Emergence of Hermitian topology from non-Hermitian knots

Abstract

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.

Figures (16)

• Figure 1: Singular-value spectrum of the Hermitian Hamiltonian $H^{I}(k,\omega)$. Panels (a), (b), and (d) show the gapped spectra for $\omega = 0.5$, $1.5$, and $34$, demonstrating that $H^{I}$ remains gapped for $\omega \neq 1$. Panel (c) corresponds to $\omega = 1$, where the gap closes at $k = \pi$, marking the topological transition between the $\nu = 0$ and $\nu = 1$ phases.
• Figure 2: Knot structures of the NH matrix $A(\omega,k)$ for Model I with $V=\sigma_x$. Panel (a) shows the unlinked knot structure for $\omega = 0.5$, while panel (b) displays the unknot for $\omega = 1.5$. These two configurations correspond to winding numbers $\nu = 0$ and $|\nu| = 1$, respectively, illustrating the change in knot topology across the transition at $\omega = 1$.
• Figure 3: Singular-value spectrum of the extended SSH Hamiltonian $H^{II}(k,\omega)$. Panels (a), (b), and (d) show the gapped spectra for $\omega = 0.5$, $1.5$, and $67$, demonstrating that $H^{II}$ remains gapped for all $\omega \neq 1$. Panel (c) corresponds to $\omega = 1$, where the gap closes at three points in $k$, marking the topological transition between the $\nu = 1$ and $\nu = 2$ phases.
• Figure 4: Knot structures of the NH matrix $A(\omega,k)$ for Model II with $V=\sigma_x$. Panel (a) shows the unknot configuration for $\omega = 0.5$, while panel (b) displays the Hopf-link structure for $\omega = 1.5$. These correspond to winding numbers $|\nu| = 1$ and $|\nu| = 2$, respectively, illustrating the knot-topology change across the transition at $\omega = 1$.
• Figure 5: Real and imaginary parts of the eigenvalues of $A(\omega,k)$ at $k=\pi$ for Model I. Panels (a) and (c) show the discontinuous jump at $\omega = 1$, indicating the first-order knot transition. Panels (b) and (d) show the behavior near the exceptional point at $\omega \simeq 34$, where the transition occurs without any discontinuity.