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Physical review. B, Solid state

Critical dynamics of ferromagnets in 6−ε dimensions: General discussion and detailed calculation

Shang-keng Ma, Gene F. Mazenko

Published Jun 1, 1975• 324 citations
Publisher

Abstract

We report in detail a study of the critical dynamics of a model ferromagnet in $6\ensuremath{-}\ensuremath{\epsilon}$ dimensions both above and below ${T}_{c}$, for small $\ensuremath{\epsilon}$. The precession of spins in the local magnetic field plays a major role in the dynamics. Its effect can be ignored at 6 or more dimensions, as many authors have previously observed. At a dimension slightly less than 6, the effect is nontrivial but tractable by perturbation theory. We carry out a renormalization-group analysis and determine the dynamic exponent $z=4\ensuremath{-}\frac{\ensuremath{\epsilon}}{2}=1+\frac{d}{2}$, accurate to $O(\ensuremath{\epsilon})$. We also solve the equation of motion to obtain the spin response function. The frequency and damping of spin waves below ${T}_{c}$ are determined. Forms of equations of motion and the extent of universality in critical dynamics are discussed.

DOI
10.1103/physrevb.11.4077

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Critical Dynamics of Ferromagnets in 6−ε Dimensions

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Physical review. B, Solid state•Mar 1, 1976•24 citations
We report in detail a study of the critical dynamics of a model ferromagnet in 6\ensuremath−\ensuremathϵ6\ensuremath{-}\ensuremath{\epsilon}6\ensuremath−\ensuremathϵ dimensions both above and below Tc{T}_{c}Tc​, for small \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ. The precession of spins in the local magnetic field plays a major role in the dynamics. Its effect can be ignored at 6 or more dimensions, as many authors have previously observed. At a dimension slightly less than 6, the effect is nontrivial but tractable by perturbation theory. We carry out a renormalization-group analysis and determine the dynamic exponent z=4\ensuremath−\ensuremathϵ2=1+d2z=4\ensuremath{-}\frac{\ensuremath{\epsilon}}{2}=1+\frac{d}{2}z=4\ensuremath−2\ensuremathϵ​=1+2d​, accurate to O(\ensuremathϵ)O(\ensuremath{\epsilon})O(\ensuremathϵ). We also solve the equation of motion to obtain the spin response function. The frequency and damping of spin waves below Tc{T}_{c}Tc​ are determined. Forms of equations of motion and the extent of universality in critical dynamics are discussed.
The critical dynamics of a model isotropic ferromagnet is studied in $6\ensuremath{-}\ensuremath{\epsilon}$ dimensions both above and below Tc{T}_{c}Tc​. The renormalization group and the characteristic frequency exponent zzz are determined to order $\ensuremath{\epsilon}$. Implications of the renormalization group are discussed. A direct calculation of the dynamic response function to order $\ensuremath{\epsilon}$ shows many interesting features, especially for T<TcT<{T}_{c}T<Tc​.
We calculate the dynamic structure factor $S(\mathbit{k},\ensuremath{\omega})$ in the paramagnetic regime of quantum Heisenberg ferromagnets for temperatures TTT close to the critical temperature Tc{T}_{c}Tc​ using our recently developed functional renormalization group approach to quantum spin systems. In d=3d=3d=3 dimensions, we find that for small momenta $\mathbit{k}$ and frequencies $\ensuremath{\omega}$ the dynamic structure factor assumes the scaling form $S(\mathbit{k},\ensuremath{\omega})=(\ensuremath{\tau}TG(\mathbit{k})/\ensuremath{\pi})\mathrm{\ensuremath{\Phi}}(k\ensuremath{\xi},\ensuremath{\omega}\ensuremath{\tau})$, where $G(\mathbit{k})$ is the static spin-spin correlation function, $\ensuremath{\xi}$ is the correlation length, and the characteristic timescale $\ensuremath{\tau}$ is proportional to ${\ensuremath{\xi}}^{5/2}$. We explicitly calculate the dynamic scaling function $\mathrm{\ensuremath{\Phi}}(x,y)$ and find satisfactory agreement with neutron scattering experiments probing the critical spin dynamics in EuO and EuS. Precisely at the critical point where $\ensuremath{\xi}=\ensuremath{\infty}$, our result for the dynamic structure factor can be written as $S(\mathbit{k},\ensuremath{\omega})={(\ensuremath{\pi}{\ensuremath{\omega}}_{k})}^{\ensuremath{-}1}{T}_{c}G(\mathbit{k}){\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\omega}/{\ensuremath{\omega}}_{k})$, where ${\ensuremath{\omega}}_{k}\ensuremath{\propto}{k}^{5/2}$. We find that ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$ vanishes as ${\ensuremath{\nu}}^{\ensuremath{-}13/5}$ for large $\ensuremath{\nu}$, and as ${\ensuremath{\nu}}^{3/5}$ for small $\ensuremath{\nu}$. While the large-frequency behavior of ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$ is consistent with calculations based on mode-coupling theory and with perturbative renormalization group calculations to second order in $\ensuremath{\epsilon}=6\ensuremath{-}d$, our result for small frequencies disagrees with previous calculations. We argue that until now, neither experiments nor numerical simulations have been sufficiently accurate to determine the low-frequency behavior of ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$. We also calculate the low-temperature behavior of $S(\mathbit{k},\ensuremath{\omega})$ in one- and two-dimensional ferromagnets and find that it satisfies dynamic scaling with exponent z=2z=2z=2 and exhibits a pseudogap for small frequencies.
We study a model equation of motion describing the dynamics of isotropic antiferromagnets near the critical point. We perform a renormalization-group analysis of this equation of motion correct to $O(\ensuremath{\epsilon})$, where $\ensuremath{\epsilon}=4\ensuremath{-}d$, finding recursion relations and analyzing their fixed points and their stability. For the one stable fixed point we obtain the dynamical index $z=\frac{d}{2}=2\ensuremath{-}\frac{\ensuremath{\epsilon}}{2}$. We also calculate the response and correlation functions in the scaling region using perturbation theory correct to $O(\ensuremath{\epsilon})$. Our calculation gives analytical forms for the correlation functions for the staggered magnetization and the magnetization. These correlation functions can be written in the dynamical scaling form. The shape function that characterizes the frequency spectrum for the staggered magnetization is Lorentzian in the hydrodynamical regime, but shows fluctuation induced peaks at and near the N\ifmmode\dagger\else\textdagger\fi{}eel temperature. The shape function for the magnetization is essentially Lorentzian for all $x=q\ensuremath{\xi}$ (where qqq is the wave number and $\ensuremath{\xi}$ is the correlation length) and shows a narrow lydrodynamical width as $x\ensuremath{\rightarrow}0$.
Critical behaviors of a ferromagnetic spin system under quenched random magnetic fields are investigated near above four space dimensions. Critical temperature Tc(m, µ, ε) as a function of the number of spin components m(≥2) and strength µ of the random field is obtained in a power series of ε ≡ d-4 (>0) by means of the Migdal-Polyakov renormalization group theory. Critical exponents ν and η are also calculated in powers of ε. It is shown that they are in agreement with those of (2 + ε)-dimensional pure system for sufficiently close to Tc. In the limit m→∞, present results reduce to d = 4 + ε case of our previous results for the spherical model.
The zero-field critical behavior of ferromagnets above Tc{T}_{c}Tc​, with both isotropic exchange coupling and dipolar interactions, is studied by renormalization-group techniques in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions with n(=d)n (=d)n(=d) component spins. The critical exponents are calculated to order ${\ensuremath{\epsilon}}^{2}$, and are found to be numerically very close to their nondipolar analogs. In particular the correlation length exponent is given by $2\ensuremath{\nu}=1+(\frac{9}{34})\ensuremath{\epsilon}+(\frac{7013}{58956}){\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$. The value of the specific-heat exponent ${\ensuremath{\alpha}}_{s} (=2\ensuremath{-}d\ensuremath{\nu})$ seems inconsistent with experimental data (at d=3d=3d=3, $\ensuremath{\epsilon}=1$). The critical scattering function is shown to have the form ${\ensuremath{\Gamma}}^{\ensuremath{\alpha}\ensuremath{\beta}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})=C{t}^{\ensuremath{-}\ensuremath{\gamma}}\stackrel{^}{D}({\ensuremath{\xi}}^{2}{q}^{2})({\ensuremath{\delta}}_{\ensuremath{\alpha}\ensuremath{\beta}}\ensuremath{-}\frac{{q}^{\ensuremath{\alpha}}{q}^{\ensuremath{\beta}}}{{q}^{2}})$ to all orders in $\ensuremath{\epsilon}$. The exponents related to the leading correction to scaling, to crossovers to anisotropic exchange behavior, and to cubic dipolar behavior, are also calculated, and are found to differ significantly from their nondipolar counterparts.
The renormalization-group equations for critical behavior in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions are generalized to the discussion of dynamic exponents at T=TcT={T}_{c}T=Tc​ and then applied to a situation recently considered by Hohenberg, Halperin, and Ma (HHM); namely, one in which an nnn-component primary field $\ensuremath{\psi}$ is coupled to a secondary field $\ensuremath{\phi}$ via a ${\ensuremath{\lambda}}_{0}(\ensuremath{\psi}\ensuremath{\psi})\ensuremath{\phi}$ coupling. In this work their analysis of relaxational fields is explicitly considered at order ${\ensuremath{\epsilon}}^{2}$. For n<4n<4n<4, we find the dynamic exponents differ in one case at $O({\ensuremath{\epsilon}}^{2})$ from that given by HHM. In contrast to HHM, the dynamic scaling postulate is also preserved by the second-order fixed points in all instances and detailed analysis is given. The fixed-point values of RRR, a dimensionless ratio of relaxation rates for the $\ensuremath{\phi}$ and $\ensuremath{\psi}$ fields, in certain cases, is found to go as ${R}^{*}\ensuremath{\sim}\frac{1}{\ensuremath{\epsilon}}$ which is a new result. This large value where it occurs is found essential to understanding dynamic scaling. Analysis of the case of a propagating conserved $\ensuremath{\phi}$ field also is given. It is found that its dynamic critical properties can be characterized by its damping term alone.