Alternatives to Eigenstate Thermalization

An isolated quantum many-body system in an initial pure state will come to thermal equilibrium if it satisfies the eigenstate thermalization hypothesis (ETH). We consider alternatives to ETH that have been proposed. We first show that von Neumann’s quantum ergodic theorem relies on an assumption that is essentially equivalent to ETH. We also investigate whether, following a sudden quench, special classes of pure states can lead to thermal behavior in systems that do not obey ETH, namely, integrable systems. We find examples of this, but only for initial states that obeyed ETH before the quench.

Figure 1

Time evolution of $\delta n_k$, after a sudden quench, for hard-core bosons (left panels) and spinless fermions (right panels) in lattices with $L=21$ and $L=24$ ($N=L/3$), and $T=3$. Results are presented for quenches from $t^{\prime }=V^{\prime }=0.04$ (a),(e), $t^{\prime }=V^{\prime }=0.16$ (b),(f), $t^{\prime }=V^{\prime }=0.32$ (c),(g), and $t^{\prime }=V^{\prime }=0.64$ (d),(h), to $t^{\prime }=V^{\prime }=0$. In all cases $t=V=1$ before and after the quench.

Figure 2

Relative difference between $n_k$ as predicted by the microcanonical ensemble and the long-time average, as a function of $t^{\prime }$, $V^{\prime }$ in the initial Hamiltonian. Results are reported for hard-core bosons (a) and spinless fermions (b), for lattices with 24 sites (main panels) and 21 sites (insets). In all systems, we considered five different initial states that resulted in five effective temperatures after the quench ($T=2$, 3, 5, 7, and 10 in the plots).

Figure 3

Normalized IPR (see text) for the same quenches depicted in Fig. 2. Results for hard-core bosons (fermions) are shown in the left (right) panels, and for $L=21$ ($L=24$) in the top (bottom) panels.