Analog dual to a 2+1-dimensional holographic superconductor
AAnalog dual to a 2+1-dimensional holographicsuperconductor
Neven Bili´c ∗ a,b and J´ulio C. Fabris † aa Departamento de F´ısica, Universidade Federal do Esp´ırito Santo (UFES)Av. Fernando Ferrari s/n CEP 29.075-910, Vit´oria, ES, Brazil b Division of Theoretical Physics, Rudjer Boˇskovi´c Institute, 10002 Zagreb,CroatiaMarch 3, 2021
Abstract
We study an analog hydrodynamic model that mimics a 3+1 AdS planar BH space-time dual to a 2+1-dimensional superconductor. We demonstrate that the AdS bulkand its holographic dual could be realized in nature in an analog gravity model basedon fluid dynamics. In particular we mimic the metric of an O holographic supercon-ductor and calculate the entanglement entropy of a conveniently designed subsystemat the boundary of the analog AdS bulk. A pseudo-Riemannian geometry of spacetime can be mimicked by fluid dynamics in Minkowskispacetime. The basic idea is the emergence of an effective metric G µν = a [ g µν − (1 − c ) u µ u ν ] , (1)which describes the effective geometry for acoustic perturbations propagating in a fluidpotential flow with u µ ∝ ∂ µ θ . The quantity c s is the adiabatic speed of sound, the conformalfactor a is related to the equation of state of the fluid, and the background spacetime metric g µν is usually assumed Minkowski. The metric of the form (1) has been exploited in variouscontexts including emergent gravity [1, 2], scalar theory of gravity [3], Einstein-aether gravity[4], acoustic geometry [5, 6, 7, 8] and euclidean gravity [9, 10, 11].The work presented here is motivated by recent development of AdS/CFT dual theoryof 2+1-dimensional superconductor [12, 13, 14, 15, 16, 17, 18, 19, 20, 21] (for a review and ∗ [email protected] † [email protected] a r X i v : . [ h e p - t h ] M a r dditional references see [22]). The AdS/CFT duality in these models is based on a corre-spondence between gravitational theory and dynamics of quantum field theory on the bound-ary of asymptotically AdS spacetime. The gravity side can be well described by classicalgeneral relativity, while the dual field theory involves the dynamics with strong interaction.This correspondence is often referred to as “holography” since a higher dimensional gravitysystem is described by a lower dimensional field theory without gravity, which resemblesoptical holography.A particularly important work in this context is the minimal model of a holographicsuperconductor by Bobev et al [19] with an Abelian gauge field embedded in the truncationof four-dimensional maximal gauged super-gravity. Besides, it is worth mentioning the workon d-wave superconductivity by Benini et al [16] in which interesting physical phenomenaare demonstrated such as the formation of Fermi arcs.The AdS spacetime as a solution to Einstein’s equations cannot actually exist in naturedue to instability problems. However, it can inspire some configurations where the underlyinggeneral gravitational structure can be studied through analogue models. The aim of thispaper is to demonstrate that AdS and its holographic dual could be realized in nature inan analog gravity model based on hydrodynamics of a physical fluid. In particular we willmimic the bulk metric of the minimal model of a holographic superconductor consisting ofthe metric, a charged scalar with a non-trivial potential and an Abelian gauge field embeddedin the truncation of four-dimensional maximal gauged super-gravity [19]. This model wasrecently studied in the context of holographic entanglement [20, 21]. Our first task is to derivean analog acoustic geometry which mimics a d +1-dimensional asymptotic AdS geometry witha general planar Black hole. Furthermore, we will apply this to a 3+1-dimensional model andcalculate the entanglement entropy for a particular geometry obtained as solution related tothe holographic O superconductor. The reason why we are specifically interested in the O type is due to its pronounced first order phase transition at finite temperature.We divide the remainder of the paper into three sections and an two appendices. We startwith section 2 in which we derive an analog metric for a d + 1-dimensional AdS planar blackhole of the form relevant for a holographic description of the superconductor. In the nextsection, Sec. 3, we apply our formalism to a 3+1-dimensional bulk related to the minimalmodel of the 2+1-dimensional holographic superconductor. For a particular geometry relatedto the O superconductor we calculate the entanglement entropy. Concluding remarks aregiven in section 4. In appendix A we outline a derivation of the relativistic acoustic metricand in appendix B we derive the effective speed of sound in a fluid with an external pressure. Geometric structures in the form of a planar black hole may have interesting applicationsin condensed matter physics [23]. In this section we construct a model of an analog planarBH hole in a general asymptotic AdS d +1 . A similar model for d = 4 was discussed in detailby Hossenfelder [24, 25] and recently in [26, 27]. We will discuss in more detail the case d = 3 which is of particular interest for 2+1-dimensional superconductor [19, 20, 22]. In ourapproach we will consider a nonisentropic fluid flow which yields the desired analog metric.We start from a general form of the AdS planar BH metric in an arbitrary number of2pace-like dimensions dds = (cid:96) z (cid:2) e − χ ( z ) γ ( z ) dt − γ ( z ) − dz − d x (cid:3) , (2)where (cid:96) is the curvature radius of AdS d +1 and d x = d − (cid:88) i =1 d x i d x i . (3)For d = 3 we will relate the functions χ and γ to the truncated Lagrangian of the four-dimensional N = 8 super-gravity [28] studied by Bobev et al [19] in the context of holographicsuperconductivity. In order to have an asymptotic AdS for z → γ (0) = 1 , χ (0) = 0 . (4)Next, the dimensionless functions χ and γ can be thought of as functions of the dimensionlessvariable z/z h , where z = z h is the location of the horizon. In other words γ ( z h ) = 0 (5)and γ has no zeros on the interval 0 < z < z h . Then, the horizon temperature is T = e − χ/ π dγdz (cid:12)(cid:12)(cid:12)(cid:12) z = z h . (6)This temperature measured in some chosen fixed units, e.g., in units of (cid:96) − is ambiguousbecause the geometry (2) is invariant under rescaling τ → ατ, z → αz, x i → αx i z h → αz h . (7)Thus, the metric (2) has a rescaled horizon z h /α with the corresponding rescaled horizontemperature ¯ T = e − χ/ π dγ ( αz ) dz (cid:12)(cid:12)(cid:12)(cid:12) z = z h /α = αT. (8)However, the temperature T expressed in units of 1 /z h is unique, i.e., the quantity T z h isinvariant under the rescaling (7). Therefore, in the following we will express the temperatureand other dimensionfull physical quantities in units of some power of z h .Now we seek a fluid analog model which would mimic the induced metric of the form(2). The basic idea is to find a suitable coordinate transformation t → ¯ t , z → ¯ z such thatthe new metric takes the form of the relativistic acoustic metric (84) derived in appendix Awith g µν replaced by the Minkowski metric η µν G µν = nm c s w [ η µν − (1 − c ) u µ u ν ] . (9)3ere n and w denote the particle number density and specific enthalpy, respectively, and anarbitrary mass scale m is introduced to make G µν dimensionless. The specific enthalpy isdefined as usual w = p + ρn , (10)where p and ρ denote the pressure and energy density, respectively. The quantity c s is theso-called “adiabatic” speed of sound defined by c ≡ ∂p∂ρ (cid:12)(cid:12)(cid:12)(cid:12) s = nw (cid:18) ∂n∂w (cid:12)(cid:12)(cid:12)(cid:12) s (cid:19) − , (11)where | s denotes that the specific entropy, i.e., entropy per particle s = S/N , is kept fixed.The second equality in (11) follows from the thermodynamic law dw = T ds + 1 n dp. (12)Following Hossenfelder [24] we transform the metric (2) by making use of a coordinatetransformation t = ¯ t + h ( z ) , z = z (¯ z ) , (13)where the functions z (¯ z ) and h ( z ) are determined by the requirement that the transformedmetric takes the form (9). By simple algebraic manipulations the line element (2) can berecast into a convenient form ds = (cid:96) z (cid:26) d ¯ t − d ¯ z − d x − (1 − ˜ γ ) d ¯ t +2(1 − ˜ γ ) / ( c − ˜ γ ) / d ¯ td ¯ z − ( c − ˜ γ ) d ¯ z ] (cid:27) , (14)where we have set dzd ¯ z = e χ/ c s , (15) dhdz = (1 − ˜ γ ) / ( c − ˜ γ ) / c s e χ/ ˜ γ , (16)and an abbreviation ˜ γ = e − χ γ. (17)From (4) and (5) it follows 0 ≤ ˜ γ ≤
1; ˜ γ ( z h ) = 0 , ˜ γ (0) = 1 . (18)Comparing (14) with the acoustic metric (9) we identify c s as the speed of sound and thenon-vanishing components of the velocity vector u ¯ t and u ¯ z in transformed coordinates as u ¯ t = (1 − ˜ γ ) / (1 − c ) / , u ¯ z = − ( c − ˜ γ ) / (1 − c ) / . (19)4hese equations imply ˜ γ ≤ c ≤ . (20)Next, by applying the potential-flow equation (see appendix A) wu µ = ∂ µ θ (21)we derive closed expressions for w , n , and c s in terms of the variable z . Since the metric isstationary, the velocity potential must be of the form θ = m ¯ t + g ( z ) , (22)where m is an arbitrary mass parameter which we can identify with the mass scale thatappears in (9) and g ( z ) is a function of ¯ z through z . Then, from (21) and (22) it follows w = mu ¯ t = m (1 − c ) / (1 − ˜ γ ) / , (23)and the function g in (22) must satisfy dgdz = wu ¯ z (cid:18) dzd ¯ z (cid:19) − = − mc s e χ/ ( c − ˜ γ ) / (1 − ˜ γ ) / . (24)The particle number density can be obtained from the condition that the conformal factorin (2) must be equal to that of (9), i.e., we require nm c s w = (cid:96) z . (25)As m is arbitrary it is natural to choose m = 1 (cid:96) , (26)so using this and (23) we find n = c s (cid:96)z (1 − c ) / (1 − ˜ γ ) / . (27)In this way, both w and n are expressed as functions of z and c s . However, c s is notindependent since by the definition (11) c = nw ∂w∂n (cid:12)(cid:12)(cid:12)(cid:12) s = nw dwdz (cid:18) dndz (cid:19) − . (28)Using (28) with (23) and (27) we obtain a differential equation for c s c s dc s dz − c (cid:20) z + 12 ddz ln(1 − ˜ γ ) (cid:21) + 12 ddz ln(1 − ˜ γ ) = 0 , (29)with solution c = 1 − z z (1 − ˜ γ ) / (cid:18) K + 2 z h (cid:90) z h z dzz (1 − ˜ γ ) − / (cid:19) . (30)5he integration constant must satisfy the constraint 1 ≥ K ≥ ≤ c ≤
1. Dimensionless physical quantities such as (cid:96)w , c s and the componentsof the fluid velocity field are functions of z/z h and are invariant under the rescaling (7).Plugging (30) into (23) and (25) one obtains w and n as functions of z . Note thatexplicit functional forms of z (¯ z ), γ ( z ), and g ( z ) can be obtained by making use of (30) andintegrating respectively (15), (16), and (24). However, the precise forms of these functionsare not really needed for obtaining a closed expression for the analog metric.It is of particular interest to discuss the above solution in the asymptotic limit, i.e., in thelimit z →
0. Motivated by the asymptotic behavior of the O holographic superconductorwith d = 3 (see section 3.1) ˜ γ ( z ) = 1 + c (cid:18) zz h (cid:19) + O ( z ) , (31)in the following we assume for general d ˜ γ ( z ) = 1 + c d (cid:18) zz h (cid:19) d + O ( z d +1 ) , (32)with c d <
0. Then, it may be easily shown that in the limit z → c → d/ ( d + 4) <
1. However, in this limit ˜ γ → z → c ≥ ˜ γ . This putsthe constraint as to how close to the boundary is our analog metric applicable. Our analogmodel breaks down at a point z = z min which is the maximal root of the equation c = ˜ γ .For the minimal value of K , K min = 0, this equation reads(1 − ˜ γ ( z )) / − z (cid:90) z h z dyy (1 − ˜ γ ( y )) − / = 0 . (33)In the case of a Schwarzschild AdS planar black hole, i.e., for χ = 0 and γ = 1 − ( z/z h ) d ,the integration in (29) can be easily performed yielding c = dd + 4 + (cid:18) d + 4 − K (cid:19) (cid:18) zz h (cid:19) d/ , (34)The condition c − γ = 0 now reads (cid:18) zz h (cid:19) d + (cid:18) d + 4 − K (cid:19) (cid:18) zz h (cid:19) d/ − d + 4 = 0 , (35)For example, for d = 4, the root z min is given by z min z h = (3 − K ) − / ≥ − / , (36)and for d = 3 we find numerically z min z h = 0 . , for K = K min = 0 . (37)6ence, the simple prescription for an analog model is only valid from the point z min up to the location of the horizon at z h . In principle we could place the boundary of ourmodel at z min and cut off the section of AdS from z = 0 to z min as it has been done in theRandall-Sundrum model [29, 30]. However, as we aim to make a connection with CFT at theboundary of AdS and calculate the boundary entanglement entropy at z = 0, we would liketo extend our model all the way down to the AdS boundary at z = 0. As we demonstratein appendix B, such an an extension can be achieved by manipulating the equation of stateby adding an external pressure. For a fluid with an external pressure of the form p ext = α ( p + ρ ) , (38)where α is a function of z , one finds the effective speed of sound˜ c = c − α α . (39)Depending on the functional form of ˜ γ we can choose α to make the quantity ˜ c satisfyequation (20) in the interval 0 ≤ z ≤ z h . For example, if ˜ γ behaves as in (32) near z = 0,we can choose α = d − ( d + 4)˜ γ ( d + 4)(1 + ˜ γ ) (40)to obtain c ≥ ˜ γ in the entire interval 0 ≤ z ≤ z h andlim z → ˜ c = 1 . (41) Here we consider a concrete example of the analog metric of the form (2) for d = 3 relatedto the holographic superconductor. Instead of solving the field equations we will implementthe already known solutions [19, 20, 21] into our analogue setup. Based on the known resultswe will construct approximate analytic expressions for γ corresponding to a chosen horizontemperature. With this we can calculate the entanglement entropy and by comparisonwith the results of Refs. [20, 21] we can also find an analytic expression for χ ( z ). Theanalog geometry which we have derived in general form can be used to mimic these analyticexpressions. Here we briefly review the minimal model of a holographic superconductor following Bobevet al [19]. We consider the minimal model of a holographic superconductor realized by an SO (3) × SO (3) invariant truncation of four-dimensional N = 8 gauged super-gravity [28].The truncated action is S = 116 πG (cid:90) d x √− G ( −R + L ) , (42)7here L involves two real dimensionless scalar fields λ and ϕ coupled to an Abelian gaugefield A µ and gravity. The Lagrangian can be written as L = − F µν F µν + 2 ∂ µ λ∂ µ λ + sinh (2 λ )2 (cid:16) ∂ µ ϕ − g A µ (cid:17) (cid:16) ∂ µ ϕ − g A µ (cid:17) − P , (43)with potential P = − g (cid:18) λ − λ sinh λ + 32 sinh λ (cid:19) . (44)The gauge coupling g sets the scale (cid:96) of AdS via the relation P (cid:96) = − λ = 0 related to SO (8) globalsymmetry [19, 28] we obtain the relation g (cid:96) = 1. The spacetime metric can be parameter-ized as ds = (cid:96) z (cid:20) γ ( z ) e − χ ( z ) dt − (cid:0) dx + dx (cid:1) − dz γ ( z ) (cid:21) , (45)where the functions γ and χ are to be determined by solving the field equations with appro-priate boundary conditions. As we have noted in section 2, the value χ ≡ χ (0) can be setto zero by rescaling the time coordinate.The field equations are derived in Ref. [19] for the gauge choice ϕ = 0 and A µ =( ψ ( z ) , , ,
0) and solved for two types of superconductors depending on the choice of bound-ary conditions, with non-trivial gauge fields and scalar condensates below some critical valueof the temperature. The solutions are characterized by the vacuum expectation values ofthe charged operators O and O (see Fig. 1). Depending on the asymptotic behavior of thefield λ we distinguish two solutions: i) λ = λ ˜ z + O (˜ z ) corresponding to an O superconductor with O ∝ λ and O = 0, and ii) λ = λ ˜ z + O (˜ z ) corresponding to an O superconductor with O ∝ λ and O = 0.Here and from here on we use the dimensionless variable ˜ z = z/(cid:96) . As functions of tem-perature, the condensates O and O exhibit the second and first order phase transitions,respectively. The typical behavior of the condensates as functions of temperature is shownin Fig. 1. The quantity ρ c which was chosen to set the units in Fig. 1 appears as a coefficientin the expansion ψ = µ(cid:96) − ρ c (cid:96)z + . . . near the AdS boundary. Physically, µ and ρ c areappropriately normalized chemical potential and charge density, respectively. From the fieldequations one can derive the following asymptotic expansions near z = 0: λ = λ ˜ z + λ ˜ z + λ (cid:0) λ − e χ ψ (cid:1) ˜ z + O (˜ z ) , (46) ψ = ψ + ψ ˜ z + ψ λ ˜ z + ψ λ λ ˜ z + O (˜ z ) , (47) γ = 1 + λ ˜ z + γ ˜ z + O (˜ z ) , (48) χ = χ + λ ˜ z + 83 λ λ ˜ z + 14 (cid:0) λ + 8 λ − e χ λ ψ (cid:1) ˜ z + O (˜ z ) . (49)8 .05 0.10 0.15 0.20 T ρ O ρ c c T c ρ T ρ O ρ c c c Figure 1: The condensates O (left panel) and O (right panel) as functions of temperature.Taken with permission from Ref. [19]. The dashed vertical line indicates a discontinuity inthe condensate value related to a first order phase transition.As we have mentioned, χ can be set to 0 and the other coefficients in the expansion arerelated to physical quantities as follows: λ = 4 (cid:96)O , λ = 4 (cid:96) O , (50) ψ = (cid:96)µ , ψ = − (cid:96)ρ c . (51)For λ = χ = 0 there are no condensates and the solution is just the Reisner-Nordstrom(RN) AdS planar black hole with γ RN = 1 − (1 + Q ) z z + Q z z (52)and ψ RN = 2 Q(cid:96)z RN (cid:18) − zz RN (cid:19) . (53)The charge squared Q ranges between 0 and 3 where Q = 0 corresponds to a SchwarzschildAdS planar BH and and Q = 3 to the maximal RN AdS planar BH. Here we present the calculation of the holographic entanglement entropy in the analoguemodel discussed in section 3. The holographic entanglement entropy S in a 2+1-dimensionalboundary CFT for a subsystem A that has an arbitrary one-dimensional boundary ∂ A isdefined by the following area law [31, 32, 33] S = Area(Σ)4 (cid:96) , (54)where Σ is the two-dimensional static minimal surface in AdS with boundary ∂ A and (cid:96) Pl is the Planck length. 9 B B y x L d z z=z * Figure 2: Strip geometry employed to calculate the entanglement entropy. Adapted illustra-tion taken with permission from Ref. [20].As we are dealing with an analog geometry we will assume that there exist a minimallength, typically of the order of the atomic separation, below which the bulk description ofthe fluid fails. This length is referred to in the condensed matter literature as the coherencelength, where the meaning of the word ”coherence” is different from that in optics. Sinceit describes the distance over which the wave function of a BE condensate tends to its bulkvalue when subjected to a localized perturbation, it is also referred to as the healing length[34]. In analog gravity systems, a healing length (cid:96) hl plays the role of the Planck length[35, 36, 37, 38, 39] and for a BE gas is typically of order (cid:96) hl (cid:39) / ( mc s ) where m is the bosonmass. Hence, to calculate the entanglement entropy we use (54) with the Planck length (cid:96) Pl replaced by the healing length (cid:96) hl . Furthermore, we will identify the arbitrary scale (cid:96) with (cid:96) hl . Next we apply the prescription (54) to the geometry of Albash and Johnson [20]illustrated in Fig. 2 and calculate the entropy S as a function of the strip width d for a fixedtemperature.Consider the bulk metric (2) with d = 3 and a surface Σ defined by the equation z − z ( x ) = 0 , (55)where z ( x ) is a function of x such that Σ extends into the bulk and is bounded by theperimeter of A as illustrated in Fig. 2. From the induced metric σ ij on Σ with line element ds = σ ij dx i dx j = (cid:96) z (cid:20) dx (cid:18) z (cid:48) γ (cid:19) + y (cid:21) , (56)we find the area of ΣArea(Σ) = (cid:90) dxdy (cid:112) det σ ij = L (cid:90) d/ − d/ dx (cid:96) z (cid:18) z (cid:48) γ (cid:19) / . (57)10igure 3: The metric function γ versus z/z h for T = 0 . × − √ ρ c .Extremization of this area with respect to z ( x ) yields the entanglement entropy expressedas an integral over z S = Area(Σ)4 (cid:96) = 2 L (cid:90) z ∗ dz z ∗ z (cid:112) ( z ∗ − z ) γ , (58)where z ∗ is the location of the bottom of the extremal surface related to the strip width d = 2 (cid:90) d/ − d/ dx = 2 (cid:90) z ∗ dz z (cid:112) ( z ∗ − z ) γ . (59)Details of the derivation of (58) can be found in Ref. [20]. The integral in (58) is divergentnear z = 0 and can be regularized by adding and subtracting a counter-term2 L (cid:90) z ∗ (cid:15) dz/z . (60)The entropy is then expressed as S = S fin + 2 L(cid:15) , (61)where the finite part reads S fin = 2 L (cid:90) z ∗ dz (cid:32) z ∗ z (cid:112) ( z ∗ − z ) γ − z (cid:33) − Lz ∗ . (62)Next we calculate the entanglement entropy using the bulk profile corresponding to an O superconductor at fixed temperature. The reason why we specifically address the O type11s that the O superconductor exhibits a first order phase transition which manifests itselfas a discontinuity depicted in Fig. 1. To calculate S fin we use a polynomial function γ = 1 + (cid:88) i =3 c i (cid:18) zz h (cid:19) i (63)with c = − , c = 118 , c = − , c = 23 . (64)We plot this function in Fig. 3. This choice is motivated by the superconductor bulk metricprofile plotted in Fig. 7(b) of Ref. [21] for a fixed horizon temperature T = 0 . × − √ ρ c where ρ c is the charge density of the O superconductor (see section 3.1). The function(63) is an analytic approximation to the bulk metric found by numerically solving the fieldequations of the holographic superconductor.In Fig. 4 we plot S fin as a function of d/
2. For comparison we plot in the same figurethe entanglement entropies of a Schwarzschild AdS planar BH hole and a maximal RN AdSplanar BH which have the same asymptotic behavior near z = 0. The metric profiles aredetermined so that the cubic terms are the same as in the O superconductor case. Hencewe have γ AdS = 1 + c (cid:18) zz h (cid:19) (65)for the Schwarzschild AdS planar BH and γ RN = 1 + c (cid:18) zz h (cid:19) + 3 (cid:16) c (cid:17) / (cid:18) zz h (cid:19) (66)for the maximal RN AdS planar BH. The coefficient of the quartic term in (66) was fixedby virtue of (52) and requirement γ RN ( z RN ) = 0, where z RN = ( − /c ) / z h is the locationof the RN BH horizon.To complete our model we still need to determine the function χ ( z ). To do this wewill compare our results with those obtained in [19, 20, 21] at a fixed temperature T =0 . × − √ ρ c . In Ref. [19] the charge density ρ c of dimension of length − is chosen to setthe scale whereas in Refs. [20, 21] the scale is set by the quantity˜ ρ c = ρ c √ πG (cid:96) . (67)The relation between ˜ ρ c and ρ c can be fixed by identifying the O phase transition tempera-ture T tr of Albash and Johnson [20] (their figure 2(b)) T tr = 0 . ρ / with that of Bobevet al [19] (their figure 2) T tr = 0 . ρ / . From this we obtain˜ ρ c = 4 ρ c . (68)In our approach the scale is set by z h so we have to find a relation between our z h and ˜ ρ c or ρ c . To this end we compare the transition point d tr / . z h (Fig. 1) with that ofChakraborty [21] d tr / .
56 ˜ ρ − / . This yields1 z h = 0 . ρ / = 0 . ρ / . (69)12igure 4: The finite part of the entanglement entropy S fin in units of 2 L/z h versus halfstrip width d/ z h at fixed temperature. Dotted and dashed lines represent theentanglement entropies of the Schwarzschild AdS planar and maximal RN AdS planar BH,respectively. The right panel shows the zoomed-in crossover region.Using this we can express the horizon temperature of our configuration depicted in Fig. 3in units of z − h , T z h ≡ π e − χ h / = 1 . × − , (70)which yields χ h ≡ χ ( z h ) = − . π . . (71)Next, we express χ as a function of z using the expression (49) from section 3.1 in which weset χ = 0, λ = 0, keep the z term and neglect the higher order terms. Hence we write χ ( z ) = 2 λ z (cid:96) + χ z (cid:96) , (72)where the coefficient λ can be fixed from Eq. (50) with the value of O deduced from Fig.1. At T = 0 . × − √ ρ c we find λ = 1 . χ ( z ) = 2 . (cid:18) zz h (cid:19) + 6 . (cid:18) zz h (cid:19) . (73)This equation together with (63) and (64) can be used to find closed expressions for thehydrodynamic functions and variables of our analog model.The considerations in this section can as well be carried out for the type O supercon-ductor. 13 Summary and conclusions
We have derived an analog acoustic geometry which mimics a d + 1-dimensional asymptoticAdS geometry with a planar Black hole. In 3+1 dimensions, this geometry has been exploitedas a holographic model for the 2+1-dimensional superconductor. We have applied thisgeneral analog geometry to a 3+1-dimensional bulk and calculated the entanglement entropyfor a particular geometry obtained as solution related to the holographic O superconductor.We have demonstrated that the entanglement entropy in our analog model exhibits the usualfirst order phase transition which characterizes the O superconductor.In this way we have confirmed the basic idea that a 3+1 AdS bulk with a planar BHcan be realized in nature as a hydrodynamic analog gravity model. Moreover, the analogbulk metric can be parameterized so that the coefficient in the asymptotic expansion inpowers of z are such that the dual AdS/CFT boundary field theory corresponds to the type O superconductor. A procedure similar to the one described in section 3.2 can easily beapplied to the case of type O superconductor. Acknowledgments
The work of N. Bili´c has been partially supported by the European Union through theEuropean Regional Development Fund - the Competitiveness and Cohesion OperationalProgramme (KK.01.1.1.06). J.C. Fabris thanks CNPq (Brazil) and FAPES (Brazil) forpartial support.
A Acoustic metric
Here we briefly review the derivation of the relativistic acoustic metric. Acoustic metricis the effective metric perceived by acoustic perturbations propagating in a perfect fluidbackground. Under certain conditions the perturbations satisfy a Klein-Gordon equation incurved geometry with metric of the form (1).We first derive a propagation equation for linear perturbations of a nonisentropic flowassuming a fixed background geometry. Following Landau and Lifshitz [40] we assume thatthe enthalpy flow wu µ is a gradient of a scalar potential, i.e., that there exist a scalar function θ such that the velocity field satisfies wu µ = ∂ µ θ, (74)where w is the specific enthalpy defined by (10). Then, from the relativistic Euler equa-tion and standard thermodynamic identities it follows [26] that the entropy gradient is alsoproportional to the gradient of the potential, i.e., s ,µ = w − u ν s ,ν θ ,µ . (75)Furthermore, instead of the continuity equation ( nu µ ) ; µ = 0, one finds( nu µ ) ; µ = 1 w ∂p∂s u µ s ,µ . (76)14n a nonisentropic flow we have u µ s ,µ (cid:54) = 0 and the above equation shows that the particlenumber is generally not conserved. As demonstrated in Ref. [26], from equation (75) andLagrangian description of fluid dynamics it follows that the specific entropy is a function ofthe velocity potential θ only. Then, using (75) equation (76) can be expressed in the form( nu µ ) ; µ = ∂p∂θ , (77)where p = p ( w, s ( θ )) is the pressure of the fluid.Given some average bulk motion represented by w , n , and u µ , following the standardprocedure [5, 6, 40], we make a replacement w → w + δw, n → n + δn, u µ → u µ + δu µ , (78)where the perturbations δw , δn , and δu µ are induced by a small perturbation δθ around abackground velocity potential θ . From (74) it follows δw = u µ δθ ,µ , (79) wδu µ = ( g µν − u µ u ν ) δθ ,ν . (80)Using this and (78) equation (77) at linear order yields( f µν δθ ,ν ) ; µ + (cid:34)(cid:18) ∂n∂θ u µ (cid:19) ; µ − (cid:18) ∂ p∂θ (cid:19)(cid:35) δθ = 0 , (81)where f µν = nw (cid:20) g µν − (cid:18) − wn ∂n∂w (cid:19) u µ u ν (cid:21) . (82)Then, it may be easily shown that equation (81) can be recast into the form1 √− G ∂ µ (cid:16) √− G G µν ∂ ν δθ (cid:17) + m δθ = 0 , (83)where the matrix G µν is the inverse of the acoustic metric tensor G µν = nm c s w [ g µν − (1 − c ) u µ u ν ] , (84)with determinant G . Here m is an arbitrary mass parameter introduced to make G µν di-mensionless and c s is the speed of sound defined by (11).The effective mass squared is given by m (cid:112) | G | m = (cid:34)(cid:18) ∂n∂θ u µ (cid:19) ; µ − ∂ p∂θ (cid:35) . (85)Hence, the linear perturbations χ propagate in the effective metric (84) and acquire aneffective mass.In an equivalent field-theoretical description [1, 26, 41] the fluid velocity u µ is derived fromthe scalar field as u µ = ∂ µ θ/ √ X , and n and c s are expressed in terms of the Lagrangian and itsfirst and second derivatives with respect to the kinetic energy term X = g µν θ ,µ θ ,ν . Obviously,the quantity √ X in this picture is identified with the specific enthalpy w . Equation (83) with(84) and (11) coincides with that of Ref. [1] derived in field theory with a general Lagrangianof the form L = L ( X, θ ). 15
Effective sound speed with external pressure
Consider a fluid with internal variables p , ρ , and n . Suppose we apply to the fluid an externalpressure p ext so that the total pressure is P = p + p ext . (86)The speed of sound is still defined by c = ∂p∂ρ (cid:12)(cid:12)(cid:12)(cid:12) s , (87)but the thermodynamic TdS equation (12) must include the external pressure, i.e., dW = T ds + 1 n dP, (88)where W = P + ρn = w + p ext n . (89)Then the sound speed is given by c = ∂ ( P − p ext ) ∂ρ (cid:12)(cid:12)(cid:12)(cid:12) s = nW ∂W∂n (cid:12)(cid:12)(cid:12)(cid:12) s − ∂p ext ∂ρ . (90)For an isentropic process from (88) it follows dP = ndW, dρ = W dn, (91)so by making use of ∂∂n = W ∂∂ρ (92)we find c = nw + p ext /n (cid:18) ∂w∂n (cid:12)(cid:12)(cid:12)(cid:12) s − p ext n (cid:19) . (93)Now we make the following ansatz p ext = α ( p + ρ ) , (94)where α = α ( z ) will be determined by the requirement that the speed of sound is well definedas z →
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