PHYSICAL REVIEW X 8, 021026 (2018)
Does a Single Eigenstate Encode the Full Hamiltonian?
James R. Garrison1,2 and Tarun Grover3,4 1Department of Physics, University of California, Santa Barbara, California 93106, USA 2Joint Quantum Institute and Joint Center for Quantum Information and Computer Science, National Institute of Standards and Technology and University of Maryland, College Park, Maryland 20742, USA 3Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA 4Department of Physics, University of California at San Diego, La Jolla, California 92093, USA
(Received 31 August 2017; revised manuscript received 14 December 2017; published 30 April 2018)
The eigenstate thermalization hypothesis (ETH) posits that the reduced density matrix for a subsystem corresponding to an excited eigenstate is “thermal.” Here we expound on this hypothesis by asking: For which class of operators, local or nonlocal, is ETH satisfied? We show that this question is directly related to a seemingly unrelated question: Is the Hamiltonian of a system encoded within a single eigenstate? We formulate a strong form of ETH where, in the thermodynamic limit, the reduced density matrix of a subsystem corresponding to a pure, finite energy density eigenstate asymptotically becomes equal to the thermal reduced density matrix, as long as the subsystem size is much less than the total system size, irrespective of how large the subsystem is compared to any intrinsic length scale of the system. This allows one to access the properties of the underlying Hamiltonian at arbitrary energy densities (or temperatures) using just a single eigenstate. We provide support for our conjecture by performing an exact diagonalization study of a nonintegrable 1D quantum lattice model with only energy conservation. In addition, we examine the case in which the subsystem size is a finite fraction of the total system size, and we find that, even in this case, many operators continue to match their canonical expectation values, at least approximately. In particular, the von Neumann entanglement entropy equals the thermal entropy as long as the subsystem is less than half the total system. Our results are consistent with the possibility that a single eigenstate correctly predicts the expectation values of all operators with support on less than half the total system, as long as one uses a microcanonical ensemble with vanishing energy width for comparison. We also study, both analytically and numerically, a particle-number conserving model at infinite temperature that substantiates our conjectures.
DOI: 10.1103/PhysRevX.8.021026 Subject Areas: Condensed Matter Physics, Quantum Physics, Statistical Physics
I. INTRODUCTION ergodic quantum many-body Hamiltonian is sufficient to determine the properties of the system at all temperatures. Given a local Hamiltonian, what information about the It is not very surprising that the ground states of quantum system is encoded in a single eigenstate? If the eigenstate many-body systems contain some information about their happens to be a ground state of the Hamiltonian, a excitations. This is because an entanglement cut often tremendous amount of progress can be made on this mimics an actual physical cut through the system, thus question for Lorentz invariant systems [1,2], especially exposing the underlying excitations along the entangling conformal field theories (CFTs) [3–6], and for topological boundary [9]. The same intuition is tied to the fact that the phases [7–9]. For example, one can read off the central “ ” – ground state entanglement satisfies a boundary law of charge of a CFT from the ground state entanglement [3 5], entanglement entropy [11,12]; that is, the von Neumann while for topological phases, essentially all “topological − ρ ρ ” entanglement entropy S1 ¼ trAð A logð AÞÞ of the ground data such as braiding statistics of anyons can be extracted A – state corresponding to a subsystem scales with the size of from the degenerate ground states [8 10]. In this paper, we the boundary of subsystem A. argue that a single finite energy density eigenstate of an How does the nature of information encoded evolve as one goes from the ground state to an excited eigenstate? Typically, there always exist eigenstates with energy E just above the ground state that continue to satisfy an area law Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- of entanglement. These are the eigenstates that have zero − 0 bution of this work must maintain attribution to the author(s) and energy density, i.e., limV→∞½ðE E0Þ=V ¼ , where E0 is the published article’s title, journal citation, and DOI. the ground state energy and V is the total volume of the
2160-3308=18=8(2)=021026(24) 021026-1 Published by the American Physical Society JAMES R. GARRISON and TARUN GROVER PHYS. REV. X 8, 021026 (2018) system. These eigenstates can often be interpreted as the often mentioned that in systems where Eq. (1) does hold, it action of a sum of local operators acting on the ground does so only for “few-body” operators [35–37], where, to our state; for example, in a system with spontaneous symmetry knowledge, the precise meaning of a few-body operator has breaking, one can construct an eigenstate consisting of a not been clarified (see Ref. [38] for related discussion). In this few magnons by a superposition of spin-flips acting on the paper, we conjecture and provide numerical evidence that ground state. Furthermore, the level spacing between two Eq. (1) holds for all operators within a subsystem A when the δ ∼ 1 α contiguous low-lying excitations scales as E =L , volume VA of subsystem A satisfies VA ≪ V (or, more α 0 where > depends on dimensionality and the phase precisely, when VA=V → 0 as V → ∞). We also explore the of matter under consideration. In this paper, we will instead validity of ETH when the subsystem fraction VA=V is taken be concerned with excited eigenstates that have a finite to be finite as VA, V → ∞. While we find in this case that energy density, i.e., limV→∞½ðE − E0Þ=V ≠ 0. For nota- Eq. (1) fails to hold for some operators, our results are tional convenience, we will set E0 ¼ 0 for the remainder of consistent with the possibility that ETH holds for all this paper. operators, as long as a microcanonical ensemble is used As argued by Srednicki [13], the question of how (and for comparison. [We will return to this case shortly, after first whether) an isolated quantum system eventually reaches considering the implications of Eq. (1).] thermal equilibrium is closely related to properties of its The satisfaction of Eq. (1) for all operators in a finite energy density eigenstates. Let us consider an subsystem A is equivalent to the statement that the reduced ρ ψ ψ ψ arbitrary (nonequilibrium) initial state, with finite energy density matrix Aðj iβÞ¼trA¯ j iββh j corresponding to an density and subextensive energy fluctuations, as might be eigenstate jψiβ is given by [23] prepared by performing a quantum quench [14]. Evolving such a state under the Hamiltonian H for sufficient time is ρ ψ ρ β Aðj iβÞ¼ A;canonicalð Þ; ð2aÞ expected to lead to the predictions dictated by the basic tenets of equilibrium statistical mechanics, if the system where thermalizes. Such an expectation leads to the “eigenstate thermalization hypothesis” (ETH) [13,15,16], which stip- −βH ρ β trA¯ ðe Þ ulates that the thermalization occurs at the level of each A;canonicalð Þ¼ −β ; trðe HÞ individual eigenstate. An alternative approach by Deutsch [15], which is based on perturbing an integrable system by A¯ being the complement of A. Note that the trace in the a small integrability breaking term, leads to the same denominator is over the whole Hilbert space. The equality suggestion. If ETH holds true, then, in the thermodynamic in Eq. (2a) means that the density matrices become limit, the equal-time correlators of an operator with respect → ∞ ψ elementwise equal in any basis as V . to a finite energy density eigenstate j i are precisely equal One immediate consequence of Eq. (2a) is that the to those derived from the canonical ensemble, i.e., thermodynamical properties of a system at arbitrary tem- −βH peratures can be calculated using a single eigenstate. ψ ψ trðOe Þ h jOj i¼ −β ; ð1Þ For example, Eq. (2a) implies that to the leading order, trðe HÞ − 1 α − 1 ρα the Renyi entropies Sα (¼ ½ =ð Þ log ½trAð AÞ ) for an eigenstate jψiβ corresponding to a subsystem A with where β is chosen such that Eq. (1) holds true when O ¼ H, VA ≪ V are given by [39,40] the Hamiltonian. Henceforth, we will use the notation jψiβ to denote an eigenstate whose energy density corresponds α −1 Sα V β f αβ − f β ; to temperature β . A notable exception to ETH is a many- ¼ α − 1 A ð ð Þ ð ÞÞ ð3Þ body localized system in the context of strongly disordered −1 interacting quantum systems [17–23], which fails to ther- where fðβÞ is the free energy density at temperature β . malize and does not satisfy Eq. (1). The possibility [24–30], The above equation allows one to access the free energy or impossibility [31–34], of the violation of ETH without density f at an arbitrary temperature by varying α. Note that disorder has also been discussed recently. Eq. (3) holds only to the leading order because Renyi In this paper, we restrict ourselves to systems where entropies Sα receive additional subleading contributions ETH, as defined by Eq. (1), holds. However, Eq. (1) alone due to the conical singularity induced at the boundary of is incomplete unless one also specifies the class of subsystem A [3–5]. In the limit α → 1, one recovers the operators for which it holds. For example, one well-known equality between the von Neumann entanglement entropy β β nonlocal operator for which Eq. (1) breaks down is the S1 and the thermal entropy Sth ¼ VAsthð Þ, where sthð Þ is projection operator jψihψj onto the eigenstate jψi that the thermal entropy density at temperature β−1, a result enters Eq. (1); the left-hand side of Eq. (1) yields unity for which was argued to hold in Ref. [41] for the special case of this operator, while the right-hand side is exponentially two weakly coupled ergodic systems. We emphasize that small in the volume, a clear disagreement. On that note, it is these results could not be derived from Eq. (1) alone if it were
021026-2 DOES A SINGLE EIGENSTATE ENCODE THE FULL … PHYS. REV. X 8, 021026 (2018) to hold only for local operators, since entanglement entropies On the other hand, we are able to construct operators for do not correspond to the expectation value of any local which Eq. (1) fails when VA=V is nonzero. These operators operator. We also note that Refs. [40,42] simulated the include powers of the Hamiltonian HA restricted to sub- thermal Renyi entropy Sα [starting with the expression on system A, as well as certain “nonequithermal” operators the right-hand side of Eq. (2a)] using quantum Monte Carlo (which we define below) when the subsystem fraction is −1 to access the properties of the system at temperature ðαβÞ . greater than a critical Oð1Þ number, f . Crucially, these Of course, quantum Monte Carlo methods are not well suited cases each correspond to operators for which the canonical to verifying ETH since they cannot access properties of a and microcanonical ensembles predict different results single eigenstate [the left-hand side of Eq. (2a)]. (they are only strictly equivalent when VA ≪ V). We will also discuss an approximate, but more intuitive In fact, the results of Refs. [49,50] suggest that a single form of ETH, given by eigenstate is expected to reproduce the results of a micro- canonical ensemble, rather than the canonical ensemble, −βH e A when 0
021026-3 JAMES R. GARRISON and TARUN GROVER PHYS. REV. X 8, 021026 (2018) finite temperature. Section VI studies the validity of ETH contain information about a system at all temperatures. when VA ≪ V by providing a close look into the entan- In terms of Eqs. (2a) and (2b), this is equivalent to asking glement Hamiltonian, focusing on its spectrum and whether a single eigenstate contains accurate information Schmidt vectors. This section also demonstrates the validity about microstates in subregion A that have thermodynami- ≪ of Eq. (2a) when VA V by considering the trace norm cally rare energy density. In this paper, we phrase this distance between both sides. Section VII explores the question in terms of which operators satisfy ETH. From this validity of ETH when VA=V is taken to be finite as perspective, it is useful to separate operators into two → ∞ V . Section VIII provides an application, by using classes with respect to a given Hamiltonian, H: the reduced density matrix from a single eigenstate to Equithermal operators: When the full system is predict correlators and free energy densities at all (finite) considered with respect to the canonical ensemble, temperatures. Section IX summarizes our results and the expectation value of an equithermal operator provides thoughts for future discussion. receives significant contribution only from microstates in subregion A with thermodynamically common II. GENERAL CONSIDERATIONS energy density (i.e., those corresponding to the tem- −1 A. Determining the Hamiltonian from microstates in perature, β ). [Here, A is the subregion spanned by classical statistical mechanics the operator, and “microstates” in subregion A are the eigenvectors of either ρ ðβÞ or H (corre- Suppose that, for an isolated system described by A;canonical A sponding to Eq. (2a) or Eq. (2b), respectively).] Local classical statistical mechanics in a total volume V,we operators fall into this class because there are no are given access to all classical microstates in a small pffiffiffiffi Δ Δ ∼ thermodynamically rare microstates on one site (or energy window ½E; E þ E , where E V is on the very few sites). This class also includes sums of local order of the energy fluctuations in the total system if it were operators, as well as some nonlocal quantities, such as coupled to a thermal bath, and, thus, all microstates the energy variance and von Neumann entropy (S1) correspond to the same energy density. We pose the within some subregion. Essentially, equithermal op- question: Does this information suffice to determine the erators are those for which the saddle point approxi- underlying Hamiltonian, assuming that the Hamiltonian is mation is valid. Crucially, a single eigenstate of energy local? The answer is indeed yes, following the standard E need not contain accurate information about energy procedure of obtaining the canonical ensemble from a densities away from E=V for Eq. (1) to hold for microcanonical ensemble. In particular, let us make a ¯ equithermal operators. fictitious division of the system into A and A, such that Nonequithermal operators: Nonequithermal operators ≪ VA VA¯ , and count the number of times a particular are those that, when considered with respect to the configuration CA appears in subsystem A. This determines canonical ensemble, receive significant contribution the probability distribution for finding a given configura- from microstates with thermodynamically rare energy tion, PðCAÞ. If all microstates are equally likely, then [51] density, i.e., energy densities other than that corre- sponding to the temperature, β−1. In many cases, these −βEðCAÞ P e are operators specifically designed to probe thermo- PðCAÞ¼ −β ; ð5Þ e EðCAÞ fCAg dynamically rare microstates in a subregion of the system. For instance, an operator that includes a factor −β0H where EðCAÞ is the energy in subsystem A.Onemaynow like e A is nonequithermal because it probes micro- −1 invert this equation to obtain the energy EðCAÞ¼ states of subregion A at temperature ðβ þ β0Þ .A − 1 β ð = Þ logðPðCAÞÞ, up to an irrelevant constant shift of projector onto the ground state (or any collection of energy. In a classical statistical mechanical system, EðCAÞ energetically rare microstates) within a macroscopic is the Hamiltonian for subsystem A. In particular, knowing subregion is also a nonequithermal operator. This EðCAÞ, one may now calculate any thermodynamic prop- class also includes all Renyi entropies Sα for α ≠ erty at any temperature. Here it is crucial to note that Eq. (5) 1 [39]. does not assume that the energy density EðCAÞ=VA equals If handed an operator as a black box, determining whether the energy density E=V of the microstates being sampled. it is most accurately characterized as equithermal or non- As discussed in the Introduction, we will provide equithermal with respect to a Hamiltonian requires analyzing evidence that the quantum mechanical analog of Eq. (5) it on the subsystem spanned by the operator. Let H be the – A is given by Eqs. (2a) (2c). We now proceed to discuss the ðAÞ conditions under which Eqs. (2a)–(2c) are valid. Hamiltonian restricted to this subregion, and let En and jnðAÞi denote the eigenvalues and eigenvectors, respectively, B. Two classes of operators of HA. Then, the expectation value hOiβ;H is equal to P A −β ðAÞ −β 1 En ðAÞ ðAÞ HA One of the central tasks of this paper is to investigate ð =ZAÞ ne hn jOjn i, where ZA ¼ trðe Þ. whether a single finite energy density eigenstate can Hence, we can create a weighted histogram of contributions
021026-4 DOES A SINGLE EIGENSTATE ENCODE THE FULL … PHYS. REV. X 8, 021026 (2018)
−β ðAÞ e En jhnðAÞjOjnðAÞij with respect to energy density operators. Indeed, as discussed below, several known results already point to the conclusion that ETH holds EðAÞ=V . If, for various β, this histogram is peaked at the n A for at least some equithermal operators, as long as V 021026-5 JAMES R. GARRISON and TARUN GROVER PHYS. REV. X 8, 021026 (2018) hðH − hHiÞ2i in the full system, unlike in the canonical knowledge of the properties of the system at all ensemble, where the variance scales proportionally with temperatures. system size. This relationship can be expressed as Now let us turn to the case in which VA=V is nonzero, which turns out to be much more subtle. First, we consider VA¯ the potential validity of Eqs. (1) and (2a) in this case, tr½ρ ðjψiβÞO β ¼ tr½ρ ðβÞO β ; ð8Þ A A; V A;canonical A; which, for the purposes of the following discussion, we refer to as “canonical ETH.” As mentioned in the previous 2 where OA;β ¼ðHA − hHAiβÞ is the energy variance oper- subsection, there is already an equithermal operator for ator. We will explore implications of the subsystem energy which this equation fails in this limit, namely, the sub- variance mismatch between a single eigenstate and the system energy variance. Thus, we do not expect that Eq. (1) canonical ensemble more carefully in Sec. VII A. will hold for all nonequithermal operators either when f is Of course, it is important to note that a suitably defined nonzero. In addition, for a given ratio VA=V with both VA, microcanonical ensemble has zero energy variance in the V → ∞, there is a physical constraint on the range of thermodynamic limit, just like a single eigenstate. This energy densities for which the spectrum of jψiβ,in ρ β provides further evidence that a microcanonical ensemble principle, can match that of A;canonicalð Þ. To appreciate is the appropriate ensemble to compare a single eigenstate this, let us consider a slightly different problem—an with when VA=V is nonzero [49,50] and, therefore, that arbitrary Hamiltonian of hardcore bosons with particle- Eq. (2c) is the most appropriate form of ETH to consider in number conservation, at infinite temperature. We will this case. We will consider this equation’s validity in consider an explicit example of such a system in the next Sec. VII B. section. Since the total particle-number operator Nˆ com- There is another issue to consider regarding the sub- mutes with the Hamiltonian and satisfies the equation system energy variance. For the case of time-evolved states, ˆ ˆ ˆ ρ N ¼ NA þ NA¯ , the reduced density matrix A for a wave the full system energy variance is independent of time. For ψ function j iβ¼0 is block diagonal in the number of particles a given initial state, this variance may indeed be different NA in subsystem A. Furthermore, if canonical ETH holds from the energy variance expected in the canonical ensem- [as given by a generalization of Eqs. (2a) and (2b)], then the ble, which implies that the energy variance for any Schmidt decomposition is given by subsystem that is a finite fraction of the total system will disagree, even at long times [60]. In this case, the system XN qffiffiffiffiffiffiffi X will relax to a subsystem energy variance given by the ψ λ u ⊗ v ; 9 j iβ¼0 ¼ NA j iiNA j iiN−NA ð Þ diagonal ensemble, which depends upon the energy histo- NA¼0 i gram of the initial state. λ In summary, while we expect that Eq. (1) is obeyed by where NA are the Schmidt coefficients in the sector NA, and many equithermal operators when V =V is nonzero, it u , v are the corresponding eigenvectors. The A j iiNA j iiN−NA cannot be satisfied by all such operators, since the sub- label i captures fluctuations of particles within a fixed system energy variance provides an important counterex- λ sector NA. Note that there is no index i on NA because we ample. Instead, a more appropriate definition of ETH in this are at infinite temperature and all Schmidt states within a case is the one given by Eq. (2c), which compares the sector NA are equally likely. reduced density matrix from a single eigenstate to a The decomposition in Eq. (9) allows one to calculate microcanonical ensemble with vanishing energy window. properties of subsystem A at infinite temperature even away from filling N=V, since the reduced density matrix ρA will 2. ETH for nonequithermal operators contain sectors with various densities NA=VA. However, there is both an upper limit and a lower limit on the density The extra ingredient introduced by nonequithermal in subsystem A, since operators is that if ETH holds for them, then taking such ’ ψ an operator s expectation value with respect to a state j iβ max ½N − ðV − V Þ; 0 ≤ N ≤ min ½N;V ; ð10Þ allows one to access the properties of the Hamiltonian A A A β−1 at a temperature different than . For example, the and, thus, the particle density N =V in subsystem A ρ ψ A A Renyi entropy Sα corresponding to Aðj iβÞ satisfies Sα ¼ satisfies ½α=ðα − 1Þ VAβðfðαβÞ − fðβÞÞ, thus allowing one to access the free energy density at temperature ðαβÞ−1. N max ½1 − ð1 − nÞ=f; 0 ≤ A ≤ min ½n=f; 1 ; ð11Þ Let us first consider the validity of ETH for nonequi- VA thermal operators when VA ≪ V. Remarkably, the results presented in the remainder of this paper demonstrate that where n ≡ N=V is the overall particle density and ETH is valid for all nonequithermal operators in this limit. f ≡ VA=V. A necessary condition for the wave function Thus, a single eigenstate of finite energy density contains in Eq. (9) to encode properties of the system at all fillings is 021026-6 DOES A SINGLE EIGENSTATE ENCODE THE FULL … PHYS. REV. X 8, 021026 (2018) f ≤ min ½n; 1 − n : ð12Þ boundary terms in HA¯ gives the inequality ≥− d−1 hvi0jHA¯ jvi0i cL , where c is a constant (recall that, The above discussion, with some modifications, carries in our convention, the ground state energy for the full to systems with (only) energy conservation, at an arbitrary Hamiltonian is set to zero). Since both E and hui0jHAjui0i d temperature. The Schmidt decomposition of an eigenstate scale as L , the only way that hui0jHAjui0i can exceed jψiβ with eigenvalue E may now be written as E is if the second term on the left-hand side of Eq. (17), pffiffiffiffiffiffiffiffiffiffiffiffi def P X pffiffiffiffi viz. E ¼ λ =λ 0hu 0j ⊗ hv 0jH ¯ ju i ⊗ jv i,is ψ λ ⊗ boundary j j i i i AA j j j iβ ¼ i juii jvii: ð13Þ negative and scales as Ld. When that happens, canonical i ETH no longer holds, as we now argue on general grounds, The physical content of canonical ETH, as approximated in and as we will also demonstrate numerically for a lattice −βEA;i Hamiltonian in Sec. VII. To see this, we reiterate that Eq. (2b), is that λi ∝ e , where EA;i is the ith energy canonical ETH requires that (i) juii’s are approximate eigenvalue of HA (the projection of the Hamiltonian to −β −β eigenstates of H , and (ii) λ ∝ e huijHAjuii ¼ e EA;i . subsystem A), while juii is the corresponding eigenstate of A i Firstly, when hu 0jH ju 0i ¼ E − hui0jHAjui0i: ð17Þ where e ¼ E=V is the energy density corresponding to the wave function and emax is the maximum energy density for Because of the variational principle, the energy density the Hamiltonian H (recall that emax can be finite for lattice- in subregion A¯ cannot be less than that of the ground regularized quantum systems, e.g., for models of fermions state. Allowing negative energy contributions due to or spins/hardcore bosons). Above, we have assumed that 021026-7 JAMES R. GARRISON and TARUN GROVER PHYS. REV. X 8, 021026 (2018) 2 2 ≪ e 021026-8 DOES A SINGLE EIGENSTATE ENCODE THE FULL … PHYS. REV. X 8, 021026 (2018) FIG. 1. Entanglement entropies S1 through S4 for a model with no conservation law [left panel, given by Eq. (30) at L ¼ 21] and a model with particle-number conservation [right panel, given by Eq. (22) at L ¼ 27, with filling N ¼ 6]. We use the parameters mentioned in the text to place each model at a nonintegrable point. In each case, we consider eigenstates in the k ¼ 1 sector, with subsystem size LA ¼ 4. The grey vertical line denotes infinite temperature (point of maximum S1), and the black circles mark the theoretical predictions for the entanglement entropies there. The brown markers denote the theoretical values of the entropies in the limit LA, L → ∞ while LA=L → 0, as given by Eqs. (26) and (27). Because of particle-number conservation, the reduced with that of the thermal reduced density matrix ρ β 0 density matrix from any pure state is block diagonal, with A;canonicalð ¼ Þ studied in Ref. [40], consistent with each block corresponding to some filling number NA of the ETH. ρðNAÞ With this, the von Neumann entropy at infinite temper- subsystem A. The block of the reduced density matrix A ature becomes corresponding to filling NA is a dNA × dNA matrix, where 1 d ≡ ðLA Þ. At infinite temperature and for L =L < , the X NA NA A 2 ρ S1 − d λ λ ; eigenvalues of A must be equal to one another within a ¼ NA NA log NA ð23Þ given block, but the eigenvalues in different blocks will be NA − different: They are proportional to ðL LA Þ, the number of N−NA microstates consistent with such a configuration in sub- and the Renyi entropies are given by ρ 1 system A. Taking into account that trð AÞ¼ , one finds ðN Þ X ≡ LA ρ A 1 that each of the dN ð Þ eigenvalues of is given by − λα A NA A Sα ¼ log dN ; ð24Þ − α − 1 A NA λ ≡ L LA L ρ ð Þ=ð Þ. The spectrum of that we find NA NA N−NA N A for a single eigenstate (as shown in Fig. 2) is in agreement where the sums over NA are restricted to subsystem particle fillings NA that satisfy the constraint in Eq. (10). The above 1 expressions are valid when LA=L < 2. Because the eigenvalues are nonuniform, the Renyi entropies Sα at infinite temperature depend on the Renyi index α, in contrast to an energy-only conserving model. The right panel of Fig. 1 shows how the actual values of S1 through S4 match those predicted by the above counting argument. [Note that if we had instead studied this model at half filling (i.e., “zero chemical potential”), the Renyi entropy densities at infinite temper- ature would again be independent of α, just like for the energy-only conserving model.] For comparison, we also calculate Sα analytically in the thermodynamic limit. For simplicity, we consider the limits FIG. 2. Eigenvalue spectrum of the reduced density matrix of L, N, L → ∞, such that n ¼ N=L is held constant, while an infinite temperature eigenstate, ρ ðjψiβ 0Þ, for the hardcore A A ¼ L =L → 0. In these limits, one can evaluate the expressions boson model Eq. (22), with L ¼ 27, LA ¼ 4, and filling N ¼ 6. A The red lines plot the theoretical value of each eigenvalue in the in Eq. (24) using Stirling’s approximation logðx!Þ ≈ − thermodynamic limit, determined from the filling NA of the sector x logðxÞ x. One finds that, in the limits considered, Sα in which it lies. receives contribution only from NA given by 021026-9 JAMES R. GARRISON and TARUN GROVER PHYS. REV. X 8, 021026 (2018) L by Eq. (22) at infinite temperature. Following the logic in N ¼ A : ð25Þ A 1 1 − 1 α the previous subsection, the subsystem occupancy is given þðn Þ by the hypergeometric distribution. While the average Thus, Sα probes the system at the filling NA=LA ¼ 1=f1 þ½ð1=nÞ − 1 αg, which is different than the actual subsystem filling is hNAi¼nLA ¼ Nf, the variance in filling n, unless α ¼ 1 (which corresponds to the von this quantity for a single eigenstate is Neumann entanglement entropy). This also immediately L hðN − hN iÞ2i¼L ð1 − fÞnð1 − nÞ : ð29Þ leads to expressions for Renyi and von Neumann entan- A A A L − 1 glement entropies in the thermodynamic limit: Both the filling and its variance are proportional to LA as 1 α α expected, but the variance includes an additional factor Sα=L ¼ − log ½n þð1 − nÞ ð26Þ A α − 1 (1 − f), which causes it to be suppressed compared with the grand canonical ensemble when f is finite. In Sec. VII Awe and will witness a similar suppression of the subsystem energy variance when the condition LA=L → 0 is relaxed. S1=LA ¼ −½n logðnÞþð1 − nÞ logð1 − nÞ : ð27Þ We plot these values in Fig. 1 for comparison. Remarkably, IV. MODEL HAMILTONIAN WITH ONLY even with the small system sizes we can access, the ENERGY CONSERVATION difference between the exact finite-size result (obtained by counting over all sectors) and the result valid in the To develop some understanding of the questions posed in thermodynamic limit is quite small. the Introduction, we study in detail throughout the remain- In the above derivation, it is also possible to relax the der of the paper a finite 1D quantum spin-1=2 chain with → 0 → ∞ the following Hamiltonian: restriction LA=L as LA, L . We then find that NA is given by the solution to XL σzσz σx σz H ¼ ð i iþ1 þ hx i þ hz i Þ: ð30Þ LA N ¼ ; ð28Þ i¼1 A 1 1−f − 1 α þ n−fN =L 0 9045 0 8090 A A We set hx ¼ . and hz ¼ . such that the model is far away from any integrable point and is thus expected to f → 0 which reduces to Eq. (25) when . satisfy ETH in the sense of Eq. (1), as shown in Ref. [63]. Let us note a few things about this equation: We use periodic boundary conditions throughout. (1) When α 1, the solution is N nL , regardless ¼ A ¼ A We diagonalized the Hamiltonian in Eq. (30) for system of f. Thus, the von Neumann entropy always probes sizes up to L ¼ 21, obtaining all eigenvalues and eigen- the system at its given filling, even when f is finite. states. As we hinted earlier, we assigned to each eigenstate Further analysis shows that Eq. (27) holds generally −1 1 a temperature β by finding the value β for which the when f<2. 1 energy expectation value in the canonical ensemble (2) When the system is at half filling (n ¼ 2), the matches the energy of the eigenstate: 1 α solution is NA ¼ 2 LA, regardless of f or . 1 1 ψ ψ −βH (3) When α > 1, 0 021026-10 DOES A SINGLE EIGENSTATE ENCODE THE FULL … PHYS. REV. X 8, 021026 (2018) limit α → 1 leads to the conclusion that the von Neumann entanglement entropy S1 satisfies β S1 ¼ VAsthð Þ; ð35Þ β ρ β where sthð Þ¼S1ð A;canonicalð ÞÞ=LA is the thermal entropy density at temperature β−1. We note that Eqs. (33) and (34) are consistent with recent work studying Renyi entropies in pure states representing thermal equilibrium [64,65]. B. Numerical results for von Neumann and Renyi entropies Figure 3 shows the scaling of the von Neumann entropy S1 as a function of subsystem size LA for the eigenstates FIG. 3. Scaling of the von Neumann entanglement entropy S1 jψiβ of our model [Eq. (30)]. As discussed in Sec. II C 1, with subsystem size for the L 20 system given in Eq. (30).Up ¼ we expect Eq. (35) to hold as long as VA 021026-11 JAMES R. GARRISON and TARUN GROVER PHYS. REV. X 8, 021026 (2018) FIG. 4. The von Neumann entropy S1 and Renyi entropies S2, S3, and S4 for the system given in Eq. (30) with L ¼ 21 and LA ¼ 4. −βHA Here, ZA ¼ trAðe Þ. The entropies of the reduced density matrix at each energy density agree remarkably with the entropies calculated from the canonical ensemble, given by Eqs. (2a) and (2b). deviation of Renyi entropy S2 from its ETH predicted value, as V → ∞. While our numerical results consider the case Δ Eq. (3). The finite-size scaling of S2 is relatively difficult, where VA is held constant as V → ∞, we expect that all because unlike S1, S2 shows oscillations as a function of LA results in this section also hold true when the limits are Δ (see, e.g., Refs. [40,66]). Despite this, S2 is less than a few taken such that f ≡ VA=V → 0 as VA, V → ∞. We focus percent of S2 itself. here on ETH in the sense of Eqs. (2a) and (2b), as the Figure 6 plots the entropy deviation ΔS1=LA for constant canonical ensemble is known to be equivalent to a micro- ratio LA=L at all available system sizes. Although it is canonical ensemble when VA ≪ V. Focusing on the difficult to do a detailed scaling analysis with so few points, canonical ensemble also allows us to avoid introducing Δ the data strongly suggest that S1=LA vanishes in the the energy width ΔE as an additional parameter in our thermodynamic limit. systematic comparisons. In Sec. VII, we will explore more The finite-size scaling of Renyi entropies at constant 1 carefully the case when f<2 is finite, using both the ratio LA=L is less conclusive, as can be seen in Fig. 7. The canonical and microcanonical ensembles for comparison. analytical argument for the particle-number conserving We begin by probing in detail the entanglement spectra α ≠ 1 model suggests that the Renyi entropies Sα for do of individual eigenstates as well as the corresponding not match their canonical counterparts when VA=V is held Schmidt states. Specifically, we compare five different fixed. The authors of Ref. [67] arrived at similar con- quantities, as shown in Fig. 8, that test the validity of clusions using a different approach. Eqs. (2a)–(2c) for the model described in Sec. IV.The agreement of the spectrum of ½−1=β log½ρAðjψiβÞ with that VI. EXTRACTING THE HAMILTONIAN FROM A −1 β ρ β of ½ = log½ A;canonicalð Þ , as well as with the actual SINGLE EIGENSTATE Hamiltonian HA in region A, implies that, essentially, the −βEA;i In this section, we will present numerical results that Schmidt eigenvalues λi satisfy λi ∝ e , where EA;i are the substantiate our conjecture that ETH is valid for all eigenvalues of HA. Similarly, the agreement with the expect- equithermal and nonequithermal operators when VA ≪ V ation value huijHAjuii shows that the Schmidt eigenvectors 021026-12 DOES A SINGLE EIGENSTATE ENCODE THE FULL … PHYS. REV. X 8, 021026 (2018) FIG. 6. Scaling of the von Neumann entropy deviation ΔS1 with 1=L, for constant ratio LA=L, averaged over all eigenstates in the range 0.28 < β < 0.32.AsinFig.5, the error bars represent one standard deviation away from the mean. Even though this plot considers the case where the subsystem size LA becomes infinite as L → ∞, the entropy deviations are going to zero rapidly as L becomes larger. between the reduced and canonical density matrices at various system sizes. The trace norm distance, defined as ρ ψ − ρ β k Aðj iβÞ A;canonicalð Þk1 1 hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii ≡ ρ ψ − ρ β 2 2 tr ð Aðj iβÞ A;canonicalð ÞÞ ; ð36Þ FIG. 5. Scaling of the entropy deviation ΔSα ≡ ρ β − ρ ψ 1 Sαð A;canonicalð ÞÞ Sαð Aðj iβÞÞ with =L, for constant LA, places an upper bound on the probability difference that averaged over all eigenstates in the range 0.28 < β < 0.32, for could result from any quantum measurement on the two S1 (top panel) and S2 (bottom panel). The error bars represent one density matrices [68]. As such, it provides an excellent standard deviation away from the mean. For S1, this deviation is measure of how distinguishable the two density matrices strictly non-negative, but, for higher Renyi entropies, it can are. If the trace norm distance between two finite-size oscillate and become negative before tending to zero as L → ∞. density matrices is zero, they are equal to each other element by element. juii have the same character as the eigenvectors of the canonical reduced density matrix. It is important to empha- size that this agreement holds throughout the spectrum: The agreement near the energy density of the eigenstate itself (denoted by the dashed grey line) suggests that ETH holds for equithermal operators, while the agreement away from this energy density suggests that Eq. (1) holds for nonequithermal operators as well. (There is a slight disagreement in the spectrum near the critical energy density e , and we explore this in the following section.) To probe the Schmidt eigenvectors further, we directly calculated the overlaps between the eigenvectors of the reduced density matrix ρAðjψiβÞ and the eigenvectors of the ρ β thermal density matrix A;canonicalð Þ (see Fig. 9). Again, we find excellent agreement. Again, the agreement throughout the spectrum suggests that Eq. (1) is valid for both FIG. 7. Scaling of the Renyi entropy deviation ΔS2 with 1=L, equithermal and nonequithermal operators. for constant ratio LA=L, averaged over all eigenstates in the range To quantify the extent to which Eq. (2a) is valid, we calcu- 0.28 < β < 0.32. As in Fig. 5, the error bars represent one ρ ψ − ρ β late the trace norm distance k Aðj iβÞ A;canonicalð Þk1 standard deviation away from the mean. 021026-13 JAMES R. GARRISON and TARUN GROVER PHYS. REV. X 8, 021026 (2018) eigenstate. Our results also imply that when VA ≪ V, ETH as specified by Eq. (2a) holds. One consequence of this is that if VA is held fixed, the density matrices ρ ψ ρ β Aðj iβÞ and A;canonicalð Þ become elementwise equal in any basis as V → ∞. VII. ETH WITH FINITE RATIO VA=V In this section, we consider to what extent ETH is valid 1 when the ratio f ≡ VA=V < 2 is held fixed and finite as VA, V → ∞. When the limits are taken in this way, the canonical and microcanonical ensembles are no longer equivalent, so we consider each ensemble independently. FIG. 8. Comparison of the five quantities defined in the inset A. Comparison with the canonical ensemble 4 21 β 0 3 for an LA ¼ subsystem at L ¼ and ¼ . . Each quantity As demonstrated in Sec. VB, the von Neumann entropy has been normalized so that the y axis has units of energy density. of ρAðjψiβÞ matches the thermal entropy in the thermody- The large blue circle markers show the spectrum of the canonical namic limit even for finite f<1. In the current section, we reduced density matrix, while the red diamond markers corre- 2 consider the extent to which other quantities match between spond to the eigenvalues of a reduced density matrix ρAðjψiβÞ for a single eigenstate at temperature β−1. The yellow triangle a single eigenstate and the canonical ensemble. markers show the spectrum of the microcanonical reduced There is one notable equithermal operator for which density matrix that consists of an equal-weighted mixture of Eq. (1) fails when f is finite. As explained in Sec. II C 1 the 50 eigenstates nearest in energy to the one under consid- [see Eq. (8)], the subsystem energy variance taken from a 1 − eration. The grey markers show the eigenvalues of HA with a shift single eigenstate is suppressed by a factor of ( f) −βH cA ≡ ð1=βÞ log ZA ¼ð1=βÞ log trAðe A Þ so that it can be di- compared with its value in the canonical ensemble. To rectly compared with −ð1=βÞ log½ρAðjψiβÞ in accordance with understand this, consider first the expectation value of the Eq. (2b) (note, also, that the combination H c is independent 2 − 2 A þ A operator HA hHAi with respect to the canonical ensem- of the shift of the spectrum of HA by an arbitrary uniform ble. This will be given by [51] constant). Finally, the orange markers represent the expectation R 2 − δ 2 value of HA, again with a shift cA, with respect to the Schmidt dðδE ÞðδE Þ e ð EAÞ =cVA ρ ψ 2 − 2 R A A eigenvector juii of Aðj iβÞ. In each case, the eigenvalues/ hHA hHAi iρ ðβÞ ¼ − δ 2 ; A;canonical δ ð EAÞ =cVA eigenvectors are ordered from smallest to largest energy density. dð EAÞe The horizontal lines plot the energy density e (dashed grey line) ð37Þ and the critical energy density e ¼ eL=LA (solid brown line) of the original eigenstate jψiβ, with respect to the ground state where c is the specific heat. Note that the probability energy density of HA þ cA (dotted black line). distribution is Gaussian because it is obtained by expanding the Boltzmann factor around its maximum. On the other If ETH holds for all operators in subsystem A, then the hand, in an eigenstate, the probability distribution will − δ 2 results of Ref. [16] imply that the trace norm distance should acquire an extra multiplicative factor of e ð EAÞ =cVA¯ go to zero as 1=L. The suggestion that the trace norm distance because a fluctuation δEA of energy in region A is between the pure state and thermal reduced density matrices necessarily accompanied by a fluctuation −δEA in the with fixed subsystem size would tend to zero was also made region A¯ since, for an eigenstate, there are no fluctuations of in Refs. [49,69]. We restrict ourselves to states in a β range energy in the total system. Thus, the expectation value of 0 28 β 0 32 2 − 2 ψ given by . < < . . In the left panel of Fig. 10, we plot HA hHAi with respect to the eigenstate j iβ is given by the trace norm distance of every eigenstate in this β range at 5 2 − 2 LA ¼ for a few select system sizes. For each system size, hHA hHAi iρ ðjψiβÞ R A the distribution of the trace norm distance is nearly constant 2 − δ 2 − δ 2 d δE δE e ð EAÞ =cVA e ð EAÞ =cVA¯ throughout the given β range. The right panel then takes these ðR AÞð AÞ ¼ − δ 2 − δ 2 : ð38Þ δ ð EAÞ =cVA ð EAÞ =cVA¯ data for each pair of L and LA and plots the mean and dð EAÞe e standard deviation of the trace norm distance against 1=L. The trace norm distance is tending toward zero at least Equations (37) and (38) imply 2 2 linearly with 1=L, perhaps even faster. hH − hH i iρ ψ A A Aðj iβÞ 1 − These results, taken together, strongly support the con- 2 2 ¼ VA=V: ð39Þ hH − hH i iρ β jecture that ETH, as given by Eq. (1), holds for all operators A A A;canonicalð Þ when VA ≪ V. The Schmidt eigenvalues and eigenvectors To demonstrate this relationship, the top panel of Fig. 11 match at all energy densities, not just the energy density of the shows scaling of the subsystem energy variance with 021026-14 DOES A SINGLE EIGENSTATE ENCODE THE FULL … PHYS. REV. X 8, 021026 (2018) FIG. 9. Overlap between the Schmidt eigenvectors juji and the eigenvectors jφii of the canonical reduced density matrix, for an L ¼ 21 system with β ¼ 0.3, and subsystem sizes LA ¼ 2, 3, 4, 5. In each case, the eigenvectors are ordered from most significant (largest eigenvalue) to least significant (smallest eigenvalue). The locations of the energy densities e (dashed grey) and e (solid brown) are denoted with lines, as in Fig. 8. subsystem size LA for both a single eigenstate and the additional term that is negative and quadratic in the canonical ensemble, for the model described in Sec. IV. subsystem size, matching the prediction in Eq. (39). While the energy variance grows linearly for LA ≪ L in The bottom panel of Fig. 11 shows, for comparison, the both cases, the single eigenstate energy variance has an variance of a different operator JA between a single ρ β ρ ψ FIG. 10. Trace norm distance between the canonical density matrix A;canonicalð Þ and the reduced density matrix Aðj iβÞ for all eigenstates jψiβ in the range 0.28 < β < 0.32. The left panel plots the trace norm distance for all such eigenstates with system sizes L ¼ 12, 15, and 20, and subsystem size LA ¼ 5. The right panel plots the mean and standard deviation of the trace norm distance in this β range for values of L up to 21 and LA up to 5. 021026-15 JAMES R. GARRISON and TARUN GROVER PHYS. REV. X 8, 021026 (2018) this more carefully, note that the trace norm distance places a bound on the difference in expectation value of any operator Λ that is bounded between zero and one as [68] jtrðρΛÞ − trðσΛÞj ≤ kρ − σk1: ð40Þ In order to calculate a lower bound on trace norm distance ρ ψ − ρ β k Aðj iβÞ A;canonicalð Þk1 due to the variance difference, we must write the energy variance as a bounded operator that maximally differs between the two density matrices. Naively, one might be tempted to consider the operator 2 2 2 OA;β=Δ ≡ ðHA − hHAiβÞ =Δ , where for the operator to be bounded, Δ must be chosen to be the largest energy available to the system. Since both OA;β and Δ scale linearly with V, the expectation value of this operator is actually zero in the thermodynamic limit for both ρ ψ ρ β Aðj iβÞ and A;canonicalð Þ. Thus, no bound can be placed on the trace norm distance due to this particular operator. Let us instead now consider a modified energy variance operator, OA;β Λ β Δ ≡ P β Δ P β Δ; A; ; A; ; Δ2 A; ; ð41Þ where Δ is an arbitrary energy scale and PA;β;Δ projects 2 onto the subspace where OA;β=Δ has eigenvalues in the FIG. 11. Top panel: Subsystem energy variance with respect to range [0, 1], thus making Λ β Δ a bounded operator. This subsystem size L for both the canonical ensemble (blue circular A; ; A operator considers the energy variance within a restricted markers) and a single eigenstate jψiβ (red diamond markers), at 2Δ L ¼ 18 and β ¼ 0.3. The inset shows the ratio between the window of width about the mean energy. energy variances at each subsystem size, which is expected to fit To arrive at an approximate bound due to this operator, 1 − let us assume that the energy histograms of ρ ðβÞ LA=L in the thermodynamic limit [Eq.P (8)]. Bottom panel: A;canonical ≡ LA ðiÞσx ðiÞσz and ρAðjψiβÞ are given by normal distributions with The variance of an operator JA i¼1ðhx i þ hz i Þþ P −1 2 2 2 LA ðiÞσzσz variance σ and σψ ¼ð1 − fÞσ , respectively. i¼1 Jz i iþ1, which includes the same terms as HA but canonical canonical does not relate to energy conservation, is plotted for comparison. Since both distributions have the same mean, the difference ðiÞ ðiÞ ðiÞ in expectation values is expected to be Here, the quantities hx , hz , and Jz are each taken from the uniform distribution ½−1; 1 . ≡ ρ β Λ − ρ ψ Λ D tr½ A;canonicalð Þ A;β;Δ tr½ Aðj iβÞ A;β;Δ Z 2 2 2 2 Δ −E =2σ −E =2σψ 1 e canonical e 2 eigenstate and the canonical ensemble. The operator JA ¼ pffiffiffiffiffiffi − pffiffiffiffiffiffi E dE: ð42Þ Δ2 σ 2π σ 2π (defined in the figure’s caption) is chosen to span the length −Δ canonical ψ of subsystem A and to include the same terms as H ; A σ Δ however, the coefficient of each term is different. The fact Given canonical and f, it is possible to find numerically such that D is maximized. Although Δ is proportional to that the variance of JA matches between a single eigenstate pffiffiffiffi V, the value of D itself is independent of V as V → ∞, and the canonical ensemble suggests that many equithermal pffiffiffiffi σ operators that do not explicitly involve energy conservation since canonical also scales with V. The maximum quantity will satisfy Eq. (1) at least approximately, even when VA=V D then provides a lower bound on the trace norm distance ρ β ρ ψ is finite. between A;canonicalð Þ and Aðj iβÞ in the thermodynamic Let us now consider an implication of the difference in limit [70]. ρ ψ subsystem energy variance between Aðj iβÞ and Let us now turn to our results on the scaling of trace ρ β A;canonicalð Þ. This difference, which occurs only when f norm distance with system size when the ratio f ¼ L =L is ρ ψ − A is finite, suggests that the trace norm distance k Aðj iβÞ held fixed as L, LA → ∞, which are shown in the left panel ρ β → 0 A;canonicalð Þk1 vanishes only when f . To explore of Fig. 12. Although there are few points available for each 021026-16 DOES A SINGLE EIGENSTATE ENCODE THE FULL … PHYS. REV. X 8, 021026 (2018) FIG. 12. Left panel: Trace norm distance between the reduced density matrices as a result of a single eigenstate and the canonical ensemble for constant ratio LA=L and 0.28 < β < 0.32. As in Fig. 10, the error bars represent one standard deviation away from the mean. The horizontal lines indicate the approximate theoretical minimum each trace norm distance can take, based on the suppressed energy variance given by maximizing Eq. (42). Right panel: Trace norm distance between the reduced density matrices from a single eigenstate and a microcanonical ensemble for constant ratio LA=L, considering all eigenstates in the range 0.28 < β < 0.32.Asin Fig. 8, the microcanonical ensemble is defined as the equal mixture of 50 eigenstates nearest in energy to the one under consideration. ratio, the trend is clearly for the trace norm distance to for all equithermal operators that span less than half decrease as L increases. The horizontal dotted lines denote the system. the theoretical minimum each trace norm distance can take, It is important to emphasize that, even if the trace norm ρ ψ − ρ given by Eq. (42). Remarkably, for each subsystem ratio, distance k Aðj iEÞ A;mcðEÞk1 indeed goes to zero in the the trace norm distance rapidly approaches this lower thermodynamic limit with VA=V fixed, this only implies bound, suggesting that the bound may actually provide that all equithermal operators satisfy Eq. (4). When VA=V the result in the thermodynamic limit. is fixed, nonequithermal operators will have a vanishing contribution to the trace norm distance because they probe B. Comparison with a microcanonical ensemble thermodynamically rare states. One way to investigate whether Eq. (4) holds even for nonequithermal operators Now we consider to what extent the reduced density for a nonzero subsystem fraction is to examine the matrix from a single eigenstate equals that of a micro- 1 entanglement spectra. canonical ensemble when the subsystem ratio f<2 is nonzero [i.e., the potential validity of Eq. (2c)]. We first C. Entanglement spectrum note that, unlike in the case of the canonical ensemble, the We now turn to results on the entanglement spectrum subsystem energy variance from a single eigenstate when f is a significant fraction of the total system size. As matches that from a microcanonical ensemble with vanish- discussed in Sec. II C 2, if the constraint in Eq. (20) is ing energy width, independent of system size. In particular, violated, the entanglement spectrum of a single eigenstate the subsystem energy variance provides an example of an cannot match that of the canonical ensemble above a operator for which ETH in the sense of Eq. (4) holds but for critical energy density e ¼ e=f [see Eq. (19)], where e which ensemble equivalence breaks down. To further is the energy density of the state jψiβ. Figure 13 shows the investigate the potential validity of Eq. (2c), we consider comparison of spectra of five different quantities consid- the trace norm distance scaling in the right panel of Fig. 12. ered in Sec. VI for several different energy densities of the Unlike in the case of the canonical ensemble, we know of reference state jψiβ with f ¼ 1=3, at four different values no examples of operators that span less than half the system of β. With f ¼ 1=3, the energy constraint Eq. (20) is for which the comparison is known to fail, and so there is violated, and, therefore, we expect that the entanglement no lower bound plotted in the figure. Indeed, the values are spectrum from a single eigenstate should deviate from the uniformly closer to zero than the comparison with the actual spectrum of the Hamiltonian at least beyond the canonical ensemble, which provides strong evidence that a critical energy density e ¼ e=f. We find for each value of microcanonical ensemble is the appropriate one to compare β that significant deviation starts to occur essentially right a single eigenstate against, in agreement with Sec. II and at this critical energy density. Refs. [49,50]. The trace norm distance appears to approach While the entanglement spectrum from a single eigen- zero at least linearly with 1=L, which suggests the state clearly differs from the actual spectrum of the possibility that ETH in the sense of Eq. (4) holds Hamiltonian and that of the canonical ensemble, it follows 021026-17 JAMES R. GARRISON and TARUN GROVER PHYS. REV. X 8, 021026 (2018) FIG. 13. Comparison of the five quantities defined in the inset of Fig. 8 for eigenstates of an L ¼ 21 system, with LA ¼ 7,atβ ¼ 0.2, 0.3, 0.4, and 0.5. Each inset plots a 12-bin histogram of the log of the density of states versus the energy density: the solid blue curve ρ ψ ρ β from a single eigenstate Aðj iβÞ and the dotted cyan curve from the canonical ensemble A;canonicalð Þ. We notice that, in each of the four plots, the eigenvalues of the reduced density matrix corresponding to a single eigenstate (red diamond markers) begin to deviate significantly from the eigenvalues of the canonical reduced density matrix (blue circle markers) as the energy density reaches the critical value e (solid brown line), indicating the breakdown of Eq. (2a) beyond e . sffiffiffiffiffiffi closely the spectrum due to a microcanonical ensemble. In X λ ≡ j ⊗ ⊗ fact, the deviation between these two spectra increases for Eboundary λ hui0j hvi0jHAA¯ juji jvji; ð43Þ lower temperatures (higher β), where finite-size effects are j i0 expected to be the largest. These results thus leave open the possibility that the entanglement spectrum matches exactly scales with the total system size. We find that this is in the thermodynamic limit when VA, V → ∞ such that indeed the case, as shown in Fig. 14. In agreement VA=V is held constant. with the general considerations in Sec. II C 2, the Surprisingly, even though the entanglement spectrum Schmidt eigenvalues deviate from their canonical ETH from a single eigenstate does not match the actual spectrum predicted values beyond e (Fig. 13) and become consid- of the Hamiltonian beyond the energy density e , the erably smaller. expectation values huijHAjuii=LA continue to match the energy eigenvalues of the actual Hamiltonian. This sug- D. Summary gests that the eigenvectors of ρAðjψiβÞ match those of ρ β In this section, we compared the reduced density A;canonicalð Þ throughout the spectrum, even though the eigenvalues do not agree everywhere. (This is also con- matrix from a single eigenstate to that of both the sistent with the observation that the eigenvector overlaps canonical ensemble and a microcanonical ensemble when match even beyond e in Fig. 9.) To understand this VA=V is fixed and nonzero as the thermodynamic phenomenon better, we analyze the different terms in limit is taken. We find evidence that a microcanonical Eq. (17). As argued in Sec. II C 2, the only way ensemble agrees well with a single eigenstate, and our huijHAjuii can exceed the total energy E of the eigenstate results are consistent with the possibility that their reduced is if the Eboundary term, density matrices match precisely in the thermodynamic 021026-18 DOES A SINGLE EIGENSTATE ENCODE THE FULL … PHYS. REV. X 8, 021026 (2018) FIG. 14. Decomposition of the energy density corresponding to an eigenstate amongst the three terms in Eq. (17) for β ¼ 0.2 (left β 0 5 21 7 panel) and ¼ . (right panel) at L ¼ and LA ¼ . The dotted magenta line marks the ground state of HA¯ . As in Fig. 8, the solid brown line denotes the critical energy density e for subsystem A. limit. On the other hand, we demonstrated that Surprisingly, while the eigenvalues between ρAðjψiβÞ ρ β while comparison with the canonical ensemble works and A;canonicalð Þ differ, their eigenvectors appear to match. for many operators, it nonetheless fails both for some (We leave the understanding of this result for future equithermal and for some nonequithermal operators. studies.) FIG. 15. Equal-time correlators for an L ¼ 21 system plotted against inverse temperature β. The blue dots denote the expectation value with respect to each eigenstate, the dashed cyan curve plots the expectation value in the canonical ensemble, and the red curve plots the expectation value predicted from a single eigenstate at β0 ¼ 0.3 (yellow dot) by raising the LA ¼ 4 density matrix to the power β=β0 and rescaling it to have unit trace. We also independently consider predictions from each of the 214608 eigenstates in the range 0.2 < β0 < 0.5: At each inverse temperature β, the pink shaded region contains predictions at most one standard deviation away from the mean prediction among all such eigenstates. 021026-19 JAMES R. GARRISON and TARUN GROVER PHYS. REV. X 8, 021026 (2018) VIII. APPLICATIONS A. Equal-time correlators as a function of temperature from a single eigenstate In the previous sections, we provided evidence that a single eigenstate encodes the full Hamiltonian. As an application of this result, we now calculate correlation functions at arbitrary temperatures using a single eigenstate jψiβ. The basic idea is similar to the relation between the Renyi entropies and the free energy densities [Eq. (3)]. In particular, consider the correlation function ρn ψ trAð Aðj iβÞOðxÞOðyÞÞ hOðxÞOðyÞiβ;n ¼ n ; ð44Þ tr ðρ ðjψiβÞÞ A A FIG. 16. Free energy density as a function of temperature for system size L ¼ 21. The green curve is the free energy density for where x, y are located in subsystem A, away from the the entire system in the canonical ensemble, given by − β boundary. Using Eqs. (2a) and (2b) to leading order in the ½hHi Sth= =L. The remaining curves each plot the free energy 4 − ρ β subsystem size, hOðxÞOðyÞiβ;n equals the expectation value density within the LA ¼ subsystem (i.e., ½hHAi Sð AÞ= =LA), −1 ρ of the operator OðxÞOðyÞ at a temperature ðnβÞ . with respect to some reduced density matrix A in subregion A: Figure 15 shows the expectation values of local operators The blue dots consider each energy eigenstate, the dashed cyan within subsystem A as a function of β, as predicted from a curve represents the canonical ensemble, and the red curve contains the values predicted from a single eigenstate at β0 ¼ single eigenstate of Eq. (30) at inverse temperature β0 0.3 (itself shown in yellow). As in Fig. 15, the “predicted” density (indicated by a yellow dot on the red curve). We choose matrix is obtained by raising the eigenstate’s reduced density operators that are as far away from the subsystem boundary matrix to the power β=β0 and rescaling it to have unit trace. In fact, as possible so as to minimize the finite-size corrections. the eigenstate at β0 ¼ 0.3 is not unique: The independent Even though the agreement with the canonical ensemble is predictions from each of the 214608 eigenstates in the range 0.2 < not perfect, the qualitative trends and the numerical values β0 < 0.5 all fall within the region shaded in pink. match incredibly well, given the modest total system sizes to which we are restricted. These predicted correlators also IX. SUMMARY AND DISCUSSION undoubtedly suffer from corrections expected due to the conical singularity at the boundary of A in Eq. (44).To In this paper, we analyzed the structure of reduced emphasize that there is nothing special about the particular density matrices corresponding to the eigenstates of eigenstate at β0 ¼ 0.3, we investigate predictions from generic, nonintegrable quantum systems. We argued that, eigenstates in a broad range of energy densities, the results given an eigenstate jψiβ with energy density e and a of which are given by the pink shaded region in Fig. 15. corresponding temperature β−1, the reduced density matrix The shape of this region suggests that a large fraction of for a subsystem A is given by eigenstates are indeed able to make correct qualitative and ρ ψ ρ β approximate predictions at all temperatures, a remarkable Aðj iβÞ¼ A;canonicalð Þ; feat considering the reduced density matrix (upon which all predictions are made) contains just four sites. where −βH ρ β trA¯ ðe Þ B. Free energy density as a function of temperature A;canonicalð Þ¼ −β ; trðe HÞ from a single eigenstate It is also possible to predict the free energy density at all if the condition VA ≪ V is satisfied. This means that, for a temperatures from a single eigenstate by raising the fixed eigenstate jψiβ, one can always extract the properties reduced density matrix to different powers β=β0 and of the corresponding Hamiltonian at arbitrary energy rescaling to have unit trace, a notion that is consistent densities by taking VA=V → 0 as the limits VA, V → ∞ with Eq. (3). As Fig. 16 shows, this quantity exhibits are taken. Remarkably, even when VA=V (< 1=2) is taken remarkable agreement with the free energy density of the to be fixed and finite, one can still access many properties system as calculated by more direct means. The prediction of the underlying Hamiltonian. However, it is important to of free energy at all temperatures from a single eigenstate be precise about which thermal ensemble is used for provides additional evidence that just one eigenstate con- comparison because ensemble equivalence breaks down tains information about the thermodynamical properties of when VA=V is nonzero. While we constructed examples of a system at all temperatures. operators for which a single eigenstate and the canonical 021026-20 DOES A SINGLE EIGENSTATE ENCODE THE FULL … PHYS. REV. X 8, 021026 (2018) ensemble disagree, our results are consistent with the allow investigation of our predictions in larger systems than possibility that ETH might hold precisely in the thermo- would be feasible with tomography. 1 dynamic limit when VA=V < 2, as long as a microcanon- We conclude by posing a few questions and future ical ensemble with vanishing energy window is used for directions. comparison [Eq. (2c)]. In this paper, we only considered contiguous subsys- We also introduced the notion of equithermal and non- tems. We leave for future work a systematic study of equithermal operators. In a canonical ensemble at temper- whether Eq. (2a) continues to hold for noncontiguous −1 ≪ ature β , the expectation value of equithermal operators subsystems when VA V. depends only on the properties of the underlying In Sec. VIII A, we predicted equal-time correlators at all Hamiltonian at temperature β−1, while the same is not true temperatures using a single eigenstate. We expect such for nonequithermal operators. The validity of ETH for both predictions to be most successful when the thermal corre- equithermal and nonequithermal operators implies the lation length is short compared to the subsystem size used ability to predict properties of a system at all temperatures to take powers of the reduced density matrix. We leave for given just a single eigenstate. We provided evidence that future work a study of how the accuracy of such predictions ETH is valid for both classes of operators, and, armed with depends upon the subsystem size in comparison with the this knowledge, successfully predicted local correlation thermal correlation lengths at both the base temperature β−1 β−1 functions and free energy densities at all temperatures from ( 0 ) and the target temperature ( ). a single eigenstate. It will also be interesting to see if a similar method works We also provided analytical results for the Renyi and von for unequal-time correlators at arbitrary temperatures. The Neumann entropies of infinite temperature eigenstates of a main difference is that this requires calculating expressions particle-number conserving model. These results substan- such as Eq. (44) at an imaginary exponent, and estimating tiate our numerical results for the energy-only conserving the effects due to the conical singularity in this case requires model. In particular, we find that the von Neumann further study. entanglement entropy for a subsystem of size V equals As mentioned above, we expect that our entire discus- A sion carries over to time-evolved states as well, since such the thermal entropy for that subsystem as long as VA < V=2 and, therefore, follows the so-called “Page curve” [58] states are also expected to have thermal behavior at long at all nonzero temperatures, thus generalizing the original times in the same sense as a single finite energy density work of Page and others [58], and in agreement with the eigenstate. If so, does the timescale for thermalization for a recent work on large central charge CFTs [53–55]. given operator (i.e., the time it takes for the expectation Let us mention some of the practical implications of our value of the operator to become equal to its canonical results. Firstly, the fact that a single eigenstate encodes expectation value) depend on whether the operator is properties of the full Hamiltonian could potentially be a equithermal or nonequithermal? useful numerical tool. For example, one could imagine Another question concerns the subleading corrections to targeting a finite energy density eigenstate of a Hamiltonian the entanglement entropy. One expects that there always H by variationally minimizing the energy of the exist subleading area-law contributions to the entanglement 2 entropy (either von Neumann or Renyi) of a single Hamiltonian ðH − EÞ with respect to trial wave functions. eigenstate. Are these contributions also captured correctly The techniques in this paper would then allow one to access in the entanglement entropies calculated via a thermal thermal properties of the Hamiltonian without directly reduced density matrix? Perhaps a more interesting ques- calculating the partition function, which could be extremely tion is whether the mutual information of two disjoint helpful for Hamiltonians that suffer from the sign problem. intervals (which cancels out both the volume law contri- Secondly, owing to the recent progress in single atom bution and the area law contribution) takes the same value imaging techniques in cold atomic systems [71], one can for a single eigenstate and its canonical counterpart. now access nonlocal operators experimentally [72–75]. This potentially allows one to check some of our predic- ACKNOWLEDGMENTS tions in cold atomic systems. For example, one can perform a quantum quench on a low entanglement state, which We are grateful to Leon Balents, John Cardy, Matthew would, at sufficiently long times, lead to a thermal state in Fisher, Tom Hartman, Patrick Hayden, Pavan Hosur, Max the same sense as Eq. (2a). In principle, this allows one to Metlitski, Olexei Motrunich, Chetan Nayak, Anatoli determine the underlying Hamiltonian of a cold atomic Polkovnikov, Marcos Rigol, Lea Santos, Brian Swingle, system by performing tomography on a small subsystem to and especially David Huse and Xiaoliang Qi for con- obtain the corresponding reduced density matrix and then versations about this work. We thank anonymous referees taking its logarithm. Recent proposals based on Ramsey for suggesting the calculation that led to Fig. 16 and spectroscopy provide methods for directly measuring the for suggesting a comparison with a microcanonical entanglement spectrum in systems of cold atoms [76] or in ensemble in Sec. VII. This research was supported in part nuclear spins [77]. Implementation of these protocols could by the National Science Foundation, under Grant 021026-21 JAMES R. GARRISON and TARUN GROVER PHYS. REV. X 8, 021026 (2018) No. DMR-14-04230 (J. R. G.), by the Caltech Institute of [16] M. 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