Probing the ergodicity breaking transition via violations of random matrix theoretic predictions for local observables

Abstract

Quantum many-body systems can exhibit distinct regimes where dynamics is either ergodic, dynamically exploring an extensive region of available state-space, or non-ergodic, where the dynamics may be restricted. An example is the many-body localization (MBL) transition, where disorder induces non-ergodic behaviour. Most measures of ergodicity notably rely on global quantities, such as level spacing statistics. We explore the ability for a subsystem to probe the ergodicity of dynamics via measurement of local observables, and use expected results from random matrix theory (RMT) as a benchmark for the ergodic regime. We exploit two predictions from RMT as ergodicity is broken: the time evolution of the quantum Fisher information, and a fluctuation-dissipation relation. These are investigated in three different ergodicity breaking mechanisms, namely, as a consequence of transition to integrability, MBL, and Quantum Many-Body Scars (QMBS). We show that the predicted behaviour from RMT can be used as a potential witness for transition to non-ergodic behaviour from the measurement of local observables alone.

Figures (4)

• Figure 1: Standard measures for quantum ergodicity. a) Energy level-spacing statistics $P(s)$ for $N=13$ and different coupling constants $J_x^{(SB)}$. The dashed line denotes Poisson distribution $P(s)=e^{-s}$ and the dash-dotted line denotes Wigner-Dyson distribution $P(s)=\frac{\pi}{2} se^{-\pi s^2/4}$. The parameters are set to $B=0.01$, $B_x^{(\rm B)}=0.3$, $J_z^{(\rm SB)}=0.2$, $J_x =1$. b) The average ratio of consecutive energy spacings $\langle r\rangle$ as a function of the disorder $W$ for different number of spins $N$. The coupling constant is set to $J_x^{(\rm SB)}=0.4$. The results for $W \neq 0$ are averaged over 30 iterations. c) Time evolution of the von-Neumann entropy $S$ for $N=11$ and initial state $|\Psi_0\rangle = |\varphi_{\alpha_0} \rangle$ with $\alpha_0=1000$ and various disorder strengths $W$. The numerical results are compared with the function $a\ln(t)+b$ (dashed lines). The results for $W \neq 0$ are averaged over 30 iterations. d) The density of states $D(E)$ as a function of the disorder $W$ for $N=13$. The results for $W \neq 0$ are averaged over 30 iterations. e) Overlap of $|\mathbb{Z}_2\rangle$ with the eigenstates of $\hat{H}_{\rm QMBS}$ as a function of the energies $E$ for $B=0.4$ and number of spins $N=20$. The scarred states are denoted in red. f) The survival probability as a function of time for different initial states. Revivals can be seen for $|\mathbb{Z}_2\rangle$.
• Figure 2: Comparison of the QFI in all three ergodicity breaking mechanisms. a) Long time evolution of the QFI for different coupling strengths $J^{(\rm SB)}_{x}$ for $N=11$ and initial state $|\Psi_0\rangle = |\uparrow_x\rangle \bigotimes|\psi_{\rm B} \rangle$ where $|\psi_{\rm B} \rangle$ is an eigenstate of $\hat{H}_{\rm B}$. The dashed lines show the functions $\alpha t$ and $\beta t^2$ for $\alpha=7.5$ and $\beta=0.0165$. The parameters are set to $B=0.01$, $B_x^{(\rm B)}=0.3$, $J_z^{(\rm SB)}=0.2$, and $J_x =1$ b) Long time evolution of the QFI for various disorders $W$ for $N=13$ and initial state $|\Psi_0\rangle = |\varphi_{\alpha_0} \rangle$ with ${\alpha_0}=5500$. The dashed lines show the functions $\alpha t$ and $\beta t^2$ for $\alpha=26$ and $\beta=0.024$. The results for $W \neq 0$ is averaged over 30 iterations. (inset) The adjusted coefficient of determination $R^2$ for the fits $\alpha t + \beta t^2$ and $\gamma t^2$ as a function of the disorder $W$. The vertical dashed line shows the dynamical phase transition point $W_c$. c) Long time evolution of the QFI for different initial states, for $N=16$ and $B=0.4$. The dashed lines show the functions $\alpha t$ and $\beta t^2$ for $\alpha=2.7$ and $\beta=0.1$.
• Figure 3: a) Shows violation of fluctuation-dissipation relation (labeled RMT) for weak couplings in the integrable-ergodic transition. Circles show $N=13$, squares show $N=14$. We see that for both system sizes deviation from the RMT prediction occurs for couplings $J_z \lesssim 0.2$. The dashed line denotes the RMT result. b) Shows the fluctuation-dissipation relation for the MBL transition rescaled by constants as in Eq. \ref{['eq:QCFDT']}. Here we similarly see a deviation from the RMT prediction for $W \gtrsim 2$, as observed for ergodicity measures in Fig. \ref{['fig2']}. c) Shows fluctuation-dissipation relation for PXP model for $B = 0$ and a central probe spin for varying $N =[14, 16, 18, 20, 22]$ (increasing from right to left) for both scarred and ergodic initial states.
• Figure 4: Shows fluctuation-dissipation relation for the MBL transition for different disorder strengths $W$ and various numbers of spins $N$.