The Emergent Fermi LiquidFermions, Fermions, Fermions...

Koenraad Schalm

Institute Lorentz for Theoretical PhysicsLeiden University

Mihailo Cubrovic, Jan Zaanen, Koenraad Schalmarxiv/0904.1993

Lee: arxiv/0809.3402Liu, McGreevy, Vegh: arxiv/0903.2477Faulkner, Liu, McGreevy, Vegh: arxiv/0907.xxxxCubrovic, Sadri, Schalm, Zaanen: arxiv/090x.xxx

AdS/CMT July 2009 KITP UCSB

Fermions at finite density

• Electrons in the real world/in the “mundane” [Hartnoll (Monday)]

Experiment Theory

Fermi Liquid adiabatic continuation Fermi-gasLandau quasi-particles

BCS superconductor BCS PairingSpontaneous symmetry breaking

Non Fermi Liquids- high Tc cuprates- heavy fermions- ...

?

Fermions at finite density

• Electrons in the real world/in the “mundane” [Hartnoll (Monday)]

Experiment Theory

Fermi Liquid adiabatic continuation Fermi-gasLandau quasi-particles

BCS superconductor BCS PairingSpontaneous symmetry breaking

Non Fermi Liquids- high Tc cuprates- heavy fermions- ...

?

Non-Fermi Liquids and the fermion-sign-problem

• The non-Fermi liquid: two important characteristics– Physics controlled by a QCP [conjecture]– Adjacent phase is regular Fermi Liquid

• Fundamental problem:

Fermion-sign-problem [original]

[Computer]

[not weakly interacting electrons:interacting CFT?strongly coupled?]

Non-Fermi Liquids and the fermion-sign-problem

• The non-Fermi liquid: two important characteristics– Physics controlled by a QCP [conjecture]– Adjacent phase is regular Fermi Liquid

• Fundamental problem:

Fermion-sign-problem [here]

How can a Fermi Liquid (defined by EF , kF )disappear into a quantum critical state?

[not weakly interacting electrons:interacting CFT?strongly coupled?]

Non-Fermi Liquids and the fermion-sign-problem

• The non-Fermi liquid: two important characteristics– Physics controlled by a QCP [conjecture]– Adjacent phase is regular Fermi Liquid

• Fundamental problem:

Fermion-sign-problem [here]

How can a Fermi Liquid (defined by EF , kF )emerge from a quantum critical state?

[not weakly interacting electrons:interacting CFT?strongly coupled?]

⇒difficult for standard QFT......”natural fit” for AdS dual

The fermion-sign-problem II

High Tc superconductors Heavy fermion systems

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The fermion-sign-problem II

High Tc superconductors Heavy fermion systems

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AdS-to-ARPES

• 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]

[Source: A. Damascelli CIAR 2003]

AdS-to-ARPES

• 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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9)-.%/:%(#.;%<*!"#$#%&'(('#)*,0%3$2#(3#0+

[Source: A. Damascelli CIAR 2003]

AdS-to-ARPES

• 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)

Caveat:dynamicalinstabilityin full theory

Fermions in AdS/CFT I

• Action: Einstein YM + charged fermions

S =∫ √−g [R+ 6− 1

4F 2 − Ψ̄eMA ΓA (DM + igAM ) Ψ−mΨ̄Ψ

]+ Sbnd

• Fermions:[1. ] constrained (first order) system:⇒ reduce to physical d.o.f.

ΓzΨ± = ±Ψ±

[2. ] action vanishes on shell.⇒ add boundary term

Sbnd =∫ √−hΨ̄+Ψ−

⇒ variation∂S

∂Ψ−well-defined

[L = κ = 1; also set g = 1 from hereon]

[Henningson,SfetsosMueck,ViswanathanHenneauxContino,Pomarol]

Fermions in AdS/CFT II

• Standard AdS/CFT: .... with some subtleties– to solve Dirac Equation...

(∂z +Az)Ψ± = ∓/T Ψ±

– ... write as second order equation

(∂2z + P∂z +Q)Ψ+ = 0

– ... compute on-shell action(bdy term only; bdy Ψ0 determined via EOM)

Ψ± = F±(z)F−1+ (z0)Ψ0

+ ⇔ Ψ0− = F−(z)F−1

+ (z0)Ψ0+

– Green’s function

G(z0) = F−(z0)F−1+ (z0)

Fermions in AdS/CFT III

• Unitarity bound Fermion mass [Contino,Pomarol;CZS;Iqbal,Liu]– 2nd order equation (∂2

z + P∂z +Q)Ψ+ = 0 has asymptoticsolutions

Ψ+ = zd+1

2 −|m+ 12 |(A+ . . .) + z

d+12 +|m+ 1

2 |(B + . . .)

– CFT Green’s function (T = 0) has a pole unless

−12< m (Sbnd breaks m⇔ −m)

⇒ Unitarity bound on CFT fermionic operator dimension

〈OΨ(−k)OΨ(k)〉 = k2∆Ψ−d ∆Ψ =d− 1

2+ |m+

12|

d− 12

< ∆

Which ∆/m?

• IR emergence of a new state– Recall from holographic superconductors

[Gubser;Hartnoll,Herzog,Horowitz]

relevant perturbations = scalars with −d2

4 < m2φ < 0

⇒ For fermions: choose mass near unitarity bound: − 12 < mΨ

– Upper bound on mass?Green’s function marginal at: mΨ = 0, ∆Ψ = d

2⇒ Interesting range to obtain new physics

−12< mΨ < 0 − d− 1

2< ∆ <

d

2

[mΨ = 0: Lee:0809.3402; Liu, McGreevy, Vegh:0903.2477][Qualitative argument (g = κ = 1): quantitative cf. Denef,Hartnoll]

Unitarity bound

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�

The emergent Fermi Liquid from AdS/CFT

• Our expectation:

T !0 µ0 !

!! d

2

Fermi-Liquid

CFT

0d! 1

2

T

|µ0| !

Remnant scaling behavior

• Boundary AdS Green’s function invariant under

ω → λω , k → λk , α→ λα

– (recall that 4πT = α(3− q2) , µ0 = −2qα)

• in terms of CFT quantities

Teff (µ0) = T

(12

+12

√1 +

µ20

T 2

)

Gµ0/T

Ψ̄Ψ(ω, k) = T 2∆Ψ−df

(ω

Teff,k

Teff;µ0

T

)

Spectral functions and a Quasiparticle peak

A(ω, k) = − 1π ImTr (iγ0GR(ω, k))

2 4 6 80

0.02

0.04

0.06

0.08

!/Teff

A (!

, k)

k = 0k = 1.5k = 1.9k = 2.3!2"−d fit

µ0

T→ 0

GR = 1(√−ω2+k2)d−2∆Ψ

∆ = 1.25

Spectral functions and a Quasiparticle peak

A(ω, k) = − 1π ImTr (iγ0GR(ω, k))

2 4 6 80

0.02

0.04

0.06

0.08

!/Teff

A (!

, k)

k = 0k = 1.5k = 1.9k = 2.3!2"−d fit

µ0

T→ 0

GR = 1(√−ω2+k2)d−2∆Ψ

0−2 2 40

0.01

0.02

0.03

0.04

!/Teff

A (!

, k)

k = 0k = 1.5k = 1.9k = 2.3

k < kF k > kF

(!−µ0)2"−d tail

EF = − 0.65

k = 0

µ0

T= 30.9

µ0

T>(µ0

T

)c

QP Peaks

∆ = 1.25

Fermi Liquid phenomenology

• Single particle Green’s functions encodes the ground-state

G(ω, k) =1

ω − µ0 − k2

2m + Σ(ω, k)

≡ Z

(ω − EF )− vF (k − kF ) + . . .

– Quasiparticle peak (QP) ω = EF , k = kF and a sharp Fermisurface |k| = kF

• Spectral function

A(ω, k) =ImΣ(ω, k)

|ω + µ0 + (k−kF )2

2m + ReΣ(ω, k)|2 + |ImΣ(ω, k)|2

List of Fermi Liquid tests:

• Quasiparticle peaks

• Zero density of states at the Fermi surface A(EF , k) = 0

• Quadratic scaling of the QP-widths with temperature

• Analytical structure of the self-energy Σ(ω, k)

• Linear dispersion of QP-excitations

Identifying EF , kF

−0−1−2−3−4−50

0.05

0.1

A(!

, k)

−1−2−3−4−50

0.05

0.1

A(!

, k)

−1−2−3−4−50

0.05

0.1

!/Teff

A(!

, k) k > kF

k " kF

k < kF

T = 0: A(EF , k) = Zδ(k − kF )⇔ ImΣ(ω, k) =12

(ω − EF )2 ∂2ImΣ∂ω2

∣∣∣∣ω=EF

Identifying EF , kF

−0−1−2−3−4−50

0.05

0.1

A(!

, k)

−1−2−3−4−50

0.05

0.1

A(!

, k)

−1−2−3−4−50

0.05

0.1

!/Teff

A(!

, k) k > kF

k " kF

k < kF

T = 0: A(EF , k) = Zδ(k − kF )⇔ ImΣ(ω, k) =12

(ω − EF )2 ∂2ImΣ∂ω2

∣∣∣∣ω=EF

Dip in the spectrum:zero density of statesat EF for k 6= kF

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BBBBBBBBBBBBBBBBN

Identifying EF , kF

−0−1−2−3−4−50

0.05

0.1

A(!

, k)

−1−2−3−4−50

0.05

0.1

A(!

, k)

−1−2−3−4−50

0.05

0.1

!/Teff

A(!

, k) k > kF

k " kF

k < kF

T = 0: A(EF , k) = Zδ(k − kF )⇔ ImΣ(ω, k) =12

(ω − EF )2 ∂2ImΣ∂ω2

∣∣∣∣ω=EF

Dip in the spectrum:zero density of statesat EF for k 6= kF

Finite T effect:dip position“smeared”

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Temperature Dependence/Self-energy structure

−4 −3 −2 −1 00

0.2

0.4

0.6

0.8

1

1.2

1.4

!/Teff

A(!

, k)

µ0/T = 10.25µ0/T = 1.04µ0/T = 0.73µ0/T = 0.47µ0/T = 0.23µ0/T = 0.12

EF

(!−EF)/Teff

−5

−2.4 −1.4 −0.4 1.6 2.6

Peak width δ vs. T : δ ∼ T 2

0 0.1 0.20.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.010

(T/Teff)2

!

Tc2

210 −1 −2 0

1

2

3

4

5

6

7

8

9

!/Teff

Im"(

!, k

)

k = 1.7k = 1.8k = 1.9

EF = − 0.24

−0.24

2.241.240.24−0.76−1.76(!−EF)/Teff

0

No constant term in ∂ImΣ/∂ω

210−1−2

0.08

0.06

0.04

0.02

0

−0.02

−0.04

−0.06

−0.08 !/Teff

d(Im

")/d

!

k = 1.8k = 1.9k = 1.7

− 0.24

−0.76 0 0.24 1.24 2.24− 1.76(!−EF)/Teff

EF = − 0.24

∆Ψ = 1.05 ∆Ψ = 1.40, µ0/T = 30.9

“ µT

”c' 4

XXXXXy

Dispersion relation

Quasiparticle dispersion: ω − EF = vF (k − kF ) +O((k − kF )2)

−8 −6 −4 −2 0 2 4 6 8

6

4

2

0

−2

−4

−6k/Teff

4.48

2.48

6.48

0.480

FL region

−2.92−3.52

−5.52

−1.52

(!−EF)/Teff

−3.4

!/Teff

µ0 = − 3.4

EF = − 0.48

kF = 1.82

vF = 0.67

Renormalized QP velocityPPPPPq

Dispersion relation

Quasiparticle dispersion: ω − EF = vF (k − kF ) +O((k − kF )2)

−8 −6 −4 −2 0 2 4 6 8

6

4

2

0

−2

−4

−6k/Teff

4.48

2.48

6.48

0.480

FL region

−2.92−3.52

−5.52

−1.52

(!−EF)/Teff

−3.4

!/Teff

µ0 = − 3.4

EF = − 0.48

kF = 1.82

vF = 0.67

Bare (Lorentz dispersion) ω = k at high ω

Renormalized QP velocity

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Dispersion relation

Quasiparticle dispersion: ω − EF = vF (k − kF ) +O((k − kF )2)

−8 −6 −4 −2 0 2 4 6 8

6

4

2

0

−2

−4

−6k/Teff

4.48

2.48

6.48

0.480

FL region

−2.92−3.52

−5.52

−1.52

(!−EF)/Teff

−3.4

!/Teff

µ0 = − 3.4

EF = − 0.48

kF = 1.82

vF = 0.67

Bare (Lorentz dispersion) ω = k at high ω EF is not AdS zero – a new scale

Renormalized QP velocity

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List of Fermi Liquid tests:

• Quasiparticle peaks X

• Zero density of states at the Fermi surface A(EF , kF ) = 0 X

• Quadratic scaling of the QP-widths with temperature X

• Analytical structure of the self-energy Σ(ω, k) X

• Linear dispersion of QP-excitations X

Charge density dependence

0 6 12 18 24 300

0.5

1

1.5

2

µ0/T

−EF/T

eff, k

F/Tef

f, vF

0 6 12 18 24 30

−10

−5

µ0/T

log

Z

− EF/TeffkF/TeffvFmF

0

2

4

(µ0/T)c

(µ0/T)cmF/Teff

0 5 10 150

2

4

6

8

10

12

kF/T

µ0/T− EF/TvFmF/T

− EF(ren)/T

36

30

24

18

6

12

µ0/T,

mF/T

−EF/T,

−EF(ren)/T,

vF

∆Ψ = 1.25

E(ren)F ≡ k2

F

2mF

mF ≡ kFvF

Fermi LiquidZ = mbarevF

kF

vF = Z(1 + ∂kReΣ|kF ) + . . .Z = 1/(1− ∂ωReΣ|EF ) + . . .

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):

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 .