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Journal of Physics Conference Series

Specific heat amplitude ratios for generic competing systems

M. I. Sena-Junior, Marcelo M. Leite

Published Jan 21, 2015• 5 citations
Publisher

Abstract

The specific heat amplitude ratios for generic anisotropic as well as isotropic Lifshitz critical behaviors for the N-vector model are computed at one-loop level using the ε L - expansion. In the anisotropic case, we show that the universal result obtained reduces very easily to that from the simpler m-axial universality class result. In the isotropic case, if n is the number of neighbors coupled via competing interactions, we demonstrate that the ratio vanishes close to n = 4 and, becomes negative for n > 4 when it is calculated exactly, which is rather odd. The evaluation using the orthogonal approximation is shown to yield positive results for arbitrary n. Explicit computations for the case n = 2, d = 3, N = 1 yield an exact amplitude ratio equal to 0.06, with the approximate amplitude ratio being 1.17.We discuss two physical mechanisms to pick out one of the amplitude ratio values. We propose an experiment in homopolymer-diblock copolymer blends in order to determine the amplitude ratio.

DOI
10.1088/1742-6596/574/1/012170

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The specific heat amplitude ratios for generic anisotropic as well as isotropic Lifshitz critical behaviors for the N-vector model are computed at one-loop level using the ε L - expansion. In the anisotropic case, we show that the universal result obtained reduces very easily to that from the simpler m-axial universality class result. In the isotropic case, if n is the number of neighbors coupled via competing interactions, we demonstrate that the ratio vanishes close to n = 4 and, becomes negative for n > 4 when it is calculated exactly, which is rather odd. The evaluation using the orthogonal approximation is shown to yield positive results for arbitrary n. Explicit computations for the case n = 2, d = 3, N = 1 yield an exact amplitude ratio equal to 0.06, with the approximate amplitude ratio being 1.17.We discuss two physical mechanisms to pick out one of the amplitude ratio values. We propose an experiment in homopolymer-diblock copolymer blends in order to determine the amplitude ratio.
Generic higher character Lifshitz critical behaviors are described using field theory and ϵL{ϵ}_{L}ϵL​ expansion renormalization-group methods. These critical behaviors describe systems with arbitrary competing interactions. We derive the scaling relations and the critical exponents at the two-loop level for anisotropic and isotropic points of arbitrary higher character. The framework is illustrated for the NNN-vector ${\ensuremath{\phi}}^{4}$ model describing a ddd-dimensional system. The anisotropic behaviors are derived in terms of many independent renormalization-group transformations, each one characterized by independent correlation lengths. The isotropic behaviors can be understood using only one renormalization-group transformation. Feynman diagrams are solved for the anisotropic behaviors using a different dimensional regularization associated to a generalized orthogonal approximation. The isotropic diagrams are treated using this approximation as well as with a different exact technique to compute the integrals. The entire procedure leads to the analytical solution of generic loop order integrals with arbitrary external momenta. The property of universality class reduction is also satisfied when the competing interactions are turned off. We show how the results presented here reduce to the usual mmm-fold Lifshitz critical behaviors for both isotropic and anisotropic criticalities.
A general derivation of the two-scale-factor universality is given using the renormalized ${\ensuremath{\varphi}}^{4}$ theory. As a consequence ten universal relations among critical amplitudes are obtained, any other universal relations being a combination of them. The situation is shown to be the same as for the scaling laws for critical exponents. The calculations up to order ${\ensuremath{\epsilon}}^{2}$ for systems with continuous symmetry are performed for some of these relations such as the specific-heat amplitudes ratio and ${R}_{\ensuremath{\xi}}^{+}$, which relates the amplitudes of the correlation length and of the specific heat, etc. Comparison with experiments and series results and, in particular, the discussion of the superfluid helium and RbMnF3{\mathrm{F}}_{3}F3​ cases, are repeated with our improved numerical results.
As one of the series of papers on 1/n expansion of the classical n-vector model up to O(1/n2), the universal amplitude ratio Rχ=1"DB8-1 is investigated up to O(1/n2) in the present paper. The 1/n2 term for Rχ is calculated numerically for d=3. In case of the critical exponent β, γ or ν, the result of the second-order calculation is not satisfactory for small values of n. On the contrary, the situation is much better for Rχ. The numerical value obtained is Rχ=1.20 for n=3 at d=3, which is almost equal to that of high-temperature expansion, Rχ=1.23.
The phase diagram in the vicinity of a Lifshitz point is analyzed using scaling arguments and renormalization-group calculations. Feynman-graph techniques in $d=4+\frac{1}{2}m\ensuremath{-}\ensuremath{\epsilon}$ dimensions are used to calculate the universal amplitude ratio governing the shape of the two branches of the critical line which intersect at an mmm-fold Lifshitz point. For m<6m<6m<6 the amplitude ratio is found to be $\frac{{A}_{H}(m)}{{A}_{F}(m)}=1\ensuremath{-}\frac{2}{m}+O(\ensuremath{\epsilon})$, where AH(AF){A}_{H}({A}_{F})AH​(AF​) is the amplitude associated with the helical (ferromagnetic) branch, so that to zeroth order in $\ensuremath{\epsilon}$, this ratio is independent of the number nnn of components of the order parameter. The calculation indicates that along the remainder (6< m< 86<~m<~86< m< 8) of the critical dimensionality line, $\ensuremath{\epsilon}=0$, no helical long-range order is expected. We also calculate AHAF\frac{{A}_{H}}{{A}_{F}}AF​AH​​ for a spherical model of a ddd-dimensional system exhibiting an mmm-fold Lifshitz point. Along the critical dimensionality line $\ensuremath{\epsilon}=0$, the spherical model result reduces to that of the renormalization-group calculation.
Monte Carlo simulations and series expansions data for the energy, specific heat, magnetization, and susceptibility of the three-state Potts model on the square lattice are analyzed in the vicinity of the critical point in order to estimate universal combinations of critical amplitudes. We estimate these amplitudes using the correction-to-scaling exponents predicted by conformal field theory. We also form effective ratios of the observables close to the critical point and analyze how they approach the universal critical-amplitude ratios. In particular, using the duality relation, we show analytically that for the Potts model with a number of states $q\ensuremath{\leqslant}4$, the effective ratio of the energy critical amplitudes always approaches unity linearly with respect to the reduced temperature. This fact leads to the prediction of relations among the amplitudes of correction-to-scaling terms of the specific heat in the low- and high-temperature phases. We present numerical and analytical support for the form of the first two correction-to-scaling terms. Our results for the amplitude ratios closely agree with the theoretical predictions and the earlier numerical estimates of the specific heat and the susceptibility amplitude ratios.
Field-theoretic renormalization-group methods are developed to describe in a unified fashion the critical exponents of an m-fold Lifshitz point at the two-loop order in the anisotropic $(m\ensuremath{\ne}d)$ and isotropic (m=d(m=d(m=d close to 8) situations. The general theory is illustrated for the N-vector ${\ensuremath{\varphi}}^{4}$ model describing a d-dimensional system. A new regularization and renormalization procedure is presented for both types of Lifshitz behavior. The anisotropic cases are formulated with two independent renormalization-group transformations. The description of the isotropic behavior requires only one type of renormalization-group transformation. We point out the conceptual advantages implicit in this picture and show how this framework is related to other previous renormalization-group treatments for the Lifshitz problem. The Feynman diagrams of arbitrary loop order can be performed analytically provided these integrals are considered to be homogeneous functions of the external momenta scales. The anisotropic universality class (N,d,m)(N,d,m)(N,d,m) reduces easily to the Ising-like (N,d)(N,d)(N,d) when m=0.m=0.m=0. We show that the isotropic universality class (N,m)(N,m)(N,m) when m is close to 8 cannot be obtained from the anisotropic one in the limit $\stackrel{\ensuremath{\rightarrow}}{d}m$ near 8. The exponents for the uniaxial case d=3,N=m=1d=3,N=m=1d=3,N=m=1 are in good agreement with recent Monte Carlo simulations for the axial next-nearest-neighbor Ising model.