tilde p) = 0$
and (\ref{iso1311111}) below is orientation reversing.}
\end{cases}$$ In the current case of virtual dimension 0, the transversality hypothesis implies that the derivative $$\label{iso1311111}
D_{\tilde p} \frak s_{\frak p,i}^{n} :
T_{\tilde p}V \to E_{\tilde p}$$ becomes an isomorphism at every point $\tilde p$ satisfying $\frak s_{\frak p,i}^{n}(\tilde p) = 0$.
2. The [*multiplicity*]{} $m_p$ of $(\frak s_{\frak p}^{n})^{-1}(0)$ at $p$ is a rational number and is defined by $$m_p = \frac{1}{\ell\#\Gamma}\sum_{i=1}^{\ell} \epsilon_{p,i}.$$
\[lem134\]
1. The multiplicity $m_p$ in Definition \[defn13223\] is independent of the choice of representative $(\frak s_{\frak p,1}^{n},\dots,
\frak s_{\frak p,\ell}^{n})$.
2. If $q \in U_{\frak p\frak q}$ and $p = \varphi_{\frak p\frak q}(q)$, then the multiplicity at $p$ is equal to the multiplicity at $q$.
This is immediate from the definition.
\[defn1355\] In Situation \[Situation132\] we assume $\dim (X,{\widetriangle{\mathcal U}}) = 0$. We define the [*virtual fundamental chain*]{} $[(X,{\widetriangle{\mathcal U}},\mathcal K',\widetriangle{\frak s^{n}})]$ of $(X,{\widetriangle{\mathcal U}},\mathcal K',\widetriangle{\frak s^{n}})$ by $$\label{defvfcdim0}
[(X,{\widetriangle{\mathcal U}},\mathcal K',\widetriangle{\frak s^{n}})]
=
\sum_{p \in \frak U(X) \cap \vert\mathcal K'\vert
\cap \bigcup_{\frak p \in \frak P}(\frak s_{\frak p}^{n})^{-1}(0)} m_p.$$ This is a rational number. Here the sum in the right hand side of (\[defvfcdim0\]) is defined as follows. Let us consider the [*disjoint*]{} union $$\bigcup_{\frak p \in \frak P}
(\frak U(X) \cap \mathcal K'_{\frak p}\cap (\frak s_{\frak p}^{n})^{-1}(0))
\times \{\frak p\}.$$ We define a relation $\sim$ on it by $(p,\frak p) \sim (q,\frak q)$ if $\frak p \le \frak q$, $q = \varphi_{\frak q\frak p}(p)$ or $\frak q \le \frak p$, $p = \varphi_{\frak p\frak q}(q)$. This is an equivalence relation by Definition \[gcsystem\] (7). The set of the equivalence classes is denoted by $\frak U(X) \cap \vert\mathcal K'\vert\cap \bigcup_{\frak p \in \frak P}(\frak s_{\frak p}^{n})^{-1}(0)$. By Lemma \[lem134\] (2) the multiplicity $m_p$ is a well-defined function on this set.
We note that in case $(X,{\widetriangle{\mathcal U}})$ has a boundary, the number $[(X,{\widetriangle{\mathcal U}},\mathcal K',\widetriangle{\frak s^{n}})]$ depends on the choice of the multivalued perturbation $\widetriangle{\frak s} =\{\frak s^{n}_{\frak p} \}$. It also depends on $n$.
The next result is a multivalued perturbation version of Proposition \[indepofukuracont\].
\[prop14777\] Let $\mathcal K^1,\mathcal K^2, \mathcal K^3$ be support systems with $\mathcal K^1 < \mathcal K^2< \mathcal K^3 = \mathcal K$. Let $X, \widetriangle{\mathcal U}, \widetriangle{\frak s} = \{\widetriangle{\frak s^{n}}\}$ and $\mathcal K$ be as in Definition \[defn1355\].
1. The number $[(X,{\widetriangle{\mathcal U}},\mathcal K',\widetriangle{\frak s^{n}})]$ in (\[defvfcdim0\]) is independent of $\frak U(X)$ for all sufficiently large $n$. Here $\mathcal K'$ is either $\mathcal K_1$ or $\mathcal K_2$.
2. We have $$[(X,{\widetriangle{\mathcal U}},\mathcal K_1,\widetriangle{\frak s^{n}})]
=
[(X,{\widetriangle{\mathcal U}},\mathcal K_2,\widetriangle{\frak s^{n}})]$$ for all sufficiently large $n$.
\(1) Let $\frak U'(X)$ be an alternative choice. By Corollary \[cor69\] we have $$\frak U(X) \cap \vert\mathcal K'\vert
\cap \bigcup_{\frak p \in \frak P}(\frak s_{\frak p}^{n})^{-1}(0)
=
\frak U'(X) \cap \vert\mathcal K'\vert
\cap \bigcup_{\frak p \in \frak P}(\frak s_{\frak p}^{n})^{-1}(0).$$ Independence of the multiplicity $m_p$ can be proved in the same way as the argument in Definition \[defn1355\]. (1) follows.
\(2) By Proposition \[lem715\] we have $$\frak U(X) \cap \vert\mathcal K_1\vert
\cap \bigcup_{\frak p \in \frak P}(\frak s_{\frak p}^{n})^{-1}(0)
=
\frak U(X) \cap \vert\mathcal K_2\vert
\cap \bigcup_{\frak p \in \frak P}(\frak s_{\frak p}^{n})^{-1}(0).$$ Independence of the multiplicity $m_p$ can be proved in the same way as the argument in Definition \[defn1355\]. (2) follows.
Since $[(X,{\widetriangle{\mathcal U}},\mathcal K',\widetriangle{\frak s^{n}})]$ is independent of $\mathcal K'$ by Proposition \[prop14777\], we will write it as $[(X,{\widetriangle{\mathcal U}},\widetriangle{\frak s^{n}})]$ hereafter.
The main result of this section is the following.
\[prop13777\] Let $(X,{\widetriangle{\mathcal U}},\widetriangle{\frak s})$ be as in Situation \[Situation132\]. We assume $\dim (X,{\widetriangle{\mathcal U}}) = 1$. We consider its normalized boundary $\partial(X,{\widetriangle{\mathcal U}}) =
(\partial X,\partial{\widetriangle{\mathcal U}})$ where $\widetriangle{\frak s}$ induces a multivalued perturbation $\widetriangle{\frak s_{\partial}}$ thereof and $(\partial X,\partial{\widetriangle{\mathcal U}},\widetriangle{\frak s_{\partial}^{n}})$ is as in Situation \[Situation132\] with $\dim (\partial X,\partial{\widetriangle{\mathcal U}}) = 0$. Then the following formula holds. $$[(\partial X,\partial{\widetriangle{\mathcal U}},\widetriangle{\frak s_{\partial}^{n}})]
=0.$$
\[rem1820moved\] Here we remark a slightly delicate point about the definition of transversality of the multivalued perturbation $\widetriangle{\frak s} = \{\widetriangle{\frak s^{n}}\}$.
We remark that for a CF-perturbation we studied a family parameterized by $\epsilon$ which is a positive real number close to $0$. For a multivalued perturbation we considered a sequence of multisections $\widetriangle{\frak s^{n}}$, where $n$ is an integer. In other words in the case of CF-perturbation the parameter space is uncountable while in the case of multivalued perturbation the parameter space is countable.
In many parts of the story of multivalued perturbation we can consider $\widetriangle{\frak s} = \{\widetriangle{\frak s^{\epsilon}}\}$ in place of $\{\widetriangle{\frak s^{n}}\}$. However we need a countable family of objects to apply Baire’s category theorem (See the end of the proof of Lemma \[lem1313\]).
When we discuss transversality of $\widetriangle{\frak s} = \{\widetriangle{\frak s^{\epsilon}}\}$ we may consider one of the following two versions:
1. Fix sufficiently small $\epsilon > 0$ and define the transversality of $\widetriangle{\frak s^{\epsilon}}$ as a multisection.
2. We consider the whole family $\widetriangle{\frak s}$ as a multisection on $(X,{\widetriangle{\mathcal U}}) \times (0,\epsilon_0)$.
Sard’s theorem implies that if $\widetriangle{\frak s} = \{\widetriangle{\frak s^{\epsilon}}\}$ is transversal to $0$ in the sense of (2) then [*for generic $\epsilon$*]{} the multisection $\widetriangle{\frak s^{\epsilon}}$ is transversal to $0$ in the sense of (1).
The transversality we need to define virtual fundamental chain is one in the sense of (1).
The point we elaborate in Remark \[rem1820moved\] is related to the $n$-dependence of the virtual fundamental chain $[(X,{\widetriangle{\mathcal U}},
\widetriangle{\frak s^{n}})]$ of a $0$ dimensional good coordinate system as follows.
This phenomenon occurs only in case when $(X,{\widetriangle{\mathcal U}})$ has a boundary.
Suppose $\widetriangle{\frak s}$ is transversal to $0$ in the sense of (2) above. Then for sufficiently small generic $\epsilon$, $\widetriangle{\frak s^{\epsilon}}$ is transversal to $0$ at $\epsilon$. So we can define the rational number $[(X,{\widetriangle{\mathcal U}},\widetriangle{\frak s^{\epsilon}})]$.
On the other hand, there is a discrete subset $S \subset (0,\epsilon_0)$ such that if $\epsilon_1 \in S$ then $\widetriangle{\frak s^{\epsilon_1}}$ may not be transversal to $0$. In particular the zero set of $\widetriangle{\frak s^{\epsilon_1}}$ may intersect with the boundary $\partial(X,{\widetriangle{\mathcal U}})$. Therefore $$\lim_{\epsilon \uparrow \epsilon_1} [(X,{\widetriangle{\mathcal U}},
\widetriangle{\frak s^{\epsilon}})]
\ne
\lim_{\epsilon \downarrow \epsilon_1}
[(X,{\widetriangle{\mathcal U}},\widetriangle{\frak s^{\epsilon}})],$$ in general. In other words, wall crossing may occur at $\epsilon_1$. By this reason the virtual fundamental chain $ [(X,{\widetriangle{\mathcal U}},\widetriangle{\frak s^{\epsilon}})]$ depends on $\epsilon$.
We remark that in the case when we use CF-perturbations a similar phenomenon happens and the integration along the fiber depends on $\epsilon$. However this phenomenon appears there in a slightly different way. Suppose we consider the zero dimensional case. Then the integration along the fiber (of the function $1$ and the map $X \to $ point) $
[(X,{\widetriangle{\mathcal U}},\widetriangle{\frak S^{\epsilon}})]
$ defines a real number for each $\epsilon$. This is defined for all sufficiently small $\epsilon$ but is not a constant function of $\epsilon$.
The number we obtain using the CF-perturbation is a real number and so changes continuously as $\epsilon$ varies. (It is easy to see from definition that $[(X,{\widetriangle{\mathcal U}},
\widetriangle{\frak s^{\epsilon}})]$ is a smooth function of $\epsilon$.) The number we obtain using multivalued perturbation is a rational number. So it jumps. In other words it is impossible to obtain virtual fundamental chain $[(X,{\widetriangle{\mathcal U}},\widetriangle{\frak s^{\epsilon}})],$ for all $\epsilon$ in case $\widetriangle{\frak s}$ is a multivalued perturbations.
A simple Morse theory on K-space {#subsec:kuramorse}
--------------------------------
We can prove this theorem by taking an appropriate perturbation $\widetriangle{\frak s^{n}}$ so that its zero set has a triangulation. This is the proof given in [@FO Theorem 6.2]. (Theorem \[prop13777\] is a special case of [@FO Theorem 6.2] where $Y$ is a point.) Here we give an alternative proof without using triangulation.
Let $(\mathcal K',\mathcal K)$ be a support pair of ${\widetriangle{\mathcal U}}$. Note $\widetriangle{\frak s}$ is a multivalued perturbation of $({\widetriangle{\mathcal U}},\mathcal K)$. We consider a strongly smooth function $f : U(\vert\mathcal K\vert) \to [0,\infty)$ of a neighborhood $U(\vert\mathcal K\vert)$ of $\vert\mathcal K\vert$ in $\vert{\widetriangle{\mathcal U}}\vert$ such that $$\label{133formula}
f^{-1}(0) = \vert\partial{\widetriangle{\mathcal U}}\vert
\cap U(\vert\mathcal K\vert).$$
We say that $f$ is [*normally positive at the boundary*]{} if the following holds.
Let $p \in \partial U_{\frak p} \cap U(\vert\mathcal K\vert)$ and we identify its neighborhood with $(W \times [0,1)^k)/\Gamma$ where $p$ corresponds to $(p_0,(0,\dots,0))$ and $p_0 \in W$ is an interior point. Let $\vec v \in T_{(p_0,0,\dots,0)}(W \times [0,1)^k)$ such that $\vec v = (\vec v_0,v_1,\dots,v_k)$ with $v_i > 0$ for all $i=1,\dots,k$. Then $$\label{normallypositive}
\vec v(f) > 0.$$
The above definition can be rephrased as follows. Consider the conormal space $$N^*_{(p_0,0,\dots,0)} (W \times \{0\}) \subset T^*_{(p_0,0,\dots,0)} (W \times [0,1]^k).$$ Then $f$ is normally positive at $(p_0,0,\dots,0)$ if and only if $df(p_0,0,\dots,0)$ is contained in the cone $$\aligned
C_+(N^*_{(p_0,0,\dots,0)} (W \times \{0\})) & := \{\alpha \in N^*_{(p_0,0,\dots,0)} (W \times \{0\})
\mid \alpha(\vec v) > 0, \, \vec v = (\vec v_0,v_1,\dots,v_k) \\
& \text{ with },
v_i > 0 \text{ for all }\, i=1,\dots,k\}.
\endaligned$$
\[139lem\] There exists a strongly smooth function $f$ as above which is normally positive at the boundary and satisfies (\[133formula\]).
We take $\mathcal K^+$ such that $(\mathcal K,\mathcal K^+)$ is a support pair. Let $x \in \vert\mathcal K\vert$. We then take a maximal $\frak p$ such that $x \in \mathcal K_{\frak p}$. We take a sufficiently small neighborhood $\Omega_x$ of $x$ in $\vert{\widetriangle{\mathcal U}}\vert$ such that $\Omega_x \cap \mathcal K_{\frak q} \ne \emptyset$ implies $\frak q \le \frak p$. Then we may slightly shrink $\Omega_x$ so that $\Omega_x \cap \vert\mathcal K\vert$ is contained in $\Omega_x \cap \mathcal K^+_{\frak p}$. Note $\Omega_x \cap \mathcal K^+_{\frak p}$ is an orbifold with corner. Therefore a neighborhood of $x$ in it is identified with a point $(x_0,(0,\dots,0))$ in $(V_x \times [0,1)^{k_x})/\Gamma_x$. Here $x \in \overset{\circ}{S}_{k_x}(\mathcal K^+_{\frak p})$. We can choose this coordinate so that the $\Gamma_x$ action on $V_x \times [0,1)^{k_x}$ is given in the form as $$(y,(t_1,\dots,t_{k_x})) \mapsto
(\gamma(y,(t_1,\dots,t_{k_x})),(t_{\sigma(1)},\dots,t_{\sigma(k_x)})),$$ that is, the action on $[0,1)^{k_x}$ factor is by permuting its components. We define a function $f_x$ on $\Omega_x \cap \mathcal K^+_{\frak p}$ by $$\label{cornertrans}
f_x(y,(t_1,\dots,t_{k_x}))
=
\begin{cases}
t_1 t_2 \cdots t_{k_x} &\text{if $k_x >0$,} \\
1 & \text{if $k_x =0$.}
\end{cases}$$ We may regard it as a strongly smooth function on a neighborhood of $W_x = \Omega_x \cap \vert\mathcal K\vert$. We may assume that if $x \in \overset{\circ}{S}_{k_x}(\mathcal K^+_{\frak p})$ then $W_x \cap \overset{\circ}{S}_{k_x+1}(\mathcal K_{\frak p})
= \emptyset$ for any $\frak p$. Let $W_{0,x}$ be a relatively compact neighborhood of $x$ in $W_x$.
We take finitely many points $x_i$, $i=1,\dots,N$ of $\vert\mathcal K\vert$ such that $$\bigcup_{i=1}^N W_{0,x_i}
\supseteq \vert\mathcal K\vert.$$ Then there exists a strongly smooth functions $\chi_i$ on a neighborhood of $\vert\mathcal K\vert$ to $[0,1]$ such that
1. The support of $\chi_i$ is in $W_{0,x_i}$.
2. $
\sum_{i=1}^N \chi_i \equiv 1.
$
We can prove the existence of such $\chi_i$ in the same way as in the proof of Proposition \[pounitexi\] by using Lemma \[bumpfunctionlemma\]. We put $$f = \sum_{i=1}^N \chi_i f_{x_i}.$$ By (\[cornertrans\]) this function has the required properties.
Let $\mathcal K''$ be a support system with $\mathcal K' < \mathcal K'' < \mathcal K$.
Let $f$ be a strongly smooth function defined on a neighborhood of $\vert\mathcal K\vert$. We say that a point $p \in \vert\mathcal K\vert$ is a [*critical point*]{} of $f$ if there exists $\frak p$ such that $p \in \mathcal K_{\frak p}$ and $p$ is a critical point of the restriction of $f$ to a neighborhood of $p$ in $U_{\frak p}$. We denote by ${\rm Crit}(f)$ the set of all critical points of $f$.
Let $f$ be a strongly smooth function defined on a neighborhood of $\vert\mathcal K\vert$. We say that $f$ is a [*Morse function*]{} if the restriction of $f$ to each $U_{\frak p}$ is a Morse function. (Namely its Hessian at all the critical points are nondegenerate.)
Suppose ${f} : \vert\mathcal K\vert \to \R$ is a strongly smooth function and $\mathcal K'$ is a support system with $\mathcal K' < \mathcal K$, then the restriction ${f}\vert_{\vert\mathcal K'\vert} : \vert\mathcal K'\vert \to \R$ is strongly smooth. If ${f}$ is Morse so is ${f}\vert_{\vert\mathcal K'\vert}$.
We use the following:
\[lem131111\] There exist finitely many orbifold charts $(V_i,\Gamma_i,\psi_i)$ of $U_{\frak p_i}$ ($\frak p_i \in \frak P$) and smooth embeddings $
h_i : [0,1] \to V_{i}
$ such that $$\frak U(X) \cap
\bigcup_{\frak p\in \frak P}
(\frak s_{\frak p}^{n})^{-1}(0) \cap \mathcal K''_{\frak p}
\subset
\bigcup_{i=1}^N [(\psi_i\circ h_i)((0,1))].$$ Here $\frak U(X)$ is as in Corollary \[cor69\].
Since all the branches of $\frak s^n_{\frak p}$ are transversal to $0$ and the (virtual) dimension is $1$, locally the zero set of $\frak s^n_{\frak p}$ is a one dimensional manifold. The lemma then follows from compactness of the left hand side. (Corollary \[cor69\].)
\[morseufnction\] Suppose that $\dim U_{\frak p}\ge 2$ for each $\frak p$. Then there exists a strongly smooth function $f$ on a neighborhood $\frak U(X)$ of $X$ in $\vert\mathcal K\vert$ such that
1. $f$ is normally positive at the boundary.
2. $f$ satisfies (\[133formula\]).
3. $f$ is a Morse function.
4. The composition $f\circ \psi_i \circ h_i : (0,1) \to \R$ is a Morse function for each $
h_i : [0,1] \to V_{i}
$ in Lemma \[lem131111\]
The proof of the lemma is a minor modification of a standard argument. We give a proof below for completeness’ sake. We will use certain results concerning denseness of the set of Morse functions, which we will prove in Subsection \[subsec:moresedense\].
We make a choice of Fréchet space we work with. Take a support system $\mathcal K^+$ such that $(\mathcal K,\mathcal K^+)$ is a support pair and define the set $$C^{\infty}(\mathcal K^+)
=
\{
(f_{\frak p})_{\frak p \in \frak P}
\in \prod_{\frak p \in \frak P}C^{\infty}{(\mathcal K^+_{\frak p})}
\mid
f_{\frak p} \circ \varphi_{\frak p\frak q}
=f_{\frak q},
\text{on $\varphi_{\frak p\frak q}^{-1}(\mathcal K^+_{\frak p})
\cap \mathcal K^+_{\frak q}$}
\}.$$ Here $C^{\infty}(\mathcal K^+_{\frak p})$ is the space of $C^{\infty}$ functions of an orbifold $U_{\frak p}$ defined on its compact subset $\mathcal K^+_{\frak p}$ and so is a Fréchet space with respect to the $C^{\infty}$ topology. Then the set $C^{\infty}(\mathcal K^+)$ is a closed subspace of a finite product of the Fréchet spaces and so is a Fréchet space.
Let $f_0 \in C^{\infty}(\mathcal K^+_{\frak p})$ be a function satisfying (1)(2) above. (Existence of such $f_0$ follows from Lemma \[139lem\].) We can take another neighborhood $\frak U_0$ of $\partial X$ in $\vert\mathcal K\vert$ such that $\text{\rm Crit}f_0 \cap \frak U_0 = \emptyset$. We take a neighborhood $\frak U_1$ of $\partial X$ in $\vert\mathcal K\vert$ such that $\overline{\frak U_1} \subset \frak U_0$. Let $C^{\infty}(\mathcal K^+)_0$ be the set of all $f \in C^{\infty}(\mathcal K^+)$ that vanish on $\overline{\frak U_1}$. Then $C^{\infty}(\mathcal K^+)_0$ itself is a Fréchet space.
\[lem1313\] For each $p \in X \setminus \frak U_0$ there exists a compact neighborhood $\frak U_p$ of $p$ such that the set of $g\in C^{\infty}(\mathcal K^+)_0$ satisfying conditions (1)(2) below is a dense subset of $C^{\infty}(\mathcal K^+)_0$.
1. $f_0+g$ is a Morse function on $ \frak U_p$.
2. The composition $(f_0+g)\circ \psi_i \circ h_i : (0,1) \to \R$ is a Morse function on $h_i^{-1}(\psi_i^{-1}(\frak U_p))$ for each $
h_i : [0,1] \to V_{i}
$ in Lemma \[lem131111\]
Let $\frak p \in \frak P$ be the maximal element of $\{\frak p \mid p \in \vert\mathcal K_{\frak p}\vert\}$. (Maximal element exists because of Definition \[gcsystem\] (5).) Let $\Omega_p$ be a neighborhood of $p$ in ${\rm Int}\,\mathcal K^+_{\frak p}$. We may choose $\Omega_p$ sufficiently small so that the following $(*)$ holds.
1. If $\mathcal K_{\frak q} \cap \Omega_p \ne \emptyset$, then $\frak q \le \frak p$ and $K^+_{\frak q} \cap \Omega_p$ is an open subset of $U_{\frak q}$.
\[sublem1314\] For each $\frak q \le \frak p$ the set of $g\in C^{\infty}(\mathcal K^+)_0$ satisfying conditions (a)(b) below is an open dense subset of $C^{\infty}(\mathcal K^+)_0$ for each $n \ge n_0$.
1. The restriction of $
f_0+g
$ to $\mathcal K_{\frak q} \cap \Omega_p$ is a Morse function on $\mathcal K_{\frak q} \subset U_{\frak q}$.
2. The composition $(f_0+g)\circ h_i : (0,1) \to \R$ is a Morse function on $h_i^{-1}(\psi_i^{-1}(\Omega_p))$ for each $
h_i : [0,1] \to V_{i}
$ in Lemma \[lem131111\]
This is an immediate consequence of Propositions \[prop1427\] and \[lem1430\], which we will prove in Subsection \[subsec:moresedense\].
Lemma \[lem1313\] follows from Sublemma \[sublem1314\] and Baire’s category theorem.
We remark that we have used the fact that we consider only countably many $n$ here.
Now Proposition \[morseufnction\] easily follows from Lemma \[lem1313\].
Now we go back to the proof of Theorem \[prop13777\]. We fix $n \ge n_0$. (We choose $n_0$ so that the compactness in Corollary \[cor69\] holds.) We take a strongly smooth function $f$ that satisfies (1)-(4) of Proposition \[morseufnction\]. We take the maps $h_i:[0,1] \to U_{\frak p_i}$ as in Lemma \[lem131111\]. We take $0 < a_i < b_i < 1$ such that $$\frak U(X) \cap
\bigcup_{\frak p\in \frak P}
(\frak s_{\frak p}^{n})^{-1}(0) \cap \mathcal K''_{\frak p}
\subset
\bigcup_{i=1}^N [(\psi_i\circ h_i)([a_i,b_i])].$$ and consider the union $$S = f({\rm Crit}(f)\cap \mathcal K''_{\frak p})
\cup \bigcup_{i=1}^N (f\circ\psi_i\circ h_i)({\rm Crit}(f \circ\psi_i\circ h_i) \cap [a_i,b_i]).$$ Here ${\rm Crit}(f \circ\psi_i\circ h_i)$ is the critical point set of the function $f \circ\psi_i\circ h_i : [0,1] \to [0,\infty)$. Proposition \[morseufnction\] implies that $S$ is a finite subset of $[0,\infty)$. ($S$ depends on $n$.)
Suppose $s \notin S$. We consider $X^s = X \cap f^{-1}(s)$ and $U^s_{\frak p} = f^{-1}(s) \cap {\rm Int} \,\mathcal K_{\frak p}$.
Then restricting $\mathcal U_{\frak p}$ to $U^s_{\frak p}$ and restricting the coordinate changes thereto, we obtain a good coordinate system on $X^s$ of dimension $0$. We write it as ${\widetriangle{\mathcal U}}\vert_{\{U^s_{\frak p}\}}$.
The restriction of $\frak s^{\epsilon}_{\frak p}$ defines a multivalued perturbation on ${\widetriangle{\mathcal U}}\vert_{\{U^s_{\frak p}\}}$. We write it as $\widetriangle{\frak s^{n,s}}
= \{ \frak s^{n,s}_{\frak p}\}$. The multisections $\widetriangle{\frak s^{n,s}}$ are transversal to $0$.
The proof is obvious from definition. For $s \in [0,\infty) \setminus S$ we have $$[(X^s,{\widetriangle{\mathcal U}}\vert_{ \{U^s_{\frak p}\}},
\widetriangle{\frak s^{n,s}})]
\in \Q$$ by Definition \[defn1355\].
\[1314\] For each $s_0 \in (0,\infty)$ there exists a positive number $\delta$ such that $[(X^s,{\widetriangle{\mathcal U}}\vert_{ \{U^s_{\frak p}\}},
\widetriangle{\frak s^{n,s}})]$ is independent of $s \in (s_0-\delta,s_0+\delta) \setminus S$.
\[lem1315\] There exists $\delta > 0$ such that $S \cap [0,\delta) = \emptyset$ and $$\label{ddd137}
[(X^s,{\widetriangle{\mathcal U}}\vert_{\{U^s_{\frak p}\}},\widetriangle{\frak s^{n,s}})]
=[(\partial X,\partial{\widetriangle{\mathcal U}},\widetriangle{\frak s^{n}_{\partial}})].$$
Let $$\{p_i \mid i=1,\dots,I\}
= \frak U(X) \cap \vert{\widetriangle{\mathcal U}} \vert_{ \{U^{s_0}_{\frak p}\}}\vert
\cap \bigcup_{\frak p}(\frak s_{\frak p}^{n,{s_0}})^{-1}(0).$$ The right hand side is a finite set by Proposition \[morseufnction\] (4).
For each $p_i$ we take $\frak p_i$ with $p_i \in {\rm Int} \,\mathcal K_{\frak p_i}$, and a representative $(\frak s^{n}_{\frak p_i,j})_{j=1,\dots,\ell_i}$ of $\frak s^{n}_{\frak p_i}$ on an orbifold chart $\frak V_{p_i} = (V_{p_i},\Gamma_{p_i},E_{p_i},\psi_{p_i},\hat\psi_{p_i})$ of $(U_{\frak p_i},\mathcal E_{\frak p_i})$ at $p_i$.
Then for all sufficiently small $\delta>0$ and $s \in (s_0-\delta,s_0+\delta) \setminus S$, we have the following. When we put $\hat f_i = f \circ \psi_{p_i} : V_{p_i} \to \R$,
1. $s$ is a regular value of $\hat f_i$,
2. $\hat{f}_i^{-1}(s)$ intersects transversally to $(\frak s^{n}_{\frak p_i,j})^{-1}(0)$.
Moreover, we can orient $(\frak s^{n}_{\frak p_i,j})^{-1}(0)$ for each $i,j$ so that $$[(X^s,{\widetriangle{\mathcal U}}\vert_{\{U^s_{\frak p}\}},\widetriangle{\frak s^{n,s}})]
=\sum_{i=1}^I
\sum_{j=1}^{\ell_i}
\frac{1}{\ell_i\#\Gamma_{\Gamma_{p_i}}} \hat{f}_i^{-1}(s)
\cdot (\frak s^{n}_{\frak p_i,j})^{-1}(0).$$ Here $\cdot$ in the right hand side is the intersection number, that is, the order of the intersection counted with sign. We use compactness of $\frak U(X) \cap \bigcup_{\frak p}(\frak s^{n}_{\frak p})^{-1}(0)$ to show that the intersection number $\hat{f}_i^{-1}(s)
\cdot (\frak s^{\epsilon}_{\frak p_i,j})^{-1}(0)$ is independent of $s \in (s_0-\delta,s_0+\delta) \setminus S$. Thus Lemma \[1314\] follows. Note we use Proposition \[prop14777\] during this argument.
The existence of $\delta$ with $S \cap [0,\delta) = \emptyset$ is an immediate consequence of (\[normallypositive\]). The formula (\[ddd137\]) can be proved in the same way as the proof of Lemma \[1314\].
Now we are ready to complete the proof of Theorem \[prop13777\]. Lemma \[1314\] implies that $[(X^s,{\widetriangle{\mathcal U}}\vert_{ \{U^s_{\frak p}\} },
\widetriangle{\frak s^{n,s}})]
$ is independent of $s \in (0,\infty) \setminus S$. (Note we did not assume $s_0 \notin S$ in Lemma \[1314\].) Therefore Lemma \[lem1315\] implies that it is equal to $[(\partial X,\partial{\widetriangle{\mathcal U}},
\widetriangle{\frak s^{n}_{\partial}})]$. On the other hand, by compactness of $X$ we find that $X^s$ is empty for sufficiently large $s$. Hence $[(\partial X,\partial{\widetriangle{\mathcal U}},
\widetriangle{\frak s^{n}_{\partial}})] = 0$ as required.
We have the following corollary of Theorem \[prop13777\]. In particular, we can use it to prove that Gromov-Witten invariants are independent of the choice of multivalued perturbations.
1. If ${\widetriangle{\mathcal U}}$ is an oriented and $0$ dimensional good coordinate system without boundary of $X$, then the rational number $[(X,{\widetriangle{\mathcal U}},\widetriangle{\frak s^{n}})]$ is independent of the multivalued perturbation $\{
\widetriangle{\frak s^{n}}\}$ for all sufficiently large $n$.
2. Let $(X,{\widehat{\mathcal U}})$ be an oriented Kuranishi space without boundary. Let ${\widetriangle{\mathcal U}}$ be an oriented good coordinate system and a KG-embedding ${\widehat{\mathcal U}} \to {\widetriangle{\mathcal U}}$. Then the rational number $[(X,{\widetriangle{\mathcal U}},\widetriangle{\frak s^{n}})]
$ is independent of the choice of ${\widetriangle{\mathcal U}}$ and $\widetriangle{\frak s^{n}}$ but depends only on the Kuranishi structure $(X,{\widehat{\mathcal U}})$ itself.
\(1) Let $\{\widetriangle{\frak s^{(k) n}}\}$, $k=0,1$ be the two choices of multivalued perturbations. We consider the direct product of the good coordinate system ${\widetriangle{\mathcal U}} \times [0,1]$ on $X \times [0,1]$. The pair thereon so such that its restriction to ${\widetriangle{\mathcal U}} \times \{0\}$ (resp. ${\widetriangle{\mathcal U}} \times \{1\}$) is $\widetriangle{\frak s^{(0) n}}$ (resp. $\widetriangle{\frak s^{(1) n}}$) and such that it is transversal to $0$. In fact, we can take it so that it is constant in $[0,1]$ direction in a neighborhood of ${\widetriangle{\mathcal U}} \times \partial [0,1]$. We apply Proposition \[prop13777\] to $(X \times [0,1],{\widetriangle{\mathcal U}} \times [0,1],
\widetriangle{\frak s^{n}})$ to obtain $
[X,{\widetriangle{\mathcal U}},\widetriangle{\frak s^{(0) n}}]
=
[X,{\widetriangle{\mathcal U}},\widetriangle{\frak s^{(1) n}}].
$
Independence of $n$ then follows by observing that $\{\widetriangle{\frak s^{n+n_0}}\}$ is a multivalued perturbation if $\{\widetriangle{\frak s^{n}}\}$ is.
\(2) Let ${\widetriangle{\mathcal U^{j}}}$, $j=1,2$ be two good coordinate systems which are compatible with ${\widehat{\mathcal U}}$. We consider the direct product Kuranishi structure $(X \times [0,1],{\widehat{\mathcal U}} \times [0,1])$ on $X \times [0,1]$. In the same way as the proof of Proposition \[relextendgood\], we may assume that ${\widetriangle{\mathcal U^{1}}}$ we put on ${\widehat{\mathcal U}} \times \{0\}$ can be extended to a neighborhood of it in ${\widehat{\mathcal U}} \times [0,1]$. Similarly ${\widetriangle{\mathcal U^{2}}}$ we put on ${\widetriangle{\mathcal U}} \times \{1\}$ can be extended to a neighborhood of it in ${\widehat{\mathcal U}} \times [0,1]$. We then use Proposition \[prop7582752\] to show that there exists a good coordinate system ${\widetriangle{\mathcal U}}$ on $X \times [0,1]$ so that its restriction to $X \times \{0\}$ (resp. $X \times \{1\}$) becomes ${\widetriangle{\mathcal U^{1}}}$ (resp. ${\widetriangle{\mathcal U^{2}}}$.) Now we use Proposition \[prop1220rev\] to find a multivalued perturbation $\widetriangle{\frak s^{n}}$ on ${\widetriangle{\mathcal U}}$. (2) now follows from Theorem \[prop13777\].
In the situation of (2) we call the rational number $[(X,{\widetriangle{\mathcal U}},\widetriangle{\frak s^n})]
$ the [*virtual fundamental class* ]{}of $(X,{\widehat{\mathcal U}})$ and write $[(X,{\widehat{\mathcal U}})] \in \Q$.
We can also prove the following analogue of Proposition \[cobordisminvsmoothcor\]
\[cobordisminvsmoothcormulti\] Let $\frak X_i = (X_i,\widehat{\mathcal U^i})$ be $K$ spaces without boundary of dimension $0$. Suppose that there exists a $K$ space $\frak Y = (Y,\widehat{\mathcal U})$ (but without corner) such that $$\partial \frak Y = \frak X_1 \cup -\frak X_2.$$ Here $-\frak X_2$ is the smooth correspondence $\frak X_2$ with opposite orientation. Then we have $$\label{chomotopyrelation2}
[(X_1,\widehat{\mathcal U^1})]
=
[(X_2,\widehat{\mathcal U^2})].$$
Using Proposition \[prop13777\] in place of Stokes’ formula the proof of Corollary \[cobordisminvsmoothcormulti\] goes in the same way as the proof of Proposition \[cobordisminvsmoothcor\].
Denseness of the set of Morse functions on orbifold {#subsec:moresedense}
---------------------------------------------------
In this subsection we review the proof of the denseness of the set of Morse functions on orbifolds. We consider the case of one orbifold chart. The case when we have several orbifold charts is the same but we do not need it. All the results of this subsection should be well-known. We include it here only for completeness’ sake.
\[shitu1426\] Let $V$ be a manifold on which a finite group $\Gamma$ acts effectively. We denote by $C^{k}_{\Gamma}(V)$ the set of all $\Gamma$ invariant $C^k$ functions on $V$. $\blacksquare$
We take and fix a $\Gamma$ invariant Riemannian metric on $V$, which we use in the proof of some of the lemmata below.
\[prop1427\] Suppose we are in Situation \[shitu1426\]. The set of all $\Gamma$ invariant smooth Morse functions on $V$ is a countable intersection of open dense subsets in $C^{\infty}_{\Gamma}(V)$.
Let $K$ be a compact subset of $V$. It suffices to prove that the set of all the functions in $C^{2}_{\Gamma}(V)$ which are Morse on $K$ is open and dense. The openness is obvious. We will prove that it is also dense.
For $p \in X$ we put $$\Gamma_p = \{ \gamma \in \Gamma \mid \gamma p = p\},$$ and define $$\aligned
\overset{\circ}{X}(n) &= \{p \in X \mid \#\Gamma_p = n\}, \\
{X}(n) &= \{p \in X \mid \#\Gamma_p \ge n\}.
\endaligned$$ Note $\overset{\circ}{X}(n)/\Gamma$ is a smooth manifold.
Let $p \in \overset{\circ}{X}(n)$. Then $p$ is a critical point of $f$ if and only if $p$ is a critical point of $f\vert_{\overset{\circ}{X}(n)}$.
This is a consequence of the fact that the directional derivative $X[f]$ is zero if $X \in T_{p}X$ is perpendicular to $\overset{\circ}{X}(n)$. This fact follows from the $\Gamma$ invariance of $f$.
We define the following sets. $$\aligned
A(n) &= \{ f \in C^{\infty}_{\Gamma}(V) \mid
\text{all the critical points of $f$ on ${X}(n) \cap K$ is Morse.}\} \\
B(n) & = A(n+1)
\cap
\{ f \in C^{\infty}_{\Gamma}(V) \mid
\text{the restriction of $f$ to $\overset{\circ}{X}(n) \cap K$ is Morse.}\}
\endaligned$$They are open sets.
\[lemlem1429\] If $A(n+1)$ is dense then $B(n)$ is dense.
Let $W$ be a relatively compact open subset of $K \cap \overset{\circ}{X}(n)$. We define a $C^1$-map $
F : W \times C^{2}_{\Gamma}(\overline W) \to T^*\overset{\circ}{X}(n)
$ by $$F(x,f) = D_xf \in T^*\overset{\circ}{X}(n).$$ (Since $f$ is $\Gamma$ invariant and $x \in \overset{\circ}{X}(n)$ it follows $D_xf \in T^*\overset{\circ}{X}(n)$.) It is easy to see that $F$ is transversal to the submanifold $\overset{\circ}{X}(n) \subset T^*\overset{\circ}{X}(n)$. Here we identify $\overset{\circ}{X}(n)$ with the zero section of $T^*\overset{\circ}{X}(n)$. We put $$\frak W = \{(x,f) \in W \times C^{2}_{\Gamma}(\overline W)
\mid F(x,f) \in \overset{\circ}{X}(n) \subset T^*\overset{\circ}{X}(n)\}.$$ $\frak W$ is a sub-Banach manifold of the Banach manifold $W \times C^{2}_{\Gamma}(\overline W)$. Moreover the restriction of the projection $${\rm pr} : \frak W \to C^{2}_{\Gamma}(\overline W)$$ is a Fredholm map. Therefore by Sard-Smale theorem the regular value of ${\rm pr}$ is dense.
\[lem1428\] If $f$ is a regular value of ${\rm pr}$ then $f\vert_{\overset{\circ}{X}(n)}$ is Morse on $W$.
Let $x \in W$ be a critical point of $f$. Then $(x,f) \in \frak W$. We consider the following commutative diagram where all the vertical and horizontal lines are exact. $$\begin{CD}
&&
&&
0
&&
0
\\
&&
&& @VVV @VVV
\\
&&
&&
T_x\overset{\circ}{X}(n)
@>>>
T^*_x\overset{\circ}{X}(n)
\\
&&
&& @ VVV @VVV
\\
0 @>>>T_{(x,f)}\frak W
@>>>T_x\overset{\circ}{X}(n) \oplus T_fC^{2}_{\Gamma}(\overline W)
@>{\overline{D_{(x,f)}F}}>>
\frac{T_{(x,o)}T^*\overset{\circ}{X}(n)}{T_x\overset{\circ}{X}(n)} = T^*_x\overset{\circ}{X}(n)
@>>> 0
\\
&& @ VVV @VVV @VVV\\
0 @>>>
T_fC^{2}_{\Gamma}(\overline W) @>>>T_fC^{2}_{\Gamma}(\overline W)
@>>>0
\\
&& && @VVV
\\
&&&& 0
\end{CD}$$ Here $\overline{D_{(x,f)}F}$ is the composition of $D_{(x,f)}F : T_x\overset{\circ}{X}(n) \oplus T_fC^{2}_{\Gamma}(\overline W)
\to T_{(x,o)}T^*\overset{\circ}{X}(n)$ and the projection. Since $f$ is a regular value the first vertical line is surjective. We can use it to show that the first horizontal line $T_x\overset{\circ}{X}(n) \to T_x^*\overset{\circ}{X}(n)$ is surjective by a simple diagram chase. This map is identified with the Hessian at $x$ of $f\vert_{\overset{\circ}{X}(n)}$. The sublemma follows.
We observe that if $f \in A(n+1)$ then the set of critical points in $\overset{\circ}{X}(n) \cap K$ is compact. (This is because it does not have accumulation points on ${X}(n+1) \cap K$.) Therefore Lemma \[lemlem1429\] follows from Sublemma \[lem1428\] and Sard-Smale theorem.
\[lem1431\] If $B(n)$ is dense then $A(n)$ is dense.
Let $f \in B(n)$. We remark that the set of critical points of $f\vert_{\overset{\circ}{X}(n)}$ on $\overset{\circ}{X}(n) \cap K$ is a finite set. This is because $f\vert_{\overset{\circ}{X}(n)}$ is a Morse function on $\overset{\circ}{X}(n) \cap K$ and $f\vert_{\overset{\circ}{X}(n)}$ does not have accumulation points on ${X}(n+1) \cap K$. Let $p_1,\dots,p_m$ be the critical points of $f\vert_{\overset{\circ}{X}(n)}$ on $\overset{\circ}{X}(n) \cap K$. Note the Hessian of $f$ at those points are non-degenerate on $T_{p_i}\overset{\circ}{X}(n)$ but may be degenerate in the normal direction to $\overset{\circ}{X}(n)$. We choose functions $\chi_i$ and $V_i$ with the following properties.
1. $V_i$ is a neighborhood of $p_i$.
2. The support of $\chi_i$ is in $V_i$
3. $\chi_i \equiv 1$ in a neighborhood of $p_i$.
4. $\overline V_i$ ($i=1,\dots,m$) are disjoint.
5. $\overline V_i \cap X(n+1) = \emptyset$.
6. If $\gamma p_i = p_j$, $\gamma \in \Gamma$, then $\gamma V_i = V_j$ and $\chi_j \circ \gamma = \chi_i$.
We use our $\Gamma$ invariant Riemannian metric and put $$f_n(x) = d(x,\overset{\circ}{X}(n))^2$$ We choose $V_i$ small so that $\chi_i f_n$ is a smooth function. (We use Item (5) above here.) Now $$\label{form1413}
f_{\epsilon} = f + \epsilon \sum_{i=1}^m \chi_i f_n$$ is a Morse function for sufficiently small positive $\epsilon$. Item (6) implies that this function is $\Gamma$ invariant. Therefore $f_{\epsilon} \in A(n)$. Moreover $f_{\epsilon}$ converges to $f$ as $\epsilon \to 0$.
Lemmata \[lemlem1429\] and \[lem1431\] imply that $A(1)$ is dense. The proof of Proposition \[prop1427\] is complete.
The set of functions whose gradient flow is Morse-Smale is [*not*]{} dense in general in the case of orbifold. This is an important point where Morse theory of orbifold is different from one of manifold. In fact we can use virtual fundamental chain technique to work out the theory of Morse homology for orbifold.
\[lem1430\] Suppose we are in Situation \[shitu1426\]. Let $h : [0,1] \to V$ be a smooth embedding. Then the set of all the functions $f$ in $C^{\infty}_{\Gamma}(V)$ such that $f \circ h$ is a Morse function on $(0,1)$ is a countable intersection of open dense subsets.
The proof is similar to Proposition \[prop1427\]. Let $0<c<1/2$. We put $$\aligned
\overset{\circ}T(n,c) &= \{t \in [c,1-c] \mid h(t) \in \overset{\circ}{X}(n)\}, \\
T(n,c) &= \{t \in [c,1-c] \mid h(t) \in {X}(n)\}, \\
S(n,c) &= \{t_0 \in \overset{\circ}T(n,c) \mid (dh/dt)(t_0) \in T_{h(t_0)}\overset{\circ}{X}(n) \}.
\endaligned$$ We also put $$\aligned
C(n,c) &=
\{ f \in C^{\infty}_{\Gamma}(V) \mid
\text{all the critical points of $f \circ h$ on $T(n,c)$ is Morse.}\} \\
D(n,c) &=
C(n+1,c) \cap
\{ f \in C^{\infty}_{\Gamma}(V) \mid
\text{all the critical points of $f \circ h$ on $S(n,c)$ is Morse.}\}
\endaligned$$
\[lem1434\] If $C(n+1,c)$ is dense then $D(n,c)$ is dense.
We fix $c$ and will prove $D(n,c)$ is dense. We take $\epsilon >0$ such that $c - \epsilon > 0$. Let $U_1$ be a sufficiently small neighborhood of $X(n+1)$ which we choose later. We take $U_2$ a neighborhood $X(n) \setminus U_1$ and $\pi : U_2 \to \overset{\circ}X(n)$ be the projection of normal bundle. We may assume $U_1$, $U_2$ and $\pi$ are $\Gamma$ equivariant. We take a compact subset $Z$ of $U_2 \cap \overset{\circ}X(n)$. We take a set $S \subset [c-\epsilon,1-c+\epsilon]$ with the following properties.
1. ${\rm Int}S \supset S(n,c) \cap h^{-1}(Z)$.
2. $S$ is a finite disjoint union of closed intervals.
3. The composition $\pi \circ h$ is an embedding on $S(n,c)$.
Note the differential of $\pi \circ h$ is injective on $S(n,c) \cap h^{-1}(Z)$. Therefore we can take such $S$. We define a map $
G : {\rm Int} S \times C^{2}_{\Gamma}(K) \to \R
$ by $$G(t_0,f) = \frac{d( f\circ \pi \circ h)}{dt}(t_0).$$ Using the fact that $\pi \circ h$ is an embedding and $\overset{\circ}X(n)/\Gamma$ is a smooth manifold we can easily show that $G$ is transversal to $0$. We put $\frak N = G^{-1}(0) \subset {\rm Int} S \times C^{2}_{\Gamma}(K)$. It is a Banach submanifold. The restriction of the projection $\frak N \to C^{2}_{\Gamma}(K)$ is a Fredholm map. In the same way as Subemma \[lem1428\] we can show that if $f \in C^{2}_{\Gamma}(K)$ is a regular value of $\frak N \to C^{2}_{\Gamma}(K)$ then $f\circ \pi \circ h$ is a Morse function on ${\rm Int}\,S$. We remark that at the point of $S(n,c)$ the Hessian of $f\circ \pi \circ h$ is the same as the Hessian of $f\circ h$. Thus by Sard-Smale theorem we find that the set of $f \in C^{2}_{\Gamma}(K)$ such that all the critical points on $f\circ h$ on $S(n,c) \cap h^{-1}(Z)$ is Morse, is dense.
\[sublem1435\] Let $f \in C(n+1,c)$. Then there exists a compact set $P \subset \overset{\circ}T(n,c)$ such that there is no critical point of $f\circ h$ on $\overset{\circ}T(n,c) \setminus P$.
If not there is a sequence $t_i \in \overset{\circ}T(n,c)$ such that $t_i$ is a critical point of $f\circ h$ and no subsequence of $t_i$ converges to an element of $\overset{\circ}T(n,c)$. We may assume that $t_i$ converges. Then the limit $t$ should be an element of $T(n+1,c)$. Since $f \in C(n+1,c)$ the function $f\circ h$ is Morse at $t$. Therefore there is no critical point of $f\circ h$ other than $p$ in a neighborhood of $t$. This is a contradiction.
Now let $f \in C(n+1,c)$. We choose $U_1$, $U_2$ and $Z$ such that $Z
\supset h(P)$ and $P$ is as in Sublemma \[sublem1435\]. Then there exists a sequence of functions $f_i$ such that $f_i$ converges to $f$ and all the critical points of $f_i \circ h$ on $S(n,c) \cap h^{-1}(Z)$ are Morse. Therefore $f_i \in D(n,c)$ for all sufficiently large $i$. The proof of Lemma \[lem1434\] is complete.
\[Dndense\] If $D(n,c)$ is dense then $C(n,c)$ is dense.
Let $f \in D(n,c)$.
The set $$Q = \{ t \in \overset{\circ}T(n,c) \mid \text{$t$ is a critical point of $f\circ h$} \}$$ is a finite set.
Suppose $Q$ is an infinite set and $t_i \in Q$ is its infinitely many points. We may also assume that $t_i$ converges to $t$. If $t \in \overset{\circ}T(n,c) \setminus S(n,c)$ then $h$ is transversal to $\overset{\circ}{X}(n)$ at $t$. Therefore $h(t_i) \notin \overset{\circ}{X}(n)$ for large $i$. This contradicts to $t_i \in Q$. Therefore $t \in S(n,c) \cup T(n+1,c)$. Since $f \in D(n,c)$ the composition $f \circ h$ is Morse at $t$. Therefore there is no critical point of $f \circ h$ other than $t$ in a neighborhood of $t$. This contradicts $\lim t_i = t$.
Let $Q \setminus S(n,c) = \{t_1,\dots,t_n\}$ and we put $p_i = h(t_i) \in \overset{\circ}T(n,c)$. We define $V_i$ and $\chi_i$ in the same way as the proof of Lemma \[lem1431\] and define $f_{\epsilon}$ in the same way as (\[form1413\]). It is easy to see that $f_{\epsilon} \in C(n,c)$ and $f_{\epsilon}$ converges to $f$ as $\epsilon \to 0$. The proof of Lemma \[Dndense\] is complete.
By Lemmata \[lem1434\] and \[Dndense\] we have proved that $C(1,c)$ is dense for all $c \in (0,1/2)$. The proof of Proposition \[lem1430\] is complete.
We remark that we do not assume $h$ to be $\Gamma$ invariant in Proposition \[lem1430\].
Appendices : Orbifold and orbibundle by local coordinate {#sec:ofd}
========================================================
In this section we describe the story of orbifold as far as we need in this document. We restrict ourselves to effective orbifolds and regard only embeddings as morphisms. The category $\mathscr{OB}_{\rm ef,em}$ where objects are effective orbifolds and morphisms are embeddings among them is naturally a $1$ category. Moreover it has the following property. We consider the forgetful map $$\frak{forget} : \mathscr{OB}_{\rm ef,em} \to \mathscr{TOP}$$ where $\mathscr{TOP}$ is the category of topological spaces. Then $$\frak{forget} : \mathscr{OB}_{\rm ef,em}(c,c')
\to \mathscr{TOP}(\frak{forget}(c),\frak{forget}(c'))$$ is injective. In other words, we can check the equality between morphisms set-theoretically. This is a nice property, which we use extensively in the main body of this document. If we go beyond this category then we need to distinguish carefully the two notions, two morphisms are equal, two morphisms are isomorphic. It will then makes the argument much more complicated. [^29]
We emphasize that there is nothing new in this section. The story of orbifold is classical and is well-established. It has been used in various branches of mathematics since its invention by Satake [@satake] in more than 50 years ago. Especially, if we restrict ourselves to effective orbifolds, the story of orbifolds is nothing more than straightforward generalization of the theory of smooth manifolds. The only important issue is the observation that for effective orbifolds almost everything work in the same way as manifolds.
Orbifolds and embedding among them {#subsec;ofds}
----------------------------------
\[2661\] Let $X$ be a paracompact Hausdorff space.
1. An [*orbifold chart of $X$ (as a topological space)*]{} is a triple $(V,\Gamma,\phi)$ such that $V$ is a manifold, $\Gamma$ is a finite group acting smoothly and effectively on $V$ and $\phi : V \to X$ is a $\Gamma$ equivariant continuous map[^30] which induces a homeomorphism $\overline\phi : V/\Gamma \to X$ onto an open subset of $X$. We assume that there exists $o \in V$ such that $\gamma o = o$ for all $\gamma \in \Gamma$. We call $o$ the [*base point*]{}. We say $(V,\Gamma,\phi)$ is an orbifold chart [*at $x$*]{} if $x = \phi(o)$. We call $\Gamma$ the [*isotropy group*]{}, $\phi$ the [*local uniformization map*]{} and $\overline\phi$ the [*parametrization*]{} .
2. Let $(V,\Gamma,\phi)$ be an orbifold chart and $p \in V$. We put $\Gamma_p = \{ \gamma \in \Gamma \mid \gamma p = p\}$. Let $V_p$ be a $\Gamma_p$ invariant open neighborhood of $p$ in $V$. We assume the map $\overline\phi : V_p/\Gamma_p \to X$ is injective. (In other words, we assume that $\gamma V_{p} \cap V_p \ne \emptyset$ implies $\gamma \in \Gamma_p$.) We say such triple $(V_{p},\Gamma_p,\phi\vert_{V_p})$ a [*subchart*]{} of $(V,\Gamma,\phi)$.
3. Let $(V_i,\Gamma_i,\phi_i)$ $(i=1,2)$ be orbifold charts of $X$. We say that they are [*compatible*]{} if the following holds for each $p_1 \in V_1$ and $p_2 \in V_2$ with $\phi_1(p_1) = \phi_2(p_2)$.
1. There exists a group isomorphism $h : (\Gamma_1)_{p_1}
\to (\Gamma_2)_{p_2}$.
2. There exists an $h$ equivariant diffeomorphism $\tilde\varphi : V_{1,p_1}
\to V_{2,p_2}$. Here $V_{i,p_i}$ is a $(\Gamma_i)_{p_i}$ equivariant subset of $V_i$ such that $(V_{i,p_i},(\Gamma_i)_{p_i},\phi\vert_{V_{i,p_i}})$ is a subchart.
3. $\phi_2 \circ \tilde\varphi = \phi_1$ on $V_{1,p_1}$.
4. A [*representative of an orbifold structure*]{} on $X$ is a set of orbifold charts $\{(V_i,\Gamma_i,\phi_i) \mid i \in I\}$ such that each two of the charts are compatible in the sense of (3) above and $
\bigcup_{i\in I} \phi_i(V_i) = X,
$ is a locally finite open cover of $X$.
\[def262220\] Suppose that $X$, $Y$ have representatives of orbifold structures $\{(V^X_i,\Gamma^X_i,\phi^X_i) \mid i \in I\}$ and $\{(V^Y_j,\Gamma^Y_j,\phi^Y_j) \mid j \in J\}$, respectively. A continuous map $f : X \to Y$ is said to be an [*embedding*]{} if the following holds.
1. $f$ is an embedding of topological spaces.
2. Let $p \in V^X_i$, $q \in V^Y_j$ with $f(\phi_i(p)) = \phi_j(q)$. Then we have the following.
1. There exists an isomorphism of groups $h_{p;ji} : (\Gamma_i^X)_p \to (\Gamma_j^Y)_q$.
2. There exist $V^X_{i,p}$ and $V^Y_{j,q}$ such that $(V^X_{i,p},(\Gamma^X_i)_p,\phi_i\vert_{V^X_{i,p}})$ is a subchart for $i=1,2$. There exists an $h_{p;ji}$ equivariant embedding of manifolds $\tilde f_{p;ji}: V^X_{i,p} \to V^Y_{j,q}$.
3. The diagram below commutes. $$\label{diag2633}
\begin{CD}
V^X_{i,p} @ >{\tilde f_{p;ji}}>>
V^Y_{j,q} \\
@ V{\phi_{i}}VV @ VV{\phi_{j}}V\\
X @ > {f} >> Y
\end{CD}$$
Two orbifold embeddings are said to be [*equal*]{} if they coincide set-theoretically.
Note that an embedding of effective orbifolds is a continuous map $f : X \to Y$ of underlying topological spaces, which has the properties (2) above.
When we study morphisms among not-necessary effective orbifolds or morphisms between effective orbifolds which is not necessarily an embedding, then such a morphism is a continuous map $f : X \to Y$ of underlying topological spaces [*plus*]{} certain additional data. For example, we consider noneffective orbifold that is a point with an action of a nontrivial finite group $\Gamma$. Then the morphism from this noneffective orbifold to itself contains a datum which is an automorphism of the group $\Gamma$. (Two such morphisms $h_1,h_2$ are equivalent if there exists an inner automorphism $h$ such that $h_1 = h \circ h_2$.)
\[lem26444\]
1. The composition of embeddings is an embedding.
2. The identity map is an embedding.
3. If an embedding is a homeomorphism, then its inverse is also an embedding.
The proof is easy and is left to the reader.
\[defn285555\]
1. We call an embedding of orbifolds a [*diffeomorphism*]{} if it is a homeomorphism in addition.
2. We say that two representatives of orbifold structures on $X$ are [*equivalent*]{} if the identity map regarded as a map between $X$ equipped with those two representatives of orbifold structures is a diffeomorphism. This is an equivalence relation by Lemma \[lem26444\].
3. An equivalence class of the equivalence relation (2) is called an [*orbifold structure*]{} of $X$. An [*orbifold*]{} is a pair of topological space and its orbifold structure.
4. The condition for $X \to Y$ to be an embedding does not change if we replace representatives of orbifold structures to equivalent ones. So we can define the notion of an [*embedding of orbifolds*]{}.
5. If $U$ is an open subset of an orbifold $X$, then there exists a unique orbifold structure on $U$ such that the inclusion $U \to X$ is an embedding. We call $U$ with this orbifold structure an [*open suborbifold*]{}.
\[defn26550\]
1. Let $X$ be an orbifold. An orbifold chart $(V,\Gamma,\phi)$ of underlying topological space $X$ in the sense of Definition \[2661\] (1) is called an [*orbifold chart of an orbifold*]{} $X$ if the map $\overline\phi : V/\Gamma \to X$ induced by $\phi$ is an embedding of orbifolds.
2. Hereafter when $X$ is an orbifold, an ‘orbifold chart’ always means an orbifold chart of an orbifold in the sense of (1).
3. In case when an orbifold structure on $X$ is given, a representative of its orbifold structure is called an [*orbifold atlas*]{}.
4. Two orbifold charts $(V_i,\Gamma_i,\phi_i)$ are said to be isomorphic if there exist a group isomorphism $h : \Gamma_1 \to \Gamma_2$ and an $h$-equivariant diffeomorphism $\tilde\varphi : V_1 \to V_2$ such that $\phi_2 \circ \tilde\varphi = \phi_1$. The pair $(h,\tilde\varphi)$ is said to be the [*isomorphism*]{} between two orbifold charts.
\[prop266\] In the situation of Definition \[defn26550\] (4), suppose $(h,\tilde\varphi)$ and $(h',\tilde\varphi')$ are both isomorphisms between two orbifold charts $(V_1,\Gamma_1,\phi_1)$ and $(V_2,\Gamma_2,\phi_2)$. Then there exists $\mu \in \Gamma_2$ such that $$\label{262}
h'(\gamma) = \mu h(\gamma) \mu^{-1},
\qquad
\tilde\varphi'(x) = \mu \tilde\varphi(x).$$
On the contrary, if $(h,\tilde\varphi)$ is an isomorphism between orbifold charts then $(h',\tilde\varphi')$ defined by (\[262\]) is also an isomorphism between orbifold charts. In particular, any automorphism of orbifolds charts $(h,\tilde\varphi)$ is given by $h(\gamma) = \mu\gamma \mu^{-1}$, $\tilde\varphi(x) = \mu x$ where $\mu$ is an element of $\Gamma$.
The proposition follows immediately from the next lemma.
\[lem21tenhatena\] Let $V_1$, $V_2$ be manifolds on which finite groups $\Gamma_1$, $\Gamma_2$ act effectively and smoothly. We assume $V_1$ is connected. Let $(h_i,\tilde\varphi_i)$ $(i=1,2)$ be pairs such that $h_i : \Gamma_1 \to \Gamma_2$ are injective group homomorphisms and $\tilde\varphi_i : V_1 \to V_2$ are $h_i$-equivariant embeddings of manifolds. Moreover, we assume that the induced maps ${\varphi}_i :V_1/\Gamma_1
\to V_2/\Gamma_2$ are embeddings of orbifolds. Furthermore we assume that the induced map ${\varphi}_1 : V_1/\Gamma_1
\to V_2/\Gamma_2$ coincides with ${\varphi}_2$ set-theoretically.
Then there exists $\mu \in \Gamma_2$ such that $$\tilde\varphi_2(x) = \mu\tilde\varphi_1(x),
\qquad
h_2(\gamma) = \mu h_1(\gamma)\mu^{-1}.$$
For the sake of simplicity we prove only the case when Condition \[convinv\] below is satisfied. Let $X$ be an orbifold. For a point $x \in X$ we take its orbifold chart $(V_x,\Gamma_x,\psi_x)$. We say $x \in {\rm Reg}(X)$ if $\Gamma_x = \{1\}$, and put ${\rm Sing}(X) = X \setminus {\rm Reg}(X)$.
\[convinv\] We assume that $\dim {\rm Sing}(X) \le \dim X -2$.
This condition is satisfied if $X$ is oriented. (In fact, Condition \[convinv\] fails only when there exists an element of $\Gamma_x$ (an isotropy group of some orbifold chart) whose action is $(x_1,x_2,\dots,x_n) \mapsto (-x_1,x_2,\dots,x_n)$ for some coordinate $(x_1,\dots,x_n)$. Therefore we can always assume Condition \[convinv\] in the study of Kuranishi structure, by adding a trivial factor which is acted by the induced representation of $t \mapsto -t$ to both the obstruction bundle and to the Kuranishi neighborhood.)
Let $x_0 \in V_1^0$. By assumption there exists uniquely $\mu \in \Gamma_2$ such that $\tilde\varphi_2(x_0) = \mu\tilde\varphi_1(x_0)$. By Condition \[convinv\] the subset $V^0_1$ is connected. Therefore the above element $\mu$ is independent of $x_0 \in V_1^0$ by uniqueness. Since $V_1^0$ is dense, we conclude $\tilde\varphi_2(x) = \mu\tilde\varphi_1(x)$ for any $x \in V_1$. Now, for $\gamma \in \Gamma_1$, we calculate $$h_1(\gamma)\tilde\varphi_1(x_0)
=
\tilde\varphi_1(\gamma x_0)
=
\mu^{-1}\tilde\varphi_2( \gamma x_0)
=
\mu^{-1}h_2(\gamma)\tilde\varphi_2(x_0)
=
\mu^{-1}h_1(\gamma)\mu \tilde\varphi_1(x_0).$$ Since the induced map is an embedding of orbifold, it follows that the isotropy group of $\tilde\varphi_1(x_0)$ is trivial. Therefore $h_1(\gamma) = \mu^{-1}h_2(\gamma)\mu$ as required.
The proof of Proposition \[prop266\] is complete.
\[defn281010\] Let $X$ be an orbifold.
1. A function $f : X \to \R$ is said to be a [*smooth function*]{} if for any orbifold chart $(V,\Gamma,\phi)$ the composition $f\circ \phi : V \to \R$ is smooth.
2. A [*differential form*]{} on an orbifold $X$ assigns a $\Gamma$ invariant differential form $h_{\frak V}$ on $V$ to each orbifold chart $\frak V = (V,\Gamma,\phi)$ such that the following holds.
1. If $(V_1,\Gamma_1,\phi_1)$ is isomorphic to $(V_2,\Gamma_2,\phi_2)$ and $(h,\tilde\varphi)$ is an isomorphism, then $\tilde\varphi^*h_{\frak V_2} = h_{\frak V_1}$.
2. If $\frak V_p = (V_p,\Gamma_p,\phi_p)$ is a subchart of $\frak V =(V,\Gamma,\phi)$, then $h_{\frak V}\vert_{V_p} = h_{\frak V_p}$.
3. An $n$ dimensional orbifold $X$ is said to be [*orientable*]{} if there exists a differential $n$-form $\omega$ such that $\omega_{\frak V}$ never vanishes.
4. Let $\omega$ be an $n$-form as in (3) and $\frak V = (V,\Gamma,\phi)$ an orbifold chart. Then we give $V$ an orientation so that it is compatible with $\omega_{\frak V}$. The $\Gamma$ action preserves the orientation. We call such $(V,\Gamma,\phi)$ equipped with an orientation of $V$, an [*oriented orbifold chart*]{}.
5. Let $\bigcup_{i\in I}U_i = X$ be an open covering of an orbifold $X$. A [*smooth partition of unity subordinate to the covering*]{} $\{U_i\}$ is a set of functions $\{\chi_i\mid i\in I\}$ such that:
1. $\chi_i$ are smooth functions.
2. The support of $\chi_i$ is contained in $U_i$.
3. $\sum_{i\in I}\chi_i = 1$.
For any locally finite open covering of an orbifold $X$ there exists a smooth partition of unity subordinate to it.
We omit the proof, which is an obvious analogue of the standard proof for the case of manifolds.
\[orbifolddefn\] An [*orbifold with corner*]{} is defined in the same way. We require the following.
1. In Definition \[2661\] (1) we assume that $V$ is a manifold with corners.
2. Let $S_k(V)$ be the set of points which lie on the codimension $k$ corner and $\overset{\circ}S_k(V) = S_k(V) \setminus \bigcup_{k' > k}S_{k'}(V)$. We require that $\Gamma$ action on each connected component of $\overset{\circ}S_k(V)$ is effective. (Compare Condition \[effectivitycorner\].)
3. For an embedding of orbifolds with corners we require that the map $\tilde f$ in Definition \[def262220\] (c) satisfies $\tilde f(\overset{\circ}S_k(V_1)) \subset \overset{\circ}S_k(V_2)$.
\[lem26999\] Let $X_i$ $(i=1,2)$ be orbifolds and $\varphi_{21} : X_1 \to X_2$ an embedding. Then we can find an orbifold atlas $\{\frak V^i_{\frak r} =
\{ (V_{\frak r}^i, \Gamma_{\frak r}^i, \phi_{\frak r}^i )\} \mid \frak r \in \frak R_i\}$ with the following properties.
1. $\frak R_1 \subseteq \frak R_2$.
2. $V^2_{\frak r} \cap \varphi_{21}(X_1) \ne \emptyset$ if and only if $\frak r \in \frak R_1$.
3. If $\frak r \in \frak R_1$ then $\varphi_{21}^{-1}(\phi_{\frak r}^2(V^2_{\frak r})) =
\phi_{\frak r}^1(V^1_{\frak r})$ and there exists $(h_{\frak r,21},\tilde\varphi_{\frak r,21})$ such that:
1. $h_{\frak r,21} : \Gamma^1_{\frak r} \to \Gamma^2_{\frak r}$ is a group isomorphism.
2. $\tilde\varphi_{\frak r,21} : V^1_{\frak r} \to V^2_{\frak r}$ is an $h_{\frak r,21}$-equivariant embedding of smooth manifolds.
3. The next diagram commutes. $$\label{diagin2611}
\begin{CD}
V^1_{\frak r} @ >{\tilde\varphi_{\frak r,21}}>>
V^2_{\frak r} \\
@ V{\phi^{\frak r}_{1}}VV @ VV{\phi^{\frak r}_{2}}V\\
X_1 @ > {\varphi_{21}} >> X_2
\end{CD}$$
4. In case $X_i$ has a boundary or corners we may choose our charts so that the following is satisfied.
1. $V^i_{\frak r}$ is an open subset of $\overline V^i_{\frak r}
\times [0,1)^{d({\frak r})}$, where $d(\frak r)$ is independent of $i$ and $\overline V^i_{\frak r}$ is a manifold without boundary.
2. There exists a point $o^i(\frak r)$ which is fixed by all $\gamma \in \Gamma^i_{\frak r}$ such that $[0,1)^{d({\frak r})}$ components of $o^i(\frak r)$ are all $0$.
3. We put $$\varphi_{\frak r,21}(\overline y',(t'_1,\dots,t'_{d({\frak r})}))
=
(\overline y,(t_1,\dots,t_{d({\frak r})})).$$ Then $t_i = 0$ if and only if $t'_i = 0$.
We may take our charts finer than given coverings of $X_1$ and $X_2$.
For each $x \in X_1$ we can find orbifold charts $\frak V^i_{x}$ for $i=1,2$, such that $\varphi_{21}^{-1}(U^2_{x}) = U^1_{x}$, $x \in U^1_{x}$ and that there exists a representative $(h_{x,21},\tilde\varphi_{x,21})$ of embedding $U^1_{x} \to U^2_{x}$ that is a restriction of $\varphi_{21}$. In case $X_i$ has a boundary or corners, we choose them so that (4) is also satisfied.
We cover $X_1$ by finitely many of such $U_{x_j}^{1}$. This is our $\{\frak V^1_{\frak r} \mid \frak r \in \frak R_1\}$. Then we have $\{\frak V^2_{\frak r} \mid \frak r \in \frak R_1\}$, satisfying (3)(4) and that covers $\varphi_{21}(X_1)$. We can extend it to $\{\frak V^2_{\frak r} \mid \frak r \in \frak R_2\}$ so that (1)(2) are also satisfied.
We call $(h_{\frak r,21},\tilde\varphi_{\frak r,21})$ a [*local representative of embedding*]{} $\varphi_{\frak r,21}$ on the charts $\frak V^1_{\frak r}$, $\frak V^2_{\frak r}$.
If $(h_{\frak r,21},\tilde\varphi_{\frak r,21})$, $(h'_{\frak r,21},\tilde\varphi'_{\frak r,21})$ are local representatives of an embedding of the same charts $\frak V^1_{\frak r}$, $\frak V^2_{\frak r}$, then there exists $\mu \in \Gamma_2$ such that $$\tilde\varphi'_{\frak r,21}(x) = \mu\tilde\varphi_{\frak r,21}(x),
\qquad
h_{\frak r,21}^{\prime}(\gamma) = \mu h_{\frak r,21}(\gamma)\mu^{-1}.$$
This is a consequence of Lemma \[lem21tenhatena\].
\[smoothstruemb\] Let $X$ be a topological space, $Y$ an orbifold, and $f : X \to Y$ an embedding of topological spaces. Then the orbifold structure of $X$ by which $f$ becomes an embedding of orbifolds is unique if exists.
Let $X_1$, $X_2$ be orbifolds whose underlying topological spaces are both $X$ and such that $f_i : X_i \to Y$ are embeddings of orbifolds for $i=1,2$. We will prove that the identity map $\rm{id} : X_1 \to X_2$ is a diffeomorphism of orbifolds. Since the condition for a homeomorphism to be a diffeomorphism of orbifolds is a local condition, it suffices to check it on a neighborhood of each point. Let $p \in X$ and $q = f(p)$. We take a representative $(h_i,\tilde\varphi_i)$ of the orbifold embeddings $f_i : X_i \to Y$ using the orbifold charts $\frak V^i_p = (V^i_p,\Gamma^i_p,\phi^i_p)$ of $X$ and $\frak V_q = (V_q,\Gamma_q,\phi_q)$ of $Y$. The maps $h_i : \Gamma^i_p \to \Gamma^i_q$ are group isomorphisms. So we have a group isomorphism $
h = h_2^{-1}\circ h_1 : \Gamma^1_p \to \Gamma^2_p.
$ Since $\tilde\varphi_1(V^1)/\Gamma_p = \tilde\varphi_2(V^2)/\Gamma_p$ set-theoretically, we have $\tilde\varphi_1(V^1_p) = \tilde\varphi_2(V^2_p) \subset V_q$. They are smooth submanifolds since $f_i$ are embeddings of orbifolds. Therefore $
\varphi = \tilde\varphi_2^{-1}\circ \tilde\varphi_1
$ is defined in a neighborhood of the base point $o^i_p$ and is a diffeomorphism. $(h,\tilde\varphi)$ is a local representative of ${\rm id}$.
Vector bundle on orbifold {#subsec:vectorbundle}
-------------------------
\[defn2613\] Let $(X,\mathcal E,\pi )$ be a pair of an orbifold $X$ and $\pi : \mathcal E \to X$ a continuous map between their underlying topological spaces. Hereafter we write $(X,\mathcal E)$ in place of $(X,\mathcal E,\pi)$.
1. An [*orbifold chart*]{} of $(X,\mathcal E)$ is $(V,E,\Gamma,\phi,\widehat\phi)$ with the following properties.
1. $\frak V = (V,\Gamma,\phi)$ is an orbifold chart of the orbifold $X$.
2. $E$ is a finite dimensional vector space equipped with a linear $\Gamma$ action.
3. $(V \times E,\Gamma,\widehat\phi)$ is an orbifold chart of the orbifold $\mathcal E$.
4. The diagram below commutes set-theoretically, $$\label{diag26399}
\begin{CD}
V \times E @ >{\widehat\phi}>>
\mathcal E \\
@ V{}VV @ VV{{\pi}}V\\
V @ > {\phi} >> X
\end{CD}$$ where the left vertical arrow is the projection to the first factor.
2. In the situation of (1), let $( V_{p},\Gamma_p,\phi\vert_{V_p})$ be a subchart of $(V,\Gamma,\phi)$ in the sense of Definition \[2661\] (2). Then $(V_{p},E,\Gamma_p,\phi\vert_{V_p},\widehat\phi\vert_{V_p \times E})$ is an orbifold chart of $(X,\mathcal E)$. We call it a [*subchart*]{}.
3. Let $(V^i,E^i,\Gamma^i,\phi^i,\widehat{\phi^i})$ $(i=1,2)$ be orbifold charts of $(X,\mathcal E)$. We say that they are [*compatible*]{} if the following holds for each $p_1 \in V^1$ and $p_2 \in V^2$ with $\phi^1(p_1) = \phi^2(p_2)$: There exist open neighborhoods $V^i_{p_i}$ of $p_i \in V^i$ such that:
1. There exists an isomorphism $(h,\tilde\varphi) : (V^1,\Gamma^1,\phi^1)\vert_{V^1_{p_1}} \to (V^2,\Gamma^2,\phi^2)\vert_{V^2_{p_2}}$ between orbifold charts of $X$, which are subcharts.
2. There exists an isomorphism $(h,\tilde{\hat{\varphi}}) : (V^1\times E^1,\Gamma^1,\phi^1)\vert_{V^1_{p_1}\times E^1} \to
(V^2\times E^2,\Gamma^2,\phi^2)\vert_{V^2_{p_2} \times E^2}$ between orbifold charts of $\mathcal E$, which are subcharts.
3. For each $y \in V^1_{p_1}$ the map: $ E^1 \to E^2$, $\xi \to \pi_{E^2}\tilde{\hat{\varphi}}(y,\xi)$ is a linear isomorphism. Here $\pi_{E^2} : V^2 \times E^2 \to E^2$ is the projection.
4. A [*representative of a vector bundle structure*]{} of $(X,\mathcal E)$ is a set of orbifold charts $\{(V_i,E_i,\Gamma_i,\phi_i,\widehat\phi_i) \mid i \in I\}$ such that any two of the charts are compatible in the sense of (3) above and $$\bigcup_{i\in I} \phi_i(V_i) = X,
\quad
\bigcup_{i\in I} \widehat\phi_i(V_i \times E_i) = \mathcal E,$$ are locally finite open covers.
\[def26222\] Suppose $(X^*,\mathcal E^*)$ $(* = a,b)$ have representatives of vector bundle structures $\{(V^*_i,E^*_i,\Gamma^*_i,\phi^*_i,\widehat\phi^*_i) \mid i \in I^*\}$, respectively. A pair of orbifold embeddings $(f,\widehat f)$, $f : X^a \to X^b$, $\widehat f : \mathcal E^a \to \mathcal E^b$ is said to be an [*embedding of vector bundles*]{} if the following holds.
1. Let $p \in V^a_i$, $q \in V^b_j$ with $f(\phi^a_i(p)) = \phi^b_j(q)$. Then there exist open subcharts $(V^a_{i,p}\times E^a_{i,p},\Gamma^a_{i,p},\widehat\phi^a_{i,p})$ and $(V^b_{j,q}\times E^b_{j,q},\Gamma^b_{j,q}\widehat\phi^b_{j,q})$ and a local representative $(h_{p;i,j},f_{p;i,j},\widehat f_{p;i,j})$ of the embeddings $f$ and $\widehat f$ such that for each $y \in V^a_i$ the map $\xi \mapsto \pi_{E^b}(\widehat f_{p;i,j}(y,\xi))$, $E^a_{i,p} \to E^b_{j,q}$ is a linear embedding. Here $\pi_{E^b} : V^b \times E^b \to E^b$ is the projection.
2. The diagram below commutes set-theoretically. $$\label{diag2633}
\begin{CD}
\mathcal E^a @ >{\widehat f}>>
\mathcal E^b \\
@ V{\pi_{E^a}}VV @ VV{\pi_{E^b}}V\\
X^a @ > {f} >> X^b
\end{CD}$$
Two orbifold embeddings are said to be [*equal*]{} if they coincide set-theoretically as pairs of maps.
\[lem26444AA1\]
1. A composition of embeddings of vector bundles is an embedding.
2. A pair of identity maps is an embedding.
3. If an embedding of vector bundles is a pair of homeomorphisms, then the pair of their inverses is also an embedding.
The proof is easy and is omitted.
\[defn2820\] Let $(X,\mathcal E,\pi)$ be as in Definition \[defn2613\].
1. An embedding of vector bundles is said to be an [*isomorphism*]{} if it is a pair of diffeomorphisms of orbifolds.
2. We say that two representatives of a vector bundle structure of $(X,\mathcal E)$ are [*equivalent*]{} if the pair of identity maps regarded as a map between $(X,\mathcal E)$ equipped with those two representatives of vector bundle structures is an isomorphism. This is an equivalence relation by Lemma \[lem26444AA1\].
3. An equivalence class of the equivalence relation (2) is called a [*vector bundle structure*]{} of $(X,\mathcal E)$.
4. A pair $(X,\mathcal E)$ together with its vector bundle structure is called a [*vector bundle*]{} on $X$. We call $\mathcal E$ the [*total space*]{}, $X$ the [*base space*]{}, and $\pi : \mathcal E \to X$ the [*projection*]{}.
5. The condition for $(f,\widehat f) : (X^a,\mathcal E^a) \to (X^b,\mathcal E^b)$ to be an embedding does not change if we replace representatives of vector bundle structures to equivalent ones. So we can define the notion of an [*embedding of vector bundles*]{}.
6. We say $(f,\widehat f)$ is an embedding [*over the orbifold embedding $f$.*]{}
<!-- -->
1. We may use the terminology ‘orbibundle’ in place of vector bundle. We use this terminology in case we emphasize that it is different from the vector bundle over the underlying topological space.
2. We sometimes simply say $\mathcal E$ is a vector bundle on an orbifold $X$.
\[defn2655\]
1. Let $(X,\mathcal E)$ be a vector bundle. We call an orbifold chart $(V,E,\Gamma,\phi,\widehat\phi)$ in the sense of Definition \[defn2613\] (1) of underlying pair of topological spaces $(X,\mathcal E)$ an [*orbifold chart of a vector bundle*]{} if the pair of maps $(\overline\phi,\overline{\widehat\phi}) : (V/\Gamma,(V\times E)/\Gamma) \to (X,\mathcal E)$ induced from $(\phi,\widehat\phi)$ is an embedding of vector bundles.
2. If $(V,E,\Gamma,\phi,\widehat\phi)$ is an orbifold chart of a vector bundle $(X,\mathcal E)$ we say a pair $(E,\widehat\phi)$ a [*trivialization*]{} of our vector bundle on $V/\Gamma$.
3. Hereafter when $(X,\mathcal E)$ is a vector bundle, its ‘orbifold chart’ always means an orbifold chart of a vector bundles in the sense of (1).
4. In case when a vector bundle structure on $(X,\mathcal E)$ is given, a representative of this vector bundle structure is called an [*orbifold atlas*]{} of $(X,\mathcal E)$.
5. Two orbifold charts $(V_i,E_i,\Gamma_i,\phi_i,\widehat \phi_i)$ are said to be [*isomorphic*]{} if there exist an isomorphism $(h,\tilde\varphi)$ of orbifold charts $(V_1,\Gamma_1,\phi_1) \to (V_2,\Gamma_2,\phi_2)$ and an isomorphism $(h,\tilde{\hat{\varphi}})$ of orbifold charts $(V_1\times E_1,\Gamma_1,\widehat\phi_1) \to (V_2\times E_2,\Gamma_2,\widehat\phi_2)$ such that they induce an embedding of vector bundles $(\varphi,\hat\varphi) : (V_1/\Gamma_1,(V_1\times E_1)/\Gamma_1) \to (V_2/\Gamma_2,(V_2\times E_2)/\Gamma_2)$. The triple $(h,\tilde\varphi,\tilde{\hat{\varphi}})$ is called an [*isomorphism*]{} of orbifold charts.
\[lem2619\] Let $(X^b,\mathcal E^b)$ be a vector bundle over an orbifold $X^b$ and $f : X^a \to X^b$ an embedding of orbifolds. Let $\mathcal E^a = X^a \times_{X^b} \mathcal E^b$ be the fiber product in the category of topological space. By the definition of the fiber product we have maps $\mathcal E^a \to X^a$ and $\mathcal E^a \to \mathcal E^b$. We write them $\pi$ and $\widehat f$ respectively. Then the exists a unique structure of vector bundle on $(X^a,\mathcal E^a)$ such that the projection is the above map $\pi$ and that $(f,\widehat f)$ is an embedding of vector bundles.
Let $\{\frak V^*_{\frak r} \mid \frak r \in \frak R_*\}$, $*=a,b$ be orbifold atlases where $\frak V^*_{\frak r} = (V^*_{\frak r},\Gamma^*_{\frak r},\phi^*_{\frak r})$. Let $(V^b_{\frak r},E^b_{\frak r},\Gamma^b_{\frak r},\phi^b_{\frak r},\widehat\phi^b_{\frak r})$ be orbifold atlas of the vector bundle $(X^b,\mathcal E^b)$. Let $(h_{\frak r,ba},\tilde\varphi_{\frak r,ba})$ be a local representative of the embedding $f$ on the charts $\frak V^a_{\frak r}$, $\frak V^b_{\frak r}$. We put $E^a_{\frak r} = E^b_{\frak r}$, on which $\Gamma^a_{\frak r}$ acts by the isomorphism $h_{\frak r,ba}$. By definition of fiber product, there exists uniquely a map $\widehat\phi^a_{\frak r} : V_{\frak r}^b \times E_{\frak r}^b \to \mathcal E^a$ such that the next diagram commutes. $$\label{diag2619diag}
\begin{CD}
V_{\frak r}^a @ <{\pi}<<
V_{\frak r}^a \times E_{\frak r}^b @>{\tilde\varphi_{\frak r,ba}\times id}>> V_{\frak r}^b \times E_{\frak r}^b\\
@ V{\phi^a_{\frak r}}VV @ VV{\widehat\phi^a_{\frak r}}V
@VV{\widehat\phi^b_{\frak r}}V\\
X^a @ < {\pi} <<\mathcal E^a @>{\hat f}>>\mathcal E^b
\end{CD}$$ In fact, $$f\circ \phi^a_{\frak r}\circ \pi
= \phi^b_{\frak r} \circ \varphi_{\frak r,ba} \circ \pi
= \phi^b_{\frak r} \circ \pi \circ (\tilde\varphi_{\frak r,ba}\times id)
= \pi \circ \widehat\phi^b_{\frak r} \circ (\tilde\varphi_{\frak r,ba}\times id).$$ Thus $\{(V^a_{\frak r},E^a_{\frak r},\Gamma^a_{\frak r},\phi^a_{\frak r},\widehat\phi^a_{\frak r}) \mid \frak r \in \frak R\}$ is an atlas of the vector bundle $(X^a,\mathcal E^a)$.
We call the vector bundle in Lemma \[lem2619\] the [*pullback*]{} and write $f^*(X^b,\mathcal E^b)$. (Sometimes we write $f^*\mathcal E^b$ by an abuse of notation.)
In case $X^a$ is an open subset of $X^b$ equipped with open substructure we call [*restriction*]{} in place of pullback of $\mathcal E^b$ and write $\mathcal E^b\vert_{X^a}$ in place of $f^*\mathcal E^b$.
\[lem2622\] In the situation of Lemma \[lem26999\] suppose in addition that $\mathcal E^i$ is a vector bundle over $X^i$ and $\widehat\varphi_{21} : \mathcal E^1 \to \mathcal E^2$ is an embedding of vector bundles over $\varphi_{21}$. Then in addition to the conclusion of Lemma \[lem26999\], there exists $\tilde{\hat{\varphi}}_{\frak r;21} :
V^{1}_{\frak r} \times E_{\frak r}^1
\to V^{2}_{\frak r} \times E_{\frak r}^2$ that is an $h_{\frak r;21}$ equivariant embedding of manifolds with the following properties.
1. The next diagram commutes. $$\label{diagin26777}
\begin{CD}
V^1_{\frak r}\times E^1_{\frak r} @ >{\tilde{\hat{\varphi}}_{\frak r,21}}>>
V^2_{\frak r} \times E^2_{\frak r} \\
@ V{\widehat\phi_{\frak r}^{1}}VV @ VV{\widehat\phi_{\frak r}^{2}}V\\
\mathcal E^1 @ > {\widehat\varphi_{21}} >> \mathcal E^2
\end{CD}$$
2. For each $y \in V^1_{\frak r}$ the map $\xi \mapsto \pi_2(\tilde{\hat{\varphi}}_{\frak r,21}(y,\xi))$ $: E^1_{\frak r} \to E^2_{\frak r}$ is a linear embedding.
The proof is similar to the proof of Lemma \[lem26999\] and is omitted.
\[def28262826\] We call $(h_{\frak r,21},\tilde\varphi_{\frak r,21},\tilde{\hat{\varphi}}_{\frak r,21})$ a [*local representative of embedding*]{} $(\varphi_{21},\widehat\varphi_{21})$ on the charts $(V^1\times E^1,\Gamma^1,\widehat\phi^1)$, $ (V^2\times E^2,\Gamma^2,\widehat\phi^2)$.
\[lem2715\] If $(h_{\frak r,21},\tilde\varphi_{\frak r,21},\tilde{\hat{\varphi}}_{\frak r,21})$, $(h'_{\frak r,21},\tilde\varphi'_{\frak r,21},\tilde{\hat{\varphi}}'_{\frak r,21})$ are local representatives of an embedding of vector bundles of the same charts $(V^1\times E^1,\Gamma^1,\widehat\phi^1)$, $ (V^2\times E^2,\Gamma^2,\widehat\phi^2)$, then there exists $\mu \in \Gamma^2$ such that $$\tilde\varphi_{\frak r,21}'(x) = \mu\tilde\varphi_{\frak r,21}(x),
\quad
\tilde{\hat{\varphi}}_{\frak r,21}'(x,\xi) = \mu\tilde{\hat{\varphi}}_{\frak r,21}(x,\xi)
\quad
h_{\frak r,21}'(\gamma) = \mu h_{\frak r,21}(\gamma)\mu^{-1}.$$
This is a consequence of Lemma \[lem21tenhatena\].
In Situation \[opensuborbifoldchart\] we introduced the notation $(h_{\frak r,21},\tilde\varphi_{\frak r,21},\breve{{\varphi}}_{\frak r,21})$ where $\breve{{\varphi}}_{\frak r,21}$ is related to $\tilde{\hat{\varphi}}_{\frak r,21}$ by the formula $$\tilde{\hat{\varphi}}_{\frak r,21}(y,\xi)
=(\tilde{{\varphi}}_{\frak r,21}(y),\breve{{\varphi}}_{\frak r,21}(y,\xi)).$$
We use the pullback of vector bundles in a different situation. Let $\mathcal E^i$, $i=1,2$, are vector bundles over an orbifold $X$. We take the Whitney sum bundle $\mathcal E^1 \oplus \mathcal E^2$. Let $\vert\mathcal E^1 \oplus \mathcal E^2\vert$ be its total space. There exists a projection $$\label{projfromWhe}
\vert\mathcal E^1 \oplus \mathcal E^2\vert
\to \vert\mathcal E^2\vert.$$
\[pullbackbyproj\] $\vert\mathcal E^1 \oplus \mathcal E^2\vert$ has a structure of vector bundle over $\vert\mathcal E^2\vert$ such that (\[projfromWhe\]) is the projection. We write it as $\pi_{\mathcal E^2}^*\mathcal E^1$ and call the [*pullback*]{} of $\mathcal E^1$ by the projection $\pi_{\mathcal E^2} : \vert\mathcal E^2\vert \to X$.
When $U$ is an open subset of $\vert\mathcal E^2\vert$ and $\pi : U \to X$ is the restriction of $\pi_{\mathcal E^2}$ to $U$, the pullback $\pi^*_{\mathcal E^2}\mathcal E^1$ is by definition the restriction of $\pi_{\mathcal E^2}^*\mathcal E^1$ to $U$.
The proof is immediate from definition.
We note that the total space $\vert\mathcal E^1 \oplus \mathcal E^2\vert$ is [*not*]{} a fiber product $\vert\mathcal E^1 \vert \times_X \vert\mathcal E^2\vert$. In fact, if $X$ is a point and $\mathcal E^1
= \mathcal E^2= \R^n/\Gamma$ with linear $\Gamma$ action, then the fiber of $\vert\pi_{\mathcal E^2}^*\mathcal E^1 \vert \to
\vert\mathcal E^2\vert \to X$ at $[0]$ is $(E^1 \times E^2)/\Gamma$. The fiber of the map $\vert\mathcal E^1 \vert \times_X \vert\mathcal E^2\vert \to X$ at $[0]$ is $(E^1/\Gamma) \times (E^2/\Gamma)$.
Let $(X,\mathcal E)$ be a vector bundle. A [*section*]{} of $(X,\mathcal E)$ is an embedding of orbifolds $s : X \to \mathcal E$ such that the composition of $s$ and the projection is the identity map (set-theoretically).
\[lem2626\] Let $\{(V_{\frak r},E_{\frak r},\Gamma_{\frak r},
\psi_{\frak r},\widehat\psi_{\frak r})\mid \frak r \in \frak R\}$ be an atlas of $(X,\mathcal E)$. Then a section of $(X,\mathcal E)$ corresponds one to one to the following object.
1. For each $\frak r$ we have a $\Gamma_r$ equivariant map $s_{\frak r} : V_{\frak r} \to E_{\frak r}$, which is compatible in the sense of (2) below.
2. Suppose $\phi_{\frak r_1}(x_1) = \phi_{\frak r_2}(x_2)$. Then the definition of orbifold atlas implies that there exist subcharts $(V_{\frak r_i,x_i},E_{\frak r_i,x_i},\Gamma_{\frak r_i,x_i},
\phi_{\frak r_i,x_i},\widehat\phi_{\frak r_i})$ of the orbifold charts $(V_{\frak r_i},E_{\frak r_i},\Gamma_{\frak r_i},
\phi_{\frak r_i},\widehat\phi_{\frak r_i})$ at $x_i \in V_{\frak r_i}$ for $i = 1,2$ and an isomorphism of charts $$\aligned
(h^{\frak r,p}_{12},\tilde\varphi^{\frak r,p}_{12},
\tilde{\hat{\varphi}}^{\frak r,p}_{12}) ~:~
&(V_{\frak r_2,x_2},E_{\frak r_2},\Gamma_{\frak r_2,x_2},
\phi_{\frak r_2,x_2},\widehat\phi_{\frak r_2})
\\
&\to (V_{\frak r_1,x_1},E_{\frak r_1,x_1},\Gamma_{\frak r_1,x_1},
\phi_{\frak r_1,x_1},\widehat\phi_{\frak r_1}).
\endaligned$$
Now we require the next equality: $$\label{sectioncompati}
\tilde{\hat{\varphi}}^{\frak r,p}_{12}(s_{\frak r_1}(y,\xi)) =
s_{\frak r_2}(\tilde\varphi^{\frak r,p}_{12}(y),\xi).$$
The proof is mostly the same as the corresponding standard result in the case of vector bundle on a manifold or on a topological space. Let $s : X \to \mathcal E$ be a section, which is an orbifold embedding. Let $p \in \phi_{\frak r}(V_{\frak r})$. Then there exist a subchart $(V_{\frak r,p},\Gamma_{\frak r,p},\phi_{\frak r,p})$ of $\frak V_{\frak r}$ and a subchart $(\widehat V_{\frak r,\tilde p},
\Gamma_{\frak r,\tilde p},\phi_{\frak r,\tilde p})$ of $(V_{\frak r}\times E_{\frak r},\Gamma_{\frak r},\phi_{\frak r,\tilde p})$ such that a representative $(h',\tilde\varphi')$ of $s$ exists on this subcharts. Since $\pi \circ s = $identity set-theoretically, it follows that $\pi_1(\tilde\varphi(y)) \equiv y \mod \Gamma_p$ for any $y \in V_{\frak r,p}$. We take $y$ such that $\Gamma_{y} = \{1\}$. Then, there exists [*uniquely*]{} $\mu \in \Gamma_p$ such that $\pi_1(\tilde\varphi'(y)) \equiv \mu y$. By continuity this $\mu$ is independent of $y$. (We use Condition \[convinv\] here.)
We replace $\tilde p$ by $\mu^{-1}\tilde p$ and $(\widehat V_{\frak r,\tilde p},\Gamma_{\frak r,\tilde p},\phi_{\frak r,\tilde p})$ by $(\mu^{-1}\widehat V_{\frak r,\tilde p},\mu^{-1}\Gamma_{\tau,\tilde p} \mu,\phi_{\frak r,\tilde p}\circ \mu)$ and $(h',\tilde\varphi')$ by $(h'\circ {\rm conj}_{\mu},\tilde\varphi' \circ \mu^{-1})$. (Here ${\rm conj}_{\mu}(\gamma) = \mu \gamma \mu^{-1}$.) Therefore we may assume $\pi_1(\tilde\varphi'(y)) = y$. Note that $\tilde\varphi'$ is $h'$-equivariant and $\pi_1$ is ${\rm id}$-equivariant. Here ${\rm id}$ is the identity map $\Gamma_{\frak r,y} \to \Gamma_{\frak r,y}$. Therefore the identity map $V_{\frak r,p} \to V_{\frak r,p}$ is $h'$ equivariant. Hence $h' = {\rm id}$.
In sum we have the following. (We put $s_{\frak r,p} = \tilde\varphi'$.) For a sufficiently small $\frak V_{\frak r,p}$ there exists uniquely a map $
s_{\frak r,p} : V_{\frak r,p} \to V_{\frak r,p} \times E_{\frak r}$ such that
1. $
\pi_1(s_{\frak r,p}(x)) =x
$
2. $s_{\frak r,p}$ is equivariant with respect to the embedding $\Gamma_{\frak r,p} \to \Gamma_{\frak r}$. (Recall $\Gamma_{\frak r,p} =
\{\gamma \in \Gamma_{\frak r} \mid \gamma p = p\}$.)
3. $({\rm id},s_{\frak r,p})$ is a local representative of $s$.
We can use uniqueness of such $s_{\frak r,p}$ to glue them to obtain a map $V_{\frak r} \to V_{\frak r} \times E_{\frak r}$. By (a) this map is of the form $x \mapsto (x,\frak s_{\frak r}(x))$. This is the map $\frak s_{\frak r}$ in (1). Since $x \mapsto \gamma^{-1}\frak s_{\frak r}(\gamma x)$ also has the same property, the uniqueness implies that $\frak s_{\frak r}$ is $\Gamma_{\frak r}$ equivariant. (\[sectioncompati\]) is also a consequence of the uniqueness.
We thus find a map from the set of sections to the set of $(s_{\frak r})_{\frak r \in \frak R}$ satisfying (1)(2). The construction of the converse map is obvious.
The next lemma is proved during the proof of Lemma \[lem2626\].
\[lem2627\] Let $(V_{\frak r},E_{\frak r},\Gamma_{\frak r},\phi_{\frak r},\widehat\phi_{\frak r})$ be an orbifold chart of $(X,\mathcal E)$ and $s$ a section of $(X,\mathcal E)$. Then there exists uniquely a $\Gamma$ equivariant map $s_{\frak r} : V_{\frak r} \to E_{\frak r}$ such that the following diagram commutes. $$\label{diagin26277XXrev}
\begin{CD}
V_{\frak r}\times E_{\frak r} @ >{\widehat\phi_{\frak r}}>>
\mathcal E_{\frak r} \\
@ A{{\rm id} \times s_{\frak r}}AA @ AA{s}A\\
V_{\frak r} @ > {\phi_{\frak r}} >> X
\end{CD}$$
\[defnlocex\] We call the system of maps $s_{\frak r}$ the [*local expression*]{} of $s$ in the orbifold chart $(V_{\frak r},E_{\frak r},\Gamma_{\frak r},\phi_{\frak r},\widehat\phi_{\frak r})$.
We next review the proof of a few well-known facts on pullback bundle etc.. Those proofs are straightforward generalization of the corresponding results in manifold theory. We include them for completeness’ sake only.
\[homotopicpulback\] Let $\mathcal E$ is a vector bundle on $X \times [0,1]$, where $X$ is an orbifold. We identify $X \times \{0\}$, $X \times \{1\}$ with $X$ in an obvious way. Then there exists an isomorphism of vector bundles $$I : \mathcal E\vert_{X \times \{0\}} \cong \mathcal E\vert_{X \times \{1\}}.$$ Suppose in addition that there exists a compact set $K \subset X$ an its neighborhood $V$ and isomorphism $$I_0 : \mathcal E\vert_{V \times [0,1]} \cong \mathcal E\vert_{V \times \{0\}} \times [0,1]$$ Then we may choose $I$ so that it coincides with the isomorphism induced by $I_0$ on $K$.
In the case $K$ is a submanifold we may take $K=V$.
To prove the proposition we use the notion of connection of vector bundle on orbifolds. Note a vector field on an orbifold is a section of the tangent bundle.
A [*connection*]{} of a vector bundle $(X,\mathcal E)$ is an $\R$ linear map $$\nabla : C^{\infty}(TX) \otimes_{\R} C^{\infty}(\mathcal E)
\to C^{\infty}(\mathcal E)$$ such that $\nabla_X(V) = \nabla(X,V)$ satisfies $$\nabla_{fX}(V) = f\nabla_X(V), \qquad
\nabla_{X}(fV) = f\nabla_X(V) + X(f)V.$$
Here $C^{\infty}(\mathcal E)$ is the vector space consisting of all the smooth sections of $\mathcal E$.
For any connection $\nabla$ and piecewise smooth map $
\ell : [a,b] \to X$ we obtain parallel transport $${\rm Pal}^{\nabla} : \mathcal E_{\ell(a)} \to \mathcal E_{\ell(b)}$$ in the same way as the case of manifold.
Here $\mathcal E_{\ell(a)}$ is the fiber of $\mathcal E$ at $\ell(a) \in X$ and is defined as follows. We take a chart $(V_{\frak r},E_{\frak r},\Gamma_{\frak r},\psi_{\frak r},\widehat\psi_{\frak r})$ of $(\mathcal E,X)$ at $\ell(a)$. Then $\mathcal E_{\ell(a)} = E_{\frak r}$. If $(V_{\frak r'},E_{\frak r'},\Gamma_{\frak r'},\psi_{\frak r'},\widehat\psi_{\frak r'})$ is another chart we can identify $E_{\frak r}$ and $E_{\frak r'}$ by $\xi \mapsto \breve{\varphi}_{\frak r'\frak r}(\xi,y)$ where $\psi_{\frak r}(y) = \ell(a)$ and $ \breve{\varphi}_{\frak r'\frak r} : V_{\frak r} \times E_{\frak r} \to E_{\frak r'}$ is a part of the coordinate change. (Situation \[opensuborbifoldchart\].)
Note the identification $\xi \mapsto \breve{\varphi}_{\frak r'\frak r}(\xi,y)$ is well-defined up to the $\Gamma_{\ell(a)}
= \{ \gamma \in \Gamma_{\frak r} \mid \gamma(y) = y\}$ action.
The parallel transport $
{\rm Pal}^{\nabla} : \mathcal E_{\ell(a)} \to \mathcal E_{\ell(b)}
$ is well-defined up to $\Gamma_{\ell(a)} \times \Gamma_{\ell(b)}$ action.
Any vector bundle $\mathcal E$ over orbifold $X$ has a connection. Moreover if a connection is given for $\mathcal E\vert_{V}$ where $V$ is an open neighborhood of a compact subset $K$ of $X$, then we can extend it without changing it on $K$. In the case $K$ is a submanifold we may take $K=V$.
The proof is an obvious analogue of the proof of the existence of connection of a vector bundle on a manifold, which uses partition of unity.
We start the proof of Proposition \[homotopicpulback\]. We take a connection of $\mathcal E\vert_V$. We then take direct product connection on $\mathcal E\vert_{V \times \{0\}} \times [0,1]$, and use $I_0$ to obtain a connection on $ \mathcal E\vert_{V \times [0,1]}$. We extend it to a connection on $\mathcal E$ without changing it on $K \times [0,1]$. Let $x \in X$ then we can use parallel transportation along the path $t \mapsto (x,t)$ to obtain an isomorphism $\mathcal E_{(x,0)} \cong \mathcal E_{(x,1)}$. We thus obtain set theoretical map $$\vert\mathcal E\vert_{X \times \{0\}}\vert \cong \vert\mathcal E\vert_{X \times \{1\}}\vert.$$ It is easy to see that it induces an isomorphism of vector bundles.
Using the fact that our connection is direct product on $K \times [0,1]$, we can check the second half of the statement.
We say two embeddings of orbifold $f_i : X \to Y$ ($i=1,2$) to be [*isotopic*]{} each other if there exists an embedding of orbifolds $H : X\times [0,1] \to Y \times [0,1]$ such that the second factor of $H(x,t)$ is $t$ and that $$H(x,0) = (f_1(x),0) \qquad
H(x,1) = (f_2(x),1).$$ Suppose $V \subset X$ and $f_1 = f_2$ on a neighborhood $V$ of $K$. We say $f_1$ is [*isotopic to $f_2$ relative to $K$*]{} if we may take $H$ such that $$\label{homoisidentity}
H(x,t) = (f_1(x),t) = (f_2(x),t)$$ for $x$ in a neighborhood of $K$.
In the case $K$ is a submanifold we may take $K=V$ and then (\[homoisidentity\]) holds for $x \in K$.
\[cor2939\] Let $f_i : X \to Y$ be two embeddings which are isotopic and $\mathcal E$ is a vector bundle on $Y$. Then the pullback bundle $f_1^*\mathcal E$ is isomorphic to $f_2^*\mathcal E$.
If $f_1 = f_2$ on a neighborhood of $K \subset X$ and $f_1$ is isotopic to $f_2$ relative to $K$ then we may choose the isomorphism $f_1^*\mathcal E \cong f_2^*\mathcal E$ so that its restriction to $K$ is the identity map.
This is an immediate consequence of Proposition \[homotopicpulback\] and the definition.
We next recall Definition \[lem123000\] which we copy below.
\[lem1230002\] Let $f : X \to Y$ be an embedding of orbifolds and $K\subset X$ a compact subset and $U$ be an open neighborhood of $K$ in $Y$. We say that a continuous map $\pi : U \to X$ is diffeomorphic to the projection of normal bundle if the following holds.
Let ${\rm pr} : N_XY \to X$ be the normal bundle. Then there exists a neighborhood $U'$ of $K$ in $ N_XY$, (Note $K\subset X \subset N_XY$.) and a diffeomorphism of orbifolds $h : U' \to U$ such that $\pi\circ h = {\rm pr}$. We also require that $h(x) = x$ for $x$ in a neighborhood of $K$ in $X$.
\[defpullbackbundlenbd\] Suppose $\pi : U \to X$ is diffeomorphic to the projection of normal bundle as in Definition \[lem1230002\] and $\mathcal E$ be a vector bundle on $X$. We define $\pi^*\mathcal E$, the pullback bundle as follows.
Let $h$, $U'$ be as in Definition \[lem1230002\]. We defined a pull back bundle ${\rm pr}^*\mathcal E$ on $N_XY$ in Definition \[pullbackbyproj\]. We put $$\pi^*\mathcal E = (h^{-1})^*{\rm pr}^*\mathcal E\vert_{U'}.$$ This is independent of the choice of $(U',h)$ in the following sense. Let $U'_i$, $h_i$ ($i=1,2$) be two choices. Then we can shrink $U$ and $U'_i$ so that the restriction of $h_i$ becomes an isomorphism between them. Then $$\label{294949}
(h_1^{-1})^*{\rm pr}^*\mathcal E\vert_{U'_1}
\cong
(h_2^{-1})^*{\rm pr}^*\mathcal E\vert_{U'_2}.$$ Moreover the isomorphism (\[294949\]) can be taken so that the following holds in addition. We regard $K \subset U$. Then by definition it is easy to see that the restriction of both sides of (\[294949\]) are canonically identified with the restriction of $\mathcal E$ to $K \subset X$. The isomorphism (\[294949\]) becomes the identity map on $K$ by this isomorphism.
We can replace $U$ by a smaller open neighborhood so that $h_1^{-1} : U \to N_XY$ is isotopic to $h_2^{-1} : U \to N_XY$. (See the proof of Proposition \[prop2942\] below.) Then (\[294949\]) follows from Corollary \[cor2939\].
The second half of the claim follows also from the second half of Corollary \[cor2939\].
Actually the pullback bundle is independent of $\pi$ but depend only on $U$ in the situation of Definition \[lem1230002\]. In fact we have
\[prop2942\] Let $\pi_i : U \to X$ be as in Definition \[lem1230002\] for $i=1,2$. Then there exists a neighborhood $U_0$ of $X$ in $Y$ and $f : U_0 \to U$ such that
1. $
\pi_2 \circ f = \pi_1
$
2. $f : U_0 \to U$ is isotopic to identity relative to $X$.
Let $h_i : U'_i \to U_i$ be as in Definition \[lem1230002\]. We put $f = h_2 \circ h^{-1}_1$ which is defined for sufficiently small $U_0$. If suffices to show that $f$ is isotopic to identity. We first prove it in the case when the following additional assumption is satisfied. (We will remove this assumption later.)
\[assym2929\] For any $x \in K \subset N_XY$ the first derivative at $x$, $D_xf : T_x(N_XY) \to T_x(N_XY)$ is the identity map.
In the case of manifold we can prove Proposition \[prop2942\] in this case, by observing $f$ is $C^1$-close to the identity map. Then for example by using minimal geodesic we can show that $f$ is isotopic to identity.
In the case of orbifold we need to work out this last step a bit more carefully since the number $$\inf \{ r \mid \text{if $d(x,y) < r$ the minimal geodesic joining $x$ and $y$ is unique}\}$$ can be $0$ in general.
We will prove certain lemmata to clarify this point. We need certain digression to state the lemmata. We can define the notion of Riemannian metric of orbifold $X$ in an obvious way. For $p\in X$ we have a geodesic coordinate $(TB_p(c_p),\Gamma_p,\psi_p)$ where $$TB_p(c_p) = \{ \xi \in T_cX \mid \Vert \xi\Vert < c\}.$$ The group $\Gamma_p$ is the isotropy group of the orbifold chart of $X$ at $p$. The uniformization map $\psi : TB_p(c) \to X$ is defined by using minimal geodesic in the same way as the usual Riemannian geometry. We remark that this map is well defined up to the action of $\Gamma_p$.
We need to take the number $c$ small so that $\psi$ induces homeomorphism $TB_p(c)/\Gamma_p \to X$. We can not choose $c$ uniformly away from $0$ even in the compact set. (This is because $d(p,q) < c_p$ must imply $\#\Gamma_q \le \# \Gamma_p$.) However we can prove the following.
Let $X$ be an orbifold and $Z$ be a compact set. Suppose $B_{c_0}(Z) = \{x \mid d(x,Z) \le c_0\}$ is complete with respect to the metric induced by the Riemannian metric.
Let $Z \subset X$ be a compact subset. Then there exists a finite set $\{p_i \mid j \in J\} \subset Z$ and $0 < c_j < c_0$ such that
1. The geodesic coordinate $(TB_{p_j}(c_j),\Gamma_{p_j},\psi_{p_j})$ exists.
2. $$Z \subset \bigcup_{j} \psi_{p_j}(TB_{p_j}(c_j/2)).$$
The proof is immediate from the compactness of $Z$. We call such $\{(TB_{p_j}(c_j),\Gamma_{p_j},\psi_{p_j}) \mid j\}$ a [*geodesic coordinate system*]{} of $(X,Z)$. We put $P = \{p_j \mid j =1,\dots, J\}$.
\[lem2945\] We fix a geodesic coordinate system of $(X,K)$. Let $Z_0 \subset Z$ be a compact subset containing $P$. Suppose $F : U \to X$ be an embedding of orbifold where $U \supset Z$ is an open neighborhood of $Z$. We say $F$ is [*$C^1$-$\epsilon$ close to identity*]{} on $Z_0$ if the following holds.
1. $F(B_{p_j}(c_j/2)) \subset B_{p_j}(c_j)$.
2. There exists $\tilde F_{j} : B_{p_j}(c_j/2) \to B_{p_j}(c_j)$ such that:
1. $
\psi_{p_j} \circ \tilde F_{j} = F \circ \psi_{p_j}$.
2. $d(x,\tilde F_{j}(x)) < \epsilon$ for $x \in TB_{p_j}(c_j/2) \cap \psi_j^{-1}(Z_0)$.
3. $d(D_x\tilde F_{j},id) < \epsilon$ for $x \in TB_{p_j}(c_j/2) \cap \psi_j^{-1}(Z_0)$.
Here $d$ in Item (b) is the standard metric on Euclidean space $T_{p_j}X$ (together with metric induce by our Riemannian metric), $d$ in Item (c) is a distance in the space of $n\times n$ matrices. ($n = \dim X$. We use our Riemannian metric to define a metric on this space of matrices, which is a vector space of dimension $n^2$ with metric.)
\[lem2946\] For each $Z$ and a geodesic coordinate system of $(X,Z)$ there exists $\epsilon$ such that the following holds for any $Z_0$ and $F : U \to X$.
If $F$ is $C^1$-$\epsilon$ close to identity on $Z_0$ then $F$ is isotopic to the identity on $Z_0$.
Moreover for any $\delta$ there exists $\epsilon(\delta)$ that that if $F$ is $C^1$-$\epsilon(\delta)$ close to identity on $Z_0$ then the isotopy from $F$ to the identity map is taken to be $C^1$-$\delta$ close to identity on $Z_0$.
We first observe that if $\epsilon$ is sufficiently small then the maps $\tilde F_j$ satisfying Definition \[lem2945\] (2) (a),(b) and (c) is unique. In fact such $\tilde F_j$ is unique up to the action of $\Gamma_{p_j}$. Since the $\Gamma_{p_j}$ action is effective, $\Gamma_{p_j}$ is a finite group and $p_j \in Z_0$, we find that at most one such $\tilde F_j$ can satisfy (c). (Note the map $\Gamma_{p_j} \to O(n)$ taking the linear part at $p_j$ of the action is injective since $\Gamma_{p_j}$ action has a fixed point and is effective.)
We define for $t \in [0,1]$ a map $$\tilde F_{t,j} : V_j \to TB_{p_j}(c_j/2)$$ as follows. Here $V_j$ is a sufficiently small neighborhood of $TB_{p_j}(c_j/2) \cap \psi_j^{-1}(Z_0)$. We take a Riemannian metric on $V_j$ which is a pullback of our Riemannian metric on $X$ by $\psi_{p_j}$. By choosing $\epsilon$ sufficiently small and using (b), there exists a unique minimal geodesic $\ell_{x,j} : [0,1] \to TB_{p_j}(c_j/2+2\epsilon)$ joining $x$ to $\tilde F_{j}(x)$. We put $$\tilde F_{t,j}(x) = \ell_{x,j}(t).$$ In the same way as the proof of the uniqueness of $\tilde F_j$ we can show that there exists $F_t$ such that $
\psi_{p_j} \circ \tilde F_{t,j} = F_t \circ \psi_{p_j}$. Using Definition \[lem2945\] (2) (b) and (c) we can show that $F_t$ together with its first derivative is close to the identity map. We can use it to show that $F_t$ is a diffeomorphism to its image. Thus $F_t$ is the required isotopy from $F$ to the identity map.
We now use Lemma \[lem2946\] to prove Proposition \[prop2942\] under Assumption \[assym2929\].
Note we put $f = h_2 \circ h^{-1}_1$ and we want to show that $f$ is isotopic to the identity map in a neighborhood of $K \subset X \subset Y$.
We take a finite cover $
K \subset \bigcup_{j} \psi_{p_j}(TB_{p_j}(c_j/2))
$ by geodesic coordinate, where $p_j \in K$. We take a compact neighborhood $Z \subset K$ such that $Z \subset \bigcup_{j} \psi_{p_j}(TB_{p_j}(c_j/2))$. We apply Lemma \[lem2945\] to obtain $\epsilon$. Note $f$ is the identity map on $K$. Moreover its first derivative is identity at $K$ by Assumption \[assym2929\]. Therefore we can find a compact neighborhood $Z_0$ of $K$ sufficiently small so that $f$ is $C^1$-$\epsilon$ close to identity on $Z_0$. Thus Lemma \[lem2945\] implies that $f$ is isotopic to the identity map. The proof of Proposition \[prop2942\] under the additional Assumption \[assym2929\] is complete.
To remove Assumption \[assym2929\], we use the next lemma.
\[lem292333\] Let $U$ be an open neighborhood of $K$ in $N_XY$ and $F : U \to N_XY$ be an open embedding of orbifold.
Assume $F =$ identity on a neighborhood of $K$ in $X$ and $D_xF(V) \equiv V \mod T_xV$ for $x$ in a neighborhood of $K$.
Then there exists a smaller neighborhood $U'$ of $K$ such that the restriction of $F$ to $U'$ is isotopic to the embedding satisfying Assumption \[assym2929\].
\[lem2947\] We take the first derivative of $F$ at points $x$ in a neighborhood of $K$ in $X$ and obtain $$D_x F : T_x N_XY \to N_XY$$ Note $T_xX \subset N_XY$ is preserved by this map and $T_x N_XY = T_xX \oplus (N_XY)_x$. Therefore there exists linear bundle map $$H : N_XY \to TX$$ on a neighborhood of $K$ such that $$D_xF (V_1,V_2) = (V_1 + H_x(V_2), V_2),$$ where $V_1 \in T_xX$ and $V_2 \in (N_XY)_x$. Now we define $G_t : U' \to N_XY$ as follows. ($U'$ is a small neighborhood of $K$ in $N_XY$.) Let $(x,V) \in U'$. Here $x$ is in a neighborhood of $K$ in $X$ and $V \in (N_XY)_x$. We take a geodesic $\ell : [0,1] \to X$ of constant speed with $\ell(0) = x$ and $D\ell/dt(0) = H_x(V)$. Let $\ell_{\le t}$ be its restriction to $[0,t]$. Then $G_t(x,V) = (\ell(t),{\rm Pal}_{\ell_{\le t}}(V))$, here ${\rm Pal}_{\ell_{\le t}}(V) \in (N_XY)_{\ell(t)}$ is the parallel transport along $\ell_{\le t}$. By construction the first derivative of $G_t$ at a point in $K$ is given by $$(V_1,V_2)
\mapsto
(V_1 + tH_x(V_2), V_2),$$ which is invertible. Therefore, if $V'$ is sufficiently small neighborhood of $K$, then the restriction of $G_t$ is an embedding $V' \to N_XY$.
Note $F\circ G_1^{-1}$ satisfies Assumption \[assym2929\]. The proof of Lemma \[lem292333\] is complete.
Using Lemma \[lem292333\] we can reduce the general case of Proposition \[prop2942\] to the case when Assumption \[assym2929\] is satisfied. The proof of Proposition \[prop2942\] is complete.
We used the next result in Subsection \[subsection:bdlextcompa\].
\[prop2949\] Let $f : X \to Y$ be an embedding of orbifolds and $K_i$ compact subsets of $X$ for $i=1,2$ such that $K_1 \subset {\rm Int}K_2$. Suppose $U_1$ is an open neighborhood $K_1$ in $Y$ and $\pi_1 : U_1 \to X$ such that it is diffeomorphic to the normal bundle.
Then there exists an open neighborhood $U_2$ of $K_2$ in $Y$ and $\pi_2 : U_2 \to X$ such that it is diffeomorphic to the normal bundle and $\pi_1 = \pi_2$ on a open neighborhood of $K_1$.
In the case of manifold this is a standard result and can be proved using isotopy extension lemma. By applying Lemma \[lem2946\] we can prove it in the same way in orbifold case. For completeness’ sake we give detail of the proof below.
We first apply [@fooooverZ Lemma 6.5] to obtain $U'_2$ and $\pi'_2 : U'_2 \to X$ such that $U'_2 \supset K_2$ and $(U'_2,\pi'_2)$ is diffeomorphic to the normal bundle.
We modify it to obtain $\pi_2$ so that $\pi_1 = \pi_2$ on an open neighborhood of $K_1$. The detail follows.
Let $W_1^{(i)}$ be a neighborhood of $K_1$ in $X$ such that $$\overline{W_1^{(1)}} \subset W_1^{(2)} \subset \overline{W_1^{(2)}}
\subset U_1 \cap X,$$ Let $\Omega$ be an open subset of $U_1$ with $$\overline{W_1^{(2)}} \subset \Omega
\subset \overline\Omega \subset U_1.$$ Later on we will choose $\Omega$ so that it is in a sufficiently small neighborhood of $X$. We put $$V_1^{(i)} = \pi_1^{-1}(W_1^{(i)}) \cap \Omega.$$ Let $\tilde\chi : {\rm Int}(K_2) \cup \Omega \to [0,1]$ be a smooth function such that $$\tilde\chi =
\begin{cases}
1 &\text{on $W_1^{(1)}$} \\
0 &\text{on the complement of $W_1^{(2)}$}.
\end{cases}$$ and put $\chi = \tilde\chi \circ \pi'_2$. We may choose $\Omega$ small so that the following holds. $$\chi =
\begin{cases}
1 &\text{on $V_1^{(1)}$} \\
0 &\text{on the complement of $V_1^{(2)}$}.
\end{cases}$$
Let $Z = \overline{W_1^{(2)}} \setminus {W_1^{(1)}}$. We take a neighborhood $U'$ of $Z$ and restrict $\pi_1$ and $\pi'_2$ there. Then we can apply Proposition \[prop2942\] to prove that there exists an isotopy $F_t : U' \to X$ such that $\pi'_2 \circ F_1 = \pi_1$ and $F_0$ is the identity map. Now we put $$\pi_2(x)
=
\begin{cases}
(\pi_2 \circ F_{\chi(x)})(x) &\text{on $U'$} \\
\pi_1 &\text{ on $V_1^{(1)}$} \\
\pi'_2(x) &\text{elsewhere on $U'_2$}.
\end{cases}$$ It is easy to see that they are glued to define a map. To complete the proof it suffices to show that $x \mapsto F_{\chi(x)}(x)$ is an embedding $: U' \to X$. We will prove it below.
We first assume that Assumption \[assym2929\] is satisfied for the map $f : U' \to X$ with $\pi'_2 \circ f = \pi_1$. In this case we may choose the isotopy $F_t$ to be arbitrary close to the identity map in $C^1$ sense by taking $\Omega$ small. (This is the consequence of the second half of Lemma \[lem2946\].) Therefore the first derivative of $x \mapsto F_{\chi(x)}(x)$ is close to identity. It follows that this map is an embedding.
We finally show that we can choose $(U'_2,\pi'_2)$ so that Assumption \[assym2929\] is satisfied for the map $f : U' \to X$ with $\pi'_2 \circ f = \pi_1$.
We consider the fiber $\pi_1^{-1}(x)$ of $\pi_1$. We may choose the Riemannian metric of $X$ in a neighborhood of $K_1$ so that this fiber is perpendicular to $X$ for any $x$ in a neighborhood of $K_1$. We now extend this Riemannian metric to the whole $X$. We use this Riemannian metric and exponential map in the normal direction to identify a neighborhood of $K_2$ with a normal bundle and to obtain $U'_2$ and $\pi'_2$. Then Assumption \[assym2929\] is satisfied.
The proof of Proposition \[prop2949\] is now complete.
[FOOOXX]{}
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[^1]: Kenji Fukaya is supported partially by JSPS Grant-in-Aid for Scientific Research No. 23224002 and NSF Grant No. 1406423, Yong-Geun Oh by the IBS project IBS-R003-D1, Hiroshi Ohta by JSPS Grant-in-Aid for Scientific Research No. 23340015 and Kaoru Ono by JSPS Grant-in-Aid for Scientific Research, Nos. 26247006, 23224001.
[^2]: In our main application, such a system is constructed from the moduli spaces of pseudo-holomorphic curves. However those results we mention here is independent of the origin of such a system.
[^3]: When teaching a course of manifold, it is important not to over-emphasize the importance of these issues so that the students will not get lost in the technicalities.
[^4]: The paper [@ChenTian] was published in 2010 but this paper had appeared in the arXiv in 2006.
[^5]: One reason why it does not seem to work is that there is an automorphism which exchange irreducible components, in general.
[^6]: We had experiences to be complained when we wrote similar sentences in our earlier papers. We agree that it is preferable to avoid it. However it is sometimes impossible to do so, especially in a research paper whose main focus is application and is not the detail of foundation.
[^7]: except the choice of smooth structure of gluing parameter.
[^8]: This is the viewpoint taken and insisted by Grotendieck.
[^9]: Theory of singularity of $C^{\infty}$ functions is one typical example.
[^10]: By this reason in many places we do not need to say much about the proof.
[^11]: Thorough knowledge of such a technicality, of course, should be shared among the people whose interest lie also on extending the technology to the extreme of its potential border and/or using the most delicate and difficult case of the technology to obtain as the sharpest results as possible.
[^12]: The authors thank D. Joyce we call attention this point to us.
[^13]: Actually this is automatic from our definition, since the submersivity from manifolds with corners implies that its restriction to each corner (of any fixed codimension) is a submersion, by definition.
[^14]: In certain situations, for example in [@fooobook2 Subsection 8.8], we discussed slightly more general case. Namely we discussed the case when $\widehat{\mathcal U}$, $M_s$ and $M_t$ are not necessarily orientable by introducing appropriate $\Z_2$ local systems. See [@fooobook2 Section A2] or an appendix to Part 2.
[^15]: In [@FO] the case when the dimension is higher is also discussed. Those cases are more nontrivial to handle. However none of the application in [@FO] uses the case when the dimension of $(X,\widetriangle{\mathcal U})$ is higher than $1$.
[^16]: Since Čech cohomology behaves nicely with respect to the projective limit, it might be the best choice if we want to define chain model directly on the topological space $X$.
[^17]: We disregard ‘objections’ directed to the points whose answers we had already provided before they were presented.
[^18]: Note however that there is an error related to the notion of ‘germ of Kuranishi neighborhood’ in [@FO], which was explained in [@foootech Subsection 34.1]. This error had been corrected in [@fooobook2 Section A1].
[^19]: The proof of continuity is the same as Remark \[remrem327\].
[^20]: The fiber product in the sense of category theory is always associative if it exists. Since we do [*not*]{} study morphism between K-spaces, the fiber product we defined is [*not*]{} a fiber product in the sense of category theory. Therefore we need to prove its associativity. However it is obvious in our case.
[^21]: We will discuss multisection in Section \[sec:multisection\]. Here we just mention it to motivate the definition we give in this section. The readers who do not know the definition of mutisection can safely skip the part before Definition \[stratadim\].
[^22]: Note the definition of multisection we use here exactly the same as one of the smooth multisection in [@FO].
[^23]: See Definition \[defn285555\] (1) for the definition of diffeomorphism here.
[^24]: We remark that actually we can make sense the equality (\[formula611\]) without taking and fixing the splitting $
\mathcal E_{\frak p}
\equiv
\mathcal E_{\frak q;\frak p} \oplus \frac{\mathcal E_{\frak q}}{\mathcal E_{\frak q;\frak p}}
$, since the projection $\mathcal E_{\frak p} \to \frac{\mathcal E_{\frak q}}{\mathcal E_{\frak q;\frak p}}$ is well-defined without fixing this splitting.
[^25]: In certain situations, for example in [@fooobook2 Subsection 8.8], we discussed slightly more general case. Namely we discussed the case when $\widehat{\mathcal U}$, $M_s$ and $M_t$ are not necessarily orientable by introducing appropriate $\Z_2$ local systems. See [@fooobook2 Section A2].
[^26]: The tubular distance function is, roughly speaking, the distance from the image $\varphi_{\frak p\frak q}(U_{\frak p\frak q})$. We do not give the precise definition here since we do not use this notion. See [@Math73].
[^27]: To find a topology which s metrizable, we consider a support system $\mathcal K$ and use the fact that $\partial X
= \bigcup \mathcal K_{\frak p} \cap (s_{\frak p}^{\partial})^{-1}(0)$ and [@foooshrink Proposition 5.1].
[^28]: The proof below is basically the same proof as written in[@foootech Section 7]. (The proof in [@foootech Section 7] is a detailed version of one given in [@FO page 957-958].) However we polish the presentation and reorganize the proof slightly so that it becomes shorter and easier to read.
[^29]: We need to use several maps between underlying topological spaces of orbifolds, such as projection of bundles or covering maps. In case we include those maps, we need to see carefully whether set-theoretical equality is enough to show various properties of them are preserved.
[^30]: The $\Gamma$ action on $X$ is trivial.
|
---
abstract: 'Let $P^k_\ell$ denote the loose $k$-path of length $\ell$ and let define $f^k_\ell(n,m)$ as the minimum value of $\Delta(H)$ over all $P^k_\ell$-free $k$-graphs $H$ with $n$ vertices and $m$ edges. In the paper we study the behavior of $f^4_2(n,m)$ and $f^3_3(n,m)$ and characterize the structure of extremal hypergraphs. In particular, it is shown that when $m\sim n^2/8$ the value of each of these functions drops down from $\Theta(n^2)$ to $\Theta(n)$.'
address:
- 'Adam Mickiewicz University, Faculty of Mathematics and Computer Science ul. Umultowska 87, 61-614 Poznań, Poland'
- 'Adam Mickiewicz University, Faculty of Mathematics and Computer Science ul. Umultowska 87, 61-614 Poznań, Poland'
author:
- 'Tomasz [Ł]{}uczak'
- Joanna Polcyn
date: 'March 22, 2017'
title: 'Paths in hypergraphs: a rescaling phenomenon'
---
[^1]
Introduction
============
In extremal graph theory we often study functions which emerge when we appropiately scale the extremal parameters of graphs. A typical example is the minimum number of triangles, scaled by $\binom n3$, in graphs on $n$ vertices and density $p$. The celebrated result of Razborov [@Ra] gives a full description of this function as a function of $p$; in particular he showed it is smooth everywhere except at the points $1-1/t$ for integer $t$ (for a similar result on cliques of larger size see Reiher [@R]). Thus, at these points a kind of the continuous phase transition takes place, which is related to the structural changes of the graph on which the minimum is attained.
It is not too hard to construct examples which exhibits a much more rapid, discontinuous change of the structure. In this paper however we give examples of two functions where not only such a transition is discontinuous but the studied function rapidly drops to zero and so requires another rescaling.
In order to state our result we need a few definitions. By a $k$-uniform hypergaph $H=(V,E)$ on $n$ vertices or, briefly, $k$-graph, we mean the family of $k$-element subsets (called edges) of a set of vertices of $H$. Let ${P^{k}_\ell}$ denote the loose $k$-uniform path of length $\ell$, i.e. the connected linear $k$-graph with $\ell$ edges and $k\ell-\ell+1$ vertices. Our aim is to exhibit a ‘rescaling phenomenon’ for the maximum degree in 4-graphs which contains no loose paths of length two and 3-graphs without loose paths of length three. In particular we prove the following two results (for more precise statements see Theorems \[th1a\], \[th2a\] below).
\[th1\] There exists $n_1$ such that for every $P_2^4$-free $4$-graph $H$ with $n\ge n_1$ vertices and $m\ge \binom{\lfloor n/2\rfloor}2+1$ edges we have $\Delta(H)\ge n^2/32-n$.
On the other hand, for every $n\ge 4$ there exists a $P_2^4$-free 4-graph $H_0$ with $m= \binom{\lfloor n/2\rfloor}2$ edges and $\Delta(H)= \lfloor n/2\rfloor-1$.
\[th2\] There exists $n_2$ such that for every $P_3^3$-free $3$-graph $H$ with $n\ge n_2$ vertices and $m\ge n^2/8 +1$ edges we have $\Delta(H)\ge n^2/32-n$.
On the other hand, for every $n\ge 4$ there exists a $P_3^3$-free 3-graph $H_0$ with $m=\lfloor n^2/8\rfloor$ edges and $\Delta(H)\le \lceil n/2\rceil $.
Paths of length two in 4-graphs
===============================
In this section we study the maximum degree of hypergraphs which contains no paths of length two. For 2-graphs the problem is trivial since each graph without paths of length two clearly consists of isolated edges. For 3-graphs the problem is also not very exciting. It is easy to see that every component of 3-graph without $P^3_2$ is either a [*2-star*]{}, i.e. consists of edges which contain two given vertices, or is a subgraph of the complete 3-graph on four vertices. Since the latter graph is denser, 3-graph without paths of length two on $n$ vertices contains at most $\lfloor (n+1)/4\rfloor+3\lfloor n/4\rfloor$ edges and this maximum number is achieved, for instance, for the 3-graph which consists of disjoint cliques of size four and, perhaps, one isolated edge (in the case $n\equiv 3$ (mod 4)). Hence the minimum maximum degree of any $P^3_2$-free graph is three.
For 4-graphs the problem starts to be interesting. Indeed, let us recall that, at least for large $n$, the maximum number of edges in $P^4_2$-free graph on $n$ vertices is $\binom {n-2}2$ and it is achieved only for 2-stars in which there is a vertex which is contained in every edge of 4-graph; more precisely the following result was proved by Keevash, Mubayi, and Wilson [@KMW].
\[KMW\] If $h(n)$ denote the maximum number of edges in a ${P^{4}_2}$-free 4-graph on $n$ vertices, then $$h(n)=\begin{cases}
\binom n4 &\quad \textrm{for}\quad n=4,5,6,\\
15 &\quad \textrm{for}\quad n=7,\\
17 &\quad \textrm{for}\quad n=8,\\
\binom {n-2}2 &\quad \textrm{for}\quad n\ge 9.\\
\end{cases}$$
In order to state our result precisely we introduce some notation. For $n$ large enough and $m\le \binom {n-2}2$ let function ${f^4_2}(n,m)$ be defined as $$\begin{gathered}
{f^4_2}(n,m)=\min\{\Delta(H): H=(V,E)\textrm{\ is a $4$-graph such that \ }\\
|V|=n, |E|=m, \textrm{\ and\ }H\not\supset P^4_2\},\end{gathered}$$ where here and below $\Delta (H)$ denotes the maximum degree of $H$. By ${\mathcal{F}_2^4}(n,m)$ we denote the ‘extremal’ family of $P^4_2$-free $4$-graphs on $n$ vertices and $m$ edges such that $\Delta(H)={f^4_2}(n,m)$. By $\tilde K^4_n$ we mean the [*thick $n$-clique*]{}, i.e. the graph on $n$ vertices (almost) partitioned into $\lfloor n/2\rfloor $ ‘dubletons’ such that any pair of dubletons form an edge of $\tilde K^4_n$. The main theorem of these section can be stated as follows.
\[th1a\] There exists $\bar n_1$ such that for every $n\ge \bar n_1$ and $$\binom{\lfloor n/2\rfloor}2-\frac n5\le m\le \binom{\lfloor n/2\rfloor}2$$ each graph from ${\mathcal{F}_2^4}(n,m)$ is a subgraph of a thick clique.
Moreover, there exist $\tilde n_1$ such that for every $n\ge \tilde n_1$ and all $m\ge \binom{\lfloor n/2\rfloor}2+1$ each graph from ${\mathcal{F}_2^4}(n,m)$ has the maximum degree at least $n^2/32-n$ and one can delete from it at most $470$ edges to obtain a union of at most four 2-stars and some number of isolated vertices.
Clearly, Theorem \[th1\] follows from Theorem \[th1a\]. Before we present its proof let us mention few of its other consequences. It is easy to see that if we want to minimize the maximum degree in union of $r$ stars for a given $n$, $m$ and $r$, we need to make the $r-1$ largest stars roughly as equal as possible. On the other hand subgraphs of a thick clique can be made almost regular, so for small $m$ the function ${f^4_2}(n,m)$ decreases linearly with $m$. This observations lead directly to the following result.
\[cor1a\] For every $x\in [0,1/4)\cup (1/4,1]$ the limit $$f(x)=\lim_{n\to\infty} \frac{{f^4_2}(n,x \binom{n-2}{2})}{\binom{n-2}2}$$ exists and $$f(x)=\begin{cases}0&\textrm{\ for\ }\quad 0\le x< 1/4,\\
(1+2x+\sqrt{12x-3})/24&\textrm{\ for\ }\quad 1/4<x< 1/3,\\
(1+3x+2\sqrt{6x-2})/18&\textrm{\ for\ }\quad 1/3<x< 1/2,\\
(x+\sqrt{2x-1})/2&\textrm{\ for\ }\quad 1/2<x\le 1.\end{cases}$$
Moreover, for every $m\le \binom{\lfloor n/2\rfloor}2$ we have $$\left\lfloor\frac{4m}{n-1}\right\rfloor\le {f^4_2}(n,m)\le \left\lceil\frac{4m}{n}\right\rceil\,.\qed$$
We note also that once the function ${f^4_2}(n,m)$ drops from $\Theta(n^2)$ to $\Theta(n)$ it becomes ‘more stable’, i.e. the following result holds.
\[cor1b\] For large enough $n$ the following holds.
1. If $\binom{\lfloor n/2\rfloor}2+4\le m\le \binom {n-2}2$, then ${f^4_2}(n,m-4)<{f^4_2}(n,m)$.
2. If $\binom{\lfloor n/2\rfloor}2-\frac n{10}\le m\le \binom{\lfloor n/2\rfloor}2$, then ${f^4_2}(n,m)=\lfloor n/2\rfloor -1$.
The main ingredient of our argument is the following decomposition lemma which is true for all ${P^{4}_2}$-free 4-graphs no matter what are their density.
\[podzialp4\] For any ${P^{4}_2}$-free 4-graph $H$ there exists a partition of its set of vertices $V=R\cup S\cup T$, such that subhypergraphs of $H$ defined as $H_R=\{h\in H:h\cap R\neq \emptyset\}$, $H_S=H[S]$ and $H_T=H\setminus(H_R\cap H_S)=\{h\in H[V\setminus R]: h\cap T\neq\emptyset\}$ satisfy:
1. $|H_R|\le 10|R|$,
2. $H_S$ is a subgraph of a thick clique, and so $|H_S|\le {\lfloor |S|/2\rfloor\choose 2} $,
3. $H_T$ is a family of disjoint 2-stars such that centers of these stars are in $S$ whereas all other vertices are in $T$. In particular, $|H_T|\le \binom{|T|}2$.
Let $H$ be a ${P^{4}_2}$-free 4-graph with the set of vertices $V$, $|V|=n$, and the set of edges $E$, $|E|=m$. We start with defining the set of exceptional vertices $R\subset V$. We put into $R$ vertices of degree at most ten one by one, until only vertices of degree at least eleven remain. Then, clearly,
$$\label{hrp4}
|H_R|\le 10|R|,$$
Let us consider the 4-graph ${\hat{H}}=H[V\setminus R]=({\hat{V}}, {\hat{E}})$. For a set $S\subseteq {\hat{V}}$ by its [*signature*]{} ${\operatorname{sg}}(S)$ we mean the projection of the edges of ${\hat{H}}$ into $S$, i.e. $${\operatorname{sg}}(S)=\{S\cap e: e\in {\hat{E}}\}\,.$$ Our argument is based on the number of facts on signatures of $e\in{\hat{E}}$.
\[si1\] The signature of each edge $e$ of ${\hat{H}}$ contains no singletons and at least one dubleton. Moreover, each vertex of $e$ is contained in at least one element of the signature.
The first part of the statement follows from the fact that ${\hat{H}}$ is ${P^{4}_2}$-free. Now take $e\in {\hat{E}}$. Since the degree of each vertex $v\in E$ is at least eleven, so it must be contained in at least one set from ${\operatorname{sg}}(e)$. Finally, if ${\operatorname{sg}}(e)$ contains no dubletons, then each edge $e'$ intersecting $e$ must share with it precisely three elements. But then, for $v'=e'\setminus e$, the vertex $v'\in {\hat{V}}$ has degree at most four, contradicting our assumption on ${\hat{H}}$.
\[si2\] If the signature of an edge $e$ from ${\hat{H}}$ contains two dubletons they are disjoint.
Let $e_1=\{x_1,x_2,x_3,x_4\}\in {\hat{H}}$ and let $\{x_1,x_2\},\{x_2,x_3\}\in {\operatorname{sg}}(e_1)$. Then, there exist in $H$ two other edges, $e_2=\{x_1,x_2,y_1,y_2\}$ and $e_3=\{x_2,x_3,y_2,y_3\}$, where $y_1,y_2,y_3\notin e_1$. Set $V_1=e_1\cup e_2\cup e_3$. We argue that at least one of vertices in the component of ${\hat{H}}$ containing $V_1$ has degree at most 10 contradicting the definition of ${\hat{H}}$.
Let us first consider the case where $y_1\neq y_3$. Since $H$ is $P^4_2$-free, the signature of $V_1$ contains no singletons, but one can easily verify that it cannot contain dubletons either. Note also that there exists an edge $e'$ not contained in $V_1$ but intersecting it, since otherwise, because of the degree restriction, $V_1$ would contain at least $11\cdot 7/4>19$ edges, contradicting Theorem \[KMW\]. Furthermore, one can check that to avoid ${P^{4}_2}$, any edge $e'$ not contained in $V_1$ can intersect $V_1$ on one of ten possible triples. But this means that the vertex $v=e'\setminus V_1$ has degree at most ten, contradicting the choice of ${\hat{H}}$.
Now let us assume that $y_1=y_3$. Note that to avoid ${P^{4}_2}$ any edge containing $x_4$ not contained in $V_1$ must be of type $\{v,y_i,x_2,x_4\}$. Since the degree of $x_4$ is at least eleven and it belongs to at most $\binom 53=10$ edges contained in $V_1$ such an edge, say, $e_4=\{v,y_1,x_2,x_4\}$ exists. But now any edge which intersect set $V_1$ on two vertices and does not contain $v$ creates a copy of ${P^{4}_2}$ and there are only five triples which added to $v'\notin V_1\cup \{v\}$ create no copy of ${P^{4}_2}$. Since $V_1\cup \{v\}$ cannot be a component of ${\hat{H}}$ (by the degree restriction such a component would contain more than $15$ edges contradicting Theorem \[KMW\]), the assertion follows.
\[si3\] The signature of no edge of ${\hat{H}}$ contains a triple and a dubleton which intersect on one vertex.
If the signature of $e_1=\{x_1,x_2,x_3,x_4\}\in {\hat{H}}$ contains a triple $\{x_1,x_2,x_3\}$ and a dubleton $\{x_3,x_4\}$, then there exist in ${\hat{H}}$ two edges, $e_2=\{x_1,x_2,x_3,y_1\}$ and $e_3=\{x_3,x_4,y_1,y_2\}$, with $y_1,y_2\notin e_1$. But then the signature of $e_3$ contains two dubletons sharing exactly one vertex contradicting Claim \[si2\].
\[si4\] Signature of each edge of ${\hat{H}}$ consists either of two disjoint dubletons or one dubleton and two triples intersecting on this dubleton.
It is a straightforward consequence of Claims \[si1\]-\[si3\].
Now we are ready to show Lemma \[podzialp4\]. We call a pair of vertices $\{x,y\}\subset {\hat{V}}$ *a twin* if there is no edge $e\in {\hat{E}}$ such that $|\{x,y\}\cap e|=1$. In other words, each edge of ${\hat{E}}$ either contains both vertices $x$ and $y$, or none of them. Now let the set $S\subset {\hat{V}}$ be the union of all twins in ${\hat{H}}$, $|S|=s$, and $T={\hat{V}}\setminus S$. Observe that due to Claim \[si4\], an edge $e\in{\hat{E}}$ is contained in $S$ (and thus belongs to $H_S$) if and only if its signature consists of two disjoint dubletons. Consequently, $H_S$ is a subgraph of a thick clique, and so $|H_S|\le {s/2\choose 2}$. Moreover, it is easy to see that if $e\in {\hat{E}}$ contains two triples intersecting on a dubleton, then the dubleton must be contained in $S$ while two other vertices of $e$, which can be seperated by some edge, lie outside $S$, i.e. they belong to $T$.
Let $H\in {\mathcal{F}_2^4}(n,m)$, $m\ge n^2/8-2n/3$, and let a partition $V=R\cup S\cup T$ and subgraphs $H_R$, $H_S$ and $H_T$ of $H$ be defined as in Lemma \[podzialp4\]. Set $|R|=r$, $|S|=s$ and $|T|=t$. By Lemma \[podzialp4\], $$\label{mp4}
|H|=|H_R| + |H_S| + |H_T|\le 10r +{\lfloor s/2\rfloor\choose 2} +\frac{t^2}2 \le 10r+\frac{s^2}8-\frac s4+\frac{t^2}2.$$ We start with the following claim.
\[tp4\] If $T\neq \emptyset$ then there exists in $H_T$ a 2-star with at least $2m^2/n^2+m/n$ edges.
Let us define a 2-graph $G_T=(T,E_T)$ on the set of vertices $T$ putting $E_T=\{h\cap T: h\in H_T\}$. Note that $\delta({\hat{H}})\ge 11$ and so $|T|\ge 12$. Then the average degree of the graph $G_T$ is bounded below by $$\begin{aligned}
\frac{2|H_T|}{|T|}&=\frac{2(m-|H_R|-|H_S|)}{n-r-s}\ge 2\frac{m-10r-s^2/8}{n-r-s}\\
&=\frac{2m}{n}+\frac {(2m/n)(r+s)-20r-s^2/4}{n-r-s}\ge \frac{2m}{n},
\end{aligned}$$ where the last inequality follows by the facts that $m\ge n^2/8-2n/3$ and $r+s \le n-|T|\le n-12$. Since any graph with average degree $d$ contains a component of at least $(d+1)d/2$ edges, the assertion follows.
As an immediate consequence of the above fact we get the following result.
\[cl41\] If $n^2/9\le m\le {\lfloor n/2\rfloor\choose 2}$, then $T=\emptyset$.
Note that a thick clique on $n$ vertices has ${\lfloor n/2\rfloor\choose 2}$ edges and the maximum degree $\Delta \le n/2$. As a consequence, if for $n^2/9\le m\le {\lfloor n/2\rfloor\choose 2}$, $H\in {\mathcal{F}_2^4}(n,m)$, then $\Delta(H)\le n/2<0.02n^2<2m^2/n^2$. Hence, by Claim \[tp4\], $T=\emptyset$.
\[cl42\] If $H_T=\emptyset$ then $m\le {\lfloor n/2\rfloor\choose 2}$. Furthermore, if in addition $H_R\neq \emptyset $, then $m \le {\lfloor n/2\rfloor\choose 2}-n/5$.
Let $H\in {\mathcal{F}_2^4}(n,m)$ be such that $H_T=\emptyset$. Then the vertex set of $H$ can be partitioned into sets $S$ and $R$, where $|S|=s$, $s$ is even, and $|R|=r=n-s$. The number of edges $m$ in such graph is bounded from above by $s^2/8+10r$. It is easy to see that if $r\ge 2$ and $n$ is large enough this number is smaller than ${\lfloor n/2\rfloor\choose 2} -{n}/5$. Let us consider now the case when $R$ consists of just one vertex $v$; note that in this case $n$ is odd. Suppose that $v$ belongs to an edge $e$. Then $e$ must separate one twin $\{w,w'\}$ in $S$. But then each edge $e'$ of $H_S$ which contains the twin $\{w,w'\}$ must intersect $e$ on at least one more vertex and consequently $w$, as well as $w'$, can be contained in at most two edges of $H_S$. Since $\deg(v)\le 10$ so, very crudely, $\deg(w), \deg (w')\le 12$. But then $$\begin{aligned}
m&\le \binom {(n-3)/2}2+10+26 =\binom{(n-1)/2} 2 -\frac{n-3}2+36\\
&\le \binom{(n-1)/2} 2-\frac {n}5\,.{}\end{aligned}$$
Note that Claim \[cl42\] immediately implies the first part of Theorem \[th1a\]. To consider the second part let us assume that $H\in {\mathcal{F}_2^4}(n,m)$, where $m>{\lfloor n/2\rfloor\choose 2}$. Then, by Claim \[cl42\], $T\neq \emptyset$. Consequently, by Claim \[tp4\], there exists in $H_T$ a 2-star with at least $$\label{bigstarp4}
\frac{2m^2}{n^2}+\frac mn> \frac{2m}{n}\cdot\frac{n-4}{8}+\frac mn =\frac m4> \frac{n^2}{32}-\frac n8$$ edges implying that $\Delta (H) > n^2/32 -n$ and $H_T$ contains at most three largest 2-stars.
\[6starsp4\] $H_T$ consists of at most seven disjoint 2-stars.
Assume for a contradiction, that $H_T$ consists of at least eight disjoint 2-stars. Denote them by $S_i$, $i\ge 1$, where $\deg_H(v_i)\ge \deg(v_j)$, for $i<j$, and $\{v_i,v'_i\}$ stands for a center of a 2-star $S_i$. Note that by (\[bigstarp4\]), $$\deg_H(v_7)+\deg_H(v_8)< \frac 27\cdot \left(m-\frac m4\right)=\frac{3}{14}m < \Delta(H)-3.$$ But then we can modify $H$ by switching edges of $S_8$ so they form one 2-star with $S_7$, removing one edge from each of the 2-stars, $S_1$, $S_2$, $S_3$ and add three edges to $S_7$ (note that since $\delta({\hat{H}})\ge 11$, both $S_7$ and $S_8$ has at least twelve vertices each). Clearly, the modified graph $H'$ is ${P^{4}_2}$-free, has $n$ vertices, $m$ edges but the maximum degree of $H'$ is smaller than the maximum degree of $H$, contradicting the fact, that $H\in {\mathcal{F}_2^4}(n,m)$.
One can delete from $H$ at most $470$ edges and get a union of at most 7 disjoint stars and some isolated vertices.
First we observe that $r+s <48$. Indeed, if this is not the case we can modify $H$ by removing $\bar{m}=|H_R|+|H_S|\le 10r+s^2/8-s/4$ edges of $H_R\cup H_S$, delete one edge from each of the three largest 2-stars of $H_T$ and on the remaining $r+s-14$ vertices disjoint from the centers of 2-stars build a new 2-star (or three 2-stars if $r+s>m/4$) with $\bar{m}+3$ edges. The ${P^{4}_2}$-free graph obtained in this way would have the same number of edges but the maximum degree smaller than $H$, contradicting the fact that $H\in {\mathcal{F}_2^4}(n,m)$.
Since $r+s \le 47$ we have $|H_R\cup H_S|\le 10r+s^2/8-s/4\le 470$ and therefore, by Claim \[6starsp4\], one can delete from a graph $H$ at most $470$ edges of $H_S\cup H_R$ to get a graph which is an union of at most seven disjoint 2-stars and some isolated vertices.
To complete the proof of Theorem \[th1a\] we need to reduce the number of 2-stars from seven to four. Let $S_1$ be a 2-star with the largest number of edges in $H_T$. By (\[bigstarp4\]), it has $t_1>n^2/32-n/8$ edges and therefore $n_1\ge n/4+2$ vertices. Note that there are no place on four such stars in the graph of $n$ vertices. In fact, the fourth 2-star must be build on at most $n/4-1$ vertices and consequently have $t_4\le n^2/32-7n/8+6\le t_1-3n/4+6$ edges.
Now suppose that a graph $H_T$ has at least 5 disjoint 2-stars. By $n_4$ and $n_5$ we denote the number of vertices in the forth and the fifth largest 2-stars of $H_T$. Without loss of generality $n_4\ge n_5$. Then we have $n_5<n/5$, so one can modify $H$ by removing from each of $S_1$, $S_2$, $S_3$ one edge, choosing two vertices of the smallest degree in $S_5$, removing $m'<2n/5$ edges incident to them, and joining them to $S_4$ increasing the number of edges in $S_4$ to at most $t_1-3n/4+6+2n/5+3 <t_1-n/3$. The resulting graph $H'$ has the maximum degree smaller than $\Delta(H)$, which contradict the assumption that $H\in {\mathcal{F}_2^4}(n,m)$. Consequently, removing $470$ edges results in a graph which contains at most four 2-stars and the assertion follows.
Paths of length three
=====================
In this section we study the maximum degree of dense ${P^3_3}$-free 3-graphs. As we see soon, both the results and their proofs are surprsingly similar to that presented in the previous section.
For 2-graphs the problem is again an easy exercise – a graph whose components are cycles of length three (except, perhaps, one isolated edge if $n\equiv 2$ (mod 3)) is the largest $P^2_3$-free graph on $n$ vertices and has the maximum degree two. Here, we concentrate on the first non-trivial case when we study the maximum degree of $P^3_3$-free 3-graphs.
The maximum number of edges in a $P^3_3$-free 3-graph on $n$ vertices for all $n$ was found by Jackowska, Polcyn and Ruciński in [@JPRt].
\[ex1\] Let $\hat h(n)$ denote the maximum number of edges in a $P^3_3$-free 3-graph on $n$ vertices. Then $$\hat h(n)=\begin{cases}
\binom n3 &\quad \textrm{for}\quad n=3,4,5,6,\\
20 &\quad \textrm{for}\quad n=7,\\
\binom {n-1}2 &\quad \textrm{for}\quad n\ge 8\,.
\end{cases}$$
Let $$\begin{gathered}
{f^3_3}(n,m)=\min\{\Delta(H): H=(V,E)\textrm{\ is a $3$-graph such that \ }\\
|V|=n, |E|=m, \textrm{\ and\ }H\not\supset P^3_3\},\end{gathered}$$ and let ${\mathcal{F}^3_3}(n,m)$ denote the ‘extremal’ family of $P^3_3$-free $3$-graphs on $n$ vertices and $m$ edges such that $\Delta(H)={f^3_3}(n,m)$. Moreover, let us call a 3-graph $H$ [*quasi-bipartite*]{} if one can partition its set of vertices into three sets: $X=\{x_1,x_2,\dots, x_s\}$, $Y=\{y_1,y_2,\dots, y_s\}$, and $Z=\{z_1,z_2,\dots, z_t\}$ in such a way that all the edges of $H$ are of type $\{x_i,y_i,z_j\}$ for some $i=1,2,\dots, s$, $j=1,2,\dots, t$. Finally, by a star with center $v$ we denote a 3-graph in which each edge contains $v$. Then the following holds.
\[th2a\] There exists $\bar n_2$ such that for every $n\ge \bar n_2$, and $$n^2/8-\frac n5\le m\le n^2/8 \,,$$ each graph from ${\mathcal{F}^3_3}(n,m)$ is quasi-bipartite.
Moreover, there exists $\tilde n_2$ such that for every $n\ge \tilde n_2$ and $$n^2/8< m\le \binom {n-1}2\,,$$ each graph from ${\mathcal{F}^3_3}(n,m)$ has the maximum degree at least $n^2/32$ and we can delete from it at most $144$ edges and get a union of at most four stars and some number of isolated vertices.
Observe that Theorem \[th2\] follows directly from Theorems \[th2a\]. Another immediate consequence of the above two statements is the following result (note that the function $f(x)$ below is the same as the one defined in Corollary \[cor1a\]).
\[cor2a\] For every $x\in [0,1/4)\cup (1/4,1]$ the limit $$f(x)=\lim_{n\to\infty} \frac{{f^3_3}(n,x \binom{n-1}{2})}{\binom{n-1}2}$$ exists and $$f(x)=\begin{cases}0&\textrm{\ for\ }\quad 0\le x< 1/4,\\
(1+2x+\sqrt{12x-3})/24&\textrm{\ for\ }\quad 1/4<x< 1/3,\\
(1+3x+2\sqrt{6x-2})/18&\textrm{\ for\ }\quad 1/3<x< 1/2,\\
(x+\sqrt{2x-1})/2&\textrm{\ for\ }\quad 1/2<x\le 1.{}\end{cases}$$
The proof of Theorems \[th2a\] follows closely the way we proved Theorem \[th1a\]. Thus, as before, we start with the following decomposition lemma.
\[podzial\] For any ${P^3_3}$-free 3-graph $H$ there exists a partition of its set of vertices $V=R\cup S\cup T$, such that subhypergraphs of $H$ defined as $H_R=\{h\in H:h\cap R\neq \emptyset\}$, $H_S=H[S]$ and $H_T=H\setminus(H_R\cap H_S)=\{h\in H[V\setminus R]: h\cap T\neq\emptyset\}$ satisfy:
1. $|H_R|\le 6|R|$,
2. $H_S$ is quasi-bipartite, and so $|H_S|\le |S|^2/8$,
3. $H_T$ is a family of disjoint stars such that centers of these stars are in $S$ whereas all other vertices are in $T$, and so $|H_T|\le \binom{|T|}2$.
Let $H=(V,E)$ be a ${P^3_3}$-free 3-graph with $|V|=n$ and $|E|=m$. We start with defining the set of ‘exceptional’ vertices $R\subseteq V$. By a triangle $C$ we mean linear 3-graph with six vertices and three edges. First we include in $R$ all the components of $H$ which contain $C$. Then, from the remaining graph we move to $R$ vertices of degree at most six one by one, until we end up with a graph ${\hat{H}}$ of minimum degree at least seven. Then we set $H_R=\{h\in H:h\cap R\neq\emptyset\}$ and define a graph ${\hat{H}}=({\hat{V}},{\hat{E}})$ by putting ${\hat{V}}=V\setminus R$, ${\hat{E}}=E\setminus H_R$.
In order to estimate the number of edges in $H_R$ we need the following simple fact from [@JPR].
\[spojny\] If $H$ is a connected ${P^3_3}$-free 3-graph on $n$ vertices containing ${C}$, then $
|E(H)| \le 4n
$.
Thus, the required bound $6|R|$ for the number of edges in $H_R$ follows.
The main tool in proving Lemma \[podzial\] is, again, an analysis of possible signatures of edges in a 3-graph ${\hat{H}}$, where as before, the signature of $e\in {\hat{E}}$ is defined as the projection of ${\hat{E}}$ onto $e$.
\[sii1\] Every vertex of $e\in {\hat{E}}$ is covered by at least one set of the signature of $e$.
It follows from the fact that $\delta({\hat{H}})\ge 7>1$.
\[sii2\] The signature of none of the edges of ${\hat{H}}$ contains two singletons.
Assume that an edge $e=\{x_1,x_2,x_3\}\in {\hat{E}}$ contains two singletons, say $x_1$ and $x_2$. Since ${\hat{H}}$ is $\{{P^3_3},{C}\}$-free, two edges that intersects $e$ on $x_1$ and $x_2$ must share two points, say $y_1$ and $y_2$.
Set $X=\{x_1,x_2,x_3,y_1,y_2\}$. Since the degree of $x_3$ is at least seven it must belong to an edge $e'$ which is not contained in $X$. If $|e'\cap X|=1$ it would lead to a ${P^3_3}$, if $e'\cap X=\{x_3,y_i\}$ it would create $C$. Hence, $e'$ must consists of $v\notin X$ and one of the vertices $x_1, x_2$. Let us assume that $e'=\{v,x_1,x_3\}$. Now consider possible candidates for edges $e''$ which contain $v$. If for such an edge $|e''\cap X|\le 1$ then it leads to ${P^3_3}$, whereas if $e''=\{v,x_i,y_j\}$ for some $i=1,2,3$, $j=1,2$, it creates a triangle $C$. Thus, the only candidates for $e''$ are triples $\{v,y_1,y_2\}$, and $\{v,x_i,x_j\}$ for $1\le i<j\le 3$. But it means that the degree of $v$ is at most four, while $\delta({\hat{H}})\ge 7$. A contradiction.
\[sii3\] If the signature of an edge $e\in {\hat{H}}$ contains two dubletons, then their intersection is a singleton of $e$.
Let $e=\{x,y,z\}\in {\hat{H}}$ and let $f=\{x,y,v_x\}$ and $f'=\{y,z,v_z\}$, denote two edges containing two dubletons $\{x,y\},\{y,z\}\in {\operatorname{sg}}(e)$. Suppose that $y$ is not a singleton of ${\operatorname{sg}}(e)$. By Claim \[sii2\] we may assume that $x$ is not a singleton either. We first argue that then there exists $f''=\{x,y,v\}$ such that $v\neq v_z$. If $v_x\neq v_z$, then we can take just $f''=f$, so let $v_x=v_z$. Note that $x$ is not a singleton in ${\operatorname{sg}}(e)$ so each edges containing it must contain some other vertex of $e$. Moreover, any edge $e'=\{w, x,z\}$ with $w\neq v_z$ is prohibited since it contains two singletons $x$ and $z$. Thus, the existence of $f''$ follows from the fact that $\deg_H(x)\ge 7>3$. Now consider possible candidates for edges $e'$ containing $v$. If $e'\cap e=\emptyset$ then it leads to either ${P^3_3}$ or $C$. If $|e'\cap e|=1$ then it creates either singletons $x$ or $y$ in $e$, or the second (next to $y$) singleton $v$ in $f''$. Thus, all edges containing $v$ are contained in $e\cup \{v\}$, contradicting the fact that $\deg(v)\ge 7$.
\[sii4\] The signature of no edge from ${\hat{H}}$ contains three dubletons.
It follows from Claims \[sii2\] and \[sii3\].
\[sii5\] The signature of an edge from ${\hat{H}}$ consists either of disjoint singleton and dubleton, or of two dubletons intersecting on a singleton.
It is a direct consequence of Claims \[sii1\]-\[sii4\].
Now we can describe the partition of ${\hat{V}}$ into $S$ and $T$. We call a pair of vertices $\{x,y\}\subset {\hat{V}}$ *a twin* if it cannot be separated by an edge $e\in {\hat{E}}$, i.e. for no such edge $|\{x,y\}\cap e|=1$. By singletons we mean all one-element sets which belong to a signature of some edge of $E$.
Now let $S\subset {\hat{V}}$ consists of all twins and singletons of ${\hat{H}}$, $|S|=s$, and $T={\hat{V}}\setminus S$. It is easy to see that an edge of ${\hat{H}}$ is contained in $S$ if and only if it has signature which consists of disjoint dubleton and singleton. All other edges belong to $H_T$. Note that each edge of $H_T$ contains a singleton which belong to $S$.
Finally, note that any quasi-bipartite 3-graph on $s$ vertices contains at most $$m\le \max\{s'(s-2s'):s'\le s\}\le s^2/8\,,$$ edges, so $|H_S|\le s^2/8$.
Since the argument is almost identical to the one from the proof of Theorem \[th1a\] we skip some technical details. Let $H\in {{\mathcal{F}^3_3}}(n,m)$, $m\ge n^2/8-n/5$, and let a partition $V=R\cup S\cup T$ and subgraphs $H_R$, $H_S$ and $H_T$ of $H$ be as defined in Lemma \[podzial\]. Set $|R|=r$, $|S|=s$ and $|T|=t$. By Lemma \[podzial\], $$\label{m}
|H|=|H_R| + |H_S| + |H_T|\le 4r+s^2/8+t^2/2.$$ We start with the following claim.
\[t\] If $T\neq \emptyset$ then there exists in $H_T$ a star with at least $2m^2/n^2+m/n$ edges.
Indeed, then the 2-graph $G_T=(T,E_T)$ defined on the set of vertices $T$ by taking $E_T=\{h\cap T: h\in H_T\}$ has the average degree bounded from below by $$\begin{aligned}
\frac{2|H_T|}{|T|}&=\frac{2(m-|H_R|-|H_S|)}{n-r-s}\ge \frac{2(m-6r-s^2/8)}{n-r-s}=\\
&\frac{2m}{n}+\frac {(2m/n)(r+s)-12r-s^2/4}{n-r-s}\ge \frac{2m}{n},
\end{aligned}$$ and so contains a component of at least $2m^2/n^2+im/n$ edges.
Since there exists a ${P^3_3}$-free quasi-bipartite graph with $n$ vertices and $\lfloor n^2/8\rfloor$ edges, the above result immediately implies the following fact.
\[granica\] If $n^2/9\le m\le n^2/8$, then $T=\emptyset$.
On the other hand, since $H_R$ is sparse, it turns out that when $H_T=\emptyset$ the number of edges $H$ is bounded from below by $\lfloor n^2/8\rfloor$ and this maximum is achieved only when $H_R=\emptyset$.
\[cl422\] If $H_T=\emptyset$ then $m\le n^2/8$. Furthermore, if in addition $H_R\neq \emptyset $, then $m \le n^2/8-n/5$.
Now the first part of Theorem \[th2a\] follows directly from Claims \[granica\] and \[cl422\]. In order to show the second part of the assertion we can repeat, almost verbatim, the argument used in the proof of Theorem \[th1a\]. Thus, from Claims \[t\] and \[cl422\], it follows that if $m> n^2/8$ then $H_T$ contains a star with more than $n^2/32+n/8$ edges. Consequently, $H$ contains at most three vertices with maximum degree. Then we infer that $H_T$ consists of at most six disjoint stars since otherwise we could decrease the maximum degree of three largest ones by merging the sixth and seventh into one and add to them three edges taken from the biggest stars.
Since $H_T$ consists of only few stars the sets $S$ and $R$ must be quite small (simple calculations show that $r+s< 25$) since otherwise we could remove all $\bar m$ edges inside it, take three edges from the largest stars, and on the set of $r+s-6$ vertices, where we excluded the centers of stars of $H_T$, build a star with $\bar m+3$ edges. Then, the ${P^3_3}$-free graph constructed in this way would have the same number of edges as $H$ but smaller maximum degree, contradicting the fact that $H\in {{\mathcal{F}^3_3}}(n,m)$. Since $r+s<25$, we have $|H_R\cup H_S|\le 6r+s^2/8<144$, i.e. we can remove from $H$ at most 144 edges and get a forest of at most 6 stars and, perhaps, some isolated vertices.
Finally, to complete the proof, it is enough to show that in fact $H_T$ consists of at most four stars. Indeed, otherwise we could modify a graph accordingly (by decreasing by one three largest stars and incorporate these three edges to small stars by shuffling their vertices) so we could keep its number of edges and ${P^3_3}$-freeness but decrease by one its maximum degree.
Final remarks and comments
==========================
It is easy to see that the constant 470 in Theorem \[th1a\] is far from being optimal. The reader can easily rewrite the proof to replace it by, say, 30. However finding the smallest possible value of this constant requires more work and studying quite a few cases of small 4-graphs. Since it is not crucial for the main result, we just give the examples of the extremal 4-graphs we have found.
Let $H_1^4$ be a 4-graph on $3k+8$ vertices, $k\ge 100$, which consists of three complete disjoint 2-stars on $k$ vertices each, three edges joining centers of these stars, and a copy of the unique ${P^{4}_2}$-free graph $F^4_{1,3}$ on the vertex set $\{x_1,\dots,x_8\}$ with $17$ edges found in [@KMW] whose set of 4-edges consists of all 4-element subsets of $\{x_1,\dots, x_8\}$ which have at least three elements in $\{x_1,\dots,x_4\}$. Then, to make $H_1^4$ a union of disjoint 2-stars, we need to remove three edges joining the centers of three large 2-stars and at least eight edges from $F^4_{1,3}$. It seems that one can always delete at most eleven 4-edges from a dense enough ${P^{4}_2}$-free graph to get a union of at most 4-stars (and, perhaps, some number of isolated vertices), so the graph $H_1^4$ defined above is in a way extremal. It is however not unique – one can modify it removing from each 2-star the same number $i\le k/10$ of edges to get another extremal example.
On the other hand, if we want to get a union of four stars instead of four 2-stars, it is enough to remove from $H_1^4$ only seven edges. However, $H_1^4$ is not extremal for the variant of this problem. A 4-graph $H_2^4$ on $3k+4$ which consists of a thick clique on $10$ vertices and three equal complete 2-stars rooted on its vertices needs at least eight edges to be deleted to become a union of at most four stars. The same is true for a 4-graph $H_3^4$ on $3k+6$ vertices which consists of three complete stars on $k$ vertices each, three edges joining their centers, and the complete clique on six vertices.
In a similar way one can try to improve the constant $144$ in Theorem \[th2a\]. Since the structure of $P^3_3$-free $3$-graphs is well studied, one can use Theorem \[th2a\] to replace 144 by just 10, and the extremal graph consists of three equal stars and the clique on six vertices.
Another, much more interesting question, is whether a similar rescaling phenomenon can be observed for other extremal problems. There is a number of candidates for such a behaviour, we just mention two possible directions which follow the line of research initiated by this work. The first one concerns linear 3-paths $P_\ell^3$ of length $\ell$, for $\ell\ge 3$. It is known [@KMV] that the largest number of edges in a $P_\ell^3$-free graph on $n$ vertices is $(1/2+o(1))n^2$ and the extremal graph contains vertices of degree $\Omega(n^2)$. Thus, since a thick clique is $P_\ell^3$-free, one can expect that this maximum degree drops to $O(n)$ at $m\sim n^2/8$.
It is also conceivable that one can generalize of our result on $P^4_2$-free 4-graphs in the following direction. For $r\ge 1$ let ${\mathcal{F}_2^4}(4r;n,m)$ be a family of $(4r)$-graphs on $n$ vertices and $m$ edges in which no two edges share precisely $2r-1$ points. Frankl and Füredi [@FF] proved that to maximize the number of edges in such a graph one needs to take the family of all sets which contain a given set on $2r$ vertices. Clearly, in such a graph the maximum degree is $m=\Theta(n^{2r})$. On the other hand a thick $(4r)$-clique on $n$ vertices, where we first partition vertex set into pairs and then choose $2r$ of them to form an edge, has $\binom{\lfloor n/2\rfloor}{2r}$ edges but its maximum degree is just $\binom{\lfloor n/2\rfloor-1}{2r-1}=\Theta(n^{2r-1})$. Thus, one expect a rapid change of the (minimum) maximum degree at $m=\binom{\lfloor n/2\rfloor}{2r}$.
[99]{}
P. Frankl and Z. Füredi, [*Forbidding just one intersection*]{}, *J. Combinat. Th. ser. A*, 36 (1985), 160–176. E. Jackowska, J. Polcyn, A. Ruciński, [*Turán numbers for 3-uniform linear paths of length 3*]{}, *Electron. J. Combin.*, 23(2) (2016), \#P2.30. E. Jackowska, J. Polcyn, A. Ruciński, [*Multicolor Ramsey numbers and restricted Turán numbers for the loose 3-uniform path of length three*]{}, [arXiv:1506.03759v1]{}, submitted. P. Keevash, D. Mubayi, R. M. Wilson, [*Set systems with no singleton intersection*]{}, *SIAM J. Discrete Math.*, 20 (2006), 1031–1041. A. V. Kostochka, D. Mubayi and J. Verstraëte, [*Turán problems and shadows I: Paths and cycles*]{}, *J. Combinat. Th. Ser. A*, 129 (2015), 57–79. A. A. Razborov, [*On the minimal density of triangles in graphs*]{}, *Combin., Probab. and Comput.*, 17(4) (2008), 603–618. C. Reiher [*The clique density theorem*]{}, *Annals Math.*, 184 (2016), 683–707.
[^1]: The first author partially supported by NCN grant 2012/06/A/ST1/00261.
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---
abstract: 'We report on the temperature dependence of microwave-induced resistance oscillations in high-mobility two-dimensional electron systems. We find that the oscillation amplitude decays exponentially with increasing temperature, as $\exp(-\alpha T^2)$, where $\alpha$ scales with the inverse magnetic field. This observation indicates that the temperature dependence originates [*primarily*]{} from the modification of the single particle lifetime, which we attribute to electron-electron interaction effects.'
author:
- 'A.T. Hatke'
- 'M.A. Zudov'
- 'L.N. Pfeiffer'
- 'K.W. West'
title: Temperature Dependence of Microwave Photoresistance in 2D Electron Systems
---
Over the past few years it was realized that magnetoresistance oscillations, other than Shubnikov-de Haas oscillations [@shubnikov:1930], can appear in high mobility two-dimensional electron systems (2DES) when subject to microwaves [@miro:exp], dc electric fields [@yang:2002a], or elevated temperatures [@zudov:2001b]. Most attention has been paid to the microwave-induced resistance oscillations (MIRO), in part, due to their ability to evolve into zero-resistance states [@mani:2002; @zudov:2003; @willett:2004; @zrs:other]. Very recently, it was shown that a dc electric field can induce likely analogous states with zero-differential resistance [@bykov:zhang].
Despite remarkable theoretical progress towards understanding of MIRO, several important experimental findings remain unexplained. Among these are the immunity to the sense of circular polarization of the microwave radiation [@smet:2005] and the response to an in-plane magnetic field [@mani:yang]. Another unsettled issue is the temperature dependence which, for the most part [@studenikin:2007], was not revisited since early reports focusing on the apparently activated behavior of the zero-resistance states [@mani:2002; @zudov:2003; @willett:2004]. Nevertheless, it is well known that MIRO are best observed at $T \simeq 1$ K, and quickly disappear once the temperature reaches a few Kelvin.
MIRO originate from the inter-Landau level transitions accompanied by microwave absorption and are governed by a dimensionless parameter $\eac\equiv\omega/\oc$ ($\omega=2\pi f$ is the microwave frequency, $\oc=eB/m^*$ is the cyclotron frequency) with the maxima$^+$ and minima$^-$ found [@miro:phase] near $\eac^{\pm}=n \mp \pac,\,\pac \leq 1/4$ ($n \in \mathbb{Z}^+$). Theoretically, MIRO are discussed in terms of the “displacement” model [@disp:th], which is based on microwave-assisted impurity scattering, and the “inelastic” model [@dorozhkin:2003; @dmp; @dmitriev:2005], stepping from the oscillatory electron distribution function. The correction to the resistivity due to either “displacement” or “inelastic” mechanism can be written as [@dmitriev:2005]: $$\delta \rho=-4\pi\rho_0\tautr^{-1}\pc\eac \taubar \delta^{2}\sin(2\pi\eac)
\label{theory}$$ Here, $\rho_0\propto 1/\tautr$ is the Drude resistivity, $\tautr$ is the transport scattering time, $\pc$ is a dimensionless parameter proportional to the microwave power, and $\delta=\exp(-\pi\eac/\omega\tauq)$ is the Dingle factor. For the “displacement” mechanism $\taubar=3\tauim$, where $\tauim$ is the long-range impurity contribution to the quantum (or single particle) lifetime $\tauq$. For the “inelastic” mechanism $\taubar=\tauin \simeq \varepsilon_F T^{-2}$, where $\varepsilon_F$ is the Fermi energy. It is reasonable to favor the “inelastic” mechanism over the “displacement” mechanism for two reasons. First, it is expected to dominate the response since, usually, $\tauin \gg \tauim$ at $T\sim 1$ K. Second, it offers plausible explanation for the MIRO temperature dependence observed in early [@mani:2002; @zudov:2003] and more recent [@studenikin:2007] experiments.
In this Letter we study temperature dependence of MIRO in a high-mobility 2DES. We find that the temperature dependence originates primarily from the temperature-dependent quantum lifetime, $\tauq$, entering $\delta^2$. We believe that the main source of the modification of $\tauq$ is the contribution from electron-electron scattering. Furthermore, we find no considerable temperature dependence of the pre-factor in Eq.(1), indicating that the “displacement” mechanism remains relevant down to the lowest temperature studied. As we will show, this can be partially accounted for by the effect of electron-phonon interactions on the electron mobility and the interplay between the two mechanisms. However, it is important to theoretically examine the influence of the electron-electron interactions on single particle lifetime, the effects of electron-phonon scattering on transport lifetime, and the role of short-range disorder in relation to MIRO.
While similar results were obtained from samples fabricated from different GaAs/Al$_{0.24}$Ga$_{0.76}$As quantum well wafers, all the data presented here are from the sample with density and mobility of $\simeq 2.8 \times 10^{11}$ cm$^{-2}$ and $\simeq 1.3 \times 10^7$ cm$^2$/Vs, respectively. Measurements were performed in a $^3$He cryostat using a standard lock-in technique. The sample was continuously illuminated by microwaves of frequency $f=81$ GHz. The temperature was monitored by calibrated RuO$_2$ and Cernox sensors.
![(color online) Resistivity $\rho$ vs. $B$ under microwave irradiation at $T$ from 1.0 K to 5.5 K (as marked), in 0.5 K steps. Integers mark the harmonics of the cyclotron resonance. []{data-label="fig1"}](tdepmiro1.eps)
In Fig.\[fig1\] we present resistivity $\rho$ as a function of magnetic field $B$ acquired at different temperatures, from $1.0$ K to $5.5$ K in 0.5 K increments. Vertical lines, marked by integers, label harmonics of the cyclotron resonance. The low-temperature data reveal well developed MIRO extending up to the tenth order. With increasing $T$, the zero-field resistivity exhibits monotonic growth reflecting the crossover to the Bloch-Grüneisen regime due to excitation of acoustic phonons [@stormer:mendez]. Concurrently, MIRO weaken and eventually disappear at higher temperatures. This disappearance is not due to the thermal smearing of the Fermi surface, known to govern the temperature dependence of the Shubnikov-de Haas oscillations.
We start our analysis of the temperature dependence by constructing Dingle plots and extracting the quantum lifetime $\tauq$ for different $T$. We limit our analysis to $\eac\gtrsim 3$ for the following reasons. First, this ensures that we stay in the regime of the overlapped Landau levels, $\delta \ll 1$. Second, we satisfy, for the most part, the condition, $T > \oc$, used to derive Eq.(1). Finally, we can ignore the magnetic field dependence of $\pc$ and assume $\pc \equiv \pc^{(0)}\eac^2(\eac^2+1)/(\eac^2-1)^2\simeq \pc^{(0)}=e^2\ec^2 v_{F}^2/\omega^4$, where $\ec$ is the microwave field and $v_F$ is the Fermi velocity.
Using the data presented in Fig.\[fig1\] we extract the normalized MIRO amplitude, $\delta \rho/\eac$, which, regardless of the model, is expected to scale with $\delta^2=\exp(-2\pi\eac/\omega\tauq)$. The results for $T=1,\,2,\,3,\,4$ K are presented in Fig.2(a) as a function of $\eac$. Having observed exponential dependences over at least two orders of magnitude in all data sets we make two important observations. First, the slope, $-2\pi/\omega\tauq$, monotonically grows with $T$ by absolute value, marking the increase of the quantum scattering rate. Second, all data sets can be fitted to converge to a single point at $\eac=0$, indicating that the pre-factor in Eq.(1) is essentially temperature independent \[cf. inset of Fig.2(a)\].
![(color online) (a) Normalized MIRO amplitude $\delta \rho/\eac$ vs. $\eac$ at $T =1.0,\,2.0,\,3.0,\,4.0$ K (circles) and fits to $\exp(-2\pi\eac/\omega\tauq)$ (lines). Inset shows that all fits intersect at $\eac=0$. (b) Normalized quantum scattering rate $2\pi/\omega\tauq$ vs. $T^2$. Horizontal lines mark $\tauq=\tauim$ and $\tauq=\tauim/2$, satisfied at $T^2=0$ and $T^2\simeq 11$ K$^2$, respectively. []{data-label="fig2"}](tdepmiro2.eps)
After repeating the Dingle plot procedure for other temperatures we present the extracted $2\pi/\omega\tauq$ vs. $T^2$ in Fig.\[fig2\](b). Remarkably, the quantum scattering rate follows quadratic dependence over the whole range of temperatures studied. This result is reminiscent of the temperature dependence of quantum lifetime in double quantum wells obtained by tunneling spectroscopy [@murphy:eisenstein] and from the analysis of the intersubband magnetoresistance oscillations [@berk:slutzky:mamani]. In those experiments, it was suggested that the temperature dependence of $1/\tauq$ emerges from the electron-electron scattering, which is expected to greatly exceed the electron-phonon contribution. Here, we take the same approach and assume $1/\tauq=1/\tauim+1/\tauee$, where $\tauim$ and $\tauee$ are the impurity and electron-electron contributions, respectively. Using the well-known estimate for the electron-electron scattering rate [@chaplik:giuliani], $1/\tauee=\lambda T^2/\varepsilon_F$, where $\lambda$ is a constant of the order of unity, we perform the linear fit to the data in Fig.\[fig2\](b) and obtain $\tauim \simeq 19$ ps and $\lambda \simeq 4.1$. We do not attempt a comparison of extracted $\tauim$ with the one obtained from SdHO analysis since the latter is known to severely underestimate this parameter.
To confirm our conclusions we now plot in Fig.3(a) the normalized MIRO amplitude, $\delta \rho/\eac$, evaluated at the MIRO maxima near $\eac=n-1/4$ for $n=3,4,5,6$ as a function of $T^2$. We observe that all data sets are well described by the exponential, $\exp(-\alpha T^2)$, over several orders of magnitude and that the exponent, $\alpha$, monotonically increases with $\eac$. The inset of Fig.3(a) shows the extension of the fits into the negative $T^2$ region revealing an intercept at $\simeq - 11$ K$^2$. This intercept indicates that at $\bar T^2 \simeq 11 $ K$^2$, $\tauee \simeq \tauim$ providing an alternative way to estimate $\lambda$. Indeed, direct examination of the data in Fig.2(b) reveals that the electron-electron contribution approaches the impurity contribution at $\bar T^2 \simeq 11 $ K$^2$, [*i.e.*]{} $1/\tauq(\bar T) = 1/\tauee(\bar T)+1/\tauim \simeq 2/\tauim=2/\tauq(0)$. Another way to obtain parameter $\lambda$ is to extract the exponent, $\alpha$, from the data in Fig.3(a) and examine its dependence on $\eac$. This is done in Fig.3(b) which shows the anticipated linear dependence, $\alpha=(2\pi\lambda/\omega\varepsilon_F)\eac$, from which we confirm $\lambda \simeq 4.1$.
![(color online) (a) Normalized MIRO amplitude, $\delta \rho/\eac$, vs. $T^2$ near $\eac=2.75,\,3.75,\,4.75,\,5.75$ (circles) and fits to $\exp(-\alpha T^2)$ (lines). Inset demonstrates that all fits intersect at $-11$ K$^2$. (b) Extracted exponent $\alpha$ vs. $\eac$ reveals expected linear dependence. []{data-label="fig3"}](tdepmiro3.eps)
To summarize our observations, the MIRO amplitude as a function of $T$ and $\eac$ is found to conform to a simple expression: $$\delta \rho \simeq A \eac \exp [-2\pi/\oc\tauq].
\label{ampl}$$ Here, $A$ is roughly independent on $T$, but $\tauq$ is temperature dependent due to electron-electron interactions: $$\frac 1 \tauq = \frac 1 \tauim+\frac 1 \tauee, \,\,\, \frac 1 \tauee \simeq \lambda \frac {T^2}{\varepsilon_F}.
\label{tauq}$$ It is illustrative to plot all our data as a function of $2\pi/\oc\tauq$, where $\tauq$ is evaluated using Eq.(\[tauq\]). As shown in Fig.4(a), when plotted in such a way, all the data collected at different temperatures collapse together to show universal exponential dependence over three orders of magnitude. The line in Fig.4(a), drawn with the slope of Eq.(\[tauq\]), confirms excellent agreement over the whole range of $\eac$ and $T$.
We now discuss observed temperature independence of $A$, which we present as a sum of the “displacement” and the “inelastic” contributions, $A=\adis+\ain$. According to Eq.(\[theory\]), at low $T$ $\adis < \ain$ but at high $T$ $\adis > \ain$. Therefore, there should exist a crossover temperature $T^*$, such that $\adis(T^*)=\ain(T^*)$. Assuming $\tauin \simeq \tauee \simeq \varepsilon_F/\lambda T^2 $ we obtain $T^*\simeq 2$ K and conclude that the “displacement” contribution cannot be ignored down to the lowest temperature studied. Next, we notice that Eq.(\[theory\]) contains transport scattering time, $\tautr$, which varies roughly by a factor of two in our temperature range. If this variation is taken into account, $\ain$ will decay considerably slower than $1/T^2$ and $\adis$ will grow with $T$, instead of being $T$-independent, leading to a rather weak temperature dependence of $A$. This is illustrated in Fig.\[fig4\](b) showing temperature evolution of both contributions and of their sum, which exhibits rather weak temperature dependence at $T\gtrsim 1.5$ K. In light of the temperature dependent exponent, we do not attempt to analyze this subtle behavior using our data.
![(color online) (a) Normalized MIRO amplitude $\delta \rho/\eac$ vs. $2\pi/\omega\tauq$ for $T =1.0,\,2.0,\,3.0,\,4.0$ K (circles). Solid line marks a slope of $\exp(-2\pi/\oc\tauq)$. (b) Contributions $\adis$ (squares), $\ain$ (triangles), and $A$ (circles) vs. $T$. []{data-label="fig4"}](tdepmiro4.eps)
Finally, we notice that the “displacement” contribution in Eq.(1) was obtained under the assumption of small-angle scattering caused by remote impurities. However, it is known from non-linear transport measurements that short-range scatterers are intrinsic to high mobility structures [@yang:2002a; @ac:dc]. It is also established theoretically that including a small amount of short-range scatterers on top of the smooth background potential provides a better description of real high-mobility structures [@mirlin:gornyi]. It is reasonable to expect that consideration of short-range scatterers will increase “displacement” contribution leading to lower $T^*$.
To summarize, we have studied MIRO temperature dependence in a high-mobility 2DES. We have found that the temperature dependence is exponential and originates from the temperature-dependent quantum lifetime entering the square of the Dingle factor. The corresponding correction to the quantum scattering rate obeys $T^2$ dependence, consistent with the electron-electron interaction effects. At the same time we are unable to identify any significant temperature dependence of the pre-factor in Eq.(1), which can be partially accounted for by the interplay between the “displacement” and the “inelastic” contributions in our high-mobility 2DES. Since this observation might be unique to our structures, further systematic experiments in samples with different amounts and types of disorder are highly desirable. It is also important to theoretically consider the effects of short-range impurity and electron-phonon scattering. Another important issue is the influence of the electron-electron interactions on single particle lifetime entering the square of the Dingle factor appearing in MIRO (which are different from the Shubnikov-de Haas oscillations where the Dingle factor does not contain the $1/\tauee \propto T^2$ term [@martin:adamov]). We note that such a scenario was considered a few years ago [@ryzhii].
We thank A. V. Chubukov, I. A. Dmitriev, R. R. Du, A. Kamenev, M. Khodas, A. D. Mirlin, F. von Oppen, D. G. Polyakov, M. E. Raikh, B. I. Shklovskii, and M. G. Vavilov for discussions and critical comments, and W. Zhang for contribution to initial experiments. The work in Minnesota was supported by NSF Grant No. DMR-0548014.
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---
abstract: 'With the prevalence of misinformation online, researchers have focused on developing various machine learning algorithms to detect fake news. However, users’ perception of machine learning outcomes and related behaviors have been widely ignored. Hence, this paper proposed to bridge this gap by studying how to pass the detection results of machine learning to the users, and aid their decisions in handling misinformation. An online experiment was conducted, to evaluate the effect of the proposed machine learning warning sign against a control condition. We examined participants’ detection and sharing of news. The data showed that warning sign’s effects on participants’ trust toward the fake news were not significant. However, we found that people’s uncertainty about the authenticity of the news dropped with the presence of the machine learning warning sign. We also found that social media experience had effects on users’ trust toward the fake news, and age and social media experience had effects on users’ sharing decision. Therefore, the results indicate that there are many factors worth studying that affect people’s trust in the news. Moreover, the warning sign in communicating machine learning detection results is different from ordinary warnings and needs more detailed research and design. These findings hold important implications for the design of machine learning warnings.'
author:
- '\'
bibliography:
- 'sample.bib'
---
Introduction
============
Fake news is a type of false information to deliberately mislead or manipulate public opinion, through traditional mass media and recent online social media. In recent years, social media platforms (e.g., Facebook, Twitter) have made it possible for individuals to produce, consume, and share different information. A report on the 2016 election indicates that fake news websites rely on online social media for 48% of traffic, which is a much higher share than of other sources [@allcott2017social]. With the blurring of the boundaries between information sources and recipients, it is difficult to control the quality of the information that people are exposed to. Especially, it must be acknowledged that people are not necessarily good at evaluating the quality of online information.
With more fact-checking work being done by machine learning algorithms [@shu2017fake], this paper studied how to communicate the detection results with users and help them make decisions subsequently. Since Twitter has become the main source of news [@broersma2013twitter], this project focuses on news and user’s behaviors on Twitter. Two research questions are proposed:
RQ1: Will a machine learning warning help users judge the authenticity of news compared with the condition in which there is no warning?
RQ2: Will a machine learning warning influence user’s subsequent behaviors such as clicking or sharing news compared with the condition in which there is no warning?
This paper investigated the two research questions by assessing the relationship between the presence of a warning sign and people’s trust in online news.
Literature Review
=================
A specific warning, which gives details about the continued influence of misinformation, succeeded in reducing the continued reliance on outdated information [@ecker2010explicit]. Clayton et al. further suggested that although the exposure to a general warning did not affect the perceived accuracy of headlines, it decreased the individual’s belief in the accuracy [@clayton2019real]. Some researchers raised the opposite view that alerted individuals may perform worse [@szpitalak2010warning]. Pennycook et al. claimed that the exposure increases subsequent perceptions of accuracy, and tagging such stories as disputed is not effective [@pennycook2018prior]. The two venues of claims motivate this project.
Warning Sign
------------
Banner or pop-up warnings are often seen when there is a potential threat [@zhang2014effects]. These warning signs can release a perceived risk from the information from social media, remind people of suspicious or false information. A logical result would be a decrease in users’ trust toward the news with the presence of a warning sign. So we propose that:
H1: With the presence of a warning sign, participants will have less trust in the information, as well as a lower tendency for sharing the information.
Methods
=======
The between-subjects online study was conducted to evaluate the effects of machine learning warnings in conveying the fake news detection results. In addition to the warning condition, a control condition, in which no warning is presented, is also conducted. Participants were asked to make detection and sharing decisions on fake and real news.
Pilot Study
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To understand the issues that participants are concerned with and the way they think about them, an initial interview with 4 individuals (2 female, 2 male) was conducted. Participants were students from The Pennsylvania State University, with a variety of majors. Through 10 minute face-to-face interviews, they were asked following open-ended questions:
Q1: Have you seen fake news on SNSs, such as Twitter, Facebook? How do you judge the authenticity of the news? What factors will affect your judgment?
Q2: When you judge the authenticity of the news, will you rely on the warning information embedded on the website?
Q3: What do you do when you find out that a piece of news is fake? Will you still comment or retweet it?
Before the interview started, I asked the participants for permission to record. Participants were compensated with \$5 Starbucks gift cards.
Participants reported varying levels of enthusiasm to browse news on the internet and SNSs. Some participants know very well about the news, while others do not care. This makes them very different in their ability to identify real and fake news. For example, By looking at the news in Figure \[fig:design\], Individual B said: *“I would think that is real.”*, while Individual A said: *“I don’t think it’s true. \[…\] Just based on instinct like, ‘donating \$1 billion’ is too generous to be real.”*
In addition to using common sense and prior knowledge to judge the authenticity of the news, some participants also mentioned that the source of the news is an important basis for their judgment of the real and fake news:
*“I look at the photo, I look at the source, and I look at where I found it. For example, if I’m reading from \[…\], I’d be very disinclined to think that’s fake. Because it’s a reputable news source.”* (Individual T)
The same applies to the warning sign. After carefully asking about the design of the warning sign, Individual S said: *“If this sign comes from a dependable or trustable source, then I would believe it and I would be cautious about that content. But if I don’t know the source of that warning sign, I will definitely be more vulnerable even if there is a warning sign.”*
Participants showed a very different attitude towards the warning sign. When asked if he would trust warning sign, Individual B answered: *“Yeah, definitely. I just whenever I see things I just kind of assume it’s real.”* Individual A had the same opinion: *“If I see it (the news in Figure \[fig:design\]), I would probably click on it to see if it is true. But if I was suspicious already and that (the warning sign) was there I would think that it (the news) is not true.”*
All participants said they would not comment or share fake news. They skip it and also not report it.
It is worth noting that political tendency also plays an important role when people are facing online news:
*“It depends on which source I believe.”* (Individual T)
Through the interview, I found individual’s frequency of reading news online and interest in political news may be related to the judgment of news. Thus these two points were included in the survey study.
Survey Study
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Based on the results of the interview, we had a general understanding of users’ perceptions of warning signs, related to their judgment of news authenticity and willingness to share. In this section, the between-subjects online study investigating the effect of a machine-learning warning sign in mitigating fake news was conducted. In addition to the warning conditions, a control group in which no warning was presented, was also included in the study.
### Participants
The study was conducted on Amazon Mechanical Turk (MTurk). 100 MTurk workers were recruited on March 25, 2019. All participants were (1) at least 18 years old; (2) located at the United States; and (3) with a human intelligence task (HIT) approval rate above 95%. Participants were allowed to participate in the study once.
### Materials
20 news headlines were created in the format of Twitter, consisting of a picture, source, header, and a short description (see Figure \[fig:design\]). 10 verified fake news headlines, and 10 verified real news headlines were chosen from “politifact.com”, which is a well-known third-party fact-checking website. Proposed machine learning warning is attached to the bottom of the fake news. Figure \[fig:design\] gives a depiction of the warning design.
The selected news were released from January to March in 2019, and the topic of news was limited to politics because political news is one type of the most popular news that most individuals read every day, so most of the people have the certain sense to judge its credibility without professional knowledge.
![Warning sign design. A piece of fake news is at the top. A warning sign which indicates that the above news is disputed by a machine learning algorithm is at the bottom.[]{data-label="fig:design"}](design.png){width="45.00000%"}
### Procedure
Participants completed a demographic questionnaire that asked for age, gender. Then we asked participants completed additional questions about their social media experience, interest in politics, factors that impact their decisions.
20 news headlines with and without warnings were shown to a participant along with the questions, respectively. 20 news were presented one at a time in a randomized order. The participants were asked to judge the accuracy and decide their willingness to share the news on a 5-point Likert scale, respectively (1 means “Very inaccurate” or “I would never share news like this one”, 5 means “Very accurate” or “I would love to share news like this one”). Each participant was compensated for \$0.5 for the completion of the task.
### Measures
Among the 100 MTurk workers, there were 68 male, 30 female. 2 people chose not to disclose. Participants came from different age groups, with 18.0% between 18 to 25 years, 35.0% between 26 to 30 years, 20.0% between 31 to 35 years, 11.0% between 36 to 45 years, 11.0% between 46 to 55 years and 5.0% above 55 years.
The social media experience was ranked into five degrees from 1 to 5 based on the frequency of browsing the web in a week: “Extremely likely (Everyday)”, “Very likely (Several times a week)”, “Moderately likely (Once or twice a week)”, “Slightly likely (Less often)” and “Not at all likely (Never)”. Most workers reported their social media experience on either “Extremely likely” (35.0%) or “Very likely” (37.0%); 20% workers reported on “Moderately likely”; and only 6.0% and 2.0% workers reported on “Slightly likely” and “Not at all likely” ($M=2.03, SD=.99$).
Similarly, the interest in politics was categorized into five types from 1 to 5: “Extremely interested”, “Very interested”, “Fairly interested”, “Not very interested” and “Not at all interested”. Most workers reported their social media experience on either “Very interested” (37.0%); 29% and 21% workers reported on “Fairly interested” and “Extremely interested” respectively; and only 9% and 4% workers were “Not very interested” and “Not at all interested” respectively ($M=2.38, SD=1.04$).
The demographic distributions were similar among the two conditions.
Participants’ trust toward the news were assessed with the aforementioned 5 points Likert scale including “Very inaccurate” as 1, “Inaccurate” as 2, etc. For fake news, participants gave lower scores with the presence of a warning sign ($M=2.88, SD=.84$) in contrast to the control condition ($M=2.70, SD=1.18$). Interestingly, with the presence of a warning sign, participants also gave lower scores to real news ($M=3.07, SD=.84$) compared with the control condition ($M=3.12, SD=.78$). Participants’ scoring results were further grouped into three categories according to the ground truth: “Correct”, “Unsure” and “Wrong”. The distribution of participants’ judgment toward fake news and real news is shown in Figure \[fig:judgment\]. After introducing the warning sign, participants’ uncertainty about the authenticity of the news decreased, with 26.2% to 24.4% toward the fake news. Interestingly, the uncertainty toward the real news was also dropped from 25.8% to 21.4%, where there was no warning sign attached to the real news in both conditions. In addition, with the presence of a warning sign, participants’ detection accuracy of fake news increased from 39% to 48.4% compared with the control group.
![Participants’ judgment toward fake news and real news.[]{data-label="fig:judgment"}](judgment.png){width="45.00000%"}
Similar patterns could be observed from participants’ sharing decision. With the presence of warning sign, participants were less willing to share both the fake news ($M=2.33, SD=1.28$) and real news ($M=2.53, SD=1.18$) in contrast to the control conditions (fake news: $M=2.79, SD=1.30$; real news: $M=2.76, SD=1.20$).
Data Analysis
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Effects of Machine Learning Warnings
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In order to test the effects of the machine learning warning sign on users’ detections of the news and their sharing behaviors, a series of analyses of covariance (ANCOVA) were conducted, controlling for age, gender, and other demographic information. Warning sign’s effects on participants’ trust toward the fake news were not significant. We could not find enough statistical evidence to support that the warning sign can lower individuals’ trust toward the fake news in contrast to the condition without the attached warning sign. However, results showed significant effects for the social media experience on participants’ trust toward the fake news, $F=4.009, p<.05$. More experience on social media indicates less trust in fake news. Thus, for RQ1, there is no sufficient evidence to support that a warning sign could help users judge the authenticity of the news.
ANCOVA also indicated that age and social media experience had significant effects on users’ sharing decision, $F=6.221, p<.05$ and $F=4.592, p<.05$. However, no statistical evidence supports the warning sign’s effects on participants’ sharing decision toward the fake news, which answered RQ2.
In sum, from the data collected, the warning sign would not impact users’ trust and sharing decision.
Modeling the Effects of Machine Learning Warnings
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The relationships between the warning sign and other variables were tested using a structural equation model (SEM) shown in Figure \[fig:sem\], which yielded a good fit $\chi^2=42.74339, df=15, p < .001$.
![SEM predicting users’ trust and sharing decision[]{data-label="fig:sem"}](Rplot.png){width="45.00000%"}
The findings suggest the presence of machine learning warning sign could not reduce users’ trust and sharing decision toward the fake news. Interestingly, users’ sharing behavior is related to their trust. Specifically, users tend to share what they think is accurate.
Discussion and Conclusions
==========================
Through data analysis, the impacts of the proposed machine learning warning sign were examined. There is no statistical evidence to support that the proposed warning sign can help people better detect fake news and decrease their willingness to share fake news. However, results showed significant effects for the social media experience on people’s trust toward the fake news. Also, age and social media experience had significant effects on people’s sharing decision.
These results might go against many people’s intuitive notion that a warning sign should lower people’s trust, and raise many questions about factors that build people’s trust, and factors that might influence people’s decisions about whether to comment on or share particular news. If a warning sign—as a common and intuitive way to communicate with users—can not let users believe the potential misinformation in the news, what other factors can? Alternatively, can we find other better design to pass the detection results to users? This study provided an open-ended question. As a result, we need further studies to inform us what factors are of major influence on people’s trust and sharing decision.
The results can be informative for researchers. Because researchers usually concentrate on how to detect the misinformation more accurately. Our results show that continuously increasing the accuracy might not be of much help in aiding people’s decision in handling misinformation, as the machine learning warnings we used were consistent with ground truth. So, researchers should look for other ways to better communicating the machine learning detection results of misinformation.
This study can also provide some design implications for designers. An important implication for warning sign design is that it has to be designed carefully and strategically in triggering appropriate cognitive heuristics. While warning signs may be appropriate for garbing users’ attention, designers may want to rethink this strategy for false information. Follow-up studies could focus on other mechanisms for persuasive appeals are linked to machine learning warning signs, as well as their effects on users’ trust and sharing decision.
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abstract: 'The *Rossi X-ray Timing Explorer (RXTE) was successfully launched on 1995 December 30 and has been operational since that time. Its three instruments are probing regions close to compact objects, degenerate dwarfs, neutron stars, stellar black holes and the central engines of AGN. Temporal studies with the ASM and PCA have already yielded rich results which pertain to the environs, evolution, and nature of the compact objects in galactic systems. Here I review some selected results from these instruments. as obtained by various RXTE observers. The ASM is providing detailed light curves of about 60 detected sources and has uncovered new temporal/spectral states of galactic binary systems. The bizarre behavior of the possibly very young binary system, Cir X-1, is being revealed in detail by both the ASM and the PCA. A rare high/soft state of the black-hole candidate, Cyg X-1 provides new insight into the nature of the low/high transitions in black-hole binaries. The PCA has made possible the discovery of oscillations near 1 kHz in ten low-mass X-ray binary systems. These are most probably direct indicators of the neutron-star spin in some cases and probably indirect indicators in others. Studies of microquasars have unveiled a host of new temporal phenomena which may provide links between accretion processes and the radio jets in these systems.*'
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=-1.2truecm =-2.5truecm
=cmbx10 =cmr10 =cmti10 =cmbx10 scaled1 =cmr10 scaled1 =cmti10 scaled1 =cmbx9 =cmr9 =cmti9 =cmbx8 =cmr8 =cmti8 =cmr7 =cmti7 \[\]
**SOME EARLY RESULTS FROM THE**
**ROSSI X-RAY TIMING EXPLORER (RXTE)**
HALE BRADT
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Room 37-587, Cambridge MA 02139-4307, USA
Objectives, Instruments, and Launch
===================================
The scientific objective of the *Rossi X-ray Timing Explorer (RXTE) is to study the nature, environs, and evolution of galactic and extragalactic compact objects/systems through the study of the temporal variation of the emerging radiation over a very wide band of X-ray energies (2–200 keV). The compact objects being studied include galactic and extragalactic objects, namely white dwarfs, neutron stars, stellar black-hole candidates, and active galactic nuclei (massive black holes).[^1]*
The three instruments on *RXTE and the spacecraft capabilities are described by Bradt, Rothschild & Swank (1993). Two pointed instruments feature, respectively, proportional counters and crystal scintillators with large effective areas, 0.70 m$^2$ and 0.16 m$^2$, respectively. Their angular resolution (1$^\circ$ FWHM, circular) is obtained with mechanical collimation. The large areas and rapid onboard data processing permit the measurement of large count rates up to high energies. The high statistics obtained from bright sources permits the study of intensity variations on time scales as short as a few microseconds.*
The proportional counter array (PCA) system consists of 5 large detectors with sensitivity over 2–60 keV. The crystal system (HEXTE) consists of two clusters of four phoswiches each with a nominal response of 15–200 keV. The latter clusters rock on or off the target every $\sim$15-s to obtain frequent background measures with phasing such that one cluster is always viewing the target.
The third instrument is an All-Sky Monitor (ASM). It surveys up to about 80% of the sky with 5–10 samples of any celestial region each day. The overall energy range of the ASM is 1.5–12 keV which is telemetered in three energy channels. It can monitor sources in uncrowded regions down to about 35 mCrab (2 $\sigma$) in one 90-s exposure or about 10 mCrab for 1-day averages. The data and light curves from the ASM are public and available on the web:
http://space.mit.edu/XTE/XTE.html
http://heasarc.gsfc.nasa.gov/docs/xte/asm\_products.html.
The spacecraft can point the PCA/HEXTE to any position on the sky anytime of the year except that directions within $30^\circ$ of the sun are excluded. A new target can be acquired within hours of a celestial event detected with the ASM or by other ground-based or satellite-borne instruments. This capability has been used to great advantage on *RXTE, though usually with a response time on the order of 24 hours because a more rapid response was not required.*
The *RXTE was launched successfully on 1995 December 30. All three instruments are currently operational with only relatively minor deficiencies; they are performing very close to their full design capabilities.*
In the following, I present some early results that are forthcoming from *RXTE . I do not attempt to present a balanced overview of the *RXTE output; the studies being carried out by hundreds of observers are too wide ranging and diverse. Rather, I select certain topics in which I have been involved or which are of particular interest to me. The results given here are mostly temporal studies from the ASM and the PCA; these results are more easily forthcoming than spectral studies (including HEXTE data) because the high quality calibrations required for spectral analysis are only now becoming available. The Guest Observing Program started in early in 1996 February and AO-2 observations began in 1996 November. Early results from *RXTE appeared in the 1996 Sept. 20 issue of Astrophysical Journal Letters.***
Light Curves from the All-Sky Monitor
=====================================
Two of the three ASM detectors developed high-voltage breakdown shortly after launch, and all three detectors were shut down for diagnostics. The working counter was reactivated on 1996 Feb. 22, and essentially complete light curves together with 3-channel hardness ratios were obtained from that date. The two failed detectors were recovered with most of their effective area by mid-March 1996.
Early results from the ASM have been reported by Levine et al. (1996). From a catalog of $\sim$170 cataloged sources, about 60 yield day-by-day light curves and an additional $\sim$40 are possibly detected when averaged over a number of days. The light curves show a wide variety of behavior. Six examples of light curves of galactic sources for the year 1996 are shown in Fig. 1. All are accreting binaries. The compact object is believed to be a neutron star for the three systems portrayed on the left side of the figure and a black hole for the three on the right. The intensity is given in ASM ct/s in which the Crab nebula yields 75 ct/s. Features of note for these sources are described below. See Levine et al. (1996) and van Paradijs (1995) for references to previous work on these sources.
Cir X-1: Circinus X-1 is believed to be an LMXB in a 16.6-d elliptical orbit as evidenced by periodic outbursts that are presumed to occur during periastron passage. (See refs. in Shirey et al. 1996.) These ASM data show it to be continuously active during 1996 with the 16.6-d outbursts highly visible. (It can be very faint or undetectable for many months at a time.) The nature of this system with its highly obscured IR counterpart is not well known (see, e.g., Glass 1994). It may be a young runaway system from a nearby supernova remnant (Stewart et al. 1993). This source is discussed further below.
SMC X-1: The periodic on-off behavior of this source confirms the $\sim$60-d cyclic period suggested by previous observers (Gruber & Rothschild 1984). If so, this is likely analogous to the 35 d cycle of Her X-1 and the 30-d cycle of LMC X-4 which are attributed to precession of the accretion disk (see Priedhorsky & Holt 1987).
Cyg X-2: This evolving double-humped light curve has an apparent 78-d period (Wijnands, Kuulkers, & Smale 1996a). This period is independently seen in Ariel 5 and Vela 5B data (e.g., Smale & Lochner 1992). This could be another precessing disk, but it does not follow the behavior observed in other known precessing-disk systems.
Cyg X-1: This black-hole candidate (BHC) entered a rarely observed high (soft) state in May 1996 with a peak intensity of $\sim$1.3 Crab. It returned to its low state in August. The spectral change was dramatically evident in the ASM hardness ratios and also in a comparison with BATSE data. The entire spectral range of the latter is above the ASM range. In both instruments, the high-energy flux was found to decrease. (Cui et al. 1997a, Zhang et al. 1997). There is more on this source below.
GRS 1915+105: This microquasar shows a remarkable series of intensity/spectral states; it is discussed below.
4U 1630–472. This soft X-ray transient may contain a black hole because its X-ray spectrum is very soft with a hard tail reminiscent of Cyg X-1 (Parmar, Stella, & White 1986). These data show the source appear and then disappear again about 150 days later. The turn on in such systems may be triggered by an accretion-disk instability (see, e.g., Mineshige & Wheeler 1989). These data show the turn-on and turn-off in previously unseen detail, which should be useful in distinguishing models.
Reports of periodic behavior from the ASM data include the confirmations of the $\sim$78-d period in Cyg X-2 and the $\sim$60-d period in SMC X-1 noted above, a new $\sim$37-d period in Sco XÐ1 (Peele & White 1996), confirmation of a 2.7-h period in 2S 0114+650 together with an 11.7-d modulation consistent with the optical period (Corbet & Finley 1996), a 5.6-d orbital period in X-rays in Cyg X-1 (Zhang, Robinson, & Cui 1996a) also reported in Ginga data, and a 24.7-d modulation in the low-mass X-ray binary GX 13+1 also apparently seen in Ariel 5 data (Corbet 1996). Most of these reports have not yet reached the refereed literature and hence some should be viewed with caution.
The ASM has not yet discovered a completely ‘new’ X-ray source. Nevertheless it is proving invaluable (1) in illuminating the character of the long-term variations of various types of sources, (2) in discovering new or unusual states of sources, (3) in guiding the observing plan of the observatory, and most important, (4) in providing a temporal/spectral context for the relatively brief observations carried out by the PCA/HEXTE of a given source.
Cir X-1: QPOs and Rapid Absorption Events
=========================================
As noted above, Circinus X-1 has been continuously active since the launch of *RXTE.. An expanded ASM light curve and plots of hardness ratios are shown for six cycles of its 16.6-d period in Fig. 2. Repeatable systematic variations of the hardness ratios are readily apparent (Shirey et al. 1996). This variation in hardness ratios had been reported previously but only as an average over many cycles in Ginga data (Tsunemi et al. 1989).*
Circinus X-1 has long been known to exhibit rapid variability on time scales down to less than 1 s (see e.g., Dower, Bradt, & Morgan 1982, Oosterbroek et al. 1995 and refs. therein). Observations with the high statistics obtainable with the *RXTE PCA add substantial insight into the character of this variability. Three segments of *RXTE data are shown in Fig. 3; dramatic dips are apparent in the two top segments. The decreased flux during a low state (presumably during a dip) and the recovery from it have been shown by Brandt et al. (1996) with ASCA data to be due to changing absorption with partial covering. The *RXTE data confirm this behavior over a wider range of intensity/temporal conditions and X-ray bandwidth. The third segment of Fig. 3 shows a steady flux but with variations well in excess of the $<$1% expected from the statistics (13,000 ct/bin), indicating the presence of rapid non-statistical fluctuations.***
Power density spectra (PDS) of this quiescent flux (Fig. 4) show substantial power in excess of Poisson fluctuations (Shirey et al. 1996). The non-Poisson variations include pronounced quasi-periodic oscillations (QPO) of relatively high Q which drift between 1.3–12 Hz, a flat-topped noise, and a broad peak (QPO) with centroid that varies from 20–100 Hz in the PDS of Fig. 4.
The evolution of source characteristics (frequency and energy spectra) during the quiescent periods between outbursts (Fig. 4) are also reported by Shirey et al. (1996). The features of the PDS (e.g., the frequencies of the two QPO and the level of the flat-topped power) are remarkably correlated with one another. These characteristics also appear to be correlated with phase of the 16-d orbit, but with some notable deviations. Some of these temporal features were previously known (e.g., Tennant 1988, Oosterbroek et al. 1995), but the high sensitivity and repeated observations of *RXTE permit one to follow their evolution with orbital phase. The conjecture is that the evolution during this quiescent phase reflects accretion from the disk while it is not being replenished from the normal companion.*
Cygnus X-1: Transitions to and from the High State
==================================================
Spectral and timing studies of Cyg X-1 were carried out periodically with the *RXTE / PCA during the transition to the high/soft state. The data are public and two groups studied them (Belloni et al. 1996; Cui et al. 1997a). Their results provide insight into the spectral-formation processes. Both groups report that the spectral softening in the transition was associated with an increase in the high-frequency cutoff in the PDS of the X-ray signal, not inconsistent with the behavior of other black-hole candidates. The latter authors note that this is consistent qualitatively with a Comptonizing corona which reduces size (but not density) as the source moves into the high state. A smaller corona means fewer inverse-Compton scatters. This results in relatively fewer high energy photons, i.e., a softening spectrum. It also leads to less smearing of the high-frequency fluctuations in the source photons; i.e., the cutoff frequency in the PDS increases.*
This scenario was reinforced dramatically with studies of the cross-correla-tion between hard and soft fluxes as the source moved into and out of the high state (Cui et al. 1997b). A delay of hard photons relative to soft photons had been reported by Miyamoto et al. (1988) for Cyg X-1. The *RXTE data exhibited this delay and further showed that the average delay decreased during the transition to the high state, becomes very small during it, and increased again during the return to the low state (Fig. 5). In general, hard-flux delays are expected from the Comptonization up-scatter process, and one would expect these delays to be reduced for a smaller corona with fewer scatters, in agreement with the data.*
Variable QPO Oscillations at $\sim$1 kHz (Sco X-1)
==================================================
The fast spin of millisecond (1–10 ms) radio pulsars has long been postulated to be due to gradual angular momentum transfer by accreting matter in low-mass X-ray binaries (Smarr & Blandford 1976; Bhattacharya 1995). However, such pulsations had not been detected from X-ray binaries before the launch of *RXTE . A prime objective of *RXTE was to find this missing link in the evolution of neutron-star systems. It now appears that *RXTE has been successful in this endeavor. There are apparently two types of such oscillations: variable-frequency quasi-periodic (QPO) oscillations in the persistent flux and coherent oscillations during X-ray bursts. Both may be ramifications of the neutron-star spin.***
At this writing, there are eight X-ray sources for which variable-frequency QPOs have been reported: 4U 1728–34 (Strohmayer et al. 1996), Sco X-1 (van der Klis et al. 1996a), 4U 1636–536 (Zhang et al. 1996b), 4U 1608–52 (Berger et al. 1996), 4U 0614+091 (Ford et al 1997), 4U 1735–44 (Wijnands et al. 1996b), 4U 1820–30 (Smale, Zhang, & White 1996), and GX 5–1 (van der Klis et al. 1996b). Here I will describe the oscillations from Sco X-1.
The brightest X-ray source in the sky is Sco X-1; hence it provides the high statistics in short periods needed in an efficient search for, and tracking of, high-frequency QPO. A double QPO feature has been found in this source (Fig. 6; van der Klis et al. 1996a); with peaks at $\sim$800 and $\sim$1100 Hz. The peaks are not always present and sometimes only the 1100-Hz peak is present. The frequencies of the two peaks increase with accretion rate. The high statistics allow the frequencies of the QPO to be tracked with time. With longer integration times, the peaks would be substantially broadened or washed out. The power in these peaks is not large; the fractional rms amplitudes are of order 1%.
The frequency variation of these QPOs precludes interpreting the period directly as the neutron-star spin frequency. Nevertheless the high frequencies suggest strongly that the rapid oscillations originate in the immediate region of the neutron star. One possibility proposed by van der Klis et al. (1996a) is that the 1100 Hz represents the Kepler frequency of matter in the inner disk and that the 800 Hz is the beat frequency between the Kepler frequency and the neutron-star spin. If so, the spin frequency is $\sim$250 Hz. Several of the other sources listed above also exhibit two QPO peaks that move in frequency as the intensity varies. It appears that a number of these sources are exhibiting the same phenomenon.
A rather explicit model for the production of kilohertz oscillations in a neutron-star system has been put forward by Miller, Lamb & Psaltis (1997). They propose that the detected oscillations are at the Kepler frequency of the marginally stable orbit, from which clumps of matter are gradually stripped to create a stream of matter falling onto the neutron star. The initial radius of the clump is the *sonic radius . The result is a hot footprint on the neutron star which rotates around the star (in the inertial frame) at the Kepler frequency of the sonic radius. This frequency is thus directly measurable. In this model, the detection of a high Keplerian frequency places an upper bound on both the mass and radius of the neutron star. In turn, this places constraints on the equation of state in all neutron stars.*
Other proposed origins of kilohz oscillations are photon-bubble oscillations in the accretion column (Klein et al. 1996) and neutron-star vibrations (e.g., McDermott, Van Horn, & Hansen 1988). The latter origin seems excluded because it does not naturally yield the substantial dependence of the QPO frequencies upon mass accretion rate.
Oscillations during Bursts at $\sim$1 kHz (4U1728–34)
=====================================================
In work that paralleled that just described, Strohmayer et al. (1996) discovered oscillations during an X-ray burst from the LMXB source 4U1728–34. This source is a well known Type I X-ray burst source; i.e., its atmosphere occasionally undergoes unstable thermonuclear He burning. Eight bursts were detected during the Strohmayer et al. (1996) observations, and six of them showed oscillations near 363 Hz.
The frequency evolution of the 363-Hz signal in one burst is shown in Fig. 7 together with the light curve of the burst itself. The frequency varied $\sim$1.5 Hz as the burst progressed becoming effectively coherent in the burst tail at 364 Hz. The pulsations were present during the rise of the burst; fell below threshold near the peak, and reappeared thereafter. The rms amplitude was $\sim$10% at the start of the rise and 3–7% after reappearing. The authors propose that these oscillations reflect directly the neutron-star spin period. The high coherence of the 364 Hz in the tail gives credence to this interpretation.
The *persistent emission from this source (4U 1728–34) exhibits two high-frequency QPO in the PDS, similar to those of Sco X-1. The higher frequency varies from 700–1100 Hz and the lower, when present, varies between 600 and 800 Hz. The rms amplitude of these pulsations is in the 5–8% range, a substantially larger portion of the flux than for Sco X-1. When both peaks are present, the difference of their frequencies is $\sim$360 $\pm$ 10 Hz, consistent with a constant frequency difference as the individual peaks vary in frequency. In the beat-frequency picture given above for Sco X-1, this would be the neutron-star spin frequency. Remarkably, it is in agreement with the frequency of the coherent frequency found during the burst. This strengthens the case for an underlying fundamental frequency of 364 Hz, and this most likely is the neutron-star spin frequency. If so, a major goal of *RXTE has indeed been accomplished.**
The $\sim$363-Hz modulation during the burst could arise from thermonuclear-flash inhomogeneities on the neutron-star surface (Bildsten 1995); the frequency drift during the burst rise could be due to the progression of the burning front on the surface (Strohmayer et al. 1996).
We note that 4U0614+091 may be another case where the difference frequency between two high-frequency QPOs also appears directly, but in this case the latter oscillations are in the persistent flux at 328 Hz at only 3.3 $\sigma$ (Ford et al. 1997). Two other cases of oscillations in bursts have been found at this writing. The transient source KS 1731–30 exhibits 524 Hz oscillations for a duration of 2 s with Q $\geq$ 900 (Smith, Morgan, & Bradt 1997; Fig. 8), and an unidentified source near GRO J1744–28 exhibited 589 Hz in a 4-s interval (Strohmayer, Lee, & Jahoda 1996).
Microquasars
============
There are at least eight galactic X-ray sources that exhibit radio jets or their temporal/spectral characteristics, namely the long known Cyg X-3, SS433, and Cir X-1 (See refs. in van Paradijs 1995), the galactic-center sources 1E1740.7–2942 and GRS 1758–258 (see refs. in Mirabel & Rodriguez 1994), the hard X-ray transient GRS 1739–278 recently discovered with Sigma/Granat (Vargas et al. 1997) and determined to be a variable radio flaring source (Hjellming et al. 1996), and finally two radio jet sources that exhibit superluminal motion: GRS 1915+105 (Mirabel & Rodriguez 1994) and GRO 1655–40 (Tingay et al. 1995; Hjellming & Rupen 1995). The latter two sources have been termed microquasars; in fact, all sources exhibiting radio jets may well deserve this name. These systems could well give insight into the relation between accretion processes and radio jets. The ASM data show that GRS 1915+105 has been active all year (see above) and that GRO 1655–40 became active on Apr. 25 of this year. Here I would like to present some remarkable *RXTE results on the two latter microquasars.*
GRS 1915+105
------------
This transient source, originally discovered in X rays with *Granat WATCH (Castro-Tirado et al. 1992) has been bright and active since the beginning of the *RXTE mission (Fig. 1). The ASM and PCA exhibit several distinct states, described and named in Morgan, Remillard, and Greiner (1997; Fig. 9). Observations with the high sensitivity of the PCA (Greiner, Morgan, & Remillard 1996) show dramatic dips, with quasi-periodic repetitions on time scales of a few minutes (Fig. 10) during the ‘chaotic’ state. These dips do not show the spectral or temporal characteristics expected of absorption events suggesting strongly that they are accretion phenomena. In the ‘bright’ state, the source exhibits intense, relatively high-Q QPOs at 0.05–1 Hz (Morgan, Remillard, & Greiner 1997; Fig. 11). These vary in frequency, disappear, and reappear at different frequencies. The stronger peaks have rms amplitudes of order 10–15%.**
The authors of the latter work have succeeded in tracking the phases of the individual cycles of some of these QPO. They find a random walk, in that the duration of one QPO cycle is uncorrelated with the duration of the adjacent cycles. Neither is it correlated with the amplitude of the cycle. These features eliminate classes of models such as those that invoke a reservoir or those predicting trains of relatively coherent (or gradually changing) pulses. Superposition of the individual pulses showed a significant delay in the hard pulsing, of $\sim$4% of the pulse period, for periods of 1/2 to 15 s. They also showed full-width pulse amplitudes up to $\sim$40% of the mean flux at the highest energies, $\sim$15 keV. The authors state “that if the high-energy photons are derived from inverse Compton scattering, then \[these characteristics\] suggest that the energy distribution of the energetic electrons must be oscillating at the QPO frequency.” Therein lies the possible connection to the radio jets.
The same work revealed a QPO at a higher frequency, 67 Hz, which reappeared six times at the same frequency (Fig. 12). It is quite a sharp feature (Q $\sim$ 20) with a relatively small rms amplitude, $\sim$1%. It too is most powerful at high energies; on May 5, it had rms amplitude of 1.5% below 4 keV and 6% above 10 keV. The authors speculate that this frequency is that of the innermost stable orbit around a non-rotating black hole, which is at 3.0 Schwarzschild radii (3.0 $R_S$). In this case, the mass of the black hole is determined to be 33 $M_{\odot}$. Of course, if the black hole has significant angular momentum, this value would be different. An alternate explanation invokes transverse ‘diskoseismic’ g-mode oscillations of the accretion disk (Nowak et al. 1997). However, this does not appear to be in accord with the very hard spectrum of the 67-Hz feature (R. Remillard, pvt. comm.).
Finally, we call attention to two other works on *RXTE data from this source during this time frame: Chen, Swank, & Taam (1997) and Belloni et al. (1997). These bring somewhat different perceptions to the same phenomenology.*
GRO 1655–40
-----------
The other microquasar, GRO J1655–40 is a transient source also discovered in X rays (Zhang et al. 1994), with CGRO/BATSE in 1994. It has exhibited subsequent outbursts, but was undetectable in the ASM in early 1996. However, it suddenly reappeared on 1996 Apr. 25 and reached $\sim$2 Crab in the next 10 days (Fig. 13). The compact object in this binary system has been found from optical studies of the companion to have a mass of 7.0 $\pm$ 0.2 $M_{\odot}$ (statistical errors only) which makes it a likely black hole (Orosz & Bailyn 1997).
The resurgence of this microquasar in April led to studies with the PCA. A high frequency QPO has been found at a frequency of $\sim$298 Hz when PDS are superimposed from the portions of the data exhibiting the hardest spectra (Remillard 1996). If, again, the periodicity is taken to be that of the innermost stable orbit of a non-rotating black hole, at 3.0 $R_S$, one obtains a black-hole mass of 7.4 $\pm$ 0.2 $M_{\odot}$ (Remillard 1996). This agrees, amazingly, with the optically measured mass of the black hole, 7.0 $\pm$ 0.2 $M_{\odot}$ quoted above. Turning the argument around, this would constitute a measurement of the angular momentum of the black hole. Unfortunately, since the scenario is not at all well established, this remains speculative, but the close agreement of the two mass determinations is quite provocative.
Just prior to the April resurgence, the source was being observed with optical photometry (Orosz et al. 1997). The multicolored ellipsoidal light curves showed a steady brightening in B, V, R, and I that began about six days before the X-ray flux began to rise. The onset began in the I band and then, sequentially, in the R, V, and B bands over a period of 1.1 d. The authors suggest that this indicates an inward moving (‘outside-in’) disturbance in the accretion disk. They further suggest that the substantial delay before the X-ray onset might provide indirect support for advection-dominated accretion flow in quiescent black-hole binaries (Narayan, McClintock, & Yi 1996).
Conclusions
===========
The All-Sky Monitor on *RXTE is exploring and revealing long term variations that were either unknown or poorly known. It has been instrumental in the productive use of the large Proportional Counter Array, and it provides a valuable spectral and temporal context for observations with the PCA and other instruments or telescopes. Several long term periodicities have been found or better established. Some of these may be other examples of accretion disk precession. The ASM light curves for the possibly uniquely young system Cir X-1 and of two superluminal microquasars show complex temporal-spectral behavior either for the first time or in remarkable new detail.*
Transitions and other variabilty in PCA data from the very different systems, Cir X-1 and Cyg X-1, are providing new insight into the conditions near the presumed neutron star or, respectively, presumed black hole.
The PCA has revealed two categories of oscillations near 1 kHz in Low-Mass X-ray Binary (LMXB) systems. Sco X-1 typifies one category, namely a relatively low-Q oscillation (i.e., a QPO) that increases in frequency with mass accretion rate and which is sometimes accompanied by a second QPO at a frequency several hundred Hz lower. The difference between these two frequencies may well be the spin frequency of the neutron star. There are currently eight examples of variable $\sim$1 kHz oscillation.
The other category consists of three sources where pulsations have been observed during X-ray Type-I (thermonuclear) bursts. These are quite coherent and are very likely a direct view of the neutron-star spin. One of these sources (4U 1728–34) exhibits both phenomena: the doublet near 1 kHz and coherent oscillations during a burst. The frequency during the burst agrees well with the difference frequency of the doublet, as expected if both represent the neutron-star spin.
It appears therefore that *RXTE has indeed demonstrated that the neutron stars in LMXB have been spun up to the high spin rates characteristic of the millisecond radio pulsars.*
The study of microquasars has become a major activity of *RXTE . In GRS 1915+105, multiple temporal/spectral states, large-amplitude ringing, strong variable QPO, and a recurrent 67-Hz signal have been observed. In GRO 1655–40, a $\sim$298-Hz signal has been found. These latter signals have occurred when the source spectrum is at its hardest. These phenomena in conjunction with observations at other wavelengths should go far toward revealing the role accretion processes play in (radio) jet formation.*
In summary, the spectral and temporal phenomena being revealed by *RXTE are informing us directly about accretion processes, the evolution of compact binary systems, and the nature of the neutron stars and black holes. I close by reminding the reader that there are many other important results from *RXTE that have not been covered here.**
Acknowledgments {#acknowledgments .unnumbered}
===============
The author is grateful for the efforts of the entire *RXTE team and the many observers whose work contributed toward this perspective. This work was supported in part by NASA under contract NAS5-30612.*
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Figure Captions
===============
Figure 1: Six light curves (1.5–12 keV) from the ASM for the year 1996. In each plot, the ordinate is ASM ct/s; 1.0 Crab = 75 ASM ct/s. From A. Levine, pvt. communication.
Figure 2: Six cycles of the 16.6-d period of Cir X-1 with hardness ratios from the ASM (1.5–12 keV). 1.0 Crab = 75 ASM ct/s. From Shirey et al. 1996, ApJ, 469, L21.
Figure 3: Segments of PCA data from Cir X-1 with 1-s time bins at three different phases of the 16-d periodicity. Note the dramatic dips in (a) and (b) and the fluctuations which greatly exceed counting statistics in (c). From Shirey et al. 1996, ApJ, 469, L21.
Figure 4: Power density spectra from Cir X-1 (PCA data) during the quiescent phases of one 16.6-d cycle. Each PDS is plotted one decade below the previous one. They are ordered by QPO frequency which approximates orbital phase. From Shirey et al. 1996, ApJ, 469, L21.
Figure 5: Effective time lags of hard photons from Cyg X-1 for three bands with effective energies, 3, 9, and 33 keV respectively from PCA data. The lags of the hard band (asterisks) and medium band (diamonds) relative to the soft band are shown. The lags are the average over 1–10 Hz in PDS frequency. From Cui et al. 1997, ApJ, in press.
Figure 6: Power density spectrum for Sco X-1 from the PCA showing the two peaks (upper) in one data set, and the single peak (lower). From van der Klis et al. 1996a, ApJ, 469, L1.
Figure 7: Frequency of QPO from 4U 1728–34 during the burst shown in the lower panel (PCA data). Note the stability of the frequency during the tail at 364 Hz. From T. Strohmayer, pvt. communication.
Figure 8: Power density spectrum (PCA data) for a 2-s period during a Type-I X-ray burst from KS 1731–260. From Smith, Morgan, & Bradt 1997, ApJLett, in press.
Figure 9: Expanded ASM light curve (1.5–12 keV) of microquasar GRS 1915+105 showing observation times of PCA (tic marks above), the several different states with their names, and hardness ratios (defined in Fig. 2). 1.0 Crab = 75 ASM ct/s. From Morgan, Remillard, and Greiner 1996, ApJ, in press.
Figure 10: PCA light curve from microquasar GRS 1915+105 showing dramatic dips and flaring. Note the narrow precursor dips in the upper figure and the softening of the spectrum during the dips. This and the excess flux after some dips argue against absorption as the cause of this variability. From Greiner, Morgan and Remillard 1996, ApJ, 473, L107.
Figure 11: Power density spectrum from GRS 1915+105 for 1996 May 5 (PCA data). The pronounced peak at 0.067 Hz (15 s) has 3 harmonics. Note also the peak at 67 Hz. From E. Morgan, pvt. communication.
Figure 12: Power density spectra for GRS 1915+105 showing the six appearances of the 67-Hz feature. From Morgan, Remillard, and Greiner 1996, ApJ, in press.
Figure 13: ASM light curve (1.5–12 keV) of the microquasar GRO 1655–40. 1.0 Crab = 75 ASM ct/s. From D. A. Smith, pvt. communication.
[^1]: This paper was completed late in 1996; some results herein reflect work postdating the conference. This paper also will appear in the Proceedings of 5th International Workshop on Data Analysis in Astronomy, CCSEM Center, Erice, Italy, (Oct. 1996), eds. L. Scarsi and C. Maccarone, World Scientific Publ. Co., with the much appreciated permission of the Editors of both proceedings.
|
---
abstract: 'We present new eclipse observations of the polar (i.e. semi-detached magnetic white dwarf + M-dwarf binary) HU Aqr, and mid-egress times for each eclipse, which continue to be observed increasingly early. Recent eclipses occurred more than 70 seconds earlier than the prediction from the latest model that invoked a single circumbinary planet to explain the observed orbital period variations, thereby conclusively proving this model to be incorrect. Using ULTRACAM data, we show that mid-egress times determined for simultaneous data taken at different wavelengths agree with each other. The large variations in the observed eclipse times cannot be explained by planetary models containing up to three planets, because of poor fits to the data as well as orbital instability on short time scales. The peak-to-peak amplitude of the O-C diagram of almost 140 seconds is also too great to be caused by Applegate’s mechanism, movement of the accretion spot on the surface of the white dwarf, or by asynchronous rotation of the white dwarf. What does cause the observed eclipse time variations remains a mystery.'
bibliography:
- 'bibliography.bib'
date: 'Accepted XX XXXX – Received XX XXXX'
title: Testing the planetary models of HU Aquarii
---
\[firstpage\]
stars: individual: HU Aquarii – white dwarfs – binaries: eclipsing
Introduction
============
HU Aqr was discovered independently by @Schwope93 and @Hakala93 as an eclipsing binary of the AM Her type, also known as polars. This subset of cataclysmic variables (CVs) contains a Roche-lobe filling M-dwarf secondary star and a strongly magnetic ($\sim$ 10 MG) white dwarf as the primary star. From its discovery onwards, HU Aqr has been studied extensively and over a wide range of wavelengths [e.g. @Glenn94; @Schwope97; @Schwope01; @Heerlein99; @Howell02; @HarropAllin01; @HarropAllin99; @Vrielmann01; @Bridge02; @Watson03; @Schwarz09]. The general picture is as follows: the M-dwarf loses matter at the L1 Lagrange point, which then follows a ballistic trajectory until the ram pressure equals the white dwarf’s magnetic pressure and the stream couples to the magnetic field lines. At that point the stream leaves the orbital plane and is guided along the field lines until it accretes onto the white dwarf’s magnetic pole, creating a luminous accretion spot.
The accretion rate in this system is highly variable, and the binary has changed from a high to a low state and back several times over the last decades. The variability causes both flickering typical of CVs on timescales of minutes, and significant changes in the overall shape of the observed light curves on timescales as short as one orbital cycle [@HarropAllin01].
One constant in the light curves is the eclipse of the white dwarf by the secondary star, the ingress and egress of which last for $\sim$ 30 seconds. In a high state the eclipse is dominated by contributions from the accretion spot and stream, while the ingress and egress of the white dwarf itself is hardly visible. Due to the geometry of the system the accretion spot and accretion stream are well separated during egress, giving this part of the light curve a fairly constant shape. During ingress the distinction between the accretion spot and stream features is less clear. From X-ray data there is also evidence of enhanced absorption by an accretion curtain along the ballistic stream at this phase [@Schwope01]. To accurately determine the eclipse times the timing has therefore been based on the egress of the accretion spot. Comparison of observed mid-egress times to expected mid-egress times, which are calculated assuming a constant orbital period, have revealed considerable variations [@Schwope14; @Gozdziewski12; @Schwarz09; @Schwope01].
Several explanations for these eclipse time variations have been offered in the literature. Similar variations have been seen in a number of other binaries, leading @Applegate92 to propose a theory that has since become known as Applegate’s mechanism. It assumes that the M-dwarf experiences Solar-like magnetic cycles that couple to the binary’s orbital period, by affecting the gravitational quadrupole moment of the M-dwarf, and therefore cause genuine modulations of the orbital period. However, for HU Aqr, the energy available in the M-dwarf is insufficient to explain the large variations observed [@Schwarz09]. A second explanation for the eclipse time variations is that they are caused by the presence of planet-like or brown dwarf-like bodies in wide orbits around the binary. The additional mass causes periodic shifts in the binary’s centre of mass, which are reflected in the eclipse times [@Marsh14]. This theory has gained popularity after the discovery of circumbinary planets around double main-sequence star binaries [e.g. @Welsh12; @Orosz12a], and models with 1, 2 and even 3 planets have been proposed for HU Aqr [@Gozdziewski12; @Qian11]. All have since been disproved on grounds of dynamical instability [@Wittenmyer12; @Horner11] or by new data [this paper, @Schwope14; @Gozdziewski12].
In this paper we present 22 new eclipse times from data taken between June 2010 and June 2014.
Observations
============
In this section we describe the technical details of the observations. The light curves themselves are discussed in Sect. \[sect:ever\_changing\_light\_curves\]. All mid-egress times are listed in Table \[tab:times\], which also summarises the technical details and contains notes on the observing conditions. Details of how we determined the mid-egress times are described in Sect. \[sect:egresstimes\].
RISE on the Liverpool Telescope
-------------------------------
We observed 12 eclipse observations of HU Aqr between 2 Aug 2011 and 26 Jun 2014 with the Liverpool Telescope [LT; @Steele04], a 2-metre robotic telescope on the island of La Palma, Spain. The instrument used was the RISE camera [@Steele08], which contains a frame transfer CCD and uses a single ‘V+R’ filter, and we used the 2x2 binned mode. The data were flatfielded and debiased in the automatic pipeline, in which a scaled dark-frame was removed as well. Aperture photometry was then performed using the ULTRACAM pipeline [@Dhillon07], and care was taken to use the same comparison star for all data reductions. This comparison is a non-variable star located 93“ South and 88” West of HU Aqr. We chose this star rather than comparison ‘C’ as in @Schwope93 [their Fig. 1] because of their relative brightnesses in RISE’s V+R filter. We did use comparison ‘C’ to calculate magnitudes for HU Aqr, as will be explained in Sect. \[sect:ever\_changing\_light\_curves\]. All stellar profiles were fitted with a Moffat profile [@Moffat69], the target aperture diameters scaled with the seeing and the comparison star was used to account for variations in the transmission. The light curves are shown in Fig. \[fig:lteclipses\].
![LT+RISE light curves of HU Aqr, taken between 2 Aug 2011 and 26 Jun 2014. The vertical axis shows the flux relative to the comparison star, which is the same star for all LT+RISE and TNT+ULTRASPEC data. Each light curve is vertically offset from the previous one by 0.2. To calculate the orbital phase we used the ephemeris in equation \[eq:ephemeris\].[]{data-label="fig:lteclipses"}](lightcurves_lt.eps)
ULTRACAM observations
---------------------
{width="\textwidth"}
Between June 2010 and October 2012 we obtained six eclipses of HU Aqr with the high-speed camera ULTRACAM, which splits the incoming light into three beams, each containing a different filter and a frame-transfer CCD [@Dhillon07]. ULTRACAM was mounted on the New Technology Telescope (NTT, 3.5m) in Chile during the first three observations and on the William Herschel Telescope (WHT, 4.2m) on La Palma, Spain, for the last three observations. The light curves are shown in Fig. \[fig:ultracameclipses\].
For the data reduction and the relative aperture photometry we used the ULTRACAM pipeline. Due to different fields of view and windowed setups during the observations we could not use the same comparison star as for the LT+RISE and TNT+ULTRASPEC data.
ULTRASPEC on the Thai National Telescope
----------------------------------------
In November 2013 we observed two HU Aqr eclipses with the 2.4-metre Thai National Telescope (TNT), located on Doi Inthanon in northern Thailand. We used the ULTRASPEC camera [@Dhillon14], which has a frame-transfer EMCCD, with an SDSS g$^{\prime}$ filter on November 10, and a Schott KG5 filter on November 13. The Schott KG5 filter is a broad filter with its central wavelength at 5075 Å and a FWHM of 3605 Å.
The data were reduced using the ULTRACAM pipeline, with which we debiased and flatfielded the data and performed aperture photometry. We used the same comparison star as for the reduction of the LT+RISE data. Both TNT light curves are shown in Fig. \[fig:tnt\_eclipses\]. During the observations on November 10, the telescope briefly stopped tracking, causing the gap seen during ingress in the light curve in Fig. \[fig:tnt\_eclipses\].
![TNT+ULTRASPEC light curves of HU Aqr, taken at 10 Nov 2013 (bottom) and 13 Nov 2013 (top, vertically offset by 0.2). To calculate the orbital phase we used the ephemeris from equation \[eq:ephemeris\]. []{data-label="fig:tnt_eclipses"}](lightcurves_ultraspec.eps)
Wide Field Camera on the Isaac Newton Telescope
-----------------------------------------------
In June 2014 we observed two HU Aqr eclipses using the Wide Field Camera (WFC) mounted at the prime focus of the 2.5-metre Isaac Newton Telescope (INT) on La Palma, Spain. The read-out time of the WHF is $\sim$ 2 seconds, and we used a Sloan-Gunn g filter.
The data were reduced using the ULTRACAM pipeline, with which we debiased and flatfielded the data and performed aperture photometry. As a comparison star we used star ‘C’ as in @Schwope93. Both light curves are shown in Fig \[fig:intwfc\_eclipses\].
![INT+WFC light curves of HU Aqr, taken at 18 Jun 2014 (bottom) and 19 Jun 2014 (top, vertically offset by 0.4). To calculate the orbital phase we used the ephemeris from equation \[eq:ephemeris\]. []{data-label="fig:intwfc_eclipses"}](lightcurves_intwfc.eps)
Ever-changing light curves {#sect:ever_changing_light_curves}
==========================
HU Aqr is known for its variable accretion rate, which significantly influences the brightness of the system and the morphology of its light curves, as is immediately clear from Figs. \[fig:lteclipses\], \[fig:ultracameclipses\], \[fig:tnt\_eclipses\] and \[fig:intwfc\_eclipses\]. HU Aqr was in a high state until mid 2012, then went into a low state, and returned to a high state in the second half of 2013, after which it dropped to a lower state again. We calculated the magnitude of HU Aqr for each dataset, using the out-of-eclipse data, and excluding obvious flares from the M-dwarf and dips due to the accretion stream. Using simultaneous g$^{\prime}$ and r$^{\prime}$ ULTRACAM data, we determined a magnitude offset for comparison ‘C’ [@Schwope93 their Fig. 1] for each filter type with respect to its V-band magnitude of = 14.65 (mean of various measurements). We then calculated magnitudes for HU Aqr as normal, using the relative flux and corrected magnitude for comparison ‘C’. These approximate magnitudes are listed in Table \[tab:times\]. Typical magnitudes were $m_{g^{\prime}}$ $\simeq$ 15.1 - 15.5 during the high states, and $m_{g^{\prime}}$ $\simeq$ 17.8 during the low state.
The out-of-eclipse data shows the characteristic flickering of a CV, caused by the irregular, blobby nature of the accretion. In these high state light curves the eclipse is dominated by both the accretion spot on the white dwarf and the accretion stream. The eclipse of the spot is characterised by the abrupt ingress when the spot goes behind the M-dwarf, and the egress when it re-emerges, each typically lasting $\sim$ 8 seconds.
Due to the small physical size of the accretion region on the white dwarf, this spot can reach very high temperatures and contributes significantly to the total light from the system when accretion rates are high. Our data shows no signs of a varying width or height of the ingress or egress feature, over time nor with changing accretion rate. Assuming for the masses $M_{\mathrm{wd}}$ = 0.80 M$_{\odot}$ and $M_2$ = 0.18 M$_{\odot}$, an inclination $i$ = 87$^{\circ}$ [@Schwope11] and the orbital period from equation \[eq:ephemeris\], we arrive at an orbital velocity of the M-dwarf of 479 km/s and a maximum spot diameter of $D_{\mathrm{spot}} \simeq$ 3829 km. Using the mass-radius relation of @VR88 for the white dwarf, this corresponds to a fractional area on the white dwarf of 0.018 and an opening angle of the accretion spot of $\sim 30^{\circ}$. This is significantly larger than the value of 3$^{\circ}$ found by @Schwope01 using ROSAT-PSPC soft X-ray data although some difference is to be expected, as only the hottest parts of the spot will radiate at X-ray wavelengths.
In the high state the ingress of the accretion stream, caused by irradiation of the stream by the hot accretion spot, is visible as an additional, shallower slope after the sharp ingress of the accretion spot. Depending on the exact geometry of the stream and its relative position to the spot, the duration and height of the stream ingress vary considerably.
Some of the light curves taken during a high state also show a narrow dip near orbital phase 0.8-0.9. This has been seen in many of the other studies on HU Aqr, and is also observed in other polars [@Watson95]. It is caused by the obscuration of the accretion spot by part of the stream if the inclination of the system is such that the hemisphere with both the spot and stream is towards the observer [@Bridge02]. The depth and width of this dip carry information about the temperature difference between the white dwarf and the spot and about the physical extent of the magnetically coupled stream respectively. The dip moves further away from the spot ingress during high accretion states, which agrees with the expectation that the ballistic stream penetrates further before coupling to the magnetic field lines in the high state. We also notice some interesting colour differences in the dip as well as in some of the other variable features in the high state ULTRACAM light curves (Fig. \[fig:ultracameclipses\]). At blue wavelengths the dip is wider than observed in the other two bands, and it possibly consists of two components. This indicates a fluffy and blobby nature of the stream, and a strongly wavelength-dependent opacity within the stream.
During a low state, the stream ingress disappears completely, flickering is less pronounced and the system is noticeably fainter.
Egress times {#sect:egresstimes}
============
------------- -------- ------------------- -------------------- ------------- -------------- ------------- -----------------------------
date cycle mid-egress time t$_{\mathrm{exp}}$ telescope + filter(s) approximate observing conditions
number BMJD(TDB) (sec) instrument magnitude
06 Jun 2010 72009 55354.2706040(4) 4, 2, 2 NTT+UCAM 15.8 clear, seeing 1.5
06 Jun 2010 72010 55354.3574451(5) 4, 2, 2 NTT+UCAM 15.7 clear, seeing 1-2
23 May 2011 76053 55705.3721585(9) 8, 4, 4 NTT+UCAM 15.5 clear, seeing 1.5
02 Aug 2011 76868 55776.1307426(20) 3 LT+RISE V+R 15.9 clear, seeing 2
04 Sep 2011 77247 55809.0356564(25) 2 LT+RISE V+R 15.8 clear, seeing 2.5
31 Oct 2011 77902 55865.9029864(22) 2 LT+RISE V+R 15.7 clear, seeing 1.6-2.5
28 May 2012 80324 56076.1818394(22) 2 LT+RISE V+R 15.9 clear, seeing 1.8-2.4
11 Jun 2012 80485 56090.1598976(19) 2 LT+RISE V+R 17.3 thin clouds, seeing 2-4
06 Sep 2012 81486 56177.0670248(8) 7, 4, 4 WHT+UCAM 16.1 clear, seeing 1-2
10 Sep 2012 81531 56180.9739470(22) 2 LT+RISE V+R 16.4 thin clouds, seeing 2.2
10 Sep 2012 81532 56191.0607721(6) 7, 4, 4 WHT+UCAM 16.6 thin clouds, seeing 2
13 Oct 2012 81910 56213.8788462(2) 6, 2, 2 WHT+UCAM 17.8 thin clouds, seeing 2-4
09 Dec 2012 82566 56270.8329602(53) 2 LT+RISE V+R 17.2 clear, seeing 2
06 May 2013 84275 56419.208883(11) 2 LT+RISE V+R 17.3 thick clouds, seeing 2-3
10 Jun 2013 84678 56454.1974374(17) 2 LT+RISE V+R 16.2 clear, seeing 2
30 Sep 2013 85965 56565.9351702(10) 2 LT+RISE V+R 14.8 clear, seeing 1.8
06 Nov 2013 86391 56602.9205819(16) 2 LT+RISE V+R 15.3 thin clouds, seeing 1.8-2.5
10 Nov 2013 86433 56606.5670216(45) 3 TNT+USPEC g$^{\prime}$ 15.1 cloudy, seeing 1.5-2
13 Nov 2013 86467 56609.5189097(21) 2 TNT+USPEC Schott KG5 15.2 thin clouds, seeing 1.5
18 Jun 2014 88973 56827.0904134(16) 5 INT+WFC g 15.7 clear, seeing 1.5
19 Jun 2014 88985 56828.1322517(37) 5 INT+WFC g 15.7 clear, seeing 2-3
26 Jun 2014 89066 56835.1647131(24) 2 LT+RISE V+R 16.1 clear, seeing 2
------------- -------- ------------------- -------------------- ------------- -------------- ------------- -----------------------------
![Egress of the LT+RISE eclipse light curve of HU Aqr, taken on 31 Oct 2011, with 2 second exposures. The solid black line shows the sigmoid+linear function that is fitted to the data. The vertical dotted line indicates the mid-egress time.[]{data-label="fig:sigmoid"}](sigmoid_20111031.eps)
As has been done for previously published times of HU Aqr, we chose to measure mid-egress times as opposed to mid-eclipse times. This because the egress feature is relatively stable, even with the variable accretion rate, whereas the shape of the ingress feature varies significantly with the changing accretion rate and even differs from cycle to cycle. For all our new data we determined the time of mid-egress by a least-squares fit of a function composed of a sigmoid and a straight line, $$y = \frac{a_1}{1 + e^{-a_2(x-a_3)}} + a_4 + a_5(x-a_3),$$ where $x$ and $y$ are the time and flux measurements of the light curve, and $a_1$ to $a_5$ are coefficients of the fit. An example of one of our fits is shown in Fig. \[fig:sigmoid\].
The straight line part allows us to fit the overall trend outside and during egress. This includes the egress of the white dwarf itself, which can have a significant contribution, especially when the system is in a low state. The sigmoid part of the function fits the sharp egress feature created by the egress of the accretion spot. To determine uncertainties, we have performed these fits in a Monte Carlo manner in which we perturb the value of the data points based on their uncertainties and we vary the number of included data points by a few at each edge, reducing any strong effects in the results caused by single datapoints.
We converted all mid-egress times to barycentric dynamical time (TDB) in the form of modified Julian days and corrected to the barycentre of the Solar System, giving what we refer to as BMJD(TDB). The times are listed in Table \[tab:times\] and we used the ephemeris of @Schwarz09 to calculate the corresponding cycle number $E$. Including the new times the best linear ephemeris is given by: $$\mathrm{BMJD(TDB)} = 49102.42039316(1) + 0.0868203980(4) E.
\label{eq:ephemeris}$$ For the ULTRACAM data we find that the times from the three individual arms agree well with each other, (Fig. \[fig:relative\_ultracam\_times\]), although the times from the blue arm are comparatively poor due to its lower time resolution (necessary to compensate for the lower flux in this band). The ULTRACAM times listed in Table \[tab:times\] are the error-weighted averages of the three individual times.
The agreement of the individual times and the absence of a particular ordering of the u$^{\prime}$, g$^{\prime}$, and r$^{\prime}$ or i$^{\prime}$ times around the weighted mean indicate that there is no significant correlation between the observed egress time and the wavelength at which the data were taken.
![Eclipse times relative to the weighted mean, with 1$\sigma$ errorbars, determined for data from the three ULTRACAM arms.[]{data-label="fig:relative_ultracam_times"}](relative_ultracam_times.eps)
Orbital period variations {#sect:orbitalperiodvariations}
=========================
Fig. \[fig:oc\_huaqr\] (referred to as an O-C diagram) shows the observed eclipse times minus times calculated assuming a constant orbital period.
A mechanism that could explain observed O-C variations in white dwarf + M-dwarf binaries was proposed by @Applegate92. He suggested that magnetic cycles in the secondary star cause quasi-periodic variations in its gravitational quadrupole moment, which couple to the binary’s orbit and cause semi-periodic variations in the orbital period. However, as orbital period variations were monitored more extensively and over longer periods of time, it became clear that, in HU Aqr and other binaries, the variations can be larger than energetically possible with Applegate’s mechanism. For the pre-CV NN Ser, @Brinkworth06 showed that the energy required for the observed variations was at least an order of magnitude larger than the energy available from the M-dwarf. Given that HU Aqr has a similarly low-mass M-dwarf star, and the O-C variations are even more extreme, a similar discrepancy exists for this system [@Schwarz09].
Planetary companions to HU Aqr
------------------------------
For eclipsing white dwarf binaries that show O-C variations too large to be explained by Applegate’s mechanism a number of models invoking circumbinary planets around close binaries have been suggested. These planets are generally in orbits with periods of years to decades, and would introduce periodic variations in the O-C eclipse times much like the ones observed. A comprehensive overview of the relevant binaries and models can be found in @Zorotovic13. Due to the long periods of the suspected circumbinary planets, and the often relatively short coverage of eclipse times, published models for planetary systems are only weakly constrained and are often proved incorrect when new eclipse times become available [@Gozdziewski12; @Beuermann12; @Parsons10]. Besides creating a model that fits the data, analysing the dynamical stability of the resulting system is a crucial step in determining the probability that circumbinary planets are present. Several published systems for which multiple planetary companions were proposed have turned out to be unstable on timescales as short as a few centuries [@Horner13; @Wittenmyer13; @Hinse12], which makes their existence unlikely. Systems for which models invoke a single planetary companion are of course dynamically stable, but not necessarily more likely to exist. Only for the post-common envelope binary NN Ser have planetary models (which include two planets) correctly predicted future eclipse times and shown long-term dynamical stability [@Marsh14; @Beuermann13; @Horner12a], while at the same time both Applegate’s mechanism [@Brinkworth06] and apsidal motion [@Parsons14] have been ruled out as the main cause of the eclipse timing variations. Despite the difficulties encountered, determining parameters of current-day planetary systems around evolved binary stars could provide a unique way to constrain uncertainties in close binary evolution, such as the common envelope phase [@PortegiesZwart13], as well as answer questions related to planet formation and evolution.
HU Aqr has been the topic of much discussion and speculation concerning possible planetary companions. A few years after the first egress times were published by @Schwope01 and @Schwarz09, @Qian11 published a model with two circumbinary planets, and a possible third planet on a much larger orbit. This model was proven to be dynamically unstable on very short timescales (10$^3$ - 10$^4$ years) by @Horner11, after which both @Wittenmyer12 and @Hinse12 reanalysed the data and found models with different parameters, which were nonetheless still dynamically unstable. @Gozdziewski12 then suggested that there may be a significant correlation between eclipse times and the wavelength at which the relevant data is obtained, and proposed a single-planet model to explain the observed O-C variations seen in data taken in white light or V-band only, thereby excluding data that was taken in X-rays (ROSAT, XMM) and at UV wavelengths (EUVE, XMM OM-UVM2, HST/FOS) and all polarimetric data. They also excluded the outliers from @Qian11, which do not agree well with other data taken at similar times.
The O-C diagram as shown in Fig. \[fig:oc\_huaqr\] includes all previously published eclipse egress times, except those from @Qian11 which we exclude for the same reason as @Gozdziewski12. Also plotted is the 1-planet model that was proposed by [@Gozdziewski12]. Our new times and the time from @Schwope14 depart dramatically from this model, and therefore we conclude that the proposed orbit is incorrect. The deviation of our new data also suggests that the times derived from satellite data (which @Gozdziewski12 argued were unreliable) should be considered alongside optical data. This is also supported by the agreement between the optical and satellite times when taken at similar epochs. As mentioned before, from the separate u$^{\prime}$, g$^{\prime}$ and r$^{\prime}$/i$^{\prime}$ data we also do not see any evidence that mid-egress times are wavelength dependent.
For completeness we fitted the O-C times with three different planetary models, containing one, two and three eccentric planets respectively. We did not include a quadratic term, since such long-term behaviour can be mimicked by a distant planet, and chose to investigate a possible secular change of the orbital period separately in Sect. \[sect:secularchange\]. Unsurprisingly, all planets in our models are forced into highly eccentric orbits in order to fit the recent steep decline in the O-C times, leading us to believe that these system would be dynamically unstable. In addition, the models leave significant residuals. We conclude that the observed variability in eclipse egress times is not caused purely by the presence of a reasonable number of circumbinary planets.
A secular change of the orbital period? {#sect:secularchange}
---------------------------------------
![O-C diagram of the mid-egress times of HU Aqr including the best quadratic fit as the dashed grey line (top panel), and the residuals relative to that fit (bottom panel). O-C times with uncertainties exceeding 3 seconds have been greyed out for clarity, but all data were used to obtain the fit.[]{data-label="fig:oc_quad"}](oc_quad.eps)
It seems that the current set of O-C times cannot be explained simply by a model that introduces one or multiple planets, but given the recent steep decline in the O-C times, we now consider whether we are seeing the long-term evolution of the binary’s orbital period. Using a quadratic model, shown in Fig. \[fig:oc\_quad\], we measure the rate of orbital period change as -5.2 $\cdot~10^{-12}$ s/s (= -4.5 $\cdot~10^{-13}$ days/cycle).
With an orbital period of only 125 minutes, HU Aqr is located just below the CV period gap [@Knigge06], so that gravitational wave emission is expected to be the main cause of angular momentum loss. Using M$_{\mathrm{wd}}$ = 0.80 M$_{\odot}$, M$_{2}$ = 0.18 M$_{\odot}$ for the masses of the two stars [@Schwope11], and the orbital period from equation \[eq:ephemeris\], we calculated the period change due to gravitational wave emission to be $\dot{P}_{\mathrm{GW}}$ = -1.9 $\cdot~10^{-13}$ s/s, 27 times smaller than the result from our best fit to the O-C times. If the measured quadratic term in the O-C times represents an actual change in the binary’s orbital period, this is not caused by gravitational wave emission alone. For binaries that lie below the period gap it is likely that some magnetic braking is still ongoing [@Knigge11]. If this occurs in short bursts of strong magnetic braking, rather than long-term steady magnetic braking, it might be possible to create large changes in the binary’s orbital period on short timescales.
Magnetic alignment of the accretion spot
----------------------------------------
A last possibility we consider is that the accretion spot, the egress of which is the feature being timed, moves with respect to the line of centres between the two stars. This could happen either because of asynchronous rotation of the white dwarf [@Cropper88], or because the spot itself migrates on the surface of the white dwarf [@Cropper89].
From eclipse data taken when HU Aqr was in a low state we know that the ingress and egress of the white dwarf last for $\sim$ 30 seconds [@Schwarz09]. Therefore the maximum libration of the spot on the surface of the white dwarf can generate O-C deviations with an amplitude of 15 seconds. For geometrical reasons, this would have to be accompanied by shifts in the time of maximum light from the accretion spot of $\sim$ 0.25 orbital phases, an effect that has not been observed. Additionally, even with a large quadratic term removed from the original O-C times, an amplitude of 15 seconds is not large enough to explain the residuals, which still fluctuate by more than 20 seconds (Fig. \[fig:oc\_quad\]). Furthermore, we find no correlation between the accretion state of the binary and the magnitude of the O-C deviations, in the original O-C diagram, nor in the residuals after removal of the best quadratic fit.
Conclusions
===========
We have presented new eclipse observations across the optical spectrum of the eclipsing polar HU Aqr while in high as well as low accretion states. From the egress feature of the accretion spot on the white dwarf we have determined the mid-egress times. Our ULTRACAM data shows that times from data taken using different SDSS filters agree well with each other. The new O-C times indicate that the eclipses are still occuring increasingly earlier than expected when using a linear ephemeris and a constant orbital period. They also confirm the result found by @Schwope14 that the circumbinary planet proposed by @Gozdziewski12 does not exist, nor can the entire set of egress times be well fitted by a model introducing one or multiple planets.
Given the large amplitude of the observed O-C times, Applegate’s mechanism can likely be excluded, and also asynchronous rotation of the white dwarf or movement of the accretion spot seem unlikely. Also a long-term orbital period change induced by gravitational wave emission or constant magnetic braking is not large enough to explain the observed O-C variations. Currently, our best guess is that more than one of these mechanisms act at the same time, working together to produce the dramatic eclipse time variations observed in this binary.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank the referee (Robert Wittenmyer) for the prompt and positive feedback. TRM acknowledges financial support from STFC under grant number ST/L000733/1. SGP acknowledges financial support from FONDECYT in the form of grant number 3140585.
\[lastpage\]
|
---
abstract: 'Stable dynamical systems are a flexible tool to plan robotic motions in real-time. In the robotic literature, dynamical system motions are typically planned without considering possible limitations in the robot’s workspace. This work presents a novel approach to learn workspace constraints from human demonstrations and to generate motion trajectories for the robot that lie in the constrained workspace. Training data are incrementally clustered into different linear subspaces and used to fit a low dimensional representation of each subspace. By considering the learned constraint subspaces as zeroing barrier functions, we are able to design a control input that keeps the system trajectory within the learned bounds. This control input is effectively combined with the original system dynamics preserving eventual asymptotic properties of the unconstrained system. Simulations and experiments on a real robot show the effectiveness of the proposed approach.'
author:
- 'Matteo Saveriano$^{1}$ and Dongheui Lee$^{2,3}$[^1][^2][^3][^4]'
bibliography:
- 'bibliography.bib'
title: '**Learning Barrier Functions for Constrained Motion Planning with Dynamical Systems**'
---
[^1]: $^{1}$Intelligent and Interactive Systems and Digital Science Center (DiSC), University of Innsbruck, Innsbruck, Austria [matteo.saveriano@uibk.ac.at]{}.
[^2]: $^{2}$Human-Centered Assistive Robotics, Technical University of Munich, Munich, Germany [dhlee@tum.de]{}.
[^3]: $^{3}$Institute of Robotics and Mechatronics, German Aerospace Center (DLR), We[ß]{}ling, Germany [dongheui.lee@dlr.de]{}.
[^4]: This work was carried out while M. Saveriano was at the Institute of Robotics and Mechatronics of the German Aerospace Center (DLR). This work has been supported by Helmholtz Association.
|
---
abstract: 'A two miniband model for electron transport in semiconductor superlattices that includes scattering and interminiband tunnelling is proposed. The model is formulated in terms of Wigner functions in a basis spanned by Pauli matrices, includes electron-electron scattering in the Hartree approximation and modified Bhatnagar-Gross-Krook collision tems. For strong applied fields, balance equations for the electric field and the miniband populations are derived using a Chapman-Enskog perturbation technique. These equations are then solved numerically for a dc voltage biased superlattice. Results include self-sustained current oscillations due to repeated nucleation of electric field pulses at the injecting contact region and their motion towards the collector. Numerical reconstruction of the Wigner functions shows that the miniband with higher energy is empty during most of the oscillation period: it becomes populated only when the local electric field (corresponding to the passing pulse) is sufficiently large to trigger resonant tunneling.'
author:
- 'M. Álvaro and L.L. Bonilla'
title: 'Two mini-band model for self-sustained oscillations of the current through resonant tunneling semiconductor superlattices'
---
Introduction {#sec:0}
============
Consider a n-doped semiconductor superlattice (SL) under a sufficiently large vertical voltage bias so that electron transport is due to resonant tunneling between minibands. For small voltage values, electron transport chiefly involves the lowest miniband and there are many appropriate kinetic theory descriptions: semiclassical Boltzmann-type equations [@kss72; @ign87; @ign91; @sib95; @bep03], density matrix formulations [@bk97; @fis98], transport equations for the nonequilibrium Green function (NGF) [@HJ08], and Wigner-Poisson (WP) equations [@be05]. Semiclassical equations are easier to handle and, in particular, can be used to describe space-charge instabilities such as self-sustained oscillations of the current (SSOC) in dc voltage biased SLs due to the formation and dynamics of electric field domains [@BGr05]. SSOC can be found by deriving and solving a drift-diffusion system from the semiclassical kinetic equation [@bep03], or by a direct numerical solution of the latter [@cbc09]. Quantum transport description based on NGFs are still limited to spatially homogeneous electric fields and therefore cannot be used to describe properly space-charge phenomena [@HJ08]. WP equations can be used to derive nonlocal drift-diffusion systems exhibiting SSOC provided collision terms are of Bhatnagar-Gross-Krook (BGK) type [@be05].
In contrast to work in one-miniband SL, much less is known about first-principles space-charge transport involving resonant tunneling in SL [@BGr05]. Most of the work on resonant tunneling SL assume a large separation between time scales such that electron density and electric field can be assumed to be constant in each SL period and the tunneling current across barriers can be assumed to be stationary. Then expressions for the stationary current in an infinitely long SL under a constant electric field can be calculated by any quantum kinetic method and inserted in discrete balance equations [@BGr05]. The resulting models have been vastly useful to understand nonlinear electron transport in SL but they have not been derived from first principles. Recently, we have found a consistent perturbation method to derive nonlocal drift-diffusion systems (NDDS) from WP descriptions of two-miniband SLs with Rashba spin-orbit interaction [@bba08]. However, coupling between minibands in that work does not contemplate resonant tunneling between them for the underlying physical description of the SL is too simple.
Some time ago, Morandi and Modugno studied a variant of the standard k-p theory in which interband coupling terms depend on the applied electric field and used it to study wave function dynamics of a resonant tunneling diode [@MM]. For the same system, multiband Wigner function approaches have also been considered [@unl04; @dem02; @mor08; @mor09]. Unlu et al [@unl04] use a nonequilibrium Green function formulation that includes scattering due to weak coupling to a phonon bath to derive equations for the multiband Wigner functions. A treatment of space-dependent but time-independent NGF and Wigner functions in MOSFET can be found in Ref. . The other works focused their attention in coherent transport under an external field and near the semiclassical limit, thereby ignoring scattering [@dem02; @mor08; @mor09]. In this paper, we present a simplified model of a two-miniband SL using a field dependent coupling between minibands similar to that introduced for resonant tunneling diodes[@MM]. We consider the corresponding WP system with BGK collision terms that include collision broadening and decay between minibands due to scattering. Electron-electron scattering is treated in the Hartree approximation through the Poisson equation. We are interested in the [*hyperbolic*]{} limit in which electric field effects, including field-dependent inter-miniband transitions, are as strong as the BGK collision terms and dominate electron transport. By using the Chapman-Enskog perturbation method, we derive nonlocal balance equations for the electron population of the minibands and the electric field that inherit the nonlocality of the quantum Wigner equation. Numerical solutions of these nonlocal equations allow us to reconstruct the time-resolved Wigner matrix and they exhibit resonant tunneling between minibands and SSOC. During SSCO, we show that the miniband with higher-energy is practically empty except when the local electric field is sufficiently large to allow resonant tunneling from the miniband with lowest energy. Our calculations provide a first-principles description of SSCO in a resonant tunneling SL under dc voltage bias.
The rest of the paper is organized as follows. Section \[sec:1\] contains the Hamiltonian we use as the basis of our kinetic theory. The governing WPBGK equations for the Wigner functions are introduced in Section \[sec:2\]. The derivation of nonlocal balance equations by the Chapman-Enskog method is given in Section \[sec:3\]. Section \[sec:4\] presents numerical results obtained by solving the nonlocal balance equations with appropriate boundary conditions for the contact regions and dc voltage bias. In particular, these solutions include SSCO. Finally Section \[sec:5\] contains our conclusions and the Appendix is devoted to technical matters.
Model Hamiltonian {#sec:1}
=================
Let us assume that the total Hamiltonian describing our system is $$\mathbf{H}_{\rm total} = \mathbf{H} + \mathbf{H}_{\rm sc},\label{ham}$$ where $\mathbf{H}_{\rm sc}$ represents scattering and $\mathbf{H}(x,-i\partial/\partial x)$ is a $2\times 2$ Hamiltonian $\mathbf{H}$ corresponding to a SL with two minibands of widths $\Delta_{1}$ and $\Delta_{2}$, gap energy $2g$ and SL period $l$, $$\begin{aligned}
\mathbf{H}(x,k) &=& \left(
\begin{array}{cc}
-\frac{\Delta_{2}}{2}(1-\cos kl) - e W(x) + g & e F l \delta\\
e F l \delta & \frac{\Delta_{1}}{2}(1-\cos kl) - e W(x) - g
\end{array}
\right), \label{1}\\
& \equiv& [h_{0}(k)- e W(x)]\boldsymbol{\sigma}_{0}+\vec{h}(k)\cdot
\vec{\boldsymbol{\sigma}} + eFl\delta\,\boldsymbol{\sigma_1} \nonumber ,
%&&\mathbf{H}(x,k) = [h_{0}(k)- e W(x)]\boldsymbol{\sigma}_{0}+\vec{h}(k)\cdot
%\vec{\boldsymbol{\sigma}} - i \frac{\partial}{\partial x}(-e W(x))\frac{(-i \hbar P)}{2 m^* g}%\boldsymbol{\sigma_1} ] \label{1}\\
%&& \quad\equiv\left(
%\begin{array}{cc}
%(\alpha+\gamma)(1-\cos kl) - e W(x) + g & - i\beta\sin kl + e F l \delta\\
% i\beta\sin kl + e F l \delta & (\alpha-\gamma)(1-\cos kl) - e W(x) - g
%\end{array} - i \frac{\partial}{\partial x}(-e W(x))\frac{(-i \hbar P)}{2 m^* g}
%\right) \nonumber .\end{aligned}$$ Here we have considered tight-binding dispersion relations for the minibands and $-e<0$, $W$ and $-F=-\partial W/\partial x$ are the electron charge, the electric potential, and the electric field, respectively. The electric potential $W$ in $\mathbf{H}$ describes electron-electron interaction in a self-consistent Hartree approximation.
The matrix Hamiltonian $\mathbf{H}$ can be written as a linear combination of the Pauli matrices $$\begin{aligned}
&&\boldsymbol{\sigma}_{0}= \left(\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}\right),
\,\boldsymbol{\sigma}_{1}= \left(\begin{array}{cc}
0 & 1 \\
1 & 0
\end{array}\right),\,
\boldsymbol{\sigma}_{2}= \left(\begin{array}{cc}
0 & -i \\
i & 0
\end{array}\right),\,
\boldsymbol{\sigma}_{3}= \left(\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right), \label{3} \nonumber\end{aligned}$$ with coefficients: $$\begin{aligned}
\begin{array}{cc}
h_{0}(k)= -\alpha\, (1-\cos kl), & h_{1}(k)=0, \\
h_{2}(k)=0, & h_{3}(k)= -\gamma\, (1-\cos kl) + g,\\
\alpha=\frac{\Delta_{2}-\Delta_{1}}{4}, & \gamma=\frac{\Delta_{2}+
\Delta_{1}}{4}.
\end{array} \label{2}\end{aligned}$$ The term $eFl\delta\,\boldsymbol{\sigma}_1$ in (\[1\]) is a field-dependent tunneling term derived by means of the k-p theory for the evolution of the Wannier envelope functions \[cf. Equations (33) of Ref. without second order terms, i.e. with $M_{n n'}=0$\]. The dimensionless parameter $\delta$ is a phenomenological parameter proportional to the interminiband momentum matrix element: $$\begin{aligned}
\delta=\frac{\hbar P}{2 m^* g l}, \quad P = \frac{\hbar}{l}\int_{-l/2}^{l/2}
{u_2^* \frac{\partial{u_1}}{\partial{ x}} dx}, \label{2bis}\end{aligned}$$ where $u_{1,2}$ are the periodic parts of the miniband Bloch functions. A related model has been used to describe coherent transport in a resonant interband tunnelling diode [@MM; @mor08; @mor09].
The miniband energies ${\cal E}^{\pm}(k)$ are the eigenvalues of the free Hamiltonian $\mathbf{H}_{0}(k)=h_{0}(k)\boldsymbol{\sigma}_{0}+\vec{h}(k)\cdot
\vec{\boldsymbol{\sigma}}$ (zero electric potential), given by $$\mathcal{E}^{\pm}(k) = h_{0}(k) \pm h_{3}(k). \label{4}$$ The corresponding spectral projections are $$\mathbf{P}^{\pm} = \frac{\boldsymbol{\sigma}_0\pm\boldsymbol{\sigma}_3}{2},
\label{5}$$ so that we can write $$\mathbf{H}_{0}(k)= \mathcal{E}^{+}(k)\mathbf{P}^{+} + \mathcal{E}^{-}(k)
\mathbf{P}^{-}. \label{6}$$
Wigner function description {#sec:2}
===========================
If $\psi_a(x,y,z,t)$, $a=1,2$, are the second quantized wave function amplitudes expressed in the Bloch basis, the Wigner matrix is [@bba08] $$\begin{aligned}
f_{ab}(x,k,t)= {2l\over S}\sum_{j=-\infty}^\infty \int_{\mathbb{R}^2}\langle
\psi^\dagger_a(x+jl/2,y,z,t)\psi_b(x-jl/2,y,z,t)\rangle e^{ijkl} d\mathbf{x}_{\perp} ,
\label{wigner}\end{aligned}$$ where $S$ is the SL cross section. Note that the Wigner matrix is periodic in $k$ with period $2\pi/l$. It is convenient to write the Wigner matrix $\mathbf{f}(x,k,t)$ in terms of the Pauli matrices: $$\mathbf{f}(x,k,t) = \sum_{i=0}^{3} f^{i}(x,k,t)\boldsymbol{\sigma}_{i} =
f^0(x,k,t)\boldsymbol{\sigma}_{0} + \vec{f}(x,k,t)\cdot\vec{ \boldsymbol{\sigma}}. \label{7}$$ The Wigner components $f^i(x,k,t)$ are real and can be related to the coefficients of the Hermitian Wigner matrix by $$\begin{aligned}
\begin{array}{cc}
f_{11}= f^0 + f^3, & f_{12}= f^1 - if^2, \\
f_{21}=f^1 + i f^2, & f_{22}= f^0 - f^3.
\end{array} \label{8}\end{aligned}$$ Hereinafter we shall use the equivalent notations $$\begin{aligned}
f= \left(\begin{array}{c}
f^0 \\
\vec{f}
\end{array}\right) =
\left(\begin{array}{c}
f^0 \\
f^1\\
f^2\\
f^3
\end{array}\right). \label{9}\end{aligned}$$ The populations of the minibands with energies $\mathcal{E}^\pm$ are given by the moments: $$n^\pm(x,t) = {l\over 2\pi}\int_{-\pi/l}^{\pi/l} \left[ f^0(x,k,t)\pm f^3(x,k,t)\right]
\, dk, \label{10}$$ and the total electron density is $n^+ + n^-$.
After some algebra, from the time-dependent Schrödinger equations for wave functions $\psi_a$ with the Hamiltonian $\mathbf{H}_{\rm tot}$ in (\[ham\]), we can obtain the following Wigner-Poisson-Bhatnagar-Gross-Krook (WPBGK) equations for the Wigner components $$\begin{aligned}
&&{\partial f^0\over\partial t} - {\alpha\over\hbar}\sin kl\,\Delta^- f^0 -
{\gamma\over\hbar}\sin kl\,\Delta^- f^3 - \Theta_1 f^0 - \Theta_2 f^1 = Q^0[f],
\label{11}\\
&&{\partial\vec{f}\over\partial t} - {\alpha\over\hbar}\sin kl\,\Delta^-
\vec{f} - {\gamma\over\hbar}\sin kl\,\Delta^- f^0\,\vec{\nu} +
\vec{\omega}\times\vec{f} - \vec{\Theta}[f] = \vec{Q}[f], \label{12}\end{aligned}$$ whose right hand sides contain collision terms $Q[f]$ arising from $\mathbf{H}_{\rm sc}$. These terms will be modeled phenomenologically and described later. Electron-electron collisions are treated in the Hartree approximation and described by the Poisson equation for the electrostatic potential: $$\begin{aligned}
\varepsilon\, {\partial^2 W\over\partial x^2} = {e\over l}\, (n^+ + n^- -N_{D}), \label{13}\end{aligned}$$ where $\varepsilon$ and $N_D$ are the SL permittivity and the 2D doping density, respectively. In (\[11\]) - (\[12\]), $$\begin{aligned}
&&\vec{\omega} = {2(g-\gamma)+\gamma\cos kl\,\Delta^+\over\hbar}\,\vec{\nu},
\quad\vec{\nu}= (0,0,1), \label{15}\\
&&\Theta_1 f^{m} (x,k,t)= {el\over i\hbar} \sum_{j=-\infty}^\infty j\langle F(x,t) \rangle_{j} e^{ijkl} f^{m}_{j}(x,t),\label{19}\\
&&\Theta_2 f^{m} (x,k,t)= -\frac{el \delta}{ i\hbar} \sum_{j=-\infty}^\infty e^{ijkl} f^{m}_{j}(x,t)\,\Delta^-_j F(x,t),\label{20}\\
&&\Theta_3 f^{m} (x,k,t)= \frac{el \delta}{ i\hbar}\sum_{j=-\infty}^\infty e^{ijkl} f^{m}_{j}(x,t)\,\Delta^+_j F(x,t),\label{21}\\
&&\vec{\Theta}[f]= \Theta_1\, \vec{f} + \left(%
\begin{array}{r}
\Theta_2\, f^0 \\
\Theta_3\, f^3 \\
-\Theta_3\, f^2 \\
\end{array}
\right) . \label{22theta}\end{aligned}$$ We have defined the operators $$\begin{aligned}
(\Delta^\pm_j u)(x,k) = u\left(x+\frac{jl}{2},k\right)\pm u\left(x-\frac{jl}{2},k\right) \label{14}\end{aligned}$$ (the subscript is omitted for $j=1$) and the spatial averages: $$\begin{aligned}
\langle F(x,t)\rangle_{j} &\equiv& {1\over jl} \int_{-jl/2}^{jl/2} F(x+s,t)\, ds \label{23average}\\
&=& \left\langle \frac{\partial W}{\partial x}(x,t)\right\rangle_j =\frac{\partial}{\partial x}\left\langle W(x,t)\right\rangle_j = \frac{\Delta^-_j W(x,t)}{jl}. \label{17}\end{aligned}$$ Our collision model is similar to that used in Ref. and it contains two terms: a BGK term which tries to send the miniband Wigner function to its local equilibrium and a scattering term that sends electrons from the miniband with higher energy (whose electron density is $n^+$) to the miniband with lower energy (whose electron density is $n^-$): $$\begin{aligned}
&& Q^0[f] = - {f^0 - \Omega^0\over\tau}, \label{20q0}\\
&&\vec{Q}[f] = - {\vec{f} - \vec{\Omega}\over\tau} -
{\vec{\nu} f^0 + \vec{f}\over\tau_{\rm sc}}, \label{21qvec}\\
&&\Omega^{0} = {\phi^+ + \phi^-\over 2}\,, \quad
\vec{\Omega} = {\phi^+ - \phi^-\over 2}\,\vec{\nu},\label{22}\\
&&\phi^{\pm}(k;n^\pm) = {m^{*}k_{B}T\over
\pi\hbar^2}\,\int_{-\infty}^\infty
\frac{\sqrt{2}\,\Gamma^3/\pi}{\Gamma^4 +[E-{\cal E}^\pm(k)]^4}\,
\ln\left(1+ e^{{\mu^\pm - E\over k_{B}T}}\right)\, dE,\label{23}\\
&& n^\pm = {l\over 2\pi}\int_{-\pi/l}^{\pi/l} \phi^\pm(k;n^\pm)\, dk. \label{24}\end{aligned}$$ The chemical potentials of the minibands, $\mu^+$ and $\mu^{-}$ are calculated in terms of $n^+$ and $n^-$ respectively, by inserting (\[23\]) in (\[24\]) and solving the resulting equations. The local equilibria $\phi^\pm$ are the integrals of collision-broadened 3D Fermi-Dirac distributions over the lateral components of the wave vector on the plane perpendicular to the growth direction $x$. [@bba08] As the broadening energy $\Gamma\to 0$, the line-width function in the integrand of (\[23\]) becomes $\delta(E-\mathcal{E}^\pm(k))$.
Our collision model should enforce charge continuity. To check this, we first calculate the time derivative of $n^\pm$ using (\[10\]) to (\[12\]): $$\begin{aligned}
{\partial n^\pm\over\partial t} - {\alpha l\Delta^-\over
2\pi\hbar} \int_{-\pi/l}^{\pi/l} \sin kl\,(f^0\pm f^3)\, dk -{\gamma l\Delta^-\over
2\pi\hbar}\int_{-\pi/l}^{\pi/l}\sin kl (f^3 \pm f^0)\, dk \nonumber\\
\pm {l\over 2\pi}\int_{-\pi/l}^{\pi/l}\Theta_3 f^2\, dk
= {l\over 2\pi}\int_{-\pi/l}^{\pi/l} (Q^0[f]\pm Q^3[f])\, dk = \mp {n^+
\over\tau_{\rm sc}}, \label{25}\end{aligned}$$ where we have employed $\int \Theta_1 f^0 dk = \int \Theta_2 f^1 dk = 0$. Then we obtain: $$\begin{aligned}
{\partial\over\partial t}(n^+ + n^-) - \Delta^-\left[ {l\over\pi\hbar}\int_{-\pi/
l}^{\pi/l} \sin kl\left(\alpha f^0 + \gamma f^3\right) dk \right] = 0. \label{26}\end{aligned}$$ Noting that $\Delta^- u(x)= l\,\partial\langle u(x)\rangle_{1}/\partial x$, we see that (\[26\]) [*is the charge continuity equation*]{}. Differentiating in time the Poisson equation (\[13\]), using (\[26\]) in the result and integrating with respect to $x$, we get the following nonlocal Ampère’s law for the balance of current: $$\begin{aligned}
\varepsilon {\partial F\over\partial t} - \left\langle {el\over\pi\hbar}
\int_{-\pi/l}^{\pi/l} \sin kl \left(\alpha f^0 + \gamma f^3\right) dk \right
\rangle_{1} = J(t). \label{27}\end{aligned}$$ Here the space independent function $J(t)$ is the total current density. Since the Wigner components are real, we can rewrite (\[27\]) in the following equivalent form: $$\begin{aligned}
\varepsilon {\partial F\over\partial t} + {2e\over\hbar}\,\left\langle \alpha\,
\mbox{Im}f^0_{1} +\gamma\, \mbox{Im}f^3_{1}\right\rangle_{1} = J(t). \label{28}\end{aligned}$$ We are using the notation $f_j^m$ for the Fourier coefficients of $f^m$: $$\begin{aligned}
f^m(x,k,t)=\sum_{j=-\infty}^\infty f^m_{j}(x,t)\, e^{ijkl}. \label{fourier}\end{aligned}$$
The Chapman-Enskog method and balance equations {#sec:3}
===============================================
In this Section, we shall derive the reduced balance equations for our two-miniband SL using the Chapman-Enskog method. Note that if we were to know the Wigner matrix as a function of $n^\pm$ and the electric field, Equations (\[25\]) and the Poisson equation (\[13\]) would be the sought balance equations and could be solved directly. As they are now, Equations (\[25\]) are not closed. However, in a limit in which collisions and electric potential terms dominate all others in the Wigner equations, it is possible to use perturbation theory to close (\[25\]). The idea is that in this so-called [*hyperbolic limit*]{}, the Wigner matrix is very close to a local equilibrium (modified by the electric field) which depends on $n^\pm$ and $F$. Using two terms in a Chapman-Enskog expansion, we show below that Equations (\[25\]) can be closed.
First of all, we should decide the order of magnitude of the terms in the WPBGK equations (\[11\]) and (\[12\]) in the hyperbolic limit. In this limit, the collision frequency $1/\tau$ and the Bloch frequency $eF_{M}l/\hbar$ are of the same order, say about 10 THz. Then $F_{M}=O(\hbar/(el\tau))$. Typically, $2g/\hbar$ is of the same order, so that the term containing $2g/\hbar$ in (\[12\]) should also balance the BGK collision term. The other terms are of order $\gamma l/(\hbar x_{0})$, where $x_{0}$ is the characteristic length over which the field varies, and they are much smaller, so that $\lambda=\gamma\tau l/(\hbar x_{0})\ll 1$. From the Poisson equation, we obtain $x_{0}/l=\varepsilon F_{M}/(eN_{D})=\varepsilon \hbar/(e^2
\tau l N_{D})$, and therefore the small dimensionless parameter is $$\lambda = \frac{e^2\tau^2\gamma l N_{D}}{\varepsilon\hbar^2}. \label{16}$$ The scattering time $\tau_{\rm sc}$ is much longer than the collision time $\tau$, and we shall consider $\tau/\tau_{\rm sc}=
O(\lambda)\ll 1$. Equations (\[11\]) and (\[12\]) can be written as the scaled WPBGK equations as follows: $$\begin{aligned}
\mathbb{L} f -\Omega = - \lambda\,\left(\tau\,{\partial f\over\partial t}+ \Lambda f\right).
\label{30}\end{aligned}$$ where we have inserted the book-keeping parameter $\lambda$ which is set equal to 1 at the end of our calculations. [@bep03; @bba08] This trick saves us from rewriting our equations in nondimensional units. Here the operators $\mathbb{L}$ and $\Lambda$ are defined by $$\begin{aligned}
&&\mathbb{L} f= f - \tau\, \Theta_1 f - \tau\, \Theta_2\left(%
\begin{array}{c}
f^1 \\
f^0 \\
0 \\
0 \\
\end{array}%
\right) - \tau\, \Theta_3\left(%
\begin{array}{c}
0 \\
0 \\
f^3 \\
-f^2 \\
\end{array}%
\right) + \eta_{1} \left(
\begin{array}{c}
0\\
- f^2\\
f^{1}\\
0
\end{array}\right), \label{31}\\
&& \Lambda f = \eta_{2}\, \left( \begin{array}{c}
0\\
\vec{f} + \vec{\nu} f^0
\end{array}\right) - {\tau\over\hbar}\sin kl\,\Delta^- \left[\alpha f
+ \gamma \left( \begin{array}{c}
f^3\\
\vec{\nu} f^0 \end{array}\right)\right] + \frac{\gamma\tau}{\hbar}
(\cos kl\, \Delta^+ -2)\left(
\begin{array}{c}
0\\
\vec{\nu}\times\vec{f} \end{array}\right), \nonumber\end{aligned}$$ where $$\eta_{1}= {2g\tau\over\hbar},\quad \eta_{2}= {\tau\over\tau_{\rm sc}}. \label{33}$$
To derive the reduced balance equations, we use the following Chapman-Enskog ansatz: $$\begin{aligned}
&& f(x,k,t;\epsilon) = f^{(0)}(k;n^+,n^-,F) + \sum_{m=1}^{\infty}
f^{(m)}(k;n^+,n^-,F)\, \lambda^{m} , \label{34}\\
&& \varepsilon {\partial F\over\partial t} + \sum_{m=0}^{\infty}
J_{m}(n^+,n^-,F)\, \lambda^{m} = J(t), \label{35}\\
&&{\partial n^\pm\over\partial t} = \sum_{m=0}^{\infty} A^{\pm}_{m}(n^+,n^-,F)\, \lambda^{m}.
\label{36}\end{aligned}$$ The functions $A_{m}^\pm$ and $J_{m}$ are related through the Poisson equation (\[13\]), so that $$\begin{aligned}
A^{+}_{m}+ A^-_{m} = - {l\over e} \, {\partial J_{m}\over\partial x}. \label{37}\end{aligned}$$ Inserting (\[34\]) to (\[36\]) into (\[30\]), we get $$\begin{aligned}
&& \mathbb{L} f^{(0)} = \Omega, \label{38}\\
&& \mathbb{L} f^{(1)} = - \left.\tau\, {\partial f^{(0)}\over\partial t}\right|_{0} - \Lambda f^{(0)},\label{39}\\
&&\mathbb{L} f^{(2)} = - \left.\tau\, {\partial f^{(1)}\over\partial t}\right|_{0} -
\Lambda f^{(1)} - \left.\tau\,{\partial f^{(0)}\over\partial t}\right|_{1}, \label{40}\end{aligned}$$ and so on. The subscripts 0 and 1 in the right hand side of these equations mean that we replace $\varepsilon\,\partial
F/\partial t|_{m}= J \delta_{0m}-J_{m}$, $\partial n^\pm/\partial t|_{m}=A^\pm_{m}$, provided $\delta_{00}=1$ and $\delta_{0m}=0$ if $m \neq 0$. Moreover, inserting (\[34\]) into (\[10\]) yields the following compatibility conditions: $$\begin{aligned}
&& f^{(1)\,0}_{0}= f^{(1)\, 3}_{0} = 0,\label{41}\\
&& f^{(2)\, 0}_{0} = f^{(2)\, 3}_{0} = 0,\label{42}\end{aligned}$$ etc.
To solve (\[38\]) for $f^{(0)}\equiv\varphi$, we first note that $$\begin{aligned}
&&-\tau\, \Theta_1\varphi = i\sum_{j=-\infty}^\infty \vartheta_{j} \varphi_{j} e^{ijkl}, \label{43}\\
&&-\tau\, \Theta_2\varphi = - i \delta \,\sum_{j=-\infty}^\infty \varphi_{j} e^{ijkl}\Delta^-_j \mathcal F , \label{44}\\
&&-\tau\, \Theta_3\varphi = - \delta \,\sum_{j=-\infty}^\infty \varphi_{j} e^{ijkl}\Delta^+_j \mathcal F , \label{45}\\
&&\mathcal F \equiv {\tau el\over\hbar}\, F,\quad \vartheta_{j} \equiv j\,\langle {\mathcal F}\rangle_{j}.\label{46}\end{aligned}$$ Then (\[38\]) and (\[22\]) yield $$\begin{aligned}
&&\varphi_{j}^0 = {\phi^+_{j} + \phi^-_{j}\over 2}\, \left[{1 \over 1+i \vartheta_j} - \eta_1 \delta^2 Z_j\, M_j^+
(\Delta_j^-\mathcal F)^2 \right] \label{47} \\
&&\quad\quad +\, i{\phi_j^+-\phi_j^- \over 2} \eta_1 \delta^2Z_j\, (\Delta_j^- \mathcal F)\, (\Delta_j^+ \mathcal F) , \nonumber\\
&&\varphi_{j}^1 = {1\over2}\eta_1\delta(1+i\vartheta_j)\, Z_j \, \left[(\phi^+_{j} + \phi^-_{j})\, i M_j^+
\Delta_j^-\mathcal F + (\phi^+_{j} - \phi^-_{j})\, \Delta_j^+ \mathcal F \right], \label{48}\\
&&\varphi_{j}^2 = -{1\over2}\eta_1\delta(1+i\vartheta_j)\, Z_j \, \left[(\phi^+_{j} + \phi^-_{j})\, i
\Delta_j^-\mathcal F - (\phi^+_{j} - \phi^-_{j})\, M_j^-\, \Delta_j^+ \mathcal F \right], \label{49}\\
&&\varphi_{j}^3 = {\phi^+_{j} - \phi^-_{j}\over 2}\, \left[{1 \over 1+i \vartheta_j} - \eta_1 \delta^2 Z_j\,M_j^-
(\Delta_j^+\mathcal F)^2 \right] \label{50}\\
&&\quad\quad +\, i{\phi_j^+ + \phi_j^- \over 2} \eta_1 \delta^2 Z_j\, (\Delta_j^- \mathcal F)\, (\Delta_j^+ \mathcal
F) . \nonumber\end{aligned}$$ Here we have used that the Fourier coefficients $$\begin{aligned}
\phi_{j}^\pm = {l\over \pi}\, \int_{0}^{\pi/l}\cos(jkl)\,\phi^\pm\, dk, \label{51}\end{aligned}$$ are real because $\phi^\pm$ are even functions of $k$. The coefficients $Z_j$ and $M_j^{\pm}$ are defined as $$\begin{aligned}
&&M_j^{\pm} \equiv {1\over \eta_1}\left[1 + i \vartheta_j + {\delta^2(\Delta_j^{\pm}\mathcal F)^2 \over 1 +
i\vartheta_j} \right], \label{52} \\
&&Z_j \equiv {1\over \eta_1^2\,(1+i\vartheta_j)^2\, (1+M_j^+\,M_j^-)}. \label{53}\end{aligned}$$ The solution $f^{(0)}=\varphi$ given by (\[47\])-(\[50\]) is essentially the local equilibrium $\Omega$ given by (\[22\])-(\[24\]) modified by the field-dependent terms $\Theta_i$ that appear in the Wigner equations (\[11\]) and (\[12\]). This solution yields convective terms in the balance equations which contain first order differences. In the semiclassical limit, these equations become a hyperbolic system which may have discontinuous solutions (shock waves). Then it is convenient to regularize such solutions by keeping diffusion-like terms (second order differences) arising from the next-order Wigner functions $f^{(1)}$.
The solution of (\[39\]) is $f^{(1)}\equiv \psi$ with $$\begin{aligned}
&& \psi_{j}^0 = {r_j^0 \over 1+i\vartheta_j}\left[1 - {\delta^2 M_j^+(\Delta_j^- \mathcal F)^2 \over \eta_1
(1+i\vartheta_j)(1+M_j^+M_j^-)}\right] \label{54}\\
&&\quad + {i\delta \Delta_j^- \mathcal F \over \eta_1(1+i\vartheta_j)(1+M_j^+M_j^-)}\left[M_j^+ r_j^1 + r_j^2 + {
\delta \Delta_j^+ \mathcal F \over 1 + i \vartheta_j}\, r_j^3 \right], \nonumber \\
&& \psi_{j}^1 = {1 \over \eta_1(1+M_j^+ M_j^-)}\left[M_j^+r_j^1 + {i\delta M_j^+\,\Delta_j^- \mathcal F \over 1 +
i \vartheta_j}\, r_j^0 + r_j^2 + {\delta\, \Delta_j^+ \mathcal F \over 1+i\vartheta_j }\,r_j^3 \right], \label{55}\\
&& \psi_{j}^2 = {1 \over \eta_1(1+M_j^+M_j^-)}\left[M_j^-\,r_j^2 + {\delta\,M_j^-\,\Delta_j^+ \mathcal F \over 1+i
\vartheta_j}\,r_j^3 - r_j^1 - {i\delta\, \Delta_j^- \mathcal F \over 1+i\vartheta_j}\,r_j^0 \right], \label{56}\\
&& \psi_{j}^3 = {r_j^3 \over 1+i\vartheta_j}\left[1 - {\delta^2 M_j^-(\Delta_j^+ \mathcal F)^2 \over \eta_1
(1+i\vartheta_j)(1+M_j^+M_j^-)}\right] \label{57}\\
&&\quad - {\delta \Delta_j^+ \mathcal F \over \eta_1(1+i\vartheta_j)(1+M_j^+M_j^-)}\left[M_j^-\,r_j^2 - r_j^1 - {i
\delta \Delta_j^- \mathcal F \over 1 + i \vartheta_j}\, r_j^0 \right]. \nonumber\end{aligned}$$ Here $r$ is the right hand side of (\[39\]).
The balance equations can be found in two ways. We can calculate $A^\pm_m$ for $m = 0,1$ in (\[36\]) by using the solvability conditions (\[41\]) and (\[42\]) in (\[39\]) and (\[40\]), respectively. More simply, we can obtain the balance equations by inserting the solutions (\[47\]) to (\[50\]) and (\[54\]) to (\[57\]) in the balance equations (\[25\]) and in the Ampère’s law (\[27\]). The result is: $$\begin{aligned}
&& {\partial n^\pm\over\partial t} + \Delta^- D_{\pm}(n^+,n^-,F)= \mp R(n^+,n^-,F),\label{58}\\
&& \varepsilon\,{\partial F\over\partial t}+ e\,\left\langle D_+(n^+,n^-,F) + D_-(n^+,n^-,F) \right\rangle_{1} = J(t)\label{59}\\
&&D_{\pm} = {\alpha\pm\gamma\over\hbar}\,\mbox{Im}(\varphi_1^0 \pm \varphi_1^3 + \psi_1^0 \pm \psi_1^3),\label{60}\\
&& R = {1 \over \tau}\left[\eta_{2}n^+ + 2\delta \mathcal F (\varphi_0^2 + \psi_0^2)\right] .\label{61}\end{aligned}$$ Note that Eq. (\[59\]) can be obtained from (\[58\]) and the Poisson equation. Equations (\[58\]) to (\[61\]) must be solved together with the Poisson equation (\[13\]), the expression for the local equilibrium Wigner densities (\[23\]) and expressions (\[24\]) for $n^\pm$. The zeroth and first order Wigner functions $\varphi_j$ and $\psi_j$ in (\[60\]) and (\[61\]) can be obtained from Equations (\[47\])-(\[50\]) and (\[54\])- \[57\]), respectively. The complete expressions for $D_{\pm}$ and $R$ are shown in Appendix \[appA\].
Numerical results {#sec:4}
=================
To solve numerically the system of equations (\[58\]) - (\[61\]), we have to add the voltage bias conditions for the electric potential and appropriate boundary conditions at the contact regions. Note that our equations involve finite differences and several one-period integral averages. This means that we need to give boundary conditions over intervals of size $2l$ before $x=0$ and after $x=Nl$, not just boundary conditions at $x=0, Nl$ as we would give for semiclassical drift-diffusion equations. At the injecting region (cathode), the usual boundary condition is that the electron current density satisfies Ohm’s law and therefore it is proportional to the electric field there. We use this condition for each point of the interval $-2l\leq x\leq 0$. Similarly, we also need the electron densities $n^\pm$ at the cathode. To avoid inconvenient boundary layer effects, we choose their values for a spatially uniform stationary state with a given value of the field. The resulting boundary conditions in $-2l\leq x \leq 0$ are: $W=0$ and $$\begin{aligned}
&& \varepsilon\,{\partial F\over\partial t} + \sigma_{cathode}\,F = J, \label{62}\\
&& n^\pm = n^{\pm}_{st}, \label{63}\end{aligned}$$ where $n^{\pm}_{st}$ are the miniband electron densities corresponding to a spatially uniform stationary state. The latter can be obtained by equating to zero the right hand sides of the rate equation (\[58\]) and the Poisson equation (\[13\]): $R(n^+,n^-,F)=0$ and $n^++n^-=N_D$, respectively. The result is $$\begin{aligned}
&& n^{\pm}_{st} = N_D\left({1\over2} \mp {\eta_2(1 + \eta_1^2 + 4\delta^2\mathcal F^2) \over 8\delta^2\mathcal F^2 + 2\eta_2(1 + \eta_1^2 +
4\delta^2\mathcal F^2)}\right).\end{aligned}$$
The boundary conditions in the anode region ($Nl\leq x \leq Nl+2l$) are: $W=V$ and $$\begin{aligned}
&& \varepsilon\,{\partial F\over\partial t} + \sigma_{anode}\,({n^++n^- \over N_D})\,F = J, \label{62anode}\\
&& n^+ = 0.\end{aligned}$$ The lower miniband electron density $n^-$ in the anode region is obtained from the Poisson equation (\[13\]).
To present numerical results, we have used the parameter values corresponding to a GaAs/AlAs SL from Table I of which has narrow minibands so that resonant tunneling plays an important role in electron transport. Our parameter values are: $d_{B}=1.5$ nm, $d_{W}=9$ nm, $l= d_{B}+d_{W} = 10.5$ nm, $N_{D}= 2.5\times 10^{10}$ cm$^{-2}$, $\tau= 0.0556$ ps, $\tau_{\rm
sc}= 0.556$ ps,[@shah] $V=9$ V, $N=200$, $\sigma_{cathode}= 1.4\, \Omega^{-1}$m$^{-1}$, $\sigma_{anode}= 0.7\, \Omega^{-1}$m$^{-1}$, $T=5$ K, $\Delta_1= 2.6$ meV, $\Delta_2= 13.2$ meV, $P/\hbar=0.2238$/nm,[@P] $\Gamma = 1$ meV.[@bba08] With these values, $\alpha = 2.6$ meV, $\gamma = 3.9$ meV, $\delta=0.12$. We have selected the following units to present our results graphically: $F_M=\hbar/(el\tau)= 11.28$ kV/cm, $x_{0} = \varepsilon F_M l /(eN_D) =31.4$ nm, $t_{0} = \hbar/\alpha = 0.25$ ps, $J_{0} = \alpha e N_D/\hbar = 1.58\times 10^4$ A/cm$^2$.\
![(a) Electron current vs field in a spatially uniform stationary state. (b) Total current density vs time. (c) Electric field profile at different times of one current self-oscillations cycle. At time $(1)$ the field is above the resonant value for the middle SL point $x=Nl/2$. (d) Electron densities $n^\pm/(n^++n^-)$ vs time for point $x=Nl/2$. When the electric field is above the resonant value (time $(1)$), the electron transport between minibands occurs. []{data-label="fig1"}](fig_J_F.eps "fig:"){width="5.5cm"}![(a) Electron current vs field in a spatially uniform stationary state. (b) Total current density vs time. (c) Electric field profile at different times of one current self-oscillations cycle. At time $(1)$ the field is above the resonant value for the middle SL point $x=Nl/2$. (d) Electron densities $n^\pm/(n^++n^-)$ vs time for point $x=Nl/2$. When the electric field is above the resonant value (time $(1)$), the electron transport between minibands occurs. []{data-label="fig1"}](fig_J_t.eps "fig:"){width="5.5cm"}
![(a) Electron current vs field in a spatially uniform stationary state. (b) Total current density vs time. (c) Electric field profile at different times of one current self-oscillations cycle. At time $(1)$ the field is above the resonant value for the middle SL point $x=Nl/2$. (d) Electron densities $n^\pm/(n^++n^-)$ vs time for point $x=Nl/2$. When the electric field is above the resonant value (time $(1)$), the electron transport between minibands occurs. []{data-label="fig1"}](fig_F_x.eps "fig:"){width="5.5cm"}![(a) Electron current vs field in a spatially uniform stationary state. (b) Total current density vs time. (c) Electric field profile at different times of one current self-oscillations cycle. At time $(1)$ the field is above the resonant value for the middle SL point $x=Nl/2$. (d) Electron densities $n^\pm/(n^++n^-)$ vs time for point $x=Nl/2$. When the electric field is above the resonant value (time $(1)$), the electron transport between minibands occurs. []{data-label="fig1"}](fig_n_tunnel_t.eps "fig:"){width="5.5cm"}
Figure \[fig1\] (b) illustrates the resulting stable self-sustained current oscillations. They are due to the periodic formation of a pulse of the electric field at the cathode $x = 0$ and its motion through the SL. Figure \[fig1\] (a) depicts the electron current vs field in a spatially uniform stationary state, with a local maximum at the field resonant value $2g/(el)$. Figure \[fig1\] (c) depicts the electric field profile at different times during one self-sustained current cycle. Figure \[fig1\] (d) shows the tunneling transport between minibands when the electric field is above the resonant value (time $(1)$) calculated at the middle point of the SL ($x=Nl/2$).
![(a)-(b) Wigner matrix off-diagonal terms $f^1$ and $f^2$ vs $k$, at time $(1)$ (tunneling transport), and time $(2)$ (no tunneling). (c)-(d) Wigner matrix diagonal terms $f^0 \pm f^3$ vs $k$ []{data-label="fig2"}](fig_f1_k.eps "fig:"){width="5.5cm"}![(a)-(b) Wigner matrix off-diagonal terms $f^1$ and $f^2$ vs $k$, at time $(1)$ (tunneling transport), and time $(2)$ (no tunneling). (c)-(d) Wigner matrix diagonal terms $f^0 \pm f^3$ vs $k$ []{data-label="fig2"}](fig_f2_k.eps "fig:"){width="5.5cm"}
![(a)-(b) Wigner matrix off-diagonal terms $f^1$ and $f^2$ vs $k$, at time $(1)$ (tunneling transport), and time $(2)$ (no tunneling). (c)-(d) Wigner matrix diagonal terms $f^0 \pm f^3$ vs $k$ []{data-label="fig2"}](fig_f0masf3_k.eps "fig:"){width="5.5cm"}![(a)-(b) Wigner matrix off-diagonal terms $f^1$ and $f^2$ vs $k$, at time $(1)$ (tunneling transport), and time $(2)$ (no tunneling). (c)-(d) Wigner matrix diagonal terms $f^0 \pm f^3$ vs $k$ []{data-label="fig2"}](fig_f0menf3_k.eps "fig:"){width="5.5cm"}
Figure \[fig2\] shows the Wigner matrix elements $f^i$, from equations (\[47\])-(\[50\]), (\[54\])-(\[57\]) and (\[fourier\]), for the middle SL point ($x=Nl/2$) vs $k$ at times $(1)$ (with tunneling transport between minibands) and $(2)$ (with no tunneling). Figure \[fig2\](a)-(b) illustrates the Wigner matrix off-diagonal terms $f^1$ and $f^2$, which are responsible for the tunneling transport between minibands. Figure \[fig2\](c)-(d) shows $f^0 \pm f^3$, which are related with the electron densities $n^{\pm}$.
![(a) Total current density (average, maximum and minimum values) vs voltage bias. (b) Current oscillation frequencies vs voltage bias. []{data-label="fig3"}](fig_J_V.eps "fig:"){width="5.5cm"}![(a) Total current density (average, maximum and minimum values) vs voltage bias. (b) Current oscillation frequencies vs voltage bias. []{data-label="fig3"}](fig_Freq_V.eps "fig:"){width="5.5cm"}
Figure \[fig3\] illustrates the effect of varying the voltage bias on the total current for a $N=60$ period SL. Figure \[fig3\] (a) depicts the total current density average, maximum and minimum values for different voltages. It can be seen that when the bias is above a critical voltage, the current self sustained oscillations appear and their amplitude increases from zero at the bifurcation point. This circumstance does not depend on whether the voltage is increasing or decreasing, therefore the critical voltage corresponds to a supercritical Hopf bifurcation. Figure \[fig3\] (b) shows that the oscillation frequencies decrease as the voltage increases above its critical value. Immediately above the critical voltage, self-oscillations are due to repeated triggering of small pulses of the electric field that die near the cathode and before they can reach the end of the SL. As the voltage increases, the pulses are able to grow and reach the anode region. Since their average velocity does not vary that much, the oscillation frequency is correspondingly smaller. In a transition region between 1.5 and 3V, the current oscillation is somewhat irregular. The region of self-oscillations ends at a larger voltage of about 5.3V. Similar phenomena are observed in models of the Gunn effect in bulk GaAs. See Chapter 6 in Ref. .
If we use parameters corresponding to a weakly coupled SL with miniband widths below 1 meV (that come from using wider quantum barriers), we run into problems of numerical convergence and, possibly, breakdown of the Chapman-Enskog perturbation scheme. To explore the limit of weakly coupled SL, a different perturbation scheme based on miniband smallness seems necessary. This is outside the scope of the present paper.
Conclusions {#sec:5}
===========
For strongly coupled SLs having two populated minibands, we have introduced a kp Hamiltonian that contains a field-dependent tunneling term and derived the corresponding Wigner-Poisson-BGK system of equations. The collision model comprises two terms, a BGK term trying to bring the Wigner matrix closer to a broadened Fermi-Dirac local equilibrium at each miniband, and a scattering term that brings down electrons from the upper to the lower miniband. By using the Chapman-Enskog method, we have derived quantum drift-diffusion equations for the miniband populations which contain generation-recombination terms. As it should be, the recombination terms vanish if there is no inter-miniband scattering and the off-diagonal terms in the Hamiltonian are zero. These terms represent miniband coupling due to the electric field and originate the resonant tunneling transport. For a superlattice under dc voltage bias in the growth direction, numerical solutions of the corresponding quantum drift-diffusion equations show self-sustained current oscillations due to periodic recycling and motion of electric field pulses, and resonant tunneling between minibands when the electric field is above the resonant value. Numerical reconstruction of the Wigner functions during self-oscillations confirms this picture.
This research has been supported by the Spanish Ministerio de Ciencia e Innovación (MICINN) through Grant No. FIS2008-04921-C02-01.
Detailed expressions for $D_{\pm}$ and $R$ {#appA}
==========================================
The recombination term $R(n^+,n^-,F)$ (\[61\]) depends on $\varphi_0^2$ and $\psi_0^2$ which can be obtained from (\[49\]) and (\[56\]) for $j=0$, taking into account that $\Delta_0^-\mathcal F =0$, $\Delta_0^+\mathcal F = 2 \mathcal F $ and $\phi_0^\pm = n^\pm$:\
$$\begin{aligned}
\varphi_0^2 &=& {\delta \mathcal F\, (n^+-n^-) \over 1+\eta_1^2 + 4\delta^2 \mathcal F^2} \nonumber \\
\psi_0^2 &=& {1 \over 1+\eta_1^2 + 4\delta^2 \mathcal F^2} \left[ {\alpha\tau \over \hbar}\left[ {\delta(1+\eta_1^2- 4\delta^2 \mathcal F^2)
(1-\eta_1^2- 4\delta^2 \mathcal F^2(n^+-n^-)) \over 1+\eta_1^2 + 4\delta^2 \mathcal F^2}\left. {\partial \mathcal F \over \partial t}\right|_0 -
\right.\right. \nonumber \\
&& {4\delta^3\mathcal F^2 \over 1+\eta_1^2 + 4\delta^2 \mathcal F^2}\left( 2(n^+-n^-)\left. {\partial \mathcal F \over \partial t}\right|_0 +
\mathcal F\, \left(\left. {\partial n^+ \over \partial t}\right|_0 - \left. {\partial n^- \over \partial t}\right|_0 \right)\right) + \nonumber \\
&& \left. \delta\,\left((n^+-n^-)\left. {\partial \mathcal F \over \partial t}\right|_0 + \mathcal F\left( \left. {\partial n^+ \over \partial
t}\right|_0 - \left. {\partial n^- \over \partial t}\right|_0 \right)\right) \right] - \nonumber \\
&& \eta_2\delta \mathcal F \left( {1-\eta_1^2-4\delta^2 \mathcal F^2 (n^+-n^-) \over 1+\eta_1^2 + 4\delta^2 \mathcal F^2} + 2n^+ \right)
- \nonumber \\
&& {\alpha\tau\over\hbar}\Delta^-\left[\mbox{Im} \varphi_1^2 - \eta_1 \mbox{Im} \varphi_1^1 + 2\delta \mathcal F \mbox{Im} \varphi_1^3\right] -
{\gamma\tau\over\hbar}\left(-{4\eta_1\delta\mathcal F (n^+-n^-)\over 1+\eta_1^2 + 4\delta^2 \mathcal F^2} + \right. \nonumber \\
&& \left. 2\delta\mathcal F \Delta^- \mbox{Im} \varphi_1^0 + \Delta^+(\mbox{Re} \varphi_1^1 + \eta_1 \mbox{Re} \varphi_1^2) \right) \nonumber\end{aligned}$$
The time derivatives $\displaystyle\left.{\partial n^\pm \over\partial t} \right|_0$ and $\displaystyle\left. {\partial F \over \partial
t}\right|_0$, are obtained from the first two terms of the Chapman-Enskog expansion of (\[58\]) and (\[59\]) respectively: $$\begin{aligned}
\left.{\partial n^\pm \over \partial t}\right|_0 &=& \mp Q^{(0)} - \Delta^-D_\pm^{(0)} = \mp{n^+
\over \tau_{sc}} \mp {2\delta^2 \mathcal F^2 (n^+-n^-) \over \tau (1+\eta_1^2+4\delta^2\mathcal F^2)} + \nonumber \\
&& {\alpha \pm \gamma \over \hbar}\Delta^-\left[\phi_1^{\pm}\left({\vartheta_1 \over 1 + \vartheta_1^2} +
\eta_1\delta^2 \left(\mp \Delta^-\mathcal F\Delta^+\mathcal F \mbox{Re} Z_1 + \right. \right. \right. \nonumber \\
&& \mbox{Re} Z_1((\Delta^-\mathcal F)^2 \mbox{Im} M_1^+ + (\Delta^+\mathcal F)^2 \mbox{Im} M_1^-) + \nonumber \\
&& \left. \left. \left. \mbox{Im} Z_1((\Delta^-\mathcal F)^2 \mbox{Re} M_1^+ + (\Delta^+\mathcal F)^2 \mbox{Re} M_1^-)\right) \right) + \right. \nonumber \\
&& \phi_1^{\mp}\eta_1 \delta^2[\mbox{Re} Z_1((\Delta^-\mathcal F)^2 \mbox{Im}
M_1^+ - (\Delta^+\mathcal F)^2 \mbox{Im} M_1^-) + \nonumber \\
&& \left. \mbox{Im} Z_1((\Delta^-\mathcal F)^2 \mbox{Re} M_1^+ - (\Delta^+\mathcal F)^2 \mbox{Re} M_1^-)] \right] \nonumber \\
\left. \varepsilon{\partial F \over \partial t}\right|_0 &=& J - e\left<D_+^{(0)}+D_-^{(0)}\right> = J -
{e\alpha \over \hbar}\left[(\phi_1^++\phi_1^-)\left({-\vartheta_1 \over 1 + \vartheta_1^2} \right.\right. - \nonumber \\
&& \eta_1 \delta^2 (\Delta^- \mathcal F)^2 (\mbox{Re} M_1^+ \mbox{Im} Z_1 + \mbox{Im} M_1^+ \mbox{Re} Z_1) + \nonumber \\
&& \left. {\gamma\over\alpha}\eta_1 \delta^2 \Delta^+ \mathcal F \Delta^- \mathcal F \mbox{Re} Z_1\right) +
(\phi_1^+-\phi_1^-)\left(\eta_1\delta^2\Delta^+\mathcal F \Delta^-\mathcal F \mbox{Re} Z_1 - \right. \nonumber \\
&& \left.\left. {\gamma\over\alpha}\left({\vartheta_1 \over 1+ \vartheta_1^2} + \eta_1\delta^2(\Delta^+ \mathcal F)^2 (\mbox{Re} M_1^- \mbox{Im} Z_1 + \mbox{Im} M_1^-
\mbox{Re} Z_1)\right)\right)\right] \nonumber\end{aligned}$$
The expression of $D_{\pm}(n^+,n^-,F)$ is based on the first two terms of the Chapman-Enskog expansion $D_{\pm}^{(0)}$ and $D_{\pm}^{(1)}$:
$$\begin{aligned}
D_{\pm}(n^+,n^-,F) &=& D_{\pm}^{(0)}(n^+,n^-,F) + D_{\pm}^{(1)}(n^+,n^-,F) \nonumber\end{aligned}$$
Where $D_{\pm}^{(0)}$ and $D_{\pm}^{(1)}$ are as follows: $$\begin{aligned}
D_{\pm}^{(0)}(n^+,n^-,F) &=& {\alpha \pm \gamma \over \hbar} \mbox{Im}(\varphi_1^0 \pm \varphi_1^3) = \nonumber \\
&& -{\alpha \pm \gamma \over \hbar}\left[ \phi_1^{\pm}\left({\vartheta_1 \over 1+\vartheta_1^2} \mp \eta_1
\delta^2 (\Delta^- \mathcal F)(\Delta^+ \mathcal F)\mbox{Re} Z_1 \right. \right. \nonumber \\
&& \left. \left. + {\eta_1 \delta^2 \over 2}((\mbox{Im} M_1^+(\Delta^- \mathcal F)^2 \nonumber + \mbox{Im} M_1^-(\Delta^+ \mathcal
F)^2)\mbox{Re} Z_1 \right. \right. \nonumber \\
&& \left.\left. + (\mbox{Re} M_1^+(\Delta^- \mathcal F)^2 + \mbox{Re} M_1^-(\Delta^+ \mathcal F)^2)\mbox{Im} Z_1) \right) \right.\nonumber\\
&& \left. + \,\phi_1^{\mp}\,{\eta_1 \delta^2 \over 2}( \mbox{Im} M_1^+(\Delta^- \mathcal F)^2 - \mbox{Im} M_1^-(\Delta^+ \mathcal F)^2 )\mbox{Re} Z_1\right. \nonumber \\
&& \left. + \,(\mbox{Re} M_1^+(\Delta^- \mathcal F)^2 - \mbox{Re} M_1^-(\Delta^+ \mathcal F)^2)\mbox{Im} Z_1) \right] \nonumber\end{aligned}$$ $$\begin{aligned}
D_{\pm}^{(1)}(n^+,n^-,F) &=& {\alpha \pm \gamma \over \hbar} \mbox{Im}(\psi_1^0 \pm \psi_1^3) = \nonumber\\
&& {\alpha \pm \gamma \over \hbar} \left[\mbox{Re} S_1^0 \, \mbox{Im} A_1^{\pm} + \mbox{Im} S_1^0\, \mbox{Re} A_1^{\pm} \pm \mbox{Re} S_1^3\, \mbox{Im}
C_1^{\pm} \pm \mbox{Im} S_1^3\, \mbox{Re} C_1^{\pm} \right. \nonumber \\
&&\left. + \eta_1 \delta (\mbox{Re} Z_1\, \mbox{Im} B_1^{\pm} + \mbox{Re} Z_1\, \vartheta_1\, \mbox{Re} B_1^{\pm} + \mbox{Im} Z_1\, \mbox{Re} B_1^{\pm} - \mbox{Im} Z_1\, \vartheta_1\, \mbox{Im}
B_1^{\pm}) \right] \nonumber\end{aligned}$$ The functionals $S_1(n^+,n^-,F)$, $A_1^\pm(F)$, $B_1^\pm(F)$ and $C_1^\pm(F)$ are as follows: $$\begin{aligned}
\mbox{Re} A_1^{\pm} &=& {1\over 1+\vartheta_1^2} - \eta_1 \delta^2 \Delta^- \mathcal F\, (\mbox{Re} Z_1\, \mbox{Re} M_1^+ \Delta^-
\mathcal F - \mbox{Im} Z_1 ( \Delta^- \mathcal F\, \mbox{Im} M_1^+ \mp \Delta^+ \mathcal F)), \nonumber \\
\mbox{Im} A_1^{\pm} &=& {-\vartheta_1 \over 1+\vartheta_1^2} - \eta_1 \delta^2 \Delta^- \mathcal F (\mbox{Re} Z_1 (\Delta^-
\mathcal F \, \mbox{Im} M_1^+ \mp \Delta^+ \mathcal F)+ \mbox{Im} Z_1 \, \Delta^- \mathcal F \, \mbox{Re} M_1^+), \nonumber \\
\mbox{Re} B_1^{\pm} &=& \mbox{Re} S_1^1 (-\Delta^- \mathcal F \mbox{Im} M_1^+ \pm \Delta^+ \mathcal F)- \mbox{Im} S_1^1 \, \Delta^- \mathcal F \,\mbox{Re}
M_1^+ \mp \mbox{Re} S_1^2 \, \Delta^+ \mathcal F \, \mbox{Re} M_1^- \nonumber \\
&& - \mbox{Im} S_1^2(\Delta^- \mathcal F \mp \Delta^+ \mathcal F \, \mbox{Im} M_1^-), \nonumber \\
\mbox{Im} B_1^{\pm} &=& \mbox{Re} S_1^1 \, \Delta^- \mathcal F \, \mbox{Re} M_1^+ + \mbox{Im} S_1^1 (\pm\Delta^+\mathcal F - \Delta^- \mathcal F \mbox{Im}
M_1^+), \nonumber \\
&& + \mbox{Re} S_1^2(\Delta^- \mathcal F \mp \Delta^+ \mathcal F \mbox{Im} M_1^-) \mp \mbox{Im} S_1^2\, \Delta^+ \mathcal F\, \mbox{Re} M_1^-
\nonumber \\
\mbox{Re} C_1^{\pm} &=& {1 \over 1+ \vartheta_1^2} - \eta_1 \delta^2 \Delta^+ \mathcal F (\mbox{Re} Z_1 \,\Delta^+ \mathcal F \, \mbox{Re}
M_1^- - \mbox{Im} Z_1 ( \Delta^+ \mathcal F\, \mbox{Im} M_1^- \mp \Delta^- \mathcal F)), \nonumber \\
\mbox{Im} C_1^{\pm} &=& {-\vartheta_1 \over 1+\vartheta_1^2} - \eta_1\delta^2 \Delta^+ \mathcal F (\mbox{Re} Z_1 (
\Delta^+ \mathcal F \mbox{Im} M_1^- \mp \Delta^- \mathcal F) + \mbox{Im} Z_1 \, \Delta^+ \mathcal F\, \mbox{Re} M_1^-) \nonumber \\
\mbox{Re} S_1^0 &=& -\tau\, \left. {\partial \mbox{Re} \varphi_1^0 \over \partial t}\right|_0 - {\alpha \tau \over 2
\hbar}\Delta^-\mbox{Im} \varphi_2^0 - {\gamma \tau \over 2 \hbar}\Delta^-\mbox{Im} \varphi_2^3 \nonumber \\
\mbox{Im} S_1^0 &=& -\tau\, \left. {\partial \mbox{Im} \varphi_1^0 \over
\partial t}\right|_0 + {\alpha \tau \over 2 \hbar}\Delta^-(\mbox{Re} \varphi_2^0 - \varphi_0^0) + {\gamma \tau \over 2 \hbar}\Delta^-(\mbox{Re} \varphi_2^3 - \varphi_0^3) \nonumber \\
\mbox{Re} S_1^1 &=& -\tau\, \left. {\partial \mbox{Re} \varphi_1^1 \over \partial t}\right|_0 - \eta_2 \mbox{Re}\varphi_1^1 - {\alpha \tau \over 2\hbar}\Delta^-\mbox{Im}
\varphi_2^1 - {2\gamma\tau\over\hbar}\mbox{Re} \varphi_1^2 + {\gamma \tau \over 2 \hbar}\Delta^+(\mbox{Re} \varphi_2^2+\varphi_0^2) \nonumber \\
\mbox{Im} S_1^1 &=& -\tau\, \left. {\partial \mbox{Re} \varphi_1^1 \over \partial t}\right|_0 - \eta_2 \mbox{Im}\varphi_1^1 +{\alpha \tau \over 2\hbar}\Delta^-(\mbox{Re}
\varphi_2^1-\varphi_0^1) - {2\gamma\tau\over\hbar}\mbox{Im} \varphi_1^2 + {\gamma \tau \over 2 \hbar}\Delta^+\mbox{Im} \varphi_2^2 \nonumber \\
\mbox{Re} S_1^2 &=& -\tau\, \left. {\partial \mbox{Re} \varphi_1^2 \over \partial t}\right|_0 - \eta_2 \mbox{Re}\varphi_1^2 - {\alpha \tau \over 2\hbar}\Delta^-\mbox{Im}
\varphi_2^2 + {2\gamma\tau\over\hbar}\mbox{Re} \varphi_1^1 - {\gamma \tau \over 2 \hbar}\Delta^+(\mbox{Re} \varphi_2^1+\varphi_0^1) \nonumber \\
\mbox{Im} S_1^2 &=& -\tau\, \left. {\partial \mbox{Re} \varphi_1^2 \over \partial t}\right|_0 - \eta_2 \mbox{Im}\varphi_1^2 +{\alpha \tau \over 2\hbar}\Delta^-(\mbox{Re}
\varphi_2^2-\varphi_0^2) + {2\gamma\tau\over\hbar}\mbox{Im} \varphi_1^1 - {\gamma \tau \over 2 \hbar}\Delta^+\mbox{Im} \varphi_2^1 \nonumber \\
\mbox{Re} S_1^3 &=& -\tau\, \left. {\partial \mbox{Re} \varphi_1^3 \over \partial t}\right|_0 - \eta_2( \mbox{Re}\varphi_1^3 + \mbox{Re} \varphi_1^0) - {\alpha \tau \over
2\hbar}\Delta^-\mbox{Im} \varphi_2^3 - {\gamma \tau \over 2 \hbar}\Delta^-\mbox{Im} \varphi_2^0 \nonumber \\
\mbox{Im} S_1^3 &=& -\tau\, \left. {\partial \mbox{Im} \varphi_1^3 \over \partial t}\right|_0 - \eta_2( \mbox{Im}\varphi_1^3 + \mbox{Im} \varphi_1^0) + {\alpha \tau \over
2\hbar}\Delta^-(\mbox{Re} \varphi_2^3-\varphi_0^3) + {\gamma \tau \over 2 \hbar}\Delta^-(\mbox{Re} \varphi_2^0 - \varphi_0^0) \nonumber\end{aligned}$$ Where the real and imaginary parts of $\varphi_j$ are: $$\begin{aligned}
\mbox{Re} \varphi_j^0 &=& {\phi_j^++\phi_j^- \over 2}\left[{1\over 1+
\vartheta_j^2}-\eta_1\delta^2(\Delta_j^-\mathcal F)^2(\mbox{Re} M_j^+ \mbox{Re} Z_j - \mbox{Im} M_j^+ \mbox{Im} Z_j)\right] - \nonumber \\
&& {\eta_1\delta^2 \over 2}(\phi_j^+-\phi_j^-)\Delta_j^+\mathcal F \Delta_j^-\mathcal F \mbox{Im} Z_j \nonumber \\
\mbox{Im} \varphi_j^0 &=& {\phi_j^++\phi_j^- \over 2}\left[{-\vartheta_j\over 1+ \vartheta_j^2}-
\eta_1\delta^2(\Delta_j^-\mathcal F)^2(\mbox{Re} M_j^+ \mbox{Im} Z_j + \mbox{Im} M_j^+ \mbox{Re} Z_j)\right] + \nonumber \\
&& {\eta_1\delta^2 \over 2}(\phi_j^+-\phi_j^-)\Delta_j^+\mathcal F \Delta_j^-\mathcal F \mbox{Re} Z_j \nonumber \\
\mbox{Re} \varphi_j^1 &=& {\eta_1\delta \over 2}\left[-(\phi_j^++\phi_j^-)\Delta_j^-\mathcal F(\mbox{Im} M_j^+ (\mbox{Re}
Z_j-\vartheta_j \mbox{Im} Z_j)\right. + \nonumber \\
&& \mbox{Re} M_j^+ (\mbox{Im} Z_j+\vartheta_j \mbox{Re} Z_j)) + \left.
(\phi_j^+-\phi_j^-)\Delta_j^+\mathcal F(\mbox{Re} Z_j - \vartheta_j \mbox{Im} Z_j)\right] \nonumber \\
\mbox{Im} \varphi_j^1 &=& {\eta_1\delta \over 2}\left[(\phi_j^++\phi_j^-)\Delta_j^-\mathcal F(\mbox{Re} M_j^+ (\mbox{Re}
Z_j-\vartheta_j \mbox{Im} Z_j)\right. - \nonumber \\
&& \mbox{Im} M_j^+ (\mbox{Im} Z_j+\vartheta_j \mbox{Re} Z_j)) + \left.
(\phi_j^+-\phi_j^-)\Delta_j^+\mathcal F(\mbox{Im} Z_j + \vartheta_j \mbox{Re} Z_j)\right] \nonumber \\
\mbox{Re} \varphi_j^2 &=& {\eta_1\delta \over 2}\left[(\phi_j^++\phi_j^-)\Delta_j^-\mathcal F(\mbox{Im} Z_j+\vartheta_j \mbox{Re} Z_j)\right. + \nonumber \\
&& \left. (\phi_j^+-\phi_j^-)\Delta_j^+\mathcal F(\mbox{Re} M_j^- (\mbox{Re} Z_j - \vartheta_j \mbox{Im} Z_j) -
\mbox{Im} M_j^-(\mbox{Im} Z_j + \vartheta_j \mbox{Re} Z_j))\right] \nonumber \\
\mbox{Im} \varphi_j^2 &=& {\eta_1\delta \over 2}\left[(\phi_j^++\phi_j^-)\Delta_j^-\mathcal F(-\mbox{Re} Z_j+\vartheta_j \mbox{Im} Z_j)\right. + \nonumber \\
&& \left. (\phi_j^+-\phi_j^-)\Delta_j^+\mathcal F(\mbox{Re} M_j^- (\mbox{Im} Z_j + \vartheta_j \mbox{Re} Z_j) +
\mbox{Im} M_j^-(\mbox{Re} Z_j - \vartheta_j \mbox{Im} Z_j))\right] \nonumber \\
\mbox{Re} \varphi_j^3 &=& {\phi_j^+-\phi_j^- \over 2}\left[{1\over 1+
\vartheta_j^2}-\eta_1\delta^2(\Delta_j^+\mathcal F)^2(\mbox{Re} M_j^- \mbox{Re} Z_j - \mbox{Im} M_j^- \mbox{Im} Z_j)\right] - \nonumber \\
&& {\eta_1\delta^2 \over 2}(\phi_j^++\phi_j^-)\Delta_j^+\mathcal F \Delta_j^-\mathcal F \mbox{Im} Z_j \nonumber \\
\mbox{Im} \varphi_j^3 &=& {\phi_j^+-\phi_j^- \over 2}\left[{-\vartheta_j\over 1+ \vartheta_j^2}-
\eta_1\delta^2(\Delta_j^+\mathcal F)^2(\mbox{Re} M_j^- \mbox{Im} Z_j + \mbox{Im} M_j^- \mbox{Re} Z_j)\right] + \nonumber \\
&& {\eta_1\delta^2 \over 2}(\phi_j^++\phi_j^-)\Delta_j^+\mathcal F \Delta_j^-\mathcal F \mbox{Re} Z_j \nonumber\end{aligned}$$ Now we can obtain the expressions for $\displaystyle \left. {\partial \varphi_j \over \partial t}\right|_0$: $$\begin{aligned}
\left. {\partial \mbox{Re} \varphi_1^0 \over \partial t}\right|_0 &=& {1\over 2}\left((\phi_1^+)' \, \left.{\partial n^+
\over \partial t}\right|_0 + (\phi_1^-)' \, \left.{\partial n^- \over \partial t}\right|_0 \right)\times \nonumber \\
&& \left[{1 \over 1+\vartheta_1^2} - \eta_1\delta^2(\Delta^- \mathcal F)^2\,(\mbox{Re} M_1^+ \mbox{Re} Z_1 - \mbox{Im} M_1^+ \mbox{Im} Z_1)\right] +\nonumber \\
&& {1\over 2}(\phi_1^+ + \phi_1^-)\left[{-2\vartheta_1 \over (1+\vartheta_1^2)^2} \left.{\partial \vartheta_1 \over \partial t}\right|_0
\right. - \nonumber \\
&& 2\eta_1\delta^2 \Delta^- \mathcal F \Delta^- \left.{\partial \mathcal F \over \partial t}\right|_0 (\mbox{Re} M_1^+ \mbox{Re} Z_1 - \mbox{Im} M_1^+ \mbox{Im} Z_1) -
\nonumber \\
&& \eta_1\delta^2 (\Delta^- \mathcal F)^2\left(\left.{\partial \mbox{Re} M_1^+ \over \partial t}\right|_0 \mbox{Re} Z_1 + \mbox{Re} M_1^+ \left.{\partial \mbox{Re} Z_1 \over
\partial t}\right|_0 - \mbox{Im} Z_1 \left.{\partial \mbox{Im} M_1^+ \over \partial t}\right|_0 \right. - \nonumber \\
&& \left.\left. \mbox{Im} M_1^+ \left.{\partial \mbox{Im} Z_1 \over \partial t}\right|_0\right) \right] - \nonumber \\
&& {\eta_1\delta^2 \over 2}\left(\left((\phi_1^+)' \left.{\partial n^+ \over \partial t}\right|_0 - (\phi_1^-)'
\left.{\partial n^- \over \partial t}\right|_0 \right) \Delta^+ \mathcal F \Delta^- \mathcal F \mbox{Im} Z_1 -\right. \nonumber \\
&& (\phi_1^+-\phi_1^-)\left(\Delta^+\left.{\partial \mathcal F \over \partial t}\right|_0\Delta^- \mathcal F \mbox{Im} Z_1 +
\Delta^-\left.{\partial \mathcal F \over \partial t}\right|_0\Delta^+ \mathcal F \mbox{Im} Z_1 \right. + \nonumber \\
&& \left.\left.\left.{\partial \mbox{Im} Z_1 \over \partial t}\right|_0\Delta^+ \mathcal F \Delta^- \mathcal F \right)\right) \nonumber \\
\left. {\partial \mbox{Im} \varphi_1^0 \over \partial t}\right|_0 &=& {1\over 2}\left((\phi_1^+)' \, \left.{\partial n^+
\over \partial t}\right|_0 + (\phi_1^-)' \, \left.{\partial n^- \over \partial t}\right|_0 \right)\times \nonumber \\
&& \left[{-\vartheta_1 \over 1+\vartheta_1^2} - \eta_1\delta^2(\Delta^- \mathcal F)^2\,(\mbox{Re} M_1^+ \mbox{Im} Z_1 + \mbox{Im} M_1^+ \mbox{Re} Z_1)\right] +\nonumber \\
&& {1\over 2}(\phi_1^+ + \phi_1^-)\left[{\vartheta_1^2-1 \over (1+\vartheta_1^2)^2} \left.{\partial \vartheta_1 \over \partial t}\right|_0
\right. - \nonumber \\
&& 2\eta_1\delta^2 \Delta^- \mathcal F \Delta^- \left.{\partial \mathcal F \over \partial t}\right|_0 (\mbox{Re} M_1^+ \mbox{Im} Z_1 + \mbox{Im} M_1^+ \mbox{Re} Z_1) -
\nonumber \\
&& \eta_1\delta^2 (\Delta^- \mathcal F)^2\left(\left.{\partial \mbox{Re} M_1^+ \over \partial t}\right|_0 \mbox{Im} Z_1 + \mbox{Re} M_1^+ \left.{\partial \mbox{Im} Z_1 \over
\partial t}\right|_0 + \mbox{Re} Z_1 \left.{\partial \mbox{Im} M_1^+ \over \partial t}\right|_0 \right. - \nonumber \\
&& \left.\left. \mbox{Im} M_1^+ \left.{\partial \mbox{Re} Z_1 \over \partial t}\right|_0\right) \right] + \nonumber \\
&& {\eta_1\delta^2 \over 2}\left(\left((\phi_1^+)' \left.{\partial n^+ \over \partial t}\right|_0 - (\phi_1^-)'
\left.{\partial n^- \over \partial t}\right|_0 \right) \Delta^+ \mathcal F \Delta^- \mathcal F \mbox{Re} Z_1 + \right.\nonumber \\
&& (\phi_1^+-\phi_1^-)\left(\Delta^+\left.{\partial \mathcal F \over \partial t}\right|_0\Delta^- \mathcal F \mbox{Re} Z_1 +
\Delta^-\left.{\partial \mathcal F \over \partial t}\right|_0\Delta^+ \mathcal F \mbox{Re} Z_1 \right. + \nonumber \\
&& \left.\left.\left.{\partial \mbox{Re} Z_1 \over \partial t}\right|_0\Delta^+ \mathcal F \Delta^- \mathcal F \right)\right) \nonumber\end{aligned}$$ $$\begin{aligned}
\left. {\partial \mbox{Re} \varphi_1^1 \over \partial t}\right|_0 &=& {\eta_1\delta\over 2}\left[-\left((\phi_1^+)' \, \left.{\partial n^+ \over
\partial t}\right|_0 + (\phi_1^-)' \, \left.{\partial n^- \over \partial t}\right|_0\right)\Delta^- \mathcal F(\mbox{Im} M_1^+ (\mbox{Re} Z_1 - \vartheta_1\mbox{Im} Z_1)
+ \right. \nonumber \\
&& \mbox{Re} M_1^+ (\mbox{Im} Z_1 + \vartheta_1 \mbox{Re} Z_1)) - (\phi_1^+ + \phi_1^-)\Delta^- \left.{\partial \mathcal F \over \partial
t}\right|_0 \times \nonumber \\
&& (\mbox{Im} M_1^+(\mbox{Re} Z_1 - \vartheta_1 \mbox{Im} Z_1) + \mbox{Re} M_1^+ (\mbox{Im} Z_1 + \vartheta_1 \mbox{Re} Z_1)) - \nonumber \\
&& (\phi_1^+ + \phi_1^-)\Delta^-\mathcal F\left(\left.{\partial \mbox{Im} M_1^+ \over \partial t}\right|_0(\mbox{Re} Z_1 - \vartheta_1 \mbox{Im} Z_1) \right. + \nonumber \\
&& \mbox{Im} M_1^+\left(\left.{\partial \mbox{Re} Z_1 \over \partial t}\right|_0-\left.{\partial \vartheta_1 \over \partial t}\right|_0\, \mbox{Im} Z_1 - \vartheta_1
\left.{\partial \mbox{Im} Z_1 \over \partial t}\right|_0\right) + \nonumber \\
&& \left.{\partial \mbox{Re} M_1^+ \over \partial t}\right|_0 (\mbox{Im} Z_1 + \vartheta_1 \mbox{Re} Z_1) + \mbox{Re} M_1^+\left(\left.{\partial \mbox{Im} Z_1 \over \partial
t}\right|_0 + \left.{\partial \vartheta_1 \over \partial t}\right|_0 \, \mbox{Re} Z_1 + \right. \nonumber \\
&& \left. \left. \vartheta_1\left.{\partial \mbox{Re} Z_1 \over \partial t}\right|_0 \right)\right) + \left((\phi_1^+)'\left.{\partial n^+ \over
\partial t}\right|_0- (\phi_1^-)'\left.{\partial n^- \over \partial t}\right|_0\right)\Delta^+\mathcal F (\mbox{Re} Z_1-\vartheta_1 \mbox{Im} Z_1) + \nonumber \\
&& (\phi_1^+ - \phi_1^-)\left(\Delta^+\left.{\partial \mathcal F \over \partial t}\right|_0 (\mbox{Re} Z_1 - \vartheta_1 \mbox{Im} Z_1) + \right.\nonumber \\
&& \left.\left.\Delta^+ \mathcal F \left(\left.{\partial \mbox{Re} Z_1 \over \partial t}\right|_0 - \left.{\partial \vartheta_1 \over \partial t}\right|_0
\mbox{Im} Z_1 - \vartheta_1 \left.{\partial \mbox{Im} Z_1 \over \partial t}\right|_0\right)\right)\right] \nonumber \\
\left. {\partial \mbox{Im} \varphi_1^1 \over \partial t}\right|_0 &=& {\eta_1\delta\over 2}\left[\left((\phi_1^+)' \, \left.{\partial n^+ \over
\partial t}\right|_0 + (\phi_1^-)' \, \left.{\partial n^- \over \partial t}\right|_0\right)\Delta^- \mathcal F (\mbox{Re} M_1^+ (\mbox{Re} Z_1 - \vartheta_1 \mbox{Im} Z_1)
- \right. \nonumber \\
&& \mbox{Im} M_1^+ (\mbox{Im} Z_1 + \vartheta_1 \mbox{Re} Z_1)) + (\phi_1^+ + \phi_1^-)\Delta^- \left.{\partial \mathcal F \over \partial t}\right|_0 \times \nonumber \\
&& (\mbox{Re} M_1^+(\mbox{Re} Z_1 - \vartheta_1 \mbox{Im} Z_1) - \mbox{Im} M_1^+ (\mbox{Im} Z_1 + \vartheta_1 \mbox{Re} Z_1)) - \nonumber \\
&& (\phi_1^+ + \phi_1^-)\Delta^-\mathcal F\left(\left.{\partial \mbox{Re} M_1^+ \over \partial t}\right|_0(\mbox{Re} Z_1 - \vartheta_1 \mbox{Im} Z_1) \right. + \nonumber \\
&& \mbox{Re} M_1^+\left(\left.{\partial \mbox{Re} Z_1 \over \partial t}\right|_0-\left.{\partial \vartheta_1 \over \partial t}\right|_0\, \mbox{Im} Z_1 - \vartheta_1
\left.{\partial \mbox{Im} Z_1 \over \partial t}\right|_0\right) - \nonumber \\
&& \left.{\partial \mbox{Im} M_1^+ \over \partial t}\right|_0 (\mbox{Im} Z_1 + \vartheta_1 \mbox{Re} Z_1) - \mbox{Im} M_1^+\left(\left.{\partial \mbox{Im} Z_1 \over \partial
t}\right|_0 + \left.{\partial \vartheta_1 \over \partial t}\right|_0 \, \mbox{Re} Z_1 + \right. \nonumber \\
&& \left. \left. \vartheta_1\left.{\partial \mbox{Re} Z_1 \over \partial t}\right|_0 \right)\right) + \left((\phi_1^+)'\left.{\partial n^+ \over
\partial t}\right|_0- (\phi_1^-)'\left.{\partial n^- \over \partial t}\right|_0\right)\Delta^+\mathcal F (\vartheta_1 \mbox{Re} Z_1+ \mbox{Im} Z_1) + \nonumber \\
&& (\phi_1^+ - \phi_1^-)\left(\Delta^+\left.{\partial \mathcal F \over \partial t}\right|_0 (\vartheta_1 \mbox{Re} Z_1 + \mbox{Im} Z_1) + \right.\nonumber \\
&& \left.\left.\Delta^+ \mathcal F \left(\vartheta_1\left.{\partial \mbox{Re} Z_1 \over \partial t}\right|_0 + \left.{\partial \vartheta_1 \over \partial
t}\right|_0 \mbox{Re} Z_1 + \left.{\partial \mbox{Im} Z_1 \over \partial t}\right|_0\right)\right)\right] \nonumber\end{aligned}$$ $$\begin{aligned}
\left. {\partial \mbox{Re} \varphi_1^2 \over \partial t}\right|_0 &=& {\eta_1\delta\over 2}\left[\left(\left((\phi_1^+)' \, \left.{\partial n^+ \over
\partial t}\right|_0 + (\phi_1^-)' \, \left.{\partial n^- \over \partial
t}\right|_0\right)\Delta^- \mathcal F + (\phi_1^++\phi_1^-)\Delta^- \left.{\partial \mathcal F \over \partial t}\right|_0 \right)\times \right. \nonumber \\
&& (\mbox{Im} Z_1 + \vartheta_1 \mbox{Re} Z_1) + (\phi_1^++\phi_1^-)\Delta^- \mathcal F \left( \left.{\partial \mbox{Im} Z_1 \over \partial t}\right|_0 + \left.{\partial
\vartheta_1 \over \partial t}\right|_0 \mbox{Re} Z_1 + \vartheta_1 \left.{\partial \mbox{Re} Z_1 \over
\partial t}\right|_0 \right) + \nonumber \\
&& \left(\left((\phi_1^+)' \, \left.{\partial n^+ \over \partial t}\right|_0 - (\phi_1^-)' \, \left.{\partial n^- \over
\partial t}\right|_0\right)\Delta^+ \mathcal F + (\phi_1^+-\phi_1^-)\Delta^+\left.{\partial \mathcal F \over \partial
t}\right|_0\right)\times \nonumber \\
&& (\mbox{Re} M_1^-(\mbox{Re} Z_1- \vartheta_1 \mbox{Im} Z_1) - \mbox{Im} M_1^-(\mbox{Im} Z_1 + \vartheta_1 \mbox{Re} Z_1)) + \nonumber \\
&&(\phi_1^+-\phi_1^-)\Delta^+ \mathcal F \left(\left.{\partial \mbox{Re} M_1^- \over \partial t}\right|_0 (\mbox{Re} Z_1 -
\vartheta_1 \mbox{Im} Z_1) + \mbox{Re} M_1^-\left(\left.{\partial \mbox{Re} Z_1 \over \partial t}\right|_0 \right.\right. - \nonumber \\
&& \left. \left.{\partial \vartheta_1 \over \partial t}\right|_0 \mbox{Im} Z_1 - \vartheta_1 \left.{\partial \mbox{Im} Z_1 \over
\partial t}\right|_0 \right) - \left.{\partial \mbox{Im} M_1^- \over \partial t}\right|_0 (\mbox{Im} Z_1+\vartheta_1 \mbox{Re} Z_1) - \nonumber \\
&&\left.\left. \mbox{Im} M_1^- \left( \left.{\partial \mbox{Im} Z_1 \over \partial t}\right|_0 + \left.{\partial \vartheta_1 \over
\partial t}\right|_0 \mbox{Re} Z_1 + \vartheta_1 \left.{\partial \mbox{Re} Z_1 \over \partial t}\right|_0 \right)\right)\right]
\nonumber\\
\left. {\partial \mbox{Im} \varphi_1^2 \over \partial t}\right|_0 &=& {\eta_1\delta\over
2}\left[\left(\left((\phi_1^+)' \, \left.{\partial n^+ \over \partial t}\right|_0 + (\phi_1^-)' \, \left.{\partial n^- \over \partial
t}\right|_0\right)\Delta^- \mathcal F + (\phi_1^++\phi_1^-)\Delta^- \left.{\partial \mathcal F \over \partial t}\right|_0 \right)\times \right. \nonumber \\
&& (-\mbox{Re} Z_1 + \vartheta_1 \mbox{Im} Z_1) + (\phi_1^++\phi_1^-)\Delta^- \mathcal F \left( -\left.{\partial \mbox{Re} Z_1 \over
\partial t}\right|_0 + \left.{\partial \vartheta_1 \over \partial t}\right|_0 \mbox{Im} Z_1 + \vartheta_1 \left.{\partial \mbox{Im}
Z_1 \over \partial t}\right|_0 \right) + \nonumber \\
&& \left(\left((\phi_1^+)' \, \left.{\partial n^+ \over \partial t}\right|_0 - (\phi_1^-)' \, \left.{\partial n^- \over
\partial t}\right|_0\right)\Delta^+ \mathcal F + (\phi_1^+-\phi_1^-)\Delta^+\left.{\partial \mathcal F \over \partial
t}\right|_0\right)\times \nonumber \\
&& (\mbox{Re} M_1^-(\mbox{Im} Z_1+ \vartheta_1 \mbox{Re} Z_1) + \mbox{Im} M_1^-(\mbox{Re} Z_1 - \vartheta_1 \mbox{Im} Z_1)) + \nonumber \\
&&(\phi_1^+-\phi_1^-)\Delta^+ \mathcal F \left(\left.{\partial \mbox{Re} M_1^- \over \partial t}\right|_0 (\mbox{Im} Z_1 +
\vartheta_1 \mbox{Re} Z_1) + \mbox{Re} M_1^-\left(\left.{\partial \mbox{Im} Z_1 \over \partial t}\right|_0 \right.\right. + \nonumber \\
&& \left. \left.{\partial \vartheta_1 \over \partial t}\right|_0 \mbox{Re} Z_1 + \vartheta_1 \left.{\partial \mbox{Re} Z_1 \over
\partial t}\right|_0 \right) + \left.{\partial \mbox{Im} M_1^- \over \partial t}\right|_0 (\mbox{Re} Z_1-\vartheta_1 \mbox{Im} Z_1) + \nonumber \\
&&\left.\left. \mbox{Im} M_1^- \left( \left.{\partial \mbox{Re} Z_1 \over \partial t}\right|_0 - \left.{\partial \vartheta_1 \over
\partial t}\right|_0 \mbox{Im} Z_1 - \vartheta_1 \left.{\partial \mbox{Im} Z_1 \over \partial t}\right|_0 \right)\right)\right] \nonumber\end{aligned}$$ $$\begin{aligned}
\left. {\partial \mbox{Re} \varphi_1^3 \over \partial t}\right|_0 &=& {1\over 2}\left((\phi_1^+)' \, \left.{\partial n^+
\over \partial t}\right|_0 - (\phi_1^-)' \, \left.{\partial n^- \over \partial t}\right|_0 \right)\times \nonumber \\
&& \left[{1 \over 1+\vartheta_1^2} - \eta_1\delta^2(\Delta^+ \mathcal F)^2\,(\mbox{Re} M_1^- \mbox{Re} Z_1 - \mbox{Im} M_1^- \mbox{Im} Z_1)\right] +\nonumber \\
&& {1\over 2}(\phi_1^+ - \phi_1^-)\left[{-2\vartheta_1 \over (1+\vartheta_1^2)^2} \left.{\partial \vartheta_1 \over
\partial t}\right|_0 \right. - \nonumber \\
&& 2\eta_1\delta^2 \Delta^+ \mathcal F \Delta^+ \left.{\partial \mathcal F \over \partial t}\right|_0 (\mbox{Re} M_1^- \mbox{Re}
Z_1 - \mbox{Im} M_1^- \mbox{Im} Z_1) - \nonumber \\
&& \eta_1\delta^2 (\Delta^+ \mathcal F)^2\left(\left.{\partial \mbox{Re} M_1^- \over \partial t}\right|_0 \mbox{Re} Z_1 - \mbox{Re} M_1^-
\left.{\partial \mbox{Re} Z_1 \over \partial t}\right|_0 - \mbox{Im} Z_1 \left.{\partial \mbox{Im} M_1^- \over \partial t}\right|_0 \right. - \nonumber \\
&& \left.\left. \mbox{Im} M_1^- \left.{\partial \mbox{Im} Z_1 \over \partial t}\right|_0\right) \right] - \nonumber \\
&& {\eta_1\delta^2 \over 2}\left(\left((\phi_1^+)' \left.{\partial n^+ \over \partial t}\right|_0 + (\phi_1^-)'
\left.{\partial n^- \over \partial t}\right|_0 \right) \Delta^+ \mathcal F \Delta^- \mathcal F \mbox{Im} Z_1 + \right.\nonumber \\
&&(\phi_1^++\phi_1^-)\left(\Delta^+\left.{\partial \mathcal F \over \partial t}\right|_0\Delta^- \mathcal F \mbox{Im} Z_1 +
\Delta^-\left.{\partial \mathcal F \over \partial t}\right|_0\Delta^+ \mathcal F \mbox{Im} Z_1 \right. + \nonumber \\
&& \left.\left.\left. {\partial \mbox{Im} Z_1 \over \partial t}\right|_0\Delta^+ \mathcal F \Delta^- \mathcal F \right)\right) \nonumber \\
\left. {\partial \mbox{Im} \varphi_1^3 \over \partial t}\right|_0 &=& {1\over 2}\left((\phi_1^+)' \, \left.{\partial n^+
\over \partial t}\right|_0 + (\phi_1^-)' \, \left.{\partial n^- \over \partial t}\right|_0 \right)\times \nonumber \\
&& \left[{-\vartheta_1 \over 1+\vartheta_1^2} - \eta_1\delta^2(\Delta^+ \mathcal F)^2\,(\mbox{Re} M_1^- \mbox{Im} Z_1 + \mbox{Im} M_1^- \mbox{Re} Z_1)\right] +\nonumber \\
&& {1\over 2}(\phi_1^+ - \phi_1^-)\left[{\vartheta_1^2-1 \over (1+\vartheta_1^2)^2} \left.{\partial \vartheta_1 \over
\partial t}\right|_0 \right. - \nonumber \\
&& 2\eta_1\delta^2 \Delta^+ \mathcal F \Delta^+ \left.{\partial \mathcal F \over \partial t}\right|_0 (\mbox{Re} M_1^- \mbox{Im}
Z_1 + \mbox{Im} M_1^- \mbox{Re} Z_1) - \nonumber \\
&& \eta_1\delta^2 (\Delta^+ \mathcal F)^2\left(\left.{\partial \mbox{Re} M_1^- \over \partial t}\right|_0 \mbox{Im} Z_1 + \mbox{Re} M_1^-
\left.{\partial \mbox{Im} Z_1 \over \partial t}\right|_0 + \mbox{Re} Z_1 \left.{\partial \mbox{Im} M_1^- \over \partial t}\right|_0 \right. + \nonumber \\
&& \left.\left. \mbox{Im} M_1^- \left.{\partial \mbox{Re} Z_1 \over \partial t}\right|_0\right) \right] + \nonumber \\
&& {\eta_1\delta^2 \over 2}\left(\left((\phi_1^+)' \left.{\partial n^+ \over \partial t}\right|_0 + (\phi_1^-)'
\left.{\partial n^- \over \partial t}\right|_0 \right) \Delta^+ \mathcal F \Delta^- \mathcal F \mbox{Re} Z_1 + \right.\nonumber \\
&& (\phi_1^++\phi_1^-)\left(\Delta^+\left.{\partial \mathcal F \over \partial t}\right|_0\Delta^- \mathcal F \mbox{Re} Z_1 +
\Delta^-\left.{\partial \mathcal F \over \partial t}\right|_0\Delta^+ \mathcal F \mbox{Re} Z_1 \right. + \nonumber \\
&& \left.\left.\left.{\partial \mbox{Re} Z_1 \over \partial t}\right|_0\Delta^+ \mathcal F \Delta^- \mathcal F \right)\right) \nonumber\end{aligned}$$ In the above expressions we have used $(\phi_1^\pm)'= \partial \phi_1^\pm / \partial n^\pm $ and $\left. \partial \vartheta_1 /
\partial t \right|_0 = \left<\left. \partial \mathcal F / \partial t\right|_0\right>_1$. We also need to calculate $Z_j$, $M_j^\pm$, $\left.{\partial Z_1 / \partial t}\right|_0$ and $\left.{\partial M_1^\pm /\partial t}\right|_0$: $$\begin{aligned}
\mbox{Re} Z_j &=& {Z_{j1} \over Z_{j1}^2 + Z_{j2}^2} \nonumber \\
\mbox{Im} Z_j &=& {Z_{j2} \over Z_{j1}^2 + Z_{j2}^2} \nonumber\end{aligned}$$ Where the functionals $Z_{j1}(F)$ and $Z_{j2}(F)$ are as follows: $$\begin{aligned}
Z_{j1} &=& 1 - 6\vartheta_j^2 +\vartheta_j^4 + (1-\vartheta_j^2)\left(\eta_1^2 + \delta^2\left((\Delta_j^- \mathcal F)^2 + (\Delta_j^+ \mathcal
F)^2\right)\right)+ \delta^4\left(\Delta_j^- \mathcal F \, \Delta_j^+ \mathcal F\right)^2 \nonumber \\
Z_{j2} &=& -2\vartheta_j\left(2 + \eta_1^2 + \delta^2\left((\Delta_j^- \mathcal F)^2 + (\Delta_j^+ \mathcal F)^2\right) - 2\vartheta_j^2\right)
\nonumber\end{aligned}$$ Therefore, the time derivative $\displaystyle\left.{\partial Z_{1} \over \partial t}\right|_0$ is as follows: $$\begin{aligned}
\left.{\partial \mbox{Re} Z_{1} \over \partial t}\right|_0 &=& { (Z_{12}^2-Z_{11}^2)\displaystyle\left.{\partial Z_{11} \over \partial t}\right|_0 -
2Z_{11}Z_{12}\left.{\partial Z_{12} \over \partial t}\right|_0 \over (Z_{11}^2 + Z_{12}^2)^2} \nonumber \\
\left.{\partial \mbox{Im} Z_{1} \over \partial t}\right|_0 &=& { (Z_{11}^2-Z_{12}^2)\displaystyle\left.{\partial Z_{12} \over \partial t}\right|_0 -
2Z_{11}Z_{12}\left.{\partial Z_{11} \over \partial t}\right|_0 \over (Z_{11}^2 + Z_{12}^2)^2} \nonumber\end{aligned}$$ Where the time derivatives $\displaystyle\left.{\partial Z_{11} \over \partial t}\right|_0$ and $\displaystyle\left.{\partial Z_{12} \over \partial
t}\right|_0$ are as follows: $$\begin{aligned}
\left.{\partial Z_{11} \over \partial t}\right|_0 &=& 4( \vartheta_1^3 - 3 \vartheta_1)\left.{\partial \vartheta_{1} \over \partial t}\right|_0 +
2(1- \vartheta_1^2)\left(\delta^2 \left(\Delta^+ \mathcal F \Delta^+ \left.{\partial \mathcal F \over \partial t}\right|_0 + \Delta^- \mathcal F
\Delta^- \left.{\partial \mathcal F \over \partial t}\right|_0\right) - \right. \nonumber \\
&&\left. \vartheta_1 \left.{\partial \vartheta_1 \over \partial t}\right|_0(\eta_1^2 + \delta^2 ((\Delta^- \mathcal F)^2 +
(\Delta^+ \mathcal F)^2)\right) + \nonumber \\
&& 2\delta^4 \Delta^+ \mathcal F \Delta^- \mathcal F \left(\Delta^+ \mathcal F\, \Delta^- \left.{\partial \mathcal F \over \partial t}\right|_0 +
\Delta^- \mathcal F\, \Delta^+ \left.{\partial \mathcal F \over \partial t}\right|_0\right) \nonumber \\
\left.{\partial Z_{12} \over \partial t}\right|_0 &=& -2\left[\left.{\partial \vartheta_{1} \over \partial t}\right|_0
(2 + \eta_1^2 + \delta^2\left((\Delta^- \mathcal F)^2 + (\Delta^+ \mathcal F)^2\right) - 2\vartheta_1^2) \, + \right. \nonumber \\
&& \left. 2\vartheta_1 \left(\delta^2\left(\Delta^+ \mathcal F\, \Delta^- \left.{\partial \mathcal F \over \partial t}\right|_0 +
\Delta^- \mathcal F\, \Delta^+ \left.{\partial \mathcal F \over \partial t}\right|_0\right) -
2\vartheta_1 \left.{\partial \vartheta_1 \over \partial t}\right|_0\right)\right] \nonumber\end{aligned}$$ The functionals $M_j^\pm(F)$ are as follows: $$\begin{aligned}
\mbox{Re} M_j^\pm &=& {1 \over \eta_1}\left[1 + {\delta^2 (\Delta_j^\pm \mathcal F)^2 \over 1+ \vartheta_j^2} \right] \nonumber \\
\mbox{Im} M_j^\pm &=& {1 \over \eta_1}\left[\vartheta_j - {\delta^2 \vartheta_j (\Delta_j^\pm \mathcal F)^2 \over 1+ \vartheta_j^2} \right] \nonumber\end{aligned}$$ Finally, we need to calculate the time derivative $\displaystyle\left.{\partial M_1^\pm \over \partial t}\right|_0$: $$\begin{aligned}
\left.{\partial \mbox{Re} M_1^\pm \over \partial t}\right|_0 &=& {2\delta^2 \Delta^\pm \mathcal F\over \eta_1 (1+\vartheta_1^2)}\left( \Delta^\pm
\left.{\partial \mathcal F \over \partial t}\right|_0 - {\vartheta_1 \Delta^\pm \mathcal F \over 1+\vartheta_1^2}\left.{\partial \vartheta_1 \over
\partial t}\right|_0 \right) \nonumber \\
\left.{\partial \mbox{Im} M_1^\pm \over \partial t}\right|_0 &=& {1 \over \eta_1}\left[ \left.{\partial \vartheta_1 \over
\partial t}\right|_0 - {\delta^2 \Delta^\pm \mathcal F \over (1+\vartheta_1^2)^2} \left( (1-\vartheta_1^2) \Delta^\pm \mathcal F \left.{\partial \vartheta_1 \over
\partial t}\right|_0 + 2\vartheta_1 (1+\vartheta_1^2) \Delta^\pm \left.{\partial \mathcal F \over \partial t}\right|_0 \right) \right] \nonumber\end{aligned}$$
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abstract: 'Black holes are spun up by accreting matter and possibly spun-down by magnetic fields. In our work we consider the effect on black hole rotation of the two electromagnetic processes, Blandford-Znajek and Direct Magnetic Link, that differ in their magnetic field configuration. The efficiency of these processes varies with mass accretion rate and accretion regime and generally result in an equilibrium spin parameter in the range from $0.35$ to $\sim 0.98$. Magnetic field loses its energy while being accreted that may lead to an increase in equilibrium Kerr parameter for the case of advection-dominated disc. We find magnetic field decay decay can decrease electromagnetic term significantly thus increasing the Kerr parameter. We have performed Monte-Carlo simulations for a supermassive black hole population. Our simulations show broad distributions in Kerr parameter ($0.1\lesssim a \lesssim0.98$) with a peak at $a\sim 0.6$. To explain the high observational Kerr parameter values of $a \gtrsim 0.9$, episodes of supercritical accretion are required. This implication does not however take into account black hole mergers (that play an important role for supermassive black hole evolution).'
author:
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Anna Chashkina [^1] and Pavel Abolmasov\
Sternberg Astronomical Institute, Moscow State University, Moscow, Russia 119992\
bibliography:
- 'mybib.bib'
date: 'Accepted —. Received —; in original form — '
title: Black Hole Spin Evolution Affected by Magnetic Field Decay
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\[firstpage\]
Physical data and processes: Black hole physics – Physical data and processes: Magnetic fields
Introduction
============
Stellar mass black holes (BH) are the final stage of the evolution of the most massive stars. Supermassive black holes (SMBH) can be formed by accretion onto stellar mass BH or by mergers of stellar mass and intermediate mass BHs, but this question now is open (see @shankar09 and @haiman13 for review). Any BH is characterized by its mass $M$ and its total angular momentum $J$. These parameters may evolve under the influence of different processes, primarily due to accretion of matter. There are two types of objects where the black hole mass grows significantly during their evolution: black holes in some X-ray binaries such as X-ray novae and SMBH in galactic nuclei.
Black holes are believed to be spun up by gas accretion. [ For the standard thin disc case ]{}the net angular momentum of the material at the innermost stable circular Keplerian orbit (having radius $R_{in}$) always exceeds this of the black hole itself, the latter limited by the value $GM/c$. Below we will use relativistic dimensionless quantities $j^\dagger =L^\dagger(R_{in}) / (GM/c)$ (specific angular momentum in $GM/c$ units) and $E^\dagger$ (dimensionless specific energy at infinity for a particle at $R_{in}$) following the designations used by @PT74. We also describe BH rotation in terms of its Kerr parameter $a = J c /GM^2 < 1$.
Inside $R_{in}$ accreting matter conserves its angular momentum, hence the black hole is spun up according to a simple law $da/d\ln M = \left(\displaystyle\frac{j^\dagger}{E^\dagger}-2a\right)
\times \mu$, where $\mu\simeq 1$ is dimensionless specific enthalpy [@BAN97] and accounts for internal energy contribution to the total (relativistic) mass accretion rate. In advection-dominated regime and supercritical accretion these assumptions are violated due to non-Keplerian rotation and transonic structure of the disc, where angular momentum can be transferred by stresses inside the last stable orbit (see @Abramowicz2010 and sec. 3).
Black hole spin evolution has been considered by more than forty years, since @bardeen70 obtained an analytical solution for black hole spin-up by accretion. Since then, several processes were proposed that may stop this spin-up at higher or lower spin values. First of all, selective capture of accretion disc radiation in Kerr metric is able to provide a strong counteracting spin-down torque if the BH rotation parameter is close to maximal. Due to this reason, it is impossible to spin up black holes to very high $a$ [@thorne74]. Dimensionless spin parameter stalls at a value $\sim 0.998$, when the spin-up is compensated by the angular momentum of the captured photons emitted by the inner parts of the flow.
The work by @Kato09 proposes that Blandford-Znajek process [@BZ77] may be responsible for black hole spin settling somewhere around $0.4-0.5$. The power of this process is determined by the Poynting flux generated by poloidal magnetic field and toroidal electric field induced by black hole rotation. Radio-bright active galactic nuclei (AGNs) and microquasars are supposed to power their jets via the Blandford-Znajek mechanism. At the end of XX century alternative mechanism known as direct magnetic link was proposed (see @uzdensky05 and references therein). Below we use the name DML (direct magnetic link) for any process of magnetic-field mediated angular momentum exchange between the BH and accretion disc.
Magnetic field in this case connects the black hole with the accretion disc. Depending on relation between the angular velocities of the black hole and the accretion disc, black hole can spin-up or spin-down. This mechanism differs from Blandford-Znajek process in its magnetic field geometry (see the next section and @uzdensky05, @krolik1999). If magnetic flux through the disc is not zero, flux can be accumulated inside the last stable orbit that leads to Blandford-Znajek geometry. This is confirmed by many simulations (see for example @Tchekhovskoy12). In our work we will consider black hole spin evolution taking into account both electromagnetic mechanisms: Blandford-Znajek process and DML. We will describe these processes in detail in the next section.
It should be noted that magnetic fields dissipate at the vicinity of the event horizon. Below, we use membrane paradigm [@membrane] to estimate “Joule losses” at the stretched horizon and the power that magnetic fields extract from black hole rotation. One actually needs accretion to supply the magnetic field. In this paper, we estimate the effect of magnetic field decay for constant mass accretion rate. Note that magnetic field can not dissipate in BZ case because magnetic flux is conserved and the field can be supported by currents in the disc.
In the following section, we discuss the basic properties of the basic electromagnetic field configurations, in section \[sec:rev\] we use membrane paradigm to estimate the angular momentum and energy extracted from a rotating black hole by magnetic fields for these configurations. In section \[sec:decay\], we discuss the influence of magnetic field decay. We present our results in section \[sec:results\] and discuss them in section \[sec:discussion\]. [ In section \[sec:results\] we also present some applications supermassive and stellar mass black holes.]{}
Power and Spin-Down Rate
========================
In our work we use Membrane paradigm [@membrane] that helps describe the black hole horizon avoiding complicated general relativity calculations.
Recent numerical results such a jet simulations support the capability of this approach to reproduce the main features of black hole magnetospheres and accretion flows
[@Narayan2013].\
In the Membrane approach framework [@membrane], the event horizon is considered surrounded by the so-called stretched horizon – viscous conducting sphere that has finite entropy and does not conduct heat. This sphere rotates with angular velocity $\Omega_H= \displaystyle\frac{a}{2}\displaystyle \frac{c}{R_H}$, here $R_H =
\left( 1+ \sqrt{1-a^2}\right) GM/c^2=r_H GM/c^2$ is horizon radius. Using this paradigm, properties of a black hole can be considered without complicated general relativistic calculations. This approach is sufficient for our needs.
There are two main electromagnetic processes that affect the rotational evolution of a black hole: Blandford-Znajek process [@BZ77] and direct magnetic link [@uzdensky05]. The main difference between these processes is in magnetic field configuration. Blandford-Znajek process works when field lines connect the black hole with a distant region such as jet [^2] (Fig. 1, right). In DML case magnetic lines connect the stretched horizon with the accretion disc (Fig. 1, left).
\[processes\] {width="0.5\columnwidth"} {width="0.5\columnwidth"}
One can describe a black hole magnetosphere as a steady and axisymmetric system of nested electric circuits consisting of a battery and a load. The battery here is the warped space region near the black hole, and the jet or accretion disc plays the role of the load.
Poloidal magnetic field:
$$\vec{B_p}=\frac{({\nabla \Psi}) \times e_{\hat{\phi}}}{2 \pi \varpi}$$
here $\Psi$ is magnetic flux, $\varpi$ is the radial cylindric coordinate. Field lines in such configuration rotate as a rigid body with Ferraro angular velocity $\Omega_F$ [@ferraro37]. In a force-free magnetosphere magnetic field is degenerate, $\vec{E}\cdot \vec{B}=0$, and one can write electric field as follows: $$\vec{E}=-\frac{(\Omega_F-\omega)}{2\pi\alpha_{l}}\nabla{\Psi}$$ where $\omega$ is Lense-Thirring precession frequency and $\alpha_{l}$ is the lapse function taking in account dilation of time.
The electromotive force produced by the gravitational field:\
$$\Delta V=\oint \alpha_l \vec{E}d\vec{l}=\frac{1}{2\pi}\Omega_H \Delta \Psi$$
This electromotive force is balanced by the potential difference along the horizon surface and potential difference in the load region: $$\label{eqq:all}
\Delta V=\Delta V_H+\Delta V_L$$
The potential difference along the horizon part of the current:
$$\label{eqq:horizon}
\Delta V_{H}=\oint \alpha_{l} \vec{E}d\vec{l}=\frac{1}{2\pi}(\Omega_H-\Omega_F )\Delta \Psi$$
One can find a potential difference within the load using eq. (\[eqq:all\]) and eq. (\[eqq:horizon\]) as follows: $$\Delta V_{L}=\frac{1}{2\pi}\Omega_F \Delta \Psi$$
Resistance of a horizon belt of latitudinal size $\Delta l$: $$\Delta R_{H}=R_{H}\displaystyle\frac{\Delta l}{2\pi \varpi}=R_H\displaystyle\frac{\Delta \Psi}{4\pi^2\varpi^2 B_n}$$ here $R_H=4\pi/c=377$ Ohm is specific horizon surface resistivity, $B_n$ is normal component of magnetic field at the horizon, $\Delta l$ is elementary distance along the horizon between two magnetic layers. Since $I=$ const inside a given flux tube, $$\displaystyle\frac{\Omega_F}{\Omega_H-\Omega_F}=\displaystyle\frac{\Delta V_L}{\Delta V_H}=\displaystyle\frac{\Delta R_L}{\Delta R_H}$$
Taking into account that $\Delta V_L=I\Delta R_L$ and $\Delta V_H=I\Delta R_H$, one can write: $$I=\displaystyle\frac{\Delta V}{\Delta R_H+\Delta R_L}=\displaystyle\frac{1}{2}(\Omega_H-\Omega_F)\varpi^2 B_n$$
Total Joule losses for an elementary circuit are $\Delta P=I\Delta V$.
Dissipative power at the horizon:
$$\Delta P_H=\displaystyle\frac{(\Omega_H-\Omega_F)^2}{4\pi}\varpi^2 B_n\Delta \Psi$$
Power transferred by Poynting vector towards the load equals:
$$\Delta P_L=\displaystyle\frac{\Omega_F(\Omega_H-\Omega_F)}{4\pi}\varpi^2 B_n\Delta \Psi$$
The total energy losses towards the load may be found by integrating $\Delta P_L$ over the magnetosphere:
$$\label{E:power:gen}
\frac{dE}{dt} = \int_{\Psi}\delta P_{L}= \displaystyle\frac{1}{8c} \Omega_F (\Omega_F - \Omega_H) r_H^4
\langle B_p^2 \rangle$$
Angular momentum losses may be calculated by integrating the $r\varphi$- component of the Maxwell stress tensor:
$$\frac{dJ}{dt}=\int_{\Psi}\displaystyle\frac{1}{4\pi}B_{\phi}B_{p}\cdot 2\pi\varpi^{2}\delta\Psi=\displaystyle \frac{1}{8c} (\Omega_F - \Omega_H) r_H^4
\langle B_p^2 \rangle$$
Since energy and angular momentum losses differ by the factor of $\Omega_{F}$, we can interpret them as extraction of rotational energy from the black hole.
Depending on the difference between $\Omega_F$ and $\Omega_H$, energy and angular momentum may be either absorbed by the black hole or extracted from it by magnetic fields. The main difference between the two configurations, Blandford-Znajek and direct magnetic link, is in the value of Ferraro frequency. For the Blandford-Znajek case, relation between $\Omega_H$ and $\Omega_F$ is determined by the unknown resistance of the distant load. Below we assume $\Omega_F=\Omega_H/2$ in the case of Blandford-Znajek configuration, that corresponds to maximal energy extraction rate from black hole rotation.
We assume that magnetic field is in equipartition with gas+radiation as $\displaystyle\frac{B^2_p}{8\pi}=\displaystyle\frac{1}{\beta}p$, where $p$ is total thermal pressure, and $\beta$ is a dimensionless coefficient about unity.\
Finally, we write the two dynamical equations (12,13) for Blandford-Znajek process as follows:
$$\label{E:power}
\frac{dE}{dt} = \displaystyle\frac{\pi}{\beta}\displaystyle \frac{\Omega_H^2 R_H^4 p}{4c}$$
$$\label{E:spin}
\frac{dJ}{dt} =\displaystyle\frac{\pi}{\beta} \displaystyle\frac{\Omega_H R_H^4 p}{2c}$$
In the case of direct magnetic link the effective black hole conductivity is much smaller than the conductivity of the ionized plasma of the disc. Therefore, magnetic field may be considered frozen into the disc at some effective radius $r_{eff}$ close to the last stable orbit radius $r_{in}$. The pressure inside a standard disc (in gas pressure dominated zone, [@SS73] is close to the maximal value at the distance of $1.3r_{in}$, hence we assume $r_{eff}=1.3r_{in}$ in our calculations. Ferraro frequency equals Kepler frequency at the effective radius, $\Omega_F=\Omega_K(r_{eff})$.
For higher rotation parameters, Keplerian frequency at the inner face of the disc is lower than the frequency $\Omega_H$, and black hole rotation energy is transferred to the disc. The innermost stable orbit corotates with the stretched horizon at $a=a_{cr} \simeq
0.3594$ [@uzdensky05], for $r_{eff}=1.3r_{in}$, the corresponding critical value is $a_{cr} \sim 0.218$. It can be checked that for $a>a_{cr}$ magnetic field slows down or stops black hole spin-up, and sometimes is proposed to stop accretion [@agolkrolik] by creating a non-zero torque at the inner boundary of the disc.
Magnetic field lines in the case of direct magnetic link connect disc to the stretched horizon, therefore the magnetic field strength should be modified by a factor determined by geometry. Magnetic flux conservation implies:
$$R_H h_H B_p^{H} = 2 h R_{eff} B_p^{disc}$$
Here, $h_H$ is the width of the equatorial band at the horizon surface that is connected to the disc, $h$ is disc half-thickness at the $r_{eff}$. Left- and right-hand sides correspond to the flux at the horizon and at the surface of the disc at $r_{eff}$. Finally, the energy and angular momentum transmitted by direct magnetic link may be expressed in a form similar to eqs. (\[E:power\],\[E:spin\]):
$$\label{E:power:dml}
\frac{dE}{dt} = \displaystyle\frac{\pi}{\beta c} \Omega_K(r_{eff}) (\Omega_K(r_{eff}) - \Omega_H) R_H^2 R^2_{eff}p \left(\displaystyle\frac{h}{h_{H}}\right)^{2}$$
$$\label{E:spin:dml}
\frac{dJ}{dt} = \displaystyle\frac{\pi}{\beta c} (\Omega_K(r_{eff}) - \Omega_H) R_H^2 R^2_{eff}p\left(\displaystyle\frac{h}{h_{H}}\right)^{2}$$
In our calculations we assume $h_H=R_H$.
[ The question may arise whether these two magnetic field configurations are the only ones allowed for accretion disks. The geometry of seed magnetic fields may be different but as long as we consider axisymmetric configurations such as accretion disks the possible zoo of magnetic field configurations is limited. One can expand magnetic field in multipoles. In general, every magnetic field configuration will consist of some uniform part and higher multipoles that should be confined inside the falling matter. Note that magnetic loop curvature leads to Lorentz force that makes some configurations unstable (such as the conventional DML case shown in figure 1, right panel). Our consideration of DML does not depend on the characteristic magnetic field spatial scales in the disk as long as radial motions are much faster inside the last stable orbit.]{}
Rotational evolution {#sec:rev}
====================
In this section we estimate the electromagnetic terms in rotational evolution for different accretion regimes. It is convenient to normalize all characteristic timescales to Eddington time $t_{Edd} = c\varkappa/4\pi G \sim
0.4{\ensuremath{\rm Gyr}}$ that is the characteristic time for Eddington-limited accretion mass gain. In this section we use time normalized to Eddington time $\tau=t/t_{Edd}$, where $\varkappa$ is Thomson (electron scattering) opacity. A general expression for Kerr parameter evolution is:
$$\displaystyle\frac{da}{d\tau}=\displaystyle\frac{c}{G}\left(\displaystyle\frac{1}{M^2}\displaystyle\frac{dJ}{d\tau}-\displaystyle\frac{2aG}{Mc^3}\displaystyle\frac{dE}{d\tau} \right)$$
Accretion disk models
---------------------
In this subsection we give a small introduction to different accretion models and try to describe some of their limitations.
Standard disc model by @SS73 now is the most widely used for intermediate accretion rates $\dot M \sim 0.01 - 1 \dot M_{Edd}$. In this model, there are at least three basic assumptions. One is “alpha-prescription”, i.e. the assumption that viscous stress is proportional to thermal (gas+radiation) pressure with some coefficient $\alpha$ constant throughout the disk, $W_{r\phi}=\alpha p$. The second assumption is that accretion disk is geometrically thin that implies that the thermal velocity and the radial velocity are much smaller than Keplerian velocity, and all the energy dissipated by viscous processes is radiated locally. The thinness of the standard disc makes it difficult to simulate numerically. We use relativistic version of the model [@NT73] in our calculations. Third, less frequently mentioned, assumption is the boundary condition at the inner edge of the disk where the viscous stress is assumed to be zero. Near the last stable orbit, some of the standard disk assumptions are violated. Firstly, near the last stable orbit the radial velocity can become comparable to Keplerian. Another thing is viscous stress tensor may deviate from zero near the last stable orbit. In a more comprehensive analysis, boundary condition is non-trivial that may be viewed as a torque applied to the inner edge of the disc. It contributes to angular momentum exchange between accretion disk and black hole that may be effectively incorporated in the DML term.
For very large and very small accretion rates one should take into account advection process that makes accretion disk geometrically thick.
For low accretion rates $\dot M\lesssim 0.01 \dot M_{Edd}$ we use ADAF model by @NY95 that describes a geometrically thick disk with radial energy advection, that has some non-negligible radial pressure gradient and radial velocity. Narayan& Yi’s model is essentially self-similar hence all the velocities are locally proportional to the Keplerian velocity. The self-similar nature is the main disadvantage of this ADAF model since boundary conditions cannot be satisfied. The other problem is its instability to thermal perturbations [@Blandford02]. This model can explain some observational results [@Mahadevan98], but other observations and simulations indicate the insufficiency of ADAF approach [@Oda2012]. The main advantage of this model is its simplicity and the existence of analytical expressions for all the quantities such as gas pressure and the thickness of the disk.
The other model applied for low accretion rates is a modification of ADAF, Advection-dominated inflow-outflow solution – ADIOS [@Begelman99]. Apart from advection, this model also takes into account isotropic wind from the disk. Therefore, only a small fraction of the accreting matter falls onto the BH. Note however that for hydrodynamical disks, radiative inefficiency does not directly imply formation of massive outflows, as it was shown by @Abramowicz00 using analytical arguments.
The only model that consistently considers the transonic nature of the flow near the last stable orbit is the slim disk by @Abramowicz88. Slim disk may be thought of as a standard disk where the assumptions of negligibility of pressure gradient and $({\mathbf v}\nabla){\mathbf v}$ terms in Euler equations are relaxed. This model can be applied to any accretion disk within a wide range of accretion rates. The main reason why we do not use this model is lack of analytical solutions for the equation set defining the slim disk.
Supercritical accretion was considered with help of the model by @lipunova99 and its modified version by @poutanen07 . This model is a generalization of the Shakura-Synaev’s model for super-Eddington accretion rates. Unlike the slim-disk approach, it takes into account the loss of accreting material in the disk wind but includes advection that is important for large accretion rates. Numerical simulations confirm existence of outflows (see @Ohsuga05, @Fukue11, @Yang2014). However, some recent numerical results [@Sadowski14] argue for a more complex structure of the flow where winds exist at distances larger than the classical non-relativistic spherization radius but mass accretion rate is practically constant with radius in the inner parts of the disk. This may imply that the real mass accretion rates are several times larger in the super-critical case. This allows black holes to grow more rapidly than expected in the naïve non-relativistic approach but does not alter significantly the expected spin-up curve that we find indistinguishable from the dust-like solution (see below section 5.2.3).
Surely, there is a lot of limitations and simplifications in the models we use. But we hope that does not contribute significantly to our results. In discussion we consider some possible consequences of the over-simplifications introduced by the models used in our work.
Standard disc
-------------
Falling matter with high angular momentum forms an accretion disc around a black hole. As we have already mentioned above we used the relativistic version of Shakura-Sunyaev’s model introduced by @NT73. This accretion regime apparently occurs in outbursts of X-ray novae and in the bright AGNs (Seyfert galaxies and QSOs). In this model, the accretion disc is considered geometrically thin and optically thick and is divided into three zones A, B and C depending on the terms dominating pressure and opacity. Since we study the hottest inner parts of the accretion disc, only zones A and B were considered, where the main opacity contribution is electron scattering. Gas and radiation pressure dominate in zones B and A, respectively.
$$\label{pressure1}
p(r) =
\left\{ \begin{array}{lc}
1.93 \cdot 10^{16} \alpha^{-1} m^{-1} r^{-3/2} {\cal A}^{-2} {\cal B}^{2} {\cal S} ~ {\rm erg \, cm^{-3}} & \mbox{zone A}\\
2.65 \cdot 10^{18} \alpha^{-9/10} m^{-17/10} {\dot m}^{4/5} r^{-51/20} {\cal A}^{-1} {\cal B}^{1/5} {\cal D}^{-2/5} {\cal S}^{1/2} {\cal F}^{4/5} ~ {\rm erg \, cm^{-3}} & \mbox{zone B}\\
\end{array} \right.$$
Here ${\cal A}, {\cal B}, {\cal C}, {\cal D}, {\cal F}, {\cal S}$ are relativistic correction factors (see @penna).\
The boundary between these zones is determined by equality between radiation and gas pressures and lies at $r=r_{ab}$:
$$r_{ab}=96 \alpha^{2/21} m^{2/21} {\dot m}^{16/21} {\cal A}^{20/21} {\cal B}^{-12/7} {\cal D}^{-8/21} {\cal S}^{-10/21} {\cal F}^{16/21}$$
For a low accretion rates, effective radius is situated in zone B, and for accretion rates higher than $\sim 0.2 \dot M_{Edd}$ (for $M=10^7 M\odot$) in zone A.
In our calculations we used viscosity parameter $\alpha$ equal to 0.1 and dimensionless coefficient $\beta$ equal to 1. Variations of these parameters will be discussed in sec. 5.2. Accretion rate onto a black hole and its mass were normalized to the critical accretion rate (Eddington limit $L_{Edd}/c^2$) and to the solar mass, respectively:
$$\dot m=\displaystyle\frac{\dot M}{\dot M_{edd}}=\displaystyle\frac{\dot M c \kappa}{4\pi GM}$$
$$m=\displaystyle\frac{M}{M \odot}$$
One can re-write the main evolutionary equations (14-17) for each of the two zones A and B in the case of standard relativistic sub-Eddington disc using eq. (18) for the pressure and Keplerian frequency as viewed from infinity $\Omega_{K}=\displaystyle\frac{c^3}{GM}(r_{eff}^{3/2}+a)^{-1}$ at effective radius $r_{eff}=1.3r_{in}$ as follows:
### Zone A
$$\label{E:vr45}
\left(\displaystyle\frac{dE}{d\tau}\right)_{em} =
\left\{ \begin{array}{lc}
0.017 M_{\odot} c^2 \times \alpha^{-1} \beta^{-1} a^2 r^2_{H} m {r_{eff}}^{-3/2} {\cal A}^{-2} {\cal B}^{2} {\cal S} & \mbox{ BZ}\\
0.322 M_{\odot} c^2 \times \alpha^{-1} \beta^{-1} m {\dot m}^{2} {r_{eff}}^{1/2} (r_{eff}^{3/2}+a)^{-1} \displaystyle\left((r_{eff}^{3/2}+a)^{-1}-\frac{a}{2r_{H}}\right) {\cal A}^{2} {\cal B}^{-4} {\cal C}^{2} {\cal D}^{-2} {\cal S}^{-1} {\Phi}^{2} & \mbox{ DML}\\
\end{array} \right.$$
$$\label{E:vr4}
\left(\displaystyle\frac{dJ}{d\tau}\right)_{em} =
\left\{ \begin{array}{lc}
0.07 \displaystyle\frac{GM_{\odot}^{2}}{c} \times \alpha^{-1} \beta^{-1} a r^3_{H} m^{2} {r_{eff}}^{-3/2} {\cal A}^{-2} {\cal B}^{2} {\cal S} & \mbox{ BZ}\\
0.322 \displaystyle\frac{GM_{\odot}^{2}}{c} \times \alpha^{-1} \beta^{-1} m^{2} {\dot m}^{2} {r_{eff}}^{1/2} \displaystyle\left((r_{eff}^{3/2}+a)^{-1}-\frac{a}{2r_{H}}\right) {\cal A}^{2} {\cal B}^{-4} {\cal C}^{2} {\cal D}^{-2} {\cal S}^{-1} {\Phi}^{2} & \mbox{ DML}\\
\end{array} \right.$$
$$\label{E:vr1}
\left(\displaystyle\frac{da}{d\tau}\right)_{em} =
\left\{ \begin{array}{lc}
0.07 \alpha^{-1} \beta^{-1} a r^3_{H} {r_{eff}}^{-3/2} \displaystyle\left(1-\frac{a^{2}}{2r_{H}}\right) {\cal A}^{-2} {\cal B}^{2} {\cal S} & \mbox{ BZ}\\
0.322 \alpha^{-1} \beta^{-1} {\dot m}^{2} {r_{eff}}^{1/2} \displaystyle\left((r_{eff}^{3/2}+a)^{-1}-\frac{a}{2r_{H}}\right) \displaystyle\left(1-2 a (r_{eff}^{3/2}+a)\right) {\cal A}^{2} {\cal B}^{-4} {\cal C}^{2} {\cal D}^{-2} {\cal S}^{-1} {\Phi}^{2} & \mbox{ DML}\\
\end{array} \right.$$
### Zone B
$$\label{E:vr}
\left(\displaystyle\frac{dE}{d\tau}\right)_{em} =
\left\{ \begin{array}{lc}
2.1 M_{\odot} c^2 \times \alpha^{-9/10} {\beta}^{-1} a^2 r^2_{H} m^{11/10} {\dot m}^{4/5} {r_{eff}}^{-51/20} {\cal A}^{-1} {\cal B}^{1/5} {\cal D}^{-2/5} {\cal S}^{1/2} {\Phi}^{4/5} & \mbox{ BZ}\\
0.0036 M_\odot c^2 \times \alpha^{-11/10} \beta^{-1} m^{9/10} {\dot m}^{6/5} r_{eff}^{31/20} (r_{eff}^{3/2}+a)\displaystyle\left((r_{eff}^{3/2}+a)^{-1}-\frac{a}{2r_{H}}\right) {\cal A} {\cal B}^{-11/5} {\cal C} {\cal D}^{-8/5} {\cal S}^{-1/2} {\Phi}^{6/5} & \mbox{ DML}\\
\end{array} \right.$$
$$\label{E:vr}
\left(\displaystyle\frac{dJ}{d\tau}\right)_{em} =
\left\{ \begin{array}{lc}
8.39 \displaystyle\frac{GM_{\odot}^{2}}{c} \times \alpha^{-9/10} {\beta}^{-1} a r^3_{H} m^{21/10} {\dot m}^{4/5} {r_{eff}}^{-51/20} {\cal A}^{-1} {\cal B}^{1/5} {\cal D}^{-2/5} {\cal S}^{1/2} {\Phi}^{4/5} & \mbox{ BZ}\\
0.0036 \displaystyle\frac{GM_{\odot}^{2}}{c} \times \alpha^{-11/10} \beta^{-1} m^{19/10} {\dot m}^{6/5} {r_{eff}}^{31/20} \displaystyle\left((r_{eff}^{3/2}+a)^{-1}-\frac{a}{2r_{H}}\right) {\cal A} {\cal B}^{-11/5} {\cal C} {\cal D}^{-8/5} {\cal S}^{-1/2} {\Phi}^{6/5} & \mbox{ DML}\\
\end{array} \right.$$
$$\label{E:vr}
\left(\displaystyle\frac{da}{d\tau}\right)_{em} =
\left\{ \begin{array}{lc}
8.39 \alpha^{-9/10} {\beta}^{-1} a r^3_{H} m^{1/10} {\dot m}^{4/5} {r_{eff}}^{-51/20} \left(1-\displaystyle\frac{a^2}{2r_{H}}\right) {\cal A}^{-1} {\cal B}^{1/5} {\cal D}^{-2/5} {\cal S}^{1/2} {\Phi}^{4/5} & \mbox{ BZ}\\
0.0036 \times \alpha^{-11/10} \beta^{-1} m^{-1/10} {\dot m}^{6/5} r_{eff}^{31/20} \displaystyle\left((r_{eff}^{3/2}+a)^{-1}-\frac{a}{2r_{H}}\right) \displaystyle\left(1-2 a (r_{eff}^{3/2}+a)^{-1}\right) {\cal A} {\cal B}^{-11/5} {\cal C} {\cal D}^{-8/5} {\cal S}^{-1/2} {\Phi}^{6/5} & \mbox{ DML}\\
\end{array} \right.$$
Here BZ and DML denote Blandford-Znajek scenario and direct magnetic link, respectively.
Advective disc {#sec:addisc}
--------------
For low accretion rates (${\ensuremath{\dot{m}}}\lesssim 10^{-2}$), standard accretion disc is predicted to be unstable to evaporation [@meyer]. In this case, accretion flows are optically thin and geometrically thick and lose inefficiently the energy released by viscous heating. In combination with efficient angular momentum transfer, this leads to a non-Keplerian flow where the heat is advected towards the black hole rather than lost with radiation. This type of accretion flows is known as Advection-Dominated Accretion Flows (ADAF).
Massive black holes in non-active (quiescent) galactic nuclei (like Sgr A\*) in general show radiatively inefficient accretion having very little in common with the standard thin disc accretion picture.
There are two main effects that characterize this accretion regime in the disc: radial velocities are non-Keplerian and the disc is geometrically thick. Since there is no comprehensive relativistic analytical model for ADAF flows similar to the standard disc model, we use the non-relativistic self-similar model proposed by @NY95. In this model, all velocity components at some radius are proportional to the Keplerian velocity.
In particular, the radial velocity component is $v_r(R)= -\displaystyle\frac{(5+2\epsilon)}{3\alpha^2}\left(\left[1+\displaystyle\frac{18\alpha^2}{(5+2\epsilon)^2}\right]^{1/2}-1\right)v_{K}(R)= -c_1\alpha v_K(R)$, where $0<c_1<1$. Angular rotational frequency $\Omega(R)=\left(\displaystyle\frac{2\epsilon(5+2\epsilon)}{9\alpha^2}\left(\left[1+\displaystyle\frac{18\alpha^2}{(5+2\epsilon)^2}\right]^{1/2}-1\right)\right)^{1/2}\displaystyle\frac{v_{K}}{R}=c_2\displaystyle\frac{v_K}{R}$. Speed of sound squared $c^2_s(R)=\displaystyle\frac{2(5+2\epsilon)}{9\alpha^2}\left(\left[1+\displaystyle\frac{18\alpha^2}{(5+2\epsilon)^2}\right]^{1/2}-1\right)v_{K}(R)= c_3 v^2_K(R)$.
In our simulations we use $\alpha=0.1$ and $\epsilon=1$, implying $c_1=0.43$, $c_2=0.53$ and $c_3=0.285$.
One more correction should be made connected to the high temperature of the accreted gas. One can estimate dimensionless specific enthalpy as follows:
$$\mu=1+\displaystyle\frac{U+\Pi}{\Sigma c^2}=1+\frac{5}{4}\frac{\Pi}{\Sigma c^2}=1+\frac{5}{4}\frac{p}{\rho c^2}=1+\displaystyle\frac{5}{4}\displaystyle\frac{c_3}{r}$$
Here, $\Sigma$ and $\Pi$ are vertically integrated mass density $\rho$ and pressure $p$. Black hole spin evolution in absence of electromagnetic terms for ADAF regime is described by the following equation:
$$\displaystyle\frac{da}{dt}=\mu \displaystyle\frac{\dot M}{M}\left(c_2 j^{\dagger}_K(r_{in})-2a E^{\dagger}(r_{in}) \right)$$
The main evolutionary equations for the case of conventional advective flow are the following: $$\label{E:vmr}
\left(\displaystyle\frac{dE}{d\tau}\right)_{em} =
\left\{ \begin{array}{lc}
\displaystyle\frac{\sqrt{5}}{320} M_{\odot} c^2 \times \alpha^{-1} {\beta}^{-1} c^{-1}_{1} c^{1/2}_{3} a^2 r^2_{H} r^{-5/2}_{eff} m \dot m & \mbox{ BZ}\\
\displaystyle\frac{\sqrt{5}}{8} M_{\odot} c^2 \times \alpha^{-1} {\beta}^{-1} c^{-1}_{1} c^{3/2}_{3}\Omega_{*} m r^{3/2}_{eff}
\displaystyle\left(\Omega_{*}-\frac{a}{2r_{H}}\right)\dot m & \mbox{ DML}\\
\end{array} \right.$$
$$\label{E:vr3}
\left(\displaystyle\frac{dJ}{d\tau}\right)_{em} =
\left\{ \begin{array}{lc}
\displaystyle\frac{\sqrt{5}}{80} \displaystyle\frac{GM_{\odot}^{2}}{c} \times \alpha^{-1} {\beta}^{-1} c^{-1}_{1} c^{1/2}_{3} a r^3_{H} r^{-5/2}_{eff} m^2 \dot m & \mbox{ BZ}\\
\displaystyle\frac{\sqrt{5}}{8} \displaystyle\frac{GM_{\odot}^{2}}{c} \times \alpha^{-1} {\beta}^{-1} c^{-1}_{1} c^{3/2}_{3}\displaystyle\left(\Omega_{*}-\frac{a}{2r_{H}}\right){r_{eff}}^{3/2} m^2 \dot m & \mbox{ DML}\\
\end{array} \right.$$
$$\label{E:vrw}
\left(\displaystyle\frac{da}{d\tau}\right)_{em} =
\left\{ \begin{array}{lc}
\displaystyle\frac{\sqrt{5}}{80}\alpha^{-1} \beta^{-1} c^{-1}_{1} c^{1/2}_{3} a r^3_{H} r^{-5/2}_{eff} \dot m \left(1-\displaystyle\frac{a^{2}}{2r_{H}}\right) & \mbox{ BZ}\\
\displaystyle\frac{\sqrt{5}}{8} \alpha^{-1} \beta^{-1} c^{-1}_{1} c^{3/2}_{3}\displaystyle\left(\Omega_{\ast}-\frac{a}{2r_{H}}\right) {r_{eff}}^{3/2} \left(1-2a \Omega_{\ast}\right) \dot m & \mbox{ DML}\\
\end{array} \right.$$
Here $\Omega_{*}=c_2\Omega_K$ is angular frequency for ADAF disc. The pressure is calculated in the following way:
$$\label{pressure2}
p=1.7\times 10^{16} \alpha^{-1} c_1^{-1} c_3^{1/2} \dot m m^{-1} r^{-5/2}_{eff} {\rm erg\, cm^{-3}}$$
There are some unconstrained coefficients and it is important to understand how much Kerr parameter evolution depends on their values. Numerical coefficient $\epsilon$ depends on the ratio of specific heats $\gamma$ and on the fraction of advected energy $f$ as $\epsilon=\displaystyle\frac{1}{f}\displaystyle\frac{5/3-\gamma}{\gamma-1}$. In non-relativistic case $\gamma=5/3$, while for relativistic flow $\gamma \to
4/3$. Fully advective disc has $f=1$. If $f\ll 1$, a large fraction of energy is locally radiated and the disc rapidly becomes thin. Hence we always use $f=1$ for ADAF.
Numerical coefficients $c_1$ and $c_3$ depend weakly on parameters $\epsilon$ and $\alpha$. In the range $0.1<\epsilon<1$, and $0.01<\alpha<1$, they vary by about 10%. Non-Keplerianity parameter $c_2$ depends on $\epsilon$ and $\alpha$ much stronger. Near the last stable orbit, no inward-directed pressure gradient can be present hence the self-similar model probably under-estimates the value of $c_2$. Matter-only term is proportional to $c_2$, electromagnetic term is proportional to $\displaystyle\frac{c_2}{c_1}c_3^{3/2}$ that varies by about 18% in DML case and to $\displaystyle\frac{c_3^{1/2}}{c_1}$ that varies by about 21% in BZ case, hence its variations due to unknown equation of state and viscosity do not exceed $\sim 0.2$. If we fix $c_2$ equilibrium Kerr parameter varies up to 0.1% for BZ case and up to 6.5% in DML case. The largest uncertainty is connected to the unknown value of $c_2$. Hopefully, further, more comprehensive studies will help constrain its value.
Supercritical disc $\dot m>>1$
------------------------------
When the accretion rate is not very low and not very high, the standard disc theory can be applied. If accretion disc luminosity reaches the Eddington limit, $H/R$ becomes close to 1 in the inner parts of the disc and the thin disc approximation breaks down. There are at least two effects influencing the properties of supercritical accretion: outflow formation (considered in [@SS73]) and photon trapping (first considered in the advective Polish doughnut model [@abram05]).
Effect of advection dominates when accretion rate is very high $\dot
m>10^3$. [ Since there is numerical support for the outflow scenario (see for example @osuga07), we use an outflow-based model by @lipunova99.]{} An outflow starts at the spherization radius where $H/R=1$. Below we will adopt spherization radius in the form $r_{sp}=5/3 \dot m$. If effective radius is less than spherization radius, accretion rate becomes smaller and effective $\dot m_{eff}$ may be written as follows (see @lipunova99 and @poutanen07):
$$\dot m_{eff}=
\left\{\begin{array}{lc}
\dot m\displaystyle\frac{r}{r_{sp}}\displaystyle\frac{1+2/3r^{-5/2}}{1+2/3r_{sp}^{-5/2}} & \mbox{$r<r_{ sp}$}\\
\dot m & \mbox{$r>r_{sp}$}\\
\end{array}\right.$$
The matter flowing from the region inside the spherization radius removes angular momentum. Radial flux of angular momentum is:
$$g(r)=
\left\{\begin{array}{lc}
\displaystyle\frac{\dot m r^{3/2}}{3r_{sp}}\displaystyle\frac{1-r^{-5/2}}{1+2/3r_{sp}^{-5/2}} & \mbox{ $r<r_{sp}$}\\
\displaystyle\frac{\dot m \sqrt{r}}{3}\displaystyle\frac{1-r_{sp}^{-5/2}}{1+2/3r_{sp}^{-5/2}}+\dot m(\sqrt{r}-\sqrt{r_{sp}}) & \mbox{$r>r_{sp}$}\\
\end{array}\right.$$
Taking into account the features of supercritical regime, mass and spin evolution is described by the following equations:
$$\label{E:vr45}
\left(\displaystyle\frac{dE}{d\tau}\right)_{em} =
\left\{
\begin{array}{lc}
0.02 M_{\odot} c^2 \times \alpha^{-1} \beta^{-1} a^2 r^2_{H} {r_{eff}}^{-1} {\dot m_{eff}} r_{in}^{-1/2} m {g(r_{eff})}^{-1} \displaystyle\frac{1-\left(\displaystyle\frac{r_{eff}}{r_{in}}\right)^{-5/2}}{1+\displaystyle\frac{2}{3}\left(\displaystyle\frac{r_{eff}}{r_{in}}\right)^{-5/2}} & \mbox{ BZ}\\
0.21 M_{\odot} c^2 \alpha^{-1} \beta^{-1} {\dot m_{eff}} ({r_{eff}}^{3/2}+a)^{-1} \displaystyle\left(({r_{eff}}^{3/2}+a)^{-1}-\frac{a}{2r_{H}}\right) g(r_{eff}) m r_{in}^{1/2} \displaystyle\frac{1-\left(\displaystyle\frac{r_{eff}}{r_{in}}\right)^{-5/2}}{1+\displaystyle\frac{2}{3}\left(\displaystyle\frac{r_{eff}}{r_{in}}\right)^{-5/2}} & \mbox{ DML}\\
\end{array}
\right.$$
$$\label{E:vr4}
\left(\displaystyle\frac{dJ}{d\tau}\right)_{em} =
\left\{
\begin{array}{lc}
0.08 \displaystyle\frac{GM_{\odot}^{2}}{c} \times \alpha^{-1} \beta^{-1} a r^3_{H} {r_{eff}}^{-1} {\dot m_{eff}} m^2 r_{in}^{-1/2} {g(r_{eff})}^{-1} \displaystyle\frac{1-\left(\displaystyle\frac{r_{eff}}{r_{in}}\right)^{-5/2}}{1+\displaystyle\frac{2}{3}\left(\displaystyle\frac{r_{eff}}{r_{in}}\right)^{-5/2}} & \mbox{ BZ}\\
0.21 M_{\odot} c^2 \alpha^{-1} \beta^{-1} {\dot m_{eff}} \displaystyle\left(({r_{eff}}^{3/2}+a)^{-1}-\frac{a}{2r_{H}}\right) g(r_{eff}) m^2 r_{in}^{1/2} \displaystyle\frac{1-\left(\displaystyle\frac{r_{eff}}{r_{in}}\right)^{-5/2}}{1+\displaystyle\frac{2}{3}\left(\displaystyle\frac{r_{eff}}{r_{in}}\right)^{-5/2}} & \mbox{ DML}\\
\end{array}
\right.$$
$$\label{E:vr1}
\left(\displaystyle\frac{da}{d\tau}\right)_{em} =
\left\{\begin{array}{lc}
0.08 \alpha^{-1} \beta^{-1} a r^3_{H} {r_{eff}}^{-1} {\dot m_{eff}} r_{in}^{-1/2} \displaystyle\left(1-\frac{a^2}{2 r_{H}}\right) {g(r_{eff})}^{-1} \displaystyle\frac{1-\left(\displaystyle\frac{r_{eff}}{r_{in}}\right)^{-5/2}}{1+\displaystyle\frac{2}{3}\left(\displaystyle\frac{r_{eff}}{r_{in}}\right)^{-5/2}} & \mbox{ BZ}\\
0.21 \alpha^{-1} \beta^{-1} {\dot m_{eff}} r_{in}^{1/2} \displaystyle\left((r_{eff}^{3/2}+a)^{-1}-\frac{a}{2r_{H}}\right) \displaystyle\left(1-2 a (r_{eff}^{3/2}+a)^{-1}\right) g(r_{eff}) \displaystyle\frac{1-\left(\displaystyle\frac{r_{eff}}{r_{in}}\right)^{-5/2}}{1+\displaystyle\frac{2}{3}\left(\displaystyle\frac{r_{eff}}{r_{in}}\right)^{-5/2}} & \mbox{ DML}\\
\end{array} \right.$$
Magnetic field decay {#sec:decay}
====================
Hall decay
----------
In an accretion disc magnetic field decay is in equilibrium with magnetic field amplification by the instabilities such as MRI [@MRI] and dynamo processes [@dynamo]. This equilibrium state is shifted inside the last stable orbit where the plasma becomes magnetically dominated (gas and radiative pressure are reduced but magnetic field stress remaines practically unchanged). In these conditions magnetic field decay is primarily due to Hall cascade [@gold]. This effect does not change the total magnetic energy, but it can transfer the energy to smaller spatial scales where Ohmic decay runs faster. We assume that magnetic field lines, connecting a black hole with the disc have a curvature radius $L$ of the order $GM/c^2$. Following @gold we can write magnetic field evolution equation for magnetic field affected by Hall effect:
$$\label{max1}
\displaystyle\frac{\partial {\ensuremath{\mathbf{B}}}}{\partial t}=\displaystyle\frac{-c}{4\pi n e}\nabla \times [(\nabla \times {\ensuremath{\mathbf{B}}})\times {\ensuremath{\mathbf{B}}}]+\displaystyle\frac{c^2}{4\pi \sigma}\nabla^2 {\ensuremath{\mathbf{B}}}$$
The first term in this equation describes Hall cascade.
For the largest loops, one may estimate $\partial {\ensuremath{\mathbf{B}}}/\partial t\sim B/\tau$, $\nabla \sim 1/L$ and obtain Hall timescale as following:
$$\label{max3}
\tau_{Hall}=\displaystyle\frac{4\pi n e L^2}{cB}$$
Here $n$ is electron concentration and $L$ is characteristic curvature scale. For geometrically thick flow electron concentration may be estimated as $n\simeq \displaystyle\frac{\dot m}{\sigma_T}\left(\displaystyle\frac{c^2}{GM}\right)$.
Characteristic Hall timescale is:
$$\tau_{Hall}=\displaystyle\frac{4\pi \dot m c e R_H^2}{\sigma_T GM B} =
\dfrac{4\pi e}{\sigma_T B} {\ensuremath{\dot{m}}}r_H^2 \times \tau_{dyn}$$
Hall decay becomes important when $\tau_{Hall} \lesssim t_{dyn}$, when mass accretion rate is very small. Using expressions for an advective disc (section \[sec:addisc\]), one arrives to the following estimate:
$$\tau_{Hall}\sim 10^7 \sqrt{{\ensuremath{\dot{m}}}m} \times \tau_{dyn}$$
Hall drift is thus unimportant for astrophysical black holes.
The second term in eq. (\[max1\]) describes Ohmic decay and its timescale $\tau_{Ohm}\propto \sigma$, but conductivity of ionized relativistic plasma is very large, hence Ohmic timescale approaches infinity.
Joule losses {#subdecay}
------------
In Blandford-Znajek case magnetic field lines connect the black hole horizon with a distant load. Magnetic flux is conserved, hence large-scale magnetic field having non-zero flux through the equatorial plain can not dissipate and accumulates up to equipartition. These magnetic fields may be supported by electric currents in the accretion disc where resistance and dissipation are negligibly low.[ When we consider classical Blandford-Znajek magnetic field configurations, there is always a non-zero magnetic flux through the equatorial plane. As long as magnetic field lines are not allowed to move outwards (that is true as long as the field is frozen-in and accretion disc is present), the magnetic flux through the region inside the last stable orbit is also conserved, and one can set a lower limit for the magnetic field strength. Hence in the BZ case, magnetic flux conservation prohibits magnetic field dissipation below certain level.]{} In the direct magnetic link case there are always tangential magnetic fields near the black hole horizon and the currents should at least partially flow close to the horizon where effective resistance appears due to general relativity effects. Hence we will consider magnetic field decay effect only for the DML case.
Classical DML scenario involving magnetic loops extending above the disc is excluded by numerical simulations (such as @Shafee10). Instead, most of the magnetic field energy is stored in frozen-in loops that are advected with the falling matter. If magnetic fields are frozen into the free-falling matter inside the ISCO, the direction of the field should alter at least once at any radius, even close to the horizon. Magnetic field configuration is that of a driven current sheet, where dissipation is concentrated towards the middle plane and may be interpreted as magnetic field reconnection. Magnetic field dissipation leads to a magnetic torque value efficiently altered by some factor of $\chi <
1$.
To estimate the magnetic field decay rate in membrane paradigm, one may consider toroidal currents responsible for radial magnetic field direction change. The amplitude of such a current is, approximately:
$$j_\varphi = \frac{c}{4\pi} [\nabla B] \simeq \frac{c}{4\pi} \frac{B^r}{D},$$
where $D$ is the vertical scale of the falling magnetized flow. The radial extent of the flow is about $R_{in}-R_{H}$. Effective surface current is thus:
$$g_\varphi \simeq \frac{c}{4\pi} \frac{R_{in}-R_H}{D} B^r$$
Magnetic field $B^r$ here is estimated at the stretched horizon and differs from the “initial” magnetic field near the ISCO (that is close to equipartition) by the factor of disc cross-section ratio $\dfrac{H_{in} R_{in}}{DR}$ Magnetic field energy dissipated near the horizon may be estimated as the following integral upon the BH horizon:
$$-\dot{E}_H \simeq \frac{4\pi}{c} \int g_\varphi^2 dS$$
$$-\dot{E}_H \simeq 4\pi RH \times \left(\frac{c}{4\pi} \frac{R_{in}-R_H}{D}
B^r\right)^2 \times \frac{4\pi}{c} = -8\pi \left(\dfrac{H_{in}}{D}\right)^2
\dfrac{r_{in}^2(r_{in}-r_H)^2}{r_H^2} \times \left(\frac{GM}{c^2}\right)^2 p_M c$$
This is the estimated loss of the magnetic field energy near the horizon. Note that it does not involve any processes connected to black hole rotation. Magnetic field energy input from the disc:
$$\dot{E}_D \simeq p_M \times u^r \times 4\pi H_{in} R_{in} \simeq
p_M \times c \alpha \dfrac{H_{in}}{R_{in}} \times 4\pi H_{in} R_{in} =
4\pi \alpha \left(\dfrac{H_{in}}{R_{in}}\right)^3 \dfrac{r_{in}}{r_{in}^{3/2}+a} \times
\left(\frac{GM}{c^2}\right)^2 p_M c$$
Practically always $-\dot{E}_H \gg \dot{E}_D$, and the two rates become comparable only if viscosity is high and the flow is geometrically thick. Equilibrium between field decay and replenishment requires $H_{in}/R_{in}\gtrsim 1$.
Presence of a decay mechanism may mean that the magnetic field responsible for radial transfer is generally smaller than one may expect. Inner flow thickness $D$ close to the horizon may be estimated as $D\simeq H_{in}$
if the flow geometry is not altered strongly by magnetic field decay itself.
$$\chi \simeq \exp\left( -t_{replenishment} / t_{decay} \right) \simeq
\exp\left( \dot{E}_H / \dot{E}_D \right)$$
Substituting all the quantities:
$$\chi \simeq \exp\left(
-\dfrac{2 (r_{in}-r_H)^2 r_H}{\alpha h_{in} d^3}\right)$$ where $H_{in}=h_{in}GM/c^2$ and $D=d GM/c^2$.
![Ratio of magnetic field decay $\dot{E}_H$ and replenishment $\dot{E}_D$ rates as a function of Kerr parameter. Viscosity parameter $\alpha=0.1$, disc initial thickness $H_{in}=0.3R_{in}$. []{data-label="fig:edeh"}](edeh.eps){width="1.0\columnwidth"}
Figure \[fig:edeh\] shows the estimated decay factor as a function of Kerr parameter.
Summarizing, magnetic field decay is an important effect whenever there is tangential magnetic field component anywhere on the stretched horizon. Since magnetic field decay rate is generally several orders of magnitude larger than the rate of magnetic field replenishment from the disc (primarily because of $v_r \ll 1$), we conclude that DML momentum transfer is strongly damped.
Results {#sec:results}
=======
Equilibrium Kerr parameter
--------------------------
Spin evolution equation may be written as:
$$\label{ono}
\dfrac{da}{d\tau} = \mu {\ensuremath{\dot{m}}}\left[ j^\dagger - 2a E^\dagger \right]+
\dfrac{c}{G} \left[ \dfrac{1}{M^2} \left(\dfrac{dJ}{d\tau}\right)_{em} -
\dfrac{2aG}{Mc} \left(\dfrac{dM}{d\tau}\right)_{em}\right] \times \chi$$
First term here describes the Kerr parameter evolution connected to the angular momentum advected by the falling matter. It is proportional to mass accretion rate and usually positive. The second term corresponds to the contribution of electromagnetic processes described in sec. 2. This term is always negative in Blandford-Znajek regime and changes sign in the DML case when corotation radius equals the effective disc radius.
Additional multiplier $\chi$ takes into account magnetic field decay (see section \[sec:decay\]).
As long as the first term is higher than the second by its absolute value, black hole is spun up toward $a\sim 1$ for a standard disc and up towards $a\sim 0.6$ for a sub-Keplerian ADAF disc. Here we used $j^\dagger=c_2j_{K}(r_{in})$, where $c_2$ is a parameter taking into account deviation from Keplerian law (see section \[sec:addisc\]) and $j_{K}(r_{in})$ is net Keplerian angular momentum at the inner radius. If the second term is balanced by the first, black hole spin-up stops at the equilibrium Kerr parameter value $a_{eq}<0.998$.
![Equilibrium Kerr parameters for different accretion rates are plotted for BZ processes by dashed lines and for DML processes without and with magnetic field decay by solid and dotted lines respectively. Black lines correspond to SMBH ($M_0=10^7 M\odot$) while red (grey) lines to the stellar mass black hole ($M_0=10 M\odot$) case.[]{data-label="fig:aeq"}](equilibrium1.eps){width="0.7\columnwidth"}
Equilibrium spin parameter dependence on accretion rate is shown in Figure \[fig:aeq\]. Solid lines correspond to the DML case and BZ is plotted by a dashed line. Red (grey) lines are hereafter used for stellar-mass black holes ($M=10{\ensuremath{\rm M_\odot}}$) and black lines for SMBH ($M=10^7{\ensuremath{\rm M_\odot}}$). In this figure equilibrium Kerr parameter for DML processes with magnetic field decay is plotted by dotted line (see sections 4 and 5.2.1). In ADAF regime both electromagnetic and accretion terms are proportional to accretion rate $\dot m$ hence the equilibrium Kerr parameter does not depend on $\dot m$. Also it does not depend on the BH mass that is why the solutions for supermassive and stellar-mass BHs are identical.
For the standard disc case, the breakpoint marks transition from B to A zones. For the effective radius in zone B, spin evolution strongly depends on BH mass, and the influence of the BZ process for SMBH is more significant than for LMBH. The boundary between A and B zones depends on mass and accretion rate. For LMBH this transition takes place for accretion rates greater than for SMBH. In zone A, the dependence on mass is insignificant and spin evolution proceeds in the similar way for the two types of BH. But electromagnetic term depends on $\dot m$ somewhat weaker than accretion term, therefore the influence of BZ processes decreases with increasing $\dot m$. DML in inefficient in the Standard disc case for both SMBH and LMBH.
Equation (41) has no more than one attractor in the range $-1<a<1$. For some parameter sets (DML+standard disc), there is no stable solution of the $da/dt=0$ equation and the BH is spun up until $a\simeq 0.998$ where other effects not considered in this work (such as radiation capture, see @thorne74) come into play. In other cases, the black hole is always spun up for $a<a_{eq}$ and spun-down for $a>a_{eq}$.
Rotational evolution of a typical SMBH
--------------------------------------
For a more detailed analysis of black hole rotational evolution we made simulations for different accretion regimes for a supermassive Kerr black hole with the following parameters: initial mass $M_0=10^7 M_{\odot}$, initial Kerr parameters $a_0=0$ and $a_0=0.7$. For arbitrary dimensionless accretion rate $\dot{m}$, both mass and spin change at characteristic timescales $\sim t_{Edd} / \dot{m}$.
For a constant non-zero accretion rate, mass grows monotonically and may be used as an independent variable.
We consider ADAF regime with accretion rate $\dot m=10^{-3}$, standard disc with $\dot m=0.4$ and supercritical regime with $\dot m=10$. We set magnetization parameter $\beta=1$ and $\alpha=0.1$. Note that if viscosity is created by chaotic magnetic field stresses $\alpha \sim \displaystyle\frac{B_{\phi}B_{r}}{4\pi p} \sim \displaystyle\frac{2}{\beta}$ hence $\alpha \beta=$const. Since the two parameters enter the equations in combination $\alpha\beta$ (save the standard disc zone B where this statement is still approximately fulfilled; see section 3), the result depends weakly on the exact value of $\beta$.
### Rotational evolution in ADAF regime
In section \[sec:rev\] it was noted that spin evolution under the matter accretion contribution for ADAF regime differs from the Bardeen solution due to non-Keplerian velocity law. In Fig. \[fig:adaf\], spin evolution for the matter-only solution is plotted with crosses, and one can see that there is an equilibrium Kerr parameter value less than unity. Spin evolution affected by the Blandford-Znajek process is plotted by a dotted line for $a_0=0$ and by a solid line for $a_0=0.7$. The dashed and dot-dashed lines show the DML case for $a_0=0$ and $a_0=0.7$, respectively . Maximal difference between BZ and sub-Keplerian matter-only solution is about $0.0067$. We adopted $\epsilon^\prime=0.5$ that implies parameter values of $c_1\simeq 0.43$, $c_2\simeq 0.53$ and $c_3\simeq 0.285$.
Electromagnetic term for DML is higher than for BZ due to the difference in numerical coefficients and in the different dependence on the effective radius. Electromagnetic terms in these two cases differ by three orders of magnitude, the DML process being more efficient. Black hole mass growth is very slow and the Kerr parameter can not increase significantly during the cosmological time if ${\ensuremath{\dot{m}}}\lesssim 10^{-2}$.
Note that the character of black hole rotation evolution at the timescales longer than the evolutionary time scale $t_{Edd}/\dot m$ is insensitive to the initial Kerr parameter.
As it was shown above (in sec.4) magnetic field decay is efficient only in the DML case. This process affects only the behaviour of the electromagnetic term in the evolutionary equation. Influence of magnetic field decay is shown in Fig. 5. Here evolution for matter contribution is plotted by a crosses, evolution affected by DML processes without and with magnetic field decay is plotted by a dotted and a solid lines, respectively. One can see that equilibrium Kerr parameter increases from $0.36$ to $0.45$ due to the magnetic field decay.
![Spin evolution of a black hole with initial mass $M_0=10^7
M_{\odot}$, initial Kerr parameters $a_0=0$ and $a_0=0.7$, accretion rate $\dot M=10^{-3}\dot M_{Edd}$ (ADAF regime). Unmagnetized case evolution is plotted by crosses, evolution affected by DML processes is plotted by dashed ($a_0=0$) and dot-dashed ($a_0=0.7$) lines. Evolution affected by BZ process by a dotted line for $a_0=0$ and by a solid line for $a_0=0.7$. The bottom panel shows difference in Kerr parameter between the unmagnetized (Bardeen) and Blandford-Znajek cases for $a_0=0$. []{data-label="fig:adaf"}](final.eps){width="0.7\columnwidth"}
![Spin evolution of a black hole with initial mass $M_0=10^7
M_{\odot}$, initial Kerr parameter $a_0=0$ accretion rate $\dot
M=10^{-3}\dot M_{Edd}$. Evolution for accretion matter contribution is plotted by crosses, evolution affected by DML processes with and without magnetic field decay are plotted by dotted line and a solid line respectively.[]{data-label="fig:stand"}](adaf_decay.eps){width="0.7\columnwidth"}
![Spin evolution of a black hole with initial mass $M_0=10^7 M_{\odot}$, initial Kerr parameters $a_0=0$ and $a_0=0.7$, accretion rate $\dot M=0.4\dot M_{Edd}$ (Standard disc regime). Evolution for accretion matter contribution is plotted by crosses, evolution affected by DML processes is plotted by dashed ($a_0=0$) and dot-dashed ($a_0=0.7$) lines. Evolution affected by BZ process by a dotted line for $a_0=0$ and by a solid line for $a_0=0.7$. The inset shows difference in Kerr parameter between the unmagnetized (Bardeen) and DML cases for $a_0=0$. []{data-label="fig:stand"}](final2.eps){width="0.7\columnwidth"}
![Spin evolution of a black hole with initial mass $M_0=10^7
M_{\odot}$, initial Kerr parameters $a_0=0$ and $a_0=0.7$, accretion rate $\dot M=3\dot M_{Edd}$ (Supercritical regime). Evolution for accretion matter contribution is plotted by grey solid line, evolution affected by DML processes is plotted by dotted ($a_0=0$) and solid ($a_0=0.7$) lines. Evolution affected by BZ process is shown by a dashed line for $a_0=0$ and by a dot-dashed line for $a_0=0.7$.[]{data-label="fig:scrit"}](combined.eps){width="0.9\columnwidth"}
### Rotational evolution in standard disc regime
The results for the standard accretion disc are presented in Fig. \[fig:stand\]. Here again spin evolution in the Bardeen’s case is shown by crosses, spin evolution affected by the Blandford-Znajek processes is plotted by dotted ($a_0=0$) and solid ($a_0=0.7$) lines and in DML case by dashed ($a_0$) and by dot-dashed ($a_0=0.7$) lines. In this case direct magnetic link is much less efficient than Blandford-Znajek processes.
Depending on magnetic field geometry, spin evolution may result either in fast rotation with $a\simeq 1$ or in an intermediate $a\simeq a_{eq}$ where matter spin-up is balanced by magnetic field spin-down. As in ADAF case, equilibrium Kerr parameter does not depend on the initial Kerr parameter value.
### Rotational evolution in supercritical regime
In Fig. \[fig:scrit\] we present our results for the case of supercritical regime. Here spin evolution of a black hole for matter contribution is plotted by a grey solid line, evolution affected by DML processes is plotted by dotted ($a_0=0$) and solid ($a_0=0.7$) lines. Evolution affected by BZ process by a dashed line for $a_0=0$ and by a dot-dashed line for $a_0=0.7$. The corrections to all Kerr parameters introduced by different electromagnetic processes are plotted in the inset by the same linestyles.
As it was shown above (in sec. \[sec:decay\]) magnetic field decay is efficient only in the DML case. This process affects only the behaviour of the electromagnetic term in the evolutionary equation (41).
Note that the character of black hole rotational evolution at the timescales longer than the evolutionary time scale $t_{Edd}/{\ensuremath{\dot{m}}}$ is insensitive to initial Kerr parameter.
Black holes in X-ray binaries. Comparison with observational data {#sec:stellar}
-----------------------------------------------------------------
X-ray binaries are usually divided into low-mass X-ray binaries (LMXB) with a donor star of several solar masses or smaller and high-mass X-ray binaries (HMXB) with the donor masses of tens of solar masses. In this section we consider spin evolution of such systems including X-ray novae and microquasars. Predictions of simple constant mass accretion rate models are compared to the observational data on several black hole X-ray binaries. We assume that initial Kerr parameters of these objects are $a_0=0$ since there are reasons to expect BHs to be born slowly rotating [@moreno11]. According to the work by @Podsiadlowski, accretion rates in these systems $\dot m \gtrsim 1$ hence we used a constant accretion rate model with $\dot m=1$ in our calculations. We plot evolutionary tracks for black holes with masses 5, 7 and 10 $M_{\odot}$ in Figure 8. Solid and dashed lines represent the black hole evolution affected by Blandford-Znajek and DML processes, respectively. Here, DML is indistinguishable from Bardeen solution.
\[fig:aevolv\] ![Spin evolution for stellar mass black holes with initial masses $M_0=5M_\odot$, $M_0=7M_\odot$ and $M_0=10M_\odot$. Observation data are plotted with error bars (see @Podsiadlowski).[]{data-label="fig:small"}](small.eps "fig:"){width="0.7\columnwidth"}
There are at least two methods used for black hole spin estimates: fitting the thermal X-ray continuum and modelling the FeK$\alpha$ line profile. Due to some effects such as comptonization the line profile can broaden and Kerr parameter value will be overestimated. Continuum fitting, on the other hand, allows to reproduce the jet luminosity proportionality to $a^2$ [@narayan13]. Therefore we used fitting results [@Gou2010] for six objects analyzed with a continuum-fitting technique. These objects are shown in Fig. \[fig:small\].
This figure can be used to constrain the possible initial masses for each object and mass gains during mass exchange, these estimates given in Table 1. Electromagnetic processes do not change initial black hole mass estimates significantly, but observational spin values may be used to constrain the contribution of electromagnetic processes and possibly the mass accretion rates.
object names $M_{d}/M_{\odot}$ $M_{BH}/M_{\odot}$ $a$ $M^{*}_{0}/M_{\odot}$ $\Delta M^{*}_{BH}/M_{\odot}$
-------------- ---------------------------------------------- ------------------------ ---------------------- ----------------------- -------------------------------
LMC X-3 $>13^{[6]}$ $5-11^{[0]}$ $<0.26^{[0]}$ $4.0-9.5$ $0.3-0.5$
GRO J1655-40 $2.36-2.94^{[3]}$ $6.30 \pm 0.27^{[0]}$ $0.65-0.75^{[0]}$ $4.5-5.0$ $\sim 1.5$
4U 1543-47 $2.3-2.6^{[7]}$ $9.4 \pm 1.0^{[0]}$ $0.75-0.85^{[0]}$ $7.0-\sim 8.0$ $\sim 1.3$
LMC X-1 $35-40^{[4]}$, $31.79 \pm 3.48^{[5]}$ $10.91 \pm 1.41^{[0]}$ $0.92\pm 0.06^{[0]}$ $6.0-7.5$ $\sim 3.5$
GRS 1915+105 $0.47 \pm 0.27^{[1]}$, $0.81 \pm 0.53^{[2]}$ $14 \pm 4.4^{[0]}$ $0.98-1^{[0]}$ $5.0-9.0$ $5-10$
M33 X-7 $\sim 54^{[8]}$ $15.65\pm 1.45$ $0.77\pm 0.05^{[0]}$ $10-12^{[0]}$ $\sim 4$
GU Mus $0.86\pm 0.075^{[9]}$ $6.95\pm 0.6^{[9]}$ $\leq 0.4^{[12]}$ $\sim 6-7$ $0.3-0.5$
A0620 $0.40\pm 0.05^{[11]}$ $6.61\pm 0.25^{[10]}$ $0.3-0.6^{[12]}$ $5.4-5.8$ $\sim 1$
: The main parameters of X-ray binary systems. Donor masses $M_d$, black hole masses $M_{BH}$, Kerr parameter $a$, estimated black hole initial masses $M^{*}_0$ and mass gains of black holes $\Delta M^{*}_{BH}$. Black holes and donor parameters were taken from: \[0\] - @Gou2010, \[1\]- @steeghs13, \[2\] - @Harlaftis2004, \[3\] - @Shahbaz2003, \[4\] - @Zi2011, \[5\] - @Orosz2009, \[6\] - @Hutchings, \[7\] - @Orosz1998, \[8\] - @Valsecchi2010, \[11\] - @Casares1997, \[12\] -@xraynovae []{data-label="tab:time"}
Results of our simulations are in accordance with the sources listed in a Table 1 except for the object GRS 1915+105. Mass gain of the black hole in this system is about 5 solar masses, but donor mass is much smaller. The black hole has a large Kerr parameter and mass, that suggests that this system could undergo component exchange with another system. Our estimates do not take into account the possibility of such exchange that can still be important for binary evolution. The other peculiar thing about this system is the extremely high value of $a\gtrsim 0.98$ that either requires super-critical accretion or excludes the contribution of Blandford-Znajek processes during the black hole spin-up. Since this system is currently an active microquasar, the supercritical stage scenario seems more probable.
While for microquasars, estimated spin values are usually $\gtrsim 0.5$, X-ray novae such as A 0620-00 and GRS 1124-68 are supposed to have small $a\lesssim 0.3$ [@xraynovae]. The only exception is GRO J1655-40 that is classified both as an X-ray nova and a microquasar. Donor masses in such systems are $\lesssim 1{\ensuremath{\rm M_\odot}}$ and the black hole mass gains are probably insufficient for significant spin-up during the evolutionary time.
Massive black hole binaries like M33 X-7 and the two objects from LMC have a broad range of spin values that conforms with the expectation that these objects should follow a nearly Bardeen spin-up track and thus can acquire $a \simeq 1$ if given sufficient time.
[ Our results are in a good agreement with the recent results by @Fragos14 who use models of stellar evolution to explain the observed correlation between black hole masses and rotation parameters in low-mass X-ray binary systems. ]{}
Monte-Carlo simulations of a SMBH population {#sec:moncar}
--------------------------------------------
All the evolutionary scenarios described in the previous sections are simplified models. In real life black hole evolution is more complicated. We have made population synthesis of black holes rotational evolution assuming that Blandford-Znajek and DML processes may affect black hole spin evolution simultaneously. We use the Monte-Carlo technique for population synthesis of rotational evolution of a reasonably realistic supermassive black hole population. We assume the initial masses distributed according to the log-uniform law ($dN/d\ln M = const$) between $10^6 M_{\odot}$ and $5\times 10^8 M_{\odot}$. All the objects are born at $z=15$. Dimensional accretion rate was chosen for each black hole and for each time bin according to a log-normal law with the dispersion of $ D(\lg \dot{M})=0.65$ and the mode [(probability maximum) at]{} $1.3\times 10^{-3}{\ensuremath{\rm M_\odot \, yr^{-1}}}$.
For these parameter values, a conventional black hole with $M \sim 10^7 M_\odot$ spends $\sim 5-10\%$ of all time in active state that conforms to the existing duty cycle estimates [@shankar13]. Initial Kerr parameters are distributed uniformly between 0 and 1.
Here we consider the black hole active if the dimensionless accretion rate exceeds $0.01$. This value is also assumed the transition mass accretion rate between standard and advective disc regimes. Supercritical disc model is applied when ${\ensuremath{\dot{m}}}> 1/\eta(a)$, where $\eta = 1-E^\dagger$ is radiative efficiency.
We assume that Blandford-Znajek process operates in the circumpolar regions and DML operates near the equatorial plane of the black hole horizon. Simulations [@Tchekhovskoy12; @barkov12] show that magnetic lines connecting a black hole with infinity occupy much larger surface area of the black hole horizon (Blandford-Znajek process works here). Therefore we assume that the DML works in a narrow equatorial region that occupies only $10\%$ of the black hole horizon and Blandford-Znajek process works on the remaining $90\%$ of the surface. Existence of such a region corotating with the inner part of the disc is also confirmed by simulations [@Narayan2013].
\[fig:aevolv\] ![Kerr parameter distribution for evolved supermassive black hole population. Distribution for all population is plotted by solid line, black hole in active nuclei are plotted by dashed line and in nonactive nuclei are plotted by dotted line. []{data-label="fig:dist1"}](kerr.eps "fig:"){width="0.7\columnwidth"}
Distribution of all the objects in Kerr parameter at $z=0$ is shown in Fig. \[fig:dist1\] with a solid line. Kerr parameter distribution for AGNs is plotted by a dashed line and quiescent nuclei are shown by a dotted line. In this figure one can see a wide distribution with a peak at intermediate Kerr parameters $a \sim 0.6-0.7$ that corresponds to accretion through a standard disc. Supercritical accretion regime creates the high Kerr parameter wing of the distribution. The trail to the left consists of unevolved black holes with sub-equilibrium rotation parameters of $a\sim 0.1-0.5$ and is produced by ADAF regime in quiescent BH like Sgr A\*.
\[fig:distr\] ![Distribution in Kerr parameter. Distribution with initial Kerr parameter $a_0=0$ is plotted with a dotted line. Distribution with uniform initial Kerr parameter $0<a<1$ is plotted with a solid line and with a dashed line with a large fraction ($\sim 10\%$) of supercritical regime, $a_0=0$. []{data-label="fig:dist2"}](kerr_2.eps "fig:"){width="0.7\columnwidth"}
We have also performed simulations in other assumptions, but the results presented above seem the most realistic. In particular, we simulated black hole population evolution affected by the BZ process only. In this case we have only one narrow peak at $a\sim 0.5$ in contradiction with observational data [@reynolds13] . Another model was evolution driven solely by the DML process. This case is more interesting: there is a lot of objects with small and intermediate Kerr parameters, but there are no objects with Kerr parameters $\gtrsim 0.7$ that disagrees with the observational data.
In the recent work of @moreno11 the author shows that in X-ray binaries initial black hole Kerr parameters should be very small $a \sim 0$ and to explain the observed large Kerr parameters, black holes must have suffered long periods of supercritical accretion. Distribution of all objects with initial Kerr parameter uniformly distributed ($a_0<1$) is shown in Fig. \[fig:dist2\]. Simulations with $a_0=0$ and a large fraction ($\sim
10\%$) of supercritical regime are plotted by a dashed line. In the same figure distribution with uniformly distributed initial Kerr parameter described above is shown by a solid line. Distribution with $a_0=0$ and with a reasonable fraction of supercritical regime ($\sim 5 \%$) is plotted by a dotted line. This figure shows the following features for the model with $a_0
\sim 0$: black hole evolution in sense of $a(M)$ is determined by standard disc accretion, but its timescales are longer and closer to the timescales of inefficient ADAF accretion. Hence, a lot of objects in the simulated population are unevolved. This model shows deficit in black holes with $a\gtrsim 0.6$ and this is in contradiction with observations (see @mcclintock11 for review). The contribution of supercritical accretion is too small to significantly spin up a black hole, since this regime is difficult to sustain for black holes with $m_0>5\times 10^7$ and the fraction of supercritical accretors with $m_0\sim 10^7$ is less than several percent. Subsequently all the objects accumulate mass and dimensionless accetion rate decreases. Supercritical accretion shifts the mode of the distribution towards higher Kerr parameters. The model with increased mass accretion rate (shown by a dashed line in Fig. \[fig:dist2\]) shows a lot of rapidly rotating black holes, but there is a lack at medium and small $a$. These two simulations are in contradiction with observations (see @reynolds13 and references therein).
Discussion {#sec:discussion}
==========
It is hard to satisfy boundary conditions in the framework of the @NY95 self-similar ADAF model. Accretion disk structure is strongly deformed near the last stable orbit where gas pressure approaches zero. In our calculations we assumed magnetic pressure proportional to thermal pressure. But while gas pressure goes to zero near the ISCO, magnetic stresses vary smoothly and remain reasonably high due to magnetic flux conservation (see @Abolmasov14). Using the self-similar model is better justified in this case than using slim-disk models together with the assumption that magnetic fields are everywhere proportional to thermal pressure. A self-consistent MHD approach would be a better approximation.
While corrections to the electromagnetic term due to transonic nature of the disk are probably slight, angular momentum of the falling matter should be more sensitive to the processes at the inner boundary of the disk because the balance between the individual terms in the radial Euler equation shifts rapidly close to the sonic surface.
The matter falling inside the last stable orbit conserves its angular momentum while outside the sonic surface, angular momentum may be easily transferred by tangential viscous stresses. Hence the angular momentum of the in-falling matter should be estimated at the sonic surface rather than at the ISCO.
Continuity equation may be written approximately as:
$$\displaystyle\frac{d(r^d \rho v_r)}{dr}=0,$$
where $d$ describes the geometry of the streamlines: $d=1$ corresponds to cylindrical radial flow, $d=2$ in case of spherical radial flow.
Let us combine it with the radial component of the Euler equation:
$$v_r \displaystyle\frac{d v_r}{dr}-\displaystyle\frac{v^2_{\phi}}{r} = -\displaystyle\frac{1}{\rho}\displaystyle\frac{dp}{dr}-\displaystyle\frac{v^2_K}{r}$$
Using these equations one can obtain the radial Grad-Shafranov equation [@Beskinbook]:
$$\displaystyle\frac{d \ln \rho}{d \ln r}= \displaystyle\frac{v^2_{\phi}- v^2_K+d\cdot v^2_r}{r c^2_s(1-\displaystyle\frac{v^2_r}{c^2_s})}$$
At the sonic surface, denominator becomes zero hence the numerator should be zero too. This yields us angular momentum in the form:
$$L=L_K\sqrt{1-\displaystyle\frac{d v^2_r r}{GM}}=L_K\sqrt{1-\displaystyle\frac{d c^2_s r}{GM}}=L_K\sqrt{1-d \cdot \displaystyle\left(\frac{H}{r}\right)^2}=c^{*}_1 L_K$$
Here $c^{*}_1$ is coefficient that describes the deviations from Keplerian angular law (see section 3.3). In the self-similar ADAF model, $(H/r)^2=2.5c_3$. In our calculations we used $c_3=0.285$. Using this value for cylindrical radial flow one can find $c^{*}_1=0.536$ that is close to the value $c_1=0.43$ that we used in our calculations. For spherical flow $c^{*}_1$ is very close to unity. We may conclude that our calculations are close to reality if disk height is constant near the sonic surface.
This may be the case if the flow inside the last stable orbit becomes magnetically supported in vertical direction.
In @Shafee08, @Abolmasov14 and other works, rotation of magnetized accretion disks was found to be close to Keplerian near the last stable orbit. However, these works aimed on reproducing relatively thin disks with $H/r \sim 0.1$. In thicker disks, deviations from Keplerian law should be larger. Besides, magnetic field configuration in fact is expected to affect the rotation of the inner disk regions through magnetic pressure and tangential stress.
In general, recent simulations support the BZ scenario [@Tchekhovskoy12] and practically exclude the DML scenario. There are two reasons for this. First, inside the last stable orbit, density becomes very small and thermal pressure can no more counteract the Maxwellian stresses that leads to deformation and expansion of the magnetic field lines toward infinity. The second reason is existence of a large magnetic field in Blandford-Znajek configuration (with geometry close to a split-monopole) that leaves very small area for accretion and for field lines connecting the stretched horizon with the accretion disc. Numerical results suggest that either DML contribution to black hole rotation is practically absent or at least operates only on a small fraction of the stretched horizon surface (see also above section \[sec:moncar\]).
However, the results of these simulations are in apparent contradiction with observational data on super-massive black holes in galactic nuclei. Observations show that only about $7\div 8\%$ of active galactic nuclei launch radio-bright jets [@Ivezi02]. The reason for prevalence of radio-quiet objects is yet unclear, it may be connected to the overall magnetic flux through the black hole vicinity or to the broken symmetry of the magnetic field geometry. A probable reason for the discrepancy between simulations and observations is that most of the recent numerical simulations consider accretion discs with large-scale poloidal magnetic fields capable for rapid accumulation of magnetic flux inside the last stable orbit. Perhaps, smaller-scale chaotic magnetic fields would produce field geometry that is closer to the DML case.
[ DML configuration plotted in Fig. 1 can not be realized because Lorentz force acting on the magnetic loops inside the free-fall region will collapse them vertically. A more realistic picture with both processes, BZ and DML, taking place, is plotted in Fig. 11. Magnetic fields are supposed to be chaotic in the accretion disc: even regular fields are converted to small-scale loops due to Hall effect and magnetic instabilities (such as the magneto-rotational instability applied for accretion discs by @MRI). Hence, in the accretion flow, substantial part of the magnetic field energy will be stored in small-scale loops having zero magnetic flux through the equatorial plane of the black hole. This part of the magnetic field is also a subject to dissipation at the horizon. As long as these chaotic magnetic fields are confined inside the accretion flow they can contribute to the DML process. [ Note that any magnetic loop with non-zero radial extension contributes to DML through frame-dragging; in full general relativity, one does not need to approach the horizon.]{} Whether this condition is normally fulfilled or not is non-trivial and requires further thorough analytical and numerical consideration. ]{}
\[fig:9\] {width="0.7\columnwidth"}
In our paper we did not consider mergers that may however have an important effect upon the evolution of SMBH. For example, in the work of @volonteri05 the authors have made population synthesis of black holes with mergers and obtained Kerr parameter distribution similar to our result: Kerr parameter distributed between 0.1 and 0.9 with a peak at $a=0.5-0.6$. Most probably, if we will perform population synthesis of a black hole population taking into consideration both mergers and electromagnetic processes, we would obtain a qualitatively similar but broader distribution. But the question remains open: why do we observe a lot of black holes with high Kerr parameter? And the most probable explanation is supercritical accretion.
For a sample of black holes accreting magnetized matter for a large enough amount of time, broad distribution in spin is expected. Some of the objects stall near the critical $a$ value, some are expected to evolve toward $a\sim 1$. However, spin and mass evolution timescales are mostly larger than the Eddington timescale, and often larger than Hubble time. Together with the importance of mergers for black holes in galactic nuclei, it makes unreasonable to expect black hole rotation to be consistent with any unique spin value. Modeling taking into account all the effects is required.
In general, any black hole that gains an amount of mass comparable to its initial mass approaches its equilibrium rotation. Higher-mass black holes in binary systems should have also higher Kerr parameters, in consistence with the observed higher masses of the black holes in microquasars in comparison with the lighter black holes in X-ray novae [@banerjee13].
Conclusions {#sec:conclusion}
===========
In our work we aimed on constraining the role of angular momentum exchange between the BH and accretion disc and that of general relativistic magnetic field decay that should affect this process. We find that DML-like processes may be important for thick advective discs, but magnetic field is damped significantly close to the horizon. Equilibrium Kerr parameter in this case is in the range $\sim 0.3-0.5$ and is primarily determined by deviations from Keplerian law in the disc. In supercritical regime, both electromagnetic processes do not influence black hole spin evolution significantly.
We have made Monte-Carlo simulations for a supermassive black hole population assuming that both processes operate simultaneously. For a reasonable parameter set, we obtain a broad distribution in Kerr parameter $0.1\lesssim a \lesssim 0.9$ with a peak at $a\sim 0.7$ that is in fair agreement with observational data. For stellar mass black holes, the observed correlation between black hole masses and Kerr parameters may be understood as a consequence of accretion mass gain of $1-10 M_{\odot}$ in most of black hole binary systems.
Our results show that black hole observational appearance is sensitive to the accretion history, primarily to the period when most of the mass is gained. In the standard disc regime, a black hole gains a larger part of its mass and Kerr parameter increases faster than in ADAF regime.
Acknowledgments {#acknowledgments .unnumbered}
===============
Anna thanks to RSF grant 14-12-00146 and Pasha also thanks to RFBR grant 14-02-91172 and Dynasty foundation. We would like to thank Professor Nikolai Shakura for inspiring of the basic idea of this work, and Professor Vasily S. Beskin and Sasha Tchekhovskoy for fruitful discussions on black hole magnetospheres. We are also grateful to the anonymous referees for valuable comments.
\[lastpage\]
[^1]: E-mail: sagitta.minor@gmail.com
[^2]: We do not need the jet to be collimated for the Blandford-Znajek process to operate. Jet collimation occurs at the distances much larger than characteristic scales where black hole rotation energy is converted into electromagnetic [@Tchekhovskoy10]. Collimation can be produced by some surrounding media such as thick disc or intense disk wind. Numerical simulations such as @Fragile12 also suggest that jet collimation is not directly connected to disc thickness but rather to existence of disc corona or wind providing the necessary pressure.
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