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Physical review. B, Condensed matter

Critical behavior at a Lifshitz point: Calculation of a universal amplitude ratio

David Mukamel, Marshall Luban

Published Oct 1, 1978• 30 citations
Publisher

Abstract

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 $m$-fold Lifshitz point. For $m<6$ the amplitude ratio is found to be $\frac{{A}_{H}(m)}{{A}_{F}(m)}=1\ensuremath{-}\frac{2}{m}+O(\ensuremath{\epsilon})$, where ${A}_{H}({A}_{F})$ is the amplitude associated with the helical (ferromagnetic) branch, so that to zeroth order in $\ensuremath{\epsilon}$, this ratio is independent of the number $n$ of components of the order parameter. The calculation indicates that along the remainder ($6<~m<~8$) of the critical dimensionality line, $\ensuremath{\epsilon}=0$, no helical long-range order is expected. We also calculate $\frac{{A}_{H}}{{A}_{F}}$ for a spherical model of a $d$-dimensional system exhibiting an $m$-fold Lifshitz point. Along the critical dimensionality line $\ensuremath{\epsilon}=0$, the spherical model result reduces to that of the renormalization-group calculation.

DOI
10.1103/physrevb.18.3631

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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+12m\ensuremath−\ensuremathϵd=4+\frac{1}{2}m\ensuremath{-}\ensuremath{\epsilon}d=4+21​m\ensuremath−\ensuremathϵ 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 AH(m)AF(m)=1\ensuremath−2m+O(\ensuremathϵ)\frac{{A}_{H}(m)}{{A}_{F}(m)}=1\ensuremath{-}\frac{2}{m}+O(\ensuremath{\epsilon})AF​(m)AH​(m)​=1\ensuremath−m2​+O(\ensuremathϵ), where AH(AF){A}_{H}({A}_{F})AH​(AF​) is the amplitude associated with the helical (ferromagnetic) branch, so that to zeroth order in \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ, 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, \ensuremathϵ=0\ensuremath{\epsilon}=0\ensuremathϵ=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 \ensuremathϵ=0\ensuremath{\epsilon}=0\ensuremathϵ=0, the spherical model result reduces to that of the renormalization-group calculation.
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 momentum-shell technique is employed to derive the recursion relations for the Lifshitz point above the lower critical dimensionality. For the uniaxial point (m=1m=1m=1) the expansion is made in $\ensuremath{\epsilon}=d\ensuremath{-}2.5$ and the critical exponents are calculated. The results are ${\ensuremath{\lambda}}_{\mathrm{t}}={\ensuremath{\nu}}_{4}^{\ensuremath{-}1}=2\ensuremath{\epsilon}$, ${\ensuremath{\eta}}_{2}=\frac{\ensuremath{\epsilon}}{(n\ensuremath{-}2)}$, ${\ensuremath{\eta}}_{4}=\frac{2\ensuremath{\epsilon}}{(n\ensuremath{-}2)}$, ${\ensuremath{\beta}}_{k}=\frac{1}{2}+\frac{3\ensuremath{\epsilon}}{4(n\ensuremath{-}2)}$, where nnn is the number of components of the spin. The critical behavior in the presence of long-range power-law interactions is also studied.
We study d-dimensional ferromagnetic spherical models with power-law, J(R)\ensuremath{\sim}1/${\mathit{R}}^{\mathit{d}+\mathrm{\ensuremath{\sigma}}}$ (\ensuremath{\sigma}2), or short-range (\ensuremath{\sigma}\ensuremath{\simeq}2) interactions and on-site terms of the form Ds2{\mathit{Ds}}^{2}Ds2-\ensuremath{\Vert}U\ensuremath{\Vert}s4{\mathit{s}}^{4}s4+Vs6{\mathit{Vs}}^{6}Vs6, and establish that the phase diagrams for all d and \ensuremath{\sigma} with \ensuremath{\sigma}d2\ensuremath{\sigma}, may exhibit critical end points terminating nonclassical critical lines; tricritical points may arise only for 3/2\ensuremath{\sigma}\ensuremath{\le}d. The phase boundary ${\mathit{D}}_{\mathrm{\ensuremath{\sigma}}}$(T,h), where h is the magnetic field, between the noncritical, spectator phase and the ordering and critical phase is shown to exhibit singularities at the end point with amplitudes obeying the universal ratio laws advanced on phenomenological grounds in Part I.
Feynman graph methods are used to analyse crossover behaviour in an n-component Heisenberg system, with a spin anisotropy that results asymptotically in m-component critical behaviour. An expression for the universal n- and m-dependent amplitude locating the singularity of the crossover scaling function is derived to second order in epsilon =4-d and found to give fair agreement with the results of recently derived series expansions. The same techniques are used to obtain, to 0( epsilon 2 ), a universal amplitude ratio characterizing the shape of the phase boundary in the vicinity of spin-flop bicritical point.
A new renormalization group treatment is proposed for the critical exponents of an m-fold Lifshitz point. The anisotropic cases (m not equal 8) are described by two independent fixed points associated to two independent momentum flow along the quadratic and quartic directions, respectively. The isotropic case is described separately. In that case, the fixed point is due to renormalization group transformations along the quartic directions. The new scaling laws are derived for both cases and generalize the ones previously reported.
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.