Non-Hermitian extended midgap states and bound states in the continuum

It is standard textbook knowledge that states within the energy continuum are generally extended, while states in the gap are spatially localized. However, almost a century ago, von Neumann and Wigner challenged this paradigm,¹ by proposing a bound state with an energy embedded within the extended states, hence providing an example of an anomalous localization phenomenon. This work was extended in the 1970s by Stillinger and Herrick² and briefly addressed in the following years.^3–5 In the past decade, however, there has been renewed interest in bound states in the continuum (BICs) across different physics communities,^6–10 featuring several potential applications in electronics,^11–13 mechanical metamaterials,¹⁴ plasmonics,^15–19 and notably in photonics.^20–24 

In Hermitian systems, BICs were realized in optical setups using trapped light.^25,26 Topological properties of BICs in the form of vortex centers in the polarization have been proposed,⁹ experimentally realized²⁷ and used for unidirectional guided resonances.²⁸ BICs can also be induced by moving flat bands²⁹ or topological edge states.^30–32 In non-Hermitian systems, impurity-induced BICs were studied,^33,34 or recently, a generalized framework to study BICs in nanophotonics lattices with engineered dissipation was introduced in Ref. 35.

In the context of non-Hermitian topological phenomena,³⁶ localization phenomena go beyond the single bound state in the continuum. In particular, the non-Hermitian skin effect (NHSE)^37–44 and biorthogonal bulk-boundary correspondence when⁴⁰ viewed as localization phenomena provide a natural framework to study the interplay of localized and extended states. Such states give rise to several anomalous localization phenomena with no known counterpart in the Hermitian systems. In addition to a single bound state, a continuum of bound states was also proposed.⁴⁵ In contrast to localized states, an extended midgap state (EGS) was realized^46,47 and analysed⁴⁸ in metamaterials and proposed using acoustoelectric effect.⁴⁹ An antipode of the bound states in the continuum, an extended state lying inside the continuum of bound states (ELC), was proposed in Ref. 14. Anomalous localization phenomena are also known to lead to a remarkable spectral sensitivity,^48,50,51 which may be harnessed in sensing devices.^50,52–54

The aforementioned anomalous (de)localization phenomena have been dispersed throughout various models and platforms. In this Letter, we (i) unify all these (de)localization phenomena by demonstrating their emergence in a single general one-dimensional non-Hermitian model realizable in several platforms; (ii) show that biorthogonal polarization $P$ predicts the gap closing and emergence of bound states in the continuum and remains relevant predictive quantity in the absence of chiral symmetry; and (iii) put these results in the context of non-Hermitian topology and the anomalous bulk-boundary correspondence of non-Hermitian systems.

In what follows, we show BIC emerging when the $P$ changes its values and the robustness of $P$ in the broken chiral symmetry phase. In addition to BIC, viewing the skin states as the localized continuum give rise to the bound states in the continuum of bound states (BICB). Contrary to localized states, we demonstrate that EGS and ELC require the boundary state to become extended at the critical point.

Figure 1 shows the energy spectrum as a function of intra-cell hopping t₁ for all k values, where the color of the energy point represents the average position $⟨ s ⟩$ of the particle described by the right eigenstate $Ψ R , Bulk ( k , n )$⁠; hence, $⟨ s ⟩ = ∑ n , α = A , B N cell s n , α | Ψ R , Bulk ( k , n ) | 2$⁠, where $s n , α$ yields the site label of the sublattice α in the unit cell n, $s n , α = 2 n − δ A , α$⁠.

$P$ also marks the emergence of the bound state in the continuum(BIC). Once the gap is closed, the gap can open again or remain closed. This is visible in the middle panels of Fig. 1, where the boundary mode at $E = Δ$ touches the bulk (green lines) or enters the continuum (blue lines) at the points marked by the change of $P$⁠.

The localization boundary of the BIC is determined by the amplitude in Eq. (6) and can be the same or opposite to the localization boundary of the skin states, see Figs. 2(b) and 2(d), respectively.

In Hermitian systems, bound states appear with discrete energies. In comparison, for the non-Hermitian system we consider here, the bulk skin states can form a continuum in the limit that $N cell → ∞$⁠, leading to a continuum of bound skin states. This is visible in the middle panels in Fig. 1, where the bulk energies start to form the continuum. A similar phenomenon was proposed using imaginary momentum and Landau-type vector potential,⁴⁵ while here we demonstrate the phenomenon using a simple non-Hermitian SSH model.

Another anomalous localization composition is from the combination of the boundary mode, Eq. (6), and the bulk skin states, Eq. (4), where the former localizes in the latter giving rise to bound state in the continuum made of bound states (BICB), see (b) in Fig. 2. Therefore, in our model, the BICs can be BICB.

In the biorthogonal basis, the emergence of the extended state is predicted by the change in $P$⁠. In particular, for the chain with the odd number of sites (the last unit cell broken), the biorthogonal-bulk-boundary⁴⁰ correspondence implies that at $P$ changing points, Eq. (8), the biorthogonal boundary mode,⁵⁵  $⟨ ψ L | Π n | ψ R ⟩$⁠, becomes extended. This delocalization is different from the extended state in the right eigenvector basis, Eq. (6) at $| r R | = 1$⁠.

In this work, we have fully solved a non-Hermitian Rice–Mele model and shown how the localization of bulk and boundary state defies usual expectations. Notably, this highlights bound states in the continuum (BICs) and extended states in the gap (EGSs) as well as in a localized continuum (ELCs). In fact, the localization of non-Hermitian bulk and boundary states can be tuned essentially independently.

We have also shown how a jump in the polarization marks the emergence (or disappearance) of BICs. This highlights a fundamental difference between the biorthogonal polarization,⁴⁰ which carries physical information also when the chiral symmetry is broken (⁠ $Δ ≠ 0$⁠), and the generalized Brillouin zone winding number³⁷ that instead relies on the aforementioned symmetry. This is notable since their predictions coincide in the SSH limit Δ = 0. The difference reflects that the (eigenvector) winding number relates to symmetry-protected topology, while the polarization reflects a general feature of diverging (biorthogonal) localization lengths at phase transitions.

The simplicity of our model makes it relevant for many experimental setups. Notably, the pertinent non-Hermitian SSH model has been realized in mechanical metamaterials,^46,47 electric circuits,^52,53,63 and photonics.^42,54 In fact, the EGS was already observed in Refs. 46 and 47. Using the same methods and observables considered in the aforementioned variety of platforms, one can now explore the rich physics of BIC, EGS, and ELC in these systems by tuning the appropriate parameters following our unified analysis.

The authors thank Evan P. G. Gale for valuable comments on the manuscript. The authors are supported by the Swedish Research Council (VR, Grant 2018-00313), the Wallenberg Academy Fellows program (2018.0460), and the Göran Gustafsson Foundation for Research in Natural Sciences and Medicine.

The authors have no conflicts to disclose.

Maria Zelenayova: Formal analysis (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Emil J. Bergholtz: Conceptualization (lead); Formal analysis (equal); Funding acquisition (lead); Supervision (lead); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.