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

Abstract

We study a two-dimensional ferromagnetic Ising model on a series of regular lattices, which are represented as a 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 the 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 toward the limit p → ∞, where the system coincides with the Ising model on the Bethe lattice.