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arXiv:cond-mat/0207443 · Physical review. B, Condensed matter

Spatial decay of the single-particle density matrix in tight-binding metals: Analytic results in two dimensions

Sergei N. Taraskin, P. A. Fry, Xiaodong Zhang, David Alan Drabold, Stephen R. Elliott

Published Dec 4, 2002• 22 citations
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Abstract

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 $\ensuremath{\rho}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})\ensuremath{\propto}|\mathbf{r}\ensuremath{-}{\mathbf{r}}^{\ensuremath{'}}{|}^{\ensuremath{-}(D+1)/2}.$

DOI
10.1103/physrevb.66.233101

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Analytical results for the asymptotic spatial decay of the density matrix \ensuremathρ(r,r\ensuremath′)\ensuremath{\rho}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})\ensuremathρ(r,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 \ensuremathρ(r,r\ensuremath′)\ensuremath∝∣r\ensuremath−r\ensuremath′∣\ensuremath−(D+1)/2.\ensuremath{\rho}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})\ensuremath{\propto}|\mathbf{r}\ensuremath{-}{\mathbf{r}}^{\ensuremath{'}}{|}^{\ensuremath{-}(D+1)/2}.\ensuremathρ(r,r\ensuremath′)\ensuremath∝∣r\ensuremath−r\ensuremath′∣\ensuremath−(D+1)/2.
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 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 $\ensuremath{\gamma}\ensuremath{\propto}T$.
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.
Fast estimation of the single-particle density matrix is key to many applications in quantum chemistry and condensed matter physics. The best numerical methods leverage the fact that the density matrix elements f(H)ij decay rapidly with distance rij between orbitals. This decay is usually exponential. However, for the special case of metals at zero temperature, algebraic decay of the density matrix appears and poses a significant numerical challenge. We introduce a gradient-based probing method to estimate all local density matrix elements at a computational cost that scales linearly with system size. For zero-temperature metals, the stochastic error scales like S−(d+2)/2d, where d is the dimension and S is a prefactor to the computational cost. The convergence becomes exponential if the system is at finite temperature or is insulating.