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

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

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
and .
The tricritical point in this model [1] is at
*U*=*T*=1.
The generating equation of the PDA [7] is

where

and we have found that the generating equation (3.2) has an exact analytic solution for

Applying PDAs to a two-variable
series of a function with the scaling form (3.1), one finds
the multi-critical point
(*T*_{c},*U*_{c})
(tricritical in our case) to be approximated as the common zero
(*T*_{0},*U*_{0})
of
*Q*_{M}(*T*,*U*) and
*R*_{N}(*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:

where

and the scaling slopes

are the solutions of

For our model, comparison of the equation (2.8) and (3.7) shows that the two roots of equation (3.8) should be and

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