Charge { Vortex Duality. in Double-Layered Josephson Junction Arrays

Charge { Vortex Duality in Double-Layere Josephson Junction Arrays Ya. M. Blanter a;b an Ger Schon c a Institut fur Theorie er Konensierten Materie, Universitat Karlsruhe, 76 Karlsruhe, Germany b Department of Theoretical Physics, Moscow Institute for Steel an Alloys, Leninskii Pr. 4, 736 Moscow, Russia c Institut fur Theoretische Festkorperphysik, Universitat Karlsruhe, 76 Karlsruhe, Germany (December, 5) Abstract A system of two parallel Josephson junction arrays couple by interlayer capacitances is consiere in the situation where one layer is in the vortexominate an the other in the charge-ominate regime. This system shows a symmetry (uality) of the relevant egrees of freeom, i.e. the vortices in one layer an the charges in the other. In contrast to single-layer arrays both contribute to the kinetic energy. The charges feel the magnetic el create by vortices, an, vice versa, the vortices feel a gauge el create by charges. For long-range interaction of the charges the system exhibits two Berezinskii-Kosterlitz-Thouless transitions, one for vortices an another one for charges. The interlayer capacitance suppresses both transition temperatures. The charge-unbining transition is suppresse alreay for relatively weak coupling, while the vortex-unbining transition is more robust. The shift of the transition temperature for vortices is calculate in the quasi-classical approximation for arbitrary relations between the capacitances (both weak an strong coupling). Two-imensional (D) Josephson junction arrays have attracte much attention because of the experimental an technological progress an the rich unerlying physics (see Ref. [] for review). Classical D Josephson junction arrays, where the Josephson coupling energy E J between the superconucting islans ominates, is a stanar example of a system exhibiting the Berezinskii-Kosterlitz-Thouless (BKT) transition { the unbining of vortex-antivortex pairs at a certain temperature T J [,3]. The transition separates a superconucting phase at T < T J / E J, where vortices are boun, from a resistive phase. It was realize later (see e.g. Refs. [4{7]) that charging eects, associate with the capacitances of the islans to the groun C an of the junctions C, lea to quantum uctuations of the phase an suppress

the BKT transition temperature. Beyon a critical value of the charging energy E C = minfe =C; e =C g the transition temperature vanishes, an the superconucting phase ceases to exist. The next step [{] was to unerstan that in the \extreme" quantum limit E C E J, where the quantum uctuations of the phases are very strong, the vortices are ill-ene objects. In this regime the charges on the islans become the relevant variables. If, furthermore, C C, the interaction between the charges is (nearly) the same as that of the vortices in the quasi-classical array. In particular, the charges can be consiere as a D Coulomb gas [3], an they unergo a BKT transition at temperature T C / E C. The phase below the transition is insulating. A nite value of the Josephson coupling between the islans suppresses this transition. As another example we mention the inuence of issipation (e.g. Ohmic issipation) on the phase transition in the array, which was rst note in Ref. []. We are not going to review these theoretical results, however it is necessary to stress that the theory of D Josephson junction arrays is far from being settle. Below we escribe another, more complicate system - two parallel D Josephson junction arrays with capacitive coupling between them (no Josephson coupling ). Probably the most interesting situation arises when one array is in the quasi-classical (vortex) regime while another one is in the quantum (charge) regime. Then the vortices in one layer an the charges in the other one are well-ene ynamical variables. Another important feature of the present system is that the strength of interaction between charges an vortices is controlle by the interlayer coupling C x an consequently this interaction may be weak or strong, whereas in usual D Josephson junction arrays the strength of charge-vortex interaction is of the same orer as either the charge-charge or the vortex-vortex interaction. We also show that the physical realization of this interaction is rather ierent from that in one array. Hence, at least for weak interlayer coupling, one shoul expect two BKT transitions, the rst for charges in one layer, an the secon for vortices in the other one. In this article we provie the theoretical escription of the couple system an calculate the shifts of the transition temperatures ue to the interlayer interaction. We consier two parallel Josephson junction arrays, i.e. (square) lattices of superconucting islans connecte by Josephson links. As usual, we suppose that the magnitue of the orer parameter in the islans is constant while its phase uctuates from islan to islan. The partition function of the system may be expresse conveniently in terms of these phases i (the inices i label the islans in each array an = ; refers to the number of array) = Y i () i () i fm i g;fm i g D i ()D i () exp(?sfg) : () Here the path integration over phases is carrie out with the bounary conitions i () = () i ; i () = () i + m i ; Multi-layere systems with Josephson coupling between layers have been iscusse in the literature (see e.g. [4,5]). The analog of the BKT transition is in this case the isruption of vortex rings. In the limit of weak Josephson couplings this system is reuce to the D Y-moel, while in the opposite case of strong coupling it is essentially the 3D Y-moel. This situation, however, is absolutely ierent from the one we escribe below.

where is the inverse temperature, an h =. The Eucliean eective action Sfg is Sfg = < : C ( _ e i ) + C ( _ e i? _ j ) + C ( _ e i ) + C ( _ e i? _ j ) + i hi + C x ( e _ i? _ i ) + E J [? cos( i? j )] + E J [? cos( i? j )] ; : () i hi Here C are the capacitances of the islans in the array relative to the groun, C are the capacitances of the junctions in the array, an C x are the interlayer capacitances. Furthermore, E J are the Josephson coupling constants in the layers. Here an below we use the symbol P hi to enote the summation over nearest neighbors only, an each pair is counte once; the symbol P stans for the summation for all values i an j (in particular, each pair except hiii is counte twice). From now on we choose parameters such that the array is in the charge (quantum) regime while the array is in the quasi-classical (vortex) regime. In terms of the phase variables this means that in the array the phases on each grain are strongly uctuating in time, while in array they are nearly time-inepenent. This regime is escribe by the conitions E J e = ~ C ; E J e = ~ C ; with ~ C = maxfc ; C ; C x g. Below we rst calculate the shift of the BKT transition temperature for vortices in the array. This oes not require the introuction of charges an vortices an may be one in the phase representation. Then, we turn to the BKT transition for charges in the array. For this purpose we move from a escription in terms of phases to one in terms of charges an vortices, an use the uality of the resulting action to investigate the transition. At the same time, we will show that charges an vortices in this system can be consiere as two-imensional ynamical particles with masses. The charge-charge an vortex-vortex interaction are essentially those of D Coulomb particles, while the charge-vortex interaction is more peculiar. hi i hi = BKT transition for vortices The shift of the BKT transition temperature for vortices in the array ue to the coupling to the array can be calculate easily if we set the small parameter E J to zero. Then the action for the phases i becomes Gaussian an the latter may be integrate out. After that the shift of the BKT temperature for vortices may be obtaine by means of the quasi-classical expansion [7,6]. The rst step requires a comment. The path integration over the phases of the islans i () in the array is, as usual, performe by a linear shift of variables in orer to eliminate This means, in particular, that the results obtaine below are vali also in the case when the array is in the normal state. Because of the e-perioicity in this case the bounary conitions in the array shoul rea i()? i() = 4m i. As we will show, this oes not change the nal result. 3

terms linear in i. However, the new (shifte) variables o not generally satisfy the bounary conitions, an consequently the integration is not possible. If, nevertheless, the array is in the quasi-classical regime, the contributions of all non-zero wining numbers m i to the partition function are exponentially small in comparison with the contribution of m i = (see, e.g. [6]). If we neglect these small contributions, the phases i become perioic, an the bounary conitions in the array are met automatically. After integration over the phases i we n Y = () i D i () exp(? Sf ~ i g) (3) i with the eective action ~Sfg = < : e _ i [ ] j _ + E J [? cos( i? j )] ; : (4) Here [ ] is the eective capacitance matrix for the layer (see also below) hi [ ] =? C x? [ ^Q ] + C [ C ^Q ] (5) = an the matrices ^Q ( = ; ) have a form [ ^Q ] = 6 4 4 + C +C x C i = j? i an j are nearest neighbors otherwise : Since the array is suppose to be in the quasi-classical regime, only weakly timeepenent perioic paths i () are important. Hence we may write the phases in the form i () = () i + f i (), where f i () =? n= [f i (! n ) exp(?i! n ) + f i (! n ) exp(i! n )]: is expresse as a sum over Matsubara frequencies! n = n?. Now the action may be expane to quaratic terms in f i (), yieling ~Sf () i ; f i ()g = E J [? cos( () i? () j )] + + hi < : f_ i [ e ] f_ j + E J hi [f i ()? f j ()] cos( () i? () j ) = ; : (6) Note that the rst term is the classical action of D Coulomb gas [3]. Finally, one performs the cumulant expansion [7,6] in the last term in brackets in Eq. (6). As a result the action has exactly the same form as the classical one, but with the renormalize temperature!?? D (f i ()? f j ()) E : (7) 4

Here the angular brackets enote the averaging with the eective action S eff = (e )? f_ i ()[ ] f_ j (): From this we obtain the shift of the transition temperature ue to the interlayer coupling where T J = T J? e A=3; () A = Re([? ] ii? [? ] hi ); an the secon term is the matrix element taken for neighboring islans. (We have assume that the matrix epens on the istance between the islans only). Here T J is the transition temperature for a classical D Josephson junction array [3] (to be of orer E J ). Eq. () is the result we were aiming at, however in orer to obtain some analytical expressions we evaluate the quantity A in some approximations. In Fourier representation the matrices Q (k) have the form Q (k) = (ka) + (C )? (C x + C ); ka ; with a being the lattice parameter. Consequently the matrix? is? (k) = C (ka) + C x + C : () (C (ka) + C x + C )(C (ka) + C x + C )? Cx If we replace the rst Brillouin zone by a circle cut-o at k < a?, the integration over the angular variable can be performe easily, an we obtain A = a =a kk? (k)[? J (ka)]: In the range of integration the Bessel function J can be approximate by its expansion J (x)? x =4: Finally, in the case C C x (this situation is the most interesting) we obtain T J? T J = e 4C eff ; C eff =? C x C x C + C x C ln : () C C C x C + C x C + C C It is seen that the eect of layer is merely the renormalization of the eective capacitance. As a result, the BKT transition for vortices in the layer is suppresse. We shoul emphasize that the result () is vali for arbitrary capacitances C x, C an C. The only restriction is the valiity of the quasi-classical approximation. The shift of the transition temperature shoul be small, or, in other wors, e =C eff E J. In particular, for C x C (weak coupling between the layers) one obtains C eff C irrespectively of C vortices in layer o not feel the presence of the layer. In the case C = C C x the eective capacitance is C eff = C, while for C = C x C one has C eff = C x =3. It is seen that in the latter case the temperature begins to feel the presence of the rst layer, however the absolute value of the shift becomes now small. 5

Charge { vortex uality an BKT transition for charges Before we turn to the escription of the BKT transition for charges, it is necessary to stress the following. As shown by the eective action (), the interlayer capacitance C x not only couples the layers, but also renormalizes the capacitances C an C of the islans to the groun. Hence the logarithmic interaction between the charges in each layer has a nite range for any non-zero C x ue to the screening, an the BKT transition, is, strictly speaking, absent. One shoul realize, however, that in the situation C C x C the screening length a(c =C x ) = can be very large. Below we assume that these inequalities are satise an the range of interaction is large enough to make it meaningful to speak about the charge-unbining transition. (Note that like any phase transition in a nite system this transition is smeare; in other wors, the resistance grows exponentially, an, strictly speaking, for nite it is impossible to istinguish between insulating an resistive phases). On the other han, alreay for relatively weak coupling C x C cr C this escription becomes meaningless an the insulating phase is absent. As we show below, in the small range C x C cr the transition temperature for charges in layer oes not feel the presence of the layer an hence is essentially the charge-unbining temperature for one Josephson junction array []. Nevertheless, the charge-vortex escription require to obtain this result gives rise to an interesting physical moel to be escribe below. Now we move from the phase escription (),() to a charge-vortex escription. First we introuce the large capacitance matrix! ^C ^C =? ^Cx : ()? ^Cx ^C Here ^C is the capacitance matrix in the array while ^C x = C x. The inverse matrix in the Fourier representation reas as C? (k) = () (C (ka) + C x + C )(C (ka) + C x + C )? Cx!! C (ka)? + C x + C C x ^ (k) ^? x (k) [ C x C (ka) + C x + C ^?? x (k) ^ (k) ^C? ] (k): Inices ; = ; again label the array. The matrix ^C? escribes the interaction of charges. We have also introuce for later convenience the matrices?, escribing the interaction of charges within layer, as well as? x referring to charges in ierent layers (cf. Eq. (). Then the eective action () can be rewritten in terms of integer charges q i of each islan an phases i Sfq; g = < : e ; q i ()[ ^C? ] q j () + i h qi () _ i () + q i () _ i () i + +E J [? cos( i? j )] + E J [? cos( i? j )] ; : (3) hi Now it is possible to introuce vortex egrees of freeom by means of the Villain transformation [7] (see also []). It is important that this proceure eals only with the phase 6 hi =

variables an oes not aect the charge interaction (the rst term in Eq. (3)). The phase terms (the secon an the thir one in Eq. (3)), however, have exactly the same form in the problem of two arrays as for a single Josephson junction layer. The proceure for a single-layer array array is iscusse in etails in Refs. [,], the generalization to the ouble-layer system is straightforwar. The partition function becomes = Y i fq i ;q i g fv i ;v i g exp(?sfq; vg); (4) where the eective action for integer charges q i an vorticities v i is Sfq; vg = + 4E J +i < : e ; q i ()[ ^C? ] q j () + _q i ()G () _q j () + E J F ( E J ) _q i () v j () + i = _q i () v j () ; : 4E J F ( E J ) v j + E J v i G () _q i ()G () _q j ()+ v i G () v j + (5) Here we introuce the iscrete time variable; the time lattice spacing in the array is of orer (E J E C )?=, E C e =C. The time integration an erivatives are continuous notations for a summation over time lattice an for a iscrete erivative respectively. The function F (x) = f() _ =? [f( + )? f()]; x ln(j (x)=j (x))! x ln(4=x) ; x ; is introuce to \correct" the Villain transformation for small E J [7]. As we see, its entire eect is to renormalize (to increase) the Josephson coupling in the layer ; the renormalize coupling ~ EJ reas as ~E J (E J E C ) = (ln(e C =E J ))? : (6) Note that for E J E C (charge regime) one obtains ~ E J E C. The kernel = arctan( y i? y j x i? x j ) escribes the phase conguration at site i aroun a vortex at site j. Finally, the kernel G () is the lattice Green's function, i.e. the Fourier transform of k?. At large istances between the sites i an j it epens only on the istance r between the sites an has a form (see e.g. [3]) G () = ln( =r); a r ; = a(c =C x ) = : (7) Later on, we assume that the linear size of each array is much less that the range of interaction. This means, in particular, that we assume C x C. 7

The action (5) epens on the charges an vorticities in both layers. However, in our situation, when the layers an are in the charge an vortex regimes, respectively, the vortices in the layer an the charges in the layer may be integrate out []. To o this we suppose the latter variables to be continuous (strongly uctuating), an neglect the kinetic term for charges in the layer ( _q G () _q ). Then after performing the Gaussian integration we obtain the eective action for charges q i in the layer an vorticities v i in the layer (to be referre below as q i an v i ) Sfq; vg = + E C < : E C _v i () " G ()? q i ()G () q j () + 4 ~ EJ Cx 4 C C kl ik G () kl lj # _q i ()G () _v j () + ic x C _q j () + E J k v i G () v j + = _v i () ik G () kj q j() ; : () To erive Eq. () we have taken into account the explicit expression for the large capacitance matrix (). The action () is the central result of this section. It looks rather similar to the eective charge-vortex action in one Josephson junction, but the most important ierence is that while in one layer either charges or vortices are well-ene, Eq. () escribes the system of well-ene ynamic variables on each site charges in the layer an vortices in the layer. We postpone the iscussion of physics in this system until the next section, however it is clear that the action shows a uality between charges an vortices. The secon term in the square brackets is small if C x C ; C. Both kinetic terms for charges an vortices violate the uality ue to the numerical coecients. However close enough to the transitions these terms prouce only small renormalization of the transition temperature, an are not important. Another interesting feature of this action is that the last term, escribing the interaction between charges an vortices, is also small, while in a single-layer array the interaction is always of the same orer of magnitue as another terms. It is obvious that for long-range interaction of the charges in the layer they also exhibit the BKT transition, an uner the conitions where the action () was obtaine the transition temperature oes not feel the presence of the layer : T C? T C = ~ E J 4 : Charge an vortex motion To unerstan the physics escribe by the action () it is instructive to map this moel onto the D Coulomb gas. For this purpose we move from the space-time lattice to the continuous meium an introuce the coorinates of the vortex centers an charges q i ()! m v i ()! n q m (r? r m ()) v n (r? R n ()): ()

Here q m = an v n = represent charges an vortices respectively; r m () an R n () are the corresponing coorinates of the charges an of the vortex centers. Now the partition function reas = M = N = Dr () : : : Dr M ()DR () : : : DR N () exp(?sfr; Rg); () an we are going to eal with the eective action Sfr; Rg, escribing the behavior of the system of M charges (of which M are positive, q =, an the other M are negative, q =?), an of N positive (v = ) an N negative (v =?) vortices. The rst an thir terms of the action () can be easily transforme by means of ecomposition (). The rst one prouces the potential energy of charge interaction, S (q) int = E C M m;n= q m q n G () (r m ()? r n ()): () In principle, the summation inclues the terms with m = n; these, however, may be exclue from this sum, giving rise to the chemical potential for charges. The thir term in Eq. () yiels the interaction of vortices S (v) int = E J N m;n= v m v n G () (R m ()? R n ()): () Here again the term with m = n gives rise to the chemical potential for vortices. The terms () an () are essentially the action for (classical) Coulomb gases of charges an vortices, respectively [3]. If we neglect the small correction proportional to the Cx=C C in the fourth term in Eq.() then the secon an fourth terms can be transforme to the kinetic energy of charges an vortices respectively []. The secon term gives S (q) kin = E ~ J We have introuce the mass tensor [] M m;n= q m q n _r mm (r m? r n ) _r n: (3) M (r) =?r r G () (r): (4) It ecreases proportional to r? for r a, an consequently may be approximate by a local function M (r) = M (r); M = a : Then the kinetic term for charges takes a simple form S (q) kin = a E ~ J M m= _r m (): (5) Similarly, the fourth term in Eq.() prouces the kinetic term for vortices

S (v) kin = a E C N m= _R m(): (6) Finally, the last term in Eq.()) is responsible for the interaction between charges an vortices. The corresponing term in Sfr,Rg is S qv = ic x v m q n C a mn The integral over r can be calculate explicitly, yieling r r Rm (R m? r )G () (r? r n ) _ Rm (): (7) with S qv =? m A m (R m ) = n iv m _R m ()A m (R m ) () q n a(r m ()? r n ()); The resulting action is a(r) =? 4a C x C ( + ln( =r))[^z r]: Sfr; Rg = S (q) int + S (v) int + S (q) kin + S (v) kin + S qv : () The action () is essentially that of two D Coulomb systems. The charges an the vortices can be viewe as particles with masses M q = a ~ E J an M v = 4a E C (3) respectively. Charges interact via the eective capacitance, vortices via the usual logarithmic interaction with strength E J. Furthermore, the vortices prouce the vector potential a for the charges 3 ; the magnetic el associate with this vector potential is B = ea C x C ln r ; a r : (3) Its sign epens of the signs of the corresponing vortex an charge. Apart from its quite peculiar functional form, another important feature of this el is the small factor C x =C. 3 This seeming asymmetry is rather articial. In Eq.() one can rewrite after a partial integration the charge-vortex interaction term in orer to obtain the vector potential for vortices, create by charges, as well. However, this vector potential contains always the small factor C x =C.

Summary We have investigate the system of two D Josephson junction arrays couple by capacitances C x, in the situation when the arrays an are in the charge an vortex regime, respectively. In the case of weak coupling C x C ; C the system shows an (approximate) uality between ynamical charges in one layer an ynamical vortices in the other one. In contrast to a single layer array, both variables are well-ene. The system is equivalent to two D Coulomb gases of massful particles. The charges feel the magnetic el create by vortices, an, vice versa, the vortices feel the gauge el create by charges. In this respect the system resembles the composite fermion moel of the fractional quantum Hall eect, however the magnetic el is now small an has another functional form, so one may expect ierent physics. In this regime the system shows two BKT transitions, one for charges an another for vortices, an the coupling between the layers suppresses both transitions. Although one coul expect the suppression of one transition an the enhancement of another one, the suppression of both transitions is rather natural, since the capacitance C x also renormalizes the capacitances of the islans to the groun. The BKT transition for charges vanishes even for very small values of C x, however, the BKT transition for vortices survives uner conition e =C eff E J irrespective of the relations between the capacitances C, C an C x. The shift of this temperature ue to the capacitance eects is calculate within the phase representation for both cases of weak an strong coupling. The eect of the layer is to renormalize the capacitance matrix in the layer. For weak coupling C x C (irrespective of C ) the vortices o not feel the presence of another layer, an the temperature remains the same as for one layer. However, for C x C ierent situations are possible. In summary, we woul like to emphasize that the system of two couple Josephson junction arrays may exhibit quite rich an interesting physics. We have investigate some limiting cases, however, the further rich behavior of this system can be expecte in other cases. In particular, the magnetic el create by vortices seems to be rather unusual an interesting. We hope that experimental stuies of this system will be performe in the near future. Acknowlegments The authors are grateful to R. Fazio, K.-H. Wagenblast, an A. D. aikin for useful iscussions. One of us (G. S.) acknowleges the hospitality of the Helsinki University of Technology, where the part of this work was one. The project was supporte by the Alexaner von Humbolt Founation (Y. M. B.) an by the DFG within the research program of the Sonerforschungbereich 5.

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