Ferroelectrics Letters Section

Phase transition in the anisotropic cubic model

Yurij V. Holovatch

Published Dec 1, 1987• 2 citations

Abstract

The critical behaviour of ferroelectrics or ferromagnets with a Hamiltonian of cubic symmetry is studied by the approximate renormalization group techniques, proposed previously in the frames of collective variables method.

DOI: 10.1080/07315178708200648

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Ferroelectrics Letters Section

Phase transition in the anisotropic cubic model

Yurij V. Holovatch

Published Dec 1, 1987• 2 citations

Publisher

Abstract

DOI: 10.1080/07315178708200648

Related papers

Critical Behavior of Anisotropic Cubic Systems

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The critical behavior in zero field above ${T}_{c}$ of ferromagnets or ferroelectrics with a Hamiltonian of cubic symmetry is studied, to order ${\ensuremath{\epsilon}}^{2}$, by exact renormalization-group techniques in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions with $n$ -component spins. For $\ensuremath{\epsilon}=1$, $n\ensuremath{\ge}3$, a crossover from isotropic (Heisenberg) to characteristic cubic behavior occurs, with the new value $2{v}^{C}=1+[\frac{(n\ensuremath{-}1)}{3n}]\ensuremath{\epsilon}+[\frac{(n\ensuremath{-}1)}{324{n}^{3}}](17{n}^{2}+290n\ensuremath{-}424){\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, and cubic symmetry appearing in the four-spin correlation function. Experiments on structural phase transitions are considered briefly.The critical behavior in zero field above ${T}_{c}$ of ferromagnets or ferroelectrics with a Hamiltonian of cubic symmetry is studied, to order ${\ensuremath{\epsilon}}^{2}$, by exact renormalization-group techniques in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions with $n$-component spins. For $\ensuremath{\epsilon}=1$, $n\ensuremath{\ge}3$, a crossover from isotropic (Heisenberg) to characteristic cubic behavior occurs, with the new value $2{v}^{C}=1+[\frac{(n\ensuremath{-}1)}{3n}]\ensuremath{\epsilon}+[\frac{(n\ensuremath{-}1)}{324{n}^{3}}](17{n}^{2}+290n\ensuremath{-}424){\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, and cubic symmetry appearing in the four-spin correlation function. Experiments on structural phase transitions are considered briefly.

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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.

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Anisotropic Heisenberg surface on semi-infinite Ising ferromagnet : renormalization group treatment

U.M.S. Costa, Ananias M. Mariz, Constantino Tsallis

We use a Migdal-Kadanoff-like renormalization group approach to study the critical behaviour of a semi-infinite simple cubic Ising ferromagnet whose (1, 0, 0) free surface contains anisotropic (in spin space) Heisenberg ferromagnetic interactions. The phase diagram presents three phases (namely the paramagnetic, the bulk ferromagnetic and the surface ferromagnetic ones) which join on a multicritical point The location of this point is calculated as a function of the anisotropy. The various universality classes of the problem are exhibited.We use a Migdal-Kadanoff-like renormalization group approach to study the critical behaviour of a semi-infinite simple cubic Ising ferromagnet whose (1, 0, 0) free surface contains anisotropic (in spin space) Heisenberg ferromagnetic interactions. The phase diagram presents three phases (namely the paramagnetic, the bulk ferromagnetic and the surface ferromagnetic ones) which join on a multicritical point The location of this point is calculated as a function of the anisotropy. The various universality classes of the problem are exhibited.

Journal de Physique LettresJan 1, 19855 citations

The critical behavior in zero field above

{T}_{c}

of ferromagnets or ferroelectrics with a Hamiltonian of cubic symmetry is studied, to order ${\ensuremath{\epsilon}}^{2}$, by exact renormalization-group techniques in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions with

n

-component spins. For $\ensuremath{\epsilon}=1$, $n\ensuremath{\ge}3$, a crossover from isotropic (Heisenberg) to characteristic cubic behavior occurs, with the new value $2{v}^{C}=1+[\frac{(n\ensuremath{-}1)}{3n}]\ensuremath{\epsilon}+[\frac{(n\ensuremath{-}1)}{324{n}^{3}}](17{n}^{2}+290n\ensuremath{-}424){\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, and cubic symmetry appearing in the four-spin correlation function. Experiments on structural phase transitions are considered briefly.

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.

The zero-field critical behavior above

{T}_{c}

of ferromagnets (or ferroelectrics) with a Hamiltonian of cubic symmetry is studied, in

d

space dimensions, in the limit of infinite spin dimensionality, $n\ensuremath{\rightarrow}\ensuremath{\infty}$, with the cubic term of order unity, but the usual

{S}^{4}

term of order ${n}^{\ensuremath{-}1}$. A diagrammatic expansion about an Ising model (instead of the normal Gaussian model) is used. Characteristic cubic behavior is discovered, with renormalized Ising-model exponents (in the Fisher sense).