Cold Holographic matter in top-down models A. V. Ramallo Univ. Santiago NumHol016, Santiago de Compostela June 30, 016 Based on 1503.0437, 160.06106, 1604.03665 with Y. Bea, G. Itsios and N. Jokela
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
(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
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)
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
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
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
Pressure p = Ω = λ ɛ Speed of sound u s = p ɛ u s = ± λ
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
Non-Susy examples Model λ p q n Sakai-Sugimoto D4-D8/D8 5 4 8 3 D3-D7 4 3 7 D-D8 5 8 Notice that for p =3 λ =n
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
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
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
Speed of zero sound for different λ and p Re[ ] 1.4 1. 1.0 0.8 0.6 D3 D5 p =3,λ=4 D D6 p =,λ=6 0.4 0. 0.5 1.0 1.5.0 k D3-D5 D-D6 Re[ ] 1.4 1. 1.0 0.8 0.6 D4 D6 p =4,λ=4 D4 D8 p =4,λ=6 0.4 0. 0.5 1.0 1.5.0 k Same curves for the same λ D4-D6 D4-D8
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
Diffusion constant for D-D6 ˆD = r 5 p h D = 4πT 7 p D 3.0 D!.5.0 1.5 1.0 4 6 8 10 1 d! ˆd = d r λ h = ( 7 p 4πT ) λ 5 p d
Lowest excitations in D1-D5 Log [ Im [ # ] Im [ # ( T = 0 ) ] ] 4 3 1-3 - - 1-1 - Log [ d! - / " ] Colisionless quantum regime Hydrodynamic diffusive regime! cr T 7 p 5 p µ k cr T 7 p 5 p µ
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
ω R for D-D6 with B field Re["! ] 0.1 0.10 0.08 0.06 0.04 B 0.0 0.05 0.10 0.15 k!
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
Dispersion relation for D3-D7 Re 0.30 m 0.5 0.0 0.15 0.10 0.05 0.1 0. 0.3 0.4 0.5 k Speed of sound&attenuation for D3-D5 1.0 0.8 c s 0. 0.4 0.6 0.8 1.0 m 0.6-0.00005 0.4-0.00010 0. 0. 0.4 0.6 0.8 1.0 m -0.00015-0.0000
Massive embeddings at different densities z m 1.0 0.8 0.6 0.4 0. 0.0 0 4 6 8 10 m d 6= 0(µ>m)! black hole embedding zero density limit (m = µ) z = m! Minkowski embedding
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
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 ) =
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
Comparison with numerics for D3-D5 Re.5.0 1.5 n =0, 1 n crit,n crit 1.0 0.5 0.5 1.0 1.5.0.5 k Re.0 1.5 1.0 n = n crit m µ =0.1, 0.5, 0.8 0.5 0.5 1.0 1.5.0.5 k
How universal are our results? Let us consider cold flavors in the ABJM model
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
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 0903.1730 Gaiotto&Jafferis 0903.175
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
Grand canonical potential at T=0 0.015 0.010 0.005-0.005-0.010 0.95 1.00 1.05 Black hole embedd. Minkowski embedd. Brane-antibrane embedd. m =1-0.015 Speed of first&zero sound u s 0.5 0.4 0.3 D3-D5 ABJM 0. 0.1 0.0 0.0 0. 0.4 0.6 0.8 1.0 m Both curves are the same if m =0 They di er if m>0
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
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 0.1 0.05 0.04 0.10 0.03 0.08 0.0 0.01 0.06 0.000 0.00 0.004 0.006 0.008 0.010 _ 0.00 0.004 0.006 0.008 0.010 _ =0.65 0.75 Other critical exponents z = =1 = 1
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
Grand canonical potential with unquenched flavor 0.0 0.01 0.85 0.90 0.95 1.00 1.05 Black hole embedd. Minkowski embedd. Brane-antibrane embedd. -0.01-0.0 m =1-0.03-0.04 The transition occurs at µ<m q with ch 6=0
Phase diagram at T 6= 0 m q 1.04 1.0 1.00 0.98 0.96 nd order 1 st order Black Hole Phase 0.94 0.9 0.90 Minkowski Phase 0.5 1.0 1.5.0.5 3.0 ˆ = 3 4 N f N
Collective excitations in other probe brane systems Higgs branch for Dp-D(p+) intersections with flux (G. Itsios, N. Jokela, AVR, 1505.069) Anyons and magnetic fields in non-relativistic Lifshitz systems (J. Järvelä, N. Jokela, AVR, 1605.09156) Anisotropic backgrounds (in progress) Collective excitations with full backreaction at non-zero density