Holographic Lattices, Metals and Insulators Jerome Gauntlett - - PowerPoint PPT Presentation

Holographic Lattices, Metals and Insulators

Jerome Gauntlett 1311.3292, 1401.5077, 1406.4742, 1407.xxxx Aristomenis Donos

At = µ(1 − r+ r )

Electric flux

T

ds2 = −Udt2 + dr2 U + r2(dx2 + dy2)

Electrically charged AdS-RN black hole (brane)

Describes holographic matter at finite charge density that is translationally invariant

µ

T=0 limit: AdS4 AdS2 × R2 UV IR d=3 CFT

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

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

[Hartnoll][Herzog]

ω ω

Re(σ)

Im(σ) More precisely

σ(ω) ∼ δ(ω) + i ω

Infinite DC conductivity arises because translation invariance implies there is no momentum dissipation

ω ∼ 0

near

J(ω) = σ(ω)E(ω) σ(ω) = σDC 1 − iωτ Re[σ] Im[σ] ω ω EF “Coherent” or “good” metal

τ −1(T)

Drude Model of transport in a metal e.g. quasi-particles and no interactions σDC = nq2τ m τ → ∞

σ(ω) ∼ δ(ω) + i ω

Arises when momentum is nearly conserved

[Hartnoll,Hofman]

Can be studied using “memory matrix” formalism

• Drude physics doesn’t require quasi-particles
• There are also “incoherent” metals without Drude peaks
• Insulators with
• Metal-insulator transitions involve dramatic reorganisation of

degrees of freedom These coherent non Fermi-liquids can be modelled in holography σDC = 0 Want to study these within holography

Holographic Lattices and metals A few examples of periodic monochromatic lattices have been studied Need to solve PDEs

[Horowitz, Santos,Tong]

E.g. add a real scalar field to Einstein-Maxwell and consider φ(r, x) ∼ λ cos(kx) r3−∆ + . . . To realise more realistic metals and/or insulators we want to consider charged black holes that explicitly break translations using a deformation of the CFT Can we simplify? Find some agreement and some differences

Plan

• Calculation of thermoelectric DC conductivity , ,

in terms of black hole horizon data

• Q-lattices can give coherent metals, incoherent metals and

insulators and transitions between them.

• Holographic Q-lattices - solve ODEs

η = s 4π For c.f. [Iqbal,Liu][Davison][Blake,Tong,Vegh][Andrade,Withers] σDC Analogous to [Policastro,Kovtun,Son,Starinets] (Aside: D=5 helical lattices [Donos,Hartnoll][Donos,Gouteraux,Kiritsis]) αDC ¯ κDC σDC

UV data IR fixed points Holographic transitions T=0 New

[Donos,Hartnoll]

Strategy: vary UV data aiming to RG destabilise AdS2 × R2 First examples M-I transitions in D=5 helical lattices We will find new M-I transitions as well as M-M transitions Comment: not necessary that is RG destabilised AdS2 × R2

AdS-RN

λ/µ

k/µ

Coherent metals with Drude peaks

[Hartnoll,Hofman][Horowitz,Santos,Tong]

AdS2 × R2

Holographic Q-lattices

• Choose so that we have an vacuum and

that AdS-RN is a solution at

• Particularly interested in cases where is periodic.

AdS4

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

Φ, V, Z ϕ = φeiχ Φ = φ2 χ

• The model has a gauge and a global symmetry

U(1) U(1) Exploit the global bulk symmetry to break translations eg if it is the phase of a complex scalar field

• Illustrative D=4 model

Analysis covers cases when is not periodic e.g. χ with φ = 0

[Azeneyagi,Takayanagi,Li][Mateos,Trancanelli][Andrade,Withers]

Homogeneous and anisotropic and periodic holographic lattices Ansatz for fields ds2 = −Udt2 + U −1dr2 + e2V1dx2

1 + e2V2dx2 2

A = a(r)dt χ = kx1, φ = φ(r) U = r2 + . . . , a = µ + q r . . . , UV expansion: e2V1 = r2 + . . . e2V2 = r2 + . . . φ = λ r3−∆ + . . . UV data: T/µ λ/µ3−∆ k/µ IR expansion: regular black hole horizon

Analytic result for DC in terms of horizon data

✓ J Q ◆ = ✓ σ αT ¯ αT ¯ κT ◆ ✓ E (rT)/T ◆ Generalised Ohm/Fourier Law: For Q-lattice black holes the DC matrices diagonal Qa = T ta − µJa Ja Electric current Heat current Apply electric fields and thermal gradients and find linear response σ, α, ¯ α, ¯ κ

Switch on constant electric field perturbation Ax = −Et + δax(r) Gauge equation of motion: Use Einstein equations and regularity at the black hole horizon to relate and to get J δgtx(r) δgrx(r) δχ(r) supplemented with E

• Calculating and

σ

¯ α J = −eV2V1Z(φ)Uδa0

x1 + qe2V1δgtx1

rµ(Z(φ)F µν) = 0 ∂r(√−gZ(φ)F rx) = 0

σ

E Perturbed metric has a timelike Killing vector Q ⇒ kµ Gµν = rµkν + . . . rµGµν = V 2 kµ Similar steps then relate and to get ¯ α ⇒

• Calculating and

Consider a source for electric and heat currents α ¯ κ gtx = tδf2(r) + δgtx1(r) Ax = tδf1(r) + δax(r) Similar steps, with a subtlety that there is both a static and time-dependent heat current

First term in is finite for AdS-Schwarzschild [Iqbal,Liu] “Pair creation” term. In general it is

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

Second term “Dissipation” term Different ground states can be dominated by first or second term

¯ κDC =  4πsT k2Φ(φ)

• r=r+

σDC =  e−V1+V2Z(φ) + q2e−V1−V2 k2Φ(φ)

• r=r+

αDC = ¯ αDC = −  4πq k2Φ(φ)

• r=r+

(J/E)Q=0 σ ≡ σ − α2¯ κ−1T

ds2 = −Udt2 + U −1dr2 + e2V1dx2

1 + e2V2dx2 2

A = adt χ = kx1

• Some general results

¯ κ α = −Ts q ¯ L ≡ ¯ κ σT ≤ s2 q2 Bound is saturated for dissipation dominated systems Define thermal conductivity at zero current κ = ¯ κ − α¯ αT/σ For dissipation dominated T=0 ground states and can have different low temperature scaling (n.b. for FL) κ ¯ κ κ = ¯ κ c.f. Wiedemann-Franz Law.

• Complementary result using memory

matrix [Mahajan,Barkeshli,Hartnoll]

UV data IR fixed point

Coherent metal phases

T=0 AdS-RN At T=0 the black holes approach in the IR AdS2 × R2

[Hartnoll, Hoffman]

Always have but κ ∼ T ¯ κ → 0, ∞ σ ∼ T 2−2∆(kIR) AdS2 × R2 perturbed by irrelevant operator and Dissipation dominated ¯ κ ∼ T 3−2∆(kIR) note: depends on RG flow kIR

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

Drude peaks at finite T Similar to what was seen by [Horowitz,Santos,Tong] Intermediate scaling?

|σ(ω)| = B ω2/3 + C

[Horowitz,Santos,Tong]

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 '

reminiscent of cuprates Scaling is not universal

Re(σ)

ω ω

1 + ω |σ|00 |σ|0

Insulating phases

AdS2 × R2 New

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

Appearance of a mid-frequency hump. Spectral weight is being transferred, consistent with sum rule What are the T=0 insulating ground states?? Focus on specific models (see also [Gouteraux])

ω

Re(σ)

New Insulating and Metallic ground states - Anisotropic

Focus on models and T=0 ground states which are solutions with 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

IR “fixed point” solutions eφ ∼ r−φ0 A ∼ radt ds2 ∼ −rudt2 + r−udr2 + rv1dx2

1 + rv2dx2 2

with exponents fixed by χ = kx1 φ → ∞ γ

• Calculate AC conductivity

Obtained using a matching argument [Faulkner,Liu,McGreevy,Vegh] with ground state correlators at T=0. Valid when T << ω << µ

• Calculate DC conductivity using analytic formula

For the scaling is obtained from the IR fixed point solutions

T << µ

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

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

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 σDC ∼ T b(γ) σAC ∼ ωc(γ) Metallic ground states are all thermal insulators: electric conductivity is due to “pair creation” ¯ κ → 0

−1

Insulators Metals

σDC ∼ T b

σAC ∼ ωc

b c

γ

Reappearance of sharp peaks not related to the charge density and Drude physics

New Insulating and Metallic ground states - Isotropic ( and )

χ1 χ2

• Holographic Q-lattices are simple and illuminating
• Coherent metallic phases with Drude peaks
• Analytic result for DC conductivity in terms of horizon data.

Summary

• Also find novel metallic phases and insulating phases
• No intermediate 2/3 scaling in AC conductivity

Metal-Insulator and Metal-Metal transitions Can be generalised to inhomogeneous lattices [to appear] Absent in another recent example [Taylor,Woodhead] We find it to be absent in inhomogeneous lattices [to appear]