# Topological Properties of a Non-Hermitian Quasi-1D Chain with a Flat Band

C. Martínez-Strasser<EMAIL_ADDRESS>Department of Physics,
University of the Basque Country UPV/EHU, Apartado 644 48080, Bilbao, Spain
Donostia International Physics Center (DIPC), 20018, Donostia–San Sebastián,
Spain  Max Planck Institute for the Science of Light, Staudtstraße 2, 91058,
Erlangen, Germany  M. A. J. Herrera<EMAIL_ADDRESS>Department
of Physics, University of the Basque Country UPV/EHU, Apartado 644 48080,
Bilbao, Spain  Donostia International Physics Center (DIPC), 20018,
Donostia–San Sebastián, Spain  A. García-Etxarri<EMAIL_ADDRESS>Donostia International Physics Center (DIPC), 20018, Donostia–San Sebastián,
Spain  IKERBASQUE, Basque Foundation for Science, Plaza Euskadi 5 48009,
Bilbao, Spain  G. Palumbo<EMAIL_ADDRESS>School of
Theoretical Physics, Dublin Institute for Advanced Studies, 10 Burlington
Road, Dublin D04 C932, Ireland  F. K. Kunst<EMAIL_ADDRESS>Max Planck
Institute for the Science of Light, Staudtstraße 2, 91058, Erlangen, Germany
D. Bercioux<EMAIL_ADDRESS>Donostia International Physics Center
(DIPC), 20018, Donostia–San Sebastián, Spain  IKERBASQUE, Basque Foundation
for Science, Plaza Euskadi 5 48009, Bilbao, Spain

(August 28, 2024)

###### Abstract

The spectral properties of a non-Hermitian quasi-1D lattice in two of the
possible dimerization configurations are investigated. Specifically, it
focuses on a non-Hermitian diamond chain that presents a zero-energy flat
band. The flat band originates from wave interference and results in
eigenstates with a finite contribution only on two sites of the unit cell. To
achieve the non-Hermitian characteristics, the system under study presents
non-reciprocal hopping terms in the chain. This leads to the accumulation of
eigenstates on the boundary of the system, known as the non-Hermitian skin
effect. Despite this accumulation of eigenstates, for one of the two
considered configurations, it is possible to characterize the presence of non-
trivial edge states at zero energy by a real-space topological invariant known
as the biorthogonal polarization. This work shows that this invariant,
evaluated using the destructive interference method, characterizes the non-
trivial phase of the non-Hermitian diamond chain. For the second non-Hermitian
configuration, there is a finite quantum metric associated with the flat band.
Additionally, the system presents the skin effect despite the system having a
purely real or imaginary spectrum. The two non-Hermitian diamond chains can be
mapped into two models of the Su-Schrieffer-Heeger chains, either non-
Hermitian, and Hermitian, both in the presence of a flat band. This mapping
allows to draw valuable insights into the behavior and properties of these
systems.

##  1 Introduction

Non-Hermitian (NH) physics is an emergent field of research that has important
implications both for quantum and classical physics [1, 2, 3, 4, 5, 6]. A
systematic study of this field started with the cornerstone works by Bender
and Boettcher on Hamiltonian systems preserving the combination of parity and
time-reversal ($\mathcal{PT}$) symmetry, ensuring a real spectrum [7, 8]. At
the moment, model Hamiltonian systems respecting $\mathcal{PT}$ symmetry are
considered excellent models for describing dissipative systems with balanced
gain and loss in an effective way [1].

The condition of reality of the spectrum can be extended by considering a more
general symmetry class known as _pseudo-Hermiticity_ that includes the
$\mathcal{PT}$-one [9]. In general, NH operators exhibit intriguing phenomena
such as non-orthogonal eigenstates and complex energy spectra containing
exceptional points (EPs), representing stable points of band degeneracies at
which not only the eigenvalues but also the eigenvectors coalesce [1, 3, 4].

Recent research has focused on the topological characterization of NH systems
[1, 3, 4], expanding upon the framework established for Hermitian condensed-
matter systems. One of the key effects of moving to the NH realm leads to the
extension of the topological classification, considering now that complex
conjugation and transposition are no longer equivalent for non-Hermitian
Hamiltonians, moving to 38 classes [10] instead of the tenfold classification
of the Hermitian counterpart [11].

A key difference between NH and Hermitian systems is the breakdown of the
traditional bulk-boundary correspondence (BBC), which predicts the appearance
of boundary modes based on bulk topological invariants. This breakdown can be
manifested through the non-Hermitian skin effect, where bulk states accumulate
at the edges of the system [12, 13, 14]. Two possible research lines have been
developed for reestablishing the BBC. The first one is based on the
biorthogonal bulk-boundary correspondence approach [15]; here, right and left
eigenvectors of the system under open boundary conditions (OBCs) are combined
to project the boundary mode localization and predict gap closings accurately.
The second method is based on the concept of the so-called _generalized
Brillouin zone_ (BZ) [12, 16, 17], in which additional information is encoded
inside the standard Bloch bands.

For the investigation of topological effects in NH systems, 1D systems are
ideal platforms for presenting the key features [18, 19, 20, 15, 21]. These
simple 1D models exhibit many of the unusual properties of NH systems. A
paradigmatic example is the Hatano-Nelson model [18], which is a one-band
system with anisotropic nearest-neighbor couplings originally proposed to
study localization transitions in superconductors. Under periodic boundary
conditions (PBCs), this model features loops in the complex spectrum resulting
in a non-trivial spectral winding number [22, 23]. When going to OBCs, this
translates into the appearance of the NH skin effect, thus establishing a new,
truly NH bulk-boundary correspondence [24, 25, 26].

Even richer NH phenomena can be observed in NH versions of the Su-Schrieffer-
Heeger (SSH) chains [27, 28, 29]. This 1D two-band system features zero-energy
end modes in the Hermitian case, which are topologically protected by a non-
trivial winding number. A NH version of this chain with asymmetric hopping has
been shown to host zero-energy boundary states as well as NH skin states [12,
15]. As such, this system breaks the conventional BBC and needs to be treated
either in the biorthogonal picture [15] or within the framework of the
generalized BZ [12]. A $\mathcal{PT}$-symmetric version of the SSH chain has
also been studied, which features an onsite complex potential with alternating
sign [20, 30]. In this case, the boundary states acquire an imaginary energy,
while the NH skin effect is absent, such that the traditional BBC applies.

In this work, we present the topological properties of a quasi-1D system: the
diamond chain (DC) [31, 32, 33, 34]. The unit cell of this quasi-1D system
contains three sites with unequal connectivity; in the following, we will name
the site with higher connectivity as H, whereas the sites with lower
connectivity will be named A and B — see Fig. 1. This lattice model has also
been studied with respect to the effects of localization due to an external
magnetic field and many-body effects [31, 35, 36]. The imbalance in the
connectivity results in the appearance of a bulk zero-energy mode, where the
wave function is localized only in the sites of lower connectivity with
opposite amplitudes, while it has zero amplitude on the remaining H sites. In
the Hermitian case, in Ref. [34], it was shown that two possible dimerizations
could be chosen for the DC, but only one of these presents topological
properties analogous to the SSH model [27, 28, 29]. Possible experimental
implementations for the Hermitian system involve cold atoms in optical
lattices [37], photonic [38], and solid-state platform [39]. Within this work,
we investigate two possible NH configurations of the DC chain. A NH version of
the DC has already been investigated with particular emphasis on possible
photonic realizations and focusing mainly on the $\mathcal{PT}$-symmetric
version [40, 41, 42, 3, 43, 44, 45]. Additional research has investigated the
possibility of obtaining lasing from the flat band [46]. Within this work, we
will relax this symmetry constriction.

We introduce non-Hermiticity by imposing a preferred hopping direction within
unit cells, resulting in two non-reciprocal tight-binding models. The
motivation beyond studying the NH diamond lattice stems from its potential for
realizing two distinct dimerization configurations within the lattice. The
first configuration displays zero-energy edge states, which can be
characterized through the evaluation of the biorthogonal polarization [15,
21]. The interest in the second configuration arises from the properties of
the flat band [47, 48, 49] resulting in a _giant boost_ of the quantum metric
properties [50]. Additionally, we show how to map these two systems into a
combination of Hermitian and NH SSH models coupled to a flat band. This
mapping allows us to draw valuable insights into the behavior and properties
of these systems, opening new avenues for further exploration of non-Hermitian
1D systems.

This article is structured as follows: in Sec. 2.1, we describe the NH DC in
the two possible dimerization configurations that we have uncovered. In Sec.
2.2, we explore the topological properties of these two NH lattice
configurations, focusing in particular on the biorthogonal polarization and
the quantum metric. In Sec. 2.3, we present the path toward the unitary
transformation of the lattices into SSH chains, plus an extra site describing
the flat band. We conclude in Sec. 3 with a summary of our findings. We
include technical appendices: App. A.1 presenting an analytical expression for
the system wave functions in the case of translational invariance, and App.
A.2 summarizing the symmetry properties is the two lattices at the end of our
work.

##  2 Results

###  2.1 Systems and Formalisms

Figure 1:  Sketch of the non-Hermitian diamond lattice in the A and B
configurations, panel (a) and (b), respectively. The gray area denotes the
unit cell of each lattice configuration. The lattice periodicity is $a_{0}$.
In both panels, the blue arrows correspond to a hopping term of strength
$t_{1}+\frac{\gamma}{2}$, whereas the red ones to $t_{1}-\frac{\gamma}{2}$.

####  2.1.1 Spectrum and symmetries

The non-Hermitian counterpart of the diamond chain under “lattice I”
configuration from Ref. [34] can be divided into two different systems
according to their NH coupling configuration — DCA and DCB. The main
differences imply an intracell hopping from site H to site B of
$t_{1}-\gamma/2$ in DCA and $t_{1}+\gamma/2$ in DCB. We present a sketch of
these two lattices in Fig. 1. In the NH version, both lattice systems still
present a zero-energy flat band in the energy spectrum; this originates in the
unequal connectivity between the three lattice sites in the unit cell — H
connected to four neighbors, and A and B connected to two. The two chains
could be thought of as two joined NH SSH models [20]; however, the sharing of
a common lattice site (H) drastically changes the spectral properties of both
the Hermitian and NH systems.

Figure 2:  The absolute value of the eigenenergies is examined as a function
of the hopping parameters, denoted as $t_{1}$, in the non-Hermitian DCA model
with a fixed asymmetry term $\gamma=3$ and a system size $N=16$. The pink
lines represent the bulk spectra obtained under periodic boundary conditions
(PBC), and the blue (red) lines correspond to the bulk (edge) spectra obtained
under open boundary conditions (OBC), signaling the phase-transition points
through arrows. The zero-energy flat band, depicted in purple, is present in
both the PBC and OBC scenarios. Moreover, the black dashed line at
$t_{1}/t_{2}=2$ indicates the hopping parameter values at which the skin
effect in Fig. 3 has been computed.

####  2.1.2 Diamond Chain A (DCA)

The tight-binding Hamiltonian of the non-Hermitian diamond chain in the DCA
configuration [see Fig. 1a)] reads:

$\displaystyle\mathcal{H}_{\mathrm{DCA}}=\sum_{n}$
$\displaystyle\left\\{\left[t_{2}(c_{\mathrm{A},n}^{\dagger}+c_{\mathrm{B},n}^{\dagger})c_{\mathrm{H},n+1}+\mathrm{h.c.}\right]\right.$
$\displaystyle\left.+\left(t_{1}-\frac{\gamma}{2}\right)(c_{\mathrm{A},n}^{\dagger}+c_{\mathrm{B},n}^{\dagger})c_{\mathrm{H},n}\right.$
(1)
$\displaystyle\left.+\left(t_{1}+\frac{\gamma}{2}\right)c_{\mathrm{H},n}^{\dagger}(c_{\mathrm{A},n}+c_{\mathrm{B},n})\right\\}$

The operators $c_{\alpha,n}^{\dagger}$ and $c_{\alpha,n}$ create and
annihilate a state on sub-lattice site $\alpha\in\\{\mathrm{A,B,H}\\}$ of unit
cell $n$, respectively. Here, $n$ ranges from $1$ to $N$, where $N$ is the
total number of unit cells. The parameters $t_{1}$ and $t_{2}$ represent the
intracell and intercell hopping parameters, respectively, while $\gamma$ is an
asymmetry term that introduces the non-Hermitian character into the system —
see Fig. 1a. Throughout this paper, we fix all these parameters to be real-
valued, i.e., $\\{t_{1},t_{2},\gamma\\}\in\mathbb{R}$.

Figure 3: Skin effect of the non-Hermitian DCA chain represented through the
sum of the squared amplitudes of each site $i$ with $N=30$, $\gamma=3$ and
$t_{1}/t_{2}=2$ To visualize the eigenstates, we are considering the base
$(\mathrm{H}_{n},\mathrm{A}_{n},\mathrm{B}_{n})$. Additionally, we are
rescaling their expectation value for comparative reasons. Inset: complex
plane with the corresponding positive(green), negative(yellow) and flat(blue)
energy bands.

Assuming translational invariance, the NH Hamiltonian operator can be written
in reciprocal space as

$h_{\mathrm{DCA}}(\kappa)=d_{x}\Sigma_{x}+d_{y}\Sigma_{y}$ (2)

after an overall rotation $\text{e}^{\text{i}\kappa/2}$ over H sites in
reciprocal space, where $\kappa=ka_{0}$ is a real and dimensionless quasi
momentum, $a_{0}$ is the lattice periodicity constant, and the momentum
$k\in\textrm{BZ}$. We have defined the matrices $\Sigma_{x}$ and $\Sigma_{y}$
as:

$\displaystyle\Sigma_{x}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&1&0\\\ 1&0&1\\\
0&1&0\\\ \end{pmatrix}\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
\Sigma_{y}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&\text{i}&0\\\
-\text{i}&0&-\text{i}\\\ 0&\text{i}&0\\\ \end{pmatrix}$ (3a) accompanied by
$\displaystyle\Sigma_{z}=\frac{1}{2}\begin{pmatrix}1&0&1\\\ 0&-2&0\\\ 1&0&1\\\
\end{pmatrix}\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
\tilde{\mathbb{I}}=\frac{1}{2}\begin{pmatrix}1&0&1\\\ 0&2&0\\\ 1&0&1\\\
\end{pmatrix}$ (3b)

Therefore, the characteristic $\bm{d}$-vector, $\bm{d}=(d_{x},d_{y},d_{z})$
reads explicitly

$\bm{d}(\kappa)=\sqrt{2}(t_{1}+t_{2}\cos(\kappa),t_{2}\sin(\kappa)+\text{i}\gamma/2,0)$
(4)

It has to be noted that the set
$\\{\tilde{\mathbb{I}},\Sigma_{x},\Sigma_{y},\Sigma_{z}\\}$ forms an
orthogonal base with SU(2) Lie algebra with
$[\Sigma_{n},\Sigma_{m}]=2\text{i}\epsilon_{nmk}\Sigma_{k}\tilde{\mathbb{I}}$
and $\\{\Sigma_{n},\Sigma_{m}\\}=2\delta_{nm}\tilde{\mathbb{I}}$, where
$\epsilon_{nmk}$ is the Levi-Civita tensor with $n,m,k\in\\{x,y,z\\}$, and
$[.,.]$ and $\\{.,.\\}$ are the commutator and the anticommutator,
respectively [34].

The representation in Eq. (2) results in a three-band model. This can be
considered equivalent to the one obtained for the NH SSH two-band model [15,
51] with the inclusion of a zero-energy flat band. This analogy will be made
more explicit in Sec. 2.3.

The energy spectrum of DCA is

$\displaystyle E^{\mathrm{DCA}}_{\alpha}$
$\displaystyle=\alpha\sqrt{2\left(t_{1}^{2}+t_{2}^{2}\right)+4t_{1}t_{2}\cos(\kappa)+2\text{i}\gamma
t_{2}\sin(\kappa)-\frac{\gamma^{2}}{2}}$ (5)

with $\alpha\in\\{0,\pm\\}$. We note in passing that this spectrum is
identical to the NH-SSH model up to a factor $\sqrt{2}$ [15], see Eq. (4). We
present in App. A.1 the analytical expression of the eigenstates of Eq. (2).

By inspecting the Jordan decomposition of the DCA Hamiltonian and the spectrum
in Eq. (5), we find four exceptional points at
$\mathrm{Im}[E_{\pm}]=\mathrm{Re}[E_{\pm}]=0$. These points are located at
$t_{1}=-t_{2}\pm\gamma/2$ for $\kappa=0$ and $t_{1}=t_{2}\pm\gamma/2$ for
$\kappa=\pi$, recognizing the similarity with the standard non-Hermitian SSH
model [15]. As $\kappa$ ranges from 0 to 2$\pi$, the system’s energy spectrum
on the complex energy plane is formed of two bands and a zero-energy point
that signals the zero-energy flat band of the system under OBC (see the inset
of Fig. 3). Depending on the choice of parameters, the two-band energies can
either braid into two separate loops (inside which a reference point can be
placed acting as a point gap) or into a single loop (also presenting a point
gap) [48]. Both phases display a qualitative change in the spectrum, in which
the phase transitions correspond to the crossings of exceptional points at
which the Hamiltonian becomes defective. We find a real and a fully imaginary
gap for $\kappa=\\{0,\pi\\}$ with $|\gamma/2|\leq|t_{1}\pm t_{2}|$ and
$|\gamma/2|\geq|t_{1}\pm t_{2}|$, respectively.

Figure 4: Absolute value of the eigenenergies as a function of the hopping
parameters $t_{1}$ of the non-Hermitian DCB model for $\gamma=3$ and $N=16$.
The pink lines depict the bulk spectra under periodic boundary conditions. The
blue (red) lines correspond to the bulk (edge) spectra obtained under open
boundary conditions (OBC), signaling the phase transition points through
arrows. The black dashed lines indicate the specific values of the hopping
parameters i) $t_{1}/t_{2}=0.3$, ii) $t_{1}/t_{2}=1.2$ and iii)
$t_{1}/t_{2}=2.7$ at which the corresponding skin effects have been computed
(see Fig. 5). The zero-energy flat band is depicted in purple, present under
both boundary conditions.

The finite-size system presents edge modes under the appropriate choice of
system parameters. In Fig. 2, we compare the energy spectrum —now denoted
through $\mathcal{E}$— of a finite-size system with open boundary conditions,
in which edge modes appear, and with periodic boundary conditions. In Sec.
2.2, we will characterize the presence of these edge states using the
biorthogonal polarization [15], enabling us to differentiate them from bulk
eigenstates that pile up at the boundaries; a topologically trivial effect in
non-Hermitian systems lacking parity symmetry. This is known under the name of
non-Hermitian _skin_ effect and corresponds to a piling of the system
eigenstates at the boundary of the system. We show in Fig. 3 the skin effect
for left and right eigenstates for the DCA system. Upon evaluating the Jordan
canonical form of the non-Hermitian DCA system at $t_{1}=\gamma/2$, we find
$N+3$ independent eigenvectors, $N$ corresponding to the zero-energy flat band
and three corresponding to EPs. Similar to the non-Hermitian SSH [52]. From
these three EPs, we obtain that two EPs are of higher order and are located at
$\mathcal{E}=\pm\sqrt{2}t_{2}$, whereas the last one is a zero-energy EP of
order two.

Figure 5: Skin effect of the non-Hermitian DCB chain represented through the
sum of the squared amplitudes of each site $i$ with $N=30$ and $\gamma=3$
considering the base $(\mathrm{H}_{n},\mathrm{A}_{n},\mathrm{B}_{n})$. The
upper diagram i) represents the sum of the squared amplitudes of each site
considering the case in which the eigenvalues are fully imaginary pairs and
where also edge states are present, like at $t_{1}/t_{2}=0.3$. The middle
diagram ii) shows the skin effect under eigenvalues coming in complex
conjugate pairs as in $t_{1}/t_{2}=1.2$, where the edge states are fading out
into the bulk and the lower one iii) represents the case in which the spectra
is fully real as it happens for $t_{1}/t_{2}=2.7$, with no edge states. Each
inset depicts the corresponding positive(green), negative(yellow) and
flat(blue) energy bands in the complex plane.

####  2.1.3 Diamond Chain B (DCB)

For the DCB lattice, the non-Hermitian tight-binding Hamiltonian will read:

$\displaystyle\mathcal{H}_{\mathrm{DCB}}=$
$\displaystyle\sum_{n}\left\\{\left[t_{2}(c_{\mathrm{A},n}^{\dagger}+c_{\mathrm{B},n}^{\dagger})c_{\mathrm{H},n+1}+\text{h.c.}\right]\right.$
$\displaystyle+\left(t_{1}+\frac{\gamma}{2}\right)(c_{\mathrm{H},n}^{\dagger}c_{\mathrm{A},n}+c_{\mathrm{B},n}^{\dagger}c_{\mathrm{H},n})$
(6)
$\displaystyle+\left.\left(t_{1}-\frac{\gamma}{2}\right)(c_{\mathrm{A},n}^{\dagger}c_{\mathrm{H},n}+c_{\mathrm{H},n}^{\dagger}c_{\mathrm{B},n})\right\\}$

in which the primary distinction compared to the DCA lattice is the
orientation of the intracell hoppings ($t_{1}\pm\gamma/2$) connecting the H
and B sites — see Fig. 1b. In the translational invariant form, the
corresponding Hamiltonian cannot be expressed anymore only as a function of
the characteristic vector $\mathbf{d}$ and the matrices in Eqs. (3), but
requires the addition of the $\lambda_{7}$ Gell-Mann matrix:

$\displaystyle
h_{\mathrm{DCB}}(\kappa)=h_{\mathrm{DCA}}(\kappa)-\text{i}\begin{pmatrix}0&0&0\\\
0&0&-\text{i}\\\ 0&\text{i}&0\end{pmatrix}\gamma$ (7)

This last term is essential in order to change the sign of the non-reciprocal
hopping term on the connections between the H and the B sites — see Fig. 1b.

The corresponding energy spectrum reads

$\displaystyle E^{\text{DCB}}_{\alpha}$
$\displaystyle=\alpha\sqrt{2\left(t_{1}^{2}+t_{2}^{2}\right)+4t_{1}t_{2}\cos(\kappa)-\frac{\gamma^{2}}{2}}$
(8)

with $\alpha\in\\{0,\pm\\}$. Contrary to the case of DCA in Eq. (5), here we
no longer find terms coupling together $\gamma$ with $\kappa$. We present in
App. A.1 the analytical expression of the eigenstates of Eq. (7).

Importantly, the expression for the energy spectrum for DCB is, up to a
constant factor, the same as the energy spectrum for the
$\mathcal{PT}$-symmetric NH-SSH model [20], with the addition of a zero energy
flat band. When the magnitudes of the first two terms in Eq. (8) exceed that
of the third term, a real gap emerges. Conversely, if the first two terms are
smaller than the third term, an entirely imaginary gap is present. When
plotted on the complex energy plane, contrary to the case of DCA, the
eigenenergies (8) collapse into fully imaginary or real lines or a combination
of both — see insets of Fig. 5.

In spite of the similarities of the energy spectrum for a finite DCB system
with a finite $\mathcal{PT}$-symmetric NH-SSH system (see Fig. 4), the DCB
model does not possess EPs under OBC due to its non-Hermitian nature being
attributed to non-reciprocal hoppings rather than on-site potentials. As a
result, there is no phase transition between a $\mathcal{PT}$-broken and
$\mathcal{PT}$-unbroken phase in the DCB model.The complex energy spectrum of
the DCB collapsing into lines should signal the disappearance of the skin
effect [53]. However, from Fig. 5ii,iii, we can clearly observe that the skin
effect is still present for the DCB case despite the shape of the energy
spectrum in the complex plane. We will give some additional insight into the
skin effect of DCB in Sec. 2.2.

###  2.2 Topological properties

In this section, we are going to explore the topological properties of DCA and
DCB. While both exhibit edge modes, DCA possesses zero-energy edge modes,
whereas DCB has them at finite energy. Additionally, the flat band eigenstate
of the system for the DCB, unlike for DCA, is $k$-dependent — see Eqs. (A5a)
and (A6a). Consequently, we will concentrate on characterizing non-trivial
topological properties of the zero-energy modes on the DCA through the
biorthogonal polarization [15, 1]. Whereas, for the case of DCB, we will focus
on the quantum metric properties of its flat band.

The DCA model exhibits several relevant symmetries considering the 38-fold
classification typical on NH systems [10]. It presents the Hermitian
conjugated particle-hole symmetry (PHSc), with eigenenergies coming as either
pure imaginary energies or pairs $(E(k),-E(-k)^{*})$. Additionally, it shows
the standard time-reversal symmetry (TRSc), where eigenenergies come as real
eigenvalues or in complex conjugate pairs, $(E(k),E(-k)^{*})$. Moreover, the
DCA model possesses the standard sublattice symmetry (SLS), for which the
eigenvalues come in $\pm$ pairs at each $k$ point. The DCB model shares the
same symmetries as DCA and, in addition, possesses the Hermitian conjugated
time-reversal symmetry (TRSt) where the eigenenergies are paired by
$(E(k),E(-k))$ and the standard pseudo-Hermiticity symmetry, which restricts
the eigenenergies to be real. For a comprehensive overview of the symmetries
discussed, please refer to the symmetry table in App. A.2.

####  2.2.1 Biorthogonal polarization for DCA

Figure 6: Phase diagram of the DCA model along with the biorthogonal
polarization of two cuts in this phase diagram. b) Phase diagram of the non-
Hermitian DCA model. In this analysis, pink lines represent the parameter
values where exceptional points (EPs) emerge in the eigenenergies under
periodic boundary conditions (PBC), while blue lines indicate the appearance
or disappearance of zero-energy edge states under open boundary conditions
(OBC). In the Hermitian system ($\gamma=0$) under OBC, the phase transition
coincides with the one observed in the system under PBC. However, in the non-
Hermitian case, the topological phase (purple area) is defined by the OBC
system, which broadens the range in which the trivial phase appears (both the
pink-shaded and white areas). The dashed lines correspond to the values at
which the polarization has been computed analytically for a DCA chain of
$N=100$. In panel (a), the calculation is performed for $\gamma/t_{2}=3$, and
in panel (c), it is done at $\gamma/t_{2}=1$, such that when the zero-energy
edge states are present (absent), the biorthogonal polarization is
$\mathcal{P}=1(0)$ for the $\mathrm{H}$-broken chain (purple lines) and
$\mathcal{P}=0(1)$ for the $\mathrm{AB}$-broken chain (orange lines). The
dashed lines in pink and blue indicate the values of the parameters where EPs
occur under PBC and phase transition points arise under OBC, respectively.

In this section, we focus on the characterization of the zero-energy boundary
modes of the non-Hermitian diamond chain and their localization over the
parameter spectrum. We start by combining the left and right non-orthogonal
eigenvectors of the non-Hermitian Hamiltonian operators to map the weight
distribution of a given band—in this case, the edge-band—over the finite
system [54]. This arrangement is obtained through the biorthogonal projection
expectation value, $\langle\Pi_{n}\rangle_{\mathrm{LR}}$, also found as the
biorthogonal density [15] where the biorthogonal projection operator is
defined as
$\Pi_{n}=\sum_{\alpha}\ket{e_{\mathrm{\alpha},n}}\bra{e_{\mathrm{\alpha},n}}$
with $\ket{e_{\alpha,n}}=c_{\alpha,n}^{\dagger}\ket{0}$ [55]. Therefore a
real-parameter space topological invariant can be defined, namely the
biorthogonal polarization:

$\mathcal{P}=M-\lim_{N\rightarrow\infty}\braket{\Psi^{\mathrm{L}}_{0}}{N^{-1}\sum_{n=1}^{N}n\hat{\Pi}_{n}}{\Psi^{\mathrm{R}}_{0}}$
(9)

where $M$ is the number of boundary modes and
$\ket{\Psi^{\mathrm{R}[\mathrm{L}]}_{0}}$ are zero-energy right and left
boundary modes. The biorthogonal polarization will therefore be
$\mathcal{P}=M(0)$ when boundary states are present (absent) at the beginning
of a quasi-1D chain. In order for the wave functions
$\ket{\Psi^{\mathrm{R}[\mathrm{L}]}_{0}}$ in Eq. (9) to provide significant
information for the appearance and the disappearance of boundary states in the
corresponding lattices, the destructive interference method is used.

This local interference is naturally present on a quasi-1D lattice that begins
and also ends with the same motif, in this case with an $\mathrm{H}$ site or
with $\mathrm{AB}$ sites, under the constraint of presenting only nearest-
neighbor hopping [56, 57]. Under these circumstances, two chains can be
obtained, the so-called broken chains, characterized by an exactly
disappearing weight on one of the motifs. As a result, the localization
factors of the broken chains’ eigenmodes, $r_{\mathrm{L}[\mathrm{R}]}$, can be
exactly solved. These factors, whose magnitude will solely depend on the
hopping terms $|r_{\mathrm{L}[\mathrm{R}]}|=f(t_{1},t_{2},\gamma)$, will
eventually give the dispersion rate of the eigenmode, signaling the parameter
regions at which the topological phase transitions occur, i.e., at which the
zero-energy wavefunctions correspond to bulk states
$|r^{*}_{\mathrm{L}}r_{\mathrm{R}}|>1$ or to boundary states
$|r^{*}_{\mathrm{L}}r_{\mathrm{R}}|<1$ [15]. In spite of the eigenstates of
the broken chains not being exact to the ones of the original unbroken chain,
the phase transition points will stay unchanged up to finite-size effects in
both broken and unbroken lattices [15].

To further understand the destructive interference method, we will now solve
the biorthogonal polarization for the DCA. The number of boundary modes
forthis model is $M=1$ for an $\mathrm{H}$-broken chain and $M=1$ for an
$\mathrm{AB}$-broken chain.

For DCA, the zero-energy eigenmodes for the $\mathrm{H}$-broken chain,
$\ket{\Psi^{\mathrm{R}[\mathrm{L}]}_{0}}_{\mathrm{H}}$, and for the
$\mathrm{AB}$-broken chain,
$\ket{\Psi^{\mathrm{R}[\mathrm{L}]}_{0}}_{\mathrm{AB}}$, will be described
through

$\displaystyle\ket{\Psi^{\mathrm{R}[\mathrm{L}]}_{0}}_{\mathrm{H}}$
$\displaystyle=\mathcal{N}_{\mathrm{\mathrm{R}[\mathrm{L}]}}\left(1,0,0,r_{\mathrm{R}[\mathrm{L}]},0,0,\cdots,r_{\mathrm{R}[\mathrm{L}]}^{N-2},0,0,r_{\mathrm{R}[\mathrm{L}]}^{N-1},0,0\right)$
(10a) $\displaystyle\ket{\Psi^{\mathrm{R}[\mathrm{L}]}_{0}}_{\mathrm{AB}}$
$\displaystyle=\mathcal{N}_{\mathrm{\mathrm{R}[\mathrm{L}]}}\left(1,1,0,\frac{1}{r_{\mathrm{R}[\mathrm{L}]}},\frac{1}{r_{\mathrm{R}[\mathrm{L}]}},0,\cdots,\frac{1}{r_{\mathrm{R}[\mathrm{L}]}^{N-2}},\frac{1}{r_{\mathrm{R}[\mathrm{L}]}^{N-2}},0,\frac{1}{r_{\mathrm{R}[\mathrm{L}]}^{N-1}},\frac{1}{r_{\mathrm{R}[\mathrm{L}]}^{N-1}},0\right)$
(10b)

where we consider a base of $(\mathrm{H}_{n},\mathrm{A}_{n},\mathrm{B}_{n})$
for the $\mathrm{H}$-broken chain and of
$(\mathrm{A}_{n},\mathrm{B}_{n},\mathrm{H}_{n})$ for the $\mathrm{AB}$-broken
chain to describe each unit cell $n$. In Eqs. (10),
$\mathcal{N}_{\mathrm{\mathrm{R}[\mathrm{L}]}}$ are the right or left
normalization factors and $r_{\mathrm{L}[\mathrm{R}]}$ are the left and right
localization factors, which are equivalent to those derived for the non-
Hermitian SSH model [15]:

$r_{\mathrm{L}}=-\frac{t_{1}+\gamma/2}{t_{2}}\quad\text{and}\quad
r_{\mathrm{R}}=-\frac{t_{1}-\gamma/2}{t_{2}}$ (11)

The presence of destructive interference in the previous eigenmodes of Eqs.
(10) can be clearly visualized through the vanishing amplitudes on
$\mathrm{A}$ and $\mathrm{B}$ sites for the $\mathrm{H}$-broken chain and in
$\mathrm{H}$ sites for the $\mathrm{AB}$-broken one.

Considering the exponential (de)localization of the eigenmodes into the
boundaries,
$\braket{n}{\Psi_{0}^{\mathrm{R}[\mathrm{L}]}}=\text{e}^{-n/\xi_{\mathrm{R}[\mathrm{L}]}}$
where
$\ket{n}=\ket{e_{\mathrm{A},n}}+\ket{e_{\mathrm{H},n}}+\ket{e_{\mathrm{B},n}}$,
the jump on the biorthogonal polarization is constructed through the right and
left penetration lengths $\xi_{\mathrm{R}[\mathrm{L}]}$ of the boundary modes.
Knowing that a biorthogonal bulk state forms from right and left states
localized at opposite ends $\xi_{\mathrm{R}}=-\xi_{\mathrm{L}}$ and
considering that under translational invariance the inverse penetration
lengths can be defined through the localization factors as

$\xi^{-1}_{\mathrm{R}[\mathrm{L}]}(\mathbf{k})=\ln|r_{\mathrm{R}[\mathrm{L}]}(\mathbf{k})|$

the condition for having a jump between bulk and boundary eigenmodes will be
given by $|r^{*}_{\mathrm{L}}r_{\mathrm{R}}|=1$ [56]. As a result, the
biorthogonal polarization acts as an indicator of the topological phase
transition using real-space parameters, $\mathcal{P}=f(t_{1},t_{2},\gamma)$,
resulting in a real-space topological invariant [54].

Despite the fact that the left and right states might be localized at opposite
or the same edges, only one of the boundaries will be considered at a time
when analyzing the broken lattices reflecting each end of our unbroken chain.
This means that the computation of $\mathcal{P}$ with the left and right
eigenstates of Eq. (10a) will represent the localization of boundary states at
$n=1$ (see purple lines of Fig. 6) and the computation of $M-\mathcal{P}$
through Eq. (10b) will represent the localization of boundary states at $n=N$
(see orange lines of Fig. 6) for the original chain (starting with an
$\mathrm{H}$ site and ending with $\mathrm{A}$ and $\mathrm{B}$ sites).
Considering the shortcut solution used in Ref. [12] for the NH SSH, we obtain
the phase transition points of the DCA unbroken chain to be defined as

$\displaystyle t_{1}$
$\displaystyle=\pm\sqrt{t_{2}^{2}+\left(\frac{\gamma}{2}\right)^{2}}\quad\text{for}\quad|t_{2}|>\frac{|\gamma|}{2}\quad\text{and}$
(12a) $\displaystyle t_{1}$
$\displaystyle=\pm\sqrt{-t_{2}^{2}+\left(\frac{\gamma}{2}\right)^{2}}\quad\text{for}\quad|t_{2}|<\frac{|\gamma|}{2}$
(12b)

which have been represented in blue lines at Fig. 6b and in blue dashed lines
at Fig. 6a,c. This shows that the appearance of the zero-energy edge states
delimited by the biorthogonal polarization is in accordance with the predicted
phase transitions from the generalized BZ method used in Ref. [12].

####  2.2.2 Band geometry and topology for DCB

Figure 7: Plot of the quantum metric associated to the flat band (14)of
lattice DCB as a function of $t_{1}$ and of the non-reciprocal coupling term
$\gamma$.

As described before, the wave function associated with the flat band of DCB
has a non-trivial dependence on the momentum $\kappa$ — see App. A.1. In the
following, we will show that this leads to non-trivial quantum metric
properties. To begin with, we briefly recall the definition and main
properties of the quantum metric $g_{\mu\nu}$ in Hermitian and non-Hermitian
systems. In the Hermitian case, the quantum metric [58] is a gauge invariant
and measurable quantity [59, 60, 61] that can be seen as a momentum-space
Riemannian metric [62, 63, 64, 65]. It has been shown that $g_{\mu\nu}$ plays
a central role in Chern insulators [66, 67, 68, 69], spreading of the Wannier
functions [70], superconducting weight in flat-band superconductors [71, 72,
73, 74], topological semimetals [75, 76, 77, 78], quantum phase transitions
[79, 80, 81] and in the semiclassical equations for wave-packets [82].
Moreover, the non-Hermitian version of the quantum metric has been originally
introduced in Ref. [83] and has been recently shown to be relevant in
different kinds of non-Hermitian topological phases [84] (see also Ref. [85]
for a more recent work on the subject). In order to define this geometric
quantity in the NH case, we first introduce the NH quantum geometric tensor
(QGT) $Q_{\mu\nu}^{n}$, given by

$\displaystyle
Q^{n}_{\mu\nu}=\frac{1}{2}[\langle\partial_{\mu}u_{n}^{\text{L}}|$
$\displaystyle(1-P_{n})|\partial_{\nu}u_{n}^{\text{R}}\rangle+$ (13)
$\displaystyle+\langle\partial_{\mu}u_{n}^{\text{R}}|(1-P_{n}^{\dagger})|\partial_{\nu}u_{n}^{\text{L}}\rangle]$

where $n$ is the band index, $|u_{n}^{\text{L/R}}\rangle$ are the left/right
Bloch wave-eigenvectors, $\partial_{\mu}=\partial_{k_{\mu}}$ and
$P_{n}=|u_{n}^{\text{R}}\rangle\langle u_{n}^{\text{L}}|$ is the projector
operator. Thus, the corresponding NH quantum metric tensor is given by the
real part of the above tensor, i.e., $g_{\mu\nu}=\text{Re}(Q_{\mu\nu}^{n})$
while the imaginary part of the NH QGT corresponds to the NH Berry curvature,
namely $F^{n}_{\mu\nu}=-2\text{Im}(Q^{n}_{\mu\nu})$. However, we notice that
in 1D systems, the imaginary part of the QGT is absent such that $g_{\mu\nu}$
is the only non-trivial geometric quantity that can be built. For the DCB, the
NH quantum metric for the flat and negative energy bands are respectively
given by

$\displaystyle g^{0}_{xx}(\kappa)$
$\displaystyle=-\frac{t_{2}^{2}\,\gamma^{2}}{\left(E^{\mathrm{DCB}}_{-}\right)^{4}}$
(14) $\displaystyle g^{-}_{xx}(\kappa)$
$\displaystyle=-\frac{t_{2}^{2}\,(\gamma^{2}-t_{1}^{2}-2t_{2}^{2}-4t_{1}t_{2}\cos\kappa-
t_{1}^{2}\cos 2\kappa)}{2\left(E^{\mathrm{DCB}}_{-}\right)^{4}}$

We can now make some relevant observations concerning the peculiar band
geometry of DCB: first, $g^{0}_{xx}$ is non-zero only in the NH regime, in
which even the flat band acquires some non-trivial geometric features
differently from the Hermitian case. In fact, due to its dependence on the
energy dispersion, $E^{\mathrm{DCB}}$, the band geometry of the flat band
contains information about the exceptional points of DCB. Thus the
corresponding quantum metric is divergent at those points, similar to what has
been shown in NH models without flat bands [86, 87]. Second, another main
consequence of the non-trivial behavior of $g^{0}_{xx}$ is related to the
conductivity at the flat band that is robust in the presence of disorder and
can even be tremendously boosted, similar to the Hermitian case discussed in
Ref. [50]. The boost is obtained for values of $\gamma$ changing the spectrum
from real to imaginary (8) — see Fig. 7. Third, in the limit $t_{1}\rightarrow
0$, DCB supports three completely flat bands, and the quantum metric becomes
constant for all the bands. This quantum geometry behavior resembles the one
related to the Hermitian two-band Creutz ladder, where the quantum metric has
been shown to be constant for both flat bands [88]. In this limit, the ratio
between $g^{0}_{xx}$ and $g^{-}_{xx}$ acquires a simple expression

$\frac{g^{0}_{xx}}{g^{-}_{xx}}=\frac{2\gamma^{2}}{\gamma^{2}-2t_{2}^{2}}$ (15)

which converges to 2 for $\gamma\gg t_{2}$, i.e., in the strongly NH regime.
On the other hand, in the opposite Hermitian limit $\gamma=0$ and for
$t_{1}\rightarrow 0$, ${g^{-}_{xx}}$ is still constant and equal to $1/4$
while $g^{0}_{xx}$ becomes identically null.

To characterize the band topology of DCB, we employ the approach originally
proposed in the Hermitian framework in Ref. [89] and then extended in the non-
Hermitian systems in Ref. [84], which is based on the construction of suitable
momentum-space scalar fields $\phi^{\text{L}}_{n}$ and $\phi^{\text{R}}_{n}$,
given by

$\phi^{\text{L}}_{n}=-\frac{\text{i}}{2}\log\prod_{\aleph}u^{\text{L}}_{n,\aleph},\hskip
8.5359pt\phi^{\text{R}}_{n}=-\frac{\text{i}}{2}\log\prod_{\aleph}u^{\text{R}}_{n,\aleph}$
(16)

where $u^{\text{L}}_{n,\aleph}$ and $u^{\text{R}}_{n,\aleph}$ denote the non-
zero components of $|u^{\text{L}}_{n}\rangle$ and $|u^{\text{R}}_{n}\rangle$,
respectively. From the above scalar fields in our 1D case, we can define a NH
Zak-like phase for the DCB as follows

$w_{n}=-\int_{0}^{2\pi}d\kappa\,\partial_{\kappa}\phi^{\text{L}}_{n}=-\int_{0}^{2\pi}d\kappa\,\partial_{\kappa}\phi^{\text{R}}_{n}$
(17)

where the integration is performed in the first Brillouin zone. From this
expression, we observe that for the flat band, $w_{0}$ is never quantized,
which implies a trivial band topology. On the other hand, the lower band
carries non-trivial band topology in the following regime

$w_{-}=\pi,\hskip 8.5359pt|t_{2}|-|t_{1}|>|\gamma|/2$ (18)

This result represents a natural but non-trivial generalization of the
topological phase supported for $|t_{2}|>|t_{1}|$ in the Hermitian model [34].

Figure 8: Rotated diamond chain from the DCA to a) a non-Hermitian SSH chain
with non-reciprocal hoppings plus disconnected $\mathcal{B}$ sites and from
the DCB to a Hermitian SSH chain coupled through b) non-reciprocal hopping
parameters to $\mathcal{B}$ sites using $U_{1}$ matrix transformation or c)
complex hopping parameters to the virtual sites using $U_{\text{i}}$,
respectively. Figure 9: Skin effect for the rotated non-Hermitian DCA and DCB
chains represented through the sum of the squared amplitudes of each site $i$
with $N=30$ and $\gamma=3$. The upper diagram (a) represents the skin effect
of the rotated non-Hermitian DCA system at $t_{1}/t_{2}=2$, the second panel
(b), shows the first case of the rotated non-Hermitian DCB system where the
eigenvalues are fully imaginary pairs like at $t_{1}/t_{2}=0.3$, the third
diagram (c) shows the skin effect of the rotated non-Hermitian DCB model skin
effect under eigenvalues coming in complex conjugate pairs as in
$t_{1}/t_{2}=1.2$ and the lower one (d) represents the rotated non-Hermitian
DCB skin effect for the case in which the spectra is fully real as it happens
for $t_{1}/t_{2}=2.7$. In the present figure, the rotated systems have been
represented considering the base
$(\mathcal{H}_{n},\mathcal{A}_{n},\mathcal{B}_{n})$. Additionally, the sites
of the open chain have been redistributed such that the skin effect of the
corresponding $\mathcal{B}$ sites are depicted in the inset of each figure,
leaving the main figure for the skin effect originated from the $\mathcal{A}$
and $\mathcal{H}$ sites, respectively.

###  2.3 Rotation into a 1D model with a flat band

In the following, we show how to perform a rotation to a new base choice of
the diamond chain, which allows us to have a better understanding of several
of the properties we have shown so far. We obtain the new base considering a
real or complex linear combination of the site with lower connectivity. The
site operators in this new base read

$\displaystyle\mathcal{A}_{n}^{\beta}$
$\displaystyle=\frac{1}{\sqrt{2}}(c_{\mathrm{A},n}+\beta c_{\mathrm{B},n})$
(19a) $\displaystyle\mathcal{H}_{n}$ $\displaystyle=c_{\mathrm{H},n}$ (19b)
$\displaystyle\mathcal{B}_{n}^{\beta}$
$\displaystyle=\frac{1}{\sqrt{2}}(c_{\mathrm{A},n}-\beta c_{\mathrm{B},n})$
(19c)

where $\beta=\\{1,\text{i}\\}$. The matrices $U_{\beta}$ associated with this
rotation result in:

$\displaystyle
U_{\mathrm{\beta}}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&0&\beta\\\
0&\sqrt{2}&0\\\ 1&0&-\beta\\\ \end{pmatrix}$ (20)

and applied to the NH Hamiltonian operator in reciprocal space of Eq. (2),
give rise to the same rotated model of the DCA. The resulting Hamiltonian in
reciprocal space reads:

$h_{\mathrm{DCA}}^{\mathrm{Rot}}(\kappa)=d_{x}\Sigma_{x}^{\mathrm{Rot}}+d_{y}\Sigma_{y}^{\mathrm{Rot}}$
(21)

being the $\Sigma_{x}^{\mathrm{Rot}}$ and $\Sigma_{y}^{\mathrm{Rot}}$ matrices
defined as:

$\displaystyle\Sigma_{x}^{\mathrm{Rot}}=\begin{pmatrix}0&1&0\\\ 1&0&0\\\
0&0&0\\\ \end{pmatrix}\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
\Sigma_{y}^{\mathrm{Rot}}=\begin{pmatrix}0&\text{i}&0\\\ -\text{i}&0&0\\\
0&0&0\\\ \end{pmatrix}$ (22)

which is analogous to the non-Hermitian Stub model of Ref. [48] with
$t_{3}=0$. The rotated DCA model, therefore, describes a non-Hermitian SSH
chain with non-reciprocal hopping parameters [see Fig. 8a] formed by
$\mathcal{A}$ and H sites decoupled to the $\mathcal{B}$ sites originating the
flat band. This rotation helps to understand that in this new base, the NH
skin effect and the topological properties should be analogous to the one of
the NH SSH model [15].

The transformation matrices defined in Eq. (20) applied to the DCB lattice in
reciprocal space described in Eq. (7) give rise to two equivalent systems for
real and complex $\beta$:

$\displaystyle h_{\mathrm{DCB}}^{\mathrm{Rot1}}(\kappa)$
$\displaystyle=U_{1}^{-1}h_{\mathrm{DCB}}(\kappa)U_{1}$
$\displaystyle=\sqrt{2}\begin{pmatrix}0&t_{1}+t_{2}\text{e}^{\text{i}\kappa}&0\\\
t_{1}+t_{2}\text{e}^{-\text{i}\kappa}&0&\frac{\gamma}{2}\\\
0&-\frac{\gamma}{2}&0\end{pmatrix}$ (23a) $\displaystyle
h_{\mathrm{DCB}}^{\mathrm{Rot2}}(\kappa)$
$\displaystyle=U_{\text{i}}^{-1}h_{\mathrm{DCB}}(\kappa)U_{\text{i}}$
$\displaystyle=\sqrt{2}\begin{pmatrix}0&t_{1}+t_{2}\text{e}^{\text{i}\kappa}&0\\\
t_{1}+t_{2}\text{e}^{-\text{i}\kappa}&0&\text{i}\frac{\gamma}{2}\\\
0&\text{i}\frac{\gamma}{2}&0\end{pmatrix}$ (23b)

Both of these models contain a Hermitian SSH chain formed by $\mathcal{A}$ and
$\mathcal{H}$ sites that are now coupled to the $\mathcal{B}$ sites in a non-
Hermitian fashion, cf. Figs. 8b,c. It is important to note that the
$\mathcal{B}$ sites are disconnected from each other, thus representing the
flat band. However, they are coupled to the Hermitian SSH model. This coupling
is found to be non-reciprocal for the first rotation in Eq. (23), where we
observe that we have broken the inversion symmetry $\mathcal{I}=\Sigma_{x}$ —
see Fig. 8b. While the case of the imaginary rotation in Eq. (23), the
coupling between the flat band and the Hermitian SSH chain breaks time-
reversal symmetry $\mathcal{T}=\mathcal{C}$ — see Fig. 8c. This rotation helps
us to unveil the nature of the NH skin effect we have presented in Fig. 5. The
dispersive part of the system is a Hermitian SSH model and should not present
any skin effect; however, it is connected in a NH fashion to the flat band
giving rise to a finite skin effect. It has to be noted that the strength of
this skin effect is smaller than in the case of DCA since it originates in the
flat band where all the $\mathcal{B}$ sites are disconnected.

In Fig. 9, we present the results for the skin effect for DCA and DCB,
considering the tight-binding representations of Eqs. (21) and (23). For DCA
in Fig. 9a, we can clearly observe a skin effect identical to the case of the
NH SSH for the $\mathcal{H}$ and $\mathcal{A}$, whereas, for the $\mathcal{B}$
sites corresponding to the flat band, the skin effect is absent. For the case
of DCB, we show in Fig. 9b-d the results for the same parameter values we used
in Fig. 5 but in the rotated base, i.e.,
$\\{\mathcal{H},\mathcal{A},\mathcal{B}\\}$. We clearly observe that now the
skin effect that arises in the flat band is transmitted by _proximity_ to the
Hermitian SSH model via the NH coupling. Most importantly, within this choice
of the base, the right, and the left eigenvectors differ only up to a phase
factor, for the reason they are identical when representing them in Fig. 9b,d.

##  3 Discussions and Conclusion

In this article, we have investigated the topological properties of a non-
Hermitian version of the diamond chain. We have shown that compared to the
Hermitian case, there exist two possible non-Hermitian configurations
presenting non-trivial topological properties. Specifically, for one of the
two configurations that we have named DCA, the system presents properties
analogous to a non-Hermitian SSH chain plus a flat band. We have characterized
this system by investigating the biorthogonal polarization. The second
configuration, which we have named DCB, presents a non-zero quantum metric for
the flat band due to the wave function’s $k$ dependency. This quantum metric
becomes highly divergent near the exceptional points, providing a significant
boost when tuning the non-Hermitian parameter $\gamma$. The band topology of
the DCB configuration indicates that the flat band possesses a trivial nature,
while the non-Hermitian Zak phase of the lower band can exhibit a non-trivial
band topology. Both systems can be mapped to the SSH model. The DCA system can
be transformed into a non-Hermitian SSH model with disconnected sites
representing the flat band. In contrast, the DCB model can be similarly
transformed into a Hermitian SSH chain, connected in a non-Hermitian fashion
to these additional sites representing the flat band. Interestingly, both
systems present a NH skin effect. This is especially surprising for DCB, which
presents a spectrum in the complex plane that is either real, imaginary, or a
combination of both.

###### Acknowledgements.

The authors acknowledged useful discussions with Duy Hoang Minh Nguyen and
Alejandro Sebastián Gómez Gómez. C.M.S, M.A.J.H, and D.B. acknowledged the
support from the Spanish MICINN-AEI through Project No. PID2020-120614GB-I00
(ENACT). D.B. further acknowledged the support from Transnational Common
Laboratory $Quantum-ChemPhys$, and the Department of Education of the Basque
Government through the project PIBA_2023_1_0007 (STRAINER). A.G.E., D.B.,
M.A.J.H., and C.M. S. acknowledge funding from the IKUR Strategy under the
collaboration agreement between Ikerbasque Foundation and DIPC on behalf of
the Department of Education of the Basque Government, Programa de ayudas de
apoyo a los agentes de la Red Vasca de Ciencia, Tecnología e Innovación
acreditados en la categoría de Centros de Investigación Básica y de Excelencia
(Programa BERC) from the Departamento de Universidades e Investigación del
Gobierno Vasco and Centros Severo Ochoa AEI/CEX2018-000867-S from the Spanish
Ministerio de Ciencia e Innovación and from the Gipuzkoa Provincial Council
within the QUAN-000021-01 project. A.G.E., M.A.J.H., and C.M.S. acknowledged
support from the Spanish Ministerio de Ciencia e Innovación
(PID2022-142008NB-I00) and the Basque Government Elkartek program
(KK-2023/00016). F.K.K. acknowledges funding from the European Union via the
ERC Starting Grant “NTopQuant”. Views and opinions expressed are, however,
those of the authors only and do not necessarily reflect those of the European
Union or the European Research Council (ERC). Neither the European Union nor
the ERC can be held responsible for them.

## Appendix A

###  A.1 Eigenvectors of the non-Hermitian diamond chain

In this appendix, we will present the analytical expressions for the
eigenvectors of the two lattice systems, DCA and DCB. We start by introducing
the following quantities:

$\displaystyle\rho_{\pm}(\kappa)$
$\displaystyle=\frac{t_{1}\pm\frac{\gamma}{2}}{t_{2}}+\cos\kappa+\text{i}\sin\kappa$
$\displaystyle=|\rho_{\pm}(\kappa)|\text{e}^{\text{i}\phi_{\pm}(\kappa)}$ (A1)

The biorthogonal base for DCA reads:

$\displaystyle|\psi_{0,\text{R}}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(-1,0,1\right)$ (A2a)
$\displaystyle|\psi_{\alpha,\text{R}}\rangle$
$\displaystyle=\frac{1}{2}\left(1,\frac{E_{\alpha}^{\text{DCA}}}{t_{2}|\rho_{-}(\kappa)|\text{e}^{\text{i}\phi_{-}}},1\right)$
(A2b)

and

$\displaystyle|\psi_{0,\text{L}}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(-1,0,1\right)$ (A3a)
$\displaystyle|\psi_{\alpha,\text{L}}\rangle$
$\displaystyle=\frac{1}{2}\left(1,\frac{E_{\alpha}^{\text{DCA}}}{t_{2}|\rho_{+}(\kappa)|\text{e}^{\text{i}\phi_{+}}},1\right)$
(A3b)

These states fulfill the biorthogonal scalar product [55]:

$\displaystyle\langle\psi_{\alpha,\beta,\text{R}}|\psi_{\alpha^{\prime},\beta^{\prime},\text{L}}\rangle=\delta_{\alpha,\alpha^{\prime}}\delta_{\beta,\beta^{\prime}}$
(A4)

Whereas the biorthogonal base for DCB reads:

$\displaystyle|\psi_{0,\text{R}}\rangle$
$\displaystyle=\mathcal{N}\left(-|\rho_{+}(\kappa)|\text{e}^{-\text{i}\phi_{+}}+\gamma,0,|\rho_{+}(\kappa)|\text{e}^{\text{i}\phi_{+}}\right),$
(A5a) $\displaystyle|\psi_{\alpha,\text{R}}\rangle$
$\displaystyle=\mathcal{N}\left(\frac{-t_{2}|\rho_{-}(\kappa)|\text{e}^{-\text{i}\phi_{-}}+\gamma}{t_{2}|\rho_{+}(\kappa)|\text{e}^{\text{i}\phi_{+}}},\frac{E_{\alpha}^{\text{DCB}}}{t_{2}|\rho_{+}(\kappa)|\text{e}^{\text{i}\phi_{+}}},1\right)$
(A5b)

and

$\displaystyle|\psi_{0,\text{L}}\rangle$
$\displaystyle=\mathcal{N}\left(-|\rho_{+}(\kappa)|\text{e}^{-\text{i}\phi_{+}}|,0,|\rho_{-}(\kappa)|\text{e}^{\text{i}\phi_{-}}|\right)$
(A6a) $\displaystyle|\psi_{\alpha,\text{L}}\rangle$
$\displaystyle=\mathcal{N}\left(\frac{t_{2}|\rho_{+}(\kappa)|\text{e}^{\text{i}\phi_{+}}+\gamma}{t_{2}|\rho_{-}(\kappa)|\text{e}^{\text{i}\phi_{-}}},\frac{E_{\alpha}^{\text{DCB}}}{t_{2}|\rho_{-}(\kappa)|\text{e}^{\text{i}\phi_{-}}},1\right)$
(A6b)

Here, we have introduced the factor of normalization $\mathcal{N}$ in order to
fulfill the biorthogonal scalar product in Eq. (A4); the normalization factor
is defined as

$\mathcal{N}=\sqrt{\frac{1}{2}+\frac{2\text{i}\gamma
t_{2}\sin\kappa}{4\left(t_{1}^{2}+t_{2}^{2}\right)-\gamma^{2}+8t_{1}t_{2}\cos\kappa}}$
(A7)

Table 1: The DCA model is subject to symmetry conditions, each with its specific constraint equation. These equations are fulfilled by applying the symmetry operator on the Hamiltonian, where $\beta=\\{1,\text{i}\\}$. Symmetry | Equation | Operator
---|---|---
PHSt, $CC^{*}=\mathbf{1}$ | $h(-k)=-Ch^{T}(k)C^{-1}$ | $-$
PHSt, $CC^{*}=-\mathbf{1}$ | $h(-k)=-Ch^{T}(k)C^{-1}$ | $-$
TRSt, $TT^{*}=\mathbf{1}$ | $h(-k)=Th^{T}(k)T^{-1}$ | $-$
TRSt, $TT^{*}=-\mathbf{1}$ | $h(-k)=Ch^{T}(k)C^{-1}$ | $-$
PHSc, $CC^{*}=\mathbf{1}$ | $h(-k)=-Ch^{*}(k)C^{-1}$ |  $\beta R_{\mathrm{A},\mathrm{B}},\beta R_{\mathrm{H}}$,
|  | $\beta G_{\mathrm{A},\mathrm{B}},\beta G_{\mathrm{H}}$
PHSc, $CC^{*}=-\mathbf{1}$ | $h(-k)=-Ch^{*}(k)C^{-1}$ | $-$
TRSc, $TT^{*}=\mathbf{1}$ | $h(-k)=Th^{*}(k)T^{-1}$ | $\beta\mathbb{I}_{3\times 3},\beta P_{\mathrm{B},\mathrm{A}}$
TRSc, $TT^{*}=-\mathbf{1}$ | $h(-k)=Th^{*}(k)T^{-1}$ | $-$
CS, $\Gamma^{2}=\mathbf{1}$ | $h(k)=-\Gamma h^{\dagger}(k)\Gamma^{-1}$ | $-$
Pseudo-Hermiticity, |  |
$\eta^{2}=\mathbf{1}$ | $h(k)=\eta h^{\dagger}(k)\eta^{-1}$ | $-$
SLS, $S^{2}=\mathbf{1}$ | $h(k)=-Sh(k)S^{-1}$ |  $R_{\mathrm{A},\mathrm{B}},R_{\mathrm{H}}$,
|  | $G_{\mathrm{A},\mathrm{B}},G_{\mathrm{H}}$
Parity, $P^{2}=\mathbf{1}$ | $h(-k)=Ph(k)P^{-1}$ | $-$
Parity-time, |  |
$(PT)(PT)^{*}=\mathbf{1}$ | $h(k)=(PT)h^{*}(k)(PT)^{-1}$ | $-$
Parity-time, |  |
$(PT)(PT)^{*}=-\mathbf{1}$ | $h(k)=(PT)h^{*}(k)(PT)^{-1}$ | $-$
Table 2: The DCB model is subject to symmetry conditions, each with its specific constraint equation. These equations are fulfilled by applying the symmetry operator on the Hamiltonian, where $\beta=\\{1,\text{i}\\}$. Symmetry | Equation | Operator
---|---|---
PHSt, $CC^{*}=\mathbf{1}$ | $h(-k)=-Ch^{T}(k)C^{-1}$ | $-$
PHSt, $CC^{*}=-\mathbf{1}$ | $h(-k)=-Ch^{T}(k)C^{-1}$ | $\beta G_{\mathrm{A},\mathrm{B}},\beta G_{\mathrm{H}}$
TRSt, $TT^{*}=\mathbf{1}$ | $h(-k)=Th^{T}(k)T^{-1}$ | $\beta P_{\mathrm{B},\mathrm{A}}$
TRSt, $TT^{*}=-\mathbf{1}$ | $h(-k)=Ch^{T}(k)C^{-1}$ | $-$
PHSc, $CC^{*}=\mathbf{1}$ | $h(-k)=-Ch^{*}(k)C^{-1}$ | $\beta R_{\mathrm{A},\mathrm{B}},\beta R_{\mathrm{H}}$
PHSc, $CC^{*}=-\mathbf{1}$ | $h(-k)=-Ch^{*}(k)C^{-1}$ | $-$
TRSc, $TT^{*}=\mathbf{1}$ | $h(-k)=Th^{*}(k)T^{-1}$ | $\beta\mathbb{I}_{3\times 3}$
TRSc, $TT^{*}=-\mathbf{1}$ | $h(-k)=Th^{*}(k)T^{-1}$ | $-$
CS, $\Gamma^{2}=\mathbf{1}$ | $h(k)=-\Gamma h^{\dagger}(k)\Gamma^{-1}$ | $G_{\mathrm{A},\mathrm{B}},G_{\mathrm{H}}$
Pseudo-Hermiticity, |  |
$\eta^{2}=\mathbf{1}$ | $h(k)=\eta h^{\dagger}(k)\eta^{-1}$ | $P_{\mathrm{B},\mathrm{A}}$
SLS, $S^{2}=\mathbf{1}$ | $h(k)=-Sh(k)S^{-1}$ | $R_{\mathrm{A},\mathrm{B}},R_{\mathrm{H}}$
Parity, $P^{2}=\mathbf{1}$ | $h(-k)=Ph(k)P^{-1}$ | $-$
Parity-time, |  |
$(PT)(PT)^{*}=\mathbf{1}$ | $h(k)=(PT)h^{*}(k)(PT)^{-1}$ | $-$
Parity-time, |  |
$(PT)(PT)^{*}=-\mathbf{1}$ | $h(k)=(PT)h^{*}(k)(PT)^{-1}$ | $-$

###  A.2 Table of symmetries

In this appendix, we present the table on the symmetries of DBA and DCB. We
introduce the following auxiliary matrices:

$\displaystyle R_{\mathrm{H}}$ $\displaystyle=\begin{pmatrix}-1&0&0\\\
0&1&0\\\ 0&0&-1\end{pmatrix}$ (A8a) $\displaystyle R_{\mathrm{A},\mathrm{B}}$
$\displaystyle=\begin{pmatrix}1&0&0\\\ 0&-1&0\\\ 0&0&1\end{pmatrix}$ (A8b)
$\displaystyle P_{\mathrm{B},\mathrm{A}}$
$\displaystyle=\begin{pmatrix}0&0&1\\\ 0&1&0\\\ 1&0&0\end{pmatrix}$ (A8c)
$\displaystyle G_{\mathrm{H}}$
$\displaystyle=R_{\mathrm{H}}P_{\mathrm{B},\mathrm{A}},\quad\text{and}$ (A8d)
$\displaystyle G_{\mathrm{A},\mathrm{B}}$
$\displaystyle=R_{\mathrm{A},\mathrm{B}}P_{\mathrm{B},\mathrm{A}},$ (A8e)

which represent different symmetry operations. Let’s consider the following
matrices representing a simplified version of the Hamiltonians of the DCA and
DCB models in reciprocal space:

$\displaystyle
h_{\mathrm{DC}}=\begin{pmatrix}0&t_{\mathrm{A}\rightarrow\mathrm{H}}&0\\\
t_{\mathrm{H}\rightarrow\mathrm{A}}&0&t_{\mathrm{H}\rightarrow\mathrm{B}}\\\
0&t_{\mathrm{B}\rightarrow\mathrm{H}}&0\end{pmatrix}$ (A9)

being $h_{\mathrm{DCB}}$ like $h_{\mathrm{DCA}}$ but changing $t\rightarrow
t^{\prime}$. The matrix $R_{\mathrm{H}}$ when applied at the left of the
Hamiltonians described in Eq. (A9) changes the sign of the
$\mathrm{H}$-outgoing hoppings,
$t^{(^{\prime})}_{\mathrm{H}\rightarrow\mathrm{A,B}}\Rightarrow-t^{(^{\prime})}_{\mathrm{H}\rightarrow\mathrm{A,B}}$.
Similarly, the matrix $R_{\mathrm{A,B}}$, when applied at the left of the
Hamiltonians, changes the sign of the $\mathrm{A}$– and $\mathrm{B}$–outgoing
hoppings,
$t^{(^{\prime})}_{\mathrm{A,B}\rightarrow\mathrm{H}}\Rightarrow-t^{(^{\prime})}_{\mathrm{A,B}\rightarrow\mathrm{H}}$.
Consequently, when the full transformations are applied, the resulting matrix
is a complete sign flip for both transformations: $h\Rightarrow
R_{\mathrm{H}}hR^{-1}_{\mathrm{H}}=-h$ and $h\Rightarrow
R_{\mathrm{A,B}}hR^{-1}_{\mathrm{A,B}}=-h$.

The permutation matrix $P_{\mathrm{B,A}}$ exchanges the hopping parameters
between the $\mathrm{A}$ and $\mathrm{B}$ sites
$t^{(^{\prime})}_{\mathrm{H}\rightarrow\mathrm{A,B}}\Rightarrow
t^{(^{\prime})}_{\mathrm{H}\rightarrow\mathrm{B,A}}$ and
$t^{(^{\prime})}_{\mathrm{A,B}\rightarrow\mathrm{H}}\Rightarrow
t^{(^{\prime})}_{\mathrm{B,A}\rightarrow\mathrm{H}}$. Consequently, the
transformation matrices $G_{\mathrm{H}}$ and $G_{\mathrm{A,B}}$ combine both
transformations, resulting in an overall sign flip and a permutation between
the $\mathrm{A}$ and $\mathrm{B}$ sites:

$\displaystyle
Gh_{\mathrm{DC}}G^{-1}=\begin{pmatrix}0&-t_{\mathrm{B}\rightarrow\mathrm{H}}&0\\\
-t_{\mathrm{H}\rightarrow\mathrm{B}}&0&-t_{\mathrm{H}\rightarrow\mathrm{A}}\\\
0&-t_{\mathrm{A}\rightarrow\mathrm{H}}&0\end{pmatrix}$ (A10)

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