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

Phase transitions in systems with a coupling to a nonordering parameter

Y Achiam, Y. Imry

Published Oct 1, 1975• 42 citations
Publisher

Abstract

The renormalization-group method is applied to the analysis of phase transitions in systems where the order parameter $s$ is coupled to a nonordering additional variable $y$. A variety of critical and tricritical behaviors or first-order transitions is found as a function of the physical variables and possible macroscopic constraints imposed on the system. For ${s}^{2}y$ coupling, the correlation function of $y$ was found to be governed by a correlation length which is proportional to that of the order parameter, and by a critical index ${\ensuremath{\eta}}_{y}\ifmmode\cdot\else\textperiodcentered\fi{}{\ensuremath{\eta}}_{y}=2\ensuremath{-}2\ensuremath{\alpha}$; $\ensuremath{\alpha}$ here is the specific-heat exponent of the appropriate unconstrained system. The singular part of the susceptibility, ${\ensuremath{\chi}}_{y}$, has a critical exponent equal to $\ensuremath{\alpha}$, the true specific-heat exponent. When the coupling is ${s}^{2}{y}^{2}$, a weaker singularity of ${\ensuremath{\chi}}_{y}$ appears. The crossover between this behavior and the one typical to ${s}^{2}y$ coupling is calculated. $〈y〉$ has a singular part with an exponent $1\ensuremath{-}\ensuremath{\alpha}$, in the unconstrained case. The breakdown of the scaling law related to the correlation function of $y$ in the constrained case is discussed.

DOI
10.1103/physrevb.12.2768

Related papers

Phase transitions in an n-component spin system with a coupling to a non-ordering parameter

Y Achiam

The renormalization-group method is applied to the analysis of the critical singularities in Gamma y (q), the correlation function of a non-ordering parameter y. The coupling between y and s, the order parameter with n components, is s 2 y. When alpha , the specific heat index, is negative, four types of tricritical behaviours are accessible within this Hamiltonian, while the other two may be accessible in other systems in which the non-ordering parameter is a vector. The scaling law gamma y =(2- eta y ) nu is not only trivially fulfilled (zero identically) between the indices characterizing the leading term of Gamma y (q), which is a constant, but is also fulfilled between the corrections to scaling, which are the leading singularities when alpha <0, although these are negligible compared with the regular terms.

Journal of Physics C Solid State Physics•May 14, 1977•12 citations

Critical dynamics of elastic phase transitions

R. Folk, Harald Iro, Franz Schwabl

The critical dynamics at elastic phase transitions of second order is studied by the renormalization-group theory. The dynamical theory is based on a stochastic equation of motion for damped phonons. For systems with one-dimensional soft sectors usual dynamical scaling holds with the critical dynamical exponent $z=2$. For two-dimensional sectors logarithmic corrections appear. Novel features are found for an isotropic-phonon model. The dynamical susceptibility and the characteristic frequency depend singularly on an irrelevant parameter, which, however, diverges at the fixed point. Consequently dynamical scaling breaks down; e.g., the characteristic frequency is no longer a homogeneous function of the wave number $k$ and the inverse correlation length ${\ensuremath{\xi}}^{\ensuremath{-}1}$. Instead it is a homogeneous function of these variables and an irrelevant parameter. Consequently, the critical dynamics can be characterized by the relaxational exponent $z=2+c\ensuremath{\eta}$ at ${T}_{c}$, while in the hydrodynamic region the sound frequency is characterized by $z=2\ensuremath{-}\frac{1}{2}\ensuremath{\eta}$ and the damping by yet another exponent. This breakdown of scaling is also reflected by the fact that different fixing conditions, i.e., different choices of the frequency, lead to different values of $z$. All of these apparently different transformations lead to the same modified dynamical scaling relations.

Physical review. B, Condensed matter•Aug 1, 1979•62 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

Scaling for first-order phase transitions in thermodynamic and finite systems

Michael Ellis Fisher, A. Nihat Berker

Scaling behavior for first-order phase transitions can be derived alternatively but consistently from renormalization-group, phenomenological, or finite-size considerations. A general analysis of densities at a renormalization-group fixed point demonstrates that if the coexistence of $p$ distinct phases is possible, then $p$ distinct eigenvalue exponents must equal the spatial dimensionality. This basic eigenvalue (or scaling) exponent condition can also be derived phenomenologically by various arguments not depending on detailed renormalization-group considerations. A scaling description of first-order phase transitions is presented and extended to finite systems with linear dimensions $L$, leading to a rounding proportional to ${L}^{\ensuremath{-}d}$, response-function maxima varying as ${L}^{d}\ensuremath{\propto}N$, and boundary-condition-dependent shifts which may be as large as $\ensuremath{\sim}{L}^{\ensuremath{-}1}$.

Physical review. B, Condensed matter•Sep 1, 1982•497 citations

Critical properties of random and constrained dipolar magnets

Amnon Aharony A

Renormalization-group techniques are used to study a model of $n$-coupled $m$-component spin systems, in the limit of large dipole-dipole interactions. As recently shown by Emery, this model describes the critical behavior of a constrained dipolar system in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ and that of a dipolar system with a quenched random perturbation in the limit $n\ensuremath{\rightarrow}0$. In both cases, the unperturbed dipolar fixed point is unstable, and there is a crossover to a new behavior. For the constrained system, this leads to another dipolar fixed point (if $\ensuremath{\alpha}<0$, $\ensuremath{\alpha}$ being the dipolar specific-heat exponent) with the same thermodynamic critical exponents, or to one with renormalized dipolar exponents (if $\ensuremath{\alpha}>0$). These results are different from those of previous "spherical" dipolar models. For the random case, the crossover is either to a new fixed point, with very different exponents, e.g., $2\ensuremath{\nu}\ensuremath{\simeq}1+1.183\ensuremath{\epsilon}$ for $m=d=4\ensuremath{-}\ensuremath{\epsilon}$, or away from all the fixed points found to order $\ensuremath{\epsilon}$. One of the new fixed points in this case has complex eigenvalues of the linearized recursion relations. This is related to the fact that the recursion-relation flow is not of a gradient type for random systems.

Physical review. B, Solid state•Aug 1, 1975•40 citations

Critical properties of many-component systems

Victor J. Emery

Critical properties are discussed for systems with order parameters given by $n$ vectors ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{\ensuremath{\alpha}}$, each with $m$ components. The Hamiltonian has an arbitrary symmetry for each vector separately, but there is a particular kind of coupling between them. It is shown that there is an integral representation for the partition function which reduces $n$ to an explicit parameter in an averaged partition function for the $m$-component model. This leads to a simple discussion of properties of the system as a function of $n$. In particular, it is possible to give a coherent derivation of several known and new results without the aid of perturbation theory or the renormalization-group method. It is shown that, in certain special cases, the exponents are Gaussian when $n$ is a negative even integer and that $n=0$ corresponds to the excluded-volume problem. The general case is shown to reduce to an arbitrary $m$-component model which is random when $n=0$ and constrained when $n\ensuremath{\rightarrow}\ensuremath{\infty}$. A direct derivation of the large-$n$ limit is given and leads to a variety of exactly solvable models. Expressions for the order ${n}^{\ensuremath{-}1}$ correction are obtained in terms of correlation functions. This expansion is valid at all temperatures and for any order of transition, so that it is particularly suitable for considering tricritical phenomena.

Physical review. B, Solid state•Jan 1, 1975•344 citations
The renormalization-group method is applied to the analysis of phase transitions in systems where the order parameter sss is coupled to a nonordering additional variable yyy. A variety of critical and tricritical behaviors or first-order transitions is found as a function of the physical variables and possible macroscopic constraints imposed on the system. For s2y{s}^{2}ys2y coupling, the correlation function of yyy was found to be governed by a correlation length which is proportional to that of the order parameter, and by a critical index \ensuremathηy\ifmmode⋅\else\textperiodcentered\fi\ensuremathηy=2\ensuremath−2\ensuremathα{\ensuremath{\eta}}_{y}\ifmmode\cdot\else\textperiodcentered\fi{}{\ensuremath{\eta}}_{y}=2\ensuremath{-}2\ensuremath{\alpha}\ensuremathηy​\ifmmode⋅\else\textperiodcentered\fi\ensuremathηy​=2\ensuremath−2\ensuremathα; \ensuremathα\ensuremath{\alpha}\ensuremathα here is the specific-heat exponent of the appropriate unconstrained system. The singular part of the susceptibility, \ensuremathχy{\ensuremath{\chi}}_{y}\ensuremathχy​, has a critical exponent equal to \ensuremathα\ensuremath{\alpha}\ensuremathα, the true specific-heat exponent. When the coupling is s2y2{s}^{2}{y}^{2}s2y2, a weaker singularity of \ensuremathχy{\ensuremath{\chi}}_{y}\ensuremathχy​ appears. The crossover between this behavior and the one typical to s2y{s}^{2}ys2y coupling is calculated. 〈y〉〈y〉〈y〉 has a singular part with an exponent 1\ensuremath−\ensuremathα1\ensuremath{-}\ensuremath{\alpha}1\ensuremath−\ensuremathα, in the unconstrained case. The breakdown of the scaling law related to the correlation function of yyy in the constrained case is discussed.
The renormalization-group method is applied to the analysis of the critical singularities in Gamma y (q), the correlation function of a non-ordering parameter y. The coupling between y and s, the order parameter with n components, is s 2 y. When alpha , the specific heat index, is negative, four types of tricritical behaviours are accessible within this Hamiltonian, while the other two may be accessible in other systems in which the non-ordering parameter is a vector. The scaling law gamma y =(2- eta y ) nu is not only trivially fulfilled (zero identically) between the indices characterizing the leading term of Gamma y (q), which is a constant, but is also fulfilled between the corrections to scaling, which are the leading singularities when alpha <0, although these are negligible compared with the regular terms.
The critical dynamics at elastic phase transitions of second order is studied by the renormalization-group theory. The dynamical theory is based on a stochastic equation of motion for damped phonons. For systems with one-dimensional soft sectors usual dynamical scaling holds with the critical dynamical exponent z=2z=2z=2. For two-dimensional sectors logarithmic corrections appear. Novel features are found for an isotropic-phonon model. The dynamical susceptibility and the characteristic frequency depend singularly on an irrelevant parameter, which, however, diverges at the fixed point. Consequently dynamical scaling breaks down; e.g., the characteristic frequency is no longer a homogeneous function of the wave number kkk and the inverse correlation length ${\ensuremath{\xi}}^{\ensuremath{-}1}$. Instead it is a homogeneous function of these variables and an irrelevant parameter. Consequently, the critical dynamics can be characterized by the relaxational exponent $z=2+c\ensuremath{\eta}$ at Tc{T}_{c}Tc​, while in the hydrodynamic region the sound frequency is characterized by $z=2\ensuremath{-}\frac{1}{2}\ensuremath{\eta}$ and the damping by yet another exponent. This breakdown of scaling is also reflected by the fact that different fixing conditions, i.e., different choices of the frequency, lead to different values of zzz. All of these apparently different transformations lead to the same modified dynamical scaling relations.
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.
Scaling behavior for first-order phase transitions can be derived alternatively but consistently from renormalization-group, phenomenological, or finite-size considerations. A general analysis of densities at a renormalization-group fixed point demonstrates that if the coexistence of ppp distinct phases is possible, then ppp distinct eigenvalue exponents must equal the spatial dimensionality. This basic eigenvalue (or scaling) exponent condition can also be derived phenomenologically by various arguments not depending on detailed renormalization-group considerations. A scaling description of first-order phase transitions is presented and extended to finite systems with linear dimensions LLL, leading to a rounding proportional to ${L}^{\ensuremath{-}d}$, response-function maxima varying as ${L}^{d}\ensuremath{\propto}N$, and boundary-condition-dependent shifts which may be as large as $\ensuremath{\sim}{L}^{\ensuremath{-}1}$.
Renormalization-group techniques are used to study a model of nnn-coupled mmm-component spin systems, in the limit of large dipole-dipole interactions. As recently shown by Emery, this model describes the critical behavior of a constrained dipolar system in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ and that of a dipolar system with a quenched random perturbation in the limit $n\ensuremath{\rightarrow}0$. In both cases, the unperturbed dipolar fixed point is unstable, and there is a crossover to a new behavior. For the constrained system, this leads to another dipolar fixed point (if $\ensuremath{\alpha}<0$, $\ensuremath{\alpha}$ being the dipolar specific-heat exponent) with the same thermodynamic critical exponents, or to one with renormalized dipolar exponents (if $\ensuremath{\alpha}>0$). These results are different from those of previous "spherical" dipolar models. For the random case, the crossover is either to a new fixed point, with very different exponents, e.g., $2\ensuremath{\nu}\ensuremath{\simeq}1+1.183\ensuremath{\epsilon}$ for $m=d=4\ensuremath{-}\ensuremath{\epsilon}$, or away from all the fixed points found to order $\ensuremath{\epsilon}$. One of the new fixed points in this case has complex eigenvalues of the linearized recursion relations. This is related to the fact that the recursion-relation flow is not of a gradient type for random systems.
Critical properties are discussed for systems with order parameters given by nnn vectors ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{\ensuremath{\alpha}}$, each with mmm components. The Hamiltonian has an arbitrary symmetry for each vector separately, but there is a particular kind of coupling between them. It is shown that there is an integral representation for the partition function which reduces nnn to an explicit parameter in an averaged partition function for the mmm-component model. This leads to a simple discussion of properties of the system as a function of nnn. In particular, it is possible to give a coherent derivation of several known and new results without the aid of perturbation theory or the renormalization-group method. It is shown that, in certain special cases, the exponents are Gaussian when nnn is a negative even integer and that n=0n=0n=0 corresponds to the excluded-volume problem. The general case is shown to reduce to an arbitrary mmm-component model which is random when n=0n=0n=0 and constrained when $n\ensuremath{\rightarrow}\ensuremath{\infty}$. A direct derivation of the large-nnn limit is given and leads to a variety of exactly solvable models. Expressions for the order ${n}^{\ensuremath{-}1}$ correction are obtained in terms of correlation functions. This expansion is valid at all temperatures and for any order of transition, so that it is particularly suitable for considering tricritical phenomena.