# Quantum Diffusion in the Lowest Landau Level of Disordered Graphene

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Tight-Binding and Effective Hamiltonian of Graphene

## 3. Topological Chern Number

## 4. The Effective Hamiltonian in Strong Magnetic Field

## 5. Single-Particle Propagator in the Lowest Landau Level

## 6. Static Conductivity of the Pristine Graphene vs. the Lowest Landau Level

## 7. Single-Particle Propagator Renormalization Due to the Disorder

## 8. Mean Squared Displacement of the Disordered System

$=\frac{1}{4{E}_{g}^{2}}{\left(\frac{{k}^{2}}{\pi}\right)}^{2}{\displaystyle \sum _{s=\pm}}{\left(2{X}_{s}^{}\right)}^{2}exp\left[-{k}^{2}{r}^{2}\right],$ | (49) | |

$=\frac{1}{4{E}_{g}^{2}}{\left(\frac{{k}^{2}}{\pi}\right)}^{2}{\displaystyle \sum _{s=\pm}}\phantom{\rule{3.33333pt}{0ex}}\frac{{\left(2{X}_{s}^{}\right)}^{4}}{2}exp\left[-\frac{{k}^{2}{r}^{2}}{2}\right],$ | (50) | |

$=\frac{1}{4{E}_{g}^{2}}{\left(\frac{{k}^{2}}{\pi}\right)}^{2}{\displaystyle \sum _{s=\pm}}\phantom{\rule{3.33333pt}{0ex}}\frac{{\left(2{X}_{s}^{}\right)}^{6}}{3}exp\left[-\frac{{k}^{2}{r}^{2}}{3}\right],$ | (51) | |

$=\frac{1}{4{E}_{g}^{2}}{\left(\frac{{k}^{2}}{\pi}\right)}^{2}{\displaystyle \sum _{s=\pm}}\phantom{\rule{3.33333pt}{0ex}}\frac{{\left(2{X}_{s}^{}\right)}^{8}}{4}exp\left[-\frac{{k}^{2}{r}^{2}}{4}\right],$ | (52) |

## 9. Equation of Motion for the Mean Squared Displacement

## 10. Discussions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Abrahams, E.; Anderson, P.W.; Licciardello, D.C.; Ramakrishnan, T.V. Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions. Phys. Rev. Lett.
**1979**, 42, 673–676. [Google Scholar] [CrossRef] - Gor’kov, L.G.; Larkin, A.I.; Khmel’nitskii, D.E. Particle conductivity in a two-dimensional random potential. JETP Lett.
**1979**, 30, 228–232. [Google Scholar] - Hikami, S.; Larkin, A.; Nagaoka, Y. Spin-Orbit Interaction and Magnetoresistance in the Two Dimensional Random System. Prog. Theor. Phys.
**1980**, 63, 707–710. [Google Scholar] [CrossRef] - Vollhardt, D.; Wölfle, P. Diagrammatic, self-consistent treatment of the Anderson localization problem in d⩽2 dimensions. Phys. Rev. B
**1980**, 22, 4666–4679. [Google Scholar] [CrossRef] - Hanein, Y.; Meirav, U.; Shahar, D.; Li, C.C.; Tsui, D.C.; Shtrikman, H. The metallic like conductivity of a two-dimensional hole system. Phys. Rev. Lett.
**1998**, 80, 1288–1291. [Google Scholar] [CrossRef] [Green Version] - Novoselov, K.S.; Geim, A.K.; Morozov, S.V.; Jiang, D.; Katsnelson, M.I.; Grigorieva, I.V.; Dubonos, S.V.; Firsov, A.A. Two-dimensional gas of massless Dirac fermions in graphene. Nature
**2005**, 438, 197–200. [Google Scholar] [CrossRef] - Tan, Y.-W.; Zhang, Y.; Bolotin, K.; Zhao, Y.; Adam, S.; Hwang, E.H.; Das Sarma, S.; Stormer, H.L.; Kim, P. Measurement of scattering rate and minimal conductivity in graphene. Phys. Rev. Lett.
**2007**, 99, 246803. [Google Scholar] [CrossRef] [Green Version] - Elias, D.C.; Nair, R.R.; Mohiuddin, T.M.G.; Morozov, S.V.; Blake, P.; Halsall, M.P.; Ferrari, A.C.; Boukhvalov, D.W.; Katsnelson, M.I.; Geim, A.K.; et al. Control of graphene’s properties by reversible hydrogenation: Evidence for graphane. Science
**2009**, 323, 610–613. [Google Scholar] [CrossRef] [Green Version] - Allen, M.J.; Tung, V.C.; Kaner, R.B. Honeycomb carbon: A review of graphene. Chem. Rev.
**2010**, 110, 132–145. [Google Scholar] [CrossRef] - Chen, L.; Liu, C.-C.; Feng, B.; He, X.; Cheng, P.; Ding, Z.; Meng, S.; Yao, Y.; Wu, K. Evidence for Dirac fermions in a honeycomb lattice based on silicon. Phys. Rev. Lett.
**2012**, 109, 056804. [Google Scholar] [CrossRef] [Green Version] - Castro Neto, A.H.; Guinea, F.; Peres, N.M.R.; Novoselov, K.S.; Geim, A.K. The electronic properties of graphene. Rev. Mod. Phys.
**2009**, 81, 109–162. [Google Scholar] [CrossRef] [Green Version] - Kotov, V.N.; Uchoa, B.; Pereira, V.M.; Guinea, F.; Castro Neto, A.H. Electron-Electron Interactions in Graphene: Current Status and Perspectives. Rev. Mod. Phys.
**2012**, 84, 1067–1125. [Google Scholar] [CrossRef] - Hasan, M.Z.; Kane, C.L. Colloquium: Topological insulators. Rev. Mod. Phys.
**2010**, 82, 3045–3067. [Google Scholar] [CrossRef] [Green Version] - Qi, X.-L.; Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys.
**2011**, 83, 1057–1110. [Google Scholar] [CrossRef] [Green Version] - Avsar, A.; Ochoa, H.; Guinea, F.; Özyilmaz, B.; Van Wees, B.J.; Vera-Marun, I.J. Colloquium: Spintronics in graphene and other two-dimensional materials. Rev. Mod. Phys.
**2020**, 92, 021003. [Google Scholar] [CrossRef] - Bernevig, B.A.; Hughes, T.L.; Zhang, S.-C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science
**2006**, 314, 1757–1761. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Shon, N.H.; Ando, T. Quantum transport in two-dimensional graphite system. J. Phys. Soc. Jpn.
**1998**, 67, 2421–2429. [Google Scholar] [CrossRef] [Green Version] - Ando, T.; Zheng, Y.; Suzuura, H. Dynamical conductivity and zero-mode anomaly in honeycomb lattices. J. Phys. Soc. Jpn.
**2002**, 71, 1318–1324. [Google Scholar] [CrossRef] [Green Version] - Suzuura, H.; Ando, T. Crossover from symplectic to orthogonal class in a two-dimensional honeycomb lattice. Phys. Rev. Lett.
**2002**, 89, 266603. [Google Scholar] [CrossRef] [Green Version] - McCann, E.; Kechedzhi, K.; Fal’ko, V.I.; Suzuura, H.; Ando, T.; Altshuler, B.L. Weak-localization magnetoresistance and valley symmetry in graphene. Phys. Rev. Lett.
**2006**, 97, 146805. [Google Scholar] [CrossRef] [Green Version] - Altshuler, B.L.; Aronov, A.G.; Larkin, A.I.; Khmel’nitskii, D.E. Anomalous magnetoresistance in semiconductors. Sov. Phys. JETP
**1981**, 54, 411–419. [Google Scholar] - Altshuler, B.L.; Simons, B.D. Universalities: From Anderson localization to quantum chaos. In Mesoscopic Quantum Physics, Les Houches 1994; Akkermans, E., Montambaux, G., Pichard, J.-L., Zinn-Justin, J., Eds.; North Holland: Amsterdam, The Netherlands, 1995; pp. 1–98. [Google Scholar]
- Efetov, K. Supersymmetry in Disorder and Chaos; Cambridge University Press: Cambridge, UK, 1997. [Google Scholar]
- Lee, P.A. Localized states in a d-wave superconductor. Phys. Rev. Lett.
**1993**, 71, 1887–1890. [Google Scholar] [CrossRef] [PubMed] - Wegner, F.J. The mobility edge problem: Continuous symmetry and a conjecture. Z. Physik B
**1979**, 35, 207–210. [Google Scholar] [CrossRef] - Schäfer, L.; Wegner, F.J. Disordered system withn orbitals per site: Lagrange formulation, hyperbolic symmetry, and Goldstone modes. Z. Physik B
**1980**, 38, 113–126. [Google Scholar] [CrossRef] - Hikami, S. Anderson localization in a nonlinear-σ-model representation. Phys. Rev. B
**1981**, 24, 2671–2679. [Google Scholar] [CrossRef] - Wegner, F.J. Disordered system with n orbitals per site: n=∞ limit. Phys. Rev. B
**1979**, 19, 783–792. [Google Scholar] [CrossRef] - McKane, A.J.; Stone, M. Localization as an alternative to Goldstone’s theorem. Ann. Phys.
**1981**, 131, 36–55. [Google Scholar] [CrossRef] - Fradkin, E. Critical behavior of disordered degenerate semiconductors. II. Spectrum and transport properties in mean-field theory. Phys. Rev. B
**1986**, 33, 3263–3268. [Google Scholar] [CrossRef] - Ando, T. Theory of quantum transport in a two-dimensional electron system under magnetic field. III. Many-site approximation. J. Phys. Soc. Jpn.
**1974**, 37, 622–630. [Google Scholar] [CrossRef] - Wegner, F.J. Exact density of states for lowest Landau level in white noise potential. Superfield representation for interacting systems. Z. Phys. B Condens. Matter
**1983**, 51, 279–285. [Google Scholar] [CrossRef] - Brézin, E.; Gross, D.J.; Itzykson, C. Density of states in the presence of a strong magnetic field and random impurities. Nucl. Phys. B
**1984**, 235, 24–44. [Google Scholar] [CrossRef] - Hikami, S. Borel-Padé analysis for the two-dimensional electron in a random potential under a strong magnetic field. Phys. Rev. B
**1984**, 29, 3726–3729. [Google Scholar] [CrossRef] - Hikami, S. Anderson Localization of the two-dimensional electron in a random potential under a strong magnetic field. Prog. Theor. Phys.
**1984**, 72, 722–735. [Google Scholar] [CrossRef] [Green Version] - Aoki, H. Quantised Hall effect. Rep. Prog. Phys.
**1987**, 50, 655–730. [Google Scholar] [CrossRef] - Tkachov, G. Topological Insulators: The Physics of Spin Helicity in Quantum Transport; Pan Stanford: Boca Raton, FL, USA, 2015. [Google Scholar]
- Sinner, A.; Ziegler, K. Two-parameter scaling theory of transport near a spectral node. Phys. Rev. B
**2014**, 90, 174207. [Google Scholar] [CrossRef] [Green Version] - Sinner, A.; Ziegler, K. Finite-size scaling in a 2D disordered electron gas with spectral nodes. J. Phys. Condens. Matter
**2016**, 28, 305701. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Goldenfeld, N. Lectures on Phase Transitions and the Renormalization Group; Perseus Books: Reading, MA, USA, 1992. [Google Scholar]
- Huang, K. Statistical Mechanics, 2nd ed.; John Wiley: New York, NY, USA, 1987. [Google Scholar]
- Chaikin, P.M.; Lubenski, T.C. Principles of Condensed Matter Physics; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Ziegler, K. Quantum diffusion in two-dimensional random systems with particle–hole symmetry. J. Phys. A Math. Theor.
**2012**, 45, 335001. [Google Scholar] [CrossRef] - Singh, R.R.P.; Chakravarty, S. A disordered two-dimensional system in a magnetic field: Borel-Padé analysis. Nucl. Phys. B
**1986**, 265, 265–292. [Google Scholar] [CrossRef] - Hikami, S.; Shirai, M.; Wegner, F.J. Anderson localization in the lowest Landau level for a two-subband model. Nucl. Phys. B
**1993**, 408, 415–426. [Google Scholar] [CrossRef] - Culcer, D.; Keser, A.C.; Li, Y.; Tkachov, G. Transport in two-dimensional topological materials: Recent developments in experiment and theory. 2D Mater.
**2020**, 7, 022007. [Google Scholar] [CrossRef] - König, M.; Buhmann, H.; Molenkamp, L.W.; Hughes, T.; Liu, C.-X.; Qi, X.-L.; Zhang, S.-C. The quantum spin Hall effect: Theory and experiment. J. Phys. Soc. Jpn.
**2008**, 77, 031007. [Google Scholar] [CrossRef] [Green Version] - Li, G.; Andrei, E.Y. Observation of Landau levels of Dirac fermions in graphite. Nat. Phys.
**2007**, 3, 623–627. [Google Scholar] [CrossRef] [Green Version] - Goswami, P.; Jia, X.; Chakravarty, S. Quantum Hall plateau transition in the lowest Landau level of disordered graphene. Phys. Rev. B
**2007**, 76, 205408. [Google Scholar] [CrossRef] [Green Version] - Ludwig, A.W.W.; Fisher, M.P.A.; Shankar, R.; Grinstein, G. Integer quantum Hall transition: An alternative approach and exact results. Phys. Rev. B
**1994**, 50, 7526–7552. [Google Scholar] [CrossRef] - Ziegler, K. Robust transport properties in graphene. Phys. Rev. Lett.
**2006**, 97, 266802. [Google Scholar] [CrossRef] [Green Version] - Ziegler, K. Minimal conductivity of graphene: Nonuniversal values from the Kubo formula. Phys. Rev. B
**2007**, 75, 233407. [Google Scholar] [CrossRef] [Green Version] - Sinner, A.; Ziegler, K. Conductivity of disordered 2d binodal Dirac electron gas: Effect of internode scattering. Philos. Mag.
**2018**, 98, 1799. [Google Scholar] [CrossRef] [Green Version] - Sinner, A.; Tkachov, G. Diffusive transport in the lowest Landau level of disordered 2d semimetals: The mean-square-displacement approach. Eur. Phys. J. B
**2022**. submitted. [Google Scholar] - Novoselov, K.S.; Jiang, Z.; Zhang, Y.; Morozov, S.V.; Stormer, H.L.; Zeitler, U.; Maan, J.C.; Boebinger, G.S.; Kim, P.; Geim, A.K. Room-temperature quantum Hall effect in graphene. Science
**2007**, 315, 1379. [Google Scholar] [CrossRef] [Green Version] - Jiang, Z.; Zhang, Y.; Tan, Y.-W.; Stormer, H.L.; Kim, P. Quantum Hall effect in graphene. Solid State Comm.
**2007**, 143, 14–19. [Google Scholar] [CrossRef] - Shemer, Z.; Barkai, E. Einstein relation and effective temperature for systems with quenched disorder. Phys. Rev. E
**2009**, 80, 031108. [Google Scholar] [CrossRef] [PubMed] - Jeckelmann, B.; Jeanneret, B. The quantum Hall effect as an electrical resistance standard. Rep. Prog. Phys.
**2001**, 64, 1603–1655. [Google Scholar] [CrossRef]

**Figure 1.**Spectrum of the tight-binding model along the line ${k}_{1}^{}=0$ with two Dirac cones at the corners of the Brilloun zone. The energy axis is scaled in units of the hopping parameter between nearest-neighbors t.

**Figure 2.**The circulation of the Berry vector potential corresponding to the occupied band of the full half filled tight-binding model in the reciprocal space with visible vortex-like structures around the position of the nodal points.

**Figure 3.**Perturbative processes contributing to the dressing of the single-particle propagator due to the disorder to order ${g}^{1}$ (one diagram), ${g}^{2}$ (three diagrams), and ${g}^{3}$ (fifteen diagrams). Some of the diagrams of order ${g}^{3}$ should be counted twice because of the degeneracy due to the mirror symmetry with respect to the imaginable vertical axis, which is accounted for by the factors 2 in front of them.

**Figure 4.**Evolution of the DOS of both Landau sublevels defined in Equation (11) (

**a**–

**d**) plotted in units of the DOS at each suband center $\frac{1}{{\pi}^{5/2}}\frac{{k}^{2}}{{E}_{g}^{}}$ with increasing disorder strength as a function of the dimensionless energy $\nu $. The following quantities are used: ${\u03f5}_{0}^{}/t=0.15$, ${\Delta}_{0}^{}/t=0.1$ and ${E}_{g}^{}/t=0.01,0.045,0.073,$ and $0.1$ in units of the hopping amplitude. Dashed lines emphasize the position of each eigenvalue.

**Figure 5.**Perturbative processes contributing to the dressing of the two-particles propagator up to the third order in disorder strength. Solid lines denote the fully dressed Wegner’s propagators and the dashed lines denote the disorder correlators.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sinner, A.; Tkachov, G.
Quantum Diffusion in the Lowest Landau Level of Disordered Graphene. *Nanomaterials* **2022**, *12*, 1675.
https://doi.org/10.3390/nano12101675

**AMA Style**

Sinner A, Tkachov G.
Quantum Diffusion in the Lowest Landau Level of Disordered Graphene. *Nanomaterials*. 2022; 12(10):1675.
https://doi.org/10.3390/nano12101675

**Chicago/Turabian Style**

Sinner, Andreas, and Gregor Tkachov.
2022. "Quantum Diffusion in the Lowest Landau Level of Disordered Graphene" *Nanomaterials* 12, no. 10: 1675.
https://doi.org/10.3390/nano12101675