Emergence of Hermitian topology from non-Hermitian knots

The non-Hermiticity of the system gives rise to a distinct knot topology in the complex eigenvalue spectrum, which has no counterpart in Hermitian systems. In contrast, the singular values of a non-Hermitian (NH) Hamiltonian are always real by definition, meaning that they can also be interpreted as the eigenvalues of some underlying Hermitian Hamiltonian. In this work, we demonstrate that if the singular values of an NH Hamiltonian are treated as eigenvalues of prototype translational invariant Hermitian models that undergo a topological phase transition between two distinct topological phases, the complex eigenvalues of the NH Hamiltonian will also undergo a {\it{first order knot transition}} between different knot structures. Unlike the usual knot transition, this transition is not accompanied by an Exceptional point (EP); in contrast, the real and complex parts of the eigenvalues of the NH Hamiltonian show a discrete jump at the transition point. We emphasize that the choice of an NH Hamiltonian whose singular values match the eigenvalues of a Hermitian model is not unique. However, our study suggests that this connection between the NH and Hermitian models remains robust as long as the periodicity in lattice momentum is the same for both. Furthermore, we provide an example showing that a change in the topology of the Hermitian model implies a transition in the underlying NH knot topology, but a change in knot topology does not necessarily signal a topological transition in the Hermitian system.

💡 Research Summary

The manuscript investigates a novel correspondence between Hermitian topological phase transitions and knot transitions in the complex energy spectrum of non‑Hermitian (NH) Hamiltonians. The key idea is to treat the singular values of an NH Hamiltonian as the eigenvalues of a Hermitian Hamiltonian that undergoes a conventional one‑dimensional topological transition when a control parameter ω is varied. By constructing the NH Hamiltonian A(k, ω) through singular‑value decomposition (SVD) of the Hermitian Hamiltonian H(k, ω), i.e., A = U Σ V† with Σ = diag(√E₁, √E₂,…), the authors generate a whole family of NH matrices that share the same singular‑value spectrum. The unitary matrix V represents a gauge freedom; different choices of V lead to different NH realizations but preserve the periodicity in crystal momentum k.

To quantify knot topology in the NH spectrum, the authors adopt a winding number defined for two‑band systems: ν_A = (1/2πi)∫₀^{2π} dk ∂_k ln det