Building a holographic liquid crystal

Transcription

1 Building a holographic liquid crystal Piotr Surówka Vrije Universiteit Brussel Ongoing work with R. Rahman and G. Compère Gauge/Gravity Duality, Munich,

2 Motivation Hydrodynamics is an effective theory that still requires better understanding Recent holographic models provided a lot of insight into relativistic hydrodynamics Low viscosity over entropy density in the hydrodynamic description of heavy-ion collision is understood Progress in the studies of QFT anomalies in hydrodynamics Superfluidity is investigated via AdS/CFT What else we can possibly shed some light on using holography?

3 Solid Liquid Crystal Fluid Solid is characterized by a structural rigidity and resistance to changes of shape or volume. Described by classical mechanics. Fluid changes its shape under applied shear stress. Described by hydrodynamics. Liquid crystal is a state of matter in between. It shares properties with fluids (eg. deforms under shear stress) and solids (eg. non-zero elasticity properties). Phase transition between isotropic an unoriented phase and an ordered liquid crystal phase. There is a number of phenomenological theories (eg. Landau-De Gennes theory). Theoretical formalism is very complex

4 Liquid crystals Texture - distribution of crystallographic orientations 7$8'.,%)9:'-$) ;8$%.,%)<)/#'=$($26)9:'-$)

5 Phase transitions A phenomenological theory of phase transitions was established by Landau. He suggested that Pressure phase transitions were manifestation of a broken symmetry. In the simplest cases through the definition of an appropriate order parameter, Q, the macroscopic behaviour of a phase may be followed. Typically Q = 0 in the more symmetric (less ordered) phases and Q = 0 in the less symmetric (more ordered) phases. Science 12, 207, vol. 315 no The theory, though originally introduced to describe continuous phase transitions in solids, appeared (as we know today) to correctly account for symmetry change observed at majority of continuous and first order phase transitions. The Landau theory generaly fails in the temperature adjacent to the transition, in which the behaviour of a system is dominated by fluctuations

6 Landau thery tailored to describe liquid crystals The LT appears as a necessary intermediate step (and also as a tool) in constructing generalized theory that include fluctuations - it is known as Landau-Ginzburg-Wilson theory: A major breakthrough in including liquid crystals in the Landau reasoning is due to de Gennes. He realized that instead of the scalar order parameter one should use a tensorial quantity. For the nematic phase it reads: where Q αβ = S 2 (3ˆn αˆn β δ αβ ) 3 cos 2 θ 1 ˆn θ S = 1 2

7 Landau-de Gennes theory Once the appropriate tensor order parameter of the system is identified we can assume, in a spirit of Landau theories, that the free energy density F is an analytic function of the order parameter. The Landau-de Gennes theory of the nematic-isotropic transition starts by assuming that a spatially invariant dimensionless, order parameter is small in the nematic phase close to the transition point. The difference in free energy density (per unit volume) of the two phases it thus expanded in powers of the order parameter. Since the free energy must be invariant under rigid rotations, all terms of the expansion must be scalar functions of the tensor. F LdG (P, T, Q αβ )=F 0 + α F Q αβ Q βα + β F Q αβ Q βγ Q γα + γ F Q αβ Q βα Q γρ Q ργ The most general form capturing the uniaxial phase is typically truncated at fourth order.

8 Elasticity Splay Twist Bend The question we would like to address here is: how much energy will it take to deform the director field? The deformation of relative orientations away from equilibrium position will manifest itself as curvature strain. The restoring forces which arise to oppose these deformations will cause curvature stresses or torques. If these changes in molecular orientation vary slowly in space relative to the molecular distance scale, we may describe the response of the liquid crystal with a version of a continuum elastic theory.

9 Isotropic phase - hydrodynamics Relativistic fluid with one conserved charge described by conservation laws: µ T µν =0 µ j µ =0 plus equations that express T µν and j µ in terms of local temperature T, chemical potential µ, and fluid velocity u µ : T µν =( + P )u µ u ν + Pg µν + τ µν j µ = nu µ + ν µ Hydrodynamic The definition of regime velocity has is a very ambiguous natural beyond description leading AdS/ order. CFT We fix and itthis by imposing description u µ is τ µν under = u µ analytic ν µ = 0. control.

10 AdS/CFT at finite temperature SYM on a stack of D3-branes AdS Schwarzschild geometry T>0 l s ds 2 = r2 L 2 ( f(r)dt2 + dx i dx i )+ where f(r) =1 R4 r 4 extremal -branes. L2 f(r)r 2 dr2 + L 2 dω 2 5 As we learned from the previous constructions of holographic models AdS Schwarzschild geometry can be viewed as a box in which black hole can develop a hair. Our aim is to have spin 2 hair.

11 Spin 2 Lagrangian Complicated plus various consistency issues S = d d+1 x g 1 2 ( µϕ αβ ) 2 +( µ ϕ µα ) ( µϕ) 2 µ ϕ µν ν ϕ 1 2 m2 ϕ(ϕ µν ϕ µν ϕ 2 ) 1 4 F µνf µν 1 2 m2 AA 2 µ + R µναβ ϕ µα ϕ νβ 1 2(d + 1) Rϕ2 α 2 A2 (ϕ µν ϕ µν ϕ 2 ), We need a quadratic coupling between the fields. A cubic coupling of the form a 1 A µ A ν ϕ µν + a 2 A 2 ϕ would not allow for spontaneous hairy black holes. The spin 1 field would just act as a source for the massive spin 2 field and all charged black holes would have a secondary hair. Instead we want to have a hair appearing below a critical temperature. The situation would be similar to trying to build a superconductor using a dilatonic coupling e φ F 2. It would not work since the electric field would source the dilaton.

12 Equations of motion Our ansatz: ϕ µν = diag(0, 0, ϕ x1 x 1 (z), ϕ x2 x 2 (z), ϕ x3 x 3 (z)) C µ dx µ = φ(z)dt where ϕ ij = l2 z 2 ψ(z) d 1 d 2 δi 1 δj 1 1 d 1 We get the following EOMs f ψ + f d 1 ψ + z φ d 3 z φ + + αφ2 f 2 m2 ϕl 2 z 2 f αl2 z 2 f ψ2 m2 A l2 z 2 f ψ =0, φ =0.

13 Looking for an instability The dynamical fields ψ, A 1 obey hypergeometric equations when there is no coupling α = 0. Their fall-off is ψ(z) =ψ D z d 2 d 2 4 +m2 ϕ l2 + ψ N z d 2 + d 2 4 +m2 ϕ l φ(z) =µz d 2 2 (d 2) 2 4 +m 2 A l2 + φ N z d (d 2) 2 4 +m 2 A l We impose regularity at the horizon and Dirichlet condition at the boundary 1.5 Instability! ΨN T c We have spin 2 condensate T T c

14 Partition function The Euclidean black hole solution is interpreted as a saddle-point in the path integral corresponding to the thermal partition function. The supergravity action evaluated for this solution is interpreted as the leading contribution to the free energy. Free energy for a stack of D3-branes F = TS sugra = π2 8 N 2 c T 4 To prove the instability for the spin-2 system we need to calculate the partition functions for the isotropic phase and for the phase with the condensate and show that the system lowers the free energy by developing the condensate.

15 Free energy analysis For simplicity, we will look at d = 4 but we will keep the masses general. S tot = S + c d d x γ A µ A µ The variation of the total action reduces for our ansatz to δs tot = d 5 x g (E µν δϕ µν + E µ δa µ ) l dtd 3 x z φ δφ + 2cl2 z 2 f φδφ l3 f z 3 ψ δψ We fix c in order to have a well define Dirichlet problem δs tot = d 5 x g (E µν δϕ µν + E µ δa µ )+ We pass to Euclidean signature and evaluate the action F (ψ = 0) = S E /β = V zh 0 dz αl3 φ 2 ψ 2 2f(z)z 3 + V dtd 3 x (J + Oδµ) z=0 lφφ 2z + cl2 φ 2 z 2 f + l3 fψψ 2z 3 z=0

16 An instability confirmed Finally we can write down an expression for the difference of the free energies in the normal and condensed phase F (ψ = 0) F (ψ = 0) = V β/π 0 dz αl3 φ 2 ψ 2 2f(z)z (O O n)µ. 0 FT Difference of free energy between the uniaxial nematic phase and the normal phase as a function of the temperature T = 1/β. The uniaxial nematic phase is favored

17 v (c) Remark on viscosity in liquid crystals n v and n v n n v (a) n v and n v η 1 η 2 In liquid crystals we can identify three different shear viscosities. This is very natural if we take into account the rodlike shape of the microscopic constituents n v (b) n v and n v η 3 To calculate viscosity in the holographic models we need to include the backreaction. For our system this will turn on new fields, which makes things very complicated.

18 Frank-Oseen free energy The six components of curvature can be defined by splay s 1 = n x x, twist t 1 = n y x, bend b 1 = n x z, s 2 = n y y t 2 = n x y b 2 = n y z n x (r) = s 1 x + t 2 y + b 1 z + O(r 2 ), n y (r) = t 1 x + s 2 y + b 2 z + O(r 2 ), n z (r) = 1+O(r 2 ). This definition can be understood by expanding the director in the Taylor series or We now postulate that the Gibbs free energy density of a liquid crystal, relative to its free energy density in the state of uniform orientation can be expanded in terms of six curvature strains. Using symmetries of the system we can reduce it to the following form F = 1 2 k 11( n s 0 ) k 22(n curl n + t 0 ) k 33(n curl n) 2 k 12 ( n)( curl n)

19 Elasticity coefficients Picture from Current Applied Physics vol. 12 no These constants are very i m p o r t a n t m a t e r i a l parameters, critically dictating the behavior of a nematic. They have the dimension of a force, and they must be positive (required for stability). In order to extract the elasticity coefficients from AdS/CFT we want to expand the director field in gradients, solve the EOMs, evaluate the holographic free energy on the solution and read off the parameters Work in progress...

20 Future directions Check if the model gives the equations of nematodynamics Understand spin 2 lagrangians couple to gravity and study backreaction Understand if there is a connection with elasticity of black holes Coupling to electromagnetic field Investigate the non-relativistic limit