# Generalized Lieb’s theorem for noninteracting non-Hermitian $n$-partite
tight-binding lattices

A. M. Marques<EMAIL_ADDRESS>Department of Physics $\&$ i3N,
University of Aveiro, 3810-193 Aveiro, Portugal  R. G. Dias  Department of
Physics $\&$ i3N, University of Aveiro, 3810-193 Aveiro, Portugal

###### Abstract

Hermitian bipartite models are characterized by the presence of chiral
symmetry and by Lieb’s theorem, which derives the number of zero-energy flat
bands of the model from the imbalance of sites between its two sublattices.
Here, we introduce a class of non-Hermitian models with an arbitrary number of
sublattices connected in a unidirectional and cyclical way and show that the
number of zero-energy flat bands of these models can be found from a
generalized version of Lieb’s theorem, in what regards its application to
noninteracting tight-binding models, involving the imbalance between each
sublattice and the sublattice of lowest dimension. Furthermore, these models
are also shown to obey a generalized chiral symmetry, of the type found in the
context of certain clock or parafermionic systems. The main results are
illustrated with a simple toy model, and possible realizations in different
platforms of the models introduced here are discussed.

###### pacs:

74.25.Dw,74.25.Bt

##  I Introduction

Lieb’s theorem [1], initially formulated to demonstrate that the ground state
magnetization at half filling of repulsive Hubbard bipartite lattices is
directly proportional to the sublattice (SL) imbalance [2, 3, 4, 5], is now
understood in a more broad sense. Concretely, it states that the number of
zero-energy flat bands (FBs) of a crystalline and bipartite tight-binding (TB)
model, of which Lieb-type lattices are a prime example [6, 7, 8, 9, 10], is
given by the SL imbalance [11, 12, 13]. In real-space, the global sublattice
imbalance of any bipartite system (including non-crystalline ones) indicates
the lower bound on the number of zero-energy states present there [14].

Within the context of non-Hermitian systems, several studies have already
addressed the formation, persistence or destruction of FBs in these models
through different approaches [15, 16], most commonly with the introduction of
parity-time ($\mathcal{PT}$) symmetric perturbations [17, 18, 19], including
in one-dimensional (1D) [20, 21, 22] and two-dimensional (2D) [23] Lieb-type
lattices. The compact localized states associated with these FBs have already
been experimentally detected in a $\mathcal{PT}$-symmetric photonic trimer
chain [24] with balanced gains and losses. Tuning the parameters of these
systems to fall on exceptional points has been shown to drive the formation of
FBs [25, 26]. Here, we introduce a certain class of non-Hermitian models with
$n\geq 2$ SLs, which we call $n$-partite systems, and show that the number of
zero-energy FBs in these models is given by a generalized version of Lieb’s
theorem, as it is understood in the specific context of the noninteracting TB
models studied here. This constitutes a novel mechanism for the formation of
FBs in non-Hermitian systems, which is not dependent on any specific
symmetries, like $\mathcal{PT}$ symmetry, even though the models studied here
have a built-in generalized chiral symmetry by default.

Bipartite models are also characterized by the presence of chiral symmetry,
which pairs eigenvalues with symmetric energies. Some extensions of the usual
chiral symmetry have already been considered, whether for $q$-deformed
Hamiltonians [27, 28, 29], in 1D models with finite energy edge states
topologically protected by a chiral-like symmetry [30], in models with
different adiabatically connected chiral symmetry representations at different
liming cases [31], or in 1D superlattices [32], not necessarily bipartite,
with point-chiral symmetry [33] whose energy spectrum is symmetric about a
finite momentum value. Models belonging to the class introduced here, on the
other hand, are shown to obey the same generalized chiral symmetry as the one
found in the generalized quantum Ising chains known as Baxter’s clock models
[34, 35]. A simple 1D toy model is introduced for the purpose of illustrating
both the generalized Lieb’s theorem and the generalized chiral symmetry.

The rest of the paper is organized as follows. In Sec. II, we define
$n$-partite models and introduce their general Hamiltonian. Then, we discuss
the symmetries of these models, with a particular emphasis on the generalized
chiral symmetry. We end this section by formally deriving a generalized
version of Lieb’s theorem, which counts the total number of zero-energy FBs in
these non-interacting $n$-partite TB lattices. In Sec. III, we introduce a toy
model that exemplifies the main results found in the previous section. We also
analyze the energy spectrum of this toy model for open boundaries, showing
that the FB states survive the emergence of the skin effect, while the
dispersive states do not. Finally, we present our conclusions in Sec. IV.

##  II $n$-partite models

Figure 1: Top: non-Hermitian bipartite model with staggered hoppings. Bottom:
tripartite trimer chain. The arrows indicate the direction of the hopping
parameters and $h\in\mathbb{R}$, corresponding to an imaginary gauge field.

We define an $n$-partite model as a system composed of $n$ SLs, where each SL,
in turn, is defined as a group of sites that can only connect between
themselves through integer multiples of $n$-hopping processes. This implies
that only bipartite (2-partite) lattices can be Hermitian (the $h=0$ case in
Fig. 1 top), while for $n>2$ the model is necessarily non-Hermitian and built
with unidirectional couplings (see Fig. 1 bottom), which further imposes the
absence of $\mathcal{PT}$ symmetry (since an inversion operation also inverts
the direction of the couplings). Let us consider the general form of an
$n$-partite Hamiltonian with $n$ SLs of arbitrary sizes, each coupled to its
neighbor in a directed fashion,

$H(\mathbf{k})=\begin{pmatrix}&h_{1}&&&\\\ &&h_{2}&&\\\ &&&\ddots&\\\
&&&&h_{n-1}\\\ h_{n}&&&&\end{pmatrix},$ (1)

where the entries not shown are zeros, the momentum vector reads as
$\mathbf{k}=(k_{1},k_{2},\dots,k_{D})$, with $D$ the dimensionality of the
system, and $h_{j}=h_{j}(\mathbf{k})$, with $j=1,2,\dots,n$, is a rectangular
matrix of size $d_{j}\times d_{j+1}$, with $j=n+1\to j=1$ from the periodic
boundary conditions. This Hamiltonian describes a periodic model composed of
unidirectional hopping terms from sites in SLj to sites in SLj-1. Note that,
according to our definition, the Hamiltonian of all possible $n$-partite
models can be written either in the form of (1) or as its conjugate transpose
version, $H(\mathbf{k})\to H^{\dagger}(\mathbf{k})$, which corresponds to a
global inversion of all coupling directions. Upon raising the Hamiltonian in
(1) to the $n^{\text{th}}$ power, one arrives at a diagonal matrix of the form

$\displaystyle H^{n}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\text{diag}(H_{1},H_{2},\dots,H_{n}),$ (2) $\displaystyle H_{j}$
$\displaystyle=$ $\displaystyle h_{j}h_{j+1}\dots h_{n-1+j}.$ (3)

Each diagonal block $H_{j}$ is a $d_{j}\times d_{j}$ square matrix, and the
set $\\{H_{j}\\}$ represents all cyclic permutations of the ordered product of
all the original $h_{j}$ matrices. We note that there is a recent study in
driven systems [36] where the authors, by considering Floquet operators with
formal properties similar to those of (1)-(3), were able to construct high-
root [37, 12, 13, 38, 39] Floquet topological insulators of any order.

We assume for convenience that the SLs are ordered in a way that obeys
$d_{1}\leq d_{j\neq 1}$, such that $H_{1}$ in (2) is the smallest block (or in
the set of smallest blocks) of dimension $d_{1}\times d_{1}$. It can then be
shown that the energy spectrum of $H_{1}$ is shared by all other $H_{j\neq
1}$, such that it is $n$-fold degenerate in $H^{n}(\mathbf{k})$. The
Schrödinger equation for the $H_{1}$ block is written as

$H_{1}\ket{u^{1}_{s}(\mathbf{k})}=E_{1,s}(\mathbf{k})\ket{u^{1}_{s}(\mathbf{k})},\
\ s=1,2,\dots,d_{1},$ (4)

where $\ket{u^{1}_{s}(\mathbf{k})}$ is the eigenstate with momentum
$\mathbf{k}$ of band $s$, and only has weight on the $d_{1}$ components of the
first sublattice. Applying $h_{n}$ on both sides of (4) and using the identity
$h_{j}H_{j+1}=H_{j}h_{j}$, derived from (3), leads to

$H_{n}\big{(}h_{n}\ket{u^{1}_{s}(\mathbf{k})}\big{)}=E_{1,s}(\mathbf{k})\big{(}h_{n}\ket{u^{1}_{s}(\mathbf{k})}\big{)},$
(5)

which, after defining the $d_{n}$-dimensional (non-normalized) eigenvector
$\ket{u^{n}_{s}(\mathbf{k})}:=h_{n}\ket{u^{1}_{s}(\mathbf{k})}$, becomes

$H_{n}\ket{u^{n}_{s}(\mathbf{k})}=E_{1,s}(\mathbf{k})\ket{u^{n}_{s}(\mathbf{k})},\
\ s=1,2,\dots,d_{1}.$ (6)

Since $d_{n}\geq d_{1}$, (6) only accounts for the $d_{1}$ energy bands that
are proven to be degenerate with the equivalent ones coming from the
diagonalization of $H_{1}$. There are, however, extra $d_{n}-d_{1}$ bands
coming from $H_{n}$ which do not belong to the shared spectrum. From a
sequential application of $h_{n-1,},h_{n-2},\dots,h_{2}$ to both sides of (6),
one can generalize this proof to show that

$H_{j}\ket{u^{j}_{s}(\mathbf{k})}=E_{1,s}(\mathbf{k})\ket{u^{j}_{s}(\mathbf{k})},\
\ s=1,2,\dots,d_{1},$ (7)

where the $d_{j}$-dimensional (non-normalized) eigenvectors are defined as
$\ket{u^{j}_{s}(\mathbf{k})}:=h_{j}\ket{u^{j+1}_{s}(\mathbf{k})}$, which only
have weight on SLj.

###  II.1 Generalized chiral symmetry

In the absence of gauge fields, the spinless fermionic Hamiltonian in (1)
obeys both complex conjugation symmetry (corresponding to the time-reversal
symmetry, $\mathscr{T}$, for Hermitian systems) and a generalized version of
particle-hole ($\mathscr{P}_{n}$) symmetry, defined respectively as

$\displaystyle\mathscr{T}$ $\displaystyle:$ $\displaystyle\ \ \ \
TH(\mathbf{k})T^{-1}_{n}=H(\mathbf{-k}),\ \ \ \ \ \ \ \ \ \ T=K,$ (8)
$\displaystyle\mathscr{P}_{n}$ $\displaystyle:$ $\displaystyle\ \ \
P_{n}H(\mathbf{k})P^{-1}_{n}=\omega_{n}^{-1}H(\mathbf{-k}),\ \ \
P_{n}=\Gamma_{n}K,$ (9) $\displaystyle\Gamma_{n}$ $\displaystyle=$
$\displaystyle\text{diag}(\mathbb{1}_{d_{1}},\omega_{n}\mathbb{1}_{d_{2}},\dots,\omega_{n}^{n-2}\mathbb{1}_{d_{n-1}},\omega_{n}^{n-1}\mathbb{1}_{d_{n}}),$
(10)

where $K=K^{-1}$ represents the complex conjugation operation, with
$KK^{-1}=1$, $\mathbb{1}_{d_{j}}$ is the identity matrix of dimension $d_{j}$,
$\Gamma_{n}\Gamma_{n}^{-1}=\Gamma_{n}^{n}=\mathbb{1}_{d_{H}}$, with
$d_{H}=\sum_{j=1}^{n}d_{j}$ the dimension of $H(\mathbf{k})$,
$\omega_{n}=e^{i\frac{2\pi}{n}}$ and $\omega_{n}^{n}=1$. From the combination
of these two symmetries one can also define a generalized chiral
($\mathscr{C}_{n}$) symmetry, whose generalized chiral operator is written as
$C_{n}=P_{n}T=\Gamma_{n}$,

$\mathscr{C}_{n}:\ \ \
\Gamma_{n}H(\mathbf{k})\Gamma^{-1}_{n}=\omega_{n}^{-1}H(\mathbf{k}),$ (11)

which constitutes another branch on the already long list of non-Hermitian
symmetries, as systematically studied in [40]. For a bipartite system, $n=2$,
(11) reduces to the usual chiral symmetry [41, 42, 43] defined as
$\Gamma_{2}H(\mathbf{k})\Gamma_{2}^{-1}=-H(\mathbf{k})$. Therefore
$\Gamma_{n}$ is the operator defining the chiral symmetry of an $n$-partite
system, defined by the presence of $n$ sublattices. It should be stressed
that, since $\Gamma_{n}$ is unitary, the system retains its $\mathscr{C}_{n}$
symmetry even in the presence of gauge fields, that is, even when both
$\mathscr{T}$ and $\mathscr{P}_{n}$ symmetries are broken (as is the case for
the usual chiral symmetry in bipartite lattices crossed by finite magnetic
fluxes [44]).

The presence of $\mathscr{C}_{n}$ symmetry imposes a constraint on the complex
energy spectrum of $H(\mathbf{k})$. Let us consider a finite energy eigenstate
of the system,

$H(\mathbf{k})\ket{\psi_{0}(\mathbf{k})}=E\ket{\psi_{0}(\mathbf{k})},$ (12)

then, by iteratively applying $\Gamma_{n}$ on both sides and using (11) at
each iteration, one arrives at

$H(\mathbf{k})\ket{\psi_{l}(\mathbf{k})}=\omega_{n}^{l}E\ket{\psi_{l}(\mathbf{k})},$
(13)

where $\ket{\psi_{l}(\mathbf{k})}:=\Gamma_{n}^{l}\ket{\psi_{0}(\mathbf{k})}$,
$l=1,2,\dots,n-1$, and
$\innerproduct{\psi_{i}(\mathbf{k})}{\psi_{j}(\mathbf{k})}=\delta_{ij}$. This
tells us that if $E$ is a finite eigenvalue of an eigenstate of the system,
then all its rotated versions, given by the $n-1$ sequential
$\phi_{n}=\frac{2\pi}{n}$ rotations in the energetic Argand plane, are also
eigenvalues of orthogonal eigenstates, i.e., the finite eigenvalues come in
sequences of the form
$\\{E,\omega_{n}E,\omega_{n}^{2}E,\dots,\omega_{n}^{n-1}E\\}$, and the values
in each sequence sum to zero. This can also be understood by directly
developing (11) as

$\Gamma_{n}^{l}H(\mathbf{k})\Gamma^{-l}_{n}=\omega_{n}^{-l}H(\mathbf{k}),\ \ \
\ l=1,2,\dots,n-1,$ (14)

which, in particular, implies that $n$-partite lattices also obey the
generalized chiral symmetry of all the divisors of $n$. For example, for a
$6$-partite lattice one not only has $\mathscr{C}_{6}$ symmetry, but also
$\mathscr{C}_{3}$ and $\mathscr{C}_{2}$ symmetries, whose operators are given
by $\Gamma_{3}=\Gamma_{6}^{2}$ and $\Gamma_{2}=\Gamma_{6}^{3}$, respectively.
As a corollary, all even-partite lattices possess the usual chiral symmetry
$\mathscr{C}_{2}$. In analogy with the colored states that can be present in
certain $XXZ$ Heisenberg models [45, 46], we can similarly identify the action
of $\Gamma_{n}$ on $H(\mathbf{k})$ as an ordered transformation between
different chiral colors of the same Hamiltonian, defined from (11) and for
$n=3$, e.g., as

$\displaystyle H_{\leavevmode\hbox to2.21pt{\vbox
to2.21pt{\pgfpicture\makeatletter\hbox{\hskip 1.10417pt\lower-1.10417pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.90417pt}{0.0pt}\pgfsys@curveto{0.90417pt}{0.49936pt}{0.49936pt}{0.90417pt}{0.0pt}{0.90417pt}\pgfsys@curveto{-0.49936pt}{0.90417pt}{-0.90417pt}{0.49936pt}{-0.90417pt}{0.0pt}\pgfsys@curveto{-0.90417pt}{-0.49936pt}{-0.49936pt}{-0.90417pt}{0.0pt}{-0.90417pt}\pgfsys@curveto{0.49936pt}{-0.90417pt}{0.90417pt}{-0.49936pt}{0.90417pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ $\displaystyle=$
$\displaystyle H(\mathbf{k}),$ (15) $\displaystyle H_{\leavevmode\hbox
to2.21pt{\vbox to2.21pt{\pgfpicture\makeatletter\hbox{\hskip
1.10417pt\lower-1.10417pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.90417pt}{0.0pt}\pgfsys@curveto{0.90417pt}{0.49936pt}{0.49936pt}{0.90417pt}{0.0pt}{0.90417pt}\pgfsys@curveto{-0.49936pt}{0.90417pt}{-0.90417pt}{0.49936pt}{-0.90417pt}{0.0pt}\pgfsys@curveto{-0.90417pt}{-0.49936pt}{-0.49936pt}{-0.90417pt}{0.0pt}{-0.90417pt}\pgfsys@curveto{0.49936pt}{-0.90417pt}{0.90417pt}{-0.49936pt}{0.90417pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ $\displaystyle=$
$\displaystyle\Gamma_{3}H_{\leavevmode\hbox to2.21pt{\vbox
to2.21pt{\pgfpicture\makeatletter\hbox{\hskip 1.10417pt\lower-1.10417pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.90417pt}{0.0pt}\pgfsys@curveto{0.90417pt}{0.49936pt}{0.49936pt}{0.90417pt}{0.0pt}{0.90417pt}\pgfsys@curveto{-0.49936pt}{0.90417pt}{-0.90417pt}{0.49936pt}{-0.90417pt}{0.0pt}\pgfsys@curveto{-0.90417pt}{-0.49936pt}{-0.49936pt}{-0.90417pt}{0.0pt}{-0.90417pt}\pgfsys@curveto{0.49936pt}{-0.90417pt}{0.90417pt}{-0.49936pt}{0.90417pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\Gamma_{3}^{-1}=\omega_{3}^{-1}H_{\leavevmode\hbox
to2.21pt{\vbox to2.21pt{\pgfpicture\makeatletter\hbox{\hskip
1.10417pt\lower-1.10417pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.90417pt}{0.0pt}\pgfsys@curveto{0.90417pt}{0.49936pt}{0.49936pt}{0.90417pt}{0.0pt}{0.90417pt}\pgfsys@curveto{-0.49936pt}{0.90417pt}{-0.90417pt}{0.49936pt}{-0.90417pt}{0.0pt}\pgfsys@curveto{-0.90417pt}{-0.49936pt}{-0.49936pt}{-0.90417pt}{0.0pt}{-0.90417pt}\pgfsys@curveto{0.49936pt}{-0.90417pt}{0.90417pt}{-0.49936pt}{0.90417pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}},$ (16)
$\displaystyle H_{\leavevmode\hbox to2.21pt{\vbox
to2.21pt{\pgfpicture\makeatletter\hbox{\hskip 1.10417pt\lower-1.10417pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@rgb@stroke{0}{1}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{1}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,1,0}\definecolor[named]{pgffillcolor}{rgb}{0,1,0}\pgfsys@color@rgb@fill{0}{1}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.90417pt}{0.0pt}\pgfsys@curveto{0.90417pt}{0.49936pt}{0.49936pt}{0.90417pt}{0.0pt}{0.90417pt}\pgfsys@curveto{-0.49936pt}{0.90417pt}{-0.90417pt}{0.49936pt}{-0.90417pt}{0.0pt}\pgfsys@curveto{-0.90417pt}{-0.49936pt}{-0.49936pt}{-0.90417pt}{0.0pt}{-0.90417pt}\pgfsys@curveto{0.49936pt}{-0.90417pt}{0.90417pt}{-0.49936pt}{0.90417pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ $\displaystyle=$
$\displaystyle\Gamma_{3}H_{\leavevmode\hbox to2.21pt{\vbox
to2.21pt{\pgfpicture\makeatletter\hbox{\hskip 1.10417pt\lower-1.10417pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.90417pt}{0.0pt}\pgfsys@curveto{0.90417pt}{0.49936pt}{0.49936pt}{0.90417pt}{0.0pt}{0.90417pt}\pgfsys@curveto{-0.49936pt}{0.90417pt}{-0.90417pt}{0.49936pt}{-0.90417pt}{0.0pt}\pgfsys@curveto{-0.90417pt}{-0.49936pt}{-0.49936pt}{-0.90417pt}{0.0pt}{-0.90417pt}\pgfsys@curveto{0.49936pt}{-0.90417pt}{0.90417pt}{-0.49936pt}{0.90417pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\Gamma_{3}^{-1}=\omega_{3}^{-2}H_{\leavevmode\hbox
to2.21pt{\vbox to2.21pt{\pgfpicture\makeatletter\hbox{\hskip
1.10417pt\lower-1.10417pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.90417pt}{0.0pt}\pgfsys@curveto{0.90417pt}{0.49936pt}{0.49936pt}{0.90417pt}{0.0pt}{0.90417pt}\pgfsys@curveto{-0.49936pt}{0.90417pt}{-0.90417pt}{0.49936pt}{-0.90417pt}{0.0pt}\pgfsys@curveto{-0.90417pt}{-0.49936pt}{-0.49936pt}{-0.90417pt}{0.0pt}{-0.90417pt}\pgfsys@curveto{0.49936pt}{-0.90417pt}{0.90417pt}{-0.49936pt}{0.90417pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}},$ (17)

with $\leavevmode\hbox to2.98pt{\vbox
to2.98pt{\pgfpicture\makeatletter\hbox{\hskip 1.49165pt\lower-1.49165pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.29166pt}{0.0pt}\pgfsys@curveto{1.29166pt}{0.71336pt}{0.71336pt}{1.29166pt}{0.0pt}{1.29166pt}\pgfsys@curveto{-0.71336pt}{1.29166pt}{-1.29166pt}{0.71336pt}{-1.29166pt}{0.0pt}\pgfsys@curveto{-1.29166pt}{-0.71336pt}{-0.71336pt}{-1.29166pt}{0.0pt}{-1.29166pt}\pgfsys@curveto{0.71336pt}{-1.29166pt}{1.29166pt}{-0.71336pt}{1.29166pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\overset{\Gamma_{3}}{\to}\leavevmode\hbox
to2.98pt{\vbox to2.98pt{\pgfpicture\makeatletter\hbox{\hskip
1.49165pt\lower-1.49165pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.29166pt}{0.0pt}\pgfsys@curveto{1.29166pt}{0.71336pt}{0.71336pt}{1.29166pt}{0.0pt}{1.29166pt}\pgfsys@curveto{-0.71336pt}{1.29166pt}{-1.29166pt}{0.71336pt}{-1.29166pt}{0.0pt}\pgfsys@curveto{-1.29166pt}{-0.71336pt}{-0.71336pt}{-1.29166pt}{0.0pt}{-1.29166pt}\pgfsys@curveto{0.71336pt}{-1.29166pt}{1.29166pt}{-0.71336pt}{1.29166pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\overset{\Gamma_{3}}{\to}\leavevmode\hbox
to2.98pt{\vbox to2.98pt{\pgfpicture\makeatletter\hbox{\hskip
1.49165pt\lower-1.49165pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@rgb@stroke{0}{1}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{1}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,1,0}\definecolor[named]{pgffillcolor}{rgb}{0,1,0}\pgfsys@color@rgb@fill{0}{1}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.29166pt}{0.0pt}\pgfsys@curveto{1.29166pt}{0.71336pt}{0.71336pt}{1.29166pt}{0.0pt}{1.29166pt}\pgfsys@curveto{-0.71336pt}{1.29166pt}{-1.29166pt}{0.71336pt}{-1.29166pt}{0.0pt}\pgfsys@curveto{-1.29166pt}{-0.71336pt}{-0.71336pt}{-1.29166pt}{0.0pt}{-1.29166pt}\pgfsys@curveto{0.71336pt}{-1.29166pt}{1.29166pt}{-0.71336pt}{1.29166pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\overset{\Gamma_{3}}{\to}\leavevmode\hbox
to2.98pt{\vbox to2.98pt{\pgfpicture\makeatletter\hbox{\hskip
1.49165pt\lower-1.49165pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.29166pt}{0.0pt}\pgfsys@curveto{1.29166pt}{0.71336pt}{0.71336pt}{1.29166pt}{0.0pt}{1.29166pt}\pgfsys@curveto{-0.71336pt}{1.29166pt}{-1.29166pt}{0.71336pt}{-1.29166pt}{0.0pt}\pgfsys@curveto{-1.29166pt}{-0.71336pt}{-0.71336pt}{-1.29166pt}{0.0pt}{-1.29166pt}\pgfsys@curveto{0.71336pt}{-1.29166pt}{1.29166pt}{-0.71336pt}{1.29166pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$, from where it can
be seen that

$H_{\leavevmode\hbox to2.21pt{\vbox
to2.21pt{\pgfpicture\makeatletter\hbox{\hskip 1.10417pt\lower-1.10417pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.90417pt}{0.0pt}\pgfsys@curveto{0.90417pt}{0.49936pt}{0.49936pt}{0.90417pt}{0.0pt}{0.90417pt}\pgfsys@curveto{-0.49936pt}{0.90417pt}{-0.90417pt}{0.49936pt}{-0.90417pt}{0.0pt}\pgfsys@curveto{-0.90417pt}{-0.49936pt}{-0.49936pt}{-0.90417pt}{0.0pt}{-0.90417pt}\pgfsys@curveto{0.49936pt}{-0.90417pt}{0.90417pt}{-0.49936pt}{0.90417pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+H_{\leavevmode\hbox
to2.21pt{\vbox to2.21pt{\pgfpicture\makeatletter\hbox{\hskip
1.10417pt\lower-1.10417pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.90417pt}{0.0pt}\pgfsys@curveto{0.90417pt}{0.49936pt}{0.49936pt}{0.90417pt}{0.0pt}{0.90417pt}\pgfsys@curveto{-0.49936pt}{0.90417pt}{-0.90417pt}{0.49936pt}{-0.90417pt}{0.0pt}\pgfsys@curveto{-0.90417pt}{-0.49936pt}{-0.49936pt}{-0.90417pt}{0.0pt}{-0.90417pt}\pgfsys@curveto{0.49936pt}{-0.90417pt}{0.90417pt}{-0.49936pt}{0.90417pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+H_{\leavevmode\hbox
to2.21pt{\vbox to2.21pt{\pgfpicture\makeatletter\hbox{\hskip
1.10417pt\lower-1.10417pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@rgb@stroke{0}{1}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{1}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,1,0}\definecolor[named]{pgffillcolor}{rgb}{0,1,0}\pgfsys@color@rgb@fill{0}{1}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.90417pt}{0.0pt}\pgfsys@curveto{0.90417pt}{0.49936pt}{0.49936pt}{0.90417pt}{0.0pt}{0.90417pt}\pgfsys@curveto{-0.49936pt}{0.90417pt}{-0.90417pt}{0.49936pt}{-0.90417pt}{0.0pt}\pgfsys@curveto{-0.90417pt}{-0.49936pt}{-0.49936pt}{-0.90417pt}{0.0pt}{-0.90417pt}\pgfsys@curveto{0.49936pt}{-0.90417pt}{0.90417pt}{-0.49936pt}{0.90417pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=0,$ (18)

that is, and in more general terms, the $n$ chiral colors of a given
$\mathscr{C}_{n}$-symmetric Hamiltonian sum to zero (or, alternatively, the
$n$ colors sum to white).

The operator of the generalized chiral symmetry was first introduced in the
context of the tripartite Hermitian breathing kagome model [47], and shown to
pin the higher-order corner modes at zero energy. However, it has been
recently demonstrated that the $\mathscr{C}_{3}$ symmetry of the model fails
to protect the corner modes against certain perturbations that preserve it
[48]. On the basis of the energetic constraints imposed by $\mathscr{C}_{n}$
symmetry on the specific non-Hermitian Hamiltonians of the form of (1), and
encapsulated in (13), we argue that the Hermitian models obeying
$\mathscr{C}_{3}$ [47, 49, 50, 51, 33] or $\mathscr{C}_{4}$ [52, 53] symmetry
studied so far fail to reveal the relevant consequences of the generalized
chiral symmetry detailed here (a more expanded discussion can be found in
Appendix A). That is because $\mathscr{C}_{n}$-symmetric Hermitian models
require $n$ applications of the generalized chiral symmetry in order to
recover the original Hamiltonian, whereas in our case the Hamiltonian is
recovered, up to a global phase factor [see (11)], after each application of
the symmetry transformation, i.e., acting with $\Gamma_{n}$ on the Hamiltonian
changes its chiral color. The class of models introduced here, namely non-
Hermitian $n$-partite models with unidirectional hopping terms between
adjacent SLs defined in a cyclic fashion, should be regarded as the first
example of a fully $\mathscr{C}_{n}$-symmetric class of TB models.

It should be noted, however, that the same kind of generalized chiral symmetry
has already been addressed in a different context, namely, that of generalized
quantum Ising chains known as Baxter’s clock model [34, 54], where the “spin”
or clock internal degree of freedom at each site can take any value
$\omega_{n}^{j}$, with $j=0,1,\dots,n-1$ (this model can be reframed in a
parafermionic language, as shown, e.g., in [35, 55]). A brief introduction to
Baxter’s clock model is provided in Appendix B, along with the analogies that
can be drawn between this model and the one introduced in this paper.

It is convenient to introduce the phase commutator between matrices $A$ and
$B$, which we define as

$[A,B]_{\theta}:=AB-e^{-i\theta}BA,\ \ \ \ \ \theta\in[0,2\pi),$ (19)

reducing to the commutation relation for $[A,B]_{0}=[A,B]$, and to the anti-
commutation relation for $\theta=\pi$, $[A,B]_{\pi}=\\{A,B\\}$. The compact
expression for $\mathscr{C}_{n}$ in (10) can be restated, through (19), as a
phase commutator of the form

$[\Gamma_{n},H(\mathbf{k})]_{\phi_{n}}=0,$ (20)

where, in particular, one recovers the known anticommutation relation for a
bipartite model as $\\{\Gamma_{2},H(\mathbf{k})\\}=0$, while also trivially
recovering the commutation relation $[\Gamma_{1},H(\mathbf{k}]=0$, since
$\Gamma_{1}=\mathbb{1}$. In the context of Baxter’s clock model analyzed in
Appendix B, the phase commutator in (20) can be viewed as the analog of the
“$\omega$ commutator” [34, 35] of the generalized Clifford algebra [36]
involving the local operators with which the Hamiltonian of this model is
constructed.

To conclude the discussion of the symmetries of $n$-partite models, let us
consider a Hermitian Hamiltonian that can be written as
$H^{\prime}(\mathbf{k})=H(\mathbf{k})+H^{\dagger}(\mathbf{k})$, where
$H(\mathbf{k})$ is given in (1). We further assume that
$H^{\prime}(\mathbf{k})$ has inversion or parity symmetry, written as
$RH^{\prime}(\mathbf{k})R^{-1}=RH(\mathbf{k})R^{-1}+RH^{\dagger}(\mathbf{k})R^{-1}=H^{\prime}(-\mathbf{k})$,
where $R$ is the inversion operator. It is then straightforward to see that
the following identity holds,

$RH(\mathbf{k})R^{-1}=H^{\dagger}(-\mathbf{k}),$ (21)

which can be seen as a modified inversion symmetry for the $n$-partite model.
More concretely, if $H^{\prime}(\mathbf{k})$, constructed from
$H(\mathbf{k})$, has inversion symmetry, then the latter can be said to enjoy
inversion symmetry also, up to a global inversion of the hopping directions.

Figure 2: Unit cell of the toy model for $n=3$. The hopping terms are
unidirectional and follow the direction of the arrows. Open site 1 at the
right belongs to the adjacent unit cell. The flow between sublattices is
depicted at the top left.

###  II.2 Generalized Lieb’s theorem

The combination of the results above leads to another important result.

(i) On the one hand, from the discussion leading to (7), we found that the
finite energy spectrum of the smallest diagonal block of $H^{n}(\mathbf{k})$,
which we set as $H_{1}$, is $n$-fold degenerate. On the other hand, this
translates in the original model $H(\mathbf{k})$, through (13), as an $n$-fold
degeneracy of all absolute finite energy values, which also leads to an
$n$-fold degeneracy of $E^{n}$. As a result, whenever $d_{1}\leq d_{j>1}$, the
extra bands of $H_{j}$, in relation to $H_{1}$, must be zero-energy FBs,
otherwise their finite energies would have to be $n$-fold degenerate, that is,
shared also by $H_{1}$, which is not the case.

(ii) Furthermore, if $H_{1}$ is itself bipartite, i.e., if there are
$\\#_{\text{FB}}^{H_{1}}$ zero-energy FBs in the spectrum of $H_{1}$ coming
from sublattice imbalance within SL1, then the same number of extra FBs
appears in the other $n-1$ diagonal blocks $H_{j>1}$, meaning that the
degenerate block spectra given by (7) actually remains valid for zero-energy
FBs, that is, when $E_{1,s}(\mathbf{k})=0$. The same reasoning of (i) can be
applied here to prove the negative is impossible. Let us suppose that we
construct $\\#_{\text{FB}}^{H_{1}}$ dispersive bands in each of the $H_{j>1}$
blocks, with global $(n-1)$-fold degeneracy for band
$E_{\alpha}(\mathbf{k})>0$, with $\alpha=1,2,\dots,\\#_{\text{FB}}^{H_{1}}$
the band index. Then, due to the $\mathscr{C}_{n}$-symmetry of the original
Hamiltonian, the finite energies appear in groups of $n$ elements of the form
$\\{\sqrt[n]{E_{\alpha}(\mathbf{k})},\omega_{n}\sqrt[n]{E_{\alpha}(\mathbf{k})},\dots,\omega_{n}^{n-1}\sqrt[n]{E_{\alpha}(\mathbf{k})}\\}$.
However, $E_{\alpha}(\mathbf{k})$ was assumed to be $(n-1)$-fold degenerate,
since it is absent from $H_{1}$, and therefore cannot originate the $n$
elements for each $\mathbf{k}$ mentioned above for the original Hamitonian
and, as a consequence, the extra $(n-1)\\#_{\text{FB}}^{H_{1}}$ bands of the
spectrum are also zero-energy FBs, that is, degenerate with the
$\\#_{\text{FB}}^{H_{1}}$ FBs present in $H_{1}$ 111When
$\\#_{\text{FB}}^{H_{1}}=2$, e.g., one could believe it would be possible to
construct, from the $2n-2$ remaining bands from the other $H_{j>1}$ blocks, an
$n$-fold degenerate finite energy band plus an $(n-2)$-fold degenerate zero-
energy FB. However, the eigenstates of a finite energy band cannot vanish at
any SL, which includes SL1 (it can be easily checked that, if there are nodes
at all sites of one SL, then, through the TB equations, these nodes propagate
sequentially to all sublattices). As such, if the eigenstates of this finite
energy band have finite weight on some sites of SL1, then it would have to be
present in $H_{1}$ also, which is assumed not to be the case. Therefore all
$2n-2$ extra bands are also zero-energy FBs..

Since the number of zero-energy FBs is the same for $H^{n}(\mathbf{k})$ and
$H(\mathbf{k})$, the results of this subsection can be summarized in the
following formula that generalizes Lieb’s theorem for a
$\mathscr{C}_{n}$-symmetric $n$-partite system:

$\\#_{\text{FB}}=\sum\limits_{j=2}^{n}(d_{j}-d_{1})+n\\#_{\text{FB}}^{H_{1}},$
(22)

that is, the total number of zero-energy FBs is given by the sum of imbalances
between each SLj>1 and the smallest sublattice SL1 [the first term on the
right, coming from point (i) above], plus $n$ times the number of zero-energy
FBs already present in the smallest $H_{1}$ block [the second term on the
right, coming from point (ii) above]. It should be noted that the second term
should be included already for non-Hermitian systems with $n=2$, as we
illustrate in Appendix C with an example, showing that Lieb’s theorem can be
generalized also for the (bipartite) lattices for which it was formulated. As
a corolary, we can also infer that if $H_{1}$ has a real energy spectrum, all
$H_{j\neq 1}$ have real spectra, given that their extra bands must be zero-
energy FBs, such that they are pseudo-Hermitian Hamiltonians [57] obeying
$H_{j>1}^{\dagger}=\eta H_{j>1}\eta^{-1}$, with $\eta$ a positive definite
unitary matrix [58] that reduces to the identity for Hermitian Hamiltonians.

One should be reminded, at this point, that $H(\mathbf{k})$ in (1) is non-
Hermitian and therefore can be defective, that is, the number of linearly
independent eigenstates (LIEs) of $H(\mathbf{k})$ can be lower than its
dimensionality $d_{H}$, if $H(\mathbf{k})$ falls into exceptional points or
lines of the parameter space [59]. Regarding the eigenstates, (22) should be
interpreted as giving the maximum possible number of LIEs within the set of
zero-energy FBs of $H(\mathbf{k})$. However, defective models can have less
LIEs in this set than $\\#_{\text{FB}}$, down to a minimum given by

$\\#_{\text{LIEs}}^{\text{min}}=\sum\limits_{j=2}^{n}\text{Max}(d_{j}-d_{j-1},0),$
(23)

which we derive in Appendix D, where an explicit example of a defective system
is also provided.

##  III Toy model

Figure 3: Complex energy spectrum as a function of the momentum obtained from
diagonalizing the Hamiltonian defined in (24)-(26) for (a) $n=3$, (b) $n=4$,
and (c) $n=5$. The different symmetric partners of state $\ket{\psi}$ are
indicated in (a). (d) Normalized energy spectrum as a function of the momentum
of the model in (a)-(c) raised to the $n^{\text{th}}$ power, which is purely
real. In all plots, ($\times j$) indicates the $j$-fold degeneracy of the
respective band, $N=\sum_{i=1}^{n}i$ is the total number of bands, and only
the zero-energy FB is degenerate in (a)-(c).

In order to illustrate the results above, we introduce the simple 1D
$n$-partite model ($\mathbf{k}\to k$), with a bulk Hamiltonian of the form of
(1), whose entries are explicitly given by

$\displaystyle h_{1}$ $\displaystyle=$ $\displaystyle t(1+e^{-ik})J_{1\times
2},$ (24) $\displaystyle h_{j}$ $\displaystyle=$ $\displaystyle tJ_{j\times
j+1},\ \ \ \ \ \ \ \ \ \ \ \ j=2,3,\dots,n-1,$ (25) $\displaystyle h_{n}$
$\displaystyle=$ $\displaystyle t(1+e^{ik})J_{n\times 1},$ (26)

where $J_{i\times j}$ is a matrix of ones of size $i\times j$, the lattice
spacing was set to $a\equiv 1$ here and everywhere below, and $t$ is the
magnitude of the unidirectional hopping terms, set as the energy unit
henceforth. The unit cell of this model for $n=3$ is depicted in Fig. 2. When
raised to the $n^{\text{th}}$ power, this Hamiltonian has the form of (2),
with the diagonal blocks reading as

$H_{j}=2\frac{n!}{j}t^{n}(1+\cos k)J_{j\times j},$ (27)

with $j=1,2,\dots,n$, such that $H_{j}$ is a matrix of size $j\times j$, that
is, $d_{j}=j$ is the number of sites in SLj. In particular, the smallest block
is already diagonal and has the form $H_{1}=2n!t^{n}(1+\cos k)$, which models
a simple uniform and Hermitian linear chain with hopping strength $n!t^{n}$
and an overall $2n!t^{n}$ energy shift. The energy band characterizing the
spectrum of $H_{1}$ is $n$-fold degenerate in $H^{n}(k)$ through (7), since it
is common to all $H_{j}$ blocks.

The energy spectrum of the model defined through (24)-(26) is shown for
different $n$ in Figs. 3(a)-3(c). The presence of the respective
$\mathscr{C}_{n}$ symmetry is apparent in all three cases, as the energy
spectra are manifestly invariant under $\phi_{n}$ rotations about the $k$
axis. We illustrate the $\mathscr{T}$, $\mathscr{P}_{3}$, and
$\mathscr{C}_{3}$ symmetric partners of an arbitrary state $\ket{\psi}$ in
Fig. 3(a). At the same time, the degeneracy of the zero-energy FBs agrees with
the generalized Lieb’s theorem expressed in (22). In Fig. 3(d), we show the
normalized energy spectrum of $H^{n}(k)$, whose diagonal blocks are given by
(27). Notice that this spectrum is purely real since the smallest block
$H_{1}$ has a real spectrum, and that the degeneracy of zero-energy FB
reconfirms the generalized Lieb’s theorem, which can also be checked against
the independent diagonalization of all $H_{j>1}$ blocks and counting the total
number of FBs each of them generates.

###  III.1 Open boundary conditions

In this section, we briefly discuss the effects of considering open boundary
conditions (OBC) for the toy model of Fig. 2, both with and without closed
loops. In Fig. 4(a), we plot the complex energy spectrum of this 3-partite toy
model under OBC and for $N=7$ unit cells. Three finite energy branches of
seven states each can be observed, with an example of a $\mathscr{C}_{3}$
symmetric triplet given at the left of Fig. 4(c), together with 21 zero-energy
FB states, in agreement with the bulk spectrum of Fig. 3(a). Interestingly,
the skin effect is absent from this system, even though it is composed of non-
Hermitian unidirectional couplings. The reason for this is that loops are
present in the configuration of the hopping terms, which prevents the
eigenstates from converging to a given edge. Even though unidirectionality is
assumed for the couplings, the toy model is built in such a way that there is
no dominant hopping direction, with a global balance between leftwards and
rightwards oriented hopping terms.

Figure 4: Complex energy spectrum, in units of $t$, of the toy model of Fig. 2
under OBC and with $N=7$ unit cells as a function of the state index $j$ for
(a) the complete model, and (b) the model without loops, that is, without all
rightwards directed hopping terms. The finite energy eigenstates of the
branches in (a) become the skin modes of (b), while the number of zero-energy
states of the FBs remains the same for both cases. (c) Examples of eigenstate
profiles of $\mathscr{C}_{3}$ symmetric partners at the left, and of the skin
mode and three different zero-energy FB states at the right, where the radius
of the circle represents the amplitude of the wavefunction at the respective
site and the color represents its phase, coded by the color bar below.

In Fig. 4(b), we plot the same complex energy spectrum as in Fig. 4(a), only
removing from the open chain all the rightwards directed hopping terms (see
Fig. 2), such that there are no loops formed by the hopping terms. Immediately
we see that only the finite energy states of the three branches are affected,
namely by having all of them collapsing into the zero-energy skin mode with
weight at the left edge site only, as depicted at the top right of Fig. 4(c).
At the same time, it can be seen that the number of zero-energy FB states is
unaltered. This is to be expected, since each FB mode can be written as a
compact state that only has weight on the SL from which it is derived, as
exemplified for the three FB state depicted at the right in Fig. 4(c). As we
discuss in Appendix D, a Gram-Schmidt orthogonalization can be applied to the
chain such that the FB modes become isolated sites in the rotated basis.
Therefore the case of Fig. 4(b) highlights the fact that the FB states are
insensitive to the boundary conditions of the system, while the eigenstates
corresponding to the dispersive bulk bands for PBC may or may not coalesce
into skin modes under OBC, depending on the presence or absence of loops in
the model.

##  IV Conclusions

We introduced a class of non-Hermitian TB models characterized by the presence
of $n\geq 2$ SLs coupled in a cyclic fashion through unidirectional couplings.
When the Hamiltonian of these models is raised to the $n^{\text{th}}$ power it
becomes block diagonal, with the energy spectrum of the smallest block,
corresponding to the $H_{1}$ block in (2) by construction, being a common
feature of all blocks. The excess energy bands of the blocks of higher
dimensionality, in relation to the smallest one, were proved to be zero-energy
FBs. The same total number of these zero-energy FBs is also present at the
original Hamiltonian, which enabled us to generalize Lieb’s theorem [1],
originally only applicable to Hermitian bipartite systems, to account for the
total number of these bands in the non-Hermitian $n$-partite models we
considered.

At the same time, we showed how these models obey a generalized chiral
symmetry $\mathscr{C}_{n}$ of the type introduced in [47]. On the basis of the
different action of this symmetry on our TB models and on those appearing in
recent literature [47, 49, 52, 50, 53, 33], we argued that only the former can
be properly characterized as $\mathscr{C}_{n}$-symmetric models, expanding
their class beyond the generalized spin systems for which this symmetry was
originally proposed [34, 35]. A toy model was introduced to illustrate the
appearance of zero-energy FBs whose cardinality is in agreement with the
generalized Lieb’s theorem, and to manifest the $n$-fold rotation symmetry of
the complex energy spectrum (see Fig. 3), which is a direct consequence of the
$\mathscr{C}_{n}$ symmetry.

Since the $n^{\text{th}}$ root of the energy spectrum of the $H_{1}$ block was
shown to be $n$-fold degenerate, in absolute value, at the level of the
original Hamiltonian, the development of techniques to control the exact form
of $H_{1}$ can open up interesting perspectives. To name only one, if $H_{1}$
is dressed with topological features by appropriately designing the original
model $H(\mathbf{k})$, then the latter will inherit its topological
characterization directly from the former. In other words, one can use this
method to construct $\mathscr{C}_{n}$-symmetric $n$-root topological
insulators, which are currently limited to $2^{n}$-root systems [37, 12, 13],
therefore extending to nondriven systems the recent results obtained for
Floquet insulators [36]. These results are being finalized and will be the
subject of a forthcoming article [60].

Concerning the experimental realization of the non-Hermitian $n$-partite
models studied here, the main challenge relates to the implementation of
unidirectional hopping terms. In this regard, electrical circuits appear to be
in a prominent position to realize these systems [61, 62, 63], since
unidirectional capacitance couplings can be designed with the use of impedance
converters with current inversion, which have already been shown to be
experimentally accessible [64, 65]. Unidirectional couplings can also be very
well approximated in systems where a strong imaginary gauge field can be
induced, since these translate as highly asymmetrical nonreciprocal couplings.
This can be achieved in quantum systems such as (i) photonic lattices, either
with ring resonators coupled by mediating auxiliary rings with balanced gains
and losses [66, 67, 68] or, as has been experimentally realized recently, with
light walks in photonic fibers [69, 70], (ii) ultracold atoms in optical
lattices, where similar protocols based on exploiting the effects of
transitions to an auxiliary lattice to generate highly asymmetric hopping
terms have been proposed [71, 72, 73]. At the same time, imaginary gauge
fields have also been implemented in classical setups, namely, by including
auxiliary acoustic cavities with air dissipative materials in acoustic
lattices [74], or even in robotic metamaterials [75], where the lattice can be
mapped into a system of masses coupled by springs with effective nonreciprocal
spring constants.

## Acknowledgments

This work was developed within the scope of the Portuguese Institute for
Nanostructures, Nanomodelling and Nanofabrication (i3N) projects No.
UIDB/50025/2020 and No. UIDP/50025/2020 and funded by FCT - Portuguese
Foundation for Science and Technology through the Project No. PTDC/FIS-
MAC/29291/2017. AMM acknowledges financial support from the FCT through the
work Contract No. CDL-CTTRI-147-ARH/2018 and from i3N through the work
Contract No. CDL-CTTRI-46-SGRH/22. The authors would like to thank David
Viedma for useful discussions and suggested bibliographic material.

##  Appendix A Comments on the generalized chiral symmetry

We start by considering a Hamiltonian of the form

$H_{0}(\mathbf{k})=\begin{pmatrix}0&h_{12}&\omega_{3}h_{13}\\\
h_{12}^{*}&0&\omega_{3}^{2}h_{23}\\\
\omega_{3}h_{13}^{*}&\omega_{3}^{2}h_{23}^{*}&0\end{pmatrix},$ (28)

where $\omega_{3}=e^{i\frac{2\pi}{3}}$ and all $h_{ij}=h_{ij}(\mathbf{k})$ are
scalars. This model was introduced in [50] to model $\mathbb{Z}_{3}$ clock
parafermions in a breathing kagome lattice. This Hamiltonian can be decomposed
as (the momentum dependence is omitted henceforth)

$\displaystyle H_{0}$ $\displaystyle=$ $\displaystyle
H_{\circlearrowright}+H_{\circlearrowleft},$ (29) $\displaystyle
H_{\circlearrowright}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0&h_{12}&0\\\ 0&0&\omega_{3}^{2}h_{23}\\\
\omega_{3}h_{13}^{*}&0&0\end{pmatrix},$ (30) $\displaystyle
H_{\circlearrowleft}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0&0&\omega_{3}h_{13}\\\ h_{12}^{*}&0&0\\\
0&\omega_{3}^{2}h_{23}^{*}&0\end{pmatrix}.$ (31)

The generalized chiral symmetry $\mathscr{C}_{3}$ defined in (11) reads here
as

$\Gamma_{3}H_{0}\Gamma^{-1}_{3}=\omega_{3}^{-1}H_{\circlearrowright}+\omega_{3}H_{\circlearrowleft}:=H_{1}.$
(32)

The action of $\Gamma_{3}$ is therefore to produce two counterpropagating
$\phi_{3}$ rotations on the Hamiltonian terms, one clockwise for
$H_{\circlearrowright}$ and another counterclockwise for
$H_{\circlearrowleft}$. Only $H_{\circlearrowright}$ or $H_{\circlearrowleft}$
independently possess the generalized chiral symmetry $\mathscr{C}_{3}$ in the
precise sense of (11), while $H_{0}$ does not. The original $H_{0}$ is
recovered after three consecutive $\Gamma_{3}$ operations
($H_{0}=\Gamma_{3}^{3}H_{0}\Gamma^{-3}_{3}$). Thus, if we define

$H_{2}:=\Gamma_{3}H_{1}\Gamma^{-1}_{3}=\omega_{3}^{-2}H_{\circlearrowright}+\omega_{3}^{2}H_{\circlearrowleft},$
(33)

it follows that $H_{0}=\Gamma_{3}H_{2}\Gamma^{-1}_{3}$ and

$H_{0}+H_{1}+H_{2}=0,$ (34)

which formally replicates (18), although $H_{0}$, $H_{1}$, and $H_{2}$ are not
different chiral colors of the same Hamiltonian, that is, they do not relate
to each other by multiples of $\omega_{3}$ as in (15)-(17). From the cyclic
property of the trace of a matrix product, we have

$\Tr(H_{1})=\Tr(\Gamma_{3}H_{0}\Gamma^{-1}_{3})=\Tr(H_{0}),$ (35)

and similarly $\Tr(H_{2})=\Tr(H_{0})$. From applying the trace to both sides
of (34) we conclude that $\Tr(H_{0})=0$, that is, the eigenvalues of $H_{0}$
sum to zero.

Let us consider a general $4\times 4$ Hamiltonian of the form

$H=\begin{pmatrix}a&b&c&d\\\ e&f&g&h\\\ i&j&k&l\\\ m&n&o&p\end{pmatrix},$ (36)

where $\\{a,b,\dots,p\\}\in\mathbb{C}^{16}$. We can decompose $H$ as

$\displaystyle H$ $\displaystyle=$ $\displaystyle
H_{A}+H_{B}+H_{C}+H_{D}+H_{E},$ (37) $\displaystyle H_{A}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}a&&&\\\ &f&&\\\ &&k&\\\
&&&p\end{pmatrix},H_{B}=\begin{pmatrix}&b&&\\\ &&g&\\\ &&&l\\\
m&&&\end{pmatrix},H_{C}=\begin{pmatrix}&&&d\\\ e&&&\\\ &j&&\\\
&&o&\end{pmatrix},H_{D}=\begin{pmatrix}&&c&\\\ &&&h\\\ &&&\\\
&&&\end{pmatrix},H_{E}=\begin{pmatrix}&&&\\\ &&&\\\ i&&&\\\
&n&&\end{pmatrix},$ (38)

where all entries not shown are zeros. The generalized chiral symmetry
operator acts in this case as

$\Gamma_{4}H\Gamma^{-1}_{4}=H_{A}+\omega_{4}^{-1}H_{B}+\omega_{4}H_{C}+\omega_{4}^{-2}H_{D}+\omega_{4}^{2}H_{E},$
(39)

which obeys

$\sum\limits_{j=0}^{3}\Gamma_{4}^{j}H\Gamma^{-j}_{4}=4H_{A},$ (40)

with $\Gamma_{4}^{0}=\Gamma_{4}^{4}=\mathbb{1}_{4}$. Note that
$\omega_{4}=e^{i\frac{\pi}{2}}$ is imposed by both
$\Gamma_{4}H_{B}\Gamma^{-1}_{4}=\omega_{4}^{-1}H_{B}$ and
$\Gamma_{4}H_{C}\Gamma^{-1}_{4}=\omega_{4}H_{C}$. Taking the trace on both
sides of (40) leads to

$\Tr(\sum\limits_{j=0}^{3}\Gamma_{4}^{j}H\Gamma^{-j}_{4})=4\Tr(H_{A})=4\Tr(H),$
(41)

which, when $H_{A}$ is traceless, $\Tr(H_{A})=a+f+k+p=0$, implies that the
eigenvalues of $H$ sum to zero [47, 52]. When $H_{A}=O_{4\times 4}$, where
$O_{j\times j}$ is the $j\times j$ zero matrix, then (40) further reduces to
the generalized chiral symmetry proposed in [52, 53]. The decomposition in
(37) can be straightforwardly generalized to a Hamiltonian of any dimension
such that, under the action of $\Gamma_{n}$, the different components rotate
by multiples of $\omega_{n}$, as illustrated in (39) for $n=4$. Therefore $n$
successive applications of $\Gamma_{n}$ will retrieve the original
Hamiltonian, implying that all Hamiltonians with a trivial main diagonal obey

$\sum_{j=0}^{n-1}\Gamma_{n}^{j}H\Gamma^{-j}_{n}=0.$ (42)

However, only a small subset of these Hamiltonians, which includes at least
the chiral colored ones with the form of (1) (and evidently also their
conjugate transposed versions, corresponding to a global inversion of the
hopping directions), can be said to enjoy $\mathscr{C}_{n}$-symmetry in the
more stringent sense of (11) [with $\omega_{n}^{-1}\to\omega_{n}$ for the
conjugate transpose versions $H(\mathbf{k})\to H^{\dagger}(\mathbf{k})$], in
light of which it can be described as an extension of chiral symmetry to
$n$-partite lattices, since it reduces to the usual chiral symmetry for a
bipartite ($n=2$) model.

##  Appendix B Analogy with Baxter’s clock model

In order to highlight the parallel than can be drawn between the non-Hermitian
$n$-partite models we are considering and the generalized version of the Ising
model, we will follow closely below the systematic analysis provided by
Fendley [35], to which we refer the reader for further details. Baxter’s clock
model [34, 54, 76] can be viewed as an extension of the 1D Ising chain where,
instead of having “up” and “down” as the internal spin degree of freedom at
each site, the “spin” or clock value at each site can take the value
$\omega_{n}^{j}=e^{i\frac{2\pi}{n}j}$, with $j=0,1,\dots,n-1$. Its Hamiltonian
for an $L$ sites chain reads as

$H_{\text{BC}}=\sum\limits_{l=1}^{L}t_{2l-1}\tau_{l}+\sum\limits_{l=1}^{L-1}t_{2l}\sigma_{l}^{\dagger}\sigma_{l+1},$
(43)

where $\\{t_{i}\\}$ is a set of $2L-1$ arbitrary complex coefficients and
$\sigma_{j}=\mathbb{1}_{n}\otimes\dots\mathbb{1}_{n}\otimes\sigma\otimes\mathbb{1}_{n}\dots$
and
$\tau_{j}=\mathbb{1}_{n}\otimes\dots\mathbb{1}_{n}\otimes\tau\otimes\mathbb{1}_{n}\dots$
are operators acting at site $l=1,2,\dots,L$ [77] through the local operators
[78]

$\sigma=\begin{pmatrix}1\\\ &\omega_{n}\\\ &&\omega_{n}^{2}\\\ &&&\ddots\\\
&&&&\omega_{n}^{n-1}\end{pmatrix},\tau=\begin{pmatrix}&&&&1\\\ 1\\\ &1\\\
&&\ddots\\\ &&&1\end{pmatrix},$ (44)

where all entries not shown are zeros and the local basis at site $l$ spans
$\\{\ket{\omega_{n}^{j}}\\}$, with $j=0,1,\dots,n-1$. Clearly, $\sigma$
measures the clock value,
$\sigma\ket{\omega_{n}^{j}}=\omega_{n}^{j}\ket{\omega_{n}^{j}}$, and $\tau$ is
the shifting operator acting as
$\tau\ket{\omega_{n}^{j}}=\ket{\omega_{n}^{j+1}}$. The Ising model is
recovered for $n=2$, where $\sigma$ and $\tau$ reduce to the $\sigma_{z}$ and
$\sigma_{x}$ Pauli matrices, respectively. For $n>2$, the one-site term of
$H_{\text{BC}}$ in (43) generalizes the spin flipping term, while the two-site
term represents a generalized nearest-neighbor interaction. Baxter’s model
also enjoys a generalized chiral symmetry defined as

$\displaystyle\mathscr{C}_{n}:\ \ \ \Sigma_{n}H_{\text{BC}}\Sigma^{-1}_{n}$
$\displaystyle=$ $\displaystyle\omega_{n}H_{\text{BC}},$ (45)
$\displaystyle\Sigma_{n}$ $\displaystyle=$
$\displaystyle\prod\limits_{j=1}^{L}\sigma_{j}\prod\limits_{i=1}^{L}\tau_{i}^{-i},$
(46)

with $\Sigma_{n}$ a unitary operator whose action cycles the chiral colors of
$H_{\text{BC}}$ 222This symmetry was labeled as a generalized charge
conjugation symmetry (also known as particle-hole symmetry) in [35]. However,
charge conjugation is defined through an antiunitary operator, whereas
$\Sigma_{n}$ is unitary and therefore is rather the operator describing a
generalized chiral symmetry.. The presence of $\mathscr{C}_{n}$-symmetry
similarly imposes that the spectrum of $H_{\text{BC}}$ be formed by sequences
of the form $\\{E,\omega_{n}E,\omega_{n}^{2}E,\dots,\omega_{n}^{n-1}E\\}$,
each summing to zero [see discussion below (13)]. Notice that (45) is
equivalent to (11) if the direction of all hopping terms is switched, such
that $H(\mathbf{k})\to H^{\dagger}(\mathbf{k})$ in (1).

An even more direct analogy can be made with the “$\omega$-commutation”
relation [34] between the local $\sigma$ and $\tau$ operators in (44), which
we write here in a slightly different fashion for comparison purposes,

$\sigma\tau^{T}\sigma^{-1}=\omega_{n}^{-1}\tau^{T},$ (47)

which can be rewritten as a phase commutator of the form of (19) as
$[\sigma,\tau^{T}]_{\phi_{n}}=0$. Notice that $\tau^{T}$ has the same general
form of $H(\mathbf{k})$ in (1), where the ones are replaced by the $h_{j}$
rectangular matrices of different sizes. Similarly, $\sigma$ closely resembles
$\Gamma_{n}$ in (10), where each diagonal power of $\omega_{n}^{l}$, with
$l=0,1,\dots,n-1$, is enlarged into a diagonal square block of dimension
$d_{l+1}$, that is, $\omega_{n}^{l}\to\omega_{n}^{l}\mathbb{1}_{d_{l+1}}$. In
the interest of keeping up with this analogy, the bulk Hamiltonian of the non-
Hermitian $n$-partite models with the form of (1) can be viewed as a
generalized version of the shift operator of Baxter’s clock model, where each
SL corresponds to a different clock value, and the dimension $d_{j}$ of SLj,
i.e., the number of sites at SLj, can be viewed as counting the internal
degrees of freedom of each clock value, connected between adjacent SLs in a
unidirectional cyclic fashion.

It should also be stressed that, when mapping $\tau^{T}\to H(\mathbf{k})$, a
new ingredient is added to the system that is at the heart of the generalized
Lieb’s theorem of (22). Namely, the finite scalar entries of $\tau^{T}$ are
converted into rectangular matrices $h_{j}$ with different dimensionalities in
general, such that diagonal square blocks $H_{j}$ of different sizes $d_{j}$
are obtained for $H^{n}(\mathbf{k})$, whereas one trivially gets
$(\tau^{T})^{n}=\mathbb{1}_{n}$. From (22), a nontrivial number of zero-energy
FBs is precisely the combined result of finite sublattice imbalances (first
term) and/or the existance of zero-energy FBs in the $H_{1}$ block (second
term), both of which vanish for $(\tau^{T})^{n}$. At this point the analogy
stops, since the generalized Lieb’s theorem cannot be revealed by the
mathematical structure of Baxter’s clock model, but can be derived from the
properties of the non-Hermitian $n$-partite models we considered.

##  Appendix C Comments on the Lieb’s theorem

Here, we show that Lieb’s theorem, which states that the number of zero-energy
FBs in a bipartite system is given by the sublattice imbalance, while correct
for Hermitian systems, fails to account for the extra zero-energy FBs that
appear in certain non-Hermitian lattices. We show below an example of such a
model, further illustrating the validity of the generalized Lieb’s theorem
expressed in (22) already at the bipartite ($n=2$) level.

Figure 5: (Left) Unit cell of the non-Hermitian four-root topological
insulator with different prefactors $\beta$ at different hopping terms. The
non-Hermiticity comes from the Peierls phases picked up by the hopping terms,
which are the same in both directions. (Right) When squared, the model at the
left leads to two decoupled models, namely a diamond chain in the blue SL1 and
a two-leg ladder in the gray SL2 (only some hopping terms are indicated for
the latter). Open sites belong to the respective adjacent unit cells. Figure
6: Complex energy spectrum as a function of the momentum obtained from
diagonalizing the Hamiltonian defined in (48) for (a) $\delta=1.1$, (b)
$\delta=1.01$, and (c) $\delta=1$. (d) Squared energy spectrum as a function
of the momentum obtained by squaring the Hamiltonian with the parameters of
(c), which is purely real. In all plots, ($\times j$) indicates the $j$-fold
degeneracy of the respective band.

The model considered here, with the unit cell depicted in Fig. 5 at the left,
can be viewed as a non-Hermitian variation on the 1D four-root topological
insulator studied in [12]. The bulk Hamiltonian, parametrized by the real
$\delta$ factor included at some hopping parameters, reads as

$\displaystyle H(k,\delta)$ $\displaystyle=$ $\displaystyle
t\begin{pmatrix}O_{3\times 3}&h(k,\delta)\\\ h^{T}(-k,\delta)&O_{5\times
5}\end{pmatrix},$ (48) $\displaystyle h(k,\delta)$ $\displaystyle=$
$\displaystyle\begin{pmatrix}1&i\delta&-i\delta
e^{-ik}&e^{-ik}&\sqrt{3(\delta^{2}-1)}e^{-ik}\\\ \delta&0&i&0&0\\\
0&-i&0&\delta&0\end{pmatrix},$ (49)

The non-Hermiticity comes from the finite Peierls phases at some of the
hopping terms, which are the same in both directions. Squaring $H(k,\delta)$
in (48) leads to

$\displaystyle H^{2}(k,\delta)$ $\displaystyle=$ $\displaystyle
t^{2}\begin{pmatrix}H_{1}(k,\delta)&O_{3\times 5}\\\ O_{5\times
3}&H_{2}(k,\delta)\end{pmatrix},$ (50) $\displaystyle H_{1}(k,\delta)$
$\displaystyle=$
$\displaystyle(\delta^{2}-1)\mathbb{1}_{3}+\delta\begin{pmatrix}0&1+e^{-ik}&1+e^{-ik}\\\
1+e^{ik}&0&0\\\ 1+e^{ik}&0&0\end{pmatrix},$ (51)

where $H_{1}(k,\delta)$ models the diamond chain with the unit cell depicted
at the blue SL1 at the right-hand side of Fig. 5, which is known to host a
flat band with the energy of its diagonal term [80, 81, 82, 83], that is,
$E_{\text{FB}}=\delta^{2}-1$. The full expression of the pseudo-Hermitian
block $H_{2}(k,\delta)$, modeling a chain of the form of the gray SL2 at the
right-hand side of Fig. 5, is omitted here for simplicity. In the language of
[12], it corresponds to a topologically featureless residual block with shared
spectral properties with the relevant $H_{1}(k,\delta)$ block.

In Figs. 6(a)-6(c), we plot the complex energy spectrum of $H(k,\delta)$ in
(48) for three decreasing values of $\delta$. For all three cases, there are
two zero-energy FBs originating from sublattice imbalance, in accordance with
Lieb’s theorem. However, two extra FBs with symmetric energies, directly
obtained by taking the square-root of the diagonal term in (51), i.e.,
$E_{\text{FB},\pm}=\pm\sqrt{\delta^{2}-1}$, are present in the spectra and can
be seen to coalesce with the other two FBs as $\delta\to 1$ (they evolve in
the imaginary energy axis for $|\delta|<1$). Therefore, in Fig. 6(c) the
system displays two extra zero-energy FBs not accounted for by Lieb’s theorem.
Their appearance comes from the fact that, for the squared Hamiltonian
$H^{2}(k,1)$, whose energy spectrum is shown in Fig. 6(d), the smaller
$H_{1}(k,1)$ block itself has a zero-energy FB ($\\#_{\text{FB}}^{H_{1}}=1$
due to the sublattice imbalance within SL1), since its diagonal term vanishes
at $\delta=1$, which must be shared by the $H_{2}(k,1)$ block also due to the
isospectral properties (up to the zero-energy FBs already accounted for by the
sublattice imbalance) between the diagonal blocks, as discussed in the main
text. For $\delta=1$, the total number of zero-energy FBs obtained from the
generalized Lieb’s theorem is

$\displaystyle\\#_{\text{FB}}=d_{2}-d_{1}+2\\#_{\text{FB}}^{H_{1}}=5-3+2=4.$
(52)

There is a simple reason why the term proportional to
$\\#_{\text{FB}}^{H_{1}}$ is absent from Lieb’s theorem. It relates to the
fact that it applies to Hermitian systems, where no zero-energy FBs can be
present in $H_{1}$, apart from the trivial case where decoupled sites are
present within the unit cell, which can allways be chosen to belong to the
larger sublattice (notice, e.g., that site 8 becomes decoupled for the left
model of Fig. 5 when $\delta=1$, at which point one can ascribe it to either
sublattice). When finite Hermitian couplings between sites in SL1 and SL2 are
considered for a bipartite system, the diagonal terms of the squared
Hamiltonian are necessarily positive [11, 12, 13]. If $H_{1}$ is itself
bipartite, then it has $\\#_{\text{FB}}^{H_{1}}>0$ coming from its sublattice
imbalance, which are replicated in $H_{2}$, but with a finite energy given by
its diagonal term $c$ [with, e.g., $c=\delta^{2}-1$ for the $H_{1}$ block in
(51)]. In the original model, this translates in the appearance of
$\\#_{\text{FB}}^{H_{1}}$-fold degenerate FBs at $E=\pm\sqrt{c}$. These FBs
are pushed in pairs to zero energy as $c\to 0$. The only way this can be
achieved is by adding negative contributions to the diagonal term $c$ of the
squared model which, in turn, requires the inclusion of non-Hermitian hopping
terms in the original model, as we exemplified in (48) by considering non-
Hermitian Peierls phases at some couplings. This demonstrates, in short, that
Lieb’s theorem needs to be generalized, not only for the $n$-partite systems
considered in the main text, with $n>2$, but also already for non-Hermitian
bipartite systems, where extra zero-energy FBs originate from those that may
be present in the $H_{1}$ squared block.

If one considers the diamond chain depicted in the middle of Fig. 5, but
introducing now a $\pi$-flux per plaquette, such that the dispersive bands
above and below the zero-energy FB [see Fig. 6(d)] also become FBs with
symmetric finite energies [80, 81, 83], then the same procedure followed for
the four-root model of Fig. 5 can be applied. Namely, by introducing carefully
selected non-Hermitian phases at some of the couplings, while keeping the
$\pi$-flux pattern per plaquette [26, 84], the symmetric finite energy bands
can be pushed to zero-energy, which corresponds, in its squared Hamiltonian,
to lowering the energy of the smallest block (a single FB) to zero. In this
scenario, one obtains a completely trivial energy landscape made of three
zero-energy FBs, as was recently shown by Ding et al. [85] in a non-Hermitian
system of coupled resonators. As for the case of the four-root model analyzed
in this appendix that led to (52), our results also provide a full account of
the two extra FBs appearing in this diamond chain system.

##  Appendix D Comments on defective Hamitonians

The minimum number of LIEs within the set of zero-energy FBs follows from a
simple argument. The adjacency graph of $H(\mathbf{k})$ is a directed graph
with only outgoing links from SLj to SLj-1, where $j=1,\cdots,n$ and $j=0\to
j=n$ from the periodicity. Assuming that the number of dispersive bands of
$H^{n}(\mathbf{k})$ is given by the dimension of the smallest block $H_{1}$,
that is , it is $nd_{1}$ [where $n$ reflects the $n$-fold degeneracy of
$H^{n}(\mathbf{k})$], then the incoming hopping terms to each site $i_{1}$ in
SL1 determine a particular state $\ket{\psi_{i_{1}}^{(2)}}$ in SL2. We can
construct a basis for SL2 applying a Gram-Schmidt orthonormalization to the
states $\\{\ket{\psi_{i_{1}}^{(2)}}\\}$ (note this set spans a subspace of SL2
of dimension $d_{1}$) and choosing an arbitrary set of orthonormal basis
states (between themselves and to the set $\\{\ket{\psi_{i_{1}}^{(2)}}\\}$)
that completes the basis of SL2. If we draw the adjacency graph of
$H(\mathbf{k})$ in this basis, the latter set of nodes will have no outgoing
links and that will generate $d_{2}-d_{1}$ zero-energy FBs with LIEs.

Figure 7: Unit cell of a tripartite defective Hamiltonian. The hopping terms
are unidirectional, acting only in the direction of the arrows.

This argument can be extended to any Hamiltonian block between a pair of
consecutive sublattices. Three situations can occur: (i) $d_{j}>d_{j-1}$, (ii)
$d_{j}=d_{j-1}$, and (iii) $d_{j}<d_{j-1}$. For the two latter cases, no nodes
of SLj without outgoing links can be obtained with the procedure described
above. When $d_{j}>d_{j-1}$, then the same reasoning will generate
$d_{j}-d_{j-1}$ nodes in SLj without outgoing links. So the minimum number of
LIEs in the set of zero-energy FBs is

$\\#_{\text{LIEs}}^{\text{min}}=\sum\limits_{j=2}^{n}\text{Max}(d_{j}-d_{j-1},0).$
(53)

Basically, the argument above states that is possible to rotate the basis
within each sublattice in such a way that the number of sites in the shortest
section (smallest sublattice) in closed loops of the adjacency graph gives the
number of dispersive bands and the number of endpoints of open paths gives the
number of LIEs of the set of zero-energy FBs, while the total number of FBs in
this set is always given by the generalized Lieb’s theorem in (22). In the
example of Fig. 7, while the model, through (22), has $\\#_{\text{FB}}=3$
zero-energy FBs, this set only counts

$\\#_{\text{LIEs}}^{\text{min}}=\text{Max}(d_{2}-d_{1},0)+\text{Max}(d_{3}-d_{2},0)=2+0=2.$
(54)

The defectiveness of the model comes in this case from the decoupled cluster
within the unit cell involving sites 3 and 5, which yields two FBs but only
one LIE, while the loop accounts for the three dispersive bands and decoupled
site 4 for the other zero-energy FB. This defectiveness can be viewed as the
skin effect that takes place for the decoupled cluster within each unit cell,
which together form a set of decoupled nonreciprocal dimers in real-space.

## References

  * Lieb [1989] E. H. Lieb, Two theorems on the hubbard model, Phys. Rev. Lett. 62, 1201 (1989).
  * Shen  _et al._ [1994] S.-Q. Shen, Z.-M. Qiu, and G.-S. Tian, Ferrimagnetic long-range order of the hubbard model, Phys. Rev. Lett. 72, 1280 (1994).
  * Gouveia and Dias [2015] J. Gouveia and R. Dias, Magnetic phase diagram of the hubbard model in the lieb lattice, Journal of Magnetism and Magnetic Materials 382, 312 (2015).
  * Gouveia and Dias [2016] J. Gouveia and R. Dias, Spin and charge density waves in the lieb lattice, Journal of Magnetism and Magnetic Materials 405, 292 (2016).
  * Tindall  _et al._ [2021] J. Tindall, F. Schlawin, M. Sentef, and D. Jaksch, Lieb’s Theorem and Maximum Entropy Condensates, Quantum 5, 610 (2021).
  * Morales-Inostroza and Vicencio [2016] L. Morales-Inostroza and R. A. Vicencio, Simple method to construct flat-band lattices, Phys. Rev. A 94, 043831 (2016).
  * Zhang  _et al._ [2017] D. Zhang, Y. Zhang, H. Zhong, C. Li, Z. Zhang, Y. Zhang, and M. Belić, New edge-centered photonic square lattices with flat bands, Annals of Physics 382, 160 (2017).
  * Madail  _et al._ [2019] L. Madail, S. Flannigan, A. M. Marques, A. J. Daley, and R. G. Dias, Enhanced localization and protection of topological edge states due to geometric frustration, Phys. Rev. B 100, 125123 (2019).
  * Mao  _et al._ [2020] X. Mao, J. Liu, J. Zhong, and R. A. Römer, Disorder effects in the two-dimensional lieb lattice and its extensions, Physica E: Low-dimensional Systems and Nanostructures 124, 114340 (2020).
  * Ni  _et al._ [2020] X. Ni, J. Yan, and F. Liu, Electronic structures of a diagonally striped lattice: Multiple $(n-1)$-fold degenerate flat bands, Phys. Rev. B 102, 235117 (2020).
  * Ezawa [2020] M. Ezawa, Systematic construction of square-root topological insulators and superconductors, Phys. Rev. Research 2, 033397 (2020).
  * Marques  _et al._ [2021] A. M. Marques, L. Madail, and R. G. Dias, One-dimensional ${2}^{n}$-root topological insulators and superconductors, Phys. Rev. B 103, 235425 (2021).
  * Marques and Dias [2021] A. M. Marques and R. G. Dias, ${2}^{n}$-root weak, chern, and higher-order topological insulators, and ${2}^{n}$-root topological semimetals, Phys. Rev. B 104, 165410 (2021).
  * Kikutake  _et al._ [2013] K. Kikutake, M. Ezawa, and N. Nagaosa, Edge states in silicene nanodisks, Phys. Rev. B 88, 115432 (2013).
  * Ge [2018] L. Ge, Non-hermitian lattices with a flat band and polynomial power increase, Photon. Res. 6, A10 (2018).
  * Maimaiti and Andreanov [2021] W. Maimaiti and A. Andreanov, Non-hermitian flat-band generator in one dimension, Phys. Rev. B 104, 035115 (2021).
  * Ge [2015] L. Ge, Parity-time symmetry in a flat-band system, Phys. Rev. A 92, 052103 (2015).
  * Lazarides and Tsironis [2019] N. Lazarides and G. P. Tsironis, Compact localized states in engineered flat-band $\mathscr{P}\mathscr{T}$metamaterials, Scientific Reports 9, 4904 (2019).
  * Jin [2019] L. Jin, Flat band induced by the interplay of synthetic magnetic flux and non-hermiticity, Phys. Rev. A 99, 033810 (2019).
  * Molina [2015] M. I. Molina, Flat bands and $\mathcal{PT}$ symmetry in quasi-one-dimensional lattices, Phys. Rev. A 92, 063813 (2015).
  * Xia  _et al._ [2021] S. Xia, C. Danieli, Y. Zhang, X. Zhao, H. Lu, L. Tang, D. Li, D. Song, and Z. Chen, Higher-order exceptional pointÂ and landau-zener bloch oscillations in driven non-hermitian photonic lieb lattices, APL Photonics 6, 126106 (2021), https://doi.org/10.1063/5.0069633 .
  * Li  _et al._ [2022a] S. Li, S. Wang, S. Chen, Y. Wu, S. Ke, B. Wang, and P. Lu, Subspace-induced dirac point and nondissipative wave dynamics in a non-hermitian optical lattice, Phys. Rev. A 105, 033512 (2022a).
  * Zhang and Jin [2019] S. M. Zhang and L. Jin, Flat band in two-dimensional non-hermitian optical lattices, Phys. Rev. A 100, 043808 (2019).
  * Biesenthal  _et al._ [2019] T. Biesenthal, M. Kremer, M. Heinrich, and A. Szameit, Experimental realization of $\mathcal{P}\mathcal{T}$-symmetric flat bands, Phys. Rev. Lett. 123, 183601 (2019).
  * Ramezani [2017] H. Ramezani, Non-hermiticity-induced flat band, Phys. Rev. A 96, 011802(R) (2017).
  * Leykam  _et al._ [2017] D. Leykam, S. Flach, and Y. D. Chong, Flat bands in lattices with non-hermitian coupling, Phys. Rev. B 96, 064305 (2017).
  * Kawarabayashi  _et al._ [2011] T. Kawarabayashi, Y. Hatsugai, T. Morimoto, and H. Aoki, Generalized chiral symmetry and stability of zero modes for tilted dirac cones, Phys. Rev. B 83, 153414 (2011).
  * Kawarabayashi  _et al._ [2016] T. Kawarabayashi, H. Aoki, and Y. Hatsugai, Lattice realization of the generalized chiral symmetry in two dimensions, Phys. Rev. B 94, 235307 (2016).
  * Kawarabayashi and Hatsugai [2021] T. Kawarabayashi and Y. Hatsugai, Bulk-edge correspondence with generalized chiral symmetry, Phys. Rev. B 103, 205306 (2021).
  * Marques and Dias [2019] A. M. Marques and R. G. Dias, One-dimensional topological insulators with noncentered inversion symmetry axis, Phys. Rev. B 100, 041104(R) (2019).
  * Dias and Marques [2022] R. G. Dias and A. M. Marques, Long-range hopping and indexing assumption in one-dimensional topological insulators, Phys. Rev. B 105, 035102 (2022).
  * Marques and Dias [2020] A. M. Marques and R. G. Dias, Analytical solution of open crystalline linear 1d tight-binding models, Journal of Physics A: Mathematical and Theoretical 53, 075303 (2020).
  * Anastasiadis  _et al._ [2022] A. Anastasiadis, G. Styliaris, R. Chaunsali, G. Theocharis, and F. K. Diakonos, Bulk-edge correspondence in the trimer su-schrieffer-heeger model, Phys. Rev. B 106, 085109 (2022).
  * Baxter [1989a] R. Baxter, A simple solvable zn hamiltonian, Physics Letters A 140, 155 (1989a).
  * Fendley [2014] P. Fendley, Free parafermions, Journal of Physics A: Mathematical and Theoretical 47, 075001 (2014).
  * Zhou  _et al._ [2022] L. Zhou, R. W. Bomantara, and S. Wu, $q$th-root non-Hermitian Floquet topological insulators, SciPost Phys. 13, 015 (2022).
  * Dias and Marques [2021] R. G. Dias and A. M. Marques, Matryoshka approach to sine-cosine topological models, Phys. Rev. B 103, 245112 (2021).
  * Bomantara [2022] R. W. Bomantara, Square-root floquet topological phases and time crystals, Phys. Rev. B 106, L060305 (2022).
  * Deng  _et al._ [2022] W. Deng, T. Chen, and X. Zhang, $n\mathrm{th}$ power root topological phases in hermitian and non-hermitian systems, Phys. Rev. Research 4, 033109 (2022).
  * Kawabata  _et al._ [2019] K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Symmetry and topology in non-hermitian physics, Phys. Rev. X 9, 041015 (2019).
  * Yin  _et al._ [2018] C. Yin, H. Jiang, L. Li, R. Lü, and S. Chen, Geometrical meaning of winding number and its characterization of topological phases in one-dimensional chiral non-hermitian systems, Phys. Rev. A 97, 052115 (2018).
  * Yao and Wang [2018] S. Yao and Z. Wang, Edge states and topological invariants of non-hermitian systems, Phys. Rev. Lett. 121, 086803 (2018).
  * Yokomizo and Murakami [2019] K. Yokomizo and S. Murakami, Non-bloch band theory of non-hermitian systems, Phys. Rev. Lett. 123, 066404 (2019).
  * Li  _et al._ [2022b] C.-A. Li, S.-B. Zhang, J. C. Budich, and B. Trauzettel, Transition from metal to higher-order topological insulator driven by random flux, Phys. Rev. B 106, L081410 (2022b).
  * Lee  _et al._ [2020] K. Lee, R. Melendrez, A. Pal, and H. J. Changlani, Exact three-colored quantum scars from geometric frustration, Phys. Rev. B 101, 241111(R) (2020).
  * Chertkov and Clark [2021] E. Chertkov and B. K. Clark, Motif magnetism and quantum many-body scars, Phys. Rev. B 104, 104410 (2021).
  * Ni  _et al._ [2019] X. Ni, M. Weiner, A. Alù, and A. B. Khanikaev, Observation of higher-order topological acoustic states protected by generalized chiral symmetry, Nature Materials 18, 113 (2019).
  * van Miert and Ortix [2020] G. van Miert and C. Ortix, On the topological immunity of corner states in two-dimensional crystalline insulators, npj Quantum Materials 5, 63 (2020).
  * Kempkes  _et al._ [2019] S. N. Kempkes, M. R. Slot, J. J. van den Broeke, P. Capiod, W. A. Benalcazar, D. Vanmaekelbergh, D. Bercioux, I. Swart, and C. Morais Smith, Robust zero-energy modes in an electronic higher-order topological insulator, Nature Materials 18, 1292 (2019).
  * Ezawa [2022] M. Ezawa, Clock-symmetric non-hermitian second-order topological insulator, The European Physical Journal B 95, 97 (2022).
  * Herrera  _et al._ [2022] M. A. J. Herrera, S. N. Kempkes, M. B. de Paz, A. García-Etxarri, I. Swart, C. M. Smith, and D. Bercioux, Corner modes of the breathing kagome lattice: Origin and robustness, Phys. Rev. B 105, 085411 (2022).
  * Li  _et al._ [2020] Z.-X. Li, Y. Cao, X. R. Wang, and P. Yan, Second-order topological solitonic insulator in a breathing square lattice of magnetic vortices, Phys. Rev. B 101, 184404 (2020).
  * Li  _et al._ [2021] Y.-Z. Li, Z.-F. Liu, X.-W. Xu, Q.-P. Wu, X.-B. Xiao, M.-R. Liu, L.-L. Chang, and R.-L. Zhang, New Journal of Physics 23, 043010 (2021).
  * Baxter [1989b] R. J. Baxter, Superintegrable chiral potts model: Thermodynamic properties, an ”inverse” model, and a simple associated hamiltonian, Journal of Statistical Physics 57, 1 (1989b).
  * Alicea and Fendley [2016] J. Alicea and P. Fendley, Topological phases with parafermions: Theory and blueprints, Annual Review of Condensed Matter Physics 7, 119 (2016), https://doi.org/10.1146/annurev-conmatphys-031115-011336 .
  * Note [1] When $\\#_{\text{FB}}^{H_{1}}=2$, e.g., one could believe it would be possible to construct, from the $2n-2$ remaining bands from the other $H_{j>1}$ blocks, an $n$-fold degenerate finite energy band plus an $(n-2)$-fold degenerate zero-energy FB. However, the eigenstates of a finite energy band cannot vanish at any SL, which includes SL1 (it can be easily checked that, if there are nodes at all sites of one SL, then, through the TB equations, these nodes propagate sequentially to all sublattices). As such, if the eigenstates of this finite energy band have finite weight on some sites of SL1, then it would have to be present in $H_{1}$ also, which is assumed not to be the case. Therefore all $2n-2$ extra bands are also zero-energy FBs.
  * Mostafazadeh [2002] A. Mostafazadeh, Pseudo-hermiticity versus pt symmetry: The necessary condition for the reality of the spectrum of a non-hermitian hamiltonian, Journal of Mathematical Physics 43, 205 (2002), https://doi.org/10.1063/1.1418246 .
  * Zhang  _et al._ [2021a] X. Zhang, K. Xu, C. Liu, X. Song, B. Hou, R. Yu, H. Zhang, D. Li, and J. Li, Gauge-dependent topology in non-reciprocal hopping systems with pseudo-hermitian symmetry, Communications Physics 4, 166 (2021a).
  * Bergholtz  _et al._ [2021] E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Exceptional topology of non-hermitian systems, Rev. Mod. Phys. 93, 015005 (2021).
  * [60] A. M. Marques and R. G. Dias, to be submitted .
  * Hofmann  _et al._ [2019] T. Hofmann, T. Helbig, C. H. Lee, M. Greiter, and R. Thomale, Chiral voltage propagation and calibration in a topolectrical chern circuit, Phys. Rev. Lett. 122, 247702 (2019).
  * Zhang  _et al._ [2022] R.-L. Zhang, Q.-P. Wu, M.-R. Liu, X.-B. Xiao, and Z.-F. Liu, Complex-real transformation of eigenenergies and topological edge states in square-root non-hermitian topolectrical circuits, Annalen der Physik , 2100497 (2022).
  * Zeng and Lü [2022] Q.-B. Zeng and R. Lü, Evolution of spectral topology in one-dimensional long-range nonreciprocal lattices, Phys. Rev. A 105, 042211 (2022).
  * Liu  _et al._ [2021] S. Liu, R. Shao, S. Ma, L. Zhang, O. You, H. Wu, Y. J. Xiang, T. J. Cui, and S. Zhang, Non-hermitian skin effect in a non-hermitian electrical circuit, Research 2021, 5608038 (2021).
  * Zou  _et al._ [2021] D. Zou, T. Chen, W. He, J. Bao, C. H. Lee, H. Sun, and X. Zhang, Observation of hybrid higher-order skin-topological effect in non-hermitian topolectrical circuits, Nature Communications 12, 7201 (2021).
  * Longhi  _et al._ [2015a] S. Longhi, D. Gatti, and G. Della Valle, Non-hermitian transparency and one-way transport in low-dimensional lattices by an imaginary gauge field, Phys. Rev. B 92, 094204 (2015a).
  * Longhi  _et al._ [2015b] S. Longhi, D. Gatti, and G. D. Valle, Robust light transport in non-hermitian photonic lattices, Scientific Reports 5, 13376 (2015b).
  * Longhi [2018] S. Longhi, Non-hermitian gauged topological laser arrays, Annalen der Physik 530, 1800023 (2018).
  * Weidemann  _et al._ [2020] S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, Topological funneling of light, Science 368, 311 (2020), https://www.science.org/doi/pdf/10.1126/science.aaz8727 .
  * Weidemann  _et al._ [2022] S. Weidemann, M. Kremer, S. Longhi, and A. Szameit, Topological triple phase transition in non-hermitian floquet quasicrystals, Nature 601, 354 (2022).
  * Gong  _et al._ [2018] Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, Topological phases of non-hermitian systems, Phys. Rev. X 8, 031079 (2018).
  * Liu  _et al._ [2019] T. Liu, Y.-R. Zhang, Q. Ai, Z. Gong, K. Kawabata, M. Ueda, and F. Nori, Second-order topological phases in non-hermitian systems, Phys. Rev. Lett. 122, 076801 (2019).
  * He  _et al._ [2021] P. He, H.-T. Ding, and S.-L. Zhu, Geometry and superfluidity of the flat band in a non-hermitian optical lattice, Phys. Rev. A 103, 043329 (2021).
  * Zhang  _et al._ [2021b] X. Zhang, Y. Tian, J.-H. Jiang, M.-H. Lu, and Y.-F. Chen, Observation of higher-order non-hermitian skin effect, Nature Communications 12, 5377 (2021b).
  * Brandenbourger  _et al._ [2019] M. Brandenbourger, X. Locsin, E. Lerner, and C. Coulais, Non-reciprocal robotic metamaterials, Nature Communications 10, 4608 (2019).
  * Fendley [2012] P. Fendley, Journal of Statistical Mechanics: Theory and Experiment 2012, P11020 (2012).
  * Albertini  _et al._ [1989] G. Albertini, B. M. McCoy, and J. H. Perk, Commensurate-incommensurate transition in the ground state of the superintegrable chiral potts model, Physics Letters A 135, 159 (1989).
  * Mittag and Stephen [1971] L. Mittag and M. J. Stephen, Dual transformations in many-component ising models, Journal of Mathematical Physics 12, 441 (1971), https://doi.org/10.1063/1.1665606 .
  * Note [2] This symmetry was labeled as a generalized charge conjugation symmetry (also known as particle-hole symmetry) in [35]. However, charge conjugation is defined through an antiunitary operator, whereas $\Sigma_{n}$ is unitary and therefore is rather the operator describing a generalized chiral symmetry.
  * Pelegrí  _et al._ [2019a] G. Pelegrí, A. M. Marques, R. G. Dias, A. J. Daley, V. Ahufinger, and J. Mompart, Topological edge states with ultracold atoms carrying orbital angular momentum in a diamond chain, Phys. Rev. A 99, 023612 (2019a).
  * Pelegrí  _et al._ [2019b] G. Pelegrí, A. M. Marques, R. G. Dias, A. J. Daley, J. Mompart, and V. Ahufinger, Topological edge states and aharanov-bohm caging with ultracold atoms carrying orbital angular momentum, Phys. Rev. A 99, 023613 (2019b).
  * Kremer  _et al._ [2020] M. Kremer, I. Petrides, E. Meyer, M. Heinrich, O. Zilberberg, and A. Szameit, A square-root topological insulator with non-quantized indices realized with photonic aharonov-bohm cages, Nature Communications 11, 907 (2020).
  * Pelegrí  _et al._ [2020] G. Pelegrí, A. M. Marques, V. Ahufinger, J. Mompart, and R. G. Dias, Interaction-induced topological properties of two bosons in flat-band systems, Phys. Rev. Research 2, 033267 (2020).
  * Zhang and Jin [2020] S. M. Zhang and L. Jin, Localization in non-hermitian asymmetric rhombic lattice, Phys. Rev. Research 2, 033127 (2020).
  * Ding  _et al._ [2021] L. Ding, Z. Lin, S. Ke, B. Wang, and P. Lu, Non-hermitian flat bands in rhombic microring resonator arrays, Opt. Express 29, 24373 (2021).