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Europhysics Letters (EPL)

Nonlocal mode-coupling interactions and phase transition near tricriticality

Rohit Kumar Singh, Kishore Dutta, Malay Kumar Nandy

Published Apr 1, 2015• 8 citations
Publisher

Abstract

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.

DOI
10.1209/0295-5075/110/16003

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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.
A renormalization-group scheme is developed for the three-dimensional O(2N)O(2N)O(2N)-symmetric Ginzburg-Landau-Wilson model, which is consistent with the use of a 1/N1/N1/N expansion as a systematic method of approximation. It is motivated by an application to the critical properties of superconductors, reported in a separate paper. Within this scheme, the infrared stable fixed point controlling critical behavior appears at z=0,z=0,z=0, where $z={\ensuremath{\lambda}}^{\ensuremath{-}1}$ is the inverse of the quartic coupling constant, and an efficient renormalization procedure consists of the minimal subtraction of ultraviolet divergences at z=0.z=0.z=0. This scheme is implemented at next-to-leading order, and the standard results for critical exponents calculated by other means are recovered. An apparently novel result of this nonperturbative method of approximation is that corrections to scaling (or confluent singularities) do not, as in perturbative analyses, appear as simple power series in the variable ${y=zt}^{\ensuremath{\omega}\ensuremath{\nu}}.$ At least in three dimensions, the power series are modified by powers of lny.\mathrm{ln}y.lny.
The phase diagrams of a variety of physical systems, among them certain antiferromagnets and ternary fluid mixtures, may include a critical line intersecting a line of first-order transitions at a critical end point rather than at the familiar tricritical point. At the critical end point the order parameter has critical fluctuations in the paramagnetic phase and a finite jump. Within Landau theory this phenomenon appears with the introduction of a negative ${\ensuremath{\varphi}}^{6}$ coupling; here we extend that theory to include fluctuations by a momentum-shell renormalization-group analysis close to four dimensions. In previous real-space analyses the critical end point was associated with a new fixed point. In contrast, the Wilson-Fisher fixed point which characterizes the entire critical line describes the critical end point here. We derive the $O(\ensuremath{\epsilon})$ corrections to the mean-field exponents which, in Landau theory, describe the approach to the critical end point from high temperatures. At the same time, the discontinuity in $〈\ensuremath{\varphi}〉$ at the critical end point predicted by Landau theory is essentially unaltered by fluctuations. It emerges from a renormalization-group calculation of the Gibbs free energy for the Ginzburg-Landau-Wilson model with a negative ${\ensuremath{\varphi}}^{6}$ term and a positive ${\ensuremath{\varphi}}^{8}$ term.
A renormalization-group analysis of critical behavior near a Lifshitz tricritical point (LTP) is presented, with emphasis on the role played by new, momentum-dependent quartic terms. These result in new stable fixed points which determine the critical behavior. For some values of n (the number of order-parameter components) and m (the dimensionality of the ``soft'' subspace, characterized by quartic momentum-dependent inverse correlation functions), the renormalization-group recursion relations have two stable and accessible fixed points. However, one of these can never be reached in practice, due to a thermodynamic instability which results in a first-order phase transition. For m=d-1, one of the fixed points describes the critical dynamics of the usual n-vector spin model in (d-1) dimensions. This dynamic fixed point also characterizes LTP behavior for large n and n=1. In all other cases, the LTP has new exponents, which are not related to the dynamic model. Our results may be relevant to Lifshitz tricritical behavior in RbCaF3{\mathrm{RbCaF}}_{3}RbCaF3​ and in some liquid-crystal systems.
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 ddd the space dimension, nnn 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 www 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.
The critical behavior of an MN-component order parameter Ginzburg-Landau model with isotropic and cubic interactions describing antiferromagnetic and structural phase transitions in certain crystals with complicated ordering is studied in the framework of the four-loop renormalization group (RG) approach in $(4-2\ve)$-dimensions. Using dimensional regularization and the minimal subtraction scheme, the perturbative expansions for RG functions are deduced for generic M and N and resummed by the Borel transformation combined with a conformal mapping. Investigation of the global structure of RG flows for the physically significant cases M=2 and N=2, N=3 shows that the model has a three-dimensionally stable fixed point different from the Bose one. The critical dimensionality is proved to be exactly two times smaller than its counterpart in the real cubic model: N_c^C ={1/2} N_c^R. The numerical value NcC=1.447±0.020N_c^C=1.447 \pm 0.020NcC​=1.447±0.020 is obtained from resumming the known five-loop $\ve$-series for N_c^R. Since N_c^C < 2, the critical thermodynamics of the model relevant to the phase transitions in real substances should be governed by the complex cubic fixed point with a new set of critical exponents: $\gm=1.404(25)$, ν=0.715(10)ν=0.715(10)ν=0.715(10), $\et=0.0343(20)$ for N=2 and $\gm=1.390(25)$, ν=0.702(10)ν=0.702(10)ν=0.702(10), $\et=0.0345(15)$ for N=3.
Tricriticdal behavior, in particular, critical exponents (η, γ, φ, ν, α), of classical N-component spin systems with a short-range or long-range interaction are studied in the dimension d (≪ 3 σ/2; σ: potential range parameter, 1 ≪ σ≤2) by means of the renormalization theory. The obtained expressions for all these critical exponents in the long-range interaction system and for the crossover exponent φ in the short-range one are new. The corrections of these tricritical exponents to the classical ones are shown to be smaller than those in the corresponding systems undergoing the second-order phase transition.