Magnetically ordered phase near transition to Bose-glass phase

We discuss magnetically ordered ("superfluid") phase near quantum transition to Bose-glass phase in a simple modeling system, Heisenberg antiferromagnet in spatial dimension $d>2$ in external magnetic field with disorder in exchange coupling constants. Our analytical consideration is based on hydrodynamic description of long-wavelength excitations. Results obtained are valid in the entire critical region near the quantum critical point (QCP) allowing to describe a possible crossover from one critical behavior to another. We demonstrate that the system behaves in full agreement with predictions by Fisher et al.\ (Phys.\ Rev.\ B {\bf 40}, 546 (1989)) in close vicinity of QCP. We find as an extension to that analysis that the anomalous dimension $\eta=2-d$ and $\beta=\nu d/2$, where $\beta$ and $\nu$ are critical exponents of the order parameter and the correlation length, respectively. The density of states per spin of low-energy localized excitations are found to be independent of $d$ ("superuniversal"). We show that many recent experimental and numerical results obtained in various 3D systems can be described by our formulas using percolation critical exponents. Then, it is a possibility that a percolation critical regime arises in the ordered phase in some 3D systems not very close to QCP.