Evolution of individual quantum Hall edge states in the presence of disorder

Abstract

By using the Bloch eigenmode matching approach, we numerically study the evolution of individual quantum Hall edge states with respect to disorder. As demonstrated by the two-parameter renormalization group flow of the Hall and Thouless conductances, quantum Hall edge states with high Chern number n are completely different from that of the n = 1 case. Two categories of individual edge modes are evaluated in a quantum Hall system with high Chern number. Edge states from the lowest Landau level have similar eigenfunctions that are well localized at the system edge and independent of the Fermi energy. On the other hand, at fixed Fermi energy, the edge state from higher Landau levels exhibit larger expansion, which results in less stable quantum Hall states at high Fermi energies. By presenting the local current density distribution, the effect of disorder on eigenmode-resolved edge states is distinctly demonstrated.

Keywords

quantum Hall edge states / Landau level / quantum phase transition

References

[1] K. V. Klitzing, G. Dorda, and M. Pepper, New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance, Phys. Rev. Lett. 45(6), 494 (1980)

[2] A. Bao, Y. H. Chen, H. F. Lin, H. D. Liu, and X. Z. Zhang, Quantum phase transitions in two-dimensional strongly correlated fermion systems, Front. Phys. 10(5), 106401 (2015)

[3] F. Evers and A. D. Mirlin, Anderson transitions, Rev. Mod. Phys. 80(4), 1355 (2008)

[4] A. M. M. Pruisken, Universal singularities in the integral quantum Hall effect, Phys. Rev. Lett. 61(11), 1297 (1988)

[5] D. E. Khmel’nitskii, Quantization of Hall conductivity, Pis’ma Z. Eksp. Teor. Fiz. 38, 454 (1983) [JETP Lett. 38(9), 552 (1983)]

[6] J. Song, and E. Prodan, Characterization of the quantized Hall insulator phase in the quantum critical regime, Eur. Phys. Lett. 105(3), 37001 (2014)

[7] M. A. Werner, A. Brataas, F. von Oppen, and G. Zaránd, Anderson localization and quantum Hall effect: Numerical observation of two-parameter scaling, Phys. Rev. B 91(12), 125418 (2015)

[8] S. Kivelson, D. H. Lee, and S. C. Zhang, Global phase diagram in the quantum Hall effect, Phys. Rev. B 46(4), 2223 (1992)

[9] S. H. Song, D. Shahar, D. C. Tsui, Y. H. Xie, and D. Monroe, New universality at the magnetic field driven insulator to integer quantum Hall effect transitions, Phys. Rev. Lett. 78(11), 2200 (1997)

[10] T. Wang, K. P. Clark, G. F. Spencer, A. M. Mack, and W. P. Kirk, Magnetic-field-induced metal-insulator transition in two dimensions, Phys. Rev. Lett. 72(5), 709 (1994)

[11] S. V. Kravchenko, W. Mason, J. E. Furneaux, and V. M. Pudalov, Global phase diagram for the quantum Hall effect: An experimental picture,Phys. Rev. Lett. 75(5), 910 (1995)

[12] D. N. Sheng and Z. Y. Weng, New universality of the metal-insulator transition in an integer quantum Hall effect system, Phys. Rev. Lett. 80(3), 580 (1998)

[13] D. N. Sheng and Z. Y. Weng, Phase diagram of the integer quantum Hall effect, Phys. Rev. B 62(23), 15363 (2000)

[14] X. C. Xie, D. Z. Liu, B. Sundaram, and Q. Niu, Transition from the integer quantum Hall state to the insulator state, Phys. Rev. B 54(7), 4966 (1996)

[15] B. I. Halperin, Quantized Hall conductance, currentcarrying edge states, and the existence of extended states in a two-dimensional disordered potential, Phys. Rev. B 25(4), 2185 (1982)

[16] M. Büttiker, Absence of backscattering in the quantum Hall effect in multiprobe conductors, Phys. Rev. B 38, 9375 (1988)

[17] Y. Ren, Z. Qiao, and Q. Niu, Topological phases in two-dimensional materials: A review, Rep. Prog. Phys. 79(6), 066501 (2016)

[18] Y. Ren, J. Zeng, K. Wang, F. Xu, and Z. Qiao, Tunable current partition at zero-line intersection of quantum anomalous Hall topologies, Phys. Rev. B 96(15), 155445 (2017)

[19] Q. Niu, D. J. Thouless, and Y. S. Wu, Quantized Hall conductance as a topological invariant, Phys. Rev. B 31(6), 3372 (1985)

[20] Y. Hatsugai, K. Ishibashi, and Y. Morita, Sum rule of Hall conductance in a random quantum phase transition, Phys. Rev. Lett. 83(11), 2246 (1999)

[21] A. M. Essin and J. E. Moore, Topological insulators beyond the Brillouin zone via Chern parity, Phys. Rev. B 76(16), 165307 (2007)

[22] Y. Y. Zhang, R. L. Chu, F. C. Zhang, and S. Q. Shen, Localization and mobility gap in the topological Anderson insulator, Phys. Rev. B 85(3), 035107 (2012)

[23] Z. G. Song, Y. Y. Zhang, J. T. Song, and S. S. Li, Route towards localization for quantum anomalous Hall systems with Chern number 2, Sci. Rep. 6(1), 19018 (2016)

[24] K. Nomura, S. Ryu, M. Koshino, C. Mudry, and A. Furusaki, Quantum Hall effect of massless Dirac fermions in a vanishing magnetic field, Phys. Rev. Lett. 100(24), 246806 (2008)

[25] J. T. Edwards, and D. J. Thouless, Numerical studies of localization in disordered systems, J. Phys. C 5(8), 807 (1972)

[26] J. S. Wang, B. K. Agarwalla, H. Li, and J. Thingna, Nonequilibrium Green’s function method for quantum thermal transport, Front. Phys. 9(6), 673 (2014)

[27] P. A. Khomyakov, G. Brocks, V. Karpan, M. Zwierzycki, and P. J. Kelly, Conductance calculations for quantum wires and interfaces: Mode matching and Green’s functions, Phys. Rev. B 72(3), 035450 (2005)

[28] T. Ando, Quantum point contacts in magnetic fields, Phys. Rev. B 44(15), 8017 (1991)

[29] L. Zhang, Y. Xing, and J. Wang, First-principles investigation of transient dynamics of molecular devices, Phys. Rev. B 86(15), 155438 (2012)

[30] Y. Xing, Q. F. Sun, and J. Wang, Influence of dephasing on the quantum Hall effect and the spin Hall effect, Phys. Rev. B 77(11), 115346 (2008)

[31] T. Fukui and Y. Hatsugai, Quantum spin Hall effect in three dimensional materials: Lattice computation of Z₂ topological invariants and its application to Bi and Sb, J. Phys. Soc. Jpn. 76(5), 053702 (2007)

[32] T. Fukui, Y. Hatsugai, and H. Suzuki, Chern numbers in discretized Brillouin zone: Efficient method of computing (spin) Hall conductances, J. Phys. Soc. Jpn. 74(6), 1674 (2005)

[33] T. Fukui and Y. Hatsugai, Topological aspects of the quantum spin-Hall effect in graphene: Z₂ topological order and spin Chern number, Phys. Rev. B 75(12), 121403 (2007)

[34] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized Hall conductance in a twodimensional periodic potential, Phys. Rev. Lett. 49(6), 405 (1982)

[35] M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. Lond. A 392(1802), 45 (1984)

[36] B. Simon, Holonomy, the quantum adiabatic theorem, and Berry’s phase, Phys. Rev. Lett. 51(24), 2167 (1983)

[37] D. Braun, E. Hofstetter, A. MacKinnon, and G. Montambaux, Level curvatures and conductances: A numerical study of the Thouless relation, Phys. Rev. B 55(12), 7557 (1997)

[38] A. P. Jauho, N. S. Wingreen, and Y. Meir, Timedependent transport in interacting and noninteracting resonant-tunneling systems, Phys. Rev. B 50(8), 5528 (1994)

[39] Y. Xing, J. Wang, and Q. F. Sun, Focusing of electron flow in a bipolar graphene ribbon with different chiralities, Phys. Rev. B 81(16), 165425 (2010)

[40] J. Li and S. Q. Shen, Spin-current-induced charge accumulation and electric current in semiconductor nanostructures with Rashba spin-orbit coupling, Phys. Rev. B 76(15), 153302 (2007)

[41] H. Jiang, L. Wang, Q. F. Sun, and X. C. Xie, Numerical study of the topological Anderson insulator in HgTe/CdTe quantum wells, Phys. Rev. B 80(16), 165316 (2009)

[42] Y. Xing, L. Zhang, and J. Wang, Topological Anderson insulator phenomena, Phys. Rev. B 84(3), 035110 (2011)

[43] D. H. Lee and J. D. Joannopoulos, Simple scheme for surface-band calculations (II): The Green’s function, Phys. Rev. B 23(10), 4997 (1981)

[44] D. H. Lee and J. D. Joannopoulos, Simple scheme for surface-band calculations (I), Phys. Rev. B 23(10), 4988 (1981)

[45] S. Sanvito, C. J. Lambert, J. H. Jefferson, and A. M. Bratkovsky, General Green’s-function formalism for transport calculations with spd Hamiltonians and giant magnetoresistance in Co- and Ni-based magnetic multilayers, Phys. Rev. B 59(18), 11936 (1999)

[46] I. Rungger and S. Sanvito, Algorithm for the construction of self-energies for electronic transport calculations based on singularity elimination and singular value decomposition, Phys. Rev. B 78(3), 035407 (2008)

[47] T. Ando, Electron localization in a two-dimensional system in strong magnetic fields (I): Case of short-range scatterers, J. Phys. Soc. Jpn. 52(5), 1740 (1983)

[48] T. Ando, Electron localization in a two-dimensional system in strong magnetic fields (II): Long-range scatterers and response functions, J. Phys. Soc. Jpn. 53(9), 3101 (1984)

RIGHTS & PERMISSIONS

2018 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature