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We study avalanches along the hysteresis loop of long-range interactingspin-glasses with continuous XY-symmetry - which serves as a toy model ofgranular superconductors with long-range and frustrated Josephson couplings. Weidentify sudden jumps in the $T=0$ configurations of the XY-phases, as anexternal field is increased. They are initiated by the softest mode of theinverse susceptibility matrix becoming unstable, which induces an avalanche ofphase updates (or spin alignments). We analyze the statistics of these events,and study the correlation between the non-linear avalanches and the soft modethat initiates them. We find that the avalanches follow the directions of asmall fraction of the softest modes of the inverse susceptibility matrix,similarly as was found in avalanches in jammed systems. In contrast to thesimilar Ising spin-glass (Sherrington-Kirkpatrick) studied previously, we findthat avalanches are not distributed with a scale-free power law, but ratherhave a typical size which scales with the system size. We also observe that theHessians of the spin-glass minima are not part of standard random matrixensembles as the lowest eigenvector has a fractal support.

Avalanches and hysteresis in frustrated superconductors and XY-spin-glasses