Phase Transition in the Density of States of Quantum Spin Glasses
László Erdős, Dominik Schröder
TL;DR
This work establishes a central limit theorem for the density of states of quantum spin glasses on general graphs and hypergraphs, proving convergence to a standard normal law under a sparsity condition that the maximal degree is vanishingly small relative to the total number of edges. It extends prior results by removing symmetry requirements and by handling hypergraph interactions, demonstrating that noncommutative effects remain subleading in sparse regimes. In the quantum $p$-spin setting, the paper uncovers a sharp phase transition at $p_n\sim\sqrt{n}$: Gaussian density for $p_n\ll\sqrt{n}$, Wigner semicircle for $p_n\gg\sqrt{n}$, and a meaningful interpolating family $\rho_\lambda$ at the critical scaling $p_n\sim\lambda\sqrt{n}$. The results unify a robust moment-method approach with detailed combinatorial analysis of Pauli-trace structures, illustrating a classical-quantum transition in the density of states while confirming the optimality of the sparsity condition.
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.