Weak correlation effects in the Ising model on triangular-tiled hyperbolic lattices

TL;DR

This paper investigates the Ising model on hyperbolic lattices formed by triangular tessellations, revealing unique correlation behaviors and confirming the absence of finite correlation length at infinite negative curvature.

Contribution

It introduces a generalized corner transfer matrix renormalization group method for hyperbolic lattices and analyzes phase transitions and correlation properties in this setting.

Findings

Correlation functions decay exponentially even at criticality

Power law behavior is absent on hyperbolic lattices

Finite correlation length does not exist at infinite negative curvature

Abstract

The Ising model is studied on a series of hyperbolic two-dimensional lattices which are formed by tessellation of triangles on negatively curved surfaces. In order to treat the hyperbolic lattices, we propose a generalization of the corner transfer matrix renormalization group method using a recursive construction of asymmetric transfer matrices. Studying the phase transition, the mean-field universality is captured by means of a precise analysis of thermodynamic functions. The correlation functions and the density matrix spectra always decay exponentially even at the transition point, whereas power law behavior characterizes criticality on the Euclidean flat geometry. We confirm the absence of a finite correlation length in the limit of infinite negative Gaussian curvature.