Sensitivity of the thermodynamics of two-dimensional systems towards the topological classes of their surfaces

Abstract

Using Monte Carlo simulations we study the two-dimensional Ising model on closed manifolds of various topologies near the critical point of the planar bulk system. To this end we consider triangular, square, and hexagonal lattices. We find that the corresponding universal scaling functions of the magnetic susceptibility χ, as well as those of the specific heat C, differ for distinct Euler characteristics K of the manifold. In particular, for each lattice decoration the maxima of the scaling functions of χ grow as K increases. For the specific heat this relationship is inverted: e.g., for each lattice decoration the maxima of the scaling functions of C on the spherical surface (Euler characteristic K=2) is smaller than on the projective plane (K=1) which, in turn, is smaller than on the torus and on the Klein bottle (both with K=0). We find that if the aspect ratio of the lattices is conform to a square geometrical shape, the magnetic susceptibility scaling functions for different lattice types differ only by a non-universal amplitude.