Bose glass transition and spin-wave localization for 2D bosons in a random potential

A spin-wave approach of the zero temperature superfluid-insulator transition for two-dimensional hard-core bosons in a random potential $\mu =\pm$ W is developed. While at the classical level there is no intervening phase between the Bose-condensed superfluid (SF) and the gapped disordered insulator, the introduction of quantum fluctuations leads to a much richer physics. Upon increasing the disorder strength W, the Bose-condensed fraction disappears first, before the SF. Then a gapless Bose-glass phase emerges over a finite region until the insulator appears. Furthermore, in the strongly disordered SF regime, a mobility edge in the spin-wave excitation spectrum is found at a finite frequency ${\mathrm{\Omega }}_c$ decreasing with W, and presumably vanishing in the Bose-glass phase.