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One-dimensional theory of the quantum Hall system
Stockholm University, Faculty of Science, Department of Physics.
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 [en]
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: urn:nbn:se:su:diva-7545ISBN: 978-91-7155-627-1 (print)OAI: oai:DiVA.org:su-7545DiVA: diva2:198562
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
List of papers
1. Density Matrix Renormalization Group Study of a Lowest Landau Level Electron Gas on a Thin Cylinder
Open this publication in new window or tab >>Density Matrix Renormalization Group Study of a Lowest Landau Level Electron Gas on a Thin Cylinder
Manuscript (Other academic)
Identifiers
urn:nbn:se:su:diva-24935 (URN)
Note
Part of urn:nbn:se:su:diva-7545Available from: 2008-05-01 Created: 2008-05-01 Last updated: 2010-01-13Bibliographically approved
2. Half-Filled Lowest Landau Level on a Thin Torus
Open this publication in new window or tab >>Half-Filled Lowest Landau Level on a Thin Torus
2005 In: Physical Review Letters, Vol. 94, no 026802Article in journal (Refereed) Published
Identifiers
urn:nbn:se:su:diva-24936 (URN)
Note
Part of urn:nbn:se:su:diva-7545Available from: 2008-05-01 Created: 2008-05-01Bibliographically approved
3. 'One-dimensional' theory of the quantum Hall system
Open this publication in new window or tab >>'One-dimensional' theory of the quantum Hall system
2006 In: Journal of Statistical Mechanics: Theory and Experiment, no L04001Article in journal (Refereed) Published
Identifiers
urn:nbn:se:su:diva-24937 (URN)
Note
Part of urn:nbn:se:su:diva-7545Available from: 2008-05-01 Created: 2008-05-01Bibliographically approved
4. Pfaffian quantum Hall state made simple: Multiple vacua and domain walls on a thin torus
Open this publication in new window or tab >>Pfaffian quantum Hall state made simple: Multiple vacua and domain walls on a thin torus
Show others...
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.

National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-24938 (URN)10.1103/PhysRevB.74.081308 (DOI)000240238900011 ()
Available from: 2008-05-01 Created: 2008-05-01 Last updated: 2017-12-13Bibliographically approved
5. 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
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
7. Quantum Hall system in Tao-Thouless limit
Open this publication in new window or tab >>Quantum Hall system in Tao-Thouless limit
2008 (English)In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 77, no 15, 155308- p.Article in journal (Refereed) Published
Abstract [en]

We consider spin-polarized electrons in a single Landau level on a torus. The quantum Hall problem is mapped onto a one-dimensional lattice model with lattice constant 2π/L1, where L1 is a circumference of the torus (in units of the magnetic length). In the Tao-Thouless limit L1→0, the interacting many-electron problem is exactly diagonalized at any rational filling factor ν=p/q≤1. For odd q, the ground state has the same qualitative properties as a bulk (L1→∞) quantum Hall hierarchy state and the lowest-energy quasiparticle excitations have the same fractional charges as in the bulk. These states are the L1→0 limits of the Laughlin and Jain wave functions for filling fractions where these exist. We argue that the exact solutions generically, for odd q, are continuously connected to the two-dimensional bulk quantum Hall hierarchy states—i.e., that there is no phase transition as L1→∞ for filling factors where such states can be observed. For even-denominator fractions, a phase transition occurs as L1 increases. For ν=1/2 this leads to the system being mapped onto a Luttinger liquid of neutral particles at small but finite L1; this then develops continuously into the composite fermion wave function that is believed to describe the bulk ν=1/2 system. The analysis generalizes to non-Abelian quantum Hall states.

National Category
Physical Sciences
Identifiers
urn:nbn:se:su:diva-24941 (URN)10.1103/PhysRevB.77.155308 (DOI)000255457400078 ()
Available from: 2008-05-01 Created: 2008-05-01 Last updated: 2017-12-13Bibliographically approved
8. 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
9. Degeneracy of non-Abelian quantum Hall states on the torus: domain walls and conformal field theory
Open this publication in new window or tab >>Degeneracy of non-Abelian quantum Hall states on the torus: domain walls and conformal field theory
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

Keyword
conformal field theory (theory), fractional QHE (theory)
National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-24943 (URN)10.1088/1742-5468/2008/04/P04016 (DOI)000255662000020 ()
Available from: 2008-05-01 Created: 2008-05-01 Last updated: 2017-12-13Bibliographically approved

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