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I am currently doing an undergraduate research project on the Bose-Hubbard model with hardcore bosons, and I am studying it theoretically using the Monte Carlo Stochastic Series Expansion method. I would very much like to ask my supervisor this question, but he seems extremely busy lately so I will try asking here first.

What order parameter does one use to determine if for a given set of Hamiltonian parameter values, the system is in a Mott Insulator or superfluid state? I've read many papers and some say to calculate compressibility, others say to calculate superfluid density. As for superfluid density, how does one tell exactly from the value that the system is in MI/SF state? Papers I've read say MI state has a low superfluid density, while SF state has a high superfluid density. What does it mean by low or high? It seems so subjective.

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# I Phase diagram of MI/SF transition Bose-Hubbard model

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