Guang-Ming Zhang Dept. of Physics, Tsinghua University, Beijing, China

1. d-wave superconductivity induced by short-range antiferromagnetic correlations in the Kondo lattice systems Guang-Ming Zhang Dept. of Physics, Tsinghua University, Beijing, China Workshop on “Heavy Fermions and Quantum Phase Transitions” 10 – 12 Nov. 2012, IOP, Beijing, China

2. OUTLINE • Basic physics in heavy fermion systems • AFM order at half-filling and relation with Kondo screening effect • FM order at small electron densities and relation with Kondo screening • Fermi surface of heavy Fermi liquid under short-range AFMcorrelations • Heavy fermion superconductivity induced by AFM short-range correlations • under the Kondo screening effect • Conclusion Collaborator: Lu YU at Institute of Physics, Chinese Academic Sciences of China.

3. Basic physics in heavy fermion systems • Kondo physics in dilute magnetic impurities – the crossover between high T and low T • At high T, free moment scatters conduction electrons → ln T resistivity. • At low T, Kondo singlet/resonance forms → local Fermi liquid. • In theKondo lattice systems,the Kondo singlets as Landau quasiparticles leads to a large Fermi surface. Y. Onuki and T. Komastubara, J. Magnetism & Magnetic Materials, 54, 433 (1986).

4. One key issue: • Nature of magnetically order and relation to the Kondo screening effect

5. Another key issue: • Mechanism of heavy fermion superconductivity and its relation to AFM correlations • Kondo temperature is a very high energy scale! • Heavy Fermi liquid state is a good starting point. • Heavy fermion SC is driven by the AFM spin fluctuations!

6. Heavy Fermi liquid state in the Kondo lattice model Model Hamiltonian: Fermion rep. of local moments: Hybridization parameter Mean field Hamiltonian: Renormalized band energies:

7. Dramatic changes of Fermi surface due to the Kondo screening ! Large Fermi surface Small Fermi surface

8. At the half-filling, the heavy Fermi liquid becomes the Kondo insulating state. The AFM long-range order can form at the small Kondo coupling regime. Can the Kondo screening coexist with AFM long-range order?

9. Transverse interaction -> spin-flip scatterings Longitudinal interaction -> polarization effect Focus on the half-filled Kondo lattice model AFM order parameters: (SDW like) Kondo screening parameter: Both antiferromagnetic correlations and Kondo screening effect can be considered on equal footing within a mean field theory ! Renormalized bands energies:

10. The numerical calculations are performed lateron a square lattice with Quasiparticle energy Kondo singlet phase J/t AFM phase Order parameters Coexistence phase J/t

11. Coexistence of Kondo screening and AFM long-range order is confirmed by QMC ! Abstract ……..

12. When the conduction electron density is far away from the half-filling, the FM long-range order can be developed in the small Kondo coupling regime. Can the FM long-range order coexist with Kondo screening effect?

13. Focus on the Kondo lattice model far away from half-filling Order parameters: Mean field Hamiltonian: Quasiparticle energy bands:

14. Two possible FM long-range order states coexisting with Kondo screening effect Spin polarized FM Spin non-polarized FM

15. The spin-polarized FM coexists with the Kondo screening has been confirmed by a recent dynamic mean field theory.

16. Recent experimental discovery

17. Our recent results on heavy fermion ferromagnet II I G. M. Zhang, et. al., in preparation.

18. The energy gap of spin-up quasiparticles

19. n=0.2

20. n=0.2

21. Dramatic changes of Fermi surface due to AFM correlations ! AFM metallic state Heavy Fermi liquid What happens to the heavy Fermi liquid in the presence of short-range antiferromagnetic correlations ?

22. Kondo-Heisenberg lattice model in the limit of Heisenberg exchange coupling Kondo exchange coupling MF order parameters: MF model Hamiltonian: Renormalized band energies:

23. Two different renormalized band structures due to different types of hybridizations On a square lattice: Hybridization between c-electrons with f-holes Hybridization between c-electrons with f-particles Ground state is unstable!

24. Low renormalized band changes as Fermi surface changes as Self-consistent MF equations:

25. Ground state energy analysis and quantum phase transitions

26. Effective mass changes

27. The electron filling factor dependence of the phase transitions HF metal phase AFM metal phase

29. Can heavy fermion superconductivity be induced by short-range antiferromagnetic correlations ?

30. Kondo-Heisenberg lattice model in the limit of Kondo singlet formation Spinon pairing attraction form MF order parameters: Kondo singlet pairing order parameter Spinon-spinon pairing order parameter MF model Hamiltonian: The local AFM short-range correlations favor the spinon-spinon pairing with d-wave symmetryon the square lattice! The ground state is a superconducting state coexisting with the Kondo screening !

31. Main result of the mean field Superconducting pairing order parameter of the conduction electrons is induced by both the spinon-spinon pairing and a finite Kondo screening !

32. Node Heavy quasiparticle band energies: (two positive energy bands) Gap

33. Spinon-spinon pairing distribution function in Brillouin zone

34. Conduction electron pairing distribution function in Brillouin zone Ground state energy density and its derivative A quantum phase transition from nodal to nodaless superconductivity occurs!

35. Possible example of quasi-two dimensional heavy fermion superconductor arXiv: 1208.3684

36. Conclusions • Kondo screening can coexist with the AFM order as a ground state of the Kondo insulating phase • Kondo screening can also coexist with the FM order in Kondo lattice model: either spin polarized or spin non-polarized phase. • AFM short-range correlations can change the Fermi surface dramatically, leading to Lifshitz transitions • Heavy fermion superconductivity can be driven by AFM short-range correlations under the Kondo screening effect. Thank you for your attention