Exact results for spatial decay of the one-body density matrix in low-dimensional insulators

TL;DR

This paper derives exact formulas for the spatial decay of the one-body density matrix in low-dimensional insulators, revealing how decay behavior depends on gap size, direction, and lattice dimension.

Contribution

It provides the first exact analytic expressions for the asymptotic decay of the density matrix in low-dimensional insulators, linking decay properties to system parameters.

Findings

Decay is a power law times exponential in all cases.

Inverse correlation length vanishes linearly with gap in diagonal directions.

Inverse correlation length scales with the square root of the gap in non-diagonal directions.

Abstract

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 $D=1,2$. 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.