Critical Phenomena and Renormalization-Group Theory

TL;DR

This work surveys critical phenomena across Ising and O($N$) universality classes, detailing RG-based scaling laws for the free energy, equation of state, and two-point functions in 2D and 3D. It synthesizes field-theoretic (fixed-dimension and $\epsilon$-expansion), numerical (HT, MC, and FT methods), and nonperturbative approaches, highlighting universal amplitudes, scaling functions, and corrections to scaling. The review also covers complex Landau-Ginzburg-Wilson Hamiltonians (cubic anisotropy, quenched disorder, frustration) and the $N\to0$ limit for self-avoiding walks/polymers, including extensive treatment of crossover phenomena and media-range interactions. By compiling high-precision exponents and amplitude ratios across multiple models and dimensions, the paper provides benchmarks for theory, numerics, and experiments and emphasizes the RG framework as the unifying language for critical phenomena. Its detailed parametric EOS representations and six-loop perturbative results for tetragonal systems broaden the scope of universality studies and cross-disciplinary applications.

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 $N\to 0$ 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 $β$-functions. Finally, we consider a Hamiltonian with symmetry $O(n_1)\oplus O(n_2)$ that is relevant for the description of multicritical phenomena.

Figures (19)

• Figure 1: The phase diagram of a magnetic system (left) and of a simple fluid (right).
• Figure 2: Estimates of $g_4^+$ obtained from an unbiased analysis (direct) of the HT series and from the analysis (RT) of the series obtained by means of the Roskies transform (\ref{['RTr']}), for the $\phi^4$ lattice model. The dashed line marks the more precise estimate (with its error) derived from the analysis of an improved HT expansion in Ref. CPRV-02, $g_4^+=23.56(2)$.
• Figure 3: MC results for the four-point coupling $g_4$ for the three-dimensional Ising model: (a) from Ref. BK-96; (b) from Ref. KL-96. For comparison we also report the extrapolation of the 18th-order HT series of Ref. CPRV-99 by means of a direct analysis ($\rm HT_{\rm direct}$) and of an analysis that uses the transformation (\ref{['RTr']}) ($\rm HT_{RT}$). For each of these extrapolations we report two lines corresponding to the one-error-bar interval.
• Figure 4: The scaling functions $f(x)$, $F(z)$, and $Q(u)$. We also plot the following asymptotic behaviors (dotted lines): $f(x)$ at the coexistence curve, i.e., $f(x)\approx f_1^{\rm coex} (1+x)$ for $x\rightarrow -1$; $F(z)$ at the HT line, i.e., $F(z)\approx z + \frac{1}{6} z^3 + \frac{1}{120} r_6 z^5$ for $z\rightarrow 0$; $Q(u)$ at the coexistence curve, i.e., $Q(u) \approx (u-1) + \frac{1}{2} v_3 (u-1)^2 + \frac{1}{6} v_4 (u-1)^3$ for $u\rightarrow 1$. Results from Ref. CPRV-02.
• Figure 5: The scaling functions $E(y)$ and $D(y)$. We also plot their asymptotic behaviors (dotted lines): $E(y)\approx R_\chi y^{-\gamma}$ for $y\rightarrow +\infty$, and $E(y)\approx (- y)^{\beta}$ for $y\rightarrow -\infty$; $D(y)\approx R_\chi y^{-\gamma}$ for $y\rightarrow +\infty$, and $D(y)\approx \beta(- y)^{-\gamma}/f_1^{\rm coex}$ for $y\rightarrow -\infty$; Results from Ref. CPRV-02.