Spatially Modulated Phases in Holography. Jerome Gauntlett Aristomenis Donos Christiana Pantelidou

Transcription

1 Spatially Modulated Phases in Holography Jerome Gauntlett Aristomenis Donos Christiana Pantelidou

2 Spatially Modulated Phases In condensed matter there is a variety of phases that are spatially modulated, spontaneously breaking translation invariance. For example: charge density waves and spin density waves The modulation is fixed by an order parameter associated with non-zero momentum O(k) =0. The modulation can be in various configurations such as stripes (as in the cuprates), in checkerboards and hexagonal lattices. Helical structure in chiral nematic liquid crystals and helimagnets e.g. MnSi There is a growing set of holographic examples described by exotic black holes (probe brane studies initiated by [Domokos,Harvey])

3 D=5 helical current phases [Nakamura,Ooguri,Park][Donos,Gauntlett] x 2 J t = const x 3 J x1 =0 J x2 cos(kx 1 ) x 1 J x3 sin(kx 1 ) D=5 helical superconducting phases [Donos,Gauntlett] D=4 stripes [Donos,Gauntlett][Donos][Withers][Rozali,Smyth,Sorkin,Stang] O cos kx J t J t cos(2kx) J x =0 J y sin kx

4 Instabilities of unbroken phase black holes driven by mass squared becoming sufficiently negative For a single mode spatially modulation is suppressed ω 2 = mass 2 + k 2 Tc k

5 However, instabilities for non-zero k can easily be achieved if there is a mixing of modes since one diagonalises a mass matrix The above examples achieve this via Chern Simons couplings and axions and one finds Tc kc k These examples all spontaneously break P and T

6 Not necessary: there are D=4,5 black holes dual CDWs that don t break P and T [Donos,Gauntlett] The above examples are for CFTS at T,µ. There are also examples of spatially modulated phases in uniform magnetic fields [Bolognesi,Tong][Donos,Gauntlett,Pantelidou][Jokela,Lifschytz,Lippert] [Cremonini,Sinkovics][Almuhairi,Polchinski][Ammon,Erdmenger,Shock,Strydom]... Closely related examples - but distinct physics - where translations broken explicitly [...][Horowitz,Santos,Tong][Donos,Hartnoll][Vegh][Erdmenger,Ge,Pang][...] Many more examples of spatially modulated black holes to be found Finding them is one way of finding new ground states

7 Plan General results on the thermodynamics of periodic AdS black branes Competing superconducting p-wave and (p+ip)-wave orders

8 Thermodynamics of Periodic AdS Black Branes with Aristomenis Donos Complementary work by [Domokos,Hoyos,Sonnenschien]

9 Consider asymptotic AdS black holes with planar topology. Assume the black holes have a Killing horizon, dual to configurations in thermal equilibrium We are interested in solutions that are periodic in the spatial directions Some of the configurations might be translationally invariant, independent of some or all of the spatial coordinates If the translation symmetry is broken it can either be spontaneous or explicit For simplicity consider bulk theory with metric g and gauge-field A g mn T µν A m J µ

10 Introduce coordinates x µ =(t, x i ) for d-dimensional boundary Behaviour at the AdS boundary: ds 2 γ µν dx µ dx ν A a µ dx µ The black hole has a Killing horizon and the Killing vector on the boundary t Bulk black brane solution is periodic in the globally defined with period L i. Might be compact or non-compact x i The solutions might be independent of some x i If solutions depend on the symmetry breaking can either be explicit if the sources γ µν or a depend periodically on x i µ or spontaneous if not x i

11 Euclideanise with and t iτ τ = τ + τ T 1 τ Focus on the non-compact case. We are interested in minimising free energy density w = w (γ µν,a µ ; τ,l i )= 1 τπ I OS (γ µν,a µ ; τ,l i ) Variation with respect to γ µν and a µ gives, as usual: Π Π i L i δw = 1 τπ τ 0 {Li } 0 dτd d 1 x γ 1 2 T µν δγ µν + J µ δa µ = 1 Π {Li } 0 d d 1 x γ 1 2 T µν δγ µν + J µ δa µ

12 To get variation with respect to periods, scale coordinates: τ = τ τ x i = L i x i with and I OS (γ µν,a µ ; τ,l i )=I OS ( γ µν, ã µ ;1, 1) ds 2 γ µν d x µ d x ν γ ττ ( τ) 2 d τ 2 +2γ τx i τl i d τd x i + γ xi x j L il j d x i d x j A ã µ d x µ a τ τd τ + a x L i i d x i Variation with respect to periods via chain rule

13 δw = Π 1 {L} where s 1 T 0 sδt i w + Π 1 {L} 0 d d 1 x γ 1 2 T µν δγ µν + J µ δa µ δl i L i X i d d 1 x γ(t tt γ tt + j T txj γ tx j + J t a t ) X i = w + Π 1 {L} Comments: 0 d d 1 x γ (T txi γ tx i + j T xi x j γ xi x j + J xi a x i) 1. For black branes invariant under translations in direction we have and this gives Smarr-type relation(s) X i =0 c.f [El-Menoufi, Ett, Kastor,Traschen] x i

14 2. Black branes that are spatially modulated in the direction, depend on wave-numbers. One has k i 2π/L i x i k i δw δk i = X i = w + Π 1 {L} 0 d d 1 x γ(t txi γ tx i + j T xi x j γ xi x j + J xi a x i) The thermodynamically preferred spatially modulated black branes in the non-compact case and where symmetry breaking is spontaneous have X i =0 The fact that this variation is simply related to T µν and J µ was obscure in a number of examples 3. The expression for S was a definition. It can be related to the area of the event horizon using bulk equations e.g. for two-derivative gravity it is 1/4 of the area of the event horizon e.g [Papadimitriou,Skenderis]

15 Further relations can be obtained! The euclidean boundary is a d dimensional torus. The circle with tangent is singled out by the black hole in the bulk τ The Sl(d, Z) transformation τ = τ + α i x i, no sum on i x i = x i with Also, α i = τ L i τ = τ parametrises the same torus with same periods. Can repeat the entire analysis in the barred coordinates - one should get the same result

16 Remarkably, this implies {L} 0 d d 1 x γ T xit γ tt + k T xi x k γ xk t + J xi a t =0 By considering other Sl(d, Z) transformations preserving τ {L} 0 d d 1 x det γ T xit γ tx j + k T xi x k γ xk x j + J xi a x j =0 i = j

17 E.g. consider CFTs in flat spacetime, γ µν = η µν, a µ =(µ, 0) Assume spontaneous symmetry breaking in the Conservation of T µν and J µ x 1 direction. T x1t, T x1 x 1,..., T x1 x d 1, J x1 are constants also δw = J t δµ sδt + δk k w + T x1 x 1 and w = T x2 x 2 = = T xd 1 x d 1 w = Ts J t µ + T xit + J xi µ =0 T tt T xi x j =0 i = j = 1 T x1 x i =0 Bars refer to quantities averaged over a period

18 Some of these were not obvious from numerics! Eg For thermodynamically preferred spatially modulated black branes T x1 x 1 = T x2 x 2 = = T xd 1 x d 1 Results can be generalised to include more matter fields Results should generalise to other holographic scaling solutions such as Lifshitz

19 Superconducting p-wave phases with Aristomenis Donos Christiana Pantelidou

20 p-wave superconductivity is seen in a number of different systems e.g., organic superconductors, He 3 Sr 2 RuO 4 Two approaches to study it holographically: 1. D=4,5 use SU(2) gauge fields. Take the background to be charged with respect to U(1) SU(2) and then spontaneously break the U(1) using the charged vector bosons [Gubser][Roberts,Hartnoll][Ammon,Erdmenger,Grass,Kerner,O Bannon] In the specific setting it was shown that p-wave is preferred over (p+ip)-wave [Gubser, Pufu] 2. D=5 use charged self-dual two-forms [Aprile,Franco,Rodriguez,Russo] Extend p-wave studies of [Donos,Gauntlett] and develop (p+ip)-wave

21 The Charged two-form model in D=5 L = (R + 12) F F 1 2 C C i 2m C H, The model admits a unit radius with F = da, H = dc + ie A C. A = C =0 AdS 5 vacuum solution which is dual to a d=4 CFT. The CFT has a global U(1) symmetry and conserved current J µ A is dual to the The two-form satisfies a self duality equation H = imc, and is dual to a d=4 self-dual tensor operator with charge (O C )=2+m e and

22 Such models arise in string theory. For example type IIB on can be truncated to this model with, e =1/ 3 m =1 S 5 Electrically charged AdS-RN black brane describes the spatially homogeneous and isotropic phase of the CFT at, and high µ T UV: r AdS 4 IR: r r + A t = µ(1 r + r ) black hole horizon topology R 2 and temp T Electric flux

23 This model has superconducting instabilities if e 2 > m2 The simplest way to see this is to find modes that violating the BF bound of the T=0 limit of the AdS-RN black brane 2 AdS 2 To determine the critical temperature at which the instability sets in, one looks for linearised normalisable zero modes in the full AdS-RN black brane solution Both p-wave and (p+ip)-wave instabilities depending on wave-number k They spontaneously break the U(1) and some of the other translations and rotations...

24 p-wave instabilities: k =0 δc = + c 3 (r) dx 1 dx 3 c 3 = c O (r r + ) c 3 (r) c c3 r m +... Order parameter points in the x 2 direction Three translations preserved Rotations in the (x 1,x 3 ) plane preserved

25 Helical p-wave instabilities: k = 0 δc = + c 3 (r)[sin(kx 1 )dx 1 dx 2 + cos(kx 1 )dx 1 dx 3 ] x 2 x 2 x 3 x 1 x 2 and x 3 preserved x 1 + k(x 3 x 2 x 2 x 3) preserved This is the homogeneous Bianchi VII 0 symmetry group

26 p+ip-wave instabilities k =0 δc = + ic 3 (r)dx 1 (dx 2 idx 3 ) The phase of the order parameter is now involved Three translations preserved Rotations in combined with gauge transformation preserved (x 2,x 3 ) p+ip-wave instabilities k = 0 δc = + e ikx 1 ic 3 (r)dx 1 (dx 2 idx 3 ) Same symmetry as - translations in now combined with gauge transformations x 1 k =0

27 P and (p+ip)-wave instabilities of AdS-RN for m =2 T 0.12 e = e = e = k Highest T instability has k = 0 Each point under bell curve corresponds to the existence of a back reacted p-wave and (p+ip)-wave black hole For both cases we should find the preferred black holes which minimises the free energy density with respect to k. Then we should compare p and (p+ip)

28 Back reacted Helical p-wave black holes Ansatz ds 2 = gf 2 dt 2 + g 1 dr 2 + h 2 ω r 2 e 2α ω e 2α ω 2 3 C =(ic 1 dt + c 2 dr) ω 2 + c 3 ω 1 ω 3 A = adt Where g, f, h, α, c i,a are functions of r and we have used the Bianchi VII 0 ω 1 = dx 1 invariant one-forms ω 2 = cos (kx 1 ) dx 2 sin (kx 1 ) dx 3, ω 3 = cos (kx 1 ) dx 2 +sin(kx 1 ) dx 3 Solve ODEs for functions with suitable boundary conditions at the black hole horizon and at the AdS boundary. Analyse the thermodynamics...

29 e =2 c c3 T k T T tt = 3M +8c h T x1 x 1 = M +8c h T x2 x 2 = M +8c α cos(2kx 1 ) T x3 x 3 = M 8c α cos(2kx 1 ) T x2 x 3 = 8c α sin(2kx 1 ) preferred branch: k w =0 T x1 x 1 = T x2 x 2 = T x3 x 3 c h =0

30 e =2 c c3 T k T T tt = 3M T x1 x 1 = M T x2 x 2 = M +8c α cos(2kx 1 ) T x3 x 3 = M 8c α cos(2kx 1 ) T x2 x 3 = 8c α sin(2kx 1 )

31 Emergent helical scaling in the IR at T=0 As T 0 we find that, with The solution interpolates between AdS5 in the UV and a helical scaling solution in the far IR: ds 2 = ( f 0 2 L 2 )r 2z dt 2 + L 2 dr2 r 2 +(k2 h 2 0)dx r 2 e 2α 0 ω2 2 + e 2α 0 ω3 2 C =... A =... s T 2/z z 2 where Scaling f 0,L,h 0,α 0 are constants t λ z t, x 2,3 λx 2,3, x 1 x 1 r λ 1 r Similar to ground states of [Iuzuka,Kachru et al]

32 e =2.8 T Similar to the previous case, but now as T 0 we have k k New anisotropic scaling solution: ds 2 = ( f 2 0 L 2 )r 2z dt 2 + L 2 dr2 r 2 + α 2 0 r2(1 γ) dx dx α 2 0 r 2(1+γ) dx 2 2 r λ 1 r, t λ z t, x 1,3 λ 1 γ x 1,3, x 2 λ 1+γ x 2 Similar to ground states of [Taylor]

33 e =3.5 T Similar to previous cases, but now we see the phenomenon of pitch inversion at T = 0 Such pitch inversion is observed in chiral nematics and helimagnets k Metric approaches AdS5 in the IR! Similar behaviour was seen in s-wave superconductor [Horowitz, Roberts] and also in p-wave black holes for su(2) model [Basu] Has been described as emergent conformal symmetry. However, correlators do not approach those of CFT for small frequency.

34 Back reacted (p+ip)-wave black holes ds 2 = gf 2 dt 2 + g 1 dr 2 + h 2 (dx 1 + Qdt) 2 + r 2 (dx dx 2 3) A = adt + bdx 1 C = e ikx 1 (ic 1 dt + c 2 dr + ic 3 dx 1 ) (dx 2 idx 3 ) Time translations preserved, but stationary metric Three space translations preserved (plus gauge) Rotations in (x 2,x 3 ) preserved (plus gauge)

35 T e = c c T k T tt = 3M +8c h, T tx1 = 4c Q, T x1 x 1 = M +8c h, T x2 x 2 = M, T x3 x 3 = M, preferred branch: k w =0 T x1 x 1 = T x2 x 2 = T x3 x 3 c h =0 Two extra Smarr relations c Q =0

36 0.015 e =2 T c c T k T tt = 3M T tx1 = 0 T x1 x 1 = M T x2 x 2 = M T x3 x 3 = M

37 e =2.8 e = T 0.12 T k k No analogue of pitch inversion For all values of the charge e we find s 0 as T 0 (in some cases at very low temperatures) The nature of the ground states is unclear...

38 Competing orders w e =2 p-wave preferred 0.10 w e =3.5 (p+ip)-wave preferred TTc TTc e =2.8 w (p+ip)-wave then first order transition to p-wave TTc

39 Final Comments Rich classes of spatially modulated black holes exist with interesting properties and novel ground states More to be discovered