Physical Review Letters

Phase Transitions in Two-Dimensional Systems

Eytan Domany, Eberhard K. Riedel

Published Feb 27, 1978• 81 citations

Abstract

Two recently studied experimental systems are shown to constitute realizations of two-dimensional anisotropic $N$-vector models. Along with a renormalization-group treatment of a general Hamiltonian encompassing these models, we discuss new kinds of multicritical points, nonuniversal critical behavior, and experiments for further study of these systems.

DOI: 10.1103/physrevlett.40.561

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Physical Review Letters

Phase Transitions in Two-Dimensional Systems

Eytan Domany, Eberhard K. Riedel

Published Feb 27, 1978• 81 citations

Publisher

Abstract

Two recently studied experimental systems are shown to constitute realizations of two-dimensional anisotropic

N

-vector models. Along with a renormalization-group treatment of a general Hamiltonian encompassing these models, we discuss new kinds of multicritical points, nonuniversal critical behavior, and experiments for further study of these systems.

DOI: 10.1103/physrevlett.40.561

Related papers

Two-dimensional anisotropic N -vector models

Eytan Domany, Eberhard K. Riedel

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

Multicritical two-dimensional vertex models

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We study the multicritical behavior of a class of two-dimensional ice-type vertex models on different lattices using renormalization-group theory. The models are classified by an integer m, with m=2 corresponding to the known square lattice case. For m>2, the specific-heat exponent is a=(m-2)/(m-1) with an upper critical dimensional confluent (lnt ${)}^{1/2}$ divergence for m=3. The nature of the transition is similar to the mth-order multicritical point, yet the exponents are not those known from c1 conformal invariance. The models are anisotropic with ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\parallel}}}$=1 and ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\perp}}}$=12. A few special features of the models are discussed.We study the multicritical behavior of a class of two-dimensional ice-type vertex models on different lattices using renormalization-group theory. The models are classified by an integer m, with m=2 corresponding to the known square lattice case. For m>2, the specific-heat exponent is a=(m-2)/(m-1) with an upper critical dimensional confluent (lnt${)}^{1/2}$ divergence for m=3. The nature of the transition is similar to the mth-order multicritical point, yet the exponents are not those known from c1 conformal invariance. The models are anisotropic with ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\parallel}}}$=1 and ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\perp}}}$=12. A few special features of the models are discussed.

Physical Review AJul 1, 19927 citations

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Journal of Applied PhysicsMar 1, 197864 citations

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Oleg Borisenko, V. Chelnokov, Gennaro Cortese +3 more

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Application of the renormalization group to phase transitions in disordered systems

G. Grinstein, Alan H. Luther

The critical behavior of spin systems with quenched disorder is studied by renormalization-group methods. For the randomly dilute $m$ -vector model, the $n=0$ limit is used to construct a translationally invariant effective Hamiltonian which describes the original disordered system. This Hamiltonian is analyzed in the $\ensuremath{\epsilon}$ expansion to order ${\ensuremath{\epsilon}}^{2}$. Sharp second-order phase transitions with exponents which do not depend continuously on impurity concentration are predicted. For $m>{m}_{c}\ensuremath{\equiv}4\ensuremath{-}4\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2})$ the isotropic $m$ -component fixed point, which characterizes the critical behavior of the pure system, is stable. For $m<{m}_{c}$ , a new random fixed point becomes stable. The exponents corresponding to this fixed point are $\ensuremath{\eta}=[\frac{(5{m}^{2}\ensuremath{-}8m)}{256{(m\ensuremath{-}1)}^{2}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, $\ensuremath{\nu}=\frac{1}{2}+[\frac{3m}{32(m\ensuremath{-}1)}]\ensuremath{\epsilon}+[\frac{m(127{m}^{2}\ensuremath{-}572m\ensuremath{-}32)}{4096{(m\ensuremath{-}1)}^{3}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$ for $m\ensuremath{\ne}1$, and $\ensuremath{\eta}=\ensuremath{-}\frac{\ensuremath{\epsilon}}{106}+O({\ensuremath{\epsilon}}^{\frac{3}{2}})$, $\ensuremath{\nu}=\frac{1}{2}+\frac{{(\frac{6\ensuremath{\epsilon}}{53})}^{\frac{1}{2}}}{4}+O(\ensuremath{\epsilon})$ for $m=1$ . More general random systems are qualitatively discussed from the effective-Hamiltonian viewpoint.The critical behavior of spin systems with quenched disorder is studied by renormalization-group methods. For the randomly dilute $m$-vector model, the $n=0$ limit is used to construct a translationally invariant effective Hamiltonian which describes the original disordered system. This Hamiltonian is analyzed in the $\ensuremath{\epsilon}$ expansion to order ${\ensuremath{\epsilon}}^{2}$. Sharp second-order phase transitions with exponents which do not depend continuously on impurity concentration are predicted. For $m>{m}_{c}\ensuremath{\equiv}4\ensuremath{-}4\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2})$ the isotropic $m$-component fixed point, which characterizes the critical behavior of the pure system, is stable. For $m<{m}_{c}$, a new random fixed point becomes stable. The exponents corresponding to this fixed point are $\ensuremath{\eta}=[\frac{(5{m}^{2}\ensuremath{-}8m)}{256{(m\ensuremath{-}1)}^{2}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, $\ensuremath{\nu}=\frac{1}{2}+[\frac{3m}{32(m\ensuremath{-}1)}]\ensuremath{\epsilon}+[\frac{m(127{m}^{2}\ensuremath{-}572m\ensuremath{-}32)}{4096{(m\ensuremath{-}1)}^{3}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$ for $m\ensuremath{\ne}1$, and $\ensuremath{\eta}=\ensuremath{-}\frac{\ensuremath{\epsilon}}{106}+O({\ensuremath{\epsilon}}^{\frac{3}{2}})$, $\ensuremath{\nu}=\frac{1}{2}+\frac{{(\frac{6\ensuremath{\epsilon}}{53})}^{\frac{1}{2}}}{4}+O(\ensuremath{\epsilon})$ for $m=1$. More general random systems are qualitatively discussed from the effective-Hamiltonian viewpoint.

Physical review. B, Solid stateFeb 1, 1976529 citations

Critical Behavior at M-Axial Lifshitz Points

Hans Werner Diehl

An introduction to the theory of critical behavior at Lifshitz points is given, and the recent progress made in applying the field-theoretic renormalization group (RG) approach to $ϕ^4$ $n$ -vector models representing universality classes of $m$ -axial Lifshitz points is surveyed. The origins of the difficulties that had hindered a full two-loop RG analysis near the upper critical dimension for more than 20 years and produced long-standing contradictory $ε$ -expansion results are discussed. It is outlined how to cope with them. The pivotal role the considered class of continuum models might play in a systematic investigation of anisotropic scale invariance within the context of thermal equilibrium systems is emphasized. This could shed light on the question of whether anisotropic scale invariance implies an even larger invariance, as recently claimed in the literature.An introduction to the theory of critical behavior at Lifshitz points is given, and the recent progress made in applying the field-theoretic renormalization group (RG) approach to $ϕ^4$ $n$-vector models representing universality classes of $m$-axial Lifshitz points is surveyed. The origins of the difficulties that had hindered a full two-loop RG analysis near the upper critical dimension for more than 20 years and produced long-standing contradictory $ε$-expansion results are discussed. It is outlined how to cope with them. The pivotal role the considered class of continuum models might play in a systematic investigation of anisotropic scale invariance within the context of thermal equilibrium systems is emphasized. This could shed light on the question of whether anisotropic scale invariance implies an even larger invariance, as recently claimed in the literature.

arXiv:cond-mat/0205284Acta Physica Slovaca Reviews and TutorialsMay 14, 200211 citations

Two recently studied experimental systems are shown to constitute realizations of two-dimensional anisotropic

N

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.

{)}^{1/2}

divergence for m=3. The nature of the transition is similar to the mth-order multicritical point, yet the exponents are not those known from c1 conformal invariance. The models are anisotropic with ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\parallel}}}$=1 and ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\perp}}}$=12. A few special features of the models are discussed.

The critical behavior of spin systems with quenched disorder is studied by renormalization-group methods. For the randomly dilute

m

-vector model, the

n=0

limit is used to construct a translationally invariant effective Hamiltonian which describes the original disordered system. This Hamiltonian is analyzed in the $\ensuremath{\epsilon}$ expansion to order ${\ensuremath{\epsilon}}^{2}$. Sharp second-order phase transitions with exponents which do not depend continuously on impurity concentration are predicted. For $m>{m}_{c}\ensuremath{\equiv}4\ensuremath{-}4\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2})$ the isotropic

m

-component fixed point, which characterizes the critical behavior of the pure system, is stable. For

m<{m}_{c}

, a new random fixed point becomes stable. The exponents corresponding to this fixed point are $\ensuremath{\eta}=[\frac{(5{m}^{2}\ensuremath{-}8m)}{256{(m\ensuremath{-}1)}^{2}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, $\ensuremath{\nu}=\frac{1}{2}+[\frac{3m}{32(m\ensuremath{-}1)}]\ensuremath{\epsilon}+[\frac{m(127{m}^{2}\ensuremath{-}572m\ensuremath{-}32)}{4096{(m\ensuremath{-}1)}^{3}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$ for $m\ensuremath{\ne}1$, and $\ensuremath{\eta}=\ensuremath{-}\frac{\ensuremath{\epsilon}}{106}+O({\ensuremath{\epsilon}}^{\frac{3}{2}})$, $\ensuremath{\nu}=\frac{1}{2}+\frac{{(\frac{6\ensuremath{\epsilon}}{53})}^{\frac{1}{2}}}{4}+O(\ensuremath{\epsilon})$ for

m=1

. More general random systems are qualitatively discussed from the effective-Hamiltonian viewpoint.

An introduction to the theory of critical behavior at Lifshitz points is given, and the recent progress made in applying the field-theoretic renormalization group (RG) approach to

ϕ^4

n

-vector models representing universality classes of

m

-axial Lifshitz points is surveyed. The origins of the difficulties that had hindered a full two-loop RG analysis near the upper critical dimension for more than 20 years and produced long-standing contradictory

ε

-expansion results are discussed. It is outlined how to cope with them. The pivotal role the considered class of continuum models might play in a systematic investigation of anisotropic scale invariance within the context of thermal equilibrium systems is emphasized. This could shed light on the question of whether anisotropic scale invariance implies an even larger invariance, as recently claimed in the literature.