Massless excitations at $θ=π$ in the $CP^{N-1}$ model with large values of $N$

Abstract

We study the instanton vacuum of the $CP^{N-1}$ model with large values of $N$ in 1+1 space-time dimensions. Unlike the longstanding claims which state that the theory always has a mass gap, we for the first time establish a complete {\em critical} theory for the transition at $θ= π$ obtained from a mapping onto the low temperature phase of the 1D Ising model. We derive a simple effective field theory in terms of 1D massless chiral fermions. Our results include, besides a diverging correlation length with an exponent $ν=1/2$, exact expressions for the $β$ functions. These expressions unequivocally demonstrate that the large $N$ expansion with varying $θ$ displays all the fundamental features of the quantum Hall effect.

Elsart class, 46 pages, 2 figures