Quantum criticality, particle-hole symmetry, and duality of the plateau-insulator transition in the quantum Hall regime

We report new experimental data on the plateau-insulator transition in the quantum Hall regime, taken from a low mobility InGaAs/InP heterostructure. By employing the fundamental symmetries of the quantum transport problem we are able to disentangle the universal quantum critical aspects of the magnetoresistance data (critical indices and scaling functions) and the sample dependent aspects due to macroscopic inhomogeneities. Our new results and methodology indicate that the previously established experimental value for the critical index (kappa = 0.42) resulted from an admixture of both universal and sample dependent behavior. A novel, non-Fermi liquid value is found (kappa = 0.57) along with the leading corrections to scaling. The statement of self-duality under the Chern Simons flux attachment transformation is verified.Comment: 4 pages, 2 figure

(Mis-)handling gauge invariance in the theory of the quantum Hall effect III: The instanton vacuum and chiral edge physics

The concepts of an instanton vacuum and F-invariance are used to derive a complete effective theory of massless edge excitations in the quantum Hall effect. We establish, for the first time, the fundamental relation between the instanton vacuum approach and the theory of chiral edge bosons. Two longstanding problems of smooth disorder and Coulomb interactions are addressed. We introduce a two dimensional network of chiral edge states and tunneling centers (saddlepoints) as a model for the plateau transitions. We derive a mean field theory including the Coulomb interactions and explain the recent empirical fits to transport at low temperatures. Secondly, we address the problem of electron tunneling into the quantum Hall edge. We express the problem in terms of an effective Luttinger liquid with conductance parameter (g) equal to the filling fraction (\nu) of the Landau band. Hence, even in the integral regime our results for tunneling are completely non-Fermi liquid like, in sharp contrast to the predictions of single edge theories.Comment: 51 pages, 8 figures; section IIA3 completely revised, section IIB and appendix C corrected; submitted to Phys.Rev.

The fractional quantum Hall effect: Chern-Simons mapping, duality, Luttinger liquids and the instanton vacuum

We derive, from first principles, the complete Luttinger liquid theory of abelian quantum Hall edge states. This theory includes the effects of disorder and Coulomb interactions as well as the coupling to external electromagnetic fields. We introduce a theory of spatially separated (individually conserved) edge modes, find an enlarged dual symmetry and obtain a complete classification of quasiparticle operators and tunneling exponents. The chiral anomaly on the edge and Laughlin's gauge argument are used to obtain unambiguously the Hall conductance. In resolving the problem of counter flowing edge modes, we find that the long range Coulomb interactions play a fundamental role. In order to set up a theory for arbitrary filling fractions $\nu$ we use the idea of a two dimensional network of percolating edge modes. We derive an effective, single mode Luttinger liquid theory for tunneling processes into the quantum Hall edge which yields a continuous tunneling exponent $1/\nu$. The network approach is also used to re-derive the instanton vacuum or $Q$-theory for the plateau transitions.Comment: 36 pages, 7 figures (eps

Theta renormalization, electron-electron interactions and super universality in the quantum Hall regime

The renormalization theory of the quantum Hall effect relies primarily on the non-perturbative concept of theta renormalization by instantons. Within the generalized non-linear sigma model approach initiated by Finkelstein we obtain the physical observables of the interacting electron gas, formulate the general (topological) principles by which the Hall conductance is robustly quantized and derive - for the first time - explicit expressions for the non-perturbative (instanton) contributions to the renormalization group beta- and gamma- functions. Our results are in complete agreement with the recently proposed idea of super universality which says that the fundamental aspects of the quantum Hall effect are all generic features the instanton vacuum concept in asymptotically free field theory.Comment: ReVTeX, 38 pages, 9 figure

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

The problem of Coulomb interactions in the theory of the quantum Hall effect

We summarize the main ingredients of a unifying theory for abelian quantum Hall states. This theory combines the Finkelstein approach to localization and interaction effects with the topological concept of an instanton vacuum as well as Chern-Simons gauge theory. We elaborate on the meaning of a new symmetry ($\cal F$ invariance) for systems with an infinitely ranged interaction potential. We address the renormalization of the theory and present the main results in terms of a scaling diagram of the conductances.Comment: 9 pages, 3 figures. To appear in Proceedings of the International Conference "Mesoscopics and Strongly Correlated Electron Systems", July 2000, Chernogolovka, Russi

The effects of macroscopic inhomogeneities on the magneto transport properties of the electron gas in two dimensions

In experiments on electron transport the macroscopic inhomogeneities in the sample play a fundamental role. In this paper and a subsequent one we introduce and develop a general formalism that captures the principal features of sample inhomogeneities (density gradients, contact misalignments) in the magneto resistance data taken from low mobility heterostructures. We present detailed assessments and experimental investigations of the different regimes of physical interest, notably the regime of semiclassical transport at weak magnetic fields, the plateau-plateau transitions as well as the plateau-insulator transition that generally occurs at much stronger values of the external field only. It is shown that the semiclassical regime at weak fields plays an integral role in the general understanding of the experiments on the quantum Hall regime. The results of this paper clearly indicate that the plateau-plateau transitions, unlike the the plateau-insulator transition, are fundamentally affected by the presence of sample inhomogeneities. We propose a universal scaling result for the magneto resistance parameters. This result facilitates, amongst many other things, a detailed understanding of the difficulties associated with the experimental methodology of H.P. Wei et.al in extracting the quantum critical behavior of the electron gas from the transport measurements conducted on the plateau-plateau transitions.Comment: 20 pages, 9 figure