1998/09/17 by Tohru Koma, Koma, Tohru
Physics and Astronomy · #FOS: Physical sciences #Mesoscale and Nanoscale Physics (cond-mat.mes-hall) #Statistical Mechanics (cond-mat.stat-mech) #cond-mat.mes-hall #cond-mat.stat-mech
paper · pdf · doi:10.48550/arxiv.cond-mat/9809228
LaTeX, 54 pages, no figures, discussions on periodic potentials and on energy and spatial cutoffs added, typos corrected, accepted for publication in J. Stat. Phys
arxiv created 1999/12/27 · arxiv updated 2009/11/30
A two-dimensional quantum Hall system without disorder for a wide class of interactions including any two-body interaction with finite range is studied by using the Lieb-Schultz-Mattis method [\it Ann. Phys. (N.Y.) \bf 16: 407 (1961)]. The model is defined on an infinitely long strip with a fixed large width, and the Hilbert space is restricted to the lowest (n\rm max+1) Landau levels with a large integer n\rm max. We proved that, for a non-integer filling ν of the Landau levels, either (i) there is a symmetry breaking at zero temperature or (ii) there is only one infinite-volume ground state with a gapless excitation. We also proved the following two theorems: (a) If a pure infinite-volume ground state has a non-zero excitation gap for a non-integer filling ν, then a translational symmetry breaking occurs at zero temperature. (b) Suppose that there is no non-translationally invariant infinite-volume ground state. Then, if a pure infinite-volume ground state has a non-zero excitation gap, the filling factor ν must be equal to a rational number. Here the ground state is allowed to have a periodic structure which is a consequence of the translational symmetry breaking. We also discuss the relation between our results and the quantized Hall conductance, and phenomenologically explain why odd denominators of filling fractions ν giving the quantized Hall conductance, are favored exclusively.