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 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.

Popular Summary

For nearly a century, physicists have sought to better understand how certain isolated quantum systems eventually reach thermal equilibrium. This has become more relevant as experimentalists create nearly isolated systems using ultracold atoms. Systems that do equilibrate possess certain remarkable features in their stationary states—special states that do not evolve in time and are analogous to the harmonics of a musical instrument. One such feature is that many equilibrium properties are already apparent in these states. This motivates us to ask how much information a single stationary state encodes about the underlying system. We argue that, surprisingly, the knowledge of a single stationary state can provide access to properties of the system at all temperatures.

We demonstrate the aforementioned claim for a quantum spin system by calculating the exact stationary states and using a single state to make predictions for all temperatures. A crucial stepping-stone in our work is a new, fundamental observation about quantum chaos: Even nonlocal equilibrium properties of a system are encoded faithfully in its stationary states, as long as the system is sufficiently large in relation to its thermal correlation length.

Our results imply that one could study the low-temperature properties of a system without actually cooling the system down to that temperature. A researcher could prepare a system at a more practical temperature and then make nonlocal observations to deduce the behavior at low temperatures. The experimental technology for implementing this already exists; the main challenge would be studying large enough systems.

Figure 1

Entanglement entropies $S_1$ through $S_4$ for a model with no conservation law [left panel, given by Eq. (30) at $L=21$] and a model with particle-number conservation [right panel, given by Eq. (22) at $L=27$, with filling $N=6$]. We use the parameters mentioned in the text to place each model at a nonintegrable point. In each case, we consider eigenstates in the $k=1$ sector, with subsystem size $L_A=4$. The grey vertical line denotes infinite temperature (point of maximum $S_1$), 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 $L_A$, $L\to \infty$ while $L_A/L\to 0$, as given by Eqs. (26) and (27).

Figure 2

Eigenvalue spectrum of the reduced density matrix of an infinite temperature eigenstate, ${\rho }_A(|\psi {⟩}_{\beta =0})$, for the hardcore boson model Eq. (22), with $L=27$, $L_A=4$, and filling $N=6$. The red lines plot the theoretical value of each eigenvalue in the thermodynamic limit, determined from the filling $N_A$ of the sector in which it lies.

Figure 3

Scaling of the von Neumann entanglement entropy $S_1$ with subsystem size for the $L=20$ system given in Eq. (30). Up to $\beta =0.5$, the scaling is linear for small $L_A$, suggesting that the states obey a volume law and are thus likely to satisfy ETH. The $\beta =1.0$ eigenstate, on the other hand, is clearly not linear, and it is too close to the ground state at this system size to exhibit ETH.

Figure 4

The von Neumann entropy $S_1$ and Renyi entropies $S_2$, $S_3$, and $S_4$ for the system given in Eq. (30) with $L=21$ and $L_A=4$. Here, $Z_A={\mathrm{tr}}_A(e^{-\beta H_A})$. 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).

Figure 5

Scaling of the entropy deviation $\mathrm{\Delta }{S}_{\alpha }\equiv S_{\alpha }({\rho }_{A,\text{canonical}}(\beta ))-S_{\alpha }({\rho }_A(|\psi {⟩}_{\beta }))$ with $1/L$, for constant $L_A$, averaged over all eigenstates in the range $0.28<\beta <0.32$, for $S_1$ (top panel) and $S_2$ (bottom panel). The error bars represent one standard deviation away from the mean. For $S_1$, this deviation is strictly non-negative, but, for higher Renyi entropies, it can oscillate and become negative before tending to zero as $L\to \infty$.

Figure 6

Scaling of the von Neumann entropy deviation $\mathrm{\Delta }{S}_1$ with $1/L$, for constant ratio $L_A/L$, averaged over all eigenstates in the range $0.28<\beta <0.32$. As in Fig. 5, the error bars represent one standard deviation away from the mean. Even though this plot considers the case where the subsystem size $L_A$ becomes infinite as $L\to \infty$, the entropy deviations are going to zero rapidly as $L$ becomes larger.

Figure 7

Scaling of the Renyi entropy deviation $\mathrm{\Delta }{S}_2$ with $1/L$, for constant ratio $L_A/L$, averaged over all eigenstates in the range $0.28<\beta <0.32$. As in Fig. 5, the error bars represent one standard deviation away from the mean.

Figure 8

Comparison of the five quantities defined in the inset for an $L_A=4$ subsystem at $L=21$ and $\beta =0.3$. Each quantity has been normalized so that the $y$ axis has units of energy density. The large blue circle markers show the spectrum of the canonical reduced density matrix, while the red diamond markers correspond to the eigenvalues of a reduced density matrix ${\rho }_A(|\psi {⟩}_{\beta })$ for a single eigenstate at temperature ${\beta }^{-1}$. The yellow triangle markers show the spectrum of the microcanonical reduced density matrix that consists of an equal-weighted mixture of the 50 eigenstates nearest in energy to the one under consideration. The grey markers show the eigenvalues of $H_A$ with a shift $c_A\equiv (1/\beta )\log Z_A=(1/\beta )\log {\mathrm{tr}}_A(e^{-\beta H_A})$ so that it can be directly compared with $-(1/\beta )\log [{\rho }_A(|\psi {⟩}_{\beta })]$ in accordance with Eq. (2b) (note, also, that the combination $H_A+c_A$ is independent of the shift of the spectrum of $H_A$ by an arbitrary uniform constant). Finally, the orange markers represent the expectation value of $H_A$, again with a shift $c_A$, with respect to the Schmidt eigenvector $|u_i⟩$ of ${\rho }_A(|\psi {⟩}_{\beta })$. In each case, the eigenvalues/eigenvectors are ordered from smallest to largest energy density. The horizontal lines plot the energy density $e$ (dashed grey line) and the critical energy density $e^{*}=eL/L_A$ (solid brown line) of the original eigenstate $|\psi {⟩}_{\beta }$, with respect to the ground state energy density of $H_A+c_A$ (dotted black line).

Figure 9

Overlap between the Schmidt eigenvectors $|u_j⟩$ and the eigenvectors $|{\phi }_i⟩$ of the canonical reduced density matrix, for an $L=21$ system with $\beta =0.3$, and subsystem sizes $L_A=2$, 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.

Figure 10

Trace norm distance between the canonical density matrix ${\rho }_{A,\text{canonical}}(\beta )$ and the reduced density matrix ${\rho }_A(|\psi {⟩}_{\beta })$ for all eigenstates $|\psi {⟩}_{\beta }$ in the range $0.28<\beta <0.32$. The left panel plots the trace norm distance for all such eigenstates with system sizes $L=12$, 15, and 20, and subsystem size $L_A=5$. The right panel plots the mean and standard deviation of the trace norm distance in this $\beta$ range for values of $L$ up to 21 and $L_A$ up to 5.

Figure 11

Top panel: Subsystem energy variance with respect to subsystem size $L_A$ for both the canonical ensemble (blue circular markers) and a single eigenstate $|\psi {⟩}_{\beta }$ (red diamond markers), at $L=18$ and $\beta =0.3$. The inset shows the ratio between the energy variances at each subsystem size, which is expected to fit $1-L_A/L$ in the thermodynamic limit [Eq. (8)]. Bottom panel: The variance of an operator $J_A\equiv \sum _{i=1}^{L_A}(h_x^{(i)}{\sigma }_i^x+h_z^{(i)}{\sigma }_i^z)+\sum _{i=1}^{L_A-1}{J}_z^{(i)}{\sigma }_i^z{\sigma }_{i+1}^z$, which includes the same terms as $H_A$ but does not relate to energy conservation, is plotted for comparison. Here, the quantities $h_x^{(i)}$, $h_z^{(i)}$, and $J_z^{(i)}$ are each taken from the uniform distribution $[-1,1]$.

Figure 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 $L_A/L$ and $0.28<\beta <0.32$. 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 $L_A/L$, considering all eigenstates in the range $0.28<\beta <0.32$. As in Fig. 8, the microcanonical ensemble is defined as the equal mixture of 50 eigenstates nearest in energy to the one under consideration.

Figure 13

Comparison of the five quantities defined in the inset of Fig. 8 for eigenstates of an $L=21$ system, with $L_A=7$, at $\beta =0.2$, 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 from a single eigenstate ${\rho }_A(|\psi {⟩}_{\beta })$ and the dotted cyan curve from the canonical ensemble ${\rho }_{A,\text{canonical}}(\beta )$. 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 $e^{*}$.

Figure 14

Decomposition of the energy density corresponding to an eigenstate amongst the three terms in Eq. (17) for $\beta =0.2$ (left panel) and $\beta =0.5$ (right panel) at $L=21$ and $L_A=7$. The dotted magenta line marks the ground state of $H_{\overline{A}}$. As in Fig. 8, the solid brown line denotes the critical energy density $e^{*}$ for subsystem $A$.

Figure 15

Equal-time correlators for an $L=21$ system plotted against inverse temperature $\beta$. 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 ${\beta }_0=0.3$ (yellow dot) by raising the $L_A=4$ density matrix to the power $\beta /{\beta }_0$ and rescaling it to have unit trace. We also independently consider predictions from each of the 214608 eigenstates in the range $0.2<{\beta }_0<0.5$: At each inverse temperature $\beta$, the pink shaded region contains predictions at most one standard deviation away from the mean prediction among all such eigenstates.

Figure 16

Free energy density as a function of temperature for system size $L=21$. The green curve is the free energy density for the entire system in the canonical ensemble, given by $[⟨H⟩-S_{\mathrm{th}}/\beta ]/L$. The remaining curves each plot the free energy density within the $L_A=4$ subsystem (i.e., $[⟨H_A⟩-S({\rho }_A)/\beta ]/L_A$), with respect to some reduced density matrix ${\rho }_A$ in subregion $A$: The blue dots consider each energy eigenstate, the dashed cyan curve represents the canonical ensemble, and the red curve contains the values predicted from a single eigenstate at ${\beta }_0=0.3$ (itself shown in yellow). As in Fig. 15, the “predicted” density matrix is obtained by raising the eigenstate’s reduced density matrix to the power $\beta /{\beta }_0$ and rescaling it to have unit trace. In fact, the eigenstate at ${\beta }_0=0.3$ is not unique: The independent predictions from each of the 214608 eigenstates in the range $0.2<{\beta }_0<0.5$ all fall within the region shaded in pink.