Infinite Range Correlations in Non-Equilibrium Quantum Systems and Their PossibleExperimental Realizations

Zohar Nussinov1, 2, ∗

1Department of Physics, Washington University, St. Louis, MO 63160, USA2Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA

(Dated: February 27, 2018)

We consider systems that start from and/or end in thermodynamic equilibrium while experiencinga finite rate of change of their energy density or of other intensive quantities q at intermediate times.We demonstrate that at these times, during which the global intensive quantities q vary, the sizeof the associated covariance, the connected pair correlator |Gij | = |〈qiqj〉 − 〈qi〉〈qj〉|, between anytwo (arbitrarily far separated) sites i and j is, on average, finite. This non-vanishing characterof the connected correlations for asymptotically distant sites also applies to theories with purelylocal interactions. In simple models, these correlations may be traced to the generic volume lawentanglement of finite temperature states. Once the global mean of q no longer changes, the averageof |Gij | over all spatial separations |i− j| may tend to zero. However, when the equilibration timesare significant (e.g., as in a glass that is not in true thermodynamic equilibrium yet in which theenergy density (or temperature) reaches a final steady state value), these long range correlationsmay persist also long after q ceases to change. We briefly discuss possible experimental implicationsof our findings and speculate on their potential realization in glasses (where a prediction of a theorybased on the effect that we describe here suggests a universal collapse of the viscosity that agreeswith all published viscosity measurements) and non-Fermi liquids.

PACS numbers: 05.50.+q, 64.60.De, 75.10.Hk

I. INTRODUCTION

In theories with local interactions, the connected cor-relations between two different sites i and j often de-cay with their spatial separation |i − j|. Indeed, con-nected correlations decay exponentially with distance insystems with finite correlation lengths. In massless (orcritical) theories, this exponential decay is typically re-placed by an algebraic drop. The detailed understandingof these decays was achieved via numerous investigationsthat primarily focused on venerable systems with fixedcontrol parameters, e.g., [1–5]. In the current work, wewish to build on these notions and ask what occurs ina general non-relativistic system, when an intensive pa-rameter such as the average energy density (set, in allbut the phase coexistence region where latent heat ap-pears, by the temperature) or external field is varied sothat, during transient times, the system is forcefully keptout of thermal equilibrium. We will illustrate that, un-der these circumstances, extensive fluctuations will gen-erally appear. These large fluctuations will imply the ex-istence of connected two point correlation functions thatwill, on average, remain finite for all spatial separations.Although our considerations are general, we will largelycouch these for theories residing on d−dimensional hyper-cubic lattices of N = Ld sites; the average energy densityε ≡ E/N with E the total energy. In theories with localinteractions, we may express (in a variety of ways) the

Hamiltonian H as a sum of N ′ = O(N) terms (HiN′

i=1)

∗ zohar@wustl.edu

that are each of finite range, H =∑N ′

i=1Hi. Our resultsapply to both lattice and continuum systems. In both ofthese cases, our principal interest lies in the thermody-namic (N 1) limit.

II. OUTLINE

The remainder of this paper is organized as follows:In Section III, we explain why, notwithstanding its strik-ing nature, our main finding of large variances (even insystems with local interactions) and the infinite rangecorrelations that they imply is quite natural. Next, inSection IV, we discuss special situations in which our re-sults do not hold- those of product states. Albeit theirsimplicity and appeal, product states do not generallydescribe systems above their ground state energy den-sity. To that end, we study in Section V, two dual sys-tems on lattices in arbitrary dimensions for which a classof finite energy density eigenstates can be exactly con-structed: (1) general rotationally symmetric spin modelsin an external magnetic field and (2) systems of itiner-ant hard-core bosons with attractive interactions. Weinvestigate the effects of “cooling/heating” and “doping”protocols on these systems and illustrate that, regard-less of the system size, after a finite amount of time, no-table energy or carrier density fluctuations will appear.Armed with these proof of principle demonstrations ofthe energy density and number density fluctuations, weexamine in Section VI the anatomy of the Magnus ex-pansion to see how generic these large fluctuations maybe. Straightforward calculations illustrate that althoughthere exist fine tuned situations in which the variance

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of intensive quantities such as the energy density remainzero (e.g., the product states of Section IV), such cir-cumstances are exceedingly rare. General evolutions thatchange the expectation values of various intensive quan-tities will, concomitantly, lead to substantial standarddeviations. In Section VII, we go one step further andestablish that under a rather mild set of constraints, in-finite range connected fluctuations are all but inevitable.Our effect has broad experimental implications: commonsystems undergoing heating/cooling and/or other evolu-tions of their intensive quantities may exhibit long rangecorrelations. In Sections VIII and IX, we turn to twoprototypical systems and ask whether our findings mayrationalize experimental (and numerical) results. In par-ticular, in Section VIII, we discuss glasses and show auniversal collapse of the viscosity data that was inspiredby considerations similar to those that we describe in thecurrent work. In Section IX, we ask whether the broad-ened distributions that we find may lead to “non-Fermi”liquid type behavior in various electronic systems. Weconclude in Section X with a synopsis of our results. InAppendix A, we show that using entangled state (simi-lar to those analyzed in Section V) reproduces the finitetemperature results for an Ising chain. Our goal in pro-viding this pedagogical one dimensional example is tofurther emphasize that general thermal states are not ofthe product state form discussed in Section VI.

III. INTUITIVE ARGUMENTS

To make our more abstract discussions clear, we firsttry to motivate why our central claim might not, at all,be surprising. Consider a system that is, initially, inthermodynamic equilibrium with a sharp energy densityε. For an initial closed equilibrium system (described bythe microcanonical ensemble), the standard deviation ofε scales as 1/N while in open systems connected to a heat

bath, the standard deviation of ε is O(1/√N). In either

of these two cases, the standard deviation of ε vanishesin the thermodynamic limit (similar results apply to anyintensive thermodynamic variable). Now imagine coolingthe system. As the system is cooled, its energy densityε drops. Various arguments hint that as ε drifts down-wards in value, its associated standard deviation also in-creases. This is analogous to the increase in width of aninitially localized “wave packet” with a non-trivial evo-lution (with the energy density itself playing the role ofthe packet location). On a rudimentary level, it might behardly surprising that the energy density obtains a finitestandard deviation when it continuously varies in time.A finite standard deviation of the energy density implieslong range correlations of the local energy terms. This is

so since the variance of the energy density

0 < σ2ε =

1

N2

∑i,j

(〈HiHj〉 − 〈Hi〉〈Hj〉)

≡ 1

N2

∑i,j

Gij ≤1

N2

∑i,j

|Gij | ≡ |G|. (1)

Thus, if σε is finite then the average |G| of |Gij | over allseparations |i − j| will be non-vanishing. More broadly,similar considerations apply to any quantity q = 1

N

∑i qi

that must have a sharp value in thermodynamic equilib-rium. Thus, generally, if q broadens as some parametersare varied, there must be finite connected correlations(〈qiqj〉 − 〈qi〉〈qj〉) even when |i− j| → ∞. Identical con-clusions to the ones presented above may be drawn forsystems that end in thermodynamic equilibrium (insteadof starting from equilibrium) while experiencing a finiterate of change of their energy density at earlier times atwhich Eq. (1) will hold. Empirically, in cases of exper-imental relevance, as in, e.g., cooling or heating a ma-terial, if that the rate of change of its temperature orenergy density is finite then Eq. (1) will hold. Althoughheat (and other currents) associated with various inten-sive quantities q traverse material surfaces, experimen-tally, even for thermodynamically large systems, the rateof change of energy density ε, and other intensive quanti-ties q can be readily made finite, i.e., dq/dt = O(1). Thisexperimentally relevant situation is the focus of our at-tention. We nonetheless remark that if the energy density(or other intensive parameter) exchange rate are domi-nated by contributions in Eq. (1) with i and j closeto the surface then the average connected correlator forarbitrarily far separated sites i and j will be boundedby |G| ≥ O(N−2/d) [6]. As we will emphasize in Sec-tion VII, in order to achieve a finite rate of change ofany intensive quantity (including that of the energy den-sity dε/dt (or, equivalently, of the measured temperaturedT/dt)), the coupling between the system and its sur-roundings must be extensive [7]. In reality, due to thesurface flow of the heat current from the surroundingenvironment to the system during periods of heating orcooling, the local energy density in the system is gener-ally spatially non–uniform and may depend on the spa-tial distance to the surrounding external bath from whichheat flows to the system. The existence of such a spa-tially non-uniform profile of the local energy density mayfurther enhance the large fluctuations that we find in thecurrent work sans an assumption of such nonuniformity.We will briefly touch on related aspects towards the endof Section VIII.

Our analysis will naturally allow for entangled quan-tum states. To highlight this aspect, we will typicallykeep all factors of ~ explicit.

3

IV. PRODUCT STATES

Prior to demonstrating that energy density broadeningnaturally accompanies a cooling or heating of the sys-tem, we first regress to a “classical” situation of statesthat may be associated with individually decoupled localsubsystems between which no entanglement exists. For adensity matrix ρ that is a direct tensor product of localdensity matrices ρlMl=1 that act on disjoint spaces, withM = O(N),

ρ = ρ1 ⊗ ρ2 ⊗ · · · ⊗ ρM , (2)

the standard deviation σE of the total energy will, inaccord with the central limit theorem, be O(

√N) even

when the rate of change of the energy dE/dt may be ex-tensive (i.e., ∝ N). As a case in point, we may considerthe initial state |ψ0

Ising〉 = |s01s

02 · · · s0

N 〉 to be a low en-

ergy eigenstate of an Ising model HI = −∑〈ij〉 JijS

zi S

zj

that is acted on during intermediate times by a trans-verse magnetic field Hamiltonian (Htr = −By(t)

∑i S

yi )

that alters the energy as measured by HI (thereby heat-ing or cooling the system). Here, any eigenstate of HI

is a product state |ψ〉 = |s1s2 · · · sN 〉 with si = ±1 cor-responding to the eigenvalues of the local spin operatorsSzi . A uniform rotation, between an initial time (t = 0)and a final time tf , of all of the N spins around the y spinaxis by the transverse field Hamiltonian Htr Hamiltonian

by an angle∫ tf

0By(t′)dt′ = π/2 will transform |ψ0

Ising〉 to

a final state |χ〉 that is an equal modulus superpositionof all Ising product states (all eigenstates of HI), viz.,

|χ〉 = 2−N/2∑s1s2···sN (−1)

∑Ni=1(δ

s0i,−1

δsi,−1)|s1s2 · · · sN 〉,with δσi,σj the Kronecker delta. We next discuss whatoccurs when the exchange constants Jij are of finite rangebut are otherwise arbitrary. The standard deviation ofthe energy (i.e., the standard deviation of HI) associatedwith this final rotated state (and any other state duringthe evolution) of the initial Ising product state scales as

O(√N) while the energy change is extensive [8]. The

state |χ〉 corresponds to the infinite temperature limitof the classical Ising model of HI (its energy density isequal to that of the system at infinite temperature andsimilarly all correlation functions vanish). A key point isthat generic finite temperature states are not of the typeof Eq. (2). In fact, general thermal states (i.e., eigen-states of either local or nonlocal Hamiltonians that are el-evated by a finite energy density difference relative to theground state) typically display volume law entanglemententropy [9–12] in agreement with the eigenstate thermal-ization hypothesis [13–21] while ground states and manybody localized states [22–28] retain area law entropies[29]. The entanglement entropy of individual quantum“thermalized” states imitates the conventional thermo-dynamic entropy of the macroscopic system that they de-scribe. In order to further flesh out these notions, in Ap-pendix A, we illustrate that correlations in finite energydensity eigenstates of the Ising chain mirror those in equi-librated Ising chains at positive temperatures. In the one

dimensional Ising model and other equilibrium systemsat temperatures T > 0, the high degree of entanglementand mixing between individual product states leads tocontributions to the two point correlation functions thatalternate in sign and ultimately lead to the usual decayof correlations with distance. Our main point is thatan external driving Hamiltonian (such as that presentin cooling/heating of a system) may lead to large exten-sive fluctuations. While these fluctuations may seem ex-pected for non-local operators (such as (Heisenberg pic-ture) time evolved local Hamiltonian terms in variousexamples), these generic fluctuations appear even for lo-cal quantities (e.g., local operators Hi in Eq. (1)). InSection V, we will study systems for which the relevant(Heisenberg picture) operators Hi are, indeed, local.

For completeness, we conclude this Section by remark-ing that even in classical systems with local interactions,broad distributions of various observables may occur inthe thermodynamic limit when these systems are disor-dered. This phenomenon is known as “non self averag-ing”, e.g., [30–33]. In these disordered classical systems,an ensemble average of a physical observable computedover different disorder realizations may differ significantlyfrom the expectation value of the same quantity in anysingle member of the ensemble. To put our work in abroader context, we note that the systems that we willfocus on in the current work need not be disordered norcritical. However, given the absence of self-averaging insuch disordered classical systems, we remark that thebroadening that we find will also apply to various sys-tems whenever the (“ensemble of”) eigenstates of thedensity matrix effectively describe these different disorderrealization of classical critical systems. This is so since,in such cases, the probability density matrix ρ averagewill emulate the average over an ensemble of these dis-ordered classical states. In instances such as the above,when all of the eigenvectors of the density matrix aretrivial classical local product states that do not exhibitentanglement, the system described by ρ is a classical sys-tem (with different classical realizations having disparateprobabilities). In general, there are both “classical” andinherently “quantum” contributions [34] to the varianceσ2ε of the energy density of Eq. (1). In the next sections,

we will demonstrate that even in a single quantum state,large fluctuations of any observable may naturally arisefor all system sizes (including systems in their thermo-dynamic limit).

V. TWO DUAL EXAMPLES

The existence of finite connected correlations |Gij | (Eq.(1)) for far separated sites |i − j| → ∞ is at odds withcommon lore. Before turning to more formal generalaspects, we illustrate how this occurs in two classes ofarchetypical systems- (i) any globally SU(2) symmetric(arbitrary graph or lattice) spin S = 1/2 model in anexternal magnetic field (Section V A) and (ii) dual hard

4

core Bose systems on the same graphs or latices (SectionV B).

A. Rotationally invariant spin models on all graphs(including lattices in general dimensions)

In what follows, we consider a general rotationallysymmetric spin model (Hsymm) of local spin-S momentsaugmented by a uniform magnetic field.

Hspin = Hsymm −Bz∑i

Szi . (3)

Amongst many other possibilities, the general rotation-ally symmetric Hamiltonian Hsymm may be a conven-tional spin interaction of the type

HHeisenberg =−∑ij

Jij ~Si · ~Sj

−∑ijkl

Wijkl(~Si · ~Sj)(~Sk · ~Sl) + · · · , (4)

with arbitrary Heisenberg spin exchange couplings Jij(whether these are of the nearest neighbor type or oflonger range or spatially uniform or not) augmented byconventional higher order rotationally symmetric terms.We wish to reiterate that the model of Eq. (3) is de-fined on any graph (including lattices in any number ofspatial dimensions). We label the eigenstates of Hspin

(and their energies) by |φα〉 (having, respectively, en-ergies Eα). In what follows, we will refer to the total

spin operator ~Stot =∑Ni=1

~Si. Since [~Stot, Hspin] = 0,it follows that all eigenstates of Hspin may be simulta-neously diagonalized with Sztot (with eigenvalue m~) and~S2tot (with eigenvalue Stot(Stot + 1)~2). Thus any eigen-

state of Eq. (3) may be written as |φα〉 = |µα;Stot, Sztot〉

with µα denoting all additional quantum numbers as-sociated in a given sector of Stot and Sztot. Althoughour results apply for local spins of any size S, in orderto elucidate certain aspects, we will often allude to spinS = 1/2 systems. For any eigenstate having a generalSztot 6= ±Smax = ±NS, the associated density matrix isnot of the local tensor product form of Eq. (2). Rather,any such eigenstate is a particular superposition of spinS = 1/2 product states having a total fixed value of Sztot.The state of maximal total spin Stot = Smax (which canbe trivially shown to be a non-degenerate eigenstate forany value of Sztot [35]) corresponds to a symmetric equalamplitude superposition of all such product states of agiven Sztot (i.e, such a sum of all product states of thetype | ↑1↑2↓3↑4↓5↑6 · · · ↑N−1↓N 〉 in which there are atotal of (N/2± Sztot/~) single spin of up/down polariza-tions along the z axis). We set an arbitrary eigenstate|φα〉 to be the initial state (at time t = 0) of the system|ψ0Spin〉. The energy density (and the global energy it-

self) will, trivially, have a vanishing standard deviationin any such initially chosen eigenstate, σε = 0. We nextevolve this initial (t = 0) state via a “cooling/heating

process” wherein the energy (as measured by Hspin) isvaried by replacing, during the period of time in whichthe system is cooled or heated, the Hamiltonian of Eq.(3) by a general time dependent transverse field Hamilto-nian Htr(t) = −By(t)

∑i S

yi . Once this “cooling/heating

process” terminates at a final time (t = tf ), the systemHamiltonian becomes, once again, the original Hamilto-nian of Eq. (3). During the evolution with Htr, the spinsglobally precess about the y axis. Thus, after a time t,the energy per lattice site is changed (relative to its ini-

tial value) by an amount BzSztotN (1−cos(

∫ tf0

By(t′) dt′)).Employing the shorthand w ≡ Sztot/Stot, the standard

deviation of the energy density ofHspinN is [36]

σε(t) =BzStot~| sin(

∫ tf0

By(t′) dt′)|N√

2

×√

1 +1

Stot− w2. (5)

When w = 1 (or −1) with Stot = Smax, the initial state|ψ0Spin〉 is a product state of all spins being maximally

up (or all spins pointing maximally down). Away fromthis singular Sz = ±Smax limit, spatial long range en-tanglement develops. When (1− |w|) = O(1), the scaledstandard deviation of the energy density is, for generaltimes, ( 1

~Bz )σε = O(1). A comparable standard devi-ation appears not only for the eigenstate but also forstates initial having an energy uncertainty of order O(1)(in units of Bz~) (e.g., c1|Stot, Sztot〉 + c2|Stot, Sztot − ~〉with c1,2 = O(1)). In the following, we briefly remarkon the simplest case of a constant (time indeodent) By.Here, the time required to first achieve 1

Bz~σε = O(1)

starting from an eigenstate of Hspin is O(1/By). Thisrequisite waiting time is independent of the system size(as it must be in this model where a finite σε is broughtabout by the sum of local decoupled transverse magneticfield terms in Htr). The large standard deviation im-plies (Eq. (1)) that long range connected correlations ofSzi emerge once the state is rotated under the evolution

with Htr. This large standard deviation of 1N

∑Ni=1 S

zi

appears in the rotated state displaying (at all sites i) auniform value of 〈Szi 〉. Even though there are no con-nected correlations of the energy densities themselves inthe initial state, the non-local entanglement enables longrange correlations of the local energy densities once thesystem is evolved with a transverse field. The varianceσε should, of course, not be confused with the spread ofenergy densities that the system assumes as it evolves(e.g., for the Sztot = 0 state, σε = O(1) while the energydensity ε(t) does not vary with time). We nonethelessremark that the standard deviation σε vanishes at thediscrete times tk = kπ/By (with k an integer)- the verysame times where the rate of change of the energy densityε(t) is zero. Indeed, in our model system, up to important

time independent multiplicative factors, σε ∝ |dε(t)dt |.We now turn to the higher order moments 〈(∆ε)p〉 ≡

1Np 〈(H

Hspin − 〈HH

spin〉)p〉 with p > 2. (The standard devi-

5

ation of Eq. (5) corresponds to p = 2.) Here, HHspin(t) =

ei~∫ t0iHtr(t′)dt′Hspine

− i~∫ t0Htr(t′)dt′ is the Heisenberg pic-

ture Hamiltonian and the expectation value is taken inthe initial state |ψ0

Spin〉. If N 1 and 1 > |w| then

Stot± |Stot, Sztot〉 = ~√Stot(Stot + 1)−m(m± 1)|Stot,m±

1〉 ∼ Stot~√

1− w2|Stot,m ± 1〉 where Sztot = m~.Trivially, for all m and m′, the matrix element ofδSztot ≡ Sztot − 〈Sztot〉 between any two eigenstates,〈Stot,m|δSztot|Stot,m′〉 = 0. Thus, the only non-vanishing contributions to 〈(∆ε)p〉 stem from 〈(Sxtot)p〉.This expectation value may be finite only for even p.Thus, in what follows, we set p = 2g. For Stot =O(N), expressing the expectation value of 〈(∆ε)2g〉 long-hand in terms of spin raising and lowering operators,one notices that, in this large N limit, each individ-ual term containing an equal number of raising andlowering operators yields an identical contribution (pro-

portional to (Stot~√

1− w2)2g) to the expectation value〈(∆ε)2g〉. Since there are

(2gg

)such contributions, we see

that for all g N in the thermodynamic (N → ∞)limit, the expectation value (∆ε)2g =

(2gg

)σ2gε . We write

the final (Schrodinger picture) state at time t = tf as|ψSpin〉 =

∑α cα|φα〉. We further define the probability

distribution of the energy density as

P (ε′) ≡∑α

|cα|2δ(ε′ −EαN

). (6)

In this example, the Heisenberg picture HamiltonianHHspin remains local for all times. Thus, the associ-

ated operators Hi are local at any time t. In othersystems, the time evolved Heisenberg picture Hamilto-nian need not be spatially local. Eq. (6) describesthe probability distribution associated with the “wavepacket” intuitively discussed in Section III (a “packet”that is now given by the amplitudes cα in our eigen-value decomposition of the final state |ψSpin〉). The mo-ments of ∆ε ≡ (ε′ − ε) are 〈(∆ε)2g〉 =

∫dε′ P (ε′) (ε′ −

ε)2g. Here, as throughout, ε = 1N 〈ψSpin|Hspin|ψSpin〉 =

−(∑ij Jij + BzS

ztot cos(

∫ tf0

By(t′) dt′))/N is the energydensity in the final state. More generally, the expec-

tation value of a general function f(HHspinN ) in the state

|ψ0Spin〉 (or, equivalently, of f(

HspinN ) in the above de-

fined final Schrodinger picture state |ψSpin〉) is given by

〈f(HHspinN )〉 =

∫dε′f(ε′)P (ε′). The mean value of each

Fourier component eiq(∆ε′) when evaluated with P (ε′) is

〈eiq(∆ε′)〉 = J0(2qσε) [37] where J0 is a Bessel function.An inverse Fourier transformation then yields

P (ε′) =θ(2σε − |∆ε′|)

2πσε

√1−

(∆ε′

2σε

)2. (7)

The Heaviside function θ(z) in Eq. (7) captures thebounded spectrum character of Hspin. Similar resultsapply to boundary couplings [38].

Qualitatively similar results appear for other “cool-ing/heating” protocols of the same type. For instance,if at intermediate times 0 ≤ t ≤ tf , the Hamiltoniangoverning the system would be that of a time indepen-dent Htr (i.e., one with a constant By(t) = By) aug-menting Hspin instead of replacing it (i.e., if at thosetimes, the total Hamiltonian is Ha = Hspin + Htr).For such an augmented (a) total Hamiltonian, the to-

tal spin ~Stot precesses around the applied external field(By ey + Bz ez) ≡ Ben. An elementary calculation anal-ogous to that leading to Eq. (5) then demonstrates thatthe corresponding standard deviation σaε of the energydensity at t = tf ,

σaε =BzByStot~NB√

2

√1 +

1

Stot− w2

×√

sin2(Btf ) +B2z (1− cos(Btf ))2

B2. (8)

We wish to stress that if Stot = O(N) and |w| < 1 then,as in Eq. (5), the standard deviation σaε = O(N) forgeneral times tf . The distribution of the energy densityfollowing an evolution with this augmented Hamiltonianwill, once again, be given by Eq. (7) for macroscopicsystems of size N →∞. The reader can readily see howsuch spin model calculations may be extended to manyother cases.

B. Itinerant hard core Bose systems

Our spin model of Section V A can be defined for localspins of any size S. The function P (ε′) of Eq. (7) char-acterizing our investigated states in this system is not avery typical probability distribution. However, the non-local entangled character of states having a finite energydensity relative to the ground state is pervasive for ther-mal states. This model can be recast in different ways.In what follows we focus on the spin S = 1/2 realiza-tion of Eq. (3). The Matsubara-Matsuda transformation[39, 40] maps the algebra of spin S = 1/2 operators ontothat of hard core bosons. Specifically, the bosonic num-

ber operator at site i is ni = b†i bi = 0, 1 with bi and b†i theannihilation and creation operators of hard core bosons.Following this transformation, the spin Hamiltonian ofEq. (3) transforms into its hard core bosonic dual,

HBose = −∑ij

Jij((b†i bj + h.c.) + ninj)

−∑i

(Bz −∑j

Jij)ni. (9)

This Hamiltonian describes hard core bosons hopping(with amplitudes Jij) on the same d−dimensional lat-tice, featuring attractive interactions and a chemicalpotential set by (Bz −

∑j Jij). Here, the transverse

field cooling/heating Hamiltonian Htr transforms into

HBose−doping = − iBy2

∑i(b†i − bi)- a Hamiltonian that

6

alters the number of the bosons (thereby “doping” thesystem). The hard core Bose states are symmetric un-der all permutations Pij of the bosons at occupied sites.The bosonic dual of, e.g., the specific spin product state| ↑1↑2↓3↑4↓5↑6 · · · ↑N−1↓N 〉 corresponds to the sym-metrized state of a fixed total number of hard corebosons that are placed on the graph (or lattice) sites(1, 2, 4, 6, · · · , (N − 1)). Thus, the bosonic dual of an ini-tial spin state |ψ0

Spin〉 with a total spin Stot = Smax =

N/2 is an initial hard core Bose state |ψ0Bose〉 that is

an equal amplitude superstition of all real space prod-uct states with the same total number of hard corebosons (

∑Ni=1 ni = m+ N

2 ) distributed over the N latticesites (an eigenstate of HBose that adheres to the fullysymmetric bosonic statistics). Evolving (during times0 ≤ t ≤ tf ) this initial state with Hdoping, the standarddeviation of Eq. (5) and the distribution of Eq. (7) areleft unchanged, apart from a trivial rescaling by ~ (e.g.,

σBoseε =Bz| sin(Bytf )|

2√

2

√1 + 2

N − w2 for Stot = N/2).

Similar to our discussion of the dual spin system, thefinite standard deviation in this energy density (and ofthe associated particle density n = 1

N

∑i ni) does not

imply that the “doping” is, explicitly, spatially inhomo-geneous (indeed, at all times, the expectation value of theparticle number 〈ni〉 stays uniform for all lattice sites i).

We conclude with two weaker statements regarding vi-able extensions of the rigorous results that we derivedthus far for hard core bosonic systems on general graphs(these graphs include lattices in general dimensions).(a) For a scalar field ϕ, the canonical Hamiltonian density

H[ϕ] =1

2(m2ϕ2 + (∇ϕ)2) + uϕ4, (10)

constitutes a continuum rendition of the hard core Boselattice model of Eq. (9). A large value of u in Eq. (10)emulates the local repulsion between hard core bosons.Thus, during various continuous changes of the Hamilto-nian, such generic scalar field theories (and myriad lat-tice system described by them) may exhibit the broad σεthat we derived for some of their lattice counterpart inthis subsection.(b) Similar results hold for spineless fermions. The mod-els of Eqs. (3,9) were defined on lattices (and graphs)in any spatial dimension. Identical results apply forspineless fermions on one dimensional chains with non-negative nearest neighbor hopping amplitudes/couplingconstants Jij and analogs of HBose−doping capturinga non-local coupling of the system to the external bath.These spinless Fermi systems may be trivially engineeredby applying the Jordan-Wigner transformation [41] toEq. (3).

VI. THE MAGNUS EXPANSION FORGENERAL EVOLUTIONS

To make progress beyond intuitive arguments and spe-cific tractable systems, we next compute the standard de-

viation of the energy density (and, by trivial extension,any other intensive quantity q). Towards that end, weemploy the Magnus expansion for a general time depen-dent Hamiltonian H(t) (of which the piecewise constantHamiltonians Hspin and Htr (or HBose and Hdoping) areparticular instances). Our analysis will demonstrate thatfor general situations, a finite σε arises unless very spe-cial conditions are satisfied. The general evolution op-

erator U(t) = T exp(− i~∫ t

0H(t′)dt′) may be written as

U = exp(Ω(t)) with Ω(t) =∑∞k=1 Ωk(t),

Ω1(t) = − i~

∫ t

0

dt1H(t1),

Ω2(t) = − 1

2~2

∫ t

0

dt1

∫ t1

0

dt2[H(t1), H(t2)],

· · · . (11)

The variance of the total energy density σ2ε (t) =

1N2

(Tr(ρ(HH(t))2)− (Tr(ρHH(t)))2

)where the Heisen-

berg picture Hamiltonian HH(t) = U†(t)HU(t) (withH ≡ H(0) the Hamiltonian of the initial system) andρ the initial density matrix the system (time t = 0) whencooling or heating commences. (In the dual examplesconsidered thus far, ρ = |ψ0〉〈ψ0| with |ψ0〉 the initialspin or hard core Bose wavefunction.) Expanding in pow-ers of H(t),

σ2ε (t) = σ2

ε (0) +1

N2

(〈[H2,Ω1]〉+ 2E0〈[Ω1, H]〉

)+

1

N2

(〈(Ω2 +

Ω21

2), H2〉 − 〈Ω1H

2Ω1〉

−2E0(〈(Ω2 +Ω2

1

2), H〉 − 〈Ω1HΩ1〉)

)+ · · · ,(12)

where 〈−〉 denotes an average computed with ρ and E0

is the initial energy 〈H〉. The brackets , denote ananticommutator. In the special case ρ = |φn〉〈φn| with|φn〉 an eigenstate of H, the initial variance of the totalenergy σE(0) = 0 and the sum of the two terms in thefirst set of brackets of Eq. (12) also vanishes identically.In this case, however, the sums appearing in the higherorder terms (the last two lines of Eq. (12) and beyond)do not cancel identically (even if H(t) is local). That acancellation cannot occur was also evident from our spinand hard core bosonic examples of Section V. Generally,when the system starts from an equilibrium state witha sharp energy density σε(0) = 0, then notwithstandingany locality of the Hamiltonian, σε may be O(1) at latertimes t. In the series of Eq. (12), the condition σε(t) = 0at an infinite number of different cooling/heating timest during the cooling/heating period, imposes stringentrestrictions on H(t) and the initial density matrix ρ thatare, in general, not satisfied.

If the variance of the energy density continues to bezero at all times t as the system evolves from its initialstate, then the derivative of the variance of the energydensity (which all too explicitly now write) will vanish,

7

ddtσ

2ε = i

~N2Tr(ρ(t)(D,H − 2 Tr(ρ(t)H)D)

)≡ J = 0

where D ≡ [H(t), H] and ρ(t) ≡ U(t)ρU†(t). Carefulattention may be paid to the scaling with N for localHamiltonians H(t) (such as those in our earlier exam-ples). In such instances, at small t > 0 (where the deriva-tive of the energy density is finite), the operator norm||D|| is bounded from below by O(N) contributions andthe derivative d

dtσ2ε for general N (including the N →∞

limit) is of order unity. Whenever the energy densityexperiences a finite rate of change dε

dt for a HamiltonianH(t) which is constant during the cooling/heating pe-riod, then the average of D with the probability densitymatrix ρ(t) will be equal to −i~N dε

dt . A vanishing Jimplies that, when evaluated with ρ(t), the two oper-ators H/N and D/N have a vanishing connected cor-relation function Tr(ρ(t)(HD))− Tr(ρ(t)H)Tr(ρ(t)D)).For Hamiltonians H(t) that are constant during the cool-ing/heating period, the single condition J = 0 at alltimes may be satisfied by an infinite number of veryspecial Hamiltonians and/or density matrices ρ. A so-lution for J = 0 is afforded by Hamiltonians H(t)that during the cooling/heating time have a commuta-tor D(t) with H that is a non-vanishing c-number (i.e.,formally, D(t) = D(t)1 with D a number). When thesystem is cooled/heated such that the energy density de-creases/increases, the time derivative of the energy den-sity dε

dt = − iN~Tr(ρ(t)D(t)) cannot vanish at all times

t during which cooling/heating occurs and thus D(t) isfinite. Under these circumstances, the evolution operatorU has the form of a shift operator for the energy (shiftingduring each interval of time [t, t+dt] the Heisenberg pic-ture Hamiltonian by HH(t) −→ (HH(t) − idt

~ D(t)1)) so

that HH(t) = H −∫ t

0dt′D(t′)1. The expectation value

of the commutator of (H − E)2 (where E = Tr(ρ(t)H)is the energy at time t) with the transverse field cool-ing/heating Hamiltonian must vanish (so that the deriva-tive of the variance vanishes and the energy density widthremains zero as it was in the initial equilibrium state priorto cooling/heating) while the commutator of H with thecooling/heating Hamiltonian must be finite (in order toenable cooling/heating). The expressions for the varianceand its time derivative bring to life the intuitive analogywith wave packets (Section III). In order to obtain a gen-eral shift of the energy without widening the width ofthe energy density distribution, one may apply a shiftoperator (an evolution with a “momentum” conjugate toH). Indeed, the above evolution with D(t) leads to ashift of the energy density with no additional changes.More comprehensive (yet, physically, exceptionally un-common) solutions to the equation d

dtσε = 0 at all timest (and thus solutions to σε(t) = 0 at all t) given thatσε = 0 at time t = 0 are afforded by combining multiple“shifts” of the above D(t) type with the product statesof Section IV. That is, we may set the initial state to bea general product of decoupled density matrices affordedby Eq. (2) with general values of 1 ≤ M ≤ N . If allof the probability density matrices are local (have their

support on regions of size O(1)) then any Hamiltonianevolution is possible. Conversely, as discussed above, ifthe density matrices cannot be factorized beyond a re-gion of size O(N) then only an innocuous shift with aconstant D will be possible. General hybrids (where forall non-local density matrices, an innocuous shift appearswhile the evolution of any local density matrices is arbi-trary) constitute solutions to the equation σε(t) = 0 atall times t. Other artificial solutions are also possible byengineering, in the eigenbasis of the original HamiltonianH(t = 0), specific matrix forms of Hamiltonians H(t > 0)and probability densities ρ that satisfy the requisite con-ditions. However, all of these situations are very special.Generally, as the system is cooled or heated, an evolu-tion from an initial sharp energy density will not onlyshift the initial delta function distribution of the energydensity but will also lead to a (non-vanishing) wideningσε. None of our above considerations were limited tothe energy density (the same results hold for any otherquantity q) nor to specific continuum or lattice systems.Quite generally, broad distributions appears in systemsdisplaying an evolution of their intensive quantities.

VII. GENERALIZED TWO-HAMILTONIANUNCERTAINTY RELATIONS

We next turn to a more specific demonstration that,generally, σε > 0 when the energy density exhibits a fi-nite rate of change. We consider non-relativistic systemsS of size N that satisfy certain general conditions:

(1) When combined with their environment (or “heatbath”) E , these systems constitute a larger isolatedclosed system I = S ∪ E .

(2) The fundamental interactions and their sums

appearing in the global Hamiltonian H describing I(including the subset of these interactions appearing inthe Hamiltonian H describing S ) are time independent.

The two Heisenberg picture Hamiltonians HH(t) =

eiHt/~He−iHt/~ and HH = H describe, respectively, theopen system S and the larger closed system I at timet. The energy of the system S is E(t) = TrI(ρHH(t))where ρ is the density matrix of I. By the uncertaintyrelations [44],

σε(t)σH ≥1

2

∣∣∣TrI(ρ[HH(t)

N, H])

∣∣∣, (13)

with σε(t) the uncertainty associated with HH(t)/N .Combined with the Heisenberg equations of motion forthe time independent H and H (assumption (2)), weobtain an extension of the time-uncertainty relations forthis two Hamiltonian realization,

σε(t)σH ≥~

2N

∣∣∣dEdt

∣∣∣. (14)

8

Eq. (1) then implies a bound on the product of the aver-age infinite range correlators in the subsystem S and thefull closed system I,

|G|S |G|I ≥~2

4N2

∣∣∣dEdt

∣∣∣2. (15)

The lower bound in Eq. (15) scales as O(N2) if theenergy E(t) of S changes at a rate proportional to thesize of S (i.e., if the energy density changes at a finiterate). Eqs. (13,14,15) will remain valid if condition (1)is relaxed, i.e., if I is an open system with a time inde-pendent Hamiltonian H that, itself, is in contact with ayet larger system. We will next further impose a slightlymore restrictive condition [43]:

(3) At asymptotically long times, the characteristicstates in the larger closed system I veer close to typicalstates in equilibrium (and thus microcanonical averagesmay be employed).

We now briefly discuss these conditions and their im-plications. Conditions (1) and (3) are often employed instandard textbook derivations of the canonical ensemblefor open systems S by applying the microcanonical en-semble averages for the larger equilibrated closed systemsI that include the relevant environments E in contact (or“entangled”) with S . The initial state of I will generally

not be an eigenstate of H and thus the system is non-stationary. If, as evinced by measurements in prototyp-ical states in the composite system I at asymptoticallylong times, ergodicity and equilibrium set in, then themicrocanonical ensemble may be invoked. In particular,in any such case described by the microcanonical ensem-ble, the uncertainty in the energy of I at asymptoticallylong times will be system size independent,

σH = O(1). (16)

Thus, since all expectation values are stationary underevolution in the larger closed system with the time inde-pendent Hamiltonian H, a corollary of (1-3) is that thestandard deviation σH of the time independent Hamilto-

nian H, at all times t, is of order unity, i.e., σH = O(1).We next turn to the scale of the righthand side of

Eqs. (13,14,15) and its consequence for systems thatare cooled/heat at finite rate. By Heisenberg’s equation,dHH

dt = i~ [H,HH ]. Therefore, in order to obtain a finite

dε/dt (or an extensive rate dE/dt), the total Hamilto-

nian H of the large system I must have a commutatorwith the Hamiltonian H of S that is of order N , i.e.,TrI(ρ[H,HH ]) = O(N). Hence, to achieve a finite global

rate of cooling/heating, H must couple to an extensivenumber of sites in the volume of S - it is not possible toobtain an extensive cooling/heating rate by a boundedstrength coupling that extends over an infinitesimal frac-tion of the system size (see also the discussion at the endof Section III). Effectively, a finite fraction of sites in the

volume of S must couple to H whenever dεdt = O(1).

In diverse situations (i.e., when cooling/heating leadsto a finite rate of change of the system energy densityor measured temperature), photons and/or other parti-cles/quasiparticles emitted/absorbed by an extensive vol-ume of the surrounding heat bath effectively couple to thesystem bulk [7]. (Similar considerations apply to any in-tensive quantity q that is changing at a finite rate dq/dt.)In the rotationally symmetric spin model of Section V,a time independent (for all times t > 0) transverse field

(By) Hamiltonian plays the role of a Hamiltonian H act-

ing on all N sites (so as to have [H,HH ] be of orderO(N)). In the examples of Section IV, the standard de-viation σH in the initial state (at time t = 0+ when the

transverse field was applied) was O(√N) when evaluated

with the wave function on S . If H emulates the effect ofa uniform transverse field in S (and does not change asa result of the interaction with the spins) then memoryof the initial state will never be lost. In particular, thecorrelation functions (even for sites arbitrarily far apart)will periodically assume the same values.

We wish to underscore yet another simple point. Theinitial state of the system S prior to its cooling/heating(or variation in its other parameters) may have a welldefined energy density ε and other state variables yetnonetheless still be far from a typical equilibrium state.The experimentalist may introduce clocks, etc., thatstart the cooling/heating process in a particular way.Thus, the initial state that the experimentalist pre-pares need not be in equilibrium but may rather bea specially crafted state. The external fields that actfrom E on S will generally lead to an effective timedependent Heisenberg picture Hamiltonian HH

eff (t) =

TrE(ρeiHt/~He−iHt/~) (akin to our earlier analysis in the

current work). However, if all of the above three condi-tions are met then at asymptotic long times, memoryof the special initial state will be lost and all observ-ables may be computed via the microcanonical ensemblewith its few thermodynamic state variables determiningall physical properties. In particular, the defining featureof closed equilibrated systems holds: σH = O(1).

With all of the above listed caveats, we now finallyturn to our result. If the energy density varies at a finiterate (i.e., if dE/dt = O(N)) then, Eqs. (14,16) implythat the standard deviation of the energy density of S ,

σε(t) = O(1). (17)

Thus, we discern from Eq. (1) that long range correla-tions must appear during the cooling or heating periodat which the energy density of the system (S ) is varied.Eq. (17) is also valid for any other intensive quantity qthat is varied at a finite rate.

If we consider an initial open thermal system compos-ite I at an assumed temperature T (instead of condition(1)), then we will obtain a bound on the cooling/heatingrate beyond which equilibrium is impossible. Our boundencompasses the physical situation of, e.g., a fluid thatis heated or cooled via contact with an external envi-ronment. In the latter case, both the subsystem ands

9

the larger (open) system may be taken to lie deep inthe fluid region (that does not involve the heat bath norcontacts to it). The inequality that we will write momen-tarily pertains to what transpires if both of these regionsof the fluid (in the subsystem and larger open system)are in equilibrium with one another at a temperature T .To derive the bound in this case, we will set, in the un-certainty relations of Eq. (14), the equilibrium valuesof standard deviations of the respective Hamiltonians inthe canonical ensemble, σH =

√kBT 2Cv,I(T ) (with kB

the Boltzmann constant and Cv,I(T ) the constant vol-ume heat capacity of the large system composite I) and

σH =√kBT 2Cv,S (T ) (where Cv,S (T ) is the heat capac-

ity of the small system at temperature T ). Repeating,mutatis mutandis, the steps that led to Eq. (17), we dis-cover that if the cooling/heating rate exceeds a thresholdvalue, ∣∣∣dE

dt

∣∣∣ > 2

~kBT

2√Cv,I(T )Cv,S (T ), (18)

then the open system composite of I will not be able tobe in equilibrium (even at asymptotically long times tso long as |dE/dt| will be extensive). That is, if we aimto keep a sharp total energy densities in open systems(by coupling to an external bath with a temperature T )[42], we will not be able to do so once Eq. (18) is satis-fied. This is so since Eq. (14) will be violated when wesubstitute the equilibrium values for σH/N and σH .

Eqs. (14, 18) apply for any rate of the energy changedE/dt. These include situations in which dE/dt scales asthe surface area of the system (O(N (d−1)/d)) for which

an extension of Eq. (1) will in turn imply that |G| ≥O(N−2/d). The central results of Eqs. (17, 18) holdfor any function f(q) of an intensive quantity q that isvaried at a finite rate. In particular, setting f(q) = qn

with n a natural number, we find that the uncertaintiesin all moments of q are, typically, finite if the rate dq/dtis finite [45].

VIII. “TO THERMALIZE OR TO NOTTHERMALIZE”

Our focus thus far has been on intermediate times tduring which the energy density (or any other intensivequantity q) varied. We showed that during these times,the standard deviation of q may be finite, σq = O(1).Thus, the variation of general quantities q (including,notably, the energy density or temperature) may triggerlong range correlations. As we explained towards the endof Section IV, this effect may be further exacerbated by“non self averaging” [30–33] in classical theories of disor-dered media. Our inequalities of Eqs. (17, 18) hold forgeneral fluctuations (regardless of the magnitude of their“classical” and “quantum” contributions [34] to the vari-ance). In most systems, after the temperature or field nolonger changes (e.g., when |dε/dt| vanishes at times t > t)thermalization rapidly ensues already at short times af-

ter t. Indeed, there are arguments (including certainrigorous results) that most “typical states” thermalizeon time scales of order O( ~

kBT) [46]. Such “Planckian”

time scales appear in a host of strongly interacting sys-tems, e.g., [47–51]. Various reaction rates are often givenby such minimal “Planckian” time scales multiplied bye∆G/(kBT ) with ∆G the effective Gibbs free energy bar-rier for the reaction or relaxation to occur [51, 52]. Somesystems such as glasses do not achieve true equilibrium:measurements on viable experimental time scales differfrom the predictions of the microcanonical or canonicalensemble averages. (The difference between the micro-canonical and canonical ensembles is irrelevant for allintensive quantities in the absence of long range inter-actions for which “ensemble inequivalence” is known toappear [53–56]). Here the system may have, as a result ofthe rapid cooling, effectively exhibit self-generated disor-der. As well known, structural glasses are disordered rel-ative to their truly thermalized crystalline counterparts.It is important to stress, however, that both structuralglasses and crystalline solids are governed by the verysame (disorder free) Hamiltonian. The effective disorderthat glasses exhibit is self-generated by the rapid (su-percooling) protocol of non-disordered liquids. Thus, thequestion remains as to whether, once the energy densityor other intensive quantity no longer varies (e.g., oncethe glass is formed and its temperature is no longer low-ered), the system will thermalize on experimental timescales or not be able to do so. Similar to our assumption(3) in Section VII, starting from a glassy state, super-cooled liquids will thermalize and may crystallize only atasymptotically long times [57]. In systems that do notthermalize on experimental time scales, the discrepancybetween equilibrium ensemble averages and empirical ob-servables hints that the width σε of the energy densitymight become smaller than it was during the cooling pro-cess yet is not vanishingly small. Indeed, if σε = 0 andno special “many body localized” states [22–27] exist (es-pecially in the most relevant physical situation of morethan one spatial dimension [28]) then the long time av-erages of all observables will be equal their microcanon-ical expectation values. In glassy systems that (by theirdefining character) cannot achieve true equilibrium onrelevant experimental time scales, the link to the exter-nal bath is effectively excised since the dynamics are soslow that little flow may appear. If, in such instances, thedensity matrix becomes time independent on measurabletime scales then an effective equilibrium may be reached.That is, at sufficiently long times, the density matrix ρmay be history independent and be a function of only afew global “state variables” yet differ from the conven-tional equilibrium statistical mechanics density matrices(in which the standard deviations of all intensive quan-tities vanish, e.g., σε = 0). In [58], we introduced thisnotion and employed it to predict the viscosity of allglass formers. This prediction was later tested [59] forthe published viscosity data of all known glass formerswhen they are supercooled below their melting temper-

10

FIG. 1. (Color Online.) Reproduced from [59]. On the vertical axis, we plot the experimentally measured viscosity data dividedby its value at the liquidus temperature, η(Tl), as a function of x, as defined along the abscissa. The viscosities of 45 liquidsof diverse classes/bonding types (metallic, silicate, organic, and others) collapse on a single curve. The underlying continuous“curve” (more clearly visible at high viscosities end where fewer data are present) is predicted by Eq. (20). Since A varies fromfluid to fluid (albeit weakly [59]), the shown collapse does not imply a corresponding collapse of the viscosity as a function ofTl/T nor as a function of Tl/(AT ) (due, relative to the latter, to an additional shift along the x axis that is set by −1/(A

√2)).

ature. A related classical theory was further advanced.Figure 1 reproduces the result. Here, the (finite) energydensity width was set to be

σε = AT (εmelt − ε)Tmelt − T

. (19)

Here, A > 0 is a liquid dependent constant (0.05 . A .0.12 for all liquids with published viscosity data [59]).The energy densities ε and εmelt in Eq. (19) are, respec-tively, those of the supercooled liquid or glass at tem-perature T < Tmelt and at the melting (or “liquidus”)temperature Tmelt. The wide distribution of Eq. (19)mirrors that in non self averaging disordered classicalsystems with an approximately linear in T standard de-viation and energy density ε(T ). (In quantum systems,all eigenstates of the density matrix may share the sameenergy while displaying a finite standard deviation σε[34].) We briefly make further contact between our fi-nite width distributions (Eq. (6)) for the energy densitywith usual thermodynamics. In the models that we in-vestigated in Section V (with the distribution of Eq. (7)that was far from the canonical normal form of equilib-rium systems), the systems were driven by an externalsource whose effect on general quantities was cyclic intime. The situation may be radically different when thesystem is no longer forcefully driven out of equilibrium

yet is still unable to fully equilibrate. In such instances,in the language of Section III, the energy density dis-tribution (P (ε′)) is no longer made to have its averagego up or down with time (i.e., the system is not heatedor cooled). The distribution may alter and narrow itsprofile as the system attempts to veer towards equilib-rium. If, as in equilibrium thermodynamics, the finalstate maximizes the Shannon entropy given a particularfinal energy then the resulting distribution for the en-

ergy density will be a Gaussian of width σε = T√kBCvN .

Since that heat capacity is extensive, Cv = O(N), thestandard 1√

Nfluctuations result in this equilibrium set-

ting. For systems that have not fully equilibrated andmaximized the Shannon entropy, we may (as we haveillustrated in some detail in the earlier Sections) find fi-nite width distributions P (ε′). If these distributions min-imally differ in form from those in equilibrium then theymay, similar to those in equilibrium systems, be normaldistributions of width σε ∝ T . The general distributionthat maximizes the Shannon entropy given that it hasa given finite standard deviation σε is a Gaussian. Thesole difference between the known Gaussian distributionof the energy density in equilibrium systems and thatassumed for systems that have reached the final temper-ature energy density (or corresponding temperature T )

11

yet have not achieved equilibrium is that in the latter sys-tems σε = O(1) (while σε = 0 for equilibrium systems intheir thermodynamic limit.) Non-rigorous considerationsfurther suggest a Gaussian distribution once the systemis no longer further cooled (or heated) [60, 61]. Assum-ing a normal distribution P (ε′) of width σε, the viscosityη of supercooled liquids at temperatures T ≤ Tmelt waspredicted to be [34, 58],

η(T ) =ηs.c.(Tmelt)

erfc(εmelt−ε(T )

σε√

2

) =ηs.c.(Tmelt)

erfc(Tmelt−TAT√

2

) . (20)

This prediction is shown by the continuous curve in Fig-ure 1. At the so-called “glass transition temperature”Tg, the viscosity η(Tg) = 1012 Pascal × second [62]. Atlower temperatures T < Tg, the viscosity is so large thatit is hard to measure it on experimental time scales. Thevery same distribution P (ε′) invoked in deriving Eq. (20)may relate other properties of supercooled liquids andglasses to those of equilibrium systems. For instance, theexperimentally measured thermal emission from super-cooled fluids may differ in a subtle manner from one thatis typical of equilibrium fluids. This deviation may befound by replacing Planck’s law for a system with welldefined equilibrium temperature T by a weighted averageof Planck’s law over effective equilibrium temperaturesT ′ that are associated with internal energy densities ofequilibrium systems that are equal to ε′. The temper-ature dependence of various response functions may beexpected to have the same increase in the time scale asthat characterizing the viscosity. Indeed, the time depen-dent heat capacity response follows exhibits a dynamicaltime that increases with temperature in a manner sim-ilar to the viscosity, e.g., [63]. We speculate that thisincrease in the relaxation time scale as the temperatureis dropped may account very naturally for the experimen-tally observed static specific heat jump [64] near the glasstransition temperature Tg. This is so since, at T ≤ Tg,on the time scales of the experiment, the system is essen-tially static (e.g., the viscosity of the Eq. (20) and theassociated measured relaxation times are large). Conse-quently, the relatively stable nearly static structures thatappear once the glass is formed need not significantly re-spond to a small amount of external heat. The situa-tion is somewhat reminiscent of the extensive latent heatthat is required to melt equilibrium crystals. A very pro-nounced thermodynamic signature appears in the tran-sition between equilibrium fluids and crystals. Once thesupercooled liquid or glass becomes effectively static atTg, it may weakly emulate the latent heat signature ofthe equilibrium liquid to solid transition.

An energy density distribution of a finite width σε al-lows for a superposition of low energy density solid typeeigenstates (that may break continuous translational androtational symmetries) and higher energy density liq-uid type eigenstates [58]. Such a general combinationof eigenstates does not imply experimentally discernibleequilibrium solid (crystalline) order. Sharp Bragg peaks

need not appear in states formed by superposing eigen-states that, individually, display order. This absence ofordering mirrors the possible lack of clear structure when,e.g., randomly superposing different Fourier modes witheach Fourier mode displaying its defining periodic order.The mixing of eigenstates of different energy densitiesover a range set by σε further suggests the appearance ofnon-uniform local dynamics. Indeed, dynamical hetero-geneities appear in supercooled fluids [65–69]. A simplecalculation indeed illustrates that the long time fluctu-ations in the local energy density given a general initialstate relate to the width of this state in the eigenbasisof the Hamiltonian governing the system [70]. We re-mark that a spatially non-uniform energy density is verynatural during the heating or cooling process of generalsystem (e.g., the exterior parts of a system being su-percooled may be colder than its interior, see also thediscussion towards the end of Section III). Once super-cooling stops, heat may diffuse through the system yetheterogeneities (that are borne in our framework from adistribution of finite σε) may still persist for a long time.

IX. POSSIBLE EXTENSIONS TO ELECTRONICAND LATTICE SYSTEMS

The electronic properties of many materials are welldescribed by Landau Fermi Liquid Theory [71–74]. Thistheory is centered on the premise of well defined quasi-particles leading to universal predictions. Recent decadeshave seen the discovery of various unconventional mate-rials displaying rich phases [71, 75–94] that often defyFermi liquid theory. Given the results of the earlier Sec-tions, it is natural to posit that as these systems are pre-pared by doping or the application of external pressureand fields (in which case the varied parameter q may bethe carrier density, specific volume, or magnetization), awidening σq will appear during the process. This widedistribution might persist also once the samples are nolonger experimentally altered. In such cases, the den-sity matrices (and associated response functions) describ-ing these systems may exhibit finite standard deviationsσq > 0. The broad distribution may trigger deviationsfrom the conventional behaviors found in systems havingsharp energy and number densities (σn = 0) or, equiv-alently, sharp chemical potentials and other intensivequantities. Theoretically, non-Fermi liquid behavior maybe generated by effectively superposing different densityFermi liquids (with each Fermi liquid having a sharp car-rier concentration n) in an entangled state. Systems har-boring such an effective distribution P (µ′) of chemicalpotentials may be described by a mixture of Fermi liq-uids of different particle densities. Any non-anomalousGreen’s function is manifestly diagonal in the total parti-cle number. Thus, the value of any such Green’s functionmay be computed in each sector of fixed particle numberand then subsequently averaged over the distribution oftotal particle numbers in order to determine its expected

12

value when σn 6= 0. In particular, this implies that theconventional jump (set by the quasiparticle weight Z~k,µ′)

of the momentum space occupancy [71–74], in the coher-ent part of the Green’s function (G = Gcoh + Gincoh)will be “smeared out” when σµ 6= 0. A distribution ofchemical potentials will lead to the replacement of thecoherent Green’s function of ordinary Fermi liquids by

Gcoh(~k, ω) =

∫dµ′P (µ′)

Z~k,µ′

ω − ε~k + µ′ + i/τ~k,µ′. (21)

Here, τ~k,µ′ is the quasi-particle lifetime in a system with

sharp µ′ at wave-vector ~k. The denominator in Eq. (21)corresponds to the coherent part of the Green’s functionof a Fermi liquid of a particular chemical potential µ′

and quasi-particle weight Z = 1 [71–74]. Qualitatively,Eq. (21) is consistent with indications of the very poorFermi liquid type behavior reported in [96]. The effectiveshift of the chemical potential in Eq. (21) is equivalentto a change in the frequency dependence while holdingthe chemical potential µ fixed; the resulting nontrivialdependence of the correlation function on the frequency(with little corresponding additional change in the mo-mentum) is, qualitatively similar to that advanced bytheories of “local Fermi liquids”, e.g., [71, 95]. Our con-siderations suggest a similar smearing with the distribu-tion P (µ′) will appear for any quantity (other than theGreen’s function of Eq. (21)) that is diagonal in theparticle number. Analogous results will appear for a dis-tribution other intensive quantities. The prediction ofEq. (21) (and similar others [58] in different arenas) maybe tested to see whether a single consistent probabilitydistribution function P accounts for multiple observables.General identities relate expectation values in interactingFermi systems to a weighted average of the same expecta-tion values in free fermionic systems [97]. These relationsraise the possibility of further related smeared averages,akin to those in Eq. (21), in numerous systems.

Numerically, in various models of electronic systemsthat display non-Fermi liquid type behaviors, the energydensity differences between contending low energy states|ψα〉 (not necessarily exact eigenstates) are often ex-ceedingly small, e.g., [98]. Since these states globallyappear to be very different from one another, the matrixelement of any local Hamiltonian between any two suchorthogonal states vanishes, 〈ψα|H|ψβ〉 = 0 for α 6= β.We notice that, given these results, arbitrary superposi-tions of these nearly degenerate states,

∑α aα|ψα〉, will

also have similar energies. Thus, a superposition of differ-ent eigenstates of various many body Hamiltonians mod-eling these systems may also be natural from energeticconsiderations.

Towards the end of Section VIII, we remarked on theviable disordered character of the states formed by su-perposing eigenstates that break continuous symmetries.We now briefly speculate on the corresponding situationfor eigenstates in electronic lattice systems that breakdiscrete point group symmetries on a fixed size unit cell.

Here, due to the existence of a finite unit cell in recip-rocal space, a superposition of eigenstates that are re-lated to each other by a finite number of discrete sym-metry operations may not eradicate all Bragg weights.In other words, order may partially persist when super-posing states on the lattice that, individually, displaydifferent distinct structures.

X. CONCLUSIONS

We demonstrated that a finite rate variation of generalintensive quantities mandates the existence of non-localcorrelations. In the simplest variant of our results, in ex-perimental systems with varying intensive observables q(such as the energy density ε) for which dq

dt = O(1), theassociated average connected two site correlation func-tions do not vanish even for sites are arbitrarily far apart.Trivial extensions of this result also hold for weaker vari-ation of the intensive quantities. For instance, if onlysurface effects are present and, in a system of N sitesin d dimensions, the rate dq

dt = O(N−1/d) then the aver-age absolute value of the connected two point correlationfunction for an arbitrary pair (i, j) of far separated sites

will be asymptotically bounded as |G| ≥ O(N−2/d).The general non-local correlations that we found orig-

inate from the long range entanglement present in typi-cal thermal states. Our results highlight that, even inseemingly trivial thermal systems, one cannot dismissthe existence of long range correlations. Our analysis ofnon-equilibrium systems does not appeal to conventionalcoarsening and spinodal decomposition phenomena (al-though the departure from a spatially uniform true equi-librium state in spinodal systems is very naturally consis-tent with a distribution of low energy solid like and higherenergy fluid like states). Cold atom systems may providea controlled testbed for our approach. We speculate thatour results may also appear in naturally occurring non-equilibrium systems. The peculiar effect that we findmay rationalize the unconventional behaviors of glassesand supercooled fluids. Our effect might further appearin electronic systems that do not feature Fermi liquidbehavior. Here, a broad distribution of effective energydensities and/or chemical potentials may appear. Thevalidity of weighted averages such as that of Eq. (21) maybe assessed by examining whether a unique distributionP simultaneously accounts for all measurable quantities.The non-local corrections that we derived do not hinge onthe existence of non-local interactions; the examples thatwe studied in Section V embodied quintessential local in-teractions. We hope our results will be further pursuedin light of their transparent mathematical generality andability to suggest new experimental behaviors (e.g., theuniversal viscosity collapse of supercooled liquids that itpredicted and is indeed empirically obeyed over sixteendecades as seen in Figure 1).

13

AcknowledgmentsI am especially grateful to S. Gopalakrishan for encour-aging me to write up these results and am very thankfulto interest by and discussions with E. Berg, A. Chan-dran, E. Fradkin, S. Gopalakrishan, A. J. Leggett, F.Nogueira, G. Ortiz, A. Polkovnikov, C. Sa de Melo, A.Seidel, and D. Sels. I also wish to express my gratitudeto N. B. Weingartner, C. Pueblo, F. S. Nogueira, andK. F. Kelton for testing the collapse of Fig. (1), a de-tailed classical phase space description, and much else inour earlier study of [59]. This work was supported theNational Science Foundation under grant NSF 1411229.Part of this work was performed at the Aspen Center forPhysics, which is supported by National Science Founda-tion grant PHY-1607761.

Appendix A: Entangled Ising chain eigenstateexpectation values produce thermal averages

In order to illustrate that large volume law type entan-glement may naturally appear in typical thermal states,we turn to a simple example- that of the uniform couplingone dimensional Ising model (the Hamiltonian HI of Sec-tion IV on an open chain with uniform nearest neighborcoupling- Jij = J). In this appendix, we will dispensewith factors of ~/2 and use the conventional definition ofthe Ising model Hamiltonian with the spin at any site rbeing Szr = ±1 (i.e., the diagonal elements of the Paulimatrix σzr ). In each single Ising state product state, thevalue of 〈SzrSzr′〉 is either 1 or (-1). This expectation valuediffers from that of the equilibrium system at finite tem-peratures It is only if we superpose many such productstates (i.e., have a highly entangled state) or average un-der uniform translations of the origin (i.e., entangle withequal weights all states related by translation) that weobtain the result from the above analysis. The Ising op-erators Szi are diagonal in the product basis and differentproduct states are orthogonal to each other. In a su-perposition of different product states, only the diagonal(i.e., weighted single Ising product expectation values)terms are of importance when computing 〈SzrSzr′〉.

We consider a highly entangled eigenstate |ψ〉 ofthe one-dimensional Ising model. Such an entangledstate emulates, in real space, entangled eigenstates|µα;Stot, S

ztot〉 with (for systems in their thermodynamic

limit) |Stotz /Smax| < 1 (i.e., not product states of all spinsmaximally polarized up or down along the field direction)of the spin models discussed in Section V. For an Isingmodel HI on a one dimensional chain of length L, givena single eigenstate of energy E, the frequency of low en-ergy nearest neighbor bonds (namely, Szr = Szr′ = ±1(“↑↑ ”or “↓↓”)) is p and that of having higher energybonds (i.e., “↑↓” or “↓↑”) is q. Clearly, p + q = 1 and(q − p) = E/(LJ) where J is the Ising model exchangeconstant and E is the total energy. In the one dimen-sional Ising model there is no constraint on the nearestneighbor bonds Szi S

zj (these products are all independent

variables that are “+1” or “-1” that sum to the scaledtotal energy E/J). Consider a spin at site r which is, say,“↑”. We may now ask what is the average value of a spinat another site r′. Evidently, if there is an even numberof domain walls (or even number of energetic bonds) be-tween sties r and r′ then the spin at site r′ is “↑” whileif there is an odd number of domain walls between thetwo sites then the spin at site r′ is “↓”. The average〈SzrSzr′〉 = (p− q)|r−r′|. That is, if we have an even num-ber of bad domain walls (corresponding to n even powerof q) then the contribution to the correlation function willbe positive while if we have an odd number of domainwalls (odd power of q) then the contribution to the cor-relation function will be negative. The prefactors in thebinomial expansion of (p− q)|r−r′| account for all of theways in which domain walls may be placed in the interval(r, r′). However, (p− q) = (−E)/(LJ). Thus, the corre-

lator 〈SzrSzr′〉 = [(−E)/(LJ)]|r−r′|. This single eigenstate

result using the binomial theorem indeed matches withthe known results for correlations in the Ising chain in thecanonical ensemble at an inverse temperature β whereE = −J(N − 1) tanhβJ and 〈SzrSzr′〉 = (tanhβJ)|r−r

′|.The agreement of the spatially long distance correlatorresult in a single eigenstate with the prediction of thefixed energy micro canonical ensemble is obvious. Theabove probabilistic derivation for general sites r and r′

will hold so long as the eigenstate |ψ〉 is a sum of numer-ous Ising product states (all having the same energy or,equivalently, the same number of domain walls). If thisresult holds for all site pairs (r, r′) then the entanglemententropy is expected to scale as the size (or “volume”) ofthis one dimensional system.

More broad than the specific example of this Ap-pendix, the coincidence between the volume law entan-gled eigenstate expectation values with the equilibriumensemble averages is expected to hold for general classi-cal systems in arbitrary dimensions. To see why this isso consider the expectation value of a general observable(including any correlation functions) that is diagonal inthe basis of degenerate classical product states. Whencomputed in a state formed by a uniform modulus su-perposition of classical degenerate states (e.g., the equalamplitude sum of all local product states of the same en-ergy), the expectation value of such an observable maynaturally emulate the microcanonical ensemble averageof this observable over all classical states of the sameenergy. Finite energy density states (i.e., states whoseenergy density is larger than that of the ground state)formed by a uniform amplitude superposition of all clas-sical product states generally obey volume law entangle-ment. As we have elaborated on in this Appendix, thisanticipation is realized for the classical Ising chain. Forthe classical Ising chains discussed above, the below twogeneral quantities are the same for a general observableO: (i) the mean of the expectation values of O in alllocal product states that are superposed to form general(not necessarily an exact uniform modulus superpositionof degenerate states) volume law entangled states and (ii)

14

the average of O as computed by a classical microcanon- ical ensemble calculation.

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[6] Along nearly identical lines, for conventional thermalsystems, the standard deviation scales as σε ∼ N−α

with α = 1/2 implying that the lattice average of the

pair correlator |G| ≥ O(N−2α) = O(1/N). At temper-atures T → ∞, this 1/N scaling merely amounts tothe existence of self-correlations (of typical magnitudeO(1)) i = j and the absence of pair correlations whenthe two lattice sites are different, i 6= j. Analogously, forcritical systems wherein Gij ∼ r−pij with p = d− 2 + η∗with η∗ the anomalous exponent, the average correlator(over all pairs of lattice sites) scales as |G| ∼ N−p/d.

[7] Given radiation traveling at a speed c, during a timeinterval ∆t, an extensive (i.e., volume proportional)amount of radiative heat ∆Qrad may flow into a hy-percubic system of spatial dimensions L × L × · · · × Lwhenever L . c(∆t). Thus, bulk effects from radiativeheat exchange may only be present only after a suffi-ciently long time t & L/c from the instant at whichradiative heating or cooling begins. Similarly, if the ef-fective radiative absorption lengths `S and `B of, re-spectively, the media comprising the system and thesurrounding heat bath both satisfy `S,B & L then thetotal radiative heat flow rate out of or into the sys-tem may be proportional to the volume of the system,dQrad/dt = O(V ). By contrast, for short absorptionlengths `S,B L and/or during times t < L/c, the ra-diative contribution to the rate of change of the energydensity dε/dt will scale as O(L−1). Our primary focusin the current article will be on the empirical situationsin which there is a measurable rate of change of theenergy density dε/dt = O(1) (or, more generally, mea-surable rates of change dq/dt = O(1) of other intensivequantities q). For completeness, we will, nonetheless,comment throughout our work also on cases in whichthe total heat transferred to (or given off by) the sys-tem is bounded by its surface area (e.g., for systems ofvery large linear dimensions L, the derivatives dε/dt anddT/dt vanish as L−1).

[8] As a concrete example if, on a hypercubic lattice ofN sites in d dimensions that is endowed with periodicboundary conditions, the exchange couplings Jij = J

for near neighbor pairs (ij) and are vanishing other-wise, and the initial state was a ground state of uniformpolarization (i.e., si = s = ±1) then the difference inenergies between the final and initial states will be givenby Eχ − Eψ(0) = NdJ . In this example, the standarddeviation of the energy in the final state |χ〉 is equal to

J√Nd.

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cos Rigol, “Entanglement Entropy of Eigenstates ofQuadratic Fermionic Hamiltonians”, https://arxiv.

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arxiv.org/pdf/1602.01874.pdf, Physics Reports 626,1 (2016).

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[19] L. F. Santos, A. Polkovnikov, and M. Rigol, “En-tropy of Isolated Quantum Systems after a Quench”,https://arxiv.org/pdf/1103.0557.pdf, Physical Re-view Letters 107, 040601 (2011).

[20] L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol,https://arxiv.org/pdf/1509.06411.pdf, “Fromquantum chaos and eigenstate thermalization to sta-

15

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[21] J. von Neumann, “Proof of the Ergodic theorem andthe H-Theorem in the new Mechanics”, Zeitschrfitfur Physik 57, 30 (1929), English translation byR. Tumulka, J. Eur. Phys. H 35, 201 (2010); Pe-ter Reimann, “Generalization of von Neumann’s Ap-proach to Thermalization”, https://arxiv.org/pdf/

1507.00262.pdf, Physical Review Letters 115, 010403(2015).

[22] D. Basko, I. Aleiner, and B. Altshuler, “Metal-insulator transition in a weakly interacting many-electron system with localized single-particle states”,https://arxiv.org/pdf/cond-mat/0506617.pdf, An-nals of Physics 321, 1126 (2006); D. Basko, I. Aleiner,and B. Altshuler, “Possible experimental manifesta-tions of the many-body localization”, https://arxiv.org/pdf/0704.1479.pdf, Physical Review B 76, 052203(2007).

[23] V. Oganesyan and D. A. Huse, “Localization of inter-acting fermions at high temperature”, https://arxiv.org/pdf/cond-mat/0610854.pdf, Physical Review B75, 155111 (2007).

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0026.pdf, Physical Review Letters 110, 067204 (2013);E. Altman and R. Vosk, “Universal Dynamics andRenormalization in Many-Body-Localized Systems”,https://arxiv.org/pdf/1408.2834.pdf, Annual Re-view of Condensed Matter Physics 6, 383 (2015).

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[27] M. Schreiber, S. S. Hodgman, P. Bordia, H. P. Luschen,M. H. Fischer, R. Vosk, E. Altman, U. Schneider,and I. Bloch, “Observation of many-body localizationof interacting fermions in a quasirandom optical lat-tice”, https://arxiv.org/pdf/1501.05661.pdf, Sci-ence 349, 842 (2015).

[28] Wojciech De Roeck and Francois Huveneers, “Stabilityand instability towards delocalization in MBL systems”,https://arxiv.org/pdf/1608.01815.pdf, Phys. Rev.B 95, 155129 (2017).

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[30] B. Derrida, “Non-Self-Averaging Effects in Sums ofRandom Variables, Spin Glasses, Random Maps, andRandom Walks” in “On Three Levels”, Edited by M.Fannes et aI., Plenum Press, New York (1994).

[31] A. Aharony and A.B. Harris, “Absence of Self-Averaging and Universal Fluctuations in Random Sys-tems near Critical Points”, Phys. Rev. Lett. 77, 3700(1996).

[32] Shai Wiseman and Eytan Domany, “Finite-Size Scal-ing and Lack of Self-Averaging in Critical Disor-

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9802095.pdf, Phys. Rev. Lett. 81, 22 (1998).[33] P. H. Lundow and I. A. Campbell, “Non-self-averaging

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[34] We denote the eigenvectors of a density matrix ρ by|ψ`〉 and their corresponding eigenvalues by p`. Theexpectation values E` = 〈ψ`|H|ψ`〉 mark the energies ofgiven density matrix eigenstates. The variance of the

energy density σ2ε = Tr[ρ(H−E)2]

N2 with E = Tr(ρH) thesystem energy may then be expressed as a sum of “clas-sical” and “quantum” contributions,

σ2ε = (σclass.

ε )2 + (σquant.ε )2. (A1)

Here,

(σclass.ε )2 ≡ 1

N2(∑`

p`E2` − E2), (A2)

(note that, generally, 〈ψk|H2|ψ`〉 6= E2` ) and

(σquant.ε )2 ≡ 1

N2(∑`

p`σ2` ), (A3)

where the variancesσ2` ≡ 〈ψ`|(H − E`)2|ψ`〉, (A4)

stem from the fluctuations of Hamiltonian in giveneigenstates of the density matrix. In classical systemswhere H and ρ commute, (σquant.

ε )2 = 0 and the solecontribution to the variance is that of Eq. (A2) as itis in, e.g., equilibrium classical systems in the canon-ical ensemble. The standard deviation of (H/N) (i.e.,σε) should not, of course, be confused with σclass.

ε . In-deed, in a general single pure state (a situation for whichσclass.ε = 0), the standard deviation of the energy den-

sity may be finite, σε = O(1). Concrete realizations ofthis maxim will be provided in the examples of Sec-tion V. Physically, one generally anticipates that whenreaching a quasi-static state, different eigenstates of thedensity matrix will exhibit similar energy densities, viz.σclass.ε = 0 (for otherwise heat may be rapidly exchanged

between states |ψ`〉 that are of higher/lower energy den-sities (E`/N) and their given surrounding heat bath).

[35] All states with maximal total spin and definite eigen-values of the total Sztot operator are eigenstates ofHspin. (These eigenstates span the basis of all ferro-magnetic spin states with spins uniformly polarizedalong different directions.) This assertion may be ex-plicitly proven by the following simple observations:(i) For any spin S = 1/2 operators, the scalar prod-

uct ~Si · ~Sj = ~2( 12Pij − 1

4) where Pij is the opera-

tor permuting the two spins, (ii) Any state of maxi-mum total spin (Stot = Smax = N~/2) is a symmet-ric state that is invariant under all permutations Pij.From properties (i) and (ii), it follows that any state|Stot = Smax = N~/2, Sztot〉 is an eigenstate of boththe first and second terms of Eq. (3) and therefore ofthe full Hamiltonian Hspin. Thus, any state of maximaltotal spin Stot = Smax that is an eigenstate of Sztot is au-tomatically an eigenstate of Hspin of Eq. (3). In general,when Stot < Smax, only some linear combination(s) ofthe multiple states of given values of Stot and Sztot areeigenstates of Hspin (hence the appearance of additionalquantum numbers ξα defining general eigenstates |φα〉).To make this clear, we can explicitly write down the

16

total spin for a system of N spin 1/2 particles. That is,

N = 2 :1

2⊗ 1

2= 1⊕ 0,

N = 3 :1

2⊗ 1

2⊗ 1

2=

3

2⊕ 1

2⊕ 1

2,

N = 4 :1

2⊗ 1

2⊗ 1

2⊗ 1

2= 2⊕ 1⊕ 1⊕ 1⊕ 0⊕ 0,

· · · . (A5)Since Hspin is defined on a (2S + 1)N dimensionalHilbert space, its eigenstates span all states in the di-rect product basis (or equivalent direct sum basis) onthe lefthand side of Eq. (A5). The sector of maximalspin (Stot = Smax) is unique. However, all other to-tal spin sectors exhibit a multiplicityMStot larger thanone. While it is, of course, possible to simultaneously di-agonalize the Hamiltonian with Stot and Sztot there aremultiple states that share the same Stot and Sztot eigen-values. One needs to diagonalizeHspin in every subspaceof given Stot and Sztot in order to find its eigenstates ineach such subspace. As an side, we may explicitly calcu-late the multiplicities appearing in Eq. (A5). Using thecharacters of the SU(2) group, we see that there are

MNStot =

N !(2Stot + 1)

(N2

+ Stot + 1)!(N2− Stot)!

(A6)

sectors of total spin 0 ≤ Stot ≤ N2

on the righthand sideof Eq. (A5). The decomposition into characters of SU(2)has a transparent physical content. Consider a global ro-tation by of all spins an arbitrary angle θ about the zaxis. The trace of the operator that implements this ro-tation is the same into the different basis appearing inEq. (A5): (1) the direct tensor product basis (the left-hand side of Eq. (A5)) and (2) the basis comprised ofthe total spin sectors (the righthand side of Eq. (A5)).When expressing the basis invariant trace of the arbi-trary rotation evolution operator in terms of the Laurentseries in eiθ/2 that arises when taking the trace of therotation operator, the series must identically match inboth of these bases of Eq. (A5). Equating the coefficientsleads to Eq. (A6).

[36] The result of Eq. (5) may readily be seen by inspec-tion. The applied transverse field leads to a global Lar-mor precession of the spins about the y−axis. Whilethe first term of Eq. (3) is manifestly invariant un-der rotations, the second term (that of (−Bz

∑i S

zi ))

will change. In the Heisenberg picture after the evolu-tion with the transverse field, each such local opera-tor (BzS

zi ) transforms into Bz(S

zi cos(

∫ tf0

By(t′) dt′) +

Sxi sin(∫ tf

0By(t′) dt′)). Since the expectation value

of Sxtot in any eigenstate of Sztot (including |ψ0Spin〉)

is zero, the only non-vanishing contributions to thevariance of the Hamiltonian of Eq. (3) will originatefrom the expectation value of the square of the sec-ond term of Hspin and thus (up to a trivial prefactor

of (B2z sin2(

∫ tf0dt′By(t′)))) from

σ2Sxtot

= 〈(Sxtot)2〉 =1

2〈(Sxtot)2 + (Sytot)

2〉

=1

2〈(~Stot)2 − (Sztot)

2〉

=1

2

(~2Stot(Stot + 1)− (Sztot)

2). (A7)

Substituting Stot = N~/2 and w ≡ Sztot/Stot (andrescaling by a factor of N2 to determine the varianceof the energy density) leads to the square of Eq. (5).

[37] Explicitly, 〈eiq(∆ε′)〉 =

∑∞g=0

(iq)2g

(2g)!

(2gg

)σ2gε =∑∞

g=0(−1)g

(g!)2(qσε)

2g = J0(2qσε).

[38] We quickly comment, in the spirit of the discussionappearing at the end of Section III, on what tran-spires if the forms of the Hamiltonian Hspin and, no-tably, the coupling of the spins to the transverse fieldas embodied by the Hamiltonian Htr apply only forspins on the surface of the system (the number of these

spins Nsurface = O(N (d−1)/d)). In such situations, thestandard deviation in Eq. (5) would naturally scale as

σε = O(N−1/d). The distribution of Eq. (7) will retainits form. If this circumstance arises then, in the discus-sions and equations in the main text concerning Hspinand Htr, the total number of spins N will be replacedby the number of surface spins Nsurface.

[39] T. Matsubara and H. Matsuda, “A Lattice Model ofLiquid Helium, I”, Prog. Theor. Physics 16, 569 (1956).

[40] Specifically, the Matsubara-Matsuda transformation[39] from spin S = 1/2 raising and lowering operatorsto the creation and annihilation operators of hard core

bosons is given byS+i~ → b†i and

S−i~ → bi (from which

it follows thatSzi~ → (ni − 1

2) where ni = b†i bi is the

hard core boson number operator).

[41] With ci and c†i denoting respectively, the spinelessFermi annihilation and creation operators (and with

ni = c†i ci), the Jordan-Wigner dual of the initial spinHamiltonian of Eq. (3) reads

HFermi =∑|i−j|=1

Jij((c†i cj + h.c.) + ninj)

−∑i

(Bz −∑|j−i|=1

Jij)ni. (A8)

We may consider the initial spineless fermion systemof Eq. (A8) is that is governed, at intermediate times,by the (now non-local) Jordan-Winger dual of Htr orHBose−doping. Following a temporal evolution with thelatter intermediate time Hamiltonian, an initial eigen-state of HFermi formed by the equal amplitude superpo-sition of all spineless Fermi states of a fixed total particlenumber (the dual of |ψ0

Spin〉) will transform into a finalstate displaying the standard deviation and distributionof Eqs. (5, 7). (In order to make the transformation lu-cid, we remark that the spineless Fermi dual of the par-ticular spin S = 1/2 state | ↑1↑2↓3↑4↓5↑6 · · · ↑N−1↓N 〉describes an antisymmetrized superposition of all prod-uct states in which the fixed total number of spinelessfermions

∑i ni are distributed over the specific sites

(1, 2, 4, 6, · · · , (N − 1)).)[42] Strictly speaking, a finite interval of the internal energy

densities corresponds to the single melting (and othercoexistence) temperature(s). This width of this inter-val is set by the latent heat of fusion at the meltingtemperature. At other temperatures in which no phasecoexistence appears, there is a unique internal energydensity ε(T ).

[43] These conditions need not always hold. For instance, one

may concoct states |ψ〉 (or density matrices ρ) in I that

exhibit arbitrarily large variances σ2H

= (〈ψ|H2|ψ〉 −(〈ψ|H|ψ〉)2) when the system size increases. (This maybe achieved by superposing eigenstates that span an ex-tensive range of H eigenvalues.) The microcaonnical en-semble (for which σE is bounded) will be rendered in-

17

compatible in states like these for which the standarddeviation scales with the system size.

[44] For a discussion of the standard uncertainty relationsfor mixed states see, e.g., L. Ballentine, “Quantum Me-chanics” (World Scientific, Singapore, 1998).

[45] Uncertainties in arbitrary order finite moments do notpreclude a Gaussian distribution P (q′).

[46] Sheldon Goldstein, Takashi Hara, and Hal Tasaki, “Onthe time scales in the approach to equilibrium of macro-scopic quantum systems”, https://arxiv.org/pdf/

1307.0572.pdf (2013), Phys. Rev. Lett. 111, 140401(2013).

[47] S. Sachdev, Quantum Phase Transitions, second Edi-tion, Cambridge University Press (2011).

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[50] Sean A. Hartnoll, “Theory of universal incoherentmetallic transport”, https://arxiv.org/pdf/1405.

3651.pdf(2014), Nature Physics 11, 54 (2015).[51] Z. Nussinov, F. Nogueira, M. Blodgett, and K. F. Kel-

ton, “Thermalization and possible quantum relaxationtimes in ”classical” fluids: theory and experiment”,https://arxiv.org/pdf/1409.1915.pdf (2014).

[52] H. Eyring, “The Activated Complex in Chemical Reac-tions”, J. Chem. Phys. 3, 107 (1935).

[53] F. Leyvraz and S. Ruffo, “Ensemble inequivalence insystems with long-range interactions” https://arxiv.

org/pdf/cond-mat/0112124.pdf, Journal of Physics A- Mathematical and General 35, 285 (2002).

[54] J. Barre, D. Mukamel, and S. Ruffo, “Inequiva-lence of ensembles in a system with long-range Inter-actions”, https://arxiv.org/pdf/cond-mat/0102036.pdf, Physical Review Letters 87, 030601 (2001).

[55] A. Campa, T. Dauxois, and S. Ruffo, “Statis-tical mechanics and dynamics of solvable modelswith long-range interactions”, https://arxiv.org/

pdf/0907.0323.pdf, Physics Reports 480, 57 (2009).[56] Y. Murata and H. Nishimori, “Ensemble Inequiva-

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[57] E. D. Zanotto and J. C. Mauro, “The glassy state ofmatter: Its definition and ultimate fate”, Journal ofNon‘Crystalline Solids 471, 490-495 (2017).

[58] Z. Nussinov, “A one parameter fit for glassy dy-namics as a quantum corollary of the liquid to solidtransition”, https://arxiv.org/pdf/1510.03875.pdf

(2015), Philosophical Magazine 97, 1509-1566; 1567-1570 (2017).

[59] Nicholas B. Weingartner, Chris Pueblo, F. S. Nogueira,K. F. Kelton, and Zohar Nussinov, “A Quantum Theoryof the Glass Transition Suggests Universality AmongstGlass Formers”, https://arxiv.org/pdf/1512.04565.pdf (2015); Nicholas B. Weingartner, Chris Pueblo,Flavio Nogueira, Kenneth F. Kelton, and Zohar Nussi-nov, “A phase space approach to supercooled liquidsand a universal collapse of their viscosity”, https:

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[60] In general, the probability distribution may be calcu-

lated along lines similar to those that led to Eq. (7) inour toy example of Eq. (3) where the system was contin-uously driven by an external transverse field. However,unlike the models studied in Section V, at long times, su-percooled liquids and glasses are no longer driven by anexternal bath Htr that continuously cools/heats themin a predetermined fashion. Instead, for supercooledliquids and glasses, at long times, the external heatbath (similar to the situation in equilibrium thermo-dynamics), becomes a source of stochastic noise (whosestrength is set by its temperature T ). Thus, the initiallydriven (i.e., continuously cooled) supercooled fluids orglasses will, at these long times, be effectively exposedto random noise. Following the same reasoning that gaverise to Eq. (7), we may compute general moments of theHeisenberg picture Hamiltonian

〈(∆ε)p〉 ≡ 1

Np〈(HH − 〈HH〉)p〉 ≡ 〈(∆HH

N)p〉 (A9)

evaluated in the initial equilibrium state prior to cooling|ψ0〉 =

∑n c

0n|φn〉. Here, c0n are the amplitudes of the

initial state |ψ0〉 in the eigenbasis of the system Hamil-tonian H. Writing Eq. (A9) longhand as a product of p

factors of ( ∆HH

N), we have

〈(∆ε)p〉 =∑n

|c0n|2( (∆HH)nm

N

)( (∆HH)mlN

)×( (∆HH)lq

N

)· · ·( (∆HH)wn

N

), (A10)

where (∆HH)ab are the matrix elements of ∆HH in theeigenbasis of H. If the matrix elements of the scaled

Heisenberg picture Hamiltonian ∆HH

N(now evolved

with the random influence of the environment at longtimes) attain random phases relative to each other thenthe only remaining contributions in Eq. (A10) will bethose in which all matrix elements come in complex con-

jugate pairs of the type(

(∆HH )abN

)((∆HH )ba

N

). Thus,

similar to the calculation that led to Eq. (7), only evenmoments p = 2g may be finite. Now, however, the num-ber of non-vanishing contributions (the number of waysin which the elements of HH may be matched in com-

plex conjugate pairs) will scale as(

(2g)!2gg!

). This, in turn,

prompts us to consider the possibility that, approxi-

mately, 〈(∆ε)2g〉 ∼(

(2g)!2gg!

)σ2gε . Thus, if these moments

of ∆ε (for all g) are equal to those evaluated with aGaussian distribution (as follows from the standard ap-plication of Wick’s theorem- the combinatorics of whichessentially reappeared in the above), then it will fol-low that the probability distribution P (ε′) for obtainingdifferent energy densities in the final state is indeed aGaussian.

[61] Since the energy density is bounded from below by itsground state value εg.s., the probability distributionP (ε′) vanishes for ε′ < εg.s.. The exact distribution P (ε′)(whether a Gaussian or of another approximate form)is, of course, cut off at such low ε′. The same holdstrue for the Gaussian distribution describing the energydensity of a large finite size system within the canonicalensemble.

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18

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[70] In this brief comment, we consider an initial state|ψ〉 =

∑n cn|φn〉 where |φn〉 are eigenstates of a local

Hamiltonian H =∑N′

i=1Hi (where H|φn〉 = En|φn〉).The system evolves with the time independent Hamil-tonian H. We define the long time average of the localenergy density fluctuations by

σR ≡ limT→∞

∫ T

0

dt(〈HR(t)〉 − HR)2, (A11)

where

HR ≡1

T

∫ T

0

dt〈HR(t)〉 =∑n

|cn|2〈φn|HR|φn〉 (A12)

is the long tme average of HR. Here, 〈HR(t)〉 denotesthe Heisenberg picture expectation value of the localenergy term HR(t) in the initial state |ψ〉. In the fol-lowing, we consider the situation when level spacings ofH are relatively incommensurate (any equation of thetype (En −Em) = (Em′ −En′) may only be satisfied ifm′ = n and n′ = m). A simple calculation (along thelines of that performed in Ref. [58] for the long timeaverage of general observables) then yields

σ2R =

∑n 6=m

|cn|2|cm|2|〈φn|HR|φm〉|2. (A13)

The fluctuations of the global energy density are givenby

σ2ε =

1

N2

(∑n

|cn|2E2n −

∑n,m

|cn|2|cm|2EnEm). (A14)

Since, in any time independent Hamiltonian system, alleigenstates are stationary, Eqs. (A13, A14) must vanishwhen |ψ〉 is an eigenstate (as they indeed do). Trivially, along time heterogeneity in the local energy (a finite σR)can only appear in non-stationary states |ψ〉. This tau-tological relation may extend to an approximate generaltrend. Qualitatively, a larger spread in the global energydensity σε may appear hand in hand with a larger stan-dard deviation of the local energy density σR. This isnatural from various viewpoints. The Schrodinger equa-tionHψ(~x) = Eψ(~x) holds for all (many body) spa-

tial coordinates ~x with a global value of E. This, ofcourse, holds true even if the Hamiltonian H is a sum oflocal operators that do not connect the wave function at~x to its values at spatially distant regions- i.e., opera-tors that do not have off diagonal matrix elements con-necting spatially far degrees of freedom. Thus, withinany eigenstate, the expectation value of any projectionof the Hamiltonian onto a local volume (i.e., the localenergy density in systems with local interactions) mustalso be spatially uniform (i.e., independent of the spa-tial positioning of this local volume). Viewed from thisperspective, this obvious single eigenstate relation sug-gests that larger deviations σR measuring spatial fluctu-ations of the local energy density may naturally appearin unison with a larger spread of the global energies inthe spectral decomposition of |ψ〉 in the eigenbasis of H(i.e., a larger standard deviation σε).

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