  In the field of symmetry-protected topological phases, a common wisdom is
that the symmetries fix the topological classifications, but they alone cannot
determine whether a system is topologically trivial or not. Here, we show that
this is no longer true in cases where symmetries are projectively represented.
Particularly, the Zak phase, a topological invariant of a one-dimensional
system, can be entirely determined by the projective symmetry algebra (PSA). To
demonstrate this remarkable effect, we propose a minimal model, termed as flux
Su-Schrieffer-Heeger (SSH) model, where the bond dimerization in the original
SSH model is replaced by a flux dimerization. We present experimental
realization of our flux SSH model in an electric-circuit array, and our
predictions are directly confirmed by experimental measurement. Our work
refreshes the understanding of the relation between symmetry and topology,
opens up new avenues for exploring PSA determined topological phases, and
suggests flux dimerization as a novel approach for designing topological
crystals.
