Holography and Mottness: a Discrete Marriage

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

Mott Problem

emergent gravity Mott Problem

What interacting problems can we solve in quantum mechanics?

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

interacting systems H = T + V

easy interacting systems H = T + V

interacting systems H = T + V easy impossible

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

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

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

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,...

UV why is strong coupling hard?

why is strong coupling hard? UV IR

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

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

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

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

Yes

L = T gϕ 4 UV QF T

L = T gϕ 4 UV IR?? QF T

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

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

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)

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)

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)

what s the geometry?

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

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

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

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

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

Holography

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

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}

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}

dual construction UV IR QF T

dual construction operators O UV IR QF T

dual construction fields φ operators O UV IR QF T

dual construction fields φ duality operators O UV IR QF T

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

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 ))

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

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

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

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

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

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)

Why is gravity holographic?

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

black hole gravitons QFT

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

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

What holography does for you?

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

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

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

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

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

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)

Can holography solve the Mott problem?

NiO insulates d 8? What is a Mott Insulator?

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

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

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

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

Half-filled band Free electrons

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

Mott Insulator-Ordering

Mott Insulator-Ordering =Mottness

Slater Mott Insulator-Ordering =Mottness

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

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

Why is the Mott problem important?

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

Experimental facts: Mottness

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, 205118 (2006)

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, 205118 (2006)

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, 205118 (2006)

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

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

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

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

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

?? Mott Insulator

?? Mott Insulator decoherence

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

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

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

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)

P=0 How is the spectrum modified?

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

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

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

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

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

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

Gauntlett, 2011

Mechanism? UV IR QF T

emergent spacetime symmetry Mechanism? UV IR QF T AdS 2

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

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

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

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

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

What does a complex scaling dimension mean?

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

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

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

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

1 = µλ

e2πin = 1 = µλ

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

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

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

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

analogy liquid solid

analogy liquid continuous translational invariance solid discrete translational invariance

example

example n iterations length 3 n number of segments 2 n

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

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

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

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

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

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!

is there an instability? y x

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

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

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

quasi-normal modes Imω < 0 no instability

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

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

G(ω, k) =G(ωλ n,k)

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

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

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

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

continuous scale invariance discrete scale invariance in energy

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

a.) yes b.) no

holography a.) yes b.) no

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

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

what else can holography do?

Finite Temperature Mott transition from Holography T/µ =5.15 10 3 T/µ =3.92 10 2

Finite Temperature Mott transition from Holography T/µ =5.15 10 3 T/µ =3.92 10 2 T crit 20 vanadium oxide

Finite Temperature Mott transition from Holography T/µ =5.15 10 3 T/µ =3.92 10 2 T crit 10 T crit 20 vanadium oxide

looks just like the experiments

Mottness looks just like the experiments

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

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

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

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

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:0905.475