Point-gap topology in amorphous non-Hermitian quantum systems

Non-Hermitian systems, which describe open quantum systems with gain and loss, often exhibit point-gap topology and the non-Hermitian skin effect (NHSE), where bulk states accumulate at system boundaries. However, in amorphous systems—where lattice sites are randomly positioned—the energy spectrum becomes highly sensitive to perturbations, system size, and boundary conditions. This spectral instability makes it nearly impossible to identify stable topological edge states using traditional eigenvalue analysis.

To overcome this, the authors propose using the singular value decomposition (SVD) of the Hamiltonian $H - E_b I$ rather than its eigenvalues. Because singular values are inherently more stable than eigenvalues, they provide a reliable way to probe topological properties. The authors construct an enlarged Hermitian Hamiltonian $\tilde{H}(E_b)$ that possesses chiral symmetry, allowing them to define a real-space topological invariant $V(E_b)$. This invariant effectively counts the number of topologically protected edge states at a given energy $E_b$.

The study demonstrates that the SVD-based winding number $V(E_b)$ remains sharply quantized even in the presence of strong structural disorder. While the conventional eigenvalue spectrum shows no clear signature of topological protection, the singular value spectrum reveals clear, localized edge states. By mapping $V(E_b)$ across the complex energy plane, the authors provide a comprehensive phase diagram for amorphous chains, effectively redefining the non-Hermitian skin effect through the lens of singular value stability. This framework is potentially applicable to experimental platforms like waveguide arrays and optical tweezers.