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 Make contact with real systems? Greatly enriched our understanding of holography and of black holes in AdS spacetime Examples • Superconducting phases - with s,p and d-wave order • Spatially modulated phases - stripes, helices,... • New ground states - Lifshitz, Schrodinger, hyperscaling violating, ...

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 ⇒ v = q τ E τ v m J = nqv σ DC = nq 2 τ J = σ DC E m

Drude Model J ( ω ) = σ ( ω ) E ( ω ) E = E ( ω ) e − i ω t = − iG J x J x ( ω ) σ DC σ ( ω ) = 1 − i ωτ J = J ( ω ) e − i ω t ω Im [ σ ] Re [ σ ] τ − 1 ( T ) ω ω E F “Good” metal

Interaction driven and strongly coupled

Holographic model of matter at finite charge density Work in D=4 L = R + 6 − 1 4 F 2 + . . . Has a unit radius vacuum with A = 0 AdS 4 which is dual to a d=3 CFT with a global U(1) symmetry

Holographic matter at finite charge density that is also translationally invariant is described by the Electrically charged AdS-RN black hole ds 2 = − Udt 2 + dr 2 U + r 2 ( dx 2 + dy 2 ) A t = µ (1 − r + r ) UV: AdS 4 r → ∞ Electric flux IR: black hole horizon r → r + topology and R 2 temp T

At T=0 AdS-RN black hole interpolates between AdS 2 × R 2 AdS 4 UV IR Interpretation: at T=0 in the far IR a locally quantum critical fixed point emerges

Conductivity calculation δ A x = e − i ω t a x ( r ) σ ( ω ) = − iG J x J x ( ω ) δ g tx = e − i ω t h tx ( r ) ω [Hartnoll] 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 0 5 10 15 20 25 0 5 10 15 20 25 ω ω w ê T Ω ê T σ ( ω ) ∼ δ ( ω ) + i Kramers-Kronig implies ω Delta function arises because translation invariance implies there is no momentum dissipation

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 Holographic lattices Probe brane constructions [Hartnoll,Polchinski,Silverstein,Tong]... and massive gravity [Vegh].... have also been used to study momentum dissipation

Holographic Lattices and metals Break translation invariance explicitly using a deformation of the CFT A few examples of periodic monochromatic lattices have been studied: [Horowitz, Santos,Tong] In Einstein-Maxwell theory construct black holes with A t ( r, x ) ∼ µ ( x ) + O (1 µ ( x ) = µ 0 + λ cos( kx ) r ) Need to solve PDEs Alternatively add a real scalar field to Einstein-Maxwell and consider φ ( r, x ) ∼ λ cos( kx ) + . . . r 3 − ∆

Key results [Horowitz, Santos,Tong] • Holographic metals with Drude peaks • Claim that there is an intermediate scaling B | σ ( ω ) | = ω 2 / 3 + C and moreover that is universal Reminiscent of the cuprates

AdS 2 × R 2 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 ∆ ( k IR , µ 0 ) O ( k IR ) Hartnoll, Hoffman: Used field theory and holographic arguments to predict that ρ DC = σ − 1 DC ∼ T 2 ∆ ( k IR ) − 2 T << µ Subtleties: • Can be more than one irrelevant operator in the IR and then it is the least irrelevant • Generically k IR 6 = k UV • Used matching argument in T << ω << µ

Holographic Metal-Insulator transition UV data AdS-RN T=0 AdS 2 × R 2 IR fixed points New AdS 2 × R 2 Strategy: vary UV data aiming to RGdestabilise First examples M-I transitions in D=5 using 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 - Part I Exploit a global bulk symmetry to break translation invariance Consider a model with gauged U(1) and global U(1) in bulk: L = R + 6 − 1 4 F 2 − | ∂φ | 2 − m 2 | φ | 2 R µ ν = g µ ν ( � 3 + m 2 2 | φ | 2 ) + ∂ ( µ φ∂ ν ) φ ∗ + 1 F 2 µ ν � 1 4 g µ ν F 2 � � , 2 r µ F µ ν = 0 , ( r 2 � m 2 ) φ = 0 , ds 2 = − Udt 2 + U − 1 dr 2 + e 2 V 1 dx 2 1 + e 2 V 2 dx 2 2 A = adt φ = e ikx 1 ϕ Homogeneous and anisotropic holographic lattice

Reminiscent of Coleman’s construction of Q-balls Equivalent to two real holographic lattices with phase shift φ 1 = cos( kx ) ϕ ( r ) φ 2 = sin( kx ) ϕ ( r ) Many generalisations are possible by allowing for more general global symmetries e.g. [Andrade,Withers] [No global symmetries expected in string theory?]

Q-lattice black holes: ds 2 = − Udt 2 + U − 1 dr 2 + e 2 V 1 dx 2 1 + e 2 V 2 dx 2 2 A = adt φ = e ikx 1 ϕ UV expansion: U = r 2 + . . . , V 1 = log r + . . . V 2 = log r + . . . a = µ + q r . . . , r 3 − ∆ + · · · + ϕ c λ ϕ = r ∆ + . . . √ m 2 = − 3 ∆ = 3 + 3 ↔ 2 2 λ /µ 3 − ∆ UV data: T/µ k/µ IR expansion: regular black hole horizon

Metallic phase √ λ /µ 3 − ∆ = 1 / 2 k/µ = 1 / 2 σ − 1 0.100 DC 0.050 0.020 Ρ 0.010 0.005 0.001 0.002 0.005 0.010 0.020 0.050 0.100 T ê Μ At T=0 the black holes approach in the IR AdS 2 × R 2

Metallic phase The irrelevant operator driving the T=0 flow from the IR has ∆ ( k ) = 1 1 q 3 + 2 m 2 + 2 k 2 2 + IR √ 2 3 There is a renormalisation of length scales from IR to UV k IR = e − v 10 k predict: [Hartnoll, Hoffman] ρ ∼ T 2 ∆ ( k IR ) − 2 Our black holes provide the first holographic verification of this prediction

Metallic phase Drude peaks 60 30 T ê m= 0.100 T ê m= 0.100 50 25 T ê m= 0.0503 T ê m= 0.0503 40 20 T ê m= 0.0154 T ê m= 0.0154 Re @ s D Im @ s D T ê m= 0.00671 T ê m= 0.00671 30 15 20 10 10 5 0 0 0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.05 0.10 0.15 0.20 0.25 w ê m w ê m Sum rule Z ∞ Re [ σ ( ω )] d ω fixed by UV data 0

Metallic phase - 0.3 T ê m= 0.100 T ê m= 0.0503 B ω 2 / 3 + C ?? - 0.4 | σ ( ω ) | = T ê m= 0.0154 - 0.5 s '' s ' T ê m= 0.00671 - 0.6 1 +w - 0.7 - 0.8 - 0.9 0.00 0.05 0.10 0.15 0.20 0.25 0.30 w ê m Intermediate scaling is NOT universal Perhaps also not seen for other lattices: Consider holographic lattices of the form 1 φ ∼ λ (cos α cos kx 1 + i sin α sin kx 1 ) r 3 − ∆ + . . . 0 ≤ α ≤ π / 4 When more fields involved and need to solve PDES. α 6 = π / 4 However higher harmonics expected to be exponentially suppressed

Insulating phase λ /µ 3 − ∆ = 2 k/µ = 1 / 2 3 / 2 0.70 0.65 σ − 1 DC 0.60 0.55 Ρ 0.50 0.45 0.40 0.35 0.00 0.02 0.04 0.06 0.08 0.10 T ê Μ

Insulating phase 1.2 3.0 T ê m= 0.100 1.0 T ê m= 0.0502 2.5 T ê m= 0.00625 0.8 T ê m= 0.00118 Im @ s D Re @ s D 2.0 0.6 T ê m= 0.100 0.4 T ê m= 0.0502 1.5 T ê m= 0.00625 0.2 T ê m= 0.00118 1.0 0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.15 0.20 0.25 w ê m w ê m 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

Holographic Q-lattices - Part 2 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 that AdS 4 Φ , V, Z 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 AdS 2 × R 2 Focus on the ground states which are solutions with φ → − ln r as r → 0 and + e φ − e γφ L → R − 3 ( ∂φ ) 2 + e 2 φ ( ∂χ ) 2 ⇤ 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 χ = kx 1 e φ ∼ r − φ 0 A ∼ r a dt with exponents fixed by k, γ Comments: • Solutions are a kind of generalisation of hyperscaling violating solutions • Can arise as T=0 limits of black holes with s=0 • 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 σ AC ∼ ω c ( γ ) • 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 σ DC ∼ T b ( γ ) In these models we have b = c (as we have for the metals) AdS 2 × R 2