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Physical Review A

Continuously varying critical exponents toO(1/n)

A. A. Inayat-Hussain, M. J. Buckingham

Published May 1, 1990• 5 citations
Publisher

Abstract

In the context of a phase transition, the critical singularity is characterized by a set of numbers, the critical exponents. It is widely believed that for transitions in systems with pair interactions on Bravais lattices (as opposed to hierarchical lattices), the critical exponents are determined solely by parameters d and n, representing the dimensionality of space and of the order parameter, respectively, and by \ensuremath{\sigma}, the interaction range. In this paper we consider a certain generalization of the Lifshitz singularity in the n-vector model in which a new parameter \ensuremath{\lambda}, describing a type of anisotropy of the interactions, is introduced. This is described by a model Hamiltonian in k space in which the leading term is of the form [${\mathrm{\ensuremath{\Sigma}}}_{\mathit{i}=1}^{\mathit{d}}$(${\mathit{k}}_{\mathit{i}}^{2}$${)}^{\mathrm{\ensuremath{\sigma}}/(2\ensuremath{\lambda})}$]\ensuremath{\lambda}. It should be noted that, as for the usual Lifshitz case, what would otherwise be the dominant ${\mathit{k}}^{2}$ term is supposed to have already been cancelled. For this model the parameters d, n, and \ensuremath{\sigma} are not sufficient to determine the value of the critical exponents to O(1/n), although they are sufficient for the first order in the \ensuremath{\epsilon} expansion.Calculations exact to O(1/n) show that, for a given set d, n, and \ensuremath{\sigma}, the exponents vary continuously with the parameter \ensuremath{\lambda} and include as a special case (for \ensuremath{\lambda}=1) the critical exponents for an anisotropic Lifshitz point. The critical exponents for this Lifshitz point differ markedly in value from those for the usually studied isotropic Lifshitz point, which is also a special case (for \ensuremath{\lambda}=\ensuremath{\sigma}/2) of our model. For \ensuremath{\sigma}=2, however, the two cases corresponding to \ensuremath{\lambda}=1 and \ensuremath{\sigma}/2 become degenerate. We provide another explicit example of the dependence of the critical exponents on \ensuremath{\lambda} by performing, in addition to the 1/n expansion, an expansion in \ensuremath{\sigma}-2. This helps to shed some light on the approach to the degeneracy at \ensuremath{\sigma}=2. The analysis of the Feynman integrals arising in the O(1/n) calculation provides interesting examples of the applications of hypergeometric functions; these integrals have themselves led to new results (published separately) in the theory of hypergeometric functions.

DOI
10.1103/physreva.41.5394

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In the context of a phase transition, the critical singularity is characterized by a set of numbers, the critical exponents. It is widely believed that for transitions in systems with pair interactions on Bravais lattices (as opposed to hierarchical lattices), the critical exponents are determined solely by parameters d and n, representing the dimensionality of space and of the order parameter, respectively, and by \ensuremath{\sigma}, the interaction range. In this paper we consider a certain generalization of the Lifshitz singularity in the n-vector model in which a new parameter \ensuremath{\lambda}, describing a type of anisotropy of the interactions, is introduced. This is described by a model Hamiltonian in k space in which the leading term is of the form [\ensuremathΣi=1d{\mathrm{\ensuremath{\Sigma}}}_{\mathit{i}=1}^{\mathit{d}}\ensuremathΣi=1d​({\mathit{k}}_{\mathit{i}}^{2}$${)}^{\mathrm{\ensuremath{\sigma}}/(2\ensuremath{\lambda})}]\ensuremath{\lambda}. It should be noted that, as for the usual Lifshitz case, what would otherwise be the dominant k2{\mathit{k}}^{2}k2 term is supposed to have already been cancelled. For this model the parameters d, n, and \ensuremath{\sigma} are not sufficient to determine the value of the critical exponents to O(1/n), although they are sufficient for the first order in the \ensuremath{\epsilon} expansion.Calculations exact to O(1/n) show that, for a given set d, n, and \ensuremath{\sigma}, the exponents vary continuously with the parameter \ensuremath{\lambda} and include as a special case (for \ensuremath{\lambda}=1) the critical exponents for an anisotropic Lifshitz point. The critical exponents for this Lifshitz point differ markedly in value from those for the usually studied isotropic Lifshitz point, which is also a special case (for \ensuremath{\lambda}=\ensuremath{\sigma}/2) of our model. For \ensuremath{\sigma}=2, however, the two cases corresponding to \ensuremath{\lambda}=1 and \ensuremath{\sigma}/2 become degenerate. We provide another explicit example of the dependence of the critical exponents on \ensuremath{\lambda} by performing, in addition to the 1/n expansion, an expansion in \ensuremath{\sigma}-2. This helps to shed some light on the approach to the degeneracy at \ensuremath{\sigma}=2. The analysis of the Feynman integrals arising in the O(1/n) calculation provides interesting examples of the applications of hypergeometric functions; these integrals have themselves led to new results (published separately) in the theory of hypergeometric functions.
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.
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.
The critical behaviour of d-dimensional n-vector models at m-axial Lifshitz points is considered for general values of m in the large-n limit. It is proven that the recently obtained large-N expansions [J. Phys.: Condens. Matter 17, S1947 (2005)] of the correlation exponents η_{L2}, η_{L4} and the related anisotropy exponent θare fully consistent with the dimensionality expansions to second order in ε=4+m/2-d [Phys. Rev. B 62, 12338 (2000); Nucl. Phys. B 612, 340 (2001)] inasmuch as both expansions yield the same contributions of order ε^2/n.
We present analytical results for the finite-size scaling in ddd-dimensional O(N)O(N)O(N) systems with strong anisotropy where the critical exponents (e.g., ${\ensuremath{\nu}}_{\ensuremath{\Vert}}$ and ${\ensuremath{\nu}}_{\ensuremath{\perp}}$) depend on the direction. Prominent examples are systems with long-range interactions, decaying with the interparticle distance rrr as ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}}$ with different exponents $\ensuremath{\sigma}$ in corresponding spatial directions, systems with space-``time'' anisotropy near a quantum critical point, and systems with Lifshitz points. The anisotropic properties involve also the geometry of the systems. We consider O(N)O(N)O(N) systems in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, confined to a ddd-dimensional layer with geometry ${L}^{m}\ifmmode\times\else\texttimes\fi{}{\ensuremath{\infty}}^{n};m+n=d$ and periodic boundary conditions across the finite mmm dimensions. The arising difficulties are avoided using a technique of calculations based on the analytical properties of the generalized Mittag-Leffler functions.
A new set of infinitesimal transformations generalizing scale invariance for strongly anisotropic critical systems is considered. It is shown that such a generalization is possible if the anisotropy exponent $\ensuremath{\theta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2/N$, with N=1,2,3…N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1,2,3\dots{}N=1,2,3…. Differential equations for the two-point function are derived and explicitly solved for all values of NNN. Known special cases are conformal invariance ( N=2N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2N=2) and Schr\"odinger invariance ( N=1N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1N=1). For N=4N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4N=4 and N=6N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}6N=6, the results contain as special cases the exactly known scaling forms obtained for the spin-spin correlation function in the axial next-nearest-neighbor spherical model at its Lifshitz points of first and second order.
The critical behavior of d-dimensional systems with an n-component order parameter is reconsidered at (m,d,n)-Lifshitz points, where a wave-vector instability occurs in an m-dimensional subspace of Rd.{R}^{d}.Rd. Our aim is to sort out which ones of the previously published partly contradictory \ensuremath{\epsilon}-expansion results to second order in $\ensuremath{\epsilon}=4+m/2\ensuremath{-}d$ are correct. To this end, a field-theory calculation is performed directly in the position space of $d=4+m/2\ensuremath{-}\ensuremath{\epsilon}$ dimensions, using dimensional regularization and minimal subtraction of ultraviolet poles. The residua of the dimensionally regularized integrals that are required to determine the series expansions of the correlation exponents ${\ensuremath{\eta}}_{l2}$ and ${\ensuremath{\eta}}_{l4}$ and of the wave-vector exponent ${\ensuremath{\beta}}_{q}$ to order ${\ensuremath{\epsilon}}^{2}$ are reduced to single integrals, which for general $m=1,\dots{},d\ensuremath{-}1$ can be computed numerically, and for special values of m, analytically. Our results are at variance with the original predictions for general m. For m=2m=2m=2 and m=6,m=6,m=6, we confirm the results of Sak and Grest [Phys. Rev. B 17, 3602 (1978)] and Mergulh\~ao and Carneiro's recent field-theory analysis [Phys. Rev. B 59, 13 954 (1999)].