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

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

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

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?

yes

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

= Z ω k Σ(ω, k)

= Z ω k Σ(ω, k) UV-IR mixing

zeros of Hubbard model: Exact Results

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

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

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)

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)

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

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

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

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

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

What are the extra states (degrees of freedom)?

Key Equation 1=2 1

Key Equation 1=2 1 e =2e e

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

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 )

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

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

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

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

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

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) 0.6 0.4 F [ϕ 0 e iq r ] F [ϕ 0 ] ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ 1.2 1 1 q y ( ) Probability (π,π) is most likely Probability( 0 ) 0.8 0.6 0.4 0 0 1 q x ( ) L charge (ϕ, c, c ) ϕc c + c.c. 0.2 0 0.4 0.6 0.8 1 1.2 1.4 0

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) 1.0 0.5 0.0-0.5 Low No Fermi Arcs (c) High 1.5 Energy (t) 1.0 0.5 0.0-0.5 Low Fermi Arcs Incoherent feature

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) 1.5 1.0 0.5 0.0-0.5 Low No Fermi Arcs (c) High 1.5 Energy (t) 1.0 0.5 0.0-0.5 Low Fermi Arcs Incoherent feature

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) 1.5 1.0 0.5 0.0-0.5 Low No Fermi Arcs Self energy (particle - hole) (c) High ϕc c (ϕc ) c Σ k = ϕ 0 2 ω q k Energy (t) 1.5 1.0 0.5 0.0-0.5 Low Fermi Arcs Incoherent feature

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 0.0 1.0 (c) x=0.12 (d) x=0.18 High Pocket size ~ x 0.5 Low 0.0 0.5 1.0 0.0 0.5 1.0 Electron pocket?

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

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

Fermi arcs: UV-IR mixing (strong coupling)

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

non-interacting interacting DMFT

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

UV QF T

UV IR?? QF T

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

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

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)

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) 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)

`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}

UV IR QF T

operators O UV IR QF T

fields φ operators O UV IR QF T

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

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 ))

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

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)

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?

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

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

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

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

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

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

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

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

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

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

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

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

Mottness

Mottness Hubbard dynamical spectral weight transfer Holography