Multiorder topological superfluid phase transitions in a two-dimensional optical superlattice

Higher-order topological superfluids have gapped bulk and symmetry-protected Majorana zero modes with various localizations. Motivated by recent advances, we present a proposal for synthesizing multiorder that support in ultracold atomic gases. For this purpose, use the two-dimensional optical superlattice introduces spatial modulation to spin-orbit coupling one direction, providing an extra degree of freedom emergent higher-order state. We find topologically trivial superfluids, first-order second-order as well different phase transitions among them respect experimentally tunable parameters. Besides conventional transition characterized Chern number associated gap closing reopening, system can corner modes, but undergoes no gap-closing bands. Instead, is refined quadrupole moment signaled out edge states. The based on $s$-wave interaction valid using existing experimental techniques, which unifies simple realistic system.