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In this paper, we study the statistics of permutation cycles of ground-statehardcore lattice bosons described by various two-dimensional Bose-Hubbard-typemodels on both square and Kagome lattices. We find that it is possible todifferentiate quantum phases by the statistics of permutations cycles. Indeed,features in the permutation cycles statistics can be used to uniquely identifycertain insulating phases, and are consistent with local resonances ofoccupation numbers in the ground-state expansion of the phase. We also confirmthat suitable quantities derived from the probability distribution of thelength of permutation cycles can be used to detect superfluid to insulatorphase transitions.

Permutation cycles of hardcore Bose-Hubbard models on square and Kagome lattices