Generalized Survival Probability

Abstract

Survival probability measures the probability that a system taken out of equilibrium has not yet transitioned from its initial state. Inspired by the generalized entropies used to analyze nonergodic states, we introduce a generalized version of the survival probability and discuss how it can assist in studies of the structure of eigenstates and ergodicity.

Keywords: disordered spin model; many-body quantum chaos; quench dynamics; spectral form factor; survival probability.

Figure 1

Generalized local density of states for GOE matrices for (a) $q=0.5$, (b) $q=1.0$, (c) $q=2.0$, and (d) $q=3.0$. Shaded areas are numerical results and the solid curves represent the semicircle law in Equation (14). A single disorder realization and a single initial state are considered. The matrix size is $\mathcal{D}=\mathrm{12,000}$.

Figure 2

Generalized local density of states for the disordered spin-$1/2$ model with $h=0.5$ for (a) $q=0.5$, (b) $q=1.0$, (c) $q=2.0$, and (d) $q=3.0$. Shaded areas are numerical results and the solid curves represent the Gaussian expression in Equation (15). A single disorder realization is considered. The system size is $L=16$ with $\mathcal{D}=\mathrm{12,870}$.

Figure 3

Width of the generalized LDoS normalized by the DoS for (a) GOE matrices, ${\tilde{\sigma }}_q^{\mathrm{GOE}}={\sigma }_q^{GOE}/{\sigma }_{DOS}^{GOE}$, and (b) the spin model, ${\tilde{\sigma }}_q^{\mathrm{Spin}}={\sigma }_q^{\mathrm{Spin}}/{\sigma }_{DOS}^{\mathrm{Spin}}$, with $h=0.5$ (circles) and $h=2$ (squares) as a function of q. Each point is an average over 10 disorder realizations and a single initial state. The dotted lines are guides for the eyes. $\mathcal{D}=\mathrm{12,000}$ for GOE and $\mathcal{D}=\mathrm{12,870}$ ($L=16$) for the spin model.

Figure 4

Generalized local density of states for the disordered spin-$1/2$ model with $h=2.0$ for (a) $q=0.5$, (b) $q=1.0$, (c) $q=2.0$, and (d) $q=3.0$. Shaded areas are numerical results and solid curves represent the Gaussian expression in Equation (15). A single disorder realization and a single initial state are considered. The system size is $L=16$ with $\mathcal{D}=\mathrm{12,870}$.

Figure 5

Generalized survival probability evolving under the GOE model for different values of q. Red curves are numerical results and the black lines correspond to the analytical expression in Equation (20). From bottom to top, $q=0.5$, $1.0$, $2.0$, and $3.0$. Matrix size is $\mathcal{D}=\mathrm{12,000}$. Averages over ${10}^4$ samples.

Figure 6

Generalized survival probability evolving under the disordered spin-$1/2$ model with (a) $h=0.5$ and (b) $h=2$ for different values of q. From bottom to top, $q=0.5$, $1.0$, $1.5$, $2.0$, and $3.0$. The system size is $L=16$. Averages over $3\times {10}^4$ samples.