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Physical review. B, Condensed matter

Two-dimensional anisotropic N -vector models

Eytan Domany, Eberhard K. Riedel

Published Jun 1, 1979• 101 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.19.5817

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Two-dimensional anisotropic NNN-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 NNN-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 NNN-vector Hamiltonians can be associated with discrete models that have the form of a generalized Potts model. Such a model, termed (N\ensuremathα{N}_{\ensuremath{\alpha}}N\ensuremathα​, N\ensuremathβ{N}_{\ensuremath{\beta}}N\ensuremathβ​) 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\ensuremathα{N}_{\ensuremath{\alpha}}N\ensuremathα​, N\ensuremathβ{N}_{\ensuremath{\beta}}N\ensuremathβ​) 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 \ensuremathα\ensuremath−\ensuremathβ\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}\ensuremathα\ensuremath−\ensuremathβ phase transition in overlayers of molecular oxygen on graphite. New experiments for the study of this system are also discussed.
Relations between two-dimensional physical systems, continuous-spin Hamiltonians, and discrete spin or Potts models are established by Landau symmetry and renormalization-group arguments. Experimental realizations of XY and Heisenberg models with cubic anisotropy as well as of different Potts models are reviewed. Then a unified model encompassing these classes of models is introduced. For the special case of a six-state discrete model a phase diagram is proposed based on duality and Migdal renormalization-group transformations.
We study the two-dimensional N-component Landau-Ginzburg Hamiltonian with cubic anisotropy. We compute and analyze the fixed-dimension perturbative expansion of the renormalization-group functions to four loops. The relations of these models with N-color Ashkin-Teller models, discrete cubic models, the planar model with fourth-order anisotropy, and the structural phase transition in adsorbed monolayers are discussed. Our results for N=2N=2N=2 (XY(XY(XY model with cubic anisotropy) are compatible with the existence of a line of fixed points joining the Ising and the O(2)O(2)O(2) fixed points. Along this line the exponent $\ensuremath{\eta}$ has the constant value 1/4,1/4,1/4, while the exponent $\ensuremath{\nu}$ runs in a continuous and monotonic way from 1 to $\ensuremath{\infty}$ [from Ising to O(2)].O(2)].O(2)]. In the four-loop approximation, for N> 3N>~3N> 3 we find a cubic fixed point in the region u,v> 0.u,v>~0.u,v> 0.
The spin 1/2 anisotropic Heisenberg model is studied by means of the real space renormalization transformations.Thermodynamic properties of one-dimensional systems are calculated by using an approximate decimation method.These results agree well with those obtained by Bonner and Fisher and by Katsura.Two-and three-dimensional systems are studied by generalizing the Migdal-Kadanoff renormalization transformations to quantum spin systems.For the Ising and Heisenberg models in two and three dimensions and for the X-V model in three dimensions, the critical properties thus obtained are in qualitative agreement with those obtained from high temperature series expansions.For the two-dimensional X-V model, our results suggest that this model undergoes some kind of phase transition and that the critical behaviour of this model is similar to that expected for the classical X-V and planar models.Contents I. Hiroshi Takano 2-3.Three-dimensional systems III.Decimation method for one-dimensional systems 3-1.Transformation function and approximation 3-2.Renormalization equations for scale factor Q= 2 3-3.Free energy IV.The Migdal-Kadanoff transformations for two-and three-dimensional systems 4-1.Fixed points and critical lines 4-2.Critical properties 4-3.Thermodynamic properties V. Summary and conclusions Appendix A. Real space renormalization transformations A-I.Renormalization transformations A-2.Critical properties A-3.Free energy
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