Conserving Schwinger boson approach for the fully-screened infinite U Anderson Model

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. ? Non-Equilibrium Kondo physics: Quantum dots Magnetically doped mesoscopic wires DC bias on Mesoscopic samples Anderson model: Moment formation 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].

4. SU(N) Kondo model in Schwinger boson representation Exactly screened Under screened Over 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: Magnetic moment where entropy And the mean field chemical potential  is determined by 2S/N

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 • Solving the saddle-point equations self consistently. • 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)  T K  0 energy 0

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(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.

10. 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)

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

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

13. 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

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

15. Gapless t-matrix - Im { t(+i)} Parameters: N=4 K=1 0 = -0.2783 D  = 0.16 D TK = 0.002 D 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.

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 T2 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 ? T PM: Gapless FL + Gapped spinons and holons Neel AF: <b>≠0 JK/I

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.