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Physical Review Letters

Critical Dynamics of Ferromagnets in 6−ε Dimensions

Shang-keng Ma, Gene F. Mazenko

Published Dec 2, 1974• 32 citations
Publisher

Abstract

The critical dynamics of a model isotropic ferromagnet is studied in $6\ensuremath{-}\ensuremath{\epsilon}$ dimensions both above and below ${T}_{c}$. The renormalization group and the characteristic frequency exponent $z$ 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 $T<{T}_{c}$.

DOI
10.1103/physrevlett.33.1383

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The critical dynamics of a model isotropic ferromagnet is studied in 6\ensuremath−\ensuremathϵ6\ensuremath{-}\ensuremath{\epsilon}6\ensuremath−\ensuremathϵ dimensions both above and below Tc{T}_{c}Tc​. The renormalization group and the characteristic frequency exponent zzz are determined to order \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ. Implications of the renormalization group are discussed. A direct calculation of the dynamic response function to order \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ shows many interesting features, especially for T<TcT<{T}_{c}T<Tc​.
We report in detail a study of the critical dynamics of a model ferromagnet in $6\ensuremath{-}\ensuremath{\epsilon}$ dimensions both above and below Tc{T}_{c}Tc​, 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 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 antiferromagnet is studied in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions above and at TN{T}_{\mathrm{N}}TN​. The renormalization-group, characteristic-frequency exponent, and shape function are determined to order $\ensuremath{\epsilon}$. In particular we find fluctuation-induced peaks in the shape function at and above the N\'eel temperature. We find results in agreement with dynamical scaling.
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.
We analyze the critical behavior of isotropic ferromagnets with a free surface by means of field-theoretic renormalization methods in d=2+\ensuremath{\epsilon} dimensions. Renormalization-group equations for surface quantities are derived which imply scaling to all orders in \ensuremath{\epsilon}. Explicit results to order ${\mathrm{\ensuremath{\epsilon}}}^{2}$ are presented for the critical surface exponents of the ordinary transition. Interpolation with the known results near four dimensions yields estimates for d=3 which seem to be remarkably accurate.
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.