Critical behavior at superconductor-insulator phase transitions near one dimension

I argue that the system of interacting bosons at zero temperature and in random external potential possesses a simple critical point that describes the proliferation of disorder-induced defects in the superfluid ground state, and that is located at weak disorder close to and above one dimension. This makes it possible to address the critical behavior at the superfluid-Bose glass transition in dirty boson systems by expanding around the lower critical dimension of $d=1.$ Within the formulated renormalization procedure near $d=1,$ the dynamical critical exponent is obtained exactly and the correlation length critical exponent is calculated as a Laurent series in the parameter √ε, with ε=d-1: $z=d,$ ν=1/√3ε+O(√ε) for the short-range, and $z=1,$ ν=√2/3ε+O(√ε), for the long-range Coulomb interaction between bosons. The identified critical point should be stable against the residual perturbations in the effective action for the superfluid, at least in dimensions $1<~d<~2,$ for both short-range and Coulomb interactions. For the superfluid-Mott insulator transition in the system in a periodic potential and at a commensurate density of bosons, I find ν=(1/2√ε)+1/4+O(√ε), which yields a result in reasonable agreement with the known XY critical exponent in $d=2.$ The critical behavior of the superfluid density, phonon velocity, and the compressibility in the system with short-range interactions is discussed.