Insulator, conductor and commensurability: a topological approach

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I discuss a topological relation of the conduction property of a many-particle system on a periodic lattice at zero temperature to the energy spectrum. When the particle number per unit cell is an irreducible fraction p/q, an insulator must have q low-lying states of energy o(1/L) in one dimension and of energy o(1) in two dimensions, where L is the linear system size.