The selfconsistency conditions inherent in the mean field approximation are generalized in several ways according to the symmetry of the Hamiltonian. A substantial improvement on the estimates of thermodynamic properties and critical parameters is obtained through rather simple calculations. Non-classical values of critical exponents are found, implementing the mean-field renormalization group method. These techniques can be applied both to classical and to quantum-mechanical systems, especially in low dimensions.
The selfconsistency conditions inherent in the mean field approximation are generalized in several ways according to the symmetry of the Hamiltonian. A substantial improvement on the estimates of thermodynamic properties and critical parameters is obtained through rather simple calculations. Non-classical values of critical exponents are found, implementing the mean-field renormalization group method. These techniques can be applied both to classical and to quantum-mechanical systems, especially in low dimensions.
The self-consistency conditions inherent in the mean-field approximation are extended to take into account some pair correlations.Ii is shown that this extension is systematically improvable and applicable to the spin systems with short-range interactions in the entire temperature region.With two correlations included and the renormalization-group ideas implemented, the critical properties of the Ising model are satisfactorily revealed.Moreover, the improved estimates of the thermodynamical quantities for the quantum linear XY model from the finite-size calculations are found down to very low temperatures.
Renormalization group recursions based on mean-field approximations [J. O. Indekeu, A. Maritan, and A. L. Stella, J. Phys.A15, L291 (1982)], commonly referred to as mean-field renormalization group methods (MFRG), have proven to be efficient and easily applicable for computing non-classical critical properties of lattice models. We give a fairly complete bibliography of applications to date, and extend previous test calculations of bulk, surface, and corner critical exponents in the two-dimensional Ising model to larger cluster sizes on triangular, square (including crossing bonds), and honeycomb lattices. Without much effort the exact value of the critical exponent ratioyH/yT is reproduced systematically with a precision of 2%. This ratio turns out to be the most accurate probe of non-classical critical behaviour that is available in the MFRG method.
Before embarking on an exploration of the modern theories of critical behaviour, it is wise to consider first the more traditional approaches, generally lumped together under the heading of mean field theory . Despite the fact that such methods generally fail sufficiently close to the critical point, there are nonetheless several good reasons for their study. Mean field theory is relatively simple to apply in most cases, and often gives a qualitatively correct picture of the phase diagram of a given model. In some cases, where fluctuation effects are suppressed for some physical reason, its predictions are also quantitatively accurate. In a sufficiently large number of spatial dimensions, it gives exact values for the various critical exponents. Moreover, it often serves as an important adjunct to the renormalization group, since the latter by itself may give direct information about the existence and location of phase transitions, but not about the nature of the phases which they separate. For this, further input is necessary, and this often may be provided by applying mean field theory in a region of the phase diagram far from the critical region, where it is applicable. The mean field free energy There are as many derivations of the basic equations of mean field theory as there are books written on the subject, all of varying degrees of rigour and completeness.
Abstract In the previous chapter we saw that it is possible to make a great deal of progress in the study of particular models by real-space renormalization methods. These methods are very elegant and can explain why critical exponents arise and how they are related to each other. However, they are also very specific: it requires considerable ingenuity to invent an accurate (and tractable) real-space scaling transformation for any given model, and even when one has succeeded it is not normally very helpful in tackling the next problem.
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.