Holographic Q-Lattices and Metal-Insulator Transitions Jerome - - PowerPoint PPT Presentation

Holographic Q-Lattices and Metal-Insulator Transitions

Jerome Gauntlett Aristomenis Donos

Holographic tools provide a powerful framework for investigating strongly coupled systems using weakly coupled theories of gravity Examples

• Superconducting phases - with s,p and d-wave order
• Spatially modulated phases - stripes, helices,...
• New ground states - Lifshitz, Schrodinger, hyperscaling violating, ...

Make contact with real systems? Greatly enriched our understanding of holography and of black holes in AdS spacetime

Metal - Insulator transition Dramatic reorganisation of degrees of freedom Furthermore, seen in strongly coupled context in Nature Can we realise them holographically and can we find new ground states? How do we realise holographic insulators?

[Hartnoll, Donos]

Drude Model of transport in a metal Assume have quasi-particles and ignore interactions

m d dtv = qE − m τ v J = nqv σDC = nq2τ m J = σDCE ⇒ v = qτE m

Drude Model

E = E(ω)e−iωt J = J(ω)e−iωt J(ω) = σ(ω)E(ω) σ(ω) = σDC 1 − iωτ Re[σ] Im[σ] ω ω EF

“Good” metal

τ −1(T)

= −iGJxJx(ω) ω

Interaction driven and strongly coupled

Holographic model of matter at finite charge density

L = R + 6 − 1 4F 2 + . . .

Has a unit radius vacuum with

AdS4 A = 0

which is dual to a d=3 CFT with a global U(1) symmetry Work in D=4

UV:

r → ∞

IR: black hole horizon topology and temp

r → r+ At = µ(1 − r+ r )

Electric flux

R2 T ds2 = −Udt2 + dr2 U + r2(dx2 + dy2)

Electrically charged AdS-RN black hole Holographic matter at finite charge density that is also translationally invariant is described by the

AdS4

At T=0 AdS-RN black hole interpolates between

AdS4 AdS2 × R2

UV IR Interpretation: at T=0 in the far IR a locally quantum critical fixed point emerges

Conductivity calculation

5 10 15 20 25 0.0 0.2 0.4 0.6 0.8 1.0 1.2 ΩêT Re@ΣD 5 10 15 20 25

• 0.5

0.0 0.5 1.0 1.5 2.0 wêT Im@sD

δAx = e−iωtax(r) δgtx = e−iωthtx(r) σ(ω) = −iGJxJx(ω) ω

Kramers-Kronig implies

σ(ω) ∼ δ(ω) + i ω

Delta function arises because translation invariance implies there is no momentum dissipation

[Hartnoll]

ω ω

Any translation invariant theory will have a delta function To realise more realistic metals and/or insulators need to consider charge black holes that break translations Probe brane constructions [Hartnoll,Polchinski,Silverstein,Tong]... and massive gravity [Vegh].... have also been used to study momentum dissipation Holographic lattices

Break translation invariance explicitly using a deformation of the CFT Holographic Lattices and metals A few examples of periodic monochromatic lattices have been studied: In Einstein-Maxwell theory construct black holes with Need to solve PDEs

[Horowitz, Santos,Tong]

At(r, x) ∼ µ(x) + O(1 r ) Alternatively add a real scalar field to Einstein-Maxwell and consider

φ(r, x) ∼ λ cos(kx) r3−∆ + . . . µ(x) = µ0 + λ cos(kx)

Key results

• Holographic metals with Drude peaks
• Claim that there is an intermediate scaling

|σ(ω)| = B ω2/3 + C

Reminiscent of the cuprates and moreover that is universal

[Horowitz, Santos,Tong]

At T=0 these holographic lattices (seem to) approach in the IR with a deformation by an irrelevant operator of the locally quantum critical theory with dimension

AdS2 × R2

Hartnoll, Hoffman:

O(kIR) ∆(kIR, µ0)

• Can be more than one irrelevant operator in the IR and then it is

the least irrelevant Subtleties: Used field theory and holographic arguments to predict that

T << µ

• Generically
• Used matching argument in T << ω << µ

kIR 6= kUV ρDC = σ−1

DC ∼ T 2∆(kIR)−2

UV data IR fixed points Holographic Metal-Insulator transition

[Donos,Hartnoll]

AdS2 × R2

Strategy: vary UV data aiming to RGdestabilise

AdS2 × R2

First examples M-I transitions in D=5 using helical lattices New We will find new M-I transitions as well as M-M transitions! T=0 Comment: not necessary that is RG destabilised

AdS2 × R2

AdS-RN

Holographic Q-lattices - Part I

L = R + 6 − 1 4F 2 − |∂φ|2 − m2|φ|2

Rµν = gµν(3 + m2 2 |φ|2) + ∂(µφ∂ν)φ∗ + 1

2

• F 2

µν 1 4gµνF 2

, rµF µν = 0, (r2 m2)φ = 0 ,

Homogeneous and anisotropic holographic lattice Exploit a global bulk symmetry to break translation invariance Consider a model with gauged U(1) and global U(1) in bulk:

ds2 = −Udt2 + U −1dr2 + e2V1dx2

1 + e2V2dx2 2

A = adt φ = eikx1ϕ

[No global symmetries expected in string theory?] Reminiscent of Coleman’s construction of Q-balls Equivalent to two real holographic lattices with phase shift

φ1 = cos(kx)ϕ(r) φ2 = sin(kx)ϕ(r)

Many generalisations are possible by allowing for more general global symmetries e.g. [Andrade,Withers]

U = r2 + . . . , V1 = log r + . . . V2 = log r + . . . a = µ + q r . . . , ϕ = λ r3−∆ + · · · + ϕc r∆ + . . .

UV data:

T/µ λ/µ3−∆ k/µ

IR expansion: regular black hole horizon

m2 = −3 2 ↔ ∆ = 3 + √ 3 2

UV expansion: Q-lattice black holes:

ds2 = −Udt2 + U −1dr2 + e2V1dx2

1 + e2V2dx2 2

A = adt φ = eikx1ϕ

0.001 0.002 0.005 0.010 0.020 0.050 0.100 0.005 0.010 0.020 0.050 0.100 TêΜ Ρ

Metallic phase

k/µ = 1/ √ 2

At T=0 the black holes approach in the IR

AdS2 × R2 λ/µ3−∆ = 1/2

σ−1

DC

The irrelevant operator driving the T=0 flow from the IR has There is a renormalisation of length scales from IR to UV

kIR = e−v10k ∆(k) = 1 2 + 1 2 √ 3 q 3 + 2m2 + 2k2

IR

ρ ∼ T 2∆(kIR)−2

[Hartnoll, Hoffman]

Metallic phase predict: Our black holes provide the first holographic verification of this prediction

Têm=0.100 Têm=0.0503 Têm=0.0154 Têm=0.00671 0.00 0.05 0.10 0.15 0.20 0.25 5 10 15 20 25 30 wêm Im@sD Têm=0.100 Têm=0.0503 Têm=0.0154 Têm=0.00671 0.00 0.05 0.10 0.15 0.20 0.25 10 20 30 40 50 60 wêm Re@sD

Z ∞ Re[σ(ω)]dω

Drude peaks Sum rule fixed by UV data Metallic phase

Têm=0.100 Têm=0.0503 Têm=0.0154 Têm=0.00671

0.00 0.05 0.10 0.15 0.20 0.25 0.30

• 0.9
• 0.8
• 0.7
• 0.6
• 0.5
• 0.4
• 0.3

wêm 1+w s '' s '

Intermediate scaling is NOT universal Consider holographic lattices of the form

φ ∼ λ (cos α cos kx1 + i sin α sin kx1) 1 r3−∆ + . . .

When more fields involved and need to solve PDES. However higher harmonics expected to be exponentially suppressed Metallic phase

0 ≤ α ≤ π/4

α 6= π/4

|σ(ω)| = B ω2/3 + C ??

Perhaps also not seen for other lattices:

Insulating phase

λ/µ3−∆ = 2 k/µ = 1/23/2

0.00 0.02 0.04 0.06 0.08 0.10 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 TêΜ Ρ

σ−1

DC

Têm=0.100 Têm=0.0502 Têm=0.00625 Têm=0.00118 0.00 0.05 0.10 0.15 0.20 0.25 0.0 0.2 0.4 0.6 0.8 1.0 1.2 wêm Im@sD

Têm=0.100 Têm=0.0502 Têm=0.00625 Têm=0.00118 0.00 0.05 0.10 0.15 0.20 0.25 0.30 1.0 1.5 2.0 2.5 3.0 wêm Re@sD

Notice the appearance of a mid-frequency hump. Spectral weight is being transferred, consistent with sum rule What are the T=0 insulating ground states?? Obscure in this model. Seem to have s=0, but apparently not simple scaling solutions Insulating phase

Holographic Q-lattices - Part 2 Choose so that we have an vacuum and that AdS-RN is a solution Want to construct black holes that approach novel ground states in the far IR at T=0 in addition to AdS2 × R2 Focus on the ground states which are solutions with

AdS4 φ → − ln r r → 0

and as

L → R − 3 2 ⇥ (∂φ)2 + e2φ(∂χ)2⇤ + eφ − eγφ 4 F 2

L = R − 1 2 ⇥ (∂φ)2 + Φ(φ)(∂χ)2⇤ + V (φ) − Z(φ) 4 F 2

Φ, V, Z

IR “fixed point” solutions

eφ ∼ r−φ0 A ∼ radt ds2 ∼ −rudt2 + r−udr2 + rv1dx2

1 + rv2dx2 2

χ = kx1

Comments:

• Can arise as T=0 limits of black holes with s=0
• Solutions are a kind of generalisation of hyperscaling

violating solutions with exponents fixed by k, γ

• Similar ground states also found by [Gouteraux]

• Calculate AC conductivity

Obtained using a matching argument with ground state correlators at T=0. Valid when T << ω << µ

• Calculate DC conductivity

We have derived an analytic result for all T in terms of horizon data! (see later) For the scaling is obtained from the IR fixed point solutions

T << µ

In these models we have (as we have for the metals)

b = c σAC ∼ ωc(γ) σDC ∼ T b(γ) AdS2 × R2

−1 < γ < 3 γ = 3 3 < γ

Have new type of insulating ground states

b = c > 0 b = c = 0

Novel metallic ground states with finite conductivity at T=0

b = c < 0

Have new type of incoherent metallic ground states not associated with Drude physics For fixed can find transitions between AdS2 metals and the new insulating and metallic ground states by varying strength

• f the lattice. In first case the AdS2 can be RG stable.

γ σDC ∼ T b(γ) σAC ∼ ωc(γ)

Generalise to models that have two axion like fields

L = R − 1 2 ⇥ (∂φ)2 + Φ1(φ)(∂χ1)2 + +Φ2(φ)(∂χ2)2⇤ + V (φ) − Z(φ) 4 F 2

Homogeneous and isotropic Q-lattice Holographic Q-lattices - Part 3

ds2 = −Udt2 + U −1dr2 + e2V1 ⇥ dx2

1 + dx2 2

⇤ A = adt χ1 = kx1, χ2 = kx2

−1

Insulators Metals

σDC ∼ T b σAC ∼ ωc b c γ

Parametric separation of thermal and momentum relaxation scales. Reappearance of delta functions not related to the charge density and Drude physics

Analytic result for DC in terms of horizon data Key steps: Switch on constant electric field from start

Ax = −Et + δax(r)

Use gauge equation of motion to solve for the current Einstein equations and demanding regularity at the black hole horizon to relate and

J E J

Related work [Iqbal,Liu][Blake,Tong,Vegh][Withers]

σDC =  e−V1+V2Z(φ) + q2e−V1−V2 k2Φ1(φ)

• r=r+

First term gives finite result of Iqbal, Liu for AdS-Schwarzschild Can use this to obtain analytic results for low T scaling of IR fixed points For the case that the IR approaches or other fixed points where we recover the prediction of Hartnoll and

• Hofman. Also works when

AdS2 × R2

L = R − 1 2 ⇥ (∂φ)2 + Φ(φ)(∂χ)2⇤ + V (φ) − Z(φ) 4 F 2

c = b c 6= b

ds2 = −Udt2 + U −1dr2 + e2V1dx2

1 + e2V2dx2 2

A = adt φ = eikx1ϕ

• Metals, insulators and transitions between them are interesting
• Holographic Q-lattices are a powerful and tractable tool to

study them

• Novel new insulating and metallic phases
• Analytic result for DC conductivity in terms of horizon data

Summary