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arXiv:2009.03965 · Physical review. E

Critical dynamics of the nonconserved strongly anisotropic permutation symmetric three-vector model

Rajiv G. Pereira

Published Dec 29, 2020• 1 citation
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Abstract

We explore, employing the renormalization-group theory, the critical scaling behavior of the permutation symmetric three-vector model that obeys nonconserving dynamics and has a relevant anisotropic perturbation which drives the system into a nonequilibrium steady state. We explicitly find the independent critical exponents with corrections up to two loops. They include the static exponents $\ensuremath{\nu}$ and $\ensuremath{\eta}$, the off equilibrium exponent $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}$, the dynamic exponent $z$, and the strong anisotropy exponent $\mathrm{\ensuremath{\Delta}}$. We also express the other anisotropy exponents in terms of these.

DOI
10.1103/physreve.102.062150

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We explore, employing the renormalization-group theory, the critical scaling behavior of the permutation symmetric three-vector model that obeys nonconserving dynamics and has a relevant anisotropic perturbation which drives the system into a nonequilibrium steady state. We explicitly find the independent critical exponents with corrections up to two loops. They include the static exponents \ensuremathν\ensuremath{\nu}\ensuremathν and \ensuremathη\ensuremath{\eta}\ensuremathη, the off equilibrium exponent \ensuremathη\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}\ensuremathη\ifmmode~\else˜\fi​, the dynamic exponent zzz, and the strong anisotropy exponent \ensuremathΔ\mathrm{\ensuremath{\Delta}}\ensuremathΔ. We also express the other anisotropy exponents in terms of these.
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Dynamical renormalization group analysis of O(n)O(n)O(n) model in steady shear flow
We study the critical behavior of the O(n) O ( n ) model under steady shear flow using a dynamical renormalization group (RG) method. Incorporating the strong anisotropy in scaling ansatz, which has been neglected in earlier RG analyses, we identify a new stable Gaussian fixed point. This fixed point reproduces the anisotropic scaling of static and dynamical critical exponents for both non-conserved (Model A) and conserved (Model B) order parameters. Notably, the upper critical dimensions are d_{\text{up}} = 2 d up = 2 for the non-conserved order parameter (Model A) and d_{\text{up}} = 0 d up = 0 for the conserved order parameter (Model B), implying that the mean-field critical exponents are observed even in both d=2 d = 2 and 3 3 dimensions. Furthermore, the scaling exponent of the order parameter is negative for all dimensions d ≥ 2 d ≥ 2 , indicating that shear flow stabilizes the long-range order associated with continuous symmetry breaking even in d = 2 d = 2 . In other words, the lower critical dimensions are d_{\rm low} < 2 for both types of order parameters. This contrasts with equilibrium systems, where the Hohenberg—Mermin—Wagner theorem prohibits continuous symmetry breaking in d = 2 d = 2 .
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The critical behavior of a semi-infinite nnn-vector model with anisotropic ferromagnetic pair interactions on the surface and isotropic interactions everywhere else is studied with the use of renormalization-group methods for dimension $d=4\ensuremath{-}\ensuremath{\epsilon}$. The effects of surface anisotropies are analyzed near the isotropic ordinary transition (where they produce corrections to scaling), near the isotropic special transition (where they are relevant perturbations), and near a new class of "anisotropic special" transitions. The latter occur if the surface exchange constants associated with me{m}_{e}me​ (easy-magnetization) components take a critical value above which me{m}_{e}me​-component surface order is possible while the interaction constants of the remaining ${m}_{h}=n\ensuremath{-}{m}_{e}$ (hard-magnetization) components are weaker. Of particular interest is the anisotropic special transition with an easy axis (me=1{m}_{e}=1me​=1) because it occurs also in three dimensions, in contrast to the isotropic special one. The associated critical, crossover, and correction-to-scaling exponents are given to second order in $\ensuremath{\epsilon}$.
This article concludes a series of papers [Folk, Holovatch, and Moser, Phys. Rev. E 78, 041124 (2008); 78, 041125 (2008); 79, 031109 (2009)] where the tools of the field theoretical renormalization group were employed to explain and quantitatively describe different types of static and dynamic behavior in the vicinity of multicritical points. Here we give the complete two-loop calculation and analysis of the dynamic renormalization-group flow equations at the multicritical point in anisotropic antiferromagnets in an external magnetic field. We find that the time scales of the order parameters characterizing the parallel and perpendicular ordering with respect to the external field scale in the same way. This holds independent whether the Heisenberg fixed point or the biconical fixed point in statics is the stable one. The nonasymptotic analysis of the dynamic flow equations shows that due to cancellation effects the critical behavior is described, in distances from the critical point accessible to experiments, by the critical behavior qualitatively found in one-loop order. Although one may conclude from the effective dynamic exponents (taking almost their one-loop values) that weak scaling for the order parameter components is valid, the flow of the time-scale ratios is quite different, and they do not reach their asymptotic values.
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