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Studies of Antiferromagnetic Spin Fluctuations in Heavy Fermion Systems.

G. Kotliar Rutgers University. Collaborators: Ping Sun, Sergej Pankov, Antoine Georges, Serge Florens, Subir Sachdev. Studies of Antiferromagnetic Spin Fluctuations in Heavy Fermion Systems. Motivation.

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Studies of Antiferromagnetic Spin Fluctuations in Heavy Fermion Systems.

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

  2. 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. . THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  3. 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) Local moments + Conduction Electrons. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  4. 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). 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). Early Treatments: Slave Bosons. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  5. Link and Bond variables.<b> 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). 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). Early Treatments: Slave Bosons. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

  7. Renewed interest: YbRh2Si2 Linear resisitivity g = a Log[b/T] T> T* g = 1/T.3 T<T* Kadowaki Woods ratio A/ g2=const (x-xc) > e A/ g2=1/(x-xc) .3 (x-xc)<e THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

  9. Gegenwart THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  10. 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 ? Critical Point THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  11. 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 g(T) . Obeyed in CeCuAu J. Benz et. al. Physica B 259-261, 380 (1999). Critical Point THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  12. 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 a eff > 1 Does the 2d spin+ 3d fermion model account for the anomalous damping of the spin fluctuations? THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

  14. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  15. 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] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

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

  19. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  20. 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, G [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 l conjugate to a, G [a, l ]. • It gives as a byproduct a additional lattice information. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  21. Observable: Local Greens function Gii (w). Exact functional G [Gii (w) ]. DMFT Approximation to the functional. Example: DMFT for lattice model (e.g. single band Hubbard). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  22. Observable: Local Greens function Gii (w). Local spin spin or charge charge correlation P (w). Exact functional G [Gii (w) P (w). ]. 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] Example: EDMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  23. 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”. DMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  24. 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 ). EDMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

  27. EDMFT equations THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

  29. 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 ? Phase diagram. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  30. Local susceptibility THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

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

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

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

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

  37. Conclusion THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

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

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

  42. 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 Advantage and Difficulties of E-DMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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