We study models with three coupled vector fields characterized by O(N_{1})⊕O(N_{2})⊕O(N_{3}) 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 N_{1},N_{2}, and N_{3}, emphasizing important differences to the related two-field 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.
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.
Abstract Chapter 7 is devoted to a discussion of the renormalization group (RG) flow when the effective field theory that describes universal properties of critical phenomena depends on several coupling constants. The universal properties of a large class of macroscopic phase transitions with short range interactions can be described by statistical field theories involving scalar fields with quartic interactions. The simplest critical systems have an O(N) orthogonal symmetry and, therefore, the corresponding field theory has only one quartic interaction. However, in more general physical systems, the flow of quartic interactions is more complicated. This chapter examines these systems from the RG viewpoint. RG beta functions are shown to generate a gradient flow. Some examples illustrate the notion of emergent symmetry. The local stability of fixed points is related to the value of the scaling field dimension.
The detailed analysis of the global structure of the renormalization-group (RG) flow diagram for a model with isotropic and cubic interactions is carried out in the framework of the massive field theory directly in three dimensions (3D) within an assumption of isotropic exchange. Perturbative expansions for RG functions are calculated for arbitrary N up to four-loop order and resummed by means of the generalized Pad\'e-Borel-Leroy technique. Coordinates and stability matrix eigenvalues for the cubic fixed point are found under the optimal value of the transformation parameter. Critical dimensionality of the model is proved to be equal to ${N}_{c}=2.89\ifmmode\pm\else\textpm\fi{}0.02$ that agrees well with the estimate obtained on the basis of the five-loop $\ensuremath{\varepsilon}$ expansion [H. Kleinert and V. Schulte-Frohlinde, Phys. Lett. B 342, 284 (1995)] resummed by the above method. As a consequence, the cubic fixed point should be stable in 3D for
N> 3, and the critical exponents controlling phase transitions in three-dimensional magnets should belong to the cubic universality class. The critical behavior of the random Ising model being the nontrivial particular case of the cubic model when
N=0 is also investigated. For all physical quantities of interest the most accurate numerical estimates with their error bounds are obtained. The results achieved in the work are discussed along with the predictions given by other theoretical approaches and experimental data.
We calculate the relaxational dynamical critical behavior of systems of O(n_||)(plus sign in circle)O(n_perpendicular) symmetry by renormalization group method within the minimal subtraction scheme in two-loop order. The three different bicritical static universality classes previously found for such systems correspond to three different dynamical universality classes within the static borderlines. The Heisenberg and the biconical fixed point lead to strong dynamic scaling whereas in the region of stability of the decoupled fixed point weak dynamic scaling holds. Due to the neighborhood of the stability border between the strong and the weak scaling dynamic fixed point to the dynamical stable fixed point a very small dynamic transient exponent of omega(Beta)_(v) =0.0044 is present in the dynamics for the physically important case n_|| =1 and n_perpendicular =2 in d=3 .
The renormalization-group recursion relations are solved for the effective Hamiltonians relative to phase transitions with four-component order parameters. For this value of n there are 22 types of Hamiltonians which can be classified into two categories according to the action of their normalizer
GN on the corresponding parameter space (
GN is the symmetry group leaving globally invariant this space). In the first case,
GN generates a finite number of isolated fixed points whose characteristics can be deduced from the detailed investigation of five Hamiltonians only.In the second category, for which
GN is a continuous group, there are, in addition to isolated fixed points, continuous manifolds of physically equivalent fixed points (the dimension of the manifolds is either one or three). In the search for a stable fixed point, the continuous manifolds can be ignored, while the isolated points are related to the five former Hamiltonians. For n=4, it is necessary to solve the recursion relations to two-loop order. The only possible stable ones among the fixed points then arise from a splitting of points which coincide, to one-loop order, with the isotropic fixed point. Extending, to two-loop order, a result recently established to the preceding order, we show that if a stable fixed point exists, it is unique. For n=4, the stable fixed point has one of three possible symmetries : di-icosahedral, hypercubic, or dicylindrical.Despite this anisotropy of the critical fluctuations, the exponents associated with any of the stable fixed points are identical to order ${\ensuremath{\epsilon}}^{2}$ to the ``isotropic'' exponents corresponding to n=4. The cubic point is destabilized by any operator of symmetry lower than cubic. The dicylindrical one remains stable with respect to certain anisotropies of lower symmetry. We examine the available experimental data in light of the preceding theoretical results concerning the critical behavior and the thermodynamic order of the transitions. On the other hand, we establish two general symmetry conditions relative to the stable fixed points determined by the renormalization-group equations. The first one specifies group theoretically, for each value of n, the possible symmetries
Gi∗ of the stable fixed points. The second one formulates a necessary condition for the occurrence of an anisotropic stable fixed point: The normalizer
GN of the considered parameter space must fulfill the condition
GN\ensuremath{\subset}@;
BGi∗ for one, at least, of the former
Gi∗ groups. These rules are shown to be very restrictive for n=4: On the basis of symmetry the lack of a stable fixed point can be asserted for 10 Hamiltonians out of 22, without solving the fixed-point equations.
We explore universal critical behavior in models with two competing order parameters, and an O(N) ⊕O(M) symmetry for dimensions d≤3. In d=3, there is always exactly one stable renormalization group fixed point, corresponding to bicritical or tetracritical behavior. Employing pseudospectral techniques to solve functional renormalization group equations in a two-dimensional field space, we uncover a more intricate structure of fixed points in d<3, where two additional bicritical fixed points play a role. Towards d=2, we discover ranges of N=M with several simultaneously stable fixed points, indicating the coexistence of several universality classes.