We introduce a fully frustrated XY model with nearest neighbor (nn) and next
nearest neighbor (nnn) couplings which can be realized in Josephson junction
arrays. We study the phase diagram for $0\leq x \leq 1$ ($x$ is the ratio
between nnn and nn couplings). When $x < 1/\sqrt{2}$ an Ising and a
Berezinskii-Kosterlitz-Thouless transitions are present. Both critical
temperatures decrease with increasing $x$. For $x > 1/\sqrt{2}$ the array
undergoes a sequence of two transitions. On raising the temperature first the
two sublattices decouple from each other and then, at higher temperatures, each
sublattice becomes disorderd.Comment: 11 pages, 5 figure

C. L. Henley

Ch. Leemann

D. J. Resnick

E. Branchina

E. Granato

G. Grest

G. Ramirez-Santiago

Giancarlo Franzese

H. A. Cerdeira

H. R. Shea

H. S. J. van der Zant

H. Weber

J. E. Mooij

J. Lee

J. M. Kosterlitz

K. Binder

M. Y. Choi

M. Yosefin

P. Chandra

P. Minnhagen

P. Olsson

P. Simon

Rosario Fazio

S. E. Korshunov

S. Teitel

T. C. Halsey

T. J. Shaw

T. Ohta

V. L. Berezinskii

Vittorio Cataudella

W. Y. Shih

English

arXiv

FAZIO, Rosario

CATAUDELLA, VITTORIO

Archivio della ricerca - Università degli studi di Napoli Federico II

Fully frustrated XY model with next-nearest-neighbor interaction

Franzese G

Cataudella V

Korshunov SE

FAZIO, ROSARIO

Archivio istituzionale della Ricerca - Scuola Normale Superiore

We investigate a fully frustrated XY model with nearest-neighbor ~NN! and next-nearest-neighbor ~NNN!
couplings which can be realized in Josephson junction arrays. We study the phase diagram for 0<x<1 (x is
the ratio between NNN and NN couplings!. When x,1/A2 an Ising transition and a Berezinskii-KosterlitzThouless
transition are present. Both critical temperatures decrease with increasing x. For x.1/A2 the array
undergoes a sequence of two transitions. On raising the temperature first the two sublattices decouple from
each other and then, at higher temperatures, each sublattice becomes disordered. The structure of phase
diagram remains the same if weak interaction with further neighbors is included

Franzese, Giancarlo

Cataudella, Vittorio

Korshunov, S. E.

Fazio, R.

Diposit Digital de la Universitat de Barcelona

RAPID COMMUNICATIONSPHYSICAL REVIEW B 1 OCTOBER 2000-IIVOLUME 62, NUMBER 14Fully frustrated XY model with next-nearest-neighbor interactionGiancarlo Franzese, 1,2,3,* Vittorio Cataudella,1,2 S. E. Korshunov,4 and Rosario Fazio1,51Istituto Nazionale per la Fisica della Materia (INFM), Unita` di Napoli e Catania, Italy2Dipartimento di Scienze Fisiche, Universita´ ‘‘Federico II,’’ Mostra d’Oltremare, Padiglione 19, 80125 Napoli, Italy3Dipartimento di Fisica ‘‘E. Amaldi,’’ Universita` Roma Tre, via della Vasca Navale 84, 00146 Roma, Italy4L. D. Landau Institute for Theoretical Physics, Kosygina 2, 117940 Moscow, Russia5Dipartimento di Metodologie Fisiche e Chimiche (DMCFI), Universita` di Catania, viale A. Doria 6, 95125 Catania, Italy~Received 24 July 2000!We investigate a fully frustrated XY model with nearest-neighbor ~NN! and next-nearest-neighbor ~NNN!couplings which can be realized in Josephson junction arrays. We study the phase diagram for 0<x<1 (x isthe ratio between NNN and NN couplings!. When x,1/A2 an Ising transition and a Berezinskii-Kosterlitz-Thouless transition are present. Both critical temperatures decrease with increasing x. For x.1/A2 the arrayundergoes a sequence of two transitions. On raising the temperature first the two sublattices decouple fromeach other and then, at higher temperatures, each sublattice becomes disordered. The structure of phasediagram remains the same if weak interaction with further neighbors is included.A variety of two-dimensional systems undergo a phasetransition without a rigorous symmetry breaking, theBerezinskii-Kosterlitz-Thouless ~BKT! transition.1 The tran-sition is driven by the thermally excited vortices which forma two-dimensional Coulomb gas.2 Josephson-junction arraysare experimental realization of the two-dimensional XYmodel where the array’s parameters can be modified in acontrolled way. In the last decade there has been a greatamount of work on the various aspects of the BKT transitionin Josephson arrays.3 Experimental studies are based on elec-trical resistance,4 two-coil inductance,5 and superconductingquantum interference device6 measurements.A magnetic field applied perpendicularly to the arrayleads to frustration.7,8 If the flux piercing the elementaryplaquette is half of the flux quantum F05hc/2e , the systemis called fully frustrated ~FF! and undergoes two phase tran-sitions related to the Z2 and U~1! symmetries. The existenceof the two critical temperatures TcZ2 and TcU(1), respectivelyhas been extensively investigated both by analyticalmethods8,9 and Monte Carlo ~MC! simulations.10–14 Thecomplete scenario is not fully understood yet. There are nu-merical evidences either supporting the existence of two veryclose critical temperatures (TcZ2.TcU(1)) with critical behav-ior typical of Ising and BKT transitions, respectively,13–15 orthe existence of a single transition with unusual criticalbehavior.11In this paper we study the properties of a two-dimensionalFF Josephson array with both nearest-neighbor ~NN! andnext-to-nearest neighbor ~NNN! couplings.16,17 The analo-gous system with unfrustrated NN interaction has been con-sidered by Henley.18 Proximity-junction arrays may be goodcandidates to experimentally probe the effects discussed inthis work. They consist of superconducting islands in goodelectric contact with a metallic substrate. Due to the proxim-ity effect there is a leakage of Cooper pairs into the normalsubstrate which extends over a temperature-dependent coher-ence length jN (jN5\vF /kBT or jN5A\D/2pkBT for theballistic and for the diffusive case, respectively, vF is thePRB 620163-1829/2000/62~14!/9287~4!/$15.00Fermi velocity, D the diffusion constant, and T the tempera-ture!. When jN becomes comparable with the lattice constantof the array the NNN coupling becomes comparable with theNN coupling. Since jN is strongly dependent on the tempera-ture, the NNN Josephson coupling may be observed coolingdown the sample. The main results of this paper are summa-rized in the phase diagram of Fig. 5.The system is defined by the HamiltonianH52 (^^i , j&&Ji j cos~u i2u j2Ai j!, ~1!where Ji j5J.0 for NN and Ji j5xJ for NNN (x>0), thesymbol ^^&& refers to the sum over NN and NNN. Herewe treat them as independent coupling. In the experimentalsituation the ratio of the couplings changes with temperaturein accordance with temperature dependence of the coherencelengthFor later convenience we introduce the gauge invariantphase difference f i j5u i2u j2Ai j . The variables u i are thephases of the superconducting order parameter of the ithisland. The magnetic field enters through Ai j5(2p/F0)* ijA"dl (A is the vector potential!. The relevantparameter which describes the magnetic frustration is f5(Ai j /(2p), where the summation runs over the perimeterof the elementary plaquette. We study the case f 51/2 on asquare lattice.Ground states. The model of Eq. ~1! combines the char-acteristics of both the FF and unfrustrated XY models. Whilethe elementary square plaquette is FF, the value of the mag-netic flux for the square plaquettes formed by NNN cou-plings is equal to F0 and therefore NNN interaction ~consid-ered by itself! is not frustrated. For x,x0[1/A2 we find thatthe ground state of this system is exactly the same as in theFF model without NNN couplings and is characterized byf i j56p/4 for all pairs of NN sites. This state combinescontinuous @U~1!# and discrete (Z2) degeneracies. Its energyper site E52A2J is independent of NNN coupling xJ .Moreover if one considers a straight domain wall separatingR9287 ©2000 The American Physical SocietyRAPID COMMUNICATIONSR9288 PRB 62FRANZESE, CATAUDELLA, KORSHUNOV, AND FAZIOthe two ground states with opposite orientations of chiralitiesit turns out that neither the form of such state nor its energychange with addition of NNN interactions.For x.x0 the ground state is the same as in the absenceof NN coupling when the system splits into two unfrustratedXY models. The relative phase shift between the two sublat-tices can be arbitrary. The energy of this state E522xJdepends only on NNN coupling xJ and does not depend onNN coupling.At the special point x5x0 the energies of both the aboveground states coincide. Moreover they can be transformedinto each other by a continuous transformation without in-creasing the energy of the system. Therefore the manifold ofthe ground states also includes an additional set of eight-sublattice ‘‘intermediate’’ states which can be parametrizedby a rotation angle x (x5p corresponding to low-x groundstate and x50 corresponding to high-x ground state with aparticular relative phase shift between the sublattices! as isshown in Fig. 1.Quite remarkably the inclusion of weak interaction withfurther neighbors ~at least those of the third and the fourthorder! does not lead to any qualitative changes, but only tothe shift of x0. In particular, all the ground states remain thesame and for x.x0 the energy remains independent of thephase shift between the sublattices.Phase diagram. We studied the finite-temperature behav-ior of the model by means of the ~low-temperature! spin-wave free energy analysis and by Monte Carlo simulations.For x,x0 neither the spectrum of spin waves ~in the long-wavelength limit! nor the domain wall energy depend on x.Therefore we can expect only a weak dependence of TcZ2 andTcU(1) on x, due to the change of the effective interactionbetween the different types of fluctuations.For x.x0 the system ~at finite temperatures! turns out tobe equivalent to two coupled XY models, the effective cou-pling between the two sublattices of the form cos 4(u2u8)provided by the free energy of small amplitude order param-eter fluctuations ~spin waves!. Although this coupling isweak ~always much smaller than the temperature! at lowtemperatures it is relevant and imposes the presence of atransition at the temperature T5TD for which both sublat-tices in absence of coupling would be in the quasiorderedFIG. 1. The family of the additional ~intermediate! ground statesat the degeneracy point x51/A2 is characterized by the angle x .The value x5p corresponds to the low-x ground state while x50 corresponds to the high-x ground state with a particular relativephase shift between the two sublattices. The angle reported in thefigure is the gauge invariant phase difference between two neigh-boring sites.state.19,20 This transition separates the phases with coupledand decoupled sublattices and is related with an additionalbreaking of a discrete symmetry group. In the low-temperature phase, where the two sublattices are locked, thespin-wave contribution to the free energy imposes a relativephase shift of 6p/4 ~or equivalently 63p/4) between thetwo sublattices. At T.TD a second phase transition of theBKT type takes place in each of the decoupled sublattices.For x@x0 the temperature of this transition depends only onNNN coupling and it is proportional to x.The spin-wave spectrum remains rigid down to x5x0.This indicates ~and it is confirmed by the MC simulations!that the critical temperature of this BKT transition remainsfinite when x→x01 . Below this temperature there is a tran-sition between the FF low-x and the unfrustrated high-xphases. We evaluated numerically the spin-wave free energyof the intermediate ground state as a function of x . Thisdependence is described by a convex function implying afirst-order phase transition line separating the low-x andhigh-x quasiordered phases.The critical properties were investigated evaluating thestaggered chiral magnetization M and the helicity modulus Gby means of standard MC simulations for different x. Theorder parameter M, which controls the Ising-like transition,is defined asM51L2 U(i ~21 ! ix1iymiU , ~2!where rW i /a5(ix ,iy) is the position vector ~in unit of latticestep a) of the site i and mi5(1/A8)(sin fi,j11sin fj1 ,j21sin fj2 ,j31sin fj3 ,i) is the chirality of the plaquette withcenter in (ix11/2,iy11/2) and with site indexes i , j1 , j2 , j3~in clockwise order!.In order to obtain a precise determination of TcZ2 and ofthe critical exponent n associated to the divergence of thecorrelation length we have calculated the Binder’scumulant21 of the staggered chiral magnetization MUM512^M 4&3^M 2&2. ~3!Since UM(TcZ2,L) does not depend on lattice size L for largesystems, TcZ2 can be identified without making any assump-tion on the critical exponents. Once a satisfactory estimationof TcZ2 is obtained the critical exponent n is estimatedthrough a data collapsing with n left as the only free param-eter. Estimations of UM have been obtained averaging, atleast, 107L2 MC configurations by using a standard Metropo-lis algorithm. The largest lattice studied is L572. The resultsfor x50.5 are shown in Figs. 2 and 3~a!. We estimatekBTcZ2/J50.40360.003. The data collapsing, shown in theinset of Fig. 2, give an estimate of 1/n51.060.1. The criti-cal temperature TcZ2 decreases with increasing x for x,x0.For x>x0, there is no sign of a Ising-like transition @see Fig.3~b!#.The helicity modulus G5]2F/]d2, used to signal the ex-istence of a BKT transition, is defined through the increaseof the free energy F due to a phase twist d imposed in oneRAPID COMMUNICATIONSPRB 62 R9289FULLY FRUSTRATED XY MODEL WITH NEXT- . . .FIG. 2. The Binder’s parameter UM vs T for several lattice sizesL at x50.5. Errors are smaller than the symbol size. Excluding thedata of smallest size (L524) one can estimate 0.400,kBTcZ2/J,0.406. Inset: Collapse of the data ~excluding the L524 points!.The scaling parameters are kBTcZ2/J50.40360.003 and 1/n51.060.1.FIG. 3. The staggered chiral magnetization M vs T for x50.5~a! and x50.8 ~b!. For large L and low T , M goes to a nonzerovalue for x50.5 and vanishes for x50.8. The errors are smallerthan the symbol size. The symbols for different lattice sizes are thesame in ~a! and ~b!.direction.22 Following the procedure proposed in Refs. 23and 24, the critical temperature TcU(1) is estimated by usingthe following ansatz for the size dependence of G:pG2Tc5gF11 12~ ln L2ln l0!G , ~4!where l0 is a fit parameter. This critical scaling is based onthe mapping between a neutral Coulomb gas and an XYmodel. Therefore Eq. ~4! can be used both as a test for theexistence of a BKT transition and for a precise evaluation ofthe critical temperature. A very good scaling was obtainedwith g51 ~the ordinary BKT transition! in the low-x phaseand g52 ~corresponding to the BKT transition on each ofthe two sublattices with lattice constant A2) in the high-xphase. In Fig. 4 we show this analysis for the cases x50.5and x51. TcU(1)(x), as well as TcZ2(x), decreases with in-creasing x up to x;x0. Our results cannot discriminate be-tween the TcZ25TcU(1) and the TcZ2.TcU(1) hypothesis sincethe two temperatures are compatible within the numericalprecision @the mean value of TcZ2(x) remains always abovethe corresponding mean value of TcU(1)(x)#. For x>0.8, in-stead, TcU(1)(x) increases, quickly tending towards the valueexpected for x→‘ , i.e., kBTcU(1)/(xJ)50.89.We finally discuss the transition related to the decouplingof sublattices in the high-x phase. The order parameter S5( j ,as j , j1ea can be defined on the bonds of the lattice andcan be chosen in the following gauge-invariant form:s j , j1ex5~21 !jx exp@ i~21 ! jx1 jyf j , j1ex# ,FIG. 4. The size dependence of the helicity modulus G for x50.5 and x51 at several temperatures T. The errors are smallerthan the symbol size. The form of the plot @gT/(pG22gT) vsln L] is chosen in such a way that the scaling behavior predicted byEq. ~4! should correspond to a straight line with the slope equal to1 ~as shown by the overimposed lines!. The estimates for the criticaltemperature TcU(1) are 0.4 and 0.7, respectively. The error on theestimates is due to the used temperature mesh.RAPID COMMUNICATIONSR9290 PRB 62FRANZESE, CATAUDELLA, KORSHUNOV, AND FAZIOs j , j1ey5i~21 !jy exp@ i~21 ! jx1 jyf j , j1ey# .This form of the order parameter is chosen in such way thatfor any of the high-x ground states the value of s will be thesame for all the bonds. At low temperatures S manifests atrue long-range order. The MC simulations confirm the p/4relative phase shift anticipated by the spin-wave analysis. ByFIG. 5. The phase diagram for the XY model with NN1NNNinteractions. Squares refer to TcZ2, circles to TcU(1) ; x is the ratiobetween NNN and NN couplings; J is the NN coupling. The errorsare smaller than the symbol size ~see text for details!. The dottedline shows the qualitative behavior of the transition associated tothe decoupling of the two NNN sublattices.increasing the temperature the long-range order in S can beexpected to disappear as a separate phase transition whereasthe unbinding of vortex pairs in each of the sublattices has tooccur at still higher temperatures. We performed MC simu-lations to evaluate the transition temperature TD . The pre-liminary results of this computation ~not reported here! indi-cate that TD is very close, but lower, than the transition to thedisordered phase TcU(1)(x). More extensive simulations areneeded to determine the critical behavior of the decouplingtransition. The dotted line in Fig. 5 shows qualitatively theexpected behavior of the decoupling transition temperatureas a function of x.In conclusion we have investigated a frustrated XY modelwith NNN interaction. The model can be experimentally re-alized in Josephson-junction arrays in a transverse magneticfield. Signatures of NNN Josephson couplings might alreadyhave been seen in specially designed setups.25 The analysispresented here leads to the phase diagram shown in Fig. 5.For 0,x,x0 the critical temperatures TcZ2 and TcU(1)(x) de-crease with increasing x. For x.x0, there is no sign of anIsing-like transition and the system behaves like the unfrus-trated XY model. At x’x0 in the low-T region there is afirst-order phase transition between the low- and high-xphases. Finally for x.x0 the array undergoes a sequence oftwo transitions. On raising the temperature, first the two sub-lattices become decoupled and then, at higher temperatures,each sublattice becomes disordered. The structure of thephase diagram remains the same if weak interaction withfurther neighbors is includedWe thank H. Courtois and B. Pannetier for fruitful discus-sions. We acknowledge the financial support of INFM ~PRA-QTMD! and the European Community ~Contract No.FMRX-CT-97-0143!. G.F. is grateful to A. Mastellone and J.Siewert for hospitality in Catania.*Present address: Center for Polymer Studies, Boston University,590 Commonwealth Avenue, Boston, MA 02215.1 V. L. Berezinskii, Zh. E´ ksp. Teor. Fiz. 59, 207 ~1970! @Sov. Phys.JETP 32, 493 ~1971!#; J. M. Kosterlitz and D. J. Thouless, J.Phys. C 6, 1181 ~1973!.2 P. Minnhagen, Rev. Mod. Phys. 59, 1001 ~1987!.3 Proceedings of the NATO ARW on Coherence in SuperconductingNetworks, edited by J. E. Mooij and G. Scho¨n @Physica B 152, 1~1988!#; Proceedings of the ICTP Workshop on Josephson Junc-tion Arrays, edited by H. A. Cerdeira and S. R. Shenoy @PhysicaB 222, 253 ~1996!#.4 D. J. Resnick et al., Phys. Rev. Lett. 47, 1542 ~1981!; H. S. J. vander Zant, H. A. Rijken, and J. E. Mooij, J. Low Temp. Phys. 82,67 ~1991!.5 Ch. Leemann et al., Phys. Rev. Lett. 56, 1291 ~1986!.6 T. J. Shaw et al., Phys. Rev. Lett. 76, 2551 ~1996!.7 W. Y. Shih and D. Stroud, Phys. Rev. B 30, 6774 ~1984!; M. Y.Choi and S. Doniach, ibid. 31, 4516 ~1985!.8 T. C. Halsey, Phys. Rev. B 31, 5728 ~1985!.9 E. Granato and M. P. Nightingale, Phys. Rev. B 48, 7438 ~1993!.10 S. Teitel and C. Jayaprakash, Phys. Rev. B 27, 598 ~1983!; Phys.Rev. Lett. 51, 1999 ~1983!.11 J. Lee, J. M. Kosterlitz, and E. Granato, Phys. Rev. B 43, 11 531~1991!.12 G. Ramirez-Santiago and J. V. Jose´, Phys. Rev. B 49, 9567~1994!.13 P. Olsson, Phys. Rev. Lett. 75, 2758 ~1995!.14 G. Grest, Phys. Rev. B 39, 9267 ~1989!.15 P. Olsson, Phys. Rev. B 55, 3585 ~1997!.16 Models with competing interaction have been widely studied inthe literature. See Magnetic Systems with Competing Interac-tions, edited by H. T. Diep ~World Scientific, Singapore, 1994!;P. Simon, J. Phys. A 30, 2653 ~1997!; E. Branchina, H. Mohr-bach, and J. Polonyi, Phys. Rev. D 60, 045006 ~1999!.17 Josephson arrays with infinite range array were investigated in P.Chandra, L. B. Ioffe, and D. Sherrington, Phys. Rev. Lett. 75,713 ~1995!; H. R. Shea and M. Tinkham, ibid. 79, 2324 ~1997!.18 C. L. Henley, Phys. Rev. Lett. 62, 2056 ~1989!.19 M. Yosefin and E. Domany, Phys. Rev. B 32, 1778 ~1985!.20 M. Y. Choi and D. Stroud, Phys. Rev. B 32, 5773 ~1985!.21 K. Binder, Z. Phys. B: Condens. Matter 43, 119 ~1981!.22 T. Ohta and D. Jasnow, Phys. Rev. B 20, 139 ~1979!.23 H. Weber and P. Minnhagen, Phys. Rev. B 37, 5986 ~1988!.24 P. Olsson, Phys. Rev. B 52, 4526 ~1995!.25 L. Glemot, B. Schmied, D. Achermann, P. Scheuzger, Ch. Lee-mann, and P. Martinoli ~unpublished!.

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Fully frustrated<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>XY</mml:mi></mml:math>model with next-nearest-neighbor interaction