Different temperature-dependence for the edge and bulk of entanglement Hamiltonian Menghan Song1Jiarui Zhao1Zheng Yan2 3 1 and Zi Yang Meng1 1Department of Physics and HKU-UCAS Joint Institute of Theoretical and Computational Physics_2

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Diﬀerent temperature-dependence for the edge and bulk of entanglement Hamiltonian

Menghan Song,1Jiarui Zhao,1Zheng Yan,2, 3, 1, ∗and Zi Yang Meng1

1Department of Physics and HKU-UCAS Joint Institute of Theoretical and Computational Physics,

The University of Hong Kong, Pokfulam Road, Hong Kong SAR, China

2Department of Physics, School of Science, Westlake University,

600 Dunyu Road, Hangzhou 310030, Zhejiang Province, China

3Institute of Natural Sciences, Westlake Institute for Advanced Study,

18 Shilongshan Road, Hangzhou 310024, Zhejiang Province, China

(Dated: July 21, 2023)

We propose a physical picture based on the wormhole eﬀect of the path-integral formulation to

explain the mechanism of entanglement spectrum (ES), such that, our picture not only explains

the topological state with bulk-edge correspondence of the energy spectrum and ES (the Li and

Haldane conjecture), but is generically applicable to other systems independent of their topological

properties. We point out it is ultimately the relative strength of bulk energy gap (multiplied with

inverse temperature β= 1/T ) with respect to the edge energy gap that determines the behavior of

the low-lying ES of the system. Depending on the circumstances, the ES can resemble the energy

spectrum of the virtual edge, but can also represent that of the virtual bulk. We design models

both in 1D and 2D to successfully demonstrate the bulk-like low-lying ES at ﬁnite temperatures,

in addition to the edge-like case conjectured by Li and Haldane at zero temperature. Our results

support the generality of viewing the ES as the wormhole eﬀect in the path integral and the diﬀerent

temperature-dependence for the edge and bulk of ES.

Introduction.— More than a decade ago, Li and Hal-

dane [1] proposed that the entanglement spectrum (ES)

is a direct measurement of the topological properties of

quantum many-body systems, in that the low-lying ES

are closely related to the true energy spectra on the edges

of open boundary systems. Entanglement entropy (EE)

as another important measure of quantum correlation

has been studied carefully over the years. It is pointed

out that EE at ﬁnite temperatures obeys a volume law

and resembles the thermal dynamic entropy of small sub-

systems [2–4]. Although the generality of Li and Hal-

dane’s statement has been questioned [5], it is still be-

lieved by many that overall the ES reveals more entan-

glement information and other non-local observables [6–

19]. Later, Qi, Katsura and Ludwig analytically demon-

strated the bulk-edge correspondence between the ES of

(2+1)D gapped topological states and the energy spec-

trum on their (1+1)D edges [20]. However, apart from

these gapped topological states, the universality of Li

and Haldane conjecture still remains an open and intrigu-

ing question. Previous works on EE [2–4] and reduced

density matrix [21] suggest the resemblance between the

bulk spectra of subsystem and the entanglement spectra

at suﬃciently high temperatures, while the relationship

between the edge mode and bulk mode is somehow un-

known so far. What’s more, besides the abstract mathe-

matical proofs, there exists no intuitive and transparent

physical picture that could penetrate the barrier between

the microscopic lattice models and the ﬁeld-theoretical

continuum description and to explain these phenomena

in simple language.

Part of the reason, that the Li and Haldane con-

jecture still remains a conjecture is because it is very

hard to compute the ES for generic 2D or 3D quantum

many-body systems. Due to the exponentially growth of

computation complexity and memory cost [6, 7, 10, 22–

26], systems with long boundaries of entanglement re-

gion and at higher dimensions are in general prohib-

ited. Besides few exactly solvable limits, most of the

ES studies by exact diagonalization (ED) and density

matrix renormalization group (DMRG) so far have fo-

cused on (quasi) 1D quantum systems. For 2D, several

ED/DMRG studies have extracted the ES of systems

with small width [21, 27, 28]. In fact, many of the ES

studies target topological phases [10, 29–37]. Methods

that can probe generic ES information in more common

systems at high dimensions and with larger sizes are still

in demand.

Quantum Monte Carlo (QMC), on the other hand,

usually stands out as the powerful tool to explore quan-

tum many-body systems with larger sizes and at higher

dimensions, as the importance sampling scheme can in

principle reduce the exponential complexity into a poly-

nomial one [38–44]. Although QMC in a path-integral

formulation accesses the partition function instead of the

ground state wave function, it has been shown that the

computation of Rényi EE can be achieved by sampling

the partition function in a replicated manifold with dif-

ferent boundary conditions for the entanglement region A

and the environment Aof the system [8, 45–50]. Conse-

quently, the entanglement signature and scaling behavior

of many novel phases and phase transitions in EE have

been reliably extracted in QMC simulations in higher di-

mension systems [15, 16, 19, 51–61].

The wormhole picture of ES.— Recently, some of us ex-

tended the QMC computation of the EE to that of the

ES and have successfully reduced the computational com-

plexity and made the computation of ES with long en-

arXiv:2210.10062v2 [quant-ph] 20 Jul 2023

tanglement boundaries and in higher dimensions possi-

ble [62, 63]. The basic idea is that in the replicated man-

ifold with partition function (as shown in Fig. 1 (a))

Z(n)

A= Tr [ρn

A] = Tr e−nHA,(1)

where ρA= TrAρis the reduced density matrix (RDM),

deﬁned as the partial trace of the total density matrix ρ

of Hamiltonian Hover a complete basis of Aand HAis

the corresponding entanglement Hamiltonian (EH), one

can deﬁne the eﬀective imaginary time βA=nfor the

EH at integer points n= 1,2,3, .... It’s worthwhile to

note that the βAshall be integer points in the QMC sim-

ulation because we can only simulate the whole RDM

ρA= TrA(e−βH )instead of a fractional one. One can

then compute the dynamic correlation functions G(τA)

at these integer time points. The ES can be readily ob-

tained from the imaginary time correlations of EH via

numeric analytic continuation methods, such as stochas-

tic analytic continuation (SAC) [64–73].

𝛽

𝐴 𝐴̅

𝛽!= 𝑛

𝜏!= 0

𝜏!= 1

……

𝜏!= 𝑛

(a) (b)

𝛽!= 𝑛

……

𝛽

𝐴 𝐴̅

FIG. 1. (a) Graphical representation of the partition func-

tion Z(n)

Ain the replicated manifold. The shaded area is the

entanglement region Ain which all the replicas are glued to-

gether along the imaginary time and has length βA=nβ. In

the environment region ¯

A, replicas are independent along the

imaginary time axis and each has length β. (b) Two diﬀerent

worldline paths along the imaginary time. Yellow one travels

inside the bulk while blue one goes into the environment and

experiences the wormhole eﬀect. Black circles are the worm-

holes which teleport a worldline to the other side of a replica

in ¯

Avia the virtual path (blue dashed line).

The physical picture of the above process of computing

the ES in the path-integral is described clearly via the

schematic diagrams in Fig. 1, where Fig. 1 (a) is the

replicated manifold of the path-integral of Z(n)

Ain Eq. (1),

and a wormhole eﬀect emerges at the entangled edges in

Fig. 1 (b). During QMC simulation, there are two typical

paths of the worldline [38, 74–78] in the path-integral

in the Fig. 1 (b): the yellow one deep in the bulk goes

through all the replicas with an imaginary time length

∼nβ; the blue one at the entangled edge can take a much

shorter path because of the periodic boundary condition

(PBC) of length βof every replica in A. The dashed blue

line shows how the wormholes (the black circles in Fig. 1

(b)) teleport the worldline from the bottom to top of one

replica without propagating through βdue to such PBC,

so that the total imaginary time length of the wormhole

path is only about ∼n. We therefore dubbed this shorter

path eﬀect cased by the diﬀerent connections of the Z(n)

in the space-time as the "wormhole" picutre of ES [62].

In the path-integral, the shorter path always leads to

more important contribution. The spectral function S(ω)

for physical observable, represented as O, can be writ-

ten by the eigenstates |n⟩with the eigenvalue Enof the

Hamiltonian H,S(ω) = 1

πPm,n e−βEn|⟨m|O|n⟩|2δ(ω−

[Em−En]). Therefore, there is a relation between en-

ergy spectrum S(ω)and imaginary time correlation G(τ)

as G(τ) = R∞

0dωK(ω, τ)S(ω). The K(ω, τ)is a kernel

with slightly diﬀerent expressions for bosonic/fermionic

O. Obviously, the imaginary time correlation G(τ)is the

summation of all the gap modes. The important modes

have larger weights in the summation. When the temper-

ature is low enough (β=∞), the S(ω)can be treated

as G(τ)∼e−τ∆, where the ∆is the ﬁrst-excited gap

and τis the length of imaginary time. In our wormhole

picture, the time length is diﬀerent for edge and bulk,

thus the wormhole path plays the important role. As

pointed out in Ref. [62], we can simply estimate the ratio

of the exponential factors deep inside the bulk and that

at the edge to be roughly β∆b: ∆e. At the ground state,

β→ ∞ renders β∆b≫∆eand edge path will domi-

nate over the path-integeral and be more important for

the low-energy ES, which is the Li and Haldane conjuc-

ture. But one immediately sees from here that the Li and

Haldane conjecture is not only limited to the topological

phase (gapped bulk and gapless edge), but also works

in the cases where both bulk and edge are gapped [79].

Moreover, since ﬁnite size systems always acquire ﬁnite

size gaps, the low-lying ES can resemble the edge spectra

when the temperature is low enough, but for systems at

ﬁnite temperatures, when the edge exponential factor is

much larger than the bulk one, and the low-lying ES will

resemble the bulk energy spectra. It is surprising that

one can also obtain the bulk-like low-lying ES at ﬁnite

temperature, hinted in Ref. [62], and the wormhole mech-

anism oﬀers a direct and lucid picture to understand the

appearance of bulk information at ﬁnite temperature im-

plied in previous works [2–4, 21]. In this article, we focus

on systematic demonstration of such wormhole picture

and the diﬀerent temperature-dependence for the edge

and bulk of entanglement Hamiltonian via QMC simula-

tions in 1D and 2D quantum spin systems.

1D Heisenberg Chains.— Here we employ the stochastic

series expansion QMC method [38, 74–78] to simulate

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Differenttemperature-dependencefortheedgeandbulkofentanglementHamiltonianMenghanSong,1JiaruiZhao,1ZhengYan,2,3,1,∗andZiYangMeng11DepartmentofPhysicsandHKU-UCASJointInstituteofTheoreticalandComputationalPhysics,TheUniversityofHongKong,PokfulamRoad,HongKongSAR,China2DepartmentofPhysics,SchoolofScience...

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Different temperature-dependence for the edge and bulk of entanglement Hamiltonian Menghan Song1Jiarui Zhao1Zheng Yan2 3 1 and Zi Yang Meng1 1Department of Physics and HKU-UCAS Joint Institute of Theoretical and Computational Physics_2

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