VII.B Topological Defects in the XY model

As stated in the previous section, thermal excitation of Goldstone modes destroys spontaneous order in two dimensional models with a broken continuous symmetry. The RG study of the non-linear σ-model confirms that the transition temperature of n-component spins vanishes as T ∗ = 2πǫ/(n − 2) for ǫ = (d − 2) → 0. However, the same RG procedure appears to suggest a different behavior for n = 2. The first indication of unusual behavior for the two dimensional XY model (n = 2), appeared in an analysis of high temperature series by Stanley and Kaplan in 1966. The series results strongly suggested the divergence of susceptibility at a finite temperature, seemingly in contradiction with the absence of symmetry breaking. It was indeed this contradiction that led Wigner to explore the pos­ sibility of a phase transition without symmetry breaking. The Z2 lattice gauge theory, discussed in sec.VI.E as the dual of the three dimensional Ising model, realizes such a possibility. The two phases of the Z2 gauge theory are characterized by different func­ tional forms for the decay of an appropriate correlation function (the Wilson loop). We can similarly examine the asymptotic behavior of the spin-spin correlation functions of the XY model at high and low temperatures. A high temperature expansion for the correlation function for the XY model on a lattice is constructed from