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 (${N}_{\ensuremath{\alpha}}$, ${N}_{\ensuremath{\beta}}$) 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 (${N}_{\ensuremath{\alpha}}$, ${N}_{\ensuremath{\beta}}$) 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 $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}$ phase transition in overlayers of molecular oxygen on graphite. New experiments for the study of this system are also discussed.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 (${N}_{\ensuremath{\alpha}}$, ${N}_{\ensuremath{\beta}}$) 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 (${N}_{\ensuremath{\alpha}}$, ${N}_{\ensuremath{\beta}}$) 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 $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}$ phase transition in overlayers of molecular oxygen on graphite. New experiments for the study of this system are also discussed.

Physical review. B, Condensed matterJun 1, 1979101 citations

Renormalization group treatment of quantum Heisenberg model

I. Golhirsch, Evgeny Levich, Victor Yakhot

Jan 1, 19770 citations

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 (${N}_{\ensuremath{\alpha}}$, ${N}_{\ensuremath{\beta}}$) 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 (${N}_{\ensuremath{\alpha}}$, ${N}_{\ensuremath{\beta}}$) 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 $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}$ phase transition in overlayers of molecular oxygen on graphite. New experiments for the study of this system are also discussed.