In non-Hermitian photonic crystals, the spectrum of the wave operator forms closed loops in the complex plane rather than lying on the real line. When these loops have nontrivial topology — when they wind around points in the complex plane — edge modes appear that are exponentially localized at interfaces. This is the non-Hermitian skin effect.
The paper on spectral topology and edge modes (arXiv: 2603.22515) provides a rigorous mathematical framework for this effect in continuous wave models, where the discrete-lattice theory (based on Toeplitz matrices) breaks down.
The transfer matrix approach describes wave propagation through a one-dimensional periodic medium. The eigenvalues of the transfer matrix — which track how waves grow or decay across each unit cell — define a new topological invariant equivalent to the winding number of the non-Hermitian spectrum. This invariant counts how many times the spectrum wraps around the origin as the quasimomentum varies.
A nonzero winding number implies edge modes: waves trapped at the boundary because the bulk spectrum's topology prevents them from propagating. The mathematical theory works for continuous media — no discretization or tight-binding approximation needed.
The through-claim: the skin effect is topological, not just a lattice artifact. In discrete models, the skin effect comes from the non-reciprocity of hopping amplitudes. In continuous models, it comes from the topology of the transfer matrix's eigenvalues. The phenomenon survives the continuum limit because it's protected by a winding number, not by the lattice.
2603.22515. Photonics / non-Hermitian physics / skin effect / topological invariants / transfer matrices.