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.
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.
We study the multicritical behavior arising from the competition of two distinct types of ordering characterized by
O(n) symmetries. For this purpose, we consider the renormalization-group flow for the most general $\mathrm{O}{(n}_{1})\ensuremath{\bigoplus}\mathrm{O}{(n}_{2})$-symmetric Landau-Ginzburg-Wilson Hamiltonian involving two fields ${\ensuremath{\varphi}}_{1}$ and ${\ensuremath{\varphi}}_{2}$ with
n1 and
n2 components, respectively. In particular, we determine in which cases, approaching the multicritical point, one may observe the asymptotic enlargement of the symmetry to
O(N) with
N=n1+n2. By performing a five-loop $\ensuremath{\epsilon}$-expansion computation we determine the fixed points and their stability. It turns out that for
N=n1+n2> 3 the
O(N)-symmetric fixed point is unstable. For
N=3, the multicritical behavior is described by the biconal fixed point with critical exponents that are very close to the Heisenberg ones. For
N> 4 and any
n1,n2 the critical behavior is controlled by the tetracritical decoupled fixed point. We discuss the relevance of these results for some physically interesting systems, in particular for anisotropic antiferromagnets in the presence of a magnetic field and for high-
Tc superconductors. Concerning the SO(5) theory of superconductivity, we show that the bicritical O(5) fixed point is unstable with a significant crossover exponent ${\ensuremath{\varphi}}_{4,4}\ensuremath{\approx}0.15;$ this implies that the O(5) symmetry is not effectively realized at the point where the antiferromagnetic and superconducting transition lines meet. The multicritical behavior is either governed by the tetracritical decoupled fixed point or is of first-order type if the system is outside its attraction domain.
We calculate the static critical behavior of systems of O(n_||)(plus sign in circle)O(n_perpendicular) symmetry by the renormalization group method within the minimal subtraction scheme in two-loop order. Summation methods lead to fixed points describing multicritical behavior. Their stability border lines in the space of the order parameter components n_|| and n_perpendicular and spatial dimension d are calculated. The essential features obtained already in two-loop order for the interesting case of an antiferromagnet in a magnetic field ( n_|| =1, n_perpendicular =2 ) are the stability of the biconical fixed point and the neighborhood of the stability border lines to the other fixed points, leading to very small transient exponents. We are also able to calculate the flow of static couplings, which allows us to consider the attraction region. Depending on the nonuniversal background parameters, the existence of different multicritical behavior (bicritical or tetracritical) is possible, including a triple point.
The critical behavior of many physical systems involves two competing n1- and n2-component order-parameters, S1 and S2, respectively, with n = n1 + n2. Varying an external control parameter g, one encounters ordering of S1 below a critical (second-order) line for g < 0 and of S2 below another critical line for g > 0. These two ordered phases are separated by a first-order line, which meets the above critical lines at a bicritical point, or by an intermediate (mixed) phase, bounded by two critical lines, which meet the above critical lines at a tetracritical point. For n = 1 + 2 = 3, the critical behavior around the (bi- or tetra-) multicritical point either belongs to the universality class of a non-rotationally invariant (cubic or biconical) fixed point, or it has a fluctuation driven first-order transition. These asymptotic behaviors arise only very close to the transitions. We present accurate renormalization-group flow trajectories yielding the effective crossover exponents near multicriticality.
We discuss the static and dynamic multicritical behavior of three-dimensional systems of
O(n∥)⊕O(n⊥) symmetry as it is explained by the field theoretical renormalization group method. Whereas the static renormalization group functions are currently know within high order expansions, we show that an account of two loop contributions refined by an appropriate resummation technique gives an accurate quantitative description of the multicritical behavior. One of the essential features of the static multicritical behavior obtained already in two loop order for the interesting case of an antiferromagnet in a magnetic field (
n∥=1,
n⊥=2) are the stability of the biconical fixed point and the neighborhood of the stability border lines to the other fixed points leading to very small transient exponents. We further pursue an analysis of dynamical multicritical behavior choosing different forms of critical dynamics and calculating asymptotic and effective dynamical exponents within the minimal subtraction scheme.
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.