Dynamic universality class of Model C from the functional renormalization group

Abstract

We establish new scaling properties for the universality class of Model C, which describes relaxational critical dynamics of a nonconserved order parameter coupled to a conserved scalar density. We find an anomalous diffusion phase, which satisfies weak dynamic scaling while the conserved density diffuses only asymptotically. The properties of the phase diagram for the dynamic critical behavior include a significantly extended weak scaling region, together with a strong and a decoupled scaling regime. These calculations are done directly in 2 ≤ d ≤ 4 space dimensions within the framework of the nonperturbative functional renormalization group. The scaling exponents characterizing the different phases are determined along with subleading indices featuring the stability properties.Dynamic properties such as transport coefficients or relaxation rates play a crucial role for a wide variety of physical systems. Irrespective of the details of the underlying microscopic dynamics, they can be grouped into dynamic universality classes close to a critical point. Following the standard classification scheme 1 , the universality class of Model C is characterized in terms of an Ncomponent order parameter with relaxational dynamics coupled to a diffusive field. Apart from being a model for the coupling of the energy density for Ising-like systems close to criticality, it is applied to the critical dynamics of mobile impurities 2 , structural phase transitions 3 , longwavelength fluctuations near the QCD critical point 4 , and out-of-equilibrium dynamics 5 .Despite its importance and a long history of discussions 6-10 , 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. Previous calculations have mainly relied on the expansion in d = 4− dimensions, while direct numerical simulations 11 still represent an exception. 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 6-9 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. In more recent work 10 this region was discarded as an artifact of the expansion, which was argued to break down for 2 < N < 4 close to d = 4.In this paper we compute the (N, d) phase diagram for the dynamic critical behavior of Model C using the functional renormalization group, which is a nonperturbative approach that does not rely on the expansion. We establish an an...

Phase diagram of Model C in the parametric space of order parameter and space dimensions

“…1a ). Therewith this outcome qualitatively supports these results of the NPRG study 29 (see black curves in Fig. 1) obtained for n > 1 and ǫ < 1, showing them trustable.…”

“…It is developed intensively now and is applied to describe scaling properties of different classical and quantum models. In particular it is successful in studies of the critical properties of O(n) models 17,[19][20][21] , models with different types of disorder 22 , systems with complex symmetries [23][24][25] , critical dynamics near equilibrium 26 , reaction-diffusion processes 27 , and obtained within the perturbative field-theoretical approach by resummation of the ǫ-expansion (a) and due to resummation of the RG functions at fixed d (b) in comparison with the non-perturbative results 29 . Solid lines separate the decoupled region D from the weak scaling region W, dashed lines separate the weak scaling region W from the strong scaling region S. Green curves (a): Padé-Borel resummation of epsilon-expansions, data of this paper; blue curves (b): fixed dimension approach based on resummed five-loop RG functions, data of this paper; black curves: nonperturbative RG analysis, data from Ref.…”

“…Recently the NPRG approach was also applied to study model C dynamics 29 . These results show changes concerning the borders separating different regimes in the (n, d) plane (shown by black curves in Fig.…”

Self-consistent spectral functions in the O(N) model from the functional renormalization group

“…The simply continued correlators have the convenient property that they can be calculated for fixed values of the imaginary part of the external momentum, Im p 0 , as the flow equation only involves propagators G(q + p) in addition to Euclidean propagators, G(q). This is no longer possible if one wants to calculate the properly continued correlators via (7) that relate to the retarded correlators via (6), as the evaluation of the pole corrections requires knowledge of the full threedimensional momentum dependence of the propagator. The application of (7) in the context of the FRG leaves two possibilities: either the pole correction can be computed just on the solution of the simply continued propagators, which is the procedure that has been applied in all LPA studies so far, or the full complex propagator including pole correction can be coupled back into all complex equations, i.e.…”

Keldysh field theory for driven open quantum systems

“…Instead, it may be viewed as an alternative and potentially more tractable tool for the analysis of the complete many-body problem, also on length scales well below the correlation length near criticality. Indeed, it has proven a very versatile tool in many different physical context, ranging from quantum dots [234][235][236], ultracold atoms [233], strongly correlated electrons [237], classical stochastic models [238,239], quantum chromodynamics [232,240], to quantum gravity [241]. Here we give a brief overview of the general concept adapted to non-equilibrium systems [26,181,[234][235][236][242][243][244][245][246][247][248][249][250][251][252][253][254].…”