The fully frustrated XY model with next nearest neighbor interaction

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.

A variety of two dimensional systems undergo a phase transition without a rigorous symmetry breaking, the Berezinskii-Kosterlitz-Thouless (BKT) transition [1]. The transition is driven by the thermally excited vortices which form a two dimensional Coulomb gas [2]. Josephson junction arrays are experimental realization of the two-dimensional XY model where the array's parameters can be modified in a controlled way. In the last decade there has been a great amount of work on the various aspects of the BKT transition in Josephson arrays [3]. Experimental studies are based on electrical resistance [4], two-coil inductance [5], and SQUID [6] measurements.
A magnetic field applied perpendicularly to the array leads to frustration [7,8]. If the flux piercing the elementary plaquette is half of the flux quantum Φ 0 = hc/2e, the system is called fully frustrated (FF) and undergoes two phase transitions related to the Z 2 and U(1) symmetries. The existence of the two critical temperatures T Z 2 c and T U (1) c respectively has been extensively investigated both by analytical methods [8,9] and Monte Carlo (MC) simulations [10][11][12][13][14]. The complete scenario is not fully understood yet. There are numerical evidences either supporting the existence of two very close critical temperatures (T Z 2 c > T U (1) c ) with critical behavior typical of Ising and BKT transitions respectively [13][14][15] or the existence of a single transition with novel critical behavior [11].
Untill recently only the Josephson arrays with nearest neighbor (nn) couplings were studied. Recent theoretical [16] and experimental [17] works on infinite range array opened a new field of investigation in these systems. In this Letter we study the properties of a two dimensional FF Josephson array with both nn and next-to-nearest neighbor (nnn) couplings [18]. Proximity-junction arrays may be good candidates to experimentally probe the effects discussed in this work. They consist of superconducting islands in good electric contact with a metallic substrate. Due to the proximity effect there is a leakage of Cooper pairs into the normal substrate which extends over a temperature-dependent coherence length ξ N (ξ N =hv F /k B T or ξ N = hD/2πk B T for the ballistic and for the diffusive case respectively, v F is the Fermi velocity, D the diffusion constant and T the temperature). When ξ N becomes comparable with the lattice constant of the array the nnn coupling becomes comparable with the nn coupling. Since ξ N is strongly dependent on the temperature, the nnn Josephson coupling may be observed cooling down the sample. The main results of this Letter are summarized in the phase diagram of Fig. 5.
The system is defined by the Hamiltonian where J ij = J > 0 for nn and J ij = xJ for nnn (x ≥ 0), the symbol << . . . >> refers to the sum over nn and nnn. For later convenience we introduce the gauge invariant phase difference φ ij = θ i − θ j − A ij . The variables θ i are the phases of the superconducting order parameter of the i-th island. The magnetic field enters through A ij = (2π/Φ 0 ) j i A·dl (A is the vector potential). The relevant parameter which describes the magnetic frustration is f = A ij /(2π), where the summation runs over the perimeter of the elementary plaquette.
We study the case f = 1/2 on a square lattice.
Ground States -The model of Eq.( 1) combines the characteristics of both the FF and unfrustrated XY models. While the elementary square plaquette is FF, the square plaquettes formed by nnn couplings are not frustrated. For x < x 0 ≡ 1/ √ 2 we find that the ground state of this system is exactly the same as in the FF model without nnn couplings and is characterized by φ ij = ±π/4 for all pairs of nn sites. This state combines continuous [U(1)] and discrete (Z 2 ) degeneracies. Its energy per site E = − √ 2J is independent of nnn coupling xJ. Moreover if one considers a straight domain wall separating the two ground states with opposite orientations of chiralities it turns out that neither the form of such state nor its energy change with addition of nnn interactions.
For x > x 0 the ground state is the same as in the absence of nn coupling when the system splits into two unfrustrated XY -models. The relative phase shift between the two sublattices can be arbitrary. The energy of this state E = −2xJ depends only on nnn coupling xJ and does not depend on nn coupling.
At the special point x = x 0 the energies of both the above ground states coincide. Morover they can be transformed into each other by a continuous transformation without increasing the energy of the system. Therefore the manifold of the ground states also includes an additional set of eight-sublattice "intermediate" states which can be parametrised by a rotation angle χ (χ = π corresponding to low-x ground state and χ = 0 corresponding to high-x ground state with a particular relative phase shift between the sublattices) as it is shown in Fig. 1.
Phase Diagram -We studied the finite temperature behaviour of the model by means of the (low temperature) spin-wave free energy analysis and by Monte Carlo simulations.
For x < x 0 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 T Z 2 c and T U (1) c on x, due to the change of the effective interaction between the different types of fluctuations.
For x > x 0 the system (at finite temperatures) turnes out to be equivalent to two coupled XY models, the effective forth-order coupling between the two sublattices provided by the free energy of spin waves. Although this coupling is weak (always much smaller than the temperature) at low temperatures it is relevant and imposes the presence of a transition at T = T D . This transition separates the phases with coupled and decoupled sublattices. In the low temperature phase, where the two sublattices are locked, the spin-wave contribution to the free energy imposes a relative phase shift of ±π/4 (or equivalently ±3π/4) between the two sublattices. At T > T D a second phase transition of the BKT type takes place in each of the decoupled sublattices. For x ≫ x 0 the temperature of this transition depends only on nnn coupling and it is proportional to x.
The spin wave spectrum remains rigid down to x = x 0 . This indicates (and it is confirmed by the MC simulations) that the critical temperature of this BKT transition remain finite when x → x + 0 . Below this temperature there is a transition between the FF low-x and the unfrustrated high-x phases. We evaluated numerically the spin wave free energy of the intermediated ground state as a function of χ. This dependence is described by a convex function implying a first order phase trasition line separating the low-x and high-x quasiordered phases.
The critical properties were investigated evaluating the helicity modulus Γ and the staggered chiral magnetization M by means of standard MC simulations for different x. The order parameter M, which controls the Ising-like transition, is defined as where r i /a = (i x , i y ) is the position vector (in unit of lattice step a) of the site i and m i = 1 √ 8 (sin φ i,j 1 + sin φ j 1 ,j 2 + sin φ j 2 ,j 3 + sin φ j 3 ,i ) is the chirality of the plaquette with center in (i x + 1/2, i y + 1/2) and with site indexes i,j 1 ,j 2 ,j 3 (in clockwise order).
The helicity modulus Γ = ∂ 2 F /∂δ 2 , used to signal the existence of a BKT transition, is defined through the increase of the free energy F due to a phase twist δ imposed in one direction [19]. In order to obtain a precise determination of T Z 2 c and of the critical exponent ν associated to the divergence of the correlation length we have calculated the Binder's cumulant [20] of the staggered chiral magnetization M Since U(T Z 2 c , L) does not depend on lattice size L for large systems, T Z 2 c can be identified without making any assumption on the critical exponents. Once a satisfactory estimation of T Z 2 c is obtained the critical exponent ν is estimated through a data collapsing with ν left as the only free parameter. Estimation of U M have been obtained averaging, at least, 10 7 · L 2 MC configurations by using a standard Metropolis algorithm. The largest lattice studied is L = 72. The result for x = 0.5 are shown in Figs.2,3a. We estimate k B T Z 2 c /J = 0.403±0.003. The data collapsing, shown in the inset of Fig.2, gives an estimate of 1/ν = 1.0 ± 0.1. The critical temperature T Z 2 c decreases with increasing x for x < x 0 . For x ≥ x 0 , there is no sign of a Ising-like transition (see Fig.3b).
Following the procedure proposed in Refs. [21,22], the critical temperature T U (1) c is estimated by using the following ansatz for the size dependence of Γ where l 0 is a fit parameter. This critical scaling is based on the mapping between a neutral Coulomb gas and a XY model. Therefore Eq.(4) can be used both as a test for the existence of a BKT transition and for a precise evaluation of the critical temperature. A very good scaling was obtained with γ = 1 (the ordinary BKT transition) in the low-x phase and γ = 2 (corresponding to BKT transition on each of the two sublattices with lattice constant √ 2) in the high-x phase. In Fig.4  This form of the order parameter is chosen in such way that for any of the high-x ground states the value of s will be the same for all the bonds. At low temperatures S manifests a true long-range order. The MC simulations confirm the π/4 relative phase shift anticipated by the spin-wave analysis. By increasing the temperature the long-range order in S can be expected to disappear as a separate phase transition whereas the unbinding of vortex pairs in each of the sublattices has to occur at still higher temperatures. We performed MC simulations to evaluate the transition temperature T D . The results of this computation (not reported here) show that T D is very close, but lower, than the transition to the disordered phase T U (1) c (x). More extensive simulations are needed to determine the critical behaviour of decoupling transition. The dotted line in Fig.5 shows the qualitative behaviour of the decoupling transition as a function of x.
In conclusion we have introduced a frustrated XY model with nnn interaction. The model can be experimentally realized in Josephson junction arrays in a transverse magnetic field. Signatures of nnn Josephson couplings might already have been seen in specially designed setups [23]. The analysis presented here leads to the phase diagram shown in Fig.5. For 0 < x < x 0 the critical temperatures T Z 2 c and T U (1) c (x) decrease with increasing x. For x > x 0 , there is no sign of a Ising-like transition and the system behaves like the unfrustrated XY model. At x ≈ x 0 in the low T region there is a first order phase transition between the low and high-x phases. Finally for x > x 0 the array undergoes a sequence of two transitions. On raising the temperature, first the two sublattices become decoupled and then, at higher temperatures, each sublattice becomes disordered.
We thank H. Courtois and B. Pannetier for fruitful discussions. We acknowledge the financial support of INFM (PRA-QTMD) and the European Community (Contract FMRX-CT-97-0143). G.F. is grateful to A. Mastellone and J. Siewert for hospitality in Catania. x is the ratio between nnn and nn couplings; J is the nn coupling. The errors are smaller than the symbols size (see text for details). The dotted line shows the qualitative behaviour of the transition associated to the decoupling of the two nnn sublattices.