Critical phenomena and renormalization-group theory

Abstract

We review results concerning the critical behavior of spin systems at equilibrium. We consider the Ising and the general O(N)-symmetric universality classes, including the N --> 0 limit that describes the critical behavior of self-avoiding walks. 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 beta-functions. Finally, we consider a Hamiltonian with symmetry 0(n(1))circle plus 0(n(2)) that is relevant for the description of multicritical phenomena. (C) 2002 Elsevier Science B.V. All rights reserved

Nonperturbative U(1) gauge theory at finite temperature

“…Although our estimates of γ/ν agree with what is expected, α/ν is entirely off. For the XY model a small negative value is established [17], while Fig. 3 shows that all our specific heat maxima increase steadily.…”

“…But for N τ = 5 and 6 the Q values are unacceptably small, although the data scatter nicely about the curves. For large N s the maxima of the specific heat curves scale like (see [17])…”

The universal equation of state near the critical point of QCD

“…The universal function f Z 2 (x) is specified by the Ising universality class [21,37]. The critical exponents are β = 0.33, δ = 4.8.…”

Functional renormalization group in the broken symmetry phase: momentum dependence and two-parameter scaling of the self-energy

“…The corresponding scaling function σ + (kξ, k/k c ) in the symmetric phase (i.e., for temperature T > T c ) has recently been discussed in [20]. The fact that the Ginzburg scale k c appears in the scaling of thermodynamic variables has been discussed in several recent works [21,22,4], but apparently has been ignored in the older RG literature [23,24,25]. For models with weak interactions, a universal regime, covering the complete crossover from the vicinity of the Gaussian fixed point to the vicinity of the Wilson-Fisher-fixed point, exists which can be described completely within a two parameter scaling theory [21,22,4].…”

“…The fact that the Ginzburg scale k c appears in the scaling of thermodynamic variables has been discussed in several recent works [21,22,4], but apparently has been ignored in the older RG literature [23,24,25]. For models with weak interactions, a universal regime, covering the complete crossover from the vicinity of the Gaussian fixed point to the vicinity of the Wilson-Fisher-fixed point, exists which can be described completely within a two parameter scaling theory [21,22,4]. For the weakly interacting Bose gas at criticality the one-parameter scaling function σ * (k/k c ) = σ − (∞, k/k c ) has been calculated in [26,27,17,18].…”