Locality, entanglement, and thermalization of isolated quantum systems

A way to understand thermalization in an isolated system is to interpret it as an increase in entanglement between subsystems. Here we test this idea through a combination of analytical and Krylov-subspace-based numerical methods applied to a quantum gas of bosons. We find that the entanglement entropy of a subsystem is rapidly generated at the initial state of the evolution, to quickly approach the thermal value. Our results also provide an accurate numerical test of the eigenstate thermalization hypothesis (ETH), according to which a single energy eigenstate of an isolated system behaves in certain respects as a thermal state. In the context of quantum black holes, we propose that the ETH is a quantum version of the classical no-hair theorem.

Figure 1

The type of lattice used in our numerical computations. A gas of bosons is initially contained in region $A$ and expands into regions $A+B$.

Figure 2

Left: Mean values $\langle {\tilde{E}}_{\ell }|n_i|{\tilde{E}}_{\ell }\rangle$ for two different sites as functions of the energy eigenvalue ${\tilde{E}}_{\ell }$ in a fixed dimension $n=1241$ Krylov subspace. The lighter colored dots correspond to a site in region A and the darker (blue) dots to a site in region B. The values of $n_i$ are larger for region A since the initial state has all the bosons there. Right: The same for two different single-particle state occupation numbers, the lighter colored curve for the lowest energy state and the darker (blue) curve for an excited state, which is occupied only when the total energy is large enough. The dimension of the full Hilbert space $N$ increases from bottom to top, the plots clearly showing that the functions become smoother as a result.

Figure 3

Left: Entanglement entropy $S_{AB}\left({\tilde{E}}_{\ell }\right)$ as a function of the (Ritz) energy eigenvalue ${\tilde{E}}_{\ell }$ in a fixed dimension $n=1241$ Krylov subspace. Right: Entanglement entropy (lower curve) and thermal entropy (upper curve) as functions of time. Inset: Initial growth of the entanglement entropy. At short times, the numerical result (monotonically increasing curve) matches the result of the series expansion.