Classical spin liquids from frustrated Ising models in hyperbolic space

TL;DR

This work investigates classical spin liquids arising from frustrated antiferromagnetic Ising models on hyperbolic lattices, focusing on the {3,7} tessellation and other {3,q} tilings. Using Metropolis Monte Carlo with parallel tempering and the Fisher–Kasteleyn–Temperley mapping, the authors characterize thermodynamics, correlations, and the ground-state manifold under open boundary conditions, revealing a finite residual entropy per spin $s_{res}=0.102(2)$. Crucially, boundary geometry acts as a control parameter: type-A boundaries sustain a spin-liquid state, while type-B boundaries drive ferrimagnetic order, illustrating boundary-driven frustration in curved space. The findings suggest deep links between curvature, boundary effects, and spin-liquid physics, with potential implications for discrete holography and future quantum generalizations involving dimer models or boundary theories.

Abstract

Antiferromagnetic Ising models on frustrated lattices can realize classical spin liquids, with highly degenerate ground states and, possibly, fractionalized excitations and emergent gauge fields. Motivated by the recent interest in many-body system in negatively curved space, we study hyperbolic frustrated Ising models. Specifically, we consider nearest-neighbor Ising models on tesselations with odd-length loops in two-dimensional hyperbolic space. For finite systems with open boundaries we determine the ground-state degeneracy exactly, and we perform extensive finite-temperature Monte-Carlo simulations to obtain thermodynamic data as well as correlation functions. We show that the shape of the boundary, constituting an extensive part of the system, can be used to control low-energy states: Depending on the boundary, we find ordered or disordered ground states. Our results demonstrate how geometric frustration acts in curved space to produce classical spin liquids.

Figures (7)

• Figure 1: Snapshot low-temperature configurations of antiferromagnetic Ising models on $\{3,7\}$ lattices. Red/blue bonds indicate parallel/antiparallel spins on the respective bond. (a) System with type-A boundary with $N=29$ spins, showing a spin-liquid configuration with antiferromagnetic alignment along the boundary. (b) Type-B boundary with $N=50$ spins, showing an ordered ferrimagnet.
• Figure 2: Entropy (main) and specific heat (inset) per site as function of temperature for $\{3,7\}$ type-A systems. The entropy, obtained from integrating the specific heat, is used to extract a residual entropy, for details see text.
• Figure 3: Uniform magnetic susceptibility $\chi$ as function of temperature $T$ for type-A systems; the inset shows $1/\chi$ at low $T$.
• Figure 4: (a) Average NN spin correlations in each layer $L_i$ as function of $1/T$ of the $N=617$ type-A lattice (four layers); the fourth layer being the boundary. (b) Average NN spin correlations between neighboring layers.
• Figure 5: Residual entropy per spin of $\{3,7\}$ type-A systems as function of inverse system size, obtained from the FKT algorithm; MC results are given for comparison.