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

Progress of Theoretical Physics

Renormalized Field Theory of Random Magnetic Mixtures with Competing Spin Anisotropies. II

Hiroshi Igarashi, M. Oku

Published Feb 1, 1985• 0 citations
Publisher

Abstract

Fixed points of /:l functions and eigenvalues of B matrix in Part I are computed in Part II. It is confirmed that all 256 fixed points in the nonsymmetric XY-Ising like mixtures are shown to be infrared unstable, and that 128 fixed points in the nonsymmetric Ising-Ising like mixtures, among which 8 fixed points are "runaway solutions", become unstable. It is concluded that tetracritical points in the studied magnetic systems are not of second-order.

DOI
10.1143/ptp.73.361

Related papers

Renormalized Field Theory of Random Magnetic Mixtures with Competing Spin Anisotropies. I

M. Oku, Hajime Igarashi

Non-symmetric multicritical points in random magnetic mixtures with competing spin anisotropies are studied by means of the renormalized field theory and the replica method. Nine β functions and two critical indices η are evaluated up to O(ε2)[ε=4-d(d: space dimensionality)]. The investigated model consists of two terms: the anisotropic exchange energy previously studied by Fishman-Aharony and the random off-diagonal anisotropy energy (RAM) recently proposed by Mukamel-Grinstein. Many randomly mixed magnets have the latter term due to the spin lattice coupling or the dipole-dipole interaction. Random off-diagonal anisotropy energy couples longitudinal and transverse spin components in this system. Owing to the existence of this term, the second-order tetracritical point predicted by Fishman-Aharony should be replaced by the unstable coupled tetracritical point, which corresponds to the first-order or smeared phase transition.

Progress of Theoretical Physics•Dec 1, 1983•14 citations

Field-theoretical approach to critical phenomena

C. Bagnuls, Claude Bervillier

The study of the Wegner-Houghton equation in the local-potential approximation yields infrared-stable renormalization-group trajectories that are rejected by field theory. The most stable one corresponds to an approach to the infrared-stable fixed point from the unusual side, yielding negative correction-to-scaling amplitudes as already observed in a binary mixture. We also show how this affects the analysis made by Baker and Kincaid, who concluded that the field-theoretic framework fails in describing the critical behavior of the spin-(1/2 Ising model at d=3.

Physical review. B, Condensed matter•Jan 1, 1990•153 citations

Exact Renormalization Group Exhibiting a Tricritical Fixed Point for a Spin-1 Ising Model in One Dimension

S. Krinsky, D. Furman

Exact renormalization-group recursion relations are studied for a spin-1 Ising model in one dimension. In this model, critical lines are marked by a double degeneracy of the largest eigenvalue of the transfer matrix, and the tricritical point by a triple degeneracy. This is directly related to the tricritical fixed point being more unstable than the critical fixed points.

Physical Review Letters•Apr 1, 1974•25 citations

Ising model with competing axial interactions in the presence of a field: A renormalization-group treatment

Carlos S. O. Yokoi, Mauricio D. Coutinho-Filho, Silvio R. A. Salinas

On the basis of an \ensuremath{\epsilon} expansion, we present a renormalization-group analysis of the anisotropic next-nearest-neighbor Ising model in a field. For small fields, the \ensuremath{\gamma} surface remains XY-like. For larger fields, we confirm the existence of lines of tricritical points which are governed by Gaussian-type exponents. We pay special attention to the many couplings between a two-component critical spin field, associated with the main harmonic component of the magnetization, and several noncritical spin fields, associated with the uniform and the higher harmonic components. We remark that it is necessary to take into account the uniform and the second-harmonic components of the magnetization to describe the tricritical behavior.

Physical review. B, Condensed matter•Apr 1, 1985•8 citations

Spin-glass transition in a magnetic field: A renormalization group study

Iveta R. Pimentel, T. Temesvári, C. De Dominicis

We study the transition of short-range Ising spin glasses in a magnetic field, within a general replica symmetric field theory, which contains three masses and eight cubic couplings, that is defined in terms of the fields representing the replicon, anomalous, and longitudinal modes. We discuss the symmetry of the theory in the limit of replica number $\stackrel{\ensuremath{\rightarrow}}{n}0,$ and consider the regular case where the longitudinal and anomalous masses remain degenerate. The spin-glass transitions in zero and nonzero fields are analyzed in a common framework. The mean-field treatment shows the usual results, that is a transition in zero field, where all the modes become critical, and a transition in nonzero field, at the de Almeida--Thouless (AT) line, with only the replicon mode critical. Renormalization group methods are used to study the critical behavior, to order $\ensuremath{\varepsilon}=6\ensuremath{-}d.$ In the general theory we find a stable fixed point associated with a spin-glass transition in a zero field. This fixed point becomes unstable in the presence of a small magnetic field, and we calculate crossover exponents, which we relate to zero-field critical exponents. In a finite magnetic field, we find no physical stable fixed point to describe the AT transition, in agreement with previous results of other authors.

arXiv:cond-mat/0204615•Physical review. B, Condensed matter•Jun 6, 2002•50 citations

Application of Renormalization-Group Techniques to Random Magnetic Systems.

Ciriyam Jayaprakash

"Renorma1ization-group methods have been applied in the study of quenched random magnetic systems in recent years. We begin with a brief review of second-order phase transitions in pure, homogeneous systems and also of the .. renorma1ization group framework. Then we provide an introduction to quenched random magnetic systems. Next, momentum-space methods and position-space techniques as applied to quenched random magnets are outlined and compared. Grinstein and Luther applied the Wilson-Fisher E-expansion to random n-vector models; Khme1'nitsky discovered that the random Ising model (n = 1) possessed a ""random"" fixed point of 0(squareroot(e)). This fixed point was found to have one marginal and one irrelevant operator. We have investigated the stability of this fixed point using Ca11an-Symanzik equations and renorma1ized perturbation theory. We find the fixed point stable in the next order; we have also obtained critical exponents to one higher order. Next, position-space techniques are used to study some simple model systems. In addition to critical exponents, global thermodynamic properties are determined. These calculations are based on the Migda1-Kadanoff approximate recursion relations suitably generalized to the inhomogeneous case. Firstly we study the randomly bond-dilute two-dimensional nearest- neighbor Ising model on a square lattice. Calculations give both thermal and magnetic exponents associated with the percolative fixed point. Differential recursion relations yield a phase diagram which is in quantitative agreement with all known results. Curves for the specific heat, percolation probability, and magnetization are displayed. The critical region of the specific heat becomes unobservably narrow well above the percolation threshold Pc. This provides a possible explanation for the apparent specific-heat rounding in certain experiments. We then study the Edwards-Anderson model of a spin glass. The current theoretical situation, which is far from satisfactory at present, is briefly reviewed. We treat the spin-l/2 Ising model with independently random nearestneighbor interactions in dimensionalities d = 2, 3, and 4. The phase diagram, which is in qualitative agreement with mean-field results, exhibits paramagnetic, ferromagnetic, antiferromagnetic, and spin-glass phases. The spinglass and paramagnetic phases meet along an extended second-order phase boundary, which terminates in two tricritical points. Critical and tricritical exponents are calculated. The spin-glass specific-heat exponent turns out to be large and negative, compatibly with recent experiments which show a rounded specific heat anomaly. Global specific-heat curves are also displayed for d = 2."

•Feb 1, 1979•0 citations
Fixed points of /:l functions and eigenvalues of B matrix in Part I are computed in Part II. It is confirmed that all 256 fixed points in the nonsymmetric XY-Ising like mixtures are shown to be infrared unstable, and that 128 fixed points in the nonsymmetric Ising-Ising like mixtures, among which 8 fixed points are "runaway solutions", become unstable. It is concluded that tetracritical points in the studied magnetic systems are not of second-order.
Non-symmetric multicritical points in random magnetic mixtures with competing spin anisotropies are studied by means of the renormalized field theory and the replica method. Nine β functions and two critical indices η are evaluated up to O(ε2)[ε=4-d(d: space dimensionality)]. The investigated model consists of two terms: the anisotropic exchange energy previously studied by Fishman-Aharony and the random off-diagonal anisotropy energy (RAM) recently proposed by Mukamel-Grinstein. Many randomly mixed magnets have the latter term due to the spin lattice coupling or the dipole-dipole interaction. Random off-diagonal anisotropy energy couples longitudinal and transverse spin components in this system. Owing to the existence of this term, the second-order tetracritical point predicted by Fishman-Aharony should be replaced by the unstable coupled tetracritical point, which corresponds to the first-order or smeared phase transition.
The study of the Wegner-Houghton equation in the local-potential approximation yields infrared-stable renormalization-group trajectories that are rejected by field theory. The most stable one corresponds to an approach to the infrared-stable fixed point from the unusual side, yielding negative correction-to-scaling amplitudes as already observed in a binary mixture. We also show how this affects the analysis made by Baker and Kincaid, who concluded that the field-theoretic framework fails in describing the critical behavior of the spin-(1/2 Ising model at d=3.
On the basis of an \ensuremath{\epsilon} expansion, we present a renormalization-group analysis of the anisotropic next-nearest-neighbor Ising model in a field. For small fields, the \ensuremath{\gamma} surface remains XY-like. For larger fields, we confirm the existence of lines of tricritical points which are governed by Gaussian-type exponents. We pay special attention to the many couplings between a two-component critical spin field, associated with the main harmonic component of the magnetization, and several noncritical spin fields, associated with the uniform and the higher harmonic components. We remark that it is necessary to take into account the uniform and the second-harmonic components of the magnetization to describe the tricritical behavior.
We study the transition of short-range Ising spin glasses in a magnetic field, within a general replica symmetric field theory, which contains three masses and eight cubic couplings, that is defined in terms of the fields representing the replicon, anomalous, and longitudinal modes. We discuss the symmetry of the theory in the limit of replica number $\stackrel{\ensuremath{\rightarrow}}{n}0,$ and consider the regular case where the longitudinal and anomalous masses remain degenerate. The spin-glass transitions in zero and nonzero fields are analyzed in a common framework. The mean-field treatment shows the usual results, that is a transition in zero field, where all the modes become critical, and a transition in nonzero field, at the de Almeida--Thouless (AT) line, with only the replicon mode critical. Renormalization group methods are used to study the critical behavior, to order $\ensuremath{\varepsilon}=6\ensuremath{-}d.$ In the general theory we find a stable fixed point associated with a spin-glass transition in a zero field. This fixed point becomes unstable in the presence of a small magnetic field, and we calculate crossover exponents, which we relate to zero-field critical exponents. In a finite magnetic field, we find no physical stable fixed point to describe the AT transition, in agreement with previous results of other authors.
"Renorma1ization-group methods have been applied in the study of quenched random magnetic systems in recent years. We begin with a brief review of second-order phase transitions in pure, homogeneous systems and also of the .. renorma1ization group framework. Then we provide an introduction to quenched random magnetic systems. Next, momentum-space methods and position-space techniques as applied to quenched random magnets are outlined and compared. Grinstein and Luther applied the Wilson-Fisher E-expansion to random n-vector models; Khme1'nitsky discovered that the random Ising model (n = 1) possessed a ""random"" fixed point of 0(squareroot(e)). This fixed point was found to have one marginal and one irrelevant operator. We have investigated the stability of this fixed point using Ca11an-Symanzik equations and renorma1ized perturbation theory. We find the fixed point stable in the next order; we have also obtained critical exponents to one higher order. Next, position-space techniques are used to study some simple model systems. In addition to critical exponents, global thermodynamic properties are determined. These calculations are based on the Migda1-Kadanoff approximate recursion relations suitably generalized to the inhomogeneous case. Firstly we study the randomly bond-dilute two-dimensional nearest- neighbor Ising model on a square lattice. Calculations give both thermal and magnetic exponents associated with the percolative fixed point. Differential recursion relations yield a phase diagram which is in quantitative agreement with all known results. Curves for the specific heat, percolation probability, and magnetization are displayed. The critical region of the specific heat becomes unobservably narrow well above the percolation threshold Pc. This provides a possible explanation for the apparent specific-heat rounding in certain experiments. We then study the Edwards-Anderson model of a spin glass. The current theoretical situation, which is far from satisfactory at present, is briefly reviewed. We treat the spin-l/2 Ising model with independently random nearestneighbor interactions in dimensionalities d = 2, 3, and 4. The phase diagram, which is in qualitative agreement with mean-field results, exhibits paramagnetic, ferromagnetic, antiferromagnetic, and spin-glass phases. The spinglass and paramagnetic phases meet along an extended second-order phase boundary, which terminates in two tricritical points. Critical and tricritical exponents are calculated. The spin-glass specific-heat exponent turns out to be large and negative, compatibly with recent experiments which show a rounded specific heat anomaly. Global specific-heat curves are also displayed for d = 2."