Novel scaling behavior of the Ising model on curved surfaces

TL;DR

This paper investigates the unique scaling behaviors of the Ising model on curved surfaces, revealing multiple critical temperatures and deviations in critical exponents due to curvature effects through Monte Carlo simulations.

Contribution

It demonstrates the nontrivial scaling behavior of the Ising model on curved surfaces, including multiple critical points and altered critical exponents caused by curvature.

Findings

Two critical temperatures on donut-shaped surface with sharp peaks in specific heat and susceptibility

Critical exponents vary depending on temperature range on the donut-shaped surface

Critical exponents deviate from flat surface values on negatively curved surfaces

Abstract

We demonstrate the nontrivial scaling behavior of Ising models defined on (i) a donut-shaped surface and (ii) a curved surface with a constant negative curvature. By performing Monte Carlo simulations, we find that the former model has two distinct critical temperatures at which both the specific heat $C(T)$ and magnetic susceptibility $\chi(T)$ show sharp peaks.The critical exponents associated with the two critical temperatures are evaluated by the finite-size scaling analysis; the result reveals that the values of these exponents vary depending on the temperature range under consideration. In the case of the latter model, it is found that static and dynamic critical exponents deviate from those of the Ising model on a flat plane; this is a direct consequence of the constant negative curvature of the underlying surface.