Holography and Mottness: a Discrete Marriage

Transcription

1 Holography and Mottness: a Discrete Marriage Thanks to: NSF, EFRC (DOE) Ka Wai Lo M. Edalati R. G. Leigh

2 Mott Problem

3 emergent gravity Mott Problem

4 What interacting problems can we solve in quantum mechanics?

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10 #of solvable interacting QM problems #of interacting QM problems =0

11 #of solvable interacting QM problems #of interacting QM problems =0

12 interacting systems H = T + V

13 easy interacting systems H = T + V

14 interacting systems H = T + V easy impossible

15 interacting systems H = T + V r s = V /T easy impossible r s < 1

16 interacting systems H = T + V r s = V /T easy impossible r s < 1 `free systems (perturbative interactions) metals,fermi liquids,...

17 interacting systems H = T + V r s = V /T easy impossible r s < 1 `free systems (perturbative interactions) metals,fermi liquids,... p 1 p 2 short-range repulsions are irrelevant

18 interacting systems H = T + V r s = V /T easy r s < 1 impossible r s 1 `free systems (perturbative interactions) metals,fermi liquids,... p 1 p 2 short-range repulsions are irrelevant strongly coupled systems (non-perturbative) QCD, high-temperature superconductivity,...

19 UV why is strong coupling hard?

20 why is strong coupling hard? UV IR

21 why is strong coupling hard? UV IR emergent low-energy physics (new degrees of freedom not in UV)

22 why is strong coupling hard? UV IR emergent low-energy physics (new degrees of freedom not in UV)

23 why is strong coupling hard? UV IR emergent low-energy physics (new degrees of freedom not in UV)

24 why is strong coupling hard? UV IR emergent low-energy physics (new degrees of freedom not in UV)

25 why is strong coupling hard? UV? IR emergent low-energy physics (new degrees of freedom not in UV)

26 What computational tools do we have for strongly correlated electron matter?

27

28 Is strong coupling hard because we are in the wrong dimension?

29 Yes

30 L = T gϕ 4 UV QF T

31 L = T gϕ 4 UV IR?? QF T

32 L = T gϕ 4 UV IR?? QF T Wilsonian program (fermions: new degrees of freedom)

33 L = T gϕ 4 UV IR?? QF T Wilsonian program (fermions: new degrees of freedom)

34 L = T gϕ 4 UV coupling constant g =1/ego IR?? QF T Wilsonian program (fermions: new degrees of freedom)

35 L = T gϕ 4 UV coupling constant g =1/ego IR?? QF T dg(e) dlne = β(g(e)) locality in energy Wilsonian program (fermions: new degrees of freedom)

36 L = T gϕ 4 implement E-scaling with an extra dimension UV coupling constant g =1/ego IR?? QF T dg(e) dlne = β(g(e)) locality in energy Wilsonian program (fermions: new degrees of freedom)

37 gauge-gravity duality (Maldacena, 1997) L = T gϕ 4 implement E-scaling with an extra dimension IR gravity IR?? UV QF T coupling constant g =1/ego dg(e) dlne = β(g(e)) locality in energy Wilsonian program (fermions: new degrees of freedom)

38 what s the geometry?

39 what s the geometry? dg(e) dlne = β(g(e)) =0 scale invariance (continuous)

40 what s the geometry? dg(e) dlne = β(g(e)) E λe x µ x µ /λ =0 scale invariance (continuous)

41 what s the geometry? dg(e) dlne = β(g(e)) E λe x µ x µ /λ =0 scale invariance (continuous) solve Einstein equations

42 what s the geometry? dg(e) dlne = β(g(e)) E λe x µ x µ /λ =0 scale invariance (continuous) solve Einstein equations ds 2 = u L 2 ηµν dx µ dx ν + L E 2 de 2 anti de-sitter space

43 Holography Λ 1 Λ 6 Λ 5 Λ 4 Λ 3 Λ 2 ds 2 = dt 2 + dx 2

44 Holography

45 Holography r 0 ds 2 = L2 r 2 dt 2 + dx 2 + dr 2 AdS d+1 r = L 2 /E

46 Holography r 0 ds 2 = L2 r 2 dt 2 + dx 2 + dr 2 AdS d+1 β(g) is local geometrize RG flow r = L 2 /E symmetry: {t, x, r} {λt, λx, λr}

47 Holography QFT in d-dimensions r 0 weakly-coupled classical gravity in d+1 ds 2 = L2 r 2 dt 2 + dx 2 + dr 2 AdS d+1 β(g) is local geometrize RG flow r = L 2 /E symmetry: {t, x, r} {λt, λx, λr}

48 dual construction UV IR QF T

49 dual construction operators O UV IR QF T

50 dual construction fields φ operators O UV IR QF T

51 dual construction fields φ duality operators O UV IR QF T

52 dual construction fields φ duality UV operators O e d d xφ O O IR QF T

53 dual construction fields φ duality UV operators O e d d xφ O O IR QF T Claim: Z QFT = e Son shell ADS (φ(φ ADS=JO ))

54 How does it work? O UV some fermionic system IR QF T

55 How does it work? fields ψ duality O some fermionic UV system IR QF T

56 How does it work? Dirac equation fields ψ duality O some fermionic UV system IR QF T

57 How does it work? Dirac equation fields ψ duality O some fermionic UV system IR QF T A t µ

58 How does it work? Dirac equation fields ψ duality O some fermionic UV system IR QF T A t µ ψ ar m + br m source response

59 How does it work? Dirac equation fields ψ duality O some fermionic UV system IR QF T A t µ ψ ar m + br m source response Green function: G O = b a =f(uv,ir)

60 Why is gravity holographic?

61 Why is gravity holographic?

62 Why is gravity holographic?

63 Why is gravity holographic?

64 Why is gravity holographic? Entropy scales with the area not the volume: gravity is naturally holographic

65 black hole gravitons QFT

66 Z QFT = e Son shell ADS (φ(φ ADS=JO )) black hole gravitons QFT

67 Z QFT = e Son shell ADS (φ(φ ADS=JO )) black hole gravitons QFT reality

68 What holography does for you?

69 Landau-Wilson What holography does for you? Hamiltonian ξ t ξ z long-wavelengths RG equations

70 What holography does for you? Landau-Wilson holography Hamiltonian ξ t ξ z long-wavelengths RG equations

71 What holography does for you? Landau-Wilson Hamiltonian holography RG=GR ξ t ξ z long-wavelengths RG equations

72 What holography does for you? Landau-Wilson Hamiltonian ξ t ξ z holography RG=GR strong-coupling is easy long-wavelengths RG equations

73 What holography does for you? Landau-Wilson Hamiltonian ξ t ξ z long-wavelengths holography RG=GR strong-coupling is easy microscopic UV model not easy (need M-theory) RG equations

74 What holography does for you? Landau-Wilson Hamiltonian ξ t ξ z long-wavelengths RG equations holography RG=GR strong-coupling is easy microscopic UV model not easy (need M-theory) so what (currents, symmetries)

75 Can holography solve the Mott problem?

76 NiO insulates d 8? What is a Mott Insulator?

77 NiO insulates d 8? What is a Mott Insulator? EMPTY STATES= METAL

78 NiO insulates d 8? What is a Mott Insulator? EMPTY STATES= METAL band theory fails!

79 NiO insulates d 8? perhaps this costs energy What is a Mott Insulator? EMPTY STATES= METAL band theory fails!

80 Mott Problem: NiO (Band theory failure) (N rooms N occupants) t U t

81 Half-filled band Free electrons

82 Half-filled band gap with no symmetry breaking!! U t µ U charge gap ε Free electrons

83 Mott Insulator-Ordering

84 Mott Insulator-Ordering =Mottness

85 Slater Mott Insulator-Ordering =Mottness

86 its all about order. No Mottness Slater Mott Insulator-Ordering =Mottness

87 its all about order. No Mottness Slater Mott Insulator-Ordering =Mottness of course it does!! order is secondary Anderson

88 its all about order. No Mottness Slater Mott Insulator-Ordering =Mottness of course it does!! order is secondary Anderson

89 its all about order. No Mottness Slater Mott Insulator-Ordering =Mottness of course it does!! order is secondary Anderson

90 Why is the Mott problem important?

91 Why is the Mott problem important?

92 Why is the Mott problem important? U/t = 10 1 interactions dominate: Strong Coupling Physics

93 Experimental facts: Mottness

94 Experimental facts: Mottness σ(ω) Ω -1 cm -1 VO 2 T=360 K T=295 K 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)

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

96 Experimental facts: Mottness = 0.6eV > dimerization (Mott, 1976) T crit 20 σ(ω) Ω -1 cm -1 VO 2 T=360 K T=295 K transfer of spectral weight to high energies beyond any ordering scale cm -1 Recall, ev = 10 4 K M. M. Qazilbash, K. S. Burch, D. Whisler, D. Shrekenhamer, B. G. Chae, H. T. Kim, and D. N. Basov PRB 74, (2006)

97 Experimental facts: Mottness = 0.6eV > dimerization (Mott, 1976) T crit 20 σ(ω) Ω -1 cm -1 VO 2 T=360 K T=295 K transfer of spectral weight to high energies beyond any ordering scale cm -1 Recall, ev = 10 4 K M. M. Qazilbash, K. S. Burch, D. Whisler, D. Shrekenhamer, B. G. Chae, H. T. Kim, and D. N. Basov PRB 74, (2006)

98 What gravitational theory gives rise to a gap in ImG without spontaneous symmetry breaking?

99 What gravitational theory gives rise to a gap in ImG without spontaneous symmetry breaking? dynamically generated gap: Mott gap (for probe fermions)

100 gi ψ(d m)ψ What has been done? AdS-RN MIT, Leiden group

101 gi ψ(d m)ψ What has been done? AdS-RN MIT, Leiden group

102 gi ψ(d m)ψ AdS-RN MIT, Leiden group What has been done? Fermi peak G(ω, k) = Z v F (k k F ) ω Σ

103 What has been done? gi ψ(d m)ψ AdS-RN MIT, Leiden group Fermi peak G(ω, k) = Z v F (k k F ) ω Σ marginal Fermi liquid

104 What has been done? gi ψ(d m)ψ AdS-RN MIT, Leiden group Fermi peak G(ω, k) = Z v F (k k F ) ω Σ marginal Fermi liquid

105 ?? Mott Insulator

106 ?? Mott Insulator decoherence

107 S probe (ψ, ψ) = d d x gi ψ(γ M D M m + )ψ

108 S probe (ψ, ψ) = d d x gi ψ(γ M D M m + )ψ what is hidden here?

109 S probe (ψ, ψ) = d d x gi ψ(γ M D M m + )ψ what is hidden here? F µν Γ µν consider gi ψ(d m ipf )ψ

110 S probe (ψ, ψ) = d d x gi ψ(γ M D M m + )ψ what is hidden here? F µν Γ µν consider gi ψ(d m ipf )ψ QED anomalous magnetic moment of an electron (Schwinger 1949)

111 P=0 How is the spectrum modified?

112 How is the spectrum modified? P=0 Fermi surface peak

113 How is the spectrum modified? P=0 P>4.2 Fermi surface peak

114 How is the spectrum modified? P=0 Fermi surface peak P > 4.2

115 How is the spectrum modified? P=0 Fermi surface peak P > 4.2 dynamically generated gap:

116 How is the spectrum modified? P > 4.2 P=0 spectral weight transfer Fermi surface peak dynamically generated gap:

117 How is the spectrum modified? P > 4.2 P=0 spectral weight transfer Fermi surface peak dynamically generated gap: confirmed by Gubser, Gauntlett, 2011

118 Gauntlett, 2011

119 Mechanism? UV IR QF T

120 emergent spacetime symmetry Mechanism? UV IR QF T AdS 2

121 emergent spacetime symmetry Mechanism? ψ O ± UV IR QF T AdS 2

122 emergent spacetime symmetry Mechanism? operators O ψ O ± UV IR QF T AdS 2

123 emergent spacetime symmetry Mechanism? operators O ψ O ± UV IR QF T AdS 2 holography within holography

124 emergent spacetime symmetry ψ IR O ± Mechanism? UV QF T operators O where is k_f? AdS 2 holography within holography

125 emergent spacetime symmetry ψ IR O ± Mechanism? UV QF T operators O where is k_f? AdS 2 holography within holography k_f moves into log-oscillatory region: IR O ± acquires a complex dimension

126 What does a complex scaling dimension mean?

127 continuous scale invariance O = µ(λ)o(λr)

128 continuous scale invariance O = µ(λ)o(λr) 1=µ(λ)λ = ln µ ln λ

129 continuous scale invariance O = µ(λ)o(λr) 1=µ(λ)λ = ln µ ln λ is real, independent of scale

130 continuous scale invariance O = µ(λ)o(λr) 1=µ(λ)λ = ln µ ln λ is real, independent of scale what about complex

131 1 = µλ

132 e2πin = 1 = µλ

133 Discrete scale invariance (DSI) e 2πin = 1=µλ n=0:csi = ln µ ln λ + 2πin ln λ

134 Discrete scale invariance (DSI) e 2πin = 1=µλ n=0:csi = ln µ ln λ + 2πin ln λ scaling dimension depends on scale λ n = λ n

135 Discrete scale invariance (DSI) e 2πin = 1=µλ n=0:csi = ln µ ln λ + 2πin ln λ scaling dimension depends on scale λ n = λ n magnification

136 Discrete scale invariance (DSI) e 2πin = 1=µλ n=0:csi = ln µ ln λ + 2πin ln λ scaling dimension depends on scale λ n = λ n magnification preferred scale

137 analogy liquid solid

138 analogy liquid continuous translational invariance solid discrete translational invariance

139 example

140 example

141 example

142 example n iterations length 3 n number of segments 2 n

143 example n iterations length 3 n number of segments 2 n scale invariance only for λ p =3 p

144 example n iterations length 3 n number of segments 2 n scale invariance only for λ p =3 p discrete scale invariance: D = ln 2 ln 3 + 2πin ln 3

145 more examples stock market crashes turbulence DLA earthquakes fractals TB Law percolation

146 discrete scale invariance hidden scale (length, energy,...)

147 G = β (0,k) α (0,k) p=-0.4 p=0.0 p=0.2 p=1.0 no poles outside log-oscillatory region for p >1/ 6

148 G = β (0,k) α (0,k) p=-0.4 p=0.0 p=0.2 p=1.0 no poles outside log-oscillatory region for p >1/ 6 Mott physics and DSI are linked!

149 is there an instability? y x

150 is there an instability? y x

151 is there an instability? E E + iγ γ > 0 y x

152 is there an instability? E E + iγ γ > 0 y φ =0 x condensate (bosonic)

153 is there an instability? E E + iγ γ > 0 y φ =0 x condensate (bosonic) does this happen for fermions?

154 quasi-normal modes Imω < 0 no instability

155 quasi-normal modes quasi-particle residue Imω < 0 no instability

156 quasi-normal modes quasi-particle residue but the residue drops to zero: opening of a gap Imω < 0 no instability

157 G(ω, k) =G(ωλ n,k)

158 G(ω, k) =G(ωλ n,k) discrete scale invariance in energy

159 G(ω, k) =G(ωλ n,k) discrete scale invariance in energy emergent IR scale

160 G(ω, k) =G(ωλ n,k) discrete scale invariance in energy emergent IR scale no condensate

161 G(ω, k) =G(ωλ n,k) discrete scale invariance in energy emergent IR scale no condensate energy gap: Mott gap (Mottness)

162 continuous scale invariance discrete scale invariance in energy

163 continuous scale invariance discrete scale invariance in energy is this the symmetry that is ultimately broken in the Mott problem?

164 a.) yes b.) no

165 holography a.) yes b.) no

166 holography VO_2, cuprates,... a.) yes b.) no

167 holography VO_2, cuprates,... a.) yes b.) no if yes: holography has solved the Mott problem

168 what else can holography do?

169 Finite Temperature Mott transition from Holography T/µ = T/µ =

170 Finite Temperature Mott transition from Holography T/µ = T/µ = T crit 20 vanadium oxide

171 Finite Temperature Mott transition from Holography T/µ = T/µ = T crit 10 T crit 20 vanadium oxide

172 looks just like the experiments

173 Mottness looks just like the experiments

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175

176 toy model: merging of UV and IR fixed points β =(α α ) (g g ) 2 g ± = g ± α α

177 toy model: merging of UV and IR fixed points β =(α α ) (g g ) 2 β g ± = g ± α α g IR g g UV g + α > α

178 toy model: merging of UV and IR fixed points β =(α α ) (g g ) 2 β g ± = g ± α α g IR g g g UV g + α > α α = α

179 toy model: merging of UV and IR fixed points β =(α α ) (g g ) 2 β g ± = g ± α α g IR g g g UV g + α > α α = α α < α Λ IR = Λ UV e π/ α α

180 toy model: merging of UV and IR fixed points β =(α α ) (g g ) 2 β g ± = g ± α α g IR g g g UV g + α > α g ± are complex (conformality lost) α = α α < α Λ IR = Λ UV e π/ α α DSI BKT transition Kaplan, arxiv: