Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. B, Solid state

Exactly solvable model for tricritical phenomena

Victor J. Emery

Published May 1, 1975• 43 citations
Publisher

Abstract

An $n$-component continuous-spin model with a tricritical point is solved exactly in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$. Its properties are derived and are compared with the results of experiments and approximate calculations. The model leads to an analytical description of the features of tricritical points and suggests the use of an approximation procedure for discrete spin systems which is a substantial qualitative and quantitative improvement on mean-field theory.

DOI
10.1103/physrevb.11.3397

Related papers

Three-component model and tricritical points: A renormalization-group study. II. General dimensions and the three-phase monohedron

Julia M. Yeomans, Michael Ellis Fisher

In Part I the global phase diagram of the general spin-1, nearest-neighbor, ferromagnetic Ising model was studied for $d=2$ dimensions using an approximate renormalization-group approach of the Migdal-Kadanoff type. Here general $d$ is considered with emphasis on $\ensuremath{\epsilon}=d\ensuremath{-}1$ expansions and on the variation of the tricritical exponents with $d<~4$. A dilution field is introduced which can be chosen to yield tricritical exponents for $d=2$, in better agreement with more reliable estimates and conjectured exact results. The three-phase monohedra, which describe the densities of coexisting phases near a tricritical point, are calculated quantitatively for $d=2$, and their shape contrasted with that predicted by classical theory: they are found, in particular, to be significantly flattened.

Physical review. B, Condensed matter•Sep 1, 1981•43 citations

Equivalence of two exactly soluble models for tricritical points

St. Sarbach, T. Schneider

An exactly soluble $n$-component continuous spin model with tricritical points is shown to be equivalent, in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$, to a one-component model with anharmonic long-range interactions.

Physical review. B, Solid state•Jan 1, 1976•9 citations

Renormalization-group theory and calculations of tricritical behavior

Bernard Nienhuis, Michael Nauenberg

The conventional renormalization-group analysis of the tricritical behavior of a metamagnet is completed by a description of the first-order transition below the tricritical temperature, and the conditions for the discontinuity in the magnetization are explicitly verified in an Ising spin model. The singularity structure of the crossover scaling function is derived from relations between the scaling fields for the critical and tricritical fixed points. Numerical calculations illustrating various aspects of the theory are given for a square-lattice Ising model in a four-cell cluster approximation.

Physical review. B, Solid state•Mar 1, 1976•49 citations

Tricritical scaling in the spherical model limit

Stéphane Sarbach, Michael Ellis Fisher

The tricritical scaling properties of the n-component continuous-spin model are investigated in the sphercal model limit, n→∞, with special emphasis on the odd fields h and h3. A full scaling description of the tricritical region for 3<d<4 necessitates the recognition of a dangerous irrelevant variable, p, related to the inverse range of the pair interaction. As the lambda line in zero field (h=h3=0) is approached, p becomes important and determines the correct spherical model exponents. Setting p=0 gives only classical theory; away from the lambda line p yields corrections to scaling of Gaussian character. The wing critical exponents arising for h≠0, h3=0 and T<Tt are classical. Furthermore, for h3≠0 classical behavior is found even on the lambda line above Tt which, in fact, continues into one of the wing critical lines. Some consequences for real tricritical behavior are reviewed.

Journal of Applied Physics•Mar 1, 1978•13 citations

Tricritical points in systems with random fields

Amnon Aharony A

Mean-field theory and renormalization-group arguments are used to show that the phase transition in a system with a random ordering field becomes first order at sufficiently low transition temperature, provided the (symmetric) random-field distribution function has a minimum at zero field. The first-order region is separated from the second-order region by a tricritical point. Both the critical and the tricritical exponents at $d>4$ dimensions are shown to be the same as for the pure system at $d\ensuremath{-}2$ dimensions. The relevance to spin glasses and other systems is discussed. The new tricritical point is very different from all previously studied tricritical points, as it deviates from mean-field theory at $d=5$, and not at $d=3$. Although quantitative results are calculated only at $d=5\ensuremath{-}\ensuremath{\epsilon}$ dimensions, the qualitative results are expected to apply at $d=3$.

Physical review. B, Condensed matter•Oct 1, 1978•402 citations

Phase Diagram and Tricritical Behavior in the Three Dimensional Heisenberg Model with a Random Field

A. Oubelkacem, K. Htoutou, A. Ainane +1 more

The random field Ising model (RFIM) has been investigated extensively both theoretically and experimentally in recent years [1–4]. Some questions, such as the lower critical dimension and the existence of a static phase transition, have already been solved from the theoretical point of view, but other questions, such as the existence of the tricritical point, are still open. However, it is now understood that the RFIM describes the essential physics of a rich class of experimentally accessible disordered systems. These include structural phase transitions in random alloys, commensurate charge-density-wave systems with impurity pinning, binary fluid mixtures in random porous media, and the melting of intercalates in layered compounds such as TiS2. The behavior of the RFIM is governed by the competing tendencies of the spins to align ferromagnetically under impetus of the exchange or to follow the local fields and thus be uncorrelated. The RFIM has been studied using different methods. One of the authors [5] of the present paper has developed a new version of effective field theory, which extends the approximation methods [6, 7] to random Ising systems, in which randomness can be described by discrete random variables and the geometry of the lattice can be easily taken into account. This method is based on the determination of the probability distribution of the discrete variables, which takes into account exactly the self-site correlations. This approach has been used to study cooperative phenomena and phase transitions in various systems, and gives useful qualitative and quantitative insights for the critical behavior of a wide variety of classical and quantum lattice spin systems [8–10]. In recent decades, the renormalization group theory of phase transitions and critical phenomena represents one of the most powerful tools of theoretical physics. Therefore, employing this theory (which can be applied for weak fields), the proof

Chinese Journal of Physics•Dec 1, 2004•6 citations
An nnn-component continuous-spin model with a tricritical point is solved exactly in the limit n\ensuremath→\ensuremath∞n\ensuremath{\rightarrow}\ensuremath{\infty}n\ensuremath→\ensuremath∞. Its properties are derived and are compared with the results of experiments and approximate calculations. The model leads to an analytical description of the features of tricritical points and suggests the use of an approximation procedure for discrete spin systems which is a substantial qualitative and quantitative improvement on mean-field theory.
In Part I the global phase diagram of the general spin-1, nearest-neighbor, ferromagnetic Ising model was studied for d=2d=2d=2 dimensions using an approximate renormalization-group approach of the Migdal-Kadanoff type. Here general ddd is considered with emphasis on $\ensuremath{\epsilon}=d\ensuremath{-}1$ expansions and on the variation of the tricritical exponents with d< 4d<~4d< 4. A dilution field is introduced which can be chosen to yield tricritical exponents for d=2d=2d=2, in better agreement with more reliable estimates and conjectured exact results. The three-phase monohedra, which describe the densities of coexisting phases near a tricritical point, are calculated quantitatively for d=2d=2d=2, and their shape contrasted with that predicted by classical theory: they are found, in particular, to be significantly flattened.
An exactly soluble nnn-component continuous spin model with tricritical points is shown to be equivalent, in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$, to a one-component model with anharmonic long-range interactions.
The conventional renormalization-group analysis of the tricritical behavior of a metamagnet is completed by a description of the first-order transition below the tricritical temperature, and the conditions for the discontinuity in the magnetization are explicitly verified in an Ising spin model. The singularity structure of the crossover scaling function is derived from relations between the scaling fields for the critical and tricritical fixed points. Numerical calculations illustrating various aspects of the theory are given for a square-lattice Ising model in a four-cell cluster approximation.
The tricritical scaling properties of the n-component continuous-spin model are investigated in the sphercal model limit, n→∞, with special emphasis on the odd fields h and h3. A full scaling description of the tricritical region for 3<d<4 necessitates the recognition of a dangerous irrelevant variable, p, related to the inverse range of the pair interaction. As the lambda line in zero field (h=h3=0) is approached, p becomes important and determines the correct spherical model exponents. Setting p=0 gives only classical theory; away from the lambda line p yields corrections to scaling of Gaussian character. The wing critical exponents arising for h≠0, h3=0 and T<Tt are classical. Furthermore, for h3≠0 classical behavior is found even on the lambda line above Tt which, in fact, continues into one of the wing critical lines. Some consequences for real tricritical behavior are reviewed.
Mean-field theory and renormalization-group arguments are used to show that the phase transition in a system with a random ordering field becomes first order at sufficiently low transition temperature, provided the (symmetric) random-field distribution function has a minimum at zero field. The first-order region is separated from the second-order region by a tricritical point. Both the critical and the tricritical exponents at d>4d>4d>4 dimensions are shown to be the same as for the pure system at $d\ensuremath{-}2$ dimensions. The relevance to spin glasses and other systems is discussed. The new tricritical point is very different from all previously studied tricritical points, as it deviates from mean-field theory at d=5d=5d=5, and not at d=3d=3d=3. Although quantitative results are calculated only at $d=5\ensuremath{-}\ensuremath{\epsilon}$ dimensions, the qualitative results are expected to apply at d=3d=3d=3.
The random field Ising model (RFIM) has been investigated extensively both theoretically and experimentally in recent years [1–4]. Some questions, such as the lower critical dimension and the existence of a static phase transition, have already been solved from the theoretical point of view, but other questions, such as the existence of the tricritical point, are still open. However, it is now understood that the RFIM describes the essential physics of a rich class of experimentally accessible disordered systems. These include structural phase transitions in random alloys, commensurate charge-density-wave systems with impurity pinning, binary fluid mixtures in random porous media, and the melting of intercalates in layered compounds such as TiS2. The behavior of the RFIM is governed by the competing tendencies of the spins to align ferromagnetically under impetus of the exchange or to follow the local fields and thus be uncorrelated. The RFIM has been studied using different methods. One of the authors [5] of the present paper has developed a new version of effective field theory, which extends the approximation methods [6, 7] to random Ising systems, in which randomness can be described by discrete random variables and the geometry of the lattice can be easily taken into account. This method is based on the determination of the probability distribution of the discrete variables, which takes into account exactly the self-site correlations. This approach has been used to study cooperative phenomena and phase transitions in various systems, and gives useful qualitative and quantitative insights for the critical behavior of a wide variety of classical and quantum lattice spin systems [8–10]. In recent decades, the renormalization group theory of phase transitions and critical phenomena represents one of the most powerful tools of theoretical physics. Therefore, employing this theory (which can be applied for weak fields), the proof