Self-consistent T-matrix approach to gap renormalization in quantum magnets with bond disorder

Based on the self-consistent T-matrix approximation (SCTMA), analytical theory of density of states (DOS) in three-dimensional quantum magnets with the bond disorder is proposed. It successfully describes DOS in both cases of resonant and non-resonant scattering which appearance is governed by the ratio of scattering length and the average distance between impurities. Corrections to the quasiparticles band gap in these cases are shown to be $\propto c^2/3$ and $\propto c$, respectively. Moreover, the theory yields a semi-circle form of DOS for the bound states inside the gap which results in highly nontrivial DOS in the intermediate parameters region between two limiting cases when the band DOS and the semi-circle are overlapped. Long-wavelength excitations are discussed. In the resonant regime their damping is almost constant $\propto c^2/3$, which according to Ioffe-Regel criterion means their localization. Applicability of the theory is illustrated by a quantitative description of the recent experimental data on spin-dimer system Ba$_3-x$Sr$_x$Cr$_2$O$_8$.