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Subsystem Eigenstate Thermalization Hypothesis Subsystem eigenstate thermalization hypothesis The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Dymarsky, Anatoly et al. "Subsystem eigenstate thermalization hypothesis." Physical Review E 97, 1 (January 2018): 012140 © 2018 American Physical Society 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 eigen- state 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) 2470-0045/2018/97(1)/012140(7) 012140-1 ©2018 American Physical Society ANATOLY DYMARSKY, NIMA LASHKARI, AND HONG LIU PHYSICAL REVIEW E 97, 012140 (2018) A = | | discussion. In particular, in case of finite-dimensional models it (ii) The “off-diagonal” matrices ρab TrA¯ Ea Eb are expo- nentially 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 = .
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