Area-law entanglement from quantum geometry Nisarga Paul Department of Physics Massachusetts Institute of Technology Cambridge MA USA

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Area-law entanglement from quantum geometry

Nisarga Paul∗

Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA

Quantum geometry, which encompasses both Berry curvature and the quantum metric, plays a

key role in multi-band interacting electron systems. We study the entanglement entropy of a region

of linear size `in fermion systems with nontrivial quantum geometry, i.e. whose Bloch states have

nontrivial kdependence. We show that the entanglement entropy scales as S=α`d−1ln `+β`d−1+

··· where the ﬁrst term is the well-known area-law violating term for fermions and βcontains the

leading contribution from quantum geometry. We compute this for the case of uniform quantum

geometry and cubic domains and provide numerical results for the Su-Schrieﬀer-Heeger model,

2D massive Dirac cone, and 2D Chern bands. An experimental probe of the quantum geometric

entanglement entropy is proposed using particle number ﬂuctuations. We oﬀer an intuitive account

of the area-law entanglement related to the spread of maximally localized Wannier functions.

I. INTRODUCTION

A central topic in current condensed matter physics

is the study of strongly correlated phases in partially

ﬁlled Chern bands, such as unconventional superconduc-

tivity, Mott insulators, and fractional Chern insulators

[1–4]. For non-topological bands, the study of strongly

correlated phases often begins with the Hubbard model,

since one can always ﬁnd a basis of exponentially local-

ized Wannier orbitals. In contrast, one cannot ﬁnd such a

basis for topological bands in greater than one dimension,

a fact known as the Wannier obstruction [5]. For exam-

ple, in a Landau level in a disordered potential, most

states are localized while a narrow band of states must

be delocalized– which is crucial for the integer quantum

Hall eﬀect.

The inability to ﬁnd exponentially localized Wannier

orbitals in topological bands should imply longer-range

entanglement. Entanglement has proven a valuable con-

ceptual tool and diagnostic in topological condensed mat-

ter physics [6–8], but there has not been a detailed

study of real-space entanglement in partially ﬁlled Chern

bands.

We ﬁnd that an essential quantity is the quantum ge-

ometry of the partially ﬁlled band. Quantum geometry

is simply the structure of Bloch state overlaps, and it en-

compasses both Berry curvature and the quantum metric

[9]. Recent works have highlighted the importance of the

full quantum geometry, and not simply Berry curvature,

in the study of interactions in Chern bands, for instance

in twisted bilayer graphene [10–12].

In particular we show that nontrivial quantum geom-

etry contributes additional area-law entanglement; that

is, the von Neumann entropy of a region of linear size `

in ddimensional systems goes as

S=α`d−1ln `+β`d−1+··· (1)

where the quantum geometry contributes at order `d−1.

The leading term is a log-enhanced area law originating

∗npaul@mit.edu

from the O(`d−1) number of independent 1d chiral modes

on the Fermi surface, each of which contributes ln `[13].

αcan be computed using Widom’s conjecture and de-

pends only on the entangling region and Fermi surface,

not on the quantum geometry [14].

We ﬁnd it useful to deﬁne the quantum geometric en-

tanglement entropy SQG by subtracting from Sthe en-

tropy obtained by setting all Bloch overlaps huk|uqito

unity. In other words,

SQG =S−S|huk|uqi→1.(2)

SQG measures the additional entropy coming from intrin-

sically multi-band eﬀects, omitting contributions shared

with continuum fermions with an identical Fermi surface.

Our main results can then be expressed as follows:

1. We show SQG =β`d−1+··· for the case of uniform

quantum geometry (i.e. huk|uk+qi=g(q)) and

provide a closed form for β

2. We show numerically that SQG is area law for

the 1D Su-Schrieﬀer-Heeger model, the 2D massive

Dirac fermion, and 2D Chern bands

3. We establish that particle number variance receives

a similar area-law contribution from quantum ge-

ometry and propose an experimental measure of

SQG

As a corollary to 3, we ﬁnd a protocol to measure the

quantum metric at the band bottom using particle num-

ber ﬂuctuations.

This paper is organized as follows. In Sec. II we de-

ﬁne fermion entanglement entropy and other preliminar-

ies. In Sec. III we provide a simple two-particle example

which captures general features of quantum geometric

entanglement entropy. In Sections III, V, and VI we cal-

culate SQG for the cases of uniform quantum geometry

in ddimensions, the 1d Su-Schrieﬀer-Heeger model, and

2d models, respectively. In Sec. VII we establish similar

results for particle number ﬂuctuations and suggest an

experimental protocol for measuring SQG. In Sec. VIII

we provide an intuitive explanation for the area-law be-

havior of SQG and its relation to quantum geometry using

Wannier orbital spread, and in Sec. IX we conclude.

arXiv:2210.13502v1 [cond-mat.str-el] 24 Oct 2022

II. PRELIMINARIES

In this section we review how entanglement entropy

can be computed for fermions, although we note for

the reader that only Eq. (15) is needed in the sec-

tions that follow. We start by considering a system of

Nnoninteracting fermions with a discrete one-particle

spectrum. The many-body wavefunction takes the form

Ψ(x1,...,xN) = det[φn(xi)]/√N! where xiincludes

both spatial and internal coordinates (e.g. spin / sub-

lattice) and the φnare the normalized wavefunctions of

the occupied states. The partial density matrix of a spa-

tial region Ais

ρA= Tr ¯

A|ΨihΨ|(3)

and the α’th R´enyi entropy is

S(α)(A) = 1

1−αln Tr ρα

A,(4)

with α→1 the von Neumann entropy. We have the

useful relation [15]

ρA= (det C) exp 

X

x,y∈A

ln C−1−Ix,yc†

xcy

(5)

where

C(x,y) = c†(x)c(y)=

n=1

φ∗

n(x)φn(y) (6)

is the two-point correlator in Ψ restricted to A. This

allows us to express the R´enyi entropies as

S(α)(A) = X

eα(λ`) (7)

where

eα(λ) = 1

1−αln(λα+ (1 −λ)α) (8)

and λ`are the eigenvalues of C. Note that the limit

α→1 gives

e1(λ) = −λln λ−(1 −λ) ln(1 −λ).(9)

Since Cgrows in size with A, computing its spectrum

can be a nontrivial task even on a lattice. To simplify

things, we use the Fredholm determinant

DA(λ) = det[λI−C] (10)

to rewrite S(α)(A) as [16, 17]

S(α)(A) = Idλ

2πi eα(λ)d ln DA(λ)

dλ(11)

where the integration contour encircles the segment [0,1].

Next we introduce the overlap matrix [17, 18]

Anm =ZA

dxφ∗

n(x)φm(x).(12)

Importantly, by RAdxwe mean an integral over the spa-

tial region Aand a sum over internal indices. This matrix

enjoys the property

Tr Ak=

n1,...,nk=1 ZA

dx1φ∗

n1(x1)φn2(x1)···ZA

dxkφ∗

nk(xk)φn1(xk)

=ZA

dx1···ZA

dxkC(x1,xk)C(xk,xk−1)···C(x2,x1)

= Tr Ck(13)

and therefore [17]

ln DA(λ) = ln λ−

∞

k=1

Tr Ck

kλk= ln λ−

∞

k=1

Tr Ak

kλk

m=1

ln(λ−am) (14)

where the amare the eigenvalues of A. We apply the

residue theorem to Eq. (11) to obtain

S(α)(A) = X

eα(am),(15)

which lends itself well to numerical calculations since Ais

an N×Nmatrix, independent of the entangling region.

III. SIMPLE TWO-PARTICLE EXAMPLE

Some implications of quantum geometry for entangle-

ment entropy can be seen in a simple two-particle ex-

ample. Consider a Slater determinant of two occupied

single-particle states in a translationally invariant m-

band system,

φi(r) = L−d/2eiki·r|ukii, i = 1,2,(16)

where Ldis the system size and |ukiiis an m-component

vector. We will parameterize the Bloch overlap as

huk1|uk2i=ueiφ.(17)

When u= 1, the quantum geometry is trivial (φcan be

gauged away). Therefore the deviation of ufrom 1 is a

measure of quantum geometry, in a way we make precise.

Let the entangling region be A. The overlap matrix is

A=I0Idkueiφ

I−dkue−iφ I0(18)

where dk≡k2−k1and

Ik=L−dZA

ddre−ik·r.(19)

The eigenvalues of the overlap matrix are

a1,2=I0± |Idk|u(20)

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Area-lawentanglementfromquantumgeometryNisargaPaulDepartmentofPhysics,MassachusettsInstituteofTechnology,Cambridge,MA,USAQuantumgeometry,whichencompassesbothBerrycurvatureandthequantummetric,playsakeyroleinmulti-bandinteractingelectronsystems.Westudytheentanglemententropyofaregionoflinearsize`infer...

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Area-law entanglement from quantum geometry Nisarga Paul Department of Physics Massachusetts Institute of Technology Cambridge MA USA

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