Eigenstate thermalization within isolated spin-chain systems

The thermalization phenomenon and many-body quantum statistical properties are studied on the example of several observables in isolated spin-chain systems, both integrable and generic nonintegrable. While diagonal matrix elements for nonintegrable models comply with the eigenstate thermalization hypothesis, the integrable systems show evident deviations and similarity to properties of noninteracting many-fermion models. The finite-size scaling reveals that the crossover between the two regimes is given by a scale closely related to the scattering length. Low-frequency off-diagonal matrix elements related to dc transport quantities also follow in a generic system a behavior analogous to the eigenstate thermalization hypothesis, however, unrelated to one of the diagonal matrix elements.

Figure 1

Distribution of diagonal matrix elements of (a) spin current $J^s$, (b) energy current $J^E$, and (c) kinetic energy $H^{\mathrm{kin}}$ vs energy $E$ for the integrable model ${\Delta }_2=0$ (left-hand side) and the nonintegrable model ${\Delta }_2=0.5$ (right-hand side). In all cases, $\Delta =0.5$, $L=20$, $M=-1$, and $k=2\pi /L$.

Figure 2

Finite-size dependence of the diagonal matrix elements fluctuations of $J^s$, $J^E$, and $H^{\mathrm{kin}}$, respectively, for $\Delta =0.5$, $M=-1$, and different ${\Delta }_2$. Fluctuations are evaluated within $E=[-1,1]$. In (b) the exact sum rule is indicated (solid line). Inset in (a): Curves for ${\Delta }_2=0.2$ and $0.5$, illustrating the onset of an exponential decrease with $L$.

Figure 3

Probability distribution for the real part of the off-diagonal matrix elements of $J^s$, $J^E$, and $H^{\mathrm{kin}}$, respectively, for a nonintegrable model with $\Delta ={\Delta }_2=0.5$, $L=18$, and $M=-1$. For comparison, Gaussian functions are indicated (dashed curves).

Figure 4

Ratio between off-diagonal and diagonal matrix elements fluctuations for $J^s$ and $J^E$, respectively, for a nonintegrable model with $\Delta =0.5,1$, ${\Delta }_2=0.5$, $L=20$, $M=-1$, and $k$ average. Inset in (a): Finite-size dependence of $r^s(E=0)$.