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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

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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.

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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).

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

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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

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

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

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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.

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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 !

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

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

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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.

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

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

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

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Recent experimental discovery

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

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The energy gap of spin-up quasiparticles

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n=0.2

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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

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

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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!

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

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Ground state energy analysis and quantum phase transitions

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Effective mass changes

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

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

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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 !

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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 !

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

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Spinon-spinon pairing distribution function in Brillouin zone

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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!

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

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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.

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