Effect of Rare Fluctuations on the Thermalization of Isolated Quantum Systems

We consider the question of thermalization for isolated quantum systems after a sudden parameter change, a so-called quantum quench. In particular, we investigate the prerequisites for thermalization, focusing on the statistical properties of the time-averaged density matrix and of the expectation values of observables in the final eigenstates. We find that eigenstates, which are rare compared to the typical ones sampled by the microcanonical distribution, are responsible for the absence of thermalization of some infinite integrable models and play an important role for some nonintegrable systems of finite size, such as the Bose-Hubbard model. We stress the importance of finite size effects for the thermalization of isolated quantum systems and discuss two scenarios for thermalization.

Figure 1

Full diagonalization results of $({\mathcal{G}}_1{)}_{\alpha }$ versus energy $E_{\alpha }$ in the even parity, $k=0$ momentum sector for $U_f/J=10$ (upper panels) and correlation of $|c_{\alpha }|$ versus $({\mathcal{G}}_1{)}_{\alpha }$ for a quench from $U_i/J=2$ (lower panel). Additionally, the average energy (dashed-dotted line) after the quench and the average value of ${\mathcal{G}}_1$ obtained from the time-dependent density matrix renormalization group method time evolution ($L=100$; see 13) (solid line) and the diagonal ensemble for $L=11$ (dashed line) are shown. The shaded region corresponds to the microcanonical average [32]. Upper-right panel: Distribution of values ${\mathcal{G}}_1$ for the kinetic energy for the values $E_{\alpha }/LJ\in [4.5;5.5]$. The average value is removed from the distribution, and the histograms are shifted vertically for visibility.

Figure 2

Left panel: Full diagonalization results of $({\mathcal{G}}_1{)}_{\alpha }$ versus energy $E_{\alpha }$ in the even parity, $k=0$ momentum sector for $U_f/J=1$. Right panel: Distribution of values ${\mathcal{G}}_1$ for $E_{\alpha }/LJ\in [-0.1,0.1]$. The linear trend of the observable in the considered window is removed from the distribution, and the histograms are shifted vertically for visibility.