Complex temperature singularities for the two-dimensional Heisenberg O(∞) model

Abstract We compute the complex temperature singularities for the two-dimensional O( N ) symmetric Heisenberg model in the limit N → ∞. The model is exactly soluble but the analysis of singularities requires a quite arduous study of analytic properties of the inverse of an elliptic function with respect to its modulus. We show that the mass gap and other physical quantities are analytic in a narrow strip around the real inverse temperature axis. This fact explains why low-temperature properties are not easily described by high-temperature expansion methods. We then use the known analyticity properties to identify a conformal transformation which improves the convergence of the high-temperature expansion and, as a consequence, the estimate of physical quantities at low temperature. The method is suitable for extension to all N > 2 models, in which a similar analytic structure presumably holds.