Holographic insulator/superconductor phase transition by matching method and thermodynamic geometry
HHolographic insulator/superconductor phasetransition by matching method and thermodynamicgeometry
Diganta Parai a ∗ , Debabrata Ghorai b † , Sunandan Gangopadhyay b ‡ a Indian Institute of Science Education and Research KolkataMohanpur, Nadia 741246, India b Department of Theoretical SciencesS.N. Bose National Centre for Basic SciencesJD Block, Sector III, Salt Lake, Kolkata 700106, India
Abstract
In this work, we have analytically analyzed the insulator/superconductor phasetransition in the presence of a 5-dimensional
AdS soliton background using match-ing method and thermodynamic geometry approach . We have first employed thematching method to obtain the critical chemical potential. We then move on toinvestigate the free energy and thermodynamic geometry of this model in 3+1 di-mensions. This investigation of the thermodynamic geometry leads to the criticalchemical potential of the system from the condition of the divergence of the scalarcurvature. We have then compared the value of the critical chemical potential µ c indimension d = 5 obtained from these two different methods, namely, the matchingmethod and the thermodynamic geometry procedure. We have also obtained anexpression for the condensation operator using the matching method. Our findingsagree very well with the numerical findings in the literature. It is very difficult to study strongly coupled systems using the standard techniques ofperturbation theory in condenced matter physics. The
AdS/CF T correspondence [1]-[4] allows us to analyze such systems. The duality claims that a 4-dimensional stronglycoupled gauge theory is related to a 5-dimensional weakly coupled gravity theory. Thisfascinating development is very important for theoretical physics, which allows one to mapthe strongly coupled systems to weakly coupled systems. The weakly coupled system can ∗ [email protected] † [email protected], [email protected] ‡ [email protected], [email protected] a r X i v : . [ h e p - t h ] F e b e tackled by perturbation theory and then using this correspondence, one can get apicture of some of the properties of the strongly coupled system.The asymptotically AdS black hole spacetime in the bulk can become unstable leadingto the condensation of scalar hair below a certain critical temperature. This instabilitycorresponds to a second order phase transition from normal to superconducting statethereby giving birth to the model of the holographic s-wave superconductor and owesits origin to the breaking of a local U (1) symmetry near the event horizon of the blackhole. A number of investigations have been made in this direction in order to understandvarious properties of holographic superconductor/metal phase transition in the frameworkof usual Maxwell electromagnetic theory [5]-[15] as well as in Born-Infeld electrodynamics[16]-[26] which is a non-linear theory of electrodynamics. It has also been realized thatthere can be a holographic superconductor model in the bulk AdS soliton backgroundwhich has the ability to describe an insulator/superconductor phase transition. In thiscase the AdS soliton background becomes unstable to form condensates of the scalar fieldwhich is then interpreted as a superconducting phase for the chemical potential µ > µ c , µ c being the critical chemical potential.Another interesting development that has taken place recently is the association of ageometrical structure with thermodynamic systems in equilibrium. This was first realizedthrough the works in [27]-[29]. It was shown that one can get a Riemannian metricwith an Euclidean signature from the equilibrium state of a thermodynamic system. TheRiemannian scalar curvature can then be computed and captures the details of interactionsof the thermodynamic system. It turns out that this framework based on a geometricalstructure gives a handle to study critical phenomena [29].In this present work, we have investigated analytically a holographic model of insulator tosuperconductor phase transition in AdS soliton background using the matching methodand the formalism of the thermodynamic geometry. We employ the matching methodto obtain the behaviour of the matter fields near the tip of the soliton. This in turn isused to compute the critical chemical potential. We obtain the critical chemical potentialfor a value of the matching point where the near tip and boundary values of the fieldsare matched. The analysis is based on the probe limit approximation which neglects theback reaction of the matter fields on the background space-time geometry, and is carriedout for a particular boundary condition. We also compute the condensation operatorusing the matching method. We then proceed to compute the free energy of this 3 + 1-dimensional holographic superconductor. The trick here is to relate the free energy of thetheory on the boundary to the value of the on-shell action of the Abelian-Higgs sector ofthe full Euclidean action with proper boundary terms [30],[31]. From this, we computethe thermodynamic metric using the formalism of [29]. The scalar curvature is computednext and the chemical potential at which the scalar curvature diverges is said to be thecritical chemical potential in this approach. This chemical potential is then comparedwith that obtained from the matching method.This paper is organized as follows. In section 2, we discuss the basic set up for the insula-tor/superconductor phase transition in AdS soliton background. In section 3, using theprobe limit approximation, we compute the critical chemical potential using the matchingmethod,where the matching has been carried out at a point between boundary and thetip of the soliton. In section 4, we analytically obtain the free energy expression in termsof the chemical potential and the charge density. In section 5, we calculate the thermo-dynamic metric and the scalar curvature. We conclude finally in section 6.2 Basic set up
The model of a holographic insulator/superconductor phase transition with the Einstein-Maxwell-scalar action in five dimensional spacetime can be constructed by consideringthe following action S = (cid:90) d x √− g (cid:104) R + 12 L − F µν F µν − ( D µ ψ ) ∗ D µ ψ − m ψ ∗ ψ (cid:105) (1)where F µν = ∂ µ A ν − ∂ ν A µ is the field strength tensor, D µ ψ = ∂ µ ψ − iqA µ ψ is the covariantderivative, A µ and ψ represent the gauge and the scalar fields, L is the radius of AdS spacetime.When the Maxwell field and scalar field are absent, the above action admits
AdS solitonsolution [32] ds = L f ( r ) dr + r ( − dt + dx + dy ) + f ( r ) dχ (2)with f ( r ) being given by f ( r ) = r (cid:18) − r r (cid:19) (3)where r is the tip of the soliton. For the smoothness at the tip, we need to impose aperiod β = πLr for the coordinate χ . This gives a dual picture of a three dimensional fieldtheory with a mass gap, which resembles an insulator in condensed matter physics [33].Making the ansatz ψ = ψ ( r ) and A t = φ ( r ), the equations of motion for the scalar field ψ and gauge field φ read ∂ r ψ + (cid:18) ∂ r ff + 3 r (cid:19) ∂ r ψ + (cid:18) − m f + q φ r f (cid:19) ψ = 0 (4) ∂ r φ + (cid:18) ∂ r ff + 1 r (cid:19) ∂ r φ − q ψ f φ = 0 . (5)By introducing a new coordinate z = r r , the above equations take the form ψ (cid:48)(cid:48) ( z ) + (cid:18) F (cid:48) ( z ) F ( z ) − z (cid:19) ψ (cid:48) ( z ) + (cid:18) q φ ( z ) r F ( z ) − m z F ( z ) (cid:19) ψ ( z ) = 0 (6) φ (cid:48)(cid:48) ( z ) + (cid:18) F (cid:48) ( z ) F ( z ) − z (cid:19) φ (cid:48) ( z ) + 2 q ψ ( z ) z F ( z ) φ ( z ) = 0 . (7)Also under the above transformation of coordinates, the spacetime metric (3) becomes f ( r ) = r F ( z ) z (8)where F ( z ) = 1 − z . The rescalings ψ → ψq and φ → ( r /q ) φ allows one to set q = 1and r = 1. For the rest of analysis we shall set L = 1. The asymptotic behaviour of thefields can be written as ψ b ( z ) = ψ − z ∆ − + ψ + z ∆ + (9)3 b ( z ) = µ − ρz (10)where ∆ ± = 2 ± √ m . (11)From the AdS/CFT dictionary, ψ ± can be interpreted as the expectation value of thecondensation operator O ± of the dual field theory living in the boundary. We are focusingon the boundary condition in which ψ − = 0 and ψ + (cid:54) = 0. In principle, one can do thesame analysis with opposite boundary condition that is ψ + = 0 and ψ − (cid:54) = 0. This type ofboundary condition is required because we want to turn on the condensate without beingexternally sourced. µ c from matchingmethod It has been shown numerically that when the chemical potential µ exceeds a critical value µ c , the condensations of the operators occur. This can be viewed as a superconductorphase. For µ < µ c , the scalar field is zero and this can be viewed as an insulator phase[33]. Therefore, the critical chemical potential µ c is the turning point of this holographicinsulator/superconductor phase transition. So near µ c , eq.(7) reduces to φ (cid:48)(cid:48) ( z ) + (cid:18) F (cid:48) ( z ) F ( z ) − z (cid:19) φ (cid:48) ( z ) = 0 . (12)Multiplying by the integrating factor F ( z ) z , the above equation can be recast asdd z (cid:26) F ( z ) z φ (cid:48) ( z ) (cid:27) = 0 ⇒ F ( z ) z φ (cid:48) ( z ) = constant . (13)Using the fact that F ( z = 1) = 0, we find that the above constant = 0. Hence the gaugefield equation finally reduces to φ (cid:48) ( z ) = 0 ⇒ φ ( z ) = C = µ (14)where the constant of integration C gets fixed from the asymptotic behaviour of the field φ ( z )(eq.(10)).The Taylor series expansions of the fields near the tip read ψ t ( z ) = ψ (1) + ψ (cid:48) (1)( z −
1) + ψ (cid:48)(cid:48) (1)2 ( z − + ... (15) φ t ( z ) = φ (1) + φ (cid:48) (1)( z −
1) + φ (cid:48)(cid:48) (1)2 ( z − + ... . (16)4e shall now determine the undetermined coefficients in eq.(s)(15, 16) using eq.(s)(6, 7).These read ψ (cid:48) (1) = ψ (1)4 (cid:8) φ (1) − m (cid:9) (17) ψ (cid:48)(cid:48) (1) = ψ (1) (cid:20) φ (1)4 (cid:26) − ψ (1)2 (cid:27) − (cid:8) φ (1) − m (cid:9) (cid:8) m − φ (1) (cid:9)(cid:21) (18) φ (cid:48) (1) = − ψ (1) φ (1) (19) φ (cid:48)(cid:48) (1) = ψ (1) φ (1)8 (cid:8) m + ψ (1) − φ (1) (cid:9) . (20)Hence the near tip expansions of these fields up to O ( z − read ψ t ( z ) = ψ (1) (cid:34) z − (cid:8) φ (1) − m (cid:9) + ( z − (cid:20) φ (1)4 (cid:26) − ψ (1)2 (cid:27) − (cid:8) φ (1) − m (cid:9) (cid:8) m − φ (1) (cid:9)(cid:21) (cid:35) (21) φ t ( z ) = φ (1) (cid:20) − ( z − ψ (1) + ( z − ψ (1) (cid:8) m + ψ (1) − φ (1) (cid:9)(cid:21) . (22)We now proceed to match the near tip expansions of the fields with the asymptotic solutionof these fields at any arbitrary point between the tip and the boundary, say z = λ , where λ lies between [1 , ∞ ].The matching conditions are ψ t (cid:18) λ (cid:19) = ψ b (cid:18) λ (cid:19) , ψ (cid:48) t (cid:18) λ (cid:19) = ψ (cid:48) b (cid:18) λ (cid:19) (23) φ t (cid:18) λ (cid:19) = φ b (cid:18) λ (cid:19) , φ (cid:48) t (cid:18) λ (cid:19) = φ (cid:48) b (cid:18) λ (cid:19) . (24)From eq.(24), we obtain the following relations ψ (1) + ψ (1) (cid:26) (cid:18) λ − λ − (cid:19) + m − φ (1) (cid:27) = 0 (25) ρ = λφ (1) ψ (1)4 (cid:20) λ − λ (cid:8) m + ψ (1) − φ (1) (cid:9)(cid:21) . (26)Near the critical chemical potential, eq.(s)(25, 26) reduce to ψ (1) = (cid:26) µ − m − (cid:18) λ − λ − (cid:19)(cid:27) (27) ρ = ( λ − µ (cid:26) λ − λ − m − µ (cid:27) . (28)5rom eq.(23), we get − λ ∆ + 1 + ∆( λ − (cid:8) φ (1) − m (cid:9) − λ − λ (cid:26) λ − (cid:27) φ (1) (cid:26) − ψ (1)2 (cid:27) + λ − λ (cid:26) λ − (cid:27) (cid:8) φ (1) − m (cid:9) (cid:8) m − φ (1) (cid:9) = 0 . (29)Now near the critical chemical potential, eq.(29) gives with the help of eq.(27)3( λ − λ (cid:26) λ − (cid:27) µ + (cid:20)(cid:26) λ − (cid:27) − λ − λ (cid:26) λ − (cid:27) (cid:26) m + 8 (cid:18) λ − λ − (cid:19)(cid:27)(cid:21) µ − (cid:20) λ ∆ + m (cid:26) λ − (cid:27) + λ − λ (cid:26) λ − (cid:27) m (8 + m ) (cid:21) = 0 . (30)To estimate the critical chemical potential we now need to solve eq.(30). For m = 0 wehave ∆ = ∆ + = 4. Setting λ = 2 the above equation reads9 µ − µ −
512 = 0 . (31)The solution of the above equation is µ = 4 .
533 which agrees with the numerical value3.404 [34].We now want to derive an expression for the condensation operator. From eq.(23), weobtain ψ + = λ ψ (1) (cid:20) φ (1)4 + 1 − λλ (cid:20) φ (1)4 (cid:26) − ψ (1)2 (cid:27) − φ (1) (cid:8) − φ (1) (cid:9)(cid:21)(cid:21) . (32)Using the map ψ + = (cid:104)O + (cid:105) and eq.(27), the above expression near the critical chemicalpotential takes the form (cid:104)O + (cid:105) = λ (cid:18) λ − (cid:19) µ (cid:20) λ − λ − m − λ − λ − µ (cid:21) (cid:112) µ − µ c (33)where the critical chemical potential µ c reads µ c = (cid:115)(cid:26) m + 4(4 λ − λ − (cid:27) . (34)For m = 0 and λ = 2, we find µ c = 4 . µ c obtained from eq.(s)(30,34). Near the criticalchemical potential, the expression of condensation operator reads using eq.(s)(33,34) (cid:104)O + (cid:105) = λ (cid:18) λ − (cid:19) µ c (cid:20) λ − λ − m − λ − λ − µ c (cid:21) (cid:112) µ c √ µ − µ c = λ √ (cid:18) λ − λ − m (cid:19) (cid:20) m + 4 (cid:18) λ − λ − (cid:19)(cid:21) √ µ − µ c ≡ γ √ µ − µ c . (35)6or m = − and λ = 2 .
7, we get (cid:104)O + (cid:105) = 1 . √ µ − µ c which agrees reasonably wellwith the Sturm-Liouville analytic result (cid:104)O + (cid:105) = 1 . √ µ − µ c in [35]. This result showsthat the phase transition between the s-wave holographic insulator and superconductorbelongs to second order and the critical exponent of the system takes the mean-field value1/2. In Table 1, we present the values of the condensation operator (cid:104)O + (cid:105) for m = − , λ .Table 1: Value of condensation operator (cid:104)O + (cid:105) = γ √ µ − µ c with different values of λλ γ for m = − , ∆ + = γ for m = 0 , ∆ + = 42.6 1.574 5.8632.7 1.940 6.856 In this section, we shall calculate the free energy at zero temperature of the field theoryliving on the boundary of the (4+1)-dimensional bulk theory. To proceed further, we firstwrite down the action for the Abelian Higgs sector S M = (cid:90) d x √− g (cid:104) − F µν F µν − ( D µ ψ ) ∗ D µ ψ − m ψ ∗ ψ (cid:105) . (36)Using the same ansatz ψ = ψ ( r ) and A t = φ ( r ) and setting r = 1 and q = 1, we get S M = (cid:90) d x (cid:20) F ( z ) φ (cid:48) ( z )2 z − F ( z ) ψ (cid:48) ( z ) z + ψ ( z ) φ ( z ) z − m ψ z (cid:21) . (37)Applying the boundary condition ( F (1) = 0) and the equations of motion (6),(7), weobtain the on-shell value of the action S o to be S o = (cid:90) d x (cid:20) − F ( z ) φ ( z ) φ (cid:48) ( z )2 z | z =0 + F ( z ) ψ ( z ) ψ (cid:48) ( z ) z | z =0 − (cid:90) φ ( z ) ψ ( z ) z dz (cid:21) . (38)Setting m = 0 and substituting the asymptotic behavior of φ ( z ) = µ − ρz and ψ ( z ) = ψ − + ψ + z in the first two diverging terms of above action, we get S o = (cid:90) d x (cid:20) µρ + 4 ψ + ψ − − (cid:90) φ ( z ) ψ ( z ) z dz (cid:21) . (39)The free energy of the 3+1-dimensional boundary field theory can now be obtained byΩ = − T S o = βT V [ − µρ − ψ + ψ − + I ]= βT V [ − µρ + I ] (40)7here in the second equality we have set (cid:82) d x = βV , V being the volume of the 3-dimensional space of the boundary, and in the last equality we have used the fact that ψ − = 0. The integral I reads I = (cid:90) φ ( z ) ψ ( z ) z = (cid:90) λ φ b ( z ) ψ b ( z ) z + (cid:90) λ φ t ( z ) ψ t ( z ) z ≡ I + I . (41)To evaluate the integral we replace ψ b , φ b from eq.(s)(9,10) and ψ t , φ t from eq.(s)(21,22).Now I equals to I = (cid:90) λ φ b ( z ) ψ b ( z ) z = (cid:90) λ ( µ − ρz ) ψ z z = (cid:90) λ (cid:0) µ − µρz + ρ z (cid:1) ψ z . (42)The evaluation of integral I gives I = (cid:90) λ φ t ( z ) ψ t ( z ) z dz = (cid:90) λ dzz ψ (1) (cid:20) z − φ (1) + ( z − (cid:20) φ (1)4 (cid:26) − ψ (1)2 (cid:27) − φ (1) (cid:8) − φ (1) (cid:9)(cid:21)(cid:21) × φ (1) (cid:20) − z − ψ (1) + ( z − ψ (1) (cid:8) ψ (1) − φ (1) (cid:9)(cid:21) . (43)Using eq.(27) and φ (1) = µ , we obtain from the above equation I = (cid:90) λ dz µ z (cid:26) µ − λ − λ − (cid:27) (cid:20) z − µ + ( z − µ (cid:26) λ − λ − − µ (cid:27)(cid:21) (cid:20) − (cid:26) µ − λ − λ − (cid:27) (cid:26) z −
12 + 2 λ − λ −
1) ( z − (cid:27)(cid:21) . (44)Substituting eq.(42) and eq.(44) in eq.(40), the analytical expression for the free energyin terms of the chemical potential and charge density readsΩ V = − µρ + (cid:90) λ dz (cid:0) µ − µρz + ρ z (cid:1) z (cid:18) λ (cid:19) (cid:18) λ − λ (cid:19) (cid:26) µ − λ − λ − (cid:27) × µ (cid:20) λ − λ − µ − (cid:21) + (cid:90) λ dz µ z (cid:26) µ − λ − λ − (cid:27) (cid:20) z − µ + ( z − µ × (cid:26) λ − λ − − µ (cid:27)(cid:21) (cid:20) − (cid:26) µ − λ − λ − (cid:27) (cid:26) z −
12 + 2 λ − λ −
1) ( z − (cid:27)(cid:21) . (45)8rom the above result, the holographic free energy in terms of the chemical potential andcharge density reads (for λ = 2)Ω V = µ ( −
20 + µ )(0 . − . µ − . µ + 0 . µ +0 . µ − . µ + 3 . × − µ ) − µρ +0 . µ (21 . − µ ) ( −
20 + µ )( µ − . µρ + 0 . ρ ) . (46)In the next section, we shall make use of these results to investigate the thermodynamicgeometry of the system. With the above results in hand, we now proceed to investigate the thermodynamic geom-etry of this holographic superconductor. The thermodynamic metric is defined as g ij = − µ ∂ ω ( µ, ρ ) ∂x i ∂x j (47)where ω = Ω V , x = µ and x = ρ . Note that we have modified the standard definitionof thermodynamic metric g ij = − T ∂ ω ( T,ρ ) ∂x i ∂x j [36]. Since the chemical potential plays themain role of the phase transition between insulator to superconductor, we have replacedtemperature T by the chemical potential µ in the definition of the thermodynamic metric.The scalar curvature of a general metric ds th = g ( dx ) + 2 g dx dx + g ( dx ) (48)is given by [29] R = − √ g (cid:20) ∂∂x (cid:18) g g √ g ∂g ∂x − √ g ∂g ∂x (cid:19) + ∂∂x (cid:18) √ g ∂g ∂x − √ g ∂g ∂x − g g √ g ∂g ∂x (cid:19)(cid:21) . (49)A singularity in R can be found by checking whether the denominator of the right-handside of eq.(49) vanishes. The condition of the divergence of R is g = 0 which reads g µµ g ρρ − g µρ = 0 . (50)The chemical potential for which the scalar curvature diverges can be obtained by solvingeq.(50) and eq.(28) simultaneously.This chemical potential is said to be the critical chemical potential. We take the greatestroot as the critical chemical potential. For the case m = 0, λ = 2, the thermodynamicmetric components are g µµ = 1 µ (16 . µ ( − . − . µ + 0 . µ − . µ − . µρ + 0 . µ ρ − . µ ρ + 0 . µ ρ + 0 . ρ + µ (1 . − . ρ ) + µ (0 . − . ρ )+ µ (0 . . ρ ))) (51)9 ρρ = − . × − µ (21 . − µ ) ( −
20 + µ ) (52) g µρ = 1 µ − . µ + 0 . µ − . µ + 0 . µ + 0 . µ ρ − . µ ρ + 0 . µ ρ − . µ ρ . (53)Substituting the metric components in eq.(50) and using eq.(28), we obtain1 − . µ + 0 . µ + 0 . µ − . µ + 0 . µ +0 . µ − . µ + 0 . µ − . × − µ +1 . × − µ − . × − µ = 0 . (54)The solution of the above equation gives the critical chemical potential, which is µ c =4 . µ c ) with different values of λ for m = 0 and ∆ =∆ + = 4 (Numerical value µ c = 3 .
404 [34]) λ µ c from matching method µ c from divergence of R From the root of eq.(30) From expression of µ c (eq.(34))2 4.533 4.472 4.7033 4.340 4.242 4.5104 4.276 4.163 4.4545 4.244 4.123 4.428 We now summarize our findings in this work. Using the formalism of matching methodand thermodynamic geometry, we have investigated analytically a holographic insula-tor/superconductor phase transition in the
AdS soliton background. The set up we haveconsidered is that of a 5-dimensional AdS soliton background with the matter field cou-pled with Maxwell electrodynamics. For µ < µ c , the AdS soliton background is stableand the dual field theory can be interpreted as an insulator where as for µ > µ c , the AdS soliton background will be unstable to forming condensates of the scalar field which isinterpreted as the superconducting phase in the dual field theory. Using this basic idea,we have calculated the critical chemical potential and condensation operator for m = 0with a particular boundary condition. The near tip expressions obtained by the matchingmethod plays a crucial role in obtaining the free energy of the holographic superconduc-tor. This in turn is used to compute the thermodynamic geometry. It is observed that theresults for the critical chemical potential obtained from the two approaches , namely, thematching method and the thermodynamic geometry method agree with each other. Wealso obtain an expression for the condensation operator using the matching procedure.The analytical results also match very well with the numerical findings in the literature.10 cknowledgments DP would like to thank CSIR for financial support.