  A tight binding network of diamond shaped unit cells trapping a staggered
magnetic flux distribution is shown to exhibit a topological phase transition
under a controlled variation of the flux trapped in a cell. A simple real space
decimation technique maps a binary flux staggered network into an equivalent
Su-Shrieffer-Heeger (SSH) model. In this way, dealing with a subspace of the
full degrees of freedom, we show that a topological phase transition can be
initiated by tuning the applied magnetic field that eventually simulates an
engineering of the numerical values of the overlap integrals in the
paradigmatic SSH model. Thus one can use an external agent, rather than
monitoring the intrinsic property of a lattice to control the topological
properties. This is advantageous from an experimental point of view. We also
provide an in-depth description and analysis of the topologically protected
edge states, and discuss how, by tuning the flux from outside one can enhance
the spatial extent of the Aharonov-Bohm caging of single particle states for
any arbitrary period of staggering. This feature can be useful for the study of
transport of quantum information. Our results are exact.
