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 \ensuremath{\varphi} are allowed; in the case of model A, \ensuremath{\varphi} 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${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}$\ensuremath{\ge}0 whose inverse plays the role of an extrapolation length for the auxiliary (Martin-Siggia-Rose) field \ensuremath{\varphi}\ifmmode \tilde{}\else \~{}\fi{}. Thus two distinct semi-infinite extensions of model B exist---one with c${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}$>0 called model ${\mathit{B}}_{\mathit{A}}$ , and one with c${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}$=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.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 \ensuremath{\varphi} are allowed; in the case of model A, \ensuremath{\varphi} 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${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}$\ensuremath{\ge}0 whose inverse plays the role of an extrapolation length for the auxiliary (Martin-Siggia-Rose) field \ensuremath{\varphi}\ifmmode \tilde{}\else \~{}\fi{}. Thus two distinct semi-infinite extensions of model B exist---one with c${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}$>0 called model ${\mathit{B}}_{\mathit{A}}$, and one with c${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}$=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.

Physical review. B, Condensed matterJan 15, 199430 citations

Role of the surface in the critical behavior of finite systems

Valentin Duflot, Ph. Chomaz, Francesca Gulminelli

Physics Letters BMar 1, 20003 citations

The effects of surfaces on dynamic critical behavior

S. Dietrich, Horst Diehl

The European Physical Journal BJun 1, 19832 citations

Aging phenomena in critical semi-infinite systems

Michel Pleimling

Nonequilibrium surface autocorrelation and autoresponse functions are studied numerically in semi-infinite critical systems in the dynamical scaling regime. Dynamical critical behavior is examined for a nonconserved order parameter in semi-infinite two- and three-dimensional Ising models as well as in the Hilhorst--van Leeuwen model. The latter model permits a systematic study of surface aging phenomena, as the surface critical exponents change continuously as function of a model parameter. The scaling behavior of surface two-time quantities is investigated and scaling functions are confronted with predictions coming from the theory of local scale invariance. Furthermore, surface fluctuation-dissipation ratios are computed and their asymptotic values are shown to depend on the values of surface critical exponents.Nonequilibrium surface autocorrelation and autoresponse functions are studied numerically in semi-infinite critical systems in the dynamical scaling regime. Dynamical critical behavior is examined for a nonconserved order parameter in semi-infinite two- and three-dimensional Ising models as well as in the Hilhorst--van Leeuwen model. The latter model permits a systematic study of surface aging phenomena, as the surface critical exponents change continuously as function of a model parameter. The scaling behavior of surface two-time quantities is investigated and scaling functions are confronted with predictions coming from the theory of local scale invariance. Furthermore, surface fluctuation-dissipation ratios are computed and their asymptotic values are shown to depend on the values of surface critical exponents.

arXiv:cond-mat/0404203Physical Review BSep 3, 200434 citations

Order-Parameter Profile in Semi-Infinite Systems at Criticality

Joseph Rudnick, David Jasnow

A description of the decay into the bulk of surface perturbations in a system at its critical point is presented. The description, justified by detailed renormalization-group calculations, yields asymptotic order-parameter decay, ${z}^{\ensuremath{-}\frac{(d\ensuremath{-}2+\ensuremath{\eta})}{2}}$, along with power-law behavior connected with surface exponents under suitable conditions. Comparison with the phenomenological theories of Widom and of Fisher and de Gennes is also made.A description of the decay into the bulk of surface perturbations in a system at its critical point is presented. The description, justified by detailed renormalization-group calculations, yields asymptotic order-parameter decay, ${z}^{\ensuremath{-}\frac{(d\ensuremath{-}2+\ensuremath{\eta})}{2}}$, along with power-law behavior connected with surface exponents under suitable conditions. Comparison with the phenomenological theories of Widom and of Fisher and de Gennes is also made.

Physical Review LettersApr 19, 198246 citations

The effects of surfaces on dynamic critical behavior

S. Dietrich, Hans Werner Diehl

The European Physical Journal BDec 1, 198359 citations

{\mathit{B}}_{\mathit{A}}

, and one with c${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}$=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.