Transcript Nearsightedness of Electronic Matter

Nearsightedness of Electronic Matter
v(r)
R
w(r’)
r0
Footprint of w
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.
The NEM principle states that, for a given unperturbed system and a given R,
the density changes at r0, Dn(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:
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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 branch point connecting the
valence and conduction bands.
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.
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Asymptotics
Insulators:
with q = imaginary part of the branch point.
Metals:
In both cases, the reflection coefficient is smaller than 1, leading to a finite
nearsightedness range:
(insulators)
where ñ is a well-defined density characteristic of the unperturbed insulator, and
(metals)
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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):
Exact asymptotic expressions were also derived for “box” perturbations.
These expression also lead to a finite nearsightedness range.
For notation see the paper: E. Prodan and W. Kohn, Nearsightedness of Electronic Matter,
PNAS, 102, no.33, 11635-11638.
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Nearsightedness Range
(Insulators)
(Metals)
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Divide and Conquer O(N) algorithm
The buffer zone size as function of Dn:
(ungapped)
(gapped)
The optimal size of the domain:
CPU time as a function of desired accuracy Dn:
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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 chargenearsighted, 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 longrange 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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