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Symmetry
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New trends on Symmetry and Topology in Quantum Mechanics
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New trends on Symmetry and Topology in Quantum Mechanics

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Physics".

Deadline for manuscript submissions: closed (29 February 2020) | Viewed by 11258

Special Issue Editor

Prof. Dr. Alberto Ibort

E-Mail Website
Guest Editor
1. ICMAT, Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Nicolás Cabrera, 13-15, Campus de Cantoblanco, UAM, 28049 Madrid, Spain
2. Departemento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganés, Madrid, Spain
Interests: quantum mechanics; functional analysis; differential geometry; geometrical mechanics

Special Issue Information

Dear Colleagues,

This Special Issue “New Trends in Symmetry and Topology in Quantum Mechanics” will focus on emergent symmetry and topological properties of quantum systems beyond the standard paradigm established by the well-known theory of unitary representation of groups and non-trivial topological systems, like the electron-monopole system.  

Non-trivial topological properties are obtained from the reduction of simpler systems or found in certain phases in condensed matter physics, in both cases with an underlying higher symmetry order.   The topological and symmetry properties are also relevant in the description of quantum systems in graphs and, in a much more general fashion, in quantum systems described by general self-adjoint extensions on manifolds with boundary. Systems whose symmetries are described by groupoids instead of just groups are called to play a relevant role both at the fundamental level and in concrete applications in condensed matter. Finally, the innovative role played by symmetries in the description of a sampling theory of quantum systems is another signature of the diversity that a whole family of new developments in the theory of symmetry and topology in quantum systems will bring.

Prof. Alberto Ibort
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Symmetries, topology, and self-adjoint extensions
  • Symmetries and topology in condensed matter
  • Symmetries and groupoids
  • Symmetries and reduction in quantum systems
  • Symmetries, graphs, and quantum systems
  • Symmetries, sampling, and quantum systems.

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Published Papers (4 papers)

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Research

21 pages, 762 KiB  
Article
Covering Graphs, Magnetic Spectral Gaps and Applications to Polymers and Nanoribbons
by John Stewart Fabila-Carrasco and Fernando Lledó
Symmetry 2019, 11(9), 1163; https://doi.org/10.3390/sym11091163 - 14 Sep 2019
Cited by 6 | Viewed by 2394
Abstract
In this article, we analyze the spectrum of discrete magnetic Laplacians (DML) on an infinite covering graph G ˜ → G = G ˜ / Γ with (Abelian) lattice group Γ and periodic magnetic potential β ˜ . We give sufficient conditions for [...] Read more.
In this article, we analyze the spectrum of discrete magnetic Laplacians (DML) on an infinite covering graph G ˜ → G = G ˜ / Γ with (Abelian) lattice group Γ and periodic magnetic potential β ˜ . We give sufficient conditions for the existence of spectral gaps in the spectrum of the DML and study how these depend on β ˜ . The magnetic potential can be interpreted as a control parameter for the spectral bands and gaps. We apply these results to describe the spectral band/gap structure of polymers (polyacetylene) and nanoribbons in the presence of a constant magnetic field. Full article
(This article belongs to the Special Issue New trends on Symmetry and Topology in Quantum Mechanics)
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21 pages, 1307 KiB  
Article
On Z -Invariant Self-Adjoint Extensions of the Laplacian on Quantum Circuits
by Aitor Balmaseda, Fabio Di Cosmo and Juan Manuel Pérez-Pardo
Symmetry 2019, 11(8), 1047; https://doi.org/10.3390/sym11081047 - 14 Aug 2019
Cited by 4 | Viewed by 2617
Abstract
An analysis of the invariance properties of self-adjoint extensions of symmetric operators under the action of a group of symmetries is presented. For a given group G, criteria for the existence of G-invariant self-adjoint extensions of the Laplace–Beltrami operator over a [...] Read more.
An analysis of the invariance properties of self-adjoint extensions of symmetric operators under the action of a group of symmetries is presented. For a given group G, criteria for the existence of G-invariant self-adjoint extensions of the Laplace–Beltrami operator over a Riemannian manifold are illustrated and critically revisited. These criteria are employed for characterising self-adjoint extensions of the Laplace–Beltrami operator on an infinite set of intervals, Ω , constituting a quantum circuit, which are invariant under a given action of the group Z . A study of the different unitary representations of the group Z on the space of square integrable functions on Ω is performed and the corresponding Z -invariant self-adjoint extensions of the Laplace–Beltrami operator are introduced. The study and characterisation of the invariance properties allows for the determination of the spectrum and generalised eigenfunctions in particular examples. Full article
(This article belongs to the Special Issue New trends on Symmetry and Topology in Quantum Mechanics)
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15 pages, 364 KiB  
Article
Sampling Associated with a Unitary Representation of a Semi-Direct Product of Groups: A Filter Bank Approach
by Antonio G. García, Miguel Angel Hernández-Medina and Gerardo Pérez-Villalón
Symmetry 2019, 11(4), 529; https://doi.org/10.3390/sym11040529 - 12 Apr 2019
Viewed by 2108
Abstract
An abstract sampling theory associated with a unitary representation of a countable discrete non abelian group G, which is a semi-direct product of groups, on a separable Hilbert space is studied. A suitable expression of the data samples, the use of a [...] Read more.
An abstract sampling theory associated with a unitary representation of a countable discrete non abelian group G, which is a semi-direct product of groups, on a separable Hilbert space is studied. A suitable expression of the data samples, the use of a filter bank formalism and the corresponding frame analysis allow for fixing the mathematical problem to be solved: the search of appropriate dual frames for ℓ 2 ( G ) . An example involving crystallographic groups illustrates the obtained results by using either average or pointwise samples. Full article
(This article belongs to the Special Issue New trends on Symmetry and Topology in Quantum Mechanics)
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28 pages, 373 KiB  
Article
On the Structure of Finite Groupoids and Their Representations
by Alberto Ibort and Miguel A. Rodríguez
Symmetry 2019, 11(3), 414; https://doi.org/10.3390/sym11030414 - 20 Mar 2019
Cited by 9 | Viewed by 3531
Abstract
In this paper, both the structure and the theory of representations of finite groupoids are discussed. A finite connected groupoid turns out to be an extension of the groupoids of pairs of its set of units by its canonical totally disconnected isotropy subgroupoid. [...] Read more.
In this paper, both the structure and the theory of representations of finite groupoids are discussed. A finite connected groupoid turns out to be an extension of the groupoids of pairs of its set of units by its canonical totally disconnected isotropy subgroupoid. An extension of Maschke’s theorem for groups is proved showing that the algebra of a finite groupoid is semisimple and all finite-dimensional linear representations of finite groupoids are completely reducible. The theory of characters for finite-dimensional representations of finite groupoids is developed and it is shown that irreducible representations of the groupoid are in one-to-one correspondence with irreducible representation of its isotropy groups, with an extension of Burnside’s theorem describing the decomposition of the regular representation of a finite groupoid. Some simple examples illustrating these results are exhibited with emphasis on the groupoids interpretation of Schwinger’s description of quantum mechanical systems. Full article
(This article belongs to the Special Issue New trends on Symmetry and Topology in Quantum Mechanics)
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