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Non-abelian quantum Hall states and fractional charges in one dimension
Stockholm University, Faculty of Science, Department of Physics.
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: urn:nbn:se:su:diva-89417ISBN: 978-91-7447-714-6 (print)OAI: oai:DiVA.org:su-89417DiVA: diva2:617865
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
List of papers
1. 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: 2013-05-02Bibliographically approved
2. 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, 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: 2013-05-02Bibliographically approved
3. Spin chain description of rotating bosons at v = 1
Open this publication in new window or tab >>Spin chain description of rotating bosons at v = 1
2009 (English)In: Journal of Statistical Mechanics: Theory and Experiment, ISSN 1742-5468, P07038- p.Article in journal (Refereed) Published
Abstract [en]

We consider bosons at Landau level filling ν = 1 on a thin torus. In analogy with previous work on fermions at filling ν = 1/2, we map the low-energy sector onto a spin-1/2 chain. While the fermionic system may realize the gapless XY phase, we show that typically this does not happen for the bosonic system. Instead, both delta function and Coulomb interaction lead to gapped phases in the bosonic system, and in particular we identify a phase corresponding to the non-Abelian Moore–Read state. In the spin language, the Hamiltonian is dominated by a ferromagnetic next-nearest-neighbor interaction, which leads to a description consistent with the non-trivial degeneracies of the ground and excited states of this phase of matter. In addition we comment on the similarities and differences of the two systems mentioned above and fermions at ν = 5/2.

Keyword
solvable lattice models, fractional QHE (theory), spin chains, ladders and planes (theory), Bose Einstein condensation (theory)
National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-33309 (URN)10.1088/1742-5468/2009/07/L07003 (DOI)000269353300038 ()
Available from: 2009-12-22 Created: 2009-12-22 Last updated: 2013-05-02Bibliographically approved
4. Fractional domain walls from on-site softening in dipolar bosons
Open this publication in new window or tab >>Fractional domain walls from on-site softening in dipolar bosons
2012 (English)In: Physical Review A. Atomic, Molecular, and Optical Physics, ISSN 1050-2947, E-ISSN 1094-1622, Vol. 85, no 3, 033607- p.Article in journal (Refereed) Published
Abstract [en]

We study dipolar bosons in a 1D optical lattice and identify a region in parameter space-strong coupling but relatively weak on-site repulsion-hosting a series of stable charge-density-wave (CDW) states whose low-energy excitations, built from fractional domain walls, have remarkable similarities to those of non-Abelian fractional quantum Hall states. Here, a conventional domain wall between translated CDW's may be split by inserting strings of degenerate, but inequivalent, CDW states. Outside these insulating regions, we find numerous supersolids as well as a superfluid regime. The mentioned phases should be accessible experimentally and, in particular, the fractional domain walls can be created in the ground state using single-site addressing, i.e., by locally changing the chemical potential.

National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-76264 (URN)10.1103/PhysRevA.85.033607 (DOI)000301104400026 ()
Note

4

Available from: 2012-05-24 Created: 2012-05-10 Last updated: 2017-12-07Bibliographically approved
5. Nontrivial ground-state degeneracies and generalized fractional excitations in the 1D lattice
Open this publication in new window or tab >>Nontrivial ground-state degeneracies and generalized fractional excitations in the 1D lattice
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We study a 1D lattice Hamiltonian, relevant for a wide range of interesting physical systems like, e.g., the quantum-Hall system, cold atoms or molecules in optical lattices, and TCNQ salts. Through a tuning of the interaction parameters and a departure from a strictly convex interaction, nontrivial ground-state degeneracies and fractionally charged excitations emerge. The excitations, being a generalization of the fractional charges known from the fractional quantum-Hall effect, appear as domain walls between inequivalent ground states and carry the charge ± e/mq , where m is an integer and associated with the specified interaction, and ν = p/q is the filling fraction in the lattice. The description points at an interesting resemblance to states connected to non-Abelian statistics, which is central for the concept of topological quantum computing. 

National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-89416 (URN)
Available from: 2013-04-24 Created: 2013-04-24 Last updated: 2013-05-02Bibliographically approved

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