Physical review. B, Condensed matter

Renormalization-group calculation of the relationship of the susceptibility and correlation length: The anomalous-dimension crossover function

Jeffrey F. Nicoll, T. S. Chang

Published Dec 1, 1978• 9 citations

Abstract

We give an exact relationship between the susceptibility and correlation length (which includes the crossover to classical behavior) in a differential renormalization-group approach. We calculate the relationship explicitly for arbitrary-order-$\ensuremath{\theta}$ Lifshitz systems to leading order. Global scaling properties of the $p$-point Green's functions are given in the differential framework.

DOI: 10.1103/physrevb.18.6324

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Crossover relationship between the magnetic susceptibility and the correlation length

Jeffrey F. Nicoll, T. S. Chang

Journal of Applied PhysicsMar 1, 19791 citation

Two-point correlation function near the coexistence surface

Jeffrey F. Nicoll

Physical review. B, Condensed matterFeb 1, 19809 citations

Crossover scaling: A renormalization group approach

Denjoe O’Connor, Christopher Rhodes Stephens

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

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

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Crossover functions by renormalization-group matching: Three-loop results

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Physical review. B, Condensed matterApr 1, 1985147 citations

Physical review. B, Condensed matter

Renormalization-group calculation of the relationship of the susceptibility and correlation length: The anomalous-dimension crossover function

Jeffrey F. Nicoll, T. S. Chang

Published Dec 1, 1978• 9 citations

Publisher

Abstract

\ensuremath{\theta}

Lifshitz systems to leading order. Global scaling properties of the

p

-point Green's functions are given in the differential framework.

DOI: 10.1103/physrevb.18.6324

Related papers

Crossover relationship between the magnetic susceptibility and the correlation length

Jeffrey F. Nicoll, T. S. Chang

In the asymptotic critical region of a ferromagnetic spin system, the magnetic susceptibility χ and the correlation length ξ are related by the power law: χ∼ξ2−η, where η is the correlation function exponent. Somewhat farther from the critical point, but still within the critical region, we expect the behavior to be mean-field in character: χ∼ξ2. We give an exact relationship between χ and ξ in terms of a differential renormalization group approach. The leading behavior in the disordered phase is explicitly calculated for the n-component Wilson-Fisher model to O(ε2), ε≡4−d. The result includes both limiting power laws in a smooth crossover function. For the Ising case, n=1, the relationship is extended to the entire critical region, including the ordered phase. In this case, even in the asymptotic limit, the susceptibility differs from ξ2−η by a factor which depends on the magnetization M. This function is calculated to leading order and correctly gives the associated amplitude ratios to O(ε). Applications to the general p-point vertex function are discussed.In the asymptotic critical region of a ferromagnetic spin system, the magnetic susceptibility χ and the correlation length ξ are related by the power law: χ∼ξ2−η, where η is the correlation function exponent. Somewhat farther from the critical point, but still within the critical region, we expect the behavior to be mean-field in character: χ∼ξ2. We give an exact relationship between χ and ξ in terms of a differential renormalization group approach. The leading behavior in the disordered phase is explicitly calculated for the n-component Wilson-Fisher model to O(ε2), ε≡4−d. The result includes both limiting power laws in a smooth crossover function. For the Ising case, n=1, the relationship is extended to the entire critical region, including the ordered phase. In this case, even in the asymptotic limit, the susceptibility differs from ξ2−η by a factor which depends on the magnetization M. This function is calculated to leading order and correctly gives the associated amplitude ratios to O(ε). Applications to the general p-point vertex function are discussed.

Journal of Applied PhysicsMar 1, 19791 citation

Two-point correlation function near the coexistence surface

Jeffrey F. Nicoll

Differential renormalization-group equations for the one-particle-irreducible vertex functions are solved analytically to provide information about the scaling properties of the correlation function at small and moderate wave vectors everywhere in the critical region. For Ising-like spin systems, this extends previous work on the anomalous dimension crossover function to nonzero magnetization. The susceptibility-correlation-length relative-amplitude ratio is correctly given to $O(\ensuremath{\epsilon})$, $\ensuremath{\epsilon}=4\ensuremath{-}d$; moreover, a fully nonlinear crossover-amplitude function is obtained. For general $n$ -component spins, the longitudinal part of the correlation function, ${\ensuremath{\Gamma}}_{2,L}$ is calculated to $O(\ensuremath{\epsilon})$ in crossover form for small and moderate $k$ . The asymptotic singularity ${\ensuremath{\Gamma}}_{2,L}\ensuremath{\sim}{k}^{\ensuremath{\epsilon}}$ is automatically incorporated. The result is exact in the spherical limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$.Differential renormalization-group equations for the one-particle-irreducible vertex functions are solved analytically to provide information about the scaling properties of the correlation function at small and moderate wave vectors everywhere in the critical region. For Ising-like spin systems, this extends previous work on the anomalous dimension crossover function to nonzero magnetization. The susceptibility-correlation-length relative-amplitude ratio is correctly given to $O(\ensuremath{\epsilon})$, $\ensuremath{\epsilon}=4\ensuremath{-}d$; moreover, a fully nonlinear crossover-amplitude function is obtained. For general $n$-component spins, the longitudinal part of the correlation function, ${\ensuremath{\Gamma}}_{2,L}$ is calculated to $O(\ensuremath{\epsilon})$ in crossover form for small and moderate $k$. The asymptotic singularity ${\ensuremath{\Gamma}}_{2,L}\ensuremath{\sim}{k}^{\ensuremath{\epsilon}}$ is automatically incorporated. The result is exact in the spherical limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$.

Physical review. B, Condensed matterFeb 1, 19809 citations

Crossover scaling: A renormalization group approach

Denjoe O’Connor, Christopher Rhodes Stephens

Abstract We derive a theory of crossover scaling based on a scaling variable gξg, where g is the anisotropy parameter inducing the crossover and ξg is the correlation length in the presence of g. Our considerations are field theoretic and based on a renormalization group with a g dependent differential generator that interpolates between qualitatively different degrees of freedom. ξg is a nonlinear scaling field for this renormalization group and interpolates between (T – Tc(g))–v0 and (T – Tc(g))–v∞ (v0 and v∞ being the isotropic and anisotropic exponents respectively). By expanding about a ‘floating’ fixed point we can make corrections to scaling small throughout the crossover. In this formulation effective scaling exponents obey standard scaling laws, e. g. γeff = veff(2 – ɳeff). We discuss its advantages giving for various crossovers explicit supporting perturbative calculations of the susceptibility, which is found to conform to the general form derived from the g dependent renormalization group.Abstract We derive a theory of crossover scaling based on a scaling variable gξg, where g is the anisotropy parameter inducing the crossover and ξg is the correlation length in the presence of g. Our considerations are field theoretic and based on a renormalization group with a g dependent differential generator that interpolates between qualitatively different degrees of freedom. ξg is a nonlinear scaling field for this renormalization group and interpolates between (T – Tc(g))–v0 and (T – Tc(g))–v∞ (v0 and v∞ being the isotropic and anisotropic exponents respectively). By expanding about a ‘floating’ fixed point we can make corrections to scaling small throughout the crossover. In this formulation effective scaling exponents obey standard scaling laws, e. g. γeff = veff(2 – ɳeff). We discuss its advantages giving for various crossovers explicit supporting perturbative calculations of the susceptibility, which is found to conform to the general form derived from the g dependent renormalization group.

Proceedings of the Royal Society of London Series A Mathematical and Physical SciencesFeb 8, 19948 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

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 functions by renormalization-group matching: Three-loop results

Jeffrey F. Nicoll, P. C. Albright

Crossover functions are obtained to three-loop order in an expansion around dimension d=4 for the ${\ensuremath{\varphi}}^{4}$ theory. Expressions for the internal energy, susceptibility, and correlation length are given for the general n-component system in zero magnetic field and above the critical temperature. The full Helmholtz free energy, magnetization equation of state, and susceptibility are given for the Ising-like n=1 case. Corrected values for the critical amplitude ratios as well as for the correction-to-scaling amplitude ratios are given to three-loop order.Crossover functions are obtained to three-loop order in an expansion around dimension d=4 for the ${\ensuremath{\varphi}}^{4}$ theory. Expressions for the internal energy, susceptibility, and correlation length are given for the general n-component system in zero magnetic field and above the critical temperature. The full Helmholtz free energy, magnetization equation of state, and susceptibility are given for the Ising-like n=1 case. Corrected values for the critical amplitude ratios as well as for the correction-to-scaling amplitude ratios are given to three-loop order.

Physical review. B, Condensed matterApr 1, 1985147 citations

\ensuremath{\theta}

Lifshitz systems to leading order. Global scaling properties of the

p

-point Green's functions are given in the differential framework.

n

-component spins, the longitudinal part of the correlation function, ${\ensuremath{\Gamma}}_{2,L}$ is calculated to $O(\ensuremath{\epsilon})$ in crossover form for small and moderate

k

. The asymptotic singularity ${\ensuremath{\Gamma}}_{2,L}\ensuremath{\sim}{k}^{\ensuremath{\epsilon}}$ is automatically incorporated. The result is exact in the spherical limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$.

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 $O({\ensuremath{\epsilon}}^{2})$.

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.