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Physical review. B, Condensed matter

Low-temperature renormalization group for the Lifshitz point

Gary S. Grest, Josef Sak

Published May 1, 1978• 34 citations
Publisher

Abstract

The momentum-shell technique is employed to derive the recursion relations for the Lifshitz point above the lower critical dimensionality. For the uniaxial point ($m=1$) 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 $n$ is the number of components of the spin. The critical behavior in the presence of long-range power-law interactions is also studied.

DOI
10.1103/physrevb.17.3607

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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 \ensuremathϵ=d\ensuremath−2.5\ensuremath{\epsilon}=d\ensuremath{-}2.5\ensuremathϵ=d\ensuremath−2.5 and the critical exponents are calculated. The results are \ensuremathλt=\ensuremathν4\ensuremath−1=2\ensuremathϵ{\ensuremath{\lambda}}_{\mathrm{t}}={\ensuremath{\nu}}_{4}^{\ensuremath{-}1}=2\ensuremath{\epsilon}\ensuremathλt​=\ensuremathν4\ensuremath−1​=2\ensuremathϵ, \ensuremathη2=\ensuremathϵ(n\ensuremath−2){\ensuremath{\eta}}_{2}=\frac{\ensuremath{\epsilon}}{(n\ensuremath{-}2)}\ensuremathη2​=(n\ensuremath−2)\ensuremathϵ​, \ensuremathη4=2\ensuremathϵ(n\ensuremath−2){\ensuremath{\eta}}_{4}=\frac{2\ensuremath{\epsilon}}{(n\ensuremath{-}2)}\ensuremathη4​=(n\ensuremath−2)2\ensuremathϵ​, \ensuremathβk=12+3\ensuremathϵ4(n\ensuremath−2){\ensuremath{\beta}}_{k}=\frac{1}{2}+\frac{3\ensuremath{\epsilon}}{4(n\ensuremath{-}2)}\ensuremathβk​=21​+4(n\ensuremath−2)3\ensuremathϵ​, 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.
Ferromagnets with the number of spin components n>2n>2n>2 and with power-law interactions ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}}$ are studied in dimensions $d=2+\ensuremath{\epsilon}$ for small $\ensuremath{\epsilon}$. The following theorem is proved: Let ${\ensuremath{\eta}}_{\mathrm{SR}}$ be the value of the critical exponent $\ensuremath{\eta}$ for short-range force (e.g., nearest-neighbor exchange). Then in the presence of the long-range power-law interaction the critical behavior is the same as for short-range interactions provided that the inequality $2\ensuremath{-}\ensuremath{\sigma}<{\ensuremath{\eta}}_{\mathrm{SR}}$ is satisfied. In this case the critical exponents depend on ddd and nnn but not on $\ensuremath{\sigma}$. In the opposite case, the value of $\ensuremath{\eta}$ is ${\ensuremath{\eta}}_{\mathrm{LR}}=2\ensuremath{-}\ensuremath{\sigma}$ and the critical exponent $\ensuremath{\nu}$ depends on $\ensuremath{\sigma}$ in addition to ddd and nnn.
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.
Uniaxial Lifshitz Point at O(εL2)O(ε_L^2)O(εL2​)
The critical exponents νL2,ηL2ν_{L2}, η_{L2}νL2​,ηL2​ and γL2γ_{L2}γL2​ of a uniaxial Lifshitz point are calculated at two-loop level using renormalization group and εLε_{L}εL​-expansion techniques. We found a new constraint involving the loop momenta along the competition axis, which allows to solve the two-loop integrals. The exponent γL2γ_{L2}γL2​ obtained using our method is in good agreement with numerical estimates based on Monte Carlo simulations.
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.
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.
Critical, crossover and correction-to-scaling exponents for isotropic Lifshitz points to order (8 ndashndashndash d)2
A two-loop renormalization group analysis of the critical behaviour at an isotropic Lifshitz point is presented. Using dimensional regularization and minimal subtraction of poles, we obtain the expansions of the critical exponents ν and η, the crossover exponent φ, as well as the (related) wave vector exponent β q and the correction-to-scaling exponent ω to second order in 8 = 8 − d . They are compared with the authors' recent -expansion results (2000 Phys. Rev. B 62 12338, 2001 Nucl. Phys. B 612 340) for the general case of an m -axial Lifshitz point. It is shown that the expansions obtained here by a direct calculation for the isotropic ( m = d ) Lifshitz point all follow from the latter upon setting m = 8 − 8 . This is so despite recent claims to the contrary by de Albuquerque and Leite (2002 J. Phys. A: Math. Gen. 35 1807).