F-invariance and its application to the quantum Hall effect

The contents of this booklet can be summarised as follows. We have found a new symmetry in the replica formalism for disorder and interactions. This F-invariance has enabled us to set up a unifying theory for the quantum Hall effect. Combined with an unusual new frequency truncation procedure it dictates the form of effective actions and allows for the inclusion of U(1) gauge fields in the theory. The proposed theory connects edge and bulk physics and combines the Finkelstein action with the topological concepts of an instanton vacuum and statistical gauge fields.

The existence of F-invariance has made it possible to do a perturbative RG analysis of the Finkelstein theory to two loop order. We have shown that the infrared behaviour of the theory can only be extracted from a limited class of ( F-invariant) correlation functions. We have identified the `order parameter' of the metal-insulator transition in 2 + 2epsilon dimensions and the `Coulomb gap'. F-invariance also has fundamental consequences for the insulating phase, where ordinary perturbation theory is no longer valid. To this end, we exploited the analogy of the Finkelstein theory with the classical Heisenberg ferromagnet. We have shown that the transport problem in this case must be dominated by the S_Y part of the action which is usually discarded on the basis of naive scaling dimensions.

Fluctuations about integer quantisation of the topological charge q[Q] give rise to massless excitations on the edge. These are equivalent to a model of disordered chiral fermions subject to inter-channel scattering. This equivalence leads to a better understanding of the Q-field theory in 1+1 dimensions, since the disorder can be gauged away in the chiral fermion theory, resulting in an exactly solvable model. In particular, several operators bilinear in Q turn out to be redundant. We have also shown that a topological principle for the quantisation of sigma_{xy} can still be formulated, because spherical boundary conditions are dynamically generated.

We have derived a complete theory of the edge, including interactions and gauge fields, in the limit of vanishing bulk density of states. Chiral edge bosons arise through a beautiful interplay between the topological term and F- and gauge invariance. The chiral anomaly at the edge provides a natural description of Laughlin's gauge argument, connecting sigma_{xy} as a bulk quantity to sigma_{xy} as an edge quantity.

Our approach to edge physics can be used to address several problems of smooth disorder and interaction effects. We model electrons in a sample with smooth disorder as a collection of spatially separated but interacting `edge modes' that live on the equipotential contours. We have pointed out that fundamental differences exist between tunnelling at the edge and electron transport. Transport experiments inject electrons directly into edge states; these electrons do not get enough time to equilibrate with the rest of the sample and are therefore effectively decoupled from the bulk. A tunnelling measurement, however, probes eigenstates of the whole system, which involve not only edge electrons but also localised bulk orbitals. Since tunnelling processes do not probe the incompressibility of the electron gas, they are generally treated incorrectly by the theory of isolated edges. By taking into account the effect of Coulomb interactions between the edge and the localised bulk states, we have derived an effective edge theory in which the neutral modes get suppressed, while the charged mode acquires a non-quantised Luttinger liquid parameter. A tunnelling exponent 1/\nu is obtained, in accordance with experiments. The Hall conductance in this theory is still quantised, as can be understood by applying the Laughlin gauge argument to all edge channels separately.

For the plateau transitions we have constructed a percolation model of interacting edges. Inelastic scattering at the `nearly saddlepoints' sets the temperature scale at which the transport coefficients cross over from mean field behaviour to critical scaling. This crossover can involve arbitrarily low temperatures and it explains the `lack of scaling' in the transport data taken from samples with long range disorder at finite temperatures. Our mean field expression for the conductances agrees with recent empirical fits to transport data at plateau transitions.

We have applied the Chern-Simons flux attachment procedure to the bulk theory, the edge theory and the percolating edge state description of plateau transitions. We studied the bulk theory at weak magnetic fields, where the C-S mapping results in states at nu approx1/2. The conductances are mapped according to SL(2,Z). For free particles or short range interactions the quasiparticle density of states becomes divergent, indicating that the C-S procedure requires long range interactions in order to be well defined.

The C-S mapping applied to the iqHe theory of chiral edge bosons leads to an action that has the well known K-matrix structure but also some new properties. It offers several new insights, in particular for the nu=2/3 state. In the absence of long range interactions, the Chern-Simons procedure is ill-defined for systems with counter-flowing edge modes. The Hamiltonian of the charged mode is unbounded from below and the theory can only be made stable by the Coulomb interaction. Our description for these systems is not plagued by non-universal behaviour of tunnelling exponents and the Hall conductance. The chiral anomaly provides an elegant way of using Laughlin's gauge argument.

Our direct way of mapping tunnelling operators has shown that the C-S procedure affects the charge and statistics of tunnelling particles. An electron operator outside the sample retains its unit charge after the mapping, but inside the charge is mapped to nu/m. The operators inside and outside are related by a T-duality transformation that inverts the compactification radius (K-matrix); thus we have found a new geometrical interpretation for this duality. Our analysis of spatially separated edge modes has revealed an even richer duality structure in the spectrum of quasiparticles, where the reversal inside/outside is generalised to a reversal of the order of edge channels. Samples with positive m and negative m are jointly described in this picture. We have shown that in the limit of large length scales, the theory of `clean', spatially separated edges describes identical physics as the `dirty' K-matrix theory. In the case of counter-flowing modes this happens in a quite nontrivial way.

We have made a short analysis of transitions between fqH plateaus, mapping the Q-field theory for percolating integer edges. The results indicate that the critical aspects of the plateau transitions are generally the same for both the integral and fractional regime. It turns out that the Chern-Simons mapping is not described by SL(2,Z).

To summarise the thesis, F-invariance has quite an impact on our understanding of disorder in the qHe and on our ability to handle calculations. It is amusing to see how much of the physics occurs in the form of edge terms, not only in real space but in the `corners' of frequency space as well.

The end of a summary is traditionally the place where one looks forward and tries to predict the future. I think it is quite safe to say that the uses we have found for F-invariance do not exhaust the full range of its possible applications. In the description of spin degrees of freedom of electrons in quantum Hall samples, for instance, neither disorder nor interactions can be neglected. It would be very interesting to see how disorder affects the formation of skyrmions. Spin indices will of course have to be included in the Q-matrix formalism for such an exercise. Another spin-related phenomenon is the nu=5/2 effect. Perhaps an edge theory in terms of Q-fields can be found that is equivalent to one of the proposed wave functions for the nu=5/2 state. Such a Q-field theory may also be used to study the corresponding quasiparticles, which are beleived to have nonabelian statistics.