Bulk-spatiotemporal vortex correspondence in gyromagnetic zero-index media

Photonic double-zero-index media, distinguished by concurrently zero-valued permittivity and permeability, exhibit extraordinary properties not found in nature^1-8. Notably, the notion of zero index can be substantially expanded by generalizing the constitutive parameters from null scalars to non-reciprocal tensors with non-zero matrix elements but zero determinants^9,10. Here we experimentally realize this class of gyromagnetic double-zero-index metamaterials possessing both double-zero-index features and non-reciprocal hallmarks. As an intrinsic property, this metamaterial always emerges at a spin-1/2 Dirac point of a topological phase transition. We discover and demonstrate that a spatiotemporal reflection vortex singularity is always anchored to the Dirac point of the metamaterial, with the vortex charge being determined by the topological invariant leap across the phase transition. This establishes a unique bulk-spatiotemporal vortex correspondence that extends the protected boundary effects into the time domain and characterizes topological phase-transition points, setting it apart from any pre-existing bulk-boundary correspondence. Based on this correspondence, we propose and experimentally demonstrate a mechanism to deterministically generate optical spatiotemporal vortex pulses^11,12 with firmly fixed central frequency and momentum, hence showing ultrarobustness. Our findings uncover connections between zero-refractive-index photonics, topological photonics and singular optics, which might enable the manipulation of space-time topological light fields using the inherent topology of extreme-parameter metamaterials.