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Physical review. B, Solid state

Critical dynamics of isotropic antiferromagnets using renormalization-group methods: T≥TN

R. Feeedman, Gene F. Mazenko

Published Jun 1, 1976• 48 citations
Publisher

Abstract

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 $q$ is the wave number and $\ensuremath{\xi}$ is the correlation length) and shows a narrow lydrodynamical width as $x\ensuremath{\rightarrow}0$.

DOI
10.1103/physrevb.13.4967

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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(\ensuremathϵ)O(\ensuremath{\epsilon})O(\ensuremathϵ), where \ensuremathϵ=4\ensuremath−d\ensuremath{\epsilon}=4\ensuremath{-}d\ensuremathϵ=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=d2=2\ensuremath−\ensuremathϵ2z=\frac{d}{2}=2\ensuremath{-}\frac{\ensuremath{\epsilon}}{2}z=2d​=2\ensuremath−2\ensuremathϵ​. We also calculate the response and correlation functions in the scaling region using perturbation theory correct to O(\ensuremathϵ)O(\ensuremath{\epsilon})O(\ensuremathϵ). 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\ensuremathξx=q\ensuremath{\xi}x=q\ensuremathξ (where qqq is the wave number and \ensuremathξ\ensuremath{\xi}\ensuremathξ is the correlation length) and shows a narrow lydrodynamical width as x\ensuremath→0x\ensuremath{\rightarrow}0x\ensuremath→0.
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 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.
Equipped with the field theory representation of non-linear Langevin equations, the tools of dynamic perturbation theory, and the dynamic renormalization group introduced in Chapters 4 and 5, we are now in the position to revisit models for dynamic critical behavior that entail reversible mode couplings and other conserved hydrodynamic modes. We have already encountered some of these in Section 3.3. In models C and D, respectively, a non-conserved or conserved n -component order parameter is coupled to a conserved scalar field, the energy density. Through a systematic renormalization group analysis, we may critically assess the earlier predictions from scaling theory, and discuss the stability of fixed points characterized by strong dynamic scaling, wherein the order parameter and conserved non-critical mode fluctuate with equal rates, and weak dynamic scaling regimes, where these characteristic time scales differ. Next we investigate isotropic ferromagnets (model J), with the conserved spin density subject to reversible precession in addition to diffusive relaxation. Exploiting rotational invariance, we can now firmly establish the scaling relation z = ( d + 2 − η )/2. Similar symmetry arguments yield a scaling relation for the dynamic exponents associated with the order parameter and the non-critical fields in the O ( n )-symmetric SSS model that encompasses model E for planar ferromagnets and superfluid helium 4 (for n = 2), and model G for isotropic antiferromagnets ( n = 3). There exist competing strong- and weak-scaling fixed points, with the former stable to one-loop order, and characterized by z = d /2 for all slow modes.
We study the near-equilibrium critical dynamics of the O(3) nonlinear sigma model describing isotropic antiferromagnets with a nonconserved order parameter reversibly coupled to the conserved total magnetization. To calculate response and correlation functions, we set up a description in terms of Langevin stochastic equations of motion, and their corresponding Janssen-De Dominicis response functional. We find that in equilibrium, the dynamics is well-separated from the statics, at least to one-loop order in a perturbative treatment with respect to the static and dynamical nonlinearities. Since the static nonlinear sigma model must be analyzed in a dimensional d=2+ɛ expansion about its lower critical dimension d_{lc}=2, whereas the dynamical mode-coupling terms are governed by the upper critical dimension d_{c}=4, a simultaneous perturbative dimensional expansion is not feasible, and the reversible critical dynamics for this model cannot be accessed at the static critical renormalization group fixed point. However, in the coexistence limit addressing the long-wavelength properties of the low-temperature ordered phase, we can perform an ε=4-d expansion near d_{c}. This yields anomalous scaling features induced by the massless Goldstone modes, namely subdiffusive relaxation for the conserved magnetization density with asymptotic scaling exponent z_{Γ}=d-2, which may be observable in neutron scattering experiments. Intriguingly, if initialized near the critical point, the renormalization group flow for the effective dynamical exponents recovers their universal critical values z_{c}=d/2 in an intermediate crossover region.
In this central and essential chapter, we develop the dynamic renormalization group approach to time-dependent critical phenomena. Again, we base our exposition on the simple O ( n )-symmetric relaxational models A and B; the generalization to other dynamical systems is straightforward. We begin with an analysis of the infrared and ultraviolet singularities appearing in the dynamic perturbation expansion. Although we are ultimately interested in the infrared critical region, we first take care of the ultraviolet divergences. Below and at the critical dimension d c = 4, only a finite set of Feynman diagrams carries ultraviolet singularities, which we evaluate by means of the dimensional regularization prescription, and then eliminate via multiplicative as well as additive renormalization (the latter takes into account the fluctuation-induced shift of the critical temperature). The renormalization group equation then permits us to explore the ensuing scaling behavior of the correlation and vertex functions of the renormalized theory upon varying the arbitrary renormalization scale. Of fundamental importance is the identification of an infrared-stable renormalization group fixed point, which describes scale invariance and hence allows the derivation of the critical power laws in the infrared limit from the renormalization constants determined in the ultraviolet regime. This program is explicitly carried through for the relaxational models A and B. For model B, we derive a scaling relation connecting the dynamic exponent to the Fisher exponent η . The critical exponents ν , η , and z are computed to first non-trivial order in an ∊ expansion around the upper critical dimension d c = 4.
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.