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39 pages, 537 KiB

Open AccessArticle

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 850

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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41 pages, 479 KiB

Open AccessFeature PaperArticle

Topological Quantization of Fractional Quantum Hall Conductivity

by J. Miller and M. A. Zubkov

Symmetry 2022, 14(10), 2095; https://doi.org/10.3390/sym14102095 - 8 Oct 2022

Cited by 1 | Viewed by 1603

Abstract

We derive a novel topological expression for the Hall conductivity. To that degree we consider the quantum Hall effect (QHE) in a system of interacting electrons. Our formalism is valid for systems in the presence of an external magnetic field, as well as for systems with a nontrivial band topology. That is, the expressions for the conductivity derived are valid for both the ordinary QHE and for the intrinsic anomalous QHE. The expression for the conductivity applies to external fields that may vary in an arbitrary way, and takes into account disorder. Properties related to symmetry and topology are revealed in the fractional quantization of the Hall conductivity. It is assumed that the ground state of the system is degenerate. We represent the QHE conductivity as

\frac{e^{2}}{h} \times \frac{N}{K}

, where K is the degeneracy of the ground state, while

N

is the topological invariant composed of the Wigner-transformed multi-leg Green functions, which takes discrete values. Full article

(This article belongs to the Special Issue Mathematical Modelling of Physical Systems 2021)

29 pages, 930 KiB

Open AccessArticle

Elastic Deformations and Wigner–Weyl Formalism in Graphene

by I.V. Fialkovsky and M.A. Zubkov

Symmetry 2020, 12(2), 317; https://doi.org/10.3390/sym12020317 - 23 Feb 2020

Cited by 17 | Viewed by 3389

Abstract

We discuss the tight-binding models of solid state physics with the

Z_{2}

We discuss the tight-binding models of solid state physics with the

Z_{2}

sublattice symmetry in the presence of elastic deformations in an important particular case—the tight binding model of graphene. In order to describe the dynamics of electronic quasiparticles, the Wigner–Weyl formalism is explored. It allows the calculation of the two-point Green’s function in the presence of two slowly varying external electromagnetic fields and the inhomogeneous modification of the hopping parameters that result from elastic deformations. The developed formalism allows us to consider the influence of elastic deformations and the variations of magnetic field on the quantum Hall effect. Full article

(This article belongs to the Special Issue Relations between Condensed Matter Physics and Relativistic Quantum Field Theory)

16 pages, 314 KiB

Open AccessArticle

Variations à la Fourier-Weyl-Wigner on Quantizations of the Plane and the Half-Plane

by Hervé Bergeron and Jean-Pierre Gazeau

Entropy 2018, 20(10), 787; https://doi.org/10.3390/e20100787 - 13 Oct 2018

Cited by 8 | Viewed by 3917

Abstract

Any quantization maps linearly function on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with the symmetry group of the phase space. Covariant means that the quantization map intertwines classical (geometric operation) and quantum (unitary transformations) symmetries. Integral means that we use all resources of integral calculus, in order to implement the method when we apply it to singular functions, or distributions, for which the integral calculus is an essential ingredient. We first review this quantization scheme before revisiting the cases where symmetry covariance is described by the Weyl-Heisenberg group and the affine group respectively, and we emphasize the fundamental role played by Fourier transform in both cases. As an original outcome of our generalisations of the Wigner-Weyl transform, we show that many properties of the Weyl integral quantization, commonly viewed as optimal, are actually shared by a large family of integral quantizations. Full article

(This article belongs to the Special Issue Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst century)

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