  Non-Hermitian Hamiltonians, and particularly parity-time (PT) and anti-PT
symmetric Hamiltonians, play an important role in many branches of physics,
from quantum mechanics to optical systems and acoustics. Both the PT and
anti-PT symmetries are specific instances of a broader class known as
anyonic-PT symmetry, where the Hamiltonian and the PT operator satisfy a
generalized commutation relation. Here, we study theoretically these novel
symmetries and demonstrate them experimentally in coupled lasers systems. We
resort to complex coupling of mixed dispersive and dissipative nature, which
allows unprecedented control on the location in parameter space where the
symmetry and symmetry-breaking occur. Moreover, tuning the coupling in the same
physical system, allows us to realize the special cases of PT and anti-PT
symmetries. In a more general perspective, we present and experimentally
validate a new relation between laser synchronization and the symmetry of the
underlying non-Hermitian Hamiltonian.
