# The Ergodicity Landscape of Quantum Theories

## TL;DR

This paper reviews the conceptual landscape of spectral universality in quantum systems, using a graph-based approach to understand ergodicity and quantum chaos across various models, including the SYK model.

## Contribution

It introduces a unified graph-based framework for analyzing ergodicity in quantum theories and proposes criteria to diagnose ergodicity in different bases.

## Key findings

- Relation between ergodicity onset and graph structure
- Heuristic model of trajectory summation on graphs
- Transition from maximal to partial ergodicity in SYK model

## 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.