We investigate the crossover of critical behavior for the dynamic phase transition (DPT) in ferromagnetic thin films using Monte Carlo simulations of the kinetic Ising model, focusing on the scaling behavior of the dynamic order parameter under a time-dependent external magnetic field. Specifically, we study the transition of the critical behavior of such multilayer film systems from two-dimensional (2D) to three-dimensional (3D) as a function of the film thickness and the distance to the critical point, which enables dimensional crossover observations. Our results indicate that the effective critical exponents exhibit a clear transition in their scaling behavior, with thinner films showing 2D-like characteristics and thicker films displaying 3D-like behavior, for both the DPT and the thermodynamic phase transitions (TPT). Quantitatively, the crossover from 2D to 3D behavior occurs at larger film thicknesses for the DPT compared to the TPT, suggesting that DPT and TPT are governed by distinctly different length scales and underlying surface effects. These findings are in agreement with experimental observations in ultrathin Co films, where dynamic and thermodynamic critical exponents were found to differ. Therefore, our study provides an in-depth explanation for critical phenomena in thin-film ferromagnets driven by a time-dependent magnetic field. By comparing the dimensional crossover properties of both TPT and DPT, we present a comprehensive understanding of how thin-film geometry and surface effects influence the scaling laws and critical behavior in nonequilibrium systems.We investigate the crossover of critical behavior for the dynamic phase transition (DPT) in ferromagnetic thin films using Monte Carlo simulations of the kinetic Ising model, focusing on the scaling behavior of the dynamic order parameter under a time-dependent external magnetic field. Specifically, we study the transition of the critical behavior of such multilayer film systems from two-dimensional (2D) to three-dimensional (3D) as a function of the film thickness and the distance to the critical point, which enables dimensional crossover observations. Our results indicate that the effective critical exponents exhibit a clear transition in their scaling behavior, with thinner films showing 2D-like characteristics and thicker films displaying 3D-like behavior, for both the DPT and the thermodynamic phase transitions (TPT). Quantitatively, the crossover from 2D to 3D behavior occurs at larger film thicknesses for the DPT compared to the TPT, suggesting that DPT and TPT are governed by distinctly different length scales and underlying surface effects. These findings are in agreement with experimental observations in ultrathin Co films, where dynamic and thermodynamic critical exponents were found to differ. Therefore, our study provides an in-depth explanation for critical phenomena in thin-film ferromagnets driven by a time-dependent magnetic field. By comparing the dimensional crossover properties of both TPT and DPT, we present a comprehensive understanding of how thin-film geometry and surface effects influence the scaling laws and critical behavior in nonequilibrium systems.

arXiv:2412.20579Physical review. ESep 5, 20251 citation

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

Hans Werner Diehl

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

Kibble-Zurek Dynamics in the Anisotropic Ising Model of the Si(001) Surface

G. Schaller, Friedemann Queißer, S. P. Katoorani +8 more

As a simplified description of the nonequilibrium dynamics of buckled dimers on the Si(001) surface, we consider the anisotropic two-dimensional (2D) Ising model and study the freezing of spatial correlations during a cooling quench across the critical point. Depending on the cooling rate, we observe a crossover from one-dimensional (1D) to 2D behavior. For rapid cooling, we find effectively 1D behavior in the strongly coupled direction, for which we provide an exact analytic solution of the nonequilibrium dynamics. For slower cooling rates, we start to see 2D behavior where our numerical simulations show an approach to the usual Kibble-Zurek scaling in 2D.As a simplified description of the nonequilibrium dynamics of buckled dimers on the Si(001) surface, we consider the anisotropic two-dimensional (2D) Ising model and study the freezing of spatial correlations during a cooling quench across the critical point. Depending on the cooling rate, we observe a crossover from one-dimensional (1D) to 2D behavior. For rapid cooling, we find effectively 1D behavior in the strongly coupled direction, for which we provide an exact analytic solution of the nonequilibrium dynamics. For slower cooling rates, we start to see 2D behavior where our numerical simulations show an approach to the usual Kibble-Zurek scaling in 2D.

arXiv:2310.18216Physical Review LettersMay 15, 20253 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.