Entropy and Temperature in nite isolated quantum systems

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Entropy and Temperature in ﬁnite isolated quantum systems

Phillip C. Burke1and Masudul Haque2, 1, 3

1Department of Theoretical Physics, Maynooth University, Maynooth, Kildare, Ireland

2Institut f¨ur Theoretische Physik, Technische Universit¨at Dresden, 01062 Dresden, Germany

3Max-Planck Institute for the Physics of Complex Systems, Dresden, Germany

(Dated: April 5, 2023)

We investigate how the temperature calculated from the microcanonical entropy compares with

the canonical temperature for ﬁnite isolated quantum systems. We concentrate on systems with sizes

that make them accessible to numerical exact diagonalization. We thus characterize the deviations

from ensemble equivalence at ﬁnite sizes. We describe multiple ways to compute the microcanonical

entropy and present numerical results for the entropy and temperature computed in these various

ways. We show that using an energy window whose width has a particular energy dependence

results in a temperature with minimal deviations from the canonical temperature.

I. INTRODUCTION

In recent years, there has been considerable inter-

est in understanding how statistical mechanics emerges

from the quantum dynamics of isolated many-body sys-

tems. Such considerations invariably require a correspon-

dence between energy, a quantity well-deﬁned in quan-

tum mechanics, and temperature, which is necessary for

a statistical-mechanical description. Since temperature

is not a priori deﬁned in quantum mechanics, assigning

temperatures to energies is a nontrivial issue.

Ideas regarding thermalization in isolated quantum

systems, e.g., the eigenstate thermalization hypothesis

(ETH) [1–9] and its extensions or variants are often

tested or veriﬁed using numerical “exact diagonalization”

calculations [5,6,10–50]. As a result, quantum sys-

tems with Hilbert space dimensions between ∼103and

∼105have acquired particular relevance. It is therefore

important to ask how meaningful various deﬁnitions of

thermodynamic quantities like temperature or entropy

are for ﬁnite systems, particularly systems of sizes typ-

ically treated by full numerical diagonalization. In this

work, we critically examine diﬀerent ways of calculating

entropy from the energy eigenvalues of ﬁnite systems and

compare the temperature derived from the entropy with

the so-called ‘canonical’ temperature.

The most common deﬁnition of temperature for ﬁnite

isolated quantum systems is the canonical temperature.

This is obtained for any energy Eby inverting the canon-

ical equation

E=hHi=tre−βH H

tr(e−βH )=Pne−βEnEn

Pne−βEn.(1)

If the eigenvalues {En}of a system are known, then

this relationship provides a map between energy and the

canonical temperature Tc= (kBβc)−1. (Here kBis the

Boltzmann constant.) The relationship (1) originates

in statistical mechanics from the context of a system

with a bath. However, it is widely used in the study

of thermalization of isolated (bath-less) quantum sys-

tems, to obtain an energy-temperature correspondence

[5,6,10,11,13,15–19,24,26,28,36,45,51–55].

An alternate way to deﬁne temperature is to use the

thermodynamic relation [56–61]

T=∂E

∂S Xi

=∂S

∂E −1

.(2)

Here Sis the (thermal) entropy, and the subscript Xi

denotes the system parameters that should be held con-

stant. For an isolated (i.e. microcanonical) quantum

system, deﬁning the entropy S(E) at a particular en-

ergy Einvolves counting the number of eigenstates (‘mi-

crostates’) within an energy window ∆Earound that en-

ergy E. This raises the question of how to choose the

width ∆E, or possibly avoiding an explicit choice of ∆E

and instead estimating the density of states. In the large-

size (thermodynamic) limit, these choices can be shown

to become inconsequential. This paper aims to explore

the consequences of these choices for ﬁnite sizes, focusing

on systems of Hilbert space dimensions ∼104typical for

full numerical diagonalization studies.

Motivated by the analysis of Ref. [62], we consider four

choices for deﬁning the entropy. In each case, we compare

the resulting temperature obtained using Eq. (2) with the

canonical temperature obtained directly from inverting

Eq. (1).

First, we consider counting eigenstates in an arbitrar-

ily chosen but constant (energy-independent) width win-

dow, as illustrated in Figure 1(top). This is the most

obvious choice but turns out to be far from optimal —

the resulting temperature deviates strongly at ﬁnite sizes

from the canonical temperature.

Second, noting that the leading correction to the large-

size limit can be made to disappear by choosing ∆E∝

pT2

cCc[62], we examine the result of using such an

energy-dependent window width. (Here Ccis the heat ca-

pacity.) Such an energy-dependent window is illustrated

in Figure 1(bottom). We show that this choice works ex-

tremely well for the sizes of interest, modulo some caveats

regarding the proportionality constant.

Finally, we can compute the microcanonical entropy

using standard numerical estimation procedures for the

d.o.s. via approximations to the integrated d.o.s. or cu-

mulative spectral function. We ﬁrst formulate the deﬁ-

arXiv:2210.02380v2 [cond-mat.stat-mech] 4 Apr 2023

FIG. 1. Illustration of two ways of choosing the energy win-

dow ∆Eused to deﬁne the microcanonical entropy. The top

ﬁgure shows the obvious choice: the width ∆Eis the same

at all energies. The lower schematic illustrates an energy-

dependent width: ∆E(E)∝√T2

cCc, where the temperature

Tcis the canonical temperature corresponding to energy E

and Ccis the heat capacity at T=Tc. The window is shown

at three energy values Ea,Eb, and Ec, which are in the re-

gions of low temperature, inﬁnite temperature, and low neg-

ative temperature. Each vertical tick marks an eigenvalue.

nition without reference to a speciﬁc energy window ∆E

and show that this choice is sub-optimal. We analyze

the reason for the strong ﬁnite-size mismatch in this

case. Secondly, we show that using the energy-dependent

∆E∝pT2

cCc, designed to account for ﬁnite-size eﬀects,

results in excellent agreement between the temperatures

even at the small sizes investigated, without any ﬁne-

tuning of proportionality constants.

This article is organized as follows. Section II recounts

the deﬁnitions and saddle-point expressions that lead to

the various choices for calculating the microcanonical en-

tropy. We also provide details (II C) of the quantum

many-body systems that we use for numerical explo-

ration — we provide results for multiple systems with

diﬀerent geometries, to ensure that the resulting conclu-

sions are not artifacts of a particular lattice or Hamilto-

nian. In the next three sections, we describe the proce-

dures for, and the results of, the four ways of calculat-

ing entropy. First, Section III describes counting eigen-

states in a constant-width energy window ∆E. Second,

Section IV describes counting eigenstates in an energy-

dependent window ∆E(E), designed to cancel out the

explicit ∆E-dependence of the resultant temperature,

following/extending the suggestion of Ref. [62]. Third,

Section Voutlines using a smoothened cumulative d.o.s.

Ω(E) to calculate the d.o.s. g(E). We can then choose ei-

ther to neglect ∆E, which we explain leads to deviations,

or make use of the derived energy dependent ∆E(E) de-

signed to account for these deviations. Section VI pro-

vides concluding discussion and some context.

II. PRELIMINARIES

We ﬁrst recall the standard deﬁnition of microcanoni-

cal entropy and highlight the roles of the density of states

g(E) and of the energy window ∆E(subsection II A). We

then recall (II B) the saddle-point formulation often used

to show the equivalence of microcanonical and canonical

ensembles in the large-size limit [62], and extend beyond

the leading order in order to analyze the eﬀect of ∆E

at ﬁnite sizes. Subsection II C describes the quantum

systems to be used in subsequent sections to provide nu-

merical examples.

A. Entropy, density of states, and the energy

window

The microcanonical entropy S(E) of a system at en-

ergy Eis [57–62]

S(E) = kBln Γ(E),(3)

where Γ(E) is the statistical weight, which is the num-

ber of microstates at energy E. For a quantum system,

microstates are to be interpreted as eigenstates. For a

quantum system with a discrete spectrum, counting the

number of eigenstates is problematic because at any par-

ticular energy there is usually zero or one eigenstate, or

perhaps a handful if there are degeneracies. Thus, S(E)

would be = −∞ for all values of energy except at a count-

able number of discrete energy values. This issue is usu-

ally resolved [59,60,62] by taking Γ(E) to be the number

of eigenstates in an energy window ∆Earound E, rather

than the number of eigenstates exactly at energy E. Thus

we deﬁne

Γ(E) = ZE+∆E/2

E−∆E/2

g(E0)dE0

=ZE+∆E/2

E−∆E/2X

δ(E0−En)dE0,(4)

where the sum is understood to to include all the eigen-

values of the system Hamiltonian that lie in the window,

i.e., all Ensatisfying En∈(E−∆E

2, E +∆E

2) [63]. It

is then common to approximate this integral [62,64–67]

via

Γ(E)≈∆Eg(E) = ∆EX

δ(E−En).(5)

Here, g(E) is the density of states — the number

of many-body eigenstates per unit energy interval. Al-

though deﬁned as a sum over delta functions, the density

of states can be thought of as a smooth function of energy

over energy scales much larger than the typical level spac-

ing. In numerical work, this is often achieved by broad-

ening the delta functions into Gaussians or Lorentzians

of ﬁnite width [24,68–75]. Alternatively, we can deﬁne

Ω(E) as the number of eigenstates with energy less than

E, i.e., the integrated density of states. Fitting a smooth

function to the staircase form of Ω(E), one can obtain a

smooth density of states as the derivative; g(E)=Ω0(E).

The entropy now depends on an energy window ∆E,

so we are thus faced with choosing an appropriate ∆E.

The purpose of introducing a ﬁnite energy width was to

smooth out the discreteness of the energy spectrum, so

∆Eshould be large enough to include a large number

of eigenstates. On the other hand, we want ∆Eto be

suﬃciently small so that the density of states (regarded

as a smooth function of energy) does not vary appreciably

within the window, i.e., ∆Eshould be much smaller than

the scale of the bandwidth of the system. Other than

these general principles, we have the freedom to choose

∆E, and in general, the entropy Swill depend on the

choice.

As we will explain next (II B), the sub-leading con-

tributions to the entropy, which contain ∆E, vanish in

the large system size limit. So, for inﬁnite systems, it

will generally not matter what ∆Eis, but the choice can

drastically aﬀect the entropy and resultant temperature

for ﬁnite quantum systems. This paper aims to investi-

gate and clarify the eﬀect of this choice for ﬁnite systems

whose Hilbert space sizes make them accessible to full

numerical diagonalization.

B. Saddle point expressions

To understand the role of ∆E, it is helpful to ex-

press the entropy as an integral over (complex) inverse

temperature and perform saddle-point approximations

[56,58,62], extending the order beyond what is necessary

in the thermodynamic limit, to account for ﬁnite sizes.

Replacing the delta function in Eq. (5) with δ(x) =

R∞

−∞(2π)−1eiβxdβ, and deﬁning the free energy as

F(β) = −β−1lnPne−βEn, we can write the entropy

eS/kB= Γ(E)=∆EZi∞

−i∞

dβ

2πeβ(E−F(β)).(6)

To apply the saddle-point approximation, one ﬁrst ﬁnds

the critical point of the exponent h(β) = β(E−F(β)).

The condition is

E−F(β)−β∂F

∂β = 0 ,(7)

which is equivalent to Eq. (1) deﬁning the canonical tem-

perature. Thus the saddle point is at β=βc, the canon-

ical inverse temperature. The leading order saddle-point

approximation is thus

=βcE−βcF(βc) (8)

with βcbeing the solution of Eq. (7) or Eq. (1). This

matches the standard thermodynamic relation, and is

consistent with the energy derivative of Sbeing the in-

verse temperature βc.

To examine the eﬀect of ∆E, one needs to extend the

calculation to the next order. One expands h(β) as a

Taylor series about βc, up to second order, as the ﬁrst

order term is zero by deﬁnition. Introducing the heat

capacity C=∂E

∂T =−T∂2F

∂T 2, one obtains [62]

eS/kB≈∆E

2πeβc(E−F(βc)) Z∞

−∞

dye−kBT2

cCcy2/2.(9)

Here, Tcand Ccare the values at the saddle point βc.

Evaluating the Gaussian integral, one obtains

=βcE−βcF(βc) + ln ∆E−ln p2πkBT2

cCc.(10)

Here βcis determined by the energy, and hence so are

Tcand Cc. The two leading terms are both extensive

in system size. The ﬁrst correction term ln ∆Eis sub-

extensive as long as ∆Eis not chosen to grow exponen-

tially or faster with system size. The second correction

term grows logarithmically with system size, as the heat

capacity is extensive. However, as it appears in the ar-

gument of a logarithm, its dependence on Ldisappears

when diﬀerentiated with respect to E. This means that

when diﬀerentiating Eq. (10) to obtain β, the contribu-

tion from the last term does not grow with L, not even

logarithmically. Thus, at large enough sizes, the leading-

order saddle-point approximation suﬃces, and the choice

of energy window ∆Eplays no role.

However, the two correction terms should be consid-

ered when calculating the entropy at ﬁnite sizes. From

Eq. (10), we see that choosing ∆Eto be an energy-

independent constant (the most obvious choice, explored

in Section III) would leave an energy-dependent correc-

tion term, causing deviations from the canonical temper-

ature. We also see (as pointed out in Ref. [62] and ex-

plored below in Section IV) that the energy dependence

of the correction terms could be canceled by a judicious

choice of energy dependence for the width ∆E.

C. Hamiltonians and numerics

To investigate the eﬀect of diﬀerent choices for deﬁning

the microcanonical entropy in ﬁnite systems, we use sev-

eral spin-1/2 lattice systems consisting of Lspins, Nof

which are up. The spins interact via XXZ interactions,

which have a U(1) symmetry conserving N. We check

all results and show data for 1D, 2D, and fully-connected

geometries, to demonstrate that our results are very gen-

eral and not particular to any model. In the case of the

1D chain, we also include magnetic ﬁelds in the zand

xdirections; the latter break the U(1) symmetry. The

Hilbert space dimension is D=L

Nwhen Nis conserved

and D= 2Lotherwise.

The system parameters are always chosen such that

the level spacing statistics match that expected of chaotic

quantum systems. This ensures there are no complica-

tions due to proximity to integrability or localization.

Staggered ﬁeld XXZ chain: Starting with the open-

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EntropyandTemperatureinniteisolatedquantumsystemsPhillipC.Burke1andMasudulHaque2,1,31DepartmentofTheoreticalPhysics,MaynoothUniversity,Maynooth,Kildare,Ireland2InstitutfurTheoretischePhysik,TechnischeUniversitatDresden,01062Dresden,Germany3Max-PlanckInstituteforthePhysicsofComplexSystems,Dresden,...

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