  Square-root topology is one of the newest additions to the ever expanding
field of topological insulators (TIs). It characterizes systems that relate to
their parent TI through the squaring of their Hamiltonians. Extensions to
$2^n$-root topology, where $n$ is the number of squaring operations involved in
retrieving the parent TI, were quick to follow. Here, we go one step further
and develop the framework for designing general $n$-root TIs, with $n$ any
positive integer, using the Su-Schrieffer-Heeger (SSH) model as the parent TI
from which the higher-root versions are constructed. The method relies on using
loops of unidirectional couplings as building blocks, such that the resulting
model is non-Hermitian and embedded with a generalized chiral symmetry. Edge
states are observed at the $n$ branches of the complex energy spectrum,
appearing within what we designate as a ring gap, shown to be irreducible to
the usual point or line gaps. We further detail on how such an $n$-root model
can be realistically implemented in photonic ring systems. Near perfect
unidirectional effective couplings between the main rings can be generated via
mediating auxiliary rings with modulated gains and losses. These induce high
imaginary gauge fields that strongly supress couplings in one direction, while
enhancing them in the other. We use these photonic lattices to validate and
benchmark the analytical predictions. Our results introduce a new class of
high-root topological models, as well as a route for their experimental
realization.
