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!