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Pfaffian quantum Hall state made simple: Multiple vacua and domain walls on a thin torus
Stockholm University, Faculty of Science, Department of Physics.
Stockholm University, Faculty of Science, Department of Physics.
Stockholm University, Faculty of Science, Department of Physics.
Stockholm University, Faculty of Science, Department of Physics.
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2006 (English)In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 74, no 8, 081308- p.Article in journal (Refereed) Published
Abstract [en]

We analyze the Moore-Read Pfaffian state on a thin torus. The known sixfold degeneracy is realized by two inequivalent crystalline states with a four- and twofold degeneracy, respectively. The fundamental quasihole and quasiparticle excitations are domain walls between these vacua, and simple counting arguments give a Hilbert space of dimension 2n−1 for 2n−k holes and k particles at fixed positions and assign each a charge ±e∕4. This generalizes the known properties of the hole excitations in the Pfaffian state as deduced using conformal field theory techniques. Numerical calculations using a model Hamiltonian and a small number of particles support the presence of a stable phase with degenerate vacua and quarter-charged domain walls also away from the thin-torus limit. A spin-chain Hamiltonian encodes the degenerate vacua and the various domain walls.

Place, publisher, year, edition, pages
2006. Vol. 74, no 8, 081308- p.
National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
URN: urn:nbn:se:su:diva-24938DOI: 10.1103/PhysRevB.74.081308ISI: 000240238900011OAI: oai:DiVA.org:su-24938DiVA: diva2:198556
Available from: 2008-05-01 Created: 2008-05-01 Last updated: 2013-05-02Bibliographically 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
3. Quasiparticles in the Quantum Hall Effect
Open this publication in new window or tab >>Quasiparticles in the Quantum Hall Effect
2006 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The fractional quantum Hall effect (FQHE), discovered in 1982 in a two-dimensional electron system, has generated a wealth of successful theory and new concepts in condensed matter physics, but is still not fully understood. The possibility of having nonabelian quasiparticle statistics has recently attracted attention on purely theoretical grounds but also because of its potential applications in topologically protected quantum computing.

This thesis focuses on the quasiparticles using three different approaches. The first is an effective Chern-Simons theory description, where the noncommutativity imposed on the classical space variables captures the incompressibility. We propose a construction of the quasielectron and illustrate how many-body quantum effects are emulated by a classical noncommutative theory.

The second approach involves a study of quantum Hall states on a torus where one of the periods is taken to be almost zero. Characteristic quantum Hall properties survive in this limit in which they become very simple to understand. We illustrate this by giving a simple counting argument for degeneracy 2n-1, pertinent to nonabelian statistics, in the presence of 2n quasiholes in the Moore-Read state and generalise this result to 2n-k quasiholes and k quasielectrons.

In the third approach, we study the topological nature of the degeneracy 2n-1 by using a recently proposed analogy between the Moore-Read state and the two-dimensional spin-polarized p-wave BCS state. We study a version of this problem where one can use techniques developed in the context of high-Tc superconductors to turn the vortex background into an effective gauge field in a Dirac equation. Topological arguments in the form of index theory gives the degeneracy 2n-1 for 2n vortices.

Place, publisher, year, edition, pages
Stockholm: Fysikum, 2006. 102 p.
Keyword
quantum Hall effect, quasiparticles, noncommutative Chern-Simons theory, nonabelian statistics, thin torus, p-wave superconductivity, topology, index theory, effective gauge fields
National Category
Condensed Matter Physics
Identifiers
urn:nbn:se:su:diva-1250 (URN)91-7155-320-7 (ISBN)
Public defence
2006-09-27, sal FD5, AlbaNova universitetscentrum, Roslagstullsbacken 21, Stockholm, 09:30
Opponent
Supervisors
Available from: 2006-09-01 Created: 2006-09-01 Last updated: 2013-04-29Bibliographically approved

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