Workshop on “Heavy Fermions and Quantum Phase Transitions”
d-wave superconductivity induced by
short-range antiferromagnetic
correlations in the Kondo lattice systems
Guang-Ming Zhang
Dept. of Physics, Tsinghua University, Beijing, China
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.
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 the Kondo 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).
One key issue:
• Nature of magnetically order and relation to the Kondo screening effect
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!
Heavy Fermi liquid state in the Kondo lattice model
Model Hamiltonian: H

 
  c k  c k   J

k
k

S i  si
i
Fermion rep. of local moments:
Hybridization parameter
Mean field Hamiltonian:
H
mf

 c

k
k

,f k
k  

 V J
2



V
2

J  c k 

  N
 f 
 k 

1
2
J V
Renormalized band energies:

Ek 
1
2

      
 k

      J V
2
k

2


2
 

Dramatic changes of Fermi surface due to the Kondo screening !
Small Fermi surface
Large Fermi surface
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?
Focus on the half-filled Kondo lattice model
Longitudinal interaction -> polarization effect
Transverse interaction -> spin-flip scatterings
AFM order parameters:
(SDW like)

Kondo screening parameter: c  d
 d i c i 
i i



c i d i   d i c i 
 V
Both antiferromagnetic correlations and Kondo screening effect can be considered
on equal footing within a mean field theory !
Renormalized bands energies:
J 11  J
Quasiparticle energy
The numerical calculations are performed later on a square lattice with
Order parameters
J/t
Kondo singlet phase
AFM phase
J/t
Coexistence phase

 J.
Coexistence of Kondo screening and AFM long-range order is confirmed by QMC !
Abstract
……..
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?
Focus on the Kondo lattice model far away from half-filling
Order parameters:
Mean field Hamiltonian:
Quasiparticle energy bands:
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
TK  8 K ,
TC  0 .1 7 K
M o rd  0 .0 5  B
Our recent results on heavy fermion ferromagnet
II
G. M. Zhang, et. al., in preparation.
I
The energy gap of spin-up quasiparticles
n=0.2
n=0.2
Dramatic changes of Fermi surface due to AFM correlations !
Heavy Fermi liquid
AFM metallic state
What happens to the heavy Fermi liquid
in the presence of short-range antiferromagnetic correlations ?
J K  J H
Kondo-Heisenberg lattice model in the limit of
Heisenberg exchange coupling
Kondo exchange coupling
MF order parameters:
MF model Hamiltonian:
Renormalized band energies:
Wk 

    k   J KV
2
k

2
Two different renormalized band structures
due to different types of hybridizations
On a square lattice:  k   2t cos k x  cos k y

4t ' cos k x cos k y -  ,
 k  J H  cos k x  cos k y


Hybridization between
c-electrons with f-holes
Hybridization between
c-electrons with f-particles
  0
  0
Ground state is unstable!
Self-consistent MF equations:
nc  0.9
For J K  J H , we always obtain the solution
Low renormalized band changes as J H / J K
with   0 .
Fermi surface changes as J H / J K
Ground state energy analysis and quantum phase transitions
nc  0.9
Effective mass changes
nc  0.9
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
JK  JH
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!
 k   0 cos k x  cos k y 
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)

Ek 
E k ,1 
E k ,2 
E k ,1 
1
2

2
k

 
k
2
2
E k ,1  E k ,2 ,
   J H k
2
2
1
4
2
J KV
2
2

2

2


1
4
2
J KV
 kJ H k
2

2
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
t ' / t  0 . 3,
J K / t  2 . 0,
n c  0.8
A quantum phase transition from nodal to nodaless superconductivity occurs!
Possible example of quasi-two dimensional heavy fermion superconductor
arXiv: 1208.3684
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.