The Emergent Fermi Liquid Fermions, Fermions, Fermions... Koenraad Schalm Institute Lorentz for Theoretical Physics Leiden University Mihailo Cubrovic, Jan Zaanen, Koenraad Schalm arxiv/0904.1993 Lee: arxiv/0809.3402 Liu, McGreevy, Vegh: arxiv/0903.2477 Faulkner, Liu, McGreevy, Vegh: arxiv/0907.xxxx Cubrovic, Sadri, Schalm, Zaanen: arxiv/090x.xxx AdS/CMT July 2009 KITP UCSB
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The Emergent Fermi LiquidFermions, Fermions, Fermions...
Koenraad Schalm
Institute Lorentz for Theoretical PhysicsLeiden University
Mihailo Cubrovic, Jan Zaanen, Koenraad Schalmarxiv/0904.1993
• Essential: Single fermionic quasi-particle spectrum encodesFermi Liquid ground state
A(ω, k) = − 1π
ImGR(ω, k)
• Experimental probe: ARPES [Source: A. Damascelli CIAR 2003]
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[Source: A. Damascelli CIAR 2003]
The AdS set-up
• Spectral density from charged AdS BH [e.g. McGreevy’s lectures]
– GΨΨR (ω, k) : infalling b.c. at BH Horizon ⇔ T > 0
– GΨΨR (ω, k): finite µF? ⇔ µU(1)
⇒ Charged AdS BH in 3+1 dim dual to 2+1 dim relativistic CFT:
ds2 =α2
z2
(−f(z)dt2 + dx2
1 + dx22
)+
1f(z)
dz
z2
A0 = 2qα(z − 1)f(z) = (1− z)(z2 + z + 1− q2z3)
4πT = α(3− q2) , µ0 = −2qα
Is µU(1) = µF?
• AdS Phenomenology:
– Do not know exact quantum theory (CFT or AdS)i.e. do not know constituents of the BH. There could be manyU(1) charged particles (fermions and bosons)
It is not a priori guaranteed that µU(1) will act as µF
• Empirical approach:
– QFT: In an interacting system µF renormalizes:µ
(IR)F is empirically determined by the pole in GR
– AdS: We will follow the same approach.Expectation µ(UV )
U(1) ≡ µ0 induces a µF ≡ EF , whose value we
read off from the spectrum. A priori µF 6= µ(UV )U(1)
restored Lorentzinvariance[cf. Randeria et alcond-mat/0307217]∂kReΣ−∂ωReΣ ∼ 0
XXXXXXXz
Coupling strength dependence
Scaling dimension ∆Ψ proxy for coupling strength
1 1.1 1.2 1.3 1.4 1.5−10
−5
0
!
log
Z
1 1.1 1.2 1.3 1.4 1.50
0.5
1
1.5
2
2.5
3
3.5
!
kF/TeffvFEF/Teff
µ0/T = 30.9
• kF ' constant for all ∆⇒ Luttinger theorem!?
• Pole strength Z dropsexponentially as ∆→ 1.5...[cf. Lawler et alcond-mat/0508747]
• ...but vF ,mF stay finite⇒ emergence of Lorentzinvariance[cf. Randeria et alcond-mat/0307217]
Comparison with previous results
• Always two peaks (are there more?) :
– standard FL,marginal FL [Faulkner,Liu,McGreevy,Vegh] ,. . .
– CFT pole = BH quasinormal mode [Kovtun,Starinets]AdS BH: No a priori reason ∃ only one Dirac quasinormal mode.
– What is the true IR? “Band structure”?
−1.5 −1 −0.5 0 0.50
1
2
3
4
5
6x 10−3
!/Teff
A(!
, k)
" = 1.35" = 1.40" = 1.45" = 1.49
µ0
T= 30.9
marg-FLstandard FL QP
@@R
@@@R
Including a magnetic field
Landau levels in a magnetic field[Preliminary....] 8
0 100 200 300 400 500 600 700 8000
0.5
1
1.5
2
2.5
3
3.5
4
4.5x 10
5
! (H
)
Erel
FIG. 7: Level broadening.
1 2 3 4 5 6 7 8 9 10!1.5
!1
!0.5
0
0.5
1x 10
9
1/H
"#
$ = 1.45
$ = 1.25
$ = 1.05
FIG. 8: Oscillating behavior of the thermodynamical quantities is shown for µ/T = 30.9 and H/T between 0.1 and 1.0. Theconformal dimension is ! = 1.25. The curves correspond to three di"erent values of the conformal dimension. It is seen thatthe period remains the same (count the number of peaks).
!2.5 !2 !1.5 !1 !0.5 0 0.5 1 1.5 2 2.50
0.5
1
1.5
2
2.5x 10
!4
%/Teff
A (
%,k
)
FIG. 9: Intermediate field high charge density system, with |µ0|/T = 30.9, ! = !5/4 and H/T = 3.7. Shown is the dispersion(EDCs) along kx with ky/Teff = 1. The x-momentum takes the values 0.58, 0.72, 0.87, 1.15, 1.44, 1.73, 1.88, 2.02, 2.17, 2.30.We see the splitting of the quasiparticle as well as two branches of gapless modes. The curves suggest disappearance of theFermi surface in favor of a set of disconnected sheets.
ρ(ω) =dN
dω, N(ω) =
∫d2~k nFD(ω, k − kF ;EF )A(ω, k)
[Cubrovic,KS,Zaanen,Sadri]
Conclusion and Outlook
• Theoretical Calculation of an emergent Fermi Liquid(CFT) = (AdS)
probing a charged black hole with electrons
• What is kF on the gravity side?– in string theory geometry depends on the probe...
• Transition to BCS superconductor• Physical interpretation of the ∆ = 3/2 transition?• Interpretation of/co-existence with ω = 0 pole?
[Faulkner,Liu,McGreevy,Vegh]
• Inclusion magnetic field/.../(better understanding of theemergence)
• ... direct connection with experimental data?! ...
Conclusion and Outlook
• Theoretical Calculation of an emergent Fermi Liquid(CFT) = (AdS)
probing a charged black hole with electrons
• What is kF on the gravity side?– in string theory geometry depends on the probe...
• Transition to BCS superconductor• Physical interpretation of the ∆ = 3/2 transition?• Interpretation of/co-existence with ω = 0 pole?
[Faulkner,Liu,McGreevy,Vegh]
• Inclusion magnetic field/.../(better understanding of theemergence)
• ... direct connection with experimental data?! ...
Conclusion and Outlook
• Theoretical Calculation of an emergent Fermi Liquid(CFT) = (AdS)
probing a charged black hole with electrons
Thank you.
• ... direct connection with experimental data?! ...
Boundary behavior of Ψ±
• Asymptotic behavior near z = 0:
Ψ+(z) = zd+1
2 −|m+ 12 |(ψ+ + . . .) + z
d+12 +|m+ 1
2 |(A+ + . . .)
Ψ−(z) = zd+1
2 −|m−12 |(ψ− + . . .) + z
d+12 +|m− 1
2 |(A− + . . .)
• The boundary value of Ψ− is not independent.
(∂z − d/2−mz
)Ψ+ = −/T |z=0Ψ− + . . .
Thus ψ− ∝ ψ+ and A− ∝ A+.• The scaling behavior of G(ω, k):
G(ω, k) =1
N F−F−1+ ∼ z
d+12 −|m−
12 |(ψ− + . . .) + z
d+12 +|m− 1
2 |(A− + . . .)
zd+1
2 −|m+ 12 |(ψ+ + . . .) + z
d+12 +|m+ 1
2 |(A+ + . . .).
• Three different regimes:
G(ω, k) ∼
8>>><>>>:z“ψ−ψ+
+ . . .”
+ z2m“A−ψ+
+ . . .”
m > 12,
z2m“ψ−ψ+
+ . . .”
+ z“A−ψ+
+ . . .”
12> m > − 1
2,
1z
“ψ−ψ+
+ . . .”
+ 1z2m
“A−ψ+
+ . . .”
− 12> m .
Boundary behavior of Ψ±
• Asymptotic behavior near z = 0:
Ψ+(z) = zd+1
2 −|m+ 12 |(ψ+ + . . .) + z
d+12 +|m+ 1
2 |(A+ + . . .)
Ψ−(z) = zd+1
2 −|m−12 |(ψ− + . . .) + z
d+12 +|m− 1
2 |(A− + . . .)
• The boundary value of Ψ− is not independent.
(∂z − d/2−mz
)Ψ+ = −/T |z=0Ψ− + . . .
Thus ψ− ∝ ψ+ and A− ∝ A+.• The scaling behavior of G(ω, k):