arXiv2004
It has recently been pointed out that the existence of massless chiral edge excitations has important strong coupling consequences for the topological concept of an instanton vacuum. In the first part of this paper we elaborate on the effective action for ``edge excitations'' in the Grassmannian $U(m+n)/U(m) \times U(n)$ non-linear sigma model in the presence of the $θ$ term. This effective action contains complete information on the low energy dynamics of the system and defines the renormalization of the theory in an unambiguous manner. In the second part of this paper we revisit the instanton methodology and embark on the non-perturbative aspects of the renormalization group including the anomalous dimension of mass terms. The non-perturbative corrections to both the $β$ and $γ$ functions are obtained while avoiding the technical difficulties associated with the idea of {\em constrained} instantons. In the final part of this paper we present the detailed consequences of our computations for the quantum critical behavior at $θ= π$. In the range $0 \leq m,n \lesssim 1$ we find quantum critical behavior with exponents that vary continuously with varying values of $m$ and $n$. Our results display a smooth interpolation between the physically very different theories with $m=n=0$ (disordered electron gas, quantum Hall effect) and $m=n=1$ (O(3) non-linear sigma model, quantum spin chains) respectively, in which cases the critical indices are known from other sources. We conclude that instantons provide not only a {\em qualitative} assessment of the singularity structure of the theory as a whole, but also remarkably accurate {\em numerical} estimates of the quantum critical details (critical indices) at $θ= π$ for varying values of $m$ and $n$.