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Degeneracy of non-Abelian quantum Hall states on the torus: domain walls and conformal field theory
Stockholm University, Nordic Institute for Theoretical Physics (Nordita).
Stockholm University, Faculty of Science, Department of Physics.
Stockholm University, Faculty of Science, Department of Physics.
2008 (English)In: Journal of Statistical Mechanics: Theory and Experiment, ISSN 1742-5468, E-ISSN 1742-5468, P04016- p.Article in journal (Refereed) Published
Abstract [en]

We analyze the non-Abelian Read–Rezayi quantum Hall states on the torus, where it is natural to employ a mapping of the many-body problem onto a one-dimensional lattice model. On the thin torus—the Tao–Thouless (TT) limit—the interacting many-body problem is exactly solvable. The Read–Rezayi states at filling ν = k/(kM+2) are known to be exact ground states of a local repulsive k+1-body interaction, and in the TT limit this is manifested in that all states in the ground state manifold have exactly k particles on any kM+2 consecutive sites. For M \neq 0 the two-body correlations of these states also imply that there is no more than one particle on M adjacent sites. The fractionally charged quasiparticles and quasiholes appear as domain walls between the ground states, and we show that the number of distinct domain wall patterns gives rise to the nontrivial degeneracies, required by the non-Abelian statistics of these states. In the second part of the paper we consider the quasihole degeneracies from a conformal field theory (CFT) perspective, and show that the counting of the domain wall patterns maps one to one on the CFT counting via the fusion rules. Moreover we extend the CFT analysis to topologies of higher genus

Place, publisher, year, edition, pages
2008. P04016- p.
Keyword [en]
conformal field theory (theory), fractional QHE (theory)
National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
URN: urn:nbn:se:su:diva-24943DOI: 10.1088/1742-5468/2008/04/P04016ISI: 000255662000020OAI: oai:DiVA.org:su-24943DiVA: diva2:198561
Available from: 2008-05-01 Created: 2008-05-01 Last updated: 2017-12-13Bibliographically approved
In thesis
1. One-dimensional theory of the quantum Hall system
Open this publication in new window or tab >>One-dimensional theory of the quantum Hall system
2008 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The quantum Hall (QH) system---cold electrons in two dimensions in a perpendicular magnetic field---is a striking example of a system where unexpected phenomena emerge at low energies. The low-energy physics of this system is effectively one-dimensional due to the magnetic field. We identify an exactly solvable limit of this interacting many-body problem, and provide strong evidence that its solutions are adiabatically connected to the observed QH states in a similar manner as the free electron gas is related to real interacting fermions in a metal according to Landau's Fermi liquid theory.

The solvable limit corresponds to the electron gas on a thin torus. Here the ground states are gapped periodic crystals and the fractionally charged excitations appear as domain walls between degenerate ground states. The fractal structure of the abelian Haldane-Halperin hierarchy is manifest for generic two-body interactions. By minimizing a local k+1-body interaction we obtain a representation of the non-abelian Read-Rezayi states, where the domain wall patterns encode the fusion rules of the underlying conformal field theory.

We provide extensive analytical and numerical evidence that the Laughlin/Jain states are continuously connected to the exact solutions. For more general hierarchical states we exploit the intriguing connection to conformal field theory and construct wave functions that coincide with the exact ones in the solvable limit. If correct, this construction implies the adiabatic continuation of the pertinent states. We provide some numerical support for this scenario at the recently observed fraction 4/11.

Non-QH phases are separated from the thin torus by a phase transition. At half-filling, this leads to a Luttinger liquid of neutral dipoles which provides an explicit microscopic example of how weakly interacting quasiparticles in a reduced (zero) magnetic field emerge at low energies. We argue that this is also smoothly connected to the bulk state.

Place, publisher, year, edition, pages
Stockholm: Fysikum, 2008. 70 p.
Keyword
fractional quantum Hall effect, thin torus, spin chains, conformal field theory, strong correlations, non-abelian states
National Category
Condensed Matter Physics
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-7545 (URN)978-91-7155-627-1 (ISBN)
Public defence
2008-05-28, sal FB53, AlbaNova universitetscentrum, Roslagstullsbacken 21, Stockholm, 13:15 (English)
Opponent
Supervisors
Available from: 2008-05-01 Created: 2008-05-01 Last updated: 2010-02-23Bibliographically approved
2. Non-abelian quantum Hall states and fractional charges in one dimension
Open this publication in new window or tab >>Non-abelian quantum Hall states and fractional charges in one dimension
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The fractional quantum Hall effect has, since its discovery around 30 years ago, been a vivid field of research—both experimentally and theoretically. In this thesis we investigate certain non-abelian quantum Hall states by mapping the two-dimensional system onto a thin torus, where the problem becomes effectively one-dimensional and hopping is suppressed, meaning that the classical electrostatic interaction dominates. The approach assists with a simplified view of ground states and their degeneracies, as well as of the nature of the fractionally charged, minimal excitations of the corresponding quantum Hall states. Similar models are also relevant for cold atoms trapped in one-dimensional optical lattices, where interaction parameters are available for tuning, which opens up for realizing interesting lattice states in controllable environments. The diverse applicability of the one-dimensional electrostatic lattice hamiltonian motivates the exploration of the systems and models treated in this work.

In the absence of hopping or tunneling, the low-energy behavior of the one-dimensional lattice system is ultimately dependent on the nature of the electrostatic interaction present. For ordinary interactions such as Coulomb, the ground state at particle filling fraction ν= p/q has a well-known q-fold center-of-mass degeneracy and the elementary excitations are domain walls of fractional charge e* = ±e/q. These appear in abelian quantum Hall systems and are known since earlier. In this work, we show how other types of interaction give rise to increased ground state degeneracies and, as a result, to the emergence of split fractional charges recognized from non- abelian quantum Hall systems. 

Place, publisher, year, edition, pages
Stockholm: Department of Physics, Stockholm University, 2013. 72 p.
National Category
Physical Sciences Condensed Matter Physics
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-89417 (URN)978-91-7447-714-6 (ISBN)
Public defence
2013-06-04, FB42, AlbaNova universitetscentrum, Roslagstullsbacken 21, Stockholm, 14:00 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 5: Manuscript.

Available from: 2013-05-13 Created: 2013-04-24 Last updated: 2013-06-10Bibliographically approved

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