Transcript 1. dia

Inelastic scattering and “dephasing” from
quantum impurities
Gergely Zaránd (BUTE, TU Karlsruhe)
Collaborators:
László Borda (BUTE)
Natan Andrei (Rutgers)
Jan von Delft (LMU)
Gergely Zaránd, László Borda, Jan von Delft, Natan Andrei
Phys. Rev. Lett. 93, (2004)
Pankaj Mehta, László Borda, Gergely Zarand, Natan Andrei, P. Coleman,
Phys. Rev. B 72, 014430 (2005)
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Theory seminar, Grenoble
Outline
• Motivation, experimental relevance
scattering off magnetic impurities is
important in many experiments
• How to define / compute inelastic scattering ?
use of reduction formulas
• Application to the Kondo problem:
Non-perturbative results using numerical
renormalization group
• Conclusions
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Theory seminar, Grenoble
Motivation, some conceptual questions,
and experimental relevance
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(My) Definition of Inelastic scattering:
A scattering process where the electron (or other particle) scatters by
changing the quantum state of the “environment”

Inelastic scattering destroys quantum interference
(AB interference, localization, UCF etc.)
: the typical time scale of inelastic scattering
Example of AB interferometer (goes back to Einstein):
electron
photon (phonon,
electron-hole pair)
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Some (unanswered) conceptual questions:
Is separation of “particle” and “environment” possible?
• Fermions, e.g.
• Quasiparticles are always “dressed”
Is one measuring quasiparticles or particles ?
Can one always describe “nature” in terms of quasiparticles
as T goes to 0 ?
Yes ! They are sometimes
Very different from electrons…
• Luttinger liquid
• some NFL impurity models
• Quantum critical points
• 1D disordered interacting electrons
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Sources of inelastic scattering
• Electron-electron interaction
• Magnetic impurities
• Tow-level systems
• Phonons
• Magnons
• …

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Expectation:
diverges as
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T 0
1   from weak localization
Pierre et al [PRB 2003]
Mohanty, Jariwala, & Webb [PRL 1997]
Saturation of
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
Theoretical proposals
• Electron-electron interaction:
Intrinsic dephasing is suppressed
Aleiner, Altshuler, Gershenson [1999]
Altshuler, Aronov, Kmelnitsky
[J. Phys. C 1982]
Golubev & Zaikin
[PRB 1999 & PRB 2000]
Saturation of

  ~ T 2 / 3
• Magnetic impurities:
Magnetic impurities mediate inelastic scattering
Kaminski & Glazman [PRL 2001]; (Sólyom and Zawadowski [Z. Phys. ,1969])
Göppert, Galperin, Altshuler, Grabert [PRB, 2002]
Kroha & Zawadowski [PRL 2002]
• Tow-level systems:
Imry, Fukuyama, Schwab [EPL, 1999]
Zawadowski, von Delft, Ralph [PRL 1999]
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Experiments
Experiments
measuring

• Weak localization experiments on wires:
Mohanty, Jariwala, & Webb [PRL 1997];
Mohanty, & Webb [PRL 2003]
Saturation of
importance of magnetic
scattering
Pierre et al [PRB 2003]
Schopfer, Bauerle, Rabaud, Saminadayar [PRL 2003]
Bauerle, et al [PRL 2005]
• Energy distribution measurements:
Pothier, Gueron, Birge, Esteve, Devoret [PRL 1997];
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1 
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Systematic study
of magnetic
scattering
Out-of equilibrium measurements
SC wires
L ~ L
Pothier, Gueron, Birge, Esteve, Devoret [PRL 1997];
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Importance of impurities
Ag(6N)
  ~ T 2 / 3
Au(6N)
• No saturation in high
purity samples
• Mn doping similar to
low purity
Ag(5N)
Cu(6N)
Pierre et al [PRB 2003]
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1   from Kondo impurities
  ~ 1/ T 2
Schopfer, Bauerle, Rabaud,
Saminadayar [PRL 2003]
  ~ 1/ T
Mohanty & Webb [PRL 2003]
Experiments
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1  ~ T
for T  TK
1   ~ cst
for T  TK
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How to define / compute
inelastic scattering from a
quantum impurity ?
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Inelastic scattering for Kondo model: T=0
Interaction of a S = ½ magnetic impurity with one band of itinerant electrons
Kondo temperature:
Ground state is a singlet ~ Fermi liquid
[Nozières, 1974]
1 /   ~ T / TK
2
Quasiparticles at T=0 DO NOT RELAX electrons DO
Electric field couples to ELECTRONS (not quasiparticles)
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Inelastic scattering for Kondo model: T=0
Consider impurity in ground state + electron wave packet far away
Elastic
scattering
energyE
Inelastic
Scattering
electron
leaves
behind
excitations
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Definition of the S-matrix
b, in Sˆ a, in  b, out a, in
in the interaction representation:

Sˆ  T exp
T-matrix:
Sˆ  1ˆ  i Tˆ



 i H int (t ) dt
Many-body operator
Note: Sˆ describes the scattering of
ELECTRONS, not quasiparticles!
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!
Scattering of single electron states
single electron states:
i.e. eigenstates of H0, but as
or, in terms of wave packets:
Electrons far away when
the interaction is off
Electron is still far away
when the interaction is on
Note: we send in ELECTRONS and watch
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outgoing electrons,
not quasiparticles!
Connection to scattering cross sections
T-matrix:
Sˆ  1ˆ  i Tˆ
Many-body operator
Total cross section (optical theorem):
forward scattering of single particles
Elastic cross section:
elastic cross section is also related to
!
total cross section
Tˆ
:
Sum over all final states with
precisely one outgoing electron
Inelastic scattering cross section:



 inel   total   el
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How to compute
p, Tˆ p' , '
Reduction formulas relate the time-ordered Green’s
function with
p, Tˆ p' , '
Follow, e.g., Itzikson & Zuber to obtain:
Full time-ordered
Green’s function
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Application to Kondo problem
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Case of Kondo model
Impurity
spin
Conduction
electron
 
~J S
~ J 2 F ; F ( )
[Costi PRL2000]
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Method: Wilson’s numerical renormalization
[Wilson75]
group
 one defines a sequence of
discretized Hamiltonians
 diagonalize iteratively
1
H  lim (1   1 ) (N 1) / 2H N
N  2
•
•
•
•
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 at N-th iteration one can
calculate physical quantities
at energy scale N~-N/2
Spectral function of ANY local operator
Hilbert transform
real part too
High precision data needed (symmetries)
Proper normalization is crucial
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Results obtained by numerical renormalization group:
 inel ()
G.Z., László Borda, Jan von Delft,
Natan Andrei, Phys. Rev. Lett. 93, (2004)
•
•
 inel roughly linear for 0.05TK    0.5TK
 inel ~  2 for   0.05TK
• Similar behavior is expected as a function of temperature
[ as indeed found by T. Micklitz et al, PRL 2005]
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Results obtained by numerical renormalization group:
[C. Bauerle, F. Mallet, F. D. Mailly,
G. Eska, and L. Saminadayar, PRL 95, 266805 (2005)]
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High energy scattering rate:
At very high frequencies
all the scattering is
inelastic !
Inelastic
scattering
Elastic
scattering
~ J2
~ J4
See also M. Garst, P. Wölfle, L. Borda, J. von Delft, L. I. Glazman, cond-mat/0507431
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Magnetic field dependence
already a very small
field
results in a strong spindependence
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2CK model

~ anisotropy
 inel (  0)   el (  0)
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Conclusions:
•  inel (, B) can be computed by exploiting reduction
formulas and using NRG
• the quadratically vanishing inelastic rate appears only well
below TK
• even a very small
of the inelastic rate
• For
  TK
 inel ( ) ~
B
results in a strong spin asymmetry
we obtain
1
ln 2 ( / TK )
  el ( ) ~
1
ln 4 ( / TK )
[confirmed in M. Garst, et al., cond-mat/0507431]
• Our formalism carries over to other quantum impurity
models
• The finite
T version of our formula describes the
dephasing from magnetic impurities in weak localization
experiments
[T. Micklitz et al, cond-mat/0509583]
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Theory seminar, Grenoble