High-Temperature and Low-Temperature Expansions

Abstract

Series expansions remain, in many cases, one of the most accurate ways of estimating critical exponents. The high-temperature expansion is an expansion in powers of the inverse temperature. In the low-temperature expansion configurations are counted in order of their importance as the temperature is increased from zero. Starting from the ground state the series is constructed by successively adding terms with and increasing number of flipped spins. Each term in the high- or low-temperature series is represented by a graph on a lattice and constructing the series amounts to counting the graphs belonging to a fixed order in the expansions. With various extrapolations, for example with Pade approximants, we extract the behavior of a lattice system near a critical point and in particular its critical exponents. The expansions are calculated to high orders for the Ising model in two and three dimensions and for the non-linear O(N) lattice model. At the end we consider polymers and self-avoiding walks. In several tables we compare predictions for the critical temperatures and critical exponents of various expansions, renormalization group techniques and lattice simulations.