177 research outputs found

### (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

### 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 Quantum Hall Effect: Unified Scaling Theory and Quasi-particles at the Edge

We address two fundamental issues in the physics of the quantum Hall effect:
a unified description of scaling behavior of conductances in the integral and
fractional regimes, and a quasi-particle formulation of the chiral Luttinger
Liquids that describe the dynamics of edge excitations in the fractional
regime.Comment: 11 pages, LateX, 2 figures (not included, available from the
authors), to be published in Proceedings of the International Summer School
on Strongly Correlated Electron Systems, Lajos Kossuth University, Debrecen,
Hungary, Sept 199

### 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

### The instanton vacuum of generalized $CP^{N-1}$ 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

### (Mis-)handling gauge invariance in the theory of the quantum Hall effect II: Perturbative results

The concept of F-invariance, which previously arose in our analysis of the
integral and half-integral quantum Hall effects, is studied in 2+2\epsilon
spatial dimensions. We report the results of a detailed renormalization group
analysis and establish the renormalizability of the (Finkelstein) action to two
loop order. We show that the infrared behavior of the theory can be extracted
from gauge invariant (F-invariant) quantities only. For these quantities
(conductivity, specific heat) we derive explicit scaling functions. We identify
a bosonic quasiparticle density of states which develops a Coulomb gap as one
approaches the metal-insulator transition from the metallic side. We discuss
the consequences of F-invariance for the strong coupling, insulating regime.Comment: 26 pages, 7 figures; minor modifications; submitted to Phys.Rev.

### (Mis-)handling gauge invariance in the theory of the quantum Hall effect I: Unifying action and the \nu=1/2 state

We propose a unifying theory for both the integral and fractional quantum
Hall regimes. This theory reconciles the Finkelstein approach to localization
and interaction effects with the topological issues of an instanton vacuum and
Chern-Simons gauge theory. We elaborate on the microscopic origins of the
effective action and unravel a new symmetry in the problem with Coulomb
interactions which we name F-invariance. This symmetry has a broad range of
physical consequences which will be the main topic of future analyses. In the
second half of this paper we compute the response of the theory to
electromagnetic perturbations at a tree level approximation. This is applicable
to the theory of ordinary metals as well as the composite fermion approach to
the half-integer effect. Fluctuations in the Chern-Simons gauge fields are
found to be well behaved only when the theory is F-invariant.Comment: 20 pages, 6 figures; appendix B revised; submitted to Phys.Rev.

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