Stochastic dynamics of a nonconserved scalar order parameter near its critical point, subject to random stirring and mixing, is studied using the field theoretic renormalization group. The stirring and mixing are modelled by a random external Gaussian noise with the correlation function
∝δ(t−t′)k4−d−y and the divergence-free (due to incompressibility) velocity field, governed by the stochastic Navier--Stokes equation with a random Gaussian force with the correlation function
∝δ(t−t′)k4−d−y′. Depending on the relations between the exponents
y and
y′ and the space dimensionality
d, the model reveals several types of scaling regimes. Some of them are well known (model A of equilibrium critical dynamics and linear passive scalar field advected by a random turbulent flow), but there are three new nonequilibrium regimes (universality classes) associated with new nontrivial fixed points of the renormalization group equations. The corresponding critical dimensions are calculated in the two-loop approximation (second order of the triple expansion in
y,
y′ and
ε=4−d).
We consider the breakdown of conformal and scale invariance in random systems with strongly random critical points. Extending previous results on one-dimensional systems, we provide an example of a three-dimensional system that has a strongly random critical point. The average correlation functions of this system demonstrate a breakdown of conformal invariance, while the typical correlation functions demonstrate a breakdown of scale invariance. The breakdown of conformal invariance is due to the vanishing of the correlation functions at the infinite disorder fixed point, causing the critical correlation functions to be controlled by a dangerously irrelevant operator describing the approach to the fixed point. We relate the computation of average correlation functions to a problem of persistence in the renormalization group flow.
Correlation-function exponents ${\ensuremath{\eta}}_{\ensuremath{\perp}}$ and ${\ensuremath{\eta}}_{\ensuremath{\parallel}}$ appropriate to the free-surface problem have been obtained by renormalization-group calculation to order ${\ensuremath{\epsilon}}^{2}$. By using the scaling relations ${\ensuremath{\gamma}}_{1}=\ensuremath{\nu}(2\ensuremath{-}{\ensuremath{\eta}}_{\ensuremath{\perp}})$ and ${\ensuremath{\gamma}}_{11}=\ensuremath{\nu}(1\ensuremath{-}{\ensuremath{\eta}}_{\ensuremath{\parallel}})$, expansions for ${\ensuremath{\gamma}}_{1}$ and ${\ensuremath{\gamma}}_{11}$ are obtained. These expansions are in agreement with the surface scaling relation $2{\ensuremath{\gamma}}_{1}\ensuremath{-}{\ensuremath{\gamma}}_{11}=\ensuremath{\gamma}+\ensuremath{\nu}$, but disagree with the relation ${\ensuremath{\gamma}}_{11}=\ensuremath{\nu}\ensuremath{-}1$ due to Bray and Moore.
The authors consider m two-dimensional semi-infinite planes of Ising spins joined together through surface spins and study the critical behaviour near to the junction. The m=0 limit of the model-according to the replica trick-corresponds to the semi-infinite Ising model in the presence of a random surface field (RSFI). Using conformal mapping, second-order perturbation expansion around the weak- and strong-coupled planes limits and a differential renormalization group, they show that the surface critical behaviour of the RSFI model is described by Ising critical exponents with logarithmic corrections to scaling, while at multiple junctions (m>2) the transition is of first-order. There is a spontaneous junction magnetization at the bulk critical point.
We study the three-state antiferromagnetic Potts model on the simple-cubic lattice, paying attention to the surface critical behaviors. When the nearest-neighboring interactions of the surface is tuned, we obtain a phase diagram similar to the XY model, owing to the emergent O(2) symmetry of the bulk critical point. For the ordinary transition, we get ${y}_{h1}=0.780(3), {\ensuremath{\eta}}_{\ensuremath{\parallel}}=1.44(1)$, and ${\ensuremath{\eta}}_{\ensuremath{\perp}}=0.736(6)$; for the special transition, we get ${y}_{s}=0.59(1), {y}_{h1}=1.693(2), {\ensuremath{\eta}}_{\ensuremath{\parallel}}=\ensuremath{-}0.391(4)$, and ${\ensuremath{\eta}}_{\ensuremath{\perp}}=\ensuremath{-}0.179(5)$; in the extraordinary-log phase, the surface correlation function ${C}_{\ensuremath{\parallel}}(r)$ decays logarithmically with decaying exponent
q=0.60(2), however, the correlation ${C}_{\ensuremath{\perp}}(r)$ still decays algebraically with critical exponent ${\ensuremath{\eta}}_{\ensuremath{\perp}}=\ensuremath{-}0.442(5)$. If the ferromagnetic next-nearest-neighboring surface interactions are added, we find two transition points, the first one is a special point between the ordinary phase and the extraordinary-log phase, the second one is a transition between the extraordinary-log phase and the
Z6 symmetry-breaking phase, with critical exponent
ys=0.41(2). The scaling behaviors of the second transition is very interesting, the surface spin-correlation function ${C}_{\ensuremath{\parallel}}(r)$, and the surface squared staggered magnetization at this point decays logarithmically with exponent
q=0.37(1); however, the surface structure factor with the smallest wave vector and the correlation function ${C}_{\ensuremath{\perp}}(r)$ satisfy power-law decaying, with critical exponents ${\ensuremath{\eta}}_{\ensuremath{\parallel}}=\ensuremath{-}0.69(1)$ and ${\ensuremath{\eta}}_{\ensuremath{\perp}}=\ensuremath{-}0.37(1)$, respectively.
With use of a novel Monte Carlo method suited for the study of surface phenomena, the authors have investigated a three-dimensional Ising model with modified surface exchange. The results verify the scaling behavior of the magnetization profile and confirm and extend renormalization-group predictions. The surface-layer magnetization exponent is ${\ensuremath{\beta}}_{1}=0.78\ifmmode\pm\else\textpm\fi{}0.02$ except at the "special" (multicritical) transition where ${{\ensuremath{\beta}}_{1}}^{m}=0.175\ifmmode\pm\else\textpm\fi{}0.025$ and the crossover exponent is $\ensuremath{\phi}=0.56\ifmmode\pm\else\textpm\fi{}0.04$.