1. Conserving Schwinger boson approach for the fully- screened infinite U Anderson Model Eran Lebanon Rutgers University with Piers Coleman, Jerome Rech, Olivier Parcollet arXiv: cond-mat/0601015 DOE grant DE-FE02-00ER45790

2. 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

3. 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].

4. SU(N) Kondo model in Schwinger boson representation Exactly screenedUnder screenedOver screened

5. 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 entropy Magnetic moment

6. 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

7. 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 ???

8. Infinite-U Anderson model in the Schwinger boson representation t-matrix (caricature) energy 0  0  T K

9. Nozieres analysis: FL properties (2S=K) Phase shift: sum of conduction electron phase shifts must be equal to the charge change K-n  +O(T K /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.

10. 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!

11. // 1/N Im ln {t(0+i  )} (K-n  )/NK Conserved charge sum rule:  /T K |ImG b | 0 --  N c -n  Phase shift Conservation of Friedel sum-rule

12. Ward identities and sum-rules for LW approaches Derivation is valid when is OK. (for NCA not OK…) [Coleman Paul Rech 05] Ward identity

13. Boson and holon spectral functions Boson spectral functionHolon spectral function  /T K  /D  0 = -0.2783 D  = 0.16 D T K = 0.002 D

14. Thermodynamics: entropy and susceptibility T/T K  imp T K S imp Parameters: N=4 K=1  0 = -0.2783 D  = 0.16 D T K = 0.002 D

15. 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 T K = 0.002 D

16. 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.

17. Transport: Resistivity and Dephasing  0 = -0.2783 D Solid lines:  =0.16 D, dashed lines  =0.1 D [Micklitz, Altland, Costi, Rosch 2005]

18. Shortcomings The T 2 term at low-T is not captured by the approach. The case of N=2 Just numerical difficulties? Gapless bosons? More fundamental problem?

19. 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

20. (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 J K /I Neel AF: ≠0 PM: Gapless FL + Gapped spinons and holons T ?

21. 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.