Autocorrelators in models with Lifshitz scaling

Transcription

1 Autocorrelators in models with Lifshitz scaling Lárus Thorlacius based on V. Keränen & L.T.: Class. Quantum Grav. 29 (202) arxiv:50.nnnn (to appear...) International workshop on holography and condensed matter systems University of Perugia, September 24-25, 205

2 Applied AdS-CFT Investigate strongly coupled quantum field theories via classical gravity - growing list of applications: hydrodynamics of quark gluon plasma jet quenching in heavy ion collisions quantum critical systems strongly correlated electron systems cold atomic gases out of equilibrium dynamics... Bottom-up approach: Look for interesting behavior in simple models

3 Classic example of a QCP Resistivity vs. temperature in thin films of bismuth T = 0 state changes from insulating to superconducting at a critical thickness From D.B. Haviland, Y. Liu and A.M. Goldman, Phys. Rev. Lett. 62 (989) 280.

4 Quantum criticality in heavy fermion materials CeCu 6 x Au x T N 0.2 YbRh 2 Si 2 H c T (K) AF T (K) 0. AF NFL FL x H (T) YbRh 2 (Si 0.95 Ge 0.05 ) 2 0 T N CePd 2 Si 2 ρ (μω cm) 5 0 T (K) 5 5 AF SC T (K) P (GPa) From P. Gegenwart, Q. Si and F. Steglich, Figure Quantum critical points innature HF metals. Phys. a,aforderingtemperaturet 4 (2008) 86. N versus Au concentration x for CeCu 6 x Au x (ref. 7), showing a doping-induced QCP.

5 Some measured c/t values in heavy fermion metals H. Lohneysen et al. PRL 72 (994) From G.Stewart, Rev.Mod.Phys. 73 (200) 797.

6 Quantum critical points Typical behavior at T =0 2nd order critical pt. characteristic energy (g g c ) z coherence length (g g c ) z z = dynamical scaling exponent Scale invariant theory at finite T : = ct /z Deformation away from fixed pt.: i (length) QCP has i = 0 Quantum critical region : = T /z (T /z i) (0) = c Physical systems with z =, 2, and 3 are known -- non-integer values of z are also possible z = scaling symmetry is part of SO(d+,) conformal group = isometries of ads d+ z > scale invariance without conformal invariance - asymptotically Lifshitz spacetime

7 Models with Lifshitz scaling t! z t, x! x, z x =(x,...,x d ) Example: Quantum Lifshitz model S = d 2 xdt (@ t ) 2 K r 2 2 z =2,d=2 Q: Can we give a gravity dual description of a strongly coupled system which exhibits anisotropic scaling? A: Look for a gravity theory in d +2 dim s with spacetime metric of the form ds 2 = L 2 which is invariant under r 2z dt 2 + r 2 d 2 x + dr2 r 2 t! z t, x! x, r! r L = Kachru, Liu, & Mulligan 08

8 Holographic models with Lifshitz scaling Einstein-Maxwell-Proca model Kachru, Liu, & Mulligan 08; Taylor 08; Brynjólfsson et al. 09 S EMP = d d+2 x p g R 2 4 F µ F µ 4 F µ F µ c 2 2 A µa µ Einstein-Maxwell-Dilaton model Taylor 08; Tarrío & Vandoren S EMD = d d+2 x p g R 2 2 (@ )2 4 NX i= e i F 2 i! we will take N = or 2 d = 3, 2, or for CM applications

9 Fixed point metric The Lifshitz metric x =(x,...,x d ) ds 2 = r 2z dt 2 + r 2 dx 2 + dr2 r 2 is a solution of both models for particular values of couplings and background fields EMP model: c = p zd, = z2 +(d )z + d 2 r 2 2(z ) A t = r z, A xi = A r =0 z A µ =0 EMD model: = r 2d z, = (d + z)(d + z ) 2, e = r r 0 p 2d(z ) F () p d+z r rt =2r0 z (d+z)(z ), F µ (2) =0 r 0

10 Gravity duals at finite temperature periodic Euclidean time: introduces an energy scale: ' +, = T scale symmetry is broken thermal state in field theory: black hole with T Hawking = T qft finite charge density in dual field theory: electric charge on BH magnetic effects in dual field theory: dyonic BH z = : AdS-Reissner-Nordström BH in d+2 dimensions z > : charged BH in d+2 dimensional z > gravity model

11 Black brane solutions in EMD model (d = 2) ds 2 = r 2z b(r)dt 2 + r 2 dx 2 + dr2 r 2 b(r) Tarrío & Vandoren b(r) = r0 + 2 z z r 4z r0 2z+2 r e = r r 0 2 p z F () p z+ r rt =2r0 z (z+2)(z ) r 0 event horizon at r = r 0 F (2) rt = z r0 z r0 z+ r electric charge density Hawking temperature: T = rz 0 4 AdS-RN black hole: z! z (from S. Hartnoll, arxiv: )

12 Euclidean action: Quantum Lifshitz model and + CFT S = 2 d 2 xd (@ ) 2 + K(r 2 ) 2 Scaling operators: i (x) O (x) =e Vacuum 2-point function G E (x 2, 2 ; x, )=ht E (x 2,t 2 ) (x,t )i = d!d 2 p e i!( 2 )+ip (x2 x) (2 ) 3! 2 + Kp 4 Equal time correlators: 2 = =0 G E (x 2,x )= d!d 2 p (2 ) 3 e ip (x2 x)! 2 + Kp 4 = 2 p K Equal to 2-point function of 2 dimensional free scalar d 2 p e ip (x2 x) (2 ) 2 p 2 S = p K d 2 x(r ) 2 ho (x )...O n (x n )i = he i (x )...e i n (x n ) i CFT

13 Vacuum autocorrelators Autocorrelators: ho (x, t )O 2 (x, t 2 )...O n (x, t n )i Vacuum 2-point function: G E ( 2, )= = d!d 2 p (2 ) 3 d!dp (2 ) 2 = 4 p K e i! 2! 2 + Kp 4 ij = i j pe i! 2! 2 + Kp 4 d!dq (2 ) 2 e i! 2! 2 + q 2 d 2 p =2 pdp q = p Kp 2 Equal to 2-point function of 2 dimensional free scalar: S CFT =2 p K dxd (@ ) 2 +(@ x ) 2

14 G E ( 2, )= d!dq 4 p e i! 2 K (2 ) 2! 2 + q 2 + µ 2 = 8 p K K 0(µ 2 ) = 8 p log(µ 2 ) + log 2 K E IR regulator + O(µ log(µ 2 )), Wick contraction gives 2 point function of scaling operators ht E e i ( 2) e i ( ) i = e 2 ( 2 G E ( +, )+ 2 G E ( 2 +, 2 )+2 G E ( 2, )) =(2µe E ) ( + ) 2 6 p K 2 8 p K 6 p K Consider + =0 definition of scaling operator: Regularized 2 point function: to avoid IR divergence and absorb UV divergence into e i R = ei ht E e i ( 2) R e i ( ) R i = 2 2 = 2 6 p K Regularized 3 point function: =0 ht E e i ( ) R e i 2 ( 2 ) R e i 3 ( 3 ) R i = 2 3 e 2 = P i,j i j G E ( i, j )

15 Thermal state autocorrelators Matsubara sum: G E ( 2, )=T X n = 4 p K T X n d 2 p (2 ) 2 e i!n 2! 2 n + Kp 4,! n =2 nt dq 2 e i! n 2! 2 n + q 2 q = p Kp 2 Thermal 2-point function: P. Ghaemi, A. Vishwanath and T. Senthil 04 ht E e i ( 2) R e i ( ) R i = 2 e 2 2 (G E( +, )+G E ( 2 +, 2 ) 2G E ( 2, )) =! 2 T sin( T 2 ) Thermal 3-point function: ht E e i ( ) R e i 2 ( 2 ) = R e i 3 ( 3 ) R i ( T) sin( T 2 ) sin( T 3 ) sin( T 23 ) 2 + 3

16 Holographic correlation functions Bulk scalar field: S = 2 d 4 x p µ + m 2 2 r! : (r)! c r c + r ± = z +2 2 s z 2 +2 ± + m 2 2 Calculate 2-pt function of operator dual to valid for large m in geodesic approximation ho(x)o(x 0 )i 2 e L(x,x 0 ) UV cutoff where L(x,x ) is the length of the shortest geodesic connecting x and x L(x, x 0 )= 2 d r g µ dx µ d dx d = 2

17 Geodesics in Lifshitz spacetime Lifshitz metric: Geodesic equations: ds 2 = d 2 u 4 + du2 + dx 2 u 2 = E, u4 (assume endpoints lie on x axis) u 2 ẋ = p, u 2 u 2 = u4 E 2 u 2 p 2 The resulting 2- point function has a scaling form: G E (r, ) / x 4 F (x2 / ) 2 =m/ F Quantum Lifshitz Holographic model G E (x, t) / 2 2 x x a = x 2 /

18 Three-point function in Lifshitz spacetime Consider a bulk theory with a 3-point vertex: 3! d d x p g (x) 2 (x) 3 (x) To leading order the 3 point function is then given by: G 3 (, 2, 3 )= d d x p gg BB (, 0; z,,x)g BB ( 2 ; 0; z,,x)g BB ( 3, 0; z,,x) Bulk-to-boundary propagator: G BB ( j, 0; z,,x) / e m jl(, j,0;z,,x) The Lifshitz metric can be rewritten: ds 2 = z 2 y 2 (d 2 + dy 2 )+ dx2 (zy) 2/z Saddle point evaluation: G 3 (, 2, 3 )= C(, 2, 3) C(, 2, 3) = ( ) ( ) 2 ( ) 3 This has the same form as a 3 point function in a + dimensional CFT!

19 Thermal autocorrelators z =2,d=2 Lifshitz black hole: ds 2 = f(u) u 4 d 2 + du2 u 2 f(u) + dx2, f(u) = u4 u2 Thermal 2-point function: G E ( 2,x;,x) /! 2 T sin( T 2 ) Identical to thermal autocorrelator in the quantum Lifshitz model! Geodesics that contribute to autocorrelator only involve Mapping to BT black hole: ds 2 2,Lif = ds 2 2,BT = y 2 " u 4 u 4 H d 2 y 2 y 2 H g,g rr u 4 + du 2 u 2 u ( 4 ) u 4 H d 2 + dy2 y 2 2 y H y = u2 2, ds2 2,Lif = 4 ds2 2,BT y H = u 2 H/2 BT geometry is dual to a thermal state in + dimensional CFT #

20 Dynamical solutions EMD model, d = 2 Rewrite static black brane solution using dv = dt + r z b(r) dr ds 2 = r 2z b(r)dv 2 +2r z dv dr + r 2 dx 2 Keränen, Keski-Vakkuri, & L.T. b(r) = m r0 z r 4z r0 2z+2 r F (2) rt = z r0 z r0 z+ r e = r r 0 p 2 z F () p z+ r rt =2r0 z (z+2)(z ) r 0 Lifshitz-Vaidya solution: m! m(v),! (v) m(v), (v) determined by incoming energy and charge density The time and radial parts of metric can be mapped to a BT-Vaidya solution 2-point autocorrelators can then be obtained from autocorrelators calculated in BT-Vaidya spacetime Balasubramanian, Bernamonti, Craps, Keränen, Keski-Vakkuri, Müller, L.T., Vanhoof 2

21 Summary Black branes in asymptotically Lifshitz spacetime provide a window onto finite temperature effects in strongly coupled models with anisotropic scaling. Analytic black brane solutions are available for all physical values of d and z in an Einstein-Maxwell-Dilaton (EMD) model, at the price of having a strongly coupled background gauge field (that does not couple directly to other gauge fields or matter). Autocorrelators of scaling operators are identical in the quantum Lifshitz model and in a strongly coupled Lifshitz model with a holographic dual. Vacuum autocorrelators in the holographic model at any value of z and d can be expressed in terms of autocorrelators of a + dimensional CFT. Thermal autocorrelators at z = d can also be expressed in terms of the + dimensional CFT. These methods can be used to study out-of-equilibrium phenomena in Lifshitz models: - mass quench in the quantum Lifshitz model - holographic quench in Lifshitz-Vaidya spacetime