  We demonstrate the inplane anisotropy of longitudinal thermal conductivities
and the weak localization of magnons in a disordered screw-type spiral magnet
on a square lattice. We consider a disordered spin system, described by a spin
Hamiltonian for the antiferromagnetic Heisenberg interaction and the
Dzyaloshinsky-Moriya interaction with the mean-field type potential of
impurities. We derive longitudinal thermal conductivities for the disordered
screw-type spiral magnet in the weak-localization regime by using the
linear-response theory with the linear-spin-wave approximation and performing
perturbation calculations. We show that the inplane longitudinal thermal
conductivities are anisotropic due to the Dzyaloshinsky-Moriya interaction.
This anisotropy may be useful for experimentally estimating the magnitude of a
ratio of the Dzyaloshinsky-Moriya interaction to the Heisenberg interaction. We
also show that the main correction term gives a logarithmic suppression with
the length scale due to the critical back scattering. This suggests that the
weak localization of magnons is ubiquitous for the disordered two-dimensional
magnets having global time-reversal symmetry. We finally discuss several
implications for further research.
