Scaling theory of fading ergodicity

TL;DR

The authors develop a two-parameter scaling framework for fading ergodicity, a precursor to many-body ergodicity breaking, and show that the ergodic regime is governed by a two-parameter beta function that yields a universal critical exponent $\nu=1$ at the ergodicity-breaking point. Using the quantum sun model, they derive $\beta_{\rm fading}(s,c)=-(1-s)/s\ln(1-s)+(1-s)/s\ln(c)$ and demonstrate that the observable entanglement entropy follows $s(L)=\frac{1}{1+a e^{-L/\eta}}$ with a correlation-length analogue $\eta$ diverging at the critical point $\alpha_c$, while $a$ depends on the interaction parameter $\alpha$. Random-matrix theory baselines (Wishart distribution) establish finite-size corrections to $s$ in the ergodic regime, and numerical scaling collapses near criticality yield consistent estimates of $\alpha_c$, $s_c$, and $\nu$, highlighting a distinct universality class from conventional one-parameter scaling. The work provides a building block for two-parameter scaling theories in interacting quantum systems and clarifies how two-parameter flows can reduce to one-parameter behavior in certain limits, with finite-size corrections characterized by a shift $L_0$.

Abstract

In most noninteracting quantum systems, the scaling theory of localization predicts one-parameter scaling flow in both ergodic and localized regimes. On the other hand, it is expected that the one-parameter scaling hypothesis breaks down for interacting systems that exhibit the many-body ergodicity breaking transition. Here we introduce a scaling theory of fading ergodicity, which is a precursor regime of many-body ergodicity breaking. We argue that the two-parameter scaling governs the entire ergodic regime; however, (i) it evolves into the one-parameter scaling at the ergodicity breaking critical point with the critical exponent $ν=1$, and (ii) it gives rise to the resilient one-parameter scaling close to the ETH point. Our theoretical framework may serve as a building block for two-parameter scaling theories of many-body systems.

Figures (8)

• Figure 1: Two-parameter scaling of fading ergodicity. Dashed blue lines are results for the beta function $\beta_s$ from Eq. \ref{['eq:beta:complete']} at different interactions $\alpha$, and the overlapping solid blue lines are results for the numerically available system sizes $L$. The solid red line is the critical one-parameter flow with the critical exponent $\nu=1$, i.e., Eq. \ref{['eq:beta:complete']} with $a=a_c$. The two-parameter flows in the ergodic phase terminate, with the same derivative, at the ETH fixed point $s=1$.
• Figure 2: Symbols: Numerical results for $1-s$ in the ergodic phase of the quantum sun model at (a) $g_0=0.5$ and (c) $g_0=2$. Dashed lines are fits using Eq. (\ref{['eq:s_Fermi']}), for system sizes $L>9$, with free parameters $a$ and $\eta$, see SM for the actual values of $a$ and $\eta$. The horizontal dash-dotted lines are the values $1-s_c$ at the critical point.
• Figure 3: (a),(b) Results for $\beta_s$ at (a) $g_0=0.5$ and (b) $g_0=2$. (c),(d) Results for $\beta^{\rm (new)}_s$ at (c) $g_0=0.5$ and (d) $g_0=2$. All results correspond to the ergodic phase $\alpha>\alpha_c$ in the quantum sun model. Solid lines are numerical results for $\beta_s$ in (a),(b), see Eq. \ref{['def_beta_s']}, and for $\beta^{\rm (new)}_s$ in (c),(d), see Eq. \ref{['eq:beta:new']}, in which the value of $L_0$ is estimated from the fits in Fig. \ref{['figM2']} (see also SM for details of numerical calculations). The arrows indicate the increase in system size $L=7,...,16$, and the colors denote the value of interaction $\alpha$. Thin dotted lines in (a),(b) show the two-parameter functions $\beta_s(s,a)$ from Eq. \ref{['eq:beta:complete']}, with values of $a$ extracted from Fig. \ref{['figM2']}, while the thick dashed (black) lines show the one-parameter function $\beta_s(s,a=a_c)$. The corresponding values of $\alpha$ are the same in all panels, and the legend in panel (b) applies to all panels.
• Figure 4: (a) Numerical results for $1-s$ vs $L$ in the ergodic phase $\alpha>\alpha_c$, and (b) for $s$ vs $L$ in the localized phase $\alpha<\alpha_c$, both at $g_0=1$. The dashed curves are fits from Eq. \ref{['eq:s_Fermi']} to sizes $L\geq7$. Inset of (b): Correlation length $\eta$ obtained from the fits in the main panel. The dotted line shows the prediction $\eta_{\rm loc}=1/\ln\qty(\alpha_c/\alpha)^2$.
• Figure 5: Results for (a) $\beta_s$ and (b) $\beta^{\rm (new)}_s$, in the ergodic phase at $g_0=1$. They complement those in Fig. \ref{['figM3']} of the main text. The thin dotted lines show the two-parameter beta function $\beta_s(s,a)$ from Eq. \ref{['eq:beta:complete']} with values of $a$ extracted from Fig. \ref{['figM5']}(a), while the thick dashed line shows the one-parameter function $\beta_s(s,a=a_c)$.