Subsystem Eigenstate Thermalization Hypothesis

Subsystem eigenstate thermalization hypothesis

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As Published http://dx.doi.org/10.1103/PhysRevE.97.012140

Publisher American Physical Society

Version Final published version

Citable link http://hdl.handle.net/1721.1/114450

Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW E 97, 012140 (2018)

Subsystem eigenstate thermalization hypothesis

Anatoly Dymarsky,1,2 Nima Lashkari,3 and Hong Liu3 1Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506, USA 2Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, Moscow 143026, Russia 3Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

(Received 27 March 2017; revised manuscript received 7 December 2017; published 25 January 2018)

Motivated by the qualitative picture of canonical typicality, we propose a reﬁned formulation of the eigenstate thermalization hypothesis (ETH) for chaotic quantum systems. This formulation, which we refer to as subsystem ETH, is in terms of the reduced density matrix of subsystems. This strong form of ETH outlines the set of observables deﬁned within the subsystem for which it guarantees eigenstate thermalization. We discuss the limits when the size of the subsystem is small or comparable to its complement. In the latter case we outline the way to calculate the leading volume-proportional contribution to the von Neumann and Renyi entanglment entropies. Finally, we provide numerical evidence for the proposal in the case of a one-dimensional Ising spin chain.

DOI: 10.1103/PhysRevE.97.012140

I. INTRODUCTION AND MAIN RESULTS Another important development was the so-called eigenstate thermalization hypothesis (ETH) [5–7] which conjectures During the last two decades there has been significant that a chaotic quantum system in a finitely excited energy progress in understanding how quantum statistical physics eigenstate behaves thermally when probed by few-body op- emerges from the dynamics of an isolated quantum many-body erators. More explicitly, for a few-body operator O, ETH system in a pure state. An important recent development was postulates that [8,9] the realization that a typical pure state, when restricted to a small subsystem, is well approximated by the microcanonical −1/2 Ea|O|Eb=fO(E)δab + (E)rab, ensemble [1–3]. More explicitly, for a system comprised of a (3) = + sufficiently small subsystem A and its complement A¯, for any E (Ea Eb)/2, random pure state from an energy shell (E,E + E), | where Ea denotes an energy eigenstate, fO(E)isasmooth = S(E) |= c |E ,E∈ (E,E + E), (1) function of E, (E) e is the density of states of the full a a a ∼ a system, and the fluctuations rab are of order 1, rab O(1). The big O here and in what follows refers to the limit when the size A ≡ | | the corresponding reduced density matrix ρ TrA¯ of the full system is taken to infinity. ··· is almost microcanonical. Taking the average over all Canonical typicality applies to all systems independent of states (1) with respect to the Haar measure one finds [3] the Hamiltonian, as opposed to ETH which only concerns chaotic systems, and does not apply to integrable or many-body 1 d 1 √ A − A √ A || || = † localized systems. It is a stronger statement, as ETH implies ρ ρmicro , O Tr OO . (2) 2 dE 2 the emergence of the microcanonical ensemble not only for random , but also for a wider class of states, including the A = Here ρmicro TrA¯ ρmicro is the reduction of the microcanonical linear combination of a few energy eigenstates. density matrix ρmicro associated with the same energy shell The fact that ETH applies only to chaotic systems can be (E,E + E), dE is the number of energy levels inside it, and heuristically understood from the general picture of CT; only dA = dim HA is the dimension of the Hilbert space of A. for chaotic systems energy eigenstates are “random enough” Equation (2) implies that, when the system is sufficiently to be typical. This perspective thus motivates us to study the large, i.e., ln dE ln dA, the subsystem of a typical pure state properties of the reduced density matrix of a subsystem in an is well approximated by that of the microcanonical ensemble energy eigenstate; see [10–12] for some earlier works. with an exponential precision. We refer to this mechanism as Now consider a chaotic many-body system in an energy “canonical typicality” (CT). It is important to note that CT eigenstate |Ea reduced to a spatial subsystem A which is is a purely kinematic statement, and provides no insight into smaller than its complement A¯. We postulate the subsystem whether or how a nonequilibrium initial state thermalizes [4]. ETH: A = | | Heuristically, canonical typicality can be understood as a (i) The reduced density matrix ρa TrA¯ Ea Ea for re- consequence of the entanglement between a sufficiently small gion A in state |Ea is exponentially close to some universal subsystem and its complement [3]. While the full system density matrix ρA(E), which depends smoothly on E, evolves unitarily, a small subsystem can behave thermally as A − A = ∼ −1/2 its complement plays the role of a large bath. ρa ρ (E Ea) O[ (Ea)]. (4)

A = | | discussion. In particular, in case of finite-dimensional models it (ii) The “off-diagonal” matrices ρab TrA¯ Ea Eb are exponentially small, immediately leads to a natural interpretation of thermal entropy as the volume part of the entanglement entropy of a subsystem A ∼ −1/2 = = 1 + ρab O[ (E)],Ea Eb,E 2 (Ea Eb). (5) (see [15,16] for recent discussions). We should caution that H The pre-exponential factors in (4) and (5) could depend on the when dim A is infinite, arbitrarily close proximity of density size of subsystem A. Importantly, these factors should remain matrices does not automatically imply equality for nonlocal ob- bounded for the fixed A. In the next section, we will give servables. For example, in such cases, higher Renyi entropies A numerical support for the exponential suppression of (i) and for ρa may be different from those of the microcanonical or (ii) using a spin system. Recently support for (4) and (5)was other thermal ensembles [13]. A given in the context of CFTs in [13]. In the case of the spin model, for all matrix elements (ρ )ij , In the thermodynamic limit, i.e., with the system size taken we find strong evidence that raa of (3) are normally distributed. | to infinity, V →∞, while keeping the size of A and the energy This is consistent with the heuristic picture of typical Ea and density E/V finite and fixed, it can be readily seen from (i) rab being a Gaussian random matrix. and (ii) that It is tempting to ask whether one could further refine pre-exponential factors in (4) and (5), especially when the A − A ∼ ρa ρmicro O(E/E). (6) subsystem A is macroscopic. Motivated by the A-dependent prefactor in (2) and the average value of the “off-diagonal” An implicit assumption here is that ρA(E) is well defined in matrices (23), it is natural to postulate that the prefactor in (4) the thermodynamic limit, i.e., it is a function of E/V 1 and the →∞ and (5) should also be given by prefactor in (4) and (5) remains bounded in the limit V . A A N −S(E)/2 Note that while the suppressions in (4) and (5) are exponential ρ − ρ (E = E ) ∼ O(e A ), a a in the system size, those in (6) are only power law suppressed. A N −S(E)/2 (9) ρ ∼ O(e A ), Using ||ρ|| = maxO Tr(Oρ)/2, where maximum is taken ab ||O|| = over all Hermitian operators of unit norm 1, we where N denotes the number of effective degrees of freedom A A conclude from (i) and (ii) that the matrix elements of ρa and in A. For a system of finite dimensional Hilbert space, such A ρ (E = Ea) are exponentially close, NA = H as a spin system, e simply corresponds to dA dim A,but ρA = ρA + O(−1/2), ρA = O(−1/2). (7) for a system with an infinite dimensional Hilbert space at each a ij ij ab ij lattice site or a continuum field theory we may view (9)as The formulation in (4) and (5) is stronger than the conventional a definition of effective number of degrees of freedom. For a form of ETH. In particular, for systems with an infinite- spin system we will give some numerical evidence for (9)in dimensional local Hilbert space (e.g., with harmonic oscillators the next section. A at each lattice site) or continuum field theories it guarantees In addition to (6) it is interesting to compare ρa with ETH for the particular class of observables, while for other the reduced density matrices for other statistical ensembles. observables ETH may not apply. In particular, subsystem Of particular interest are the reduced state on the canonical ETH implies the exponential proximity between expectation ensemble for the whole system E |O|E values in an eigenstate a a and the universal value −βH = O A O TrA¯ e fO(Ea) Tr[ ρ (Ea)] for any observable with the support ρA = , (10) in A.2 This immediately follows from the Cauchy-Schwarz C Tr e−βH inequality,3 and the local canonical ensemble for the subsystem A, A A Tr ρ − ρ (E ) O − a a e βHA ρA = . (11) 1/2 G −βH 1/2 A + A O2 A − A TrA e A 2 Tr ρa ρ (Ea) ρa ρ (Ea) . (8) Here, the Hamiltonian of the subsystem is the restriction of the Moreover, the subsystem ETH can be applied directly to Hamiltonian H = Tr ¯ H .In(10), β is to be chosen so that nonlocal measures which are defined in terms of reduced A A the average energy of the total system is E .In(11), β can be density matrices, such as entanglement entropy, Renyi en- a interpreted as a local temperature of A (seealso[17–19]). There tropies, negativity, and so on. See, e.g., [12] for a recent is no canonical choice for β in this case. Below, we choose it to be the same as in (10). In the thermodynamic limit, V →∞ with the subsystem A and E/V kept fixed, the standard saddle 1At a technical level (6) requires a weaker condition of finite Lips- point approximation argument provides equality between the || A − A || chitz constant κ in the thermodynamic limit, ρ (E1) ρ (E2) canonical and the microcanonical ensembles leading to κ|E1 − E2|/V.In(6) we also use that for finitely excited states A − A = −1 ⇒ A − A = −1 E ∼ V . ρmicro ρC O(V ) ρa ρC O(V ), 2 In case of the continuous quantum field theory, when the full Hilbert (12) space does not admit a tensor product structure of the Hilbert space of A and of its compliment A¯, operators O should be defined in terms A A wherewehavealsoused(6). The reduced states ρC and ρG of the net of local operator algebras; see, e.g., [14]. always remain different at the trace distance level, including 3 For any physically sensible observable O the fluctuations of thermodynamic limit [17]. Hence, AO O2 A expectation value Tr(ρa ), which are given by Tr( ρa ), must be A = A →∞ finite. ρa ρG,V . (13)

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Finally, it is interesting to investigate whether (4) and (5) II. NUMERICAL RESULTS remain true in an alternative thermodynamic limit when the Now we examine hypothesis (4) and (5) of the subsystem size of subsystem A scales proportionally with the full system. formulation of ETH by numerically simulating an Ising spin In this limit both the system volume V and the volume V for A chain with a transverse and longitudinal magnetic field, A go to infinity, but we keep the ratio fixed, n−1 n n VA 1 =− k ⊗ k+1 + k + k 0

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(a) (b)

FIG. 1. (a) Values of ln(σm,n) with superimposed linear fit functions −αmn + βm for m = 1 ...8, n = 12 ...17, and E = 0.1n, g = = 2 1.05,h 0.1. The slope of linear functions αm for all m is within 5% close to the theoretical value ln(2)/2. (b) The maximum value of Tr(ρa ) over all eigenstates inside the central band |Ea| E = 0.1n. not suppressed. This is the case for integrable systems. To superimposed normal distribution (continuous blue line) that eliminate this possibility, we further examine the following is fitted to have the same variance (and the mean value, which −1 quantity: is of order dE, i.e., exponentially small), ≡ A † A Mm max max Tr ρab ρab , (24) ij 1 ij 2 |Ea |<E b σ = R . (26) n d a E a where for a given Ea we first scan all Eb = Ea to find the ≡ A † A maximal value LA(a) maxb Tr[(ρab) ρab], and then find Figure 3(a) shows that P (R) is well approximated by the = MA maxa LA(a) by scanning all values of Ea within the normal distribution. The situation for all other matrix elements | | = | | window Ea <E 0.1n. The restriction to Ea <Eis for m = 1,2,3 is very similar. necessary as ETH is only expected to apply to the finitely Numerically, the standard deviation σn shows a robust excited states, not to the states from the edges of the spectrum. independence of the width of the energy shell E that includes This is manifest from the plot of Fig. 2(a). Figure 2(b) indicates a large number of states. We plot ln(σn)asafunctionofn in that MA decreases exponentially with n. This provides strong Fig. 3(b). We find that σn decreases exponentially with the numerical support for (5). A = system size n for all matrix elements of ρa for m 1,2,3 To study the fluctuations raa of individual matrix elements and values of h which are not too close to the integrable point A E of ρE around the mean value we introduce eigenstates a˜ of h = 0. The exponential suppression of Ra follows from the the local Hamiltonian HA and define exponential suppression of ||ρ||.But(4) does not guarantee 1 that different matrix elements of ρa would converge to those of ij = √ E A − AE Ra i ρa+1 ρa j . (25) ρ(E ) with the same rate. Numerics show that the convergence 2 a rate α = d ln σn/dn is approximately the same, fluctuating −1/2 In terms of the fluctuations Rab = √rab of (3), Ra is around the numerical value − ln(2)/2, for all matrix elements − m=1 simply the difference (R(a+1)(a+1) Raa)/ 2. In Fig. 3(a),we of ρa ; see Fig. 3(b). The behavior for all matrix elements m=2,3 show the distribution (histogram) P (R)forEa from the of ρa is very similar. The numerical proximity of α to central band |Ea| <E and one particular choice of i, j, − ln(2)/2 provides a strong numerical support for the form of and A consisting of m = 1 spin. The plot also contains a the exponentially suppressed factor in (3), which was originally

(a) (b)

FIG. 2. (a) Plot of LA(E)vs = E/n for n = 15 and n = 17. (b) Mm=1,2,3 all decrease exponentially with n. Here E is chosen to be equal to 0.1n and h = 0.1.

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(a) (b)

= 11 E | m=1|E = FIG. 3. (a) Probability distribution P (R) of the deviation R Ra corresponding to the matrix element 1 ρa 1 for E 0.1n and h = 0.1. It is superimposed with a Gaussian distribution fit. The vertical axis is the number of energy eigenstates within the energy shell | | m=1,2,3 Ea <Ewith a particular value of R. All matrix elements of ρE show almost identical behavior. (b) Linear behavior of ln(σn)asa 11 12 function of system size n for two matrix elements R and R for m = 1andh = 0.1. Because of the approximate equality ρC ≈ ρG the typical magnitude of the diagonal terms of ρa is much larger than the off-diagonal ones. There is no qualitative difference between different matrix elements. Results for m = 2,3 are similar. introduced in [8]. Provided that P (R) is well described by actual slope of the linear fit of ln(σm,n) as a function of m is ∼ normal distribution, the probability of√ a given Raa to be of order 0.219. This is substantially smaller than ln(2)/2 0.347, −2nR2/R2 R or larger is given by 1 − Erf(R/ 2σn) ∼ e 0 , where providing numerical support for (9). A R0 is some constant. If the total number of eigenstates grows Finally, we consider the behavior of ρa when A becomes n ¯ as 2 , the probability of finding an energy eigenstate Ea which comparable to A to probe the validity of (16) in the regime of n −2nR2/R2 does not satisfy ETH and has large Raa is given by 2 e 0 . fixed p. This is numerically more challenging. Nevertheless, This probability quickly goes to zero with n, which explains our numerical results are still quite suggestive. We consider the strong version of ETH recently discussed in [21]. subspace A consisting of eight leftmost consecutive spins with Next, we investigate the prefactor in (4) to test the n = 17 and h = 0.1. The numerical results comparing one di- E | A|E bound behavior outlined in (9). Namely, we are interested agonal matrix element 1 ρa 1 corresponding to the lowest A in the dependence of the exponential suppression factor energy level of HA isgiveninFig.4(b). It shows that ρa follows A ≈ A on subsystem size m. To illustrate this behavior we plot (16) pretty well while it differs from ρC ρG significantly. ln σm,n for a fixed value of n = 17 and different m in Fig. 4(a). In terms of the spin chain, the bound (9) means the || A − A || ACKNOWLEDGMENTS trace distance ρa ρ (Ea) should not grow faster than m ln(2)−n ln(2)/2 O (e ). This follows from (21) if the second norm This work was supported by funds provided by MIT- A − A 2 m ln(2)/2−n ln(2)/2 Tr[ρa ρ (Ea)] is bounded by O(e ). The Skoltech Initiative. We would like to thank the University of

(a) (b)

FIG. 4. (a) Dependence of ln σm,17 on the subsystem size m with the superimposed linear fit −4.455 + 0.219m. (b) Comparison of matrix A A A A E | m=8|E elements of ρa ,ρC ,ρG and the quasiclassical result (16) which we refer to as ρQ. Blue dots are matrix elements 1 ρa 1 as a function of = = = E | m=8|E E | m=8|E energy per site Ea/n for h 0.1andn 17. We see that 1 ρa 1 follows the semiclassical result 1 ρQ 1 as given by (16) well, E | m=8|E ≈E | m=8|E while it differs significantly from 1 ρC 1 1 ρG 1 , which lie on top of each other. Quasiclassical result (16) is calculated using density of states specified in Appendix Sec. 1. Other matrix elements show similar behavior.

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for some fixed a and d being the dimension of the full Hilbert space. Since |Ea is a complete basis, Eb|12|Eb = 2|1. (A5) b Now let us introduce a basis in the Hilbert space |i,j¯=|i⊗ | ¯ H = H ⊗ H j associated with the decomposition A A¯ . Then A = A = ¯| | ¯ ρab ij i ρab j i,k Ea Eb j,k (A6) k¯ FIG. 5. The density of states of the spin chain system for g = and 1.05,h= 0.1,n= 17. The horizontal axis is energy per site = 1 E/n. The yellow bars which fill the plot are the histogram for the 2 = i,k¯|E E |j,k¯j, ¯|E E |i, ¯. a d a b b a density of states calculated using direct diagonalization. The blue b i,j k,¯ ¯ solid line is a theoretical fit by the binomial distribution function (A2)withκ ≈ 0.3489; see Appendix Sec. 1. Now we use (A5) to get d d 2 = A | ¯ ¯| = A a Ea i,j i,j Ea . (A7) Kentucky Center for Computational Sciences for computing d d i j¯ time on the Lipscomb High Performance Computing Cluster. 3. Semiclassical expression APPENDIX We now discuss the properties of (16), 1. Density of states ¯ (E − E ) E ρAE = E ρA E = A a i , (A8) A spin chain without nearest neighbor interactions exhibits i a i i micro i (E ) a degenerate spectrum with the level spacing of order 1. In this a case the density of states is given by the binomial distribution. in the limit when Once the nearest neighbor interaction term is introduced, the V 1 0

012140-6 SUBSYSTEM EIGENSTATE THERMALIZATION HYPOTHESIS PHYSICAL REVIEW E 97, 012140 (2018) relation between the inverse temperature β and the mean With the help of (A14) one can rewrite (16) and (A8)as energy E, follows: S S −E −S −E¯ −S E¯ ∂ (Ea) E AE A¯ (E ) A¯ (E A) A( A) β = . (A16) ρa e , (A19) ∂E Together with (A14) and (A15)thisimplies while off-diagonal matrix elements are negligible. Then the corresponding entropy SA is given by S S a = ∂ A = ∂ A¯ β . (A17) A =− A A = S E¯ = A ∂E E¯ ∂E E¯ Sa Tr ρa ln ρa A( A) SG. (A20) A A¯ A A Then it follows in a standard way that the entropy SG associated Applying saddle point approximation to powers of (A18) A with the diagonal density matrix ρG and (A19) one can readily calculate the leading volume- A A A −β(E−E¯ )−S (E¯ ) proportional contribution to Renyi entropies for ρ and ρ and E ρ E e A A A (A18) a G G see that they are different. This is consistent with the numerical A = S E¯ is simply SG A( A). results of [12].

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