Symmetry-protected topological phase transitions and robust chiral order on a tunable zigzag lattice

Symmetry fractionalization, generating a large amount of symmetry-protected topological phases, provides scenarios for continuous phase transitions different from spontaneous symmetry breaking. However, it is hard to detect these symmetry-protected topological phase transitions experimentally. Motivated by the recent development of highly tunable ultracold polar molecules, we show that the setup in a zigzag optical lattice of this system provides the perfect platform to realize symmetry-protected topological phase transitions. By using infinite time-evolving block decimation, we obtain the phase diagram in a large parameter region and find another scheme to realize the long-sought vector chiral phase, which is robust from quantum fluctuations. We discuss the existence of the chiral phase by an effective field analysis.