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Kondo Problem to Heavy Fermions and Local Quantum Criticality - - PowerPoint PPT Presentation

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Kondo Problem to Heavy Fermions and Local Quantum Criticality Qimiao Si Rice University Advanced School Developments and Prospects in Quantum Impurity Physics, MPI-PKS, Dresden, May 30, 2011 Introduction to quantum critical point

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SLIDE 1

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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SLIDE 2
  • 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 “Understanding

Quantum Phase Transitions”, ed. L. D. Carr (2010).

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SLIDE 3
  • S. Friedemann
  • C. Krellner
  • Y. Tokiwa
  • P. Gegenwart
  • S. Paschen
  • S. Wirth
  • N. Oeschler
  • T. Westerkamp
  • R. Küchler
  • T. Lühmann
  • T. Cichorek
  • K. Neumaier
  • O. Tegus
  • O. Trovarelli
  • C. Geibel
  • F. 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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SLIDE 4

Phases and Phase Transitions

Disorder (T>Torder) Order (T<Torder)

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SLIDE 5

Continuous Phase Transitions: Criticality

Disorder (T>Torder) Order (T<Torder) Criticality -- fluctuations of order parameter in d dimensions

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SLIDE 6
  • A: every spin (spontaneously) points up

Order parameter:

  • B: every microstate equally probable:

m=0

σ σ I

  • H

z j ij z i

∑ =

> <

  • rdered

state temperature T T=0

A B

1 N / M lim lim m

site h

site

= =

∞ → → + N

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SLIDE 7
  • A: every spin (spontaneously) points up

Order parameter:

  • B: every microstate equally probable:

m=0

σ σ I

  • H

z j ij z i

∑ =

> <

  • rdered

state control parameter δ temperature T T=0

A B C

∑

i x i

) (I

  • σ

δ

1 N / M lim lim m

site h

site

= =

∞ → → + N

  • C: every spin points along the transverse field:

m=0

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SLIDE 8

Quantum Phase Transition

  • A: every spin (spontaneously) points up

Order parameter:

  • B: every microstate equally probable:

m=0

σ σ I

  • H

z j ij z i

∑ =

> < QCP quantum critical

  • rdered

state control parameter δ temperature T T=0

A B C

∑

i x i

) (I

  • σ

δ

1 N / M lim lim m

site h

site

= =

∞ → → + N

  • C: every spin points along the transverse field:

m=0

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SLIDE 9
  • 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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SLIDE 10

YbRh2Si2

  • H. v. Löhneysen et al
  • J. Custers et al

CePd2Si2

  • N. Mathur et al

TN Linear resistivity

Heavy fermion metals as prototype quantum critical points

CeCu6-xAux

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SLIDE 11

Torder

QCP

temperature δ -- control parameter

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SLIDE 12

Following Landau -- fluctuations of order parameter, , but in d+z dimensions

Torder

QCP

temperature δ -- control parameter

 ∞ @QCP

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SLIDE 13

scaling T no ω

Gaussian z d deff , 4 > + =

T=0 spin-density-wave transition

exponent MF

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SLIDE 14

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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SLIDE 15

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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SLIDE 16
  • Introduction to quantum critical point
  • Kondo problem to heavy Fermi liquid
  • Heavy fermion quantum criticality
  • Perspective and outlook
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SLIDE 17

Single-impurity Kondo Model:

fermion bath Local moment

S: spin-1/2

moment at site 0

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SLIDE 18

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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SLIDE 19

Single impurity Kondo model

  • Kondo temperature:
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SLIDE 20

Single impurity Kondo model

  • Kondo entanglement: singlet ground state
  • Kondo temperature:
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SLIDE 21

Single impurity Kondo model

  • Kondo effect (emergence of Kondo resonance):

– Kondo-singlet ground state yields an electronic resonance – local moment acquires electron quantum number due to Kondo entanglement

  • Kondo entanglement: singlet ground state
  • Kondo temperature:
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SLIDE 22

Kondo lattices:

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SLIDE 23

Kondo lattices:

heavy Fermi liquid:

  • Kondo singlet
  • Kondo resonance
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SLIDE 24

JK >>W>>I

  • xNsite tightly bound local singlets

(cf. If x were =1, Kondo insulator)

  • (1-x)Nsite lone moments:
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SLIDE 25

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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SLIDE 26

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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SLIDE 27
  • 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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SLIDE 28
  • Kondo resonance

…

  • …

heavy electron bands

Heavy Fermi Liquid (Kondo Lattice)

) , (

  • 1

ω ε ω ω k k G

k c

Σ = ) , (

pole in Σ

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SLIDE 29
  • Kondo resonance

…

  • …

heavy electron bands

Heavy Fermi Liquid (Kondo Lattice)

) , (

  • 1

ω ε ω ω k k G

k c

Σ = ) , (

pole in Σ k-independent

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SLIDE 30
  • Cond. electron band

Heavy Fermi Liquid Heavy electron bands E1,2(k) Kondo resonance

) (k ε

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SLIDE 31
  • Cond. electron band

Heavy Fermi Liquid Heavy electron bands E1,2(k) Kondo resonance

) (k ε

Large Fermi surface

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SLIDE 32
  • Kondo lattices:

heavy Fermi liquid:

  • Kondo singlet
  • Kondo resonance

No symmetry breaking, but macroscopic order

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SLIDE 33

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 “Understanding

Quantum Phase Transitions”, ed. L. D. Carr (2010).

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SLIDE 34
  • Introduction to quantum critical point
  • Kondo problem to heavy Fermi liquid
  • Heavy fermion quantum criticality
  • Perspective and outlook
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SLIDE 35

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 “Understanding

Quantum Phase Transitions”, ed. L. D. Carr (2010).

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SLIDE 36

Kondo lattices: δ= TK

0/I

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SLIDE 37
  • 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/TK

0, 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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SLIDE 38

T=0 spin-density-wave transition

smooth q-independence

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SLIDE 39

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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SLIDE 40

Extended-DMFT of Kondo Lattice

fermion bath fluctuating magnetic field Local moment

Kondo Lattice Bose-Fermi Kondo

Jk g

+ self-consistency

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SLIDE 41

ε-expansion of Bose-Fermi Kondo Model

JK

Kondo Critical Kondo breakdown

g

ε

δ

− 1

− ∑ ω w ω

p p

~ ) (

0<ε<1: sub-ohmic dissipation Kondo breakdown

QS, Rabello, Ingersent, Smith, Nature ’01; PRB ’03;

  • L. Zhu & QS, PRB ’02
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SLIDE 42

ε-expansion of Bose-Fermi Kondo Model

JK

Kondo Critical Kondo breakdown

g

ε

δ

− 1

− ∑ ω w ω

p p

~ ) (

Critical: Crucial for LQCP solution 0<ε<1: sub-ohmic dissipation Kondo breakdown

QS, Rabello, Ingersent, Smith, Nature ’01; PRB ’03;

  • L. Zhu & QS, PRB ’02
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SLIDE 43

Role of Berry phase

is a geometrical phase and equals the area

  • n 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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SLIDE 44

Dynamical Scaling

  • f Local Quantum Critical Point
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SLIDE 45

Continuous phase transition

δ ≡ IRKKY / TK

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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SLIDE 46

Dynamical Scaling

  • f 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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SLIDE 47

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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SLIDE 48
  • Introduction to quantum critical point
  • Kondo problem to heavy Fermi liquid
  • Heavy fermion quantum criticality
  • Perspective and outlook
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SLIDE 49

Experiments in CeCu6-xAux

  • A. Schröder et al., Nature (’00);
  • O. Stockert et al; M. Aronson et al.
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SLIDE 50

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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SLIDE 51

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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SLIDE 52
  • 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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SLIDE 53
  • T. Park et al., Nature 440, 65 (’06);
  • G. Knebel et al., PRB74, 020501 (’06)

Quantum Criticality vs Superconductivity

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SLIDE 54

Dynamical Scaling

  • f 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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SLIDE 55

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