Holography and Mottness: a Discrete Marriage
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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
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