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On Aspects of Anyons and Quantum Graphs
Stockholm University, Faculty of Science, Department of Physics. Nordita.
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of two distinct parts. The first part, based on the first two accompanied papers, is in the field of topological phases of matter and the second part, based on the third accompanied paper, looks at a problem in the field of quantum graphs, a rapidly growing field of mathematical physics.

First, we investigate the entanglement property of the Laughlin state by looking at the rank of the reduced density operator when particles are divided into two groups. We show that the problem of determining this rank translates itself into a  question about symmetric polynomials, namely, one has to determine the lower bound for the degree in each variable of the symmetric polynomials that vanish under a transformation that clusters the particles into groups of equal size and then brings the particles in each group together. Although we were not able to prove this, but we were able to determine the lower bound for the total degree of symmetric polynomials that vanish under the  transformation described. Moreover, we were able to characterize all symmetric polynomials that vanish under this transformation.

In the second paper, we introduce a one-dimensional model of interacting su(2)k anyons. The specific feature of this model is that, through pairing terms present in the Hamiltonian,  the number of anyons of the chain can fluctuate. We also take into account the possibility that anyons hop to empty neighboring sites. We investigate the model in five different points of the parameter space. At one of these points, the Hamiltonian of the model becomes a sum of projectors and we determine the explicit form of all the zero-energy ground states for odd values of k. At the other four points, the system is integrable and we determine the behavior of the model at these integrable points. In particular, we show that the system is critical and determine the CFT describing the system at these points.

It is known that there are non-Hermitian Hamiltonians whose spectra are entirely real. This property can be understood in terms of a certain symmetry of these Hamiltonians, known as PT-symmetry. It is also known that the spectrum of a non-Hermitian PT-symmetric Hamiltonian has reflection symmetry with respect to the real axis. We then ask the reverse question whether or not the reflection symmetry of a non-Hermitian Hamiltonian necessarily implies that the Hamiltonian is PT-symmetric. In the context of quantum graphs, we introduce a model for which the answer to this question is positive.

Place, publisher, year, edition, pages
Stockholm: Department of Physics, Stockholm University , 2017. , p. 107
National Category
Condensed Matter Physics
Research subject
Theoretical Physics
Identifiers
URN: urn:nbn:se:su:diva-142319ISBN: 978-91-7649-813-2 (print)ISBN: 978-91-7649-814-9 (electronic)OAI: oai:DiVA.org:su-142319DiVA, id: diva2:1092010
Public defence
2017-06-13, sal FB42, AlbaNova universitetscentrum, Roslagstullsbacken 21, Stockholm, 13:00 (English)
Opponent
Supervisors
Available from: 2017-05-19 Created: 2017-04-28 Last updated: 2017-05-22Bibliographically approved
List of papers
1. On the particle entanglement spectrum of the Laughlin states
Open this publication in new window or tab >>On the particle entanglement spectrum of the Laughlin states
2015 (English)In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 48, no 28, article id 285205Article in journal (Refereed) Published
Abstract [en]

The study of the entanglement entropy and entanglement spectrum has proven to be very fruitful in identifying topological phases of matter. Typically, one performs numerical studies of finite-size systems. However, there are few rigorous results in this regard. We revisit the problem of determining the rank of the 'particle entanglement spectrum' (PES) of the Laughlin states. We reformulate the problem into a problem concerning the ideal of symmetric polynomials that vanish under the formation of several clusters of particles. We introduce an explicit generating set of this ideal, and we prove that polynomials in this ideal have a total degree that is bounded from below. We discuss the difficulty in proving the same bound on the degree of any of the variables, which is necessary to determine the rank of the PES.

Keywords
entanglement spectrum, quantum Hall effect, Laughlin state
National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-119287 (URN)10.1088/1751-8113/48/28/285205 (DOI)000357281400009 ()
Available from: 2015-08-04 Created: 2015-08-03 Last updated: 2017-05-11Bibliographically approved
2. Anyon chains with pairing terms
Open this publication in new window or tab >>Anyon chains with pairing terms
2017 (English)In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 50, no 13, article id 135201Article in journal (Refereed) Published
Abstract [en]

In this paper we introduce a one-dimensional model of $su(2)_k$ anyons in which the number of anyons can uctuate by means of a pairing term. The model can be tuned to a point at which one can determine the exact zero-energy ground states, in close analogy to the spin-1 AKLT model. We also determine the points at which the model is integrable and determine the behavior of the model at these integrable points. 

Keywords
anyon chain, pairing term, integrable
National Category
Condensed Matter Physics
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-142317 (URN)10.1088/1751-8121/aa5fba (DOI)000396120700001 ()
Available from: 2017-04-28 Created: 2017-04-28 Last updated: 2017-05-11Bibliographically approved
3. Quantum graphs: PT -symmetry and reflection symmetry of the spectrum
Open this publication in new window or tab >>Quantum graphs: PT -symmetry and reflection symmetry of the spectrum
2017 (English)In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 58, no 2, article id 023506Article in journal (Refereed) Published
Abstract [en]

Not necessarily self-adjoint quantum graphs-differential operators on metric graphs-are considered. Assume in addition that the underlying metric graph possesses an automorphism (symmetry) P. If the differential operator is PT -symmetric, then its spectrum has reflection symmetry with respect to the real line. Our goal is to understand whether the opposite statement holds, namely, whether the reflection symmetry of the spectrum of a quantum graph implies that the underlying metric graph possesses a non-trivial automorphism and the differential operator is PT symmetric. We give partial answer to this question by considering equilateral stargraphs. The corresponding Laplace operator with Robin vertex conditions possesses reflection-symmetric spectrum if and only if the operator is PT -symmetric with P being an automorphism of the metric graph.

National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-142541 (URN)10.1063/1.4975757 (DOI)000395290100029 ()
Available from: 2017-05-08 Created: 2017-05-08 Last updated: 2017-05-22Bibliographically approved

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