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

Abstract

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.

Renormalization-group picture of the Lifshitz critical behavior

“…The RG analysis by Mergulhão and Carneiro has been used in a attempt to extend the calculation for all m. Using the fact that the quartic momenta scale including σ and the quadratic external momenta scales are equal Diehl and Shpot considered the anisotropic problem for general m [31,32]. In their first work [31], they worked directly in position space.…”

“…The RG analysis by Mergulhão and Carneiro has been used in a attempt to extend the calculation for all m. Using the fact that the quartic momenta scale including σ and the quadratic external momenta scales are equal Diehl and Shpot considered the anisotropic problem for general m [31,32]. In their first work [31], they worked directly in position space. After that, using a hybrid approach, going to coordinate or momentum space using the free propagator (scaling function) in coordinate space to make the transition according to the necessity, they calculated the critical exponents using a minimal subtraction procedure which sets the external quartic momenta scales to zero.…”

Scaling in a temperature quench in systems with a Lifshitz point: nonconserved and conserved order parameters

“…A well-known model is the Anisotropic Next Nearest Neighbour-interaction Ising (ANNNI) model. There have been a great deal of work on the static critical proprties at the Lifshitz point; recent references include Frisch, Kimball and Binder [7], Diehl, Shpot and Zia [8], Shpot and Diehl [9], Leite [10], Pleimling and Henkel [11]. There have been few studies on the dynamical properties at the Lifshitz point; see, e.g., Huber [12], Folk and Selke [13], Selke [14], and Selke [15].…”

Specific-heat amplitude ratio for anisotropic Lifshitz critical behaviors

“…Recent investigations of the associated critical exponents for the m-axial Lifshitz universality class have been put forward using numerical Monte Carlo simulations [3] and fieldtheoretic approaches [4][5][6]. Within the perturbative ǫ L -expansion, there are two proposals in order to unravel the higher loop structure of this sort of critical behavior.…”

“…Within the perturbative ǫ L -expansion, there are two proposals in order to unravel the higher loop structure of this sort of critical behavior. One of them makes use of a semi-analytic ǫ L -expansion for the critical exponents, where some loop integrals are evaluated through numerical integration [4]. Another alternative is the purely analytical treatment of all loop integrals involved, such that new renormalization group as well as ǫ Lexpansion ideas have been developed in order to determine those critical indices [5,6].…”