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Physical review. E

Critical dynamics of a nonlocal model and critical behavior of perovskite manganites

Rohit Singh, Kishore Dutta, Malay Kumar Nandy

Published May 18, 2016• 5 citations
Publisher

Abstract

We investigate the nonconserved critical dynamics of a nonlocal model Hamiltonian incorporating screened long-range interactions in the quartic term. Employing dynamic renormalization group analysis at one-loop order, we calculate the dynamic critical exponent $z=2+\ensuremath{\epsilon}{f}_{1}(\ensuremath{\sigma},\ensuremath{\kappa},n)+O({\ensuremath{\epsilon}}^{2})$ and the linewidth exponent $w=\ensuremath{-}\ensuremath{\sigma}+\ensuremath{\epsilon}{f}_{2}(\ensuremath{\sigma},\ensuremath{\kappa},n)+O({\ensuremath{\epsilon}}^{2})$ in the leading order of $\ensuremath{\epsilon}$, where $\ensuremath{\epsilon}=4\ensuremath{-}d+2\ensuremath{\sigma}$, with $d$ the space dimension, $n$ the number of components in the order parameter, and $\ensuremath{\sigma}$ and $\ensuremath{\kappa}$ the parameters coming from the nonlocal interaction term. The resulting values of linewidth exponent $w$ for a wide range of $\ensuremath{\sigma}$ is found to be in good agreement with the existing experimental estimates from spin relaxation measurements in perovskite manganite samples.

DOI
10.1103/physreve.93.052132

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We investigate the nonconserved critical dynamics of a nonlocal model Hamiltonian incorporating screened long-range interactions in the quartic term. Employing dynamic renormalization group analysis at one-loop order, we calculate the dynamic critical exponent z=2+\ensuremathϵf1(\ensuremathσ,\ensuremathκ,n)+O(\ensuremathϵ2)z=2+\ensuremath{\epsilon}{f}_{1}(\ensuremath{\sigma},\ensuremath{\kappa},n)+O({\ensuremath{\epsilon}}^{2})z=2+\ensuremathϵf1​(\ensuremathσ,\ensuremathκ,n)+O(\ensuremathϵ2) and the linewidth exponent w=\ensuremath−\ensuremathσ+\ensuremathϵf2(\ensuremathσ,\ensuremathκ,n)+O(\ensuremathϵ2)w=\ensuremath{-}\ensuremath{\sigma}+\ensuremath{\epsilon}{f}_{2}(\ensuremath{\sigma},\ensuremath{\kappa},n)+O({\ensuremath{\epsilon}}^{2})w=\ensuremath−\ensuremathσ+\ensuremathϵf2​(\ensuremathσ,\ensuremathκ,n)+O(\ensuremathϵ2) in the leading order of \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ, where \ensuremathϵ=4\ensuremath−d+2\ensuremathσ\ensuremath{\epsilon}=4\ensuremath{-}d+2\ensuremath{\sigma}\ensuremathϵ=4\ensuremath−d+2\ensuremathσ, with ddd the space dimension, nnn the number of components in the order parameter, and \ensuremathσ\ensuremath{\sigma}\ensuremathσ and \ensuremathκ\ensuremath{\kappa}\ensuremathκ the parameters coming from the nonlocal interaction term. The resulting values of linewidth exponent www for a wide range of \ensuremathσ\ensuremath{\sigma}\ensuremathσ is found to be in good agreement with the existing experimental estimates from spin relaxation measurements in perovskite manganite samples.
Abstract The dynamic critical behavior at the paramagnetic-antiferromagnetic (PM-AFM) transition in manganites has recently been studied experimentally (Niermann et al 2015 Phys. Rev. Lett. 114 037204). We extend the Hamiltonian of paper I by incorporating an energy field, and study the corresponding Model C of critical dynamics. We use the dynamic renormalization group approach and calculate the dynamic critical exponents z, ν z and the line-width exponent Δ to leading order in the small expansion parameters ϵ = 4 − d + 2 σ and ϵ ′ = 4 − d . Here, d is the space dimension and σ is the long-range exponent. Using σ as an adjustable parameter, the theory gives us a good match to the experimentally available static and dynamic critical exponents at the PM-AFM transition.
Employing Wilson's renormalization group scheme, we investigate the critical behaviour of a modified Ginzburg-Landau model with a nonlocal mode-coupling interaction in the quartic term. Carrying out the calculations at one-loop order, we obtain the critical exponents in the leading order of , where ρ is an exponent occurring in the nonlocal interaction term and d is the space dimension. Interestingly, the correlation exponent η is found to be non-zero at one-loop order and the expansion corresponds to an expansion about the tricritical mean-field theory in three dimensions, unlike the conventional theory. The ensuing critical exponents are in good agreement with experimental values for samples close to tricriticality. Our analysis indicates that tricriticality is a feature only in three dimensions.
We investigate the critical dynamics of the time-dependent Landau-Ginzburg model with nonconserved n-component order parameter (model A) in the presence of long-range correlated quenched impurities. We use a special kind of long-range correlations $g(|r\ensuremath{-}{r}^{\ensuremath{'}}|)\ensuremath{\sim}|r\ensuremath{-}{r}^{\ensuremath{'}}{|}^{\ensuremath{-}a},$ $a=4\ensuremath{-}\ensuremath{\delta},$ previously introduced by Weinrib and Halperin. The parameter $\ensuremath{\delta}\ensuremath{\sim}\ensuremath{\epsilon},$ where $\ensuremath{\epsilon}=4\ensuremath{-}d$ is the usual small parameter within the Wilson-Fisher renormalization group theory. Using a double expansion in $\ensuremath{\epsilon}$ and $\ensuremath{\delta}$ we calculate the critical exponent zzz up to second order on the small parameters. We show that quenched impurities of this kind affect the critical dynamics already in first order of $\ensuremath{\epsilon}$ and $\ensuremath{\delta},$ leading to a relevant correction for the mean field value of the exponent z.z.z.
The relaxational models introduced in a previous treatment of critical dynamics are studied in detail using renormalization-group methods. The earlier results are justified by an analysis to all order in $\ensuremath{\epsilon}=4\ensuremath{-}d$, where ddd is the dimensionality. The diagrammatic formalism of the full dynamic renormalization group is presented, and applied to the earlier models. A generalization of Wilson's Feynman-graph expansion method is used to calculate the exponents to second order in $\ensuremath{\epsilon}$. In model CCC, where a nonconserved order parameter is coupled to a conserved energy field, ambiguities were found in the earlier recursion-relation treatment for 2<n<42<n<42<n<4, $d\ensuremath{\approx}4$ (nnn is the number of components of the order parameter). These ambiguities are discussed in the present work, but are not fully resolved.
A generalization of the Ising model is solved, qualitatively, for its critical behavior. In the generalization the spin ${S}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{n}}}$ at a lattice site $\stackrel{\ensuremath{\rightarrow}}{\mathrm{n}}$ can take on any value from $\ensuremath{-}\ensuremath{\infty} \mathrm{to} \ensuremath{\infty}$. The interaction contains a quartic term in order not to be pure Gaussian. The interaction is investigated by making a change of variable ${S}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{n}}}=\ensuremath{\Sigma}{m}^{}{\ensuremath{\psi}}_{m}(\mathrm{n}){S}_{m}^{\ensuremath{'}}$, where the functions ${\ensuremath{\psi}}_{m}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{n}})$ are localized wavepacket functions. There are a set of orthogonal wave-packet functions for each order-of-magnitude range of the momentum $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$. An effective interaction is defined by integrating out the wave-packet variables with momentum of order 1, leaving unintegrated the variables with momentum 0.5. Then the variables with momentum between 0.25 and 0.5 are integrated, etc. The integrals are computed qualitatively. The result is to give a recursion formula for a sequence of effective Landau-Ginsberg-type interactions. Solution of the recursion formula gives the following exponents: $\ensuremath{\eta}=0$, $\ensuremath{\gamma}=1.22$, $\ensuremath{\nu}=0.61$ for three dimensions. In five dimensions or higher one gets $\ensuremath{\eta}=0$, $\ensuremath{\gamma}=1$, and $\ensuremath{\nu}=\frac{1}{2}$, as in the Gaussian model (at least for a small quartic term). Small corrections neglected in the analysis may make changes (probably small) in the exponents for three dimensions.
Ferromagnets with the number of spin components n>2n>2n>2 and with power-law interactions ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}}$ are studied in dimensions $d=2+\ensuremath{\epsilon}$ for small $\ensuremath{\epsilon}$. The following theorem is proved: Let ${\ensuremath{\eta}}_{\mathrm{SR}}$ be the value of the critical exponent $\ensuremath{\eta}$ for short-range force (e.g., nearest-neighbor exchange). Then in the presence of the long-range power-law interaction the critical behavior is the same as for short-range interactions provided that the inequality $2\ensuremath{-}\ensuremath{\sigma}<{\ensuremath{\eta}}_{\mathrm{SR}}$ is satisfied. In this case the critical exponents depend on ddd and nnn but not on $\ensuremath{\sigma}$. In the opposite case, the value of $\ensuremath{\eta}$ is ${\ensuremath{\eta}}_{\mathrm{LR}}=2\ensuremath{-}\ensuremath{\sigma}$ and the critical exponent $\ensuremath{\nu}$ depends on $\ensuremath{\sigma}$ in addition to ddd and nnn.