An introduction into the theory of boundary critical phenomena and the application of the field-theoretical renormalization group method to these is given. The emphasis is on a discussion of surface critical behavior at bulk critical points of magnets, binary alloys, and fluids. Yet a multitude of related phenomena are mentioned. The most important distinct surface universality classes that may occur for a given universality class of bulk critical behavior are described, and the respective boundary conditions of the associated field theories are discussed. The short-distance singularities of the order-parameter profile in the diverse asymptotic regimes are surveyed.Comment: 24 pages, Latex with 2 figures as eps-files and World Scientific stylefile sprocl.sty, minor additions, updates and correction

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.

The critical behavior of a semi-infinite n-vector model in 4 -e dimensions near the special transition is studied. The renormalization of the relevant surface operators is discussed and explicitly carried out in two-loop order, It is found that the crossover exponent @, related to the square of the order parameter in the surface does not satisfy the relation re, =1 -v due to Bray and Moore.Renormalization-group (RG) methods of field theory have been applied with considerable success in the study of bulk critical phenomena. ' In a recent paper3 (hereafter referred to as I) similar methods were used to investigate the effects of free surfaces on critical behavior. The analysis in I was based on a semi-infinite n-vector model and restricted to the ordinary transition. "' This transition occurs in ferromagnetic systems when the coupling between the magnetic moments in the surface is less or only slightly stronger than in the bulk. However, if the surface coupling is sufficiently strong, the surface may order at a higher temperature than the bulk.The bulk then orders in the presence of the ordered surface. This is the extraordinary transition. The lines of ordinary, surface, and extraordinary transitions meet in a multicritical point that describes the special transition.In this Communication we will analyze the critical behavior of the semi-infinite n-vector model in d =4 -e dimensions near the special transition. Previous work on this transition was based on meanfield theory, ' position-space RG methods6 '0 (for the Ising case n = I), and e-expansion techniques. '" '3 Bray and Moore'4 also discussed the n =~c ase. In addition, they proposed' the scaling relation Q, =1 -u (where v is the bulk correlationlength exponent) for the crossover exponent $, related to the square of the order parameter in the sur-face. An analogous prediction y~~= v -1 was made for the susceptibility exponent y~~a t the ordinary transtition. Both relations are satisfied by the semiinfinite n-vector model to order e. %e will show that the relation $, =1 -v is in generai not valid. For the other one this has been shown already. '"With the exception of Reeve's work, ' previous analyses of the special transition based on the e expansion were limited to first order. Reeve presented results to order e2 for surface exponents, but did not calculate $, . He renormalized the usual bulk operators and identified surface exponents by exponentiation. However, a systematic RG analysis of critical phenomena in semi-infinite systems requires that one incorporates al/relevant, i.e. , bulk and surface operators into the renormalization program. This requirement is less important in the case of the bulk-driven ordinary transition discussed in I where a single surface operator (the normal derivative of the order parameter) had to be renormalized.The inclusion of this operator into the RG was partly a matter of convenience only, but needed to derive the surface scaling laws. Since the special transition is described by a rnulitcritical point that involves relevant...

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.