Critical exponents for the Lifshitz point:<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>ε</mml:mi></mml:math>expansion

Sak, J.; Grest, Gary S.

doi:10.1103/physrevb.17.3602

Cited by 33 publications

(45 citation statements)

References 7 publications

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“…The resulting series expansions, given in Eqs. ͑84͒-͑88͒, correct earlier results by Mukamel 28 and Hornreich and Bruce; 29 for the special values mϭ2 and mϭ6, we recovered Sak and Grest's 30 findings. To clarify this long-standing controversy, it proved useful to work directly in position space and to compute the Laurent expansion of the dimensionally regularized distributions associated with the Feynman diagrams.…”

Section: Discussionsupporting

confidence: 78%

“…2,32,33 Unfortunately, the ⑀-expansion results to order ⑀ 2 one group of authors 1,28,29 obtained for the correlation exponents l2 and l4 and the wave-vector exponent ␤ q are in conflict with those of Sak and Grest 30 for the cases mϭ2 and mϭ6. In order to resolve this long-standing controversy, Mergulhão and Carneiro 17,18 recently presented a reanalysis of the problem based on renormalized field theory and dimensional regularization.…”

Section: Introductionmentioning

confidence: 53%

“…The problem has been studied decades ago by means of an ⑀ expansion about the upper critical dimension 1,[28][29][30] d*͑m ͒ϭ4ϩ m 2 , mр8. ͑1͒…”

Section: Introductionmentioning

confidence: 99%

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Critical behavior atm-axial Lifshitz points: Field-theory analysis and ε-expansion results

2000

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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 R d . Our aim is to sort out which ones of the previously published partly contradictory ⑀-expansion results to second order in ⑀ϭ4ϩm/2Ϫd are correct. To this end, a field-theory calculation is performed directly in the position space of dϭ4ϩm/2Ϫ⑀ 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 l2 and l4 and of the wave-vector exponent ␤ q to order ⑀ 2 are reduced to single integrals, which for general mϭ1, . . . ,dϪ1 can be computed numerically, and for special values of m, analytically. Our results are at variance with the original predictions for general m. For mϭ2 and mϭ6, we confirm the results of Sak and Grest ͓Phys. Rev. B 17, 3602 ͑1978͔͒ and Mergulhão and Carneiro's recent field-theory analysis ͓Phys. Rev. B 59, 13 954 ͑1999͔͒.

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Section: Discussionsupporting

confidence: 78%

Section: Introductionmentioning

confidence: 53%

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Critical behavior atm-axial Lifshitz points: Field-theory analysis and ε-expansion results

2000

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“…It should not be surprising that the approach fails to describe the d = m = 8 case, for the exponent η L2 is divergent as can be seen from equation (21). The approximation made is not suitable for general isotropic cases d = m = 8 as well, since there is no longer any preferred direction.…”

mentioning

confidence: 99%

“…The constraint introduced along the m-dimensional subspace is equivalent to expanding around the theory without competition, by varying the space dimension d, with m kept fixed. This is the main difference between the approach described here and other proposals [21][22][23]. The first calculation for the critical index η L2 for general m( =d) was performed [20] using the method of momentum shell integration based on the Hamiltonian formalism.…”

mentioning

confidence: 99%

Permissible overheating of the gaseous medium in a gas-flow laser excited by fission fragments from uranium nuclei

Prikhod'ko

Sizov

1999

Quantum Electron.

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We present the critical exponents ν L2 , η L2 and γ L for an m-axial Lifshitz point at second order in an L expansion. We introduce a constraint involving the loop momenta along the m-dimensional subspace in order to perform two-and three-loop integrals. The results are valid in the range 0 m < d. The case m = 0 corresponds to the usual Ising-like critical behaviour.PACS numbers: 7540, 0550, 1110, 6460K, 7540C Lifshitz multicritical points appear at the confluence of a disordered phase, a uniformly ordered phase and a modulated ordered phase [1,2]. The spatially modulated phase is characterized by a fixed equilibrium wavevector k 0 . In this phase, k 0 goes continuously to zero as the system approaches the Lifshitz point. If this wavevector has m components, the critical system under consideration presents an m-fold Lifshitz critical behaviour. This sort of critical behaviour is present in a variety of real physical systems including high-T c superconductors [3][4][5], ferroelectric liquid crystals [6,7], magnetic compounds and alloys [8-10], among others.In magnetic systems [11], the m-fold Lifshitz point can be described by a spin-1 2 Ising model on a d-dimensional lattice with nearest-neighbour ferromagnetic interactions as well as next-nearest-neighbour competing antiferromagnetic couplings along m directions. This system can be described in a field-theoretic setting using a modified φ 4 theory with higherorder derivative terms, which arises as an effect of the competition along the m directions. The Lifshitz universality class is defined by the parameters (N, d, m), where N is the number of components of the order parameter, d is the space dimension of the system and m is the number of competing directions.Other examples of field theories containing higher derivative terms have been investigated in different physical scenarios. In cosmology, the recently proposed model known as 'kinflation' describes inflation driven by higher-order kinetic terms for the inflaton scalar field [12]. Another instance which arises in quantum field theory in curved spacetime is the quantization of scalar fields with a high-frequency dispersion relation around a classical gravitational background [13,14]. In this case, the higher-order term accounts for deviations

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A Feynman integral in Lifshitz‐point and Lorentz‐violating theories in

Paris

Shpot

2018

Math Methods in App Sciences

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We evaluate a 1‐loop, 2‐point, massless Feynman integral ID,m(p,q) relevant for perturbative field theoretic calculations in strongly anisotropic d=D+m dimensional spaces given by the direct sum RD⊕Rm. Our results are valid in the whole convergence region of the integral for generic (noninteger) codimensions D and m. We obtain series expansions of ID,m(p,q) in terms of powers of the variable X:=4p2/q4, where p=|p|, q=|q|, bold-italicp∈RD, bold-italicq∈Rm, and in terms of generalised hypergeometric functions 3F2(−X), when X<1. These are subsequently analytically continued to the complementary region X≥1. The asymptotic expansion in inverse powers of X1/2 is derived. The correctness of the results is supported by agreement with previously known special cases and extensive numerical calculations.

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Critical exponents for the Lifshitz point:εexpansion

Cited by 33 publications

References 7 publications

Critical behavior atm-axial Lifshitz points: Field-theory analysis and ε-expansion results

Critical behavior atm-axial Lifshitz points: Field-theory analysis and ε-expansion results

Permissible overheating of the gaseous medium in a gas-flow laser excited by fission fragments from uranium nuclei

A Feynman integral in Lifshitz‐point and Lorentz‐violating theories in

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