Critical phenomena and renormalization-group theory

doi:10.1016/S0370-1573(02)00219-3

Physics Reports

Volume 368, Issue 6, October 2002, Pages 549-727

https://doi.org/10.1016/S0370-1573(02)00219-3 Get rights and content

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 β-functions. Finally, we consider a Hamiltonian with symmetry O(n₁)⊕O(n₂) that is relevant for the description of multicritical phenomena.

Section snippets

Plan of the review

The main issue of this review is the critical behavior of spin systems at equilibrium.

In Section 1 we introduce the notations and the basic renormalization-group results for the critical exponents, the equation of state, and the two-point function of the order parameter, which are used throughout the paper.

In Section 2 we outline the most important methods that are used in the study of equilibrium spin systems: high-temperature expansions, Monte Carlo methods, and field-theoretical methods. It

Numerical determination of critical quantities

In this section we review the numerical methods that have been used in the study of statistical systems at criticality. In two dimensions many nontrivial models can be solved exactly, and moreover there exists a powerful tool, conformal field theory, that gives exact predictions for the critical exponents and for the behavior at the critical point. In three dimensions there is no theory providing exact predictions at the critical point. Therefore, one must resort to approximate methods. The

Physical relevance

The Ising model is one of the most studied models in the theory of phase transitions, not only because it is considered as the prototype of statistical systems showing a nontrivial power-law critical behavior, but also because it describes several physical systems. Indeed, many systems characterized by short-range interactions and a scalar order parameter undergo a critical transition belonging to the Ising universality class. We mention the liquid–vapor transition in simple fluids, the

Physical relevance

The three-dimensional XY universality class is characterized by a two-component order parameter and effective short-range interactions with U(1) symmetry. The most interesting representative of this universality class is the superfluid transition of $^{4} He$ along the λ-line T_λ(P). It provides an exceptional opportunity for a very accurate experimental test of the RG predictions, because of the weakness of the singularity in the compressibility of the fluid, of the purity of the samples, and of the

The three-dimensional Heisenberg universality class

The three-dimensional Heisenberg universality class is characterized by a three-component order parameter, O(3) symmetry, and short-range interactions. It describes the critical behavior of isotropic magnets, for instance the Curie transition in isotropic ferromagnets such as Ni and EuO, and of antiferromagnets such as RbMnF₃ at the Néel transition point. Moreover, it describes isotropic magnets with quenched disorder, see also Section 11.4. Indeed, since α<0, the Harris criterion [508] states

Critical behavior of N-vector models with N⩾4

Among the three-dimensional N-vector models with N⩾4, the physically most relevant ones are those with N=4 and 5. The N=4 case is relevant for high-energy physics because it describes the finite-temperature transition in the theory of strong interactions, i.e., quantum chromodynamics (QCD), with two light degenerate flavored quarks. The case N=5 might be relevant for superconductivity: indeed, an SO(5) theory has been proposed to explain the critical properties of high-T_c superconductors [1144]

The Kosterlitz–Thouless critical behavior

The two-dimensional XY universality class is characterized by the Kosterlitz–Thouless (KT) critical behavior [668], [670] (see, e.g., [570], [1152] for reviews on this issue). According to the KT scenario, the free energy has an essential singularity at T_c and the correlation length diverges as $ξ∼ exp (b/t^{σ})$ for t≡T/T_c−1→0⁺. The value of the exponent is σ=1/2 and b is a nonuniversal positive constant. At the critical temperature, the asymptotic behavior for r→∞ of the two-point correlation function

Two-dimensional N-vector models with N⩾3

Two-dimensional N-vector models with N⩾3 are somewhat special since in this case there is no phase transition at finite values of T. The correlation length is always finite and a critical behavior is observed only when T→0.³¹

The limit N→0, self-avoiding walks, and dilute polymers

In this section we discuss the limit N→0 of the N-vector model. This is not an academic problem as it may appear at first. Indeed, in this limit, the N-vector model describes the statistical properties of linear polymers in dilute solutions and in the good-solvent regime, i.e., above the Θ temperature [325], [334], [432], [442]. Note that FT methods can also be applied to describe the full crossover from the dilute to the semidilute regime, and, in particular, to compute the universal scaling

Critical crossover as a two-scale problem

Every physical situation of experimental relevance has at least two scales: one scale is intrinsic to the system, while the second one is related to experimental conditions. In statistical mechanics the correlation length ξ is related to experimental conditions (it depends on the temperature), while the interaction length (related to the Ginzburg parameter G) is intrinsic. The opposite is true in quantum field theory: here the correlation length (inverse mass gap) is intrinsic, while the

Acknowledgements

We thank Tomeu Allés, Pasquale Calabrese, Massimo Campostrini, Sergio Caracciolo, José Carmona, Michele Caselle, Serena Causo, Alessio Celi, Robert Edwards, Martin Hasenbusch, Gustavo Mana, Victor Martı́n-Mayor, Tereza Mendes, Andrea Montanari, Paolo Rossi, Alan Sokal, for collaborating with us on some of the issues considered in this review.

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Critical phenomena and renormalization-group theory

Abstract

Section snippets

Plan of the review

Numerical determination of critical quantities

Physical relevance

Physical relevance

The three-dimensional Heisenberg universality class

Critical behavior of N-vector models with N⩾4

The Kosterlitz–Thouless critical behavior

Two-dimensional N-vector models with N⩾3

The limit N→0, self-avoiding walks, and dilute polymers

Critical crossover as a two-scale problem

Acknowledgements

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