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Quantum graphs: PT -symmetry and reflection symmetry of the spectrum
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Physics.
Number of Authors: 2
2017 (English)In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 58, no 2, 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.

Place, publisher, year, edition, pages
2017. Vol. 58, no 2, 023506
National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
URN: urn:nbn:se:su:diva-142541DOI: 10.1063/1.4975757ISI: 000395290100029OAI: oai:DiVA.org:su-142541DiVA: diva2:1093604
Available from: 2017-05-08 Created: 2017-05-08 Last updated: 2017-05-22Bibliographically approved
In thesis
1. On Aspects of Anyons and Quantum Graphs
Open this publication in new window or tab >>On Aspects of Anyons and Quantum Graphs
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. 107 p.
National Category
Condensed Matter Physics
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:su:diva-142319 (URN)978-91-7649-813-2 (ISBN)978-91-7649-814-9 (ISBN)
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

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Kurasov, PavelMajidzadeh Garjani, Babak
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