  The interface separating a liquid from its vapor phase is diffuse: the
composition varies continuously from one phase to the other over a finite
length. Recent experiments on dynamic jamming fronts in two dimensions
[Waitukaitis et al., Europhysics Letters 102, 44001 (2013)] identified a
diffuse interface between jammed and unjammed discs. In both cases, the
thickness of the interface diverges as a critical transition is approached. We
investigate the generality of this behavior using a third system: a model of
cyclically-sheared non-Brownian suspensions. As we sediment the particles
towards a boundary, we observe a diffuse traveling front that marks the
interface between irreversible and reversible phases. We argue that the front
width is linked to a diverging correlation lengthscale in the bulk, which we
probe by studying avalanches near criticality. Our results show how diffuse
interfaces may arise generally when an incompressible phase is brought to a
critical point.
