Estimate of equilibration times of quantum correlation functions in the thermodynamic limit based on Lanczos coefficients

Abstract

We study the equilibration times $T_\text{eq}$ of local observables in quantum chaotic systems by considering their auto-correlation functions. Based on the recursion method, we suggest a scheme to estimate $T_\text{eq}$ from the corresponding Lanczos coefficients that is expected to hold in the thermodynamic limit. We numerically find that if the observable eventually shows smoothly growing Lanczos coefficients, a finite number of the former is sufficient for a reasonable estimate of the equilibration time. This implies that equilibration occurs on a realistic time scale much shorter than the life of the universe. The numerical findings are further supported by analytical arguments.

Figures (16)

• Figure 1: Lanczos coefficients $B_N$ (red square) related to the autocorrelation function $\mathcal{C}^2$ versus $2 b_{N/2}$ (blue dot) with $b_n$ the Lanczos coefficients related to $\mathcal{C}$, for different models, observables and system parameters. (a): Ising ladder, energy difference operator ${\cal A}_\Delta$, $\lambda = 0.5$; and tilted field Ising model, energy density wave operator ${\cal A}_{q}$ for (b): $q = \pi/12$, $\lambda = 0.5$; (c): $q = \pi$, $\lambda = 0.5$ and (d): $q= \pi$, $\lambda = 2.0$. $q$ indicates the wavenumber of the density-wave operator ${\cal A}_q$. $\delta_S$ is the smoothness indicator defined in Eq. \ref{['eq-smooth']}.
• Figure 2: Equilibration time$T^{\text{rc}}_{\text{eq}}$ (given in Eq. \ref{['eq-T-rc']}) for different numbers$R$of included Lanczos coefficients, for different models, observables and system parameters. (a): Ising ladder, energy difference operator ${\cal A}_\Delta$, $\lambda = 0.5$; and tilted field Ising model, energy density wave operator ${\cal A}_{q}$ for (b):$q = \pi/12$, $\lambda = 0.5$; (c):$q = \pi$, $\lambda = 0.5$ and (d): $q= \pi$, $\lambda = 2.0$.
• Figure 3: Comparison between the equilibration time estimated using the recursion method $T^\text{rc}_\text{eq}$ (Eq. \ref{['eq-T-rc']}), and that obtained from exact dynamics $T^\text{typ}_\text{eq}$ for different observables and $\lambda$ ($\lambda \in [0.1, 2.0]$ for ${\cal A}_{\pi}$ and ${\cal A}_{\pi/12}$, and $\lambda \in [0.1, 4.0]$ for ${\cal A}_\Delta$). (a)-(d) refer to the corresponding cases illustrated in Figs. \ref{['Fig1']} and \ref{['Fig2']}.
• Figure 4: Lanczos coefficients $B_N$ (red square) related to the autocorrelation function $\mathcal{C}^2$ versus $2 b_{N/2}$ (blue dot) with $b_n$ the Lanczos coefficients related to $\mathcal{C}$, for the spin density wave operators in XXZ model. The results are shown for different $\lambda$ and wave number $q$. (a): $q = \pi,\ \lambda = 1.0$; (b) $q = \pi/14,\ \lambda = 1.0$; (c): $q = \pi,\ \lambda = 10.0$ and (d): $q = \pi/14,\ \lambda = 1.5$. $\delta_S$ is the smoothness indicator defined in Eq. \ref{['eq-smooth']}.
• Figure 5: Equilibration time $T^{\text{rc}}_{\text{eq}}$ (given in Eq. \ref{['eq-T-rc']}) for different numbers $R$ of included Lanczos coefficients, for the spin density wave operators in XXZ model. The results are shown for different $\lambda$ and wave number $q$. (a): $q = \pi,\ \lambda = 1.0$; (b) $q = \pi/14,\ \lambda = 1.0$; (c): $q = \pi,\ \lambda = 10.0$ and (d): $q = \pi/14,\ \lambda = 1.5$.