   Next: Analytic Solution by PDA Up: Series Analysis of Tricritical Previous: Introduction

# The mean-field model

The models of Glasser, Privman , and Schulman avoid some pathologies common to other infinite range models. They have a soft'' temperature (T) dependence which is an artifact of the infinite-range model, and there are some other artificial features near T=0, but the series is well defined and can be derived in closed form to any fixed order given sufficiently powerful computational facilities. The model has interacting scalar spins, , with energy where N is the number of spins, and J>0. The Gaussian-integral method  was used to show that in the limit the dimensionless free energy, f, can be obtained as where The spins are weighed with measure in the partition function from whence the free energy is obtained, and the variable x  is equal to y(2J/kTN)1/2, where y is the variable introduced in the Gaussian integral. If the minimum in (2.2) is at some x=xm, then one can further show that the magnetization m is where the last equality follows from the fact that the global minimum is obtained at one (or more) roots of Thus, we note that m=kTxm/J, i.e., for low temperatures. This is one of those artificial infinite-range-model features. It turns out convenient to work with xm directly rather than with m, as the order-parameter-like quantity for series analysis. Of course, the actual critical-tricritical-first-order behavior is at T>0 so the difference only affects the form of analytic corrections to scaling. In order to have a solvable model with tricritical behavior, Q(x) was taken as an even, six-degree polynomial in x, the actual series was conveniently generated  for where U is a dimensionless coupling constant just like T is a dimensionless temperature in this model. Using Mathematica , the order 50 double series in T and U for this order-parameter quantity was derived in . It is expressed as the first 2601 coefficients cij, for , in Analysis of the exact solution  showed that Uc=Tc=1, and that near the tricritical point, one can write the low-T-side scaling form in terms of the scaling variables  where the scaling form (2.9) applies for and the scaling function is    Next: Analytic Solution by PDA Up: Series Analysis of Tricritical Previous: Introduction

1998-12-30