Fading ergodicity and quantum dynamics in random matrix ensembles

Abstract

Recent work has proposed fading ergodicity as a mechanism for many-body ergodicity breaking. Here, we show that two paradigmatic random matrix ensembles -- the Rosenzweig-Porter model and the ultrametric model -- fall within the same universality class of ergodicity breaking when embedded in a many-body Hilbert space of spins-1/2. By calibrating the parameters of both models via their Thouless times, we demonstrate that the matrix elements of local observables display similar statistical properties, allowing us to identify the fractal phase of the Rosenzweig-Porter model with the fading-ergodicity regime. This correspondence is further supported through the analyses of quantum-quench dynamics of local observables, their temporal fluctuations and power spectra, and survival probabilities. Our findings reveal that local observables thermalize within the fading-ergodicity regime on timescales shorter than the Heisenberg time, thus providing a unified framework for understanding ergodicity breaking across these distinct models.

Figures (15)

• Figure 1: Sketch of the parameter regimes in both models in which fading ergodicity is observed. In the Rosenzweig-Porter model where the tuning parameter is $\gamma$, see Eq. \ref{['eq:RP model']}, fading ergodicity is observed at $1<\gamma<2$, while in the ultrametric model where the tuning parameter is $\alpha$, see Eq. \ref{['eq:um_def']}, fading ergodicity is observed at $1/\sqrt{2}<\alpha<1$. In both cases, the observable relaxation time, denoted as the Thouless time $t_{\rm Th}$, is shorter than the Heisenberg time $t_{\rm H}$.
• Figure 2: Coarse grained spectral function $\mathcal{O}(\omega)$, see Eq. \ref{['eq:f_function:coarse']}, of the observable ${\hat{O} = \hat{S}^z_L}$ and $L=15$ for (a) the RP model at different $\gamma$ and (b) the UM model at different $\alpha$. The symbols denote the position of the Thouless energy extracted using Eq. \ref{['eq:integrated_spec']}. The dashed lines represent a characteristic $1/\omega^2$ decay.
• Figure 3: The logarithm of the Thouless energy $\Gamma$ for the UM [RP] model as function of control parameter $\alpha$ [$\gamma$] for two system sizes $L$. We determine the distance $\delta\gamma$, see the sketch in the main panel, by calculating the deviation of the numerical result for the UM model from a linear fit to the RP model results within the same range of $\Gamma$. Inset: Finite size scaling of $\delta\gamma$, extracted from the main panel, versus $1/L$, for two values of $\alpha$. Dashed lines are fits of the linear function to the results.
• Figure 4: Scaling of the off-diagonal matrix elements with the system size $L$ for the (a) RP model and (b) UM model in the frequency range ${\omega\in\qty[\sqrt{\Gamma\Delta}/2,2\sqrt{\Gamma\Delta}]}$. The solid lines show fits of the function $\propto 2^{2L/\eta}$ to the results. (c) Fluctuation exponent $\eta$ extracted from the fits in panels (a,c) for the RP model (circles) and UM model (squares) as function of $\gamma$ [$\alpha$] on the lower [upper] x-axis. The parameters $\alpha$ and $\gamma$ are related through Eq. \ref{['eq:RP model_UM_relation']}. The dash-dotted line denotes the analytical prediction for $\eta$ from Eq. \ref{['eq:eta:prediction']}.
• Figure 5: (a) Averaged autocorrelation function $C(t)$ from Eq. \ref{['eq:autocorrelation:av']} at short times ($t\ll t_H$) for the UM [RP] model at $\alpha=0.82, 0.88$ [$\gamma=1.4, 1.6$] and ${L}=15$, marked with solid [dashed] lines. The time is rescaled by the Thouless energy $\Gamma$, numerically extracted from the spectral function as described in Sec. \ref{['sec:anomalous_dynamics']}. The dash-dotted line shows a decay following $e^{-\Gamma t}$. Inset: Long-time dynamics of $C(t)$ as a function of the rescaled time $\Gamma t$. The horizontal solid [dashed] lines show the diagonal ensemble values, $C_{\rm DE} = {\rm Avr}_\mu\{C_{\rm DE}^{(\mu)}\}$, see also Eq. \ref{['eq:autocorrelation:DE']}, for the UM [RP] at the same values of $\alpha$ [$\gamma$]. (b,c) Scaling of $\Delta Q$ as function of system size $L$ for the RP model [panel (b)] and UM model [panel (c)] for various values of $\gamma$ and $\alpha$ in the fading ergodicity regime, respectively. The solid lines in panels (b,c) show exponential fits of $a2^{-2L/\eta_Q}$ to the results.