Thanks to: NSF, EFRC (DOE)

Transcription

1 From Fermi Arcs to Holography Thanks to: NSF, EFRC (DOE) Seungmin Hong and S. Chakraborty T.-P. Choy M. Edalati R. G. Leigh

2 Fermi Arcs P. Johnson, PRL 2011 Na-CCOC (Shen, Science 2006) La-Bi2201 (Meng, Nature 2009)

3 Fermi Arcs P. Johnson, PRL 2011 seen not seen Na-CCOC (Shen, Science 2006) La-Bi2201 (Meng, Nature 2009) E F k

4 Fermi Arcs P. Johnson, PRL 2011 seen not seen Na-CCOC (Shen, Science 2006) La-Bi2201 (Meng, Nature 2009) E F k meaning? can we explain this?

5 yes

6 yes poles zeros E F k degree of freedom orthogonal to an electron

7 = Z ω k Σ(ω, k)

8 = Z ω k Σ(ω, k)

9 = Z ω k Σ(ω, k)

10 = Z ω k Σ(ω, k) UV-IR mixing

11 zeros of Hubbard model: Exact Results

12 zeros of Hubbard model: Exact Results atomic limit G(ω, k) = 1+x ω µ + U/2 + 1 x ω µ U/2

13 zeros of Hubbard model: Exact Results atomic limit ω = µ x =0 G(ω, k) = 1+x ω µ + U/2 + 1 x ω µ U/2

14 zeros of Hubbard model: Exact Results atomic limit ω = µ x =0 (0, π) 1/2-filling p-h symmetry (π, π) G(ω, k) = 1+x ω µ + U/2 + 1 x ω µ U/2 G R > 0 G R < 0 (0, 0) (π, 0)

15 zeros of Hubbard model: Exact Results atomic limit ω = µ x =0 (0, π) 1/2-filling p-h symmetry (π, π) G(ω, k) = 1+x ω µ + U/2 + 1 x ω µ U/2 G R > 0 No Fermi arcs G R < 0 (0, 0) (π, 0)

16 zeros of Hubbard model: Exact Results atomic limit ω = µ x =0 (0, π) 1/2-filling p-h symmetry (π, π) G(ω, k) = 1+x ω µ + U/2 + 1 x ω µ U/2 G R > 0 No Fermi arcs G R < 0 (0, 0) (π, 0) dynamical spectral weight transfer

17 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

18 beyond the atomic limit: any real system density of states 1 + x + α(t/u, x) α = t U c iσ c jσ > 0 ij 1 x α Harris & Lange, 1967 dynamical spectral weight transfer

19 beyond the atomic limit: any real system density of states 1 + x + α(t/u, x) α = t U c iσ c jσ > 0 ij 1 x α Harris & Lange, 1967 dynamical spectral weight transfer Intensity>1+x

20 beyond the atomic limit: any real system density of states 1 + x + α(t/u, x) α = t U c iσ c jσ > 0 ij 1 x α Harris & Lange, 1967 dynamical spectral weight transfer Intensity>1+x # of charge e states

21 beyond the atomic limit: any real system density of states 1 + x + α(t/u, x) α = t U c iσ c jσ > 0 ij 1 x α Harris & Lange, 1967 dynamical spectral weight transfer Intensity>1+x # of charge e states # of electron states in lower band

22 beyond the atomic limit: any real system density of states 1 + x + α(t/u, x) α = t U c iσ c jσ > 0 ij 1 x α Harris & Lange, 1967 dynamical spectral weight transfer Intensity>1+x # of charge e states # of electron states in lower band not exhausted by counting electrons alone?

23 What are the extra states (degrees of freedom)?

24 Key Equation 1=2 1

25 Key Equation 1=2 1 e =2e e

26 Key Equation 1=2 1 e =2e e composite (bound) excitation 2e(boson) + hole `` `` Hubbard bands are not rigid: Mottness

27

28 Key Idea: Extended Hilbert Space H c H D + Integrate out D s U D i Fermionic (one per site) D i Unphysical Hilbert space µ δ(d θc c )

29 Key Idea: Extended Hilbert Space H c H D + Integrate out D s U D i Fermionic (one per site) D i Unphysical Hilbert space µ δ(d θc c ) θϕ i charge 2e boson

30 1+x L = L projected (c)+l spin + L charge (ϕ, c, c ) t-j model

31 charge dynamics due to mixing 1 + x +α 1 x α(t/u, x) 1+x L = L projected (c)+l spin + L charge (ϕ, c, c ) t-j model

32 charge dynamics due to mixing 1 + x +α 1 x α(t/u, x) 1+x L = L projected (c)+l spin + L charge (ϕ, c, c ) t-j model quadratic spectral function

33 charge dynamics due to mixing 1 + x +α 1 x α(t/u, x) 1+x L = L projected (c)+l spin + L charge (ϕ, c, c ) t-j model quadratic spectral function

34 charge dynamics due to mixing 1 + x +α 1 x α(t/u, x) 1+x L = L projected (c)+l spin + L charge (ϕ, c, c ) t-j model quadratic spectral function ϕ i ϕ 0 e iq r i F = 1 β ln Z = F [ϕ 0 e iq r ] q ϕ 0

35 0.8 ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ (π,π) Free energy minimization ϕ i ϕ 0 e iq r i (0,0) F [ϕ 0 e iq r ] F [ϕ 0 ] ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ q y ( ) Probability (π,π) is most likely Probability( 0 ) q x ( ) L charge (ϕ, c, c ) ϕc c + c.c

36 Green Function and Self Energy ϕ i ϕ 0 e iq r i q =(π, π) G(ω, k) = FT T τ c iσ (τ) c j (0) = g t ω k Σ k + O(t/U) (b) High 1.5 Energy (t) Low No Fermi Arcs (c) High 1.5 Energy (t) Low Fermi Arcs Incoherent feature

37 Green Function and Self Energy ϕ i ϕ 0 e iq r i q =(π, π) G(ω, k) = FT T τ c iσ (τ) c j (0) = g t ω k Σ k + O(t/U) Self energy (particle - particle) (b) High ϕc c ϕc c Σ k = ϕ 0 2 ω + q k Energy (t) Low No Fermi Arcs (c) High 1.5 Energy (t) Low Fermi Arcs Incoherent feature

38 Green Function and Self Energy ϕ i ϕ 0 e iq r i q =(π, π) G(ω, k) = FT T τ c iσ (τ) c j (0) = g t ω k Σ k + O(t/U) Self energy (particle - particle) (b) High ϕc c ϕc c Σ k = ϕ 0 2 ω + q k Energy (t) Low No Fermi Arcs Self energy (particle - hole) (c) High ϕc c (ϕc ) c Σ k = ϕ 0 2 ω q k Energy (t) Low Fermi Arcs Incoherent feature

39 Results 1.0 (a) x=0.05 (b) x=0.08 Fermi Arc (1 n σ )c σ 0.5 Shadow band ϕc c (ϕc ) c (c) x=0.12 (d) x=0.18 High Pocket size ~ x 0.5 Low Electron pocket?

40 Pseudogap=`confinement More `e states at lower temper ature composite or bound states not in UV theory

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

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

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

44 Fermi arcs: UV-IR mixing (strong coupling)

45 Fermi arcs: UV-IR mixing (strong coupling) What computational tools do we have for strongly correlated electron systems?

46

47 non-interacting interacting DMFT

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

49 UV QF T

50 UV IR?? QF T

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

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

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

54 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)

55 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)

56 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)

57 `Holography RN black hole r 0 ds 2 = L 2 dt 2 r 2 + dx2 r 2 + dr2 r 2 AdS d+1 β(g) is local geometrize RG flow symmetry: {t, x, r} {λt, λx, λr}

58 UV IR QF T

59 operators O UV IR QF T

60 fields φ operators O UV IR QF T

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

62 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 ))

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

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

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

66 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

67 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)

68 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?

69 How is the spectrum modified? P=0, MIT, Leiden, McMaster Fermi surface peak

70 How is the spectrum modified? P=0, MIT, Leiden, McMaster P Fermi surface peak

71 How is the spectrum modified? P>4.2 P P=0, MIT, Leiden, McMaster Fermi surface peak

72 How is the spectrum modified? P=0, MIT, Leiden, McMaster Fermi surface peak P > 4.2 P

73 How is the spectrum modified? P=0, MIT, Leiden, McMaster Fermi surface peak P > 4.2 P dynamically generated gap:

74 How is the spectrum modified? P > 4.2 P P=0, MIT, Leiden, McMaster spectral weight transfer Fermi surface peak dynamically generated gap:

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

76 Mechanism log-oscillatory k_f moves into log-oscillatory region: O ± acquires a complex dimension G(ω, k) =G(ωλ n,k) discrete scale invariance

77 Finite Temperature Mott transition T/µ = T/µ =

78 Finite Temperature Mott transition T/µ = T/µ = T crit 20 vanadium oxide

79 Finite Temperature Mott transition T/µ = T/µ = T crit 10 T crit 20 vanadium oxide

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

81

82 Mottness

83 Mottness Hubbard dynamical spectral weight transfer Holography