Systematic approximation method for the critical properties of lattice spin systems

We present an extension of past approximations such as the Bethe approximation where we look at generalized Husimi trees. In doing so we obtain a sequence of ever more accurate approximations. The sequence is very systematic, and sequences of as few as three points can be extrapolated to give critical temperature estimates to within 0.1% or better of the exact value for our test cases. In the case of the square lattice, Ising ferromagnet, where five levels of approximation are obtained, estimates within 0.003% are obtained. Critical exponents can also be approximated but with much less accuracy. Much of our attention is devoted to the critical line of the phase diagram of the antiferromagnetic Ising model on the square lattice, where our results compare very favorably with other approximations. The method is general enough to be applicable to any spin lattice system where the spin takes on a discrete set of values.