Unveiling the Kondo cloud: unitary RG study of the Kondo model
Kondo Problem to Heavy Fermions and ... - Max Planck Society
Transcript of Kondo Problem to Heavy Fermions and ... - Max Planck Society
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Qimiao Si
Rice University
Advanced School – Developments and Prospects in Quantum Impurity
Physics, MPI-PKS, Dresden, May 30, 2011
Kondo Problem to Heavy Fermions
and Local Quantum Criticality
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• Introduction to quantum critical point
• Kondo problem to heavy Fermi liquid
• Heavy fermion quantum criticality
• Perspective and outlook
Q. Si, arXiv:1012.5440, a chapter in the book “UnderstandingQuantum Phase Transitions”, ed. L. D. Carr (2010).
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S. Friedemann C. Krellner Y. Tokiwa P. Gegenwart S. Paschen S. WirthN. Oeschler T. Westerkamp R. KüchlerT. Lühmann T. Cichorek K. NeumaierO. Tegus O. Trovarelli C. GeibelF. Steglich P. Coleman E. Abrahams
Pallab Goswami, Jed Pixley, Jianda Wu (Rice University) Stefan Kirchner (MPI-PKS, CPfS)Seiji Yamamoto (NHMFL, FSU)Jian-Xin Zhu, Lijun Zhu (Los Alamos)Kevin Ingersent (Univ. of Florida)
Jianhui Dai (Zhejiang U.)Daniel Grempel (CEA-Saclay)Ralf Bulla (U. Cologne)
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Phases and Phase Transitions
Disorder(T>Torder)
Order(T<Torder)
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Continuous Phase Transitions: Criticality
Disorder(T>Torder)
Order(T<Torder)
Criticality -- fluctuations of order parameter in d dimensions
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• A: every spin (spontaneously) points up
Order parameter:
• B: every microstate equally probable: m=0
σσ I- H zj
ij
zi∑=
><
orderedstate
tem
pera
ture
T
T=0A
B
1N /M limlim m site0h site
==∞→→ + N
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• A: every spin (spontaneously) points up
Order parameter:
• B: every microstate equally probable: m=0
σσ I- H zj
ij
zi∑=
><
orderedstate
control parameter δ
tem
pera
ture
T
T=0A
B
C∑i
xi) (I- σδ
1N /M limlim m site0h site
==∞→→ + N
• C: every spin points along the transverse field: m=0
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Quantum Phase Transition
• A: every spin (spontaneously) points up
Order parameter:
• B: every microstate equally probable: m=0
σσ I- H zj
ij
zi∑=
><
QCP
quantumcritical
orderedstate
control parameter δ
tem
pera
ture
T
T=0A
B
C∑i
xi) (I- σδ
1N /M limlim m site0h site
==∞→→ + N
• C: every spin points along the transverse field: m=0
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• Heavy fermion metals• Iron pnictides• Cuprates• Organic charge-transfer salts• Weak magnets (eg Cr-V, MnSi, Ruthenates)• Mott transition (eg V2O3)• Insulating Ising magnet (eg LiHoF4)• Field-driven BEC of magnons• MIT/SIT/QH-QH in disordered electron systems
• Tunable systems (eg quantum dots, cold atoms)
Materials (possibly) showing Quantum Phase Transitions
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YbRh2Si2
H. v. Löhneysen et al
J. Custers et al
CePd2Si2
N. Mathur et al
TN
Linearresistivity
Heavy fermion metals as prototype quantum critical points
CeCu6-xAux
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Torder
QCP
tem
pera
ture
δ -- controlparameter
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Following Landau -- fluctuations of order parameter, , but in d+z dimensions
Torder
QCP
tem
pera
ture
δ -- controlparameter
∞ @QCP
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scalingT
no ω
Gaussianzddeff ,4>+=
T=0 spin-density-wave transition
exponent MF
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Beyond the Order-parameter Fluctuations
Inherent quantum modes may be important -- need to identify the additional critical modes before constructing the critical field theory.
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Critical Kondo Destruction-- Local Quntum Critical Point
Kondo Destruction (f-electron Mott localization)at the T=0 onset of antiferromagnetism
QS, S. Rabello, K. Ingersent, & J. L. Smith, Nature 413, 804 (2001);Phys. Rev. B68, 115103 (2003)
P. Coleman et al, JPCM 13, R723 (2001)
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• Introduction to quantum critical point
• Kondo problem to heavy Fermi liquid
• Heavy fermion quantum criticality
• Perspective and outlook
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Single-impurity Kondo Model:
fermion bathLocal
moment
S: spin-1/2 moment at site 0
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Single-impurity Kondo Model:
– resistivity minimum (scattering increases as T is lowered!)
– asymptotic freedom
– Kondo screening (process of developing Kondo singlet correlations as T is lowered)
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Single impurity Kondo model
• Kondo temperature:
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Single impurity Kondo model
• Kondo entanglement: singlet ground state
• Kondo temperature:
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Single impurity Kondo model
• Kondo effect (emergence of Kondo resonance):– Kondo-singlet ground state yields an electronic resonance
– local moment acquires electron quantum numberdue to Kondo entanglement
• Kondo entanglement: singlet ground state
• Kondo temperature:
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Kondo lattices:
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Kondo lattices:
heavy Fermi liquid:
•Kondo singlet
•Kondo resonance
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JK >>W>>I
• xNsite tightly bound local singlets
(cf. If x were =1, Kondo insulator)
• (1-x)Nsite lone moments:
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JK >>W>>I
• xNsite tightly bound local singlets
(cf. If x were =1, Kondo insulator)
• (1-x)Nsite lone moments:– projection:
– (1-x)Nsite holes with U=∞
(C. Lacroix, Solid State Comm. ’85)
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JK >>W>>I
• 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!
(C. Lacroix, Solid State Comm. ’85)
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• The large Fermi surface applies to the paramagnetic phase, when the ground state is a Kondo singlet.
• This can be seen through adiabatic continuity of a Fermi liquid.
• It can also be seen, microscopically, through eg slave-boson MFT (Auerbach & Levin, Millis & Lee, Coleman, Read & Newns)
Heavy Fermi Liquid(Kondo Lattice)
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• Kondo resonance …
• … heavy electron bands
Heavy Fermi Liquid(Kondo Lattice)
),(--1
ωεωω
kkG
kc Σ
=),(
pole in Σ
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• Kondo resonance …
• … heavy electron bands
Heavy Fermi Liquid(Kondo Lattice)
),(--1
ωεωω
kkG
kc Σ
=),(
pole in Σ
k-independent
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Cond. electron band
Heavy Fermi LiquidHeavy electron bands
E1,2(k)
Kondo resonance
)(kε
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Cond. electron band
Heavy Fermi LiquidHeavy electron bands
E1,2(k)
Kondo resonance
)(kε
Large Fermi surface
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• Kondo lattices:
heavy Fermi liquid:
•Kondo singlet
•Kondo resonance No symmetry breaking, but macroscopic order
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Critical Kondo Destruction-- Local Quantum Critical Point
Kondo Destruction (f-electron Mott localization)at the T=0 onset of antiferromagnetism
Q. Si, arXiv:1012.5440, a chapter in the book “UnderstandingQuantum Phase Transitions”, ed. L. D. Carr (2010).
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• Introduction to quantum critical point
• Kondo problem to heavy Fermi liquid
• Heavy fermion quantum criticality
• Perspective and outlook
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Critical Kondo Destruction-- Local Quantum Critical Point
Kondo Destruction (f-electron Mott localization)at the T=0 onset of antiferromagnetism
Q. Si, arXiv:1012.5440, a chapter in the book “UnderstandingQuantum Phase Transitions”, ed. L. D. Carr (2010).
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Kondo lattices:
δ= TK0/I
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• In the paramagnetic phase, Eloc* is finite:
– Ground state is a Kondo singlet– Fermi surface is large– Call this “PL” phase
• Increasing RKKY interaction, I/TK0, leads to
AF order, yielding AF QCP
• What happens to the Eloc* scale as the AF QCP is approached from the PL side?
Kondo Effect at the AF QCP
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T=0 spin-density-wave transition
smooth q-independence
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Extended-DMFT* of Kondo Lattice
Mapping to a Bose-Fermi Kondo model:
(* Smith & QS; Chitra & Kotliar)
+ self-consistency conditions
– Electron self-energy Σ (ω) G(k,ω)=1/[ω – εk - Σ(ω)]
– “spin self-energy” M (ω) χ(q,ω)=1/[ Iq + M(ω)]
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Extended-DMFT of Kondo Lattice
fermion bath
fluctuating magnetic field
Localmoment
Kondo Lattice
Bose-Fermi Kondo Jk
g
+ self-consistency
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ε-expansion of Bose-Fermi Kondo Model
JK
Kondo
Critical Kondo breakdown
g
ε δ −1 −∑ ωwω pp
~)(
0<ε<1:sub-ohmicdissipation
Kondo breakdown
QS, Rabello, Ingersent, Smith, Nature ’01; PRB ’03;
L. Zhu & QS, PRB ’02
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ε-expansion of Bose-Fermi Kondo Model
JK
Kondo
Critical Kondo breakdown
g
ε δ −1 −∑ ωwω pp
~)(
Critical:
Crucial for LQCP solution
0<ε<1:sub-ohmicdissipation
Kondo breakdown
QS, Rabello, Ingersent, Smith, Nature ’01; PRB ’03;
L. Zhu & QS, PRB ’02
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Role of Berry phase
is a geometrical phase and equals the area on the unit sphere enclosed by
For ½<ε<1: Retaining Berry phase yields ω/T scaling Dropping Berry phase violates ω/T scaling
S. Kirchner & QS, arXiv:0808.2647
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Dynamical Scaling of Local Quantum Critical Point
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Continuous phase transition
δ ≡ IRKKY / TK0
J.-X. Zhu, D. Grempel, and QS,Phys. Rev. Lett. (2003)
J.-X. Zhu, S. Kirchner, R. Bulla & QS, PRL 99, 227204 (2007);M. Glossop & K. Ingersent, PRL 99, 227203 (2007)
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Dynamical Scaling of Local Quantum Critical Point
α = 0.72
α = 0.83
α = 0.78
J-X Zhu, D. Grempel and QS, PRL (2003)
J-X Zhu, S. Kirchner, R. Bulla, and QS, PRL (2007)
M. Glossop & K. Ingersent, PRL (2007)
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Local Quantum Critical Point
• ω/T scaling in χ(ω,T) and G(ω,T)
•Collapse of a large Fermi surface
•Multiple energy scales
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• Introduction to quantum critical point
• Kondo problem to heavy Fermi liquid
• Heavy fermion quantum criticality
• Perspective and outlook
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Experiments in CeCu6-xAux
A. Schröder et al., Nature (’00); O. Stockert et al; M. Aronson et al.
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Experiments in YbRh2Si2
S. Friedemann, N. Oeschler, S. Wirth, C. Krellner, C. Geibel, F. Steglich, S. Paschen, S. Kirchner, and QS, PNAS 107, 14547 (2010)
S. Paschen et al, Nature (2004); P. Gegenwart et al, Science (2007)
T *
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Global Phase Diagram
S. Friedemann et al, Nat. Phys. 5, 465 (2009)
also J. Custers et al, PRL 104, 186402 (2010)
Pure and doped YbRh2Si2
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T. Park et al., Nature 440, 65 (’06);G. Knebel et al., PRB74, 020501 (’06) _
Superconductivity near Kondo-destroying AF QCP in CeRhIn5
H. Shishido, R. Settai, H. Harima, & Y. Onuki, JPSJ 74, 1103 (’05)
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T. Park et al., Nature 440, 65 (’06);G. Knebel et al., PRB74, 020501 (’06)
Quantum Criticality vs Superconductivity
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Dynamical Scaling of Local Quantum Critical Point
AdS/CMT:
N. Iqbal, H. Liu, M. Mezei and QS, PRD 82, 045002 (’10)
T. Faulkner, G. T. Horowitz and M. M. Roberts, arXiv:1008.1581
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SUMMARY
• Heavy fermions -- prototype quantum critical points
• Heavy Fermi liquid – Kondo entaglement in the ground state– quantum order without broken symmetry, supports
Kondo resonances
• Local quantum critical point: Kondo destruction at antiferromagnetic QCP