  The notion of Higgs-de Rham flows was introduced by Lan-Sheng-Zuo, as an
analogue of Yang-Mills-Higgs flows in the complex nonabelian Hodge theory. In
this short note we investigate a small part of this theory, and study those
Higgs-de Rham flows which are of level zero. We improve the original definition
of level-zero Higgs-de Rham flows (which works for general levels), and
establish a Hitchin-Simpson-type correspondence between such objects and
certain representations of fundamental groups in positive characteristic, which
generalizes the classical results of Katz. We compare the deformation theories
of two sides in the correspondence, and translate the Galois action on the
geometric fundamental groups of algebraic varieties defined over finite fields
into the Higgs side.
