Precise Wigner–Weyl Calculus for the Honeycomb Lattice
Abstract
:1. Introduction
- Star product identity:
- First trace identity:Here, by , we understand the trace of the Weyl symbol, while is the trace of an operator itself.
- Second trace identity:
- Weyl symbol of the identity operator:
2. Statement of the Main Results
2.1. Definition of Weyl Symbol and Its Properties
- Trace property:
- Second trace identity:
- Star property:
- Weyl symbol of unity:
- Star product without differentiation:
2.2. Quantum Hall Effect in Condensed Matter System Defined on Honeycomb Lattice
- Weyl symbols and obey Groenewold equationHere, the Moyal product ∗ is defined asIt is worth mentioning that, for the complete description of the system, we need the values of the Weyl symbols defined on spatial phase space . For such values of spatial momenta and coordinates, the Weyl symbol of unity is equal to 1.
- We express the DC conductivity (in units of , averaged over the lattice area) of the two-dimensional non-interacting systems asHere, means anti-symmetrization.
- We show that the above expression for the conductivity (in units of , averaged over the system area) is reduced to the following expression in the case of the equilibrium system at zero temperature:Here, the is the Weyl symbol of the Matsubara Green function, while is the Weyl symbol of its inverse. Momentum space is a Euclidean one; its points are denoted by . is the Matsubara frequency.
- One can check that Equation (27) is a topological invariant. For that, we need the system to have been in thermal equilibrium originally, and need the thermal equilibrium to correspond to zero temperature. Moreover, we need the Hamiltonian to not depend on time. The value of the average conductivity is then robust to smooth variations of the system. The sum over x is important for the topological invariance of this quantity.
3. Wigner–Weyl Calculus of Felix Buot (1D)
3.1. The Hilbert Space
3.2. The Symbol
3.3. Moyal Product
3.4. Trace and Its Properties
3.5. The Symbol of the Identity Operator
4. Wigner–Weyl Calculus of Felix Buot in Graphene
4.1. The Hilbert Space (Physical Properties)
4.2. Symbol
4.3. Moyal Product
4.4. Trace and Its Properties
4.5. Symbol of the Identity Operator
4.6. Star Product without Differentiation
5. Precise Wigner–Weyl Calculus in Graphene
5.1. The Hilbert Space (Extended Properties)
5.2. W Symbol
5.3. Moyal Product
5.4. Trace and Its Properties
5.5. W Symbol of the Identity Operator
5.6. Star Product without Differentiation
6. Dynamics of Systems Defined on Honeycomb Lattice, and Hall Conductivity
6.1. Keldysh Technique of Field Theory
6.2. Electric Conductivity and Wigner–Weyl Calculus
6.3. Gauge Transformation of Weyl Symbol
6.4. Equilibrium Limit of Hall Conductivity
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Chobanyan, R.; Zubkov, M.A. Precise Wigner–Weyl Calculus for the Honeycomb Lattice. Symmetry 2024, 16, 1081. https://doi.org/10.3390/sym16081081
Chobanyan R, Zubkov MA. Precise Wigner–Weyl Calculus for the Honeycomb Lattice. Symmetry. 2024; 16(8):1081. https://doi.org/10.3390/sym16081081
Chicago/Turabian StyleChobanyan, Raphael, and Mikhail A. Zubkov. 2024. "Precise Wigner–Weyl Calculus for the Honeycomb Lattice" Symmetry 16, no. 8: 1081. https://doi.org/10.3390/sym16081081