Noncompact sigma-models: Large N expansion and thermodynamic limit

Abstract

Noncompact SO ( 1 , N ) sigma-models are studied in terms of their large N expansion in a lattice formulation in dimensions d ⩾ 2 . Explicit results for the spin and current two-point functions as well as for the Binder cumulant are presented to next to leading order on a finite lattice. The dynamically generated gap is negative and serves as a coupling-dependent infrared regulator which vanishes in the limit of infinite lattice size. The cancellation of infrared divergences in invariant correlation functions in this limit is nontrivial and is in d = 2 demonstrated by explicit computation for the above quantities. For the Binder cumulant the thermodynamic limit is finite and is given by 2 / ( N + 1 ) in the order considered. Monte Carlo simulations suggest that the remainder is small or zero. The potential implications for “criticality” and “triviality” of the theories in the SO ( 1 , N ) invariant sector are discussed.