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Muon-spin rotation measurements of the vortex state in Sr$_2$RuO$_4$: type-1.5 superconductivity, vortex clustering and a crossover from a triangular to a square vortex lattice

Muon-spin rotation has been used to probe vortex state in Sr$_2$RuO$_4$. At moderate fields and temperatures a lattice of triangular symmetry is observed, crossing over to a lattice of square symmetry with increasing field and temperature. At lower fields it is found that there are large regions of the sample that are completely free from vortices which grow in volume as the temperature falls. Importantly this is accompanied by {\it increasing} vortex density and increasing disorder within the vortex-cluster containing regions. Both effects are expected to result from the strongly temperature-dependent long-range vortex attractive forces arising from the multi-band chiral-order superconductivity.