Ising model on hyperbolic lattice studied by corner transfer matrix renormalization group method

TL;DR

This paper investigates the critical behavior of the two-dimensional ferromagnetic Ising model on hyperbolic lattices with polygonal tessellations using the corner transfer matrix renormalization group method, revealing mean-field like phase transitions.

Contribution

It applies the CTMRG method to hyperbolic lattices with p≥5 sides, providing new insights into phase transitions on negatively curved surfaces.

Findings

Identifies mean-field like phase transitions for all p≥5.

Calculates critical temperatures and scaling exponents.

Shows convergence towards Bethe lattice behavior as p approaches infinity.

Abstract

We study two-dimensional ferromagnetic Ising model on a series of regular lattices, which are represented as the tessellation of polygons with p>=5 sides, such as pentagons (p=5), hexagons (p=6), etc. Such lattices are on hyperbolic planes, which have constant negative scalar curvatures. We calculate critical temperatures and scaling exponents by use of the corner transfer matrix renormalization group method. As a result, the mean-field like phase transition is observed for all the cases p>=5. Convergence of the calculated transition temperatures with respect to p is investigated towards the limit p->infinity, where the system coincides with the Ising model on the Bethe lattice.