  Topological spin liquids in two spatial dimensions are stable phases in the
presence of a small magnetic field, but may give way to field-induced phenomena
at intermediate field strengths. Sandwiched between the low-field spin liquid
physics and the high-field spin-polarized phase, the exploration of magnetic
phenomena in this intermediate regime however often remains elusive to
controlled analytical approaches. Here we numerically study such
intermediate-field magnetic phenomena for two representative Kitaev models (on
the square-octagon and decorated honeycomb lattice) that exhibit either Abelian
or non-Abelian topological order in the low-field limit. Using a combination of
exact diagonalization and density matrix renormalization group techniques, as
well as linear spin-wave theory, we establish the generic features of Kitaev
spin liquids in an external magnetic field. While ferromagnetic models
typically exhibit a direct transition to the polarized state at a relatively
low field strength, antiferromagnetic couplings not only substantially
stabilizes the topological spin liquid phase, but generically lead to the
emergence of a distinct field-induced intermediate regime, separated by a
crossover from the high-field polarized regime. Our results suggest that, for
most lattice geometries, this regime generically exhibits significant spin
canting, antiferromagnetic spin-spin correlations, and an extended proximate
spin liquid regime at finite temperatures. Notably, we identify a symmetry
obstruction in the original honeycomb Kitaev model that prevents, at least for
certain field directions, the formation of such canted magnetism without
breaking symmetries -- consistent with the recent numerical observation of an
extended gapless spin liquid in this case.
