Phase Transitions and Topological Protection in Anyonic-PT-Symmetric Lattices

Parity-time (PT) symmetry and anti-PT symmetry have attracted extensive interest for their non-Hermitian spectral properties, particularly the emergence of purely real and imaginary eigenvalues in their symmetry-unbroken regime, respectively. Recently, these two scenarios have been unified under a more general framework known as anyonic-PT symmetry, yet its physical implications in waveguide platforms and corresponding topological features in extended lattice systems remain largely unexplored. Here, the phase transitions and topological protection in anyonic-PT-symmetric systems are systematically investigated in waveguide lattices. In the symmetry-unbroken regime, the arguments of all bulk eigenvalues are constrained to two discrete values separated by {\pi}, leading to distinctive oscillatory propagation dynamics accompanied by controlled amplification or dissipation. In the case of one-dimensional lattice, the energy bands exhibit a gap closing and reopening during phase transition. Moreover, in the symmetry-unbroken regime, the topological edge states emerge within the bulk gap and are protected by a generalized pseudo-anyonic-Hermiticity (PAH) symmetry. Our results establish anyonic-PT symmetry as a new tunable degree of freedom for non-Hermitian waveguide systems, where the eigenvalue argument provides a natural quantity for information encoding. This work broadens the conceptual foundation of topological protection under generalized non-Hermitian symmetries.