Phase Transition in the Density of States of Quantum Spin Glasses

TL;DR

This paper proves that the density of states in quantum spin glasses converges to a normal distribution under certain graph conditions, revealing a phase transition at a critical hypergraph threshold between classical and quantum spectral behaviors.

Contribution

It extends previous results to arbitrary hypergraphs and identifies a sharp phase transition in the spectral distribution of quantum spin glasses.

Findings

Density of states converges to normal distribution for graphs with negligible maximal degree.

Identifies a phase transition at p=n^{1/2} in hypergraph models.

Distinguishes classical and quantum spectral regimes via distribution types.

Abstract

We prove that the empirical density of states of quantum spin glasses on arbitrary graphs converges to a normal distribution as long as the maximal degree is negligible compared with the total number of edges. This extends the recent results of [6] that were proved for graphs with bounded chromatic number and with symmetric coupling distribution. Furthermore, we generalise the result to arbitrary hypergraphs. We test the optimality of our condition on the maximal degree for $p$-uniform hypergraphs that correspond to $p$-spin glass Hamiltonians acting on $n$ distinguishable spin-$1/2$ particles. At the critical threshold $p=n^{1/2}$ we find a sharp classical-quantum phase transition between the normal distribution and the Wigner semicircle law. The former is characteristic to classical systems with commuting variables, while the latter is a signature of noncommutative random matrix theory.