Mean-Field Theory and the Gaussian Approximation

G. Puddu

David A. Lavis

Nigel D. Goldenfeld

Mean field theory works well when R/a is large or the co-ordination number is large. In both cases, each degree of freedom is coupled to many neighbors. Mean field theory is exact when each spin is coupled to all other spins. In some complex situations, a systematic and relatively easy way to generate mean field theory is by extending the Hamiltonian to be infinite range. Another way of generating mean field theory is to allow the order parameter to have n components and to consider the case n → ∞. The chapter examines the limitations of the phenomenological theory. It provides the analysis which has been for the Landau theory of the Ising universality class. The chapter discusses the Gaussian approximation to the functional integral, and shows how the critical exponents are calculated the lowest order systematic correction to mean field theory.

Nikolai Borisovich Melnikov, Boris Il'ich Reser

Jean Zinn‐Justin

Abstract In Chapter 14, the singular behavior of ferromagnetic systems with O(N) symmetry and short-range interactions, near a second order phase transition has been determined in the mean-field approximation, which is also a quasi-Gaussian approximation. The mean-field approximation predicts a set of universal properties, properties independent of the detailed structure of the microscopic Hamiltonian, the dimension of space, and, to a large extent, of the symmetry of systems. However, the leading corrections to the mean-field approximation, in dimensions smaller than or equal to four, diverge at the critical temperature, and the universal predictions of the mean-field approximation cannot be correct. Such a problem originates from the non-decoupling of scales and leads to the question of possible universality. In Chapter 9, the question has been answered in four dimensions using renormalization theory, and related renormalization group (RG) equations. Moreover, below four dimensions, in an expansion around the mean-field, the most singular terms near criticality can be also formally recovered from a continuum, low-mass φ4 field theory. More generally, following Wilson, to understand universality beyond the mean-field approximation, it is necessary to build a general renormalization group in the form of flow equations for effective Hamiltonians and to find fixed points of the flow equations. Near four dimensions, the flow equations can be approximated by the renormalization group of quantum field theory (QFT), and the fixed points and critical behaviours derived within the framework of the Wilson-Fisher ϵ expansion.