Non-relativistic Josephson Junction from Holography

We construct a Josephson junction in non-relativistic case with a Lifshitz geometry as the dual gravity. We investigate the effect of the Lifshitz scaling in comparison with its relativistic counterpart. The standard sinusoidal relation between the current and the phase difference is found for various Lifshitz scalings characterised by the dynamical critical exponent. We also find the exponential decreasing relation between the condensate of the scalar operator within the barrier at zero current and the width of the weak link, as well as the relation between the critical current and the width. Nevertheless, the coherence lengths obtained from two exponential decreasing relations generically have discrepancies for non-relativistic dual.


Introduction
The purpose of the present work is to investigate the Josephson junction of the nonrelativistic theory with the Lifshitz geometry as a dual gravity. We aim at the effects due to the Lifshitz scaling in comparison with the relativistic case z = 1. More specifically, we construct holographic junctions in the Lifshitz black branes with z = 1, 2 and 3. Following ref. [13], we consider the Abelian-Higgs model for holographic superconductors with inhomogeneous boundary conditions breaking translational invariance. This model typically require us to solve complicated coupled partial differential equations (PDEs). Taking advantage of the Chebyshev spectral methods to solve those PDEs numerically, we find that the famous sinusoidal relation between the current and the phase difference across the weak link do exist no matter what z is. The condensate of the operator at zero current in the middle of the link has an exponential decreasing relation with respect to the width of the link ; Meanwhile, the critical current J max also has an exponential decreasing relation to . From the above exponential decreasing relations, one can extract the coherence length ξ independently. In relativistic cases [13,14,15,16,17], the value of the coherence length ξ fitted from critical current and condensate is consistent to each other within acceptable errors. However, for general z = 1, this result is violated. A typical example exhibiting this violation is the case with z = 3.
The paper is organised in the following: In Section (2) we derive the equations of motions in the Lifshitz black brane background; We show our numerical technique for dealing with non-trivial boundary conditions and numerical results in Section (3); In Section (4), we draw our conclusion and give some comments to the Lifshitz Josephson junction.

The Gravity Setup
We adopt the black brane background in d+2 dimensional spacetime as [31] where z is the dynamical critical exponent, r 0 is the radius of horizon, and d is the spacial dimension of the boundary. The asymptotical Lifshitz boundary is located at r → ∞. This geometry for z = 1 is nothing but the AdS-Schwarzschild black brane, while it is a gravity dual with the Lifshitz scaling as z > 1. The Hawking temperature of this black brane is In the probe limit of the above background, we consider the model of a U (1) gauge field A µ coupled a charged scalar field ψ . The corresponding action reads in which F µν is a U (1) gauge field strength with F µν = ∂ µ A ν − ∂ ν A µ . The equations of motions (EoMs) can be obtained from the above action and the background as We choose the ansatz of the fields as where |ψ|, ϕ, A t , A r , A x 1 are all real functions of r and x 1 . We would like to work with the gauge-invariant combination M µ = A µ − ∂ µ ϕ. Substituting the Lifshitz black brane background (1) and the ansatz (6) into the EoMs (4) and (5), we can obtain the following coupled PDEs: 2 where a prime denotes the derivative with respect to r. It is clear that the phase function ϕ has been absorbed into the gauge invariant quantity M µ . The second equation (7b) is a constraint equation which can be obtained from the algebraic combinations of (7d) and (7e) as Therefore, in fact there are four independent EoMs with four fields, i.e., |ψ|, M t , M r and M x . In order to solve the above coupled PDEs, we need to impose suitable boundary conditions. First, we demand the regularity of the fields at the horizon. Since the metric component g tt is zero at the horizon, the field M t should be vanishing at the horizon, while other fields are finite at the horizon.
Near the infinite boundary r → ∞, the fields |ψ|, M r and M x have the following asymptotic expansions, ), However, the asymptotic behaviour of M t is more sophisticated depending on the values of z and d, 2 For convenience, we will define x ≡ x1 in the following context.
The conformal dimension of the scalar field |ψ| is ∆ ± = (z + d ± (z + d) 2 + 4m 2 )/2. In the following, we focus on the case ψ (1) ≡ 0, which means there is no source term of the dual scalar operator. We will always regard µ as the chemical potential, although for z > d it is not the largest mode near the boundary [24]. According to the holographic dictionary, the coefficients ψ (2) , ρ, ν and J correspond to the condensate of the dual scalar operator O , charge density, superfluid velocity and current in the boundary field theory, respectively. 3 Furthermore, the gauge invariant phase difference γ = ∆ϕ − A x across the weak contact can be recast as [13] In order to mimic a SNS Josephson junction, we choose the profile of the chemical potential µ(x) similar to that in ref. [13], which is given by where µ ∞ = µ(+∞) = µ(−∞) is the chemical potential at x = ±∞, while , σ and control the width, steepness and depth of the junction, respectively. Note that the coupled PDEs (7) exhibit the following scaling symmetry: with λ an arbitrary constant. Following ref. [13], we define the critical temperature of the junction T c identical to the critical temperature of a homogenous superconductor with vanishing current. 4 Therefore, T c is proportional to µ ∞ = µ(+∞) = µ(−∞) corresponding to the scaling symmetry (12): where µ c is the critical chemical potential for a homogenous superconductor without current at temperature T = z+d 4π r z 0 . Inside the junction, x ∼ (− 2 , 2 ), the effective critical temperature reads Therefore, for T 0 < T < T c , the in-between junction is in the normal metallic phase, while the region outside the junction is in the superconducting phase. It is in this way one models the SNS Josephson junction by holography.

Numerical Results
We take advantage of the Chebyshev spectral methods [36] to numerically solve the EoMs (7a)-(7e). We first set r 0 = 1 by using of the scaling symmetry (12). For the convenience, we also 3 We also notice that there is a relation r (x) + (d + z − 2)∂xJ(x) = 0, which can be used to set J=const by imposing ∂xM (1) r = 0, in the numerical calculations in the next section. 4 Lifshitz holographic superconductors in homogenous case have studied, for example, in refs. [32,33,34,35]. make the coordinate transformation in the following way u = 1/r and y = tanh( x 4σ ), as well as |ψ| → |ψ| In the following, we will consider the case with d = 2, but it can be straightforwardly generalised to other dimensions. Specifically, we choose the dynamical critical exponent z as z = 1, 2 and 3. It is well-known that z = 1 is no other than the relativistic dual while z = 2 and z = 3 are for the non-relativistic theories. Physically, we would like to investigate the properties of the Josephson junctions with the same conformal dimension of each dual scalar operator, hence we set ∆ + = 3 as we vary z. Therefore, in this sense the mass square are m 2 = 0, −3 and −6 with respect to z = 1, 2 and 3. The values of critical chemical potential µ c (or in the sense of the critical temperature T c explained above) for the homogeneous superconductors are µ c ≈ 7.5877, 9.0445 and 9.7667 with respect to (z, m 2 ) = (1, 0), (2, −3) and (3, −6). Therefore, we choose a unified chemical potential µ(x) for the junction with the parameters µ ∞ = 10.5, σ = 0.7 and = 0.7. The profile of the chemical potential would satisfy the requirement of the Josephson junctions for the three cases.
Near the spacial boundary x = ±∞, we demand that all the fields are independent of x because of the flat µ(x) near |x| → ∞. There is also a symmetry of the fields when we flip the sign of x → −x, Therefore, M r is an odd function of x while others are even. Thus we can set M r (x = 0) = 0, and other fields have vanishing first order derivative with respect to x at x = 0. From the scaling symmetry (12) and the UV asymptotic expansions (8) and (9), it is easy to see that the quantities J/T

The case of z = 1
For z = 1, the asymptotic expansion of M t near the boundary is M t ∼ µ(x) − ρ(x) r . It can be easily calculated as before [13]. The relation between the current and the phase difference is shown in figure. (1). The blue dots are for the data from numerical calculations while the red curve is fitted by the sinusoidal relation. We can read from the plots that J/T 2 c ≈ 1.18436 sin(γ), for z = 1.
For our choosing parameters in figure. (1) the critical current is J max /T 2 c ≈ 1.18436. It has been uncovered in refs. [13,14,15,16,17] that for the asymptotic AdS geometry the relation between the condensate within the barrier at zero current O x=0 and the width of the junction , as well as the relation between the maximal current (or critical current) J max and the width of the junction behave as  Those behaviour is in good agreement with condensed matter physics [37], according to which ξ is identified as the normal metal coherence length.
We indeed reproduce similar results. On the left panel of figure. (2), we plot the relation of O x=0 /T 3/z c and for z = 1 and find that they satisfy a decreasing exponential relation as The relation between J max /T (1+z)/z c and can be found on the right panel of figure. (2). The fitting curve satisfy the following relation, We can find that for z = 1 the fitted value of ξ from the two relations (21) and (22) are very close to each other, with the error of 12.77%. 5

The case of z = 2
For z = 2, the asymptotic expansion of M t near the boundary is M t ∼ ρ(x) − µ(x) log(r). For convenience of the numerical calculation, we make a transformation The reason for dividing (1 − 1/r) in the denominator is that at the horizon r 0 = 1, we need to impose the coefficients log(r)/(1 − 1/r) be non-vanishing, thus the new fields M t at horizon can have a specific vanishing boundary condition. This step of scaling M t is crucial for the numerics, and we find it is much feasible for the codes. The relation between the current and the phase difference can be found in figure. (3) in which the green dots are from the numerics while the red line is the best fitted curve. In this case, the asymptotic behaviour is much more different from the previous one. However, we find that the famous sinusoidal relation between current and phase difference is still satisfied very well. The numerical calculation shows that Let us consider the behaviour of the condensate at the centre of the contact at zero current. As one can see in figure (4) that the condensate as a function of the length of the link can be fitted very well by the exponential decreasing function, which reads Compared to the relation (19), maybe with a little abuse of terminology, this result encourages us to identify ξ o as the coherence length. The dependence of J max on is shown in figure (5). Once again, this cure can be fitted by an the exponential decreasing relation Comparing to (20), one may also consider ξ j as the coherence length. We can see that the discrepancy between the value of coherence length ξ obtained from (25) and (26) is consistent with each other within the error 6.9%.

The case of z = 3
For z = 3, the asymptotic expansion of M t near the boundary is M t ∼ µ(x) − ρ(x)r. In this case, we introduce a transformation in our numerical calculation. Note that although now µ(x) is the subleading term in the expansion, we can still regard it as the chemical potential according to the explanation in ref. [24]. and for Lifshitz scaling. The upper curve and the lower curve correspond to z = 2 and z = 3, respectively. In both cases, the points come from numerics and red lines are best fit curves. We use µ ∞ = 10.5, σ = 0.7 and = 0.7 in the plot.
The relation between the current and the phase difference is again shown in figure. (3), where the black dots are from the numerics while the red line is the best fitted curve using sinusoidal function. It satisfies the relation as J/T 4/3 c ≈ 0.01906 sin(γ), for z = 3.
From above relation one can read off the maximum current J max ≈ 0.01906 for the choosing parameters.
Meanwhile, the dependence of O x=0 at zero current on as well as J max on can be found in figure. (4) and figure. (5), respectively. Both can be fitted very well by exponential decreasing functions and the final results read Surprisingly, we see that the values of ξ o and ξ j exhibit enormous discrepancy. This large discrepancy can neither be explained by numerical error, nor by the disagreement corresponding to the profile of µ(x) as commented in footnote (5). We shall discuss this issue in the next section.

Conclusion and Discussion
In this work, we holographically studied the properties of SNS Josephson junction in nonrelativistic case with Lifshitz scaling. It can be carried out in terms of the Abelian-Higgs model [4] coupled with an asymptotic Lifshitz black brane solution in gravity side. Due to the presence of the dynamical critical exponent z, the asymptotic expansions of the fields behave distinctly from each other for different z. Therefore, it was expected that the properties of the Josephson junctions would depend on z as well. By virtue of the Chebyshev spectral methods, we could solve the coupled PDEs (7) successfully. We found that the famous sinusoidal relations between the current J and phase difference γ across the weak link still exists for various z. Furthermore, our results showed that there is indeed an exponential decreasing relation between the condensate O at the middle of the barrier with vanishing current and the width of the link . Similar relation also holds between the critical current J max and . As a consistent check for our numerics, let us consider the behaviour of the coefficient A 1 form the relation (19) with respect to z. Note that we set the same profile of the chemical potential µ(x) for various z and the critical chemical potentials µ c for a homogeneous superconductor would increase with z increased. Therefore, the chemical potential at x = 0 in the link would be much smaller compared to µ c if z is much larger. Hence the condensate O at x = 0 would be much smaller as z increases. This would in turn made A 1 smaller when z becomes bigger. This is nothing but we found in our numerical calculation.
In order to compare with the relativistic case z = 1, we also reproduced the holographic junction in AdS-Schwarzschild black brane. Similar to z = 1 case, we found that for z = 2 the coherence length ξ obtained from the condensate within the link (see equation (25)) and the one from the critical current (see equation (26)) were close to each other within acceptable error. However, for z = 3 the ξ's got from the two relations were no longer consistent. Although we calculated the case of z = 3 by using much higher precision, we still could not render the two ξ's consistent. One should keep in mind that the relations (25) and (26) are deduced form conventional superconductivity under some additional approximations. In contrast, our holographic construction is, in principle, only applicable for the superconductors at strong coupling, thus, a priori, would far deviate form the conventional one which is weakly interacted. An instructive example is to consider the well-known Abelian-Higgs model in AdS case. As the temperature decreases, there exists a gap frequency ω g from the optical conductivity in the superconducting phase, and one can also read off the energy gap ∆ g from the low temperature behaviour of normal contribution to the DC conductivity. In a standard weak coupling picture of superconductivity, the gap ω g is understood as the energy required to break a Cooper pair into its constitutive electrons and the energy of the constituent quasiparticles is given by ∆ g . In BCS theory ω g = 2∆ g , while it does not hold in holographic setup [38,39], indicating that we are clearly not in a weak coupling regime and that such a quasiparticle picture is not applicable. Therefore, we expect that our results for Josephson junctions with Lifshitz scaling may suggest new mechanism compared to AdS case. It is desirable to understand our results form condensed matter theory point view. It will be interesting to see whether one can construct, for example, a Landau-Ginzberg like theory with Lifshitz scaling that exhibits similar deviation in this paper.
Similar discussion can be straightforwardly generalized to include hyperscaling violation characterised by θ, another important exponent in low energy physics of condensed matter system. In cases with general θ and z, the dual non-relativistic theory is not even scale invariant, qualitatively different from its Lifshitz counterpart. Nevertheless, we expect that the main results would be similar to the Lifshitz geometry. There are various kinds of junctions, and the properties of these junctions can be considerably different. It is, however, known that a sinusoidal current-phase relation is only a special case in Josephson tunneling, which is attainable only for such as temperatures sufficiently close to critical temperature or sufficiently high and wide potential barriers between two superconductors [40,41]. So far, similar studies initiated from ref. [13] all produced the sinusoidal relation between current and phase difference, including the case with Lifshitz scaling in current paper. To obtain the non-sinusoidal relation, one is suggested to consider the case with much lower temperatures. Therefore, it is natural to include the back reaction of matter fields to the geometry [42]. It will be also interesting to extend similar study to other types of junctions and to cases with competing orders. We hope to report related issues in the future.