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Exact results for spatial decay of the one-body density matrix in low-dimensional insulators
We provide a tight-binding model of insulator, for which we derive an exact
analytic form of the one-body density matrix and its large-distance asymptotics
in dimensions . The system is built out of a band of single-particle
orbitals in a periodic potential. Breaking of the translational symmetry of the
system results in two bands, separated by a direct gap whose width is
proportional to the unique energy parameter of the model. The form of the decay
is a power law times an exponential. We determine the power in the power law
and the correlation length in the exponential, versus the lattice direction,
the direct-gap width, and the lattice dimension. In particular, the obtained
exact formulae imply that in the diagonal direction of the square lattice the
inverse correlation length vanishes linearly with the vanishing gap, while in
non-diagonal directions, the linear scaling is replaced by the square root one.
Independently of direction, for sufficiently large gaps the inverse correlation
length grows logarithmically with the gap width.Comment: 4 pages, 2 figure
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