A criterion for the nature of the superconducting transition in strongly interacting field theories : Holographic approach
aa r X i v : . [ h e p - t h ] A p r A criterion for the nature of the superconducting transitionin strongly interacting field theories : Holographic approach
Ki-Seok Kim , , Kyung Kiu Kim , Youngman Kim , and Yumi Ko Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk 790-784, Republic of Korea Department of Physics, Pohang University of Science and Technology, Pohang, Gyeongbuk 790-784, Korea Institute for the Early Universe, Ewha Womans University, Seoul 120-750, KOREA
It is beyond the present techniques based on perturbation theory to reveal the nature of phasetransitions in strongly interacting field theories. Recently, the holographic approach has providedus with an effective dual description, mapping strongly coupled conformal field theories to classicalgravity theories. Resorting to the holographic superconductor model, we propose a general crite-rion for the nature of the superconducting phase transition based on effective interactions betweenvortices. We find “tricritical” points in terms of the chemical potential for U(1) charges and aneffective Ginzburg-Landau parameter, where vortices do not interact to separate the second order(repulsive) from the first order (attractive) transitions. We interpret the first order transition as theColeman-Weinberg mechanism, arguing that it is relevant to superconducting instabilities aroundquantum criticality.
PACS numbers:
Interactions between vortices contain information onthe nature of the superconducting transition. Theychange from repulsive to attractive, decreasing theGinzburg-Landau parameter κ , the ratio between thepenetration depth of an electromagnetic field and theCooper-pair coherence length [1–3]. Combined with ei-ther the ǫ = 4 − d expansion or the 1 /N approximation inthe Abelian-Higgs model [4, 5], one finds that the nonin-teracting point for vortices at κ = κ t ( ∼ / √
2) is identi-fied with the tricritical point, where the nature of thesuperconducting transition changes from second order( κ > κ t ) to first order ( κ < κ t ) [4]. Quantum correctionsdue to electromagnetic fluctuations are the mechanism,referred as the fluctuation-induced first-order transition[5] or Coleman-Weinberg mechanism [6].The situation is much more complicated when cor-related electrons are introduced. In particular, super-conducting instabilities are ubiquitous in the vicinity ofquantum critical points [7], where quantum critical nor-mal states are often described by strongly interactingconformal field theories. Although one can integrate oversuch interacting fermions, the resulting effective field the-ory contains a lot of singularly corrected terms for Higgsfields, which originate from quantum corrections due toabundant soft modes of particle-hole and particle-particleexcitations near the Fermi surface [8, 9]. Furthermore,the Fermi surface problem turns out to be out of control[10, 11] since not only self-energy corrections but alsovertex corrections should be introduced self-consistently.It is far from reliability to evaluate effective interactionsbetween vortices in this problem.Recently, it has been clarified that strongly coupledconformal field theories in d -dimension can be mappedinto classical gravity theories on anti-de Sitter space in d + 1-dimension (AdS d +1 ) [12, 13]. This framework hasbeen developed in the context of string theory, refereed asthe AdS/CFT correspondence. See Ref. [14] for a review.Immediately, it has been applied to various problems beyond techniques of field theories: non-perturbativephenomena in quantum chromodynamics (AdS/QCD orholographic QCD) [15], non-Fermi liquid transport nearquantum criticality [16–18] and superconductors [19, 20]in condensed matter physics (AdS/CMP), and etc.In this letter we propose a general criterion for thefirst-order superconducting transition based on the holo-graphic approach. We take the holographic superconduc-tor model [20] as an effective low-energy model in the dualdescription for certain classes of strongly interacting fieldtheories. The asymptotic vortex solution [21] turns outto play a central role in the nature of the superconduct-ing transition. We suggest “tricritical” points in terms ofthe chemical potential for U(1) charges and an effectiveGinzburg-Landau parameter, where vortices do not inter-act to separate the second order (repulsive) from the firstorder (attractive). We interpret the first-order transitionas the Coleman-Weinberg mechanism [5, 6], arguing to berelevant to superconducting instability around quantumcriticality.We start from the holographic superconductor modelin AdS with radius LS = 12 κ p Z d x √− g h R + L − L F −
12 ( D η ) − η ( eA µ − D µ φ ) − m η i , (1)where the complex scalar field is decomposed into the am-plitude η and the phase φ , and A µ is the bulk gauge po-tential with the field strength F = dA . κ p is the Planck’sconstant. In this work we set e = 1 and m = − /L and consider the probe limit. The background metric isgiven by ds = L z (cid:16) − α f ( z ) dt + dx + dx + dz f ( z ) (cid:17) , (2)with f ( z ) = 1 − z . The Hawking temperature is givenby T = α π .Equations of motion read D η − m η − η Q µ = 0 ,L D µ B µν − η Q ν = − L D µ X µν , (3)where Q µ ≡ A µ − D µ φ is the gauge invariant superfluidfour-velocity and B µν is its field strength. X µν is ∂ [ µ ∂ ν ] φ which can be replaced with delta functions for centers ofvortices.Now we calculate effective interactions between vor-tices. The effective interaction will be determined bythe change of a single vortex solution in a widely sepa-rated vortex-lattice configuration with a lattice spacing d L [1, 2]. The variation of the single vortex solutionoccurs dominantly around the boundary of two vortices ∼ d L /
2, proven to coincide with an asymptotic solutionof the single vortex. In this respect we proceed as follows.First, we find the asymptotic solution of a single vortexaway from the vortex core. Second, we show that thevariation of the vortex solution is given by the asymp-totic solution. Third, we represent the vortex interactionin terms of this solution.We introduce the following ansatz for an asymptoticsolution of the single vortex configuration s ( ~x, z ) ≡ η v ( r, z ) − η s ( z ) = A ( z ) R ( r ) ,q t ( ~x, z ) ≡ Q vt ( r, z ) − Q st ( z ) = B ( z ) R ( r ) ,q i ( ~x, z ) ≡ Q vi ( r, z ) − Q si ( z ) = C ( z ) R i ( r ) , (4)where the superscript v represents the single vortex solu-tion and s denotes the uniform solution with the radialcoordinate r or rectangular coordinates x i in two dimen-sion. Then, Eq. (3) becomes (cid:0) ∇ − κ (cid:1) R ( ~x ) = 0 , (cid:0) ∇ − κ (cid:1) R i ( ~x ) = 0 ,A ′′ + (cid:16) f ′ f − z (cid:17) A ′ + (cid:16) κ f + 2 z f + Q st f (cid:17) A + 2 η s Q st f B = 0 ,B ′′ + (cid:16) κ f − η s z f (cid:17) B − η s Q st z f A = 0 ,C ′′ + f ′ f C ′ + (cid:16) κ f − η s z f (cid:17) C = 0 , (5)in q z = 0 gauge. It is straightforward to see R ( r ) = K ( κ r ), R θ ( r ) = K ( κ r ), and R r ( r ) = 0 in polar coor-dinates. Here, κ and κ are constants for separation ofequations. Scaling the radial coordinate by ¯ r = κ r , wefind that Eq. (5) can be rewritten in terms of only a singleparameter κ = √ κ κ . For other equations, we need tosolve them numerically, taking the regularity conditionsat the horizon. It turns out that resulting solutions de-pend on κ and a , b , c , which are defined at the horizon, a = A (1), b = B ′ (1), and c = C (1). In addition, we findthat such solutions are characterized only by b/a and κ due to the scaling symmetry of Eq. (5). See appendix Afor the numerical analysis [22].Having the asymptotic solution, we evaluate the ef-fective interaction between vortices in the dilute vortex- lattice configuration [1, 2]. We introduce η = η v + δη , Q t = Q vt + δQ t ,Q i = Q vi + δQ i , φ = nθ + δφ , (6)where the solution with the superscript v represents thesingle vortex configuration in a Wigner-Seitz cell, whilethe “ δ ” part expresses the variation of the single vor-tex configuration around the boundary of the Wigner-Seitz cell. n is the winding number of the vortex. δφ = P ˆ i =0 n arg( ~x − ~x ˆ i ) is chosen for a multi-vortex con-figuration, where ~x ˆ i is the core position of each vortex. δη and δQ t would be much smaller than η v and Q vt insidethe Wigner-Seitz cell, respectively. On the other hand,it is not obvious if δQ i is much smaller than Q vi nearthe boundary of the Wigner-Seitz cell because Q vi will bealso small. However, it is natural to expect that δη and δQ t are much larger than δQ i near the boundary [1]. Asa result, we obtain the following linearized equations ofmotion near the boundary δη ′′ + (cid:16) f ′ f − z (cid:17) δη ′ + (cid:16) ∇ − Q vi f + 2 z f + Q vt f (cid:17) δη +2 η v Q vt f δQ t − η v Q vi f δQ i = 0 ,δQ t ′′ + (cid:16) ∇ f − η v z f (cid:17) δQ t − η v Q vt z f δη = 0 ,δQ ′′ i + f ′ f δQ ′ i + (cid:16) ∇ f − η v z f (cid:17) δQ i − η v Q vi z f δη = 0 . (7)An important aspect is that these equations are essen-tially the same with those for the asymptotic configu-ration of the single vortex, valid when η v ≫ δη and Q vt ≫ δQ t with η v or Q vt ≫ δQ i . This property leads usto write down the variation of the solution in terms of theasymptotic solution for the single vortex configuration δη = X ˆ i =0 s ( ~x − ~x ˆ i , z ) , δQ µ = X ˆ i =0 q µ ( ~x − ~x ˆ i , z ) . (8)Expanding the action (1) around a vortex solution tosecond order and using equations of motion (3), we arriveat δ Ω (2) = 1 T Z ˆ i dzd x∂ µ n √− g h δη D µ (cid:16) η v + 12 δη (cid:17) + δQ ν (cid:16) B vµν + 12 δB µν (cid:17)io . (9)We observe that only surface terms contribute to the cor-rection for the grand potential, where these boundariescorrespond to the AdS boundary ( z = 0), the horizon( z = 1), and the boundary of the Wigner-Seitz cell. Theregularity condition on the horizon does not allow con-tributions from the horizon. In addition, the dilute vor-tex configuration guarantees that the contribution alongthe AdS boundary is much smaller than that from theboundary of the Wigner-Seitz cell [22]. Therefore, the < O > Ρ B z FIG. 1: Asymptotic vortex solutions for dimension 2 conden-sation, charge density, and magnetic flux, respectively, where κ = 1 is used. Both the blue line with [ a/b = − a/c = 2 ]and the green line with [ a/b = 1, a/c = 6 ] correspond to re-pulsive interactions between vortices while the dashed red linewith [ a/b = − a/c = 6 ] results in attractive interactions. relevant contribution is from the correction at the bound-ary of the Wigner-Seitz cell, which is attainable by theasymptotic solution (8). Finally, we obtain the change ofthe grand potential for the i th cell, δ Ω (2) ∼ αT Z dz I ˆ i dl ˆ n · n z δη ∇ ( η v + 12 δη ) − f ( z ) δQ t ∇ ( Q vt + 12 δQ t ) + δ ~Q ×∇× ( ~Q v + 12 δ ~Q ) o = 4 π X ˆ i =0 h C K (¯ r ˆ i ) − ( A − B ) K ( √ κ ¯ r ˆ i ) i (10)with A = R dz A ( z ) z , B = R dz B ( z ) f ( z ) and C = R dzC ( z ) .For the analytic expression in the last line of Eq. (10),we used an identity in Ref. [23].It is possible to understand the physical meaning ofEq. (10). The interaction potential consists of both firstorder and second order contributions in “ δ ”, where theformer represents interactions between the i th = 0 vortexand others i th = 0, and the latter expresses those betweenother vortices i th = 0 except for the i th = 0 vortex.This expression is formally identical to the effectiveinteraction between vortices in the Abelian-Higgs model,where the first term results from the variation of the su-percurrent while the second originates from that of theHiggs field around the boundary [1, 2]. An important in-gredient is that coefficients of the vortex interaction aregiven by integrals in the z -direction. In addition, the κ dependence of the interaction potential is much morecomplicated since such coefficients are functions of theparameter κ . In this respect the role of the parameter κ is not completely clear yet although tuning κ results inthe change of the vortex interaction.Figure 1 shows dimension 2 condensation, charge den-sity, and magnetic flux for the asymptotic single-vortexconfiguration, respectively. It is interesting to observethat when U(1) charge density decreases rapidly near thevortex core, the effective interaction between vortices be-comes more repulsive. As long as b/a remains positive,we do not see any change from repulsive to attractiveinteractions. In this case the system might lie in a deeptype II regime. Therefore, we focus on b/a < - - Κ ∆ W - - Κ ∆ W FIG. 2: Effective interactions between vortices as a function of κ . With decreasing κ the effective potential changes from re-pulsive to attractive. The nearest interaction was consideredwhere d L = 3. Left: T /T c = 0 . b/a = − a/c = 3 (BlueSolid), and a/c = 10 (Red Dashed). Right: T /T c = 0 . b/a = − a/c = 3 (Blue Solid), and a/c = 10 (Red Dashed). condition of b/a <
0, depending on the density of U(1)charges and the ratio of a/c . First, we fix the densityof U(1) charges, determining the chemical potential. Weexpect that the regime with a/c ≪ a/c ≫ κ ≫ κ ≪
1. Notice that κ is introducedinto the second term, reducing it with κ ≫ κ ≪
1. Both A and B are positive definite,decreasing monotonically as we increase κ . We uncoverthat the regime with b/a < A − B > κ < κ t , where κ t can be regarded as the tricritical point.Next, we consider cases with a fixed κ . When κ israther large, it is difficult to find the tricritical point µ t ,originating from smallness of the second term. In thisrespect it is better to start from a small enough κ . Then,the effective interaction is attractive when µ > µ t whileit becomes repulsive when µ < µ t . Figure 3 shows a sur-face of tricritical points in the space of ( κ, µ/T ) with afixed T , a/c >
1, and b/a <
0, where effective interac-tions between vortices vanish exactly. The vortex inter-action is attractive inside the ellipse while it is repulsiveoutside the ellipse. We claim that this ellipse serves ageneral criterion for the fluctuation-driven first-order su-perconducting transition in strongly coupled conformalfield theories, possibly occurring in the vicinity of quan-tum criticality.In this study we try to answer how to classify stronglyinteracting field theories, considering the nature of thesuperconducting transition. The holographic supercon-ductor model is our main ansatz as an effective low energytheory, expected to describe certain classes of stronglycoupled conformal field theories. The effective interac-tion between vortices is our central object, allowing usto distinguish the type II superconductor from type I,where the former will show the second order transition
FIG. 3: Tricritical surfaces in ( b/a, a/c, κ ) with a fixed µ forthe left panel and ( µ, a/c, κ ) with a fixed b/a for the rightpanel, respectively, where vortices do not interact with eachother. The interaction potential is attractive inside the ellipsewhile repulsive outside it. while the latter will display the first order. As shown,an asymptotic solution for a single vortex configurationplays an essential role for the effective interaction. Theeffective interaction between vortices turns out to be acomplicated function of both κ and µ/T , where the pa-rameter κ is introduced to play basically the same roleas the Ginzburg-Landau parameter. We find a surfaceof tricritical points in the parameter space of ( κ, µ/T ),where the effective interaction vanishes, which separatesthe first order from the second order, proposed to be a general criterion in classifying quantum critical metals.There are various unsolved questions in this direction.First of all, a possible topological term such as the ax-ion term [24] may play an important role in the vortexinteraction. It can assign the U(1) charge to a vortex,modifying their interactions. We suspect the possibilityof the BKT transition [25], resulting from their Coulombinteractions due to the assigned U(1) charge, where 1 /q in the momentum space becomes ln r in two space dimen-sions. In addition to this problem, the role of the pairingsymmetry is not investigated, where non s -wave super-conductivity arises in strongly interacting electrons [7].Furthermore, it should be studied the role of fermions inthe vortex interaction.We would like to thank T.Albash for providing uswith details in his work. K.Kim would like to thankKi-myeong Lee and Chanju Kim for helpful discussions.K.-S. Kim was supported by the National Research Foun-dation of Korea (NRF) grant funded by the Korea gov-ernment (MEST) (No. 2010-0074542). Y.Kim acknowl-edges the Max Planck Society(MPG), the Korea Min-istry of Education, Science, and Technology(MEST),Gyeongsangbuk-Do and Pohang City for the support ofthe Independent Junior Research Group at the Asia Pa-cific Center for Theoretical Physics(APCTP). K.Kim wassupported by KRF-2007-313- C00150, WCU Grant No.R32-2008-000- 10130-0. [1] L. Kramer, Phys. Rev. B , 3821 (1971).[2] L. Jacobs and C. Rebbi, Phys. Rev. B , 4486 (1979).[3] S. Mo, J. Hove, and A. Sudbo, Phys. Rev. B , 104501(2002); F. Mohamed, M. Troyer, G. Blatter, and I.Lukyanchuk, Phys. Rev. B , 224504 (2002); A. Chaves,F. M. Peeters, G. A. Farias, and M. Milosevic, Phys. Rev.B , 054516 (2011).[4] F. S. Nogueira and H. Kleinert,arXiv:cond-mat/0303485, to appear in the WorldScientific review volume ”Order, Disorder, and Critical-ity”, Edited by Y. Holovatch.[5] B. I. Halperin, T. C. Lubensky, and S.-K. Ma, Phys. Rev.Lett. , 292 (1974); J.-H. Chen, T. C. Lubensky, andD. R. Nelson, Phys. Rev. B , 4274 (1978).[6] S. Coleman and E. Weinberg, Phys. Rev. D , 1888(1983).[7] H. v. Lohneysen, A. Rosch, M. Vojta, and P. Wolfle, Rev.Mod. Phys. , 1015 (2007).[8] D. Belitz, T. R. Kirkpatrick, and T. Vojta, Rev. Mod.Phys. , 579 (2005).[9] J. Rech, C. P´epin, and A. V. Chubukov, Phys. Rev. B , 195126 (2006).[10] Sung-Sik Lee, Phys. Rev. B , 165102 (2009).[11] Max A. Metlitski and S. Sachdev, Phys. Rev. B ,075127 (2010).[12] J. M. Maldacena, Adv. Theor. Math. Phys. , 231 (1998);Int. J. Theor. Phys. , 1113 (1999).[13] E. Witten, Adv. Theor. Math. Phys. , 253 (1998); S. S.Gubser, I. R. Klebanov, and A. M. Polyakov, Phys. Lett. B , 105 (1998).[14] O. Aharony, S.S. Gubser, J. Maldacena, H. Ooguri, Y.Oz, Phys. Rept. , 183 (2000).[15] For reviews, see J. Erdmenger, N. Evans, I. Kirsch andE. Threlfall, Eur. Phys. J. A 35 , 81 (2008); J. Mc-Greevy, Adv. High Energy Phys. , 723105 (2010);J. Casalderrey-Solana, H. Liu, D. Mateos, K. Rajagopal,and U. A. Wiedemann, arXiv:1101.0618 [hep-th].[16] C. P. Herzog, P. Kovtun, S. Sachdev, and D. T. Son,Phys. Rev. D , 085020 (2007); S. A. Hartnoll, P. K.Kovtun, M. Muller, and S. Sachdev, Phys. Rev. B ,144502 (2007).[17] Sung-Sik Lee, Phys. Rev. D , 086006 (2009); M.Cubrovic, J. Zaanen, and K. Schalm, Science , 439(2009).[18] T. Faulkner, N. Iqbal, H. Liu, J. McGreevy, and D. VeghScience , 1043 (2010).[19] S. S. Gubser, Phys. Rev. D , 065034 (2008).[20] S. A. Hartnoll, C. P. Herzog, G. T. Horowitz, Phys. Rev.Lett. , 031601(2008).[21] Vortex solutions in the holographic superconductormodel can be found in T. Albash, C. V. Johnson Phys.Rev. D , 126009 (2009); G. Tallarita, S. Thomas,JHEP 1012 090 (2010); V. Keranen, E. Keski-Vakkuri,S. Nowling, K. P. Yogendran, Phys. Rev. D , 126012(2010); K. Maeda, M. Natsuume, T. Okamura, Phys.Rev. D , 026002 (2010); M. Montull, A. Pomarol, P.J. Silva, arXiv:0906.2396.[22] See appendices. [23] We used P i =0 h i · ˆ z = P i =0 H d~Sπ · (cid:16) h + P j =0 ,i h j (cid:17) × ( ∇ × h i ) = − P i =0 H d~Sπ · h i × ∇ × (cid:16) h + P j =0 ,i h j (cid:17) ,where h i ≡ K ( | ~x − ~x ˆ i | )ˆ z . See appendix C.[24] F. Chandelier, Y. Georgelin, M. Lassaut, T. Masson, and J. C. Wallet, Phys. Rev. D , 065016 (2004).[25] V. L. Berezinskii, Sov. Phys. JETP , 493 (1971); J. M.Kosterlitz and D. J. Thouless, J. Phys. C , 1181 (1973). Appendix A: Numerical analysis for Eq. (5)
We start with discussion about a uniform solution in the original holographic superconductor model [S. A. Hartnoll,C. P. Herzog, G. T. Horowitz, Phys. Rev. Lett. , 031601(2008)], where Q st ( z ) and η s ( z ) depend only on the z -coordinate. It is straightforward to derive equations of motion from Eq. (3) η ′′ + (cid:18) f ′ f − z (cid:19) η ′ + (cid:18) z f + Q t f (cid:19) η = 0 , Q t ′′ − η z f Q t = 0 . (A1)The regularity at horizon ( z = 1) gives the following conditions Q t (1) = 0 , Q ′ t (1) = v, Q ′′ t (1) = − u v , (A2) η (1) = u, η ′ (1) = 23 u, η ′′ (1) = 12 u (cid:18) − − v (cid:19) . (A3)Using the above conditions, one can find solutions in terms of u and v , based on the shooting method. Near theboundary ( z = 0), the solutions behave such as η ( z ) ∼ O ( u, v ) z + O ( u, v ) z + · · · , Q t ( z ) ∼ µ ( u, v ) + ρ ( u, v ) z + · · · . (A4)When we are considering operators with dimension 1 or 2, we should constrain solutions with either O = 0 or O = 0,respectively. Therefore u and v are not independent but related with each other. As a result, the space of solutionsbecomes one dimensional, allowing us to take “ v ” as a parameter for the solution of the holographic superconductingstate. In other words, v controls either temperature or charge density of the system. According to the AdS/CFTdictionary, the total charge is given by Q ∼ − R d x α Q ′ t (0). When T = α π is fixed, the charge density varies as afunction of v . If the charge density is fixed, temperature changes as a function of v . One can say a similar statementfor the chemical potential, α Q t (0). Inserting the uniform solution of Q st ( z ) and η s ( z ) into Eq. (5), we can solve themnumerically.Equation (5) has two parameters of κ and κ . Performing the scaling as discussed in the manuscript, we obtainthe following regularity conditions near the horizon for A ( z ), B ( z ) and C ( z ), A (1) = a, A ′ (1) = a (cid:18)
23 + 2 κ (cid:19) , A ′′ (1) = − a − buv − av
18 + 4 aκ aκ , (A5) B (1) = 0 , B ′ (1) = b, B ′′ (1) = − auv + 13 b (cid:0) − u + 2 κ (cid:1) , (A6) C (1) = c, C ′ (1) = c (cid:18) − u (cid:19) , C ′′ (1) = c (cid:18) −
518 + u u (cid:19) . (A7)It is straightforward to see the scaling symmetries in Eq. (5). Equation for C ( z ) remains invariant after scaling as c ˜ C ( z ) with a parameter c . In this case c = 1 is allowed due to the boundary condition in Eq. (A7). Equations for A ( z )and B ( z ) also allow scaling, unchanged after a ˜ A ( z ) and a ˜ B ( z ). Therefore b/a and κ are only relevant parameters,governing equations for A ( z ) and B ( z ). As a result one may regard the asymptotic solution of the single vortexconfiguration as s = a ˜ A (cid:16) v, κ, ba , z (cid:17) K (cid:0) √ κ ¯ r (cid:1) , q t = a ˜ B (cid:16) v, κ, ba , z (cid:17) K (cid:0) √ κ ¯ r (cid:1) , q θ = c ˜ C ( v, z ) K (¯ r ) , (A8)where ¯ r means a re-scaled coordinate with κ . We emphasize arguments in each function. Appendix B: Derivation of the variation for the grand potential
Inserting Eq. (6) into Eq. (3), we find the following linearized equations D δη − m δη − δηQ vµ − η v Q vµ δQ µ = 0 , (B1) D µ δB µν − η v δηQ ν − η v δQ ν = −D µ δX µν , (B2)proven to be valid near the center of a vortex. One can see that this approximation is reasonable only when δη and δQ t are both larger than δQ i and smaller than η v and Q vt . The boundary of a Wigner-Seitz cell also satisfies theseconditions. In this respect the linearized equations are valid not only near a vortex but also the boundary of the cell.This is a simple extension of the observation in L. Kramer, Phys. Rev. B , 3821 (1971).Inserting the vortex solution [Eq. (6) with Eq. (8)] into the effective gravity action [Eq. (1)] and expanding theaction to the second order, we obtain the following expression for the change of the grand potential in a cell δ Ω = 1 T Z ˆ i dzd x∂ µ (cid:26) √− g (cid:20) δη D µ (cid:16) η v + 12 δη (cid:17) + δQ ν (cid:16) B vµν + 12 δB µν (cid:17)(cid:21)(cid:27) , (B3)where Eqs. (B1) and (B2) are utilized. In this derivation we need to worry about singular parts from X µν . δX µν vanishes identically because the singularity appears completely outside the cell ˆ i . The only term that we have toconcern is δB µν X µν , however this turns out to vanish when we are considering the configuration of δB µν = 0 at theorigin of the cell.Changing Eq. (B3) into surface integrals, we have three kinds of boundaries. The first is the boundary at thehorizon ( z = 1) of the black hole and the second is that of the AdS space ( z = 0). The last is the boundary of theWigner-Seitz cell. The first contribution vanishes identically thanks to regularity conditions at the horizon. For the z = 0 boundary, the contribution must be considered carefully. Actually, this contribution could be important, whena distance between vortices is comparable to a size of a vortex. However, we are taking the dilute gas limit, thus thevariation from the single vortex solution will be concentrated on boundaries of Wigner-Seitz cells.The surface integral for z = 0 is given as follows δ Ω z =0 = 1 T Z ˆ i d x √− g (cid:20) δη ∇ z (cid:16) η v + 12 δη (cid:17) + δQ ν (cid:16) B v zν + 12 δB zν (cid:17)(cid:21) z =0 . (B4)In the dilute limit δη and δQ µ have nonzero values only near the boundary of a cell. Thus, the integration range iseffectively small. As positions of vortices are far from each other, this contribution almost vanishes and it is muchsmaller than the third contribution given by the integration along the z direction at the boundary of Wigner-Seitzcells. This dilute approximation serves the validity of our calculation. Therefore, our correction of the grand potentialis well approximated as δ Ω ∼ T Z dz I ˆ i dl ˆ n i √− g (cid:20) δη ∇ i (cid:16) η v + 12 δη (cid:17) + δQ ν (cid:16) B v iν + 12 δB iν (cid:17)(cid:21) , (B5)where ˆ n i is a unit vector orthogonal to the boundary of the Wigner-Seitz cell. This leads to Eq. (9). Appendix C: Derivation of Eq. (9)
In this section we will derive the following formulae X i =0 I dl ˆ n · K ( | r − r i | )ˆ θ ( ~r − ~r i ) × ∇ × (cid:16) K ( r )ˆ θ + 12 X j =0 ,i K ( | r − r j | )ˆ θ ( ~r − ~r j ) (cid:17) = π X i =0 K ( | ~r i | ) , (C1) X i =0 I dl ˆ n · K ( | r − r i | ) ∇ (cid:16) K ( r ) + 12 X j =0 ,i K ( | r − r j | ) (cid:17) = − π X i =0 K ( | ~r i | ) . (C2)For convenience, we define a fictitious coordinate w and vector fields, h i ( ~r ) ≡ K ( | r − r i | ) ˆ w , where r i is a center of alattice. Then, the above equations can be written as follows F ≡ X i =0 I d~S · (cid:16) h + 12 X j =0 ,i h j (cid:17) × (cid:16) ∇ × h i (cid:17) = π X i =0 h i · ˆ w, (C3) F ≡ X i =0 I d~S · h i × ∇ × (cid:16) h + 12 X j =0 ,i h j (cid:17) = − π X i =0 h i · ˆ w , (C4)where d~S is an area element orthogonal to the boundary surface of a cell, i.e, d~S = dl dw ˆ n .Using the divergence theorem and taking integration by parts, one can rearrange F F X i =0 Z d x ( ∇ × (cid:16) h + 12 X j =0 ,i h j (cid:17) · ( ∇ × h i ) − (cid:16) h + 12 X j =0 ,i h j (cid:17) · ( ∇ × ∇ × h i ) ) = X i =0 Z d x ( ∇ × (cid:16) h + 12 X j =0 ,i h j (cid:17) · ( ∇ × h i ) + h i · h πδ ( r ) ˆ w − ∇ × ∇ × (cid:16) h + 12 X j =0 ,i h j (cid:17)i) = 2 π X i =0 h i (0) + X i =0 I d~S · ( h i × ∇ × (cid:16) h + 12 X j =0 ,i h j (cid:17)) , (C5)where we have used ∇ × ∇ × h i + h i = 2 πδ ( ~r − ~r i ) ˆ w with some algebra. Actually, the second term in the last is equalto − F