Holography for Ising spins on the hyperbolic plane

TL;DR

This study uses Monte Carlo simulations to explore the Ising model on hyperbolic tessellations, revealing persistent boundary conformal symmetry and scale invariance across temperatures, supporting holographic principles.

Contribution

It demonstrates that boundary conformal symmetry persists in a discrete hyperbolic setting, extending holographic insights to interacting field theories.

Findings

Boundary-boundary correlators exhibit power-law scaling at all temperatures.

Boundary susceptibility scaling aligns with boundary correlation function analysis.

Boundary conformal symmetry appears robust in discretized hyperbolic geometries.

Abstract

Motivated by the AdS/CFT correspondence, we use Monte Carlo simulation to investigate the Ising model formulated on tessellations of the two-dimensional hyperbolic disk. We focus in particular on the behavior of boundary-boundary correlators, which exhibit power-law scaling both below and above the bulk critical temperature indicating scale invariance of the boundary theory at any temperature. This conclusion is strengthened by a finite-size scaling analysis of the boundary susceptibility which yields a scaling exponent consistent with the scaling dimension extracted from the boundary correlation function. This observation provides evidence that the connection between continuum boundary conformal symmetry and isometries of the bulk hyperbolic space survives for simple interacting field theories even when the bulk is approximated by a discrete tessellation.