Abstract
Numerical simulations were done to find low energy states in frustrated large square Josephson Junction arrays in a perpendicular magnetic field using simulated annealing on the coupled RSJ model. These simulations were made possible by a new algorithm suitable for parallel gpu computing and reduced complexity. Free boundary conditions were used so that values of the frustration factor f that are incommensurate with the array size are permitted. The resulting energy as a function of f is continuous with logarithmic discontinuities in the derivative dE/df at rational frustration factors f=p/q with small q, substantiating the mathematical proof that this curve is continuous and further showing that the staircase state hypothesis is incorrect. The solution shows qualitative similarities with the lowest energy branch of the Hofstadter butterfly, which is a closely related problem. Furthermore, it is found that at the edge of an array there are either extra vortices or missing vortices depending the frustration factor, and the width of this region is independent of the array size.
Highlights
The ground state configurations of infinite square Josephson Junction arrays in a uniform perpendicular magnetic field have been studied in great detail in the past [1,2,3,4,5,6,7,8,9,10,11,12], but the problem is not fully solved
It is known that the mean number of vortices per unit cell is equal to the frustration factor f = Φ/Φ0 where Φ is the magnetic flux threading a unit cell
At rational values of the frustration factor f = p/q it was suspected that ground states have vortex configurations which are periodic with a unit cell of size q by q [1], but it was later shown that for some fractions a lower energy state can be found with a tile of size 2q by 2q [4,11]
Summary
The ground state configurations of infinite square Josephson Junction arrays in a uniform perpendicular magnetic field have been studied in great detail in the past [1,2,3,4,5,6,7,8,9,10,11,12], but the problem is not fully solved. The behavior at frustration factors close to these small fractions is not fully understood except around f = 0 where the problem reduces to that of a Coulomb gas [15] and the vortex configurations are strained triangular lattices superimposed on a square lattice [11]. Knowledge of these ground state configurations can help understanding the recent experiments in the current driven regime [16,17]. After that the vortex configurations and the edge effects are discussed, and a comparison is made with the lowest branch of the Hofstadter butterfly
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