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Conserving Schwinger boson
approach for the fullyscreened infinite U Anderson
Model
Eran Lebanon
Rutgers University
with Piers Coleman,
Jerome Rech, Olivier Parcollet
arXiv: cond-mat/0601015
DOE grant DE-FE02-00ER45790
Outline
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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
?
Anderson model:
Quantum dots
DC bias on
Mesoscopic
samples
Moment formation
Non-Equilibrium
Kondo physics:
Magnetically doped
mesoscopic wires
?
Kondo physics
Mixed valance imp.
Impurity
lattice
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].
SU(N) Kondo model in Schwinger
boson representation
Exactly screened
Under screened
Over screened
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
Magnetic moment
entropy
And the mean field chemical
potential  is determined by
2S/N
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
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
???
Infinite-U Anderson model in the Schwinger
boson representation
t-matrix (caricature)

TK
0
0
energy
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.
Conserving Luttinger-Ward approach
F is stationary with respect to variations of G:
LW approximation: Y[G] → subset of diagrams (full green function): Conserving!
O(N)
O(1)
O(1/N)
Conservation of Friedel sum-rule
Phase shift
Im ln {t(0+i)}
/
(K-n)/NK
1/N
Conserved charge sum rule:
0
-
Nc-n
|ImGb|
/TK
Ward identities and sum-rules for LW
approaches
[Coleman Paul Rech 05]
Ward identity
Derivation is valid when
is OK. (for NCA not OK…)
Boson and holon spectral functions
Boson spectral function
Holon spectral function
/D
/TK
/TK
0 = -0.2783 D
 = 0.16 D
TK = 0.002 D
Thermodynamics: entropy and susceptibility
Simp
Parameters:
N=4 K=1
0 = -0.2783 D
 = 0.16 D
impTK
TK = 0.002 D
T/TK
Gapless t-matrix
- Im { t(+i)}
Parameters:
N=4 K=1
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.
0 = -0.2783 D
 = 0.16 D
TK = 0.002 D
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.
Transport: Resistivity and Dephasing
0 = -0.2783 D
Solid lines: =0.16 D, dashed lines =0.1 D
[Micklitz, Altland, Costi, Rosch 2005]
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?
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
(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
T
?
Neel AF:
<b>≠0
PM: Gapless FL +
Gapped spinons and holons
JK/I
Summary
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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.