1. Qimiao Si Rice University
Kondo Lattices: What do we learn from microscopics?
Qimiao Si
Rice University
Lijun Zhu, Stefan Kirchner,
Tae-Ho Park, Eugene Pivovarov, (Rice University)
Silvio Rabello, J. L. Smith
Kevin Ingersent (Univ. of Florida)
Daniel Grempel (CEA-Saclay)
Jianxin Zhu (Los Alamos)
KIAS, Oct 24, 2005

2. A: every spin (spontaneously) points upB
temperature T C
T=0 A
control parameter 
A: every spin (spontaneously) points up
Order parameter:
B: every microstate equally probable: m=0
C: every spin points along the transverse field: m=0

3. Quantum Phase Transition
QCP
Quantum Critical
ordered
state
control parameter 
temperature T
T=0 A B C
A: every spin (spontaneously) points up
Order parameter:
B: every microstate equally probable: m=0
C: every spin points along the transverse field: m=0

4. Heavy fermion metals near a magnetic QCP:
YbRh2Si2
Linear
resistivity TN TN
J. Custers et al, Nature 2003

5. T=0 SDW Transition order parameter fluctuations
in space and (imaginary) time

6. T=0 SDW Transition order parameter fluctuations
in space and (imaginary) time
fermions are integrated out

7. T=0 SDW Transition order parameter fluctuations
in space and (imaginary) time
fermions are integrated out

8. Quantum Critical Electron Systems
temperature T
Non-Fermi Liquid
magnetic
order
T=0
QCP
control parameter 
Do non-Fermi liquid electronic excitations in turn
change the nature of quantum criticality?

9. + Kondo Lattice Model a lattice of s=1/2 local moments, one per site
a conduction-electron band

10. Pre-History I: Kondo resonance (one local moment in a conduction electron bath)
Kondo temperature:
Singlet ground state:
Kondo resonance:
local moment acquires electron quantum number
due to entanglement

11. Pre-History II: Heavy Fermi Liquid (Kondo Lattice)
Slave fermions:
w/ constraint:
Slave boson:

12. Pre-History II: Heavy Fermi Liquid (Kondo Lattice)
Mean field theory …
k-independent
pole in Σ

13. Pre-History II: Heavy Fermi Liquid (Kondo Lattice)
Mean field theory …
k-independent
pole in Σ
… heavy electron band
Beyond mean field: gauge theory in its Higgs phase

14. Pre-History II: Heavy Fermi Liquid (Kondo Lattice)
Mean field theory …
k-independent
pole in Σ
… heavy electron band
Beyond mean field: gauge theory in its Higgs phase
Magnetic ordering: SDW out of the heavy quasiparticles

15. DMFT* of Kondo Lattice Mapping to a self-consistent Kondo model
(* Georges and Kotliar, Metzner and Vollhardt, … )
Mapping to a self-consistent Kondo model
+ self-consistency conditions
Correctly describes Kondo screening: heavy fermion phase
But: no competing mechanism against Kondo effect: Kondo screening is too robust
No dynamical competition between Kondo and RKKY

16. Extended-DMFT* of Kondo Lattice
(* Smith & QS; Chitra & Kotliar; Sengupta & Georges )
Mapping to a Bose-Fermi Kondo model:
+ self-consistency conditions
Electron self-energy Σ () G(k,ω)=1/[ω – εk - Σ(ω)]
“spin self-energy” M () (q,ω)=1/[ Iq + M(ω)]

17. Extended-DMFT of Kondo Lattice
Bose-Fermi Kondo
fermion bath Jk
Local moment
fluctuating
magnetic field g
+ self-consistency
Cf. QS, S. Rabello, K. Ingersent and J.L.Smith, Phys. Rev. B ’03 for details

18. ε-expansion of Bose-Fermi Kondo model:JK
Critical g LM
Order ε: J. L. Smith & QS ’97; A. M. Sengupta ’97; Higher orders in ε and spin anisotropies: L. Zhu & QS ’02; G. Zarand & E. Demler ’02
J K = 0: S. Sachdev & J. Ye ’93 (large N); M. Vojta, C. Buragohain & S. Sachdev ‘00

19. ε-expansion of Bose-Fermi Kondo model:
Ising
SU(2) & XY
Kondo
Kondo JK JK
Critical
Critical g g
Critical:
Crucial for LQCP solution
Order ε: J. L. Smith & QS ’97; A. M. Sengupta ’97; Higher orders in ε and spin anisotropies: L. Zhu & QS ’02; G. Zarand & E. Demler ’02
J K = 0: S. Sachdev & J. Ye ’93 (large N); M. Vojta, C. Buragohain & S. Sachdev ‘00

20. E-DMFT solution to the Kondo lattice
The self-consistent fluctuating field bath:
Destruction of Kondo screening:
Kondo JK
Critical
Divergent χloc(ω) locates the local problem on the critical manifold g
QS, S. Rabello, K. Ingersent, & J. L. Smith, Nature 413, 804 (2001)

21. Local Quantum Critical Point
Destruction of Kondo screening (Eloc*  0)
at the QCP
Critical Kondo screening characterizes non-Fermi liquid excitations
QS, S. Rabello, K. Ingersent, & J. L. Smith, Nature 413, 804 (2001)
QS, J. L. Smith, and K. Ingersent, IJMPB 13, 2331 (1999)

22. Local Quantum Critical Point
Destruction of Kondo effect (Eloc*  0)
at the QCP
Local susceptibility also diverges:
where
“spin self-energy” has anomalous exponent
QS, S. Rabello, K. Ingersent, & J. L. Smith, Nature 413, 804 (2001)

23. Kondo lattice with Ising anisotropy
EDMFT of
(Quantum Monte Carlo algorithm of Grempel and Rozenberg ’99)
Eloc* TN
d ≡ IRKKY / TK0
The destruction of Kondo resonances (Eloc*  0)
meets with the vanishing of the Néel temperature
J.-X. Zhu, D. Grempel, & QS, Phys.Rev.Lett. ’03

24. EDMFT of Anderson lattice with Ising anisotropy
( P. Sun and G. Kotliar, Phys.Rev.Lett. ’03 )
EDMFT of
Jc1
Jc2
d ≡ IRKKY / TK0
First order transition results from double-counting of RKKY interaction:
QS, J-X Zhu, & D. R. Grempel, Journ. Phys. Cond. Matter ‘05
P. Sun and G. Kotliar, Phys.Rev. B ‘05

25. Kondo lattice with Ising anisotropy: Evidence for 2nd-order transition at T=0 (cont’d)
mAF
Eloc*
@ T=0.01TK0
d ≡ IRKKY / TK0
mAF  0: continuous AF transition
Eloc*  0: destruction of Kondo resonances

26. Quantum Critical Dynamics
Local spin susceptibility
at I ≈ Ic ≈ 1.2 T0K :
cloc (wn)
@ T=0.01TK0 wn
Calculated  ≈ 0.7
D. Grempel and QS, Phys. Rev. Lett. ’03

27. Fractional exponent in the dynamics
Inverse peak susceptibility at I ≈ Ic
D. Grempel and QS,
Phys. Rev. Lett. ’03
c --1(Q,wn)
c --1(Q)
a = 0.72
a = 0.72 d wn
(T, Ic)  T;
(T = 0)  (Ic – I)

28. Fermi Surface Evolution

29. In what sense is the QCP local?
Localization of f-electrons
Reconstruction of the Fermi surface across QCP
m*  ∞ over the entire Fermi surface as   QCP
Anomalous spin dynamics everywhere in q.
Destruction of Kondo effect
Non-Fermi liquid excitations part of the quantum-critical spectrum.

30. M. E. Fisher, S-K Ma, & B. G. Nickel, PRL ,76
Inherent quantum nature of the Kondo-destruction critical point (single-impurity Bose-Fermi Kondo model)
Order parameter fluctuations: local Φ4 theory
with
ε=0.5 would be the upper critical “dimension” …
M. E. Fisher, S-K Ma, & B. G. Nickel, PRL ,76
J. M. Kosterlitz, PRL ‘76
… for ε>0.5, the QCP would be Gaussian; should see violation of ω/T scaling

31. Inherent quantum nature of critical Kondo effect
(S. Kirchner, T-H Park, QS, & D. R. Grempel, to be published ’05)
ε=0.8
Related observations in related models:
L. Zhu, S. Kirchner, QS, & A. Georges, Phys. Rev. Lett. ’04;
M. Vojta, N-H Tong, & R. Bulla, Phys. Rev. Lett. , ’05
M. Glossop and K. Ingersent, cond-mat/

32. Dynamical large-N limit of Bose-Fermi Kondo
(Parcollet & Georges, PRL ‘97;
Cox & Ruckenstein, PRL ‘93)
Leading term:
T(,T) = f(/T),
with f(0) ≠f(∞)
Cf. f(0) =f(∞) for Gaussian f.p.
(Damle & Sachdev ’97)
L. Zhu, S. Kirchner, QS, & A. Georges, Phys. Rev. Lett. ’04;
S. Kirchner, L. Zhu, QS, & D. Natelson, cond-mat/

33. Beyond microscopcs What is the field theory?
For a<1, Smag is Gaussian; the q-dependence of M(q,ω) would be smooth.
The coupling to Scritical-kondo makes a contribution to M(q,ω) which is presumably also smoothly q-dependent.
The spatial anomalous dimension ηspatial=0.

34. Kondo Lattice in One Dimension
(E. Pivovarov, & QS, Phys.Rev. B ’04)
Earlier work on spin gap of the Kondo phase:
O. Zachar, & A. M. Tsvelik, Phys. Rev. B ’01;
E. Sikkema, I. Affleck, & S. R. White, Phys. Rev. Lett. ’97;
O. Zachar, S. A. Kivelson, & V. J. Emery, Phys. Rev. Lett. ’96

35. SUMMARY
Microscopic results of Kondo lattices: two types of quantum critical points
T=0 SDW transition (Gaussian)
Locally quantum-critical: destruction of Kondo effect exactly at the magnetic QCP (interacting)
Plausible argument for robustness
What is the field theory?