Power-law decay exponents: A dynamical criterion for predicting thermalization

From the analysis of the relaxation process of isolated lattice many-body quantum systems quenched far from equilibrium, we deduce a criterion for predicting when they are certain to thermalize. It is based on the algebraic behavior $\propto t^{-\gamma }$ of the survival probability at long times. We show that the value of the power-law exponent $\gamma$ depends on the shape and filling of the weighted energy distribution of the initial state. Two scenarios are explored in detail: $\gamma \ge 2$ and $\gamma <1$. Exponents $\gamma \ge 2$ imply that the energy distribution of the initial state is ergodically filled and the eigenstates are uncorrelated, so thermalization is guaranteed to happen. In this case, the power-law behavior is caused by bounds in the energy spectrum. Decays with $\gamma <1$ emerge when the energy eigenstates are correlated and signal lack of ergodicity. They are typical of systems undergoing localization due to strong onsite disorder and are found also in clean integrable systems.

Figure 1

Local density of states (a), survival probability decay (b), and scaling analysis of the ${\text{IPR}}_0$ (c) for the Néel state evolving under $H$ (27), with $h=0,\mathrm{\Delta }=1/2,\lambda =1$, and open boundaries. In (a) the shaded area is the numerical result and the solid line is a Gaussian with $E_0$ and ${\sigma }_0$ from Eq (28); $L=16$. In (b) the solid line is the numerical result obtained with expokit [90, 91], circles indicate the analytical Gaussian decay with ${\sigma }_0$ from Eq. (28), the dashed line is the time average coinciding with $t^{-2}$, and the thick horizontal line marks the saturation $\overline{F}={\text{IPR}}_0;L=24$. In (c) the solid line is ${\mathrm{IPR}}_0=6/\mathcal{D}$. The first three points are obtained with exact diagonalization, and the last four are infinite-time averages computed with expokit.

Figure 2

$|A_{\mathrm{G}}{\left(t\right)|}^2,{\left|A_{\mathrm{R}}\left(t\right)\right|}^2$, and the interference term $A_{\mathrm{Int}}\left(t\right)\equiv 2\text{Re}\left[A_{\mathrm{G}}^{*}\left(t\right)A_{\mathrm{R}}\left(t\right)\right]$ for different time scales: short times (a), long times (b), and the vicinity of $t_P$ (c). The dot-dashed line in (b) corresponds to $F\left(t\right)\propto t^{-2}$. The vertical solid line in (c) indicates $t_P\sim 1.98$. The data are obtained analytically for a Gaussian ${\rho }_0\left(E\right)$ using the values of $E_0$ and ${\sigma }_0$ from Eq. (28) for a Néel state under $H$ with $h=0,\mathrm{\Delta }=1/2,\lambda =1,L=16$. The values of $E_{\mathrm{low}}$ and $E_{\mathrm{up}}$ are obtained from exact diagonalization.

Figure 3

Normalized survival probability $f\left(t\right)$ for the Néel state evolving under the chaotic $H$ (27) with $h=0,\lambda =1$, and open boundary conditions. In (a) $\mathrm{\Delta }=1/2$, and in (b) $\mathrm{\Delta }=0$. Light solid lines indicate $L=22$ and dark lines $L=24$. The dashed line is $c-(1/L)ln\left(t^{-2}\right)$, where $c$ is a fitting constant.

Figure 4

Survival probability (a) and normalized survival probability (b),(c) for initial states corresponding to site-basis vectors and evolving according to $H$ (27) with $h$ in $[0.2,1],\mathrm{\Delta }=1,\lambda =0$, and closed boundaries. The average is performed over ${10}^5$ disorder realizations and initial states with energies $E_0$ close to the middle of the spectrum; $L=16$. In (a) the curves from bottom to top have $h=0.2,0.3,...,1$. The thick black line corresponds to $h=1$. In (b) and (c), the dashed lines are $c-(1/L)ln\left(t^{-\gamma }\right)$, where $c$ is a fitting constant and $\gamma$ is indicated.

Figure 5

Components $|C_{\alpha }^{\left(0\right)}{|}^2$ (a) and $f\left(t\right)$ (b) for the Néel state evolving under the $XX$ Hamiltonian [Eq. (27) with $h,\mathrm{\Delta },\lambda =0$, and open boundaries]. In (a), $L=12$. In (b), $f\left(t\right)$ (solid line) and $-(1/L)ln\left(t^{-2}\right)$ (dashed line); $L=2000$.

Figure 6

LDOS (a),(b) and survival probability [Eq. (42)]; (c),(d) for the domain wall state evolving under the $XX$ Hamiltonian [Eq. (27) with $h,\mathrm{\Delta },\lambda =0$, and open boundaries]. Two system sizes are shown in all panels. The (red) light lines indicate $L=24$; the (black) dark lines $L=30$; and the (blue) circles the Gaussian curves ${\rho }_0\left(E\right)=e^{-2E^2}/\sqrt{\pi /2}$ and $F\left(t\right)=exp(-t^2/4)$.