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

Maria Zelenayova<EMAIL_ADDRESS>Department of Physics,
Stockholm University, AlbaNova University Center, 10691 Stockholm, Sweden
Emil J. Bergholtz<EMAIL_ADDRESS>Department of Physics, Stockholm
University, AlbaNova University Center, 10691 Stockholm, Sweden

###### Abstract

We investigate anomalous localization phenomena in non-Hermitian systems by
solving a class of generalized Su-Schrieffer-Heeger/Rice-Mele models and by
relating their provenance to fundamental notions of topology, symmetry-
breaking and biorthogonality. We find two flavours of bound states in the
continuum, both stable even in the absence of chiral symmetry. The first being
skin bulk states which are protected by the spectral winding number. The
second flavour is constituted by boundary modes associated with a quantized
biorthogonal polarization. Furthermore, we find the extended state stemming
from the boundary state that delocalizes while remaining in the gap at bulk
critical points. This state may also delocalize within a continuum of
localized (skin) states. These results clarify fundamental aspects of
topology, and symmetry in the light of different approaches to the anomalous
non-Hermitan bulk-boundary correspondence – and are of direct experimental
relevance for mechanical, electrical and photonic systems.

Introduction. It is standard textbook knowledge that states within the energy
continuum are generally extended while states in the gap are spatially
localized. However, already almost a century ago, von Neumann and Wigner
challenged this paradigm [1], by proposing a bounded state with an energy
embedded within the extended states, hence providing an example of an
anomalous localization phenomenon. This work was later extended in 1970s by
Stillinger and Herrick [2], and briefly addressed in the following years [3,
4, 5]. However, only during the past decade there has been a renewed interest
in bound states in the continuum (BICs) across different physics communities
[6, 7, 8, 9, 10], featuring several potential applications in electronics [11,
12, 13], mechanical metamaterials [14], plasmonics [15, 16, 17, 18, 19] and,
saliently, in photonics [20, 21, 22, 23, 24].

In Hermitian systems, BICs were realized in optical set-ups using the trapped
light [25, 26]. Topological properties of BICs in the form of vortex centres
in the polarization have been proposed [9], experimentally realised [27], and
used for unidirectional guided resonances [28]. BICs can be also induced by
moving flat bands [29] or topological edge states [30, 31, 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 [35]. However, in the context of non-Hermitian
topological phenomena [36], localization phenomena go beyond the single
bounded state in the continuum. In particular, the non-Hermitian skin effect
(NHSE) [37, 38, 39, 40, 41, 42, 43, 44] and biorthogonal bulk boundary
correspondence [40] viewed as localization phenomena provide a natural
framework to study the interplay of localized and extended states, giving rise
to several new anomalous localization phenomena with no known counterpart in
the Hermitian systems. In addition to a single bounded state, a continuum of
bounded states was also proposed [45]. In contrast to localized states, an
extended mid-gap state (EGS) was realized [46, 47] and analysed [48] in
metamaterials, and proposed using acoustoelectric effect [49]. An antipode of
the boundary states in the continuum, an extended state lying inside the
continuum of bound states (ELC), was also proposed in Ref. 14. Anomalous
localization phenomena is also know to lead to a remarkable spectral
sensitivity [48, 50, 51] that may be harnessed in sensing devices [50, 52, 53,
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;
show that (ii) biorthogonal polarization ${\mathcal{P}}$ predicts the gap
closing and emergence of bounded states in the continuum and remains relevant
in the absence of chiral symmetry; (iii) put these results in the context of
non-Hermitian topology and the anomalous bulk-boundary correspondence of non-
Hermitian systems.

Exactly solvable model. We consider open non-Hermitian Su–Schrieffer–Heeger
chain (SSH) with nonreciprocal $\gamma$ term in the intra-cell hopping, and
placed in the staggering potential $\Delta$ breaking the chiral symmetry
turning it into a non-Hermitian Rice-Mele model,

$\displaystyle H$ $\displaystyle=$ $\displaystyle\sum_{n=1}^{N_{\rm
cell}-1}\big{[}(t_{1}-\gamma)a^{\dagger}_{B,n}a_{A,n}+(t+\gamma)a^{\dagger}_{A,n}a_{B,n}$
(1) $\displaystyle+$ $\displaystyle
t_{2}(a^{\dagger}_{B,n}a_{A,n+1}+a^{\dagger}_{A,n+1}a_{B,n})+\Delta(n_{A,n}-n_{B,n})\big{]}$
$\displaystyle+$ $\displaystyle t_{2}(a^{\dagger}_{B,{N_{\rm
cell}-1}}a_{A,{N_{\rm cell}}}+a^{\dagger}_{A,N_{\rm cell}}a_{B,{N_{\rm
cell}-1}}),$

where $N_{\rm cell}$ is the number of cells,
$n_{\alpha,n}=a_{\alpha,n}^{\dagger}a_{\alpha,n}$ is the number of particles,
and $a^{\dagger}_{\alpha,n}(a)$ creates (annihilates) a bosonic particle in a
sublattice $\alpha$ in a unit cell $n$. The open boundaries are applied such
that the last unit cell is broken, and the model has $2N_{\rm cell}-1$ sites
in total.

Diagonalizing Eq. (1), generalizing results in [55, 56, 57, 58, 40, 59, 60],
we obtain the complete eigensystem with $2N_{\rm cell}-2$ bulk modes and one
edge state [55]. The energy spectrum reads

$\\{E_{-}(k),\Delta,E_{+}(k)\\},$ (2)

where the bulk energy spectrum is derived from the Hamiltonian with periodic
boundaries by shifting the momentum $E(k)=E^{\rm PBC}(k-i{\rm\log}(r))$, where
$r=\frac{\sqrt{t_{1}-\gamma}}{\sqrt{t_{1}+\gamma}}$ [60, 57]. Hence, we find
that

$E_{\pm}(k)=\pm\sqrt{t_{1}^{2}+t_{2}^{2}+\Delta^{2}-\gamma^{2}+2t_{2}\sqrt{t_{1}-\gamma}\sqrt{t_{1}+\gamma}\cos{k}}$
(3)

where $k=\frac{\pi m}{N_{\rm cell}}$, with $m=1,2,...(N_{\rm cell}-1)$.

Figure 1: The energy spectrum of the non-Hermitian SSH chain, Eq. (1) with
$t_{2}=1$, $\gamma=3/2$, $N_{\rm cell}=60$ for $\Delta=0$ and $\Delta=1.2$
(Rice-Mele model) in the top and bottom bottom panels, respectively. The
coloring corresponds to the average position of the given mode
$\left<s\right>$ described in the main text. Focusing on the energy of the
edge mode, Eq. (6), lying at $E=\Delta$, the figure illustrates i)the change
of the localization boundary from the right (pink) to the left (dark blue) at
$t_{1}$ given by Eq. (9); ii) at the transition point, Eq. (9), the edge mode
becomes extended, see (d), (e) in Fig. 2 for comparison; iii) the gap closes
where ${\cal P}$ changes, Eq. (8), marked by the yellow and green lines at
$r_{l}^{*}r_{r}=1$ and $r_{l}^{*}r_{r}=-1$, respectively. For the latter,
polarization also marks the emergence of BIC (BICB).

Fig. 1 shows the energy spectrum as a function of intra-cell hopping $t_{1}$
for all $k$ values, where the color of the energy point represents the average
position $\left<s\right>$ of the particle described by the right eigenstate
$\Psi_{R,{\rm Bulk}}(k,n)$ corresponding to the eigenvalue $E(k)$ evaluated at
$t_{1}$, hence $\left<s\right>=\sum_{n,\alpha=A,B}^{N_{\rm
cell}}s_{n,\alpha}|\Psi_{R,{\rm Bulk}}(k,n)|^{2}$, where $s_{n,\alpha}$ yields
the site label of the sublattice $\alpha$ in the unit cell $n$,
$s_{n,\alpha}=2n-\delta_{A,\alpha}$.

While the periodic Bloch energies of our model lacks spectral inversion
symmetry, the open boundary energies (Eq. (3)) have the symmetry $E(k)=E(-k)$
which makes it possible to derive all eigenstates generalizing Refs. 55, 56.
The right bulk modes are $\Psi_{R,\mathrm{Bulk},\pm(k)}$ read

$\Psi_{R,\mathrm{Bulk},\pm}(k)=\mathcal{N}\begin{pmatrix}\Psi_{R,\mathrm{Bulk},\pm,A}(k,1)\\\
\Psi_{R,\mathrm{Bulk},\pm,B}(k,1)\\\ \Psi_{R,\mathrm{Bulk},\pm,A}(k,2)\\\
\vdots\\\ \Psi_{R,\mathrm{Bulk},\pm,A}(k,n)\\\
\Psi_{R,\mathrm{Bulk},\pm,B}(k,n)\\\ \vdots\\\
\Psi_{R,\mathrm{Bulk},\pm,B}(k,N_{\rm cell}-1)\\\
\Psi_{R,\mathrm{Bulk},\pm,A}(k,N_{\rm cell})\\\ \end{pmatrix},$ (4)

where $n$ labels the unit cell, and the amplitudes read

$\begin{split}&\Psi_{R,\mathrm{Bulk},\pm,A}(k,n)=r^{n}\left[(t_{1}\\!+\\!\gamma)\sin{(kn)}\\!+\\!\frac{t_{2}}{r}\sin{(k[n-\\!1])}\right],\\\
&\Psi_{R,\mathrm{Bulk},\pm,B}(k,n)=r^{n}\left[E_{\pm}(k)-\Delta\right]\sin{(kn)},\end{split}$
(5)

where $r=\sqrt{\frac{t_{1}-\gamma}{t_{1}+\gamma}}$. From these expressions, we
see that states are exponentially localized to one of the boundaries, the
phenomenon known as the non-hermitian skin effect (NHSE) [40, 41, 61], where
the value of the spectral winding number [43] predicts the boundary to which
states pile-up. In Fig. 1, the average position $\left<s\right>$ shows the
localization to the right (pink) boundary for $t_{1}<0$ and to the left (blue)
boundary for $t_{1}>0$. Comparing the top and bottom panels with $(\Delta=0)$
and without $(\Delta\neq 0)$ chiral symmetry, respectively, we see that the
NHSE is carried over to the broken chiral symmetry phase.

The remaining eigenstate, the edge mode, with energy $\Delta$ is an
exponentially localized boundary mode

$\ket{\psi_{R/L}}=\mathcal{N}_{R/L}\sum_{n=1}^{N_{\rm
cell}}r_{R/L}^{n}a_{n,A}^{\dagger}\ket{0},$ (6)

where $\mathcal{N}_{R/L}$ is the normalization, $r_{R}=-(t_{1}-\gamma)/t_{2}$,
and $r_{L}^{*}=-(t_{1}+\gamma)/t_{2}$ which dictate the behaviour of the right
($R$) and left ($L$) boundary modes [55, 40]. (Note that right/left here
refers to the eigenvectors fulfilling $H\Psi_{R}=E\Psi_{R}$ and
$\Psi_{L}H=E\Psi_{L}$, respectively, rather than where the state may be
localized.)

While the emergence of NHSE is related to a spectral winding number [43], the
behaviour of boundary $\Delta$-mode can be associated with the biorthogonal
polarization ${\cal P}$ [55, 40]

$\mathcal{P}=\lim_{N_{\rm cell}\rightarrow\infty}\frac{1}{N_{\rm
cell}}\sum_{s}\frac{\braket{\psi_{L}}{s\hat{\Pi}_{s}}{\psi_{R}}}{{\braket{\psi_{L}}{\psi_{R}}}},$
(7)

where $\ket{\psi_{R/L}}$ are the boundary modes in Eq. (6),
$\hat{\Pi}_{s}~{}=~{}\ket{e_{s}}\bra{e_{s}}$ is the projection operator on
each unit site $s$ and $\ket{e_{s}}$ is the basis vector in the position
basis. Note that in the reference [40] the biorthogonal polarization is
defined as $1-{\mathcal{P}}$.

Remarkably, the change in ${\cal P}$ implies the gap closings [55, 40]. To see
this, we evaluate the limit in Eq. (7) and find that the biorthogonal
polarization ${\mathcal{P}}$ changes when $|r_{L}^{*}r_{R}|=1$ [55], i.e.,
when

$\displaystyle t_{1}=\pm\sqrt{\gamma^{2}+t_{2}^{2}},$ $\displaystyle
t_{1}=\pm\sqrt{\gamma^{2}-t_{2}^{2}}.$ (8)

Evaluating Eq. (3) for $\lim_{N_{\rm cell}\rightarrow\infty}$ at $t_{1}$ in
Eq. (8) we find $E(k=N_{\rm cell}/2)=\Delta$, i.e., the gap between the bulk
modes and the $\Delta$-mode closes as predicted by the change in the
polarization number ${\cal P}$. This is also visible in Fig. 1, where the
colored dashed lines highlight the polarization ${\cal P}$ accompanied by the
gap closings, and discrepancies are due to the finite system size.

Bound states in the continuum. Biorthogonal polarization $\mathcal{P}$ 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 $E=\Delta$ touches the
bulk (yellow lines) or enters the continuum (green lines) at the points marked
by the change of ${\cal P}$, which remains to be predictive quantity also in
the phase with broken chiral symmetry phase $\Delta\neq 0$ (bottom panels).

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 Fig. 2 (b) and (c) for comparison.

Figure 2: The wave-function squared of the eigenstates in Eq. (4) and Eq. (6),
where $s$ labels the sites, $t_{2}=1$, $\gamma=3/2$, $\Delta=1.2$, $N_{\rm
cell}=60$ and the coloring of the mode corresponds to the absolute value of
the energy of that particular mode as shown in (a). Bounded states in the
continuum with opposite localization boundaries at $t_{1}^{(b)}=-0.3$ and
$t_{1}^{(c)}=0.3$ are shown in (b) and (c), respectively. For the former, the
edge state is localized among the bulk modes, leading to BICB, whereas for the
latter, it localizes on the opposite as the skin states. An extended state at
$t_{1}^{d}$ with energy lying inside the localized continuum of bound states
(ELC) is shown in (d), while EGC emerging at $t_{1}^{(e)}$ is visible in (e).

In Hermitian systems, bounded states appear with quantized energies. In
comparison, here, the bulk skin states can form a continuum when $\lim_{N_{\rm
cell}\rightarrow\infty}$, see Eq. (3), 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 them using a simple non-Hermitian SSH model.

Another anomalous localization composition surfaces 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 bounded state in the continuum
made of bounded states (BICB), see (b) in Fig. 2. Therefore, in our model, the
BICs can be BICB.

Anomalous extended states. We now turn to anomalous delocalization phenomena.
From Eq. (6), we see that the edge mode can transition from being localized to
the left (right) boundary for $|r_{R}|<1$ ($|r_{R}|>1$) to the extended state
at $|r_{R}|=1$, i.e. when

$\displaystyle t_{1\pm}=\gamma\pm t_{2}.$ (9)

At these points, the otherwise localized boundary state is now extended
throughout the lattice while all the bulk skin states are pilled to the
boundary, see (d) and (e) in Fig 2. The energy of these extended states can
lie inside or outside of the continuum. When it lies in the continuum, an
extended state in the localized continuum (ELC) is formed, see (d) in Fig. 2,
where the localized continuum is formed by the bulk skin states. An analogue
of ELC was realized in a mechanical two-dimensional system [14] while we show
that it can also emerge in a one-dimensional model. The extended state lying
outside of the continuum creates extended mid-gap state (EGS), see middle
pannels in Fig. 1 and (e) in Fig. 2, which is localized at zero for
$\Delta=0$, while for $\Delta\neq 0$ it is shifted by $\Delta$. A similar EGS
was realised in metamaterials [46, 47], the transition to EGS was also
analysed [48], and recently a similar phenomenon was addressed in solitons
[62].

In the biorthogonal basis, the emergence of the extended state is predicted by
the change in $\mathcal{P}$. In particular, for the chain with the odd number
of sites (the last unit cell broken), the biorthogonal-bulk boundary[40]
correspondence implies that at ${\cal P}$ changing points, Eq. 8, the
biorthogonal boundary mode [55] defined as
$\braket{\psi_{L}}{\Pi_{n}}{\psi_{R}}$, becomes extended. This delocalization
is different from the extended state in the right eigenvector basis, Eq. (6)
at $|r_{R}|=1$.

Discussion. In this work we have fully solved a non-Hermitian Rice-Mele model
and showcased how the localization of bulk and boundary states defy 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 ($\Delta\neq 0$), and the generalized Brillouin
zone winding number [37] that instead relies on the aforementioned symmetry.
This is notable since their predictions coincide in the SSH limit $\Delta=0$.
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, in
particular in photonics which naturally feature a high degree of tunability
and where non-Hermiticity is an ubiquitous feature.

###### Acknowledgements.

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.

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