Spatial decay of the one-body density matrix in insulators revised

TL;DR

This paper revises the understanding of the spatial decay of the one-body density matrix in insulators, showing that previous models have unphysical properties and deriving new asymptotic formulas valid across different dimensions and gap widths.

Contribution

It demonstrates that certain existing insulator models exhibit unphysical behavior and derives improved asymptotic decay formulas for the one-body density matrix in various dimensions.

Findings

Previous models show no dependence of the density matrix on gap width.

New asymptotic formulas are valid for all gap widths in 1D and diagonal directions.

Exponential decay rate vanishes linearly with the gap, with numerical evidence for square root scaling in non-diagonal directions.

Abstract

In the framework of the band theory, we consider two tight-binding models of insulators. The first one, proposed recently by Taraskin et al, is a translationally invariant system, built out of two independent non-overlapping bands of single-particle orbitals that are coupled by a weak inter-band hybridization. This kind of insulator exhibits unphysical properties: we show, in particular, that the one-body density matrix does not depend on the width of the gap between the bands. Consequently, there is no delocalization effect with increasing metallicity. In the second model there are also two bands. However, they are not imposed by construction but are created from a band of single-particle orbitals due to the breaking of the translational symmetry by a periodic potential. These bands are separated by a gap for all nonzero values of the unique energy parameter of the model. We demonstrate that the one-body density matrix has the same structure as in the first model. As a result, the large distance asymptotic formulae derived by Taraskin et al in dimensions $D=1,2,3$, apply as well, but only for very large gap widths. In D=1 and in the diagonal direction of D=2 cases, we derive a stronger asymptotic formula, valid for all gap widths. The both kinds of asymptotic formulae are composed of a dimension-dependent power-law factor and a gap-dependent exponentially decaying factor. The latter asymptotics implies that the exponential decay rate vanishes linearly with the vanishing gap. In non-diagonal directions, we have found numerically that the linear scaling is replaced by the square root one. Independently of the direction, the exponential decay rate grows logarithmically with the gap width, for sufficiently large gaps.