Vortex in holographic two-band superfluid/superconductor

We construct numerically static vortex solutions in a holographic model of two-band superconductor with an interband Josephson coupling in both the superfluid and superconductor regime. We investigate the effects of the interband coupling on the order parameter of each superconducting band in the vortex solution, and we find that it is different for each of the two bands. We compute also the free energy, critical magnetic field, magnetic penetration length and coherence lengths for the two bands, and we study their dependence on the interband coupling and temperature. Interestingly, we find that the coherence lengths of the two bands are close to identical.


I. INTRODUCTION
Multiband superconductors have attracted much attention since the discovery of the first twoband superconductor in MgB 2 [1], and more recently the first iron-based superconductor [2]. Many novel features were discovered in MgB 2 , such as having a large critical current, an anisotropy in the Fermi velocity, and an anomalous vortex dynamics [3], while iron-based superconductors may give rise to a new class of high-temperature superconductors, given the similarity of their planar structures and their phase diagrams to the cuprates [4].
Multiband superconductivity has been studied using Ginzburg-Landau (GL) theory now generalized from having just a single superconductor condensate to having multiple ones [5]. Interesting phenomena such as the formation of interband phase difference soliton [6], fractional flux quanta [6], and possibly type-1.5 superconductivity [7] in which vortex clusters can coexist with the Meissner domain are seen. Our goal in this paper is to study multiband superconductivity beyond the regime of validity of the GL theory, i.e. fields are not assumed to be small. In particular, we shall study vortices in two-band superconductors that are strongly coupled.
The tool we use to study strongly-coupled/correlated systems is the AdS/CFT correspondence [8,9] or "holography", which has proven to be very useful in a variety of different areas, including QCD [10], heavy ion physics [11], and superconductivity [12][13][14][15][16][17]. In Ref. [18], a holographic model of two-band superconductor was constructed 1 . The model takes into account fully the back-reaction from the matter sector on the gravity background, and emphasizes the effects of the interband Josephson coupling, which was realized by a Josephson-like coupling between two bulk complex scalar fields. The transport properties of the holographic model were studied, and were shown to have the same qualitative features as seen in experiments.
In this paper, we continue the study of two-band superconductor initiated in Ref. [18] by two of the present authors. In particular, we shall look for vortex solutions as a response to the magnetic field. It is known that the type of AdS-boundary conditions imposed on the bulk U (1) gauge field determines the kind of vortices found: Dirichlet type give rises to superfluid vortices, Neumann type to superconductor vortices [22,23]. Here we shall study both types. We shall also check whether the purported type-1.5 superconductivity -which were seen in some studies, but not allexists in our holographic model. A quantitative indicator for a type-1.5 two-band superconductor is when the coherence lengths for the two bands, ξ 1 and ξ 2 , and the magnetic penetration length, λ, satisfy the relation ξ 1 < √ 2λ < ξ 2 [7]. By extracting the coherence and penetration lengths from our holographic model, we can test for type-1.5 superconductivity.
The paper is organized as follows. In Sec. II, we describe the set-up for finding vortex solutions in the holographic model of two-band superconductor of Ref. [18]. We give the ansatz for the vortex solution and we specify the boundary conditions for both superfluid and superconductor type vortices. In Sec. III, we study the vortex solutions in detail, and we compute the coherence lengths for both types of vortices. In the case of superconductor vortex, we compute also the magnetic penetration length as the magnetic field is dynamical, showing that the holographic two-band superconductor is always type II. We conclude in Sec. IV with a summary.

II. THE HOLOGRAPHIC MODEL
We consider the minimal holographic model of two-band superconductor in AdS 4 given in Ref. [18]: where ψ 1,2 are complex scalar fields with masses m 1,2 respectively, A µ is the U (1) gauge field with F = dA the field strength, and q is the U (1) charge of the complex scalar fields 2 . In the potential V , denotes an interband Josephson coupling, and η a density-density coupling.
We shall work in the probe limit, where the matter sector does not cause backreaction on the background metric. We take the background to be an AdS-Schwarzchild black hole, whose metric is given by where z h is the location of horizon. For convenience, we have used polar coordinates (ρ, φ) for the two-dimensional (2D) plane in the spatial field theory directions.

A. The vortex solution
A consistent ansatz respecting the global U (1) symmetry and rotational symmetry on the 2D plane is given by The winding or "vortex" number n i ∈ Z distinguishes between different topological solutions. With the above ansatz, the equations of motion obtained from the action given in Eq. (3) are We will consider a finite system with radius R, which we take to be much larger than the vortex radius. The boundary conditions at ρ = R for superfluid vortices are given by For superconductor vortices, the same boundary conditions apply except now A φ | ρ=R = n. Boundary conditions at ρ = 0 are the same for both superfluid and superconductor vortices. For n = 0, they are For n = 0, the boundary condition on the scalar changes to ∂ ρ ϕ| ρ=0 = 0. In order to avoid the divergence in energy from multiple fractional magnetic flux [24], we shall set n 1 ≡ n 2 = n ∈ Z henceforth.

III. SUPERFLUID/SUPERCONDUCTOR VORTICES
To find the vortex solutions, we numerically solve the equations of motion (EoMs) given by Eqs. (8), (9), (10), and (11), employing the pseudo-spectral Chebyshev method. For the discretization, we use a Gauss-Lobatto grid, and we set 20 grid points for the bulk z-direction, and 40 for the radial ρ-direction. After translating the EoMs as well as the boundary conditions into a system of non-linear algebraic equations, which we set up as a matrix equation using the Chebyshev differential matrices, we then solve by using the Newton-Raphson method; the error tolerance is set at 10 −6 . Our numerics is implemented using matlab.
In our numerical calculations, we work in units where L = 1. We set q = 1, n 1 = n 2 = n, m 2 1 = −2, m 2 2 = −5/4, and R = 8. We have checked that the solution obtained with R extended up to 24 differ little with that obtained with R = 8, the fractional difference being less than 10 −4 . We shall also set η = 0 since we will not consider the effects of the density coupling here. Below we compute various properties of the superfluid vortices, varying the parameters that include the Josephson coupling, , the constant external angular velocity of the rotating superfluid, B, and the dimensionless ratio of chemical potential to temperature,μ ≡ µ/T 4 .

A. Superfluid vortex solutions at various Josephson couplings
We show in Fig. 1 order parameters of a superfluid, n = 1 single vortex solution for various values of the Josephson coupling , with the external angular velocity set at B = 0.03125, and µ = 6.2. Note that when = 0, only the scalar ψ 2 condenses but not ψ 1 , i.e. O 1 ≡ 0 while O 2 forms at a critical temperature given byμ c = 5.81. It is only when is nonzero that ψ 1 also condenses, and both condensates form at the same critical temperature [18]. So we see in Fig Here and below we work atμ = 6.2, which translates to T = 0.937T c . We have obtained superfluid vortex solutions at other values of the temperature from just below T c to T = 0.5T c . We have checked that the features we show here and below persist at other values of the temperature. Fig. 1 show the usual radial behavior of the order parameter: it is zero at ρ = 0, the vortex core, and tends to a constant far away from the core. Note that while O 2 seem to decrease monotonically as increases, O 1 does not behave monotonically at all. To better demonstrate the dependence of the scalar condensates on , we plot in Fig. 2 the values of the condensates at the system boundary O 1,2 ρ=R as functions of . We have plotted this dependence for two values of B, and we see that B has little effect qualitatively. From Fig. 2, we see clearly that O 2 ρ=R decreases monotonically as increases; the rate of decrease is quite slow for 0.1. In contrast, O 2 ρ=R first increases as is increased from zero, turns at ≈ 0.26, and then decreases with increasing . Note that there is a critical value, c ≈ 0.5 5 , above which both scalar condensates vanish and not only at ρ = R, i.e. we can no longer find a superfluid solution for a Josephson coupling above c , only the normal state solution with both O 1,2 ≡ 0.
B. Free energy and the critical angular velocity B c1 The free energy can be calculated holographically from the properly renormalized on-shell action. For the holographical two-band model, the (bare) on-shell action is given by Note that terms involving have been removed by the equations of motion. The first term in Eq. (18) produces a surface integral. To remove the divergence coming from it, we need to add the counterterm where γ is a reduced metric on the boundary with √ −γ = ρ/z 3 . Adding all the contributions together, we obtain the free energy from the finite, regularized on-shell action Note that we have kept the density coupling, η, for completeness above. Since we do not consider its effect here, it is set to zero below in our numerical calculations. We show in Fig. 3 the temperature dependence of the free energy for the normal state (nonsuperfluid) solution, and the superfluid, n = 0 and n = 1 vortex solutions at = 0.05 and B = 0.03125. As already mentioned above, the critical temperature at which the scalar condensate forms is given byμ c = 5.81.
From the left panel of Fig. 3, we see that below T c when superfluid forms, the superfluid solutions have lower free energy than the normal state solution, as expected of the superfluid being thermodynamically favored below T c . Next, to distinguish which is thermodynamically favored, the n = 0 or the n = 1 vortex solution, we plot in the right panel of Fig. 3 for each winding configuration the free energy difference between the superfluid vortex solution and the normal state solution where F (ϕ i = 0) (F (ϕ i = 0)) denotes the free energy of the normal state (superfluid vortex) solution with both scalars vanishing (condensing). We see that for B = 0.03125 and = 0.05, the n = 0 is preferred over the n = 1 vortex solution. Note that we have displayed only the region close to T c so that the two solution curves can be clearly distinguished. We show in Fig. 4 the B dependence of the free energy of the normal state and the superfluid vortex solutions at = 0.05 and T = 0.937T c . We see that there is a critical value, B c1 = 0.09, where the free energy for both the n = 1 and the n = 0 configurations coincide, and so marks the beginning for which the n = 1 vortex solution becomes thermodynamically favored over the n = 0 one. We show in Fig. 5 the dependence of the free energy of the normal state and the superfluid vortex solutions on the Josephson coupling for both B < B c1 and B > B c1 at T = 0.937T c . We see that for the range of shown, when B < B c1 , the n = 0 solution is favored over the n = 1 one, and so has the lower free energy and thus larger |∆F |, while when B < B c1 , the reverse is true. Next, we see that when approaches c (B) ≈ 0.5 for the values of B used here, the free energy of both the n = 0 and n = 1 vortex solutions approach that of the normal state solution. This reflects the fact that above c (B) no superfluid solution can be found in our numerics, only the normal state solution. This feature was already seen in Fig. 2.

C. Superfluid density and coherence lengths
By the AdS/CFT correspondence, the superfluid density, n s , can be obtained from the conjugate current, J φ , as [21] where a φ = 1 2 ρ 2 B with B the external angular velocity. Note that the denominator n − a φ is the gauge-invariant superfluid velocity along the angular direction, v φ = (∇ arg[ψ i ]) φ − a φ [25]. We show in Fig. 6 the profile of n s and J φ in the radial ρ-direction for the n = 1 configuration at B = 0 and 0.03125. For n s , we see that external rotation has a little effect on the superfluid density. But for J φ , when there is external rotation (B = 0), after rising from zero at the vortex core, instead of approaching a nonzero, finite constant far away from the core, J φ drops back to zero at some distance from the core. This reflects the fact that n s stays finite and nonzero whether there is external rotation or not, but the superfluid angular velocity v φ = n − a φ = n − 1 2 ρ 2 B will become zero at some ρ > 0 when there is external rotation.
For a two-band superfluid, we expect there to be two condensates circulating around the vortex core, and thus two coherence lengths, ξ i , corresponding to each condensate. The coherence length can be extracted from the condensate itself [26]: where O i (∞) denotes the asymptotic value of the condensate. In Fig 7, we show the dependence of the coherence lengths on and the temperature for the n = 1 vortex configuration. We see that coherence lengths increases as both and temperature increases, and as T approaches T c , the coherence length diverges as it should. We see also that the two coherence lengths are very close to each other throughout the range of we looked at, whether for small 0.1 or for close to c . We have checked that these features persist for other values of B, both above and below B c1 . In Fig 7, close to T c the coherence lengths have the form ξ i (T ) = 0.2604(1 − T /T c ) −1/2 , which is the expected temperature dependence from the GL theory. Another feature we see immediately is that the two coherence lengths differ very little from each other (barring numerical errors). Close to T c , this is expected from the GL theory. But it is surprising to find that this behavior persists down to low temperatures. A possible reason for this may be that the Josephson coupling is locking the growth and the saturation of the condensates together. We will investigate the mechanism behind this in future works.

D. Superconductor vortex
We consider now superconductor vortices. In this case, the external magnetic field B is dynamical, and we can thus see the screening of B. We show in the left panel of Fig. 8 the profile of the superconducting order parameters (scaled to have unit mass dimension) inside the superconductor for the n = 1 configuration at T = 0.937T c and = 0.05. The coherence lengths, ξ i , can be extracted as in the superfluid case using the form given in Eq. (23), and we show in the right panel of Fig. 8 their dependence on . We see that the two superconductor coherence lengths stay very close to each other throughout the range of we looked at.
We show in Fig. 9 the profile of the magnetic field B inside the superconductor for the n = 1 configuration at T = 0.937T c and = 0.05. The magnetic penetration length can be extracted from B = be −ρ/λ . At T = 0.937T c and = 0.05, we obtain ξ 1 = 1.02106, ξ 2 = 1.01662, and λ = 2.06235. Calculating the GL parameters, κ i = λ/ξ i , we get κ 1 = 2.01981 and κ 2 = 2.02862, which are within 0.4% to each other. Note that κ 1,2 > 1/ √ 2, which indicates that we have a type II superconductor. We show in Fig. 10 the temperature dependence of ξ i and λ. Near T c , we have good fits from ξ(T ) = 0.2495/ 1 − T /T c and λ(T ) = 0.4820/ 1 − T /T c . We see that there are very little difference between ξ 1 and ξ 2 down to T ∼ 0.5T c . Computing the GL parameter κ i for temperature range considered here, we find κ 1,2 > 1/ √ 2 over the entire range, indicating a type II superconductor down to T ∼ 0.5T c .

IV. SUMMARY AND OUTLOOK
In this paper, we have studied the magnetic response of a holographic two-band superconductor that has an interband Josephson coupling between the two bulk complex scalars. We have constructed the single vortex solution and study the effects of the Josephson coupling. By imposing appropriate boundary conditions, we can consider both superfluid and superconductor vortices. For superfluid vortices, we find one condensate is insensitive to the Josephson coupling when it is below 0.1. By comparing the free energy of n = 0 and n = 1 vortex configurations, we have estimated the first critical magnetic field. We have also extracted coherence lengths from the condensates for both the superfluid and superconductor cases, as well as the magnetic penetration length in the superconductor case where the magnetic field is dynamical, and we can see explicit screening. Near the critical temperature, we have checked that the temperature dependence of the coherence lengths are consistent with GL theory. Surprisingly, for both the superfluid and superconductor vortices we find there is effective only one coherence length in the range of parameters we consider, leading to the virtually the same GL parameter for both bands. Furthermore, the GL parameters are all greater than 1/ √ 2 for the whole temperature range that our numerics is reliable, indicating that our holographic two-band superconductor is type II, and the absence of type-1.5 superconductivity.
The paper is a fist step in the study of vortex dynamics in strongly-coupled/correlated multiband superconductors employing holography. There are many interesting future directions to take. An immediate one is to scan over a larger parameter space by going to larger Josephson coupling and different bulk scalar masses. Another would be to go beyond the static case studied here and construct dynamical vortex solutions. This would allow us to study interactions between vortices at different distances, and would allow a direct check on the dynamical mechanism of the purported type-1.5 superconductivity. It would also be very useful to generalize to a threeband model. There one can study the existence of chiral and time-reversal symmetry breaking state, interband phase difference induced domain walls, fractional quantum flux vortices [27,28] and frustrated superconductors [29]. Lastly, it would be interesting to clarify issues surrounding hidden criticality [30] using a holographic model of multiband superconductivity.