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Journal of Physics C Solid State Physics

Critical behaviour in a driven diffusive system with an odd induced current

Juha Honkonen, Antti J. Kupiainen

Published Apr 30, 1987• 0 citations
Publisher

Abstract

The authors present a field theoretic renormalisation group analysis of the critical behaviour of a diffusive system subject to an external field for the case of an odd induced current. The fixed-point equations are shown to impose constraints on the critical exponents. Below D c =3, the solution that is stable at one-loop order corresponds to a fixed point with anomalous dimensions vanishing to all orders.

DOI
10.1088/0022-3719/20/12/004

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The authors present a field theoretic renormalisation group analysis of the critical behaviour of a diffusive system subject to an external field for the case of an odd induced current. The fixed-point equations are shown to impose constraints on the critical exponents. Below D c =3, the solution that is stable at one-loop order corresponds to a fixed point with anomalous dimensions vanishing to all orders.
We present a field theoretic renormalization group study for the critical behaviour of a uniformly driven diffusive system with quenched disorder, which is modelled by different kinds of potential barriers between sites. Due to their symmetry properties, these different realizations of the random potential barriers lead to three different models for the phase transition to transverse order and to one model for the phase transition to longitudinal order all belonging to distinct universality classes. In these four models that have different upper critical dimensions dc we find the critical scaling behaviour of the vertex functions in spatial dimensions d < dc. Its deviation from purely diffusive behaviour is characterized by the anomaly–exponent η that we calculate at first and second order, respectively in ǫ = dc − d. In each model η turns out to be positive which means superdiffusive spread of density fluctuations in the driving force direction. 1 1
This article concludes a series of papers [Folk, Holovatch, and Moser, Phys. Rev. E 78, 041124 (2008); 78, 041125 (2008); 79, 031109 (2009)] where the tools of the field theoretical renormalization group were employed to explain and quantitatively describe different types of static and dynamic behavior in the vicinity of multicritical points. Here we give the complete two-loop calculation and analysis of the dynamic renormalization-group flow equations at the multicritical point in anisotropic antiferromagnets in an external magnetic field. We find that the time scales of the order parameters characterizing the parallel and perpendicular ordering with respect to the external field scale in the same way. This holds independent whether the Heisenberg fixed point or the biconical fixed point in statics is the stable one. The nonasymptotic analysis of the dynamic flow equations shows that due to cancellation effects the critical behavior is described, in distances from the critical point accessible to experiments, by the critical behavior qualitatively found in one-loop order. Although one may conclude from the effective dynamic exponents (taking almost their one-loop values) that weak scaling for the order parameter components is valid, the flow of the time-scale ratios is quite different, and they do not reach their asymptotic values.
Renormalization group (RG) flows for systems subject to an external field can be derived either by fixing the field or by fixing the conjugate order parameter. We show how to implement these boundary conditions within the functional renormalization group approach. We derive and compare the corresponding flow equations for ϕ4ϕ^4ϕ4 theory and show that some scaling variables and scaling exponents associated with the linearized flow of the relevant three-point vertex in the vicinity of the Ising fixed point depend on the boundary conditions. We also discuss the effect of elasticity on Ising criticality. In this case two of the fixed points (namely the renormalized Ising fixed point and the spherical fixed point) obtained for constant strain are shifted to physically inaccessible regimes of parameter space when we perform the RG at constant stress.
The fixed points of the exact renormalisation group equations with smooth cut-off are determined in the epsilon -expansion for systems characterised by mn-component vector order-parameters. The critical behaviour of quenched random n-component spin systems is obtained by taking the limit m=O. The critical exponent is obtained to order epsilon 2 , and the other exponents to order epsilon . A derivation of Nelson's trajectory-integral expression for the free energy, appropriate for the case of smooth cut-off renormalisation group equations, is presented.
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\proptoδ(t-t') k^{4-d-y}∝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′\proptoδ(t-t') k^{4-d-y'}∝δ(t−t′)k4−d−y′. Depending on the relations between the exponents yyy and y′y'y′ and the space dimensionality ddd, 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 yyy, y′y'y′ and ε=4−dε=4-dε=4−d).