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International Journal of Modern Physics B

SPATIAL MODULATION OF ORDER PARAMETERS AND CRITICAL DIMENSIONS

А.В. Бабич, S. V. Berezovsky, Vyacheslav F. Klepikov

Published Mar 20, 2008• 3 citations
Publisher

Abstract

Critical phenomena in systems which have a joint multicritical and Lifshitz-point-like behavior are investigated. An expression for the upper critical dimension of such systems is found. A condition of the variational scale invariance of the considered model is specified. If the space dimension coincides with the upper critical dimension, then the critical behavior of the fluctuation correction to the heat capacity is similar to that of the heat capacity without order parameter modulation.

DOI
10.1142/s0217979208038922

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Critical phenomena in systems which have a joint multicritical and Lifshitz-point-like behavior are investigated. An expression for the upper critical dimension of such systems is found. A condition of the variational scale invariance of the considered model is specified. If the space dimension coincides with the upper critical dimension, then the critical behavior of the fluctuation correction to the heat capacity is similar to that of the heat capacity without order parameter modulation.
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An introduction to the theory of critical behavior at Lifshitz points is given, and the recent progress made in applying the field-theoretic renormalization group (RG) approach to ϕ4ϕ^4ϕ4 nnn-vector models representing universality classes of mmm-axial Lifshitz points is surveyed. The origins of the difficulties that had hindered a full two-loop RG analysis near the upper critical dimension for more than 20 years and produced long-standing contradictory εεε-expansion results are discussed. It is outlined how to cope with them. The pivotal role the considered class of continuum models might play in a systematic investigation of anisotropic scale invariance within the context of thermal equilibrium systems is emphasized. This could shed light on the question of whether anisotropic scale invariance implies an even larger invariance, as recently claimed in the literature.
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We calculate the critical behavior of systems having a multicritical point of a new type, hereafter called a Lifshitz point, which separates ordered phases with $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}=0$ and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}\ensuremath{\ne}0$ along the $\ensuremath{\lambda}$ line. For anisotropic systems, the correlation function is described in terms of four critical exponents, whereas for isotropic systems two exponents suffice. Critical exponents are calculated using an $\ensuremath{\epsilon}$-type expansion.
Abstract ‘The critical point’ in a phase diagram denotes the conditions under which a continuous transition between two states of matter occurs. Surprisingly, near the critical point diverse systems can have the same dependence of physical properties on parameters such as temperature. The only details that determine the critical exponents are the symmetry of the order parameter and the spatial dimensionality of the system. This independence from chemical and structural details, known as universality, presented a major challenge to theoretical physics for decades. Theory must deal with the large fluctuations in the amount of order that occurs close to the critical point. The successful theory developed by Ken Wilson exploits the fact that different states of a system and different systems can be related to one another by mathematically rescaling the length scales in the system. The mathematics involved can be made tractable by considering the abstract idea of a system with a variable number of spatial dimensions. The idea of rescaling is illustrated with the results of computer simulations of an Ising model and with images of fractals.
We study a novel type of quantum criticality of the Lifshitz ${\ensuremath{\varphi}}^{4}$ theory below the upper critical dimension du=z+dc=8{d}_{u}=z+{d}_{c}=8du​=z+dc​=8, where the dynamic critical exponent z=4z=4z=4 and the spatial upper critical dimension dc=4{d}_{c}=4dc​=4. Two fixed points, one Gaussian and the other non-Gaussian, are identified with zero and finite interaction strengths, respectively. At zero temperature the particle density exhibits different power-law dependences on the chemical potential in the weak- and strong-interaction regions. At finite temperatures, critical behaviors in the quantum disordered region are mainly controlled by the chemical potential. In contrast, in the quantum critical region critical scalings are determined by temperature. The scaling ansatz remains valid in the strong-interaction limit for the chemical potential, correlation length, and particle density, while it breaks down in the weak-interaction one. Approaching the upper critical dimension, physical quantities develop logarithmic dependence on dimensionality in the strong-interaction region. These results are applied to spin-orbit coupled bosonic systems, leading to predictions testable by future experiments.