Holographic Lattices

Transcription

1 Holographic Lattices Jerome Gauntlett with Aristomenis Donos Christiana Pantelidou

2 Holographic Lattices CFT with a deformation by an operator that breaks translation invariance Why? Translation invariance ) momentum is conserved, hence no dissipation and hence DC response are infinite. To model more realistic metallic behaviour or insulating behaviour we can use a lattice The lattice deformation can lead to novel ground states at T=0. Can also model metal-insulator transitions Formal developments: thermo-electric DC conductivities in terms of black hole horizon data [Donos,Gauntlett] Analogous to = s 4 [Policastro,Kovtun,Son,Starinets]

3 Plan Drude physics Lattice with global U(1) symmetry and Maxwell theory. Coherent metals. µ(x). In Einstein- Q-lattices, using scalars and global symmetry. Can give coherent metals, incoherent metals and insulators and transitions between them. Helical lattices in D=5 pure gravity. Universal deformation. Coherent metals. Comments on calculating Greens functions

4 Drude Model of transport in a metal Quasi-particle interactions ignored m d dt v = qe J = nqv m v ) v = q E m J = DC E DC = nq2 m

5 E = E(!)e i!t J(!) = (!)E(!) J = J(!)e i!t (!) = DC 1 i! DC = q2 m Re[ ] 1 (T ) Coherent or good metal E F! When!1 (!) (!)+ i!

6 Drude physics doesn t require quasi-particles Coherent metals arise when momentum is nearly conserved [Hartnoll,Hofman] Pole on negative imaginary axis near origin! = i Similar comments apply to thermal conductivity Q = applert There are also incoherent metals without Drude peaks Not dominated by single time scale Of particular interest to realise these in holography Insulators with DC = apple DC =0 at T=0

7 Holographic CFTs at finite charge density Focus on d=3 CFT and consider D=4 Einstein-Maxwell theory: Z S = d 4 x p h i 1 g R +6 4 F Admits AdS 4 vacuum $ d=3 CFT with global U(1)

8 Electrically charged AdS-RN black hole (brane) Describes holographic matter at finite charge density that is translationally invariant ds 2 = Udt 2 + dr2 U + r2 (dx 2 + dy 2 ) A t = µ(1 r + r ) d=3 CFT µ T Electric flux T=0 limit: AdS 2 R 2 AdS 4 IR UV

9 By perturbing the black hole and using holographic tools we can calculate the electric conductivity and find a delta function at! =0 [Hartnoll] Construct lattice black holes dual to CFT with µ(x) A t (x, r) µ(x)+o( 1 r ) r!1 g µ (x, r) Need to solve PDEs in two variables e.g. Monochromatic lattice: µ(x) =µ + A cos kx [Horowitz, Santos,Tong] [Donos,Gauntlett] After constructing black holes, one can perturb, again solving PDEs, to extract thermo-electric conductivities

10 Find Drude physics at finite T Re(σ) T/μ=0.47 T/μ=0.22 T/μ=0.039 T/μ=0.14 T/μ=0.025 T/μ=0.097 T/μ=0.02 T/μ=0.08 T/μ=0.015 T/μ= ω/μ

11 Coherent metal phases Can be understood by analysing T=0 solutions: UV data k/µ A/µ AdS-RN IR fixed point AdS 2 R 2 At T=0 the black holes approach AdS 2 R 2 in the IR perturbed by irrelevant operator with (k IR ) 1 Don t find exceptions to this behaviour even for dirty lattices e.g. µ(x) =1+A 10 X n=1 cos(nkx+ n ),

12 Holographic Q-lattices Illustrative D=4 model L = R V ( ' ) Z( ' ) 4 [Donos,Gauntlett] F 2 Choose V,Z so that AdS-RN is a solution at ' =0 Now ' $ O in CFT. Want to build a holographic lattice by deforming with the operator O The model has a gauge U(1) and a global U(1) symmetry Exploit the global bulk symmetry to break translations so that we only have to solve ODEs

13 Ansatz for fields ds 2 = Udt 2 + U 1 dr 2 + e 2V 1 dx 2 + e 2V 2 dy 2 A t = a(r) '(r, x) = (r)e ikx UV expansion: U = r , a = µ + q r..., e 2V 1 = r e 2V 2 = r = r Homogeneous and anisotropic and periodic holographic lattices UV data: T/µ /µ 3 k/µ

14 For small deformations from AdS-RN we find Drude peaks corresponding to coherent metals. k/µ /µ This can be understood by examining T=0 behaviour of solutions: AdS-RN New AdS 2 R 2 For larger deformations, for specific models, we find a transition to new behaviour. The new ground states can be both insulators and also incoherent metals! See also: [Gouteraux][Andrade,Withers]

15 D=4 CFTs with a Helical Twist [Donos,Gauntlett,Pantelidou] Study a universal helical deformation that applies to all d=4 CFTS First recall the Bianchi VII 0 Lie algebra [L 1,L 2 ]= kl 3 [L 1,L 3 ]=kl 2 [L 2,L 3 ]=0 L 1 x1 + k(x x2 x x3 ) L 2 x2 L 3 x3 x 2 x 3 x 1

16 Useful to introduce the left-invariant one-forms! 1 = dx 1! 2 = cos (kx 1 ) dx 2 sin (kx 1 ) dx 3,! 3 = cos (kx 1 ) dx 2 +sin(kx 1 ) dx 3 We want to explicitly break the ISO(3) spatial symmetry of the CFT down to Bianchi VII 0 Achieve by introducing suitable sources for the stress tensor Equivalently, consider CFT not on R 1,3 but on ds 2 = dt 2 +! e 2 0! e 2 0! 2 3 with k, 0 parametrising the deformation

17 Study in holography by considering Z S = d 5 x p g(r + 12) This is a consistent truncation of all AdS 5 M solutions in string/m-theory. Hence analysis applies to entire class of CFTs Ansatz ds 2 = gf 2 dt 2 + g 1 dr 2 + h 2! r 2 e 2! e 2! 2 3 Equations of motion f 0 =..., g 0 =..., h 00 =..., 00 =... AdS-Schwarzschild: f =1, g = r 2 r+ 4, h = r, =0 r2

18 Expand functions at UV boundary f =1 + g =r 2 1 h =r 1 k2 = 0 k 2 12r 2 (1 cosh 4 0) k 2 6r 2 (1 cosh 4 0) k 2 4r 2 (1 c h r 4 + k4 96r 4 (3 + 4 cosh cosh 8 0 ) + log r() +..., + log r() +..., 4, M r cosh 4 0)+ c h + log r() +... r4 4r 2 sinh c + log r() +... r4 Source parameters: 0,k Vev parameters: c h,c,m Together these give T µ of helically deformed CFT Log terms arise because of conformal anomaly T µ µ = k4 3 (cosh(8 0) cosh(4 0 ))

19 Boundary conditions in the IR - smooth black hole horizon g = g + (r r + )+, f = f + +, h = h + +, = + + with g + =4r + Parameter count: expect two parameter family of black holes labelled by, (for fixed dynamical scale) k/t 0

20 Thermodynamics from Free energy density T µ w [Donos,Gauntlett] Boundary metric $ ds 2 = dt 2 +! e 2 0! e 2 0! 3 2 Killing t w = Ts k 2 Z 2 /k 0 dx 1 p T tt tt Killing x2 x3 w = k 2 Z 2 /k 0 dx 1 p T x2 x 2 x 2 x 2 + T x2 x 3 x 2 x 3, w = k 2 Z 2 /k 0 dx 1 p T x3 x 2 x 3 x 2 + T x3 x 3 x 3 x 3

21 First law w = s T 2 k Z 2 /k 0 dx 1 p 1 2 T µ µ Z 2 /k! + k k w + 2 k dx 1 p (T x 1 x 2 x1 x 2 + T x 1x 3 x1 x 3 ) 0

22 Results of numerics Ts'/s T α 0 = 1 4 α 0 = 1 2 α 0 =1 At T=0 the solution might be approaching AdS5?

23 T=0 interpolating solutions Consider small perturbation of about AdS5 which one solve in terms of Bessel functions Suggests the IR expansion as r! 0 g = r 2 + k3 2 + r e 4k/ h + r (1 + 5 h + 8k r + O(r2 )) +, f = f k 3 2 f r 3 e 4k/ h + r (1 + 5 h + 8k r + O(r2 )) +, k 3 + h 2 + h = h + r = +2k 2 p h+ r 2 K 2 2r 2 e 4k/ h + r ( h + 8k r + O(r2 )) +, 2k +, h + r Note that there can be a renormalisation of length scales

24 Length scale renormalisation s g x1 x 1 (r! 0) g x1 x 1 (r!1) l a 0 Note similar T=0 ground states have been seen before Chemical potential lattice [Chesler,Lucas,Sachdev] µ(x) s-wave superconductors [Horowitz,Roberts] with no zero-mode p-wave superconductors [Basu,He,Mukherjee,Rozali,Shieh] [Donos,Gauntlett,Pantelidou]

25 Greens functions for thermal conductivity at finite T Perturb black hole (ds 2 )=2 g tx1 (t, r)dtdx 1 +2 g 23 (t, r)! 2! 3 with Z d! g tx1 (t, r) = 2 e i!t h tx1 (!,r) Z d! g 23 (t, r) = 2 e i!t h 23 (!,r) We obtain linear ODEs: h =... h 00 tx =... and a constraint equation involving h 0 tx and h 0 23

26 IR boundary conditions Ingoing at black hole horizon [Son,Starinets] UV boundary conditions h tx1 = r 2 s 1 + i!k 2 sinh 2 0s 2 + v 1 r 2 + h 23 = r 2 s 2 +( 1 2 i!k sinh 2 0s 1 +!2 4 s 2 k 2 cosh s 2 )+ v 2 r with the constraint 64i!v k sinh 2 0 v 2 + i!( 128c h + 16k 4 sinh )s 1 4k 64c cosh k 2 sinh (2k 2! 2 +2k 2 cosh 4 0 ) s 2 =0

27 Greens function with J i = G ij s j J 1 = ht tx 1 i = lim r!1 J 2 = ht! 2! 3 i = lim r!1 1 S (2) p g1 h tx1 (r), 1 S (2) p g1 h 23 (r). Calculate the variation of the action and discard a horizon contribution with S (2) 1 = Z d 2 x Z! 0 d! 2 J 1 = s 1 (...)+s 2 (...) 4v 1 J 2 = s 1 (...)+s 2 (...)+4v 2 s i (!)J i (!)+ s i (!) J i (!)! dots are fixed by! and black hole background: 0,k,M,c,c h

28 To obtain G ij j we need to work j This is subtle due to residual gauge invariance Take the background black hole with x 1! x 1 + e i!t 0 which induces s 1! s 1 i 0!, v 1! v 1 2i 0!(c h + k4 8 sinh4 2 0 ) s 2! s k sinh 2 0, v 2! v 2 0 k cosh 2 0 (4c + k 4 cosh 2 0 sinh ) with some work one can j consistent with these

29 Alternative procedure: work in a gauge with s 1 =0 and calculate G i2 = J i s 2 s 1 =0 Then work in a gauge with s 2 =0 and G i1 = J i s 1 s 2 =0

30 Comment: suppose we calculate on-shell action S (2) 1 = Z! 0 d! 2 s 2 s 2 (...)+s 1 s 1 (...)+s 1 s 2 (...)+ s 1 s 2 (...) + 2(s 2 v 2 + s 2 v 2 s 1 v 1 s 1 v 1 ) dots are fixed by! and black hole background: 0,k,M,c,c h Here we did NOT discard any terms arising from the black hole horizon Using the same derivatives j as above we 2 S s j = G ij + G ij

31 Numerical results Focus on G 11 (!) =ht tx 1 T tx 1 i and recall that T apple(!) G 11 i! ImG 11 /ω k T/k=0.14 T/k=0.3 T/k=0.5 T/k=0.7 T/k=0.9 ReG 11 /k T/k=0.14 T/k=0.3 T/k=0.5 T/k=0.7 T/k= ω/k ω/k

32 Summary/Final Comments Holographic lattices are interesting d=3,4 CFTs with global U(1) symmetry: Einstein-Maxwell theory and µ(x) deformation (PDEs) Q-lattice: Einstein-Maxwell plus scalar field with global symmetry in the bulk (ODEs) d=4 CFTs with universal helical deformation (ODEs)

33 All of these included a realisation of strongly coupled Drude physics at finite T, at least for small deformations The Drude physics can be understood by the appearance of translationally invariant ground states in the far IR: or AdS 2 R 2 AdS 5 For larger deformations the Q-lattices realised incoherent metallic an insulating phases The T=0 ground states break translation invariance The phases have novel thermoelectric transport properties (not determined by memory matrix formalism) What is the landscape of such spatially modulated ground states?