Outline of a theory of the two-dimensional hall effect in the quantum limit

Summary

The ground state of two-dimensional electrons of density N/L² in a strong transverse magnetic field B is discussed in terms of localized magnetic functions. For all « commensurate » fractional fillings of the n = 0 Landau level, occurring at B_st = (s² + t² + st) 2πhcN/eL², with s, t integers, it is found that the ground state is a triangular lattice. This lattice has unusual properties, because it is tied to the magnetic functions. In particular, it has a finite Hall conductivity σ_xy = e²/2πh(s² + t² + st) and it also exhibits perfect diagmagnetism relative to B^st. It does, however, display no proper Meissner effect, because the London depth is macroscopically large. The excess field B — B_st gives rise instead to defects in the lattice, where the extra electrons (holes) become « interstitials » (« vacancies »). If the defects are free to move, the Hall conductivity will not stay quantized. On the other hand, if all defects are pinned by inhomogeneities, Hall plateaux are expected around each B_st. This picture, while providing a natural explanation for the quantized Hall effect at both integer and fractional filling, leads to a simple understanding of the plateau width vs. temperature and simple quality, and can also explain, at finite temperatures, the behaviour of the longitudinal conductivity σ_yy and its observed asymmetry for integer filling.