Hyperbolic lattices and two-dimensional Yang-Mills theory

TL;DR

This paper connects hyperbolic lattice models in quantum systems with two-dimensional Yang-Mills theory, providing a unified framework that links spectral properties to gauge theory observables, especially in the large-N limit.

Contribution

It establishes an equivalence between hyperbolic band theory and $U(N)$ Yang-Mills theory on higher-genus surfaces, unifying different analytical approaches.

Findings

Moments of the density of states correspond to Wilson loop expectation values.

The equivalence becomes exact in the large-N limit.

Reconciliation of real-space and reciprocal-space methods for hyperbolic lattices.

Abstract

Hyperbolic lattices are a new type of synthetic quantum matter emulated in circuit quantum electrodynamics and electric-circuit networks, where particles coherently hop on a discrete tessellation of two-dimensional negatively curved space. While real-space methods and a reciprocal-space hyperbolic band theory have been recently proposed to analyze the energy spectra of those systems, discrepancies between the two sets of approaches remain. In this work, we reconcile those approaches by first establishing an equivalence between hyperbolic band theory and $U(N)$ topological Yang-Mills theory on higher-genus Riemann surfaces. We then show that moments of the density of states of hyperbolic tight-binding models correspond to expectation values of Wilson loops in the quantum gauge theory and become exact in the large-$N$ limit.