Wigner Cat Phases: A finely tunable system for exploring the transition to quantum chaos

TL;DR

The paper introduces a finely tunable mixed Gaussian Orthogonal Ensemble (mGOE) to simulate the transition from quantum chaos to Many-Body Localization (MBL) and to explore ETH-related behavior. By parameterizing a mixture degree $bc$ and enforcing spectral alignment through periodicity, the authors generate Wigner Cat Phases whose empirical spectral densities diverge from the semicircle and whose spectral statistics evolve continuously from GOE to localized regimes. They quantify transitions using spectral densities, nearest-neighbor spacings, and adjacent-gap ratios, with bootstrapped confidence intervals, uncovering heavy-tailed localized phases where the mean adjacent-gap ratio can resemble integrable values. The work provides a versatile, pedagogical toy model for studying quantum chaos, ETH, and MBL, and suggests new spectral phenomena in localized regimes that can inform real many-body systems and disorder-control studies.

Abstract

The transition to chaos for quantum dynamics is quantified via a finely tunable mixed random matrix ensemble. The {\it mixed Gaussian Orthogonal Ensemble (mGOE)} forms a pedagogically accessible family of systems in simulating {\it Many-Body Localization (MBL)} transitions. It can be tuned from chaotic to localized and heavy-tailed localized phases in a continuous fashion, providing an opportunity to explore new phases. We numerically study how the spectral properties of mGOE evolve during these transitions. Characterization of transition to quantum chaos is computed and analyzed via empirical spectral density, nearest-neighbor spacing, and adjacent gap ratios with statistical uncertainty quantifications that strengthens the robustness of evidence of transitions. The transition is identified as {\it Wigner Cat Phases}, because of the shape of empirical spectral densities, which depens on the tuneable parameter. These simulated phases in mGOE appear to be an ideal tool to study {\it Eigenstate Thermalization Hypothesis (ETH)} and its related transitions, representing a family of physical systems under different localisation and disorder strengths.

Figures (4)

• Figure 1: Spectral densities are numerically identified for different tuning parameters, at (\ref{['fig:s1']}) $\mu=0.54$, (\ref{['fig:s2']}) $\mu=0.70$, (\ref{['fig:s3']}) $\mu=0.88$ and (\ref{['fig:s4']}) $\mu=0.98$. These are so-called Wigner Cat Phases due to their M-shaped densities deviating from Wigner's semi-circle law. Uncertainties are computed over mGOE ensemble via bootstrapped $95\%$ confidence intervals appear as error bars. We see that semicircle law is recovered at small mixtures, i.e., higher $\mu$ values.
• Figure 2: Nearest-neighbour spacing for different $\mu$ tuning are shown, (\ref{['fig:nn1']}) $\mu=0.54$, (\ref{['fig:nn2']}) $\mu=0.70$, (\ref{['fig:nn3']}) $\mu=0.86$, and (\ref{['fig:nn4']}) $\mu=0.98$. Deviation from Wigner-Dyson distribution at smaller $\mu$ values are demonstrated with lower values indicating heavy-tailed distribution. Uncertainties are computed over mGOE ensemble via bootstrapped $95\%$ confidence intervals appear as error bars. We see that Wigner-Dyson distribution is recovered at small mixtures, i.e., higher $\mu$ values.
• Figure 3: Density of adjacent gap ratios with mean values marked at different degree of mixtures, (\ref{['fig:ag1']}) $\mu=0.98$, (\ref{['fig:ag2']}) $\mu=0.92$, (\ref{['fig:ag3']}) $\mu=0.86$ and (\ref{['fig:ag4']}) $\mu=0.70$. Uncertainties are computed over mGOE ensemble via bootstrapped $95\%$ confidence intervals appear as error bars. We see that tail is changing at higher mixtures, i.e., lower $\mu$ values.
• Figure 4: Average (Mean) gap ratios over different mixture $\mu$ strengths. These values inform us how localisation changes. Horizontal lines describe reaching to full ETH and full integrability respectively.