VII. Continuous Spins at Low Temperatures

VII.A The non-linear σ-model Previously we considered low temperature expansions for discrete spins (Ising, Potts, etc.), in which the low energy excitations are droplets of incorrect spin in a uniform background selected by broken symmetry. These excitations occur at small scales, and are easily described by graphs on the lattice. By contrast, for continuous spins, the lowest energy excitations are long-wavelength Goldstone modes, as discussed in section II.C. The thermal excitation of these modes destroys the long-range order in dimensions d ≤ 2. For d close to 2, the critical temperature must be small, making low temperature expansions a viable tool for the study of critical phenomena. As we shall demonstrate next, such an approach requires keeping track of the interactions between Goldstone modes. Consider unit n-component spins on the sites of a lattice, i.e.