Onset of Quantum Chaos and Ergodicity in Spin Systems with Highly Degenerate Hilbert Spaces

TL;DR

The paper investigates how quantum chaos and ergodicity emerge in spin systems with highly degenerate spectra, such as the 2D transverse-field Ising model in the strong-field limit, where finite-size systems can display chaos for arbitrarily small perturbations due to extensive quasiconserved quantities. It develops a Schrieffer-Wolff framework to derive an effective Hamiltonian and analyzes chaos indicators (level-spacing statistics and bipartite entanglement) together with fidelity susceptibility and spectral functions to characterize the crossover from nonergodic to ergodic behavior. The key findings show that in finite systems, chaos appears for any nonzero perturbation, but ergodicity can be delayed by quasiconserved magnetization, with the crossover scale $J^{*}$ scaling polynomially with system size; the typical fidelity susceptibility diverges at the crossover and the low-frequency spectral function exhibits universal behavior consistent with Fermi’s golden rule. In the thermodynamic limit, ergodicity is recovered for arbitrarily small perturbations, although thermalization times can diverge as the perturbation vanishes. Overall, the work clarifies the finite-size vs thermodynamic-limit interplay in the onset of chaos and thermalization for highly degenerate spin models and reveals universal scaling signatures across 1D and 2D variants.

Abstract

We show that in systems with highly degenerate energy spectra, such as the 2D transverse-field Ising model (2DTFIM) in the strong-field limit, quantum chaos can emerge in finite systems for arbitrary small perturbations. In this regime, the presence of extensive quasiconserved quantities can prevent finite systems from becoming ergodic. We study the ensuing crossover to ergodicity in a family of models that includes the 2DTFIM, in which the onset of ergodic behavior exhibits universality and occurs for perturbation strengths that decrease polynomially with increasing system size. We discuss the behaviors of quantum chaos indicators, such as level spacing statistics and bipartite entanglement, and of the fidelity susceptibilities and spectral functions across the crossover.

Figures (6)

• Figure 1: (a) Average ratio of consecutive level spacings $r_\text{ave}$ vs the perturbation strength $J$ for $L=20$ and $22$. We average over all quasimomentum $k$ sectors with $k\ne0,\pi$ for $L=20$, $k=6\pi/11$ for $L=22$, and the two $Z_{2}$ (in the $x$ direction) symmetry sectors. The horizontal dotted line shows $r_\text{ave}$ for the Gaussian orthogonal ensemble (GOE) Atas_13. The left (right) inset shows the magnetization $S_{z}$ in the energy eigenstates vs their energy density $\varepsilon$ at $J=0.1$ ($J=1$) for $L=22$, $k=6\pi/11$, and $Z_{2}=-1$. (b) Normalized average bipartite entanglement entropy $s_\text{ave}$ in the sector with quasimomentum $k=\pi/2$, $Z_{2}=1$ for $L=16$, and $k=4\pi/9$, $Z_{2}=-1$ for $L=18$. Results are reported for $\hat{H}_\text{1D}$ (solid symbols), and $\hat{H}_\text{1DSW}$ (open symbols), and were obtained averaging over the central 20% of the spectrum of each symmetry subspace.
• Figure 2: (a),(b) Normalized bipartite entanglement entropy of energy eigenstates $s$ vs $\varepsilon$ in the central $\sim 80\%$ of the energy spectrum. (a) Exact and SW results for $J=0.1$ and $L=18$ (b) Exact (main panel, $L=16$ and 18) and SW (inset, $L=18$) results for $J=1$. The results for $L=18$ ($L=16$) are from the sector with $k=4\pi/9$, $Z_{2}=-1$ ($k=\pi/2$, $Z_{2}=1$). (c),(d) Same as (a),(b) but for the nearest-neighbor $z$-$z$ correlations $z_\text{nn}$ in the energy eigenstates of a chain with $L=22$ ($L=20$) in the sector with $k=4\pi/9$, $Z_{2}=-1$ ($k=\pi/2$, $Z_{2}=1$).
• Figure 3: (a) Rescaled typical fidelity susceptibility $\chi^v_\text{typ}$ vs the perturbation strength $J$. The left inset shows the position $J^*$ of the maximum of $\chi^v_\text{typ}$ vs the chain size $L$, and the outcome of a fit to $aL^b$ with $a$ and $b$ as fitting parameters. The right inset shows the maximum $\chi^*$ of $\chi^v_\text{typ}$ vs $\omega_H$ (mean level spacing), and the outcome of a polynomial fitting to $a\omega_H^b$. ($J^*$ and $\chi^*$ are computed via a quadratic fit of the data about the maxima.) (b) Spectral function $F^{u}_\text{ave}$ vs $\omega/J$ for $J\!\approx\!J^{*}$ (main panel), $J\!=\!0.1$ (bottom inset), and $J\!=\!1$ (top inset). The straight dashed line in the main panel shows $(\omega/J)^{-2}$ behavior. We show results for $L=20$ and $21$ ($L=22$) computed as a weighted average over the two $Z_{2}$ sectors and the quasimomentum sectors with $k\ne 0,\pi$ ($k=6\pi/11$).
• Figure 4: Rescaled typical fidelity susceptibility $\chi^u_\text{typ}$ vs the perturbation strength $J$ for 2D lattices with $L_x=6,\,L_y=3$ and $L_x=5,\,L_y=4$. We report results obtained in the quasimomentum $k=(0,0)$ sector, averaged over all states in the $Z_{2}$, $M_{x}$, and $M_{y}$ subsectors ($\hat{M}_{x}$ and $\hat{M}_{y}$ stand for mirror symmetry in $x$ and $y$, respectively). Inset: Normalized average bipartite entanglement entropy $s_\text{ave}$ in the $k=(0,0)$ subsector with $Z_{2}=-1$, $M_{x}=-1$, and $M_{y}=-1$ ($Z_{2}=1$, $M_{x}=1$ and $M_{y}=1$) for $L_x=6,\,L_y=3$ ($L_x=5,\,L_y=4$). Results are reported for $\hat{H}_\text{2DTFIM}$ (solid symbols), and $\hat{H}_\text{2DPT}$ (open symbols), and were obtained averaging over the central 20% of the spectrum of each symmetry subspace.
• Figure S1: The spectral function $F^{v}_\text{ave}$ vs $\omega/J$ for (a) $J\!=\!0.1$, (b) $J\!=\!0.3$, and (c) $J\!=\!1$ obtained using Gaussian and Lorentzian broadenings. We show results for $L=22$ computed as a weighted average over the two $Z_{2}$ sectors and the quasimomentum sector with $k=6\pi/11$.