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

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

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].

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

And the mean field chemical potential is determined by

2S/N

entropyMagnetic moment

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)

energy0 0

TK

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:

O(N) O(1) O(1/N)

LW approximation: Y[G] → subset of diagrams (full green function): Conserving!

/

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

Ward identities and sum-rules for LW approaches

Derivation is valid when is OK. (for NCA not OK…)

[Coleman Paul Rech 05]

Ward identity

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

Thermodynamics: entropy and susceptibility

T/TK

impTK

Simp

Parameters:

N=4 K=1

0 = -0.2783 D

= 0.16 D

TK = 0.002 D

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

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

JK/I

Neel AF: <b>≠0 PM: Gapless FL +

Gapped spinons and holons

T ?

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.