aa r X i v : . [ h e p - t h ] A p r Vortex and droplet in holographic D-wave superconductors
Dongfeng Gao ∗ State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China (Dated: October 7, 2018)
ABSTRACT
We investigate non-trivial localized solutions of the condensate in a (2+1)-dimensional D-waveholographic superconductor model in the presence of a background magnetic field. The calculationis done in the context of the (3+1)-dimensional dual gravity theory of a charged massive spin-2field in an AdS black hole background. By using numeric techniques, we find both vortex anddroplet solutions. These solutions are important for studying the full phase diagram of D-wavesuperconductors.
Keywords: D-wave superconductor, AdS/CFT correspondence, vortex, droplet
I. INTRODUCTION
The holographic principle (also known as the AdS/CFT correspondence) is one of the most significant results fromstring theory [1], [2] and [3]. It relates a d-dimensional quantum field theory to a d+1-dimensional gravity theory.One of the exciting aspects of the holographic principle is that it can provide us a powerful tool to study stronglycoupled quantum field theories by dealing with their dual classical gravity theories. One application of this tool is tothe high temperature superconductor physics.High temperature superconductors can be classified into two classes. The first class were discovered in 1986 [4].They are cuprates with layered structure and the superconductivity is associated with the copper-oxygen planes. Thesecond class were discovered in 2008 [5]. They are iron-based superconductors with layered structure as well. Thesuperconductivity is again associated with the two-dimensional planes. However, after more than two decades ofexperimental and theoretical investigation, the mechanism of high temperature superconductors remains an unsolvedmystery [6]. There are experimental evidences showing that electron pairs still form in cuprate superconductors andthat the pairing symmetry is a D-wave symmetry [7]. Unlike in the case of conventional superconductors, the pairingmechanism here involves strong couplings. Therefore, people hope that the holographic correspondence can give ussome insights in high temperature superconductors.The first holographic model for superconductors was constructed in [8]. It is an S-wave model because the orderparameter is represented by the condensate of a complex scalar field. The temperature is introduced by the Hawkingtemperature of an AdS black hole. Below some critical temperature, the condensate develops. Above it, the condensatevanishes. Thus, the superconducting-to-normal phase transition is produced. Following [8], various aspects of S-wavemodels have been extensively studied (see [9], [10], [11] and [12] for reviews). The construction of S-wave models instring theory and M-theory was given in [13] and [14]. The behavior of the S-wave superconducting condensate ina magnetic field background was investigated in [15], [16], [17], [18], [19] and [20]. In particular, the vortex solutionwas studied in [16], [17], [18] and [21], and the droplet solution was discussed in [16] and [17]. Motivated by studiesof S-wave models, holographic models of P-wave superconductors were investigated in [22], [23], [24], [25] and [26],where an SU(2) Yang-Mills field is coupled to an AdS black hole, and the order parameter is represented by a vectorfield.Recently, holographic models for D-wave superconductors were constructed in [27], [28] and [29]. In these models, acharged massive spin-2 field was put in an AdS black hole background. The condensate of this spin-2 field signals theD-wave superconducting phase transition. Some properties of these models were studied in [30], [31] and [32]. In thiswork, we will study the non-trivial localized solutions of the condensate in the presence of a background magnetic field,based on the action given in [28] and [29]. Especially, we are interested in finding the vortex and droplet solutions.The organization of our paper is as follows. In section II, we will first introduce the model in [28] and [29]. Forconvenience, the AdS black hole metric is written in terms of polar coordinates. Then, equations of motion for thespin-2 field and the gauge field are derived. In section III, we obtain the vortex and droplet solutions by using numerictechniques. Properties of these solutions are discussed as well. In the final section, we conclude with some remarks. ∗ Email: [email protected]
II. THE D-WAVE HOLOGRAPHIC SUPERCONDUCTOR MODEL
The full 3+1-dimensional gravity theory which is dual to a 2+1-dimensional D-wave superconductor contains thegravity sector and the charged massive spin-2 field sector S = 12 κ Z d x √− g { ( R − L m } , (1)where R is the Ricci scalar and Λ = − /L is the negative cosmological constant with the AdS radius L . κ = 8 πG is the gravitational coupling. L m is the Lagrangian for the charged massive spin-2 field and the gauge field. It is wellknown that writing down a consistent L m in a curved spacetime is a very difficult problem. The authors in [28, 29]have constructed such a Lagrangian in an AdS spacetime, which will be used in this paper L m = −| D ρ ϕ µν | + 2 | D µ ϕ µν | + | D µ ϕ | − (cid:2) D µ ϕ ∗ µν D ν ϕ + c.c. (cid:3) − m (cid:0) | ϕ µν | − | ϕ | (cid:1) + 2 R µνρλ ϕ ∗ µρ ϕ νλ − d + 1 R| ϕ | − iqF µν ϕ ∗ µλ ϕ νλ − F µν F µν , (2)where ϕ µν represents the spin-2 field, which is symmetric. We introduce the notation ϕ ≡ ϕ µµ . D µ = ∇ µ − iqA µ is thecovariant derivative, which reduces to the familiar D µ = ∂ µ − iqA µ in flat spacetimes. d is the bulk spatial dimension,which is 3 in our case. F µν is the U(1) gauge field strength tensor. As discussed in [29], this Lagrangian works wellonly when the background spacetime is an Einstein manifold, which means R µν = 2Λ d − g µν . (3)This condition restricts us to do calculations in the probe limit [8], where the background metric is not perturbed bythe spin-2 field and the gauge field.It is easy to write down the 3+1-dimensional AdS-Schwarzschild planar black hole solution to (3) ds = − L α z f ( z ) dt + L z ( dr + r dφ ) + L z f ( z ) dz , (4)where f ( z ) = 1 − z . (5)The black hole horizon is at z = 1, and the boundary is at z = 0. They are 2-dimensional planes. We have introducedthe polar coordinate system ( r , φ ) on the planes. In our notations, r and z are now dimensionless coordinates. TheHawking temperature of this black hole is T = 3 α π . (6) III. LOCALIZED SOLUTIONS OF THE D-WAVE CONDENSATE
According to the AdS/CFT correspondence, the bulk massive spin-2 field ϕ ij is mapped to a spin-2 operator O ij in the boundary conformal field theory. The conformal dimension ∆ ij of O ij is determined by m L = ∆ ij (∆ ij − . (7)The following ansatz is used in our work. For the gauge field A = A µ dx µ = A t ( r, z ) dt + A φ ( r, z ) dφ, (8) Strictly speaking, there is no dynamical gauge field in the dual boundary theory. But as discussed in many papers, e.g. [10], thisproblem does not prevent us from introducing spatially dependent magnetic field. and for the spin-2 field, ϕ µν = L α z ψ ( r, z ) ir L α z ψ ( r, z ) 00 ir L α z ψ ( r, z ) − r L α z ψ ( r, z ) 00 0 0 0 e i ( n +2) φ . (9)Obviously, the ansatz for ϕ µν satisfies ϕ = 0. We also require that D µ ϕ µν = 0 in later calculation. The equations ofmotion for A t , A φ and ψ are found to be0 = − q L r ψ A t + rz ( r ∂ z A t + ∂ r A t + r∂ r A t ) , (10)0 = 4 q L rψ A φ − z ( rf ′ ∂ z A φ + rf ∂ z A φ − ∂ r A φ + r∂ r A φ ) , (11)0 = q r z A t ψ + α f (cid:0) − ψ ( L m r + n z + q z A φ + qrz ∂ r A φ )+ rz (cid:0) f ( − r∂ z ψ + rz∂ z ψ ) + z ( rf ′ ∂ z ψ + ∂ r ψ + r∂ r ψ ) (cid:1)(cid:1) . (12)The behavior of A t , A φ and ψ , near the AdS boundary at z = 0, contains important information. The seriesexpansions around z = 0 take the following form: A t ( r, z ) = µ − ρ z + A t ( r ) z + A t ( r ) z + · · · , (13) A φ ( r, z ) = a φ ( r ) + J φ ( r ) z + A φ ( r ) z + A φ ( r ) z + · · · , (14) ψ ( r, z ) = z − ∆ ( ψ ( s ) ( r ) + O ( z )) + z ∆ (cid:0) < O ( r ) > − O ( z ) (cid:1) . (15)From the expansion of A t , we can read out the chemical potential µ and the related charge density ρ . Similarly, fromthe first two terms in the expansion of A φ , we read out the boundary magnetic field B = r F rφ | z =0 = ∂ r a φ /r and theazimuthal current density J φ . In the expansion of ψ , ψ ( s ) is identified as the source and < O ( r ) > is identified as theD-wave condensate, whose conformal dimension ∆ is determined by (7). Only one of them can exist at a time, so wefocus on < O ( r ) > by setting ψ ( s ) = 0.Since we work in the probe limit, we can fix q = 1 without losing generality. For convenience, we also set theAdS radius L = 1. Then, the free parameters in our model are the temperature T (through the relation (6)), thecharge density ρ and the mass m . Finding solutions to (10), (11) and (12) is highly non-trivial. We have to resort tonumerical techniques. To solve the equations numerically, the boundary conditions have to be specified. At the AdSboundary ( z = 0), we impose the conditions ψ | z =0 = 0, ∂ z A t | z =0 = − ρ , and A φ = r B/
2. At the black hole horizon( z = 1), we require that ψ and A φ are finite and that A t = 0. We also have to consider how to determine the criticaltemperature T c . The condensate only exists below T c . Above it, the superconducting state changes to the normalstate ψ = 0 , A t = µ − ρz, A φ = 12 r B. (16)Therefore, by observing when a non-zero condensate begins to form, we can determine the value of T c in terms of ρ and m . In the following, the Maple 15 package [33] is used to numerically solve the system of the above three partialdifferential equations.Two types of non-trivial localized solutions of the condensate will be discussed, characterized by their behavior at r → ∞ . The first type is called the vortex solution, where < O ( r ) > → a non-zero constant at r → ∞ . The other oneis called the droplet solution, where < O ( r ) > → r → ∞ . A. The vortex solutions
In FIG. 1 and 2, we display the vorticity n = 1 sample solutions for cases of m = 0 .
025 and m = 2 .
0, respectively.For brevity, only figures for the condensate and the magnetic field are shown in each case. In each figure, forcomparison, curves for two different temperatures are drawn. Our result is similar to that of the S-wave holographicsuperconductors in [16], [17] and [18]. We can see that the condensate goes to zero at the origin where the core ofthe vortex is located, and runs to a constant quickly when approaching to the infinity. Therefore, the U(1) gaugesymmetry is broken at large r as expected for vortex solutions. The magnetic field B starts from a non-zero valueat the core of the vortex, and drops down to zero at large r . It is obvious that the magnetic field penetrates thesuperconductor only through the small region around the core of the vortex. The magnetic flux, Φ = R dφ R rBdr ,is calculated to be 2 π . In both cases, the solutions behave more like vortices at lower temperatures than at highertemperatures.As pointed out in [29], the unitarity requires that m ≥ m forcomparison. We find that the vortex solution begins to form at lower temperature for smaller masses than that forlarger masses. The condensate grows more quickly for m = 2 . m = 0 . m . B. The droplet solutions
Now, let us discuss the droplet solutions. In FIG. 3 and 4, we show the n = 0 sample solutions for cases of m = 0 . m = 2 .
0, respectively. Opposite to vortex solutions, the condensate is non-zero at the origin where the core ofthe droplet is located, and drops down to zero when approaching to the infinity. So the U(1) gauge symmetry is notbroken at large r . The magnetic field exhibits new behavior. For high temperatures, it decreases from a non-zero valueat the core and down to a constant value at large r . For low temperatures, it increases from a non-zero value at thecore and all the way to a constant value at large r . Similar behavior has also been found for S-wave superconductorsin [16] and [17]. The magnetic field penetrates almost all of the superconductor except the small region around thecore of the droplet. The solutions again behave more like droplets at lower temperatures than at higher temperatures.Compare vortex solutions with droplet solutions, one can find that droplet solutions form at higher values ofthe magnetic field. This is consistent to the fact that our D-wave holographic superconductor model is of Type IIsuperconductors. Type II superconductor has two critical magnetic fields, the lower one B c and the higher one B c . Below B c , it is completely in the superconducting state. Above B c , it is completely in the normal state. Inbetween B c < B < B c , it is in a mixed state of superconducting and normal states. As B increase from B c , thesuperconducting state dominates the mixed state, it is preferable for vortex solutions to form first. When B becomeswell greater than B c , the normal state dominates the mixed state, then it is expected to find droplet solutions. Ofcourse, it is also possible that the vortex lattice maybe form for some values of the magnetic field. The details remainfor future work. IV. CONCLUSIONS
We have studied the non-trivial localized solutions of a D-wave superconductor model of [28] and [29] in the presenceof a background magnetic field. Numerically, two types of solutions are found, the vortex and the droplet solutions.The properties of these solutions are discussed. The forming of these solutions depends on the temperature and themagnetic field. They are important configurations in the phase diagram. Of course, a complete investigation of thefull phase diagram is missing in this work, which deserves a future study on it. Another interesting project for futurestudy is to try to derive more realistic D-wave holographic models directly from string theory or M-theory, just aswhat have been tried for S-wave and P-wave superconductors. This method is called the top-down approach. It isinteresting to see what new insights of D-wave superconductors the top-down approach may provide us.
Acknowledgements
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Maity, d + id holographic superconductors, JHEP < (r) > / r T/T C =0.39 T/T C =0.24 (a) The condensate B (r) r T/T C =0.39 T/T C =0.24 (b) The magnetic field FIG. 1: The vortex solution for m=0.025. < (r) > / r T/T C =0.96 T/T C =0.49 (a) The condensate B (r) r T/T C =0.96 T/T C =0.49 (b) The magnetic field FIG. 2: The vortex solution for m=2.0. < (r) > r T/T C =0.95 T/T C =0.59 (a) The condensate B (r) r T/T C =0.95 T/T C =0.59 (b) The magnetic field FIG. 3: The droplet solution for m=0.025. < (r) > r T/T C =0.95 T/T C =0.59 (a) The condensate B (r) r T/T C =0.95 T/T C =0.59 (b) The magnetic field(b) The magnetic field