Real-space entanglement spectra of projected fractional quantum Hall states using Monte Carlo methods Abhishek Anand and G. J. Sreejith

2025-04-29 0 0 1.03MB 14 页 10玖币

侵权投诉

Real-space entanglement spectra of projected fractional quantum Hall states using

Monte Carlo methods

Abhishek Anand and G. J. Sreejith

Indian Institute of Science Education and Research, Pune 411008, India

Real-space entanglement spectrum (RSES) of a quantum Hall (QH) wavefunction gives a natural

route to infer the nature of its edge excitations. Computation of RSES becomes expensive with

an increase in the number of particles and included Landau levels (LL). RSES can be eﬃciently

computed using Monte Carlo (MC) methods for trial states that can be written as products of

determinants such as the composite fermion (CF) and parton states. This computational eﬃciency

also applies to the RSES of lowest Landau level (LLL) projected CF and parton states; however,

LLL projection to be used here requires approximations that generalize the Jain Kamilla (JK)

projection. This work is a careful study of how this approximation should be made. We identify the

approximation closest in spirit to JK projection and perform tests of the approximations involved

in the projection by comparing the MC results with the RSES obtained from computationally

expensive but exact methods. We present the techniques and use them to calculate the exact RSES

of the exact LLL projected bosonic Jain 2/3 state in bipartition of systems of sizes up to N= 24

on the sphere. For the lowest few angular momentum sectors of the RSES, we present evidence to

show that MC results closely match the exact spectra. We also discuss other plausible projection

schemes. We also calculate the exact RSES of the unprojected fermionic Jain 2/5 state obtained

from the exact diagonalization of the Trugman-Kivelson Hamiltonian in the two lowest LLs on the

sphere. By comparing with the RSES of the unprojected 2/5 state from Monte Carlo methods, we

show that the latter is practically exact.

I. INTRODUCTION

Fractional quantum Hall (FQH) eﬀect1,2 provides a

rich set of experimentally realizable interacting topolog-

ically ordered quantum phases. Conceptual insights into

the microscopic structure of these phases have been aided

by careful numerical studies of variational wavefunctions

describing the phases. The many aspects of the FQH

states like fractional charge,3,4 non-trivial statistics4–8

and edge modes have been understood in the language of

such variational wavefunctions. Entanglement spectrum

(ES)9of a variational state is a key tool in the char-

acterization of the order in the state.10–14 The notion

of entanglement spectra was introduced as a means to

get more insights into the topological order beyond what

is provided by more succinct measures like the entan-

glement entropy.15–18 Entanglement spectra of a many-

body state is deﬁned as the eigenvalue spectrum (typi-

cally represented in a negative log scale) of the reduced

density matrix after a bipartition of the system. Cor-

respondence between the entanglement Hamiltonian and

the edge spectrum has been explored both in the FQH

contexts and elsewhere.13,19–24 Being a quantity natu-

rally obtained during DMRG calculations, entanglement

spectrum has been used to characterize the phases of in-

teracting Hamiltonians in such studies.25–28

In quantum Hall systems, entanglement spectrum can

be deﬁned by bipartitioning the system in the momen-

tum orbital space or in the real space producing the or-

bital space entanglement spectrum (OES) or real space

entanglement spectrum(RSES) respectively. Since under

Landau quantization, the single particle momentum or-

bitals are spatially localized, they contain closely related

information. Both OES and RSES have been studied

extensively for many QH states.29–34 Typically OES is

easier to compute for states obtained from exact diago-

nalization or DMRG calculations whereas RSES is easier

to compute for states described by variational wavefunc-

tions expressed in terms of particle coordinates. In this

paper, we focus on the RSES estimations using Monte

Carlo methods focusing on the Jain composite fermion

(CF) states.35

ES of CF wavefunctions have been studied but meth-

ods using exact evaluation of the LLL projected state

is computationally expensive, allowing studies only in

small systems (N∼10).30 On the other hand, entan-

glement spectrum of the unprojected CF state and par-

ton states35 can be eﬃciently obtained using Monte Carlo

methods.33,34 The bottleneck in extending the same tech-

nique to the projected CF states (which are energetically

more favorable) is the diﬃculty in implementing the LLL

projection. In a large number of studies involving en-

ergetics of the CF states, an approximate method in-

troduced by Jain and Kamilla (JK) to perform lowest

Landau level (LLL) projection has been found to pro-

vide computationally eﬃcient and reliable results.36–38

Combining the JK projection with the MC methods

for RSES evaluation involves further approximations but

this can be used for calculations in systems upto hundred

particles.32–34 Testing the approximations require com-

parison with computationally expensive but exact RSES

calculated using alternate methods. This is the goal of

the paper.

We identify cases where alternate exact methods can

be employed to calculate RSES in relatively large sys-

tems. For the fermionic Jain 2/5th state, the Trugman-

Kivelson (TK) Hamiltonian39 can be diagonalized to pro-

duce an exact expansion of the unprojected CF state.

arXiv:2210.11514v1 [cond-mat.str-el] 20 Oct 2022

RSES of the state can be obtained using a generalization

of the method presented by Chandran et al. in Ref. 29 We

use this to show that the MC method indeed produces

practically exact results for the RSES.

Construction of exact Hamiltonians for the projected

state is an open problem,40 so a similar strategy cannot

be employed to test the RSES for the projected state.

Instead we consider the case of the bosonic Jain 2/3rd

state, where we employ exact projection in a manner

that allows us to exactly compute the low momentum

entanglement spectrum in systems as large as N= 24.

We ﬁnd that as system size increases the results from

approximate projection employed in the MC method ap-

proach the exact RSES, at least in the low momentum

sectors.

We emphasize that the results presented in this work

is not a comparison of the RSES of the exact projected

CF states (ψEX) and the RSES of the JK projected CF

states (ψJK). Since the two states are nearly identically

to each other, we expect their RSES to be nearly the

same. Instead what we are testing is the eﬀect of the

approximate projection used while implementing the MC

method for RSES calculation. The approximation in the

latter is similar in spirit to the JK projection but is not

the same. Secondly, the results presented compare the

approximate results from MC estimates of RSES with

the RSES of the ψEX. The comparison is done with ψEX

rather than with ψJK because, using methods described

in Sec. III B, RSES of ψEX can be computed exactly at

least for small systems. Doing the same for ψJK is harder.

This paper is structured as follows. We begin by pre-

senting the numerical techniques involved. We present,

in Sec II, a strategy for a numerically exact computation

of the RSES of the unprojected Jain 2/5th state obtained

from exact diagonalization of the TK Hamiltonian. We

will later use this to demonstrate that the RSES using

MC method is practically exact for the unprojected state.

In Sec. III, we give a summary of the method for com-

puting RSES of variational states by expanding them in

terms of entanglement wavefunctions (EWFs). The de-

tails of the method, originally introduced in Ref. 33, can

be found in Refs. 32 and 34. Section III B provides

the details for using this method with exactly projected

Jain 2/3rd state of bosons. We could use this to ob-

tain numerically exact RSES in systems up to size 24.

Section III D details the approximations that are made

to perform LLL projection of the EWFs, which makes

accessing large systems possible. All numerical results

benchmarking the methods are given in Sec. IV, and ﬁ-

nally we conclude with Sec. V.

Notations: All calculations in the paper are performed

for systems in the spherical geometry where the sin-

gle particle Landau orbitals for a particle in the nth

LL and with angular momentum mare the monopole

harmonics,41–43 given by

YQnm =NQnm(−1)Q+n−mvQ−muQ+m

s=0 n

s 2Q+n

Q+n−m−s|v|2(n−s)|u|2s(1)

where uand vare given by u= cos(θ/2)eιφ/2and v=

sin(θ/2)e−ιφ/2in terms of coordinates 0 ≤θ < π and 0 ≤

φ < 2πon the sphere, and Qquantiﬁes the strength of

monopole which produces a radial magnetic ﬁeld of ﬂux

2Qin units of ﬂux quanta φ0=hc/e. The normalization

factor is given by

NQnm =

(2Q+ 2n+ 1)

4π

(Q+n−m)!(Q+n+m)!

n!(2Q+n)! 1/2

(2)

II. RSES OF STATE EXPANDED IN SLATER

DETERMINANT BASIS

In this section, we describe the method to calculate

RSES for fermionic quantum Hall states which are ex-

pressed as linear combinations

ψ(r1,...,rN) = X

cλMλ(r1,...,rN) (3)

where the basis states Mλare Slater determinants of

single particle momentum orbitals. The coeﬃcients cλ’s

can, for instance, be from exact diagonalization. The

basis states Mλare parametrized by the ordered list of

occupied single particle orbitals λ≡(λ1, λ2. . . ) and can

be expanded as

Mλ(r1,...,rN) = 1

√N!X

σ∈SN

(σ)

i=1

φλσ(i)(ri) (4)

where φλi(ri)’s are the normalized single particle Landau

orbitals λi≡(ni, mi) speciﬁed by the LL-index niand

the angular momentum mi, and SNis the set of all per-

mutations of (1,2, . . . , N). In the spherical geometry, the

single particle orbitals are given by monopole harmonics

given in Eq.(1).

We present a method which enables us to compute

RSES for states with particles occupying diﬀerent LLs.

For a real-space cut which respects the rotational symme-

try of the system, angular momentum states (projected

onto either subsystem) remain orthogonal to each other

as long as they are in same LL. However states with same

angular momentum but diﬀerent LLs have non-zero over-

lap due to restricted limits of integration within each sub-

system. The method presented below extends the one

given in Ref. 29 which works for states restricted to the

LLL, by incorporating non-orthogonal momentum states.

In this work, we will use this method to compute

the RSES of unprojected Jain 2/5-state, which we get

as the ground state (using ED) of Trugman-Kivelson

Hamiltonian39 projected into the lowest 2 LLs treated

as degenerate.

For any wavefunction ψ(r1,...,rN) for Nparticles,

the density matrix is given by

ρ(r0

1, . . , r0

N;r1, . . , rN) = ¯

ψ(r1, . . , rN)ψ(r0

1, . . , r0

RQid2ri|ψ(r1, . . , rN)|2

(5)

We partition the system into two subsystems Aand B

using an azimuthally symmetric cut (Fig.1) and con-

sider the sector where region Acontains NAparticles

(and Bhas NB=N−NAparticles). We use the

following shorthands for collections of particle coordi-

nates R≡(r1,...,rN), RA≡(r1,...,rNA) and RB≡

(rNA+1,...,rN). The reduced density matrix for subsys-

FIG. 1. Azimuthally symmetric cut for the spherical geom-

etry.

tem Ais then given by

ρNA(R

A;RA) = RBdRBρ(RA,RB;R0

A,RB)

RAdRARBdRBρ(RA,RB;RA,RB)

pNAN

NAZB

dRBρ(RA,RB;R

A,RB)

(6)

where pNAis the probability that subsystem Acontains

exactly NAparticles. Using Eqs.(3) and (4), we can

rewrite the Eq.(6) as

ρNA=1

pNAN

NAX

λ,λ0

¯cλcλ0ZB

dRBMλMλ0(7)

To simplify further, we will use the following property

Mλ(R) = rNA!NB!

N!X

µ,ν

hµ;νi=λ

µν Mµ(RA)Mν(RB).

(8)

Here the Slater determinant of Nparticles is expanded

as anti-symmetrization of products of Slater determi-

nants corresponding to ordered set of orbitals µ(of size

NA) and ν(of size NB) such that the ordered com-

bination of two is equal to λ( this constraint is rep-

resented by hµ;νi=λ). The sign corresponding to

the permutation σ, which makes (λσ(1), . . . , λσ(N)) =

(µ1, . . . , µNA, νNA+1, . . . , νN) is given by µν =(σ).

This allows us to rewrite ρNAas

ρNA(R

A,RA) = X

µ,µ0

Qµ,µ0Mµ(RA)Mµ0(R

A) (9)

where

Qµ,µ0=1

pNAX

ν,ν0ZB

dRB¯

Fµ

ν(RB)Fµ0

ν0(RB)

Fµ

ν(RB) = µν chµ;νiMν(RB) (10)

This integral contains the overlaps between the angular

momentum states which are restricted in Bsubsystem.

As mentioned before, overlap between single particle mo-

mentum orbitals in subsystem Bis non-zero only when

they have same angular momentum. Hence, only those

ordered sets ν,ν0contribute in the sum in Eq.(II) which

have identical set of angular momentum quantum num-

bers. Note that the LL-indices need not be the same.

The matrix Qin Eq.(10) can be numerically computed

by summing over such ordered sets νand ν0.

The reduced density matrix ρNAis block-diagonal

in angular momentum. Its LA

z-sectors, represented as

ρNA,LA

zis

ρNA,LA

z(R

A,RA) =

µ,µ0Mµ(RA)Qµ,µ0Mµ0(R

A).(11)

where we the restricted sum is over ordered sets µ,µ0of

size NAwhich have the correct total angular momentum

i.e. Lz(µ) = Lz(µ0) = LA

z. As the Slater determinant

states Mµs span the entire Hilbert space of A-subsystem,

any eigenvector χof ρNAwith eigenvalue kcan be written

as following linear combination

χ(RA) = X

aµMµ(RA) (12)

where the basis Mµs are generally not orthogonal if they

contain states from higher LLs (n > 1). Using the eigen-

value equation, given by

dRAρNA(RA,R

A)χ(RA) =kχ(R

A) (13)

and Eq.(12), it can be shown that a matrix Mcan be

constructed such that it has same set of non-zero eigen-

values as that of ρNA, where M=QP where Qis deﬁned

in Eq.(10) and Pis given by

Pµ,µ0=ZA

dRAMµ(RA)Mµ0(RA) (14)

which is the overlap matrix (inside A) for EWFs, which

can be computed using accurate numerical integration.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Real-spaceentanglementspectraofprojectedfractionalquantumHallstatesusingMonteCarlomethodsAbhishekAnandandG.J.SreejithIndianInstituteofScienceEducationandResearch,Pune411008,IndiaReal-spaceentanglementspectrum(RSES)ofaquantumHall(QH)wavefunctiongivesanaturalroutetoinferthenatureofitsedgeexcitations.C...

展开>> 收起<<

Real-space entanglement spectra of projected fractional quantum Hall states using Monte Carlo methods Abhishek Anand and G. J. Sreejith.pdf

共14页,预览3页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Real-space entanglement spectra of projected fractional quantum Hall states using Monte Carlo methods Abhishek Anand and G. J. Sreejith

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: