qimiao si rice university
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Quantum Criticality and Fermi Surfaces. Qimiao Si Rice University. KITPC, July 6, 2007. Stefan Kirchner, Seiji Yamamoto, (Rice University) Silvio Rabello, J. L. Smith Lijun Zhu (UC Riverside) Eugene Pivovarov (UC San Diego) Kevin Ingersent (Univ. of Florida) - PowerPoint PPT PresentationTRANSCRIPT
Qimiao Si
Rice University
KITPC, July 6, 2007
Quantum Criticalityand
Fermi Surfaces
S. Paschen P. Gegenwart R. KüchlerT. Lühmann S. Wirth Y. Tokiwa,N. Oeschler T. Cichorek K. NeumaierO. Tegus O. Trovarelli C. GeibelJ. A. Mydosh F. Steglich P. ColemanE. Abrahams
Stefan Kirchner, Seiji Yamamoto, (Rice University)Silvio Rabello, J. L. SmithLijun Zhu (UC Riverside)Eugene Pivovarov (UC San Diego) Kevin Ingersent (Univ. of Florida)Daniel Grempel (CEA-Saclay)Jian-Xin Zhu (Los Alamos)Jianhui Dai (Zhejiang U.)
• 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
Materials (possibly) showing Quantum Criticality
Heavy fermions near an antiferromagnetic QCP:
CeCu6-xAux
H. v. Löhneysen et al, PRL 1994
AF Metal
TN
TN
Linearresistivity
(modern era of the heavy fermion field)
Heavy fermions near an antiferromagnetic QCP:
CeCu6-xAux
YbRh2Si2
H. v. Löhneysen et al, PRL 1994
J. Custers et al, Nature 2003
CePd2Si2N. Mathur et al, Nature 1998
AF Metal
TN
TN
TN
AF MetalSupercond.
Linearresistivity
(modern era of the heavy fermion field)
scalingT no
) x,m(
Gaussian
zddeff ,4
T=0 spin-density-wave transition
order parameter fluctuationsin space and (imaginary) time
exponent MF
Dynamical and Static Susceptibilities in CeCu5.9Au0.1
A. Schröder et al., Nature ’00; PRL ’98; O. Stockert et al., PRL ’98; M. Aronson et al., PRL ‘95
• /T scaling
•Fractional exponent =0.75
• =0.75 `everywhere’ in q.
INS @ q=Q
q=Q
q=0
INS and M/H
T0.75
1/(q)
...
ω/T
• 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 singlet
• Kondo temperature TK0
• 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
• Spin damping: anomalous exponent and /T scaling
• Fermi surface jumps from “large” to “small”
• 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)
• 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
Local Moments: Quantum non-Linear Sigma Model
Heisenberg model + coherent spin path integral QNLMHaldane (1983)Affleck (1985) :
Conduction electrons
Not important insideordered phase
Chakravarty, Halperin, NelsonPRB 39, 2344 (1989)
Effective fermi-magnoncoupling
N.B.: coupling irrelevant (see next slide)S. Yamamoto & QS, PRL (July 5, 2007);
cond-mat/0610001
Effective Kondo coupling w/ magnons
e),( xQ nmxS i
Qq Q
Large momentum transfer scattering NOT allowed
kinematically.with AF order
Forward scatteringALLOWED.
RELEVENT ???
0q
Combined bosonic/fermionic* RGTree-level scaling (*ferminoc RG: Shankar, RMP ’94)
marginal!
note: remember for forward scattering
From NLM
From electrons near FS
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.
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
Large N:
RG at 1-loop & beyond:
Marginal even at 1-Loop and beyond.
AF phase w/ “small” Fermi surface!
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]
• 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 lattice
JK
G
G ~ frustration, reduceddimensionality, etc.
Loc
al-m
omen
t m
agn
etis
m, I
rkk
y
Kon
do
cou
pli
ng
J K
JK=0
JK
G
PML
paramagnet, w/ Kondo screening
JK >>W>>Irkky
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!
JK
G
AFS
PML
AFL
QS, Physica B 378, 23 (2006);
QS, J.-X. Zhu, D. Grempel, JPCM ‘05
I
II
Global phase diagram
SDW of PML
Type I transition
Type II transition
Hertz-Moriya-Millis fixed point for T=0 SDW transition
Second order if
Destruction of Kondo screening where magnetism sets in
• 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
Extended-DMFT* of Kondo Lattice
• Mapping to a Bose-Fermi Kondo model:
(* Smith & QS; Chitra & Kotliar)
+ 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)]
E-DMFT solution to the Kondo lattice
1 )/(1ln
1Im~~)(
iωwω p
p
QS, S. Rabello, K. Ingersent, & J. L. Smith, Nature 413, 804 (2001)
• The self-consistent fluctuating field bath:
1~
1~loc
• Destruction of Kondo screening:
Divergent χloc(ω) locates the local problem on the critical manifold
JK
g
Critical
Kondo
Kondo lattice with Ising anisotropy: Quantum Monte Carlo of EDMFT
• At the QCP, I ≈ Ic
D. Grempel and QS, Phys. Rev. Lett. ’03
loc (n)
n
Critical
KondoJK
g
Kondo lattice with Ising anisotropy: Quantum Monte Carlo of EDMFT
• At the QCP, I ≈ Ic
= 0.72D. Grempel and QS, Phys. Rev. Lett. ’03
-1(Q,n)
n
loc (n)
n
Critical
KondoJK
g
Kondo lattice with Ising anisotropy
≡ IRKKY / TK0
Quantum Monte Carlo
J.-X. Zhu, D. Grempel, and QS,
PRL (2003)
≡ IRKKY / TK0
EDMFT of Anderson lattice with Ising anisotropy
EDMFT of
P. Sun and G. Kotliar, Phys.Rev.Lett. ’03
Jc1Jc2
Avoiding double-counting in EDMFT
QS, J.-X. Zhu, & D. Grempel, J. Phys.: Condens. Matter 17, R1025 (2005)
see also:
P. Sun & G. Kotliar, Phys. Rev. B71, 245104 (2005)
Quantum transition is second order only when ΔIq is kept the same on the paramagnetic and magnetic sides
Kondo lattice with Ising anisotropy
≡ IRKKY / TK0
Numerical RG (T=0)
Quantum Monte Carlo
J.-X. Zhu, S. Kirchner, QS, & R. Bulla,
cond-mat/0607567 (2006);
See also, M. Glossop & K. Ingersent, cond-mat/0607566 (2006)
J.-X. Zhu, D. Grempel, and QS,
Phys. Rev. Lett. (2003)
(T=0.01 TK0)
• 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
T. Park et al., Nature 440, 65 (‘06);
G. Knebel et al., PRB74, 020501 (‘06)
_
dHvA across Magnetic Transition in CeRhIn5
H. Shishido, R. Settai, H. Harima,
& Y. Onuki, JPSJ 74, 1103 (2005)
• 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)
Hall Effect in YbRh2Si2
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
ε-expansion of Bose-Fermi Kondo model:
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
JK
Kondo
Critical
g
ωwω pp
~)(
Critical:
ε-expansion of Bose-Fermi Kondo model:
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
JK
Kondo
Critical
g
ωwω pp
~)(
JKKondo
Critical
g
IsingSU(2) & XY
Critical:
Crucial for LQCP solution
Bose-Fermi Kondo model in a large-N limit
S. Kirchner and QS,
arXiv:0706.1783
At the Kondo-destroying QCP (g=gc)
Of the form of a boundary CFT
Bose-Fermi Kondo model in a large-N limit
T=0:
Finite T:
S. Kirchner and QS,
arXiv:0706.1783
Spin susceptibilityat g=gc:
All these suggest: an underlying boundary CFT; enhanced symmetry at the QCP
Bose-Fermi Kondo modelwith Ising anisotropy
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
.1
~)( 2
1
0const
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 & under-estimates 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)