We discuss several examples of three-dimensional critical phenomena that can be described by Landau–Ginzburg–Wilson ϕ4 theories. We present an overview of field-theoretical results obtained from the analysis of high-order perturbative series in the frameworks of the ∊ and of the fixed-dimension d=3 expansions. In particular, we discuss the stability of the O (N)-symmetric fixed point in a generic N-component theory, the critical behaviors of randomly dilute Ising-like systems and frustrated spin systems with noncollinear order, and the multicritical behavior arising from the competition of two distinct types of ordering with symmetry O (n1) and O (n2) respectively.
We study the nature of the multicritical point in the three-dimensional $\mathrm{O}(3)\ensuremath{\bigoplus}\mathrm{O}(2)$ symmetric Landau-Ginzburg-Wilson theory, which describes the competition of two order parameters that are O(3) and O(2) symmetric, respectively. This study is relevant for SO(5) theory of high-
Tc superconductors, which predicts the existence of a multicritical point in the temperature-doping phase diagram, where the antiferromagnetic and superconducting transition lines meet. We investigate whether $\mathrm{O}(3)\ensuremath{\bigoplus}\mathrm{O}(2)$ symmetry gets effectively enlarged to O(5) approaching the multicritical point. For this purpose, we study the stability of the O(5) fixed point. By means of a Monte Carlo simulation, we show that the O(5) fixed point is unstable with respect to the spin-4 quartic perturbation with the crossover exponent ${\ensuremath{\phi}}_{4,4}=0.180(15)$, in substantial agreement with recent field-theoretical results. This estimate is much larger than the one-loop
ϵ-expansion estimate ${\ensuremath{\phi}}_{4,4}=1∕26$, which has often been used in the literature to discuss the multicritical behavior within SO(5) theory. Therefore, no symmetry enlargement is generically expected at the multicritical transition. We also perform a five-loop field-theoretical analysis of the renormalization-group flow. It shows that bicritical systems are not in the attraction domain of the stable decoupled fixed point. Thus, in these systems---high-
Tc cuprates should belong to this class---the multicritical point corresponds to a first-order transition.
We review results concerning the critical behavior of spin systems at equilibrium. We consider the Ising and the general O(N)-symmetric universality class. For each of them, we review the estimates of the critical exponents, of the equation of state, of several amplitude ratios, and of the two-point function of the order parameter. We report results in three and two dimensions. We discuss the crossover phenomena that are observed in this class of systems. In particular, we review the field-theoretical and numerical studies of systems with medium-range interactions. Moreover, we consider several examples of magnetic and structural phase transitions, which are described by more complex Landau-Ginzburg-Wilson Hamiltonians, such as N-component systems with cubic anisotropy, O(N)-symmetric systems in the presence of quenched disorder, frustrated spin systems with noncollinear or canted order, and finally, a class of systems described by the tetragonal Landau-Ginzburg-Wilson Hamiltonian with three quartic couplings. The results for the tetragonal Hamiltonian are original, in particular we present the six-loop perturbative series for the β-functions and the critical exponents.
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 study the combined effect of cubic anisotropy and quenched uncorrelated impurities on multicomponent spin models. For this purpose, we consider the field-theoretical approach based on the Ginzburg-Landau-Wilson ${\ensuremath{\varphi}}^{4}$ Hamiltonian with cubic-symmetric quartic interactions and quenched randomness coupled to the local energy density. We compute the renormalization-group functions to six loops in the fixed-dimension
(d=3) perturbative scheme. The analysis of such high-order series provides an accurate description of the renormalization-group flow. The results are also used to determine the critical behavior of three-dimensional antiferromagnetic three- and four-state Potts models in the presence of quenched impurities.
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.