Vortex and Droplet Engineering in Holographic Superconductors
VVortex and Droplet EngineeringinHolographic Superconductors
Tameem Albash, Clifford V. Johnson
Department of Physics and AstronomyUniversity of Southern CaliforniaLos Angeles, CA 90089-0484, U.S.A. talbash, johnson1, [at] usc.edu
Abstract
We give a detailed account of the construction of non–trivial localized solutions in a2+1 dimensional model of superconductors using a 3+1 dimensional gravitational dualtheory of a black hole coupled to a scalar field. The solutions are found in the presenceof a background magnetic field. We use numerical and analytic techniques to solve thefull Maxwell–scalar equations of motion in the background geometry, finding conden-sate droplet solutions, and vortex solutions possessing a conserved winding number.These solutions and their properties, which we uncover, help shed light on key featuresof the (
B, T ) phase diagram. a r X i v : . [ h e p - t h ] A ug Introduction
An holographic model of some of the key phenomenological attributes of superconductivity in2+1 dimensions was proposed in ref.[1]. It works roughly as follows (a more detailed reviewwill follow in the next section). The dual is a simple model of gravity in four dimensions(with negative cosmological constant) coupled to a U (1) gauge field and a minimally coupledcharged complex scalar Ψ. Asymptotic values of the scalar on the boundary correspond tothe vacuum expectation value (vev) of a charged operator in the 2+1 dimensional theory.For high temperatures (relative to a scale set by non–zero charge density in the model) thesystem is in a normal phase represented by a charged black hole solution in the gravitationaldual with the scalar set to zero. At a critical temperature the system undergoes a phasetransition, the U (1) getting spontaneously broken by a non–zero vev of the charged operator .The gravitational description of this is a charged black hole with a non–trivial scalar profilethat gives the vev on the boundary. This means that the black hole has “scalar hair” inthis regime. (For a discussion of violations of no–hair theorems in this context, see ref.[3].There, it was shown that it is possible in this context, for large enough charges, equivalentto the low temperature regime here.) The authors of ref.[1] showed using linear responsetheory that the DC conductivity of this new phase diverges in a manner consistent with theexpectation that the system is in a superconducting phase . The authors carried out furtherstudy of the system in ref.[2].Our focus in this paper is the system in an external magnetic field, continuing thework we began in ref.[6]. Generically, for non–zero magnetic field B filling the two spatialdimensions, it is inconsistent to have non–trivial spatially independent solutions on theboundary, and we present and study two classes of localized solutions in some detail. Thefirst is a “droplet” solution, the prototype of which was found in our earlier work[6] as astrip in 2D (straightforwardly generalized to circular symmetry in ref.[2]), and the secondis a vortex solution, with integer winding number ξ ∈ Z , which is entirely new. We obtainthese as full solutions of the Maxwell–scalar sector in a limit, and determine a number oftheir properties.Our analysis in various connected limits shows where these solutions can exist in the( B, T ) plane. There is a critical line below which droplets are not found, while vortices can Strictly speaking, the U (1) that is broken is global on the boundary, but it can be gauged in a numberof ways without affecting the conclusions. See e.g. ref.[2]. Note that a global U (1) does not restrict us to aspatially independent magnetic field. Other holographic superconductors are available. See e.g. refs. [4, 5].
2e found there. Our interpretation is that this region is the superconducting phase, and thatfor non–zero B , the vortices develop, trapping the magnetic flux into filaments, as is familiarin type II superconductors. Above the critical line, the system leaves the superconductingphase, and either forms droplets of condensate or simply reverts to the normal phase (dualto a dyonic black hole with zero scalar everywhere, which may well yield lower action thanthe droplets if we had back–reacting solutions to work with).In section 2 we review the model, and discuss the two limits in which most of ourstudies will be carried out. Section 3 reviews the spatially independent solution correspond-ing to the prototype superconducting solution. We briefly discuss our numerical approachto finding the solution as a warmup for the more difficult problems in the sequel. Section 4presents our search for and construction of the non–trivial spatially dependent solutionscorresponding to condensate droplets and to vortices. We discuss the numerical methodswe used to find them, and then examine a number of their properties. Section 5 examinesaspects of the solutions’ stability. We conclude in section 6, and we also present two ap-pendices. One appendix establishes the normalisation of our gauge/gravity dual dictionary,while the other discusses the flux quantization in our vortices. The holographic model of superconductivity in 2+1 dimensions proposed in ref.[1] is a modelof gravity in four dimensions coupled to a U (1) gauge field and a minimally coupled chargedcomplex scalar Ψ with potential V ( | Ψ | ) = − | Ψ | /L . There is a negative cosmologicalconstant that defines a scale L via Λ = − /L . The action is: S bulk = 12 κ (cid:90) d x √− G (cid:26) R + 6 L + L (cid:18) − F − | ∂ Ψ − igA Ψ | − V ( | Ψ | ) (cid:19)(cid:27) , (1)where κ = 8 πG N is the gravitational coupling and our signature is ( − + ++).We will use coordinates ( t, z, r, φ ) for much of our discussion, with t time, ( r, φ )forming a plane, and z a “radial” coordinate for our asymptotically AdS spacetimes suchthat z = 0 is the boundary at infinity. The AdS metric is: ds = L z (cid:0) dz − dt + dr + r dφ (cid:1) . (2)Note that the mass of the scalar m = − /L is above the Breitenlohner–Freedman stability3ound[7] m = − / L for scalars in AdS . We will write the scalar as:Ψ = ˜ ρ √ L exp( iθ ) . (3)Near the boundary z = 0 we have: ˜ ρ → ˜ ρ z + ˜ ρ z , (4)where ˜ ρ i ( i = 1 ,
2) sets the vacuum expectation value (vev) of an operator O i with dimension∆ = i [8]. Only one of these vevs can be non–zero at a time, and we will choose to study thecase of i = 1, for much of the paper. Our charged operator will be the order parameter forthe spontaneous breaking of the U (1) symmetry. A gauge field of the form A = A t dt doesnot give an electric field in the dual theory on ( r, φ ), but defines instead[9] a U (1) chargedensity, ρ and its conjugate chemical potential µ , as we will recall below.Black holes in this study will be planar, i.e., their horizons are an ( r, φ ) plane atsome finite z = z h . In the familiar manner, their Hawking temperature T and mass per unithorizon area ε = M/ V , corresponds to the dual 2+1 dimensional system at temperature T and with energy density ε . Generically, the black hole will couple to the gauge sector, havingsome profile for the field A t . The temperature T will have dependence on the charge densityparameter ρ . This is quite natural since without the charge density there is no other scale inthe theory, and there would be no meaning to a high or low temperature phase, and henceno possibility of a phase transition.The high T phase of the theory is simply the charged black hole (Reissner–Nordstr¨om(AdS–RN)) with the scalar Ψ vanishing. This corresponds to the non–superconducting or“normal” phase of the theory, where the order parameter vanishes. The mass of the scalar Ψis set not just by V ( | Ψ | ) but by the density ρ through the coupling to the gauge field. In fact m decreases with T until at T c it goes below m , becoming tachyonic. The theory seeks anew solution, in which the black hole is no longer AdS–RN, but one that has a non–trivialprofile for Ψ.In studying the system in a magnetic field background, there are some generic expec-tations to consider. The magnetic field B (which fills the two dimensions of the supercon-ducting theory), also contributes to m , via its square, but contributes with opposite sign to the electric contribution of the background. It therefore lowers the temperature T c atwhich m falls below m , triggering the phase transition. On these grounds alone one then(naively) expects a critical line in the ( B, T ) plane connecting (0 , T c ) to some ( B c , After a field redefinition: A µ → g A µ , Ψ → g Ψ , (5)our action (1) becomes: S bulk = 12 κ (cid:90) d x √− G (cid:26) R + 6 L + L g (cid:18) − F − | ∂ Ψ − iA Ψ | + 2 L ¯ΨΨ (cid:19)(cid:27) . (6)If we consider the limit g → ∞ , then the Maxwell–scalar sector decouples from gravity.This allows us to work with a fixed uncharged background, which we take to be the AdS –Schwarzschild (AdS–Sch) black hole, given by: ds = L α z (cid:0) − f ( z ) dt + dr + r dφ (cid:1) + L z f ( z ) dz , (7)where f ( z ) = 1 − z . The coordinate z is a dimensionless parameter scaled such that theevent horizon is at z h = 1. The Hawking temperature is given by the usual Gibbons–Hawkingcalculus [10]: T = 34 π α . (8)Note that α is related to the mass of the black hole: ε = M V = L α κ , (9)where V is the volume of the ( r, φ ) plane. In terms of the two real fields ( ˜ ρ, θ ) into which wedecomposed Ψ into in equation (3) we have: − L | ∂ Ψ − iA Ψ | + 2 ¯ΨΨ = − G µν (cid:2) ∂ µ ˜ ρ∂ ν ˜ ρ + ˜ ρ ( ∂ µ θ∂ ν θ − A µ ∂ ν θ + A µ A ν ) (cid:3) + 1 L ˜ ρ . (10)From the action, we derive the equation of motion for the fields ˜ ρ , θ , and A µ : √− G ∂ µ (cid:0) √− GG µν ∂ ν ˜ ρ (cid:1) − G µν ˜ ρ ( A µ − ∂ µ θ ) ( A ν − ∂ ν θ ) + L ˜ ρ = 0 , − √− G ∂ µ (cid:0) √− GG µν ˜ ρ ( A ν − ∂ ν θ ) (cid:1) = 0 , (11) √− G ∂ ν (cid:0) √− GG νλ G µσ F λσ (cid:1) − G µν L ˜ ρ ( A ν − ∂ ν θ ) = 0 . A , and for an electric background A = A t dt , we will have, as z → A t α ≡ ˜ A t → µ − ρ z , (12)defining a chemical potential µ and a charge density ρ . Sometimes we will also work in a probe limit, where we take the scalar in the Maxwell–scalarsector to be small, and hence not back–reacting on either the geometry. In general, we cando this at arbitrary g . (We will combine this with the decoupling limit ( g → ∞ ) for onecase, as we shall see later.) For finite g we will consider our small non–backreacting scalarto be moving in a dyonic Reissner–Nordstr¨om background, given by [11]: ds = L α z (cid:0) − f ( z ) dt + dr + r dφ (cid:1) + L z dz f ( z ) , (13) F = 2 hα rdr ∧ dφ + 2 qαdz ∧ dt ,f ( z ) = 1 + (cid:0) h + q (cid:1) z − (cid:0) h + q (cid:1) z = (1 − z ) (cid:0) z + z + 1 − (cid:0) h + q (cid:1) z (cid:1) . The temperature and charge density are given by: T = 1 β = α π (cid:0) − h − q (cid:1) , ρ = 1 V β δS on − shell δA t ( z = 0) = − L κ qα . (14)We choose a gauge such that the gauge field is written as: A = hα r dφ + 2 qα ( z − dt . (15) We begin by considering a spatially independent solution, reviewing the original presentationof ref.[1], working in the decoupling limit of section 2.2. We take an ansatz for the fieldsgiven by: θ ≡ const , ˜ ρ = ˜ ρ ( z ) , A t ≡ α ˜ A t ( z ) , A φ = 0 . (16)where ˜ ρ and ˜ A t are dimensionless fields. The equations of motion are given by: ∂ z ˜ ρ + (cid:16) f (cid:48) f − z (cid:17) ∂ z ˜ ρ + f ˜ ρ ˜ A t + z f ˜ ρ = 0 ,∂ z ˜ A t − z f ˜ ρ ˜ A t = 0 . (17)6e can study the equations’ behaviour near the event horizon ( i.e. as z → (cid:2) ∂ z ˜ ρ − ˜ ρ (cid:3) z =1 = 0 , (cid:20) ∂ z ˜ ρ − ∂ z ˜ ρ + ˜ ρ + ˜ ρ (cid:16) ∂ z ˜ A t (cid:17) (cid:21) z =1 = 0 , ˜ A t (cid:12)(cid:12)(cid:12) z =1 = 0 , (cid:104) ∂ z ˜ A t + ˜ ρ ∂ z ˜ A t (cid:105) z =1 = 0 . We note that the only free variables (to be chosen) at the event horizon are ˜ ρ (1) (or ∂ z ˜ ρ (1))and ∂ z ˜ A t (1). The other limit to study is to consider the behavior near the AdS boundary( z → (cid:20) ∂ z ˜ ρ − z ∂ z ˜ ρ + 2 z ˜ ρ (cid:21) z =0 = 0 , (cid:20) ∂ z ˜ A t − z ˜ ρ ˜ A t (cid:21) z =0 = 0 , which has as solutions˜ ρ ( z → → ˜ ρ z + ˜ ρ z , ˜ A t ( z → → µ − ρ z , (18)where ˜ ρ , ˜ ρ , µ , and ρ are constants related to the vev of a ∆ = 1 operator, the vev ofa ∆ = 2 operator, the chemical potential, and the charge density of the dual field theoryrespectively. The solution for ˜ ρ at the AdS boundary admits two normalisable modes, andtherefore the constants are associated with vevs of two separate operators. Only one of thesevevs is to be non–zero at a time, and the two different gauge theories are related to eachother via a Legendre transformation [8]. To simplify the numerical analysis, it is convenient to define a new field ˜ R ( z ) such that:˜ R ( z ) = z ˜ ρ ( z ) . (19)With this redefinition, the boundary condition of having either ˜ ρ or ˜ ρ in equation (18)to be zero becomes the requirement of having either a Dirichlet or a Neumann boundarycondition on ˜ R at the AdS boundary. The equations in the bulk of AdS are given by: ∂ z ˜ R + f (cid:48) f ∂ z ˜ R + f ˜ R ˜ A t + (cid:16) f (cid:48) zf − z + z f (cid:17) ˜ R = 0 ,∂ z ˜ A t − f ˜ R ˜ A t = 0 , (20)and we solve them using a shooting method (discretizing using finite differences) with shoot-ing conditions:˜ R (1) = const , ∂ z ˜ R (1) = − ˜ R (1) , ∂ z ˜ R (1) = − ˜ R (1) − (cid:16) ∂ z ˜ A t (1) (cid:17) ˜ R (1) . ˜ A t (1) = 0 , ∂ z ˜ A t (1) = const , ∂ z ˜ A t (1) = − ˜ R (1) ∂ z ˜ A t (1) . (21)7he solution at z = 0 goes as:˜ R ( z →
0) = ˜ R + ˜ R z , ˜ A t ( z →
0) = µ − ρ z . (22)We fix ˜ R (1) and then tune ∂ z ˜ A t (1) until the solution satisfies the necessary Dirichlet orNeumann boundary condition at z = 0. We then read off the scalar and also the valueof ρ for that solution, which defines the temperature. We can determine T c since there is aminimum charge density (over temperature squared) needed for the scalar field to condense.Note that there are multiple choices for ∂ z ˜ A t (1) that give the necessary boundary conditionat the AdS boundary, sample solutions of which we present in figure 1. Solutions with agreater number of nodes are associated with higher chemical potential/charge density. Thesesolutions are of a higher energy and so are thermodynamically unfavorable, therefore we onlypresent results of the zero–node solutions in what follows. In figure 2 we show the solutions (cid:45) R (cid:142) (cid:72) z (cid:76) Figure 1:
Three solutions with the same ˜ R (1) but different ∂ z ˜ A t (1) that satisfy Dirichlet boundaryconditions at z = 0. The solutions are distinguished by the number of nodes (times they cross the z –axis) they have. for the scalar values ˜ ρ , at the boundary which give the vevs of the operators O , . Asanticipated, in each case, the vev of the operator is zero above T /T c = 1. Below T /T c = 1,it is not zero, showing the spontaneous breaking of the U (1) symmetry. A non–zero magnetic field B in the ( r, φ ) plane will correspond to some non–zero A φ ( r ). Insuch a case, consistency of the solution requires the fields to have some spatial dependencein the plane. This situation was studied in the linear case in ref. [6], (see also ref.[2]) buthere we consider the full non–linear problem of equation (11). First, notice that the U (1)gauge transformation acts as: ρ → ρ , θ → θ + Λ , A µ → A µ + ∂ µ Λ . (23)8 .2 0.4 0.6 0.8 1.0 TT c Α Ρ(cid:142) T c (a) TT c Α Ρ(cid:142) T c (b) Figure 2:
Vaccuum expectation values for the scalar. Here T c is defined to be 0 . α √ ρ and0 . α √ ρ for the ∆ = 1 and ∆ = 2 operator respectively. In the previous section we chose θ to have no non–trivial dependence. Naively, it would seemthat we can freely shift θ by gauge transformations. However, this freedom is only availableif the gauge symmetry is not broken. We will return to this once we have constructed thesolutions. This motivates us to consider the following ansatz: θ ≡ ζ + ξ φ , ˜ ρ = ˜ ρ (˜ r, z ) , A t = α ˜ A t (˜ r, z ) , A φ = ˜ A φ (˜ r, z ) , (24)where we have defined a dimensionless radial coordinate ˜ r = αr , dimensionless fields ˜ ρ , ˜ A t ,and ˜ A φ , and ( ζ, ξ ) are constants where ξ is an integer. Under this ansatz, the equations ofmotion reduce to: ∂ z ˜ ρ + (cid:16) f (cid:48) f − z (cid:17) ∂ z ˜ ρ + f (cid:18) ∂ r ˜ ρ + r ∂ ˜ r ˜ ρ − r ˜ ρ (cid:16) ˜ A φ − ξ (cid:17) (cid:19) + f ˜ ρ ˜ A t + z f ˜ ρ = 0 ,∂ z ˜ A φ + f (cid:48) f ∂ z ˜ A φ + f (cid:16) ∂ r ˜ A φ − r ∂ ˜ r ˜ A φ (cid:17) − z f ˜ ρ (cid:16) ˜ A φ − ξ (cid:17) = 0 , (25) ∂ z ˜ A t + f (cid:16) ∂ r ˜ A t + r ∂ ˜ r ˜ A t (cid:17) − z f ˜ ρ ˜ A t = 0 , where the equation of motion for the field θ is trivially satisfied by our ansatz. Near theevent horizon, these equations reduce to the following conditions that must be satisfied: (cid:20) ∂ r ˜ ρ + r ∂ ˜ r ˜ ρ − r ˜ ρ (cid:16) ˜ A φ − ξ (cid:17) + 2 ˜ ρ = 3 ∂ z ˜ ρ (cid:21) z =1 , (cid:20) ∂ z ˜ ρ = − ˜ ρ − ˜ ρ (cid:16) ∂ z ˜ A t (cid:17) (cid:21) z =1 , (cid:104) ∂ r ˜ A φ − r ∂ ˜ r ˜ A φ − (cid:16) ˜ A φ − ξ (cid:17) ˜ ρ = 3 ∂ z ˜ A φ (cid:105) z =1 , (cid:104) ∂ z ˜ A φ = (cid:16) ˜ A φ − ξ (cid:17) ˜ ρ − ∂ z ˜ A φ (cid:105) z =1 , ˜ A t (˜ r, z = 1) = 0 , (cid:104) ∂ z ˜ A t = ∂ z (cid:16) ∂ r ˜ A t + r ∂ ˜ r ˜ A t − ˜ ρ ˜ A t (cid:17) + 2 ˜ A t ˜ ρ (cid:105) z =1 , (26)9here in the first equation we have used that ˜ A t (˜ r, z = 1) = 0. We now have three freefunctions to fix in these equations, ∂ z ˜ ρ (˜ r, z = 1) , ∂ z ˜ A φ (˜ r, z = 1) , and ∂ z ˜ A t (˜ r, z = 1), whichdetermine the spatial profile of the solutions at the event horizon. Note that in order toavoid a divergence in the equation for ˜ ρ at ˜ r = 0, we must have that near ˜ r = 0, the field ˜ ρ must go as ˜ r ξ . This motivates the following field redefinitions:˜ ρ = z ˜ r ξ ˜ R (˜ r, z ) , ˜ A φ = ˜ r ˜ A , (27)where ˜ R near ˜ r = 0 is a non–zero value. The particular redefinition of ˜ A φ simplifies thenumerical analysis. Our new equations of motion for these fields are: ∂ z ˜ R + f (cid:48) f ∂ z ˜ R + 1 f (cid:18) ∂ r ˜ R + 2 ξ + 1˜ r ∂ ˜ r ˜ R + ξ ˜ r ˜ R − r ˜ R (cid:16) ˜ r ˜ A − ξ (cid:17) (cid:19) + 1 f ˜ R ˜ A t + (cid:18) f (cid:48) zf − z + 2 z f (cid:19) ˜ R = 0 ,∂ z ˜ A + f (cid:48) f ∂ z ˜ A + 1 f (cid:18) ∂ r ˜ A + 3˜ r ∂ ˜ r ˜ A (cid:19) − ˜ r ξ f ˜ R (cid:18) ˜ A − ξ ˜ r (cid:19) = 0 ,∂ z ˜ A t + 1 f (cid:18) ∂ r ˜ A t + 1˜ r ∂ ˜ r ˜ A t (cid:19) − ˜ r ξ f ˜ R ˜ A t = 0 . (28) (cid:20) ∂ r ˜ R + 2 ξ + 1˜ r ∂ ˜ r ˜ R + ξ ˜ r ˜ R − r ˜ R (cid:16) ˜ r ˜ A − ξ (cid:17) − ˜ R = 3 ∂ z ˜ R (cid:21) z =1 , (cid:20) ∂ z ˜ R = −
13 ˜ R − ∂ z ˜ R −
19 ˜ R (cid:16) ∂ z ˜ A t (cid:17) (cid:21) z =1 , (cid:20) ∂ r ˜ A + 3˜ r ∂ ˜ r ˜ A − (cid:18) ˜ A − ξ ˜ r (cid:19) ˜ r ξ ˜ R = 3 ∂ z ˜ A φ (cid:21) z =1 , (cid:104) ∂ z ˜ A = − ∂ z ˜ A (cid:105) z =1 , (29) (cid:20) ∂ r ˜ T + 1˜ r ∂ ˜ r ˜ T − ˜ r ξ ˜ R ˜ T = 6 ∂ z ˜ T (cid:21) z =1 , (cid:104) ∂ z ˜ T = − ∂ z ˜ T (cid:105) z =1 , where we have used that: lim z → A t (˜ r, z ) = lim z → (1 − z ) ˜ T (˜ r, z ) . (30)In particular, at the event horizon, we can expand the fields near ˜ r = 0 as:lim ˜ r → ˜ R (˜ r, z ) = R ( z ) (cid:18) a ˜ r + O (˜ r ) (cid:19) , lim ˜ r → ˜ A (˜ r, z ) = A ( z ) (cid:18) b ˜ r + O (˜ r ) (cid:19) , (31)lim ˜ r → ˜ T (˜ r, z ) = T ( z ) (cid:18) c ˜ r + O (˜ r ) (cid:19) . a = R (1) + 3 ∂ z R (1) − ξA (1) R (1)2( ξ + 1) R (1) ,b = A (1) R (1) +3 ∂ z A (1)4 A (1) , ξ = 0 − ξR (1) +3 ∂ z A (1)4 A (1) , ξ = 1 ∂ z A (1)4 A (1) , ξ ≥ , (32) c = (cid:40) T (1) R (1) +6 ∂ z T (1)2 T (1) , ξ = 0 ∂ z T (1)2 T (1) , ξ ≥ . The solutions we study are characterized by the value of ξ and the ˜ r asymptotic behaviorof the field ˜ R . For any allowed value of ξ , the solution for ˜ R can asymptote to zero or to aconstant non–zero value.We will be extracting non–trivial profiles for the fields at the boundary at z = 0 asfollows: ˜ A t (˜ r, z ) = µ (˜ r ) − ρ (˜ r ) z , ˜ A φ (˜ r, z ) ≡ ˜ r ˜ A (˜ r, z ) = a φ (˜ r ) + J φ (˜ r ) z , ˜ ρ (˜ r, z ) ≡ z ˜ R (˜ r, z ) = ˜ R (˜ r, z + ∂ z ˜ R (˜ r, z , (33)where µ is related to the chemical potential, ρ is related to the charge density, J φ is relatedto the azimuthal current, and a φ is related to the magnetic field via ˜ B z = ( ∂ ˜ r a φ ) / ˜ r . For theexact relationships, please consult Appendix A. We consider the case of the case of ξ = 0, and use O as our order parameter. For this choiceof ξ the solution that asymptotes to a constant value is simply the spatially–independentsolution described earlier in section 3. In this section we consider solutions that asymptoteto zero. To that end, we fix the following functions to: ∂ z ˜ R (˜ r, z = 1) = −
13 (1 + γ ) ˜ R (˜ r, , ∂ z ˜ A (˜ r,
1) = 0 , (34)where γ is a positive number. With these choices, the equations of motion at the eventhorizon reduce to: (cid:104) ∂ r ˜ R + r ∂ ˜ r ˜ R − ˜ r ˜ A ˜ R + γ ˜ R (cid:105) z =1 = 0 , (cid:104) ∂ z ˜ R = (cid:0) + γ (cid:1) ˜ R − ˜ R ˜ T (cid:105) z =1 , (cid:104) ∂ r ˜ A + r ∂ ˜ r ˜ A − ˜ R ˜ A (cid:105) z =1 = 0 , (cid:104) ∂ z ˜ A (cid:105) z =1 = 0 , (35) (cid:104) ∂ r ˜ T + r ∂ ˜ r ˜ T − ˜ R ˜ T = 6 ∂ z ˜ T (cid:105) z =1 , (cid:104) ∂ z ˜ T = − ∂ z ˜ T (cid:105) z =1 . a = − γ , b = R (1) , c = T (1) R (1) + 6 ∂ z T (1)2 T (1) . (36) We begin by solving equations (29), which are at the event horizon. For a given R (1), wefind that there is a specific value for A (1) and T (1) that gives regular solutions for thethree functions ˜ R (˜ r, , ˜ A (˜ r, , ˜ T (˜ r, A (1)and T (1) using a shooting method. By this we mean that we pick values for A (1) and T (1) at the origin and “shoot” towards ˜ r → ˜ r max , where ˜ r max is the largest radius out towhich we will construct our solutions. Typically, this leads to a divergence in the functions,and therefore we iterate the procedure, fine–tuning our initial conditions such that a regularsolution is found. This has now determined our initial conditions for the bulk problem.The initial conditions at the event horizon having been determined, we solve the bulkequations of motion (28) and shoot towards the boundary at z = 0. The coupled partialdifferential equations are discretized using a finite difference method, and we adjust the meshspacings ∆ z and ∆ r until we achieve stability for our code.In order to satisfy the necessary boundary conditions at the AdS boundary, we tryto minimize the positive area under the curve of ∂ z ˜ R (˜ r, ∂ z T (˜ r,
1) at the event horizon. This is acheived by expanding itin an appropriate basis of functions in ˜ r and using a Monte–Carlo method to determine thecoefficients (with the area playing the role of energy). To give a sense of how the solutions behave, we present multiple solutions for multiple valuesof γ and ˜ R (1). The solutions are presented in figure 3. From figures 3(g) and 3(h), we seethat for a given value of γ , for various initial condition values of the scalar field, we get thesame asymptotic charge density. We learned from the spatially independent solution thatthe ratio of T /T c is determined by the charge density, and therefore we learn that γ fixes thevalue of T /T c . In fact, as γ →
0, we have
T /T c → γ → ∞ , we have T /T c →
0. Fromfigures 3(c) and 3(d), we see that the magnetic field asymptotes to a constant value, whichindicates that the solutions “live” in a background magnetic field. As the magnitude of thescalar field increases, the value of this background magnetic field rises. Note also that how12 r (cid:142) Ρ(cid:142) (cid:72) r (cid:142) (cid:76) R (cid:61) R (cid:61) R (cid:61) (a) Scalar Field for γ = 0 . r (cid:142) Ρ(cid:142) (cid:72) r (cid:142) (cid:76) R (cid:61) R (cid:61) R (cid:61) (b) Scalar Field for γ = 2 r (cid:142) B (cid:142) z (cid:72) r (cid:142) (cid:76) R (cid:61) R (cid:61) R (cid:61) (c) Magnetic Field for γ = 0 . r (cid:142) B (cid:142) z (cid:72) r (cid:142) (cid:76) R (cid:61) R (cid:61) R (cid:61) (d) Magnetic for γ = 2 r (cid:142)(cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) J Φ (cid:72) r (cid:142) (cid:76) R (cid:61) R (cid:61) R (cid:61) (e) Current density for γ = 0 . r (cid:142)(cid:45) (cid:45) (cid:45) J Φ (cid:72) r (cid:142) (cid:76) R (cid:61) R (cid:61) R (cid:61) (f) Current density for γ = 2 r (cid:142) Ρ (cid:72) r (cid:142) (cid:76) R (cid:61) R (cid:61) R (cid:61) (g) Charge density for γ = 0 . r (cid:142) Ρ (cid:72) r (cid:142) (cid:76) R (cid:61) R (cid:61) R (cid:61) (h) Charge density for γ = 2 Figure 3:
Droplet solutions for γ = 0 . γ = 2 on the right. They correspond to T /T c ≈ .
84 and
T /T c ≈ .
67 respectively. the magnetic field behaves in the core of the droplet varies considerably between low andhigh temperatures. At low temperatures (figure 3(d)), the magnetic field is enhanced by the13roplet, whereas as high temperatures (figure 3(c)), the droplet weakens the magnetic field inthe “core.” This variation in behavior suggests that these are perhaps not superconductingdroplets as was thought[6, 2].We can try to study the minimum magnetic field needed to first form these dropletsolutions. This would correspond to studying the problem in the limit where the magnitudeof the scalar field is approaching zero, i.e. the perturbative or probe limit. In this limit, itis consistent to take ˜ A to be a constant in both ˜ r and ˜ z and to take ˜ A t to only depend on z .The equation of motion for ˜ R with ˜ A = γ/ (cid:20) ∂ r ˜ R + 1˜ r ∂ ˜ r ˜ R −
14 ˜ r γ ˜ R + γ ˜ R (cid:21) z =1 = 0 . (37)This equation has (lowest energy) solution given by:˜ R (˜ r, z = 1) = R (1) exp (cid:18) − γ ˜ r (cid:19) . (38)Next, we can solve for the z dependence by solving: ∂ z ˜ R + f (cid:48) f ∂ z ˜ R − f γ ˜ R + f ˜ R ˜ A t + (cid:16) f (cid:48) zf − z + z f (cid:17) ˜ R = 0 ,∂ z ˜ A t = 0 , (39)with the appropriate boundary conditions at the AdS boundary. This in turn fixes the valueof the temperature of the solution. The value of the magnetic field found here correspondsto the critical magnetic field at which the droplet solutions first form. We draw the corre-sponding diagram in figure 4. The source of the divergence in the critical magnetic field as TT c B z T c Figure 4:
The limiting droplet line in the g → ∞ limit. Below this, droplets disappear. T /T c → g → ∞ limit, and further taking the probe limit gives a limiting line at whichthey drop to zero height, ceasing to exist for lower B .We can study this further (and extend to lower T /T c ) by working again in the probelimit, but at arbitrary g , as outlined in section 2.3. Here, the background is now a chargedblack hole solution, our method of taking into account some of the back–reaction of thegauge fields. In this limit the ˜ ρ equation of motion is given by: ∂ z ˜ ρ + (cid:18) f (cid:48) f − z (cid:19) ∂ z ˜ ρ + 1 f (cid:18) ∂ r ˜ ρ + 1˜ r ∂ ˜ r ˜ ρ − g h ˜ r ˜ ρ (cid:19) + (1 − z ) f g q ˜ ρ + 2 z f ˜ ρ = 0 , (40)This equation has a separable solution that we write as:˜ ρ = zZ ( z ) R (˜ r ) , (41)and we write their respective equations of motion as: ∂ r R + 1˜ r ∂ ˜ r R −
14 ˜ r (cid:0) g h (cid:1) R − ghR = 0 , (42) ∂ z Z + f (cid:48) f ∂ z Z − f ghZ + (1 − z ) f g q Z + (cid:18) z f + f (cid:48) f z − z (cid:19) Z = 0 . (43)The solution for R (˜ r ) is given by: R (˜ r ) = exp (cid:0) − gh ˜ r / (cid:1) . (44)We can solve the equation for Z ( z ) using the same arguments as before, and we get thesolutions shown in figure 5. Our claim is that the perturbative scalar field on the dyonic TT (cid:142) B z T (cid:142) g (cid:61) (cid:61) (cid:61) Figure 5:
Limiting droplet line for three values of g with ˜ T = (3 − q c ) / π . black hole describes the entire phase transition line in the phase diagram, and the problem15tudied earlier is simply the g → ∞ limit near T /T c →
1. To prove this, we do several non–trivial checks. First, we check that the dyonic theory can predict the critical temperature ofthe g → ∞ theory. In order to do this, we make the identification:2 qg = ρ g →∞ , (45)by comparing the charge density of both theories. Next, we know that as the magnetic fieldapproaches zero, the droplets appear at the critical temperature (or T /T c = 1) for boththeories. Given that we defined T c in different ways for both theories, by setting them equal,we should be able to calculate the relationship between T c and √ ρ that we saw in the g → ∞ theory. In particular: T c /α = 14 π (cid:0) − q c (cid:1) = σ (cid:112) gq c . (46)Therefore, we can solve for σ as g → ∞ . We present the results in figure 6. As one cansee, in the limit of g → ∞ we indeed recover the values 0.226 and 0.118 respectively, whichwere obtained earlier in the g → ∞ probe case (see caption of figure 2). We can also check g Σ (cid:68) (cid:61) (cid:68) (cid:61) Figure 6:
Convergence to the g → ∞ critical temperature. The curves asymptote to 0.225492 and0.118412 respectively. See text for discussion. whether the phase diagrams coincide. This is presented in figure 7. As we see, the dyonicblack hole results very quickly approach our results for the g → ∞ case for a range of T /T c near one. We are now in a position to answer what happens when T /T c → g → ∞ limit. The zero temperature limit requires us to take: q + h = 3 . (47)As g → ∞ , we find that regularity of any solution at zero temperature requires us to take: h → √ −
32 1 g , q → / √ g . (48)Therefore, the Gaussian profile in equation (44) vanishes in the limit of zero temperatureand g → ∞ (note that this does not happen at finite g ). Therefore, the droplet no longer16 .2 0.4 0.6 0.8 1.0 TT c g B z T c g (cid:61) (cid:165) g (cid:61) (cid:61) (cid:61) Figure 7:
The droplet limiting curves for a range of couplings, after rescaling to include the g → ∞ case. T c = ασ √ qg . exists in that limit. Another way to see this is that in this limit, the dimensionless quantity gB z /T c as used in figure 7 diverges. We now consider solutions for the scalar that asymptote to a constant non–zero value. Theseare the vortex solutions.
The numerical procedure for the vortex is almost identical to that of the droplet (see section4.1.1). We found it much more difficult to solve for the initial functions on the horizon usinga shooting method, and so we inserted an initial guess function for an approximation to thescalar field ˜ R (˜ r,
1) at the event horizon, parameteriszed by two constants R and R . Wethen use that function to solve for the field ˜ A (˜ r, A (1)and R , there is a specific R that leads to a regular solution; we again use a shooting methodto determine this constant. The constant T (1) is determined in the same way. With both˜ R (˜ r,
1) and ˜ A (˜ r,
1) determined, we have fully fixed ∂ z ˜ R (˜ r, ∂ z ˜ T is the constant alreadydetermined in the spatially independent problem using equations (21), since our vorticesasymptote to that case. Here, we illustrate the case of ξ = 1 and ξ = 2 and again use O as our order parameter. Weagain consider the equations of motion given in equations (28) and (29). For simplicity, we17ocus on the case of ∂ z ˜ A = 0. We note that as the solutions approach a constant value, theyshould asymptote to the spatially–independent solutions we have presented earlier. Thisin turn allows us to define the temperature at which a given solution exists. We presentexamples of such solutions in figures 8 and 9. In figure 8, we see that very far away from r (cid:142) Ρ(cid:142) (cid:72) r (cid:142) (cid:76) TT c (cid:61) TT c (cid:61) (a) Scalar Field r (cid:142) Ρ(cid:142) (cid:72) r (cid:142) (cid:76) TT c (cid:61) TT c (cid:61) (b) Scalar Field Figure 8:
Vortex Solutions for ξ = 1 (LHS) and ξ = 2 (RHS). the origin, the scalar field has a constant vev, but as it approaches the core of the vortex, itdecreases to zero value. The behavior at the origin is of course determined by the choice of ξ .The choice of ξ also influences how the gauge field ˜ A φ behaves. In figures 9(g) and 9(h), wesee that the value of ˜ A φ asymptotes to the value of ξ . This is exactly the behavior requiredfor the magnetic flux penetrating the vortex to be quantized with value 2 πξ . We reviewthis briefly in Appendix B. Indeed, ξ defines a non–trivial topological winding number: Thescalar ˜ ρ becomes constant at infinity, breaking the U (1). Therefore the gauge symmetry ofequation (23) cannot be used to unwind θ . Gauge symmetry is unbroken at infinity for thedroplets, so ξ is not a winding number for them. The current density J φ (˜ r ) (figures 9(e) and9(f)) is zero asymptotically and peaks in a ring around the core, supporting the magneticfield, as expected for a vortex.In figure 9(b), we find that the charge density near the origin begins to oscillate asthe density drifts downwards. It is difficult to say whether or not this is a physical attributeof the solutions or whether it is an artifact of our scheme to find the appropriate shootingfunctions that satisfy the z = 0 boundary condition. If they are physical, they may be causedby screening effects being strong in the core of the vortex. It is interesting to note that thisbehavior appears to be absent for the ξ = 1 vortex (see figure 9(a)), although we do see thatthere is a transition from the charge density increasing in the core to decreasing in the coreas the temperature is lowered. Another curiosity for the ξ = 1 vortex is the behavior of themagnetic field near the origin (see figure 9(c). Instead of flattening out as is the case for the ξ = 2 solutions, it dips slightly downwards. We expect this also to be a numerical artifact,18 r (cid:142) Ρ (cid:72) r (cid:142) (cid:76) TT c (cid:61) TT c (cid:61) (a) Charge Density r (cid:142)(cid:45) (cid:45) Ρ (cid:72) r (cid:142) (cid:76) TT c (cid:61) TT c (cid:61) (b) Charge Density r (cid:142) B (cid:142) z (cid:72) r (cid:142) (cid:76) TT c (cid:61) TT c (cid:61) (c) Magnetic Field r (cid:142) B (cid:142) z (cid:72) r (cid:142) (cid:76) TT c (cid:61) TT c (cid:61) (d) Magnetic Field r (cid:142) J Φ (cid:72) r (cid:142) (cid:76) TT c (cid:61) TT c (cid:61) (e) Current density r (cid:142) J Φ (cid:72) r (cid:142) (cid:76) TT c (cid:61) TT c (cid:61) (f) Current density r (cid:142) A (cid:142) Φ (cid:72) r (cid:142) (cid:76) TT c (cid:61) TT c (cid:61) (g) A φ r (cid:142) A (cid:142) Φ (cid:72) r (cid:142) (cid:76) TT c (cid:61) TT c (cid:61) (h) A φ Figure 9:
Vortex Solutions for ξ = 1 (LHS) and ξ = 2 (RHS). since the ξ = 1 case is more numerically sensitive because of its sharper profile near ˜ r = 0.19 Stability and Quasinormal Modes
In this section we consider the stability of our solutions to fluctuations in the fields. We workin the decoupling limit and proceed by considering fluctuations about our classical solutions: A µ = A (0) µ + A (1) µ , ˜ ρ = ˜ ρ (0) + ˜ ρ (1) , θ = θ (0) + θ (1) , (49)where the fields with superscripts (0) are the background fields and the fields with super-scipts (1) are the fluctuation fields. Note in particular that because we are working in the g → ∞ limit, we do not consider fluctuations of the metric since those come at O (1 /g ).The action to quadratic order in the fluctuations is given by: S = 12 g κ (cid:90) d x √− G (cid:18) − G µν (cid:104) ∂ µ ˜ ρ (1) ∂ ν ˜ ρ (1) + (cid:0) ˜ ρ (1) (cid:1) (cid:0) A (0) µ − ∂ µ θ (0) (cid:1) (cid:0) A (0) ν − ∂ ν θ (0) (cid:1) +4 ˜ ρ (0) ˜ ρ (1) (cid:0) A (0) µ − ∂ µ θ (0) (cid:1) (cid:0) A (1) ν − ∂ ν θ (1) (cid:1) + (cid:0) ˜ ρ (0) (cid:1) (cid:0) A (1) µ − ∂ µ θ (1) (cid:1) (cid:0) A (1) ν − ∂ ν θ (1) (cid:1)(cid:105) + 1 L (cid:0) ˜ ρ (1) (cid:1) − L F (1) µν F µν (1) (cid:19) . (50)The resulting equations of motion are given by: √− G ∂ µ (cid:0) √− GG µν ∂ ν ˜ ρ (1) (cid:1) − ˜ ρ (1) G µν (cid:16) A (0) µ − ∂ µ θ (0) (cid:17) (cid:16) A (0) ν − ∂ ν θ (0) (cid:17) − ρ (0) G µν (cid:16) A (0) µ − ∂ µ θ (0) (cid:17) (cid:16) A (1) ν − ∂ ν θ (1) (cid:17) + L ˜ ρ (1) = 0 , √− G ∂ µ (cid:16) √− G (cid:16) ρ (0) ˜ ρ (1) G µν (cid:16) A (0) ν − ∂ ν θ (0) (cid:17) + (cid:0) ˜ ρ (0) (cid:1) G µν (cid:16) A (1) ν − ∂ ν θ (1) (cid:17)(cid:17)(cid:17) = 0 , L √− G ∂ ν (cid:16) √− GG νλ G µσ ˆ F λσ (cid:17) − G µν ˜ ρ (0) ˜ ρ (1) (cid:16) A (0) ν − ∂ ν θ (0) (cid:17) − (cid:0) ˜ ρ (0) (cid:1) G µν (cid:16) A (1) ν − ∂ ν θ (1) (cid:17) = 0 . We can consider the ansatz: A (1) x = αe − iωt ˜ A (1) x (˜ ω, z ) . (51)The equation of motion for the fluctuation field is given by: ∂ z ˜ A (1) x + f (cid:48) f ∂ z ˜ A (1) x + ˜ ω f ˜ A (1) x − f (cid:16) ˜ R (0) (cid:17) ˜ A (1) x = 0 . (52)Near the event horizon, the field satisfies an equation of the form: ∂ z ˜ A (1) x − − z ∂ z ˜ A (1) x + ˜ ω − z ) ˜ A (1) x = 0 . (53)20his has solutions given by ingoing (negative) and outgoing (positive) waves:˜ A (1) x ∝ (1 − z ) ± i ˜ ω/ . (54)The appropriate condition to have at the event horizon is of ingoing waves, and therefore,we make the following field redefinition:˜ A (1) x = (1 − z ) − i ˜ ω/ χ (˜ ω, z ) . (55)The equation of motion is now given by: ∂ z χ − ˜ ω − z ) χ + f (cid:48) f ∂ z χ + ˜ ω f χ − f (cid:16) ˜ R (0) (cid:17) χ + i ˜ ω (cid:18) − z ∂ z χ + 1(1 − z ) χ + f (cid:48) f − z χ (cid:19) = 0 . (56)Expanding near the event horizon, we get the following restrictions on the initial conditions: ∂ z χ = 3 i ˜ ω + 2˜ ω − (cid:16) ˜ R (0) (cid:17) − i ˜ ω χ z =1 . (57)By requiring that χ ( z = 0) = 0, we find that this condition is only satisfied for discretevalues of ˜ ω . We present some of the values in table 1. In particular, since the imaginarypart of ˜ ω is negative, this corresponds to having fluctuations that decay away. Therefore,the constant solutions are stable under fluctuations. n ˜ ω i i Table 1: Values of ˜ ω that give regularizable solutions at T /T c = 0 . As an aside, we can push our analysis a little more to compute the DC conductivity (alreadydone in ref.[1]). We proceed by studying the problem in the limit where ˜ ω approaches zero.We therefore consider solving for the field χ (˜ ω, z ) as an expansion in ˜ ω : χ (˜ ω, z ) = χ ( z ) + ˜ ωχ ( z ) + . . . (58)21s explained in ref. [12], in the hydrodynamic limit and where the field is solved as anexpansion in ˜ ω , the normalizibility condition cannot be satisfied by the terms χ i . We proceedby focusing on the χ term. The equation of motion is given by: ∂ z χ + f (cid:48) f ∂ z χ − f (cid:16) ˜ R (0) (cid:17) χ = 0 . (59)Note that at the AdS ( z = 0) boundary the solution for χ is given by:lim z → χ = (cid:26) a x ; T > T c ,a x + j x z ; T < T c , (60)where a x and j x are constants and j x is proportional to the current. In particular, if wedefine the conductivity using: σ = J x E x = J x iωA x . (61)Therefore for T < T c , Im( σ ) ∝ ˜ ω − , and therefore by the Kramers–Kronig relations, we havethat Re( σ ) ∝ δ (˜ ω ) as shown in ref. [1]. We begin by slightly simplifying by considering the following field behavior: ρ (1) = e − iωt ˜ ρ (1) (˜ ω, ˜ r, z ) , A (1) t = αe − iωt ˜ A (1) t (˜ ω, ˜ r, z ) , A (1) φ = e − iωt ˜ A (1) φ (˜ ω, ˜ r, z ) , (62) A (1) r = αe − iωt ˜ A (1) r (˜ ω, ˜ r, z ) , A (1) z = αe − iωt ˜ A (1) z (˜ ω, ˜ r, z ) , θ (1) = 0 . (63)where we have defined dimensionless fields and variables ω = α ˜ ω . This reduces the equationsto: ∂ r ˜ A (1) z − ∂ ˜ r ∂ z ˜ A (1) r + 1˜ r (cid:16) ∂ ˜ r ˜ A (1) z − ∂ z ˜ A (1) r (cid:17) − f (cid:16) − ˜ ω ˜ A (1) z + i ˜ ω∂ z ˜ A (1) t (cid:17) − z (cid:0) ˜ ρ (0) (cid:1) ˜ A (1) z = 0 ,∂ z ˜ A (1) r − ∂ ˜ r ∂ z ˜ A (1) z + f (cid:48) f (cid:16) ∂ z ˜ A (1) r − ∂ ˜ r ˜ A (1) z (cid:17) − f (cid:16) − ˜ ω ˜ A (1) r + i ˜ ω∂ ˜ r ˜ A (1) t (cid:17) − z f (cid:0) ˜ ρ (0) (cid:1) ˜ A (1) r = 0 , (64)22nd ∂ z ˜ A (1) t + i ˜ ω∂ z ˜ A (1) z + 1 f (cid:18) ∂ r ˜ A (1) t + i ˜ ω∂ ˜ r ˜ A (1) r + 1˜ r (cid:16) ∂ ˜ r ˜ A (1) t + i ˜ ω ˜ A (1) r (cid:17)(cid:19) − z f ˜ ρ (0) ˜ ρ (1) ˜ A (0) t − z f (cid:0) ˜ ρ (0) (cid:1) ˜ A (1) t = 0 ,∂ z ˜ A (1) φ + f (cid:48) f ∂ z ˜ A (1) φ + 1 f (cid:18) ∂ r ˜ A φ − r ∂ ˜ r ˜ A (1) φ (cid:19) + 1 f ˜ ω ˜ A (1) φ − z f ˜ ρ (0) ˜ ρ (1) (cid:16) ˜ A (0) φ − ξ (cid:17) − z f (cid:0) ˜ ρ (0) (cid:1) ˜ A (1) φ = 0 ,∂ z ˜ ρ (1) + (cid:18) f (cid:48) f − z (cid:19) ∂ z ˜ ρ (1) + 1 f (cid:18) ∂ r ˜ ρ (1) + 1˜ r ∂ ˜ r ˜ ρ (1) − r (cid:16) ˜ A (0) φ − ξ (cid:17) ˜ ρ (1) − r (cid:16) ˜ A (0) φ − ξ (cid:17) ˜ A (1) φ ˜ ρ (0) (cid:19) + 1 f ˜ ω ˜ ρ (1) + 1 f ˜ ρ (1) (cid:16) ˜ A (0) t (cid:17) + 2 f ˜ A (0) t ˜ A (1) t ˜ ρ (0) + 2 z f ˜ ρ (1) = 0 . (65)Consider two possible ans¨atze. First, let us consider the radial gauge choice ˜ A (1) z = 0. Inthis gauge, the equation for ˜ A (1) z gives the following restriction: ∂ z (cid:18) ∂ ˜ r ˜ A (1) r + 1˜ r ˜ A (1) r (cid:19) = − i ˜ ωf ∂ z ˜ A (1) t . (66)This result is interesting because it suggests that in the limit of ˜ ω →
0, there is a consistentsolution with ˜ A (1) r = 0. This suggests that for this ansatz, ˜ A (1) r ∝ ˜ ω . However, implementingthe restriction in equation (66) is not trivial, and therefore, we consider a different ansatz.For lack of a better name, we call the temporal gauge, ˜ A (1) t = 0. In this gauge, the equationfor ˜ A (1) t gives the following restriction: i ˜ ω (cid:18) ∂ z ˜ A (1) z + 1 f (cid:18) ∂ ˜ r ˜ A (1) r + 1˜ r ˜ A (1) r (cid:19)(cid:19) = 2 z f ˜ ρ (1) ˜ ρ (0) ˜ A (0) t . (67)In particular, for this ansatz, in the limit of ˜ ω →
0, we see that we must have ˜ ρ (1) ∝ ˜ ω whichin turn means that we must have ˜ A (1) φ ∝ ˜ ω for consistency. This particular restriction ismore straightforward to implement since we can directly insert it into the equations for ˜ A (1) z A (1) r . The resulting equations of motion are given by: ∂ z ˜ A (1) z + f (cid:48) f ∂ z ˜ A (1) z − iωf ∂ z (cid:16) z ˜ ρ (0) ˜ ρ (1) ˜ A (0) t (cid:17) + f (cid:16) ∂ r ˜ A (1) z + r ∂ ˜ r ˜ A (1) z (cid:17) + ˜ ω f ˜ A (1) z − fz (cid:0) ˜ ρ (0) (cid:1) ˜ A (1) z = 0 ,∂ z ˜ A (1) r + f (cid:48) f ∂ z ˜ A (1) r + f (cid:16) ∂ r ˜ A (1) r + r ∂ ˜ r ˜ A (1) r − r ˜ A (1) r (cid:17) − f (cid:48) f ∂ ˜ r ˜ A (1) z − z fi ˜ ω ∂ ˜ r (cid:16) ˜ ρ (0) ˜ ρ (1) ˜ A (0) t (cid:17) + ˜ ω f ˜ A (1) r − z f (cid:0) ˜ ρ (0) (cid:1) ˜ A (1) r = 0 ,∂ z ˜ A (1) φ + f (cid:48) f ∂ z ˜ A (1) φ + f (cid:16) ∂ r ˜ A φ − r ∂ ˜ r ˜ A (1) φ (cid:17) + f ˜ ω ˜ A (1) φ − z f ˜ ρ (0) ˜ ρ (1) (cid:16) ˜ A (0) φ − ξ (cid:17) − z f (cid:0) ˜ ρ (0) (cid:1) ˜ A (1) φ = 0 , (68)and ∂ z ˜ ρ (1) + (cid:18) f (cid:48) f − z (cid:19) ∂ z ˜ ρ (1) + 1 f (cid:18) ∂ r ˜ ρ (1) + 1˜ r ∂ ˜ r ˜ ρ (1) − r (cid:16) ˜ A (0) φ − ξ (cid:17) ˜ ρ (1) − r (cid:16) ˜ A (0) φ − ξ (cid:17) ˜ A (1) φ ˜ ρ (0) (cid:19) + 1 f ˜ ω ˜ ρ (1) + 1 f ˜ ρ (1) (cid:16) ˜ A (0) t (cid:17) + 2 z f ˜ ρ (1) = 0 . (69)What is of particular interest is that we find that the equations for ˜ ρ (1) and ˜ A (1) φ only dependon each other, and so we can solve for these two fields independently of ˜ A (1) r and ˜ A (1) z . Forsimplicity, we therefore set these fields to zero. We now note that under this assumption, theequation of motion for ˜ A (1) z appears to be completely independent of ˜ A (1) r . In particular, onecould imagine first solving the equation for ˜ A (1) z and then substituting the solution into theequation for ˜ A (1) r and solving for ˜ A (1) r under the constraint described above. Unfortunately, wedo not know how to solve the partial differential equation with the constraint, and thereforewe choose to solve only for the field ˜ A (1) z . To proceed, we would like to note that near theevent horizon, all the fields satisfy the same equation: ∂ z X + f (cid:48) f ∂ z X + ˜ ω f X = 0 . (70)This equation has solutions given by in–going (negative) and out–going (positive) waves: X ∝ (1 − z ) ± i ˜ ω/ . (71)The correct boundary condition to have at the event horizon is of in–going waves, so weredefine the field to reflect this: ˜ A (1) z = (1 − z ) − ˜ ω/ χ (˜ r, z ) . (72)24he equation of motion for Z is now given by: ∂ z χ − ˜ ω − z ) χ + f (cid:48) f ∂ z χ + ˜ ω f χ + i ˜ ω (cid:18) − z ∂ z χ + 1(1 − z ) χ + f (cid:48) f − z χ (cid:19) = 0 , f (cid:18) ∂ r χ + 1˜ r ∂ ˜ r χ − (cid:16) ˜ R (0) (cid:17) χ (cid:19) = 0 . (73)At the event horizon, this gives: ∂ r χ + 1˜ r ∂ ˜ r χ − (cid:16) ˜ R (0) (cid:17) χ + i ˜ ωχ + 23 ˜ ω χ + (2 i ˜ ω − ∂ z χ = 0 ,∂ z χ + (cid:18) − i ˜ ω (cid:19) ∂ z χ + 127 ˜ ω (˜ ω − i ) χ = 0 . (74)The first point we need to address when solving these equations is what the appropriatechoice for ˜ ω is. Since for the droplet solutions the scalar field asymptotes to zero, we expectthat in order to satisfy the normalizibility conditions for large ˜ r , the correct choice for ˜ ω arethe quasi–normal solutions when there is no scalar field present. We present some of thevalues in table 2. Using these values, we try to find normalizable solutions for the droplet n ˜ ω i i i i i Table 2: Values of ˜ ω that give regularizable solutions for zero scalar field.scalar field solutions. What is interesting is that we can only find solutions for (cid:60) (˜ ω ) > (cid:60) (˜ ω ) do not exist, just that ourextensive numerical search could not find them). We present an example of this solutionin figure 10. Therefore, these results suggest (note that this result is not conclusive sincewe have not solved for the field ˜ A (1) r ) that the droplet solutions are stable under quadraticfluctuations. The situation for the vortex solutions is very similar, except that now, the ˜ ω values we use are those of the spatially independent solutions described earlier. Once again,we find that solutions can be easily found above the lower lying modes, although we findthem to be much higher above as illustrated by the example shown in figure 10(b).25 r (cid:142) (cid:182) z Χ (cid:72) z (cid:61) (cid:76) Re (cid:72) Χ (cid:76) Im (cid:72) Χ (cid:76) (a) Droplet Fluctuation with ˜ ω =6 . − . i and T /T c = 0 . r (cid:142)(cid:45) (cid:45) (cid:182) z Χ (cid:72) z (cid:61) (cid:76) Re (cid:72) Χ (cid:76) Im (cid:72) Χ (cid:76) (b) Linear Vortex Fluctuation with˜ ω = 13 . − . i and T /T c = 0 . Figure 10:
Solutions for the fluctuation field χ . We have constructed two broad families of localized solution to the equations of motion forref.[1]’s holographic model of a superconductor, and considered several of their key properties.The vortices, with winding number ξ , contain 2 πξ units of magnetic flux, and arecandidates to fill out the superconducting part of the phase diagram in the presence of anexternal magnetic field. This is because a lattice of them in the ( r, φ ) plane can trap flux linesof an applied external B –field into filaments as it passes through the two dimensional sample.Vortices presumably repel each other, and so such a lattice will cost energy. Therefore atsome critical B c ( T ) the system will seek a lower energy phase, possibly returning to thenormal phase. We have not constructed such a lattice, and further study to understand sucha configuration is a very interesting avenue of research to pursue. Forming such a lattice isa method by which, at a given T /T c <
1, the superconducting phase can be made to persistin some constant background B , even though the system cannot eject the magnetic fieldentirely `a la the Meissner effect (there is a nice energetic argument in ref.[2] as to why theMeissner effect is not possible in this two dimensional case). This vortex phase is of coursethe same method by which a standard type II superconductor can persist beyond the (lower)critical line at which the Meissner effect disappears, and we expect that it applies here. Thestudy of multi–vortex solutions needed to establish this is left for further study.Crucially, we’ve established that the droplets do not exist below a certain critical valueof B , dropping to zero height on a family of lines that we were able to compute explicitly. Forthis and a variety of other stated reasons, and also considering the fact that they are of finitesize and hence a lattice or gas of them would not give a connected superconducting path forcharge transport, we believe that they do not represent a superconducting phase. (Hence, we26isagree with the statements made about the phase diagram in ref.[2]. The authors find thecritical line, but state (similarly to our ref.[6]) that the droplets exist below the line, and aresuperconducting. As they did not have the full droplet solutions, nor the vortex solutions,their analyses are not sufficient to make these determinations.). They seem to represent anon–superconducting phase that is inhomogeneous. Whether or not the droplets are thefavoured solution for arbitrarily large B is an interesting question. There is the possibilitythat the system may prefer to return to the normal phase represented by a dyonic blackhole with zero scalar. Our partial stability analysis showed that the droplets we studied arestable against fluctuations of the fields, but there is the possibility that other fluctuationmodes may be undamped. We mention here that we also noticed the curious fact that thedroplet solution at higher magnetic field (that we presented earlier) contains regions wherethe local value of the squared scalar mass is below the Breitenlohner–Freedman bound.It would also be of interest to establish whether the critical line where the vortex phasewould disappear coincides with the limiting line where the droplets’ existence begins. Weconjecture this to be likely on the grounds that we have found no other candidate solutionsto fill an intermediate region. While this is the simplest possibility, further study is neededto establish it firmly. Acknowledgments
We would like to thank Arnab Kundu and Rob Myers for conversations. CVJ thanks theAspen Center for Physics for a stimulating working atmosphere while this manuscript wasprepared.
A Normalizations in the Holographic Dictionary
We recall the AdS dictionary (working in Euclidean metric): (cid:28) exp (cid:90) φ O (cid:29) = exp ( − S on − shell [ φ ]) = Z . (75)Taking derivatives on both sides with respect to the boundary source φ gives us our AdSdictionary: (cid:104)O ( x ) . . . O ( x n ) (cid:105) = ( β V ) − n lim φ → Z − δδφ ( x ) . . . δδφ ( x n ) Z . (76)We define the the free energy density of the dual theory to be given: F = 1 β V S on − shell . (77)27here β is the inverse temperature and V is the “spatial volume” of the dual theory. If weuse the notation that we are working in AdS d +1 , then V has mass dimension d −
1. For us, d = 3. In the Euclidean language, our action is given by: S bulk = 12 κ (cid:90) d x √− G (cid:26) − R − L + L (cid:18) F + | ∂ Ψ − igA Ψ | + V ( | Ψ | ) (cid:19)(cid:27) . (78)where we emphasize that although the metric is now purely positive, we are still using A t and now a Wick rotated version of it. In the dual theory, the charge density is given by: ρ ( x ) = − δ F δµ ( x ) . (79)where µ is the chemical potential and x represents the space–time coordinates in the dualfield theory. Using the AdS/CFT dictionary, we can write: δ F δµ ( x ) = 1 β V δS on − shell δA t ( x, . (80)Using the action given in equation (78), we find: gβ V δS on − shell δA t ( x,
0) = L κ g α∂ z A t ( x,
0) = L κ g α ∂ z ˜ A t ( x, , (81)where we have used that: δA t ( x (cid:48) ) δA t ( x ) = β V δ ( d +1) ( x (cid:48) − x ) . (82)and we have dropped the contribution coming from the event horizon. Therefore, the endresult is given by: ρ ( x ) = − L κ g α ∂ z ˜ A t ( x, . (83)A similar procedure allows us to calculate the vev of the azimuthal current as well: J φ ( x ) = − gβ V δS on − shell δA φ ( x ) = L κ g α ˜ r ∂ z ˜ A φ . (84)Note that here we are using the vector field that has been rescaled by g , hence why the factorof g appears at the beginning of our definition. Next, we can calculate the form of the vevsof the ∆ = 1 and ∆ = 2 operators. To proceed, we take the ∆ = 1 operator to be the sourceof the ∆ = 2 operator [8]. Therefore, we write: (cid:104)O ( x ) (cid:105) = − δ F δ (cid:104)O ( x ) (cid:105) ∝ − δ F δρ ( x ) , (85)28here we are using the notation that: ρ ( x, z →
0) = zρ ( x ) + z ρ ( x ) . (86)To proceed, we calculate the variation of the bulk action (keeping only divergent and finiteterms): δS bulk = − lim z → L κ g (cid:90) d x L α (cid:18) z ρ ( x ) δρ ( x ) + 2 ρ ( x ) δρ ( x ) + ρ ( x ) δρ ( x ) (cid:19) . (87)The first term in parentheses is divergent, but it is removed by an appropriate counterterm: S CT = − L κ g lim z → √− γL (cid:90) d x ρ ( x, z ) , (88)which leaves us with: δS bulk + δS CT = − L κ g (cid:90) d x L α ρ ( x ) δρ ( x ) . (89)Therefore, we find: − αL δ F δρ ( x ) = L κ g Lα ρ ( x ) = L κ g α ˜ ρ ( x ) . (90)To calculate the vev of the ∆ = 1 operator, we need to perform a Legendre transform on F [8]: G = −F − β V L κ L α g (cid:90) d x ρ ( x ) ρ ( x ) , (91)and we now have: − Lα δ G δρ ( x ) = L κ g Lαρ ( x ) = L κ g α ˜ ρ ( x ) . (92)Therefore, in order to satisfy the conditions: (cid:104)O ( x ) (cid:105) = − δ F δ (cid:104)O ( x ) (cid:105) , (cid:104)O ( x ) (cid:105) = − δ G δ (cid:104)O ( x ) (cid:105) , (93)we choose: (cid:104)O ( x ) (cid:105) = L √ gκ Lαρ ( x ) = L √ gκ α ˜ ρ ( x ) , (94) (cid:104)O ( x ) (cid:105) = L √ gκ Lα ρ ( x ) = L √ gκ α ˜ ρ ( x ) . (95)29 Flux Quantization
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