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arXiv:cond-mat/0404142 · Physical Review B

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

Janusz Jędrzejewski, Taras Krokhmalskii

Published Oct 12, 2004• 8 citations
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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 formulas imply that in the diagonal direction of the square lattice the inverse correlation length vanishes linearly with the vanishing gap, while in nondiagonal 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.

DOI
10.1103/physrevb.70.153102

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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=1D=1D=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 formulas imply that in the diagonal direction of the square lattice the inverse correlation length vanishes linearly with the vanishing gap, while in nondiagonal 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.
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=1,2,3D=1,2,3D=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.
We consider tight-binding electrons on a square lattice, with an anisotropic hopping intensity, and in a periodic external potential. Breaking of the translational symmetry of the system results in two bands, separated by a gap of width proportional to the unique energy parameter of the model. The off-diagonal matrix elements of the one-body reduced density matrix are exactly expressed by product of Euler's Γ-functions and Appell function. Their large-distance (σ) decay rate is derived to be of the form σ −1 exp (−σ/ξ). A control of the crossover from 2D to 1D system is achieved. In comparison with the isotropic case, the correlation length ξ varies with the gap in a strikingly different manner. In particular, it remains nonzero as the gap vanishes, in all the directions except the direction of stronger hopping intensity.
We study decay rates of one-body reduced density matrices in insulators, described by a tight-binding model, where not only an external potential but also hoppings are spatially modulated. We determine analytically the power in the power law and the correlation length in D=1D=1D=1 case and in several lattice directions in D=2D=2D=2 case. Unlike the uniform hopping case, in the D=1D=1D=1 system and in some directions of the D=2D=2D=2 system the correlation length is not determined uniquely by the gap. Moreover, a crossover from D=2D=2D=2 decay rates to D=1D=1D=1 ones is investigated.
Analytical results for the asymptotic spatial decay of the density matrix $\ensuremath{\rho}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})$ in the tight-binding model of the two-dimensional metal are presented. In various dimensions D, it is found analytically and numerically that the density matrix decays with distance according to the power law $\ensuremath{\rho}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})\ensuremath{\propto}|\mathbf{r}\ensuremath{-}{\mathbf{r}}^{\ensuremath{'}}{|}^{\ensuremath{-}(D+1)/2}.$
We present an analytical study of the spatial decay rate $\ensuremath{\gamma}$ of the one-particle density matrix $\ensuremath{\rho}(\stackrel{\ensuremath{\rightarrow}}{r},{\stackrel{\ensuremath{\rightarrow}}{r}}^{\ensuremath{'}})\ensuremath{\sim}\mathrm{exp}(\ensuremath{-}\ensuremath{\gamma}|\stackrel{\ensuremath{\rightarrow}}{r}\ensuremath{-}{\stackrel{\ensuremath{\rightarrow}}{r}}^{\ensuremath{'}}|)$ for systems described by single-particle orbitals in periodic potentials in arbitrary dimensions. This decay reflects electronic locality in condensed matter systems and is also crucial for O(N)O(N)O(N) density functional methods. We find that $\ensuremath{\gamma}$ behaves contrary to the conventional wisdom that generically $\ensuremath{\gamma}\ensuremath{\propto}\ensuremath{\surd}\ensuremath{\Delta}$ in insulators and $\ensuremath{\gamma}\ensuremath{\propto}\ensuremath{\surd}T$ in metals, where $\ensuremath{\Delta}$ is the direct band gap and TTT is the temperature. Rather, in semiconductors $\ensuremath{\gamma}\ensuremath{\propto}\ensuremath{\Delta}$, and in metals at low temperature $\ensuremath{\gamma}\ensuremath{\propto}T$.