#
Vortex Dynamics of Charge Carriers in the Quasi-Relativistic Graphene Model: High-Energy
k
→
·
p
→
Approximation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

## 3. Band Structure and Non-Abelian Zak Phase Simulations

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Definition of a change $\overrightarrow{\gamma}\left[\overrightarrow{l}\left({k}_{y}\right)\right]$ in the flux of non-Abelian gauge field along the paths $\overrightarrow{l}\left({k}_{y}\right)$ to define Wilson loops. A rhombic first Brillouin zone (BZ) of the honeycomb lattice consisting of two triangular sublattices A (green) and B (red) is labelled with a dashed line. High-symmetry points are $\Gamma ,\phantom{\rule{4pt}{0ex}}K\left({K}^{\prime}\right),\phantom{\rule{4pt}{0ex}}M$. Occupied half-BZ is gray shaded. A reference point of coordinates $({k}_{x},{k}_{y})$ is in $\Gamma $. (

**b**,

**e**) Non-Abelian phases ${\mathsf{\Phi}}_{1},\dots ,{\mathsf{\Phi}}_{4}$ of the Wilson-loop eigenvalues in the units of $\pi $ at non-zero (

**b**) and zero (

**e**) gauge fields. (

**c**) A spin–orbit texture of the bands on momentum scales $q/K=0.002$ in contour plots and a model vortex of the precessing orbitals in inset to (

**d**). The angle $\alpha ,0\le \alpha \le \pi /2$ is a variable precession angle of p${}_{z}$-orbital. (

**d**,

**f**) Sketch of topological defects for the quasi-relativistic graphene model at non-zero (

**d**) and zero (

**f**) gauge fields; a bypass over each contour in (

**d**,

**f**) gives phase shift value $4\pi $; $\overrightarrow{q}=\overrightarrow{k}-\overrightarrow{K}$.

**Figure 2.**(

**a**) A vortex texture in contour plots of electron (

**left**) and hole (

**right**) bands calculated without (

**top**) and with (

**bottom**) Majorana-like mass term on momentum scales $q/{K}_{A}=0.002$. (

**b**) A mass correction to the hole (

**top**) and electron (

**bottom**) bands on momentum scales $q/{K}_{A}=0.02$. Contour plots in (

**b**) are displayed in color gradations.

**Figure 3.**Band structure calculated with (

**a**) and without (

**b**) Majorana-like mass term on momentum scale $q/{K}_{A}=0.02$ for quasi-relativistic model of graphene with non-zero gauge field.

**Table 1.**Topological characterization of the graphene models: Model 1 is the massless pseudo-Dirac fermion model, and Models 2 and 3 are the quasi-relativistic graphene model in the approximations of zero- and nonzero-gauge field, respectively. the second column from the left: arguments of the Wilson-loop eigenvalues $\mathcal{W}\left({q}_{y}\right)$.

Type of the Graphene Model | $\mathbf{Arg}\phantom{\rule{4pt}{0ex}}\mathcal{W}\left({\mathit{q}}_{\mathit{y}}\right)$ |
---|---|

Model 1 | $\{0,\phantom{\rule{4pt}{0ex}}\pm \pi \}$; |

Model 2 | $\{0,\phantom{\rule{4pt}{0ex}}\pm \pi /2,\phantom{\rule{4pt}{0ex}}\pm \pi \}$; |

$\{0,-\pi /6,-2\pi /6,-3\pi /6,\dots ,-\pi \}$ at ${q}_{y}\to 0$, | |

Model 3 | $\{0,-\pi /4,-\pi /2,-3\pi /4,-\pi \}$ at ${q}_{y}>0.2\left|K\right|$, |

$\{0,-\pi /4,-\pi /2,-3\pi /4,-\pi \}$ and $\{0,\pi /2,\pi \}$ at ${q}_{y}>0.24\left|K\right|$ |

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Grushevskaya, H.; Krylov, G.
Vortex Dynamics of Charge Carriers in the Quasi-Relativistic Graphene Model: High-Energy *Symmetry* **2020**, *12*, 261.
https://doi.org/10.3390/sym12020261

**AMA Style**

Grushevskaya H, Krylov G.
Vortex Dynamics of Charge Carriers in the Quasi-Relativistic Graphene Model: High-Energy *Symmetry*. 2020; 12(2):261.
https://doi.org/10.3390/sym12020261

**Chicago/Turabian Style**

Grushevskaya, Halina, and George Krylov.
2020. "Vortex Dynamics of Charge Carriers in the Quasi-Relativistic Graphene Model: High-Energy *Symmetry* 12, no. 2: 261.
https://doi.org/10.3390/sym12020261