Renormalized Gaussian approach to critical fluctuations in the Landau–Ginzburg–Wilson model and finite-size scaling

Abstract

This paper investigates the application of the renormalized Gaussian approach (RGA) on spatial fluctuations and their manifestations in the vicinity of the critical point. The basis for calculation is the Landau–Ginzburg–Wilson (LGW) Hamiltonian (HLGW). Within the framework of this approach, the model predictions are examined in some detail and the importance of fluctuations is established. Analytic expressions for the correlation length, susceptibility, correlation function as well as the specific heat are calculated and an approximate Ginzburg criterion is deduced. The approach also establishes the limitations of the mean-field theory description, since it is found that the mean-field critical temperature Tc is a characteristic scale of temperature linked to the thermal fluctuations rather than the transition temperature. By applying the RGA, the concept of dimensionality appears essential in the manifestation and analysis of phase transitions since the approach suggests the dependence of some critical parameters and exponents on the dimensionality, as well as the quantum character. Also, it is shown that at low temperature, the correlation length of 1D systems becomes infinite at T = 0 K in accord with the Landau and Mermin–Wagner theorems. Finally, three illustrative examples are provided as analysis support in order to demonstrate the usefulness of the approach for a variety of phase transitions: (1) non-conventional superconductors, (2) localized magnetism, and (3) ferroelectricity.