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Using Monte Carlo simulations we study the two-dimensional Ising model ontriangular, square, and hexagonal lattices with various topologies. We focus onthe behavior of the magnetic susceptibility and of the specific heat near thecritical point of the planar bulk system. We find that scaling functions ofthese quantities on the spherical surface (Euler characteristic K = 2) differfrom the scaling functions on the projective plane (K = 1) which, in turn,differ from the scaling functions on the torus and on the Klein bottle (both K= 0). This provides strong evidence that phase transitions of the Ising modelon two-dimensional surfaces depend on their topologies.

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