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Quasielectrons in Abelian and non-Abelian Quantum Hall States
Stockholm University, Faculty of Science, Department of Physics. (Quantum and Field Theory)
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 [en]
quantum Hall effect
National Category
Condensed Matter Physics
Research subject
Theoretical Physics
Identifiers
URN: urn:nbn:se:su:diva-37203ISBN: 978-91-7447-016-1 (print)OAI: oai:DiVA.org:su-37203DiVA: diva2:294410
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
List of papers
1. Microscopic theory of the quantum Hall hierarchy
Open this publication in new window or tab >>Microscopic theory of the quantum Hall hierarchy
2007 (English)In: Physical Review Letters, ISSN 0031-9007, E-ISSN 1079-7114, Vol. 99, no 25, 256803-1-256803-4 p.Article in journal (Refereed) Published
Abstract [en]

We solve the quantum Hall problem exactly in a limit and show that the ground states can be organized in a fractal pattern consistent with the Haldane-Halperin hierarchy, and with the global phase diagram. We present wave functions for a large family of states, including those of Laughlin and Jain and also for states recently observed by Pan et al., and show that they coincide with the exact ones in the solvable limit. We submit that they establish an adiabatic continuation of our exact results to the experimentally accessible regime, thus providing a unified approach to the hierarchy states.

National Category
Subatomic Physics
Research subject
Physics
Identifiers
urn:nbn:se:su:diva-37206 (URN)10.1103/PhysRevLett.99.256803 (DOI)000251887100047 ()
Available from: 2010-02-17 Created: 2010-02-17 Last updated: 2017-12-12Bibliographically approved
2. Quantum Hall hierarchy wave functions: From conformal correlators to Tao-Thouless states
Open this publication in new window or tab >>Quantum Hall hierarchy wave functions: From conformal correlators to Tao-Thouless states
Show others...
2008 (English)In: Physical Review B Condensed Matter, ISSN 0163-1829, E-ISSN 1095-3795, Vol. 77, no 16, 165325-1-165325-9 p.Article in journal (Refereed) Published
Abstract [en]

Laughlin’s wave functions, which describe the fractional quantum Hall effect at filling factorsν=1/(2k+1), can be obtained as correlation functions in a conformal field theory, and recently, this construction was extended to Jain’s composite fermion wave functions at filling factors ν=n/(2kn+1). Here, we generalize this latter construction and present ground state wave functions for all quantum Hall hierarchy states that are obtained by successive condensation of quasielectrons (as opposed to quasiholes) in the original hierarchy construction. By considering these wave functions on a cylinder, we show that they approach the exact ground states, which are the Tao-Thouless states, when the cylinder becomes thin. We also present wave functions for the multihole states, make the connection to Wen’s general classification of Abelian quantum Hall fluids, and discuss whether the fractional statistics of the quasiparticles can be analytically determined. Finally, we discuss to what extent our wave functions can be described in the language of composite fermions.

National Category
Physical Sciences
Identifiers
urn:nbn:se:su:diva-37220 (URN)10.1103/PhysRevB.77.165325 (DOI)000255457500082 ()
Available from: 2010-02-17 Created: 2010-02-17 Last updated: 2017-12-12Bibliographically approved
3. Conformal Field Theory Approach to Abelian and Non-Abelian Quantum Hall Quasielectrons
Open this publication in new window or tab >>Conformal Field Theory Approach to Abelian and Non-Abelian Quantum Hall Quasielectrons
2009 (English)In: Physical Review Letters, ISSN 0031-9007, E-ISSN 1079-7114, Vol. 102, no 16, 166805-1-166805-4 p.Article in journal (Refereed) Published
Abstract [en]

The quasiparticles in quantum Hall liquids carry fractional charge and obey fractional quantum statistics. Of particular recent interest are those with non-Abelian statistics, since their braiding properties could, in principle, be used for robust coding of quantum information. There is already a good theoretical understanding of quasiholes in both Abelian and non-Abelian quantum Hall states. Here we develop conformal field theory methods that allow for an equally precise description of quasielectrons and explicitly construct two- and four-quasielectron excitations of the non-Abelian Moore-Read state.

Place, publisher, year, edition, pages
American Physical Society, 2009
National Category
Physical Sciences
Identifiers
urn:nbn:se:su:diva-37221 (URN)10.1103/PhysRevLett.102.166805 (DOI)000265479300056 ()
Available from: 2010-02-17 Created: 2010-02-17 Last updated: 2017-12-12Bibliographically approved
4. Quantum Hall quasielectron operators in conformal field theory
Open this publication in new window or tab >>Quantum Hall quasielectron operators in conformal field theory
2009 (English)In: Physical Review B Condensed Matter, ISSN 0163-1829, E-ISSN 1095-3795, Vol. 80, no 16, 165330-1-165330-22 p.Article in journal (Refereed) Published
Abstract [en]

In the conformal field theory (CFT) approach to the quantum Hall effect, the multielectron wave functions are expressed as correlation functions in certain rational CFTs. While this approach has led to a well-understood description of the fractionally charged quasihole excitations, the quasielectrons have turned out to be much harder to handle. In particular, forming quasielectron states requires nonlocal operators, in sharp contrast to quasiholes that can be created by local chiral vertex operators. In both cases, the operators are strongly constrained by general requirements of symmetry, braiding, and fusion. Here we construct a quasielectron operator satisfying these demands and show that it reproduces known good quasiparticle wave functions, as well as predicts additional ones. In particular, we propose explicit wave functions for quasielectron excitations of the Moore-Read Pfaffian state. Further, this operator allows us to explicitly express the composite fermion wave functions in the positive Jain series in hierarchical form, thus settling a long-time controversy. We also critically discuss the status of the fractional statistics of quasiparticles in the Abelian hierarchical quantum Hall states and argue that our construction of localized quasielectron states sheds new light on their statistics. At the technical level we introduce a generalized normal ordering that allows us to “fuse” an electron operator with the inverse of an hole operator and also an alternative approach to the background charge needed to neutralize CFT correlators. As a result we get a fully holomorphic CFT representation of a large set of quantum Hall wave functions.

Place, publisher, year, edition, pages
American Physical Society, 2009
National Category
Physical Sciences
Identifiers
urn:nbn:se:su:diva-37222 (URN)10.1103/PhysRevB.80.165330 (DOI)000271352100105 ()
Available from: 2010-02-17 Created: 2010-02-17 Last updated: 2017-12-12Bibliographically approved
5. Condensing Non-Abelian Quasiparticles
Open this publication in new window or tab >>Condensing Non-Abelian Quasiparticles
2010 (English)In: Physical Review Letters, ISSN 0031-9007, E-ISSN 1079-7114, Vol. 104, no 5, 056803-1-056803-4 p.Article in journal (Refereed) Published
Abstract [en]

A most interesting feature of certain fractional quantum Hall states is that their quasiparticles obey non-Abelian fractional statistics. So far, candidate non-Abelian wave functions have been constructed from conformal blocks in cleverly chosen conformal field theories. In this work we present a hierarchy scheme by which we can construct daughter states by condensing non-Abelian quasiparticles (as opposed to quasiholes) in a parent state, and show that the daughters have a non-Abelian statistics that differs from the parent. In particular, we discuss the daughter of the bosonic, spin-polarized Moore-Read state at ν=4/3 as an explicit example.

Place, publisher, year, edition, pages
American Physical Society, 2010
National Category
Physical Sciences
Identifiers
urn:nbn:se:su:diva-37224 (URN)10.1103/PhysRevLett.104.056803 (DOI)000274336800036 ()
Available from: 2010-02-17 Created: 2010-02-17 Last updated: 2017-12-12Bibliographically approved
6. Quantum Hall wave functions on the torus
Open this publication in new window or tab >>Quantum Hall wave functions on the torus
Show others...
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
National Category
Physical Sciences
Research subject
Physics
Identifiers
urn:nbn:se:su:diva-37219 (URN)10.1103/PhysRevB.77.125321 (DOI)000254543000093 ()
Available from: 2010-02-17 Created: 2010-02-17 Last updated: 2017-12-12Bibliographically approved

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