Holographic Metals. Valentina Giangreco Marotta Puletti Chalmers Institute of Technology. XIII Marcel Grossmann Meeting Stockholm, July 5th, PDF Free Download

Holographic Metals Valentina Giangreco Marotta Puletti Chalmers Institute of Technology XIII Marcel Grossmann Meeting Stockholm, July 5th, 2012 in collaboration with S. Nowling, L. Thorlacius, and T. Zingg based on 1110.4601 [hep-th] + work in progress (and also on 1011.6261 [hep-th])

Overview Holography + metals: apply AdS/CFT to Condensed Matter physics (AdS/CMT) What do I mean by AdS/CFT here? - gravity in (asymptotical) (d+1)-ads (BULK) QFT in d R (BOUNDARY) - weakly/strongly coupled duality - radial coordinate z energy scale Why? - strongly coupled QFT and perturbative methods fail here: in the large N limit and when the QFT is strongly coupled: where L = AdS curvature radius and Newton s constant is - AdS/CFT can give a geometrical translation of these systems - new geometries L 2 apple 2 N # >> 1 apple 2 =8 G N

Outline Introductions & Motivations (Fermi liquids vs Non-Fermi liquids) Introduction to Friedel oscillations Introduction to electron star geometry Friedel oscillations in the electron star geometry Electron star vs AdS hard wall geometry Summary plus Future

Intro and motivations: The broad picture Goal? - Holographic description of (2+1) strongly correlated charged fermions at finite density (µ) and at very low temperature (or zero) T (T<< µ) - Universality class for such systems via AdS/CFT - Which are the good ingredients in the bulk? Bottom-Up approach QFT at µ? z 0 Bulk (IR) Boundary (UV)

Introduction and Motivations Why these systems? Non-Fermi liquid (NFL): - they are not described by Landau-Fermi theory - strange metallic behavior: resistivity T (ex. in high-tc superconductors) (vs. Fermi liquid: resistivity T 2 ) picture by Hartnoll 09, and by S. Kasahara et al. 10 (in the 2nd picture: resistivity T α )

Introduction and Motivations Why these systems? Non-Fermi liquid (NFL): - they are not described by Landau-Fermi theory - strange metallic behavior: resistivity T (ex. in high-tc superconductors) (vs. Fermi liquid: resistivity T 2 ) BaFe 2 (As 1-x P x ) 2 T 0 T SD T c! " 2.0 AF SDW 1.0 Superconductivity 0 picture by Hartnoll 09, and by S. Kasahara et al. 10 (in the 2nd picture: resistivity T α )

Intro and Motivations: Intermezzo Recall: Landau-Fermi theory describes Fermi liquids (FL): - Weakly interacting quasi-particles ( dressed electrons ) - Fermi surface - very robust (up to BCS): IR free fixed point But... not Non-Fermi liquids - are not described by quasi-particles - but still central role of Fermi surface Recall: Fermi surface is a locus of points where the fermionic Green s function has a pole in the momentum space: G 1 (! =0,k = k F )=0 G : fermionic Green s function kf

Intro and Motivations What s behind? - Quantum Critical Point (QCP) and Quantum Phase Transition (ex. FL can be tuned to a QCP by doping the material) [Broun 08] - What s special about QCT? 1) strongly interacting 2) scale invariant... it sounds good! - RG flow from free fixed point (Landau-Fermi theory) to a non-trivial fixed point (Non-Fermi liquid) - NFL: anisotropic scaling behavior: where s = dynamical critical exponent k! s!!!,k! s k

Intro and motivations: Our work: Goals... Recall the Big Goal: geometries to holographically model Non-Fermi liquids Goal: - Fermi surface is our key but how is it encoded in a holographic geometry? - Which are the good ingredients in the bulk? Test holographic models on the market! - How do we test the holographic models? - When good features in the bulk stay good in the boundary (see Focus I)? Focus 1: - When do bulk Fermi features (FS) induce boundary Fermi features (FS)?? QFT at µ Bulk (IR) z Boundary (UV)

Intro: Friedel oscillations Strategy? - isolate properties related to Fermi surface: Friedel oscillations - search for such a signal in holographic models What? - oscillations in configuration space present in static response functions, like current current correlation functions, at very low temperature (also T=0) - due to the presence of a sharp Fermi surface - present in FL andnfl Example: sin x x! FT : (k) 2 kf singularity!

Things you have to remember about AdS/CFT Minimal Dictionary [Maldacena 97] [Witten 98] [Gubser et al 98] Bulk Boundary (z = 0) - Dynamical field with boundary value - Maxwell field A µ A µ (z) A (0) µ + za (1) µ +... z! 0 O 0 0 - Operator sourced by - U(1) global current with µ A (0) µ = µ, A (1) µ =< J t > - Partition Function - Expectation value Z bulk [! 0 ]=hexp Z i - Linear response - Linear response 0O i

Electron star [Hartnoll et al. 09] [Arsiwala et al. 10] [Hartnoll&Tavanfar 10] IR Lifshitz geometry Bulk (IR) + + + + + + z QFT at µ 0 Boundary (UV) Ingredients in the bulk: - Maxwell gauge field: A t = el apple h(z) lim A t = µ @ z!0 - High density of fermions (Oppenheimer-Volkoff approx) - T=0 perfect fluid of free charged fermions with mass m and charge e in the bulk - fermions are in a local Lorentz frame (LL) at each value of z: - fluid thermodynamic variables: p(z), ρ(z) and σ(z) with - asymptotically AdS metric: µ loc (z) ds 2 = L2 z 2 ( f(z)dt2 + g(z)dz 2 + dx 2 + dy 2 ) µ loc (z) = A t p gtt

Electron star [Hartnoll et al. 09] [Arsiwala et al. 10] [Hartnoll&Tavanfar 10] Ingredients in the bulk: - Maxwell gauge field: A t = el apple h(z) lim A t = µ @ z!0 - High density of fermions (Oppenheimer-Volkoff approx) - T=0 perfect fluid of free charged fermions with mass m and charge e in the bulk - fermions are in a local Lorentz frame (LL) at each value of z: - fluid thermodynamic variables: p(z), ρ(z) and σ(z) with - asymptotically AdS metric: µ loc (z) ds 2 = L2 z 2 ( f(z)dt2 + g(z)dz 2 + dx 2 + dy 2 ) µ loc (z) = A t p gtt

Electron star [Hartnoll et al. 09] [Arsiwala et al. 10] [Hartnoll&Tavanfar 10] IR Lifshitz geometry Bulk (IR) + + + + + + z QFT at µ 0 Boundary (UV) The action: [Hartnoll &Tavanfar 10] [Schutz 70] S = S HE + S M + S fluid = 1 2apple 2 Z d 4 x p G R + 6L 2 1 4e 2 Z d 4 x p GF µ F µ + Z d 4 x p Gp Properties of ES: Why this model? - back-reaction of the metric w.r.t. fluid in a controlled approx - in the interior IR emergent Lifshitz scaling in the boundary emergent critical exponent s f = z 2s+2,g = g 1,h= h 1 ) (z,x,y)! (z,x,y),t! s t - s is determined by the eom: s = s(e, m)

Electron star [Hartnoll et al. 09] [Arsiwala et al. 10] [Hartnoll&Tavanfar 10] s, r, p 1.4 1.2 1.0 0.8 0.6 0.4 0.2 z 2 4 6 8 10 The action: [Hartnoll &Tavanfar 10] [Schutz 70] S = S HE + S M + S fluid = 1 2apple 2 Z d 4 x p G R + 6L 2 1 4e 2 Z d 4 x p GF µ F µ + Z d 4 x p Gp Properties of ES: Why this model? - back-reaction of the metric w.r.t. fluid in a controlled approx - in the interior IR emergent Lifshitz scaling in the boundary emergent critical exponent s f = z 2s+2,g = g 1,h= h 1 ) (z,x,y)! (z,x,y),t! s t - s is determined by the eom: s = s(e, m)

Go ahead I! Summary so far: - Friedel oscillations as a diagnostic in order to detect signals of Fermi surfaces in electron star - Compute static current-current correlation functions, which are bosonic observables - but we want to detect a fermionic structure... 1-loop diagram = (1/N corrections)! Focus 1I: - we will use internal d.o.f. to study the low-energy dynamics: - very different from the probe fermion approximation [Liu,McGreevy,Vegh 09],[Hartnoll, Hofman, Tavanfar 11],[Cubrovic, Liu, Schalm, Sun, Zaanen 11]

Things you have to remember about AdS/CFT II Minimal Dictionary II [Maldacena 97] [Witten 98] [Gubser et al 98] [Son et al. 02] QFT Gravity Z bulk [! 0 ]=hexp Z i O(x) 0O i O(y) ho(x)o(y)i compute the on-shell action and run to the boundary (bulk-to-bdry propagator) How do we proceed? - Insert a disturbance in the system: shear modes ( A x, g tx, g yx, u x )e i!t+ik yy - Compute the correlators with 1/N corrections (induced magnetic effect): hj x (k y )J x (k y )i - introduce an effective term in the action to take into account loop effects: polarization tensor Π

Go ahead II! Set-up - the effective action Z S Pol = e2 dzdz 0 d 2 k p g(z) g(z 2 0 ) A µ (z, k) µ CG,EC (z,z0,k) A (z 0,k) µ CG,EC is the coarse grained polarization tensor - the fermionic loop: Recall the fermions live in a local Lorentz frame! µ CG,flat (z,z0,k L.L. )= (z z 0 ) µ (0,k L.L. ) - we need to project all the quantities entering in Π into the LL frame: k L.L. = k p gyy = kz - use for Π the expression for (3+1) QED singular part: µ (0,k L.L. ) N µ (k L.L. ) ln kl.l. 2k f k L.L. +2k f crucial point here: the quantities entering in the singular part k L.L. = zk, k F (z) = p s µ loc (z) 2 m 2 (za t (z)) = 2 f(z) m 2 +... the Fermi momentum depends on z: there is a different bulk FS for each point in the radial direction!

(Absence of!) Friedel oscillations in the electron star Results we do NOT have boundary Fermi surfaces! :( 2.0 1.5 E. C. HTêT c =.03L B. H. HTêT c =.03L E. C. HTêT c =.51L B. H. HTêT c =.51L 1.0 0.5 0.0 0.0 0.5 1.0 1.5 2.0 Key-points: - continuum of bulk Fermi surfaces, each at each point z - each bulk Fermi surface is different: set by different value of the local µ loc (z) - the bulk FS are very localized:... this is the ultimate meaning of the OV approx! Lesson: each different FS at a different radius will not act coherently summing from the deep interior (IR) to the boundary (UV) they are just smeared out! cf. [Kulaxizi, Parnachev 08] for D4 D8 D8

Friedel oscillations in the AdS hard wall geometry Ingredients in the bulk: [Sachdev 11] - Maxwell gauge field: A t = el apple h(z) lim A t = µ @ z!0 - low density of fermions = bulk single-particle wave function z m - AdS metric frozen and truncated at :? ds 2 = L2 z 2 ( dt2 + dz 2 + dx 2 + dy 2 ) QFT at µ Bulk (IR) z Boundary (UV) The action S = S M + S D = 1 4e 2 Z d 4 x p GF µ F µ + Z d 4 x p Gi µ D µ + m

Friedel oscillations in the AdS hard wall geometry Ingredients in the bulk: [Sachdev 11] - Maxwell gauge field: A t = el apple h(z) lim A t = µ @ z!0 - low density of fermions = bulk single-particle wave function z m - AdS metric frozen and truncated at : ds 2 = L2 z 2 ( dt2 + dz 2 + dx 2 + dy 2 ) The action S = S M + S D = 1 4e 2 Z d 4 x p GF µ F µ + Z d 4 x p Gi µ D µ + m

Friedel oscillations in the AdS hard wall geometry -2.005 XrH-k x LrHk x L\ Tot -2.035-2.065 1 2 3 k x êk f Results: we do have boundary Fermi surface! :) Key-points: - deconfining geometry - different role of the fermions: here treated QM! - discreteness of the bulk Fermi surfaces (due to the gapped spectrum) - delocalization of bulk Fermi surfaces: wave-functions fill all the way the space-time, they know all the geometry!

Friedel oscillations in the AdS hard wall geometry XrH-k x LrHk x L\ Cor -0.002-0.0035-0.005 1 2 3 k x êk f Results: we do have boundary Fermi surface! :) Key-points: - deconfining geometry - different role of the fermions: here treated QM! - discreteness of the bulk Fermi surfaces (due to the gapped spectrum) - delocalization of bulk Fermi surfaces: wave-functions fill all the way the space-time, they know all the geometry!

Summary and Future Summary: - introduction to Non-Fermi liquid theory and Friedel oscillations - review of electron star geometry - strategy to compute current-current correlators including 1-loop effect (effective action with Π) - results: static current-current correlators do NOT show boundary FS in the electron star geometry (yes in the AdS hard wall geometry) - lesson: in order that bulk FS induces boundary FS in all correlation functions, it is necessary for the bulk FS to be non-local and discrete Future: - better stringy embedding for AdS hard wall geometry? Yes: AdS soliton! It works the same! - Gravitational back-reaction in AdS hard wall? - Non-Fermi liquid?

Thanks!

= 2 µ (, 2 )Π µν (,, ) ν (, ). 1 Π µν (,, ν, ω 2 )= (2π) 3 [ µ (ω,,, ) ν (ω + ν, +,, ) ], µ = Γ 0 Γ µ ν =0 (ω,,, )= ( ) 1 χ l, ( )χ l, ω l 0 l ( )+ η ( l ( )) ( ), l ( ) Π ( ) Π µν (,, ) Γ µ χχ Γν χχ + P, 2 (2π) 2 ( ) θ( + )θ( ) 1 ( + ) 1 ( )

( 2 2 ) 0 (, ) =0. 0 (, ) = ( ) ( ) ( ) ρ( )ρ( ) 0 = ( ) ( 2 2 ) 0 (, ) = 2 [ Π 00 (,, ) 0 (, )+Π 0 (,, ) (, ) ] λ 1 2 (, ) = 0 (, )+λ 1 (, )+... ( ) 0 = ( 2 2 0 = ( 2 2 ) 0 (, ), ) 1 (, )+ 2 λ Π 00 (,, ) 0 (, )

Π µν (,, )= Π µν = Π µν + Π µν. ω 2 [ (2π) µ (ω,,, ) ν (ω, ] +,, ) 3 [ µ 0 (ω,,, ) ν 0 (ω, ] +,, ). (ω,,, ) = ( ) 1 χ l, ( )χ l, ω l 0 l ( ) + η ( l ( )) ( ), Π µν (,, ) =Π µν + µ χ 1, ( )χ 1, ( ) ν χ 1, + ( )χ 1, + ( ) 2 ( ) θ( + )θ( ) (2π) 2 1 ( + ) 1 ( ) + η ( 1 ( )) θ( + )θ( ) 1 ( + ) 1 ( ) η ( 1 ( )) 1 ( ) 1 + 1 2 Π µν (,, ) = = 1 1 2 [ (2π) 2 θ( 1 ( + ))θ( 1 ( )) ( ) +2 (θ) + η ( 1 ( )) µ χ 1, ( )χ 1, ( ) ν χ 1, + ( )χ 1, + ( ) θ( 1( ))θ( 1 ( )) ( )] 2 (θ) η ( 1 ( )) µ χ 1, ( )χ 1, ( ) ν χ 1, ( )χ 1, ( )

Π µν (, 1, ) = 4 1 π 2 P θ θ( ) ( ) /2 + (θ) µ χ 1, ( )χ 1, ( ) ν χ 1, + ( )χ 1, + ( ) ) ) ( 0 χ 1, ( )χ 1, ( ) 0 χ 1, + ( )χ 1, + ( ) ( 0 χ 1,0 ( )χ 1,0 ( ) 0 χ 1, ( )χ 1, ( ). Π µν (,, ) = 1 ( ) 4 1 π 2 µ χ 1,0 ( )χ 1,0 ( ) ν χ 1, ( )χ 1, ( ) P θ θ( ) /2 + (θ) λ 1 4π 1 ) Π µν (,, ) = λ ( µ χ 1,0 ( )χ 1,0 ( ) ν χ 1, ( )χ 1, ( ) 1 1 ( 2 ) 2 θ( 2 )

About the parameters Let us check the validity of the approximations made. For let s write what the rescaled parameters are: ˆm = apple e m, ˆ = e 4 L 2 = 1 el 2 apple ˆ, = 1 L 2 apple 2 ˆ, p = 1 L 2 apple 2 ˆp, apple - classical gravity: L << 1 - is of order 1 free parameter (e.g. for the electrons is 1 2,then ˆ 1! 1 ˆ e 4 L 2 = apple 2! e 2 << 1, and,e 2 apple L << 1 open/closed string coupling!! in the second step I used the fact that we are in classical gravity. - the other rescaled parameter ˆm 2 = apple2 e 2 m2! ˆm 2 L2 L 2 = apple2 L 2 e 2 L 2 m2 = L 2 e 2 m 2! ˆm eml - For a dual operator ml >> 1 ml ml applee2 applee 2 m apple e 2 = ˆm e >> 1 and e2 << 1! ˆm 1 - high density compared to the scale of curvature L 3 ( µ 3 )L 3 apple 2 apple2 e 4 L ˆ( e 2 apple )3 L 3 L applee 1 >> 1 (21) e3 where I have used that ˆ 1 and (E 2 m 2 ) 3/2 µ µ = el apple hv/(lp f). and the definition of µ, - The last equation (21) is also consistent with the Compton wavelength, since: ml >> 1! C L << 1 with C = h mc el apple Notice that the control of the loop parameter is really 1 m h el apple it controls also the WKB approx and the localization approx. So is large!

Holographic Metals. Valentina Giangreco Marotta Puletti Chalmers Institute of Technology. XIII Marcel Grossmann Meeting Stockholm, July 5th, 2012