Dynamics of a coupled-mode system: Explicit analysis at orderε2

Abstract

The renormalization-group equations for critical behavior in 4a dimensions are generalized to the discussion of dynamic exponents at T = T, and then applied to a situation recently considered by Hohenberg, Halperin, and Ma (HHM); namely, one in which an n-component primary field Q is coupled to a secondary field q via a Q(QQ)cp coupling. In this work their analysis of relaxational fields is explicitly considered at order &'. For n (4, we find the dynamic exponents differ in one case at 0(&') from that given by HHM. In contrast to HHM, the dynamic scaling postulate is also preserved by the second-order fixed points in all instances and detailed analysis is given. The fixed-point values of R, a dimensionless ratio of relaxation rates for the q and Q fields, in certain cases, is found to go as R» -1/a, which is a new result. This large value where it occurs is found essential to understanding dynamic scaling. Analysis of the case of a propagating conserved q field also is given. It is found that its dynamic critical properties can be characterized by its damping term alone.

Critical Dynamics of ModelCResolved

“…Halperin, Hohenberg, and Ma then compared their second and higher order results with results known at the time [5] and found agreement with the boundary curves of [3]. On the other hand, the peculiarities in the two-loop order and the results of [4] led them to the supposition that the anomalous region might not exist, but they were unable to draw a definite conclusion.…”

“…While in region I the OP dynamical critical exponent takes its model A value, and in region II the value z 2 = ( being the static exponent of the specific heat and the exponent of the correlation length), in the anomalous region the value of z and the validity of dynamical scaling behavior remained unclear. Model C, as it has been named since then, was subsequently treated within the field theoretical version of renormalization group theory up to two-loop order by Brezin and De Dominicis [3] and Murata [4].…”

“…While in region I the OP dynamical critical exponent takes its model A value, and in region II the value z 2 = ( being the static exponent of the specific heat and the exponent of the correlation length), in the anomalous region the value of z and the validity of dynamical scaling behavior remained unclear. Model C, as it has been named since then, was subsequently treated within the field theoretical version of renormalization group theory up to two-loop order by Brezin and De Dominicis [3] and Murata [4].The results of Brezin and De Dominicis corroborated a scenario where the anomalous region exists, and they obtained from their calculations at small values of the boundary curve separating region III from region II, although the extension of the anomalous region to the whole phase space could not be given. Further peculiarities, namely, a nonuniformity, appeared in the two-loop calculation when taking the limits !…”

Dynamic universality class of Model C from the functional renormalization group

“…While the existence of the so-called weak, strong, and decoupled scaling regions is undebated, there have been conflicting claims on important quantitative properties and even on the possible existence of another distinctive region in the phase diagram of Model C as a function of N and d. Earlier results [6][7][8][9] found evidence for such a region, however, it was unclear whether it persists to higher orders in the expansion. Other results to second order showed that for the ratio of kinetic coefficients an essential singularity occurs in this region 8 . It was speculated that this property might even restore critical behavior with a dynamic scaling exponent identical to the strong scaling exponent.…”

“…Despite its importance and a long history of discussions [6][7][8][9][10] , parts of the phase diagram for the dynamic critical behavior of Model C are still controversial. The reason for this uncertainty is that the physics is nonperturbative and only a few theoretical approaches apply.…”