Curvature-induced frustration in the $XY$ model on hyperbolic surfaces

TL;DR

This paper investigates how negative surface curvature induces frustration in the $XY$ model, leading to glass-like behavior and high-energy spin clusters, even without disorder.

Contribution

It demonstrates that geometric curvature alone can cause glassy states in the $XY$ model on hyperbolic surfaces, revealing a new mechanism for frustration.

Findings

Zero-temperature spin-glass transition suggested by susceptibility analysis

Formation of high-energy spin clusters due to curvature-induced frustration

Glass-like behavior without disorder on negatively curved surfaces

Abstract

We study low-temperature properties of the $XY$ spin model on a negatively curved surface. Geometric curvature of the surface gives rise to frustration in local spin configuration, which results in the formation of high-energy spin clusters scattered over the system. Asymptotic behavior of the spin-glass susceptibility suggests a zero-temperature glass transition, which is attributed to multiple optimal configurations of spin clusters due to nonzero surface curvature of the system. It implies that a constant ferromagnetic spin interaction on a regular lattice can exhibit glasslike behavior without possessing any disorder if the lattice is put on top of a negatively curved space such as a hyperbolic surface.