Series expansions from the finite lattice method

Some Algebraic Techniques for obtaining Low-temperature Series Expansions

“…In Section 4 resummations of the strong graph expansion are described, showing how it is possible to obtain low-temperature expansions for square lattice systems by combining the partition functions for finite rectangles. This extends the methods of de Neef and Enting (1977) and Enting and Baxter (1977) to low-temperature series. Because algebraic techniques are almost always implemented on digital computers, Section 5 is devoted to considering several technical computational simplifications which should make these series expansion techniques more efficient.…”

“…The combinatorial arguments given by de Neef (1975) and de Neef and Enting (1977) then show that there is a combination of contributions from rectangles which brings in each connected graph with its correct combinatorial weight.…”

“…For models defined on a square lattice there have been a number of descriptions of techniques for obtaining high-temperature series by using only the partition functions for rectangular subgraphs (de Neef 1975;de Neef and Enting 1977;Enting and Baxter 1977;Enting 1978a). The advantage of such a formulation is that the finite lattice partition function can be easily calculated by transfer matrix techniques.…”

“…de Neef and Enting (1977) have pointed out that in recent years techniques of series expansion have, to an increasing degree, involved substituting algebraic complexity for combinatorial complexity. One reason for this trend appears to be the increasing use of digital computers at all stages of series derivation.…”

“…This 'counting theorem' represents probably the earliest of the algebraic techniques of series expansion. Other techniques making extensive use of algebraic manipulation have been described by Sykes et al (1965Sykes et al ( , 1975, Wortis (1974) and de Neef and Enting (1977). Enting (1978b) gives a more extensive analysis of algebraic techniques vis-a-vis combinatorial techniques, using the framework of general theories of computational complexity.…”