Two-dimensional anisotropicN-vector models

Two-dimensional anisotropic $N$-vector models are discussed in three contexts. (i) A comprehensive approach to the description of phase transitions in two-dimensional physical systems is outlined. It involves the identification of discrete models for critical phenomena in two-dimensional systems (such as adsorbed thin films) and their investigation by symmetry, duality, and Migdal renormalization-group methods. The identification is based on the Landau-Ginzburg-Wilson Hamiltonian concept and universality arguments. (ii) Relations among anisotropic continuous-spin Hamiltonians and discrete models are established by the Hubbard transformation and the Migdal renormalization-group transformation. Discrete models are conjectured to be equivalent to $N$-component continuous-spin models with local anisotropies. For example, it is shown that the Migdal recursion relations map the continuous-spin, cubic Heisenberg Hamiltonian onto the discrete cubic model. (iii) Many of the anisotropic $N$-vector Hamiltonians can be associated with discrete models that have the form of a generalized Potts model. Such a model, termed ($N_{\alpha }$, $N_{\beta }$) model, is defined in terms of two interacting Potts-like variables associated with each lattice site, and is analyzed by duality and renormalization-group methods. The ($N_{\alpha }$, $N_{\beta }$) Hamiltonian provides a unified description for large classes of discrete models. The concepts are exemplified by a detailed discussion of the two-dimensional Heisenberg model with cubic anisotropy, which has applications to the magnetic $\alpha -\beta$ phase transition in overlayers of molecular oxygen on graphite. New experiments for the study of this system are also discussed.