Interpolating Between $CP(N-1)$ and $S^{2N-1}$ Target Spaces

Abstract

Some magnetic phenomena in correlated electron systems were recently shown to be described in the continuum limit by a class of sigma models which present a U(1) Hopf fibration over CP(1). In this paper we study a generalization of such models with a target space given by a U(1) fibration over Grassmannian manifolds, of which CP($N-1$) is a special case. The metric of our target space is shown to be left-symmetric which implies that it is fully parametrized by two constants: the first one -- the conventional coupling constant -- is responsible for the overall scale while the second constant $\kappa$ parametrizes the strength of a deformation. In two dimensions these sigma models are perturbatively renormalizable. We calculate their $\beta$ functions to two loops and find the RG flow of the coupling constants. We calculate the two-point function in the UV limit, which has a power law dependence with an exponent dependent on the RG trajectory.