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

We study avalanches along the hysteresis loop of long-range interacting spin-glasses with continuous XY symmetry - which serves as a toy model of granular superconductors with long-range and \ frustrated Josephson couplings. We identify sudden jumps in the $T=0$ configurations of the XY phases, as an external field is increased. They are initiated by the softest mode of the inverse\ susceptibility matrix becoming unstable, which induces an avalanche of phase updates (or spin alignments). We analyze the statistics of these events, and study the correlation between the no\ n-linear avalanches and the soft mode that initiates them. We find that the avalanches follow the directions of a small fraction of the softest modes of the inverse susceptibility matrix, sim\ ilarly as was found in avalanches in jammed systems. In contrast to the similar Ising spin-glass (Sherrington-Kirkpatrick) studied previously, we find that avalanches are not distributed with\ a scale-free power law, but rather have a typical size which scales with the system size.