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Physical review. B, Condensed matter

Quantum critical dynamics with mode coupling. I. Ferromagnets

W. L. Ridgway, John Hertz

Published Jul 1, 1980• 2 citations
Publisher

Abstract

We examine the low-temperature critical dynamics of two ferromagnetic models of the disordered state having "mode-mode coupling." A method is developed to incorporate mode-mode coupling contributions in an $O(\ensuremath{\epsilon})$ momentum-space renormalization-group analysis of a quantum system. For each model, both quantum and classical critical regions may exist. The characteristic dimensionality changes from $d=6$ classically to $d=10$ in the quantum limit. The dynamic critical index associated with the quantum fixed point is $z=4\ensuremath{-}\frac{1}{3}(10\ensuremath{-}d)$. The scaling behavior of the spin transport coefficient $\ensuremath{\Gamma}$ is examined in the $T=0$ limit. We discuss the properties of the quantum-to-classical crossover, including the "metastable" nature of the quantum fixed points.

DOI
10.1103/physrevb.22.236

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We examine the low-temperature critical dynamics of two ferromagnetic models of the disordered state having "mode-mode coupling." A method is developed to incorporate mode-mode coupling contributions in an O(\ensuremathϵ)O(\ensuremath{\epsilon})O(\ensuremathϵ) momentum-space renormalization-group analysis of a quantum system. For each model, both quantum and classical critical regions may exist. The characteristic dimensionality changes from d=6d=6d=6 classically to d=10d=10d=10 in the quantum limit. The dynamic critical index associated with the quantum fixed point is z=4\ensuremath−13(10\ensuremath−d)z=4\ensuremath{-}\frac{1}{3}(10\ensuremath{-}d)z=4\ensuremath−31​(10\ensuremath−d). The scaling behavior of the spin transport coefficient \ensuremathΓ\ensuremath{\Gamma}\ensuremathΓ is examined in the T=0T=0T=0 limit. We discuss the properties of the quantum-to-classical crossover, including the "metastable" nature of the quantum fixed points.
We extend a previ\'ous study of quantum critical dynamics in ferromagnets to the isotropic antiferromagnet. The model we consider here also includes so-called "mode-mode coupling" terms. The quantum limit is found to have the characteristic dimensionality dQ=6{d}_{Q}=6dQ​=6 versus the classical value dC=4{d}_{C}=4dC​=4. A renormalization-group expansion to order $\ensuremath{\epsilon}=6\ensuremath{-}d$ is used to evaluate spin and staggered spin transport quantities ${\ensuremath{\Gamma}}_{M}$ and ${\ensuremath{\Gamma}}_{N}$ in the limit of low critical temperature TN{T}_{N}TN​. Dynamic scaling is found to break down in the quantum case and scaling regions of quantum, classical, and "mixed" statistics are discussed in light of this breakdown.
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 discuss renormalization-group and mode-coupling theories of critical dynamics for several models. We show that to lowest order in $\ensuremath{\epsilon}={d}_{c}\ensuremath{-}d$, where for d>dcd>{d}_{c}d>dc​ conventional theory is valid, the renormalizationgroup differential equations are related by a simple transformation to those obtained from mode-coupling theory. Furthermore, the values of the dynamical fixed-point parameters essentially coincide with the critical amplitudes of the diverging transport coefficients, as determined by mode-coupling theory. In addition, we present a correct renormalization-group treatment for dynamical systems with more than one time scale, as for the binary liquid. We stress the fact that there are two distinct classes of critical dynamics, as illustrated by the results for the models studied here. We also discuss the instability of the conventional fixed point (time-dependent Ginzburg-Landau like) against mode-coupling perturbations for d<dcd<{d}_{c}d<dc​, with particular emphasis on a model for superfluid helium.
The low-temperature critical properties and crossovers of a spin-½ planar ferromagnet in a longitudinal magnetic field are explored in terms of an anisotropic bosonic action, suitable to describe the spin model in the low-temperature regime. This is performed adopting a procedure which combines an averaging over dynamic degrees of freedom and the classical Wilson renormalization group transformation. Within this framework we get the phase boundary, ending in a quantum critical point, and general expressions for the correlation length and susceptibility as functions of the temperature and the applied magnetic field within the disordered phase. In particular, two crossovers occur decreasing the temperature with the magnetic field fixed at its quantum critical point value, which might be actually observable in complex magnetic compounds, as suggested by recent experiments.
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}Tc{T}_{c}Tc​) 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.
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