  We enquire into the quasi-many-body localization in topologically ordered
states of matter, revolving around the case of Kitaev toric code on ladder
geometry, where different types of anyonic defects carry different masses
induced by environmental errors. Our study verifies that random arrangement of
anyons generates a complex energy landscape solely through braiding statistics,
which suffices to suppress the diffusion of defects in such multi-component
anyonic liquid. This non-ergodic dynamic suggests a promising scenario for
investigation of quasi-many-body localization. Computing standard diagnostics
evidences that, in such disorder-free many-body system, a typical initial
inhomogeneity of anyons gives birth to a glassy dynamics with an exponentially
diverging time scale of the full relaxation. A by-product of this dynamical
effect is manifested by the slow growth of entanglement entropy, with
characteristic time scales bearing resemblance to those of inhomogeneity
relaxation. This setting provides a new platform which paves the way toward
impeding logical errors by self-localization of anyons in a generic, high
energy state, originated in their exotic statistics.
