We present a theory describing spiral magnets with Dzyaloshinskii-Moriya interaction (DMI) subject to bond disorder at small concentration c of defects. It is assumed that both DMI and exchange coupling are changed on imperfect bonds. Qualitatively the same physical picture is obtained in two models which are considered in detail: B20 cubic helimagnets and layered magnets in which DMI leads to a long-period spiral ordering perpendicular to layers. We find that the distortion of the spiral magnetic ordering around a single imperfect bond is long-range: values of additional turns of spins decay with the distance r to the defect as 1/r 2 being governed by the Poisson's equation for electric dipole. At finite concentration of randomly distributed imperfect bonds, we calculate correction to the spiral vector. We show that this correction can change the sign of spin chirality even at c ≪ 1 if defects are strong enough. It is demonstrated that impurities lead to a diffuse elastic neutron scattering which has power-law singularities at magnetic Bragg peaks positions. Then, each Bragg peak acquires power-law decaying tails. Corrections are calculated to the magnon energy and to its damping caused by scattering on impurities.

Using the conventional T -matrix approach, we discuss gapped phases in 1D, 2D, and 3D spin systems (both with and without a long range magnetic order) with bond disorder and with weakly interacting bosonic elementary excitations. This work is motivated by recent experimental and theoretical activity in spin-liquid-like systems with disorder and in the disordered interacting boson problem. In particular, we apply our theory to both paramagnetic low-field and fully polarized highfield phases in dimerized spin-1 2 systems and in integer-spin magnets with large single-ion easy-plane anisotropy D with disorder in exchange coupling constants (and/or D). The elementary excitation spectrum and the density of states are calculated in the first order in defects concentration c ≪ 1. In 2D and 3D systems, the scattering on defects leads to a finite damping of all propagating excitations in the band except for states lying near its edges. We demonstrate that the analytical approach is inapplicable for states near the band edges and our numerical calculations reveal their localized nature. We find that the damping of propagating excitations can be much more pronounced in considered systems than in magnetically ordered gapless magnets with impurities. In 1D systems, the disorder leads to localization of all states in the band, while those lying far from the band edges (short-wavelength excitations) can look like conventional wavepackets.

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 40, 546 (1989)) in close vicinity of QCP. We find as an extension to that analysis that the anomalous dimension η = 2 − d and β = νd/2, where β and ν 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.