The Random-Bond Ising Model and its dual in Hyperbolic Spaces

TL;DR

This paper investigates the thermodynamic phase transitions of the random-bond Ising model and its dual on hyperbolic surfaces, revealing distinct transition types and their implications for hyperbolic surface codes.

Contribution

It provides a detailed analysis of the dual RBIM on hyperbolic spaces, clarifies the duality anomaly, and links phase transition behavior to quantum error correction.

Findings

RBIM exhibits second-order transition to ferromagnet or spin-glass phase.

Dual-RBIM undergoes a first-order transition from paramagnet to ferromagnet.

The ferromagnetic phase extent relates to hyperbolic surface code error correction.

Abstract

We analyze the thermodynamic properties of the random-bond Ising model (RBIM) on closed hyperbolic surfaces using Monte Carlo and high-temperature series expansion techniques. We also analyze the dual-RBIM, that is the model that in the absence of disorder is related to the RBIM via the Kramers-Wannier duality. Even on self-dual lattices this model is different from the RBIM, unlike in the euclidean case. We explain this anomaly by a careful re-derivation of the Kramers--Wannier duality. For the (dual-)RBIM, we compute the paramagnet-to-ferromagnet phase transition as a function of both temperature $T$ and the fraction of antiferromagnetic bonds $p$. We find that as temperature is decreased in the RBIM, the paramagnet gives way to either a ferromagnet or a spin-glass phase via a second-order transition compatible with mean-field behavior. In contrast, the dual-RBIM undergoes a strongly first order transition from the paramagnet to the ferromagnet both in the absence of disorder and along the Nishimori line. We study both transitions for a variety of hyperbolic tessellations and comment on the role of coordination number and curvature. The extent of the ferromagnetic phase in the dual-RBIM corresponds to the correctable phase of hyperbolic surface codes under independent bit- and phase-flip noise.