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Quantum Hall wave functions on the torus
Stockholm University, Faculty of Science, Department of Physics.
Stockholm University, Faculty of Science, Department of Physics.
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2008 (English)In: Physical Review B Condensed Matter, ISSN 0163-1829, E-ISSN 1095-3795, Vol. 77, no 12, 125321-1-125321-16 p.Article in journal (Refereed) Published
Abstract [en]

We present explicit expressions for a large set of hierarchy wave functions on the torus. Included are the Laughlin states, the states in the positive Jain series, and recently observed states at, e.g., ν=4∕11. The techniques we use constitute a nontrivial extension of the conformal field theory methods developed earlier to construct the corresponding wave functions in disk geometry.

Place, publisher, year, edition, pages
American Physical Society , 2008. Vol. 77, no 12, 125321-1-125321-16 p.
National Category
Physical Sciences
Research subject
Physics
Identifiers
URN: urn:nbn:se:su:diva-37219DOI: 10.1103/PhysRevB.77.125321ISI: 000254543000093OAI: oai:DiVA.org:su-37219DiVA: diva2:295419
Available from: 2010-02-17 Created: 2010-02-17 Last updated: 2017-12-12Bibliographically approved
In thesis
1. Quasielectrons in Abelian and non-Abelian Quantum Hall States
Open this publication in new window or tab >>Quasielectrons in Abelian and non-Abelian Quantum Hall States
2010 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Strongly correlated electron systems continue to  attract a lot of interest. Especially two-dimensional electron systems have shown many surprising behaviours. A fascinating example is the quantum Hall effect, which arises when electrons are confined to two dimensions, subjected to a very large magnetic field, and cooled to very low temperatures. Under these conditions, the electrons form new states of matter - the strongly correlated quantum Hall liquids. A hallmark of these quantum liquids is the precise quantization of the Hall conductance, but in recent years more attention has been focused on their exotic excitations. These have fractional electric charge, and fractional exchange statistics. The latter implies that the wave function is multiplied by a phase factor containing a fractional phase when two quasiparticles are moved around each other. Thus, they are neither bosons nor fermions, but so-called anyons. In recent years, there has accumulated theoretical and experimental evidence  for some of the quantum Hall liquids having even more exotic excitations with not only fractional but non-Abelian exchange statistics. The quasiparticles of such systems could be used to build topologically protected quantum bits, which are much more robust than presently available quantum bits; they would be the ideal building blocks of a quantum computer.

 We use conformal field theory to describe the quantum Hall liquids, as well as their excitations. In particular, we represent the electrons and quasiparticles by conformal field theory operators. Even though the operator description for quasiholes is very well understood, it was for a long time unclear how to describe their antiparticles, the quasielectrons. We found an operator  that describes quasielectron excitations correctly and shares many of the useful properties of the corresponding quasihole operator. For instance, many of the topological properties of the particles are manifest in the operator. This not only adds a missing piece to the quantum Hall puzzle,  but it also opens up new and exciting possibilities. For instance, we were able to extend this construction to the non-Abelian states. A highly non-trivial application of our approach is the condensation of non-Abelian quasielectrons, which yields new non-Abelian quantum Hall states with non-Abelian properties that differ from those of their parent states.

Place, publisher, year, edition, pages
Stockholm: Department of Physics, Stockholm University, 2010. 62 p.
Keyword
quantum Hall effect
National Category
Condensed Matter Physics
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-37203 (URN)978-91-7447-016-1 (ISBN)
Public defence
2010-03-30, FB52, AlbaNova University Center, Roslagstullsbacken 21, Stockholm, 10:00 (English)
Opponent
Supervisors
Available from: 2010-03-08 Created: 2010-02-17 Last updated: 2010-02-24Bibliographically approved
2. 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

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