holography for condensed matter phyiscselftrfsz/adscft/kormos_adscondmat.pdf · holography for...

Quantum criticalityAdS/CFT

Transport coefficientsHolographic superconductors

Conclusions

Holography for condensed matter phyiscs

Márton Kormos

SISSA, Trieste

AdS/CFT Correspondence and its ApplicationsTihany, August 28 2009

M. Kormos Holography in cond-mat

Conclusions

Meta-slide: Why do I talk about this?

I think it’s interestingMaybe the best chance to get something measurable outof holography. . . (e.g. you don’t need a heavy ion collider)very fresh

Sources: C.P. Herzog: arXiv:0904.1975 [hep-th]S.A. Hartnoll: arXiv:0903.3246 [hep-th]

Conclusions

Outline

Quantum criticalityHolographic duals for finite T ,B, µ systemsApplication: transport coefficients (conductivities andNernst effect)Holographic superconductorsConclusions: potential and limitations of the AdS/CFTapproach

Conclusions

Outline

Conclusions

Outline

Conclusions

Outline

Conclusions

Outline

Conclusions

Why condensed matter?

strongly correlated (e−-)systems; quantum criticality→ traditional approach(weakly coupled q.particles, classical order parameter)does NOT workthere are many Hamiltonians!engineering −→ experimental AdS/CFT?insights on both sides (“Which one is more fundamental?”)

Conclusions

General definitionsExamples

Quantum criticality

phase transition at T = 0

ξ ∼ (g − gc)−ν ,

∆ ∼ (g − gc)νz

At gc :t → λz t ,x → λx

Natural place to start: scale invariance, lack of q.particlesz = 1: Lorentz-invariance and spec. conform trf’smostly 2+1 D (layered superconductors)

Conclusions

Quantum criticality

ξ ∼ (g − gc)−ν ,

∆ ∼ (g − gc)νz

Conclusions

Quantum criticality

ξ ∼ (g − gc)−ν ,

∆ ∼ (g − gc)νz

Conclusions

Quantum criticality

ξ ∼ (g − gc)−ν ,

∆ ∼ (g − gc)νz

Conclusions

Quantum criticality

ξ ∼ (g − gc)−ν ,

∆ ∼ (g − gc)νz

Conclusions

Quantum criticality

ξ ∼ (g − gc)−ν ,

∆ ∼ (g − gc)νz

Conclusions

Phase diagram

phase 2

phase 1

Conclusions

Example: Insulating quantum magnet

HAF =∑〈ij〉 JijSi · Sj , Jij = J or J/g

g → 1 : Néel-order, spin wavesg →∞ : spin-singlet dimers, triplons

S[Φ] =∫

(∂Φ)2 + rΦ2 + u(Φ2)2

)M. Kormos Holography in cond-mat

Conclusions

Example: Insulating quantum magnet

HAF =∑〈ij〉 JijSi · Sj , Jij = J or J/g

g → 1 : Néel-order, spin wavesg →∞ : spin-singlet dimers, triplons

S[Φ] =∫

(∂Φ)2 + rΦ2 + u(Φ2)2

)M. Kormos Holography in cond-mat

Conclusions

Example: Bose–Hubbard model with integer filling

HBH = −t∑〈ij〉

(b†i bj + b†j bi

∑i ni (ni − 1)

U(1) symmetry: bi → eiφbi

U/t →∞ : decoupled sites, < bi >= 0, insulatorU/t → 0 : < bi >6= 0, superfluid

InsulatorSuperfluid

Quantumcritical

Classical vortices and wave oscillations of the condensate

CFT3 at T>0

Dilute Boltzmann/Landau gas of particle and holes

Conclusions

∑i ni (ni − 1)

InsulatorSuperfluid

Quantumcritical

CFT3 at T>0

Conclusions

∑i ni (ni − 1)

InsulatorSuperfluid

Quantumcritical

CFT3 at T>0

Conclusions

∑i ni (ni − 1)

InsulatorSuperfluid

Quantumcritical

CFT3 at T>0

Conclusions

QC in the real world

Nonconventional superconductorsheavy fermion metals (Kondo lattice)high Tc superconductors (?)

Conclusions

Scale invariant caseFinite T , B, µAds/CFT recipes

Scale invariant geometry

Realize the symmetries of the field theory as geometricalsymmetries

ds2 = L2(−dt2

r2z +dx idx i

r2 +dr2

)The physics of our strongly coupled field theory in the largeN limit is captured by classical dynamics about thisbackground metric.For z = 1 AdS space: more symmetries and solution of

∫dd+1x

√−g(

R +d(d − 1)

)z 6= 1 also possible. . .

Conclusions

ds2 = L2(−dt2

r2z +dx idx i

r2 +dr2

∫dd+1x

√−g(

R +d(d − 1)

Conclusions

ds2 = L2(−dt2

r2z +dx idx i

r2 +dr2

∫dd+1x

√−g(

R +d(d − 1)

Conclusions

ds2 = L2(−dt2

r2z +dx idx i

r2 +dr2

∫dd+1x

√−g(

R +d(d − 1)

Conclusions

Finite T at equilibrium

µ,T break dilatation symmetryfor large E scales it is restoredenergy scale is an extra dimension⇒ Asymptotically AdS!

Schwarzschild–AdS solution

ds2 =L2

(−f (r)dt2 +

f (r)+ dx idx i

)f (r) = 1−

Boundary at r → 0, horizon at r = r+

Conclusions

ds2 =L2

(−f (r)dt2 +

f (r)+ dx idx i

)f (r) = 1−

Conclusions

ds2 =L2

(−f (r)dt2 +

f (r)+ dx idx i

)f (r) = 1−

Conclusions

ds2 =L2

(−f (r)dt2 +

f (r)+ dx idx i

)f (r) = 1−

Conclusions

ds2 =L2

(−f (r)dt2 +

f (r)+ dx idx i

)f (r) = 1−

Conclusions

ds2 =L2

(−f (r)dt2 +

f (r)+ dx idx i

)f (r) = 1−

Conclusions

Thermodynamics of the Schwarzschild–AdS blackhole

temperature: T = d4πr+

Euclidean action: SE = − Ld−1

2κ2rd+

Vd−1T = − (4π)d Ld−1

2κ2dd Vd−1T d−1

free energy: F = −T log Z = TSE [g?] = − (4π)d Ld−1

2κ2dd Vd−1T d

entropy S = −∂F∂T = (4π)d Ld−1

2κ2dd−1 Vd−1T d−1

Conclusions

2κ2rd+

Vd−1T = − (4π)d Ld−1

2κ2dd Vd−1T d

Conclusions

2κ2rd+

Vd−1T = − (4π)d Ld−1

2κ2dd Vd−1T d

Conclusions

2κ2rd+

Vd−1T = − (4π)d Ld−1

2κ2dd Vd−1T d

Conclusions

Finite chemical potential and magnetic field

Additional structure: global U(1)

gauged U(1) in the bulk:subgroup of large gauge trf’s act non-trivially on theboundary −→ gives the global symmetry group for the FTEinstein–Maxwell theory

∫dd+1x

√−g[

d(d − 1)

)− 1

4g2 F 2]

for d = 3 consistent truncation of M-theory on AdS4 × X 7

Conclusions

∫dd+1x

√−g[

d(d − 1)

)− 1

4g2 F 2]

Conclusions

∫dd+1x

√−g[

d(d − 1)

)− 1

4g2 F 2]

Conclusions

∫dd+1x

√−g[

d(d − 1)

)− 1

4g2 F 2]

Conclusions

for d = 3 B does not break isotropy −→dyonic Reissner–Nordström–AdS black hole

f (r) = 1−

(r2+µ

2 + r4+B2)

+µ2 + r4

A = µ

[1− r

]dt + Bx dy

γ2 = (d−1)g2L2

(d−2)κ2

T = 14πr+

(3− r2

γ2 −r4+B2

)Ω = − L2

2κ2r3+

γ2 −3r4

)ρ = − 1

∂Ω∂µ = 2L2

r+γ2 , m = − 1V2

∂Ω∂B = −2L2

κ2r+Bγ2

Conclusions

f (r) = 1−

(r2+µ

2 + r4+B2)

+µ2 + r4

A = µ

[1− r

]dt + Bx dy

γ2 = (d−1)g2L2

(d−2)κ2

T = 14πr+

(3− r2

γ2 −r4+B2

)Ω = − L2

2κ2r3+

γ2 −3r4

)ρ = − 1

∂Ω∂µ = 2L2

r+γ2 , m = − 1V2

∂Ω∂B = −2L2

κ2r+Bγ2

Conclusions

f (r) = 1−

(r2+µ

2 + r4+B2)

+µ2 + r4

A = µ

[1− r

]dt + Bx dy

γ2 = (d−1)g2L2

(d−2)κ2

T = 14πr+

(3− r2

γ2 −r4+B2

)Ω = − L2

2κ2r3+

γ2 −3r4

)ρ = − 1

∂Ω∂µ = 2L2

r+γ2 , m = − 1V2

∂Ω∂B = −2L2

κ2r+Bγ2

Conclusions

f (r) = 1−

(r2+µ

2 + r4+B2)

+µ2 + r4

A = µ

[1− r

]dt + Bx dy

γ2 = (d−1)g2L2

(d−2)κ2

T = 14πr+

(3− r2

γ2 −r4+B2

)Ω = − L2

2κ2r3+

γ2 −3r4

)ρ = − 1

∂Ω∂µ = 2L2

r+γ2 , m = − 1V2

∂Ω∂B = −2L2

κ2r+Bγ2

Conclusions

Operators, expectation values

Zbulk[φ→ δφ(0)] = 〈exp(

ddx√−g(0)δφ(0)O

)〉F.T.

If (d −2)/2 < ∆ = dim[O] < d scalar, then φ near the boundary

φ(r) =( r

)d−∆φ(0) +

)∆φ(1) + . . . as r → 0 ,

(Lm)2 = ∆(∆− d)

〈O〉 = −iδZbulk[φ(0)]

δφ(0)

N→∞=

δS[φ(0)]

δφ(0)=

2∆− dL

Conclusions

)〉F.T.

φ(r) =( r

)d−∆φ(0) +

)∆φ(1) + . . . as r → 0 ,

(Lm)2 = ∆(∆− d)

δφ(0)

N→∞=

δS[φ(0)]

δφ(0)=

2∆− dL

Conclusions

)〉F.T.

φ(r) =( r

)d−∆φ(0) +

)∆φ(1) + . . . as r → 0 ,

(Lm)2 = ∆(∆− d)

δφ(0)

N→∞=

δS[φ(0)]

δφ(0)=

2∆− dL

Conclusions

General techniqueConductivity at B = 0Conductivity at B 6= 0Beyond the hydrodynamic regimeThe Nernst effect

Linear response

Next step: time and space-dependent perturbationsBroken time reversal symmetry⇔ irreversibility of fallinginto a BH

δ〈OA〉(ω, k) = GROAOB

(ω, k)δφB(0)(ω, k)

GROAOB

=δ〈OA〉δφB(0)

∣∣∣∣δφ=0

=2∆A − d

LδφA(1)

δφB(0)

Conclusions

Linear response

(ω, k)δφB(0)(ω, k)

GROAOB

=δ〈OA〉δφB(0)

∣∣∣∣δφ=0

=2∆A − d

LδφA(1)

δφB(0)

Conclusions

Linear response

(ω, k)δφB(0)(ω, k)

GROAOB

=δ〈OA〉δφB(0)

∣∣∣∣δφ=0

=2∆A − d

LδφA(1)

δφB(0)

Conclusions

Thermal and electric conductivities (B = 0)

Generalised Ohm’s Law (nonzero net charge):(〈Jx〉〈Qx〉

(σ αTαT κT

−(∇xT )/T

where the heat current is Qx = Ttx − µJx

σ(ω) =−iGR

Jx Jx(ω)

ω=−1g2L

δAx(1)

δAx(0),

α(ω)T =−iGR

Qx Jx(ω)

iρω− µσ(ω) ,

κ(ω)T =−iGR

Qx Qx(ω)

+ µ2σ(ω)

Conclusions

Thermal and electric conductivities (B = 0)

Generalised Ohm’s Law (nonzero net charge):(〈Jx〉〈Qx〉

(σ αTαT κT

−(∇xT )/T

where the heat current is Qx = Ttx − µJx

σ(ω) =−iGR

Jx Jx(ω)

ω=−1g2L

δAx(1)

δAx(0),

α(ω)T =−iGR

Qx Jx(ω)

iρω− µσ(ω) ,

κ(ω)T =−iGR

Qx Qx(ω)

+ µ2σ(ω)

Conclusions

Electric conductivity for B = 0 from AdS/CFT

0 5 10 15 20 250.0

Re@ΣD

0 5 10 15 20 25

Im@ΣD

Figure: The real and imaginary parts of the electrical conductivitycomputed via AdS/CFT. The conductivity is shown as a function offrequency. Different curves correspond to different values of thechemical potential at fixed temperature. The gap becomes deeper atlarger chemical potential.

Conclusions

Electric conductivity for B = 0 in graphene

Figure: Experimental plots of the real and imaginary parts of theelectrical conductivity in graphene as a function of frequency. Thedifferent curves correspond to different values of the gate voltage.

Conclusions

Conductivities with B 6= 0

B 6= 0 ⇒ off-diagonal elements: σ± = σxy ± iσxx etc.Ward identities

±α±Tω = (B ∓ µω)σ± − ρ ,

±κ±Tω =

(Bω∓ µ

)α±Tω − sT + mB

at scales 1/T l 1/√

B momentum is approx.conserved⇒ magneto-hydrodynamics

Conclusions

±α±Tω = (B ∓ µω)σ± − ρ ,

±κ±Tω =

(Bω∓ µ

Conclusions

±α±Tω = (B ∓ µω)σ± − ρ ,

±κ±Tω =

(Bω∓ µ

Conclusions

σxx = σQω(ω + iγ + iω2

(ω + iγ)2 − ω2c

σxy = − ρB−2iγω + γ2 + ω2

(ω + iγ)2 − ω2c

σQ =(sT )2

(ε+ P)21g2 AdS/CFT!

ωc =Bρε+ P

, γ =σQB2

Conclusions

(ω + iγ)2 − ω2c

σQ =(sT )2

ωc =Bρε+ P

, γ =σQB2

Conclusions

(ω + iγ)2 − ω2c

σQ =(sT )2

ωc =Bρε+ P

, γ =σQB2

Conclusions

Special cases

µ = 0, B = 0 : σxx = 1/g2 frequency-independent!DC limit (ω = 0) : σxx = 0, σxy = ρ

B Hall conductivity

B = 0 limit : σxx = σQ + ρ2

(ε+P)iω , σxy = 0

with ρ 6= 0 diverges for ω → 0with ω → ω + i/τ becomes finite

Conclusions

Special cases

B Hall conductivity

Conclusions

Special cases

B Hall conductivity

Conclusions

Special cases

B Hall conductivity

Conclusions

Beyond hydro: large B

Figure: Location of the cyclotron pole in the retarded Green’s functionin the complex frequency plane as a function of the dimensionlessmagnetic field and at a fixed charge density.

Conclusions

Beyond hydro: S-duality

B → ρ , ρ→ −B , σQ →1σQ

=⇒ σ+(ω)→ −1σ+(ω)

a) -4 -2 0 2 4ReHwL

b) -4 -2 0 2 4ReHwL

c) -4 -2 0 2 4ReHwL

Figure: A density plot of |σ+| as a function of complex ω. a) h = 0 andq = 1, b) h = q = 1/

√2, c) h = 1 and q = 0.

Conclusions

Beyond hydro: S-duality

B → ρ , ρ→ −B , σQ →1σQ

=⇒ σ+(ω)→ −1σ+(ω)

a) -4 -2 0 2 4ReHwL

b) -4 -2 0 2 4ReHwL

c) -4 -2 0 2 4ReHwL

Figure: A density plot of |σ+| as a function of complex ω. a) h = 0 andq = 1, b) h = q = 1/

√2, c) h = 1 and q = 0.

Conclusions

The Nernst effect

constant voltage in response to a temperature gradientwith no currentθ = −σ−1 · αin typical metal: vanishingin high Tc superconductors: large! /2006/⇒ effective d.o.f. are not particles/holes? (vortices?)

Conclusions

The Nernst effect

Conclusions

The Nernst effect

Conclusions

The Nernst effect

Conclusions

hydrodynamical limit:

θxy = −BT

iω(ω + iω2

c/γ)2 − ω2c

DC limit (ω → 0) vanishes due to translation invarianceω → ω + i/τimp:

limω→0

θxy = −BT

1/τimp

(1/τimp + ω2c/γ)2 + ω2

captures the qualitative B and T dependencewith more explicit holographic model: strong dependenceof τimp on B, ρ

Conclusions

θxy = −BT

iω(ω + iω2

c/γ)2 − ω2c

limω→0

θxy = −BT

1/τimp

(1/τimp + ω2c/γ)2 + ω2

Conclusions

θxy = −BT

iω(ω + iω2

c/γ)2 − ω2c

limω→0

θxy = −BT

1/τimp

(1/τimp + ω2c/γ)2 + ω2

Conclusions

θxy = −BT

iω(ω + iω2

c/γ)2 − ω2c

limω→0

θxy = −BT

1/τimp

(1/τimp + ω2c/γ)2 + ω2

Conclusions

θxy = −BT

iω(ω + iω2

c/γ)2 − ω2c

limω→0

θxy = −BT

1/τimp

(1/τimp + ω2c/γ)2 + ω2

Conclusions

Instability of the BHConductivity

Holographic superconductors

describe a symmetry breaking phase transitiontraditional theories: charged q.particles glued together byother q.particleshigh Tc anomalies −→ “superconductivity withoutelectrons”

Conclusions

Ingredients

spontaneous breaking of U(1) symmetry by thecondensation of a charged operator⇒ need a charged scalar field in the bulk!(or U(1)→ SU(2))

L = 12κ2

(R + d(d−1)

)− 1

4g2 F 2−|∇φ− iqAφ|2−m2|φ|2−V (|φ|)

How to embed this into string theory? Now V is arbitrary. . .to have Tc : Reissner–Nordström BH

Conclusions

Ingredients

L = 12κ2

(R + d(d−1)

)− 1

4g2 F 2−|∇φ− iqAφ|2−m2|φ|2−V (|φ|)

Conclusions

Ingredients

L = 12κ2

(R + d(d−1)

)− 1

4g2 F 2−|∇φ− iqAφ|2−m2|φ|2−V (|φ|)

Conclusions

Ingredients

L = 12κ2

(R + d(d−1)

)− 1

4g2 F 2−|∇φ− iqAφ|2−m2|φ|2−V (|φ|)

Conclusions

bulk scalar turns on continuously: instability of the BHunder perturbations of the scalar fieldtwo effects:

effective mass from interaction with the EM filed:m2

eff = m2 + gttA2t = m2 − r2

L2 (1− r/r+)2

near-horizon AdS2 throat

(don’t need a φ4 term!)Breitenlohner–Freedman bound −→

q2γ2 ≥ 3 + 2∆(∆− 3)