Universality classes for the dynamic surface critical behavior of systems with relaxational dynamics

We investigate the problem of which universality classes of dynamic surface critical behavior exist for semi-infinite systems whose dynamic bulk critical behavior and static surface critical behavior are representative of a given dynamic bulk universality class and a given static surface universality class. To this end systems whose bulk dynamics are described by either model B or model A of Halperin, Hohenberg, and Ma are considered, where the dynamics are allowed to be modified near the surface as follows: In the case of model B, surface terms that locally break the conservation law for the order-parameter density φ are allowed; in the case of model A, φ is assumed to be locally conserved near the surface. Semi-infinite field-theory models are constructed that are (i) compatible with the requested bulk dynamics and the requirements of causality, detailed balance, and relaxation to thermal equilibrium and (ii) minimal in the sense that no redundant or irrelevant surface terms are included. The boundary conditions to which the surface terms of the action correspond are worked out. For model B it is shown that nonconservative surface terms are relevant. Their strength can be parametrized by a coupling constant c${\mathrm{̃}}_0$≥0 whose inverse plays the role of an extrapolation length for the auxiliary (Martin-Siggia-Rose) field φ̃. Thus two distinct semi-infinite extensions of model B exist—one with c${\mathrm{̃}}_0$>0 called model ${\mathit{B}}_{\mathit{A}}$, and one with c${\mathrm{̃}}_0$=0 called model ${\mathit{B}}_{\mathit{B}}$—and each static surface universality class splits up into two dynamic surface universality classes.

The static and dynamic surface critical exponents of these dynamic surface universality classes as well as the crossover exponent associated with the nonconservative surface term are shown to be expressible in terms of static bulk and surface exponents. Model ${\mathit{B}}_{\mathit{A}}$ and its results should apply, e.g., to uniaxial ferromagnets whose rotational spin symmetry is broken in the vicinity of a surface plane. For model A the considered local modification of the dynamics near the surface is irrelevant in the renormalization-group sense and hence does not lead to new dynamic surface universality classes.