Noncompact sigma-models: Large N expansion and thermodynamic limit

TL;DR

This paper investigates noncompact SO(1,N) sigma-models in dimensions d ≥ 2, analyzing their large N expansion, infrared behavior, and finite-size effects, with explicit calculations and Monte Carlo simulations to understand their critical properties.

Contribution

It provides explicit large N expansion results for correlation functions and the Binder cumulant, demonstrating the cancellation of infrared divergences and analyzing the thermodynamic limit behavior.

Findings

The dynamically generated gap is negative and acts as an infrared regulator.

Infrared divergences cancel in invariant correlation functions in the thermodynamic limit.

The Binder cumulant approaches 2/(N+1), with small or zero finite-size corrections.

Abstract

Noncompact SO(1,N) sigma-models are studied in terms of their large N expansion in a lattice formulation in dimensions d \geq 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.