Critical phenomena and renormalization-group theory
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 along the λ-line Tλ(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-Tc 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 Tc and the correlation length diverges asfor t≡T/Tc−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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