Does a single eigenstate encode the full Hamiltonian?

TL;DR

This work strengthens the Eigenstate Thermalization Hypothesis by proposing a strong, subsystem-level form: a small subregion of a single finite-energy-density eigenstate yields a reduced density matrix indistinguishable from the thermal one, enabling extraction of Hamiltonian properties at any temperature from just one eigenstate. It introduces Class I (equithermal) and Class II (non-equithermal) operators and shows, both analytically and numerically in a non-integrable 1D model, that ETH holds for VA<<V for broad operator classes; for finite subsystem fractions, ETH persists for many but not all operators, with a notable energy-constraint e* governing deviations. The paper derives explicit entropy predictions for Renyi and von Neumann entropies, and demonstrates that equal-time correlators at various temperatures can be computed from a single eigenstate via appropriate manipulation of the reduced density matrix. Collectively, these results imply that a single eigenstate encodes the full Hamiltonian’s thermodynamics across temperatures, providing a potentially powerful tool for both numerical studies and experimental inferences of quantum many-body systems.

Abstract

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.

Figures (15)

• Figure 1: Entanglement entropies $S_1$ through $S_4$ for a model with no conservation law (left panel, given by Eq. \ref{['eq:tfi']} at $L=21$), and a model with particle number conservation (right panel, given by Eq. \ref{['eq:mu_model']} 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 $L_A, L \rightarrow \infty$ while $L_A/L \rightarrow 0$, as given by Eqs. \ref{['eq:mu_model_exact_S_alpha']} and \ref{['eq:mu_model_exact_S_1']}. Notice that the Renyi entropies all match at infinite temperature if and only if there are no additional conservation laws besides energy.
• Figure 2: Eigenvalue spectrum of the reduced density matrix of an infinite temperature eigenstate, $\rho_A(\ket{\psi}_{\beta=0})$ for the hardcore boson model Eq. \ref{['eq:mu_model']} 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. \ref{['eq:tfi']}. Up to $\beta=0.5$, the scaling is linear for small $L_A$, suggesting that the states are volume-law and are thus likely to satisfy ETH. The $\beta=1.0$ eigenstate, on the other hand, is clearly not linear, and is too close to the ground state at this system size to exhibit ETH.
• Figure 4: The von Neumann entropy $S_1$ and Renyi entropies $S_2$, $S_3$, and $S_4$ for the system given in Eq. \ref{['eq:tfi']} with $L=21$ and $L_A=4$. Here, $Z_A = \textrm{tr}_A(e^{-\beta H_A})$. The entropies of the reduced density matrix at each energy density agree remarkably with the the entropies calculated from the canonical ensemble, given by Eqs. \ref{['eq:eth2']} and \ref{['eq:eth3']}.
• Figure 5: Scaling of the entropy deviation $\Delta S_\alpha \equiv S_\alpha(\rho_{A,\mathrm{th}}(\beta)) - S_\alpha(\rho_A(\ket{\psi}_\beta))$ 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 $L \rightarrow \infty$.