Critical magnetic field in AdS/CFT superconductor
aa r X i v : . [ h e p - t h ] M a y Critical magnetic field in AdS/CFT superconductor
Eiji Nakano
1, 2, ∗ and Wen-Yu Wen † Department of Physics and Center for Theoretical SciencesNational Taiwan University, Taipei 106, Taiwan Gesellschaft f¨ur Schwerionenforschung,GSI, D-64291 Darmstadt, Germany
Abstract
We have studied a holographically dual description of superconductor in (2 + 1)-dimensions inthe presence of applied magnetic field, and observed that there exists a critical value of magneticfield, below which a charged condensate can form via a second order phase transition. ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION The holographic correspondence between a gravitational theory and a quantum fieldtheory, first emerged under the AdS/CFT correspondence[1], has been proved useful tostudy various aspects of nuclear physics such as RHIC and condensed matter phenomena,particularly in those recent studies [2, 3, 4, 5, 6].In the papers [7, 8], the author proposed a gravity model in which Abelian symmetry ofHiggs is spontaneously broken by the existence of black hole. This mechanism was recentlyincorporated in the model of superconductivity and critical temperature was observed[9],and later on non-Abelian gauge condensate[10]. In this paper, we would like to extend thework to include the magnetic field and will show the existence of critical magnetic field asexpected from physics of superconductor.To implement a magnetic field at finite temperature, we introduce a Reissner-Nordstromcharged black hole and a condensate through a charged scalar field. In the superconductingphase, the scalar field takes different values at the horizon for different condensate expec-tation value at the boundary, indicating the existence of a scalar hair; while in the normalphase, vanishing scalar field tells the ordinary tale of a black hole with no hair.
II. THE MODEL WITH APPLIED MAGNETIC FIELD
Several important unconventional superconductors, such as the cuprates and organics, arelayered in structure and interesting physics can be captured by studying a (2+1) dimensionalsystem. We are now interested in building up a gravity model (in coupled with other matterfields) in (3 + 1) dimensions which is holographically dual to the desired planar system whichdevelops superconductivity below critical temperature and critical magnetic field. We startwith a model composed of the gravity sector and the matter sector. The gravity sector isgiven by the following Lagrangian density, e − L g = R − L − F µν F µν , (1)2ogether with a solution of magnetically charged black hole in AdS , where[11] ds = − f ( r ) dt + dr f ( r ) + r ( dx + dy ) , (2) f ( r ) = r L − Mr + H r . (3)Through the paper we set radius of curvature L = 1 for numerical computation. By assump-tion the only nonzero electro-magnetic field is the magnetic component F xy = Hr , of whichthe energy density at any fixed radius coordinate r is always finite and constant, that is, F µν F µν ∝ H . This serves the purpose of constant applied magnetic field at the boundary.The black hole is censored by a horizon provided the condition 27 M − H ≥ M and H , is determined via the relation T = f ′ ( r + )4 π , (4)where r + is the most positive root of f ( r ) = 0 (outer horizon). We expect that the gravitysector, implied by its given name, can be easily obtained from a pure gravity theory of higherdimensions by appropriate reduction.For the matter sector, we will use the Ginzburg-Landau (GL) action for a Maxwell fieldand a charged complex scalar, which does not back react on the metric [8, 9], e − L m = − F ab F ab + 2 L | Ψ | − | ∂ Ψ − iA Ψ | . (5)This action differs from the usual GL theory by two places: the coefficient of | Ψ | termappears to be negative in both ordinary and superconducting phase, and a | Ψ | term is notincluded. The AdS bulk geometry, however, plays the role of stabilization and we still expectsome kind of Higgs mechanism triggered outside the horizon[8]. Enough for our purpose,we will also assume the planar symmetry ansatz for the scalar potential A t = Φ( r ) and thecomplex scalar Ψ( r ), where we have already fixed the phase to be constant. Then we needto solve a pair of coupled second order differential equationsΨ ′′ + ( f ′ f + 2 r )Ψ ′ + Φ f Ψ + 2 L f Ψ = 0 , Φ ′′ + 2 r Φ ′ − f Φ = 0 (6)with appropriate boundary conditions at the horizon and at asymptotic infinity. They canbe solved numerically regardless of difficulty which appears in finding nontrivial analytic3 = = = r - - - - - - m eff + L FIG. 1: The effective mass m eff evaluated at fixed temperature and boundary conditions atthe horizon. From bottom up, the curves are with H = 0 , . solutions. In particular, for normalizable scalar potential, we require at the horizon[8, 9]Ψ ′ Ψ (cid:12)(cid:12)(cid:12)(cid:12) r = r + = − r + r − H r , Φ( r + ) = 0 . (7)Nevertheless we still have freedom for two parameter family of solutions by assigning Φ ′ and Ψ at the horizon, therefore we have a scalar hair from black hole for non-vanishing Ψ.At the boundary, the solutions behave likeΨ = Ψ (1) r + Ψ (2) r + · · · , Φ = µ − ρr + · · · , (8)where µ and ρ are interpreted as chemical potential and charge density in the dual fieldtheory. We are interested in the case where either Ψ (1) or Ψ (2) vanishes for stability concernat asymptotic AdS region, then read off the pairing operator O dual to Ψ from the bulk-boundary coupling [9], hO i i = √ ( i ) . (9)To gain a better intuition of how a condensate is realized in this gravity setup, we mayinvestigate the effective mass of Ψ field along the radius direction, that is m eff ( r ) = − L − Φ f . (10)4 .0 0.2 0.4 0.6 0.8 1.002468 T/T √ 〈 O 〉 / T H˜ =0H˜ =2H˜ =4H˜ =6
FIG. 2: We plot order parameter hO i as a function of temperature. The critical temperature T c decreases as applied magnetic field increases. Here ˜ H is the normalized H given by H / /T , where T = T c at H = 0. We recall that there exists the Bretenlohner-Freedman (BF) bound[12], i.e. m L > − / r for smaller magnetic field. Inthe other words, condensate happens more easily while the magnetic field is smaller. Thisimplies the existence of critical magnetic field below which the condensation can take place. III. CRITICAL MAGNETIC FIELD
In the normal phase, we always have solutions to the equations (6), that is Ψ = 0and Φ = µ − ρr ; while in the superconducting phase, we may have nontrivial Ψ( r ) and itsboundary value serves as an order parameter for condensate. In the absence of appliedmagnetic field, for any fixed ρ , there exists a critical temperature T c , above which thereis no more nontrivial solution[9]. In the presence of applied magnetic field, however, theMeissner effect is expected and there exists both T c and a critical magnetic field H c , abovewhich the nontrivial solution is again not admissible. As argued in the previous section, weexpect that the stronger applied magnetic field H is, the lower is critical temperature T c .This statement is supported by our numerical results for hO i as shown in the Figure 2. Theoperator O corresponds to a pair of fermions, while O to a pair of bosons[9]. We have also5 H (cid:144) (cid:144) T T c (cid:144) T FIG. 3: The phase diagram of T c against H c . The superconducting phase where hO i 6 = 0( hO i 6 = 0)exists in the lower left part below the solid (dashed) curve, while normal phase in the upper rightpart above the curve. found similar results for hO i only at a small H region.In the Figure 3 we also plot the phase diagram of critical magnetic field against criticaltemperature. IV. DISCUSSION
In this paper, we have considered a hybrid model for AdS/CFT superconductors in thepresence of magnetic field. Several comments are in order: At first, a magnetic field isprovided in the gravity sector as a background, independent of the probed sector. Weargue that this is perfectly fine as long as we only consider a constant magnetic field at theboundary. Secondly, the matter sector has no back reaction to the gravity sector, thereforethe equation of motion for total Lagrangian is not satisfied. Although this may not becrucial to the occurrence of superconducting phase, it is still interesting to investigate afully back-reacted action which can be derived from some higher-dimensional theory such asString theory or M-theory. Thirdly, in order to discuss possible formation of vortex latticeand distinguish between type I and II superconductors, one is tempted to relax the ansatz ofplanar symmetry. This will complicate the construction and analysis and we hope to report itin the near future. At last, this construction is a tractable model of strongly coupled systemwhich may capture some physics of unconventional superconductors, in contrast to theconventional superconductors well described by GL theory macroscopically and BCS theory6icroscopically. Though we do not see fermionic degree of freedom from this macroscopicconstruction, the complex scalar, serving as order parameter, seems sufficient to explain sucha critical phenomenon as good as the usual GL theory. In order to pursue a microscopicmodel along this line of reasoning, one may still need to understand better how to realizeunderlying fermionic degree of freedom in the context of AdS/CFT correspondence.
Acknowledgments
The authors are partially supported by the Taiwan’s National Science Council and Na-tional Center for Theoretical Sciences under Grant No. NSC96-2811-M-002-018, NSC97-2119-M-002-001, and NSC96-2811-M-002-024. [1] J. M. Maldacena, “The large N limit of superconformal field theories and supergrav-ity,” Adv. Theor. Math. Phys. , 231 (1998) [Int. J. Theor. Phys. , 1113 (1999)][arXiv:hep-th/9711200].[2] C. P. Herzog, P. Kovtun, S. Sachdev and D. T. Son, “Quantum critical transport, duality,and M-theory,” Phys. Rev. D , 085020 (2007) [arXiv:hep-th/0701036].[3] S. A. Hartnoll, P. K. Kovtun, M. Muller and S. Sachdev, “Theory of the Nernst effect nearquantum phase transitions in condensed Phys. Rev. B , 144502 (2007) [arXiv:0706.3215[cond-mat.str-el]].[4] S. A. Hartnoll and C. P. Herzog, “Ohm’s Law at strong coupling: S duality and the cyclotronresonance,” Phys. Rev. D , 106012 (2007) [arXiv:0706.3228 [hep-th]].[5] S. A. Hartnoll and C. P. Herzog, “Impure AdS/CFT,” arXiv:0801.1693 [hep-th].[6] D. Minic and J. J. Heremans, “High Temperature Superconductivity and Effective Gravity,”arXiv:0804.2880 [hep-th].[7] S. S. Gubser, “Phase transitions near black hole horizons,” Class. Quant. Grav. , 5121(2005) [arXiv:hep-th/0505189].[8] S. S. Gubser, “Breaking an Abelian gauge symmetry near a black hole horizon,”arXiv:0801.2977 [hep-th].
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