Orthogonality catastrophe and the X-ray edge problem in mesoscopic physics

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Fermi Edge Singularities in the Mesoscopic X-Ray
Edge Problem
Martina Hentschel, Denis Ullmo, and Harold U.
Baranger Duke University
NIRT program
Metals
Mesoscopic Systems

• finite number of electrons
• on discrete level
• coherent, chaotic geometry
• fluctuations

?
core level
2
Martina Hentschel Yu und Cardona Fund.of SC,
p.477
The classical X-Ray Edge Problem
Singularities at the Fermi edge threshold in
X-Ray Emission or Absorption Spectra of, e.g.,
metals
e.g. Peaked Edge
L2,3-edge simple metals like Al, Mg, (Na)
from K. Othaka, Y. Tanabe, RMP 62 2929
(1990) GaAs-AlxGa1-xAs Quantum well, Lee et al.
(1987)
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What happens when a core electron is excited?
1023 cond. electrons respond

1 of 1023 electrons Many-body ground state
Fgt made of single particle wf. jigt
Anderson Orthogonality Catastrophe
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Anderson Orthogonality Catastrophe (AOC)
P. W. Anderson, Phys. Rev. Lett. 18 1049 (1967)
ground state initially
ground state under perturbation V
Or any state entirely described in terms of plane
waves
Perturbation can be small ? there is NO
ADIABATICITY in those systems!
Orthogonality Block

• Important in
• Fermi edge singularities of x-ray and
  photoluminescence spectra
• Kondo physics
• Tunneling (e.g. in double quantum dots)
• Similar phenomenon in particle physics

CHECK zero-bias anomaly (in dosordered systems)
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Peaked or rounded edge ?
Mesoscopic effects
Orthogonality block due to AOC
Many-body effect Mahans enhancement
Competition
screening dipole selection rules
acts universal
Sample-to-sample fluctuations
6
Peaked or rounded edge ?
Citrin, PRB (1979) Tanabe and Othaka (1990)
counteracting (Mahan) many- body process (dlo
only)
Anderson orthogonality catastrophe (all dl )
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Outline of talk

• Introduction
• Mesoscopic Anderson Orthogonality Catastrophe
  
• X-Ray Photoabsorption Spectra Mesoscopic vs.
  Bulk-like
• Conclusion, Experimental Realizations

• Model, numerical method, results

• Fermi golden rule approach, role of dipole
  matrix elements

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II. Anderson Orthogonality Catastrophein
Mesoscopic Systems
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AOC for a rank-1 perturbation V
Tanabe and Othaka, RMP (1990) Aleiner and
Matveev, PRL (1998)
e.g. core hole left behind at r0
? overlap between perturbed and unperturbed
ground states
f (eigenvalues only)
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Example for a rank-1 perturbation

• unperturbed level ek equidistant (picket
  fence, bulk-like)
• perturbed level lk Schrödinger equation

d
Martina Hentschel Check this der ist gar nicht
constant!!!
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Rank-1 perturbation in the mesoscopic case

• Fluctuations ek, fk(ro) ? lk

• Assumptions
• ek ? GOE / GUE
  distribution
• fk(r0)2 ? Porter-Thomas
  distribution
• Motivation Random
  matrix theory
• chaotic systems quantum dots,
  nanoparticles
• Joint probability distribution
• N ni const. V-1
  b 1 (2) for GOE (GUE)

(Aleiner/Matveev, PRL 1998)
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Boundary effects
d

• run-away level
• pressure from far away level
• ? level-dependent potential and phase
  shift

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Workhorse Metropolis algorithm on the circle

• Start picket fence
• (N1 level e, N1 level l, mean
  level spacing d2p/N, shift db)
• Random number in (0, 2N1) ? level ei or li
  shifted within interval given by neighboring
  levels
• Every third step move pair (ei, li)
• Memory lost after N steps
• Metropolis step accept / reject change with
  PMmin(1, P(ei,li))

d
Circle constant DOS

• generate many ensembles ek,lk Þ D2
• ? distribution of overlaps D2

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Results 1. Ground state overlap distribution
P(D2)
Onset of AOC

1. as perturbation V vc increases b) as
   particle number N increases

N
Vc
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1. P(D2) cont.
Scaling and role of phase shift dF at Fermi energy
dF -p/2
? P(D2) determined by phase shift dF at
Fermi energy (as in metallic x-ray edge
problem)
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Results2. Origin of Fluctuations in P(D2)
Reference Db2 - evaluate D2 starting at the
Fermi edge EF
range 1
empty j
filled i
e l
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2. Fluctuations in P(D2) cont.
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2. Fluctuations in P(D2) cont.

• analytically understanding
• of overlap fluctuations
• consider two level i 0,1 around EF
• in the mean field of other level
• s (e1-e0) ? Wigner surmise
• u02 , u12 ? Porter-Thomas
• i ³2 ei , ui2, li ? one random variable

RMT justified !
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Summary part II
AOC in mesoscopic systems
bulk-like
mesoscopic chaotic

• e,l fluctuating
• (GOE/GUE)
• RMT treatment justified
• broad distribution P(D2)
• fluctuations dominated by
• levels around EF
• analytic treatment of
• range-1 approximation

• e equidistant
• e,l fix
• single value D2 º Db2
• bulk N , Db2 0

AOC in disordered systems Gefen et al. PRB
2002 AOC in parametric random matrices Vallejos
et al. PRB 2002
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III. Mesoscopic X-ray Edge Problem
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Approaching the Mesoscopic X-Ray Edge Problem
Fermi edge singularities in x-ray spectra of
metals
core
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Model Fermi golden rule approach
Tanabe and Othaka, RMP 1990
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Model Fermi golden rule approach
wjc2 D 2
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Dipole matrix element wjc
d(ro)
s-like
V d(r-r0) l0 s-like cond. el.
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Results1. Average Photoabsorption K-edgea)
Contributions from the various processes
vc -10 d, K-edge N 100, M 50, GOE

• peaked edge
• replacement processes near EF dominate
• one-pair shake-up processes dominate

wjc2 D2
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Results1. Average Photoabsorption K-edgeb)
Taking spin into account
vc -10 d, K-edge N 100, M 50, GOE
active
spectator
? width of F0ñ in basis of perturbed
final states YFñ
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Comparison with bulk-like case
vc -10 d, GOE
L-edge bulk-like
K-edge mesoscopic
K-edge bulk-like

• Rounded edge goes into a (slightly) peaked
• edge as the system becomes coherent

M.H., D.Ullmo, H.U. Baranger, cond-mat/0402207,
subm. to PRL
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Dependence on the number of electrons
Mesoscopic K-edge
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Results2. Average Photoabsorption L-edge
? Coupling to the wave function wjo yj

• bound state l0
• y0 (r0) piles up
• screens core hole
• s-like

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Average Photoabsorption L-edge cont.
vc -10 d, GOE, active spin
Mesoscopic L-edge
Bulk-like L-edge

• small differences
• mesoscopic vs. bulk-like,
• and GOE vs. GUE
• edge peak with N

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Results 3. Mesoscopic fluctuations in A(w)
f ( D2, Drepl ,Dshup wjc)
K-edge ? wjc y large Porter-Thomas
like fluctuations ? overwhelm overlap
correlations and dominate
fluctuations of A(w)
K-edge
L-edge ? wjc y narrowly distributed
? Symmetry replacement through bound
state acts like a ground state overlap
with dF p - dF , results in highly
peaked edge
L-edge
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Experimental Realizations
Fermi sea of electrons subject to a rank-1
perturbation

• x-ray photoabsorption with metallic
  nanoparticles feasible in few years
• double quantum dots constriction
  Abanin/Levitov, cond-mat/0405383
• photoabsorption via impurity states in
  semiconductor heterostructures

Control Experiment bulk-like no dots, just
2DEG with impurities - already done?
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Summary part III
Mesoscopic X-ray Edge Problem
s-like conduction electrons d0 p/2, d1 0
bulk-like
mesoscopic

• rounded K-edge
• peaked L-edge

• (slightly) peaked K-edge
• peaked L-edge

Average áA(w)ñ

• Dipole coupling changed because mesoscopic system
  is
• - chaotic (loose l as quantum number)
• coherent confinement
• wave function and derivative independent

Mesoscopic fluctuations

• individual spectra can even zig-zag

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IV. Conclusions

• AOC in Mesoscopic Systems
• - broad distribution P(D2)
• - scaling with Db2, dF
• Mesoscopic Photoabsorption Spectra and X-Ray Edge
  Problem
• - K-edge áA(w)ñ from rounded to
• peaked as system becomes coherent,
• Porter-Thomas fluctuations
• - L-edge strongly peaked,
• same fluctuations as D2
• Experimental realizations
• - array of quantum dots, impurity
• level takes role of core electron
• - nanoparticles, double dots

M. Hentschel, D. Ullmo, H.U. Baranger,
cond-mat/0402207