We employ the nonperturbative functional renormalization group to study models with an O(N(1) ⊕O(N)(2)) symmetry. Here different fixed points exist in three dimensions, corresponding to bicritical and tetracritical behavior induced by the competition of two order parameters. We discuss the critical behavior of the symmetry-enhanced isotropic, the decoupled and the biconical fixed point, and analyze their stability in the N(1),N(2) plane. We study the fate of nontrivial fixed points during the transition from three to four dimensions, finding evidence for a triviality problem for coupled two-scalar models in high-energy physics. We also point out the possibility of noncanonical critical exponents at semi-Gaussian fixed points and show the emergence of Goldstone modes from discrete symmetries.

In these lectures we introduce the functional renormalization group out of equilibrium. While in thermal equilibrium typically a Euclidean formulation is adequate, nonequilibrium properties require real-time descriptions. For quantum systems specified by a given density matrix at initial time, a generating functional for real-time correlation functions can be written down using the Schwinger-Keldysh closed time path. This can be used to construct a nonequilibrium functional renormalization group along similar lines as for Euclidean field theories in thermal equilibrium. Important differences include the absence of a fluctuation-dissipation relation for general out-of-equilibrium situations. The nonequilibrium renormalization group takes on a particularly simple form at a fixed point, where the corresponding scale-invariant system becomes independent of the details of the initial density matrix. We discuss some basic examples, for which we derive a hierarchy of fixed point solutions with increasing complexity from vacuum and thermal equilibrium to nonequilibrium. The latter solutions are then associated to the phenomenon of turbulence in quantum field theory.

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...

We present a functional renormalization group investigation of an Euclidean three-dimensional matrix Yukawa model with U (N ) symmetry, which describes N = 2 Weyl fermions that effectively interact via a short-range repulsive interaction. This system relates to an effective low-energy theory of spinless electrons on the honeycomb lattice and can be seen as a simple model for suspended graphene. We find a continuous phase transition characterized by large anomalous dimensions for the fermions and composite degrees of freedom. The critical exponents define a new universality class distinct from Gross-Neveu type models, typically considered in this context.

We study models with three coupled vector fields characterized by O(N1) ⊕ O(N2) ⊕ O(N3) symmetry. Using the nonperturbative functional renormalization group, we derive β functions for the couplings and anomalous dimensions in d dimensions. Specializing to the case of three dimensions, we explore interacting fixed points that generalize the O(N ) Wilson-Fisher fixed point. We find a symmetry-enhanced isotropic fixed point, a large class of fixed points with partial symmetry enhancement, as well as partially and fully decoupled fixed point solutions. We discuss their stability properties for all values of N1, N2, and N3, emphasizing important differences to the related twofield models. For small numbers of field components we find no stable fixed point solutions, and we argue that this can be attributed to the presence of a large class of possible (mixed) couplings in the three-field and multifield models. Furthermore, we contrast different mechanisms for stability interchange between fixed points in the case of the two-and three-field models, which generically proceed through fixed-point collisions.