Efficient simulation of low-temperature physics in one-dimensional gapless systems

PHYSICAL REVIEW B

110

, L041122 (2024)

Letter

Yuya Kusuki

Kotaro Tamaoka

Zixia Wei

and Yasushi Yoneta

Walter Burke Institute for Theoretical Physics,

California Institute of Technology

, Pasadena, California 91125, USA

Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS),

RIKEN

, Wako, Saitama 351-0198, Japan

Department of Physics, College of Humanities and Sciences, Nihon University, Tokyo 156-8550, Japan

Center for the Fundamental Laws of Nature and Society of Fellows,

Harvard University

, Cambridge, Massachusetts 02138, USA

Yukawa Institute for Theoretical Physics, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan

Center for Quantum Computing

RIKEN

, Wako, Saitama 351-0198, Japan

Department of Basic Science,

The University of Tokyo

, Meguro, Tokyo 153-8902, Japan

(Received 23 February 2024; revised 28 May 2024; accepted 24 June 2024; published 29 July 2024)

We discuss the computational efficiency of the finite-temperature simulation with minimally entangled typical

thermal states (METTS). To argue that METTS can be efficiently represented as matrix product states, we present

an analytic upper bound for the average entanglement Rényi entropy of METTS for a Rényi index 0

In particular, for one-dimensional (1D) gapless systems described by conformal field theories, the upper bound

scales as

(

log

)where

is the central charge and

is the system size. Furthermore, we numerically find

that the average Rényi entropy exhibits a universal behavior characterized by the central charge and is roughly

given by half of the analytic upper bound. Based on these results, we show that METTS can provide a speedup

compared to employing the purification method to analyze thermal equilibrium states at low temperatures in 1D

gapless systems.

DOI:

10.1103/PhysRevB.110.L041122

Introduction

. Simulating quantum many-body systems at

finite temperature is an everlasting topic in statistical physics

and computational physics. While thermal equilibrium states

at inverse temperature

of a quantum system with Hamilto-

nian

can be described by the canonical Gibbs state

(can)

≡

−

(

−

)

(1)

the computational cost to construct it directly diverges ex-

ponentially with system size, which makes it unrealistic in

practice to be accurately prepared.

The minimally entangled typical thermal states (METTS)

algorithm [

] is one of the methods developed to efficiently

simulate finite-temperature quantum systems. Considering a

spin-1

2 chain of length

, the orthogonal 2

classical prod-

uct states

=⊗

sites

form a complete basis of the

whole Hilbert space, where

}

is an orthogonal basis of

each site. METTS are a class of states defined as

−

√

−

(2)

By construction,

(can)

can be decomposed as a classical mix-

ture of METTS,

(can)

∑

≡

−

∑

−

(3)

To recover

(can)

accurately, one may sum over 2

METTS

with weight

, but performing such a calculation is usually

intractable. However, the METTS algorithm [

] enables one

Contact author: zixiawei@fas.harvard.edu

to generate a Markov chain of METTS and sample METTS

according to the probability distribution

. Thus, there is

no need to calculate

, and one can automatically sample

only METTS with large

. Empirically, it is known that one

only needs to sample roughly 10–100 METTS to get accurate

enough thermal expectation values of local observables. As a

result, the most computationally costly part of this algorithm

is to prepare each METTS by the imaginary-time evolution of

the product state. As evaluated in Refs. [

], the computation

time to prepare a METTS with a matrix product state (MPS)

ansatz [

] scales as

, where

is the bond dimension of

the MPS.

The METTS algorithm is indeed very efficient for simulat-

ing low-temperature physics in one-dimensional (1D) gapped

systems because it is expected that

can be upper bounded

by the bond dimension necessary to approximate the ground

state [

], which does not scale with

[

On the other hand, how well METTS behaves in 1D gap-

less systems is not clear at low temperature, while at high

enough temperature it is expected to be similarly efficient to

gapped systems. There are many reasons for this. First, the

arguments presented for gapped systems do not apply to gap-

less systems. At the low-temperature limit, METTS converge

to the ground state, whose entanglement scales as

(log

)

for a half of the whole 1D gapless system. This implies that

can be upper bounded by a polynomial of

, which does not

manifest the efficiency of METTS when

is large. There-

fore, one needs to understand how entanglement grows under

imaginary-time evolution for a generic METTS [

Can we give a bound of entanglement growth under

imaginary-time evolution for generic METTS? Is METTS

efficient for simulating low-temperature physics of 1D gapless

systems? To answer these questions, we focus on 1D gapless

2469-9950/2024/110(4)/L041122(6)

L041122-1

KUSUKI, TAMAOKA, WEI, AND YONETA

PHYSICAL REVIEW B

110

, L041122 (2024)

systems whose low-energy sectors are well described by 2D

conformal field theories (CFTs). After presenting some hints

from CFT analyses, we prove that for the Rényi index 0

1, the average entanglement Rényi entropy of a METTS

decomposition is upper bounded by that of the thermofield

double (TFD) state. Furthermore, we numerically show that

the average entanglement Rényi entropy is about half of the

analytic upper bound. Based on these results, we evaluate

the bond dimension required to approximate METTS and

the computation time for generating METTS. In the end, we

demonstrate the superiority of the METTS algorithm over the

TFD algorithm [

] (also called the ancilla method or purifi-

cation method) at low temperatures in 1D gapless systems.

Hints from CFT computation.

While it is difficult to ana-

lytically compute Rényi entropies for all the METTS, we can

do it for a special class of METTS with a boundary conformal

field theory (BCFT) computation. A BCFT is a CFT defined

on a manifold with boundaries whose boundary conditions

maximally preserve the conformal symmetries. Such bound-

ary conditions, when regarded as quantum states, are called

boundary states. One crucial feature of boundary states is that

they are product states [

]. Therefore, for a boundary state

−

turns out to be a METTS.

For simplicity, let us consider a 1D system defined on an

infinite line. The density matrix

−

is real-

ized by considering a path integral of an infinite strip with

width

.The

th entanglement Rényi entropy [

]ofahalf

line

can be computed by evaluating the expectation value

of a so-called twist operator inserted at the center of the strip

[

] and turns out to be

(

)

(

)

log

(

)

const

(4)

where

is the CFT central charge, and

is the UV cutoff.

The constant term does not depend on

but on the details

. Note that, except for this constant part, the above

result is universal for any CFT and any

. Also note that

roughly counts the degrees of freedom included in the CFT.

For example, a free boson CFT has

1 and an

-copy of

free-boson CFTs has

From this result, one may expect that the average Rényi

entropy of METTS has a similar scaling. With these as hints,

let us move on to rigorous analyses.

Analytic upper bound.

From now on, we will present a the-

orem which provides a rigorous upper bound for the average

Rényi entropy of a METTS decomposition.

We will utilize the TFD state. Consider two identical copies

and

of the original system (

∼

). Let

be an

antiunitary operator that commutes with the Hamiltonian

Then the TFD state

TFD

∈

⊗

is defined as

TFD

∑

−

⊗|

(5)

where

are eigenstates of

with energy

and

. Since classical product states

}

form a complete

orthonormal basis of

,wehave

TFD

(

−

⊗

)

∑

⊗|

(6)

The TFD state is a (symmetric) purification of the canon-

ical Gibbs state, i.e.,

(can)

∝

[

TFD

]

Therefore,

one can also simulate finite-temperature quantum systems

with TFD states. Dividing the original system into

and its

complement

, then each of the two copies

and

contains

a copy of

. We label them

and

, respectively.

We find that the average entanglement Rényi entropy of

over METTS is upper bounded by the entanglement Rényi

entropy of

TFD

associated with the same

,fora

Rényi index 0

Theorem 1.

For 0

1, there holds

(

)

≡

∑

(

)

(

)

(

)

(

TFD

)

(7)

are product states for

and

For generic quantum many-body systems, it is plausible to

expect that

≡

is also a product state. For the models

presented in this Letter,

is simply the complex conjugate

with respect to the spin basis, and this condition is satisfied.

A detailed proof is presented in the Supplemental Material

[

], which utilizes the concavity of

(

)

for 0

1 since

we used the concavity of

(

)

. This also means our proof is not

valid for

1. However, we numerically find that the upper

bound (

)isvalidevenfor

1 in all the examples we study

in this Letter. It will be interesting to answer whether the result

(

) is valid for

1 in general.

In fact, the

1 case of this theorem follows directly

from Theorem 7 of Ref. [

], which was considered in a

different context related to black hole physics. However, the

extension to 0

1 is necessary and sufficient to evaluate

the required bond dimensions for an MPS to approximate

METTS.

Let us then go back to 1D gapless systems defined on an

infinite line and take the subsystem

as a half of it. With CFT

computations (see, e.g., Ref. [

]), one finds

(

)

(

)

(

TFD

)

(

)

log

(

)

(8)

for any CFT. This rigorous upper bound turns out to be twice

the right-hand side of (

) at leading order.

Numerical study in spin systems.

After obtaining the

rigorous bound (

), let us numerically study the average en-

tanglement Rényi entropy of METTS in specific models. Our

results show that it exhibits a universal behavior characterized

only by the central charge of the associated CFT, similar to (

)

for a regularized boundary state.

Numerical simulations are performed for two models with

different central charges, symmetries, and integrabilities. We

take the spin quantization axis along the

direction (

) and choose eigenvectors of

}

(which appears

in the definition of

). Here,

are the spin operators

associated to the site

The first model is the transverse-field Ising chain (TFI)

with the open boundary condition,

=−

−

∑

−

∑

(9)

which is integrable. We set

2 so that the model is

critical [

]. This theory is well described by CFT with a

L041122-2

EFFICIENT SIMULATION OF LOW-TEMPERATURE ...

PHYSICAL REVIEW B

110

, L041122 (2024)

FIG. 1. Average entanglement entropy of METTS between the

left part (sites

) and the right part (sites

) as a function

of the size

of the subsystem

for the critical transverse-field

Ising chain with

and

4. Imaginary-time evolution is car-

ried out using second-order Trotter decomposition with a time step

δτ

04. The data are averaged over 10 000 samples.

central charge

2. Since this Hamiltonian does not have

SU(2) spin rotation symmetry, the entanglement properties

of METTS depend on the choice of the quantization axis

However, as will be seen later, the behavior of entanglement

entropy at large

is independent of

The second model is the spin-1

2 Heisenberg chain with

the next-nearest-neighbor interaction,

−

∑

(

)

−

∑

(

)

(10)

Again, we take the open boundary condition. We set

241 167 so that the model is critical [

–

]. This theory

is well described by CFT with a central charge

This model is known to be not integrable [

As shown in Fig.

, the entanglement profiles are insen-

sitive to the length of the whole spin chain

or length

of the interval subsystem. Therefore, it is sufficient to con-

sider the case where the interval

consists of the half chain

,...,

}

for sufficiently large

to investigate

the

dependence (see Supplemental Material [

] and also

Refs. [

] therein). Then, in Fig.

, we show the average

entanglement Rényi entropy for the half chain as a function of

.For

1, we find

(

)

(

)

log

(

)

(11)

where

(

)

is a nonuniversal constant. Furthermore, the

standard deviation of the entanglement entropy of METTS

is bounded by a constant and is negligible in the low-

temperature limit as compared to the average. That is, at

low temperatures, the entanglement Rényi entropy of METTS

coincides with the entropy of the regularized boundary state

(

) and a half of our upper bound by the entropy of the

TFD state (

) at the leading order of

. This result is con-

sistent with the intuition obtained from the discussion in the

low-temperature limit pointed out in Ref. [

]. However, it is

not immediately evident that the entanglement of METTS is

half of the TFD state even in the finite-temperature region

since the entanglement of the imaginary-time evolved state

significantly depends on the choice of the initial state.

Computation time in the METTS algorithm.

The area law

and the logarithmically slow growth of the entanglement

Rényi entropy (

)for0

1 implies that METTS can

be efficiently represented by an MPS ansatz [

]. Hence it is

expected that the METTS algorithm can be used to reach con-

siderably low temperatures even for critical spin chains. With

the results obtained above, let us evaluate the computation

time in the METTS algorithm for simulating the canonical

Gibbs state.

Consider a Schmidt decomposition of the METTS

∑

with respect to the bipartition

{

,...,

}

and its complement

. Suppose that the

Schmidt coefficients

’s are sorted in decreasing order, i.e.,

···

. The truncation error for an MPS approxi-

mation of bond dimension

is defined as

≡

∑

As shown in Ref. [

measures an error in an MPS

FIG. 2. Growth of the average entanglement Rényi entropy of METTS for the half chain

,...,

}

(a), (b) in the critical

transverse-field Ising model of length

640 and (c) in the critical Heisenberg model with the next-nearest-neighbor interaction of

length

1280. The insets show the standard deviation of the entanglement Rényi entropy. Imaginary-time evolution is carried out using

second-order Trotter decomposition with a time step

δτ

04 for the Ising model and

δτ

01 for the Heisenberg model. The data are

averaged over 10 000 samples for the Ising model and 5000 samples for the Heisenberg model.

L041122-3

KUSUKI, TAMAOKA, WEI, AND YONETA

PHYSICAL REVIEW B

110

, L041122 (2024)

FIG. 3. Minimum bond dimension

necessary to approximate

the TFD state and METTS with a truncation error of less than 10

−

for the critical transverse-field Ising chain with

640 sites. The

data points for METTS are the average of

of 10 000 samples. The

error bars indicate the standard deviation. We set the quantization

axis

. Imaginary-time evolution is carried out using second-

order Trotter decomposition with a time step

δτ

04.

approximation in the following sense: There exists an MPS

of bond dimension

such that

−|

∑

−

. Furthermore,

can be bounded from above by

log

−

(

)

−

log

−

)

(12)

for any 0

]. By inverting this inequality, we find an

upper bound on the bond dimension necessary to approximate

the METTS within the truncation error

log

(

)

−

log

log(1

−

)

(13)

Substituting the numerical result (

), we get an upper bound

for generic METTS scales as

(

−

)

(14)

for fixed

and

. Note that the constant factor depends on

and

, and hence it does not necessarily mean that the highest

efficiency is achieved at

1. On the other hand, also for the

TFD state, there is an upper bound on

that scales as [

]

(

−

)

(15)

Then we numerically check the bond dimension necessary

to approximate the TFD state and METTS. In Fig.

,weshow

the minimum bond dimension needed to achieve a truncation

error

of less than 10

−

for the TFD state and METTS as a

function of

. We can confirm that the exponent on

for the

TFD state is just twice that for generic METTS in the low-

temperature limit. This is consistent with Eqs. (

) and (

According to Ref. [

], the computation time of the

imaginary-time evolution for

scales as

. Therefore,

we can efficiently produce METTS at sufficiently low temper-

atures with the computation time that scales as a polynomial

with an exponent smaller than that of the TFD state.

We finally demonstrate the superiority of the METTS

algorithm over the TFD algorithm at low temperatures

by comparing the computation time in actual numerical

FIG. 4. Typical trajectory of the cumulative moving average of

the energy per site

in the Markov chain of the METTS sim-

ulation for the critical transverse-field Ising chain with

640 sites

128. We set the quantization axis

for the odd-numbered

steps and

for the even-numbered steps to reduce correlations be-

tween successive samples [

]. To avoid initial transients, we discard

the first five samples when calculating the sample averages. The solid

line shows the result of the TFD approach, considered exact, and

the shaded region shows a range of 0.1 times the standard deviation

(can)

from the exact value. The inset shows the total

computation time in the METTS simulation as a function of the

number of steps in the Markov chain. The solid line shows the

computation time in the TFD simulation.

simulations. To calculate expectation values in the canonical

Gibbs state

(can)

via the METTS algorithm, we generate

a Markov chain of METTS so that the long-time average

coincides with

(can)

and then calculate the average of the ex-

pectation values in METTS sampled from that chain. Figure

depicts a typical trajectory of the cumulative moving average

of the energy per site

in METTS and the compu-

tation time in the METTS simulation. It can be seen that it

only takes about 20 steps for sample averages to converge

within a range of 0.1 times the standard deviation

(can)

from the exact value. In other words, with only

a few samples, we can correctly obtain the energy with an

accuracy sufficiently smaller than the uncertainty inherent in

the ensemble. Furthermore, we also observe that the total

computation time for reaching that step is only about 1

of that for the TFD algorithm. We therefore conclude that

the METTS algorithm can provide a speedup over the TFD

algorithm when analyzing low-temperature states in 1D gap-

less systems. However, in both algorithms, it is possible to

reduce the computation time by increasing the Trotter error or

truncation error within an allowable error range, and in such a

case the TFD algorithm outperform the METTS algorithm [

Nevertheless, these errors are difficult to control, compared to

statistical errors, which can be easily controlled by simply in-

creasing the number of samples. In addition, in the case of the

METTS algorithm, it is possible to parallelize the sampling to

reduce the computation time.

Conclusions.

In this Letter, we focus on evaluating the

computational efficiency of the METTS algorithm for sim-

ulating low-temperature thermal equilibrium states in 1D

gapless systems. We find it is efficient in the sense that the

computation time scales as a polynomial function of

and has

L041122-4

EFFICIENT SIMULATION OF LOW-TEMPERATURE ...

PHYSICAL REVIEW B

110

, L041122 (2024)

a significant superiority compared to the TFD method. These

are based on both the analytic upper bound and numerical

investigations on the average entanglement Rényi entropy of

METTS, which are motivated by a hint obtained in a BCFT

computation.

We would like to emphasize that, although it was well

known in the experience that a METTS usually requires fewer

bond dimensions than a TFD state and the METTS algorithm

takes less time than the TFD method in some cases, why and

to what extent this is true was not well understood so far, both

for gapped systems and gapless systems. More than justify-

ing this experimental rule, our results also demonstrated that

the speedup of METTS compared to TFD for simulating 1D

gapless systems at low temperatures is parametrically large,

i.e., the speedup gets larger as

gets larger, different from 1D

gapped systems.

From a theoretical point of view, the average entangle-

ment entropy of METTS is also capable of upper bounding

the entanglement of formation for the canonical Gibbs state,

which is defined as the minimum average entanglement over

all of its possible pure state decompositions. This further

bounds operational entanglement measures, such as distillable

entanglement and entanglement cost. It is expected that the

METTS decomposition is quite close to being optimal. Thus,

it is possible to obtain a tight bound on the entanglement of

formation with the average entanglement of METTS. There-

fore, by utilizing the results of this work, one can investigate

the amount of entanglement in canonical Gibbs states.

Last but not least, METTS are low-entanglement atypical

states mimicking thermal equilibrium. Translated to the lan-

guage of quantum gravity via the anti–de Sitter (AdS)

CFT

correspondence, this means METTS may be understood as

black hole microstates which have nontrivial structures near

the horizon [

]. Therefore, the statistical features of METTS

may play an important role in understanding the microstate

physics of black holes.

Acknowledgments

. We are grateful to Tomotaka Kuwahara,

Hiroyasu Tajima, Tadashi Takayanagi, and Yantao Wu for

useful discussions. We would like to especially thank Yoshi-

fumi Nakata and Yantao Wu for careful reading and valuable

comments on a draft of this manuscript. Y.K. is supported

by the Brinson Prize Fellowship at Caltech and the U.S. De-

partment of Energy, Office of Science, Office of High Energy

Physics, under Award No. DE-SC0011632. K.T. is supported

by Grant-in-Aid for Early-Career Scientists No. 21K13920

and Grant-in-Aid for Transformative Research Areas (A) No.

22H05265. Z.W. is supported by the Society of Fellows at

Harvard University.

[1] S. R. White, Minimally entangled typical quantum states at

finite temperature,

Phys. Rev. Lett.

102

, 190601 (2009)

[2] E. M. Stoudenmire and S. R. White, Minimally entangled

typical thermal state algorithms,

New J. Phys.

, 055026

(2010)

[3] U. Schollwöck, The density-matrix renormalization group,

Rev.

Mod. Phys.

, 259 (2005)

[4] F. Verstraete and J. I. Cirac, Matrix product states represent

ground states faithfully,

Phys. Rev. B

, 094423 (2006)

[5] N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, Entropy

scaling and simulability by matrix product states,

Phys. Rev.

Lett.

100

, 030504 (2008)

[6] It is known that

for

general

METTS is upper bounded

O

(

√

)[

], which does not manifest the efficiency of

METTS at low temperatures. However, we only need to upper

bound the entanglement growth rate in

generic

METTS to eval-

uate the efficiency of the METTS algorithm. Therefore, there is

large room for improvement.

[7] A. E. Feiguin and S. R. White, Finite-temperature density ma-

trix renormalization using an enlarged Hilbert space,

Phys. Rev.

, 220401(R) (2005)

[8] M. Binder and T. Barthel, Minimally entangled typical thermal

states versus matrix product purifications for the simulation of

equilibrium states and time evolution,

Phys.Rev.B

, 125119

(2015)

[9] M. Miyaji, S. Ryu, T. Takayanagi, and X. Wen, Boundary states

as holographic duals of trivial spacetimes,

J. High Energy Phys.

05 (2015) 152.

[10] For a density matrix

defined on subsystem

, its

th en-

tanglement Rényi entropy is defined as

(

)

Tr(

)

−

,andits

entanglement entropy is defined as

=−

Tr(

log

). Note

that

lim

→

(

)

[11] P. Calabrese and J. L. Cardy, Entanglement entropy and quan-

tum field theory,

J. Stat. Mech. (2004) P06002

[12] T. Hartman and J. Maldacena, Time evolution of entanglement

entropy from black hole interiors,

J. High Energy Phys. 05

(2013) 014

[13] See Supplemental Material at

http://link.aps.org/supplemental/

10.1103/PhysRevB.110.L041122

for a detailed proof of The-

orem 1 and an analysis on finite-size effects on the average

entanglement entropy of METTS are presented in the Supple-

mental Material.

[14] Z. Wei and Y. Yoneta, Counting atypical black hole microstates

from entanglement wedges,

J. High Energy Phys. 05 (2024)

251

[15] T. Barthel, One-dimensional quantum systems at finite tem-

peratures can be simulated efficiently on classical computers,

arXiv:1708.09349

[16] S. Sachdev,

Quantum Phase Transitions

(Cambridge University

Press, Cambridge, UK, 2011).

[17] K. Okamoto and K. Nomura, Fluid-dimer critical point in

2 antiferromagnetic Heisenberg chain with next nearest

neighbor interactions,

Phys. Lett. A

169

, 433 (1992)

[18] K. Nomura and K. Okamoto, Critical properties of

antiferromagnetic XXZ chain with next-nearest-neighbour in-

teractions,

J. Phys. A: Math. Gen.

, 5773 (1994)

[19] S. Eggert, Numerical evidence for multiplicative logarithmic

corrections from marginal operators,

Phys.Rev.B

, R9612

(1996)

[20] T. C. Hsu and J. C. Anglès d’Auriac, Level repulsion in inte-

grable and almost-integrable quantum spin models,

Phys. Rev.

, 14291 (1993)

[21] A. E. Rastegin, Some general properties of unified entropies,

J. Stat. Phys.

143

, 1120 (2011)

L041122-5

KUSUKI, TAMAOKA, WEI, AND YONETA

PHYSICAL REVIEW B

110

, L041122 (2024)

[22] M. Binder and T. Barthel, Symmetric minimally

entangled typical thermal states for canonical and

grand-canonical ensembles,

Phys. Rev. B

, 195148

(2017)

[23] T. Kuwahara, Á. M. Alhambra, and A. Anshu, Im-

proved thermal area law and quasilinear time algorithm

for quantum Gibbs states,

Phys. Rev. X

, 011047

(2021)

L041122-6