  To a significant extent, the rich physical properties of photonic crystals
are determined by the underlying geometry, in which the composed symmetry
operators and their combinations contribute to the unique topological invariant
to characterize the topological phases. Particularly, the inter- and
intra-coupling modulation in the two-dimensional (2D) Su-Schrieffer-Heeger
model yields the topological phase transition, and exhibit first-order edge
localized states and second-order corner localized corner states. In this work,
we use the geometric anisotropy into the 2D square lattice composed of four
rectangle blocks. We show a variety of topological phase transitions in
designed nonsymmorphic photonic crystals (PCs) and these transitions shall be
understood in terms of the Zak phase and Chern number in synthetic space, as
well as the pseudospin-2 concept, combinationally. Furthermore, Zak phase
winding in the periodic synthetic parameter space yields high-order Chern
number and double interface states. Based on the extended Zak phase and
pseudo-spin Hall effect, higher-order topological insulator is constructed in
the PC system. The intriguing and abundant topological features are also
sustained in the corresponding three-dimensional PC slab, which makes it a very
interesting platform to control the flow of optical signals.
