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 Seungmin Hong

2 Mott Problem

3 Mott Problem + # +

4 Mott Problem + # + σ(ω) Ω -1 cm -1 VO 2 T=360 K T=295 K ev = 10 4 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)

5 Mott Problem + # + σ(ω) Ω -1 cm -1 VO 2 T=360 K T=295 K = 0.6eV > dimerization (Mott, 1976) T crit 20 ev = 10 4 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)

6 Mott Problem + # + σ(ω) Ω -1 cm -1 VO 2 T=360 K T=295 K = 0.6eV > dimerization (Mott, 1976) T crit 20 ev = 10 4 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)

7 Mott Problem σ(ω) Ω -1 cm -1 + # VO 2 T=360 K T=295 K + = 0.6eV > dimerization (Mott, 1976) T crit 20 transfer of spectral ev = 10 4 K weight to 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) Mottness

8 emergent gravity Mott problem

9 What computational tools do we have for strongly correlated electron systems?

10

11 non-interacting interacting DMFT

12 non-interacting interacting DMFT why does this work at long wavelengths?

13 non-interacting interacting the only difference between this and a theory is that this is not a theory DMFT why does this work at long wavelengths?

14 Is there an alternative?

15 UV QF T

16 UV IR?? QF T

17 UV IR?? QF T Wilsonian program (fermions: new degrees of freedom)

18 UV IR?? QF T Wilsonian program (fermions: new degrees of freedom)

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

20 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)

21 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)

22 gauge-gravity duality (Maldacena, 1997) 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)

23 what s the geometry?

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

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

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

27 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

28 `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}

29 `Holography finite density: RN black hole 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}

30 UV IR QF T

31 operators O UV IR QF T

32 fields φ operators O UV IR QF T

33 fields φ IR UV QF T operators O e d d xφ O O

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

35 What holography does for you?

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

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

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

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

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

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

42 What holography does for you? Landau-Wilson Hamiltonian ξ t ξ z long-wavelengths holography RG=GR strong-coupling is easy give up fixation with Hamiltonians (UV models) RG equations perturbative

43 What holography does for you? Landau-Wilson Hamiltonian ξ t ξ z long-wavelengths RG equations perturbative holography RG=GR strong-coupling is easy give up fixation with Hamiltonians (UV models) so what (currents, symmetries)

44 What bulk gravitational theory gives rise to a gap in ImG without `spontaneous symmetry breaking?

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

46 Charged system J µ S 0 UV probe J ψ O ψ what has been done? MIT, Leiden, McMaster,...

47 Charged system J µ S 0 UV probe J ψ O ψ what has been done? MIT, Leiden, McMaster,... RN-AdS ds 2,A t

48 Charged system J µ S 0 UV probe J ψ O ψ what has been done? MIT, Leiden, McMaster,... RN-AdS ψ Dirac Eq. ds 2,A t

49 Charged system J µ S 0 UV probe J ψ O ψ what has been done? MIT, Leiden, McMaster,... RN-AdS ψ Dirac Eq. ds 2,A t in-falling boundary conditions ψ(r ) ar m + br m Retarded Green function: G = b a =f(uv (k_f), IR (q,m)) a=0 defines FS

50 gi ψ(d m)ψ AdS-RN MIT, Leiden group G(ω, k) = Z v F (k k F ) ω hω ln ω

51 gi ψ(d m)ψ AdS-RN MIT, Leiden group G(ω, k) = Z v F (k k F ) ω hω ln ω marginal Fermi liquid

52 gi ψ(d m)ψ AdS-RN MIT, Leiden group G(ω, k) = Z v F (k k F ) ω hω ln ω marginal Fermi liquid?? Mott gap

53 gi ψ(d m)ψ AdS-RN MIT, Leiden group G(ω, k) = Z v F (k k F ) ω hω ln ω marginal Fermi liquid?? Mott gap How to Destroy the Fermi surface?

54 gi ψ(d m)ψ AdS-RN MIT, Leiden group G(ω, k) = Z v F (k k F ) ω hω ln ω marginal Fermi liquid?? Mott gap How to Destroy the Fermi surface? decoherence

55 gi ψ(d m)ψ AdS-RN MIT, Leiden group G(ω, k) = Z v F (k k F ) ω hω ln ω marginal Fermi liquid?? Mott gap How to Destroy the Fermi surface? decoherence log-oscillatory regime

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

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

58 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

59 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)

60 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) what happens at the boundary?

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

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

63 How is the spectrum modified? P P= <p< >ν kf > 1/2 ω k k F ω (k k F ) 2ν k F `Fermi Liquid Fermi surface peak

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

65 How is the spectrum modified? P P=0 p = 0.53 ν kf =1/2 MFL 0.53 <p<1/ 6 1/2 >ν kf > 0 ω = ω (k k F ) 1/(2ν k F ) NFL Fermi surface peak

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

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

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

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

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

71 How is the spectrum modified? P > 4.2 P P=0 spectral weight transfer Fermi surface peak dynamically generated gap: Gubser, Gauntlett, 2011 `similar results

72 Mechanism? ψ ar + br AdS 4

73 ψ ar + br Mechanism? AdS 4 operators O where is k_f

74 Mechanism? ψ ar + br AdS 2 AdS 4 operators O where is k_f emergent IR CFT

75 Mechanism? ψ ar + br AdS 2 AdS 4 operators O where is k_f emergent IR CFT log-oscillatory k_f moves into log-oscillatory region: IR O ± acquires a complex dimension

76 near horizon radial Dirac Equation ψ I± (ζ) = ψ (0) I± (ζ)+ωψ(1) I± (ζ)+ω2 ψ (2) I± (ζ)+ ψ (0) I± (ζ) =iσ 2 1+ qe d ζ L 2 ζ mσ 3 + pe d ± kl r 0 σ 1 ψ (0) I± (ζ), m 2 k = m 2 + pe d ± kl 2 r 0 e d =1/ 2d(d 1) p: time-reversal breaking mass term (in bulk) ν ± k = m 2 k± L2 2 q2 e 2 d i, δ ± = ν ± k +1/2

77 near horizon radial Dirac Equation ψ I± (ζ) = ψ (0) I± (ζ)+ωψ(1) I± (ζ)+ω2 ψ (2) I± (ζ)+ ψ (0) I± (ζ) =iσ 2 1+ qe d ζ L 2 ζ mσ 3 + pe d ± kl r 0 σ 1 ψ (0) I± (ζ), m 2 k = m 2 + pe d ± kl 2 r 0 e d =1/ 2d(d 1) p: time-reversal breaking mass term (in bulk) ν ± k = m 2 k± L2 2 q2 e 2 d i, δ ± = ν ± k +1/2 scaling dimension is complex!

78 What does a complex scaling dimension mean?

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

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

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

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

83 1 = µλ

84 e2πin = 1 = µλ

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

86 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

87 example

88 example

89 example

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

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

92 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

93 our problem G(ω, k) =G(ωλ n,k) discrete scale invariance hidden scale (length, energy,...)

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

95 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

96 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

97 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

98 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:

99 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

100 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!

101 is there an instability? (violation of BF bound) y x

102 is there an instability? (violation of BF bound) y x

103 is there an instability? (violation of BF bound) E E + iγ γ > 0 y x

104 is there an instability? (violation of BF bound) E E + iγ γ > 0 y φ =0 x condensate (bosonic)

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

106 quasi-normal modes Imω < 0no instability

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

108 quasi-normal modes but the residue drops to zero: opening of a gap quasi-particle residue G R (ω, k) = b(0) + ωb 1 G k (ω) a (0) + ωa (1) G k (ω) Imω < 0no instability

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

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

111 p 1 p 3 T σ(ω) Ω -1 cm -1 VO 2 spectral weight transfer UV-IR mixing

112

113 is CSI->DSI the symmetry that is ultimately broken in the Mott problem?

114 a.) yes b.) no

115 holography a.) yes b.) no

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

117 holography VO_2, cuprates,... a.) yes b.) no Does scaling in VO_2 obey: Λ IR =Λ UV e π/ g c g

118 holography VO_2, cuprates,... a.) yes b.) no Does scaling in VO_2 obey: Λ IR =Λ UV e π/ g c g if yes: holography has solved the Mott problem