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Exclusion statistics for quantum Hall states in the Tao-Thouless limit
Stockholm University, Faculty of Science, Department of Physics.
Stockholm University, Faculty of Science, Department of Physics.
2011 (English)In: Journal of Statistical Mechanics: Theory and Experiment, ISSN 1742-5468, E-ISSN 1742-5468, P02037Article in journal (Refereed) Published
Abstract [en]

We consider spin-polarized Abelian quantum Hall states in the Tao-Thouless limit, i.e. on a thin torus. For any filling factor nu = p/q a well-defined sector of low energy states is identified and the exclusion statistics of the excitations is determined. We study numerically, at and near nu = 1/3 and 2/5, how the low energy states develop as one moves away from the Tao-Thouless limit towards the physical regime. We find that the lowest energy states in the physical regime develop from states in the low energy sector but that the exclusion statistics is modified.

Place, publisher, year, edition, pages
2011. P02037
Keyword [en]
solvable lattice models, fractional QHE (theory)
National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
URN: urn:nbn:se:su:diva-67482DOI: 10.1088/1742-5468/2011/02/P02037ISI: 000287802800041OAI: oai:DiVA.org:su-67482DiVA: diva2:470603
Note

authorCount :2

Available from: 2011-12-29 Created: 2011-12-28 Last updated: 2017-12-08Bibliographically approved
In thesis
1. Quantum Hall Wave Functions on the Torus
Open this publication in new window or tab >>Quantum Hall Wave Functions on the Torus
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The fractional quantum Hall effect (FQHE), now entering it's fourth decade, continues to draw attention from the condensed matter community. New experiments in recent years are raising hopes that it will be possible to observe quasi-particles with non-abelian anyonic statistics. These particles could form the building blocks of a quantum computer.

The quantum Hall states have topologically protected energy gaps to the low-lying set of excitations. This topological order is not a locally measurable quantity but rather a non-local object, and it is one of the keys to it's stability. From an early stage understanding of the FQHE has been facilitate by constructing trial wave functions. The topological classification of these wave functions have given further insight to the nature of the FQHE.

An early, and successful, wave function construction for filling fractions ν=p/(2p+1) was that of composite fermions on planar and spherical geometries. Recently, new developments using conformal field theory have made it possible to also construct the full Haldane-Halperin hierarchy wave functions on planar and spherical geometries. In this thesis we extend this construction to a toroidal geometry, i.e. a flat surface with periodic boundary conditions.

One of the defining features of topological states of matter in two dimensions is that the ground state is not unique on surfaces with non trivial topology, such as a torus. The archetypical example is the fractional quantum Hall effect, where a state at filling fraction ν=p/q, has at least a q-fold degeneracy on a torus. This has been shown explicitly for a few cases, such as the Laughlin states and the the Moore-Read states, by explicit construction of candidate electron wave functions with good overlap with numerically found states. In this thesis, we construct explicit torus wave functions for a large class of experimentally important quantum liquids, namely the chiral hierarchy states in the lowest Landau level. These states, which includes the prominently observed positive Jain sequence at filling fractions ν=p/(2p+1), are characterized by having boundary modes with only one chirality.

Our construction relies heavily on previous work that expressed the hierarchy wave functions on a plane or a sphere in terms of correlation functions in a conformal field theory. This construction can be taken over to the torus when care is taken to ensure correct behaviour under the modular transformations that leave the geometry of the torus unchanged. Our construction solves the long standing problem of engineering torus wave functions for multi-component many-body states. Since the resulting expressions are rather complicated, we have carefully compared the simplest example, that of ν=2/5, with numerically found wave functions. We have found an extremely good overlap for arbitrary values of the modular parameter τ, that describes the geometry of the torus.

Having explicit torus wave functions allows us to use the methods developed by Read and Read \& Rezayi to numerically compute the quantum Hall viscosity. Hall viscosity is conjectured to be a topologically protected macroscopic transport coefficient characterizing the quantum Hall state. It is related to the shift of the same QH-fluid when it is put on a sphere. The good agreement with the theoretical prediction for the 2/5 state strongly suggests that our wave functions encodes all relevant topologically information.

We also consider the Hall viscosity in the limit of a very thin torus. There we find that the viscosity changes as we approach the thin torus limit. Because of this we study the Laughlin state in that limit and see how the change in viscosity arises from a change in the Hamiltonian hopping elements. Finally we conclude that there are both qualitative and quantitative difference between the thin and the square torus. Thus, one has to be careful when interpreting results in the thin torus limit.

Place, publisher, year, edition, pages
Stockholm: Department of Physics, Stockholm University, 2015
Keyword
Fractional Quantum Hall Effect
National Category
Condensed Matter Physics
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-115616 (URN)978-91-7649-158-4 (ISBN)
Public defence
2015-05-06, FD5, AlbaNova University Center, Roslagstullsbacken 21, Stockholm, 13:00 (English)
Opponent
Supervisors
Funder
Swedish Research Council
Note

At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 4: Manuscript.

Available from: 2015-04-14 Created: 2015-03-26 Last updated: 2015-04-17Bibliographically approved

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