Quantum quenches in disordered systems: Approach to thermal equilibrium without a typical relaxation time

We study spectral properties and the dynamics after a quench of one-dimensional spinless fermions with short-range interactions and long-range random hopping. We show that a sufficiently fast decay of the hopping term promotes localization effects at finite temperature, which prevents thermalization even if the classical motion is chaotic. For slower decays, we find that thermalization does occur. However, within this model, the latter regime falls in an unexpected universality class, namely, observables exhibit a power-law (as opposed to an exponential) approach to their thermal expectation values.

Figure 1

Scaling variable $\eta$ [see Eq. (3)] as a function of $\alpha$ for different system sizes but the same filling factor, $1/3$. For $\alpha \lesssim 1$ ($\alpha \gtrsim 1.2$), $\eta$ decreases (increases) with the system size. This is a signature of a metallic (insulating) phase. The number of realizations is $10000$, $10000$, and 400 for $L=12$, 15, and 18, respectively.

Figure 2

${\langle \Delta n\rangle }_{\mathrm{dis}}$ and ${\langle \Delta N\rangle }_{\mathrm{dis}}$ [see Eq. (5)], as a function of the system size, for $\alpha =0.6$, $0.8$, $1.0$, $1.2$, and $1.4$. Points correspond to $(L,p)=(9,3)$, $(12,4)$, $(15,5)$, and $(18,6)$. The disorder average is performed over at least 8500 different realizations for $L\le 15$, and 1020 for $L=18$. The classical dynamics is chaotic but there is no thermalization for large $\alpha$ due to localization effects.

Figure 3

Microcanonical and diagonal results for $n(k=0)$ in 51 different quenches. The systems have 18 sites and six particles, with $\alpha =0.6$, $1.0$, and $1.4$.

Figure 4

${\langle \Delta n_{ii}^{\max }\rangle }_{\mathrm{dis}}$ and ${\langle \Delta N_{ii}^{\max }\rangle }_{\mathrm{dis}}$ (main panels), and ${\langle \Delta n_{ii}^{\mathrm{avg}}\rangle }_{\mathrm{dis}}$ and ${\langle \Delta N_{ii}^{\mathrm{avg}}\rangle }_{\mathrm{dis}}$ (insets) vs system size. Lines are the same as in Fig. 2. As a consequence of localization effects, ETH does not hold for large values of $\alpha$.

Figure 5

Time evolution of ${\langle \delta n\left(t\right)\rangle }_{\mathrm{dis}}$ and ${\langle \delta N\left(t\right)\rangle }_{\mathrm{dis}}$ [see Eq. (7)] for different $\alpha$'s. Thick lines are for 18 sites and six particles. Thin solid lines are power-law fits to the data. Other thin lines are the corresponding results for 15 sites and five particles. An average over 8500 (1020) realizations has been carried out for the 15-site (18-site) system.