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

TL;DR

This paper explores a generalized class of sigma models with target spaces as U(1) fibrations over Grassmannian manifolds, analyzing their renormalization group flow and two-point functions in two dimensions.

Contribution

It introduces a new family of sigma models with a deformed target space metric parametrized by two constants and computes their two-loop beta functions and RG flow.

Findings

The target space metric is left-symmetric and fully parametrized by two constants.

The models are perturbatively renormalizable in two dimensions.

Two-point functions exhibit power-law behavior with exponents depending on RG trajectories.

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.