Inching Towards Strange Metallic Holography David Tong Imperial, Feb 2010 Based on “Towards Strange Metallic Holography”, with Sean Hartnoll, Joe Polchinski and Eva Silverstein

Plan of the Talk Motivation  Remedial Introduction to Conductivity  Anomalous Properties of Strange Metals Conductivity from Gauge/Gravity Duality  Lifshitz Geometry and Probe Branes  DC and Hall Conductivity  Optical Conductivity

Drude Model: DC Conductivity Ohm’s Law: ~ E = ½ ~ j ½ ½ = m ne 2 ¿ Resistivity: Conductivity: ¾ = 1 = ½ Scattering Time

Scattering Mechanisms The scattering time depends on the scattering mechanism. This gives rise to different temperature dependencies Bardeen, 1940 Resistivity in metals is typically due to phonons and impurities. ½ » T 0 ½ » T 2 ½ » T ½ » T 5 Impurities: Phonons: Electrons: ( T < £ D ) ( T > £ D )

Drude Model: Hall Conductivity ¾ xx ¾ xy = 1 ! c ¿ B ~ j = ¾ ~ E Now a 2x2 matrix ¾ xy = 1 ½ ! c ¿ 1 + ! 2 c ¿ 2 ¾ xx = 1 ½ ! 2 c ¿ 2 x y ¡ ! c = e B mc ¢

Drude Model: AC Conductivity Fourier Transform: ~ j ( ! ) = ¾ ( ! ) ~ E ( ! ) ¾ ( ! ) ! i = ! Note: as ! ! 1 ~ j = ~ j ( ! ) e ¡ i ! t ~ E = ~ E ( ! ) e ¡ i ! t ¾ ( ! ) = 1 ½ 1 1 ¡ i !¿

Strange Metals High Tc Superconductors at optimal doping  e.g. cuprates Suggestions that behaviour is governed by quantum critical point. Heavy Fermion materials have similar properties

Anomalous Properties Resistivity: Hall Conductivity AC Conductivity Mackenzie, 1997 Van der Marel et al., 2003 Mackenzie, 1997 ½ » T ¾ xx = ¾ xy » T 2 ¾ ( ! ) ! ( i = ! ) º º ¼ 0 : 65

Challenge Reproduce this anomalous behaviour in conductivity from a (presumably strongly coupled) field theory.

Strategy Use AdS/CFT  Pros: Simple to compute transport properties in strongly interacting theories  Cons: Don’t know how the theory is related to microscopic degrees of freedom Low energy physics: meV not MeV The theory is defined by the gravitational dual Necessarily “large N”

The Set-Up Charged particles moving through a strongly coupled soup We will take the strongly coupled soup to obey Lifshitz Scaling Gravitational Background Probe Brane

Lifshitz Scaling Non-Relativistic Conformal Invariance t ! ¸ z t ~ x ! ¸ ~ x ~ x = ( x ; y ) : d=2 spatial dimensions z: dynamical critical exponent A weakly coupled example with z=2: S = R d t d d x h _ Á 2 ¡ ( r 2 Á ) 2 i

Lifshitz Geometry Kachru, Liu, Mulligan ‘08 d s 2 = L 2 ³ ¡ d t 2 r 2 z + d r 2 r 2 + d x 2 + d y 2 r 2 ´ r = 0 boundary r ! 1

Hot Lifshitz Geometry r = 0 boundary d s 2 = L 2 ³ ¡ f ( r ) r 2 z d t 2 + d r 2 f ( r ) r 2 + d x 2 + d y 2 r 2 ´ f ( r + ) = 0 T » f 0 ( r + )= r z ¡ 1 + » 1 = r z + r = r + horizon f ( r ) = 1 ¡ ( r = r + ) z + 2 When needed, we take

Adding Probe Branes r = 0 boundary r = r + horizon r = r 0 µ Brane wraps contractible cycle in internal space  (r) profile looks like cigar When needed, we take this space to be a sphere

Probe Branes: Holographic Dictionary r = 0 boundary r = r + horizon r = r 0 Dual to mass of charge carriers U(1) gauge field, dual to current, J S DBI = ¡ ¿ R d t d 3 ¾ p ¡ d e t [ g a b a b µ + F a b ] In this talk, we’ll drop powers of ¿ ; L 2 ; ® 0

Charge Carriers r = 0 boundary r = r + horizon string r = r 0 Charge carrier = string: Note: (, ) m = E gap » 1 = r z 0 m À T E gap » 1 e V T » 10 ! 10 3 K

Charge Carriers r = 0 boundary r = r + horizon Finite charge density: Brane now stretches to horizon.  (r) profile looks like tube. A 0 ! ¹ ¡ J t r 2 ¡ z + ::: chemical potential charge density Electric field c.f. Mateos, Myers et. al. ‘06

DC Conductivity Compute full non-linear conductivity Without doing any work….! A x ! E t + J x r z + ::: A 0 ! ¹ ¡ J t r 2 ¡ z + ::: electric field current J x = ¾ ( E ; T ) E Karch, O’Bannon ‘07

DC Conductivity ¾ ( E ; T ) = p cons t : + r 4 ? ( J t ) 2 due to pair creationdue to background charge f ( r ? ) = E 2 r 2 z + 2 ? where when.……… E gap ; ( J t ) z = 2 À T ½ = 1 ¾ » T 2 = z J t

DC Conductivity: Comments Comments:  Finite DC conductivity only because we’re ignoring backreaction  Linear for z=2  This follows from dimensional analysis if we assume linearity in charge density. Unclear if this linearity is seen experimentally (charge density = doping) ½ = 1 ¾ » T 2 = z J t

Hall Conductivity Similar technique: This is the Drude result: no sign of anomalous behaviour.  Caveat: The term from pair creation actually scales as  Buts… This term should be subdominant Experimental results show this ratio is independent of doping O’Bannon ‘07 ¾ xx ¾ x y » ½ ¾ xx ¾ xy » T 4 = z J t B

AC Conductivity Our results:  Numerical for all   Analytic for large   Analytic results in two regimes Small density, with probe string computation Finite density, with brane computation A x ( ! ) = E x ( ! ) i ! + J x ( ! ) r z + ::: T ¿ ! ¿ E gap ! » 0 : 1 ! 1 e V (c.f. )

AC Conductivity Note: z=3 matches data! Linearity in J t is now dynamical Agreement between small and large densities due to brane spike ¾ ( ! ) » 8 < : ( J t ) z = 2 ! ¡ 1 z < 2 J t ( ! l og! ) ¡ 1 z = 2 J t ! ¡ 2 = z z > 2

Comment on the z=2 Crossover Dimensional Analysis: Operators are relevant if Examples:  Charge density: relevant for z>d  Probe Inertia: irrelevant for z>2 For z>2, all inertia due to stuff that particle drags around with it R d t d d x O [ t ] = ¡ z [ x ] = ¡ 1 [ O ] · d + z [ J t ] = d ( J t ) 2 R d t _ x 2

Summary Charge carriers moving in strongly coupled Lifshitz backgrounds yields:  DC conductivity:  Hall conductivity:  AC conductivity for z>2 Starting point for model building or merely failure?! ½ » T 2 = z ¾ ( ! ) » 1 = ! 2 = z ¾ xx = ¾ xy » ½