Increasing localization

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Bennett Maxwell
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2 4f Rare Earth Increasing localization 5f Actinide Localized Moment 3d Transition metal

3 Outline: 1. Chemistry of Kondo. 2. Key Properties. 3. Tour of Heavy Fermions. 4. Anderson model: the atomic limit. 5. Adiabaticity: the heavy Fermi liquid. 6. Scaling: the Kondo effect. 7. Doniach s Kondo Lattice Concept.

4 Moment Formation Fe x<0.4

5 Moment Formation Fe x<0.4 Clogston, Mathias et al, 1962

6 Moment Formation Fe x<0.4 x>0.4 Clogston, Mathias et al, 1962

7 Moment Formation Fe Curie Behavior x<0.4 x>0.4 Clogston, Mathias et al, 1962

8 Moment Formation Fe RESISTIVITY ρ/ρ4.2k Curie Behavior x<0.4 x>0.4 Clogston, Mathias et al, 1962

9 Moment Formation Fe RESISTIVITY ρ/ρ4.2k Curie Behavior x<0.4 x>0.4 Clogston, Mathias et al, 1962 How?

10 Anderson Model E f +U E f 4

11 Anderson Model E f +U H = E f n f + Un f n f }{{} H atomic E f 4

12 Anderson Model E f +U H = E f n f + Un f n f }{{} H atomic E f 4

13 Anderson Model H = H resonance {}}{ ɛ k n kσ + ] V (k) [c kσ f σ + f σc kσ k,σ k,σ E f +U + E f n f + Un f n f }{{} H atomic E f 4

14 Anderson Model V (k) = k V atomic f =4πi l 0 r 2 drj l (kr)v (r)r f (r) H = H resonance {}}{ ɛ k n kσ + ] V (k) [c kσ f σ + f σc kσ k,σ k,σ E f +U + E f n f + Un f n f }{{} H atomic E f 4

15 Anderson Model V (k) = k V atomic f =4πi l 0 r 2 drj l (kr)v (r)r f (r) H = H resonance {}}{ ɛ k n kσ + ] V (k) [c kσ f σ + f σc kσ k,σ k,σ E f +U + E f n f + Un f n f }{{} H atomic E f U = d 3 xd 3 x V (x x ) ψ f (x) 2 ψ f (x ) 2 4

16 5

17 Atomic approach : Start with V(k)=0, then dial up the hybridization 5

18 Atomic approach : Start with V(k)=0, then dial up the hybridization Local moment, states at Ef and Ef+U 5

19 Atomic approach : Start with V(k)=0, then dial up the hybridization Local moment, states at Ef and Ef+U Adiabatic approach: Start with U=0, then dial up the interaction. 5

20 Atomic approach : Start with V(k)=0, then dial up the hybridization Local moment, states at Ef and Ef+U Adiabatic approach: Start with U=0, then dial up the interaction. Friedel-Abrikosov-Suhl Resonance. 5

21 Atomic approach : Start with V(k)=0, then dial up the hybridization Local moment, states at Ef and Ef+U Adiabatic approach: Start with U=0, then dial up the interaction. Friedel-Abrikosov-Suhl Resonance. How to reconcile the two approaches? 5

22 Anderson Model of Moment Formation

23 Anderson Model of Moment Formation

24 Anderson Model of Moment Formation

25 Anderson Model of Moment Formation Local Moment Forms

26 A f (ω) = 1 π ImG f (ω iδ) Allen et al (1983)

27 A f (ω) = 1 π ImG f (ω iδ) G f (ω) = i dt Tf σ (t)f σ(0) e iωt Allen et al (1983)

28 E f A f (ω) = 1 π ImG f (ω iδ) G f (ω) = i dt Tf σ (t)f σ(0) e iωt Allen et al (1983)

29 E f A f (ω) = 1 π ImG f (ω iδ) E f +U G f (ω) = i dt Tf σ (t)f σ(0) e iωt Allen et al (1983)

30 E f A f (ω) = 1 π ImG f (ω iδ) E f +U G f (ω) = i dt Tf σ (t)f σ(0) e iωt A f (ω) = to { }}{ λ f σ φ 0 2 δ(ω [E λ E 0 ]), (ω > 0) Energy distribution of state formed by adding one f-electron. λ λ λ f σ φ 0 2 δ(ω [E 0 E λ ]), }{{} Energy distribution of state formed by removing an f-electron (ω < 0) Allen et al (1983)

31 Adiabatic approach & Kondo Resonance 8

32 Adiabatic approach & Kondo Resonance Adiabatic invariant (Langreth 1966) 8

33 Adiabatic approach & Kondo Resonance Adiabatic invariant (Langreth 1966) A f (ω = 0) = sin2 δ f π 8

34 Adiabatic approach & Kondo Resonance Adiabatic invariant (Langreth 1966) A f (ω = 0) = sin2 δ f π σ δ fσ π =2δ π = n f 8

35 Hierachies of Energy scales: Renormalization concept. (Anderson, Wilson,...) Λ H(Λ) 1

36 Hierachies of Energy scales: Renormalization concept. (Anderson, Wilson,...) Λ Λ = Λ/b H(Λ) 1

37 Hierachies of Energy scales: Renormalization concept. (Anderson, Wilson,...) Λ Λ = Λ/b H(Λ) = [ HL V V H H ], H(Λ) 1

38 Hierachies of Energy scales: Renormalization concept. (Anderson, Wilson,...) H H Λ Λ = Λ/b H(Λ) = [ HL V V H H ], 1000 V 100 V 10 H L 1

39 Hierachies of Energy scales: Renormalization concept. (Anderson, Wilson,...) H H Λ Λ = Λ/b H(Λ) = [ HL V V H H ], H(Λ) UH(Λ)U = [ HL 0 0 H H ] 10 H L 1

40 Hierachies of Energy scales: Renormalization concept. (Anderson, Wilson,...) H H Λ Λ = Λ/b H(Λ) = [ HL V V H H ], [ HL H(Λ) UH(Λ)U = 0 H L = H L + δh = H(Λ ) 0 H H ] 10 H(Λ ) 1

41 Hierachies of Energy scales: Renormalization concept. (Anderson, Wilson,...) H H Λ Λ = Λ/b H(Λ) = [ HL V V H H ], [ HL H(Λ) UH(Λ)U = 0 H L = H L + δh = H(Λ ) 0 H H ] 10 1 H(Λ ) δh = P L V ( 1 E H H ) VP L E EL

43 Outline: 1. Chemistry of Kondo. 2. Key Properties. 3. Tour of Heavy Fermions. 4. Anderson model: the atomic limit. 5. Adiabaticity: the heavy Fermi liquid. 6. Scaling: the Kondo effect. 7. Doniach s Kondo Lattice Concept.

44 Schrieffer-Wolff Transformation. 4f 2 4f 0 Ef+U -Ef 4f 1 12

45 Schrieffer-Wolff Transformation. 4f 2 4f 0 Ef+U -Ef 4f 1 H int = kσ,k σ V f 1 +e f [{}} 2 { (c kσ k V f σ)(f σ c k σ ) k E f + U + f 1 f 0 +e {}}{] (f σ c k σ )(c kσ f σ) E f 12

46 Schrieffer-Wolff Transformation. 4f 2 4f 0 Ef+U -Ef 4f 1 δ ab δ cd + σ ab σ cd =2δ ad δ bc H int = kσ,k σ V f 1 +e f [{}} 2 { (c kσ k V f σ)(f σ c k σ ) k E f + U + f 1 f 0 +e {}}{] (f σ c k σ )(c kσ f σ) E f 12

47 Schrieffer-Wolff Transformation. 4f 2 4f 0 Ef+U -Ef δ ab δ cd + σ ab σ cd =2δ ad δ bc 4f 1 H int = Vk V k kσ,k σ f 1 +e d 2 [{}}{ (c kσ f σ)(f σ c k σ ) E f + U f 1 f 0 +e {}}{] (f σ c + k σ )(c kσ f σ) E f 13

48 Schrieffer-Wolff Transformation. 4f 2 4f 0 Ef+U -Ef δ ab δ cd + σ ab σ cd =2δ ad δ bc 4f 1 H int = H int = kα,k β Vk V k kσ,k σ f 1 +e d 2 [{}}{ (c kσ f σ)(f σ c k σ ) E f + U J k,k c kα σ c k β S f + H f 1 f 0 +e {}}{] (f σ c + k σ )(c kσ f σ) E f 13

49 Schrieffer-Wolff Transformation. 4f 2 4f 0 Ef+U -Ef δ ab δ cd + σ ab σ cd =2δ ad δ bc 4f 1 H int = J k,k H int = kα,k β = V k V k [ Vk V k kσ,k σ f 1 +e f 2 {}}{ f 1 +e d 2 [{}}{ (c kσ f σ)(f σ c k σ ) E f + U J k,k c kα σ c k β S f + H 1 E f + U + f 1 f 0 +e {}}{ ] 1 E f f 1 f 0 +e {}}{] (f σ c + k σ )(c kσ f σ) E f 13

50 Mathias and Clogston (62) Sarachik et al, (1964) White and Geballe, (1979)

Local moment physics in heavy electron systems

Local moment physics in heavy electron systems P. Coleman Center for Materials Theory, Department of Physics and Astronomy, Rutgers University, 136 Frelinghausen Road, Piscataway, NJ 08855, USA Abstract.