Non-Hermitian topological invariant of photonic band structures undergoing inversion

The interplay between symmetry and topology led to the discovery of symmetry-protected topological phases in Hermitian systems, including topological insulators and topological superconductors. However, the intrinsic symmetry-protected topological characteristics of non-Hermitian systems still await exploration. Here, we investigate experimentally the topological transition associated with the inversion of non-Hermitian band structures in an optical lattice. Intriguingly, we demonstrate that the winding number associated with the symmetry-protected bound state in the continuum is not a conserved quantity after band inversion. To define a topological invariant, we propose the skyrmion number given by spawning in momentum space a pseudo-spin with the polarisation vortex as the in-plane component and the band-index as the pseudo-spin direction at the origin. This leads to a topological transition from an antimeron to an meron-like texture through band inversion, while always conserving the half-charge skyrmion number. We foresee the use of skyrmion number to explore exotic singularities in various non-Hermitian physical system.