Holographic Transport and the Hall Angle Mike Blake - DAMTP - PowerPoint PPT Presentation

Holographic Transport and the Hall Angle Mike Blake - DAMTP arXiv:1406.1659 with Aristomenis Donos arXiv:1310.3832 with David Tong and David Vegh arXiv:1308.4970 with David Tong

Part 1: Holographic Transport

AdS/CMT 101 RN Solution

AdS/CMT 102 J x ( ω ) = σ ( ω ) E x ( ω ) δ A x ( ω ) = E x ( ω ) + h J x ( ω ) i r + ... i ω 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 ê T w ê T Hartnoll DC conductivity diverges!

• To get finite DC conductivity need to have momentum dissipation. • Break translational invariance of boundary theory weakly using irrelevant operator. A t → µ + λ cos( k L x ) Horowitz, Santos & Tong Solve very complicated PDEs!

Numerical Conductivity Horowitz, Santos & Tong ρ ∼ T 2 ∆ Jt ( k L ) σ DC σ ( ω ) = 1 − i ωτ ∆ J t ( k L ) = 1 1 + ( k L /µ ) 2 − 1 q p 5 + 2( k L /µ ) 2 − 4 2 2 Result for resistivity agrees with memory matrix prediction Hartnoll and Hofman

Analytic Conductivity • Make progress by working perturbatively in lattice strength MB, Tong and Vegh • Simplest thing is to add a background scalar lattice on top of the RN solution • To leading order can ignore the backreaction on the metric and gauge field backgrounds.

• At leading order the conductivity calculation simplifies enormously. • In radial gauge we need only δ g rx = 0 consider ( δ A x , δ g tx , δφ ) • The novel ingredient is scalar perturbation δφ ( r, x, t ) = δφ ( r, t )sin( k L x ) • This is simply the phonon mode of the lattice �� = ✏ k L � 0 ( r ) ⇡ ( r, t ) � ( r, x, t ) = ✏� 0 ( r )cos( k L [ x − ⇡ ( r, t )])

Connection with Massive Gravity • After eliminating equations take the δ g tx form • These are the nothing but the perturbations equations of massive gravity with a radially dependent graviton mass M 2 ( r ) = 1 2 ✏ 2 k 2 L � 2 0 ( r ) MB and Tong

Universal Conductivity • Surprise is the existence of a massless mode even at finite density. MB and Tong ◆ � 1  1 + µ 2 r 2 ✓ � δ A x − µf π 0 δλ 1 = M 2 r 2 i ω r h h • Whenever you have massless mode you can use the Iqbal/Liu trick to show the membrane conductivity is constant. σ DC ( r ) • Evaluating at the horizon gives Q 2 r 2 h σ DC = M 2 ( r h )

Locally Critical Scaling • Key result is that resistivity is determined by value of the graviton mass at the horizon ⇢ ∼ M 2 ( r h ) ∼ ✏ 2 k 2 L � 0 ( r h ) 2 • Zero temperature near horizon geometry is AdS 2 × R 2 φ 0 ∼ ξ − ∆ O ( k L ) • At finite temperature the geometry terminates at giving a resistivity ξ h ∼ T − 1 ρ ∼ T 2 ∆ O ( k L ) MB, Tong and Vegh

Related Work • Can break momentum conservation in other (simpler) ways: Massive Gravity Vegh, Davison Andrade and Withers, Linear Axions χ ∼ kx Gouteraux Q-lattices φ 1 ∼ sin( k L x ) φ 2 ∼ cos( k L x ) Donos and Gauntlett • In these models our method gives the exact DC conductivity in terms of horizon data. • Can use a similar approach to calculate thermal and electrothermal conductivity. Donos and Gauntlett

Part II: The Hall Angle

Drude model j ( ! ) = � ( ! ) ~ ~ E ( ! ) md ~ dt + m v v = q ( ~ v × ~ ~ E + ~ B ) ⌧ ~ j = nq ~ v

A puzzle... B 6 = 0 B = 0 θ H = σ xy = qB τ σ DC = nq 2 τ Drude σ xx m m Strange σ DC ∼ 1 θ H ∼ 1 Metal T 2 T The strange metal experiments seem to imply different scattering times for electric and Hall currents. Anderson Coleman, Schofield & Tsvelik

Q-Lattices • Can use ‘Q-lattices’ to obtain analytic expression for transport even when momentum dissipation is strong. • Build lattices out of two complex scalar fields ψ 1 ∼ φ e i χ 1 ψ 2 ∼ φ e i χ 2 Donos and Gauntlett χ 1 → kx χ 2 → ky • Stress tensor is homogeneous: can study exactly using ODEs.

DC Transport 4 π Q 2  � σ DC = Z ( φ ) + k 2 Φ ( φ ) s r h • is a new term that did not appear in Z ( φ ) | r h our perturbative lattice calculation. • Compare with the electrothermal conductivity  4 π Q � α = ¯ k 2 Φ ( φ ) r h Donos and Gauntlett • Hence corresponds to excitations Z ( φ ) | r h that carry a current but no momentum.

Weak Coupling Intuition Holes Particles P x J x Sachdev and Damle • This is analogous to what happens at `charge-conjugation symmetric’ critical points. • Hence we define σ ccs = Z ( φ ) | r h

DC Conductivity Q 2 σ DC = σ ccs + E + P τ L k 2 Φ ( φ ) � = s τ − 1 � σ ccs = Z ( φ ) | r h � L 4 π E + P � r h • At finite density there are two additive contributions to the conductivity - `Inverse Matthiessen Law’. • In holography, is present even at low σ ccs energies. This is not true for weakly coupled particles at finite density.

Hall angle  B 2 Z 2 + Q 2 + 8 π Zk 2 Φ /s �� θ H = 4 π B Q � � B 2 Z 2 + Q 2 + 4 π Zk 2 Φ /s k 2 Φ s � r h

Hall angle B Q θ H ∼ E + P τ L

Hall angle B Q θ H ∼ E + P τ L No analogous term to σ ccs Holes Particles B MB and Donos

• Weak momentum dissipation - τ L → ∞ Q 2 B Q θ H = E + P τ L σ DC = E + P τ L c.f. reproduces Drude-like results. Hartnoll & Hofman etc • Strong momentum dissipation - τ L → 0 θ H = 2 B Q E + P τ L σ DC = σ ccs can now get different scalings!

Comments • Story can be applied more generally than to the specific lattice models studied here e.g. to hydro, probe branes. Karch • Would be exciting to understand whether mechanism can be applied to the cuprates or other experimental systems. σ DC ∼ 1 /T + 1 /T 2 σ ccs ∼ 1 /T ⇒ = σ diss ∼ 1 /T 2 θ H ∼ 1 /T 2 • Supports recent suggestion that strange metals are governed by incoherent transport. Hartnoll

`` Over broad regions of doping, the two kinds of relaxation rates, the one for the conductivity and the one for the Hall rotation, seem to add as inverses: Conductivity is proportional to 1/ T + 1/ T 2 —that is, it obeys an anti-Matthiessen law.’’ P. W. Anderson - Physics Today

Thank you!