1. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Studies of Antiferromagnetic Spin Fluctuations in Heavy Fermion Systems. G. Kotliar Rutgers University. Collaborators: Ping Sun, Sergej Pankov, Antoine Georges, Serge Florens, Subir Sachdev

2. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS.  Motivation.  Spin fermion model of Rosch et. al. does it describe the data ? ( S. Pankov, S. Florens, A. Georges )  EDMFT-QMC calculations for the Anderson Lattice model ( P. Sun).  Conclusion.

3. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Local moments + Conduction Electrons. High temperatures local moments and conduction electrons. Low temperatures, TK >> JRKKY, a heavy Fermi liquid forms. The quasiparticles are composites of conduction electrons and spins. Heavy quasiparticles absorb the spin entropy.  Low temperatures TK << JRKKY the moments order. AF state. Spin ordering absorbs the spin entropy.  What happens in between? 2 impurity mode, Varma and Jones (PRL 1989)

4. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Early Treatments: Slave Bosons.  Link and Bond variables.  Crossover from weak to strong coupling as Jrkky/Tk increase. [M. Grilli G. Kotliar and A. MillisMean Field Theories of Cuprate Superconductors: A Systematic Analysis, M. Grilli, G. Kotliar and A. Millis, Phys. Rev. B. 42, 329-341 (1990).Phys. Rev. B. 42, 329-341 (1990).   Analogy with bose condensation. Strong Correlation Transport and Coherence, G. Kotliar, Int. Jour. of Mod. Phys. B5 (1991) 341- 352.  Two states: one with doubled unit cell, one with Luttinger fermi surface (no AF) Mean Field Phase Diagram of the Two Band Model for CuO Layers, C. Castellani, M. Grilli and G. Kotliar, Phys. Rev. B43, 8000-8004, (1991).Phys. Rev. B43, 8000-8004, (1991).

5. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Early Treatments: Slave Bosons.  Link and Bond variables. coherence order parameter.  Crossover from weak to strong coupling as the bqndwith of the conduction band is varied. Jrkky/Tk increase. [A., M. Grilli, G. Kotliar and A. Millis, Phys. Rev. B. 42, 329-341 (1990).Phys. Rev. B. 42, 329-341 (1990).   Analogy with bose condensation., G. Kotliar, Int. Jour. of Mod. Phys. B5 (1991) 341-352.  Finite temperature study, within large N. Bourdin Grempel and Georges PRL (2000).  N. Andrei and P. Coleman, staggered flux vs Kondo state.  Two states: one with doubled unit cell, one with Luttinger fermi surface (no AF) C. Castellani, M. Grilli and G. Kotliar, Phys. Rev. B43, 8000-8004, (1991).Phys. Rev. B43, 8000-8004, (1991).

6. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Renewed interest: CeCu6- xAux YbRh2Si2 Schroeder et.al. Nature (2000)  Functional form for DMFT, cf marginal fermi liquid.

7. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Renewed interest: YbRh2Si2 Linear resisitivity  = a Log[b/T] T> T*  = 1/T.3 T<T* Kadowaki Woods ratio A/  2=const (x-xc) > e A/  2=1/(x-xc).3 (x-xc)<e

8. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS YbRh2Si2, Gegenwart et. al. Susceptility C = 14 times the Yb moment. T0.=-.3 K

9. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Gegenwart

10. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Critical Point Can one integrate the Fermions? Is the Kondo-RKKY transition relevant to the magnetic critical point? Rosch et. al. 2d spin fluctuations and 3d electrons. Motivated by experiments. Explain linear resistivity, logarithimic enhancement of specific heat, Kadowaki Woods ratio ?

11. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Critical Point Almost local self energy. Internal consistency: vertex corrections are finite I Paul and GK Phys. Rev. B 64, 184414 (2001) Internal consistency: boson and fermion self energy scale the same way. Thermoelectric power. [Indranil Paul and GK S (T) /T scales with  Obeyed in CeCuAu J. Benz et. al. Physica B 259-261, 380 (1999).

12. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Does the 2d spin+ 3d fermion model account for the anomalous damping of the spin fluctuations? ds+z=4 marginally irrelevant coupling. Strictly speaking no E/T scaling, and Asymptotically scaling functions are all mean field like but can the corrections to scaling mimmick and effective exponent ? Answer: S. Pankov, S. Florens A. Georges and GK NO. The leading correction to scaling produce an effective exponent  eff  1

13. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Corrections to scaling

14. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

15. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spin self energy in a self consistent large N solution of the EMDFT equations of the spin fermion model. [Pankov Florens Georges and GK 2003]

16. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Introduction to DMFT.

17. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Impurity cavity construction: A. Georges, G. Kotliar, PRB, (1992)] Weiss field

18. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Impurity cavity construction

19. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

20. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT: Effective Action point of view. R. Chitra and G. Kotliar Phys Rev. B. (2000), (2001). Identify observable, A. Construct an exact functional of =a,  [a] which is stationary at the physical value of a. Example, density in DFT theory. (Fukuda et. al.) When a is local, it gives an exact mapping onto a local problem, defines a Weiss field. The method is useful when practical and accurate approximations to the exact functional exist. Example: LDA, GGA, in DFT. It is useful to introduce a Lagrange multiplier  conjugate to a,  [a,  It gives as a byproduct a additional lattice information.

21. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Example: DMFT for lattice model (e.g. single band Hubbard). Observable: Local Greens function G ii (  ). Exact functional  [G ii (  )  DMFT Approximation to the functional.

22. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Example: EDMFT Observable: Local Greens function G ii (  ). Local spin spin or charge charge correlation P (  ). Exact functional  [G ii (  ) P (  ).  EDMFT Approximation by keeping only local graphs in the Baym Kadanoff functional. “Best” “local “ approximation, targeted to the observable that one wants to compute. Natural extension to treat phases with long range order. [Chitra and Kotlar PRB 2000]

23. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Top to bottom approach. Captures the physics of Kondo and the magnetism. To treat the dispersion of the spin fluctuations, add Bose field. DMFT in the Bose field.  Functional formulation, ordered and disordered phases. “Optimal Choice of local spin and electron self energies”.

24. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS EDMFT Application to the Kondo lattice. Q. Si S Rabello K Ingersent and J Smith Nature 423 804 (2001). Remarkable agreement with the experimental observation of a quantum critical point with non trivial Landau damping. P. Sun and GK: approach the problem from high temperatures, with a different model (Anderson model ).

25. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Model and parameters U = 3 : 0, V = 0 : 6, Ef = - 0 : 5

26. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS EDMFT equations.

27. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS EDMFT equations

28. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phase Diagram. (P. Sun )

29. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phase diagram. First order of the transition. At high temperatures, artifact of EDMFT, Pankov et. al. PRB 2002. At low temperatures ? Fluctuation driven First order transition in CeIn3 ?

30. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Local susceptibility

31. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Evolution of the magnetic structure. In this parameter regime, the QP are formed Before the magnetic transition?

32. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Size of the jump

33. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Evolution of the quasiparticles parameters. (P. Sun 2003)

34. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Evolution of the electronic structure System becomes more incoherent as the transition is approached. On the antiferromagnetic side : Majority spins are more incoherent than the minority spins.

35. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS F electron Weiss field (P. Sun 2003)

36. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spin self energy.

37. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion

38. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Extended DMFT electron phonon

39. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Extended DMFT e.ph. Problem

40. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS E-DMFT classical case, soft spins

41. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS E-DMFT classical case Ising limit

42. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Advantage and Difficulties of E-DMFT The transition is first order at finite temperatures for d< 4 No finite temperature transition for d less than 2 (like spherical approximation) Improved values of the critical temperature

43. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS E-DMFT test in the classical case[Bethe Lattice, S. Pankov 2001]