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Superfluid clusters, percolation and phase transitions in the disordered, two dimensional Bose-Hubbard model

The Bose glass (BG) phase is the Griffiths region of the disordered Bose Hubbard model (BHM), characterized by finite, quasi-superfluid clusters within a Mott insulating background. We propose to utilize this characterization to identify the complete zero-temperature phase diagram of the disordered BHM in $d\ge2$ dimensions by analyzing the geometric properties of what we call superfluid (SF) clusters, which are defined to be clusters of sites with non-integer expectation values for the local boson occupation number. The Mott insulator (MI) phase then is the region in the phase diagram where no SF clusters exist, and the SF phase the region, where SF clusters percolate - the BG phase is in between: SF clusters exist, but do not percolate. This definition is particularly useful in the context of local mean field (LMF, or Gutzwiller-Ansatz) calculations, where we show that an identification of the phases on the basis of global quantities like the averaged SF order parameter and the compressibility are misleading. We apply the SF cluster analysis to the LMF ground states of the two dimensional disordered BHM to produce its phase diagram and find a) an excellent agreement with the phase diagram predicted on the basis of quantum Monte Carlo simulations for the commensurate density $n=1$, and b) large differences to stochastic mean field and other mean field predictions for fixed disorder strength. The relation of the percolation transition of the SF clusters with the onset of non-vanishing SF stiffness indicating the BG to SF transition is discussed.