Holographic Lattices Jerome Gauntlett with Aristomenis Donos - PowerPoint PPT Presentation

Holographic Lattices Jerome Gauntlett with Aristomenis Donos Christiana Pantelidou

Holographic Lattices CFT with a deformation by an operator that breaks translation invariance Why? • Translation invariance momentum is conserved, ⇒ hence no dissipation and hence DC response are infinite. To model more realistic metallic behaviour or insulating behaviour we can use a lattice • The lattice deformation can lead to novel ground states at T=0. Can also model metal-insulator transitions • Formal developments: thermo-electric DC [Donos,Gauntlett] conductivities in terms of black hole horizon data η = s Analogous to [Policastro,Kovtun,Son,Starinets] 4 π

Plan • Drude physics • Lattice with global U(1) symmetry and . In Einstein- µ ( x ) Maxwell theory. Coherent metals. • Q-lattices, using scalars and global symmetry. Can give coherent metals, incoherent metals and insulators and transitions between them. • Helical lattices in D=5 pure gravity. Universal deformation. Coherent metals. Comments on calculating Greens functions

Drude Model of transport in a metal Quasi-particle interactions ignored m d dtv = qE − m ⇒ v = q τ E τ v m J = nqv σ DC = nq 2 τ J = σ DC E m

E = E ( ω ) e − i ω t J ( ω ) = σ ( ω ) E ( ω ) J = J ( ω ) e − i ω t σ DC = q 2 τ σ DC σ ( ω ) = 1 − i ωτ m Re [ σ ] “Coherent” or “good” metal τ − 1 ( T ) ω E F σ ( ω ) ∼ δ ( ω ) + i When τ → ∞ ω

• Drude physics doesn’t require quasi-particles Coherent metals arise when momentum is nearly conserved [Hartnoll,Hofman] ω = − i Pole on negative imaginary axis near origin τ • Similar comments apply to thermal conductivity Q = � ¯ κ r T • There are also “incoherent” metals without Drude peaks Not dominated by single time scale τ Of particular interest to realise these in holography • Insulators with at T=0 σ DC = ¯ κ DC = 0

Holographic CFTs at finite charge density Focus on d=3 CFT and consider D=4 Einstein-Maxwell theory: Z R + 6 − 1 h i 4 F 2 + . . . d 4 x √− g S = Admits vacuum d=3 CFT with global U(1) AdS 4 ↔

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

By perturbing the black hole and using holographic tools we can calculate the electric conductivity and find a delta function at [Hartnoll] ω = 0 Construct lattice black holes dual to CFT with µ ( x ) A t ( x, r ) ∼ µ ( x ) + O (1 r ) r → ∞ g µ ν ( x, r ) Need to solve PDEs in two variables e.g. Monochromatic lattice: [Horowitz, Santos,Tong] µ ( x ) = µ + A cos kx [Donos,Gauntlett] After constructing black holes, one can perturb, again solving PDEs, to extract thermo-electric conductivities

Find Drude physics at finite T 60 ▽ ▽ ▽ ▽ ▽ ▽ ▽ T / μ = 0.47 T / μ = 0.058 ▽ ● ○ ▽ ▽ △ △ △ ▽ 50 △ △ ▽ △ △ ▽ △ T / μ = 0.22 T / μ = 0.039 △ ■ □ ▽ △ ◇ ◇ △ ▽ ◇ ◇ ◇ △ ◇ ▽ ◇ △ ◇ ▽ △ ◇ ◇ △ ▽ ◇ T / μ = 0.14 T / μ = 0.025 △ ◇ ◆ ◇ 40 ▽ ◇ △ ◇ △ ◇ ◇ △ ◇ ▽ Re ( σ ) ◇ □ □ □ □ □ ◇ □ □ △ T / μ = 0.097 T / μ = 0.02 □ □ ▲ △ □ □ ▽ □ ◇ □ □ △ □ □ □ 30 ◇ □ ▽ □ △ ○ ○ ○ ○ ○ ○ ○ □ ◇ T / μ = 0.08 T / μ = 0.015 ○ ○ ○ ▼ ▽ ○ ○ ▽ ○ △ ○ □ ○ ○ ◇ ○ ○ □ ○ ○ ○ ▼ ▽ △ ▼ ▼ ▼ ▼ ▼ ◇ ▼ ▼ ▼ ▼ □ ▼ ▼ ▼ ▼ ▼ △ ▽ ◇ ▼ □ ▲ ▼ ▲ ▲ ▲ ▼ ○ ▲ ▲ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▲ ▲ ▲ 20 ▲ ▲ ▲ △ ○ ○ ○ ○ ○ ○ ○ □ ▲ ◇ ▲ ▽ ▲ ▲ ▲ ▲ □ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ◇ △ ▽ □ ◇ △ ◆ ◆ □ □ ◆ ◆ ◆ ▽ ◆ ◆ ◆ ◆ ◆ ◆ ◇ ◆ △ ◆ ◆ ◆ ◆ ▽ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◇ △ □ ▽ ○ ■■■■■■■■■■■■■■■■■■■ ◇ □ △ ○ ▽ ▼ 10 □ ◇ △ ○ ▼ ▲ □ □ □ □ □ □ □ ▽ ■ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ○ ▼ △ △ △ △ △ △ △ ■ ▲ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ▽ ▽ ▽ ▽ ▽ ▽ ▽ ■ ○ ▼ ▲ ■ ○ ▼ ◆ ■ ▲ ○ ■ ▼ ▲ ◆ ○ ■ ▼ ▲ ◆ ■ ○ ▼ ■ ▲ ◆ ○ ▼ ■ ▲ ◆ ▼ ▲ ■ □ ○ ◆ ▲ ◇ ▼ ■ ◆ △ □ ◆ ● ○ ▲ ■ ▽ ◇ ▼ ◆ △ □ ○ ■ ◆ ● ▲ ▽ ◇ ▼ ■ △ □ ● ○ ▲ ◆ ■ ▽ ◇ □ ○ ▼ ■ △ ● ▼ ▲ ◆ ■ ▽ ◇ □ ○ ● ▲ ◆ ■ △ ◇ □ ○ ● ▼ ■ ▽ △ ◇ ○ ● ▼ ▲ ◆ 0 ▽ △ □ ○ ● ▼ ▲ ▽ △ ◇ ◇ □ □ ○ ● ▼ ● ▲ ▽ △ ◇ □ ○ ▼ ▽ ▽ △ △ ◇ ◇ □ □ ○ ○ ▽ ▽ △ △ ◇ ◇ □ □ ▽ ▽ △ ▽ △ ◇ △ ◇ ◇ □ ▽ ▽ △ ▽ △ ◇ △ △ ◇ ▽ ▽ 0.00 0.01 0.02 0.03 0.04 0.05 ω / μ

Coherent metal phases UV data Can be understood by A/µ k/µ analysing T=0 solutions: AdS-RN 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 with ∆ ( k IR ) ≥ 1 Don’t find exceptions to this behaviour even for dirty lattices e.g. 10 X µ ( x ) = 1 + A cos( n k x + θ n ) , n =1

Holographic Q-lattices [Donos,Gauntlett] • Illustrative D=4 model L = R − 1 2 | ∂ϕ | 2 + V ( | ϕ | ) − Z ( | ϕ | ) F 2 4 • Choose so that AdS-RN is a solution at ϕ = 0 V, Z • Now in CFT. Want to build a holographic lattice ϕ ↔ O by deforming with the operator O • The model has a gauge and a global symmetry U (1) U (1) Exploit the global bulk symmetry to break translations so that we only have to solve ODEs

Ansatz for fields ds 2 = − Udt 2 + U − 1 dr 2 + e 2 V 1 dx 2 + e 2 V 2 dy 2 A t = a ( r ) ϕ ( r, x ) = φ ( r ) e ikx UV expansion: U = r 2 + . . . , e 2 V 2 = r 2 + . . . e 2 V 1 = r 2 + . . . λ a = µ + q φ = r 3 − ∆ + . . . r . . . , Homogeneous and anisotropic and periodic holographic lattices UV data: T/µ λ /µ 3 − ∆ k/µ

For small deformations from AdS-RN we find Drude peaks corresponding to coherent metals. k/µ λ /µ This can be understood AdS-RN by examining T=0 behaviour of solutions: AdS 2 × R 2 New For larger deformations, for specific models, we find a transition to new behaviour. The new ground states can be both insulators and also incoherent metals! See also: [Gouteraux][Andrade,Withers]

D=4 CFTs with a Helical Twist [Donos,Gauntlett,Pantelidou] Study a universal helical deformation that applies to all d=4 CFTS First recall the Bianchi Lie algebra V II 0 [ L 1 , L 2 ] = − kL 3 [ L 1 , L 3 ] = kL 2 [ L 2 , L 3 ] = 0 L 1 = ∂ x 1 + k ( x 3 ∂ x 2 − x 2 ∂ x 3 ) L 3 = ∂ x 3 L 2 = ∂ x 2 x 2 x 3 x 1

Useful to introduce the left-invariant one-forms ω 1 = dx 1 ω 2 = cos ( kx 1 ) dx 2 − sin ( kx 1 ) dx 3 , ω 3 = cos ( kx 1 ) dx 2 + sin ( kx 1 ) dx 3 We want to explicitly break the spatial symmetry ISO (3) of the CFT down to Bianchi V II 0 Achieve by introducing suitable sources for the stress tensor Equivalently, consider CFT not on but on R 1 , 3 ds 2 = − dt 2 + ω 2 1 + e 2 α 0 ω 2 2 + e − 2 α 0 ω 2 3 with parametrising the deformation k, α 0

Study in holography by considering Z d 5 x √− g ( R + 12) S = This is a consistent truncation of all solutions in AdS 5 × M string/M-theory. Hence analysis applies to entire class of CFTs Ansatz ds 2 = − g f 2 dt 2 + g − 1 dr 2 + h 2 ω 2 1 + r 2 � e 2 α ω 2 2 + e − 2 α ω 2 � 3 Equations of motion f 0 = . . . , g 0 = . . . , h 00 = . . . , α 00 = . . . f = 1 , g = r 2 − r 4 AdS-Schwarzschild: + r 2 , h = r, α = 0

Expand functions at UV boundary k 2 k 4 12 r 2 (1 − cosh 4 α 0 ) − c h f =1 + r 4 + 96 r 4 (3 + 4 cosh 4 α 0 − 7 cosh 8 α 0 ) + log r () + . . . , 1 − k 2 ✓ ◆ 6 r 2 (1 − cosh 4 α 0 ) − M g = r 2 r 4 + log r () + . . . , 1 − k 2 ✓ ◆ 4 r 2 (1 − cosh 4 α 0 ) + c h h = r r 4 + log r () + . . . , α = α 0 − k 2 4 r 2 sinh 4 α 0 + c α r 4 + log r () + . . . . Source parameters: α 0 , k Vev parameters: c h , c α , M Together these give of helically deformed CFT T µ ν Log terms arise because of conformal anomaly µ = k 4 3 (cosh(8 α 0 ) − cosh(4 α 0 )) T µ

Boundary conditions in the IR - smooth black hole horizon g = g + ( r − r + ) + · · · , f = f + + · · · , h = h + + · · · , α = α + + · · · with g + = 4 r + Parameter count: expect two parameter family of black holes labelled by , (for fixed dynamical scale) k/T α 0