Topological protection requires a gap. This is the standard understanding: edge states survive because the bulk spectrum has an energy range where no states exist, and the edge modes sit inside this forbidden zone. Perturbations can't destroy them because there are no bulk states to scatter into. The gap acts as an energetic moat. At phase transitions — the critical points where the gap closes — the moat drains, edge states merge into the continuum, and topological protection vanishes.
Zhou (arXiv 2602.12588, February 2026) finds edge modes that survive exactly where they shouldn't: at the gapless critical points of a periodically driven non-Hermitian lattice.
The system is a bipartite chain with nonreciprocal hopping — leftward and rightward tunneling amplitudes differ — and periodic driving that alternates between two configurations. The non-Hermiticity produces the skin effect, where eigenstates pile up at boundaries instead of distributing uniformly. The periodic driving creates a Floquet spectrum with quasienergies defined modulo 2pi, opening the possibility of topological phases at both quasienergy zero and quasienergy pi, the latter having no static analog.
At the critical points where the bulk spectrum becomes gapless at both quasienergies simultaneously, the standard framework says topological invariants are ill-defined and edge modes should disappear. What Zhou shows is that they don't. The winding numbers remain quantized integers, computed not from conventional Bloch momentum but from the generalized Brillouin zone — a contour in complex momentum space whose radius deviates from unity to account for the skin effect. Cauchy's argument principle counts the zeros and poles of characteristic functions enclosed by this contour, and the count remains integer-valued whether or not the spectrum has a gap.
The edge modes at criticality are exponentially localized despite having energies that overlap with extended bulk states. The mechanism is sublattice symmetry. The chiral operator decomposes the Hilbert space into two sectors, and the edge states occupy a subspace that doesn't hybridize with the bulk critical modes. The protection is structural — a constraint on which states can couple to which — rather than energetic. The moat is gone, but the walls are built from symmetry rather than energy, and the symmetry walls hold even when the energetic ones don't.
The quantized invariants at criticality are nonlocal: they require winding around the entire generalized Brillouin zone contour and can't be determined from local measurements. This is consistent — the protection itself is nonlocal, depending on the global structure of the momentum-space geometry rather than a local gap condition.
The result extends to a broad class of Floquet bipartite lattices, not just the specific model studied. Wherever sublattice symmetry constrains the coupling structure, topological edge modes can persist through criticality. The authors identify acoustic lattices, photonic structures, electrical circuits with directional amplifiers, and superconducting processors as platforms where the prediction could be tested.
The assumption that topological protection requires a gap is correct for the standard mechanism — energetic isolation of edge states from bulk states. But it's not the only mechanism. When symmetry provides structural isolation — preventing hybridization through selection rules rather than energy barriers — edge modes survive the closing of the gap. The protection isn't in the moat. It's in the walls.