Zoekopdracht:
faculteit: "FNWI" en publicatiejaar: "1999"
| Auteur | B. Skoric | | Titel | F-invariance and its application to the quantum Hall effect |
| Promotor | A.M.M. Pruisken |
| Datum | 20-MAY-1999 |
| Jaar | 1999 |
| Pagina's | 148 |
| ISBN | 9057760223 |
| Faculteit | Faculteit der Natuurwetenschappen, Wiskunde en Informatica |
| Instituut/afd. | FNWI: Institute for Theoretical Physics (ITF) |
| Samenvatting | 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 F-invariance also has fundamental consequences for the insulating phase, where ordinary perturbation theory is no longer valid.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 temperatur |
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