Analytic Perturbation Theory for Bound States in the Transfer Matrix Spectrum of Weakly Correlated Lattice Ferromagnetic Spin Systems

Abstract

We consider general d-dimensional lattice ferromagnetic spin systems with nearest neighbor interactions in the high temperature region (β 1). Each model is characterized by a single site apriori spin distribution taken to be even. We also take the parameter α = s 4 − 3 s 2 2 > 0, i.e. in the region which we call Gaussian subjugation, where s k denotes the kth moment of the apriori distribution. Associated with the model is a lattice quantum field theory known to contain a particle of asymptotic mass − ln β and a bound state below the two-particle threshold. We develop a β analytic perturbation theory for the binding energy of this bound state. As a key ingredient in obtaining our result we show that the Fourier transform of the two-point function is a meromorphic function, with a simple pole, in a suitable complex spectral parameter and the coefficients of its Laurent expansion are analytic in β.Keywords Transfer matrix spectrum · Decay of correlations · Bound states · High-temperature ferromagnetic spin systems · Binding energy analyticity Low lying excitations in lattice models have been the subject of much study in statistical mechanics. Some of the models are lattice approximations to continuum quantum field theory ones [1, 2], such as those of ferromagnetic scalar and vector spin, Gross-Neveu, gauge and gauge-matter (QCD) [3]. The excitations correspond to particles and there may be excitations below the two-particle threshold corresponding to bound states. Particularly in the region of parameters where the variables at different sites are weakly correlated there are a wide variety of methods available to determine the excitations.In this paper we develop an analytic perturbation theory for the two-particle binding energy of one of the simplest models, the weakly correlated (high temperature) ferromagnetic scalar spin model. It is expected that these methods can be adapted to treat other models.