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

The mean-field model

The models of Glasser, Privman [5], 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, $\sigma_i$, with energy

\begin{displaymath}E=- \frac{J}{2N} \left( \sum\limits_{i=1}^N \sigma_i \right)^2
\;\; , \eqno(2.1) \end{displaymath}

where N is the number of spins, and J>0. The Gaussian-integral method [5] was used to show that in the limit $N \to \infty$ the dimensionless free energy, f, can be obtained as

\begin{displaymath}f=\min_x \left[ {kT x^2 \over 2 J} -Q(x) \right] \;\; ,
\eqno(2.2) \end{displaymath}


\begin{displaymath}Q(x)=\ln \int e^{x\sigma} d \mu (\sigma)\;\; . \eqno(2.3)\end{displaymath}

The spins are weighed with measure $d \mu (\sigma)$ in the partition function from whence the free energy is obtained, and the variable x [5] 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

\begin{displaymath}m=\left( {d Q \over d x} \right)_{x=x_m} =
{kT x_m \over J} \;\; , \eqno(2.4)\end{displaymath}

where the last equality follows from the fact that the global minimum is obtained at one (or more) roots of

\begin{displaymath}{d Q \over d x} = {kT x \over J} \;\; . \eqno(2.5)\end{displaymath}

Thus, we note that m=kTxm/J, i.e., $m\propto T$ 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 [1] for

\begin{displaymath}x_m \sqrt{3} = \sqrt{ \sqrt{ (U-1)^2-3(T-1) } +U -1 } \;\; \eqno(2.6)\end{displaymath}

where U is a dimensionless coupling constant just like T is a dimensionless temperature in this model. Using Mathematica [6], the order 50 double series in T and U for this order-parameter quantity was derived in [1]. It is expressed as the first 2601 coefficients cij, for $i,j=0, \ldots , 50 $, in

\begin{displaymath}\sqrt{3} x_m = \sum_{i=0}^{50} \sum_{j=0}^{50} c_{ij} T^i U^j \;\; .\eqno(2.7)\end{displaymath}

Analysis of the exact solution [1] 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

\begin{displaymath}t=T-1 < 0 \qquad {\rm and} \qquad u=U-1 \;\; , \eqno(2.8)\end{displaymath}

\begin{displaymath}\sqrt{3} x_m \simeq (u)^{1/2} Z_- \left( {-t \over u^{2}} \right)
\;\; , \eqno(2.9)\end{displaymath}

where the scaling form (2.9) applies for $t, u \to 0$ and the scaling function is

\begin{displaymath}Z_-(\zeta) = \sqrt{1+\sqrt{1+3 \zeta }} \;\; . \eqno(2.10)\end{displaymath}

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