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

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1 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 [hep-th] + work in progress (and also on [hep-th])

2 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

3 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

4 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)

5 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 )

6 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 )

7 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

8 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

9 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 T QCT h i =0 h i6=0 gc g

10 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)

11 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)

12 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!

13 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: Non relativistic degenerate fermions in (2+1) dimensions c HqL m e 2 = density current qêk f h (k)i = (k) A t (k), (k) h ( k) (k)i 2 kf singularity!

14 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 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

15 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 = 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

16 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 = 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

17 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)

18 Electron star [Hartnoll et al. 09] [Arsiwala et al. 10] [Hartnoll&Tavanfar 10] s, r, p z 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)

19 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]

20 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

21 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 the Fermi momentum depends on z: there is a different bulk FS for each point in the radial direction!

22 (Absence of!) Friedel oscillations in the electron star Results we do NOT have boundary Fermi surfaces! :( E. C. HTêT c =.03L B. H. HTêT c =.03L E. C. HTêT c =.51L B. H. HTêT c =.51L 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

23 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 = 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

24 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 = 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

25 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 = 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

26 Friedel oscillations in the AdS hard wall geometry XrH-k x LrHk x L\ Tot 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!

27 Friedel oscillations in the AdS hard wall geometry XrH-k x LrHk x L\ Cor 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!

28 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?

29 Thanks!

30 = 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 ( )

31 ( 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 (, ), ) 1 (, )+ 2 λ Π 00 (,, ) 0 (, )

32 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 ( ) Π µν (,, ) = = [ (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, ( )

33 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 )

34 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!