The Ergodicity Landscape of Quantum Theories

TL;DR

This work addresses how quantum ergodicity and spectral universality emerge across the space of quantum theories with fixed Hilbert space dimension. It develops a graph-based, operator-algebra framework and diagnostics based on infrared diversity to characterize ergodicity beyond eigenvalue statistics, linking ergodic behavior to the structure of state graphs and trajectory sums. Through surveys of 1D/2D quantum systems, random matrices, spin chains, and the SYK/Gurau-Witten models, the authors map out an ergodicity landscape with two extremal regimes—maximal ergodicity and localization—and describe transitions driven by perturbations of the underlying graph structure. In the SYK case, maximal ergodicity in parity sectors persists under substantial disorder, while decreasing disorder reveals a drift toward nonergodic behavior, illustrating the landscape’s nuanced crossovers. The results offer a framework for understanding how complexity and universality arise in quantum many-body dynamics and point to future directions linking ergodicity to quantum chaos, complexity growth, and potential connections to holographic ideas.

Abstract

This paper is a physicist's review of the major conceptual issues concerning the problem of spectral universality in quantum systems. Here we present a unified, graph-based view of all archetypical models of such universality (billiards, particles in random media, interacting spin or fermion systems). We find phenomenological relations between the onset of ergodicity (Gaussian-random delocalization of eigenstates) and the structure of the appropriate graphs, and we construct a heuristic picture of summing trajectories on graphs that describes why a generic interacting system should be ergodic. We also provide an operator-based discussion of quantum chaos and propose criteria to distinguish bases that can usefully diagnose ergodicity. The result of this analysis is a rough but systematic outline of how ergodicity changes across the space of all theories with a given Hilbert space dimension. As a special case, we study the SYK model and report on the transition from maximal to partial ergodicity as the disorder strength is decreased.

Figures (17)

• Figure 1: Distributions of IR diversities $\Omega$ of eigenvectors of random matrices from the GOE and GUE ensembles. The characteristic Gaussian distribution around $\Omega \approx 0.48$ and $\Omega \approx 0.65$ is clearly visible. We present aggregated results for 2000 instances of $512 \times 512$ matrices drawn from each ensemble.
• Figure 2: Eigenvalue statistics for a particle moving on a circle in the presence of a random potential, $H = L + L^{-1} + \lambda \sum_n \zeta_n \delta(U - e^{2\pi i n/D})$, aggregated over 2000 instances of disorder. The random strengths $\zeta_n$ are independently drawn from the Gaussian distribution $\mathcal{N}(0, 1)$. Eigenvalue statistics are shown for $\lambda = 1/D \approx 0.002$, $\lambda = 1$, and $\lambda = D = 512$, and they show how the system transitions from a free theory with uniform eigenvalue spacing to an Anderson-localized one with Poisson-distributed eigenvalue gaps.
• Figure 3: Distributions of IR diversities $\Omega$ for the particle in a random potential, with the coupling $\lambda$ is varied from $1/D$ to $D$. The left plot shows IR diversities calculated in the position basis, and sharp localization at both strong and weak coupling is clearly visible. The localization at $\Omega < 1$ in the free theory limit is due to the arbitrary choice of polarization between degenerate basis vectors. The right plot shows the distribution of diversities calculated in the momentum basis, in which a different polarization of degenerate states ensures that $\Omega = 0$ for the free theory states.
• Figure 4: Eigenvalue statistics and IR diversities below and above the critical coupling in the Aubry-André model with potential $V(e^{2\pi i n/D}) = 2\lambda \cos (2\pi \omega n + \phi)$, with $\omega = \frac{\sqrt 5 - 1}{2}$ and $\phi$ chosen uniformly from $[0, 2\pi]$. The plots show aggregate eigenvectors of $1,024,000/D$ realizations of $\phi$. The theory transitions from free (momentum-localized) at $\lambda \rightarrow 0$ to position-localized with Poisson statistics at $\lambda \rightarrow \infty$. The regime at finite $\lambda < 1$ is not universal, and does not appear to smooth out as $D$ is increased; this agrees with the known self-similarity of the eigenstates.
• Figure 5: IR diversities of the Aubry-André model near the transition location ($\lambda_c = 1$), calculated for the same model as on Fig. \ref{['fig AA']} and shown for an aggregate of 1000 runs at each $D$. A quick crossover from momentum to position space localization is observed. At $\lambda = 0.90$ and $\lambda = 0.95$, the $\Omega$-distributions appear to keep roughly the same shape as the system size is increased. This may be a sign of an unusual kind of criticality.