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

Electrically charged AdS-RN black hole (brane) Describes holographic matter at finite charge density that is translationally invariant ds 2 = − Udt 2 + dr 2 U + r 2 ( dx 2 + dy 2 ) A t = µ (1 − r + r ) d=3 CFT µ T Electric flux T=0 limit: AdS 2 × R 2 AdS 4 IR UV

Conductivity calculation σ ( ω ) = − iG J x J x ( ω ) ω δ A x = e − i ω t a x ( r ) δ g tx = e − i ω t h tx ( r ) [Hartnoll][Herzog] Re( σ ) Im( σ ) 1.2 2.0 1.0 1.5 0.8 1.0 Re @ Σ D 0.6 Im @ s D 0.5 0.4 0.0 0.2 - 0.5 0.0 ω ω 0 5 10 15 20 25 0 5 10 15 20 25 w ê T Ω ê T σ ( ω ) ∼ δ ( ω ) + i near More precisely ω ∼ 0 ω Infinite DC conductivity arises because translation invariance implies there is no momentum dissipation

Drude Model of transport in a metal e.g. quasi-particles and no interactions J ( ω ) = σ ( ω ) E ( ω ) σ DC = nq 2 τ σ DC σ ( ω ) = 1 − i ωτ m Im [ σ ] Re [ σ ] τ − 1 ( T ) ω ω E F σ ( ω ) ∼ δ ( ω ) + i “Coherent” or “good” metal τ → ∞ ω

• Drude physics doesn’t require quasi-particles Arises when momentum is nearly conserved Can be studied using “memory matrix” formalism [Hartnoll,Hofman] These coherent non Fermi-liquids can be modelled in holography • There are also “incoherent” metals without Drude peaks • Insulators with σ DC = 0 • Metal-insulator transitions involve dramatic reorganisation of degrees of freedom Want to study these within holography

Holographic Lattices and metals 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 A few examples of periodic monochromatic lattices have been studied [Horowitz, Santos,Tong] E.g. add a real scalar field to Einstein-Maxwell and consider φ ( r, x ) ∼ λ cos( kx ) + . . . r 3 − ∆ Need to solve PDEs Can we simplify? Find some agreement and some differences

Plan • Holographic Q-lattices - solve ODEs (Aside: D=5 helical lattices [Donos,Hartnoll][Donos,Gouteraux,Kiritsis] ) • Calculation of thermoelectric DC conductivity , , ¯ σ DC α DC κ DC in terms of black hole horizon data η = s Analogous to [Policastro,Kovtun,Son,Starinets] 4 π For c.f. [Iqbal,Liu][Davison][Blake,Tong,Vegh][Andrade,Withers] σ DC • Q-lattices can give coherent metals, incoherent metals and insulators and transitions between them.

Holographic transitions UV data k/µ λ /µ AdS-RN T=0 New IR fixed points AdS 2 × R 2 Coherent metals with Drude peaks [Hartnoll,Hofman][Horowitz,Santos,Tong] Strategy: vary UV data aiming to RG destabilise AdS 2 × R 2 First examples M-I transitions in D=5 helical lattices [Donos,Hartnoll] We will find new M-I transitions as well as M-M transitions Comment: not necessary that is RG destabilised AdS 2 × R 2

Holographic Q-lattices • Illustrative D=4 model L = R − 1 + V ( φ ) − Z ( φ ) ( ∂φ ) 2 + Φ ( φ )( ∂χ ) 2 ⇤ F 2 ⇥ 2 4 • Choose so that we have an vacuum and AdS 4 Φ , V, Z that AdS-RN is a solution at φ = 0 • Particularly interested in cases where is periodic. χ ϕ = φ e i χ eg if it is the phase of a complex scalar field with Φ = φ 2 Analysis covers cases when is not periodic e.g. χ [Azeneyagi,Takayanagi,Li][Mateos,Trancanelli][Andrade,Withers] • The model has a gauge and a global symmetry U (1) U (1) Exploit the global bulk symmetry to break translations

Ansatz for fields ds 2 = − Udt 2 + U − 1 dr 2 + e 2 V 1 dx 2 1 + e 2 V 2 dx 2 2 A = a ( r ) dt χ = kx 1 , φ = φ ( r ) UV expansion: U = r 2 + . . . , e 2 V 1 = r 2 + . . . e 2 V 2 = r 2 + . . . λ a = µ + q φ = r 3 − ∆ + . . . r . . . , IR expansion: regular black hole horizon Homogeneous and anisotropic and periodic holographic lattices UV data: T/µ λ /µ 3 − ∆ k/µ

Analytic result for DC in terms of horizon data Apply electric fields and thermal gradients and find linear response Generalised Ohm/Fourier Law: ✓ J ◆ ✓ ◆ ✓ ◆ α T E σ = Q α T ¯ κ T ¯ � ( r T ) /T Electric current J a Heat current Q a = T ta − µJ a For Q-lattice black holes the DC matrices diagonal σ , α , ¯ α , ¯ κ

• Calculating and ¯ α σ Switch on constant electric field perturbation A x = − Et + δ a x ( r ) supplemented with δ g rx ( r ) δ g tx ( r ) δχ ( r ) Gauge equation of motion: ∂ r ( √− gZ ( φ ) F rx ) = 0 r µ ( Z ( φ ) F µ ν ) = 0 ⇒ J = − e V 2 � V 1 Z ( φ ) U δ a 0 x 1 + qe � 2 V 1 δ g tx 1 Use Einstein equations and regularity at the black hole horizon to relate and to get E J σ Perturbed metric has a timelike Killing vector k µ r µ G µ ν = � V G µ ν = r µ k ν + . . . 2 k µ ⇒ Similar steps then relate and to get ¯ Q E α

• Calculating and ¯ κ α Consider a source for electric and heat currents g tx = t δ f 2 ( r ) + δ g tx 1 ( r ) A x = t δ f 1 ( r ) + δ a x ( r ) Similar steps, with a subtlety that there is both a static and time-dependent heat current

L = R − 1 + V ( φ ) − Z ( φ ) ( ∂φ ) 2 + Φ ( φ )( ∂χ ) 2 ⇤ F 2 ⇥ 2 4 ds 2 = − Udt 2 + U − 1 dr 2 + e 2 V 1 dx 2 1 + e 2 V 2 dx 2 2 χ = kx 1 A = adt  4 π sT �  4 π q � κ DC = ¯ α DC = ¯ α DC = − k 2 Φ ( φ ) k 2 Φ ( φ ) r = r + r = r + e − V 1 + V 2 Z ( φ ) + q 2 e − V 1 − V 2  � σ DC = k 2 Φ ( φ ) r = r + First term in is finite for AdS-Schwarzschild [Iqbal,Liu] σ “Pair creation” term. In general it is ≡ σ − α 2 ¯ κ − 1 T ( J/E ) Q =0 Second term “Dissipation” term Different ground states can be dominated by first or second term

• Some general results Define thermal conductivity at zero current α T/ σ κ = ¯ κ − α ¯ For dissipation dominated T=0 ground states and can ¯ κ κ have different low temperature scaling (n.b. for FL) κ = ¯ κ Bound is saturated for dissipation σ T ≤ s 2 ¯ κ • dominated systems ¯ L ≡ q 2 c.f. Wiedemann-Franz Law. Complementary result using memory matrix [Mahajan,Barkeshli,Hartnoll] α = − Ts ¯ κ • q

Coherent metal phases UV data Dissipation dominated AdS-RN T=0 IR fixed point AdS 2 × R 2 At T=0 the black holes approach in the IR AdS 2 × R 2 perturbed by irrelevant operator [Hartnoll, Hoffman] σ ∼ T 2 − 2 ∆ ( k IR ) note: depends on RG flow k IR and κ ∼ T 3 − 2 ∆ ( k IR ) Always have but κ → 0 , ∞ ¯ κ ∼ T ¯

Drude peaks at finite T Re( σ ) 60 Similar to what was seen T ê m= 0.100 50 by [Horowitz,Santos,Tong] T ê m= 0.0503 40 T ê m= 0.0154 Re @ s D T ê m= 0.00671 30 20 10 ω 0 1 + ω | σ | 00 0.00 0.05 0.10 0.15 0.20 0.25 w ê m | σ | 0 Intermediate scaling? - 0.3 T ê m= 0.100 T ê m= 0.0503 - 0.4 [Horowitz,Santos,Tong] T ê m= 0.0154 - 0.5 B s '' s ' T ê m= 0.00671 | σ ( ω ) | = ω 2 / 3 + C - 0.6 1 +w - 0.7 reminiscent of cuprates - 0.8 - 0.9 Scaling is not universal 0.00 0.05 0.10 0.15 0.20 0.25 0.30 w ê m ω

Insulating phases 3.0 Re( σ ) T ê m= 0.100 T ê m= 0.0502 2.5 T ê m= 0.00625 T ê m= 0.00118 Re @ s D 2.0 1.5 AdS 2 × R 2 1.0 New 0.00 0.05 0.10 0.15 0.20 0.25 0.30 w ê m ω 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] )

New Insulating and Metallic ground states - Anisotropic L = R − 1 + V ( φ ) − Z ( φ ) ( ∂φ ) 2 + Φ ( φ )( ∂χ ) 2 ⇤ F 2 ⇥ 2 4 Focus on models and T=0 ground states which are solutions with as φ → ∞ r → 0 + e φ − e γφ L → R − 3 ( ∂φ ) 2 + e 2 φ ( ∂χ ) 2 ⇤ and 4 F 2 ⇥ 2 IR “fixed point” solutions ds 2 ∼ − r u dt 2 + r − u dr 2 + r v 1 dx 2 1 + r v 2 dx 2 2 e φ ∼ r − φ 0 χ = kx 1 A ∼ r a dt with exponents fixed by 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 σ AC ∼ ω c ( γ ) • Calculate DC conductivity using analytic formula For the scaling is obtained from the IR fixed point solutions T << µ σ DC ∼ T b ( γ ) In these models we have b = c (as we have for the coherent metals) AdS 2 × R 2

σ AC ∼ ω c ( γ ) σ DC ∼ T b ( γ ) Have new type of insulating ground states b = c > 0 Have new type of incoherent metallic b = c < 0 ground states not associated with Drude physics Novel metallic ground states with b = c = 0 finite conductivity at T=0 Metallic ground states are all thermal insulators: κ → 0 ¯ electric conductivity is due to “pair creation”

New Insulating and Metallic ground states - Isotropic ( and ) χ 2 χ 1 σ AC ∼ ω c σ DC ∼ T b b c γ − 1 Metals Insulators Reappearance of sharp peaks not related to the charge density and Drude physics