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

Exact Haldane mapping for all SS and super universality in spin chains

The low energy dynamics of the anti-ferromagnetic Heisenberg spin $S$ chain in the semiclassical limit $S\to\infty$ is known to map onto the O(3) nonlinear $\sigma$ model with a $\theta$ term in 1+1 dimension. Guided by the underlying dual symmetry of the spin chain, as well as the recently established topological significance of "dangling edge spins," we report an {\em exact} mapping onto the O(3) model that avoids the conventional large $S$ approximation altogether. Our new methodology demonstrates all the super universal features of the $\theta$ angle concept that previously arose in the theory of the quantum Hall effect. It explains why Haldane's original ideas remarkably yield the correct answer in spite of the fundamental complications that generally exist in the idea of semiclassical expansions

Super universality of the quantum Hall effect and the "large NN picture" of the ϑ\vartheta angle

It is shown that the "massless chiral edge excitations" are an integral and universal aspect of the low energy dynamics of the $\vartheta$ vacuum that has historically gone unnoticed. Within the $SU(M+N)/S(U(M) \times U(N))$ non-linear sigma model we introduce an effective theory of "edge excitations" that fundamentally explains the quantum Hall effect. In sharp contrast to the common beliefs in the field our results indicate that this macroscopic quantization phenomenon is, in fact, a {\em super universal} strong coupling feature of the $\vartheta$ angle with the replica limit $M=N=0$ only playing a role of secondary importance. To demonstrate super universality we revisit the large $N$ expansion of the $CP^{N-1}$ model. We obtain, for the first time, explicit scaling results for the quantum Hall effect including quantum criticality of the quantum Hall plateau transition. Consequently a scaling diagram is obtained describing the cross-over between the weak coupling "instanton phase" and the strong coupling "quantum Hall phase" of the large $N$ theory. Our results are in accordance with the "instanton picture" of the $\vartheta$ angle but fundamentally invalidate all the ideas, expectations and conjectures that are based on the historical "large $N$ picture."Comment: 40 pages, 9 figure

Topological oscillations of the magnetoconductance in disordered GaAs layers

Oscillatory variations of the diagonal ($G_{xx}$) and Hall ($G_{xy}$) magnetoconductances are discussed in view of topological scaling effects giving rise to the quantum Hall effect. They occur in a field range without oscillations of the density of states due to Landau quantization, and are, therefore, totally different from the Shubnikov-de Haas oscillations. Such oscillations are experimentally observed in disordered GaAs layers in the extreme quantum limit of applied magnetic field with a good description by the unified scaling theory of the integer and fractional quantum Hall effect.Comment: 4 pages, 4 figure

Coulomb Blockade and Super Universality of the Theta-Angle

Based on the Ambegaokar-Eckern-Schon approach to the Coulomb blockade we develop a complete quantum theory of the single electron transistor. We identify a previously unrecognized physical observable q^\prime in the problem that, unlike the usual average charge (Q) on the island, is robustly quantized for any finite value of the tunneling conductance as the temperature goes to absolute zero. This novel quantity is fundamentally related to the non-symmetrized noise of the system. We present a unifying scaling diagram in the q^\prime - g^\prime plane where g^\prime denotes the conductance of the system. The results display all the super universal topological features of the theta-angle concept that previously arose in the theory of the quantum Hall effect.Comment: RevTeX, 4 pages, 1 figur

Renormalization of the vacuum angle in quantum mechanics, Berry phase and continuous measurements

The vacuum angle $\theta$ renormalization is studied for a toy model of a quantum particle moving around a ring, threaded by a magnetic flux $\theta$. Different renormalization group (RG) procedures lead to the same generic RG flow diagram, similar to that of the quantum Hall effect. We argue that the renormalized value of the vacuum angle may be observed if the particle's position is measured with finite accuracy or coupled to additional slow variable, which can be viewed as a coordinate of a second (heavy) particle on the ring. In this case the renormalized $\theta$ appears as a magnetic flux this heavy particle sees, or the Berry phase, associated with its slow rotation.Comment: 4 pages, 2 figure