The instanton vacuum of generalized CPN−1CP^{N-1} models

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 $\theta$ 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 $\beta$ and $\gamma$ 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 $\theta = \pi$. 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 $\theta = \pi$ for varying values of $m$ and $n$.Comment: Elsart style, 87 pages, 15 figure

Comment on ``Topological Oscillations of the Magnetoconductance in Disordered GaAs Layers''

In a recent Letter, Murzin et. al. [Phys. Rev. Lett., vol. 92, 016802 (2004)] investigated "instanton effects" in the magneto resistance data taken from samples with heavily Si-doped GaAs layers at low temperatures. This topological issue originally arose in the development of a microscopic theory of quantum Hall effect some 20 years ago. The investigations by Murzin et. al., however, do not convey the correct ideas on scaling that have emerged over the years in the general theory of quantum transport.Comment: comment on Phys. Rev. Lett., vol. 92, 016802 (2004

Two-instanton approximation to the Coulomb blockade problem

We develop the two-instanton approximation to the current-voltage characteristic of a single electron transistor within the Ambegaokar-Eckern-Sch\"on model. We determine the temperature and gate voltage dependence of the Coulomb blockade oscillations of the conductance and the effective charge. We find that a small (in comparison with the charging energy) bias voltage leads to significant suppression of the Coulomb blockade oscillations and to appearance of the bias-dependent phase shift