Josephson Junctions and AdS/CFT Networks
aa r X i v : . [ h e p - t h ] O c t Preprint typeset in JHEP style - HYPER VERSION
CCTP-2011-13
Josephson Junctions and AdS/CFT Networks
Elias Kiritsis , and Vasilis Niarchos † Crete Center for Theoretical Physics,Department of Physics, University of Crete, 71003, Greece; Laboratoire APC, Universit´e Paris-Diderot Paris 7, CNRS UMR 7164,10 rue Alice Domon et L´eonie Duquet, 75205 Paris Cedex 13, France; † [email protected] Abstract:
We propose a new holographic model of Josephson junctions (and networksthereof) based on designer multi-gravity, namely multi-(super)gravity theories on productsof distinct asymptotically AdS spacetimes coupled by mixed boundary conditions. Wepresent a simple model of a Josephson junction (JJ) that exhibits the well-known current-phase sine relation of JJs. In one-dimensional chains of holographic superconductors wefind that the Cooper-pair condensates are described by a discretized Schr¨odinger-type equa-tion. Such non-integrable equations, which have been studied extensively in the past incondensed matter and optics applications, are known to exhibit complex behavior that in-cludes periodic and quasiperiodic solutions, chaotic dynamics, soliton and kink solutions.In our setup these solutions translate to holographic configurations of strongly-coupledsuperconductors in networks with weak site-to-site interactions that exhibit interestingpatterns of modulated superconductivity. In a continuum limit our equations reduce togeneralizations of the Gross-Pitaevskii equation. We comment on the many possible ex-tensions and applications of this new approach.
Keywords:
Superconductivity, Josephson junctions, Josephson junction networks,AdS/CFT, designer (multi)gravity, discrete dynamical systems, complex dynamics. ontents
1. Introduction 22. Networks of large- N QFTs and designer multigravity 4 N QFTs 42.2 Networks of asymptotically AdS spacetimes 72.3 Designer multigravity 82.4 A holographic superconductor with vanishing charge density 11
3. A holographic model of Josephson junctions 12
4. Holographic Josephson junction arrays 17
5. Outline of future directions 31Acknowledgements 33Appendices 33A. On the linear stability of DNLS 33References 35 – 1 – . Introduction
There has been recent interest in the potential applicability of AdS/CFT-inspired methodsto traditional condensed matter problems, which are not amenable to a weak couplingquasiparticle description. This has led to the formulation and study of a large varietyof models in classical gravity which aspire to capture holographically the characteristicfeatures of some condensed matter system. A notable example is [1, 2] which providesa simple gravity dual for an s-wave superconductor. More recent developments in thissubject are reviewed in [3, 4, 5]. Ultimately one hopes that holographic techniques willprovide a new efficient description of high- T c superconductors that goes well beyond theBCS theory.Many technological applications of superconductors and superconducting devices in-volve Josephson junctions (JJs). The basic junction consists of two superconductors sep-arated by a weak link. The precise type of JJ depends on the specifics of the constituentsuperconductors and the nature of the link. The link can be an insulator (SIS junctions),a normal conductor (SNS junctions) or another superconductor. The coupled supercon-ductors can be of the same or different type. For example, one can consider sIs , sId , or dId junctions ( s denoting an s-wave superconductor, d a d-wave superconductor and I aninsulator). The properties of these junctions can be considerably different. For instance,quantum tunneling in conventional SNS and SIS junctions implies a current I across thelink, even in the absence of external voltage, which depends on the phase difference ϑ ofthe condensates of the two superconductors in the following way [6] I = I max sin ϑ . (1.1)This simple sinusoidal relation can be substantially different in other (more general) typesof junctions (see for example [7, 8, 9, 10, 11]).Another reason to be interested in JJs is the nature of high- T c superconductivity itself.Many high- T c superconductors enjoy a layered structure [12] that can be viewed as a naturalstack of atomic scale intrinsic JJs [13] with interlayer spacing of about 15.5 ˚A. In fact,such high quality SIS-type intrinsic JJs can be fabricated [14, 15] and pose as attractivecandidates of cryoelectronics.Therefore, assuming holography can provide a new window to high- T c superconduc-tor physics, there is an obvious interest to construct the holographic dual of Josephsonjunctions and more generally the dual of Josephson junction networks (JJNs).A first step towards the construction of a holographic SNS junction has been takenrecently in Ref. [16] (see also [17, 18, 19, 20, 21, 22] for related setups). In this approach– 2 –ne is looking for solutions of the equations of motion of the standard holographic su-perconductor setups that are inhomogeneous across one of the field theory directions (thedirection along which the SNS stack is arranged). In the present paper we will proposea distinctly different way to construct a holographic Josephson junction (one that is notnecessarily restricted to SNS types).The basic idea is to view each of the superconductors that compose the junction asa separate supergravity (or superstring) theory on its own asymptotically AdS spacetimeand to model the weak link between them as a mixed boundary condition that relates theboundary conditions of the condensing symmetry-breaking field on one spacetime to theboundary conditions of the condensing symmetry-breaking field on the other spacetime.This may look like a contrived operation on the gravitational side but it is a rather naturalone from the perspective of a dual large- N quantum field theory. On the field theory sidethis operation amounts to a multi-trace deformation that involves products of single-traceoperators from both theories. This multi-trace interaction is the only term that mediatesinteractions between the two theories. The single-trace operators that appear in theseinteractions are charged under the broken U (1) symmetries and thus mimic naturally thecharge quantum tunneling effects that are present in a JJ.Using multi-trace interactions to model the tunneling effects is natural in a regimewhere the mass scales of the modes that mediate the interactions between the two super-conductor field theories in the full system are large compared to the typical energy scalesthat we consider. In that case one can integrate out these higher mass modes to obtain aneffective theory at low energies. The separate gauge invariance in each of the two boundarytheories implies that the effective interactions between them can only be of the multi-tracetype. Abstracting from this picture the main features of the bulk description we will pro-ceed to employ them freely in more generic situations where explicit knowledge about theboundary description is very limited or altogether absent.The precise ingredients of our construction are presented in section 2. A simple char-acteristic example of a holographic JJ at zero temperature is discussed in section 3. Weshow that the standard sine expression for the Josephson current (1.1) is naturally repro-duced in this model in a few lines and determine I max explicitly in terms of the parametersof the system. We briefly comment on the extensions and modifications that can alterthis standard current-phase relation. We also discuss how this system differs from a typi-cal Josephson junction and what kind of extensions can be used to describe more typicalJosephson junctions.Another appealing feature of the above approach is its versatility in describing very– 3 –eneral configurations of Josephson junction networks. JJNs is a much studied subjectwith diverse applications. The great wealth of possibilities that they pose and the greatreliability of the fabrication technologies developed for their construction makes them aprototype of complex physical systems that exhibits a variety of interesting physical be-haviors. Among their many applications: they are used widely as microwave sources (see e.g. [23]), they provide controllable settings to investigate properties of granular or high- T c superconductors [24], they are frequently used as model analogs of physical systemswith complex dynamics. For instance, they have been used to model biologically realisticneurons [25].As a first simple application of our proposal in this direction we consider in section4 a holographic Josephson junction array that can also be viewed as a special case of thehoney-comb network of Ref. [26]. Using the gravitational description we find that theCooper-pair condensates are described by a discretized Schr¨odinger-type equation, whichhas been studied extensively in the past (for a special value of one of our parameters) inradically different condensed matter and optics applications [27]. Using well known factsabout this equation we show that the system in question exhibits complex behavior that in-cludes periodic and quasiperiodic solutions, chaotic dynamics, solitons and kink solutions.In our setup these solutions translate to one-dimensional configurations of holographic su-perconductor layers that exhibit interesting patterns of modulated superconductivity. Ina continuum limit, the discretized Schr¨odinger equation becomes naturally a generaliza-tion of the Gross-Pitaevskii (GP) equation, a well-known long-wavelength description ofsuperfluids. In this limit we recover some of the previously discrete solutions analytically.We conclude in section 5 with an outline of further possible extensions and applications.
2. Networks of large- N QFTs and designer multigravity N QFTs
Consider a set (network) of k d -dimensional QFTs with a large- N limit —for example, k potentially different large- N conformal field theories (CFTs). We will label each CFTby an index i ( i = 1 , . . . , k ). Equivalently, i is an index that labels a site (vertex) in ournetwork. The links of the network are provided by interactions coupling the CFT i ’s witheach other. The only kind of coupling respecting the individual gauge structure of eachsite is one mediated by multi-trace operators, so this is the only kind that we will considerhere. For example, if O i is a single-trace operator in the CFT i , then a double-trace link– 4 – ertexlink (a) (c)(b) Figure 1:
Examples of simple networks. In our context, these graphs will represent large- N CFTs coupled by double-trace interactions. between the CFT i and the CFT j corresponds to a Lagrangian interaction of the form δ L = h O i O j . (2.1)Assuming that the large- N scaling of the single-trace operators O i goes like O ( N ), thedouble-trace interaction (2.1) respects the large- N expansion when h is taken to scale asa constant, namely h ∼ O ( N ). Similarly, an ℓ -trace coupling that involves the product Q ℓs =1 O i s should have a coefficient that scales as O ( N − ℓ ).It will be useful to set up a convenient notation to denote graphically networks con-structed in this way. For CFTs coupled by double-trace interactions, like that in eq. (2.1),we will denote the corresponding link by a single line. Such a link is undirected and joinstwo different CFTs, or circles back to the same CFT. The latter denotes that the corre-sponding CFT has itself a double-trace interaction turned on. Hence, Fig. 1(b) exhibits asimple network of two CFTs, call them CFT and CFT , with the Lagrangian interaction δ L = h O + h O O + h O . (2.2)Similarly, Fig. 1(c) exhibits nine CFTs coupled pairwise by double-trace interactions in allpossible ways. This example, analyzed in [28], plays a role in quenched disorder calculationsusing the replica trick [29]. – 5 – a) (b)(c) Figure 2:
Figures (a) and (b) exhibit a triple-trace and four-trace link respectively. In order tomake the notation more transparent we have denoted a multi-trace link using a double-line notationreserving the single-line notation for the simpler double-trace link. Accordingly, Figure (c) exhibitsfour CFTs linked pairwise with two double-trace interactions whereas Figure (b) exhibits a four-trace link.
Networks formulated in this way can have different types of links. We may use adouble-line notation to denote the more general possibility of links mediated by multi-traceinteractions. Figs. 2(a) and 2(b) exhibit a triple-trace and four-trace coupling respectively.In network literature the corresponding graphs are sometimes called hypergraphs. Theuse of such more general couplings opens up many interesting possibilities. Most of ourdiscussion in this paper will be focused, however, on networks with double-trace couplingsonly.Being comprised by sites that correspond to interacting QFTs the above networks havein general a rich and complicated internal structure. This structure can evolve in time,vary in space, or vary from site to site. Moreover, under renormalization group flow thenetwork graphs may change with the appearance of new links or even new sites. Indeed, itis well-known that multi-trace couplings are naturally generated under renormalization (seefor instance [28]). Hence, renormalization effects in field theory can affect the quantitativefeatures of the links. They can also change the number of sites in the following way. Many– 6 –upersymmetric QFTs, like the N = 4 super-Yang-Mills theory, have a Coulomb branch.At generic points of this branch the original gauge group is Higgsed into product gaugegroups and in the far infrared one is left with a product of QFTs giving rise to a networkwith more sites than those of the UV theory. This is an example of an RG flow witha different number of sites in the UV and IR. In what follows, when we draw a graphrepresenting a network we will implicitly assume that this description refers to the bareUV Lagrangian of the corresponding theory. There are situations where a large- N QFT (typically at strong coupling) has a dual descrip-tion in terms of a supergravity theory on an asymptotically AdS background. Accordingly,a network of such QFTs has a dual description as a multi-gravity network where each siteis some supergravity theory on an asymptotically AdS background and each link is a mixedboundary condition for supergravity fields residing on different space-times [30, 31, 28] (see[32] for a stringy setup that involves multi-string theory networks and [33] for a brief reviewof the main idea and its implications for massive multi-gravity). Let us recall how such amulti-gravity network comes about in the AdS/CFT correspondence. For clarity, we willrestrict to the case where the boundary QFTs are conformal.Before the addition of multi-trace links, the CFT i s are independent field theories thatdo not talk to each other and the dual gravitational theory is a direct product of super-gravity theories on product AdS space-times of the form Q ki =1 [ AdS i ⊗ M i ], where M i isthe internal manifold of the spacetime of the dual of CFT i .Assume that the single-trace operators O i stated above are scalar operators withscaling dimension ∆ i . The AdS/CFT correspondence maps each O i to a dual scalar field ϕ i with AdS asymptotics ϕ i ≃ α i r ∆ i i + . . . + β i r d − ∆ i i + . . . . (2.3) r i is the radial distance in the i -th AdS space that corresponds to CFT i . We use conventionswhere the i -th AdS boundary lies at r i → ∞ . In later applications, we will assume ∆ i < d ,in which case the α i ’s should be interpreted as the vacuum expectation values (VEVs) ofthe dual operators and the β i ’s as the sources. It is not necessary to restrict ourselves to scalar operators. We will make this assumption here forreasons of simplicity and concreteness. In fact, some of the applications of this framework that we willpropose later also involve vector operators. – 7 –n this framework, and to leading order in the 1 /N expansion, it is well known that amulti-trace coupling in the boundary CFT δ L = W ( O , . . . , O k ) (2.4)imposes mixed boundary conditions to the asymptotic coefficients α i , β i of the form β i = ∂ i W ( α , . . . , α k ) . (2.5)Combined with requirements of regularity these boundary conditions fix completely theprofile of the bulk solution. The inter-theory coupling induced by the relations (2.5) leadsto a network of supergravity theories that can exhibit interesting collective phenomena.We will explore these phenomena in section 4. Let us recall how multi-trace deformations affect the profile of the bulk asymptoticallyAdS solution in a single theory; a subject that usually goes under the title of designergravity [34]. In this paper we want to consider the natural extension of this framework tomulti-AdS spaces, which we will suggestively call ‘designer multigravity’.As one of the simplest illustrations of the idea consider the ‘network’ (2.6)which contains a single vertex with a self-adjoining link. The graph (2.6) exhibits a double-trace link, but we can equally well consider any multi-trace link. This theory has beenthe main focus of most previous investigations of multi-trace interactions in the AdS/CFTcorrespondence and designer gravity. For self-completeness and in order to set the notation,we briefly review some of the most pertinent properties of this system.At the single site of this network resides a d -dimensional large- N CFT with a super-gravity dual. Assume that the CFT has a single-trace complex scalar operator O withscaling dimension ∆. This operator is dual to a charged bulk scalar field ϕ . To furthersimplify the discussion we will also assume that we can consistently reduce the dynamicsof the dual bulk supergravity to a ( d + 1)-dimensional Einstein-Abelian Higgs model of theform S bulk = Z d d +1 x √− g (cid:20) R − G ( | ϕ | ) F − ( ∇| ϕ | ) − J ( | ϕ | ) ( ∇ θ − qA ) − V ( | ϕ | ) (cid:21) (2.7)– 8 –here θ is the phase of ϕ , namely ϕ = | ϕ | e iθ , and G, J, V are model-dependent functionsof | ϕ | . A is an Abelian gauge field in the bulk and F its field strength. q is the U (1) chargeof the field ϕ .We are interested in asymptotically AdS d +1 solutions of this system. In units wherethe AdS radius is set to one, the potential V ( ϕ ) has the small- ϕ expansion V ( | ϕ | ) = − d ( d −
1) + m | ϕ | + O ( | ϕ | ) + . . . . (2.8)When the mass m lies within the range m BF < m < m BF + 1 , m BF = − d O can have two possible scaling dimensions∆ ± = d ± r d m . (2.10)We will assume that our theory lies in the window (2.9) and will pick O to have the smallerscaling dimension ∆ ≡ ∆ − that satisfies the inequality d − < ∆ − < d (the lower bound inthis inequality is the standard unitarity bound in field theory).Near the asymptotic boundary, r → ∞ , the metric is that of AdS d +1 ds ≃ r dx µ dx µ + dr r (2.11)and the scalar field ϕ exhibits two independent branches ϕ ≃ αr ∆ + . . . + βr d − ∆ + . . . . (2.12)The boundary condition β = d W dα , (2.13)for a generic smooth function W ( α ) corresponds at the boundary CFT to the multi-tracedeformation [35] δ L = W ( O ) . (2.14)When we solve the bulk equations of motion we are looking for solutions that respectthe boundary conditions (2.13). In addition, we require that these solutions are every-where regular. It turns out that regularity imposes an extra constraint on the asymptoticcoefficients α , β . We will denote this additional relation as β = − d f W dα (2.15)– 9 –here f W is a function determined by the specific dynamics of the theory, e.g. the detailsof the scalar potential V ( | ϕ | ) in the bulk action (2.7).Combining the boundary condition (2.13) and the regularity condition (2.15) we findthat α and β are completely fixed and determined as a solution to the equation d V dα = 0 , V ( α ) = W ( α ) + f W ( α ) (2.16)which therefore can be viewed as an extremum of the function V .For example, for boost invariant, planar configurations with vanishing gauge field ds = r (cid:0) − dt + dx i dx i (cid:1) + dr g ( r ) , ϕ = ϕ ( r ) , A = 0 (2.17)one finds a solution with an acceptable naked singularity at r = 0 that has f W ( α ) = s ∆ d | α | d/ ∆ . (2.18)The existence of this solution and the precise value of the parameter s depends on thedetails of the bulk potential V ( ϕ ) (see [36] for additional information). For most potentials, s turns out to be a positive number. The energy density of this solution is EV ol = ( d − V . (2.19)One can also consider analogous solutions with spherical topology. It is possible toshow that the energy (2.19), (2.18) provides a lower energy bound for all of these solutions[36]. Moreover, in the case of vanishing gauge field, one can argue [36] that • the theory with boundary conditions β = d W dα has a stable ground state provided thefunction V has a global minimum V min , and • that the minimum energy solution is the spherical soliton associated with V min .Note that because of the presence of f W , it is possible to have a stable ground state evenfor functions W that have no minimum. In general, this minimum involves a condensateof the charged scalar that higgses the corresponding U (1).What we have discussed so far applies to the case of zero temperature. By study-ing hairy black holes in the bulk it is possible to generalize the discussion to non-zerotemperature [37]. It is also possible to consider the case of non-vanishing charge density[38].It is not hard to generalize this discussion to the case of multiple CFTs coupled togetherby a multi-trace interaction of the general form W ( O , . . . , O k ). In this case (and to leading– 10 –rder in the 1 /N expansion), the bulk equations of motion are the same as before in each(super)gravity member of the product, but the boundary conditions change. For example,the bulk boost-invariant planar solutions in the i -th spacetime still retain the form (2.17)and each β i is still given by an equation of the form β i = − d f W i ( α i ) dα i (2.20)with f W i ( α i ) = s i ∆ i d | α i | d/ ∆ i . (2.21)The new ingredient, responsible for the coupling between different AdS theories, lies in themixed boundary conditions (2.5). The analog of equation (2.16) that determines the VEVs α i is ∂ V ( α , α , . . . , α k ) ∂α i = 0 (2.22)with V = W ( α , . . . , α k ) + k X i =1 f W i ( α i ) . (2.23)In general, these equations lead to a non-linear discrete system which can exhibit intricatebehavior. This behavior includes solutions that can be periodic, quasi-periodic, chaoticor even soliton-like with energy pinned around a central site. Examples of each of thesebehaviors will be discussed in section 4. A novel type of holographic superconductor with symmetry breaking induced by double-trace deformations was recently proposed in [38]. Since this setup will provide the basicbuilding block of the discussion to come, it will be beneficial to recall some of its mainproperties.Returning to the single-site example of (2.6) consider the case of a double-trace defor-mation W = g |O| (2.24)implemented with the use of a single-trace operator O with scaling dimension ∆. Thefunction V in (2.16) becomes in this case (for boost-invariant planar solutions) V ( α ) = g | α | + sδ | α | δ , δ ≡ d ∆ . (2.25)The extrema of this function obey the algebraic equation α (cid:16) g + s | α | δ − (cid:17) = 0 . (2.26)– 11 –ssuming s >
0, there is one or two possible solutions to this equation depending onthe sign of the constant g . For g >
0, the only solution is α = 0, which is a minimum anddoes not exhibit any condensate of the field ϕ . For g <
0, there are two possible solutions α = 0 , α θ = (cid:18) − gs (cid:19) δ − e iθ . (2.27)The first one, α , is a local maximum of V and therefore an unstable vacuum of the system.The other solutions, α θ , are degenerate stable minima labeled by an angular variable θ .The non-vanishing condensate of the charged scalar field ϕ in this case higgses the U (1)gauge symmetry and leads to a new type of holographic superconductor.
3. A holographic model of Josephson junctions
A Josephson junction consists of two superconductors separated by a link that mediatesweak interactions between them (several possibilities for the weak link were reviewed inthe introduction). In analogy, consider a setup where two holographic superconductors(each described by a gravitational theory on an asymptotically AdS space) are interactingweakly via mixed boundary conditions on the boundary. From the gauge theory pointof view two initially separate large- N gauge theories are brought into contact via multi-trace interactions. As we discussed in the introduction, in certain cases one can think of themulti-trace interactions as an effective description below the typical mass gap associated tothe actual interactions between the two systems. In a real Josephson junction these wouldbe the weak tunneling interactions across the material in between the superconductors.As a concrete illustration of the idea consider a pair of two identical holographic super-conductors of the type described in the previous subsection 2.4. The parameters g, s, δ, d, ∆,which will be treated here as phenomenological parameters of the model, are chosen to becommon in the two systems. Each of them has a charged scalar field O i ( i = 1 ,
2) and acorresponding dual complex scalar field ϕ i with asymptotics (2.12). We can think of O i as the ‘Cooper pair’ operator in the holographic superconductor CFT i . Holographically,the VEV of these operators are given by the leading branch coefficients α i in the asymp-totic expansion (2.12). We assume that both operators have the same scaling dimension∆ < d and employ the alternative quantization for the dual scalar fields ϕ i in the bulkAdS spacetime. We will soon discuss how similar this system is to the typical Josephson junction considered in thelaboratory and the extensions it suggests. – 12 –n order to mediate an interaction that exchanges charge between the two theories wecouple them via a double-trace interaction of the form W ( O , O ) = h (cid:16) e iϑ O O † + e − iϑ O † O (cid:17) . (3.1) h is a real number controlling the strength of the interaction and ϑ is an angular variable.The latter is an additional tunable parameter of the interaction whose physical meaningwill become clear in a moment.In the bulk this coupling implies mixed boundary conditions. One can determinethe VEVs of the dual operators O i by solving the scalar-gravity equations of motion.Equivalently, one can extremize the function V (see eqs. (2.22), (2.23)). In the case athand V ( α , α ) = X i =1 (cid:16) g | α i | + sδ | α i | δ (cid:17) + h (cid:0) e iϑ α α ∗ + e − iϑ α ∗ α (cid:1) . (3.2)Extremizing with respect to α and α we obtain two algebraic equations gα + he − iϑ α + s α | α | δ − = 0 ,gα + he iϑ α + s α | α | δ − = 0 (3.3)which determine the α i ’s uniquely in terms of the parameters g, h, s and δ up to an overallphase common in α and α . The relative phase between α and α is fixed in terms of ϑϑ ≡ ϑ − ϑ = ϑ mod π . (3.4)For instance, when δ = 4 the solutions are(1) α = 0 , (2) ± | α | = 2 s ( ± h − g ) , (3) ± | α | = − s (cid:16) g ± p g − h (cid:17) . (3.5)In all cases α = − h − e iϑ (cid:16) g + s | α | (cid:17) α . (3.6)Which solutions survive in cases (2) and (3) depends on the specific range of the parameters s, g, h . Assuming s > • | h | < g : vacuum (1), • −| h | < g < | h | : vacua (1), (2) sgn( h ) , This is consistent with ∆ ∈ (cid:0) d − , d (cid:1) iff ∆ <
1, which is possible only if d = 2 , – 13 – − | h | < g < −| h | : vacua (1), (2) ± , • g < − | h | : vacua (1), (2) ± , (3) ± .Diagonalizing the Hessian of the energy functional V (see eq. (2.19)) we find that (1)has the eigenvalues 2( g + h ) , g − h ) . (3.7)Hence, when phases with a non-zero condensate exist ( g < | h | ) the vacuum (1) of thenormal phase is unstable (one of the eigenvalues of the Hessian is negative). In thatcase there is always at least one superconducting vacuum that is stable. Such a phaseprovides a holographic model of two weakly interacting superconducting materials at zerotemperature.As a simple illustration, by setting h = 1 we find that • when − < g <
1, the only stable vacuum is (2) + , • when − < g < −
1, there are two stable degenerate vacua (2) ± , and • when g < −
2, the vacua (3) ± are also unstable and the only stable vacua are again(2) ± . A characteristic feature of conventional Josephson junctions is a transverse supercurrent I which is related to the condensate phase difference ϑ in the following way I = I max sin ϑ . (3.8)In this section we have considered a system of two sites that represents two infinitelythin layers of superconducting material at zero charge density coupled through a weak linkexpressed by an interaction of the form (3.1), which can be suggestively rewritten as W ( O , O ) = h (cid:16) e iϑ O O † + e − iϑ O † O (cid:17) = W E + W J ext (3.9)with the definition W E = h cos ϑ (cid:16) O O † + O † O (cid:17) , W J ext = ih sin ϑ (cid:16) O O † − O † O (cid:17) . (3.10) W E is an interaction that mediates no interlayer charge transfer. In contrast, W J ext is aninteraction based on the charge-transferring operator J ext = i (cid:16) O O † − O † O (cid:17) . (3.11)– 14 –he coupling A of this operator in W J ext , namely W J ext = AJ ext , A = h sin ϑ (3.12)can be interpreted as a new interlayer background gauge potential component. In thissense, for non-vanishing h, ϑ our two-site system lies in an external transverse gauge fieldand J ext is an externally imposed current.The total current running across the sites of the junction can be determined in thestandard way from an infinitesimal relative U (1) gauge transformation of the action. Foran interaction of the form (3.9) the infinitesimal transformation δ O = iǫ O , δ O = − iǫ O (3.13)gives the Lagrangian variation δ L = 2 ihǫ (cid:16) e iϑ O O † − e − iϑ O † O (cid:17) = ǫ ( J site − J site ) = 2 ǫJ tot . (3.14)The second equality is the discretized version of the gradient ∂J across the interlayerdirection. In the third equality we used charge conservation to set J site = − J site = J tot .Consequently, J tot = h (cid:16) e iϑ O O † − e − iϑ O † O (cid:17) . (3.15)Hence, in the vacuum governed by the algebraic equations (3.3) one finds (at leading orderin the 1 /N expansion) J tot = h (cid:0) e iϑ α α ∗ − e − iϑ α ∗ α (cid:1) = 0 . (3.16)This is precisely what one expects. Since our system is kept at zero charge density, in theequilibrium state charge cannot flow across the junction between the two sites. As a result,irrespective of the initial configuration, once the interaction (3.9) is turned on the systembackreacts and evolves to a new vacuum, which can be conveniently determined with theholographic techniques of subsection 3.1. In the new vacuum, which is characterized bythe solutions of the algebraic equations (3.3), the condensate phase difference is ϑ = − ϑ and the magnitude of the condensate has adapted accordingly and in direct relation to thestrength of the interlayer couplings h, ϑ . This component of the gauge field is not an inherent quantity of the ( d + 1)-dimensional theoriesliving in each site of our network. In general, a linear array of sites ‘deconstructs’ an extra spacetimedimension and quantities, like A , associated with this extra direction arise in the field theory space of thelower-dimensional multi-AdS/CFT network (quiver) as new interlayer interactions. – 15 –t is interesting to consider the VEV of the externally forced current operator J ext inthe new vacuum. At leading order in the 1 /N expansion h J ext i = i ( α ∗ α − α α ∗ ) . (3.17)Using eq. (3.6) we obtain h J ext i = I max sin ϑ = − I max sin ϑ (3.18)with I max = 2 h − | α | (cid:16) g + s | α | (cid:17) . (3.19)In the algebraically simple case of δ = 4 I max = ± s ( ± h − g ) (3.20)on the vacua (2) ± and I max = − hs (3.21)on the vacua (3) ± (interestingly in this case I max is independent of g ).The vanishing of the total interlayer current J tot implies that the backreaction hascreated an equal and opposing ‘Josephson current’ J josephson (due to the condensate phasedifference), which cancels the contribution of the externally imposed h J ext i . Specifically, J josephson = −h J ext i = I max sin ϑ (3.22)in agreement with the expected sine law (3.8).It is clear that the system we have just described is a peculiar Josephson junctionunlike the typical Josephson junction commonly discussed in the literature and engineeredin the laboratory. In contrast to our system in a typical junction the superconductor com-ponents have finite spatial thickness in the transverse junction direction, charge can flowin this direction, the condensate phase difference is dialed by choice (and not determineddynamically) and a Josephson current arises without having to apply a gauge field acrossthe weak link. In section 4 we will describe how such more conventional junctions can beengineered in our framework as simple extensions of the setup we have described in thissubsection. We briefly comment on a few possible extensions of this framework that may ultimatelylead to different current-phase relations. – 16 – possible alternative is to consider the same superconductor models as above, butmodify the double-trace interactions. For example, one could consider other types ofdouble-trace interactions, e.g. instead of (or in addition to) (3.1) a double-trace deformationof the form h ( e iω O O + e − iω O † O † ) . (3.23)This will modify the equations (3.3) and the Josephson current (3.22) in an obvious fashion.Another possibility is to consider higher multi-trace deformations. An example of atriple-trace interaction is h ( e iω O O † + e − iω O † O ) . (3.24)Finally, it is interesting to consider coupling different types of asymptotically AdStheories. For example, one can try to couple the holographic s -wave superconductor of Ref.[2] to the holographic p -wave superconductor of Ref. [39]. In the s -wave superconductor itis a bulk complex scalar field, dual to a complex scalar operator, that condenses. In the p -wave superconductor there is an SU (2) gauge field in the bulk and the U (1) ⊂ SU (2)is broken by the condensation of the remaining two components of the SU (2) gauge field,which are dual to the corresponding components of an SU (2) current in the boundarytheory. One can consider the possibility of scalar-current double-trace deformations on theboundary which presumably translate to mixed boundary conditions for the dual scalarand gauge field components.
4. Holographic Josephson junction arrays
A lot of theoretical and experimental work has been performed on Josephson junctionarrays (and more generally networks of Josephson junctions). Some of the main motivationsand results in this field were summarized in the introduction. A natural generalization ofthe holographic construction of the previous section can be used to model networks ofholographic superconductors.In what follows we will concentrate on a simple network with the topology of a chainas depicted in Fig. 3. In this network M sites (labeled by an index n , each of themrepresenting a d -dimensional holographic superconductor of the type of subsection 2.4) arelinked by the mixed boundary conditions corresponding to the double-trace interactions W ( {O n } ) = g X n O n O † n + h X n (cid:16) e iϑ O n O † n +1 + e − iϑ O † n O n +1 (cid:17) . (4.1)– 17 – Figure 3:
A chain of holographic superconductors labeled by an index n ∈ Z . Each link denotesan interaction mediated on the field theory side by a double-trace deformation. For simplicity, we are assuming the same constants g, h, ϑ, δ for all sites and links. Ac-cordingly, the potential function V that we have to extremize in gravity to obtain thecondensates α n is V ( { α n } ) = X n (cid:16) g | α n | + sδ | α n | δ + h (cid:0) e iϑ α n α ∗ n +1 + e − iϑ α ∗ n α n +1 (cid:1)(cid:17) . (4.2)The extrema of this function are sequences of complex numbers obeying recursion relationswith rich features. The solutions can be periodic, quasiperiodic (chaotic) or solitonic, andprovide interesting new examples of spatially modulated, namely lattice site-dependent,superconductivity.We will organize the discussion according to the number of boundary conditions. In the case of no boundaries, the extremization of the potential function V (4.2) withrespect to all the α i ’s gives the recursion relations gα n + h (cid:0) e iϑ α n − + e − iϑ α n +1 (cid:1) + s α n | α n | δ − = 0 , n ∈ Z . (4.3)Setting α n ≡ e inϑ ϕ n , ˜ g ≡ gh , ˜ s ≡ sh (4.4)we can recast (4.3) into the form˜ gϕ n + ϕ n − + ϕ n +1 + ˜ s ϕ n | ϕ n | δ − = 0 . (4.5)The generic solution of these equations is parameterized by two complex numbers(these could be for instance the values of ϕ , ϕ at the vertices 0 and 1). There arevarious ways to analyze this system of equations. In fact, such equations have appearedin the past in a variety of applications and discussions of discrete dynamical systems. A– 18 –otable example, that arises for the special case of δ = 4, is that of the discrete non-linearSchr¨odinger (DNLS) equation − i ddt ψ n + ψ n − + ψ n +1 + s ψ n | ψ n | = 0 . (4.6)This equation is a discretized version of the Schr¨odinger equation with a non-linear quarticpotential. Setting ψ n ≡ ϕ n e iEt (4.7)we recover our set of equations (4.5) with E = ˜ g and δ = 4.The DNLS equation has a long history (for an extensive review and references we referthe reader to [27]). In solid state physics it appeared first in the context of the Holsteinpolaron model for molecular crystals [40]. In an optics context, DNLS describes wavemotion in coupled nonlinear waveguides. In the basic nonlinear coupler model introducedin [41] two waveguides made of similar optical material are embedded in a different hostmaterial. DNLS, and generalizations, have also appeared in studies of nonlinear electricallattices [42].As a concrete algebraically simple example, in the rest of this section we will concen-trate on the case of δ = 4. Qualitatively similar results are expected for generic δ . Thelinear stability analysis is treated for generic δ in appendix A. For most purposes we followclosely the analysis of [27] which we recommend for additional details. The recursion relation (4.5) (with the ansatz δ = 4)˜ gϕ n + ϕ n − + ϕ n +1 + ˜ s ϕ n | ϕ n | = 0 (4.8)can be viewed as a four-dimensional mapping from C → C . Using polar coordinates ϕ n = r n e iθ n (4.9)we obtain the following two sets of equations r n +1 cos(∆ θ n +1 ) + r n − cos(∆ θ n ) = − (cid:18) ˜ g + ˜ s r n (cid:19) r n , (4.10) r n +1 sin(∆ θ n +1 ) − r n − sin(∆ θ n − ) = 0 , (4.11)where ∆ θ n ≡ θ n − θ n − . (4.12)– 19 –quation (4.11) is equivalent to the conservation of current J ≡ r n r n − sin(∆ θ n ) . (4.13)It is convenient to introduce the real-valued variables x n ≡ = ϕ ∗ n ϕ n − + ϕ n ϕ ∗ n − = 2 r n r n − cos(∆ θ n ) , (4.14a) y n ≡ i (cid:0) ϕ ∗ n ϕ n − − ϕ n ϕ ∗ n − (cid:1) = 2 J , (4.14b) z n ≡ | ϕ n | − | ϕ n − | = r n − r n − . (4.14c)In terms of these variables the recursion equations (4.8) become x n +1 + x n = − (cid:18) ˜ g + ˜ s w n + z n ) (cid:19) ( w n + z n ) , (4.15a) z n +1 + z n = 12 x n +1 − x n w n + z n , (4.15b) w n = p x n + z n + 4 J (4.15c)thus reducing our 4D map to a 2D map M : R → R .This map depends on two parameters: (˜ g, ˜ s ). The dependence on J can be scaled awayby setting x n → J x n , z n → J z n , ˜ s → J ˜ s (4.16)so that w n = p x n + z n . A linear stability analysis shows (see appendix A for details)that, depending on the precise parameters, there are both bounded and diverging solutions.In certain regimes, e.g. when˜ s > , ˜ g < , − g + 2)˜ s < | ϕ n | δ − < − ˜ g )˜ s ( δ −
1) (4.17)the solutions are regular and bounded and the Lyapunov exponent vanishes [27]. Recallthat the original parameter s that appears in the AdS/CFT context (2.18) is positive, but˜ s = sh can be both positive or negative. Moreover, one can tune s freely by adding on thefield theory dual quartic-trace interactions of the form |O| .The general structure of the space of solutions is organized by a hierarchy of periodicorbits surrounded by quasi-periodic orbits [27]. The periodic orbits can be traced on theintersection of any two symmetry lines S n = M n S , S n = M n S , n = 0 , , . . . [43, 44],where the fundamental symmetry lines are defined as S : z = 0 , (4.18a)– 20 – : x = − (cid:18) ˜ g + ˜ s w + z ) (cid:19) ( w + z ) . (4.18b)For example, in the intersection of the lines S and S one locates the fixed point ( x = x ∗ , z = 0) with x ∗ = − (cid:18) ˜ g + ˜ s p x ∗ (cid:19) p x ∗ . (4.19)The linear stability of an orbit with period q is conveniently characterized by the valueof Greene’s residue [45] ρ = 14 " − Tr q Y n =1 D M ( n ) ! (4.20)where D M is the linearization of the map M . The period orbit is linearly stable when0 < ρ < ρ > ρ < x ∗ ,
0) the residue is ρ = 1 − (cid:18) ˜ g + ˜ s p x ∗ (cid:19) (˜ g + ˜ s p x ∗ ) . (4.21)For ˜ s = 0 eq. (4.19) has one root, x ∗ = ˜ g − ˜ g , which is real when | ˜ g | <
2. In that case,0 < ρ = 1 − ˜ g <
1, so one obtains an elliptic fixed point. This conclusion continues tohold for generic ˜ s > | ˜ g | < x ∗ ,
0) form the largest basins of sta-bility among all elliptic orbits. These stable orbits, which include both periodic and quasi-periodic solutions, encircle the fixed point forming the main island on the map plane. Thequasi-periodic orbits, which lie on closed curves (the Kolmogorov-Arnold-Moser (KAM)tori), densely fill the island. One can show that the map M is topologically equivalentto an area-preserving map ensuring the existence of such KAM-tori near the symmetricelliptic fixed points [46].An illustration of the main island of elliptic orbits around the fixed point ( x ∗ ,
0) for˜ g = 1 .
6, ˜ s = 0 . a ) of Fig. 4. Outside this island one finds regularquasiperiodic orbits and orbits that diverge.Interesting changes in the structure of the solution space can occur as we vary theparameters of the system, here (˜ g, ˜ s ). In particular, the residue of periodic orbits canchange. When the residue changes from a positive value to a negative value then a tangentbifurcation occurs where an elliptic point converts into a hyperbolic point. Whenever theresidue exceeds the value of one from below a stable elliptic orbit converts into an unstablehyperbolic point with reflection accompanied by the creation of two new stable elliptic– 21 – - - - - - x - - z ( a ) - - x - - - z ( b ) Figure 4:
Plot (a) depicts in the ( x, z ) -plane the main island of elliptic orbits that developsaround the elliptic fixed point ( x ∗ , with x ∗ ≃ − . for ˜ g = 1 . , ˜ s = 0 . . Plot (b) depicts acharacteristic example of period-doubling bifurcation for ˜ g = − . , ˜ s = 1 . Orbits with differentinitial conditions are depicted with different colors. points. This kind of bifurcation is known as period-doubling bifurcation —a 1-periodorbit (fixed point) converts into a 2-period orbit. The new elliptic orbits remain stableuntil another period-doubling bifurcation occurs. After a cascade of such bifurcations localchaos appears.In our system, the residue ρ in eq. (4.21), is always less than one assuming ˜ s > g >
0. In that case, only tangent bifurcations can occur and global chaos can arise throughthe so-called resonance overlap. Period-doubling bifurcation can instead occur when ˜ g < g < − ˜ s < x ± = ± r g ˜ s − , z = 0 . (4.22)The map M acts on these points by sending ( x ± , → ( x ∓ , ρ = 12 (cid:18) ˜ g − ˜ s (cid:19) . (4.23)As we decrease ˜ g further the period-2 orbit loses its stability and a new period-doublingbifurcation occurs which gives rise to a period-4 orbit. This cascade terminates at a criticalparameter (see [27] and references therein)˜ g ∞ = − ˜ s + r ˜ s − | C ∞ | ! , C ∞ ≃ − . b ) of Fig.4. At ˜ g = − .
62, ˜ s = 1, the originally stable fixed point of plot ( a ) has become unstableand two new elliptic fixed points have been created giving rise to elliptic period-2 orbits.Besides the issue of linear stability, that was discussed above, one can also ask aboutthe local and global thermodynamic stability of the above solutions. Local thermodynamicstability requires a positive definite Hessian of the multi-gravity energy functional (2.23)(see also (2.19)). It would be interesting to examine the extent to which local thermody-namic stability is equivalent to linear stability. On the other hand, global thermodynamicstability implies that the more stable solutions have less energy. In the following subsec-tions 4.1.2 and 4.2 we will see that chains with a finite number of sites have a discrete setof solutions. In that case, the solutions with the minimum energy are thermodynamicallyfavored. We hope to return to a more detailed examination of these issues in the future. From the above discussion it should be clear that, for given values of the parameters andperiod, one is left with at most a discrete finite set of solutions to the recursion relations(4.8) in the case of periodic boundary conditions. Indeed, in regimes that allow for ellipticperiodic orbits the choice of a prescribed period picks the sequence of ϕ n ’s in generaluniquely. In other regimes of parameters periodic solutions do not even exist. Moreover,through period-doubling bifurcation it is interesting to note that it is possible to haveperiodic solutions where the ϕ n ’s arrange themselves in more than one different domainsof values. If there is a boundary, say at n = 0 with n valued only on non-negative integers, the n = 0version of the equation (4.8) is modified to˜ gϕ + ϕ + ˜ s ϕ | ϕ | δ − = 0 . (4.25)In that case, the whole solution is fixed by the choice of one parameter, for example ϕ .If the chain has finite size and there is also a second boundary, then the analog of (4.25)at the second boundary will fix ϕ as well and the solution will be discretely unique andexpressed completely in terms of the parameters of the system g , s , h . A specific exampleof this situation appeared in the dimer case of section 3.A special consequence of eq. (4.25) is the fact that ϕ n are all real-valued up to acommon n -independent phase. Equivalently, the phases θ n in (4.9) are all equal modulo– 23 – . This property can be deduced by using the boundary equation (4.25) to compute theconserved ( i.e. n -independent) current J (4.14b). One finds J = 0 from which the abovestatement follows immediately.Taking care to satisfy the conditions from the boundary equations one can proceed asbefore to analyze the solutions. Depending on the precise parameters and the boundaryvalue ϕ one finds again in the case of a semi-infinite chain regular or diverging solutions.The regular solutions can be periodic or quasi-periodic. In the case of a chain with finitesize only a discrete subset of the above regular solutions survives. We are not aware ofany tractable analytic method that determines these regular solutions for generic sets ofparameters. It is known that the DNLS equation admits also another interesting kind of solutions:soliton and kink solutions. A detailed discussion of these solutions in the real domainand related references can be found in [27] whose main points can be summarized brieflyas follows. A priori one might expect that the DNLS equation does not admit such so-lutions. Soliton-like solutions are typically associated to integrable systems and DNLSis not integrable. It exhibits irregular chaotic behavior which in principle may preventperfect localization. Nevertheless, it can be shown that non-integrability and discretenessappropriately combine to make such solutions possible.The solutions of interest have the following characteristics. They are solutions wherethe amplitude ϕ n is exponentially localized around a single site, say at n = 0. Followingthe nomenclature of [27] one can distinguish between two situations:(1) Bright solitons : in this case, | ϕ n | > | ϕ n +1 | for n > | ϕ n | < | ϕ n +1 | for n < | n |→∞ | ϕ n | = 0.(2) Dark solitons : in this case | ϕ n | < | ϕ n +1 | for n > | ϕ n +1 | > | ϕ n | for n < | n |→∞ | ϕ n | >
0. It turns out that lim n → + ∞ ϕ n = − lim n →−∞ ϕ n , so these solutionsare really kink solutions.In our context, where each site labeled by an index n , models a (1+1)- or (2+1)-dimensional layer of a superconducting material, such configurations would correspond incase (1) to situations where in a chain of layers the interlayer interactions work in such away that energy and superconductivity are strongly localized around a central site. In case(2) the opposite happens. Energy and superconductivity are modulated in such a way that– 24 –hey are almost uniform along the chain except around a central site where the condensatevanishes as it changes sign and superconductivity becomes very weak.In a continuum limit (see next subsection) the above configurations appear to recon-struct a junction of three materials with one dimension higher. In case (2) we recover aconfiguration that is very similar to the dark soliton of [17] and reminds of an SNS junctionof (2+1)- or (3+1)- dimensional superconductors. From this point of view the configura-tion of case (1) resembles a junction of two materials in the normal state separated by athin superconducting layer in the middle.Since ϕ n are now real it is convenient to view the equations (4.8) as a two-dimensionalmap f M : R → R by defining a new set of R coordinates ( x n , y n ) = ( ϕ n , ϕ n − ). Then, f M : (cid:26) x n +1 = − (cid:0) ˜ g + ˜ s x n (cid:1) x n − y n y n +1 = x n . (4.26)The identification of the soliton-like solutions is closely related to the structure of the fixedpoints of this map.The fixed points, which by definition obey the relation x = y , are located at x = 0 , x ± = ± r − g + 2)˜ s . (4.27)The x ± fixed points exist only when sgn(˜ g + 2) = − sgn(˜ s ). Greene’s residue ρ for the fixedpoint at the origin is [27] ρ = 14 (˜ g + 2) . (4.28)Consequently, for | ˜ g | <
2, we obtain 0 < ρ < s < x ± are unstable hyperbolic fixed points.For | ˜ g | > i ) ˜ g < −
2, ˜ s >
0. In that case ρ < x ± are stable elliptic fixed points.( ii ) ˜ g >
2, ˜ s <
0. The fixed point at the origin becomes unstable and through period-doubling bifurcation a new period-2 orbit appears located on the line x = − y .Before proceeding to explain the main idea underlying the existence of soliton-likesolutions it will be useful to introduce some language which is common in the study ofdynamical systems. – 25 – set W is called an invariant manifold of a dynamical system if for any point x ∈ W the dynamical evolution of x for any amount of time t continues to belong in W . Everyfixed point p comes with its invariant manifolds. Such manifolds are called stable, anddenoted as W s ( p ), if all points that belong on them approach asymptotically the fixedpoint p under dynamical evolution (namely p is an attractor on W s ( p )). In contrast, aninvariant manifold is called unstable, and denoted as W u ( p ), if all points that belong toit move asymptotically away from the fixed point p under dynamical evolution (in otherwords, p is a repellor on W u ( p )).For generic non-integrable maps it is known that the stable and unstable invariantmanifolds of hyperbolic fixed points cross each other. Points that reside on the intersectionof stable and unstable invariant manifolds of the same fixed point are called homoclinic points. Accordingly, points that reside on the intersection of stable and unstable invariantmanifolds of two different hyperbolic fixed points are called heteroclinic points.Having made this short introduction, we are now in position to describe what happensin our specific system provided by the map (4.26). First consider the case ( i ) with ˜ g < − s >
0. The origin is an unstable hyperbolic point. Moving along an orbit on an unstablemanifold W u of the origin and then crossing through a homoclinic point to a stable manifold W s gives rise to a soliton-like solution of the type (1) above. An explicit computation ofthe stable and unstable manifolds in the case of DNLS as well as specific examples can befound in [27] (see, for instance, Fig. 7 in [27]).In case ( ii ) with ˜ g >
2, ˜ s < ϕ n , ϕ n +1 have alternating signs. Such solutions areknown as staggered solitons [47]. The solitons of the previous paragraph are also knownas unstaggered solitons.Finally, in the case of | ˜ g | <
2, ˜ s < x ± . These are kink solutions of the type (2) above. It is interesting to consider the continuum limit of the chain configuration described above.In this limit M → ∞ , nM → x , ϕ n → ϕ ( x ) , g → GM , s → SM (4.29)with the new parameters x , G , and S kept finite. In this limit the recursion relations (4.5)turn into the second order non-linear Schr¨odinger differential equation ϕ ′′ + ϕ (cid:18) G + S | ϕ | δ − (cid:19) = 0 (4.30)– 26 –here ′ ≡ ddx .For δ = 4 we recover directly a well-known equation in the context of superfluids; theGross-Pitaevskii equation [48, 49] which gives a coarse-grained description of superfluidsat long wavelengths. The GP equation is typically relevant for weakly-interacting Bose-Einstein condensates or strongly bound fermionic superfluids at low temperature. It isinteresting that in our formalism, which has a radically different point of departure, thesame description emerges naturally (as a candidate description of strongly coupled super-conductor physics) out of a framework designed specifically to deal with layered structures.It would be worth exploring further parallels that may exist between our formalism (andthe generalizations of the GP equation that it suggests) and the known applications of theGP equation to superfluidity and superconductivity.For simplicity and concreteness let us continue to concentrate on the case of δ = 4.Two well-known solutions of this equation are:(1) Bright solitons : for
G < , S > ϕ ( x ) = ± r − GS (cid:0) √− G x (cid:1) , (4.31)(2) Dark solitons : for
G > , S < ϕ ( x ) = ± r GS tanh r G x ! . (4.32)For real ϕ we can find a more general class of solutions expressed in terms of the Jacobielliptic function sn ( u | m ) ϕ ( x ) = ± i s G − √ G + SC ) S sn ( x + x ) s G + √ G + SC (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G − √ G + SCG + √ G + SC . (4.33) C and x are integration constants.Reinstating the phase (4.4) we obtain α ( x ) = e ixθ ϕ ( x ) (4.34)where θ is the finite rescaled version of the angular inter-layer coupling ϑθ ≡ M ϑ . (4.35)– 27 –
10 15 20 x - j H x L ( a ) x - - - j H x L ( b ) - - x j H x L ( c ) - - x - - - j H x L ( d ) Figure 5:
Plot (a) depicts a generic sine-like periodic configuration for
G > . The specificplot has G = S = 1 . Plot (b) depicts a generic periodic configutation for G < , S > (in thisparticular case G = − , S = 1 ). By suitably tuning G, S one obtain the bright-soliton of eq. (4.31) , here depicted in plot (c) for G = − , S = 1 . A dark soliton (or kink) solution can beobtained by tuning G, S so that the second argument in the Jacobi sine function becomes 1. Anexample of this case appears in plot (d) for G = − S = 1 . At a boundary point x b the discrete equation (4.25) becomes in the continuum limit(4.29) a Dirichlet boundary condition ϕ ( x b ) = 0 . (4.36)By varying the parameters of the solution (4.33) one obtains qualitatively differentbehaviors. For concreteness, set C = 1, x = 0. For G > S = 0 one findsperiodic sine-like solutions like the one depicted in plot ( a ) in Fig. 5. For G <
S > b ) in Fig. 5, can be suitably tunedto obtain the bright-soliton solution (4.31) (see also plot ( c ) in Fig. 5). For G >
S <
G, S so that the secondargument in the Jacobi sine function becomes 1. In that case, we recover the dark solitonsolution (4.32) (an example of such a configuration appears in plot ( d ) of Fig. 5).By suitably truncating any of the solutions depicted in plots ( a ) , ( b ), or ( c ) of Fig.5 within an interval bounded by the location of two zeros of ϕ one obtains trivially a– 28 –nite-size Josephson junction chain in a continuum limit. For the plot ( d ) there is a singlepoint where ϕ vanishes (the core of the kink solution), hence solutions of this type thatare consistent with the Dirichlet boundary conditions (4.36) do not exist. Following the discussion of subsection 3.2 we can write the total current across the chain(evaluated across two adjacent superconductors at positions n − n ) as a sum of twocontributions J tot = h J n − ,n i + J josephsonn − ,n . (4.37) J n − ,n is the current operator J n − ,n = i ( O n − O † n − O † n − O n ) (4.38)associated to the charge-transferring part of the ( n − , n ) interlayer interaction in eq.(4.1), and J josephsonn − ,n is the Josephson current associated to the backreaction of the systemdiscussed in subsection 3.2. The total current J tot is the total conserved current of thesystem, which, up to a potential multiplicative constant that we will keep implicit, equalsthe quantity J in eq. (4.13). This is a site-independent quantity. In a chain with finitelength the boundary conditions set J = 0 and h J n − ,n i = − J josephsonn − ,n as in the two-sitesystem of subsection 3.2. More generally, however, J = h J n − ,n i + J josephsonn − , . (4.39)Using the definition (4.38) we find that the current h J n − ,n i = i ( α ∗ n − α n − α n − α ∗ n ) = i ( e iϑ ϕ ∗ n − ϕ n − e − iϑ ϕ n − ϕ ∗ n ) . (4.40)In terms of the polar coordinates (4.9) we further obtain h J n − ,n i = − r n − r n sin( ϑ + ∆ θ n ) = − J sin( ϑ + ∆ θ n )sin(∆ θ n ) (4.41)where ∆ θ n and J were defined in eqs. (4.12) and (4.13) respectively.For a general solution of the recursion equations (4.3) and ϑ = 0 , π this is a link-dependent current. It is periodically or chaotically modulated in periodic or quasi-periodicsolutions. In real solutions, where ∆ θ n = 0 mod π , h J n − ,n i = ± r n − r n sin ϑ . (4.42)For instance, in soliton-like solutions this current is very weak except around a central site.– 29 – left superconductor right superconductorweak link interface Figure 6:
An (un)conventional JJ constructed as two semi-infinite AdS/CFT arrays of (2+1)-dimensional holographic superconductors linked to each other at a two-dimensional weak-link in-terface.
In the special case where ϑ = 0 , π ( i.e. when no external interlayer gauge field isapplied), eq. (4.41) gives the site-independent current h J n − ,n i = ± J . (4.43)This current vanishes for a chain with boundaries, but can be non-zero in chains withoutboundaries, e.g. in a circular chain with periodic boundary conditions. This is a simpleexample of how the topology of the network can affect the qualitative features of theconfiguration.
In subsection 3.2 we discussed the similarities and differences between a two-site systemand typical Josephson junctions. Here we discuss how the two-site system can be extendedto look more like the typical Josephson junction.Assume we want to describe a junction of two superconductors in three spatial dimen-sions linked weakly across the third direction z at a two-dimensional interface. In previoussubsections we described how to deconstruct (3 + 1)-dimensional layered superconductorsfrom an array of (2 + 1)-dimensional holographic superconductors using linear AdS/CFTarrays. To construct an (un)conventional Josephson junction of two superconductors of– 30 –his type a possible strategy is described in Fig. 6. A layered superconductor on the left(right) is deconstructed as an array of cites linked through interactions of the form X n h L ( R ) (cid:16) O L ( R ) n O L ( R ) † n +1 + O L ( R ) † n O L ( R ) n +1 (cid:17) . (4.44)With a real coupling h L ( R ) no external transverse gauge field is applied along the z direction.Across the two-dimensional interface the right-most black site of the left chain can be linkedto the left-most red site of the right chain through a link of a different type depending onthe specific nature of the left and right sites. For s -wave holographic superconductors bothon the left and the right a simple example of a double-trace weak link is h link (cid:0) O L O R † + O L † O R (cid:1) . (4.45)Then one can solve the analog of the equations (4.3) and determine the Josephson currentas was described in the previous subsection. SNS-type solutions of a uniform array with h L = h R = h link were described in subsections 4.1, 4.2, 4.3 (in a discrete or continuumlimit). In general, the asymptotic difference of the phase of the condensates, ∆ ϑ = ϑ L − ϑ R ,is a dialed quantity in these systems. For conventional SNS or SIS-type JJs we anticipatethe presence of a Josephson current that follows the sine law relation I max sin ∆ ϑ . We hopeto return to a detailed survey of such systems in future work.
5. Outline of future directions
We have proposed a novel holographic way to model the physics of Josephson junctionsusing networks of (super)gravity theories on asymptotically AdS spacetimes coupled viamixed boundary conditions. One of the advantages of this approach, compared to previousholographic approaches, is the versatility by which it can incorporate many different typesof Josephson junctions and networks with limitless possibilities in their architecture. Forconventional SNS or SIS-type superconductors we presented a simple two-site model thatexhibits some of the standard features of Josephson junction, e.g. the sine relation betweenthe Josephson current and condensate phase difference. We explained in what sense thissystem is different from the typical Josephson junctions and how one can use AdS/CFTarrays to describe the more typical systems. We have also seen how a simple network ona chain produces complex dynamics with a variety of interesting features.Our preliminary analysis opens the possibility for a diverse set of calculations andextensions. Some of the most prominent ones are the following.– 31 – igure 7:
The architecture of the Y-Josephson junction network. (a) Finite temperature and a more complete analysis of phenomenological implications
We have so far considered simple examples of holographic JJs at vanishing temperatureand charge densities. It is of obvious interest to extend the setup to finite temperatureusing hairy black holes in designer multi-gravity (see e.g. [37] for related work) and toexplore possible phase transitions as we vary the temperature and/or charge densities.Extending the list of examples it is desirable to consider the explicit properties ofother holographic JJs (or JJNs) built from different types of holographic superconductors( e.g. s -wave, or p -wave). For example, it will be interesting to define and study interlayertransport coefficients in such models. The ultimate goal is to explore the extent to whichthese constructions reproduce known phenomenological features of JJNs or layered super-conductor physics. For example, it would be interesting to reproduce previously observednon-sinusoidal current-phase relations in unconventional JJs. (b) Other network architectures and complex behavior It has been pointed by many authors (see e.g. [26]) that the architecture of a JJN can haveimportant implications on the physical properties of the system. It is interesting to exploreother configurations and examine how they affect the collective and local properties of thesites. Networks with double-trace or higher multi-trace interactions can be constructed.An example that has been studied previously in the condensed matter literature is theY-Josephson junction (see for instance [50]). The architecture of a Y-Josephson junctionnetwork appears in Fig. 7. – 32 –t may also be interesting to explore the existence of vortex solutions in two- or three-dimensional AdS/CFT lattices. This becomes even more interesting in view of the observedconnection to the GP equation in the continuum limit. A recent discussion of vortexsolutions to the GP equation (and a related AdS/CFT application from a different pointof view) can be found in [51]. (c) Continuous limits and deconstruction
In subsection 4.3 we considered a continuum limit of a one-dimensional holographic JJN.In this limit the number of sites is scaled to infinity with an appropriate scaling of the otherparameters of the system to zero. For a specific set of parameters we recover in this limitthe Gross-Pitaevskii equation. It would be interesting to explore further relations betweenthe generalizations of this equation suggested by our formalism and known applications ofthe GP methodology in superfluidity and superconductivity.In addition, it would be interesting to explore similar continuous limits of other JJNswith more complicated topology and different ingredients. Such limits may be used tosimplify some aspects of the analysis of the network or in order to attempt a novel decon-struction of one or more extra spacetime dimensions (more comments on this aspect canbe found in [28]).
Acknowledgements
We would like to thank Nikos Flytzanis, Christos Panagopoulos and George Tsironis forexplaining pertinent aspects of their work and for providing a useful guide through thevast condensed matter literature on the subject. VN would also like to thank the GalileoGalilei Institute for Theoretical Physics for the hospitality and the INFN for partial supportduring the completion of this work. In addition, this work was partially supported by theEuropean Union grants FP7-REGPOT-2008-1-CreteHEPCosmo-228644 and PERG07-GA-2010-268246.
AppendicesA. On the linear stability of DNLS
In this appendix we discuss in more detail the linear stability analysis of the DNLS equation(4.5) ˜ gϕ n + ϕ n − + ϕ n +1 + ˜ s ϕ n | ϕ n | δ − = 0 . (A.1)– 33 –ntroducing a small perturbation u n around a solution ϕ (0) n ϕ n = ϕ (0) n + u n (A.2)we obtain (at first order) the equation u n +1 + u n − + (cid:18) ˜ g + ˜ sδ | ϕ (0) n | δ − (cid:19) u n + ˜ s ( δ − ϕ (0)2 n | ϕ (0) n | δ − u ∗ n = 0 . (A.3)Next we decompose ϕ (0) n and u n into their real and imaginary parts ( x n , y n here should notbe confused with the corresponding variables in eqs. (4.14a), (4.14b) in the main text) ϕ (0) n = X n + iY n , u n = x n + iy n . (A.4)For these variables we obtain the following two coupled sets of equations x n +1 + x n − + (cid:18) ˜ g + ˜ sδ X n + Y n ) δ − (cid:19) x n + ˜ s ( δ − (cid:0) X n + Y n (cid:1) δ − (cid:0) ( X n − Y n ) x n + 2 X n Y n y n (cid:1) = 0 , (A.5a) y n +1 + y n − + (cid:18) ˜ g + ˜ sδ X n + Y n ) δ − (cid:19) y n + ˜ s ( δ − (cid:0) X n + Y n (cid:1) δ − (cid:0) − ( X n − Y n ) y n + 2 X n Y n x n (cid:1) = 0 . (A.5b)Introducing the notation M xn = − (cid:20) ˜ g + ˜ sδ (cid:0) X n + Y n (cid:1) δ − + ˜ s ( δ − (cid:0) X n + Y n (cid:1) δ − ( X n − Y n ) (cid:21) , (A.6) M yn = − (cid:20) ˜ g + ˜ sδ (cid:0) X n + Y n (cid:1) δ − − ˜ s ( δ − (cid:0) X n + Y n (cid:1) δ − ( X n − Y n ) (cid:21) , (A.7) N n = − ˜ s ( δ − (cid:0) X n + Y n (cid:1) δ − X n Y n (A.8)we can rewrite the set of equations (A.5a), (A.5b) as a matrix equation x n +1 x n y n +1 y n = M xn − N n
01 0 0 0 N n M yn −
10 0 1 0 x n x n − y n y n − ≡ J n x n x n − y n y n − . (A.9)The characteristic polynomial for the eigenvalues λ of the matrix J n is λ − ( M xn + M yn ) (1 + λ ) λ + λ (cid:0) M xn M yn + 2 − N n (cid:1) + 1 = 0 . (A.10)– 34 –y further setting µ n = − ˜ g − ˜ sδ X n + Y n ) δ − , ν n = ˜ s ( δ − X n + Y n ) δ − (A.11)we can recast (A.10) into the more convenient form (cid:0) λ − µ n λ + 1 (cid:1) = λ ν n (cid:0) X n + Y n (cid:1) (A.12)which yields four roots labeled by two Z indices ε , ε = ± λ ε ,ε = µ n + ε ν n ( X n + Y n ) + ε q ( µ n + ε ν n ( X n + Y n )) − . (A.13)The discriminant∆ = (cid:0) µ n + ε ν n ( X n + Y n ) (cid:1) − (cid:18) ˜ g + ˜ s δ − ε ( δ − X n + Y n ) δ − (cid:19) − <
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