Criticality senses topology

Abstract

We numerically investigate the Ising model near quasi-criticality on finite two-dimensional (2D) manifolds formed by triangular and square lattices which exhibit the topology of the surface of a sphere, the projective plane, the torus, and the Klein bottle. We find that the finite-size scaling function of the magnetic susceptibility is the same for the surface of the torus and of the Klein bottle (Euler's characteristic 0 for both of them) but differs from the one for the projective plane (Euler's characteristic 1) and for the surface of the sphere (Euler's characteristic 2). This indicates that the universal properties of the continuous order-disorder phase transition in the Ising model for 2D manifolds depend on their topology.