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Symmetry
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Vortex, Topology and Singularity in Quantum Systems
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Vortex, Topology and Singularity in Quantum Systems

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

Deadline for manuscript submissions: closed (31 October 2020) | Viewed by 14399

Special Issue Editor

Prof. Dr. Yanagisawa Takashi

E-Mail Website
Guest Editor
National Institute of Advanced Industrial Science and Technology, Ibaraki 305-8568, Japan
Interests: mechanism of high temperature superconductivity; unconventional non-bcs superconductivity; quantum simulation method; kondo effect and its related phenomena; field theory and quantum many-body systems; quantum field theory

Special Issue Information

Dear Colleagues,

Symmetry plays an important role in physics. When the system is invariant under a symmetry transformation, there is a conserved current and a conserved quantity. It is often the case that an asymmetric state is realized in a symmetry system, and in general the realized state is not invariant under a transformation, while the Hamiltonian is invariant under this symmetry transformation. Symmetry is closely related to the topological structure of a physical system. A non-trivial topology of a quantum system may originate from a singularity that exists in the system. Singularity is universally present in the natural world at all levels. Singularities exist as a topological defect, classical vortex, quantum vortex, or topological structure such as a skyrmion. A singularity causes a topological effect, and universal physics appears from it. We expect that new and universal physics will emerge from the study of quantum singularity. It is important, in order to understand nature, to clarify the properties and dynamics of phenomena that are caused by the presence of singularities.

In quantum condensates such as superconductors and quantum spin systems, quantum vortices appear as a singularity and determine the property of quantum systems, where quantum means that the vorticity is quantized to be integer. In spin systems, the skyrmion emerges as a topological defect. A fractional quantization also will appear in some quantum systems. On the scale of the universe, the black hole is a singular point of the space, and there is a strong vortex around the black hole. Galaxy is a vortex and can be regarded as a singularity in the space.

The topological system is characterized by the existence of topological numbers. The topological number exists along with a singular point. Mathematically, the topological number is given by cohomology. The typical quantum numbers are the winding number and the Chern number, where the Chern number is given by the integral of a gauge field.

This Special Issue of Symmetry is devoted to theories and experiments that reveal or predict new phenomena emerging from quantum structures such as vortices and skyrmions in physical systems. Special emphasis is put on novel phenomena and the dynamics of quantum vortex states in classical and quantum systems and topological structures with non-zero quantum numbers such as skyrmion. This Special Issue will contain a wide spectrum of topics, and mathematical works on the topology and vortex are also welcome.

Prof. Dr. Takashi Yanagisawa
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

  • quantum vortex
  • topological excitation
  • topological materials
  • singularity
  • dynamics of vortices
  • fractional quantization
  • condensed systems

Published Papers (5 papers)

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Research

8 pages, 1518 KiB  
Article
Possible Three-Dimensional Topological Insulator in Pyrochlore Oxides
by Izumi Hase and Takashi Yanagisawa
Symmetry 2020, 12(7), 1076; https://doi.org/10.3390/sym12071076 - 01 Jul 2020
Cited by 6 | Viewed by 2372
Abstract
A Kene–Mele-type nearest-neighbor tight-binding model on a pyrochlore lattice is known to be a topological insulator in some parameter region. It is an important task to realize a topological insulator in a real compound, especially in an oxide that is stable in air. [...] Read more.
A Kene–Mele-type nearest-neighbor tight-binding model on a pyrochlore lattice is known to be a topological insulator in some parameter region. It is an important task to realize a topological insulator in a real compound, especially in an oxide that is stable in air. In this paper we systematically performed band structure calculations for six pyrochlore oxides A2B2O7 (A = Sn, Pb, Tl; B = Nb, Ta), which are properly described by this model, and found that heavily hole-doped Sn2Nb2O7 is a good candidate. Surprisingly, an effective spin–orbit coupling constant λ changes its sign depending on the composition of the material. Furthermore, we calculated the band structure of three virtual pyrochlore oxides, namely In2Nb2O7, In2Ta2O7 and Sn2Zr2O7. We found that Sn2Zr2O7 has a band gap at the k = 0 (Γ) point, similar to Sn2Nb2O7, though the band structure of Sn2Zr2O7 itself differs from the ideal nearest-neighbor tight-binding model. We propose that the co-doped system (In,Sn)2(Nb,Zr)2O7 may become a candidate of the three-dimensional strong topological insulator. Full article
(This article belongs to the Special Issue Vortex, Topology and Singularity in Quantum Systems)
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16 pages, 336 KiB  
Article
Possible Occurrence of Superconductivity by the π-flux Dirac String Formation Due to Spin-Twisting Itinerant Motion of Electrons
by Hiroyasu Koizumi
Symmetry 2020, 12(5), 776; https://doi.org/10.3390/sym12050776 - 07 May 2020
Cited by 8 | Viewed by 1561
Abstract
We show that the Rashba spin-orbit interaction causes spin-twisting itinerant motion of electrons in metals and realizes the quantized cyclotron orbits of conduction electrons without an external magnetic field. From the view point of the Berry connection, the cause of this quantization is [...] Read more.
We show that the Rashba spin-orbit interaction causes spin-twisting itinerant motion of electrons in metals and realizes the quantized cyclotron orbits of conduction electrons without an external magnetic field. From the view point of the Berry connection, the cause of this quantization is the appearance of a non-trivial Berry connection A fic = − ℏ 2 e ∇ χ ( χ is an angular variable with period 2 π ) that generates π flux (in the units of ℏ = 1 , e = 1 , c = 1 ) inside the nodal singularities of the wave function (a “Dirac string”) along the centers of spin-twisting. Since it has been shown in our previous work that the collective mode of ∇ χ is stabilized by the electron-pairing and generates supercurrent, the π -flux Dirac string created by the spin-twisting itinerant motion will be stabilized by the electron-pairing and produce supercurrent. Full article
(This article belongs to the Special Issue Vortex, Topology and Singularity in Quantum Systems)
15 pages, 306 KiB  
Article
Zero-Energy Modes, Fractional Fermion Numbers and The Index Theorem in a Vortex-Dirac Fermion System
by Takashi Yanagisawa
Symmetry 2020, 12(3), 373; https://doi.org/10.3390/sym12030373 - 02 Mar 2020
Cited by 2 | Viewed by 2583
Abstract
Physics of topological materials has attracted much attention from both physicists and mathematicians recently. The index and the fermion number of Dirac fermions play an important role in topological insulators and topological superconductors. A zero-energy mode exists when Dirac fermions couple to objects [...] Read more.
Physics of topological materials has attracted much attention from both physicists and mathematicians recently. The index and the fermion number of Dirac fermions play an important role in topological insulators and topological superconductors. A zero-energy mode exists when Dirac fermions couple to objects with soliton-like structure such as kinks, vortices, monopoles, strings, and branes. We discuss a system of Dirac fermions interacting with a vortex and a kink. This kind of systems will be realized on the surface of topological insulators where Dirac fermions exist. The fermion number is fractionalized and this is related to the presence of fermion zero-energy excitation modes. A zero-energy mode can be regarded as a Majorana fermion mode when the chemical potential vanishes. Our discussion includes the case where there is a half-flux quantum vortex associated with a kink in a magnetic field in a bilayer superconductor. A normalizable wave function of fermion zero-energy mode does not exist in the core of the half-flux quantum vortex. The index of Dirac operator and the fermion number have additional contributions when a soliton scalar field has a singularity. Full article
(This article belongs to the Special Issue Vortex, Topology and Singularity in Quantum Systems)
14 pages, 7957 KiB  
Article
Vortex Dynamics of Charge Carriers in the Quasi-Relativistic Graphene Model: High-Energy k → · p → Approximation
by Halina Grushevskaya and George Krylov
Symmetry 2020, 12(2), 261; https://doi.org/10.3390/sym12020261 - 08 Feb 2020
Cited by 7 | Viewed by 1835
Abstract
Within the earlier developed high-energy- k → · p → -Hamiltonian approach to describe graphene-like materials, the simulations of non-Abelian Zak phases and band structure of the quasi-relativistic graphene model with a number of flavors N = 3 have been performed in approximations [...] Read more.
Within the earlier developed high-energy- k → · p → -Hamiltonian approach to describe graphene-like materials, the simulations of non-Abelian Zak phases and band structure of the quasi-relativistic graphene model with a number of flavors N = 3 have been performed in approximations with and without gauge fields (flavors). It has been shown that a Zak-phases set for non-Abelian Majorana-like excitations (modes) in Dirac valleys of the quasi-relativistic graphene model is the cyclic group Z 12 . This group is deformed into Z 8 at sufficiently high momenta due to deconfinement of the modes. Since the deconfinement removes the degeneracy of the eightfolding valleys, Weyl nodes and antinodes emerge. We offer that a Majorana-like mass term of the quasi-relativistic model affects the graphene band structure in the following way. Firstly, the inverse symmetry emerges in the graphene model with Majorana-like mass term, and secondly the mass term shifts the location of Weyl nodes and antinodes into the region of higher energies. Full article
(This article belongs to the Special Issue Vortex, Topology and Singularity in Quantum Systems)
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19 pages, 2817 KiB  
Article
Relation of Superconducting Pairing Symmetry and Non-Magnetic Impurity Effects in Vortex States
by Yasuaki Sera, Takahiro Ueda, Hiroto Adachi and Masanori Ichioka
Symmetry 2020, 12(1), 175; https://doi.org/10.3390/sym12010175 - 18 Jan 2020
Cited by 3 | Viewed by 3394
Abstract
Non-magnetic impurity scattering effects on the vortex core states are theoretically studied to clarify the contributions from the sign-change of the pairing function in anisotropic superconductors. The vortex states are calculated by the Eilenberger theory in superconductors with p x -wave pairing symmetry, [...] Read more.
Non-magnetic impurity scattering effects on the vortex core states are theoretically studied to clarify the contributions from the sign-change of the pairing function in anisotropic superconductors. The vortex states are calculated by the Eilenberger theory in superconductors with p x -wave pairing symmetry, as well as the corresponding anisotropic s-wave symmetry. From the spatial structure of the pair potential and the local electronic states around a vortex, we examine the differences between anisotropic superconductors with and without sign-change of the pairing function, and estimate how twofold symmetric vortex core images change with increasing the impurity scattering rate both in the Born and the unitary limits. We found that twofold symmetric vortex core image of zero-energy local density of states changes the orientation of the twofold symmetry with increasing the scattering rate when the sign change occurs in the pairing function. Without the sign change, the vortex core shape reduces to circular one with approaching dirty cases. These results of the impurity effects are valuable for identifying the pairing symmetry by observation of the vortex core image by the STM observation. Full article
(This article belongs to the Special Issue Vortex, Topology and Singularity in Quantum Systems)
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