Quantized Hall conductivity of Bloch electrons: topology and the Dirac fermion

Details Quantized Hall conductivity of Bloch electrons: topology and the Dirac fermion Quantized Hall conductivity of Bloch electrons: topology and the Dirac fermion

1994-09-19

arxiv.org/abs/cond-mat/9409079v1

Phys. Rev. B 50, 17357 (1994)

Physics

Condensed Matter

13pages (REVTeX) + 3 figures (appended as a postscript file), To appear in Phys. Rev. B, ISSP Tech. Rep. Ser.A No.2850

Scientific paper

10.1103/PhysRevB.50.17357

We consider the Hall conductivity of two-dimensional non-interacting Bloch electrons when the magnetic flux per unit cell is a rational number $p/q$ where $p$ and $q$ are mutually coprime. We present a counter-example for the naive expectation that the Hall conductivity carried by a band is given by treating gap minima as Dirac fermions. Instead of the above expectation, we show that the {\em change\/} of the Hall conductivity at a gap-closing phenomenon is given by the Dirac fermion argument. Comparing with the Diophantine equation, our result implies that a band-gap closes at $q$ points simultaneously. Furthermore, we show that the dispersion relation is $q$-fold degenerate in the magnetic Brillouin zone.

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