Holography and condensed matter

Francesco Benini

Princeton University

XVII European Workshop on String TheoryPadova (Italy) 5-9 September 2011

Introduction● Q: what gapless phases from finite charge density states?● Q: what physics at quantum critical points?● In absence of sharp quasiparticle or weakly coupled

effective d.o.f.: conventional FT methods might fail

Introduction● Q: what gapless phases from finite charge density states?● Q: what physics at quantum critical points?● In absence of sharp quasiparticle or weakly coupled

effective d.o.f.: conventional FT methods might fail

→ AdS/CFT or gauge/gravity correspondence● Focus of this talk:

Low temperature limit of simple holographic modelsSome models of non-Fermi liquidsStrange metal phaseImpurity models

Quantum critical points & AdS/CFT● IR quantum critical point UV less important● d conformal theory ↔ d+1 gravity theory on AdS

Maldacena 97; Witten 98Gubser, Klebanov, Polyakov 98

Quantum critical points & AdS/CFT● IR quantum critical point UV less important● d conformal theory ↔ d+1 gravity theory on AdS

● Operators: Tμν, Jμ , Φ, Ψ ↔ gμν, Aμ, φ, ψ

● Partition function:● Large N Correlators:● U(1) symmetry chemical potential→ finite density→

O ∼ O(s) z d−Δ+⟨O⟩ zΔ

L ⊃ μ J t

Maldacena 97; Witten 98Gubser, Klebanov, Polyakov 98

Z FT [J ]=Z gravity[ϕ(r→0)=J ]

Holographic theories● Boundary theory is some “large N” gauge theory + matter● Eg: model of emergent gauge fields

Square lattice spin-half model:

H=J ∑⟨ij ⟩S i⋅S j − Q∑⟨ijkl ⟩

(S i⋅S j−14

)(S k⋅S l−14

)

Senthil, Vishwanath, Balents,Sachdev, Fisher 03; 04

Sachdev 07

Holographic theories● Boundary theory is some “large N” gauge theory + matter● Eg: model of emergent gauge fields

Square lattice spin-half model:

● Q/J 0: → isotropic Heisemberg antiferromagnetNeel order:

● Change of variables:

● Q/J → ∞: VBS order, breaking of topological symmetry

H=J ∑⟨ij ⟩S i⋅S j − Q∑⟨ijkl ⟩

(S i⋅S j−14

)(S k⋅S l−14

)

⟨ S i ⟩=(−1)i Φ

Φa= zα σαβa zβ

Leff =∣(∂−iA) z∣2+s∣z∣2+u∣z∣4+ 12e2 F

2

Ψ=(−1) jx S j⋅S j+x + i(−1) j yS j⋅S j+y ∼ V

Senthil, Vishwanath, Balents,Sachdev, Fisher 03; 04

Sachdev 07

Minimal setup● 4d theory of gravity, U(1) gauge field:

● Finite density (chemical potential), finite temperature:Reissner-Nordstrom-AdS black hole

Parameter:

● Zero temperature, near horizon limit AdS→ 2 x R2:

L= 12 κ2(R+ 6

L2)− 14 e2 Fμ ν F

μ ν+…

ds2= L2

r2 (− f (r)dt2+ dr2

f (r)+dx2+dy2) , A=μ(1− r

r+)dtγ=eL /κ

ds2= L2

6 (−dt2+dr2

r2 )+dx2+dy2 , A=γ√6

dtr

AdS2 x R2

● Emergent “local” scaling symmetry:

Eg: z → ∞ limit of Lifshiz scaling.

Possible realization: impurity model (DMFT)

r→ λ r , t →λ t , x , y→ x , y

AdS2 x R2

● Emergent “local” scaling symmetry:

Eg: z → ∞ limit of Lifshiz scaling.

Possible realization: impurity model (DMFT)● Entropy has non-vanishing zero-temperature limit:

● Density of states IR divergent:

● Possible instabilities: Bose-Einstein condensation, Fermi sea

r→ λ r , t →λ t , x , y→ x , y

s= 2π Aκ2V 2

=πμ2

3 e2

ρ(E )∼eS δ(E )+E−1

Jensen, Kachru, Karch, Polchinski, Silverstein 11

Probe fermions● Bulk probe fermion, charge q, mass m operator of dimension↔

● Lagrangian:

● Schwinger pair production for (mL)2 < q2/2 finite density of →fermions hoovering outside charged horizon

● Single-particle retarded Green's and spectral function:

Sharp peaks in A dispersion relation→ω(k) of quasinormal modes

● Direc connection with ARPES experiments

Δ= 32

+m L

GR(t , x )=iΘ(t)⟨{OΨ (t , x) ,OΨ+ (0)}⟩ , A(ω , k )= 1

π Im GR(ω , k )

L⊃−ψΓμ(∂μ+14

ωμabΓab−i q Aμ)ψ−m ψψ

S.S. Lee 08; Faulkner, Liu, McGreevy, Vegh 09; Cubrovic, Zaanen, Schalm 09

Pioline, Troost 05

Fermionic spectral functions● IR CFT has operators of momentum k and dimension

and the Green's function in the IR CFT is

δk=12

+νk , νk=1√6 √m2 L2+ 3 k 2

μ2 − q2

2

ςk (ω)=c(k ) ω2νk

Faulkner, Liu,McGreevy, Vegh 09

Fermionic spectral functions● IR CFT has operators of momentum k and dimension

and the Green's function in the IR CFT is

● For (mL)2 < q2/3: Dirac eq. has static normalizable solutions Fermi surface at k→ F

● kF depends on UV physicsPhysics around Fermi surfacedoes not.

δk=12

+νk , νk=1√6 √m2 L2+ 3 k 2

μ2 − q2

2

ςk (ω)=c(k ) ω2νk

Liu, McGreevy, Vegh 09

Faulkner, Liu,McGreevy, Vegh 09

Fermionic spectral functions● Small frequency expansion around Fermi surface:

● Compare with Landau Fermi liquid theory:

Peak dispersion relation:→

GR(ω , k )=h1

k−k F− 1vF

ω−Σ(ω , k ), Σ(ω , k )=h2 ςk (ω)

GR(ω , k )= Zω−vF k ⊥+iΓ

+… with Γ ∼ ω∗2

ωc(k )=ω∗(k )−iΓ(k )

Holographic (non)-Fermi liquids

● νkF > ½: Fermi liquid. Sharp quasiparticles:

νkF = 1 similar to Landau Fermi liquid.

ω∗(k )=vF (k−k F)+… , Γ(k )ω∗(k )

∝ k ⊥2νk F

−1 → 0 , Z=h1 vF

Faulkner, Liu,McGreevy, Vegh 09

Cubrovic, Zaanen, Schalm 09

Holographic (non)-Fermi liquids

● νkF > ½: Fermi liquid. Sharp quasiparticles:

νkF = 1 similar to Landau Fermi liquid.

● νkF < ½: non-Fermi liquid. No sharp quasiparticles:

ω∗(k )=vF (k−k F)+… , Γ(k )ω∗(k )

∝ k ⊥2νk F

−1 → 0 , Z=h1 vF

ω∗(k )∼(k−k F )z , z= 12 νk F

>1 , Γ(k )ω∗

→const , Z →0

Faulkner, Liu,McGreevy, Vegh 09

Cubrovic, Zaanen, Schalm 09

Holographic (non)-Fermi liquids

● νkF > ½: Fermi liquid. Sharp quasiparticles:

νkF = 1 similar to Landau Fermi liquid.

● νkF < ½: non-Fermi liquid. No sharp quasiparticles:

● νkF = ½: marginal Fermi liquid

ω∗(k )=vF (k−k F)+… , Γ(k )ω∗(k )

∝ k ⊥2νk F

−1 → 0 , Z=h1 vF

ω∗(k )∼(k−k F )z , z= 12 νk F

>1 , Γ(k )ω∗

→const , Z →0

GR ≃h1

k ⊥+cRω log ω+c1 ω, Z ∼ 1

∣logω∗∣→ 0

Faulkner, Liu,McGreevy, Vegh 09

Cubrovic, Zaanen, Schalm 09

Varma et al. 89

Holographic (non)-Fermi liquids● Numerics of spectral function:

● Finite temperature (T«μ): Green's function pole never reaches real axis

ω«T: ω»T:

Faulkner et al, Science 239 (2010) 1045

Σ(ω , k ) ∝ T 2νk Σ(τ , k ) ∼∣ πTsin(πT τ)∣

2Δk

One-loop contribution to conductivity

● Fermi surface contribution to conductivity appears at 1-loop

● DC conductivity:

● νkF = ½: linear resistivity (strange metal, cuprates, ...)

σ(ω)= Ciω∫d k

d ω1

2 πd ω2

2πf (ω1)− f (ω2)

ω1−ω2−ω−i ϵA (ω1 , k )Λ(ω1 ,ω2 ,ω , k )Λ(ω2 ,ω1 ,ω , k )A(ω2 , k )

σ(ω→0) ∼ T−2 νk F for νk F⩽1/2

Faulkner et alScience 239 (2010) 1045

Semi-holographic Fermi liquids

● IR spectral function reproduced by simple effective model:Fermi liquid ψ coupled to fermionic fluctuations of a critical system with large z (or local critical):

Resumming the series:

L = ψ(ω−vF k )ψ + ψχ + ψχ + χ ς−1 χ with ς=⟨χ χ ⟩=c(k )ω2ν

Faulkner, Polchinski 10

⟨ ψψ⟩= 1ω−vF k−ς

Futher interactions

● Beyond probe approximation, extra fields can be responsible for instabilities

● Bosons: Bose-Einstein condensation

● Fermions: population of Fermi sea

Superconductors● Charged scalar field φ in the bulk:

Electric flux = chemical potential expect BE condensation→

L= 12 κ2(R+ 2

L2)− 14e2 Fμ ν F

μ ν−∣∇ ϕ−i Aϕ∣2−m2∣ϕ∣2−V (∣ϕ∣)

Gubser 08

ds2

L2 =− f (r)dt 2+g (r)dr2+dx2+dy2

r2 , A=γh(r)dt , ϕ=ϕ(r)

Superconductors● Charged scalar field φ in the bulk:

Electric flux = chemical potential expect BE condensation→

● At T = 0, φ condenses below AdS2 BF bound:

(Schwinger pair production)

Condensation for T < Tc ~ μ

● Macroscopically occupied ground state: U(1) broken superconductor→

L= 12 κ2(R+ 2

L2)− 14e2 Fμ ν F

μ ν−∣∇ ϕ−i Aϕ∣2−m2∣ϕ∣2−V (∣ϕ∣)

16

(m2 L2−γ2)⩽−14

Gubser 08

Pioline, Troost 05

Hartnoll, Herzog, Horowitz 08Gubser, Nellore 08Denef, Hartnoll 09

ds2

L2 =− f (r)dt 2+g (r)dr2+dx2+dy2

r2 , A=γh(r)dt , ϕ=ϕ(r)

Some properties● Condensate: mean-field second order phase transition:

● Conductivity:

● Gap:

Hartnoll, Herzog, Horowitz 08Horowitz, Roberts 08Gubser, Rocha 08

O /T c ∼ (T−T c)1/2

ωg /T c ≈ 8

Fermions in superconductors

● Coupling of bulk fermions to the scalar condensate:

● Majorana-like coupling: leading for Fermi surface gapping.Couples positive- with negative-frequency modes, as in BCS s-wave SC

● Ansatz:

● Response:

L ⊃ i Ψ(Γμ Dμ−m)Ψ+η5∗ ϕ∗ ΨcΓ5 Ψ+h.c.

Faulkner, Horowitz, McGreevy, Roberts, Vegh 09Gubser, Rocha, Talavera 09

Ψ=e−iω t+i k⋅xΨ(ω , k )(r)+eiω t−i k⋅x Ψ(−ω ,−k )(r )

R ,k =M

S ,k M

S − ,−k c GR ,k =−i M t

Fermions in superconductors● η5 = 0: Fermi surface

0=D(1) Ψ1

0=D(2)Ψ2

⇒ Ψ : ω=ω∗( k )Ψc : ω=−ω∗(−k)

Fermions in superconductors● η5 = 0: Fermi surface

● η5 ≠ 0: gap

Ψc: Ψ:

0=D(1) Ψ1+η5 ϕ Γ5 Ψ2∗

0=D(2)Ψ2+η5 ϕΓ5 Ψ1∗

0=D(1) Ψ1

0=D(2)Ψ2

⇒ Ψ : ω=ω∗( k )Ψc : ω=−ω∗(−k)

Fermions in superconductors● Numerics:

Faulkner, Horowitz, McGreevy, Roberts, Vegh 09

p-wave superconductors

● Eg: Sr2RuO4 (p+ip), 3He (p+ip at ambient pr., p at high pr.)

● Lagrangian: Einstein gravity + SU(2) gauge field

● Chemical potential ~ τ3 breaks SU(2) massive charged W→ μ

● Two ansatze: p and p+ip

Large g: p+ip unstable p. Small g ? (+CS→ )

L= 12 κ2(R+ 6

L2)− 14g2 F μ ν

a F aμ ν

A=Φ(r)τ3dt+w (r)τ1dx or τ1dx+τ2dy

Gubser, Pufu 08Ammon, Erdmenger, Grass, Kerner, O'Bannon 09Ammon, Erdmenger, Kaminski, O'Bannon 10

Pando Zayas, Reichmann 11

d-wave superconductors● Eg: cuprates. Phase diagram:

● Superconducting phase:Fermi surface is gapped

● d-wave:anisotropic gap ~ |cos 2θ|

● 4 nodes● Dirac cones at the nodes● In pseudo-gap phase: nodes open into Fermi arcs

d-wave superconductors

● The order parameter is d-wave massive charged spin-2 →field in the bulk (graviton: massless neutral)

Benini, Herzog, Yarom 10

d-wave superconductors

● The order parameter is d-wave massive charged spin-2 →field in the bulk (graviton: massless neutral)

● 1) KK reduction tower of condensing spin-2 fields→● 2) large q limit: “Fierz-Pauli” action for single spin-2 field

where

Lspin 2=−∣D ∣22∣∣

2∣D ∣2−∗ D c.c.−m2∣ ∣2−∣∣2

2R ∗ − 1d1

R∣∣2−i q F ∗

φμ≡∂ν φνμ and φ≡φμμ

Benini, Herzog, Yarom 10

Fermions in d-wave● Majorana coupling:● Spectral function: anisotropic gap, 4 nodes (Dirac cones) or

Fermi arcs

LΨ=iΨ(ΓμDμ−m)Ψ+η∗ φμ ν∗ ΨcΓμDν Ψ+h.c.

kx

kyω = 0

Benini, Herzog, Yarom 10

Bi2Sr2CaCuO8Kanigel et al, PRL 99 (2009) 157001

IR fixed point

● Look for IR fixed point of the gravity solution● Liftshitz scaling:

Existence of such solution depends on m2 and V.● IR solution has Lifshitz scaling

Electric flux emanates from scalar field, not from horizon

ds2

L2 =−dt 2

r2z +g∞dr2

r2 +dx2+dy2

r2 , A=γ h∞dtr z

, ϕ=ϕ∞

r→ λ r , t →λ z t , (x , y)→λ(x , y)

Electron stars● Bulk fermions:

● Schwinger pair production:

→ population of Fermi sea.Unbroken U(1) (Pauli exclusion) & bulk Fermi surface

L= 12 κ2(R+ 6

L2)− 14e2 Fμ ν F

μ ν−ψΓμ(Dμ−m)ψ

(mL)2⩽γ2 Pioline, Troost 05

Arsiwalla, de Boer, Papadodimas, Verline 09; Hartnoll, Polchinski, Silverstein, Tong 09Hartnoll, Tavanfar 10; Hartnoll, Hofman, Vegh 11

Electron stars● Bulk fermions:

● Schwinger pair production:

→ population of Fermi sea.Unbroken U(1) (Pauli exclusion) & bulk Fermi surface

● mL ~ γ » 1: WKB (Thomas-Fermi-Oppenheimer-Volkov) approx

T=0 equation of state:

L= 12 κ2(R+ 6

L2)− 14e2 Fμ ν F

μ ν−ψΓμ(Dμ−m)ψ

(mL)2⩽γ2 Pioline, Troost 05

Arsiwalla, de Boer, Papadodimas, Verline 09; Hartnoll, Polchinski, Silverstein, Tong 09Hartnoll, Tavanfar 10; Hartnoll, Hofman, Vegh 11

L= 12 κ2(R+ 6

L2)− 14e2 F

2+(μlocσ−ρ)

μloc=At

√g tt

, p=μloc σ−ρ , ρ=∫m

μ loc E g (E)dE , σ=∫m

μloc g (E )dE , g (E )= Eπ2 √E2−m2

Schutz 70Hartnoll, Tavanfar 10

Lifshitz scaling

● Look for IR fixed point of the gravity solution:

● Limits: e2γ2→∞ , z 1 (AdS→ 4); e2γ2→0 , z→∞ (AdS2 × R2)

● Entropy density: with large coefficient● Lifshitz geometry not geodesically complete.

Production of excited string states in IR ?

ds2

L2 =−dt 2

r2z +g∞dr2

r2 +dx2+dy2

r2 , A=γ h∞dtr z

, p= p∞ , ρ=ρ∞

S ∝ T 2/ z

AdS2 x R2 & impurity problem● What realization of AdS2×R2? Impurity problem

Spin impurity + CFT3 (Neel/VBS antiferromagnetic transition)

Spin impurity via slave fermions:

Z=∫Dzα(x , τ)DAμ(x , τ)D χ(τ)exp(−∫d τ Limp−∫ d 2 x d τ Lz , A)Limp=χ(∂

∂ τ−i Aτ(0, τ))χ

S a=1 /2 χασαβa χβ

Sachdev 10

AdS2 x R2 & impurity problem● What realization of AdS2×R2? Impurity problem

Spin impurity + CFT3 (Neel/VBS antiferromagnetic transition)

Spin impurity via slave fermions:● Impurity correlators decay with power-law in time (ω»T)

● Finite zero-temperature ground-state entropy Simp

● Same properties of local quantum critical theory of AdS2!

Z=∫Dzα(x , τ)DAμ(x , τ)D χ(τ)exp(−∫d τ Limp−∫ d 2 x d τ Lz , A)Limp=χ(∂

∂ τ−i Aτ(0, τ))χ

S a=1 /2 χασαβa χβ

⟨ S a(τ) Sb(0)⟩ ∼ δab∣ πTsin(πT τ)∣

γ

→ δab∣τ∣−γ

Sachdev 10

AdS2 x R2 & impurity problem● Impurity + Wilson-Fisher fixed point CFT3 (φa)

● Impurity + SYM CFT4:

● Similarity with semi-holographic FL description:

● DMFT

S=∫d τ Limp+∫d 2 x d τ LSYM

Limp=χa(δab ∂∂ τ

−i (Aτ(0, τ))ab−i v I (ϕI (0, τ))a

b)χb

Kachru, Karch, Yaida 09; 10

H=∫d 2 k (ϵk−μ)ck+ck + g∫d 2k d k

+ck+h.c. + H AdS2, ⟨d k (τ)d k

+(0)⟩ ∼∣τ∣−2 Δk

¿G0 ¿

G0(ω , k ) ∼ 1ω−vF∣k−k F ( k )∣

→ G g(ω , k ) ∼ 1ω−vF∣k−k F ( k )∣−g 2 ς(ω , k )

Georges, Kotliar, Krauth, Rozemberg 96Sachdev 10

Conclusions● The minimal holographic model provides an emergent local

quantum critical point.One possible realization: impurity models.

● New models of non-Fermi liquids (including margal FL)● Local quantum criticality instabilities→

Bosons BE condensation→Fermions population of Fermi sea→

● IR emergent Lifshitz scaling→● Still unstable (creation of excited string states)? What IR?

Thank you!