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Next: Numerical Analysis Up: Series Analysis of Tricritical Previous: The mean-field model

Analytic Solution by PDA

The PDA method was developed by M. E. Fisher and his group [7,8,9] for two-variable series analysis and extrapolation. It allows us to estimate the critical parameters for functions with the following asymptotic form

\begin{displaymath}f(t,u) \approx \vert u\vert^{-\gamma} Z \left( \frac{-t}{\vert u\vert^{\phi}} \right) +B \;\; \eqno(3.1) \end{displaymath}

near the critical region, which is satisfied at both bicritical points [7,8], and tricritical points [10]. For bicritical points, analyses were made by Fisher [7,11] for the Ising-Heisenberg-XY model, and for the 2D Ising model by Styer [9]. The numerical analysis of the Ising-Heisenberg-XY model [7,11] yielded good results for the bicritical point, the critical exponents, and the scaling slopes. For the 2D Ising model, we have an analytic solution for the magnetization [9], and PDAs have an exact solution in this case; the numerical results were very accurate in this case and these will be given in [12]. Just as for the simpler single-variable Padé approximants, our aim is to calculate approximants to f(t,u) and then obtain estimates of the critical parameters from these approximants.

From comparison between equations (3.1) and (2.9) we can see that for the model defined above that we will study, we expect to obtain the results $\gamma=-0.5$ and $\phi=2$. The tricritical point in this model [1] is at U=T=1. The generating equation of the PDA [7] is

V_{J}(T,U)+P_{L}(T,U) F(T,U) &=& Q_{M}(T,...{\partial F(T,U)}{\partial U} \\
\end{array}\;\; \eqno(3.2)

where F(T,U) is the solution of the generating equation, VJ(T,U),PL(T,U),QM(T,U), and RN(T,U) are polynomials in the two variables U and T. These have non-zero coefficients on the sets J,L,M, and N, respectively, of coefficient values. (It is more usual to use UJ, rather than VJ but since U was used as a variable for our model, we use V for the polynomial.). There is another set K which is called the matching set which determines the matching coefficients of f and F, i.e. if the series expansion of the solution $F(T,U)=\sum_{i,j=0}^{\infty}
c_{ij}T^{i}U^{j}$ and that of $f(T,U)=\sum_{i,j=0}^{\infty} f_{ij}T^{i}U^{j}$, then cij=fij for all $(i,j) \in K$. In our case

\begin{displaymath}f(T,U)= x_m \sqrt{3} = \sqrt{ \sqrt{ (U-1)^2-3(T-1) }
+U -1 } \;\;, \eqno(3.3)\end{displaymath}

and we have found that the generating equation (3.2) has an exact analytic solution for f(T,U) in equation (3.3). This is the PDA that will be referred to hereafter as the minimal approximant:

V(T,U)&=&0 \\
P(T,U)&=&1 \\
\end{array}\;\; \eqno(3.4)

Applying PDAs to a two-variable series of a function with the scaling form (3.1), one finds the multi-critical point (Tc,Uc) (tricritical in our case) to be approximated as the common zero (T0,U0) of QM(T,U) and RN(T,U). Solving the generating equation (3.2) near the multi-critical point for the scaling form (3.1) one can see [7] that the critical parameters of (3.1) satisfy the relations:

...Q_{2}}&,&B=-\frac{V_{0}}{P_{0}} \\
\end{array}\;\; \eqno(3.5)


...l R(T_{0},U_{0})}{\partial U}&.\\
\end{array}\;\; \eqno (3.6)

and the scaling slopes e1,e2 which can be defined from

t&=&(T-T_{0})-\frac{(U-U_{0})}{e_{1}} \\
u&=&(U-U_{0})-e_{2}(T-T_{0}) \\
\end{array}\;\; \eqno(3.7)

are the solutions of

\begin{displaymath}Q_{2} e^{2}+(Q_{1}-R_{2})e-R_{1}=0 \;\;. \eqno(3.8) \end{displaymath}

For our model, comparison of the equation (2.8) and (3.7) shows that the two roots of equation (3.8) should be $e_{1}=- \infty$ and e2=0. Using the exact solution (3.4) to substitute the numerical values of (3.6) in (3.5), and substituting the solution of (3.8) in (3.7) one can easily see that for the case we are dealing with indeed we obtain the exact results

T_{0}&=&1&, &U_{0}&= \\
...hi&= \\
e_{1}&=&-\infty&, &e_{2}&=
\end{array}\;\; \eqno(3.9)

These are in agreement with the parameters of the full solution as given above.

next up previous
Next: Numerical Analysis Up: Series Analysis of Tricritical Previous: The mean-field model