Conserving Schwinger boson approach for the fully- screened infinite U Anderson Model Eran Lebanon...
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Conserving Schwinger boson approach for the fully-
screened infinite U Anderson Model
Eran LebanonRutgers University
with Piers Coleman, Jerome Rech, Olivier ParcolletarXiv: cond-mat/0601015 DOE grant DE-FE02-00ER45790
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Outline
• Introduction: - Motivation - Kondo model in Schwinger boson
representation - Large-N approach• Anderson model in Schwinger boson
representation • Conserving Luttinger-Ward treatment• Results of treatment• Extensions to non-equilibrium and the lattice
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Anderson model:
Moment formation
Kondo physics
Mixed valance imp.
DC bias on Mesoscopic samples
Impurity lattice
Non-Equilibrium Kondo physics:
Quantum dots
Magnetically doped mesoscopic wires
Quantum criticality:
mixed valent and heavy fermion materials
?
?
Wanted: good approach which is scalable to the Lattice and to nonequilibrium.
Schwinger bosons: Exact treatment of the large-N limit for the Kondo problem [Parcollet Georges 97] and for magnetism [Arovas Auerbach 88].
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SU(N) Kondo model in Schwinger boson representation
Exactly screened
Under screened Over screened
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Large N scheme [Parcollet Georges 97]
Taking N to infinity while fixing K/N an J, the actions scales with N, and the saddle point equations give:
where
And the mean field chemical potential is determined by
2S/N
entropyMagnetic moment
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Correct thermodynamics: need conduction electons self energy [Rech
et.al. 2005]
c = O(1/N) but contributes to the free energy leading order O(N).
conduction electrons × NK, holons × K, and Schwinger bosons × N
1. Solving the saddle-point equations self consistently.
2. Calculating conduction electrons self energy: N c → F
Exact screening (K=2S):
• Saturation of susceptibility
• Linear specific heat C=T
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Problem:
• Describes physics of the infinite N limit – which in this case is qualitatively different from physics of a realistic finite N impurity (zero phase shift, etc…)
Question: • How to generalize to a simple finite-N approach?
Possible directions:1. A brute force calculation of the 1/N corrections
2. An extension of large-N to a Luttinger-Ward approach
???
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Infinite-U Anderson model in the Schwinger boson representation
t-matrix (caricature)
energy0 0
TK
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Nozieres analysis: FL properties (2S=K)
Phase shift:
sum of conduction electron phase shifts must
be equal to the charge change K-n+O(TK/D):
In response to a perturbation the change of phase shift is:
Analysis of responses gives a generalized “Yamada-Yoshida” relation
Agrees with: [Yamada Yoshida 75] for K=1, [Jerez Andrei Zarand 98] for Kondo lim.
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Conserving Luttinger-Ward approach
F is stationary with respect to variations of G:
O(N) O(1) O(1/N)
LW approximation: Y[G] → subset of diagrams (full green function): Conserving!
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/
1/N
Im ln {t(0+i)}
(K-n)/NK
Conserved charge sum rule:
/TK
|ImGb|
0
-
Nc-n
Phase shift
Conservation of Friedel sum-rule
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Ward identities and sum-rules for LW approaches
Derivation is valid when is OK. (for NCA not OK…)
[Coleman Paul Rech 05]
Ward identity
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Boson and holon spectral functions
Boson spectral function Holon spectral function
/TK/TK
/D
0 = -0.2783 D = 0.16 D TK = 0.002 D
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Thermodynamics: entropy and susceptibility
T/TK
impTK
Simp
Parameters:
N=4 K=1
0 = -0.2783 D
= 0.16 D
TK = 0.002 D
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Gapless t-matrix
Main frame: T/D = 0.1, 0.08, 0.06, 0.04, 0.02, and 0.01
Inset: T/(10-4 D)= 10, 8, 6, 4, 2, 1, and 0.5.
- Im { t(+i)}
Parameters:
N=4 K=1
0 = -0.2783 D
= 0.16 D
TK = 0.002 D
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Gapless magnetic power spectrum
Diagrammatic analysis of the susceptibility’s vertex
shows that the approach conserves the Shiba relation
Since the static susceptibility is non-zero the
magnetization’s power spectrum is gapless.
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Transport: Resistivity and Dephasing
0 = -0.2783 D
Solid lines: =0.16 D, dashed lines =0.1 D
[Micklitz, Altland, Costi, Rosch 2005]
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Shortcomings
• The T2 term at low-T is not captured by the approach.
• The case of N=2
Just numerical difficulties?
Gapless bosons?
More fundamental problem?
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Extension to nonequilibrium environment
Keldysh generalization of the self-consistency equations
• Correct low bias description
• Correct large bias description
• A large-bias to small-bias crossover
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(Future) extension to the lattice
• Heavy fermions: Anderson (or Kondo) lattice – additional momentum index.
• Anderson- (or Kondo-) Heisenberg: the Heisenberg interaction
should be also treated with a large-N/conserving approach.
• Boson pairing - short range antiferromagnetic correlations?
boson condensation - long range antiferromagnetic order?
• Friedel sum-rule is replaced with Luttinger sum-rule
JK/I
Neel AF: <b>≠0 PM: Gapless FL +
Gapped spinons and holons
T ?
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Summary
• LW approach for the full temperature regime.• Continuous crossover from high- to low-T
behavior.• Captures the RG beta function.• It describes the low-T Fermi liquid.• Conserves the sum-rules and FL relations.• Describes finite phase shift.• Can be generalized to non-equilibrium and
lattice.