Qimiao Si

1
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
B
temperature T
C
T0
A
control parameter ?

• A every spin (spontaneously) points up
• Order parameter
• B every microstate equally probable m0

• C every spin points along the transverse
  field m0

3
Quantum Phase Transition

• A every spin (spontaneously) points up
• Order parameter
• B every microstate equally probable m0

• C every spin points along the transverse
  field m0

4
Heavy fermion metals near a magnetic QCP

YbRh2Si2
Linear resistivity
TN
TN
J. Custers et al, Nature 2003
5

• T0 SDW Transition

order parameter fluctuations in space and
(imaginary) time
6

• T0 SDW Transition

order parameter fluctuations in space and
(imaginary) time
fermions are integrated out
7

• T0 SDW Transition

order parameter fluctuations in space and
(imaginary) time
fermions are integrated out
8
Quantum Critical Electron Systems
Quantum Critical
temperature T
Non-Fermi Liquid
magnetic order
T0
QCP
control parameter ?

• Do non-Fermi liquid electronic excitations in
  turn
• change the nature of quantum criticality?

9
Kondo Lattice Model
a lattice of s1/2 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 S
13
Pre-History II Heavy Fermi Liquid (Kondo
Lattice)

• Mean field theory

k-independent
pole in S

• 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 S

• 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
( 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 S (?)
  G(k,?)1/? ek - S(?)
• spin self-energy M (?) ?(q,?)1/ Iq
  M(?)

17
Extended-DMFT of Kondo Lattice
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
e-expansion of Bose-Fermi Kondo model
Kondo
JK
Critical
g
LM

• Order e J. L. Smith QS 97 A. M. Sengupta
  97 Higher orders in e 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
e-expansion of Bose-Fermi Kondo model
Ising
SU(2) XY
Kondo
Kondo
JK
JK
Critical
Critical
g
g
Critical
Crucial for LQCP solution

• Order e J. L. Smith QS 97 A. M. Sengupta
  97 Higher orders in e 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 ?loc(?) 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 T0 (contd)
mAF
Eloc
_at_ T0.01TK0
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)
_at_ T0.01TK0
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
?(T, Ic) ? T ?(T 0) ? (Ic I)
wn
28
Fermi Surface Evolution
29
In what sense is the QCP local?

• Localization of f-electrons
• Reconstruction of the Fermi surface across ?QCP
• m ? 8 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
Inherent quantum nature of the Kondo-destruction
critical point (single-impurity Bose-Fermi Kondo
model)

• Order parameter fluctuations local F4 theory

with

• e0.5 would be the upper critical dimension

M. E. Fisher, S-K Ma, B. G. Nickel, PRL ,76 J.
M. Kosterlitz, PRL 76

• for egt0.5, the QCP would be Gaussian should
  see violation of ?/T scaling

31
Inherent quantum nature of critical Kondo effect
(S. Kirchner, T-H Park, QS, D. R. Grempel, to
be published 05)
e0.8
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/0501601
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(8)
Cf. f(0) f(8) 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/0507215
33
Beyond microscopcs

• What is the field theory?

• For alt1, Smag is Gaussian the q-dependence of
  M(q,?) would be smooth.

• The coupling to Scritical-kondo makes a
  contribution to M(q,?) which is presumably also
  smoothly q-dependent.

• The spatial anomalous dimension ?spatial0.

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
• T0 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?