# Elastic Deformations and Wigner–Weyl Formalism in Graphene

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## Abstract

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## 1. Introduction

## 2. Hamiltonian for the Nonlocal Tight–Binding Model

#### 2.1. General Case

#### 2.2. The ${Z}_{2}$ Sublattice Symmetry

## 3. Weyl Symbol for the Lattice Dirac Operator

#### 3.1. Lattice Dirac Operator

#### 3.2. The Definition of the Weyl Symbol in Momentum Space

#### 3.3. Elastic Deformation and Modification of Hoping Parameters

## 4. Green’s Function and the Groenewold Equation

#### 4.1. Appearance of the Moyal Product

#### 4.2. Lattice Groenewold Equation

#### 4.3. Expression for the Electric Current

## 5. Calculation of the Green’s Function in the Inhomogeneous Lattice Models

#### 5.1. Calculation of the Wigner Transformation of the Green’s Function

#### 5.2. Reconstruction of Fermion Propagator from Its Wigner Transformation

## 6. Total Hall Conductance as the Topological Invariant in Phase Space

#### 6.1. Derivation in the Framework of Wigner–Weyl Formalism

#### 6.2. From Topological Invariant in Phase Space Expressed Through ${G}_{W},{Q}_{W}$ to the Standard Expression for Hall Conductance

## 7. Integer Quantum Hall Effect in the Presence of Varying Magnetic Field and Elastic Deformations

#### 7.1. Constant Magnetic Field and Constant Hopping Parameters

#### 7.2. Constant Magnetic Field and Weakly Varying Hopping Parameters

#### 7.3. Weak Variations of Magnetic Field and Hopping Parameters

#### 7.4. Analytical Elastic Deformations In Graphene

## 8. Conclusions and Discussions

- We calculated the Weyl symbol of the lattice Dirac operator (i.e., the operator $\widehat{Q}$ that enters the action ${\sum}_{\mathit{x},\mathit{y}}{\overline{\Psi}}_{x}{Q}_{\mathit{x},\mathit{y}}{\Psi}_{y}$) in the presence of both elastic deformations and slowly varying external electromagnetic field:$${Q}_{W}=i\omega -t\sum _{j}\left(1-\beta {u}_{kl}(\mathit{x}){b}_{k}^{(j)}{b}_{l}^{(j)}\right)\left(\begin{array}{cc}0& {e}^{\mathrm{i}(\mathit{p}{\mathit{b}}^{(j)}-{A}^{(j)}(\mathit{r}))}\\ {e}^{-i(\mathit{p}{\mathit{b}}^{(j)}-{A}^{(j)}(\mathit{r}))}& 0\end{array}\right)$$$${A}^{(j)}(\mathit{x})={\int}_{\mathit{x}-{\mathit{b}}^{(j)}/2}^{\mathit{x}+{\mathit{b}}^{(j)}/2}\mathit{A}(\mathit{y})d\mathit{y}.$$It was assumed that the variation of electromagnetic field $A(\mathit{x})$ at the distances of order of the lattice spacing may be neglected. In practice this corresponded to magnetic fields B that obeyed $B{a}^{2}\ll 1$. In practice this bound read $B\ll 1000$ Tesla. Additionally, we required that the typical wavelenth of the external electromagnetic field was much larger than the lattice spacing. This did not allow the use of Equation (92) for matter interacting with the X-rays with the wavelengths of the order of several Angstroms and smaller;
- The Wigner transformation of the electron propagator in the presence of the slowly varying magnetic field and arbitrary elastic deformations may be calculated using the following expression:$$\begin{array}{cc}\hfill {G}_{W}(x,p)=& \sum _{M=0\dots \infty}\phantom{\rule{0.166667em}{0ex}}\underset{\u23df}{\left[\dots \left[{Q}_{W}^{-1}(1-{e}^{\overleftrightarrow{\Delta}}){Q}_{W}\right]{Q}_{W}^{-1}(1-{e}^{\overleftrightarrow{\Delta}}){Q}_{W}\right]\dots (1-{e}^{\overleftrightarrow{\Delta}}){Q}_{W}]}{Q}_{W}^{-1}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{142.26378pt}{0ex}}M\phantom{\rule{0.166667em}{0ex}}brackets\hfill \end{array}$$$$\overleftrightarrow{\Delta}=\frac{i}{2}\left({\overleftarrow{\partial}}_{x}\overrightarrow{{\partial}_{p}}-\overleftarrow{{\partial}_{p}}{\overrightarrow{\partial}}_{x}\right);$$
- The electron propagator in the presence of a slowly varying electromagnetic field and elastic deformations may be expressed through the Wigner transformed Green’s function as follows:$$\begin{array}{ccc}\hfill G({x}_{1},{x}_{2})& \approx & \frac{1}{2\pi \left|\mathcal{M}\right|}\int dp{G}_{W}(({x}_{1}+{\mathit{x}}_{2})/2,p){e}^{-ip({x}_{1}-{\mathit{x}}_{2})}\hfill \end{array}$$
- The total average Hall conductivity (i.e., the Hall conductivity integrated over the area of the sample and divided by this area) in the presence of varied weak magnetic field $\mathcal{B}\ll 1/{a}^{2}$ and elastic deformations had the form of:$${\sigma}_{xy}=\frac{\mathcal{N}}{2\pi}-\frac{{\mathcal{N}}^{(0)}}{2\pi}$$$$\mathcal{N}=\frac{T}{\mathcal{A}\phantom{\rule{0.166667em}{0ex}}3!\phantom{\rule{0.166667em}{0ex}}4{\pi}^{2}}\phantom{\rule{0.166667em}{0ex}}{\u03f5}_{ijk}\phantom{\rule{0.166667em}{0ex}}{\int}_{{\mathbb{R}}^{D+1}}\phantom{\rule{-0.166667em}{0ex}}dx\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\int}_{\mathbb{R}\otimes \mathcal{M}}\phantom{\rule{-0.166667em}{0ex}}dp\phantom{\rule{0.166667em}{0ex}}\mathrm{tr}\left[{G}_{W}(p,x)\ast \frac{\partial {Q}_{W}(p,x)}{\partial {p}_{i}}\ast \frac{\partial {G}_{W}(p,x)}{\partial {p}_{j}}\ast \frac{\partial {Q}_{W}(p,x)}{\partial {p}_{k}}\right],$$$${I}_{xy}=\frac{\mathcal{N}-{\mathcal{N}}^{(0)}}{2\pi}\phantom{\rule{0.166667em}{0ex}}U$$(Recall that we used the relativistic system of units. To obtain expression for the Hall current in an ordinary system of units we have to multiply the above expression by the unity of conductance $\frac{{e}^{2}}{\hslash}$);
- The above mentioned representation of the average Hall conductivity through the topological invariant in phase space allowed us to prove that in graphene it was robust to both sufficiently weak variations of magnetic field and sufficiently weak elastic deformations. It is worth mentioning that both mentioned variations of magnetic field and elastic deformations were to be concentrated within the finite region far from the boundary of the sample. Under these conditions Equation (96) was not changed for the smooth variations of lattice Hamiltonian (for the proof see Appendix D in [57]);
- The special case of elastic deformations was considered, when the emergent gauge field in graphene was absent. It was shown that the corresponding deformations were given by the arbitrary analytical functions of coordinates. Namely, the condition of the absence of emergent gauge field was equivalent to the Riemann–Cauchy conditions for the displacement function ${u}_{i}$, $i=1,2$. As a result the function $u(z)={u}_{1}(z)+i{u}_{2}(z)$ appeared to be an analytical function of $z={x}_{1}+i{\mathit{x}}_{2}$, where ${\mathit{x}}_{i}$ were the coordinates of the carbon atoms in the unperturbed honeycomb lattice. Under these circumstances for the constant magnetic field B the Hall current was given by:$${I}_{xy}=-\frac{{N}^{\prime}\phantom{\rule{0.166667em}{0ex}}U}{2\pi}\phantom{\rule{0.166667em}{0ex}}\mathrm{sign}\phantom{\rule{0.166667em}{0ex}}B$$

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Fialkovsky, I.V.; Zubkov, M.A.
Elastic Deformations and Wigner–Weyl Formalism in Graphene. *Symmetry* **2020**, *12*, 317.
https://doi.org/10.3390/sym12020317

**AMA Style**

Fialkovsky IV, Zubkov MA.
Elastic Deformations and Wigner–Weyl Formalism in Graphene. *Symmetry*. 2020; 12(2):317.
https://doi.org/10.3390/sym12020317

**Chicago/Turabian Style**

Fialkovsky, I.V., and M.A. Zubkov.
2020. "Elastic Deformations and Wigner–Weyl Formalism in Graphene" *Symmetry* 12, no. 2: 317.
https://doi.org/10.3390/sym12020317