Demystifying the Strange Metal in High-temperature. Superconductors: Composite Excitations

Transcription

1 Demystifying the Strange Metal in High-temperature Superconductors: Composite Excitations Thanks to: T.-P. Choy, R. G. Leigh, S. Chakraborty, S. Hong PRL, 99, (2007); PRB, 77, (2008); ibid, 77, (2008) ); ibid, 79, (2009); RMP, 82, 1719 (2010)..., DMR/NSF

2 doped Mott insulator energy

3 doped Mott insulator energy

4 doped Mott insulator energy =hole =2e mode bound

5 doped Mott insulator energy =hole =2e mode strange metal bound pseudogap

6 Strong Coupling composite or bound states not in UV theory QCD vulcanization

7 Strong Coupling composite or bound states not in UV theory QCD vulcanization emergent (collective) low-energy physics

8 AF 0 Mott insulator ρ at x

9 AF 0 Mott insulator ρ at LaSrCuO x

10 AF 0 Mott insulator ρ at x

11 How does Fermi Liquid Theory Breakdown? $6,000,000 question? AF 0 Mott insulator ρ at x

12 knee-jerk explanation 5

13 knee-jerk explanation quantum criticality 5

14 knee-jerk explanation quantum criticality Standard story 5

15 knee-jerk explanation quantum criticality Standard story 5

16 General Result 1.) one critical length scale 2.) charge carriers are critical 3.) charge conservation

17 Vector potential current

18 Vector potential current Charge conservation: d A = 1

19 scaling

20 scaling

21 PP, CC, PRL, vol. 95, (2005) σ(t ) T (d 2)/z E p z dynamical exponent

22 PP, CC, PRL, vol. 95, (2005) σ(t ) T (d 2)/z d=3 E p z dynamical exponent only if z<0

23 PP, CC, PRL, vol. 95, (2005) σ(t ) T (d 2)/z d=3 E p z dynamical exponent only if z<0

24 One-parameter Scaling breaks down new length scale new degree of freedom

25 NFL? P MI-BG = (0,D/6) P = (0,0) P MI-SF = (1,0 FL How to break Fermi liquid theory in d=2+1?

26 Polchinski, Shankar, others p 1 p = k + l, p 2 1.) e- charge carriers 2.) Fermi surface dt d 2 k 1 dl 1 d 2 k 2 dl 2 d 2 k 3 dl 3 d 2 k 4 dl 4 V (k 1, k 2, k 3, k 4 ) ψ σ(p 1 )ψ σ (p 3 )ψ σ (p 2 )ψ σ (p 4 )δ 3 (p 1 + p 2 p 3 p 4 ). No relevant short-range 4-Fermi terms in d 2

27 Polchinski, Shankar, others p 1 p = k + l, p 2 1.) e- charge carriers 2.) Fermi surface dt d 2 k 1 dl 1 d 2 k 2 dl 2 d 2 k 3 dl 3 d 2 k 4 dl 4 V (k 1, k 2, k 3, k 4 ) ψ σ(p 1 )ψ σ (p 3 )ψ σ (p 2 )ψ σ (p 4 )δ 3 (p 1 + p 2 p 3 p 4 ). No relevant short-range 4-Fermi terms in Exception: Pairing d 2

28 So what does one add to break this correspondence? Interacting system Free system low-energy: one-to-one correspondence

29 So what does one add to break this correspondence? Free system low-energy: one-to-one correspondence Interacting system new degrees of freedom!

30 So what does one add to break this correspondence? Free system low-energy: one-to-one correspondence new degrees of freedom!

31 So what does one add to break this correspondence? UV-IR mixing Free system low-energy: one-to-one correspondence new degrees of freedom!

32 So what does one add to break this correspondence? UV-IR mixing Free system low-energy: one-to-one correspondence new degrees of freedom! dynamically generated

33 classical Mottness is forbidden

34 classical Mottness

35 classical Mottness

36 atomic limit: x holes density of states 1-x 2x 1-x PES IPES G σ (ω, k) = 1 2 (1 + x) (1 x) ω µ + U 2 ω µ U 2

37 atomic limit total weight=1+x= # of ways electrons can be added in lower band intensity of lower band=# of electrons the band can hold no problems yet!

38 quantum Mottness: U finite U t c iσ n i σ µ U c iσ (1 n i σ )

39 quantum Mottness: U finite U t c iσ n i σ µ U c iσ (1 n i σ ) double occupancy in ground state!!

40 quantum Mottness: U finite U t c iσ n i σ µ U c iσ (1 n i σ ) double occupancy in ground state!! W PES > 1 + x

41 Harris & Lange, 1967 α = t U c iσ c jσ > 0 ij 1 + x + α(t/u, x)

42 Harris & Lange, 1967 α = t U c iσ c jσ > 0 ij 1 + x + α(t/u, x) the rest of this state lives at high energy

43 Harris & Lange, 1967 α = t U c iσ c jσ > 0 ij 1 + x + α(t/u, x) the rest of this state lives at high energy Intensity>1+x New non-electron degrees of freedom

44 Harris & Lange, 1967 α = t U c iσ c jσ > 0 ij 1 + x + α(t/u, x) the rest of this state lives at high energy Intensity>1+x New non-electron degrees of freedom Can this system be a Fermi liquid?

45 what are the low-energy fermionic excitations? 2 1 x α 1 + x + α

46 what are the low-energy fermionic excitations? 2 1 x α 1 + x + α

47 what are the low-energy fermionic excitations? 2 re-definition: x = x + α 1 x α 1 + x + α

48 what are the low-energy fermionic excitations? 2 re-definition: x = x + α 1 x α 1 + x + α get rid of dynamical mixing (CPH, PRB 2010)

49 what are the low-energy fermionic excitations? 2 re-definition: x = x + α 2(x + α) 1 x α x α 1 + x + α get rid of dynamical mixing (CPH, PRB 2010)

50 what are the low-energy fermionic excitations? 2 re-definition: x = x + α 2(x + α) 1 x α x α 1 + x + α n qp < n e FLT breaks down get rid of dynamical mixing (CPH, PRB 2010)

51 2(x + α) 1 x α 1 + x + α

52 2(x + α) 1 x α new degrees of freedom 1 + x + α number of fermionic qp < number of bare electrons=> FL theory breaks down

53 2(x + α) 1 x α 1 + x + α new degrees of freedom # of addition states per e per spin>1 number of fermionic qp < number of bare electrons=> FL theory breaks down

54 2(x + α) 1 x α 1 + x + α new degrees of freedom # of addition states per e per spin>1 number of fermionic qp < number of bare electrons=> FL theory breaks down ways to add a particle but not an electron (gapped spectrum)

55 2(x + α) 1 x α 1 + x + α new degrees of freedom # of addition states per e per spin>1 number of fermionic qp < number of bare electrons=> FL theory breaks down ways to add a particle but not an electron (gapped spectrum) breakdown of electron quasi-particle picture: Mottness

56 Same Physics at half-filling No proof exists? Mottness is ill-defined

57 Same Physics at half-filling No proof exists? Mottness is ill-defined

58

59 A Critique of Two Metals R. B. Laughlin idea is either missing or improperly understood. Another indicator that something is deeply wrong is the inability of anyone to describe the elementary excitation spectrum of the Mott insulator precisely even as pure phenomenology. Nowhere can one find a quantitative band structure of the elementary particle whose spectrum becomes gapped. Nowhere can one find precise information about the particle whose gapless spectrum causes the paramagnetism. Nowhere can one find information about the interactions among these particles or of their potential bound state spectroscopies. Nowhere can one find precise definitions of Mott insulator terminology. The upper and lower Hubbard bands, for example, are vague analogues of the valence and conduction bands of a semiconductor, except that they coexist and mix with soft magnetic excitations no one knows how to describe very well.

60 Church of weak coupling A Critique of Two Metals R. B. Laughlin idea is either missing or improperly understood. Another indicator that something is deeply wrong is the inability of anyone to describe the elementary excitation spectrum of the Mott insulator precisely even as pure phenomenology. Nowhere can one find a quantitative band structure of the elementary particle whose spectrum becomes gapped. Nowhere can one find precise information about the particle whose gapless spectrum causes the paramagnetism. Nowhere can one find information about the interactions among these particles or of their potential bound state spectroscopies. Nowhere can one find precise definitions of Mott insulator terminology. The upper and lower Hubbard bands, for example, are vague analogues of the valence and conduction bands of a semiconductor, except that they coexist and mix with soft magnetic excitations no one knows how to describe very well.

61 Church of weak coupling A Critique of Two Metals R. B. Laughlin idea is either missing or improperly understood. Another indicator that something is deeply wrong is the inability of anyone to describe the elementary excitation spectrum of the Mott insulator precisely even as pure phenomenology. Nowhere can one find a quantitative band structure of the elementary particle whose spectrum becomes gapped. Nowhere can one find precise information about the particle whose gapless spectrum causes the paramagnetism. Nowhere can one find information about the interactions among these particles or of their potential bound state spectroscopies. Nowhere can one find precise definitions of Mott insulator terminology. The upper and lower Hubbard bands, for example, are vague analogues of the valence and conduction bands of a semiconductor, except that they coexist and mix with soft magnetic excitations no one knows how to describe very well. Beliefs: Mott gap is heresy? HF is the way! No UHB and LHB!

62 = 0.6eV > dimerization (Mott, 1976) σ(ω) Ω -1 cm -1 VO 2 T=360 K T=295 K transfer of spectral weight to Recall, ev = 10 4 K high energies beyond any ordering scale cm -1

63 = 0.6eV > dimerization (Mott, 1976) σ(ω) Ω -1 cm -1 VO 2 T=360 K T=295 K transfer of spectral weight to Recall, ev = 10 4 K high energies beyond any ordering scale cm -1 M. M. Qazilbash, K. S. Burch, D. Whisler, D. Shrekenhamer, B. G. Chae, H. T. Kim, and D. N. Basov PRB 74, (2006)

64 Fermi-liquid analogy L FL (ω k ) ψ k 2 Mott Problem? L MI = (ω E LHB (k)) η k 2 + (ω E UHB (k)) η k 2 N N PES IPES

65 composite excitation: bound state half-filling: Mott gap doping: SWT, pseudogap? charge 2e boson

66 How?

67 perturbative interlude: standard approach H Hubb ( ) block diagonal in `double occupancy H tj ( ) each blue fermion blocks 2 states=> 2x sum rule

68 Transform Operators = e S e S = particle states bare fermions fictive fermions that conserve `double occupancy

69 What is P 0 P 0 P 0 P 0

70 } What is P 0 P 0 hole P P t t ( ) + ( ) U U } dynamical spectral weight transfer >2x

71 } What is P 0 P 0 hole P P t t ( ) + ( ) U U } dynamical spectral weight transfer >2x

72 What is lost if we ignore the dynamical spectral weight transfer? [H, ] = 0 new conserved quantity Local SU(2) symmetry at half-filling

73 What is lost if we ignore the dynamical spectral weight transfer? [H, ] = 0 new conserved quantity Local SU(2) symmetry at half-filling Not in original model (oops)

74 Ψ = c c c c S = Ψ Ψ local SU(2) Ψ = hψ h = α β β α α 2 + β 2 = 1 S > Ψ h hψ = S

75 Operator Transform t U j (1 n i σ )a iσ + t U V σa i σ b i+ g ij n j σ a jσ + n i σ (1 n j σ )a jσ + (1 n j σ ) a jσ a iσ a jσ a iσ a i σ

76 Operator Transform t U j (1 n i σ )a iσ + t U V σa i σ b i+ g ij n j σ a jσ + n i σ (1 n j σ )a jσ + (1 n j σ ) a jσ a iσ a jσ a iσ a i σ Is there an easier approach?

77 Operator Transform t U j (1 n i σ )a iσ + t U V σa i σ b i+ g ij n j σ a jσ + n i σ (1 n j σ )a jσ + (1 n j σ ) a jσ a iσ a jσ a iσ a i σ Is there an easier approach? Collective Phenomena

78 yes

79 EffectiveTheories: S(φ) at half-filling Integrate φ=φ L +φ H Out high Energy fields Low-energy theory of M I

80 Half-filling U N(ω) ε Integrate out both

81 "I'm not into this detail stuff. I'm more concepty."

82 "I'm not into this detail stuff. I'm more concepty." New Concept

83 Key idea: similar to Bohm/Pines Extend the Hilbert space: Associate with U-scale new Fermionic oscillators N(ω) U D U 2 i D U i 2 D i D i D i D i ε

84 D i Fermionic constraint one per site (fermionic) transforms as a boson

85 D i Fermionic constraint one per site (fermionic) transforms as a boson

86 D i Fermionic constraint one per site (fermionic) transforms as a boson `supersymmetry

87 D i Fermionic constraint one per site (fermionic) transforms as a boson `supersymmetry δ(d i θc i c i ) Grassmann

88 D i Fermionic constraint one per site (fermionic) transforms as a boson `supersymmetry δ(d i θc i c i ) Grassmann θϕ constraint field

89 Dual Theory solve constraint ϕ (Q ϕ = 2e) integrate over heavy fields UV limit d 2 θ θθl Hubb = i,σ c i,σċi,σ + H Hubb, Exact low-energy theory (IR limit)

90 Dual Theory solve constraint ϕ (Q ϕ = 2e) integrate over heavy fields UV limit d 2 θ θθl Hubb = i,σ c i,σċi,σ + H Hubb, Exact low-energy theory (IR limit) L hf UV = d 2 θ id Ḋ i D D U 2 (D D D D ) + t 2 D θb + t 2 θb D + h.c. + s θϕ (D θc c ) + s θ ϕ ( D θc c ) + h.c.

91 dynamics of ϕ

92 Exact low-energy Lagrangian L = #L bare (electrons) + #L bare (bosons) +f(ω)l int (c, ϕ) + f(ω)l int (c, ϕ) Ψ Ψ Ψ Ψ quadratic form: composite or bound excitations of ϕ c iσ

93 Exact low-energy Lagrangian L = #L bare (electrons) + #L bare (bosons) f(ω) = 0 +f(ω)l int (c, ϕ) + f(ω)l int (c, ϕ) dispersion Ψ Ψ quadratic form: Ψ Ψ of propagating light modes composite or bound excitations of ϕ c iσ

94 composite excitations determine spectral density γ ( k) p γ ( k) p U tε(k) p (ω) = U U + tε(k) p (ω) = U 2ω 1 + 2ω/U + 2ω 1 2ω/U. = U 4dt γ = 0 (0, 0) γ = 0 (π, π) each momentum has SD at two distinct energies

95 H(ϕ, ϕ, c, c ) 1.) no derivative couplings with respect to bosonic field 2.) no bare dynamics consistent with exact argument bosonic field can be treated as spatially uniform A(k,ω) bosonic field contains all the Mott physics (numerically verified that spatial variation of bosonic field does nothing) spin part irrelevant

96 hole-doping?

97 Extend the Hilbert space: Associate with U-scale a new Fermionic oscillator N(ω) U D_i ε UD i D i

98 charge 2e boson field (non-propagating) Non-projective

99 charge 2e boson field (non-propagating) Non-projective ϕ ι is an emergent degree of freedom Not made out of the elemental fields

100 Spectral function Bosonic field ~ independent of space

101

102 Electron spectral function t 2 /U~60meV

103 Electron spectral function t 2 /U~60meV

104 Electron spectral function $!"*#!,$!+!! -!!"*!!")#!") t 2 /U~60meV!"(#!"(!"'#!!"!#!"$!"$#!"%!"%#!"&!"&# +

105 Electron spectral function $!"*#!,$!+!! -!!"*!!")#!") t 2 /U~60meV!"(#!"(!"'#!!"!#!"$!"$#!"%!"%#!"&!"&# + Conserved charge: Q = i c i c i + 2 i ϕ i ϕ i

106 Graf, et al. PRL vol. 98, (2007). Spin-charge Two bands!! separation?

107 What is the electron operator?

108 What is the electron operator? UV

109 What is the electron operator? UV IR + σ s U ϕ i c i σ

110 What is the electron operator? UV IR + σ s U ϕ i c i σ P 0 P 0 =

111 } What is the electron operator? UV IR + σ s U ϕ i c i σ hole P 0 P 0 = t P 0 P 0 ( ) U t + + ( ) U

112 } What is the electron operator? UV IR + σ s U ϕ i c i σ hole P 0 P 0 = t P 0 P 0 ( ) U t + + ( ) U s=t

113 Origin of two bands Two charge e excitations c iσ ϕ i c i σ ϕ i is confined (no kinetic energy) New bound state Pseudogap

114 two types of charges free bound

115 direct evidence

116 direct evidence charge carrier density:

117 direct evidence charge carrier density: 100 La 2-x Sr x CuO 4 Ono, et al., Phys. Rev. B 75, (2007) 1/n eff 10 x= (a) Temperature (K) n Hall (x, T ) = n 0 (x) + n 1 (x) exp( (x)/t ), PRL, vol. 97, (2006).

118 direct evidence charge carrier density: 100 La 2-x Sr x CuO 4 Ono, et al., Phys. Rev. B 75, (2007) 1/n eff 10 x= (a) Temperature (K) n Hall (x, T ) = n 0 (x) + n 1 (x) exp( (x)/t ), PRL, vol. 97, (2006). exponentially suppressed: confinement

119 Our Theory x = 0.05 x = 0.10 x = 0.15 x = 0.2 n Hall (T) n 0 (x) n 0 = x x T!1 (ev!1 ) exponential T-dependence

120 Our Theory gap x = 0.05 x = 0.10 x = 0.15 x = Theory LSCO LSCO n Hall (T) n 0 (x) n 0 = x x!(x) (ev) T!1 (ev!1 ) exponential T-dependence x no model-dependent free parameters: just t/u

121 strange metal: breakup (deconfinement) of bound states T * LSCO T m LSCO T m LSCO Theory T * (K) x QCP:

122 T-linear resistivity New scenario as x increases E B µ E

123 T-linear resistivity New scenario as x increases E B µ E

124 T-linear resistivity New scenario as x increases E B µ E

125 Experiments

126 µ L =Area/# of sites Λ µ N(ω)dω L=# of single-particle addition states (qp) n h = # of ways electrons can be added= # of holes created by the dopants

127 µ L =Area/# of sites Λ µ N(ω)dω L=# of single-particle addition states (qp) n h = # of ways electrons can be added= # of holes created by the dopants Equivalent in Fermi liquid

128 Experimental Prediction: L/n h > 1 L/n h = 1 More states at lower temper ature E B = 0 E B > 0 ρ T

129 J. Y. Lin (arxiv: ) More addition states in PG: new charge e states

130 J. Y. Lin (arxiv: ) More addition states in PG: new charge e states

131 J. Y. Lin (arxiv: ) More addition states in PG: new charge e states

132 J. Y. Lin (arxiv: ) More addition states in PG: new charge e states

133 J. Y. Lin (arxiv: ) J. Lee, P. Abbamonte (here) More addition states in PG: new charge e states

134 J. Y. Lin (arxiv: ) J. Lee, P. Abbamonte (here) More addition states in PG: new charge e states Probe by NMR

135

136 Predictions: QO oscillations: V hp V ep = 2(x + α) x x + α Superfluid density: optical conductivity: ρ s x + α n eff (x + α) > x Thanks to R. G. Leigh and Ting- Pong Choy, Shiladitya Chakraborty, Seungmin Hong and DMR-NSF

137 Transformed electron operator 2x +terms in m=1 sector >2x

138 Transformed electron operator 2x +terms in m=1 sector >2x

139 Transformed electron operator 2x +terms in m=1 sector Projection >2x

140 What is the meaning of? organizes degrees of freedom at low energy: creates new charge e states

141 What is the meaning of? Double occupancy bound to a hole organizes degrees of freedom at low energy: creates new charge e states