Quantum Criticality
and
Fermi Surfaces
Qimiao Si
Rice University
KITPC, July 6, 2007
Stefan Kirchner, Seiji Yamamoto,
Silvio Rabello, J. L. Smith
Lijun Zhu
Eugene Pivovarov
Kevin Ingersent
Daniel Grempel
Jian-Xin Zhu
Jianhui Dai
S. Paschen
T. Lühmann
N. Oeschler
O. Tegus
J. A. Mydosh
E. Abrahams
P. Gegenwart
S. Wirth
T. Cichorek
O. Trovarelli
F. Steglich
(Rice University)
(UC Riverside)
(UC San Diego)
(Univ. of Florida)
(CEA-Saclay)
(Los Alamos)
(Zhejiang U.)
R. Küchler
Y. Tokiwa,
K. Neumaier
C. Geibel
P. Coleman
Materials (possibly) showing Quantum Criticality
•
Insulating Ising magnet
–
LiHoF4: transverse field Ising model
•
Heavy fermion metals
•
‘‘Simple’’ magnetic metals
–
•
•
Cr1-xVx, Sr3Ru2O7, MnSi (1st order, but …) …
High Tc superconductors
Mott transition
–
–
V2O3, …: QCP? (magnetic ordering intervenes at low T!)
ultracold atoms: 2nd order?
•
Field-driven BEC of magnons
•
MIT/SIT/QH-QH in disordered electron systems
Heavy fermions near an antiferromagnetic QCP:
(modern era of the heavy fermion field)
CeCu6-xAux
H. v. Löhneysen
et al, PRL 1994
TN
AF Metal
Linear
resistivity
TN
Heavy fermions near an antiferromagnetic QCP:
(modern era of the heavy fermion field)
CeCu6-xAux
H. v. Löhneysen
et al, PRL 1994
N. Mathur et al,
Nature 1998
CePd2Si2
TN
TN
AF Metal
AF Metal
Supercond.
Linear
resistivity
YbRh2Si2
TN
J. Custers et al,
Nature 2003
T=0 spin-density-wave transition
m(x, )
order parameter fluctuations
in space and (imaginary) time
T=0 spin-density-wave transition
m(x, )
order parameter fluctuations
in space and (imaginary) time
deff  d  z  4,
Gaussian
no

T
scaling
MF exponent
Dynamical and Static Susceptibilities in CeCu5.9Au0.1
• /T scaling
• a=0.75 `everywhere’ in q.
•Fractional exponent a=0.75
1/c(q)
q=0
..
q=Q
INS
@ q=Q
T0.75
ω/T
INS and M/H
A. Schröder et al., Nature ’00; PRL ’98;
O. Stockert et al., PRL ’98; M. Aronson et al., PRL ‘95
•
•
Quantum critical heavy fermions
–
prototype for quantum criticality
–
local quantum criticality – destruction of Kondo entanglement
Fermi surfaces of Kondo lattice
–
Global phase diagram
•
Extended-DMFT of Kondo Lattice
•
Some recent experiments
•
Bose-Fermi Kondo Model
–
Kondo-destroying QCP
Single-impurity Kondo model
• Kondo temperature TK0
• Kondo singlet
• Kondo resonance: spin ½ charge e
• Kondo lattices:
Historical development of heavy Fermi liquid:
Single-impurity: Anderson, Wilson, Nozières, Andrei, Wiegmann,
Coleman, Read & Newns, …
Lattice:
Varma, Doniach, Auerbach & Levin, Millis & Lee,
Rice & Ueda, …
Local Quantum Critical Point
• Destruction of Kondo screening (entanglement):
energy scale Eloc*  0 marks an electronic slowing
down at the magnetic QCP
QS, S. Rabello, K. Ingersent, & J. L. Smith, Nature 413, 804 (2001)
QS, J. L. Smith, and K. Ingersent, IJMPB 13, 2331 (1999)
Local Quantum Critical Point
• Destruction of Kondo screening (entanglement):
energy scale Eloc*  0 marks an electronic slowing
down at the magnetic QCP
• Spin damping: anomalous exponent and /T scaling
• Fermi surface jumps from “large” to “small”
QS, S. Rabello, K. Ingersent, & J. L. Smith, Nature 413, 804 (2001)
QS, J. L. Smith, and K. Ingersent, IJMPB 13, 2331 (1999)
•
•
Quantum critical heavy fermions
–
prototype for quantum criticality
–
local quantum criticality – destruction of Kondo entanglement
Fermi surfaces of Kondo lattice
–
Global phase diagram
•
Extended-DMFT of Kondo Lattice
•
Some recent experiments
•
Bose-Fermi Kondo Model
–
Kondo-destroying QCP
Fermi surface inside
the antiferromagnetic
part of the Kondo
lattice phase diagram
Kondo lattice
JK<<Irkky<<W
JK=0 as the reference point of expansion
Local Moments:
Quantum non-Linear Sigma Model
QNLM
Heisenberg model + coherent spin path integral
Haldane (1983)
Affleck (1985)
:
Chakravarty, Halperin, Nelson
PRB 39, 2344 (1989)
Not important inside
ordered phase
Conduction electrons
Effective fermi-magnon
coupling
S. Yamamoto & QS, PRL (July 5, 2007);
cond-mat/0610001
N.B.:
coupling
irrelevant (see next slide)
Kondo lattice
At JK=0:
Local-moment
antiferromagnetism
Conduction electrons
ω
k
0
Q
q
Effective Kondo coupling w/ magnons
qQ
with AF order
S ( x , )  m  n e
Large momentum transfer
scattering NOT allowed
kinematically.
i Q x
q0
Forward scattering
ALLOWED.
RELEVENT ???
Q
Combined bosonic/fermionic* RG
Tree-level scaling
(*ferminoc RG: Shankar, RMP ’94)
marginal!
From NLM
From electrons near FS
note: remember
for forward scattering
RG at 1-loop & beyond:
Marginal even at 1-Loop and beyond.
RG at 1-loop & beyond:
~(dΛ)3/2
RG at 1-loop & beyond:
Marginal even at 1-Loop and beyond.
AF phase w/ “small”
Fermi surface!
S. Yamamoto & QS,
PRL (July 5, 2007);
cond-mat/0610001
RG at 1-loop & beyond:
Marginal even at 1-Loop and beyond.
Large N:
=
~
No pole in self energy.
Fermi surface remains “small”.
AF phase w/ “small”
Fermi surface!
S. Yamamoto & QS,
PRL (July 5, 2007);
cond-mat/0610001
RG at 1-loop & beyond:
Marginal even at 1-Loop and beyond.
Side:
AFS: “hybridization” is finite at finite energies; it
only goes to zero upon reaching the fixed point.
Implications for the choice of variational
wavefunctions [H. Watanabe and M. Ogata,
arXiv:0704.1722]
AF phase w/ “small”
Fermi surface!
•
•
Quantum critical heavy fermions
–
prototype for quantum criticality
–
local quantum criticality – destruction of Kondo entanglement
Fermi surfaces of Kondo lattice
–
Global phase diagram
•
Extended-DMFT of Kondo Lattice
•
Some recent experiments
•
Bose-Fermi Kondo Model
–
Kondo-destroying QCP
Kondo coupling JK
Kondo lattice
JK=0
JK
Kondo lattice
G ~ frustration, reduced
dimensionality, etc.
Local-moment
magnetism, Irkky
Kondo coupling JK
G
JK=0
JK
JK<<Irkky<<W
G
AFS
Néel, without
Kondo screening
JK
JK >>W>>Irkky
G
paramagnet, w/
Kondo screening
PML
JK
Cf. A. C. Hewson, The Kondo Problem to Heavy Fermions
(Cambridge Univ. Press, 1993)
JK >>W>>Irkky
• xNsite tightly bound local singlets
(cf. If x were =1, Kondo insulator)
• (1-x)Nsite lone moments:
JK >>W>>Irkky
• xNsite tightly bound local singlets
(cf. If x were =1, Kondo insulator)
• (1-x)Nsite lone moments:
– projection:
– (1-x)Nsite holes with U=∞
JK >>W>>Irkky
• xNsite tightly bound local singlets
(cf. If x were =1, Kondo insulator)
• (1-x)Nsite lone moments:
– projection:
– (1-x)Nsite holes with U=∞
• Luttinger’s theorem:
(1-x) holes/site in the Fermi surface
(1+x) electrons/site
---- Large Fermi surface!
Global phase diagram
G
I
II PML
AFS
AFL
JK
QS, Physica B 378, 23 (2006);
QS, J.-X. Zhu, D. Grempel,
JPCM ‘05
SDW of PML
Type II transition
Hertz-Moriya-Millis fixed point for T=0 SDW transition
Type I transition
Second order if
Destruction of Kondo screening where magnetism sets in
Type I quantum transition (cont’d)
G
I
PML
AFS
AFL
•
•
Quantum critical heavy fermions
–
prototype for quantum criticality
–
local quantum criticality – destruction of Kondo entanglement
Fermi surfaces of Kondo lattice
–
Global phase diagram
•
Extended-DMFT of Kondo Lattice
•
Some recent experiments
•
Bose-Fermi Kondo Model
–
Kondo-destroying QCP
• Kondo lattices:
Extended-DMFT* of Kondo Lattice
(* Smith & QS; Chitra & Kotliar)
• 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 ()
c(q,ω)=1/[ Iq + M(ω)]
E-DMFT solution to the Kondo lattice
• The self-consistent fluctuating field bath:
 1

  (ω  w ) ~ ω
p
p
• Destruction of Kondo screening:
c loc   ~
1


~
1

1
~ Im
ln 1 /( i )
Kondo
JK

Critical
Divergent χloc(ω) locates the local
problem on the critical manifold
QS, S. Rabello, K. Ingersent, & J. L. Smith, Nature 413, 804 (2001)
g
Kondo lattice with Ising anisotropy:
Extended-DMFT
JK
Kondo
Critical
g
Kondo lattice with Ising anisotropy:
Quantum Monte Carlo of EDMFT
• At the QCP, I ≈ Ic
JK
cloc (n)
Kondo
Critical
g
n
D. Grempel and QS,
Phys. Rev. Lett. ’03
Kondo lattice with Ising anisotropy:
Quantum Monte Carlo of EDMFT
• At the QCP, I ≈ Ic
JK
Kondo
cloc (n)
Critical
g
c-1(Q,n)
n
D. Grempel and QS,
Phys. Rev. Lett. ’03
a= 0.72
n
Kondo lattice with Ising anisotropy
Quantum Monte Carlo
 ≡ IRKKY / TK0
J.-X. Zhu, D. Grempel, and QS,
PRL (2003)
EDMFT of Anderson lattice with Ising anisotropy
EDMFT of
Jc1
Jc2
 ≡ IRKKY / TK0
P. Sun and G. Kotliar, Phys.Rev.Lett. ’03
Avoiding double-counting in EDMFT
Quantum transition is second order
only when ΔIq is kept the same on
the paramagnetic and magnetic sides
QS, J.-X. Zhu, & D. Grempel, J. Phys.: Condens. Matter 17, R1025 (2005)
see also:
P. Sun & G. Kotliar, Phys. Rev. B71, 245104 (2005)
Kondo lattice with Ising anisotropy
Numerical RG
(T=0)
J.-X. Zhu, S. Kirchner, QS, & R. Bulla,
cond-mat/0607567 (2006);
See also, M. Glossop & K. Ingersent,
cond-mat/0607566 (2006)
Quantum Monte Carlo
(T=0.01 TK0)
 ≡ IRKKY / TK0
J.-X. Zhu, D. Grempel, and QS,
Phys. Rev. Lett. (2003)
•
•
Quantum critical heavy fermions
–
prototype for quantum criticality
–
local quantum criticality – destruction of Kondo entanglement
Fermi surfaces of Kondo lattice
–
Global phase diagram
•
Extended-DMFT of Kondo Lattice
•
Some recent experiments
•
Bose-Fermi Kondo Model
–
Kondo-destroying QCP
dHvA across M agnetic Transition in CeRhIn5
T. Park et al., Nature 440, 65 (‘06);
G. Knebel et al., PRB74, 020501 (‘06)
H. Shishido,
R. Settai, H. Harima,
_
& Y. Onuki, JPSJ 74, 1103 (2005)
Hall Effect in YbRh2Si2
• crossover width smaller
as T decreases
• T=0 (extrapolation):
sharp jump @ QCP
S. Paschen, T. Lühmann, S. Wirth, P.Gegenwart, O.Trovarelli, C. Geibel,
F. Steglich, P.Coleman, & QS, Nature 432, 881 (2004)
Multiple Energy Scales of QCP
P. Gegenwart, T. Westerkamp, C. Krellner, Y. Tokiwa, S. Paschen, C. Geibel,
F. Steglich, E. Abrahams, & QS, Science 315, 969 (2007)
•
•
Quantum critical heavy fermions
–
prototype for quantum criticality
–
local quantum criticality – destruction of Kondo entanglement
Fermi surfaces of Kondo lattice
–
Global phase diagram
•
Extended-DMFT of Kondo Lattice
•
Some recent experiments
•
Bose-Fermi Kondo Model
–
Kondo-destroying QCP
Bose-Fermi Kondo model
  (ω  w ) ~ ω
p
p

c
1

( ) ~
0
1

2 
ε-expansion of Bose-Fermi Kondo model:



(
ω

w
)
~
ω

p
p
JK
Kondo
Critical
g
Critical:
Order ε: J. L. Smith & QS ’97; A. M. Sengupta ’97; Higher orders in ε:
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
ε-expansion of Bose-Fermi Kondo model:



(
ω

w
)
~
ω

p
p
Ising
SU(2) & XY
JK
Kondo
JKKondo
Critical
Critical
g
Critical:
g
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
Bose-Fermi Kondo model in a large-N limit
At the Kondo-destroying QCP (g=gc)
S. Kirchner and QS,
arXiv:0706.1783
Bose-Fermi Kondo model in a large-N limit
T=0:
S. Kirchner and QS,
arXiv:0706.1783
Bose-Fermi Kondo model in a large-N limit
Finite T:
g=gc:
S. Kirchner and QS,
arXiv:0706.1783
Bose-Fermi Kondo model in a large-N limit
T=0:
Finite T:
Of the form of
a boundary CFT
S. Kirchner and QS,
arXiv:0706.1783
Bose-Fermi Kondo model
with Ising anisotropy
Spin susceptibility
at g=gc:
All these suggest:
an underlying boundary CFT;
enhanced symmetry at the QCP
Bose-Fermi Kondo model at ε=2:
exact solution
• General arguments and indirect calculations imply that, at
ε=1-, the QCP of BFKM has zero residual entropy.
• However, large-N limit yields a finite residual entropy for the
QCP of any 0<ε≤2, including at ε=1- (M. Kircan and M. Vojta,
PRB69, 174421 ’04).
• Shedding light from an exact solution at ε=2
c
1

( ) ~
0

1
2 
 const .
J-H Dai, C. Bolech, and
QS, to be published
Bose-Fermi Kondo model at ε=2:
exact solution
Bose-only:
SU(2):
Ising:
J-H Dai, C. Bolech, and
QS, to be published
Bose-Fermi Kondo model at ε=2:
exact solution
Bose-only:
SU(2):
Ising:
Contrast to large N:
J-H Dai, C. Bolech, and
QS, to be published
Bose-Fermi Kondo model at ε=2:
exact solution
Bose-Fermi Kondo
(Ising anisotropic, Kondo Toulouse point):
J-H Dai, C. Bolech, and
QS, to be published
Bose-Fermi Kondo model at ε=2:
exact solution
• General arguments and indirect calculations imply that, at
ε=1-, the QCP of BFKM has zero residual entropy.
• However, large-N limit yields a finite residual entropy for
any 0≤ε≤2, including at ε=1- (M. Kircan and M. Vojta, PRB69,
174421 ’04).
• Shedding light from an exact solution at ε=2:
The large N limit misses the change of the fixed-point
structure across ε=1: for singular bosonic bath, large N
over-estimates transverse fluctuations & underestimates longitudinal fluctuation.
SUMMARY
•
Quantum critical heavy fermions metals
–
–
•
Non-Fermi liquid behavior
Beyond the order parameter fluctuation picture
Local quantum criticality
–
Critical Kondo screening at the magnetic QCP
• Global phase diagram
–
An antiferromagnetic metal with “small” Fermi surface
•
Experiments: dynamical scaling, Fermi surface
reconstruction, multiple energy scales.
•
Relevance to other strongly correlated systems?
Beyond microscopcs
•
What is the field theory?
•
For α<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 expected to be smoothly q-dependent.
•
The spatial anomalous dimension ηspatial=0.
QS, S. Rabello, K. Ingersent and J.L.Smith, Nature 413, 804 (2001);
Phys. Rev. B 68, 115103 (2003)
Small Fermi surface in the AF metal phase
e.g. CeRh2Si2
TN
TN
S. Araki, R. Settai, T. C. Kobayashi, H. Harima, & Y. Onuki,
Phys Rev. B 64, 224417 (2001)