Physical Review Letters

Locality of the Density Matrix in Metals, Semiconductors, and Insulators

Sohrab Ismail‐Beigi, T. A. Arias

Published Mar 8, 1999• 217 citations

Abstract

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 $O(N)$ 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 $T$ is the temperature. Rather, in semiconductors $\ensuremath{\gamma}\ensuremath{\propto}\ensuremath{\Delta}$, and in metals at low temperature $\ensuremath{\gamma}\ensuremath{\propto}T$.

DOI: 10.1103/physrevlett.82.2127

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Physical Review Letters

Locality of the Density Matrix in Metals, Semiconductors, and Insulators

Sohrab Ismail‐Beigi, T. A. Arias

Published Mar 8, 1999• 217 citations

Publisher

Abstract

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)

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

T

is the temperature. Rather, in semiconductors

\ensuremath{\gamma}\ensuremath{\propto}\ensuremath{\Delta}

, and in metals at low temperature

\ensuremath{\gamma}\ensuremath{\propto}T

DOI: 10.1103/physrevlett.82.2127

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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)

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

T

is the temperature. Rather, in semiconductors

\ensuremath{\gamma}\ensuremath{\propto}\ensuremath{\Delta}

, and in metals at low temperature

\ensuremath{\gamma}\ensuremath{\propto}T

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.

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.