Absence of Thermalization in Nonintegrable Systems

We establish a link between unitary relaxation dynamics after a quench in closed many-body systems and the entanglement in the energy eigenbasis. We find that even if reduced states equilibrate, they can have memory on the initial conditions even in certain models that are far from integrable. We show that in such situations the equilibrium states are still described by a maximum entropy or generalized Gibbs ensemble, regardless of whether a model is integrable or not, thereby contributing to a recent debate. In addition, we discuss individual aspects of the thermalization process, comment on the role of Anderson localization, and collect and compare different notions of integrability.

Figure 1

The subsystem is taken to be the first site $S=1$ in the spin chain of $n$ sites, other choices give qualitatively the same results. For each of the product eigenvectors $|E_k^0⟩$ of $H_0$ we compare the equilibration properties of ${\psi }_0^{(1)}=|E_k^0⟩$ with that of ${\psi }_0^{(2)}={\sigma }_S^X|E_k^0⟩$ (i.e., the same state but with the first spin flipped) under the dynamics of $H$ with ${\sigma }_1=0.4{\sigma }_0$. Panels (a),(b) display averages over eigenstates. (a) Average geometric measure of entanglement $\mathbb{E}({\delta }_k)$ with respect to the $H_0$ eigenbasis and average distance of the reduced time averaged states $\mathbb{E}\mathbf{(}D({\omega }^{S(1)},{\omega }^{S(2)})\mathbf{)}$. (b) Average effective dimension and equilibration coefficient [see Eq. (2)]. Panels (c),(d) show quantities optimized over eigenstates. (c) Maximum distinguishability $\underset{k}{max}\Delta (|E_k^{(0)}⟩)$, where $\Delta (|E_k^{(0)}⟩)=\mathcal{D}({\omega }^{S(1)},{\omega }^{S(2)})-C_{\mathrm{eq}}({\psi }_0^{(1)})-C_{\mathrm{eq}}({\psi }_0^{(2)})$ ($\Delta >0$ ensures distinguishability for almost all times. See the inset for an artist’s impression). (d) Effective dimension and equilibration coefficient of the state maximizing $\Delta (|E_k^{(0)}⟩)$. All quantities have been averaged over 100 samples from the random Hamiltonian ensemble (4). The error bars represent the standard deviation. $d^{\mathrm{eff}}$ increases rapidly with $n$; hence, equilibration gets better, while the time averaged states on $S$ remain well distinguishable. Remarkably the observable that best distinguishes the equilibrium states is ${\sigma }_S^Z$. We find “equilibration without thermalization” for a very natural observable.