Journal of Physics C Solid State Physics

Cross-over effects in anisotropic spin systems

J. Michael Kosterlitz

Published Feb 14, 1976• 13 citations

Abstract

The thermodynamic properties of an anisotropic multicomponent spin system are investigated near the bicritical point in the disordered phase using a renormalization group approach. Explicit forms for the cross-over scaling functions for the specific heat and the susceptibilities are found to lowest orders in epsilon =4-d in terms of z=gt - phi where g is the non-ordering field and t=(T-T b )/T b with T b the bicritical temperature. The scaling functions have the expected properties in the limits z to O and z to zt/dt, the value on the critical lines, and interpolate smoothly between these limits.

DOI: 10.1088/0022-3719/9/3/017

Related papers

Renormalized perturbation theory for anisotropic spin systems: Crossover behavior of the susceptibility

W. K. Theumann

Physical review. B, Condensed matterAug 1, 19812 citations

Crossover theory for disordered ferromagnets

Hans-O Heuer, D. Wagner

Physical review. B, Condensed matterAug 1, 198920 citations

Equations of state for bicritical points. II. Ising-like ordered phases

Eytan Domany, David R. Nelson, Michael Ellis Fisher

Calculations of bicritical crossover scaling functions are extended into the Ising-like bicritical ordered phases. Closed-form expressions are derived to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ for the specific heat, nonordering susceptibility, and longitudinal, susceptibilities in this regime. Scaling functions in ordered and disordered regions are displayed graphically and compared with series-expansion results.

Physical review. B, Solid stateApr 1, 197721 citations

Dynamic scaling and corrections to scaling near bicritical and tetracritical points

V. Dohm, H. K. Janssen

Journal of Applied PhysicsMar 1, 19786 citations

Crossover functions by renormalization-group matching: O(ε2) results

Jeffrey F. Nicoll, Jayanta K. Bhattacharjee

Physical review. B, Condensed matterJan 1, 198190 citations

Equations of state for bicritical points. III. Cubic anisotropy and tetracriticality

Eytan Domany, Michael Ellis Fisher

Physical review. B, Solid stateApr 1, 197736 citations

Journal of Physics C Solid State Physics

Cross-over effects in anisotropic spin systems

J. Michael Kosterlitz

Published Feb 14, 1976• 13 citations

Publisher

Abstract

DOI: 10.1088/0022-3719/9/3/017

Related papers

Renormalized perturbation theory for anisotropic spin systems: Crossover behavior of the susceptibility

W. K. Theumann

A cutoff-dependent formulation of renormalized perturbation theory is extended to study the crossover of the parallel susceptibility in systems with quadratic symmetry breaking near a bicritical point, to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$. In terms of the true temperature variable $t=T\ensuremath{-}{T}_{c}(g)$, for anisotropy $g$ , and crossover variable $x=\frac{g}{{t}^{\ensuremath{\varphi}}}$, the effective susceptibility exponent ${\ensuremath{\gamma}}_{\mathrm{eff}}(t,x)$ is calculated for two ${\ensuremath{\varphi}}^{4}$-field models, one with a sharp momentum cutoff $\ensuremath{\Lambda}$ and the other with a smooth term ${\ensuremath{\Lambda}}^{\ensuremath{-}2}{({\ensuremath{\nabla}}^{2}\ensuremath{\varphi})}^{2}$. Corrections to the first model due to insertions of composite operators of canonical dimension six, inside the scaling regime, are explicitly calculated and shown to yield small but not negligible contributions to crossover behavior. The role of new, finite renormalizations is discussed and it is shown that either a fairly monotonic ${\ensuremath{\gamma}}_{\mathrm{eff}}(t,x)$ is obtained or a displaced maximum may appear depending on the outcome of a higher-order renormalization-group study, which is not done here.A cutoff-dependent formulation of renormalized perturbation theory is extended to study the crossover of the parallel susceptibility in systems with quadratic symmetry breaking near a bicritical point, to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$. In terms of the true temperature variable $t=T\ensuremath{-}{T}_{c}(g)$, for anisotropy $g$, and crossover variable $x=\frac{g}{{t}^{\ensuremath{\varphi}}}$, the effective susceptibility exponent ${\ensuremath{\gamma}}_{\mathrm{eff}}(t,x)$ is calculated for two ${\ensuremath{\varphi}}^{4}$-field models, one with a sharp momentum cutoff $\ensuremath{\Lambda}$ and the other with a smooth term ${\ensuremath{\Lambda}}^{\ensuremath{-}2}{({\ensuremath{\nabla}}^{2}\ensuremath{\varphi})}^{2}$. Corrections to the first model due to insertions of composite operators of canonical dimension six, inside the scaling regime, are explicitly calculated and shown to yield small but not negligible contributions to crossover behavior. The role of new, finite renormalizations is discussed and it is shown that either a fairly monotonic ${\ensuremath{\gamma}}_{\mathrm{eff}}(t,x)$ is obtained or a displaced maximum may appear depending on the outcome of a higher-order renormalization-group study, which is not done here.

Physical review. B, Condensed matterAug 1, 19812 citations

Crossover theory for disordered ferromagnets

Hans-O Heuer, D. Wagner

We show that the critical behavior of disordered ferromagnets can be understood in terms of an effective, translational invariant Hamiltonian with renormalized coupling parameters which exhibit a tricritical as well as a critical point. The stable critical fixed point describes the critical behavior of weakly disordered systems, whereas the tricritical fixed point is responsible for the critical behavior of strongly disordered systems at least in the experimentally accessible range of temperature. This necessitates a crossover theory to describe the crossover between these two fixed points and to calculate critical properties as a function of concentration and temperature. In this paper we study the symmetric phase. Susceptibility and specific heat are calculated in scaling form using nonlinear scaling fields to verify that disordered ferromagnets exhibit universal critical behavior. The corresponding effective exponents exhibit nonmonotonous concentration- and temperature-dependent behavior. According to our theory, Fisher-renormalized tricritical behavior dominates in the experimentally accessible temperature range which leads to extreme values of the effective exponents ${\ensuremath{\gamma}}_{\mathrm{eff}}$=2 and ${\ensuremath{\alpha}}_{\mathrm{eff}}$=-1 in the limit of strong dilution. However, in accord with the Harris criterion we verify that the asymptotic behavior (T\ensuremath{\rightarrow} ${T}_{c}$ ) is not changed by dilution if the critical exponent of the specific heat of the pure system ${\ensuremath{\alpha}}_{\mathrm{pure}<0}$, independent of the value of the concentration.We show that the critical behavior of disordered ferromagnets can be understood in terms of an effective, translational invariant Hamiltonian with renormalized coupling parameters which exhibit a tricritical as well as a critical point. The stable critical fixed point describes the critical behavior of weakly disordered systems, whereas the tricritical fixed point is responsible for the critical behavior of strongly disordered systems at least in the experimentally accessible range of temperature. This necessitates a crossover theory to describe the crossover between these two fixed points and to calculate critical properties as a function of concentration and temperature. In this paper we study the symmetric phase. Susceptibility and specific heat are calculated in scaling form using nonlinear scaling fields to verify that disordered ferromagnets exhibit universal critical behavior. The corresponding effective exponents exhibit nonmonotonous concentration- and temperature-dependent behavior. According to our theory, Fisher-renormalized tricritical behavior dominates in the experimentally accessible temperature range which leads to extreme values of the effective exponents ${\ensuremath{\gamma}}_{\mathrm{eff}}$=2 and ${\ensuremath{\alpha}}_{\mathrm{eff}}$=-1 in the limit of strong dilution. However, in accord with the Harris criterion we verify that the asymptotic behavior (T\ensuremath{\rightarrow}${T}_{c}$) is not changed by dilution if the critical exponent of the specific heat of the pure system ${\ensuremath{\alpha}}_{\mathrm{pure}<0}$, independent of the value of the concentration.

Physical review. B, Condensed matterAug 1, 198920 citations

Equations of state for bicritical points. II. Ising-like ordered phases

Eytan Domany, David R. Nelson, Michael Ellis Fisher

Physical review. B, Solid stateApr 1, 197721 citations

Dynamic scaling and corrections to scaling near bicritical and tetracritical points

V. Dohm, H. K. Janssen

We study the critical dynamics of stochastic models appropriate for describing bicritical and tetracritical points in anisotropic antiferromagnetic systems. The dynamic exponents and the transient exponents are calculated by renormalized field theory up to two-loop order. In the presence of reversible mode-coupling terms, two-loop contributions establish bicritical dynamic scaling in the restricted sense and invalidate recent predictions based on mode-coupling arguments. In the case of an n-component relaxational model total dynamic scaling is found to O (ε2) both at the bicritical (n≲3) and the tetracritical (4≲n<11) points.We study the critical dynamics of stochastic models appropriate for describing bicritical and tetracritical points in anisotropic antiferromagnetic systems. The dynamic exponents and the transient exponents are calculated by renormalized field theory up to two-loop order. In the presence of reversible mode-coupling terms, two-loop contributions establish bicritical dynamic scaling in the restricted sense and invalidate recent predictions based on mode-coupling arguments. In the case of an n-component relaxational model total dynamic scaling is found to O (ε2) both at the bicritical (n≲3) and the tetracritical (4≲n<11) points.

Journal of Applied PhysicsMar 1, 19786 citations

Crossover functions by renormalization-group matching: O(ε2) results

Jeffrey F. Nicoll, Jayanta K. Bhattacharjee

By considering the relationship of the matching techniques of Bruce and Wallace to the differential renormalization-group generators, we find that a restatement of the former gives improved results with the same number of perturbative terms. In particular, the vertex functions and specific heat of a $n$ -component spin system are given exactly in the spherical limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ even at first order in perturbation theory ( $T>{T}_{c}$ ). The nature of the nonlinear scaling variables is clarified, and the results are generally expressed in a more compact form. The general $n$ -component disordered phase functions are rederived to $O({\ensuremath{\epsilon}}^{2})$, where $\ensuremath{\epsilon}\ensuremath{\equiv}4\ensuremath{-}d$. The cross-over equations for the $n=1$ Ising-like case are derived for the Helmholtz potential $A(M)$ , the magnetic field $\frac{h}{M}$ , the inverse susceptibility ${\ensuremath{\Gamma}}_{2}$, and the correlation length $\ensuremath{\xi}$ to $O({\ensuremath{\epsilon}}^{2})$.By considering the relationship of the matching techniques of Bruce and Wallace to the differential renormalization-group generators, we find that a restatement of the former gives improved results with the same number of perturbative terms. In particular, the vertex functions and specific heat of a $n$-component spin system are given exactly in the spherical limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ even at first order in perturbation theory ($T>{T}_{c}$). The nature of the nonlinear scaling variables is clarified, and the results are generally expressed in a more compact form. The general $n$-component disordered phase functions are rederived to $O({\ensuremath{\epsilon}}^{2})$, where $\ensuremath{\epsilon}\ensuremath{\equiv}4\ensuremath{-}d$. The cross-over equations for the $n=1$ Ising-like case are derived for the Helmholtz potential $A(M)$, the magnetic field $\frac{h}{M}$, the inverse susceptibility ${\ensuremath{\Gamma}}_{2}$, and the correlation length $\ensuremath{\xi}$ to $O({\ensuremath{\epsilon}}^{2})$.

Physical review. B, Condensed matterJan 1, 198190 citations

Equations of state for bicritical points. III. Cubic anisotropy and tetracriticality

Eytan Domany, Michael Ellis Fisher

Crossover scaling functions for a two-component spin system with quadratic anisotropy $g$ and cubic anisotropy $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}$ are calculated to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ and in $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}$, near the transitions within the ordered phases below the multicritical point ( $T={T}_{m},g=0$ ). Emphasis is placed on the mechanism by which the system is driven to bicritical or tetracritical behavior by the dangerous irrelevant variable $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}$. Explicit expressions and graphs for the various thermodynamic scaling functions are presented.Crossover scaling functions for a two-component spin system with quadratic anisotropy $g$ and cubic anisotropy $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}$ are calculated to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ and in $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}$, near the transitions within the ordered phases below the multicritical point ($T={T}_{m},g=0$). Emphasis is placed on the mechanism by which the system is driven to bicritical or tetracritical behavior by the dangerous irrelevant variable $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}$. Explicit expressions and graphs for the various thermodynamic scaling functions are presented.

Physical review. B, Solid stateApr 1, 197736 citations

g

, and crossover variable $x=\frac{g}{{t}^{\ensuremath{\varphi}}}$, the effective susceptibility exponent ${\ensuremath{\gamma}}_{\mathrm{eff}}(t,x)$ is calculated for two ${\ensuremath{\varphi}}^{4}$-field models, one with a sharp momentum cutoff $\ensuremath{\Lambda}$ and the other with a smooth term ${\ensuremath{\Lambda}}^{\ensuremath{-}2}{({\ensuremath{\nabla}}^{2}\ensuremath{\varphi})}^{2}$. Corrections to the first model due to insertions of composite operators of canonical dimension six, inside the scaling regime, are explicitly calculated and shown to yield small but not negligible contributions to crossover behavior. The role of new, finite renormalizations is discussed and it is shown that either a fairly monotonic ${\ensuremath{\gamma}}_{\mathrm{eff}}(t,x)$ is obtained or a displaced maximum may appear depending on the outcome of a higher-order renormalization-group study, which is not done here.

{T}_{c}

) is not changed by dilution if the critical exponent of the specific heat of the pure system ${\ensuremath{\alpha}}_{\mathrm{pure}<0}$, independent of the value of the concentration.

n

-component spin system are given exactly in the spherical limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ even at first order in perturbation theory (

T>{T}_{c}

). The nature of the nonlinear scaling variables is clarified, and the results are generally expressed in a more compact form. The general

n

-component disordered phase functions are rederived to $O({\ensuremath{\epsilon}}^{2})$, where $\ensuremath{\epsilon}\ensuremath{\equiv}4\ensuremath{-}d$. The cross-over equations for the

n=1

Ising-like case are derived for the Helmholtz potential

A(M)

, the magnetic field

\frac{h}{M}

, the inverse susceptibility ${\ensuremath{\Gamma}}_{2}$, and the correlation length $\ensuremath{\xi}$ to $O({\ensuremath{\epsilon}}^{2})$.

Crossover scaling functions for a two-component spin system with quadratic anisotropy

g

and cubic anisotropy $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}$ are calculated to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ and in $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}$, near the transitions within the ordered phases below the multicritical point (

T={T}_{m},g=0

). Emphasis is placed on the mechanism by which the system is driven to bicritical or tetracritical behavior by the dangerous irrelevant variable $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}$. Explicit expressions and graphs for the various thermodynamic scaling functions are presented.