Thermalization of closed chaotic many-body quantum systems

Abstract

A closed quantum system thermalizes if for time $t \to \infty$, the function ${\rm Tr} (A \rho(t))$ tends asymptotically to ${\rm Tr} (A \rho_{\rm eq})$. Here $A$ is an operator that represents an observable, $\rho(t)$ is the time-dependent density matrix, and $\rho_{\rm eq}$ its equilibrium value. We investigate thermalization of a chaotic many-body quantum system by combining the Hartree-Fock (HF) approach and the Bohigas-Giannoni-Schmit (BGS) conjecture. The HF Hamiltonian defines an integrable system and the gross fatures of the spectrum. The residual interaction locally mixes the HF eigenstates. The BGS conjecture implies that the statistics of the resulting eigenvalues and eigenfunctions agrees with random-matrix predictions. In that way, the Hamiltonian $H$ of the system acquires statistical features. The agreement of the statistics with random-matrix properties is local, i.e, confined to an interval $\Delta$ (the correlation width). With $\rho(t) = \exp \{ - i t H / \hbar \} \rho(0) \exp \{ i H t / \hbar \}$, the statistical properties of $H$ define the statistical properties of ${\rm Tr} (A \rho(t))$. Using these we show that in the semiclassical regime, ${\rm Tr} (A \rho(t))$ decays with time scale $\hbar / \Delta$ towards an asymptotic value. If the energy spread of the system is of order $\Delta$, that value corresponds to statistical equilibrium. The correlation width $\Delta$ is the central parameter of our approach. It defines the interval within which the spectral fluctuations agree with random-matrix predictions. It defines the maximum energy spread of the system that permits thermalization. And it defines the time scale $\hbar / \Delta$ within which ${\rm Tr}(A \rho(t))$ approaches the value ${\rm Tr}(A \rho_{\rm eq})$.