Ergodicity breaking transition beyond qubits

Abstract

The local Hilbert space dimension is a fundamental parameter in quantum many-body systems, but its role in controlling the ergodicity-breaking transition remains less explored. We systematically study this question by introducing toy models of qudits in zero dimensions that exhibit an ergodicity-breaking transition. Using the hybridization condition, we analytically obtain the dependence of the critical value of the interactions on the local Hilbert space dimension. Numerical studies of spectral correlation measures show agreement with the theoretical predictions, hence confirming the existence of the ergodicity-breaking transition in these models. We further introduce a distinct higher spin version of the model and numerically obtain the same critical behavior, demonstrating that the observed critical point is independent of the microscopic structure of interactions considered.

How does local Hilbert-space dimension affect avalanche-driven ergodicity breaking? This paper extends the zero-dimensional quantum sun model to $q$-state particles: a fixed-size GOE quantum dot couples to external qudits through interactions decaying as $g_0\alpha^{u_j}$. It studies both clock/Potts-type interactions and conventional higher-spin couplings. An ETH-based hybridization argument balances coupling matrix elements against the level spacing of the growing thermal subsystem. With level spacing scaling as $q^{-L}$, the hybridization parameter scales as $(\alpha\sqrt q)^L$, predicting $\alpha_c=1/\sqrt q$. Exact-diagonalization spectral diagnostics test this prediction for $q=3,4$, with a two-particle dot and up to eight or six external particles, respectively. Gap-ratio finite-size collapses give $\alpha_c\approx0.610$ and $0.523$, compared with predicted values $0.577$ and $0.500$. Crossings of the Heisenberg-to-Thouless-time ratio, extracted from the spectral form factor, give approximately $0.610$ and $0.516$. Spin-$1$ and spin-$3/2$ versions exhibit nearby crossings, supporting similar dimension dependence for these different interactions. The numerical transition estimates remain shifted from the hybridization prediction. Computational limitations prevent testing whether changing the dot size accounts for this shift, and the evidence concerns finite-size spectral correlations rather than a spatially extended localized system. The proposed implication—that larger local dimension makes many-body localization more vulnerable to avalanches—is presented as a motivation for further theoretical and experimental tests.