Edge states in the integer quantum Hall effect and the Riemann surface of the Bloch function.

摘要: We study edge states in the integral quantum Hall effect on a square lattice rational magnetic field \ensuremath{\varphi}=p/q. The system is periodic y direction but has two edges x direction. have found that energies of are given by zero points Bloch function some Riemann surface (RS) (complex energy surface) when size commensurate with flux. genus RS, g=q-1, number gaps. move around holes RS as momentum conductance ${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}\mathit{y}}$ winding holes, which gives Thouless, Kohmoto, Nightingale, and den Nijs integers infinite system. This topological RS. can check this treatment same Diophantine equation numerically. Effects random potential also discussed.