Non-relativistic Josephson junction from holography

Abstract

Abstract

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.

Published for SISSA by Springer Received: October 30, 2014 Accepted: November 21, 2014 Published: December 15, 2014

Non-relativistic Josephson junction from holography

Huai-Fan Li,a Li Li,b Yong-Qiang Wangc and Hai-Qing Zhangd

aInstitute of Theoretical Physics, Department of Physics,

Shanxi Datong University, Datong, 037009, China

bCrete Center for Theoretical Physics, Department of Physics, University of Crete, 71003 Heraklion, Greece

cInstitute of Theoretical Physics, Lanzhou University,

Lanzhou, 730000, China

dCFIF, Instituto Superior Tcnico, Universidade de Lisboa,

Av. Rovisco Pais, 1049-001 Lisboa, PortugalE-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We construct a Josephson junction in non-relativistic case with a Lifshitz geometry as the dual gravity. We investigate the e ect of the Lifshitz scaling in comparison with its relativistic counterpart. The standard sinusoidal relation between the current and the phase di erence is found for various Lifshitz scalings characterised by the dynamical critical exponent. We also nd 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.

Keywords: AdS-CFT Correspondence, Holography and condensed matter physics (AdS/CMT)

ArXiv ePrint: 1410.5578

cOpen Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP12(2014)099

Web End =10.1007/JHEP12(2014)099

JHEP12(2014)099

Contents

1 Introduction 1

2 The gravity setup 3

3 Numerical results 53.1 The case of z = 1 63.2 The case of z = 2 83.3 The case of z = 3 9

4 Conclusion and discussion 10

1 Introduction

Traditional condensed matter paradigms with weakly interacting quasiparticles are challenged by strongly correlated electron systems. One of the profound examples is the high temperature superconductivity. The basic idea of the BCS theory, like the weak-coupling mean eld approximation and phonon-mediated electron pairing mechanism is no longer applied without modications. Therefore, to develop new theoretical framework and concepts is desirable to understand those strongly coupled many-body systems. On the other hand, holography [13], as a framework to access the strongly coupled regime of quantum eld theory by its gravity dual living in a spacetime with higher dimensionality, has been useful in addressing the physical properties of strongly interacted condensed matter systems, such as high Tc cuprates and heavy feimions. Within this context, models for unconventional superconductors have been widely studied holographically. The rst holo-graphic superconductor, known as Abelian-Higgs model, has been introduced in ref. [4] in terms of a charged scalar eld in the bulk whose condensate corresponds to a s-wave superconducting order. This gravity setup was soon generalised to holographic p-wave models [57] and d-wave models [8, 9], see refs. [1012] for good reviews.1

Josephson junctions possess very important features in both theoretical and practical elds of superconductivity. A typical Josephson junction consists of two superconductors separated by a week contact. Depending on the specimen of the constituent superconductors and the nature of the contact, there are various kinds of junctions. The contact can be a normal conductor, an insulator, or a narrow superconductor. The corresponding junctions are referred to as SNS, SIS and SSS junctions, respectively. Moreover, the

1Holographic superconductors have been studied usually in the absence of dynamical electromagnetic elds, thus in the limit in which they coincide with holographic superuids. The dynamics of the electromagnetic eld is very relevant for, such as, the Meissner e ect and the exponential damping of the magnetic eld in vortices. The authors of ref. [13] explained for the rst time how to introduce a dynamical gauge eld in holographic superconductors.

1

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coupled superconductors can be of di erent types. The authors of ref. [14] constructed a holographic SNS junction by using the simplest holographic superconductor [4]. This junction exhibits the standard relation between the current J across the junction and the phase di erence of the condensate, i.e. J = Jmax sin( ). The dependence of the maximum current (or critical current) Jmax on the temperature and size of the junction also reproduces familiar results. Soon after, this setup has been generalised to other types of Josephson junctions [1519] as well as superconducting quantum interference device (SQUID) [20, 21]. A distinctly di erent way to construct a holographic model of Josephson junctions based on designer multi-gravity has been proposed in ref. [22] in which Josephson junction arrays were discussed.2

The above studies focused on gravity duals with asymptotic AdS boundary, which indicates that the dual theory is a relativistic conformal eld theory. However, there exist many scale-invariant systems without the Lorentz invariance especially near the critical points [24, 25]. In particular, the electrons in real materials are in general non-relativistic, thus it is natural to ask whether one can develop a similar model with non-relativistic kinematics. The Lifshitz geometry as a dual gravity is a very natural candidate to describe those non-relativistic theories. Lifshitz geometry is characterised by the so-called dynamical critical exponent z which governs the anisotropy between spatial and temporal scaling t ! zt, [vector]x ! [vector]x. The case z = 1 is nothing but the usual relativistic scaling. The Lifshitz

holography has been used to address various aspects of non-relativistic systems, such as strange metal transport [2628], thermalization [29], (non-)Fermi liquid [3032] and so on.

The purpose of the present work is to investigate the Josephson junction of the non-relativistic theory with the Lifshitz geometry as a dual gravity. We aim at the e ects due to the Lifshitz scaling in comparison with the relativistic case z = 1. More specically, we construct holographic junctions in the Lifshitz black branes with z = 1, 2 and 3. Following ref. [14], 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 di erential equations (PDEs). Taking advantage of the Chebyshev spectral methods to solve those PDEs numerically, we nd that the famous sinusoidal relation between the current and the phase di erence 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 [lscript]; meanwhile, the critical current Jmax also has an exponential decreasing relation to [lscript]. From the above exponential decreasing relations, one can extract the coherence length independently. In relativistic cases [1418], the value of the coherence length tted from critical current and condensate is consistent to each other within acceptable errors. However, for general z [negationslash]= 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

2Holographic Josephson junctions from D-branes have been considered in ref. [23] aiming at providing a geometrical picture for the holographic dual. Through this way non-Abelian Josephson junctions and AC Josephson e ect have been naturally realized.

2

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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.

2 The gravity setup

We adopt the black brane background in d+2 dimensional spacetime as [33]

ds2 = L2 r2zf(r)dt2 +

!, f(r) = 1

rz+d0rz+d , (2.1)

where z is the dynamical critical exponent, r0 is the radius of horizon, and d is the spacial dimension of the boundary. The asymptotical Lifshitz boundary is located at r ! 1. 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

T = z + d

4 rz0. (2.2)

In the probe limit of the above background, we consider the model of a U(1) gauge eld A[notdef] coupled a charged scalar eld . The corresponding action reads

S = [integraldisplay]

dd+2xpg [bracketleftbigg][notdef]r

d

dr2 r2f(r) + r2

Xi=1 dx2i

, (2.3)

in which F[notdef] is a U(1) gauge eld strength with F[notdef] = @[notdef]A @ A[notdef]. The equations of

motions (EoMs) can be obtained from the above action and the background as

0 = (r[notdef] iA[notdef])(r[notdef] iA[notdef]) m2 , (2.4) r F [notdef] = i[ (r[notdef] iA[notdef]) (r[notdef] + iA[notdef]) ]. (2.5)

We choose the ansatz of the elds as

= [notdef] [notdef]ei', A = Atdt + Ardr + Ax1dx1, (2.6)

where [notdef] [notdef], ', At, Ar, Ax1 are all real functions of r and x1. We would like to work with the

gauge-invariant combination M[notdef] = A[notdef] @[notdef]'.

Substituting the Lifshitz black brane background (2.1) and the ansatz (2.6) into the EoMs (2.4) and (2.5), we can obtain the following coupled PDEs:3

@2r[notdef] [notdef]+

1r4f @2x[notdef] [notdef]+(

d + z + 1r +

iA [notdef]2 m2[notdef] [notdef]2

14F[notdef] F [notdef] [bracketrightbigg]

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f[prime]

f )@r[notdef] [notdef]+

1 r2f (

M2t

r2zf r2fM2r

M2x

r2 L2m2)[notdef] [notdef] = 0, (2.7a)

@rMr + 1r4f @xMx +

2 (Mr@r[notdef] [notdef] +

Mx

r4f @x[notdef] [notdef]) + (

d + z + 1r +

f[prime]

f )Mr = 0, (2.7b)

2L2[notdef] [notdef]2r2f Mt = 0, (2.7c)

@2xMr @x@rMx 2L2r2[notdef] [notdef]2Mr = 0, (2.7d)

@2rMx @x@rMr + (

@2rMt + 1

r4f @2xMt +

d z + 1

r @rMt

f[prime]

f +

d + z 1

r )(@rMx @xMr)

2[notdef] [notdef]2

L2r2f Mx = 0, (2.7e)

3For convenience, we will dene x x1 in the following context.

3

where a prime [prime] denotes the derivative with respect to r. It is clear that the phase function ' has been absorbed into the gauge invariant quantity M[notdef]. The second equation (2.7b) is a constraint equation which can be obtained from the algebraic combinations of (2.7d) and (2.7e) as 2r2[notdef] [notdef]2 [notdef] Eq.(2.7b) + @r[Eq.(2.7d)] + @r[Eq.(2.7e)] + [f[prime]/f + (d + z 1)/r] [notdef]

Eq.(2.7d) 0. Therefore, in fact there are four independent EoMs with four elds, i.e., | [notdef], Mt, Mr and Mx.

In order to solve the above coupled PDEs, we need to impose suitable boundary conditions. First, we demand the regularity of the elds at the horizon. Since the metric component gtt is zero at the horizon, the eld Mt should be vanishing at the horizon, while other elds are nite at the horizon.

Near the innite boundary r ! 1, the elds [notdef] [notdef], Mr and Mx have the following

asymptotic expansions,

| [notdef] =

(1)(x)

r(z+dp(z+d)2+4m2)/2

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+ (2)(x) r(z+d+p(z+d)2+4m2)/2

+ O(

1

r(z+d+p(z+d)2+4m2)/2+1

),

Mr =M(1)r(x) rd+z1 + O(

1 rd+z ),

(2.8)

Mx = (x) + J(x) rd+z2 + O(

1 rd+z1 ).

However, the asymptotic behaviour of Mt is more sophisticated depending on the values of z and d,

Mt =(x) [notdef](x)log(r) + O(

1r ), for (d z = 0),

Mt =[notdef](x)

(x)

rdz + O(

1rdz+1 ), for (d z < 0 or 0 < d z < 2),

Mt =[notdef](x)

(x)

r2 +

@2x[notdef](x)

2r2 log(r) + O(

1r3 ), for (d z = 2),

Mt =[notdef](x)

(x) rdz +

@2x[notdef](x) 2(d z 2)r2

+ O(

1rdz+1 ), for (d z > 2).

(2.9)

p(z + d)2 + 4m2)/2.

In the following, we focus on the case (1) 0, which means there is no source term of

the dual scalar operator. We will always regard [notdef] as the chemical potential, although for z > d it is not the largest mode near the boundary [26]. According to the holographic dictionary, the coe cients (2), , and J correspond to the condensate of the dual scalar operator [angbracketleft]O[angbracketright], charge density, superuid velocity and current in the boundary eld theory,

respectively.4 Furthermore, the gauge invariant phase di erence = '

[integraltext]

Ax across the

The conformal dimension of the scalar eld [notdef] [notdef] is [notdef] = (z + d [notdef]

weak contact can be recast as [14]

=

dx[ (x) ([notdef]1)]. (2.10)

4We also notice that there is a relation @2xM(1)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.

[integraldisplay]

+1

1

4

In order to mimic a SNS Josephson junction, we choose the prole of the chemical potential [notdef](x) similar to that in ref. [14], which is given by

[notdef](x) = [notdef]1

(1 1 [epsilon1]2 tanh( [lscript]2 )

"tanh(x + [lscript]2 ) tanh(x [lscript]2 )

[bracketrightBigg][bracerightBigg]

, (2.11)

where [notdef]1 = [notdef](+1) = [notdef](1) is the chemical potential at x = [notdef]1, while [lscript], and [epsilon1]

control the width, steepness and depth of the junction, respectively.Note that the coupled PDEs (2.7) exhibit the following scaling symmetry:

t ! zt, xi ! xi, r !

1 r, Mt !

1 Mx, Mr ! Mr, (2.12)

with an arbitrary constant. Following ref. [14], we dene the critical temperature of the junction Tc identical to the critical temperature of a homogenous superconductor with vanishing current.5 Therefore, Tc is proportional to [notdef]1 = [notdef](+1) = [notdef](1) corresponding

to the scaling symmetry (2.12):

Tc = (z + d)rz04[notdef]c [notdef](1), (2.13)

where [notdef]c is the critical chemical potential for a homogenous superconductor without current at temperature T = z+d

4 rz0. Inside the junction, x ( [lscript]2, [lscript]2), the e ective critical

temperature reads

T0 = (z + d)rz04[notdef]c [notdef](0). (2.14)

Therefore, for T0 < T < Tc, 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.

3 Numerical results

We take advantage of the Chebyshev spectral methods [38] to numerically solve the EoMs (2.7a)(2.7e). We rst set r0 = 1 by using of the scaling symmetry (2.12). For the convenience, we also make the coordinate transformation in the following way u = 1/r and y = tanh( x4 ), as well as

| [notdef] ! [notdef] [notdef]

r(z+dp(z+d)2+4m2)/2

, (3.1)

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1 z Mt, Mx !

Mr

rd+z1 . (3.2)

In the following, we will consider the case with d = 2, but it can be straightforwardly generalised to other dimensions. Specically, 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

5Lifshitz holographic superconductors in homogenous case have studied, for example, in refs. [3437].

5

Mr !

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 m2 = 0, 3 and 6 with respect to z = 1, 2 and 3.

The values of critical chemical potential [notdef]c (or in the sense of the critical temperature Tc explained above) for the homogeneous superconductors are [notdef]c 7.5877, 9.0445 and 9.7667

with respect to (z, m2) = (1, 0), (2, 3) and (3, 6). Therefore, we choose a unied chemical

potential [notdef](x) for the junction with the parameters [notdef]1 = 10.5, = 0.7 and [epsilon1] = 0.7. The

prole of the chemical potential would satisfy the requirement of the Josephson junctions for the three cases.

Near the spacial boundary x = [notdef]1, we demand that all the elds are independent of

x because of the at [notdef](x) near [notdef]x[notdef] ! 1. There is also a symmetry of the elds when we

ip the sign of x ! x,

| [notdef] ! [notdef] [notdef], Mt ! Mt, Mr ! Mr, Mx ! Mx. (3.3)

Therefore, Mr is an odd function of x while others are even. Thus we can set Mr(x = 0) = 0, and other elds have vanishing rst order derivative with respect to x at x = 0. From the scaling symmetry (2.12) and the UV asymptotic expansions (2.8) and (2.9), it is easy to see that the quantities J/T (1+z)/zc and [angbracketleft]O[angbracketright]/T 3/zc are dimensionless.

3.1 The case of z = 1

For z = 1, the asymptotic expansion of Mt near the boundary is Mt [notdef](x) (x)r. It can be

easily calculated as before [14]. The relation between the current and the phase di erence is shown in gure. (1). The blue dots are for the data from numerical calculations while the red curve is tted by the sinusoidal relation. We can read from the plots that

J/T 2c 1.18436 sin( ), for z = 1. (3.4)

For our choosing parameters in gure. (1) the critical current is Jmax/T 2c 1.18436.

It has been uncovered in refs. [1418] that for the asymptotic AdS geometry the relation between the condensate within the barrier at zero current [angbracketleft]O[angbracketright]x=0 and the width of the

junction [lscript], as well as the relation between the maximal current (or critical current) Jmax

and the width of the junction [lscript] behave as

[angbracketleft]O[angbracketright]x=0/T 3/zc A1e

Those behaviour is in good agreement with condensed matter physics [39], according to which is identied as the normal metal coherence length.

We indeed reproduce similar results. On the left panel of gure. (2), we plot the relation of [angbracketleft]O[angbracketright]x=0/T 3/zc and [lscript] for z = 1 and nd that they satisfy a decreasing exponential

relation as

[angbracketleft]O[angbracketright]x=0/T 3c 253.896 [notdef] e

6

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[lscript]2 , (3.5)

[lscript]

. (3.6)

Jmax/T (1+z)/zc A0e

[lscript]21.49478 , for z = 1. (3.7)

1.0

z 1

0.5

J[Slash1]T c

2

0.0

[Minus]0.5

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[Minus]1.0

[Minus]1.5 [Minus]1.0 [Minus]0.5 0.0 0.5 1.0 1.5

Figure 1. Relation between J/T (1+z)/zc and for z = 1. The dots are from the numerics while the the red line is the tted sin curves of these dots. We use [notdef]1 = 10.5, [lscript] = 3, = 0.7 and [epsilon1] = 0.7.

2.0 2.5 3.0 3.5 4.0 4.5

2.5

120

z[Equal]1

z[Equal]1

2.0

[Less]O[Greater] x [Equal] 0[Slash1]T c

3

J max[Slash1]T c

2

100

1.5

80

1.0

60

0.5

2.0 2.5 3.0 3.5 4.0 4.5

l

l

Figure 2. [angbracketleft]O[angbracketright]x=0/T 3/zc (left) and Jmax/T (1+z)/zc (right) as functions of [lscript] for z = 1. The

parameters are [notdef]1 = 10.5, = 0.7, [epsilon1] = 0.7 and 2 [lscript] 4.4. The dots are from the numerics while

the red lines are the tted curves.

The relation between Jmax/T (1+z)/zc and [lscript] can be found on the right panel of gure. (2).

The tting curve satisfy the following relation,

Jmax/T 2c 11.2449 [notdef] e

[lscript]1.30389 , for z = 1. (3.8)

We can nd that for z = 1 the tted value of from the two relations (3.7) and (3.8) are very close to each other, with the error of 12.77%.6

6 Due to the limitation of numerics, one can not choose a too steep prole for (x). Thus the normal and superconducting phases in the junction are not cleanly separated. This is argued to justify the disagreement between the two estimates of [14].

7

0.02

0.01

0.00

[Minus]0.01

[Minus]0.02

z 2

z 3

1 z[RParen1][Slash1]z

J[Slash1]T c[LParen1]

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[Minus]1.5 [Minus]1.0 [Minus]0.5 0.0 0.5 1.0 1.5

Figure 3. The behaviour of J/T (1+z)/zc as a function of for z = 2 (green dots) and z = 3 (black

dots). The parameters we use are [notdef]1 = 10.5, [lscript] = 3, = 0.7 and [epsilon1] = 0.7. The dots are from the

numerics while the the red lines are the best t sin curves of these dots.

3.2 The case of z = 2

For z = 2, the asymptotic expansion of Mt near the boundary is Mt (x) [notdef](x) log(r).

For convenience of the numerical calculation, we make a transformation

Mt !

8

log(r) 1 1/r

Mt, (3.9)

The reason for dividing (1 1/r) in the denominator is that at the horizon r0 = 1, we need to impose the coe cients log(r)/(1 1/r) be non-vanishing, thus the new elds Mt

at horizon can have a specic vanishing boundary condition. This step of scaling Mt is

crucial for the numerics, and we nd it is much feasible for the codes.

The relation between the current and the phase di erence can be found in gure. (3) in which the green dots are from the numerics while the red line is the best tted curve. In this case, the asymptotic behaviour is much more di erent from the previous one. However, we nd that the famous sinusoidal relation between current and phase di erence is still satised very well. The numerical calculation shows that

J/T 3/2c 0.02372 sin( ), for z = 2. (3.10)

Let us consider the behaviour of the condensate at the centre of the contact at zero current. As one can see in gure (4) that the condensate as a function of the length of the link can be tted very well by the exponential decreasing function, which reads

[angbracketleft]O[angbracketright]x=0/T 3/2c 11.6122 [notdef] e

[lscript]2o , o 0.72835 for z = 2. (3.11)

3.0

2.5

2.0

1.5

1.0

0.5

0.0

z[Equal]2

z[Equal]3

3[Slash1]z

[Less]O[Greater] x [Equal] 0[Slash1]T c

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2.0 2.5 3.0 3.5 4.0 4.5

l

Figure 4. Relations between [angbracketleft]O[angbracketright]x=0/T 3/zc at zero current and [lscript] for Lifshitz scaling. The upper

curve and the lower curve correspond to z = 2 and z = 3, respectively. In both cases, the points

are from numerics and red lines are tted curves. We choose [notdef]1 = 10.5, = 0.7 and [epsilon1] = 0.7.

Compared to the relation (3.5), maybe with a little abuse of terminology, this result encourages us to identify o as the coherence length.7

The dependence of Jmax on [lscript] is shown in gure (5). Once again, this cure can be tted by an the exponential decreasing relation

Jmax/T 3/2c 1.10613 [notdef] e

Comparing to (3.6), one may also consider j as the coherence length. We can see that the discrepancy between the value of coherence length obtained from (3.11) and (3.12) is consistent with each other within the error 6.9%.

3.3 The case of z = 3

For z = 3, the asymptotic expansion of Mt near the boundary is Mt [notdef](x) (x)r. In

this case, we introduce a transformation

Mt ! rMt, (3.13) in our numerical calculation. Note that although now [notdef](x) is the subleading term in the expansion, we can still regard it as the chemical potential according to the explanation in ref. [26].

The relation between the current and the phase di erence is again shown in gure. (3), where the black dots are from the numerics while the red line is the best tted curve using sinusoidal function. It satises the relation as

J/T 4/3c 0.01906 sin( ), for z = 3. (3.14)

7A priori, Lifshitz case would be di erent form its AdS counterpart. To stress this issue, we use o and j to denote di erent coherence lengths extracted form the condensate and critical current, respectively.

9

[lscript]

j , j 0.782017 for z = 2. (3.12)

0.08

0.06

0.04

0.02

0.00

z[Equal]2

z[Equal]3

1[Plus]z[RParen1][Slash1]z

J max[Slash1]T c[LParen1]

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2.0 2.5 3.0 3.5 4.0 4.5

l

Figure 5. Relations between Jmax/T (1+z)/zc and [lscript] 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 t curves. We use [notdef]1 = 10.5, = 0.7 and [epsilon1] = 0.7 in the plot.

From above relation one can read o the maximum current Jmax 0.01906 for the choosing parameters.

Meanwhile, the dependence of [angbracketleft]O[angbracketright]x=0 at zero current on [lscript] as well as Jmax on [lscript] can be

found in gure. (4) and gure. (5), respectively. Both can be tted very well by exponential decreasing functions and the nal results read

[angbracketleft]O[angbracketright]x=0/Tc 1.9632 [notdef] e

Surprisingly, we see that the values of o and j exhibit enormous discrepancy. This large discrepancy can neither be explained by numerical error, nor by the disagreement corresponding to the prole of [notdef](x) as commented in footnote (6). We shall discuss this issue in the next section.

4 Conclusion and discussion

In this work, we holographically studied the properties of SNS Josephson junction in non-relativistic 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 elds behave distinctly from each other for di erent 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 (2.7) successfully. We found that the famous sinusoidal relations between the current J and

10

[lscript]2o , o 0.740657 for z = 3, (3.15)

[lscript]

j , j 1.73365 for z = 3. (3.16)

Jmax/T 4/3c 0.110907 [notdef] e

phase di erence across the weak link still exists for various z. Furthermore, our results showed that there is indeed an exponential decreasing relation between the condensate [angbracketleft]O[angbracketright]

at the middle of the barrier with vanishing current and the width of the link [lscript]. Similar relation also holds between the critical current Jmax and [lscript]. As a consistent check for our numerics, let us consider the behaviour of the coe cient A1 form the relation (3.5) with respect to z. Note that we set the same prole of the chemical potential [notdef](x) for various z and the critical chemical potentials [notdef]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 [notdef]c if z is much larger. Hence the condensate [angbracketleft]O[angbracketright] at x = 0 would be

much smaller as z increases. This would in turn made A1 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 holo-graphic junction in AdS-Schwarzschild black brane. Similar to z = 1 case, we found that for z = 2 the coherence length obtained from the condensate within the link (see equation (3.11)) and the one from the critical current (see equation (3.12)) were close to each other within acceptable error. However, for z = 3 the 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 s consistent. One should keep in mind that the relations (3.11) and (3.12) 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 !g from the optical conductivity in the superconducting phase, and one can also read o 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 !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 !g = 2 g, while it does not hold in holographic setup [40, 41], 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 , another important exponent in low energy physics of condensed matter system. In cases with general and z, the dual non-relativistic theory is not even scale invariant, qualitatively di erent 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 di erent. 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 su ciently close to

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critical temperature or su ciently high and wide potential barriers between two superconductors [42, 43]. So far, similar studies initiated from ref. [14] all produced the sinusoidal relation between current and phase di erence, 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 elds to the geometry [44]. 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.

Acknowledgments

The authors are grateful to Rong-Gen Cai for his valuable discussions, and the hospitality and partial support from the Kavli Institute for Theoretical Physics China (KITPC) during the completion of this work. HFL was supported in part by the Young Scientists Fund of the National Natural Science Foundation of China (No.11205097), the Program for the Innovative Talents of Higher Learning Institutions of Shanxi, the Natural Science Foundation for Young Scientists of Shanxi Province, China (No.2012021003-4) and the Shanxi Datong University doctoral Sustentation Fund(No.2011-B-04), China; LL was supported in part by European Unions Seventh Framework Programme under grant agreements (FP7-REGPOT-2012-2013-1) no 316165, the EU-Greece program Thales MIS 375734 and was also co-nanced by the European Union (European Social Fund, ESF) and Greek national funds through the Operational Program Education and Lifelong Learning of the National Strategic Reference Framework (NSRF) under Funding of proposals that have received a positive evaluation in the 3rd and 4th Call of ERC Grant Schemes; HQZ was partially supported by a Marie Curie International Reintegration Grant PIRG07-GA-2010-268172 and the Young Scientists Fund of the National Natural Science Foundation of China (No.11205097).

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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