A simple holographic model of momentum relaxation Tom as Andrade - PowerPoint PPT Presentation

A simple holographic model of momentum relaxation Tom´ as Andrade (Durham U) January 28th, 2014, Oxford in collaboration with Ben Withers (Southampton)

“Hasn’t it occurred to you to suspect that behind that Mondrian could a Viera da Silva reality start?” Hopscotch, J. Cort´ azar.

Intro1: AdS/CMT Use AdS/CFT to understand condensed matter systems. Gravity in M with AdS boundary conditions � Field Theory that lives on ∂ M

Intro1: AdS/CMT Use AdS/CFT to understand condensed matter systems. Gravity in M with AdS boundary conditions � Field Theory that lives on ∂ M ◮ Access to strongly coupled regime, include T , ρ , etc

Intro1: AdS/CMT Use AdS/CFT to understand condensed matter systems. Gravity in M with AdS boundary conditions � Field Theory that lives on ∂ M ◮ Access to strongly coupled regime, include T , ρ , etc ◮ Limitations: it’s a conjecture, large N limit, only generic features (bottom-up), hard to implement (top-down).

Intro1: AdS/CMT Use AdS/CFT to understand condensed matter systems. Gravity in M with AdS boundary conditions � Field Theory that lives on ∂ M ◮ Access to strongly coupled regime, include T , ρ , etc ◮ Limitations: it’s a conjecture, large N limit, only generic features (bottom-up), hard to implement (top-down). ◮ examples: superconductors, QGP, non-relativistic FT, etc.

Intro1: AdS/CMT Use AdS/CFT to understand condensed matter systems. Gravity in M with AdS boundary conditions � Field Theory that lives on ∂ M ◮ Access to strongly coupled regime, include T , ρ , etc ◮ Limitations: it’s a conjecture, large N limit, only generic features (bottom-up), hard to implement (top-down). ◮ examples: superconductors, QGP, non-relativistic FT, etc. ◮ Motivation from condensed matter to study gravitational systems [AdS, hairy black holes, etc]

r = 0 boundary r = ∞ Intro2: RN black hole Gravitational solutions ⇔ states of the field theory, e.g. pure AdS is the vacuum of the CFT.

r = 0 boundary r = ∞ Intro2: RN black hole Gravitational solutions ⇔ states of the field theory, e.g. pure AdS is the vacuum of the CFT. Finite T : BH solution; finite ρ charged BH

Intro2: RN black hole Gravitational solutions ⇔ states of the field theory, e.g. pure AdS is the vacuum of the CFT. Finite T : BH solution; finite ρ charged BH ds 2 = − f ( r ) dt 2 + dr 2 f ( r ) + r 2 ( dx 2 + dy 2 ) + µ 2 r 2 f = r 2 − m 0 r = 0 0 4 r 2 r 1 − r 0 � � A = µ dt r boundary r = ∞

Intro3: Conductivity Compute conductivity at non-zero charge density: J = σ E

Intro3: Conductivity Compute conductivity at non-zero charge density: J = σ E Turn on δ A x = a x ( r ) e − i ω t , couples to δ g tx (but can be eliminated)

Intro3: Conductivity Compute conductivity at non-zero charge density: J = σ E Turn on δ A x = a x ( r ) e − i ω t , couples to δ g tx (but can be eliminated) + 1 a x ( r ) = a (0) r a (1) E x = i ω a (0) � J x � = a (1) + . . . x x x x a (1) x σ ( ω ) = i ω a (0) x

Intro3: Conductivity Compute conductivity at non-zero charge density: J = σ E Turn on δ A x = a x ( r ) e − i ω t , couples to δ g tx (but can be eliminated) + 1 a x ( r ) = a (0) r a (1) E x = i ω a (0) � J x � = a (1) + . . . x x x x a (1) x σ ( ω ) = i ω a (0) x Ingoing bc’s for retarded 2-pt a x ≈ ( r − r 0 ) − i ω/ 4 π T

Intro3: Conductivity Compute conductivity at non-zero charge density: J = σ E Turn on δ A x = a x ( r ) e − i ω t , couples to δ g tx (but can be eliminated) + 1 a x ( r ) = a (0) r a (1) E x = i ω a (0) � J x � = a (1) + . . . x x x x a (1) x σ ( ω ) = i ω a (0) x Ingoing bc’s for retarded 2-pt a x ≈ ( r − r 0 ) − i ω/ 4 π T For small ω , σ ( ω ) ≈ µ 2 � � δ ( ω ) + i r 0 ω

Intro3: Conductivity cont’d σ ( ω ) ≈ µ 2 � � δ ( ω ) + i r 0 ω Consequence of translational invariance of the background.

Intro3: Conductivity cont’d σ ( ω ) ≈ µ 2 � � δ ( ω ) + i r 0 ω Consequence of translational invariance of the background. Finite ρ , apply a constant E , charge carriers can’t dissipate p .

Intro3: Conductivity cont’d σ ( ω ) ≈ µ 2 � � δ ( ω ) + i r 0 ω Consequence of translational invariance of the background. Finite ρ , apply a constant E , charge carriers can’t dissipate p . In more realistic situations, p dissipates due to break translation invariance (lattice).

Intro3: Conductivity cont’d σ ( ω ) ≈ µ 2 � � δ ( ω ) + i r 0 ω Consequence of translational invariance of the background. Finite ρ , apply a constant E , charge carriers can’t dissipate p . In more realistic situations, p dissipates due to break translation invariance (lattice). Studied in holography introducing a hol. lattice [Horowitz, Santos, Tong] and breaking diff inv in the bulk (MG) [Vegh]

Intro3: Conductivity cont’d σ ( ω ) ≈ µ 2 � � δ ( ω ) + i r 0 ω Consequence of translational invariance of the background. Finite ρ , apply a constant E , charge carriers can’t dissipate p . In more realistic situations, p dissipates due to break translation invariance (lattice). Studied in holography introducing a hol. lattice [Horowitz, Santos, Tong] and breaking diff inv in the bulk (MG) [Vegh] Goal here: present a simple model of momentum relaxation in the holographic setup.

Outline

Outline ◮ Ward identity for ∇ i � T ij �

Outline ◮ Ward identity for ∇ i � T ij � ◮ The model

Outline ◮ Ward identity for ∇ i � T ij � ◮ The model ◮ (Finite) DC conductivity

Outline ◮ Ward identity for ∇ i � T ij � ◮ The model ◮ (Finite) DC conductivity ◮ Comparison with Massive Gravity

Outline ◮ Ward identity for ∇ i � T ij � ◮ The model ◮ (Finite) DC conductivity ◮ Comparison with Massive Gravity ◮ Conclusions

Ward identity Theory with scalar operator O and U (1) current ∇ i � T ij � = ∇ j ψ (0) � O � + F ij � J i � Basic idea: turn on sources (provided vevs are non-zero)

Ward identity Theory with scalar operator O and U (1) current ∇ i � T ij � = ∇ j ψ (0) � O � + F ij � J i � Basic idea: turn on sources (provided vevs are non-zero) Holographically, consider g µν , ψ I , A µ , ds 2 = d ρ 2 ρ 2 + 1 ρ 2 ( g (0) + . . . + ρ d τ ij + . . . ) dx i dx j ij A = ( A (0) + . . . + ρ d − 2 ˜ A i + . . . ) dx i i ψ I = ρ ∆ − ψ (0) + . . . + ρ ∆ + ˜ ψ I + . . . I

Ward identity cont’d Then, the variation of the on-shell action reads � 1 � � � 2 � T ij � δ g (0) + � O I � δψ (0) + � J i � δ A (0) − g (0) δ S ren = ij I i ∂ M

Ward identity cont’d Then, the variation of the on-shell action reads � 1 � � � 2 � T ij � δ g (0) + � O I � δψ (0) + � J i � δ A (0) − g (0) δ S ren = ij I i ∂ M � O I � ∝ ˜ � J i � ∝ ˜ � T ij � ∝ τ ij ψ I A i Ward identity is asympt. eom (bulk diff inv)

Ward identity cont’d Then, the variation of the on-shell action reads � 1 � � � 2 � T ij � δ g (0) + � O I � δψ (0) + � J i � δ A (0) − g (0) δ S ren = ij I i ∂ M � O I � ∝ ˜ � J i � ∝ ˜ � T ij � ∝ τ ij ψ I A i Ward identity is asympt. eom (bulk diff inv) Generically, spatially dependent sources introduce explicit anisotropies and non-homogeneities (solve PDE’s)

Ward identity cont’d Then, the variation of the on-shell action reads � 1 � � � 2 � T ij � δ g (0) + � O I � δψ (0) + � J i � δ A (0) − g (0) δ S ren = ij I i ∂ M � O I � ∝ ˜ � J i � ∝ ˜ � T ij � ∝ τ ij ψ I A i Ward identity is asympt. eom (bulk diff inv) Generically, spatially dependent sources introduce explicit anisotropies and non-homogeneities (solve PDE’s) Take ψ ∝ x with m 2 ψ = 0 makes bulk geometry homogeneous, can arrange more than one scalar to have isotropy. Makes use of the shift symmetry ψ I → ψ I + c I .

The model � d − 1 � √− g � R − 2Λ − 1 ( ∂ψ I ) 2 − 1 � 4 F 2 d d +1 x S 0 = 2 M I

The model � d − 1 � √− g � R − 2Λ − 1 ( ∂ψ I ) 2 − 1 � 4 F 2 d d +1 x S 0 = 2 M I Take the ansatz � � ds 2 = − f ( r ) dt 2 + dr 2 1 − r d − 2 f ( r )+ r 2 δ ab dx a dx b , 0 ψ I = α Ia x a , A = µ dt , r d − 2

The model � d − 1 � √− g � R − 2Λ − 1 ( ∂ψ I ) 2 − 1 � 4 F 2 d d +1 x S 0 = 2 M I Take the ansatz � � ds 2 = − f ( r ) dt 2 + dr 2 1 − r d − 2 f ( r )+ r 2 δ ab dx a dx b , 0 ψ I = α Ia x a , A = µ dt , r d − 2 Find the solution [Bardoux, Caldarelli, Charmousis, ’12] r 2( d − 2) d − 1 α 2 r d − 2 + ( d − 2) µ 2 2( d − 2) − m 0 1 α 2 ≡ � f = r 2 − 0 r 2( d − 2) , α a · � � α a , 2( d − 1) d − 1 a =1 provided α b = α 2 δ ab � α a · � ∀ a , b . (1)

The model cont’d � � ds 2 = − f ( r ) dt 2 + dr 2 1 − r d − 2 f ( r )+ r 2 δ ab dx a dx b , 0 ψ I = α Ia x a , A = µ dt , r d − 2 Geometry is isotropic and homogenous but solution is not.