Cold Holographic matter in top-down models

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1 Cold Holographic matter in top-down models A. V. Ramallo Univ. Santiago NumHol016, Santiago de Compostela June 30, 016 Based on , , with Y. Bea, G. Itsios and N. Jokela

2 Motivation We want to explore new phases of matter at finite density by using the holographic gauge/gravity duality Holography allows to study strongly interacting systems with no quasiparticle description Physical examples: Quark-gluon plasma Strange metals Heavy electron systems They are non-fermi liquids

3 (Top-down) Holographic model SYM theory at strong coupling and large N Stack of color Dp-branes There is a 10d metric associated Fields in the adjoint rep. Charge carriers Flavor Dq-branes They add fields in the fundamental rep. (quarks) living in the intersection mass of the quarks distance between the Dp and Dq branes

4 Brane setup Dp-Dq brane intersection of the type (n p q) x 1 x n x n+1 x p y 1 y q n y q n+1 y 9 p Dp : Dq : Dp N c color branes (p + 1-dimensional gauge theory on the bulk) Dq N f flavor branes (fundamental hypermultiplets) Probe approximation (N f << N c ) Dp represented by a gravity solution Dq a probe in the Dp-brane background Coordinates transverse to both branes z =(z 1,,z 9+n p q ) embedding functions (z =0 massless quarks)

5 Probe action S = T Dq d q+1 ξe φ det(g + F ) No WZ term g induced metric φ dilaton F worldvolume gauge field For the Dp-brane background (massless quarks) ds q+1 = ρ 7 p [ fp (ρ)dt +(dx 1 ) + (dx n ) ] + ρ p 7 [ dρ f p (ρ) + ρ dω q n 1 ] f p =1 ( rh ρ ) 7 p e φ = ( R ρ ) (7 p)(p 3) r h is related to the temperature T = 7 p 4π r 5 p h

6 Baryonic charge density dual the DBI gauge field A t J t = δs δa t A t ρ A t The dynamics of the probe depends on p and λ λ =n + 1 (p 3) (p + q n 8) Ansatz for F Action F = A t dρ dt + B dx 1 dx S Dq = N V R (n,1) dρ ρ λ + B ρ λ+p 7 1 A t A t is a cyclic variable ρλ + B ρ λ+p 7 1 A A t = d ρλ + B ρ λ+p 7 + d t A t = d J t = N d

7 Thermodynamics at zero T Chemical potential (for B =0) µ = A t ( ) = 0 dρ A t = γd λ γ = 1 ( 1 Γ π λ) 1 ( Γ 1+ 1 ) λ Grand Canonical potential = S reg on shell = N Z 1 0 " p + d 1 # d = + N d1+ = + N µ 1+ ρ = Ω µ = N d Energy density ɛ =Ω+µρ = λ λ + N γd1+ λ

8 Pressure p = Ω = λ ɛ Speed of sound u s = p ɛ u s = ± λ

9 Values of λ SUSY intersections (n p q) with n = p+q 4 λ = q p + Dp D(p +4) (p p (p +4)) λ =6 Examples D3 D7, D D6 Dp D(p +) (p 1 p (p +)) λ =4 Examples D3 D5, D4 D6 Dp Dp (p p p)) λ = Example D3 D3

10 Non-Susy examples Model λ p q n Sakai-Sugimoto D4-D8/D D3-D D-D8 5 8 Notice that for p =3 λ =n

11 Energy-radius relation Energy rescaling E! E! = Density&magnetic field Scaling behavior E 5 5 p d! d d B! B B d = 5 p B = 7 p 5 p p! related to the scaling dimension of d p =3! =n d = n B = SUSY intersections canonical dimensions SUSY d = q p + 5 p = 5 p n +3 p

12 Excitations quasinormal modes Poles of the retarded Green's functions density waves in the dual field theory Perturb as Define G and J as A ν = A (0) ν + a ν (ρ, x µ ) ( g (0) + F (0)) 1 = G 1 + J G open string metric (symmetric part) J antisymmetric part Lagrangian L ρ λ + B ρ λ+p 7 ρλ + B ρ λ+p 7 + d (G ac G bd J ac J bd + 1 J cd J ab) f cd f ab Take a ν = a ν (ρ, t, x) and Fourier transform a ν (ρ, t, x) = dωdk (π) a iω t + ikx ν(ρ, ω, k)e Solve the equations with the conditions: In-falling boundary conditions at the horizon No sources at the UV boundary Low ω, k

13 For T =0,B =0 Holographic zero sound Take ω, k small and of the same order ω O(ɛ), k O(ɛ) ω(k) =ω R (k) iγ(k) ω R (k) real part Γ(k) attenuation (decay rate) ω R = ± λ k Same speed as the first sound Γ= π µ (5 p) p 3 5 p ( [ Γ 1 5 p )] ( λ ) 7 p (5 p) Zero sound for D3-D7 (λ =6,p =3) ω = ± k 3 i 6 Karch, Son, Starinets k 7 p 5 p Zero sound for D3-D5 (λ =4,p =3) ω = ± k i 4 Different cases considered in Brattan et al., Kulaxizi &Parnachev, Goykhman et al,... k µ k µ

14 Speed of zero sound for different λ and p Re[ ] D3 D5 p =3,λ=4 D D6 p =,λ= k D3-D5 D-D6 Re[ ] D4 D6 p =4,λ=4 D4 D8 p =4,λ= k Same curves for the same λ D4-D6 D4-D8

15 For T 0,B =0 Hydrodynamic charge diffusion mode Purely imaginary pole with ω O(ɛ ), k O(ɛ) ω = idk Fick s law D diffusion constant D = 7 p π(λ ) (1 + ˆd ) 1 T ( 3 F, 1 1 λ ; 3 1 λ ; ˆd ) ˆd = d r λ h = ( 7 p 4πT ) λ 5 p d D T 1 D T 7 p 5 p T large T small

16 Diffusion constant for D-D6 ˆD = r 5 p h D = 4πT 7 p D 3.0 D! d! ˆd = d r λ h = ( 7 p 4πT ) λ 5 p d

17 Lowest excitations in D1-D5 Log [ Im [ # ] Im [ # ( T = 0 ) ] ] Log [ d! - / " ] Colisionless quantum regime Hydrodynamic diffusive regime! cr T 7 p 5 p µ k cr T 7 p 5 p µ

18 T = 0 and B 0 Zero sound with B field Gapped dispersion relation ω R = ± λ k + B µ Γ= π µ ( (5 p) p 3 5 p ( )] [ Γ 1 5 p ) p 3 λ k + B (5 p) [k ] µ λ + B µ

19 ω R for D-D6 with B field Re["! ] B k!

20 Massive quarks in SUSY intersections The speed&attenuation of the zero sound depends on the reduced mass m m = m µ ω R = λ 1 m 1 m λ k Γ= π µ (5 p) p 3 5 p ( [ Γ 1 5 p )] ( λ ) 7 p (5 p) (1 m ) 6 p 5 p 1 λ (1 m λ 7 p k ) 7 p 5 p +1 (5 p) Kulaxizi &Parnachev, Davison&Starinets Speed of the zero sound=speed of the first sound The speed of the zero-sound vanishes at m = µ

21 Dispersion relation for D3-D7 Re 0.30 m k Speed of sound&attenuation for D3-D c s m m

22 Massive embeddings at different densities z m m d 6= 0(µ>m)! black hole embedding zero density limit (m = µ) z = m! Minkowski embedding

23 There is a quantum phase transition when m µ (Ammon et al.) Non-relativistic energy density e = d phys m = Ndm near the quantum critical point e = n z P z! dynamical critical exponent! hyperscaling violating exponent In our case near m = µ e = 4 P Exponents z = θ = p

24 Non-relativistic free energy at T 6= 0 f non rel (µ, m, T )=f(µ, m, T ) d phys m = e + d phys T + O(T ) Near the critical point it should scale as f non rel µ T g µ z µ = µ m For our system we get = 6 4 =1 q 4 p = 1 The exponents satisfy the hyperscaling violating relation (n + z ) =

25 Anyonic excitations in +1 d Anyons Charges with a magnetic flux attached in +1d Fractional statistics by Aharonov-Bohm effect Holographic realization Alternative quantization with Dirichlet-Neumann b. c. lim!1 h n! constant n f µ 1 µ f =0 lim!1 E = in lim!1 a 0 y lim a y = i!1 n! k lim!1 E 0 Zero sound spectrum! 0 = 1 m m k + 1 µ dn B The alternative quantization is like an internal magnetic field gappless spectrum n crit B d

26 Comparison with numerics for D3-D5 Re n =0, 1 n crit,n crit k Re n = n crit m µ =0.1, 0.5, k

27 How universal are our results? Let us consider cold flavors in the ABJM model

28 ABJM ABJM Field Theory Chern-Simons-matter theories in +1 dimensions gauge group: U(N) k U(N) k The ABJM model has N = 6 SUSY in 3d It has two parameters N rank of the gauge groups k CS level (1/k gauge coupling) t Hooft coupling λ N k It is a CFT in 3d with very nice properties It is the 3d analogue of N=4 SYM

29 AdS 4 CP 3 + fluxes Sugra description in type IIA ds = L ds AdS 4 +4L ds CP 3 L 4 =π N k F =kj F 4 = 3π ( kn ) 1 Ω AdS4 e φ = L k = π ( N ) 1 4 k 5 Effective description for N 1 5 << k << N Flavor branes D6-branes extended in AdS 4 and wrapping RP 3 CP 3 Hohenegger&Kirsch Gaiotto&Jafferis

30 Worldvolume action of flavor D6-branes in ABJM S = S DBI + S WZ S DBI = T D6 ZM 7 d 7 e p det(g + F ) S WZ = T D6 Z M 7 Ĉ 7 + Ĉ5 ^ F + 1 Ĉ3 ^ F ^ F + 1 ^ F ^ F ^ F 6Ĉ1 Now the WZ term contributes and produces non-trivial effects when the quarks are massive

31 Grand canonical potential at T= Black hole embedd. Minkowski embedd. Brane-antibrane embedd. m = Speed of first&zero sound u s D3-D5 ABJM m Both curves are the same if m =0 They di er if m>0

32 Quantum critical behavior µ = µ m C µ n+z z log µ m near µ =0! new exponent n = in ABJM Charge density near µ =0 ch C µ n z log µ m h 1+ n z + log µ m i e/p ratio e P n z + log µ m The numerical results show that e/p! 0 near µ =0 = n = 6= 0since ch = 0 at the critical point

33 ch C log µ m e P log µ m u s 1 µ m log µ m 0.14 e P 0.06 u s _ _ = Other critical exponents z = =1 = 1

34 Including the backreaction Consider ABJM+unquenched smeared flavors (m=0) AdS 4 (squashed) CP 3 (E. Conde, AVR) The backreaction is a very mild deformation which does not include the effects of the charge density Drastic effects in the BH-Mink. transition Occurs at non-zero density It is of first-order

35 Grand canonical potential with unquenched flavor Black hole embedd. Minkowski embedd. Brane-antibrane embedd m = The transition occurs at µ<m q with ch 6=0

36 Phase diagram at T 6= 0 m q nd order 1 st order Black Hole Phase Minkowski Phase ˆ = 3 4 N f N

37 Collective excitations in other probe brane systems Higgs branch for Dp-D(p+) intersections with flux (G. Itsios, N. Jokela, AVR, ) Anyons and magnetic fields in non-relativistic Lifshitz systems (J. Järvelä, N. Jokela, AVR, ) Anisotropic backgrounds (in progress) Collective excitations with full backreaction at non-zero density

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