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Quantum Hall Wave Functions on the Torus
Stockholm University, Faculty of Science, Department of Physics.ORCID iD: 0000-0002-4158-3041
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The fractional quantum Hall effect (FQHE), now entering it's fourth decade, continues to draw attention from the condensed matter community. New experiments in recent years are raising hopes that it will be possible to observe quasi-particles with non-abelian anyonic statistics. These particles could form the building blocks of a quantum computer.

The quantum Hall states have topologically protected energy gaps to the low-lying set of excitations. This topological order is not a locally measurable quantity but rather a non-local object, and it is one of the keys to it's stability. From an early stage understanding of the FQHE has been facilitate by constructing trial wave functions. The topological classification of these wave functions have given further insight to the nature of the FQHE.

An early, and successful, wave function construction for filling fractions ν=p/(2p+1) was that of composite fermions on planar and spherical geometries. Recently, new developments using conformal field theory have made it possible to also construct the full Haldane-Halperin hierarchy wave functions on planar and spherical geometries. In this thesis we extend this construction to a toroidal geometry, i.e. a flat surface with periodic boundary conditions.

One of the defining features of topological states of matter in two dimensions is that the ground state is not unique on surfaces with non trivial topology, such as a torus. The archetypical example is the fractional quantum Hall effect, where a state at filling fraction ν=p/q, has at least a q-fold degeneracy on a torus. This has been shown explicitly for a few cases, such as the Laughlin states and the the Moore-Read states, by explicit construction of candidate electron wave functions with good overlap with numerically found states. In this thesis, we construct explicit torus wave functions for a large class of experimentally important quantum liquids, namely the chiral hierarchy states in the lowest Landau level. These states, which includes the prominently observed positive Jain sequence at filling fractions ν=p/(2p+1), are characterized by having boundary modes with only one chirality.

Our construction relies heavily on previous work that expressed the hierarchy wave functions on a plane or a sphere in terms of correlation functions in a conformal field theory. This construction can be taken over to the torus when care is taken to ensure correct behaviour under the modular transformations that leave the geometry of the torus unchanged. Our construction solves the long standing problem of engineering torus wave functions for multi-component many-body states. Since the resulting expressions are rather complicated, we have carefully compared the simplest example, that of ν=2/5, with numerically found wave functions. We have found an extremely good overlap for arbitrary values of the modular parameter τ, that describes the geometry of the torus.

Having explicit torus wave functions allows us to use the methods developed by Read and Read \& Rezayi to numerically compute the quantum Hall viscosity. Hall viscosity is conjectured to be a topologically protected macroscopic transport coefficient characterizing the quantum Hall state. It is related to the shift of the same QH-fluid when it is put on a sphere. The good agreement with the theoretical prediction for the 2/5 state strongly suggests that our wave functions encodes all relevant topologically information.

We also consider the Hall viscosity in the limit of a very thin torus. There we find that the viscosity changes as we approach the thin torus limit. Because of this we study the Laughlin state in that limit and see how the change in viscosity arises from a change in the Hamiltonian hopping elements. Finally we conclude that there are both qualitative and quantitative difference between the thin and the square torus. Thus, one has to be careful when interpreting results in the thin torus limit.

Place, publisher, year, edition, pages
Stockholm: Department of Physics, Stockholm University , 2015.
Keyword [en]
Fractional Quantum Hall Effect
National Category
Condensed Matter Physics
Research subject
Theoretical Physics
Identifiers
URN: urn:nbn:se:su:diva-115616ISBN: 978-91-7649-158-4 (print)OAI: oai:DiVA.org:su-115616DiVA: diva2:798522
Public defence
2015-05-06, FD5, AlbaNova University Center, Roslagstullsbacken 21, Stockholm, 13:00 (English)
Opponent
Supervisors
Funder
Swedish Research Council
Note

At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 4: Manuscript.

Available from: 2015-04-14 Created: 2015-03-26 Last updated: 2015-04-17Bibliographically approved
List of papers
1. Exclusion statistics for quantum Hall states in the Tao-Thouless limit
Open this publication in new window or tab >>Exclusion statistics for quantum Hall states in the Tao-Thouless limit
2011 (English)In: Journal of Statistical Mechanics: Theory and Experiment, ISSN 1742-5468, E-ISSN 1742-5468, P02037Article in journal (Refereed) Published
Abstract [en]

We consider spin-polarized Abelian quantum Hall states in the Tao-Thouless limit, i.e. on a thin torus. For any filling factor nu = p/q a well-defined sector of low energy states is identified and the exclusion statistics of the excitations is determined. We study numerically, at and near nu = 1/3 and 2/5, how the low energy states develop as one moves away from the Tao-Thouless limit towards the physical regime. We find that the lowest energy states in the physical regime develop from states in the low energy sector but that the exclusion statistics is modified.

Keyword
solvable lattice models, fractional QHE (theory)
National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-67482 (URN)10.1088/1742-5468/2011/02/P02037 (DOI)000287802800041 ()
Note

authorCount :2

Available from: 2011-12-29 Created: 2011-12-28 Last updated: 2017-12-08Bibliographically approved
2. Coherent state wave functions on a torus with a constant magnetic field
Open this publication in new window or tab >>Coherent state wave functions on a torus with a constant magnetic field
2013 (English)In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 46, no 27, 275302Article in journal (Refereed) Published
Abstract [en]

We study two alternative definitions of localized states in the lowest Landau level (LLL) on a torus. One definition is to construct localized states, as the projection of the coordinate delta function onto the LLL. Another definition, proposed by Haldane, is to consider the set of functions which have all their zeros at a single point. Since a LLL wave function on a torus, supporting N-phi magnetic flux quanta, is uniquely defined by the position of its N-phi zeros, this defines a set of functions that are expected to be localized around the point maximally far away from the zeros. These two families of localized states have many properties in common with the coherent states on the plane and on the sphere, viz a resolution of unity and a self-reproducing kernel. However, we show that only the projected delta function is maximally localized. Additionally, we show how to project onto the LLL, functions that contain holomorphic derivatives and/or anti-holomorphic polynomials, and apply our methods in the description of hierarchical quantum Hall liquids.

National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-92377 (URN)10.1088/1751-8113/46/27/275302 (DOI)000320758400011 ()
Funder
Swedish Research Council
Note

AuthorCount:1;

Available from: 2013-08-01 Created: 2013-08-01 Last updated: 2017-12-06Bibliographically approved
3. Hall viscosity of hierarchical quantum Hall states
Open this publication in new window or tab >>Hall viscosity of hierarchical quantum Hall states
2014 (English)In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 89, no 12, 125303Article in journal (Refereed) Published
Abstract [en]

Using methods based on conformal field theory, we construct model wave functions on a torus with arbitrary flat metric for all chiral states in the abelian quantum Hall hierarchy. These functions have no variational parameters, and they transform under the modular group in the same way as the multicomponent generalizations of the Laughlin wave functions. Assuming the absence of Berry phases upon adiabatic variations of the modular parameter tau, we calculate the quantum Hall viscosity and find it to be in agreement with the formula, given by Read, which relates the viscosity to the average orbital spin of the electrons. For the filling factor nu = 2/5 Jain state, which is at the second level in the hierarchy, we compare our model wave function with the numerically obtained ground state of the Coulomb interaction Hamiltonian in the lowest Landau level, and find very good agreement in a large region of the complex t plane. For the same example, we also numerically compute the Hall viscosity and find good agreement with the analytical result for both the model wave function and the numerically obtained Coulomb wave function. We argue that this supports the notion of a generalized plasma analogy that would ensure that wave functions obtained using the conformal field theory methods do not acquire Berry phases upon adiabatic evolution.

National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-102770 (URN)10.1103/PhysRevB.89.125303 (DOI)000332631500004 ()
Note

AuthorCount:3;

Available from: 2014-04-24 Created: 2014-04-22 Last updated: 2017-12-05Bibliographically approved
4. Analytical Fock coefficients of the Laughlin state on the torus
Open this publication in new window or tab >>Analytical Fock coefficients of the Laughlin state on the torus
(English)Manuscript (preprint) (Other academic)
Keyword
Fractional Quantum Hall Effect
National Category
Condensed Matter Physics
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-115615 (URN)
Available from: 2015-03-26 Created: 2015-03-26 Last updated: 2016-01-29Bibliographically approved

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