Abstract
A two-dimensional quantum Hall system is studied for a wide class of potentials including single-body random potentials and repulsive electron–electron interactions. We assume that there exists a nonzero excitation gap above the ground state(s), and then the conductance is derived from the linear perturbation theory with a sufficiently weak electric field. Under these two assumptions, we prove that the Hall conductance σ xy and the diagonal conductance σ yy satisfy |σ xy+e 2 ν/h|≤const·L −1/2 and |σ yy|≤const·L −1/12. Here e 2/h is the universal conductance with the charge −e of the electron and the Planck constant h; ν is the filling factor of the Landau level; and L is the linear dimension of the system. In the thermodynamic limit, our results show σ xy=−e 2 ν/h and σ yy=0. The former implies that integral and fractional filling factors ν with a gap lead to, respectively, integral and fractional quantizations of the Hall conductance.
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REFERENCES
R. E. Prange and S. M. Girvin (eds.), The Quantum Hall Effect, 2nd ed. (Springer, 1990).
T. Chakraborty and P. Pietiläinen, The Quantum Hall Effects: Fractional and Integral, 2nd 6 updated ed. (Springer, 1995).
S. Das Sarma and A. Pinczuk (eds.), New Perspectives in Quantum Hall Effects (Wiley, 1996).
S. Kawaji, T. Igarashi, and J. Wakabayashi, Prog. Theor. Phys. Suppl. 57:176 (1975).
T. Igarashi, J. Wakabayashi, and S. Kawaji, J. Phys. Soc. Jpn. 38:1549 (1975).
S. Kawaji, Surf. Sci. 73:46 (1978); J. Wakabayashi and S. Kawaji, J. Phys. Soc. Jpn. 44:1839 (1978); Surf. Sci. 98:299 (1980).
T. Ando, Y. Matsumoto, and Y. Uemura, J. Phys. Soc. Jpn. 39:279 (1975).
K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45:494 (1980).
S. Kawaji and J. Wakabayashi, in Physics in High Magnetic Fields, S. Chikazumi and N. Miura, eds. (Springer, 1981), p. 284.
D. C. Tsui, H. L. Störmer, and A. C. Gossard, Phys. Rev. Lett. 48:1559 (1982).
H. L. Störmer, A. M. Chang, D. C. Tsui, J. C. M. Hwang, A. C. Gossard, and W. Wiegmann, Phys. Rev. Lett. 50:1953 (1983).
H. Aoki and T. Ando, Solid State Commun. 38:1079 (1981); R. E. Prange, Phys. Rev. B 23:4802 (1981); R. B. Laughlin, Phys. Rev. B 23:5632 (1981); D. J. Thouless, J. Phys. C 14:3475 (1981); B. I. Halperin, Phys. Rev. B 25:2185 (1982); R. E. Prange and R. Joynt, Phys. Rev. B 25:2943 (1982).
B. A. Dubrovin and S. P. Novikov, Soviet Math. Dokl. 22:240 (1980); Soviet Phys. JETP 52:511 (1980); S. P. Novikov, Soviet Math. Dokl. 23:298 (1981).
D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49:405 (1982).
M. Kohmoto, Ann. Phys. 160:343 (1985).
J. E. Avron, R. Seiler, and B. Simon, Phys. Rev. Lett. 51:51 (1983); B. Simon, Phys. Rev. Lett. 51:2167 (1983); H. Kunz, Commun. Math. Phys. 112:121 (1987); K. Shizuya, Phys. Rev. 45:11143 (1992); Y. Hatsugai, Phys. Rev. B 48:11851 (1993); J. E. Avron, R. Seiler, and B. Simon, Commun. Math. Phys. 159:399 (1994); J. Bellissard, A. Van Elst, and H. Schulz-Baldes, J. Math. Phys. 35:5373 (1994); F. Nakano, J. Math. Sci. Univ. Tokyo 4:351 (1997); Calculation of the Hall conductivity by Abel limit, Preprint.
Q. Niu, D. J. Thouless, and Y.-S. Wu, Phys. Rev. B 31:3372 (1985).
J. E. Avron and R. Seiler, Phys. Rev. Lett. 54:259 (1985).
J. E. Avron and L. G. Yaffe, Phys. Rev. Lett. 56:2084 (1986).
K. Ishikawa, N. Maeda, K. Tadaki, and S. Uchiyama, Phys. Lett. A 210:312 (1996).
R. B. Laughlin, Phys. Rev. Lett. 50:1359 (1983); R. Tao and D. J. Thouless, Phys. Rev. B 28:1142 (1983); B. I. Halperin, Helv. Phys. Acta. 56:75 (1983); D. Yoshioka, B. I. Halperin, and P. A. Lee, Phys. Rev. Lett. 50:1219 (1983); Surf. Sci. 142:155 (1984); R. Tao, Phys. Rev. B 29:636 (1984); D. Yoshioka, Phys. Rev. B 29:6833 (1984); W. P. Su, Phys. Rev. B 30:1069 (1984); R. Tao and Y.-S. Wu, Phys. Rev. B 30:1097 (1984); K. Ishikawa and N. Maeda, Prog. Theor. Phys. 97:507 (1997). 458 Koma.
Q. Niu and D. J. Thouless, Phys. Rev. B 35:2188 (1987).
T. Koma, J. Stat. Phys. 99:313 (2000).
T. Koma, Localization and conductance plateaus in quantum Hall systems, in preparation.
J. Zak, Phys. Rev. 134:A1602, A1607 (1964).
T. Kato, Perturbation Theory for Linear Operators, 2nd ed. (Springer, 1980).
P. Horsch and W. von der Linden, Z. Phys. B 72:181 (1988).
T. Koma and H. Tasaki, J. Stat. Phys. 76:745 (1994).
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Koma, T. Insensitivity of Quantized Hall Conductance to Disorder and Interactions. Journal of Statistical Physics 99, 383–459 (2000). https://doi.org/10.1023/A:1018657009561
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DOI: https://doi.org/10.1023/A:1018657009561