Does a single eigenstate encode the full Hamiltonian?

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 non-local, 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/temperatures using just a $\textit{single}$ eigenstate. We provide support for our conjecture by performing an exact diagonalization study of a non-integrable 1D lattice quantum 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 find that even in this case, a large class of operators continue to match their canonical expectation values. Specifically, the von Neumann entanglement entropy equals the thermal entropy as long as the subsystem is less than half the total system. We also study, both analytically and numerically, a particle number conserving model at infinite temperature which substantiates our conjectures.

Given a local Hamiltonian, what information about the system is encoded in a single eigenstate? If the eigenstate happens to be a ground state of the Hamiltonian, tremendous amount of progress can be made on this question for Lorentz invariant systems 1,2 , especially conformal field theories (CFTs) [3][4][5][6] , and for topological phases [7][8][9] . For example, one can read off the central charge of a CFT from the ground state entanglement [3][4][5] , while for topological phases, essentially all 'topological data' such as braiding statistics of anyons can be extracted from the degenerate ground states [8][9][10] . In this paper we argue that a single finite energy density eigenstate of an ergodic quantum many-body Hamiltonian is sufficient to determine the properties of the system at all temperatures.
It is not very surprising that the ground states of quantum many-body systems contain some information about their excitations. This is because an entanglement cut often mimics an actual physical cut through the system, thus exposing the underlying excitations along the entangling boundary 9 . The same intuition is tied to the fact that the ground state entanglement satisfies a "boundary law" of entanglement entropy 11,12 , that is, the von Neumann entanglement entropy S 1 = −tr A (ρ A log(ρ A )) of the ground state corresponding to a subsystem A scales with the size of the boundary of subsystem A.
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 which continue to satisfy an area law of entanglement. These are the eigenstates which have a zero energy density, i.e. lim V →∞ E−E0 V = 0 where E 0 is the ground state energy and V is the total volume of the system. These eigenstates can often be interpreted as the action of a sum of local operators acting on the ground state; for example, in a system with spontaneous symmetry breaking one can construct an eigenstate consisting of a few magnons by a superposition of spin-flips acting on the ground state. Furthermore, the level spacing between two contiguous low-lying excitations scales as δE ∼ 1/L α where α > 0 depends on dimensionality and the phase of matter under consideration. In this paper, we will instead be concerned with excited eigenstates that have a finite energy density, i.e. lim V →∞ E−E0 V = 0. For notational convenience, we will set E 0 = 0 for the remainder of this paper.
As argued by Srednicki 13 , a typical finite energy density state (i.e. a typical state in the Hilbert space that satisfies ψ|H|ψ = V e where e is the energy density) when time-evolved with the Hamiltonian H for sufficient time is expected to lead to predictions dictated by the basic tenets of equilibrium statistical mechanics, if the system thermalizes. Such an expectation leads to the "Eigenstate Thermalization Hypothesis" (ETH) [13][14][15] , which stipulates that the thermalization occurs at the level of each individual eigenstate. An alternative approach by Deutsch 14 , which is based on perturbing an integrable system by a small integrability breaking term, leads to the same suggestion. If ETH holds true, then in the thermodynamic limit the equal-time correlators of an operator with respect to a finite energy density eigenstate |ψ are precisely equal to those derived from a thermal ensemble, i.e.
In this paper, we restrict ourselves to systems where ETH, as defined by Eqn. 1, holds. However, Eqn. 1 alone is incomplete unless one also specifies the class of operators for which it holds. For example, one simple non-local operator for which Eqn. 1 breaks down is the projection operator |ψ ψ| onto the eigenstate |ψ that enters Eqn. 1; the left hand side of Eqn. 1 yields unity for this operator, while the right hand side is exponentially small in the volume, a clear disagreement. On that note, it is often mentioned that in systems where Eqn. 1 does hold, it does so only for "few body" operators [34][35][36] where, to our knowledge, the precise meaning of few-body operator has not been clarified. In this paper, we conjecture and provide numerical evidence that Eqn. 1 holds for all operators within a subsystem A, where the volume V A of subsystem A can be arbitrarily large as long as it satisfies V A < f * V , where f * = 0 (< 1/2) is an O(1) number, to be defined later. In fact, we will make the case that for a large class of non-local operators, as well as all local operators (which we define as the operators whose support does not scale with the subsystem size V A ), the condition V A < V /2 suffices. On that note, we should mention that the questions such as which Hamiltonians (and which operators) satisfy ETH is now entering the realm of experimental physics (see e.g. Ref. 37 ) due to advances in high resolution imaging techniques 38 .
The satisfaction of Eqn. 1 for all operators in a subsystem A is equivalent to the statement that the reduced density matrix ρ A (|ψ β ) = tr A |ψ β β ψ| corresponding to an eigenstate |ψ β is given by A being the complement of A. Note that the trace in the denominator is over the whole Hilbert space. When V A is held constant, the equality in Eqn. 2a means the density matrices become elementwise equal in any basis as V → ∞. When the ratio V A /V is held constant, however, the number of matrix elements increases exponentially as V → ∞. In this case we consider the validity of Eqn. 2a in terms of the trace norm distance of the density matrices on either side raised to any power-a point which is further explained in Sec. VII. One immediate consequence of Eqn. 2a is that the thermodynamical properties of a system at arbitrary temperatures can be calculated using a single eigenstate. For example, Eqn. 2a implies that to the leading order, the Renyi entropies for an eigenstate |ψ β corresponding to a subsystem A with V A < V /2 are given by where f (β) is the free energy density at temperature β −1 . The result for V A > V /2 follows from the constraint S A = S A . The above equation allows one to access the free energy density f at an arbitrary temperature by varying α. Note that Eqn. 3 holds only to the leading order because Renyi entropies S α receive additional subleading contributions due to the conical singularity induced at the boundary of subsystem A 3-5 . In the limit α → 1, one recovers the equality between the von Neumann entanglement entropy S 1 and the thermal entropy S th = V A s th (β) where s th (β) is the thermal entropy density at temperature β −1 , a result which was argued to hold in Ref. 39 for the special case of two weakly coupled ergodic systems. We emphasize that these results cannot be derived from Eqn. 1 alone were it to hold only for local operators, since entanglement entropies do not correspond to the expectation value of any local operator. We also note that Refs. 40 We will also discuss an approximate, but more intuitive form of ETH, given by where A central task of this paper is to check the validity of Eqns. 2a and 2b and their consequences for model nonintegrable systems. As already mentioned, we will argue that ETH allows one to calculate thermodynamical quantities as well as correlators at all temperatures/energy densities using only a single eigenstate. We will demonstrate this explicitly by studying a quantum 1D model numerically.
As mentioned above, we find evidence that Eqn. 2a holds even when V A /V is held constant with V A /V less than some number f * > 0. In particular, as we discuss later, our results strongly indicate that f < 1/2 is sufficient to guarantee equivalence between the von Neumann entropy density of a pure eigenstate, and the thermal entropy density at the corresponding temperature. This is in contrast to Ref. 42 where it was argued that such an equivalence holds only in the limit f * → 0. Recently 43,44 , the requirement f * → 0 was substantiated using analytical and large scale numerical calculations for free fermions, an integrable system. Our results indicate that the f * → 0 requirement is likely a consequence of the integrable nature of the models in Refs. 43,44 .
The paper is organized as follows. Sec. II discusses general considerations for the validity of ETH, and introduces a division of all operators in a given subsystem into two distinct classes, which have different requirements for ETH to hold. Sec. III illustrates some general features of ETH by studying the entanglement entropies of a hardcore boson model with global particle number conservation for infinite temperature eigenstates. Sec. IV introduces the model we study in the remainder of the paper, the transverse field Ising model with longitudinal field. Sec. V focuses on the entanglement entropies at finite temperature. Sec. VI provides a close look into the entanglement Hamiltonian, focusing on its spectrum and Schmidt vectors. Sec. VII studies the validity of Eqn. 2a in the thermodynamic limit by considering the trace norm distance of both sides. Sec. VIII provides an application, by using the reduced density matrix from a single eigenstate to predict correlators at all (finite) temperatures. Sec. IX summarizes our results and provides thoughts for future discussion.

A. Determining Hamiltonian from Microstates in Classical Statistical Mechanics
Suppose, for an isolated system described by classical statistical mechanics in a total volume V , we are given access to all classical microstates in a small energy win- V is on the order of the energy fluctuations in the total system were the system coupled to a thermal bath, and thus all microstates correspond to the same energy density. We pose the question: does this information suffice to determine the underlying Hamiltonian, assuming that the Hamiltonian is local? The answer is indeed yes, following the standard procedure of obtaining canonical ensemble from a microcanonical ensemble. In particular, let us make a fictitious division of the system into A and A such that V A V A , and count the number of times a particular configuration C A appears in subsystem A. This determines the probability distribution for finding a given configuration, P (C A ). If all microstates are equally likely, then 45 where E(C A ) is the energy in subsystem A. One may now invert this equation to obtain the energy E(C A ) = − 1 β log(P (C A )), up to an irrelevant constant shift of energy. In a classical statistical mechanical system E(C A ) is the Hamiltonian for subsystem A. In particular, knowing E(C A ), one may now calculate any thermodynamic property at any temperature. Here it is crucial to note that Eqn. 4 does not assume that the energy density E(C A )/V A equals the energy density E/V of the microstates being sampled.
As discussed in the introduction, we will provide evidence that the quantum mechanical analog of Eqn. 4 is given by Eqns. 2a,2b. We now proceed to discuss the conditions under which Eqns. 2a,2b are valid.

B. Two Classes of Operators
For reasons soon to be discussed, we find it useful to separate operators in a given Hilbert space into two classes: Class I ("Equithermal Operators"): If the reduced density matrix takes the thermal form (i.e. the right hand side of Eqn. 2b), then in the limit V A → ∞, the expectation value of equithermal operators receives contribution only from eigenstates of H A at energy density corresponding to the temperature β −1 . One might have thought that this is true for all operators, however, there exist operators such as e −nβH A , whose expectation value includes contribution from eigenstates of H A at temperature (n + 1)β −1 in addition to the temperature β −1 . Clearly, local operators fall into this class, as do sums of local operators. Several non-local operators, including the von Neumann entropy S 1 , also fall into this class.
Class II ("Non-equithermal Operators"): We dub all operators not in Class I as "non-equithermal operators", or Class II operators. All Renyi entropies S α (for α = 1) fall into this class.

C. ETH: Class I Vs Class II Operators
Let us first consider the relationship between Eqn. 1 and Eqns. 2a,2b. Eqn. 1 may be rewritten as, If this equation holds for all operators in a subsystem A, hermitian as well as non-hermitian, then one obtains Eqn. 2a, ρ A (|ψ β ) = ρ A,th (β). This is because one may expand both the ρ A and ρ A,th in terms of the complete set of operators in subsystem A, and by choosing appropriate O prove that they are equal to each other elementby-element. One of the most important consequences of this equality is that it allows one to extract properties of the Hamiltonian at arbitrary temperatures using a single eigenstate, which is one of the central points of this paper.

ETH for Class I operators
In contrast to classical statistical mechanics, we expect that quantum mechanically, one does not require the constraint V A V A for ETH to hold for Class I operators. Indeed, as discussed below, several known results point to the conclusion that Eqn. 1 holds for all operators in Class I, as long as V A < V A with both V A , V A → ∞. Therefore, the number of operators that satisfy ETH scales exponentially not only with the subsystem size V A , but also with total system size V (in contrast to just "few-body" operators).
The strongest evidence for the sufficiency of the condition V A < V A so that Eqn. 1 holds for Class I operators comes from the study of quantum quenches in conformal field theories (CFTs). As shown in Ref. 46 , the timedependent reduced density matrix ρ A (t) of a system initially prepared in a low-entanglement state, and evolved with a CFT Hamiltonian, approaches the thermal density matrix, as long as Ref. 46 characterized the closeness between ρ A (t) and the thermal density matrix ρ A,th (Eqn. 2a) in terms of the operator overlap I(t) = tr(ρ A (t)ρ A,th ) (tr(ρ 2 A (t))tr(ρ 2 A,th )) 1/2 , which is exponentially close to unity for V A /2 < t < V A /2. It is important to note that in the thermodynamic limit, I only gets contribution from eigenstates at temperature β −1 , so this only guarantees that operators in Class I will satisfy Eqn. 1.
Recently studied large central charge theories also support the same conclusion [47][48][49] . In particular, Refs. 47,49 studied the entanglement entropy of pure states in finite temperature conformal field theories with large central charge. In the limit V A , V 1/T , while keeping V A /V fixed, it was found that the entanglement entropy becomes equal to the thermal entropy at all non-zero temperatures as long as Lastly, another piece of evidence for the sufficiency of the condition V A < V A comes from studying random states, which mimic the behavior of an eigenstate at infinite temperature (i.e. |ψ β=0 ). The entanglement entropy for a random state is given by [50][51][52] : where |H A |, |H A |, |H| are the sizes of the Hilbert spaces of subsystems A, A and the total system (= A ∪ A) respectively. Thus, as soon as V A < V A , one obtains S = − log(|H A |), which is indeed the thermal entropy for subsystem A at infinite temperature. The above results support the notion that for ETH to hold for Class I operators, it is sufficient (and necessary) that the ratio of the Hilbert space sizes of subsystem A to A, |H A | |H A | , goes to zero in the thermodynamic limit. Our results are in agreement with, and substantiate this conclusion.

ETH for Class II operators
The extra ingredient introduced by Class II operators is that if ETH holds for them, then taking such an operator's expectation value with respect to a state |ψ β allows one to access the properties of the Hamiltonian at a temperature different than β −1 . For example, the Renyi entropy S α corresponding to ρ A (|ψ β ) satis- , thus allowing one to access the free energy density at temperature (αβ) −1 .

For a given ratio V
there is a physical constraint on the range of energy densities for which the spectrum of |ψ β in principle can match that of ρ A,th (β). To appreciate this, let us consider a slightly different problem-an arbitrary Hamiltonian of hardcore bosons with particle number conservation, at infinite temperature. We will consider an explicit exam-ple of such a system in the next section. Since the total particle number operatorN commutes with the Hamiltonian and satisfies the equationN =N A +N A , the reduced density matrix ρ A for a wavefunction |ψ β=0 is block diagonal in the number of particles N A in subsystem A. Furthermore, if ETH holds (as given by a generalization of Eqns. 2a and 2b), then the Schmidt decomposition is given by where λ N A are the Schmidt coefficients in the sector N A , and |u i N A , |v i N −N A are the corresponding eigenvectors. The label i captures fluctuations of particles within a fixed sector N A . Note that there is no index i on λ N A because we are at infinite temperature and all Schmidt states within a sector N A are equally likely.
The decomposition in Eqn. 7 allows one to calculate properties of subsystem A at infinite temperature even away from filling N/V since the reduced density matrix ρ A will contain sectors with various densities N A /V A . However, there is both an upper limit and a lower limit on the density in subsystem A, since And thus the particle density where n ≡ N/V is the overall particle density and f ≡ V A /V . Thus, a necessary condition for the wavefunction in Eqn. 7 to encode properties of the system at all fillings is The above discussion, with some modifications, carries to systems with (only) energy conservation, at an arbitrary temperature. The Schmidt decomposition of an eigenstate |ψ β with eigenvalue E may now be written as: The physical content of ETH, as approximated in Eqn. 2b, is that Denoting the ground state energy to be zero, one naively expects that u i |H A |u i ≤ E ∀ |u i since the energy density in the subsystem A cannot be less than the ground state energy density. However, this argument has a loophole since in contrast to the particle number operatorN , the total Hamiltonian is not separable into subsystems A and A: H = H A + H A + H AA , which actually allows u i |H A |u i to exceed E as we will see in Sec. VI in the context of the model Hamiltonian in Eqn. 27 below. To understand the constraint on u i |H A |u i precisely, let us derive an expression which encapsulates the classical notion that the sum of energies in subsystem A and A equals E. We first note: The above expression can be re-evaluated using the decomposition Equating the two ways to calculate the same expression, one finds: Due to the variational principle for the ground state, v i0 |H A |v i0 ≥ −cL d−1 where c is a constant (recall that in our convention, the ground state energy for the full Hamiltonian is set to zero). Since both E and u i0 |H A |u i0 scale as L d , the only way for u i0 |H A |u i0 to exceed E is that the second term on the left hand side of Eqn. 16, viz. E boundary def = j λj λi0 u i0 |⊗ v i0 |H AA |u j ⊗ |v j , is negative and scales as L d . When that happens, ETH no longer holds, as we now argue on general grounds, and will also demonstrate numerically for a lattice Hamiltonian in Sec. VI. To see this, we reiterate that ETH requires that (i) |u i 's are approximate eigenstates of H A , and (ii) λ i ∝ e −β ui|H A |ui = e −βE A,i . Firstly, when u i0 |H A |u i0 < E so that ETH could in principle hold, the E boundary term can be neglected because the 'diagonal term' in E boundary (i.e. the term corresponding to j = i0) scales as the boundary (∝ L d−1 ) and is thus subleading, while the off diagonal terms scale as e −L d and thus vanish in the thermodynamic limit (recall that V A > V A ). On the other hand, when u i0 |H A |u i0 > E, the |v i0 's now correspond to states of zero energy density, and the aforementioned argument for neglecting offdiagonal terms is no longer valid. So, let us assume that u i0 |H A |u i0 > E and each |u i0 continues to be an eigenstate of H A . Thus, one requires that where we have taken the continuum limit and λ(e) denotes the Schmidt eigenvalue corresponding to an eigenvector |u at energy density e, while M (e, e ) = u(e)| ⊗ v(e)|H AA |u(e ) ⊗ |v(e ) and g(e) = e − u(e)|H A |u(e) /L d . It is obvious from Eqn. 17 that λ(e) ∝ e −βE A = e −βef L d is no longer the solution.
In fact, the only way for the integral on the left hand side of Eqn. 17 not to have any exponential dependence on L (as required by the right hand side) is that the integrand itself does not have such dependence, i.e.
M (e,e ) e −S(e ) . This implies a breakdown of ETH when u i0 |H A |u i0 > E.
The above discussion implies that for a given wavefunction and bipartition, the maximum energy density that is potentially accessible in a subsystem A, such that the corresponding Schmidt weight satisfies ETH is, where e = E/V is the energy density corresponding to the wavefunction and e max is the maximum energy density for the Hamiltonian H (recall that e max can be finite for lattice-regularized quantum systems, e.g. for models of fermions or spins/hardcore bosons). Above, we have assumed that e < e max /2. In the case when e > e max /2, the range of available energies is instead bounded from below by max [0, e max (1 − 1/f ) − e/f ]. If our goal is to capture the fluctuations in the system for all energy den-sities so that all Class II operators satisfy ETH, we obtain an analog of Eqn. 10 for the energy: E/V A ≥ e max , and, (e max V − E)/V A ≥ e max . Expressed in terms of the fraction V A /V , and the energy density of the eigenstate e = E/V , this constraint is,

D. Summary
Let us summarize the discussion in this section.
1. ETH holds for local as well as non-local Class I operators as long as V A < V A . This implies that ETH is not restricted only to few-body operators (as can be seen in the limit V A , V → ∞). It also follows that the number of operators that satisfy ETH actually grows exponentially with the total system size V .
2. Determining the full Hamiltonian from a single eigenstate is equivalent to the satisfaction of Eqn. 1 for both Class I and Class II operators. A necessary condition for this to be possible is the constraint (19), which in the presence of additional symmetries will be supplemented by constraints such as Eqn. 10.
Our numerical results described in the following sections strongly suggest that the constraint, Eqn. 19, is not only necessary, but sufficient as well. Therefore, one should be able to extract information about the full Hamiltonian at arbitrary energy densities/temperatures using a single eigenstate as long as these constraints are satisfied.
Note that even if the constraint (Eqn. 19) is not satisfied, the above arguments combined with our results in the subsequent sections indicate that a single eigenstate still encodes the properties of the Hamiltonian within the energy density window defined by e * in Eqn. 18. We will elaborate on this further while discussing the numerical results in Sec. VI.

III. A WARMUP: VON NEUMANN AND RENYI ENTROPY OF EIGENSTATES AT INFINITE T
By definition, the thermal entropy reaches a maximum at infinite temperature. Together with Eqn. 32, this implies that when ETH holds, eigenstates at "infinite temperature" are ones where the entanglement entropy is at its maximum. Consider a 1D transverse field Ising model with longitudinal field, Here the von Neumann entropy S 1 takes its maximum possible value when the eigenvalues of the reduced density matrix are all equivalent to one another. Thus, from counting the basis size of the reduced Hilbert space, we expect for infinite temperature eigenstates that each eigenvalue of the reduced density matrix will approach 2 −L A in the thermodynamic limit. From this, it follows that the Renyi entropies at infinite temperature satisfy that is, they are independent of Renyi index α. The left panel of Fig. 1 shows how the entropies S 1 through S 4 together match this predicted value at the infinite temperature point for a L = 21 system with periodic boundary conditions and subsystem size L A = 4. In general, as L → ∞ the T = ∞ entropy density is given by S α /L A = log 2. Now let us instead consider a model with an additional conservation law, namely particle number conservation. Consider a 1D chain of hardcore bosons where n i ≡ b † i b i . We focus on this system with periodic boundary conditions at the non-integrable point t = V = 1 and t = V = 0.96. This model was previously studied and shown to exhibit ETH in Refs. 53,54 .
Due to particle number conservation, the reduced density matrix from any pure state is block diagonal, with each block corresponding to some filling number N A of the subsystem A. The block of the reduced density matrix ρ At infinite temperature, the eigenvalues of ρ A must be equal to one another within a given block, but the eigenvalues in different blocks will be different: they are in fact proportional to L−L A N −N A , the number of microstates consistent with such a configuration in subsystem A. Taking into account that tr(ρ A ) = 1, one finds that each of the The spectrum of ρ A we find for a single eigenstate is in agreement with that of the thermal reduced density matrix ρ A,th (β = 0) studied in Ref. 40 , consistent with ETH.
With this, the von Neumann entropy at infinite temperature becomes and the Renyi entropies are given by where the sums over N A are restricted to subsystem particle fillings N A that satisfy the constraint in Eqn. 8. Because the eigenvalues are non-uniform, the Renyi entropies S α at infinite temperature depend on the Renyi index α, in contrast to an energy-only conserving model.  Figure 3: Scaling of the von Neumann entanglement entropy S1 with subsystem size for the L = 20 system given in Eqn. 27. Up to β = 0.5, the scaling is linear for small LA, suggesting that the states are volume-law and are thus likely to satisfy ETH. The β = 1.0 eigenstate, on the other hand, is clearly not linear, and is too close to the ground state at this system size to exhibit ETH.
The right panel of Fig. 1 shows how the actual values of S 1 through S 4 match those predicted by the above counting argument. For comparison, we also calculate S α analytically in the thermodynamic limit. For simplicity, we consider the limits, L, N, L A → ∞ such that n = N/L is held constant, while L A /L → 0. In these limits, one can evaluate the expressions in Eqn. 23 using Stirling's approximation log(x!) ≈ x log(x) − x. One finds that in the limits considered, S α receives contribution only from N A given by Thus, S α probes the system at a filling N * α , which is different than the actual filling n, unless α = 1 (which corresponds to the von Neumann entanglement entropy). This also immediately leads to expressions for Renyi and von Neumann entanglement entropies in the thermodynamic limit: and We plot these values in Fig. 1 for comparison. Remarkably, even with the small system sizes we can access, the difference between the exact finite size result (obtained by counting over all sectors) and the result valid in the thermodynamic limit is quite small.

IV. MODEL HAMILTONIAN WITH ONLY ENERGY CONSERVATION
To develop some understanding of the questions posed in the introduction, we study a finite 1D quantum spin-1/2 chain with the following Hamiltonian: We set h x = 0.9045 and h z = 0.8090 such that the model is far away from any integrable point, and is expected to satisfy ETH in the sense of Eqn. 1 as shown in Ref. 55 . We use periodic boundary conditions throughout.
We diagonalized the Hamiltonian in Eqn. 27 for system sizes up to L = 21, obtaining all eigenvalues and eigenstates. As hinted earlier, to each eigenstate we assigned a temperature β −1 by finding the value β for which the energy expectation value in the canonical ensemble matches the energy of the eigenstate: By definition, β = +∞ for the ground state and β = −∞ for the highest excited state. In practice, the range of available β values on a finite size system is much smaller. With L = 21, for instance, the first excited state has β ≈ 4.0, and the second-to-highest excited state has β ≈ −0.6 (as determined from Eqn. 28). It follows that eigenstates outside the range 4.0 β −0.6 will not appear fully thermal due to the large thermal correlation length expected at low temperatures. (This can be seen for instance in Fig. 3, where the finite size corrections to the linear scaling of the entanglement entropy become more prominent as temperature decreases.) Another thing to consider is that the infinite temperature eigenstate |ψ β=0 is completely random and contains no information about the Hamiltonian. In a finite size system, states near infinite temperature will also contain little information about the Hamiltonian and will therefore be unable to predict properties of the system at other energy densities. As a result of these finite size considerations, we typically study values of β between 0.2 and 0.5 in the remainder of this paper.

A. ETH Prediction for von Neumann and Renyi Entropy
Let us consider the Renyi Entropy S α = − 1 α−1 log(tr ρ α A (|ψ β )) corresponding to an eigenstate |ψ β at inverse temperature β. Assuming that ETH, as encoded in Eqn. 2a, holds, S α may be reexpressed as: where Z(A, α, β) is the partition function of the system on an α-sheeted Riemann surface, such that subsystem A has an effective temperature (αβ) −1 while subsystem A has an effective temperature β −1 . Z(1, β) is the regular partition function of the system 3-5 . Therefore, keeping terms only to the leading order in the subsystem size, the above expression leads to Eqn. 3 advertised in the Introduction, where f is the free energy density. Therefore, the wavefunction at temperature β −1 can be used to calculate the free energy at temperature (αβ) −1 . Indeed, the same result also follows using the approximate form in Eqn. 2b. Taking the limit α → 1 leads to the conclusion that the von Neumann entanglement entropy S 1 satisfies  Figure 5: Scaling of the entropy deviation ∆Sα ≡ Sα(ρ A,th (β)) − Sα(ρA(|ψ β )) with 1/L for constant LA averaged over all eigenstates in the range 0.28 < β < 0.32, for S1 (top panel) and S2 (bottom panel). The error bars represent one standard deviation away from the mean. For S1 this deviation is strictly non-negative, but for higher Renyi entropies it can oscillate and become negative before tending to zero as L → ∞.
where s th (β) = S 1 (ρ A,th (β))/L A is the thermal entropy density at temperature β −1 . Fig. 3 shows the scaling of von Neumann entropy S 1 as a function of subsystem size L A for the eigenstates |ψ β of our model (Eqn. 27). As discussed in Sec. II C 1, since S 1 is the expectation value of a Class I operator in our nomenclature, we expect Eqn. 32 to hold as long as V A < V A , in the limit V A , V A → ∞. This implies that in the thermodynamic limit, the function S 1 (V A ) is ex- pected to form an inverted triangle shape, similar to the behavior of a random pure state (Eqn. 6). However, in a finite total system at any non-infinite temperature, S 1 is an analytic function of the ratio V A /V with a negative sign for d 2 S1 dV 2 A , as shown in Fig. 3 (note that the sign of the curvature is fixed by the strong subadditivity of entanglement). However, even in finite system, the volume law does hold to a good accuracy when V A V /2, and the finite size scaling, discussed below, indicates that the inverted triangle shape is recovered in the thermodynamic limit. Fig. 4 shows the comparison of S 1 , S 2 , S 3 , and S 4 calculated for each individual eigenstate for a subsystem size L A = 4 in a L = 21 system, with their ETH predicted canonical counterparts, Eqns. 32 and 3. We use two different canonical counterparts corresponding to Eqns. 2a and 2b, the latter version being susceptible to boundary errors, which nevertheless are expected to vanish as V A , V A → ∞. The agreement for each entropy is remarkable. It is worth re-iterating that the Renyi entropies for an eigenstate |ψ β encode the free energy densities at temperatures different than β −1 (Eqn. 3), and these results provide an instance of non-local Class II operators satisfying ETH. The condition (19) does not come into play while calculating S α because the energy density corresponding to temperature (αβ) −1 is lower than the critical energy density (Eqn. 18), up to which the entanglement spectrum matches the actual spectrum of the Hamiltonian. We will discuss this in detail in the next section. Also note that as α becomes larger, finite size effects become more pronounced because S α probes the system at lower temperatures (αβ) −1 .

B. Numerical Results for von Neumann and Renyi Entropies
We also studied finite-size scaling of the von Neumann entropy and Renyi entropies by keeping L A constant and varying the total system size. The top panel of Fig. 5 shows the deviation ∆S1 L A = S1(|ψ β ) L A − s th (β) for eigenstates in a range of temperatures. The difference ∆S 1 /L A seemingly goes to zero faster than any inverse power of L, and is consistent with an exponential dependence ∆S 1 /L A ∼ e −L , or at the very least, a power-law decay ∆S 1 /L A ∼ 1/L x with x 1 (although we should caution that inferring the precise asymptotic finite size scaling behavior using exact diagonalization studies is an inherently difficult task). The bottom panel shows a similar plot for the deviation of Renyi entropy S 2 from its ETH predicted value, Eqn. 3. The finite size scaling of ∆S 2 is relatively difficult because unlike S 1 , S 2 shows oscillations as a function of L A (see e.g. 40,56 ). Despite this, ∆S 2 is less than a few percent of S 2 itself. Fig. 6 plots the entropy deviation ∆S 1 /L A for constant ratio L A /L at all available system sizes. Although it is difficult to do a detailed scaling analysis with so few points, the data strongly suggests that ∆S 1 /L A vanishes in the thermodynamic limit. A further, and perhaps more robust indication of this result follows from the discussion in the next section where we provide evidence that the energy constraint in Eqns. 18 and 19 is sufficient for the satisfaction of Eqn. 2a. We also substantiate this conclusion by utilizing the notion of trace norm distance in Sec. VII.

VI. ENTANGLEMENT HAMILTONIAN VS ACTUAL HAMILTONIAN
We now probe in detail the entanglement spectra of individual eigenstates as well as the corresponding Schmidt states. As discussed in Sec. II C 2, a necessary condition for a full agreement between the entanglement spectrum of the reduced density matrix ρ A (|ψ β ) of a single eigenstate and the thermal reduced density matrix (Eq.2a) is the constraint in Eqn. 19 on the energy density of the state |ψ β . Remarkably, we find that not only is this condition necessary but sufficient as well. Furthermore, when this condition is not satisfied, the entanglement spectra still matches with the actual spectra up to the critical energy density e * = e/f in Eqn. 18, where e is the energy density of the state |ψ β and f = V A /V . This is the reason that the Renyi entropies discussed in the previous section show an agreement with their canonical counterparts even when L A /L is kept constant. Specifically, we compare four different quantities, as shown in Fig. 7  To probe the Schmidt eigenvectors further, we directly calculated the overlaps between the eigenvectors of the reduced density matrix ρ A (|ψ β ) and the eigenvectors of the thermal density matrix ρ A,th (β) (see Fig. 8). Again, we find excellent agreement.
Next, we discuss how the energy constraint Eqn. 19 comes into play as the energy density of an eigenstate is varied for fixed ratio f = V A /V . Fig. 9 shows the comparison of spectra of the aforementioned four different quantities for several different energy densities of the reference state |ψ β with f = 1/3, at four different values of β. At this value of f , the energy constraint Eqn. 19 is violated, and therefore we expect that the entanglement spectrum should deviate from the actual spectrum of the Hamiltonian at least beyond the critical energy density e * = e/f . We find for each value of β that the deviation starts to occur essentially right at this critical energy density, below which the agreement continues to hold.  Figure 7: Comparison of the four quantities defined in the inset for an LA = 4 subsystem at L = 21 and β = 0.3. Each of the quantity has been normalized so that the y-axis has units of energy density. The blue markers show the spectra of the canonical (i.e. thermal) reduced density matrix while the red diamond markers correspond to the eigenvalues of a reduced density matrix ρA(|ψ β ) for a single eigenstate at temperature β; the grey markers show the eigenvalues of HA with a shift cA ≡ 1 β log ZA = 1 β log trA(e −βH A ) so that it can be directly compared with − 1 β log[ρA(|ψ β )] in accordance with Eq.2b (note also that the combination HA + cA is independent of the shift of the spectra of HA by an arbitrary uniform constant). Finally, the orange markers represent the expectation value of HA, again with a shift cA, with respect to the Schmidt eigenvector |ui of ρA(|ψ β ). In each case, the eigenvalues/eigenvectors are ordered from smallest to largest energy density. The horizontal lines plot the energy density e (dashed, grey) and the critical energy density e * = eL L A (solid, brown) of the original eigenstate |ψ β , with respect to the ground state energy density of HA + cA (dotted, black). Surprisingly, even though the entanglement spectrum does not match the actual spectrum beyond the energy density e * , the expectation values u i |H A |u i /L A continue to match the energy eigenvalues of the actual Hamiltonian! To understand this phenomenon better, we analyze the different terms in Eqn. 16. As argued in Sec. II C 2, the only way u i |H A |u i can exceed the total energy E of the eigenstate is, if the E boundary term, scales with the total system size. We find that this is indeed the case, as shown in Fig. 10. In agreement with the general considerations in Sec.II C 2, Schmidt eigenvalues deviate from their ETH predicted value beyond e * (Fig.  9) and become considerably smaller.

VII. TRACE NORM DISTANCE BETWEEN REDUCED AND CANONICAL DENSITY MATRICES
To quantify the extent to which Eqn. 2a is valid, we measure the trace norm distance ||ρ A (|ψ β ) − ρ A,th (β)|| 1 between the reduced and canonical density matrices at various system sizes. The trace norm distance, defined as places an upper bound on the probability difference that could result from any quantum measurement on the two density matrices 57 . As such, it provides an excellent measure of how distinguishable the two density matrices are. If the trace norm distance between two finite sized density matrices is zero, they are equal to each other element Each inset plots a 12-bin histogram of the log of the density of states versus the energy density: the solid blue curve from a single eigenstate ρA(|ψ β ) and the dotted cyan curve from the canonical ensemble ρ A,th (β). 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 from the other markers (in particular, the eigenvalues of the thermal reduced density matrix i.e. the blue markers), as the energy density reaches the critical value e * (denoted by the solid brown line), indicating breakdown of ETH beyond e * .
by element.
We first consider scaling of the trace norm distance with system size L while the subsystem size L A is kept constant; that is, in the thermodynamic limit we will assume that L A L. If ETH holds for all operators in subsystem A, then the results of Ref. 15 imply that the trace norm distance should go to zero as 1/L. The suggestion that the trace norm distance between pure state and thermal reduced density matrix with fixed subsystem size would tend to zero was also made in Ref. 58 . We restrict ourselves to states in a β range given by 0.28 < β < 0.32. In the left panel of Fig. 11, we plot the trace norm distance of every eigenstate in this β range at L A = 5 for a few select system sizes. For each system size, the distribution of the trace norm distance is nearly constant throughout the given β range. The right panel then takes this data for each pair of L and L A and plots the mean and standard deviation of the trace norm distance against 1/L. The trace norm distance is tending toward zero at least linearly with 1/L, perhaps even faster.
Next we consider the stronger form of ETH that we propose, namely that the reduced and canonical density matrices are equivalent when keeping the ratio L A /L (< 1 2 ) fixed as L A , L → ∞. Fig. 12 plots the scaling of the trace norm distance in this case with 1/L. Although there are few points available for each ratio, the trend is clearly toward zero as L increases, and a linear extrapolation suggests that the trace norm distance does indeed go to zero in the thermodynamic limit. Remark-  ably, the definition of the trace norm distance does not require that the operators used to distinguish between the reduced and canonical density matrices be "local" in any sense. As such, these numerics strongly suggest that given a single eigenstate, all possible operators in subsystem A will assume their canonical expectation values in the thermodynamic limit, even though subsystem A is of infinite spatial extent.
At this point it is crucial to emphasize that in the limit when L A /L is kept fixed while both L A , L → ∞, the trace norm distance ||ρ A (|ψ β )−ρ A,th (β)|| 1 can go to zero even if the reduced density matrix does not contain information about temperatures away from β −1 . This is because in the thermodynamic limit, energy densities away from β −1 do not contribute to the trace norm distance, as one can evaluate any measurable operator in the saddle point approximation, which becomes exact in the thermodynamic limit. In other words, no conceivable quantum measurement could determine whether the information in the two density matrices away from temperature β −1 are equivalent. A more direct test is necessary to determine whether the reduced density matrix contains information about temperatures away from the original eigenstate's energy density.
To test this strongest form of ETH, which is already strongly suggested by results in Figs. 7, 8, and 9, we use the trace norm distance of modified density matrices to determine whether a single eigenstate at temperature 1/β can predict the bulk properties of the canonical ensemble at a different temperature 1/β . Specifically, we consider the trace norm distance whereρ (n) A ≡ ρ n A /tr A (ρ n A ). Raising a thermal density matrix to the nth power adjusts its saddle point so that it is dominated by properties at temperature (nβ) −1and thus if this trace norm distance vanishes as L → ∞, Eqn. 2a holds for properties at temperature (nβ) −1 . (For n = 1, this comparison reduces to the case discussed in the paragraph above.) In Fig. 13, we plot the trace norm distance between these density matrices for n = 1/2 (left panels) and n = 3/2 (right panels). For these values of n, the energy density corresponding to the shifted saddle point lies within the window given by Eqn. 18. Remarkably, as L increases, the trace norm distance decreases, both when holding L A fixed (top panels) and when keeping the ratio L A /L fixed (bottom panels). For n = 1, ρ (n) A,th (β) corresponds to a thermal density matrix with a conical singularity. 5 In line with the reasoning behind Eqn. 2b, we expect that ρ A,th (β) ∼ e −βH A , from which it follows thatρ (n) A,th (β) ≈ ρ A,th (nβ); by this we mean that all correlators with support away from the boundary are equal to each other for these density matrices. The Renyi entropies will also be correct. Together with the results in Sec. VI, this builds the case that the reduced density matrix taken from a single eigenstate can predict equal time correlators at all energy densities that lie within the window given by Eqn. 18. We study this point further in the next section.

VIII. AN APPLICATION: 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 long as the constraints in Eqn. 19 are satisfied. As an application of this result, we now calculate correlation functions at arbitrary temperatures using a single eigenstate |ψ β . The basic idea is similar to the relation between the Renyi entropies and the free energy densities (Eqn. 3).
In particular, consider the correlation function, where x, y are located in subsystem A, away from the boundary. Using Eqns. 2a, 2b to the leading order in the subsystem size, O(x)O(y) β,n equals the expectation value of the operator O(x)O(y) at a temperature (nβ) −1 . Fig. 14 shows the expectation values of local operators within subsystem A as a function of β, as predicted from a single eigenstate at inverse temperature β 0 (indicated by a yellow dot on the red curve). We choose operators that are as far away from the subsystem boundary as possible, and choose the bipartition size and β 0 so that the energy constraint Eqn. 19 is satisfied for |ψ β0 . Even though the agreement with the canonical ensemble is not perfect, the qualitative trends and the numerical values match incredibly well, given the modest total system sizes to which we are restricted. These predicted correlators also undoubtedly suffer from corrections expected due to the conical singularity at the boundary of A in Eqn. 36.

IX. SUMMARY AND DISCUSSION
In this paper, we analyzed the structure of reduced density matrices corresponding to the eigenstates of generic, non-integrable quantum systems. We argued that given an eigenstate |ψ β with energy density e and a corresponding temperature β −1 , the reduced density matrix for a subsystem A is given by if the condition f ≤ min e emax , 1 − e emax is satisfied, where e max is the maximum energy density for the given A ≡ ρ n A /trA(ρ n A ) with constant ratio LA/L, for n = 1/2 (top left), n = 2/3 (top right), n = 3/4 (bottom left), and n = 3/2 (bottom right). The grey lines indicate linear fits for each ratio LA/L. The fit is done by following the same procedure as in Fig.12.
Hamiltonian and f = V A /V is the ratio of the subsystem size to the total system size. This means that for a fixed eigenstate |ψ β , one can always extract the properties of the corresponding Hamiltonian at arbitrary energy densities by choosing sufficiently small f . Similarly, for a fixed bipartition ratio f (< 1/2), if the above condition on f is not satisfied, one can still access the properties of the underlying Hamiltonian for a range of energy densities in the interval described in Eqn. 18.
We also introduced the notion of "equithermal" (Class I) and "non-equithermal" (Class II) operators. In a canonical ensemble at temperature T , the expectation value of Class I operators depends only on the properties of the underlying Hamiltonian at temperature T , while the same is not true for Class II operators. Our results strongly suggest that Class I operators, local or non-local, satisfy Eqn. 1 as long as V A < V /2. On the other hand, all Class II operators, local or non-local, satisfy Eqn. 1 as long as the energy densities they receive contribution from lies in the range of energy densities mentioned in the previous paragraph.
In the paper we only considered contiguous subsystems. It seems reasonable to conjecture that Eqn. 2a continues to hold as long as the support of operator O can be chosen to lie in a subsystem which is not ncessarily contiguous and whose volume satisfies V A /V ≤ is localized at point x and | x| can be greater than L/2 (where L is the linear dimension of the total system).
In a lattice-regularized quantum field theory with lattice cutoff a, the scaling limit corresponds to a → 0, which implies e max → ∞. Therefore, for a continuum quantum field theory, our results imply that in the limit f → 0 the reduced density matrix takes the thermal form (Eqn. 2a), allowing one to obtain properties of the Hamil- tonian at arbitary energy densities using a single finite energy density wavefunctional. However, in line with the discussion above, the Renyi entropies S α corresponding to a wavefunctional at temperature T will continue to follow Eqn. 3 as long as e(T /α) < e/f , where f ≡ V A /V and e(T ) is the energy density at temperature T . For example, this implies that in a conformal field theory in d spatial dimensions, where e ∝ T d+1 , one should see a deviation from the ETH predicted value when α < f 1 d+1 . It will be interesting if such a 'phase transition' can be detected using the methods developed in Refs. 47,49 .
We note the uncanny resemblance between the Schmidt decomposition of an individual finite energy density eigenstate (Eqn. 11) and the 'thermofield double' where Z A (β) = E A e −βE A . Coventionally, |ψ T D,β is thought of as a wavefunction for two copies of a sys-tem (denoted by an absence or prsence of a bar over the eigenvectors), where each copy has the same set of eigenvectors {|E A ≡ |E A } and the corresponding eigenvalues {E A }. Within thermofield double formalism, the reduced density matrix for one copy of the system equals the thermal density matrix . Eigenstate thermalization, as defined in this paper, provides a new viewpoint on the thermofield double formalism wherein |ψ T D,β can be reinterpreted as a finite energy density eigenstate |ψ β of a Hamiltonian, and the two copies correspond to the subsystems obtained by equal bipartition of the total system. There is one problem however with this identification. As discussed in our paper, the reduced density matrix for an equal bipartition of a subsystem is expected to be thermal only up to an energy density e/f = 2e, where e is the energy density corresponding to the eigenstate |ψ β . Nevertheless, the reinterpretation of the thermofield double state as a Schmidt decomposition of a finite energy density eigenstate is a valid description at least for all equithermal (= Class I) operators since they explore the equality between the pure state and thermal reduced density matrix only at the energy density corresponding to the eigenstate.
Let us mention some of the practical implications of our results. Firstly, the fact that a single eigenstate encodes properties of the full Hamiltonian could potentially be a useful numerical tool. For example, one could imagine targeting a finite energy density eigenstate of a Hamiltonian H by variationally minimizing the energy of the Hamiltonian (H − E) 2 with respect to trial wavefunctions. The techniques in this paper would then allow one to access thermal properties of the Hamiltonian without directly calculating the partition function, which could be extremely helpful for Hamiltonians that suffer from the sign problem.
Secondly, owing to the recent progress in single atom imaging techniques in cold atomic systems 38 , one can now access non-local operators experimentally 37,65,66 . This potentially allows one to check some of our predictions pertaining to the violation of ETH in cold atomic systems. For example, one can perform a quantum quench on a low entanglement state which would at sufficiently long times lead to a thermal state in the same sense as Eqn. 2a. In such a state, one can then employ high resolution imaging to measure string or membrane operators that span more than half the system size to detect violation of ETH. A more ambitious goal, beyond current experimental techniques, would be to measure Renyi entropies for fractional indices, as this would allow one to access the critical energy density e * in Eqn. 18.
We conclude by posing a few questions and future directions.
In the paper we extracted equal-time correlators at different temperatures using a single eigenstate. It will be interesting to see if a similar method also works for unequal time correlators at arbitrary temperatures. The main difference is that this requires calculating expressions such as Eq.36 at an imaginary exponent, and estimating the effects due to the conical singularity in this case requires further study.
Presumably, all our discussion carries over to timeevolved product states as well since such states are expected to also have thermal behavior at long times in the same sense as a single finite energy density eigenstate. If so, does the time scale for thermalization for a given operator (i.e. the time it takes for the expectation value of the operator to become equal to its canonical expectation value) depend on whether the operator is Class I (equithermal) or Class II (non-equithermal)?
Another question concerns the subleading corrections to the entanglement entropy. One expects that there always exist subleading area-law contributions to the entanglement entropy (either von Neumann or Renyi) of a single eigenstate. Are these contributions also captured correctly in the entanglement entropies calculated via a thermal reduced density matrix? Perhaps a more interesting question is whether the mutual information of two disjoint intervals (which cancels out both the volume law contribution and the area law contribution) takes the same value for a single eigenstate and its canonical counterpart.
Finally, we studied the particle number conserving model only at the infinite temperature. It will be worthwhile to extend this discussion to finite temperatures. Do all our conclusions continue to hold when energy is not the only conserved quantity? Presumably, at least in the limit f → 0, the reduced density matrix for a single eigenstate takes the form ρ where O i are the operators corresponding to the conserved quantities (e.g. particle number) and µ i are the corresponding chemical potentials. But does this form continue to hold even when f (< 1/2) is an order unity number, analogous to the case of the energy-only conserving model? A natural extension of this question is to consider integrable systems which have an infinite number of conservation laws. Finally, in the opposite limit, it also seems interesting to study the structure of the reduced density matrix for systems with no conservation laws (e.g. Floquet systems or arbitary time-dependent Hamiltonians) along lines similar to this paper.