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Symmetry
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Symmetry and Quantum Orders
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Symmetry and Quantum Orders

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

Deadline for manuscript submissions: 31 March 2025 | Viewed by 8345

Special Issue Editor

Prof. Dr. Alexander Seidel

E-Mail Website
Guest Editor
Department of Physics, Washington University in St. Louis, CB 1105, One Brookings Drive, St. Louis, MO 63130-4899, USA
Interests: theoretical condensed matter physics; quantum-many-body physics; strongly correlated systems; topological orders; fractional quantum hall effect; quantum magnetism; exactly solvable models; field theories of condensed matter

Special Issue Information

Dear Colleagues,

The classification of phases of matter and the transitions between them is the main pursuit of condensed matter physics. Symmetry has traditionally been the central idea to introduce order into the enormous complexity of the quantum many-body problem governing the behavior of matter. Through much of the 20th century, the Landau paradigm of broken symmetry shaped our understanding of how to classify phases of matter and how to characterize the critical phenomena that separate them. In recent years, many exciting developments have deepened our understanding of new invariants that, in part, depend on the presence of symmetry but do not rely on the breaking of it, thereby separating (quantum) phases of matter with identical symmetry groups. Concurrent developments have emphasized the need for higher form symmetries, describing conservation laws for extended objects, to accomplish a more complete understanding of all possible quantum phases of matter. 

The present Special Issue is broadly dedicated to the role of symmetry in constraining and shaping the possible equilibrium and non-equilibrium behaviors of quantum matter. Relevant topics include, but are not limited to symmetry-protected and symmetry-enriched topological phases, continuous phase transition, classification and detection schemes for quantum orders, entanglement measures, lattice models, higher-form symmetries, effective field theory, and quantum magnetism.

Dr. Alexander Seidel
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

  • topological phases
  • quantum orders
  • quantum magnetism
  • global symmetries
  • gauge symmetries
  • higher-form symmetries
  • emergent symmetries
  • fractons
  • quantum phase transitions
  • lattice models
  • effective field theory
  • tensor network states
  • entanglement
  • broken symmetry

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Published Papers (5 papers)

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Research

39 pages, 537 KiB  
Article
Precise Wigner–Weyl Calculus for the Honeycomb Lattice
by Raphael Chobanyan and Mikhail A. Zubkov
Symmetry 2024, 16(8), 1081; https://doi.org/10.3390/sym16081081 - 20 Aug 2024
Viewed by 429
Abstract
In this paper, we propose a precise Wigner–Weyl calculus for the models defined on the honeycomb lattice. We construct two symbols of operators: the B symbol, which is similar to the one introduced by F. Buot, and the W (or, Weyl) symbol. The [...] Read more.
In this paper, we propose a precise Wigner–Weyl calculus for the models defined on the honeycomb lattice. We construct two symbols of operators: the B symbol, which is similar to the one introduced by F. Buot, and the W (or, Weyl) symbol. The latter possesses the set of useful properties. These identities allow us to use it in physical applications. In particular, we derive topological expression for the Hall conductivity through the Wigner-transformed Green function. This expression may be used for the description of the systems with artificial honeycomb lattice, when magnetic flux through the lattice cell is of the order of elementary quantum of magnetic flux. It is worth mentioning that, in the present paper, we do not consider the effect of interactions. Full article
(This article belongs to the Special Issue Symmetry and Quantum Orders)
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14 pages, 284 KiB  
Article
Solvable Two-Dimensional Dirac Equation with Matrix Potential: Graphene in External Electromagnetic Field
by Mikhail V. Ioffe and David N. Nishnianidze
Symmetry 2024, 16(1), 126; https://doi.org/10.3390/sym16010126 - 21 Jan 2024
Cited by 1 | Viewed by 1269
Abstract
It is known that the excitations in graphene-like materials in external electromagnetic field are described by solutions of a massless two-dimensional Dirac equation which includes both Hermitian off-diagonal matrix and scalar potentials. Up to now, such two-component wave functions were calculated for different [...] Read more.
It is known that the excitations in graphene-like materials in external electromagnetic field are described by solutions of a massless two-dimensional Dirac equation which includes both Hermitian off-diagonal matrix and scalar potentials. Up to now, such two-component wave functions were calculated for different forms of external potentials, though as a rule depending on only one spatial variable. Here, we shall find analytically the solutions for a wide class of combinations of matrix and scalar external potentials which physically correspond to applied mutually orthogonal magnetic and longitudinal electrostatic fields, both depending really on two spatial variables. The main tool for this progress is provided by supersymmetrical (SUSY) intertwining relations, specifically, by their most general—asymmetrical—form proposed recently by the authors. This SUSY-like method is applied in two steps, similar to the second order factorizable (reducible) SUSY transformations in ordinary quantum mechanics. Full article
(This article belongs to the Special Issue Symmetry and Quantum Orders)
20 pages, 451 KiB  
Article
Sequencing the Entangled DNA of Fractional Quantum Hall Fluids
by Joseph R. Cruise and Alexander Seidel
Symmetry 2023, 15(2), 303; https://doi.org/10.3390/sym15020303 - 21 Jan 2023
Cited by 2 | Viewed by 1617
Abstract
We introduce and prove the “root theorem”, which establishes a condition for families of operators to annihilate all root states associated with zero modes of a given positive semi-definite k-body Hamiltonian chosen from a large class. This class is motivated by fractional [...] Read more.
We introduce and prove the “root theorem”, which establishes a condition for families of operators to annihilate all root states associated with zero modes of a given positive semi-definite k-body Hamiltonian chosen from a large class. This class is motivated by fractional quantum Hall and related problems, and features generally long-ranged, one-dimensional, dipole-conserving terms. Our theorem streamlines analysis of zero-modes in contexts where “generalized” or “entangled” Pauli principles apply. One major application of the theorem is to parent Hamiltonians for mixed Landau-level wave functions, such as unprojected composite fermion or parton-like states that were recently discussed in the literature, where it is difficult to rigorously establish a complete set of zero modes with traditional polynomial techniques. As a simple application, we show that a modified V1 pseudo-potential, obtained via retention of only half the terms, stabilizes the ν=1/2 Tao–Thouless state as the unique densest ground state. Full article
(This article belongs to the Special Issue Symmetry and Quantum Orders)
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20 pages, 369 KiB  
Article
The Franke–Gorini–Kossakowski–Lindblad–Sudarshan (FGKLS) Equation for Two-Dimensional Systems
by Alexander A. Andrianov, Mikhail V. Ioffe, Ekaterina A. Izotova and Oleg O. Novikov
Symmetry 2022, 14(4), 754; https://doi.org/10.3390/sym14040754 - 6 Apr 2022
Cited by 3 | Viewed by 1610
Abstract
Open quantum systems are, in general, described by a density matrix that is evolving under transformations belonging to a dynamical semigroup. They can obey the Franke–Gorini–Kossakowski–Lindblad–Sudarshan (FGKLS) equation. We exhaustively study the case of a Hilbert space of dimension 2. First, we find [...] Read more.
Open quantum systems are, in general, described by a density matrix that is evolving under transformations belonging to a dynamical semigroup. They can obey the Franke–Gorini–Kossakowski–Lindblad–Sudarshan (FGKLS) equation. We exhaustively study the case of a Hilbert space of dimension 2. First, we find final fixed states (called pointers) of an evolution of an open system, and we then obtain a general solution to the FGKLS equation and confirm that it converges to a pointer. After this, we check that the solution has physical meaning, i.e., it is Hermitian, positive and has trace equal to 1, and find a moment of time starting from which the FGKLS equation can be used—the range of applicability of the semigroup symmetry. Next, we study the behavior of a solution for a weak interaction with an environment and make a distinction between interacting and non-interacting cases. Finally, we prove that there cannot exist oscillating solutions to the FGKLS equation, which would resemble the behavior of a closed quantum system. Full article
(This article belongs to the Special Issue Symmetry and Quantum Orders)
22 pages, 392 KiB  
Article
Classification of Metaplectic Fusion Categories
by Eddy Ardonne, Peter E. Finch and Matthew Titsworth
Symmetry 2021, 13(11), 2102; https://doi.org/10.3390/sym13112102 - 5 Nov 2021
Cited by 2 | Viewed by 1875
Abstract
In this paper, we study a family of fusion and modular systems realizing fusion categories Grothendieck equivalent to the representation category for so(2p+1)2. These categories describe non-abelian anyons dubbed ‘metaplectic anyons’. We obtain explicit expressions [...] Read more.
In this paper, we study a family of fusion and modular systems realizing fusion categories Grothendieck equivalent to the representation category for so(2p+1)2. These categories describe non-abelian anyons dubbed ‘metaplectic anyons’. We obtain explicit expressions for all the F- and R-symbols. Based on these, we conjecture a classification for their monoidal equivalence classes from an analysis of their gauge invariants and define a function which gives us the number of classes. Full article
(This article belongs to the Special Issue Symmetry and Quantum Orders)
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