The NEM principle states that, for a given unperturbed system and a given R,

the density changes at r0, n(r0), due to all admissible w(r'), have a finite maximum

magnitude, , which, of course, depends on r0, R, and the unperturbed system.

From this definition, one can see that decays monotonically as a function of R.

In this paper we prove, for broad classes of systems, that in fact:

Schematic of Nearsightedness of Electronic Matter: v(r) is the unperturbed external potential, w(r') is the perturbing potential outside a sphere of radius R, which is centered on the point of interest r0.

Nearsightedness of Electronic Matter

R w(r’)r0

Footprint of w

v(r)

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One dimension, periodic v(x)

When adding a perturbation vanishing for x < 0 and reflection coefficient R (k),

with the contour appropriately chosen (see figure below).

The contours C1 (insulators) and C2 (metals) on the Riemann sheet of the last occupied band, used in the above contour integral.

C1 surrounds the branch point and the corresponding branch cut (dashed line).C2 intersects the real k axis at kF and reappears at - kF.

= the branch point connecting the valence and conduction bands.

..

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In both cases, the reflection coefficient is smaller than 1, leading to a finite nearsightedness range:

AsymptoticsInsulators:

with q = imaginary part of the branch point.

Metals:

(insulators)

(metals)

where ñ is a well-defined density characteristic of the unperturbed insulator, and

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Higher Dimensions

- 2D fermions in a potential with square symmetry- perturbations w(x,y) vanishing for y > 0 and periodic along x.

Insulators (asymptotics):

Metals (asymptotics):

For notation see the paper: E. Prodan and W. Kohn, Nearsightedness of Electronic Matter,PNAS, 102, no.33, 11635-11638.

Exact asymptotic expressions were also derived for “box” perturbations.These expression also lead to a finite nearsightedness range.

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(Insulators)

(Metals)

Nearsightedness Range

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Divide and Conquer O(N) algorithm

CPU time as a function of desired accuracy n:

The buffer zone size as function of n:

(gapped)

(ungapped)

The optimal size of the domain:

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Interacting Fermions

(i) Distant perturbing potentials, w(r′), with |r 0 – r′| ≥ R. The simplest description of many-body interaction effects is the random phase approximation (RPA). Within RPA, we found that, in all dimensions, the many-body interaction leads to a decrease of R in typical metals but an increase of R in typical insulators due to a reduction in the gap. (ii) Distant perturbing charge densities ρ(r′). In analogy with R(r 0, Δn), we define a charge-nearsightedness range, Rc(r 0, Δn) as the smallest distance such that any charge perturbation ρ(r′) lying entirely outside this range produces a density change at r 0, Δn(r 0), smaller than Δn.

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As is well known, the long-range Coulomb potential, because of perturbing electric charges, is screened out by metallic electrons. Preliminary model calculations for metallic electrons, in the Thomas–Fermi approximation, indicate that they are charge-nearsighted, i.e., have a finite Rc. However, charged insulating fermions are “classically farsighted,” in the sense that, at sufficiently large distances, the fermions “see” the classical long-range total potential ∫ ρt(r′)/|r 0 – r′|dr′, where ρt is the total perturbing charge density, including depolarization. Thus, for example, in metals, replacing a neutral atom or ion by another atom or ion always has short-range electronic consequences, whereas in an insulator ions lead to classical long-range electronic effects.

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