Hidden zero modes and topology of multiband non-Hermitian systems

TL;DR

This work addresses the breakdown of bulk-boundary correspondence in finite one-dimensional non-Hermitian systems by linking topology to the singular value spectrum rather than eigenvalues. It introduces the reflected Hamiltonian \\tilde{H} and uses truncated Toeplitz operator theory to show that the winding number \\mathcal{I} dictates the number of exact or hidden zero modes, with at least |\\mathcal{I}| such modes in the semi-infinite or finite limits; in multiband systems, the total count is governed by K = D(ker(H)) + D(ker(\\tilde{H})), and protected singular values correspond to long-lived states even when zero eigenvalues are not observed. A detailed sublattice-symmetric multiband model demonstrates how hidden zero modes can be topologically protected or not, depending on block windings and adiabatic connections to Hermitian limits. The results provide a robust bulk-boundary framework for non-Hermitian topology, with experimental implications for detecting hidden modes via long-lived dynamics such as Loschmidt echoes.

Abstract

In a finite one-dimensional non-Hermitian system, the number of zero modes does not necessarily reflect the topology of the system. This is known as the breakdown of the bulk-boundary correspondence and has led to misconceptions about the topological protection of edge modes in such systems. Here we show why this breakdown does occur and that it typically results in hidden zero modes, extremely long-lived zero energy excitations, which are only revealed when considering the singular value instead of the eigenvalue spectrum. We point out, furthermore, that in a finite multiband non-Hermitian system with Hamiltonian $H$, one needs to consider also the reflected Hamiltonian $\tilde H$, which is in general distinct from the adjoint $H^\dagger$, to properly relate the number of protected zeroes to the winding number of $H$.

Figures (5)

• Figure 1: (a) The Hamiltonian $H$ in Eq. \ref{['example2']} and (b) its adjoint $H^\dagger$; (c) its reflection $\tilde{H}$ and (d) its adjoint $\tilde{H}^\dagger$. The arrows indicate unidirectional hopping and the open circles the localized edge modes.
• Figure 2: Left column: Case $\alpha=1$, $x=0.8$, $y=1$ with total winding $\mathcal{I}=1$. Right column: Case $\alpha=1$, $x=1.2$, $y=1$ with $\mathcal{I}=2$. The top row shows the PBC and the OBC spectra for $L=200$, the bottom row the smallest two singular values as a function of $L$. The OBC eigenspectrum is the same in both cases and insensitive to the change of topology.
• Figure 3: Left column: Case $\alpha=-1$, $x=0.8$, $y=1$ with $\mathcal{I}=1$. Right column: Case $\alpha=-1$, $x=1.2$, $y=1$ with $\mathcal{I}=0$. The top row shows the PBC and the OBC spectra for $L=200$, the bottom row the smallest two singular values (and smallest two eigenvalues in the right panel). The zero mode in the left example is hidden while there are two eigenvalues going to zero in the right example.
• Figure 4: Example from Fig. 2 in Ref. YaoWang for $40$ unit cells. Top row: The two smallest singular values, bottom row: The two eigenvalues of smallest magnitude. The left column shows the unperturbed system, the right column the perturbed system where a random matrix $M$ with elements $|m_{ij}|<10^{-2}$ has been added. At each point $t_1$, an average has been taken over ten such random matrices. The singular values are completely stable while the eigenvalues are only stable in $-1/3<t_1<1/3$.
• Figure 5: Upper panel: $\mathcal{L}(t)$ for the Hamiltonian in example 1 of the main paper. Lower panel: $\mathcal{L}(t)$ for the SSH chain, Eq. \ref{['F_PRL']}. In both cases the chain has $L=70$ sites.