Annealed Low Energy States in Frustrated Large Square Josephson Junction Arrays

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/d f 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.

Cumulative geometric frustration and superextensive energy scaling in a nonlinear classical XY-spin model

“…The frustrated Ising anti-ferromagnet on a triangular lattice does not exhibit any of the exotic morphological response properties described above. Additional types of frustrated spin systems include the XY model frustrated by a uniform magnetic flux through the lattice [15][16][17] and the XY model embedded on hyperbolic surfaces that is frustrated due to the curvature of the embedding space [18]. These works do not exhibit super-extensive energy scaling nor long-range cooperativity in the ground-state solutions in the studied regimes of large geometric frustration.…”

Non-trivial attenuation of geometric frustration by discretization of the degrees of freedom: A study of a Potts-like frustrated spin system

“…Geometric frustration arises whenever the short-range interactions between the constituents of a system favor the formation of local motifs that are incompatible with long range propagation in the ambient space geometry or topology. Frustrated assemblies occur naturally in a wide variety of systems from spin systems [1][2][3][4], to liquidcrystals [5][6][7][8], to twisted molecular crystals [9,10] and biological fibrillar assemblies [11]. As the locally favored arrangement of the constituents cannot be realized, any finite assembly must exhibit some compromise of these locally favored tendencies.…”