Entanglement entropy and Berezin-Toeplitz operators

TL;DR

This paper establishes a Weyl law for Berezin-Toeplitz operators with characteristic function symbols on compact Kahler manifolds, linking boundary geometry to entanglement entropy and point process statistics.

Contribution

It proves a two-term Weyl law for these operators, connecting boundary volume to spectral asymptotics, and applies results to quantum Hall states and determinantal point processes.

Findings

Weyl law with boundary term proportional to boundary volume

Area law for entanglement entropy in quantum Hall states

Asymptotic normality of point counts in determinantal processes

Abstract

We consider Berezin-Toeplitz operators on compact Kahler manifolds whose symbols are characteristic functions. When the support of the characteristic function has a smooth boundary, we prove a two-term Weyl law, the second term being proportional to the Riemannian volume of the boundary. As a consequence, we deduce the area law for the entanglement entropy of integer quantum Hall states. Another application is for the determinantal processes with correlation kernel the Bergman kernels of a positive line bundle : we prove that the number of points in a smooth domain is asymptotically normal.

Equations

$$⟨T_{f,k}s,t⟩=⟨fs,t⟩,\forall s,t\in {\mathcal{H}}_k.$$

$$\dim {\mathcal{H}}_k={\left(\frac{k}{2\pi }\right)}^n\left(\mathrm{vol}(M)+\mathcal{O}\left(k^{-1}\right)\right)$$

$${\mathcal{H}}_k\simeq {\mathbb{C}}_k[\tilde{M}],$$

$$\displaystyle\begin{split}\bigl{|}\operatorname{sp}(T_{A,k})\cap[0,\epsilon]\bigr{|}=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}\bigl{(}\mu(A^{c})+\operatorname{o}(1)\bigr{)}\\ \Bigl{|}\operatorname{sp}(T_{A,k})\cap[1-\epsilon,1]\bigr{|}=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}\bigl{(}\mu(A)+\operatorname{o}(1)\bigr{)}\end{split}$$

$$|s(x)|⩽C_{K,N}{k}^{-N}\Vert s\Vert .$$

$$\displaystyle\Bigl{|}\operatorname{sp}(T_{A,k})\cap[a,b]\Bigr{|}=k^{n-1/2}\frac{\operatorname{vol}(\partial A)}{(2\pi)^{n}}(\operatorname{er}^{-1}(b)-\operatorname{er}^{-1}(a))+\operatorname{o}(k^{n-1/2})$$

$$I(f):={\int }_0^{1}f(x)dm(x)={\int }_{-\infty }^{\infty }f(\mathrm{er}(x))dx$$

$$\delta(t)\sim\frac{1}{2}t^{-1}\bigl{(}\ln(1/t)\bigr{)}^{-1/2}$$

$$\displaystyle\operatorname{tr}(f(T_{A,k}))=k^{n-1/2}\frac{\operatorname{vol}(\partial A)}{(2\pi)^{n}}I(f)\bigl{(}1+\operatorname{o}(1)\bigr{)}.$$

$$\mathrm{tr}(f(T_{A,k}))=k^{n-1/2}\sum _{\ell =0}^N{c}_{\ell }(f)k^{-\ell /2}+\mathcal{O}(k^{n-N-1}),\forall N\in \mathbb{N}$$

$$\displaystyle\operatorname{tr}(g(T_{A,k}))=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}\bigl{(}$$

$$\displaystyle+\operatorname{o}(k^{-1/2})\bigr{)}$$

$$\displaystyle\Bigl{|}\operatorname{sp}(M_{k}(A))\cap[0,a]\Bigr{|}=k\frac{\ell(S^{1}\setminus A)}{2\pi}+\mathcal{O}(\ln k),$$

$$\displaystyle\Bigl{|}\operatorname{sp}(M_{k}(A))\cap[b,1]\Bigr{|}=k\frac{\ell(A)}{2\pi}+\mathcal{O}(\ln k)$$

$$\Bigl{|}\operatorname{sp}(M_{k}(A)\cap[a,b]\Bigr{|}\sim\ln k\frac{|\partial A|}{2\pi^{2}}(m^{-1}(b)-m^{-1}(a))$$

$$\begin{array}{rl}\mathrm{tr}(T_{A,k}^p-T_{A,k}^{p+1})=& {\int }_{A^p\times A^c}{\Pi }_k(x_1,x_2){\Pi }_k(x_2,x_3)\dots {\Pi }_k(x_p,x_{p+1})\\ & {\Pi }_k(x_{p+1},x_1)\mu (x_1)\dots \mu (x_{p+1})\end{array}$$

$$I(k)={\int }_D{e}^{-k(t^2/2+t^{3}f(t,s))}g(t,s)dtds$$

$$I(k)=k^{-1}\sum _{\ell \in \mathbb{N}}{b}_{\ell }{k}^{-\ell /2},$$

$$\operatorname{tr}\bigl{(}T^{p}_{A,k}(1-T_{A,k})^{p}\bigr{)}\leqslant C_{p}k^{n-1/2}$$

$$\Psi :=s_1\wedge \dots \wedge s_{d_k}\in {\bigwedge }^{d_k}{\mathcal{E}}_k.$$

$$\displaystyle{\textstyle{\bigwedge}}^{d_{k}}\mathcal{E}_{k}=\bigoplus_{\ell=0}^{d_{k}}\bigl{(}{\textstyle{\bigwedge}}^{\ell}\mathcal{E}_{k}(A)\bigr{)}\otimes\bigl{(}{\textstyle{\bigwedge}}^{d_{k}-\ell}\mathcal{E}_{k}(A^{c})\bigr{)}$$

$$\mathrm{Pr}(N_A^{\Psi }=\ell )=\Vert {\Psi }_{\ell }{\Vert }^2,\ell =0,\dots ,d_k$$

$$\mathbb{E}(N_{A}^{\Psi})=\operatorname{tr}(T_{A}),\qquad\mathbb{E}\bigl{(}(N_{A}^{\Psi})^{2}\bigr{)}-\bigl{(}\mathbb{E}(N_{A}^{\Psi})\bigr{)}^{2}=\operatorname{tr}(T_{A}-T_{A}^{2}).$$

$$\displaystyle\mathbb{E}(N_{A}^{\Psi})=k^{n}\Biggl{(}\frac{\mu(A)}{(2\pi)^{n}}+\sum_{m=1}^{N}{\alpha}_{m}(A)k^{-m}+\mathcal{O}(k^{-N-1})\Biggr{)},$$

$$\displaystyle{\kappa}_{2\ell}(N_{A}^{\Psi})=k^{n-1/2}\Biggl{(}\frac{\operatorname{vol}(\partial A)}{(2\pi)^{n}}I(P_{2\ell})+\sum_{m=1}^{N}{\beta}_{\ell,m}(A)k^{-m}+\mathcal{O}(k^{-N-1})\Biggr{)}$$

$$\displaystyle{\kappa}_{2\ell+1}(N_{A}^{\Psi})=k^{n-1}\Biggl{(}\sum_{m=1}^{N}\eta_{\ell,m}(A)k^{-m}+\mathcal{O}(k^{-N-1})\Biggr{)}.$$

$${\kappa }_2(N_A^{\Psi })=k^{n-1/2}\frac{\mathrm{vol}(\partial A)}{(2\pi {)}^n}I(P_2)+\mathcal{O}(k^{n-3/2}).$$

$${\kappa }_{2\ell }(X(\alpha ))=\sum _{m\in \mathbb{Z}}{P}_{2\ell }(\mathrm{er}(\alpha k))={\alpha }^{-1}I(P_{2\ell })+\mathcal{O}({\alpha }^{\infty }).$$

$${\kappa }_{\ell }(N_A^{\Psi })={\kappa }_{\ell }(X({\alpha }_k))+\mathcal{O}(k^{n-1}).$$

$$S(\rho )=-\mathrm{tr}(\rho \ln \rho )\in [0,\infty ).$$

$$\operatorname{tr}(\rho_{1}A_{1})=\operatorname{tr}\bigl{(}\rho(A_{1}\otimes\operatorname{id}_{H_{2}})\bigr{)},\qquad\forall A_{1}\in\mathcal{L}(H_{1}).$$

$$\displaystyle S(\rho_{A})=\operatorname{tr}\bigl{(}f(T_{A})\bigr{)}\qquad\text{ with }\quad f(x)=-x\ln(x)-(1-x)\ln(1-x)$$

Full text

The area laws for entanglement entropy have been widely discussed in recent years in condensed matter and quantum field theories. Typically, one considers a many-particle state and a geometric partition of the space in two sub-regions. The von Neumann entropy of the reduced state of a sub-region measures the degree of entanglement between the two regions. The area law states that this entanglement entropy is proportional to the volume of the boundary of the sub-region. These area laws have been verified in many different systems, such as harmonic and spin chains, strongly correlated fermionic systems and quantum field theories, see for instance the surveys [PE09], [ECP10], [AFOV08].

So far, in the mathematical literature, only the case of the free Fermi gas has been considered [Gio06], [HLS11], [LSS16], [LSS17]. The goal of the present paper is to address the case of Integer Quantum Hall (IQH) states. Unlike the Fermi gas, these systems have a spectral gap above the ground state energy and the corresponding area law has no logarithmic correction.

As it was observed in [GK06], the area law for a Fermi gas is related to a conjecture by Widom on the correction terms in the Weyl law for pseudo-differential operators with singular symbols. Similarly, for IQH states, the area law amounts to estimating a weighted spectral average of a Berezin-Toeplitz operator $T$ whose symbol is the characteristic function of the sub-region. The theory of Berezin-Toeplitz operators is well-developed but here we have to face several difficulties: first, the symbol of the operator $T$ is singular not even continuous; second, the expectation we have to estimate is the trace of $f(T)$ with a function $f$ having a logarithmic singularity at [math] and $1$ precisely where the concentration of eigenvalues is highest; third, the relevant term in this estimate is a correction to the leading order term.

The results we obtain are quite general, for any domain with a smooth boundary of a compact Riemann surface, or in higher dimension of a compact Kähler manifold. This generality was a surprise to us, because even if the physics papers provide a lot of numerical evidence and heuristic arguments, they treat only very particular integrable geometries with a lot of symmetries, which allows one to compute explicitly an eigenbasis for the reduced state [RS09, RS10].

Besides the entanglement entropy estimate, our Weyl law have applications to determinantal processes whose correlation kernel is the Bergman kernel of a positive line bundle. In this context, very general results are already known for linear statistics, cf. [Ber08], [Ber14]. We give new estimates by providing complete asymptotic expansions for the variance and the higher cumulants of the number of points in a smooth domain. In particular, the variance is proportional to the volume of the boundary at first order, a variant of the area law. This leads after a convenient rescaling to new convergences to the normal distribution. Our result could also find applications in the context of random matrix theory, in particular the distribution of eigenvalues in a smooth domain of the complex plane for the complex Ginibre ensemble. To compare with the existing litterature, similar central limit theorems hold more generally for Coulomb gases [LS18], [Ser17], our setting corresponding to the inverse temperature ${\beta}=2$. But to our knowledge, all theses results are for smooth enough linear statistics, the case of characteristic functions being new.

Our proofs are based on Bergman kernel asymptotics. The technical part consists of estimating singular oscillatory integrals. To do this, we develop a generalisation of the Laplace method which is of independent interest.

Before we state our results precisely, let us mention that partial closed results in the Bargmann space were already obtained by Oldfield [Old15], using Weyl quantization. Bergman kernel techniques have already been used with success for the study of quantum Hall effect, cf. the survey [Kle16] for instance. In the context of Berezin-Toeplitz operators in Kähler manifolds, two recent papers on different but related subjects are [ZZ17] on partial Bergman kernels and [PU18] on the quantization of submanifolds. Finally, the appendix of [Pol18] is devoted to multiplicative properties of the Toeplitz operators with a characteristic function symbol.

1 Statements of the results

1.1 Toeplitz operators with characteristic function symbol

Let $L\rightarrow M$ be a positive holomorphic Hermitian line bundle on a compact manifold $M$. For any positive integer $k$, let $\mathcal{H}_{k}$ be the space of holomorphic sections of $L^{k}$. For any function $f$ on $M$, the Toeplitz operator with multiplicator $f$ is the endomorphism $T_{f,k}$ of $\mathcal{H}_{k}$ such that

[TABLE]

Here the scalar product $\langle\cdot,\cdot\rangle$ of sections of $L^{k}$ is defined by integrating the pointwise scalar product against the Riemannian measure $\mu$, the Riemannian metric $g$ of $M$ being determined by the curvature $\Theta(L)$ of the Chern connection of $L$, that is $g(X,Y)=-i\Theta(L)(X,jY)$ with $j$ the complex structure.

The definition of the Toeplitz operator $T_{f,k}$ makes sense for any integrable function $f$. So in particular we can consider the characteristic function of a mesurable subset $A$ of $M$. We denote by $T_{A,k}$ the corresponding Toeplitz operator.

All the results we will prove concern the semi-classical limit $k\rightarrow\infty$. The dimension of $\mathcal{H}_{k}$ is equal to first order

[TABLE]

where $n$ is the complex dimension of $M$ and $\operatorname{vol}(M)$ is the Riemannian volume of $M$.

For the reader not familiar with this geometric setting, it can be interesting to consider the example of the projective space $\mathbb{P}^{N}$ with $L_{\mathbb{P}^{N}}$ the dual of the tautological line bundle. More generally, let $M$ be a closed complex submanifold of $\mathbb{P}^{N}$ and $L$ be the restriction of $L_{\mathbb{P}^{N}}$ to $M$. In this case, the spaces $\mathcal{H}_{k}$ have a simple concrete description. Let $\pi$ be the projection from the unit sphere of ${\mathbb{C}}^{N+1}$ onto $\mathbb{P}^{N}$ and let $\tilde{M}:=\pi^{-1}(M)$. Then when $k$ is sufficiently large the holomorphic sections of $L^{k}$ lifts to polynomial functions of ${\mathbb{C}}^{N+1}$ and we have a natural identification

[TABLE]

where ${\mathbb{C}}_{k}[\tilde{M}]$ consists of the restriction to $\tilde{M}$ of homogeneous polynomials of ${\mathbb{C}}^{N+1}$ with degree $k$. Furthermore, the scalar product is given in ${\mathbb{C}}_{k}[\tilde{M}]$ by $\langle f,g\rangle=C_{n,N}\int_{\tilde{M}}f(x)\overline{g}(x)\;d\tilde{\mu}(x)$ where $\tilde{\mu}$ is the Riemannian measure of $\tilde{M}$ for the Riemannian metric of $\tilde{M}$ induced by the Euclidean scalar product of ${\mathbb{C}}^{N+1}$ and $C_{n,N}$ is a positive constant independent of $k$ and $M$.

Let us return to the general situation and consider a subset $A$ of $M$ with an empty boundary. Then $M$ is the disjoint union of two open sets: $A$ and its complementary set $A^{c}$. So $\mathcal{H}_{k}=\mathcal{H}_{k}(A)\oplus\mathcal{H}_{k}(A^{c})$ where $\mathcal{H}_{k}(A)$ (resp. $\mathcal{H}_{k}(A^{c})$) consists of the sections vanishing on $A^{c}$ (resp. on $A$). Furthermore $T_{A,k}$ is the projector onto $\mathcal{H}_{k}(A)$ with kernel $\mathcal{H}_{k}(A^{c})$. This property does not hold in general. Actually, if $M$ is connected and $A$ and $A^{c}$ have non empty interiors, then [math] and $1$ are not even in the spectrum of $T_{A,k}$. Nevertheless, $T_{A,k}$ looks like a projector in the sense that its eigenvalues concentrate at [math] and $1$. Let us explain this more precisely.

First of all, for any measurable set $A$ of $M$, $T_{A,k}$ is a Hermitian endomorphism, whose spectrum $\operatorname{sp}(T_{A,k})$ is contained in $[0,1]$. Then it is proved in [Ber03], [Lin01] that if the boundary of $A$ has a null measure, then for any $\epsilon\in(0,1/2)$ we have

[TABLE]

in the limit $k\rightarrow\infty$, where the eigenvalues are counted with multiplicity. Here and in the sequel we denote by $\mu(B)=\int_{B}\mu$ the measure of a subset $B$ of $M$.

The next question is about the possible eigenvalues in $[\epsilon,1-\epsilon]$. As a consequence of (1) and (2), we have \bigl{|}\operatorname{sp}(T_{A,k})\cap[\epsilon,1-\epsilon]\bigr{|}=\operatorname{o}(k^{n}). Also it is not difficult to see that the corresponding eigenstates are concentrated on the boundary of $A$. More precisely, let $\mathcal{H}_{k}(\epsilon)$ be the sum of the eigenspaces of $T_{A,k}$ with eigenvalue in $[\epsilon,1-\epsilon]$. Then for any compact subset $K$ of $M$ not intersecting $\partial A$, for any $N>0$, there exists $C_{N,K}$ such that for any $s\in\mathcal{H}_{k}(\epsilon)$ and $x\in K$, we have

[TABLE]

So we can consider that the states in $\mathcal{H}_{k}(\epsilon)$ live in the interface of $A$ and its complementary set.

Our first result shows that the number of eigenvalues in $[\epsilon,1-\epsilon]$ has a remarkable simple universal behavior when the boundary of $A$ is smooth. For any closed submanifold $B$ of $M$, denote by $\operatorname{vol}(B)$ the Riemannian volume of $B$.

Theorem 1.1**.**

Let $A$ be a subset of $M$ with a smooth boundary. Then for any $a,b$ such that $0<a<b<1$, we have

[TABLE]

where $\operatorname{er}:{\mathbb{R}}\rightarrow[0,1]$ is the function defined by $\operatorname{er}(x)=\pi^{-1/2}\int_{-\infty}^{x}e^{-t^{2}}\;dt$.

The function $\operatorname{er}$ is increasing from [math] to $1$, it is given in terms of the standard error function $\operatorname{erfc}$ by $\operatorname{er}(x)=1-\operatorname{erfc}(x)/2$. The transition between Theorem 1.1 and estimates (2) is rather subtle because $k^{n}$ is replaced by $k^{n-1/2}$ and the volumes of $A$ and $A^{c}$ are replaced by the volume of the boundary. So not surprisingly, the quantity $|\operatorname{er}^{-1}(b)-\operatorname{er}^{-1}(a)|$ diverges as $a\rightarrow 0$ or $b\rightarrow 1$, which prevents us to reach (2) from Theorem 1.1. Nevertheless, we can improve Theorem 1.1 and say something on the limits $a\rightarrow 0$, $b\rightarrow 1$ by considering weighted sums of eigenvalues as follows. Introduce the measure $m$ of $(0,1)$ such that $m([a,b])=|\operatorname{er}^{-1}(b)-\operatorname{er}^{-1}(a)|$ and the corresponding integral

[TABLE]

for functions $f:(0,1)\rightarrow{\mathbb{C}}$. We have $dm(t)=\delta(t)\;dt$ with a smooth density $\delta:(0,1)\rightarrow{\mathbb{R}}$ such that $\delta(t)=\delta(1-t)$ and

[TABLE]

as $t\rightarrow 0$. Consequently, the integral $I(f)$ converges for any continuous function $f:[0,1]\rightarrow{\mathbb{C}}$ which is Hölder continuous at [math] and $1$ with $f(0)=f(1)=0$.

Theorem 1.2**.**

Let $A$ be a subset of $M$ with a smooth boundary. For any continuous $f:[0,1]\rightarrow{\mathbb{C}}$ satisfying $|f(t)|=\mathcal{O}(t^{p})$ and $|f(1-t)|=\mathcal{O}(t^{p})$ for some positive $p$, we have

[TABLE]

If $f$ is polynomial with $f(0)=f(1)=0$, then we have a complete asymptotic expansion

[TABLE]

where $c_{0}(f)=(2\pi)^{-n}\operatorname{vol}(\partial A)I(f)$.

It is likewise that the expansion (5) holds also for smooth functions but our proof works only for polynomial functions. It is important that we prove (4) under the Hölder assumption, because for our application to entanglement entropy, we will consider the function $f(x)=-x\ln x-(1-x)\ln(1-x)$.

Estimate (4) may be viewed as a subprincipal term in the Weyl law for $T_{A,k}$, as stated in the following result.

Corollary 1.3**.**

For any continuous function $g:[0,1]\rightarrow{\mathbb{C}}$ which is Hölder-continuous at [math] and $1$, we have

[TABLE]

where $f(x)=g(x)-g(0)(1-x)-g(1)x$.

Remark 1.4**.**

The previous results hold as well in the Bargmann space $\mathcal{B}_{\hbar}$. Recall that for any $\hbar>0$, $\mathcal{B}_{\hbar}$ is the subspace of $L^{2}({\mathbb{C}}^{n},e^{-\hbar^{-1}|z|^{2}}\mu)$ consisting of holomorphic functions. Here $\mu$ is the Lebesque measure of ${\mathbb{C}}^{n}$. For any measurable subset $A$ of ${\mathbb{C}}^{n}$, the Toeplitz operator $T_{A}(\hbar)$ is the bounded operator of $\mathcal{B}_{\hbar}$ defined by the same formula $\langle T_{A}(\hbar)s,t\rangle=\langle 1_{A}s,t\rangle$ as previously. Assume that $A$ is bounded, so that $T_{A}(\hbar)$ is a trace class operator. Then setting $\hbar=k^{-1}$, the second estimate of (2), Theorem 1.1 and Theorem 1.2 hold. The proof is essentially the same.

In this particular case, the results are not completely new: in [Old15], Oldfield proved Theorem 1.1 and Equation (4) of Theorem 1.2 for smooth $f$. Previously, De Mari, Feichtinger and Nowak had obtained lower and upper bounds for $\operatorname{tr}(f(T_{A}(\hbar)))$ in [DMFN02]. Actually, the volume of $\partial A$ does not appear explicitly in the work of Oldfield, but Equations (3) and (4) with smooth $f$ can be deduced from it. However, (4) with a Hölder continuous $f$ does not follows from [Old15], neither the expansion (5). The proof of [Old15] is based on Weyl quantization, an approach which is not available in our geometric setting. Instead we will use Bergman kernel techniques. ∎

Remark 1.5**.**

Similar results to Theorem 1.1 are known for truncated Toeplitz matrices. Let $V_{k}$ be the subspace of $L^{2}(S^{1})$ spanned by the Fourier modes $e_{0},\ldots,e_{k}$ where $e_{\ell}(x)=e^{i\ell x}$. Let $A\subset S^{1}$ be a finite union of intervals and consider the endomorphism $M_{k}(A)$ of $V_{k}$ given by $\langle M_{k}(A)s,t\rangle=\langle 1_{A}s,t\rangle$ for any $s,t\in V_{k}$. Then by [Bas86], [BW83], [LW80], for any $0<a<b<1$, we have

[TABLE]

where we denote by $\ell(B)$ the length of a subset $B$ of $S^{1}$. Furthermore

[TABLE]

where $|\partial A|$ is the number of boundary points of $A$ and $m(x)=\frac{1}{2}(1+\tanh(x/2))$. To compare with (3), the $k^{-1/2}$ is replaced by $\ln k$ and the function $\operatorname{er}$ by the function $m$. A generalisation of this result is known for Wiener-Hopf operators, with the same logarithm and function $m$. This is the so-called Widom conjecture, proved in dimension 1 by Widom itself [Wid82] and in higher dimension by Sobolev [Sob13]. ∎

Let us sketch the main step of our proof. We first show the asymptotic expansion (5) for polynomial $f$. To do this we use that for any positive integer $p$

[TABLE]

where $\Pi_{k}$ is the Bergman kernel of $L^{k}$. The asymptotic behavior of the Bergman kernel is well-known [Zel98], [MM07], [Cha03]. From this we estimate the integral (6) by adapting the Laplace method. Typically, we have to handle integrals of the form

[TABLE]

where $f$ and $g$ are smooth functions, $g$ being compactly supported. When $D={\mathbb{R}}^{2}$, this is easily done by viewing $s$ as a parameter and applying the standard Laplace method. We get in this case $I(k)=k^{-1/2}\sum_{\ell\in{\mathbb{N}}}a_{\ell}k^{-\ell}$. We actually have a different kind of domain $D$ which is conic and contained in $\{|s|\leqslant C|t|\}$ for some $C>0$. In this case,

[TABLE]

so there are two important differences: even if the phase is degenerate only in the $t$ direction, the variable $s$ has to be considered as non-degenerate in the sense that $I(k)$ is of order $k^{-1}$. Furthermore, the expansion of $I(k)$ is in power of $k^{-1/2}$. We will develop a Laplace method for generalisations of the integral (7) in any dimension and deduce the asymptotic expansion of (6).

Once this is done, we can show the estimate (4) for smooth $f$ and Theorem 1.1. To prove (4) for Hölder-continuous $f$, we need the additional estimate:

[TABLE]

for any $p>0$, that we deduce from a Berezin-Lieb inequality.

The proof in the Bargmann space requires a little extra work because ${\mathbb{C}}^{n}$ is not bounded, but it is also simpler because the Bergman kernel is given by an explicit formula.

1.2 Application to IQH states

Free fermions

Let $\mathcal{E}_{k}$ be the space of square integrable sections of $L^{k}$. It is a Hilbert space with the scalar product defined as previously by integrating the pointwise scalar product of sections against the Riemannian volume element. Let $d_{k}$ be the dimension of $\mathcal{H}_{k}$ and $(s_{i},\;i=1,\ldots,d_{k})$ be an orthonormal basis of $\mathcal{H}_{k}$. Define

[TABLE]

As we will explain at the end of this Section, $\mathcal{H}_{k}$ is the first Landau level of a magnetic Laplacian. So $\Psi$ represents a Fermionic state in which the first Landau level is fully occupied. $\Psi$ has norm $1$ for the natural scalar product of $\bigwedge^{d_{k}}\mathcal{E}_{k}$. This space is not complete but it does not matter for what we will do.

Since we work with square integrable sections instead of holomorphic sections, for any measurable set $A$ of $M$, we have the decomposition $\mathcal{E}_{k}=\mathcal{E}_{k}(A)\oplus\mathcal{E}_{k}(A^{c})$, where $\mathcal{E}_{k}(A)$, $\mathcal{E}_{k}(A^{c})$ are the image and kernel of the multiplication by the characteristic function $1_{A}$. Consequently

[TABLE]

Let $N_{A}$ be the endomorphism of $\bigwedge^{d_{k}}\mathcal{E}_{k}$ acting by multiplication by $\ell$ in the $\ell$-th summand of (9). In statistical quantum mechanic, $N_{A}$ is the observable for the the number of particles in $A$. From the state $\Psi$ and the observable $N_{A}$, we obtain a probability distribution $N_{A}^{\Psi}$ defined by

[TABLE]

where $\Psi=\sum\Psi_{\ell}$ is the decomposition of $\Psi$ in the sum (9).

In the mathematical litterature, the Slater determinants and their associated distributions appeared as determinantal processes [Ber14], [AHM11] defined as follows. Identify first $\wedge^{d_{k}}\mathcal{H}_{k}$ with the subspace of antisymmetric vectors of $\mathcal{H}^{\otimes d_{k}}$. Realize $\mathcal{H}^{\otimes d_{k}}$ as the space of holomorphic sections of $(L^{k})^{\boxtimes d_{k}}\rightarrow M^{d_{k}}$. Then each normalized vector $\Phi$ of $H^{0}(M^{d_{k}},(L^{k})^{\boxtimes d_{k}})$ defines a probability measure $\mu_{\Phi}$ of $M^{d_{k}}$ given by $\mu_{\Phi}=|\Phi|^{2}\mu^{\boxtimes d_{k}}$. The measure $\mu_{\Psi}$ associated to $\Psi$ is a determinantal process whose number of points in a given set $A$ has the same distribution as $N_{A}^{\Psi}$.

The probability distribution $N_{A}^{\Psi}$ can be completely described in terms of spectral invariants of the Toeplitz operator $T_{A}=T_{A,k}$. Let ${\lambda}_{1}\leqslant{\lambda}_{2}\leqslant\ldots\leqslant{\lambda}_{d_{k}}$ be the eigenvalues of $T_{A}$. Recall that for $p\in[0,1]$, the Bernoulli random variable $B(p)$ takes the value $1$ with probability $p$ and the value [math] with probability $1-p$. Then $N_{A}^{\Psi}$ has the same distribution has the sum of independent random variables $B({\lambda}_{i})$, $i=1,\ldots,d_{k}$. It is actually a general property of determinantal processes that their number of points in a given set is a sum of Benoulli random variables, [HKPV06]. This important fact appeared also implicitely in the physics litterature [Kli06], [AI08]. We provide a short proof for our Fermionic states in section 7.

An easy consequence is the computation of the cumulants of $N_{A}^{\Psi}$ as spectral invariants of $T_{A}$. In particular, the expectation and variance of $N_{A}^{\Psi}$ are given by

[TABLE]

More generally, the $\ell$-th cumulant is given by ${\kappa}_{\ell}(N_{A}^{\Psi})=\operatorname{tr}P_{\ell}(T_{A})$ where the $P_{\ell}$ are defined recursively by $P_{1}(X)=X$ and $P_{\ell+1}(X)=X(1-X)P^{\prime}_{\ell}(X)$. An application of Theorem 1.2 gives the following asymptotics expansion.

Theorem 1.6**.**

Assume that $A$ has a smooth boundary. Then we have the complete asymptotic expansions:

[TABLE]

for any $N$ and $\ell\in{\mathbb{N}}^{*}$.

We will also provide estimates for the generating function of $N_{A}^{\Psi}$, cf. Proposition 8.1. By Theorem 1.6, the variance of $N_{A}^{\Psi}$ is equal to

[TABLE]

A consequence is the following tail estimate and convergence to the normal distribution. Introduce the fluctuation $\widetilde{N}_{A}^{\Psi}:=N_{A}^{\Psi}-\mathbb{E}(N_{A}^{\Psi})$ and the critical exponent ${\alpha}_{c}=n/2-1/4$.

Corollary 1.7**.**

**

there exists $k_{0}$ and $C>0$ such that for any $k\geqslant k_{0}$, for any $\beta>0$, we have \Pr\,\bigl{(}|\widetilde{N}_{A}^{\psi}|\geqslant k^{{\alpha}_{c}+{\beta}}\bigr{)}\leqslant e^{-k^{\min({\alpha}_{c}+{\beta},2{\beta})}/C}. 2. 2.

$k^{-{\alpha}_{c}}\widetilde{N}_{A}^{\psi}$* converges in distribution to a centered normal random variable with variance $(2\pi)^{-n}\operatorname{vol}(\partial A)I(P_{2})$.*

The variance estimate (11) can be compared with earlier results. Consider the linear statistics of $\mu_{\Psi}$, that is the random variables of $(M^{d_{k}},\mu_{\Psi})$ of the form $\mathcal{N}[u](x_{1},\ldots,x_{d_{k}})=\sum_{i=1}^{d_{k}}u(x_{i})$ with $u:M\rightarrow{\mathbb{R}}$ a general function. When $u$ is the characteristic function $1_{A}$, $\mathcal{N}[u]$ has the same distribution as $N_{A}^{\Psi}$. By [Ber08, Theorem 1.5], when $u$ is Lipschitz, the variance of $\mathcal{N}[u]$ is of order $k^{n-1}$. So the regularity of $u$ brings a smaller variance.

The relevance of the estimates of the higher cumulants is not obvious. Still, we can define a model probability distribution whose cumulants are the leading order terms appearing in Theorem 1.6. Let $\widetilde{B}(p)=B(p)-p$ be the fluctuation of the Bernoulli distribution. For any $n\in{\mathbb{N}}$ and ${\alpha}>0$, let $X_{n}({\alpha})$ be the sum of $2n+1$ independent random variables $\widetilde{B}(\operatorname{er}({\alpha}m))$, $m=-n,-n+1,\ldots,n$. Then $X_{n}({\alpha})$ converges as $n\rightarrow\infty$ to a random variable $X({\alpha})$ with vanishing odd cumulants and even ones given by

[TABLE]

Choosing ${\alpha}^{-1}_{k}=k^{n-1/2}\operatorname{vol}(\partial A)/(2\pi)^{n}$, we recover the leading order term of Theorem 1.6, that is for any $\ell\geqslant 2$,

[TABLE]

An interpretation for this result is that the spectrum of $T_{A}$ can be approximated by the numbers $\operatorname{er}({\alpha}_{k}m)$, $m\in{\mathbb{Z}}$ for what concerns the cumulant computations.

Entanglement entropy

Before we present our last application let us introduce the notion of entanglement, cf [BLPY16], [HR06] for detailed presentations. Let $H$ be a complex Hilbert space that we assume finite dimensional to simplify the exposition. Denote by $\mathcal{L}(H)$ the space of self-adjoint endomorphisms of $H$ and by $\mathcal{S}(H)$ the subset of all positive trace 1 endomorphisms. The elements of $\mathcal{L}(H)$ are the observables, the elements of $\mathcal{S}(H)$ are the mixed states. To any observable $A$ and mixed state $\rho$ of $H$ is associated a probability distribution whose moments are $\operatorname{tr}(\rho^{n}A)$, $n=1,2,\ldots$. $\mathcal{S}(H)$ is a convex set whose extreme points are the rank one projectors, which are called the pure state. A nonzero vector $\phi$ of $H$ defines a pure state $P(\phi)$ which is the orthogonal projection onto ${\mathbb{C}}\phi$. Furthermore any mixed state $\rho$ may be considered as a classical statistical mixture of pure states. The von Neumann entropy of a mixed state $\rho\in\mathcal{S}(H)$ is defined by

[TABLE]

It measures the amount of “mixedness” of a state. Typically, $S(\rho)=0$ if and only if $\rho$ is pure, whereas $S(\rho)$ is maximal for $\rho=(\operatorname{dim}H)^{-1}\operatorname{id}_{H}$.

Consider now two finite dimensional Hilbert spaces $H_{1}$, $H_{2}$ and let $H=H_{1}\otimes H_{2}$. Given a mixed state $\rho\in\mathcal{S}(H)$, the partial trace $\rho_{1}:=\operatorname{tr}_{H_{2}}(\rho)$ is a mixed state of $H_{1}$, called the reduced state. It is characterized amongst the mixed states of $H_{1}$ by

[TABLE]

So $\rho_{1}$ represents the state $\rho$ in the subsystem $H_{1}$ since it allows to compute the expectation values of all the observables of $H_{1}$. In contrast to classical systems, it is possible that $\rho$ is pure and $\rho_{1}$ is not, so $\rho$ is not a product of pure states. For such entangled pure state, we define the entanglement entropy as the von Neumann entropy of $\rho_{1}$. This quantity is a measure of the degree of entanglement of $\rho$.

Back to our application, the Fermionic state $\Psi$ defined in (8) may be viewed as a state of a bipartite system \bigwedge\mathcal{E}_{k}=\bigl{(}\bigwedge\mathcal{E}_{k}(A)\bigr{)}\otimes\bigl{(}\bigwedge\mathcal{E}_{k}(A^{c})\bigr{)}. The reduced state $\rho_{A}$ of $\Psi$ is a finite rank endomorphism and its von Neumann entropy is given in terms of the Toeplitz operator $T_{A}$ by

[TABLE]

This relation is generally deduced from Wick’s Theorem [Pes03]. For the convenience of the reader, we will present an elementary alternative proof in Section 7. Now we can deduce from Theorem 1.2 the asymptotic behavior of $S(\rho_{A})$.

Theorem 1.8**.**

If $A$ has a smooth boundary, we have

[TABLE]

in the limit $k\rightarrow\infty$, where $f(x)=-x\ln x-(1-x)\ln(1-x)$.

Landau level

For an electron in a plane with an external perpendicular magnetic field $B$, the Hamiltonian is $H=\frac{1}{2}(P_{x}^{2}+P_{y}^{2})$ with $P_{x}=\frac{1}{i}\partial_{x}+\frac{1}{2}By$ and $P_{y}=\frac{1}{i}\partial_{y}-\frac{1}{2}Bx$, cf. for instance [Eza08, Chapter 10]. Here we have put all the physical constants equal to 1 except the perpendicular component $B$ of the magnetic field. Then it is customary to introduce the operators $a=\frac{1}{\sqrt{2B}}(P_{x}+iP_{y})$, $a^{*}=\frac{1}{\sqrt{2B}}(P_{x}-iP_{y})$ so that

[TABLE]

So the lowest eigenvalue of $H$ is $B/2$ and the corresponding eigenspace, the lowest Landau level, consists of the $\psi$ satisfying $a\psi=0$. Introduce the complex coordinate $z=(x+iy)/\sqrt{2}$, then $(P_{x}+iP_{y})/\sqrt{2}=\frac{1}{i}(\partial_{\overline{z}}+Bz)$. We conclude that $a\Psi=0$ if and only if $\psi=fe^{-B|z|^{2}}$ with $f$ holomorphic. So the lowest Landau level is identified with the Bargmann space $\mathcal{B}_{\hbar}$ with $\hbar=B^{-1}$.

In the geometric setting, starting from a Riemannian manifold $(M,g)$ and a Hermitian line bundle $L\rightarrow M$ equipped with a connection $\nabla$, we define the magnetic Laplacian

[TABLE]

acting on ${\Gamma}(M,L)$. The magnetic field $B$ is equal to $i$ multiplied by the curvature of $\nabla$. If $M={\mathbb{R}}^{2}$ and $L$ is the trivial line bundle with connection $\nabla=d+\frac{i}{2}B(ydx-xdy)$, we recover the previous Hamiltonian.

Suppose now that $L$ is a positive holomorphic Hermitian line bundle $L\rightarrow M$, with the connection $\nabla$ being the Chern connection and the Riemannian metric $g$ being the metric determined by the curvature of $\nabla$ as explained in the beginning of Section 1.1. Then for any positive integer $k$, we have a magnetic Laplacian $H_{k}$ acting on sections of $L^{k}$.

We have another natural Laplacian acting on sections of $L^{k}$ which is the holomorphic Laplacian $\Delta_{k}=\overline{\partial}^{*}\overline{\partial}$. The Bochner-Kodaira formula relates these two Laplacians

[TABLE]

It is a generalization of (13). A proof may be found in [Dem96, 7.3] or [BGV04, Proposition 3.71]. By definition $\Delta_{k}$ is non-negative and its kernel is the space $\mathcal{H}_{k}$ of holomorphic sections of $L^{k}$. So the lowest eigenvalue of $H_{k}$ is $nk$, and the lowest Landau level is $\mathcal{H}_{k}$. The magnetic field is $i\Theta(L^{k})=ik\Theta(L)$. So the semiclassical limit $k\rightarrow\infty$ corresponds to a large magnetic field.

Acknowledgements

The authors would like to express their warm gratitude to Benoît Douçot for bringing about this collaboration and for pointing out that the IQH entanglement entropy can be computed in terms of the spectrum of a convenient Toeplitz operator. B.E. also thanks Nicolas Regnault and Semyon Klevtsov for valuable discussions.

2 Kähler quantization

Let $L\rightarrow M$ be a holomorphic Hermitian line bundle on a complex compact manifold $M$. Recall that $L$ has a natural connection $\nabla$, called the Chern connection and determined by the condition that it is compatible with the holomorphic and Hermitian structures. If ${\sigma}$ is a local holomorphic frame of $L$, then $\nabla{\sigma}=-(\partial\varphi)\otimes s$ where $\varphi=-2\ln|{\sigma}|$. The curvature of $\nabla$ is given by $\Theta(L)=\partial\overline{\partial}\varphi$. So ${\omega}:=i\Theta(L)$ is a real two form given in local complex coordinates by

[TABLE]

We assume that $L$ is positive, meaning that $(\partial^{2}\varphi/\partial z_{i}\partial\overline{z}_{j})$ is a positive definite matrix at any point. So ${\omega}$ is a Kähler form. We also have a Riemannian metric defined by $g(X,Y)={\omega}(X,jY)$. The Riemannian volume density will be denoted by $\mu$.

Let $A\rightarrow M$ be another holomorphic Hermitian line bundle not necessarily positive. Since $M$ is compact, the space $\mathcal{H}_{k}$ of holomorphic sections of $L^{k}\otimes A$ is finite dimensional. Its dimension is given at first order in the limit $k\rightarrow\infty$ by

[TABLE]

where $\mu(M)=\int_{M}\mu$. $\mathcal{H}_{k}$ has a natural scalar product

[TABLE]

where $(s,t)(x)$ denotes the pointwise scalar product at $x$. To any function $f$ of $M$ and $k\in{\mathbb{N}}$ is associated a Toepliz operator $T_{f}:\mathcal{H}_{k}\rightarrow\mathcal{H}_{k}$ defined by

[TABLE]

where the scalar product on the right-hand side is still defined by (15). Basic properties of Toeplitz operators are

[TABLE]

for any smooth functions $f,g$, where the $\mathcal{O}$ depends on $f$ and $g$. Observe that the Toeplitz operator $T_{f}$ is well-defined by (16) for any integrable multiplicator $f$. The trace estimate (18) still holds in this case, whereas (17) is not true for integrable functions as we will see.

The Bergman kernel of $\mathcal{H}_{k}$ is the holomorphic section $\Pi_{k}$ of $(L^{k}\otimes A)\boxtimes(\overline{L}^{k}\otimes\overline{A})$ defined by

[TABLE]

where $(s_{i,k})_{i=1,\ldots,d_{k}}$ is any orthonormal basis of $\mathcal{H}_{k}$. The restriction of the Bergman kernel to the diagonal can be considered as a function on $M$ because we have natural identification $L_{x}\otimes\overline{L}_{x}\simeq{\mathbb{C}}$ and $A_{x}\otimes\overline{A}_{x}\simeq{\mathbb{C}}$ given by the metrics of $L$ and $A$. We have

[TABLE]

where the $\mathcal{O}$ is uniform in $x\in M$. Furthermore for any compact set $K$ of $M^{2}$ not intersecting the diagonal, for any $N$, there exists $C_{N}$ such that

[TABLE]

The transition between (19) and (20) may be described as follows. Define a distance on $M$ by embedding $M$ into ${\mathbb{R}}^{N}$ and restricting the Euclidean distance. Denote by $|x-y|$ the distance between $x$ and $y$. Then there exists constants $C>0$, $(C_{N})_{N\in{\mathbb{N}}}$ such that for any $x,y\in M$ and $N\in{\mathbb{N}}$ we have

[TABLE]

We can actually characterize the Bergman kernel up to a $\mathcal{O}(k^{-\infty})$ in terms of geometric datas as follows. For any tangent vector $X$ of $M$, we denote by $X^{1,0}$ and $X^{0,1}$ its holomorphic and antiholomorphic parts, so $X^{1,0}=\frac{1}{2}(X-ijX)$ and $X^{0,1}=\frac{1}{2}(X+ijX)$, where $j$ is the complex structure.

Theorem 2.1**.**

We have for any $k\in{\mathbb{N}}$

[TABLE]

where

$E$* is a section of $L\boxtimes\overline{L}$ independent of $k$. $|E|<1$ outside the diagonal, $E=1$ on the diagonal. Furthermore $\nabla E=\frac{1}{i}{\alpha}_{E}\otimes E$ on a neighborhood of the diagonal, where ${\alpha}_{E}$ is a one form of $M^{2}$ vanishing along the diagonal and such that for any vector fields $X_{1}$, $X_{2}$, $Y_{1}$ and $Y_{2}$ of $M$, we have*

[TABLE] 2. 2.

$a(\cdot,k)$* is a sequence of sections of $A\boxtimes\overline{A}$ having an asymptotic expansion*

[TABLE]

where the $\mathcal{O}$ is uniform on $M^{2}$, the coefficients $a_{\ell}$ are sections of $A\boxtimes\overline{A}$, the restriction of $a_{0}$ to the diagonal being constant equal to $1$. 3. 3.

$R_{k}$* is a section of $(L^{k}\otimes A)\boxtimes(\overline{L}^{k}\otimes\overline{A})$ whose pointwise norm is uniformly in $\mathcal{O}(k^{-N})$ for any $N$.*

Estimates (19), (20) and (21) are all consequences of Theorem 2.1. In particular, to deduce (21), observe that the conditions satisfied by $E$ imply that $\varphi_{E}=-2\ln|E|$ vanish along the diagonal and is positive outside the diagonal. A short computation from (22) gives the Hessian of $\varphi_{E}$ along the diagonal: for any vector fields $X_{1}$, $X_{2}$ of $M$,

[TABLE]

where we denote by $\mathcal{L}$ the Lie derivative and by $g$ the Riemannian metric as above.

Theorem 2.1 has been deduced in [Cha03] from the Szegö kernel description of [BdMS76]. For other results on the Bergman kernel asymptotic behaviour, we refer the reader to [Zel98] and [MM07].

3 First spectral estimates

The Toeplitz quantization is real and positive in the sense that for any integrable function $f$ of $M$, the corresponding Toeplitz operators $T_{f}$ is Hermitian when $f$ is real and non negative when $f$ is non negative. Since for any mesurable set $A$ of $M$, the characteristic function of $A$ satisfies $0\leqslant 1_{A}\leqslant 1$, we deduce that $T_{A}:=T_{1_{A}}$ is Hermitian with its spectrum in $[0,1]$.

As mentioned in the introduction, if $A\subset M$ has a non empty interior and $M$ is connected, then [math] is not an eigenvalue of $T_{A}$. Indeed, if $T_{A}s=0$