The 2d Ising Model on Hyperbolic Lattices and the Role of Boundary Conditions

Abstract

We investigate the role of boundary conditions in the critical behaviour of the ferromagnetic Ising model on regular hyperbolic tessellations. Using worm and Metropolis simulations, we compare open boundaries with wired boundaries implemented through a common auxiliary spin. For open boundaries, the susceptibility exhibits volume-based finite-size scaling with non-mean-field scaling exponents that are approximately stable across four tessellations, supporting the possibility of a common universality class. Comparisons with Binder crossings and susceptibility peaks reveal substantial finite-size effects. For wired boundaries, the central-to-auxiliary spin correlator exhibits radial scaling consistent with a mean-field magnetization exponent. The central-spin response to a uniform magnetic field provides a further consistency test of $δ=3$. Layer-resolved responses to a weak boundary field also reveal an enhancement of the induced magnetization towards the interior. Our results highlight the importance of boundary conditions and the distinction between interior and whole-system observables in characterizing critical phenomena on non-amenable lattices.