Absence of continuous symmetry breaking in two-dimensional models of statistical physics

Numerous examples of discrete symmetry breaking are well-known in the statistical physics models, such as the Ising model, anisotropic Heisenberg model and so on. In these models the interaction is represented by the function invariant under some discrete group acting in the configuration space of the system. However the action of the group on the set of the Gibbs states might still be nontrivial. It is proved that the situation is different in the case of two-dimensional models and connected symmetry groups: the action of the group on the space of states turns out to be trivial.