Vortex condensation and mass gap generation in two-dimensional principal chiral models

Abstract

We consider the two-dimensional $\mathrm{SU}(N)$ principal chiral model and discuss a vortex condensation mechanism which could explain the existence of a non-zero mass gap at arbitrarily small values of the coupling constant. The mechanism is an analogue of the vortex condensation mechanism of confinement in 4D non-Abelian gauge theories. We formulate a sufficient condition for the mass gap to be non-vanishing in terms of the behavior of the vortex free energy. The $\mathrm{SU}(2)$ model is studied in detail. In one dimension we calculate the vortex free energy exactly. An effective model for the center variables of the spin configurations of the 2D $\mathrm{SU}(2)$ model is proposed and the $Z(2)$ correlation function is derived in this model. We define a $Z(2)$ mass gap in both the full and effective model and argue that they should coincide whenever the genuine mass gap is non-zero. We show via Monte Carlo simulations of the $\mathrm{SU}(2)$ model that the $Z(2)$ mass gap reproduces the full mass gap with perfect accuracy. We also test this mechanism in the positive link model which is an analogue of the positive plaquette model in gauge theories and find excellent agreement between the full and the $Z(2)$ mass gap.