  Here, we introduce and apply non-Abelian tensor Berry connections to
topological phases in multi-band systems. These gauge connections behave as
non-Abelian antisymmetric tensor gauge fields in momentum space and naturally
generalize Abelian tensor Berry connections and ordinary non-Abelian (vector)
Berry connections. We build these novel gauge fields from momentum-space Higgs
fields, which emerge from the degenerate band structure of degenerate-band
models. Firstly, we show that the conventional topological invariants of
two-dimensional topological insulators and three-dimensional Dirac semimetals
can be derived from the winding number associated to the Higgs field. Secondly,
through the non-Abelian tensor Berry connections we construct
higher-dimensional Berry-Zak phases and show their role in the topological
characterization of several gapped and gapless systems, ranging from
two-dimensional Euler insulators to four-dimensional Dirac semimetals.
Importantly, through our new theoretical formalism, we identify and
characterize a novel class of models that support space-time inversion and
chiral symmetries. Our work provides an unifying framework for different
multi-band topological systems and sheds new light on the emergence of
non-Abelian gauge fields in condensed matter physics, with direct implications
on the search for novel topological phases in solid-state and synthetic
systems.
