Mean-field universality class induced by weak hyperbolic curvatures

Abstract

Order-disorder phase transition of the ferromagnetic Ising model is investigated on a series of two-dimensional lattices that have negative Gaussian curvatures. Exceptional lattice sites of coordination number seven are distributed on the triangular lattice, where the typical distance between the nearest exceptional sites is proportional to an integer parameter $n$. Thus, the corresponding curvature is asymptotically proportional to $-n_{}^{-2}$. Spontaneous magnetization and specific heat are calculated by means of the corner transfer matrix renormalization group method. For all the finite $n$ cases, we observe the mean-field-like phase transition. It is confirmed that the entanglement entropy at the transition temperature is linear in $(c/6)\ln n$, where $c=1/2$ is the central charge of the Ising model. The fact agrees with the presence of the typical length scale $n$ being proportional to the curvature radius.

8 pages, 7 figures

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