The massive field-theory approach for studying critical behavior in fixed space dimensions d < 4 is extended to systems with surfaces. This enables one to study surface critical behavior directly in dimensions d < 4 without having to resort to the ǫ expansion. The approach is elaborated for the representative case of the semi-infinite |φ| 4 n-vector model with a boundary term This also holds for the surface crossover exponent Φ, for which we obtain the values Φ(n = 0) ≃ 0.52 and Φ(n = 1) ≃ 0.54 considerably lower than the previous ǫ-expansion estimates.

We investigate the critical behavior that d-dimensional systems with short-range forces and a n-component order parameter exhibit at Lifshitz points whose wavevector instability occurs in an m-dimensional isotropic subspace of R d . Utilizing dimensional regularization and minimal subtraction of poles in d = 4 + m 2 − ǫ dimensions, we carry out a two-loop renormalization-group (RG) analysis of the field-theory models representing the corresponding universality classes. This gives the beta function β u (u) to third order, and the required renormalization factors as well as the associated RG exponent functions to second order, in u. The coefficients of these series are reduced to m-dependent expressions involving single integrals, which for general (not necessarily integer) values of m ∈ (0, 8) can be computed numerically, and for special values of m analytically. The ǫ expansions of the critical exponents η l2 , η l4 , ν l2 , ν l4 , the wave-vector exponent β q , and the correction-toscaling exponent are obtained to order ǫ 2 . These are used to estimate their values for d = 3. The obtained series expansions are shown to encompass both isotropic limits m = 0 and m = d.

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.

Abstract. -Systems described by n-component φ 4 models in a ∞ d−1 × L slab geometry of finite thickness L are considered at and above their bulk critical temperature Tc,∞. The renormalization-group improved perturbation theory commonly employed to investigate the fluctuation-induced forces ("thermodynamic Casimir effect") in d = 4 − ǫ bulk dimensions is re-examined. It is found to be ill-defined beyond two-loop order because of infrared singularities when the boundary conditions are such that the free propagator in slab geometry involves a zero-energy mode at bulk criticality. This applies to periodic boundary conditions and the special-special ones corresponding to the critical enhancement of the surface interactions on both confining plates. The field theory is reorganized such that a small-ǫ expansion results which remains well behaved down to Tc,∞. The leading contributions to the critical Casimir amplitudes ∆per and ∆sp,sp beyond two-loop order are ∼ (u * ) (3−ǫ)/2 , where u * = O(ǫ) is the value of the renormalized φ 4 coupling at the infrared-stable fixed point. Besides integer powers of ǫ, the small-ǫ expansions of these amplitudes involve fractional powers ǫ k/2 , with k ≥ 3, and powers of ln ǫ. Explicit results to order ǫ Fluctuations associated with long wave-length, low-energy excitations play a crucial role in determining the physical properties of many macroscopic systems. When such fluctuations are confined by boundaries, walls, or size restrictions along one or several axes, important effective forces may result. In those cases where the continuous mode spectrum that emerges as the system becomes macroscopic in all directions is not separated from zero energy by a gap, these fluctuation-induced forces are longranged, decaying algebraically as a function of the relevant confinement length L (separation of walls, thickness of the system, etc).A prominent example of such forces are the Casimir forces [1] induced by vacuum fluctuations of the electromagnetic field between two metallic bodies a distance L apart [2][3][4]. Analogous long-range effective forces occur in condensed matter systems as the result of either (i) thermal fluctuations at continuous phase transitions or else (ii) Goldstone modes and similar "massless" excitations [5][6][7][8][9][10][11][12]. In particular the former ones, frequently called "critical Casimir forces," have attracted considerable theoretical and experimental attention recently. Beginning with the seminal paper by Fisher and de Gennes [5], they have been studied theoretically for more than a decade using renormalization group (RG) [6][7][8][9][10] and conformal field theory methods [13], exact solutions of models [12], as well as Monte c EDP Sciences

The large-n expansion is developed for the study of critical behaviour of d-dimensional systems at m-axial Lifshitz points with an arbitrary number m of modulation axes. The leading non-trivial contributions of O(1/n) are derived for the two independent correlation exponents η L2 and η L4 , and the related anisotropy index θ. The series coefficients of these 1/n corrections are given for general values of m and d with 0 ≤ m ≤ d and 2 + m/2 < d < 4 + m/2 in the form of integrals. For special values of m and d such as (m, d) = (1, 4), they can be computed analytically, but in general their evaluation requires numerical means. The 1/n corrections are shown to reduce in the appropriate limits to those of known large-n expansions for the case of d-dimensional isotropic Lifshitz points and critical points, respectively, and to be in conformity with available dimensionality expansions about the upper and lower critical dimensions. Numerical results for the 1/n coefficients of η L2 , η L4 and θ are presented for the physically interesting case of a uniaxial Lifshitz point in three dimensions, as well as for some other choices of m and d. A universal coefficient associated with the energy-density pair correlation function is calculated to leading order in 1/n for general values of m and d.