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

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 AFM correlations 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.

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 the Kondo lattice systems, the Kondo singlets as Landau quasiparticles leads to a large Fermi surface. Basic physics in heavy fermion systems Y. Onuki and T. Komastubara, J. Magnetism & Magnetic Materials, 54, 433 (1986).

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

Another key issue: 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! Mechanism of heavy fermion superconductivity and its relation to AFM correlations

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

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

Can the Kondo screening coexist with AFM long-range order? 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.

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

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

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

Can the FM long-range order coexist with Kondo screening effect? 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.

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

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

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

Recent experimental discovery

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

The energy gap of spin-up quasiparticles

n=0.2

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

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

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

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

Ground state energy analysis and quantum phase transitions

Effective mass changes

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

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

Kondo-Heisenberg lattice model in the limit of MF order parameters: MF model Hamiltonian: Kondo singlet formation Spinon pairing attraction form Kondo singlet pairing order parameter Spinon-spinon pairing order parameter 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 !

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 !

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

Spinon-spinon pairing distribution function in Brillouin zone

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!

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

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.