LOCALITY OF THE DENSITY MATRIX IN METALS, SEMICONDUCTORS, AND INSULATORS

We present an analytical study of the spatial decay rate γ of the one-particle density matrix ρ(~r, ~r ) ∼ exp(−γ|~r−~r ′|) 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 γ behaves contrary to the conventional wisdom that generically γ ∝ √ ∆ in insulators and γ ∝ √ T in metals, where ∆ is the band gap and T the temperature. Rather, in semiconductors γ ∝ ∆, and in metals at low temperature γ ∝ T . Hohenberg-Kohn-Sham density functional theory (DFT)[1] allows many-body systems to be described by a single-particle formalism. This theory is the basis for modern, large-scale calculations in solid-state systems, which have been quite successful in the prediction of material properties[2]. The one-particle density matrix ρ̂ ≡ ∑ n |ψn〉fn〈ψn|, which describes the state of a single-particle quantum system, is the key quantity which is needed for the computation of physical observables: total system energies, forces on atoms, and phonons can all be computed directly from ρ̂. Remarkably, despite the de-localized nature of the single particle states |ψn〉, which may extend across an entire solid, the physics of the electronic states in a given region of a material is affected only by the local environment. Reflecting this, the force on an atom depends mostly on the positions of its nearest neighbors. This electronic localization is manifest in the “nearsightedness”[3] of ρ̂: ρ(~r, ~r ) ≡ 〈~r |ρ̂|~r ′〉 ∼ exp(−γ|~r − ~r ′|) where γ > 0. This exponential decay has been verified both numerically[4, 5] and analytically[8, 9, 10, 14]. The locality of ρ not only is important for understanding the nearsightedness of effects arising from electronic structure but also has direct practical impact for DFT calculations. Recently, several methods have been proposed [16, 17, 18, 19, 20, 21, 22, 23, 24, 25] that use ρ directly and exploit its locality. The computational load of these methods scales as O(N), where N is the number of atoms in the simulation cell. However, the computational burden of these algorithms depends strongly on γ: some scale asN/γ[18, 19, 24] and others asN/γ[4, 22]. Knowledge of how γ depends on the system under study is thus critical for carrying out such calculations. Generally, solid-state systems have an underlying periodic structure. The introduction of localized defects[8] or surfaces[9] does not change the spatial range of ρ from that of the underlying periodic lattice, so that understanding the locality of ρ even for perfectly periodic systems is of direct relevance for realistic material studies. To date, the generic behavior of γ is poorly understood. For insulators in one dimension, Kohn has shown that γ ∝ √ −En in the tight-binding limit where En is an atomic ionization energy[6]. Motivated by this, it has been assumed[7, 4, 23] and argued[15] that γ ∝ √ ∆ in multiple dimensions and more general conditions, where ∆ is the band gap. For metals, it has been assumed[4] and argued[15] that γ ∝ √ T , where T is the electronic temperature. However, the results in[15], which to date represent the only effort to determine γ generically, are based on the assumption that the inverse of the overlap matrix of a set of Gaussian orbitals decays in a Gaussian manner. On the contrary, the inverses of such overlap matrices decay only exponentially, and thus the behavior of γ warrants further study. Here, we show that the behavior of γ is more complex than previously assumed. In this letter, we do not consider systems with indirect gaps, for which the relation between the global gap and localization is haphazard.