Commensurability, excitation gap, and topology in quantum many-particle systems on a periodic lattice

In combination with Laughlin's treatment of the quantized Hall conductivity , the Lieb-SchuItz-Mattis argument is extended to quantum many-particle sys tems (including quantum spin systems) with a conserved particle number on a periodic lattice in arbitrary dimensions. Regardless of dimensionality, in teraction strength, and particle statistics (Bose or Fermi), a finite excit ation gap is possible only when the particle number per unit cell of the gr ound state is an integer.