Two distinct Mott-Insulator to Bose-glass transitions and breakdown of self averaging in the disordered Bose-Hubbard model

We show that two fixed points govern the Mott insulator to Bose glass transition in the disordered Bose-Hubbard model. At incommensurate fillings, the correlation length and the inverse compressibility diverge with exponents of $\nu=1/D$ and $\gamma=4/D-1$, respectively, $D$ the spatial dimension. We show that it is the breakdown of self-averaging (rare-region Griffiths physics) in the Bose glass that leads to a violation of the bound $\nu\ge 2/D$. At commensurate fillings, the transition is controlled by a different fixed point at which both the disorder and interaction vertices are relevant.