#  $2^{n}$-root weak, Chern, and higher-order topological insulators, and
$2^{n}$-root topological semimetals

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

Recently, we have introduced in [A. M. Marques et al., Phys. Rev. B 103,
235425 (2021)] the concept of $2^{n}$-root topology and applied it to one-
dimensional (1D) systems. These models require $n$ squaring operations to
their Hamiltonians, intercalated with different constant energy downshifts at
each level, in order to arrive at a decoupled block corresponding to a known
topological insulator (TI) that acts as the source of the topological features
of the starting $2^{n}$-root TI ($\sqrt[2^{n}]{\text{TI}}$). In the process,
$n$ non-topological residual models with degenerate spectra and in-gap
impurity states appear, which dilute the topologically protected component of
the starting edge states. Here, we generalize this method to several two-
dimensional (2D) models, by finding the 4-root version of lattices hosting
weak and higher-order boundary modes (both topological and non-topological) of
a Chern insulator and of a topological semimetal. We further show that a
starting model with a non-Hermitian region in parameter space and a complex
energy spectrum can nevertheless display a purely real spectrum for all its
successive squared versions, allowing for an exact mapping between certain
non-Hermitian models and their Hermitian lower root-degree counterparts. A
comment is made on the possible realization of these models in artificial
lattices.

##  I Introduction

Square-root topology is one of the newest additions to the taxonomy of
topological systems [1]. It classifies a class of models whose conventional
topological characterization, based on quantized topological invariants, fails
when applied directly, but can be recovered when applied to the squared
Hamiltonian. An exact mapping between the eigenstates of the original and
squared Hamiltonians allows one to track precisely how the topological
features of the latter are inherited by the former [2]. After its discovery
[3], a growing body of literature dedicated to $\sqrt{\text{TIs}}$ has been
developing [4, 5, 6, 7, 8, 9], with some reports providing experimental
evidence for these models in different platforms already available [2, 10,
11]. Recently, a recipe for the identification of a new subset of
$\sqrt{\text{TIs}}$ directly in reciprocal-space has been proposed [12], and a
modified Altland-Zirbauer table [13] for their topological classification,
with all classes projected onto the A group of the conventional table [1], was
put forth.

The rules for the construction of square-root models have been introduced in
[5] and systematized in [6], and heavily rely on the bipartition of the
resulting square-root model, even when the original model is not bipartite
itself. The two main ingredients consist of subdividing the lattice by
introducing additional sites in the middle of each link (hopping term) and
renormalizing the magnitudes and phases of the doubled hopping terms, as
schematically depicted in Figs. 1(a) and 1(b). In a previous work [14], we
showed that the inclusion of an extra rule, invoked to ensure a uniform on-
site potential at the sites in the topological block of the squared
Hamiltonian [see Fig. 1(c)], enables an iterative application of the method
such that the higher root-degree versions (the $\sqrt[2^{n}]{\text{TI}}$ with
$n\geq 2$) can be derived from the $\sqrt{\text{TI}}$. This method was applied
there to one-dimensional (1D) models, including the Kitaev chain [15], a well-
known 1D topological superconductor, and to a 1D model with self-similar
properties after successive squaring operations in another work [16].

Figure 1: Schematic depiction of the main ingredients for the construction of
a higher root-degree version ($\sqrt[2^{n}]{\text{TI}}$) from its lower-root
degree chain ($\sqrt[2^{n-1}]{\text{TI}}$). (a) Subdivision of the
$\sqrt[2^{n-1}]{\text{TI}}$ by introducing a site in the middle of each
hopping term. (b) Renormalization of the new doubled hopping terms, both in
amplitude and in phase. (C) Uniformization of the squared on-site energy at
the relevant blue sublattice the $\sqrt[2^{n}]{\text{TI}}$ which, if not
achieved by default, requires the introduction of extra sites connected to the
blue sites with lower squared on-site energies with properly tuned hopping
strengths. Additional subtleties of this method can be found in [14] and here
at the end of Sec. IV.

Here, we extend the results of [14] to several relevant two-dimensional (2D)
models with different topological behavior. More concretely, we systematically
construct the quartic-root versions of a system with simultaneous weak and
higher-order topological phases, a Chern insulator, a triangular lattice
hosting conventional edge and corner modes, and a topological semimetal in a
cylindrical geometry. We further outline the general method for
straightforwardly deriving the higher root-degree versions of these lattices.
The higher-order $\sqrt[2^{n}]{\text{TIs}}$ are seen to be characterized by
the presence of finite-energy in-gap corner modes in their spectra with a
fractional topological component inherited from the zero-energy corner states
of the original TI. The former have a weaker form of topological protection
involving the chiral symmetry present at the sublattice of the original TI
[17]. For the $\sqrt[2^{n}]{\text{TIs}}$ with $\mathbb{Z}_{2}$ topological
indices, these are seen to display non-quantized values at the energy gaps
hosting the boundary modes, and whose quantization is only recovered for the
original TI obtained after $n$ squaring operations.

The rest of the paper is organized as follows. In Sec. II, we introduce and
characterize a 2D lattice hosting weak and higher-order topological states. In
Sec. III, we carefully construct its square-root version, showing how the
topological invariants lose their quantization and how it can be recovered by
squaring the Hamiltonian in order to arrive at the original model. In Sec. IV,
we go beyond square-root topology and construct the quartic-root version of
the lattice model, and finish this section with the general recipe for the
construction of the $2^{n}$-root version for any $n\geq 2$. The following
three sections are devoted to the application of the method outlined at the
end of Section IV to find the 4-root versions of different 2D systems, namely,
a Chern insulator with chiral pairs of edge bands in Sec. V, a triangular
lattice model hosting edge and corner modes in Sec. VI, and a semimetal with
topological flat bands of edge states in Sec. VII. Finally, we conclude in
Section VIII.

##  II Edge and corner states in the asymmetric 2D Su-Schrieffer-Heeger model

A simple model exhibiting a simultaneous weak and higher-order topological
phase is provided by the asymmetric 2D Su-Schrieffer-Heeger (SSH) model,
composed of coupled vertical and horizontal stacks of SSH chains [18], with
non-equivalent dimerized hoppings between orthogonal directions, as depicted
in Fig. 2(a).

Figure 2: (a) Asymmetric 2D SSH lattice. Red (green) shaded region indicates
the localization of the horizontal (vertical) edge states, while the gray
shaded region indicates the localization of the corner states. The purple
shaded square in the middle delimits the unit cell. (b) Energy spectrum for
the model in (a) for an open square lattice of $8\times 8$ unit cells, with
$t_{2x}=1$ (the energy unit) and $t_{2y}=0.25$, as a function of a linear
variation of $\sqrt[4]{t_{1x}}=\sqrt[4]{t_{1y}}$. Solid blue curves correspond
to bulk states, solid red (green) curves to horizontal (vertical) edge states
and the four-fold degenerate dashed black zero-energy curve to the corner
states. (c) Polarization profile, in units of $e$, along the $x$ (red squares)
and $y$ (green dots) direction for the bulk band $j=1,2,3,4$ (it is the same
for all bands).

The bulk Hamiltonian of this higher-order TI (HOTI), written in the ordered
$\\{\ket{j,\mathbf{k}}\\}$ basis, with $j=1,2,3,4$, $\mathbf{k}=(k_{x},k_{y})$
and lattice constants set to $a_{x}=a_{y}\equiv 1$, is given by

$\displaystyle H_{\text{HOTI}}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0&h_{\text{HOTI}}^{\dagger}\\\
h_{\text{HOTI}}&0\end{pmatrix},$ (1) $\displaystyle h_{\text{HOTI}}^{\dagger}$
$\displaystyle=$
$\displaystyle\begin{pmatrix}t_{1x}+t_{2x}e^{-ik_{x}}&t_{1y}+t_{2y}e^{ik_{y}}\\\
t_{1y}+t_{2y}e^{-ik_{y}}&t_{1x}+t_{2x}e^{ik_{x}}\end{pmatrix},$ (2)

where we assume $t_{i\mu}\geq 0$, with $i=1,2$ and $\mu=x,y$. Diagonalization
of this Hamiltonian yields four energy bands,

$\displaystyle E_{1}(\mathbf{k})$ $\displaystyle=$ $\displaystyle-
E_{4}(\mathbf{k})=-D_{x}(k_{x})-D_{y}(k_{y}),$ (3) $\displaystyle
E_{2}(\mathbf{k})$ $\displaystyle=$ $\displaystyle-
E_{3}(\mathbf{k})=-D_{x}(k_{x})+D_{y}(k_{y}),$ (4)

with $D_{\mu}(k_{\mu})=\sqrt{t_{1\mu}^{2}+t_{2\mu}^{2}+2t_{1\mu}t_{2\mu}\cos
k_{\mu}}$.

In the Benalcazar-Bernevig-Hughes (BBH) model [19, 20], consisting of a 2D SSH
model [21] with a $\pi$-flux per plaquette, the flux is responsible for
opening a zero-energy gap where corner states can be found in the topological
phase under open boundary conditions (OBC). Without the inserted flux, the
corner modes form bound states in the continuum buried within the bulk bands
at zero-energy [22], which were recently shown to keep their topological
protection, given that they do not hybridize with the surrounding bulk modes
if chiral symmetry is preserved [23, 24]. Instead of inserting a finite flux
per plaquette [25], a gap between the two middle bands can be opened at zero
flux either by including next-nearest-neighbor (NNN) hoppings [26, 27] or
setting asymmetric hopping dimerizations between the $x$ and $y$ directions,
as we are considering here.

The energy spectrum for the model in Fig. 2(a) for an open square lattice of
$8\times 8$ unit cells, with $t_{2x}=1$ (the energy unit) and $t_{2y}=0.25$,
as a function of a linear variation of $\sqrt[4]{t_{1x}}=\sqrt[4]{t_{1y}}$
(for reasons that will become clear below), is shown in Fig. 2(b). All gaps
are open between the bulk bands (blue curves) for $\sqrt[4]{t_{1x}}\lesssim
0.6$, with 1D vertical edge states 111We label the edge states vertical or
horizontal according to the direction over which they decay to the bulk.
(green curves) lying within the top and bottom gaps, and 1D horizontal edge
states (symmetric red curves) in the middle gap which, simultaneously, hosts
four degenerate zero-energy corner states (dashed black curves). Notice that
the corner states disappear when the gap between the horizontal edge states
closes, at $t_{1x}=t_{2y}=0.25$ (or $\sqrt[4]{t_{1x}}\approx 0.7$), before the
bulk gap closing at $\sqrt[4]{t_{1x}}\approx 0.8$. This closing and reopening
of the middle edge bands is highlighted in the triptych of energy spectra in
Fig. 3, where a ribbon geometry open in $x$ and periodic in $y$ is considered.
Such higher-order states, whose topological phase is protected by a boundary
energy gap have been recently termed extrinsic [29] or boundary-obstructed TIs
[30, 31]. The corner modes cannot be gapped out by chiral-symmetry preserving
perturbations, both at the bulk and boundaries, as long as the energy gap
between the bands of horizontal edge states remains open.

The polarization vector per unit length for each band $j$,
$\mathbf{P}^{j}=(P_{x}^{j},P_{y}^{j})$, serves as the weak topological index
of the system characterizing the 1D edge states, and can be computed from

$\displaystyle P_{\mu}^{j}$ $\displaystyle=$
$\displaystyle\frac{e}{2\pi}\gamma_{\mu}^{j},$ (5)
$\displaystyle\gamma_{\mu}^{j}$ $\displaystyle=$
$\displaystyle\frac{i}{2\pi}\int_{BZ}d\mathbf{k}\bra{u_{j}(\mathbf{k})}\frac{d}{dk_{\mu}}\ket{u_{j}(\mathbf{k})},$
(6)

where $e$ is the electron charge, $\gamma_{\mu}^{j}$ is the 2D Zak’s phase
[32] for the $\mu=x,y$ direction defined modulo $2\pi$, the integral goes over
the 2D Brillouin zone and $\ket{u_{j}(\mathbf{k})}$ is the eigenvector of band
$j$. The polarization along each direction is quantized to
$P_{\mu}^{j}=0,\frac{e}{2}$ by the presence of a ($\mathbf{k}$-independent)
reflection symmetry along $\mu$, whose explicit expressions are given by

$\displaystyle M_{x}H_{\text{HOTI}}(k_{x},k_{y})M_{x}^{-1}$ $\displaystyle=$
$\displaystyle H_{\text{HOTI}}(-k_{x},k_{y}),$ (7) $\displaystyle M_{x}$
$\displaystyle=$ $\displaystyle\sigma_{x}\otimes\sigma_{0},$ (8)
$\displaystyle M_{y}H_{\text{HOTI}}(k_{x},k_{y})M_{y}^{-1}$ $\displaystyle=$
$\displaystyle H_{\text{HOTI}}(k_{x},-k_{y}),$ (9) $\displaystyle M_{y}$
$\displaystyle=$ $\displaystyle\sigma_{x}\otimes\sigma_{x},$ (10)

with $\sigma_{x}$ being the $x$ Pauli matrix and $\sigma_{0}$ the $2\times 2$
identity matrix. A successive application of the reflection symmetries yields
a two-fold rotation symmetry written as

$\displaystyle C_{2}H_{\text{HOTI}}(\mathbf{k})C_{2}^{-1}$ $\displaystyle=$
$\displaystyle H_{\text{HOTI}}(-\mathbf{k}),$ (11) $\displaystyle
C_{2}=M_{x}M_{y}$ $\displaystyle=$ $\displaystyle\sigma_{0}\otimes\sigma_{x}.$
(12)

All four bands exhibit the polarization spectrum of Fig. 2(c). Naively, one
might be tempted to define the polarization along $\mu$ at each gap by the
usual expression

$P_{\mu}=\sum\limits_{j\in``occ"}P_{\mu}^{j}\mod e,$ (13)

where $``occ"$ is the set of occupied bands lying below the gap (assuming a
spinless fermionic picture). However, noting from (3)-(4) that the asymmetric
2D SSH model is factorizable in the $x$ and $y$ directions, such that
$\ket{u_{j}(\mathbf{k})}=\ket{u_{j,x}(k_{x})}\otimes\ket{u_{j,y}(k_{y})}$,
then (6) can be reduced to a 1D Zak’s phase [33],

$\gamma_{\mu}^{j}=i\int_{BZ_{\mu}}dk_{\mu}\bra{u_{j,\mu}(k_{\mu})}\frac{d}{dk_{\mu}}\ket{u_{j,\mu}(k_{\mu})},$
(14)

where BZμ is the 1D Brillouin zone along $\mu$. In other words,
$\gamma_{\mu}^{j}$ is $\bar{\mu}$ independent, where $\bar{\mu}$ is the
direction orthogonal to $\mu$. It follows that the pair of bands with energies
$\pm D_{\mu}(k_{\mu})-D_{\bar{\mu}}(k_{\bar{\mu}})$ can be decoupled from the
pair with energies $\pm D_{\mu}(k_{\mu})+D_{\bar{\mu}}(k_{\bar{\mu}})$ [see
(3) and (4)], such that each pair defines an energy gap whose topological
characterization is independent to that of the other pair. Specifically, the
computation of the polarization $P_{x}$ ($P_{y}$) considers each band pair
$\\{E_{1},E_{3}\\}$ and $\\{E_{2},E_{4}\\}$ ($\\{E_{1},E_{2}\\}$ and
$\\{E_{3},E_{4}\\}$) independently, where only the lowest band of the pair is
considered occupied when the gap is opened between the two, in which case (13)
can be recast as

$P_{\mu}^{\\{i,j\\}}=P_{\mu}^{i},\ \ i<j,$ (15)

where $\\{i,j\\}$ is the pair index. In agreement with Fig. 2(c), one readily
finds

$\displaystyle P_{x}^{\\{1,3\\}}$ $\displaystyle=$ $\displaystyle
P_{x}^{\\{2,4\\}}=\begin{cases}\frac{e}{2},\ \ \sqrt[4]{t_{1x}}<1,\\\ 0,\ \
\sqrt[4]{t_{1x}}>1,\end{cases}$ (16) $\displaystyle P_{y}^{\\{1,2\\}}$
$\displaystyle=$ $\displaystyle P_{y}^{\\{3,4\\}}=\begin{cases}\frac{e}{2},\ \
\sqrt[4]{t_{1x}}\lesssim 0.7,\\\ 0,\ \ \sqrt[4]{t_{1x}}\gtrsim
0.7,\end{cases}$ (17)

where, in the topological phase and under OBC, the non-trivial
$P_{x}^{\\{1,3\\}}$ ($P_{x}^{\\{2,4\\}}$) has a correspondence with the lower
(higher) red band of horizontal edge states in Fig. 2(b), whereas the non-
trivial $P_{y}^{\\{1,2\\}}$ ($P_{y}^{\\{3,4\\}}$) has a correspondence with
the lower (higher) green band of vertical edge states.

Figure 3: Triptych of energy spectra for the asymmetric 2D SSH model open
along $x$ and periodic along $y$ with 15 supercells, as a function of $k_{y}$.
Parameters: $t_{1x}=1$, $t_{2y}=0.25$ and $t_{1x}=t_{1y}=0.2$ for the left
panel, $t_{1x}=t_{1y}=0.25$ for the middle panel and $t_{1x}=t_{1y}=0.3$ for
the right panel. From one side to the other there is a gap closing and
reopening between the two red bands of horizontal edge states.

The four zero-energy corner states, given by the dashed black curves in Fig.
2(b), are present until $t_{1x}=0.25$ ($\sqrt[4]{t_{1x}}\approx 0.7$), which
marks the point where the gap between the horizontal red edge bands closes
and, at the same time, the point where the bulk gap between both the two lower
and higher bands closes, that is, the point where the vertical green edge
states vanish into the bulk. Clearly, the corner states are a consequence of
the confluence of $x$ and $y$ edge polarizations. The quadrupole moment per
unit area, $q_{xy}$, can be identified as the higher-order $\mathbb{Z}_{2}$
topological index characterizing the presence of absence or corner states
under OBC. However, the formula of $q_{xy}$ used for the BBH model [19, 20]
cannot be used here, as it relies on the non-commutativity of the reflection
symmetries, whereas we have $[M_{x},M_{y}]=0$ from (8) and (10). An
alternative formula for an ad hoc computation of $q_{xy}$ for the middle
energy gap of the asymmetric 2D SSH model was provided in Ref. [34], and reads
as

$\displaystyle q_{xy}$ $\displaystyle=$
$\displaystyle\frac{1}{2e}\tilde{P}_{x}\tilde{P}_{y},$ (18)
$\displaystyle\tilde{P}_{\mu}$ $\displaystyle=$
$\displaystyle\sum\limits_{j=1,2}P_{\mu}^{j},$ (19)

from where we find

$q_{xy}=\begin{cases}\frac{e}{2},\ \ \sqrt[4]{t_{1x}}\lesssim 0.7,\\\ 0,\ \
\sqrt[4]{t_{1x}}\gtrsim 0.7,\end{cases}$ (20)

and the corresponding presence (absence) of corner states under OBC for the
parameter region with non-trivial (trivial) $q_{xy}$ can be checked against
Fig. 2(b).

##  III Square root of the asymmetric 2D SSH model

The square-root version of the asymmetric 2D SSH model can be constructed by
direct application of the recipe in Ref. [6]: firstly, one constructs the
split graph [35] of the model in Fig. 2(a) by including a site at the middle
of each link representing a hopping term, such that each link gets divided in
two; secondly, the correspondence between the original and square-root
versions of a model is kept by renormalizing each pair of subdivided hopping
terms, in relation to the common original one, as
$t_{i\mu}\to\sqrt{t_{i\mu}}$. These two steps lead to the model in Fig. 4(a),
where the new sites created by subdivision of the original model form the gray
sublattice and the blue sites form the sublattice corresponding to the sites
in the original asymmetric 2D SSH lattice. After this procedure, an ambiguity
manifests itself regarding the definition of the unit cell, with the choice in
Fig. 4(a) being one among several others (e.g., sites 6 and 10 could be placed
at the left in the unit cell and/or sites 11 and 12 could be placed at the top
in the unit cell). A way to neutralize this ambiguity in real space, which
will be adopted in the next sections, is by including extra sites along the
edges such that all choices of unit cell become equivalent. In the case of
Fig. 4(a), this would translate in adding extra gray sites 6 and 10 along the
left edge and sites 11 and 12 along the top edge.

Figure 4: (a) Square-root version of the asymmetric 2D SSH lattice. The
meaning of the shaded regions is explained in Fig. 2(a). (b) Energy spectrum
for the model in (a) for an open square lattice of $8\times 8$ unit cells for
the same parameters and color scheme for the curves as in Fig. 2(b). (c)
Polarization profile, in units of $e$, along the $\mu=x,y$ direction for the
lowest bulk band. The dashed line at $P_{\mu}^{1}=0.5$ is a guide to the eye.

The bulk Hamiltonian of the model in Fig. 4(a), written in the ordered
$\\{\ket{j,\mathbf{k}}\\}$ basis, with $j=1,2,\dots,12$, is given by

$\displaystyle H_{\sqrt{\text{HOTI}}}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0&h_{\sqrt{\text{HOTI}}}^{\dagger}\\\
h_{\sqrt{\text{HOTI}}}&0\end{pmatrix},$ (21) $\displaystyle
h_{\sqrt{\text{HOTI}}}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\sqrt{t_{1x}}&0&\sqrt{t_{1x}}&0\\\
\sqrt{t_{2x}}e^{ik_{x}}&0&\sqrt{t_{2x}}&0\\\
\sqrt{t_{1y}}&0&0&\sqrt{t_{1y}}\\\ 0&\sqrt{t_{1y}}&\sqrt{t_{1y}}&0\\\
0&\sqrt{t_{1x}}&0&\sqrt{t_{1x}}\\\
0&\sqrt{t_{2x}}&0&\sqrt{t_{2x}}e^{ik_{x}}\\\
\sqrt{t_{2y}}e^{-k_{y}}&0&0&\sqrt{t_{2y}}\\\
0&\sqrt{t_{2y}}&\sqrt{t_{2y}}e^{-ik_{y}}&0\end{pmatrix}.$ (22)

In contrast with $H_{\text{HOTI}}$ in (1), the spatial symmetries of
$H_{\sqrt{\text{HOTI}}}$ are $\mathbf{k}$ dependent, as a consequence of non-
centered reflection-axes within the unit cell [17],

$\displaystyle M_{x}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}M_{x}^{\text{BS}}&0\\\
0&M_{x}^{\text{GS}}\end{pmatrix},$ (23) $\displaystyle M_{x}^{\text{BS}}$
$\displaystyle=$ $\displaystyle\sigma_{x}\otimes\sigma_{0},$ (24)
$\displaystyle M_{x}^{\text{GS}}$ $\displaystyle=$
$\displaystyle\text{diag}\big{(}1,e^{-ik_{x}},\sigma_{x},1,e^{-ik_{x}},\sigma_{x}\big{)},$
(25)

and

$\displaystyle M_{y}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}M_{y}^{\text{BS}}&0\\\
0&M_{y}^{\text{GS}}\end{pmatrix},$ (26) $\displaystyle M_{y}^{\text{BS}}$
$\displaystyle=$ $\displaystyle\sigma_{x}\otimes\sigma_{x},$ (27)
$\displaystyle M_{y}^{\text{GS}}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0&0&0&0&1&0&0&0\\\ 0&0&0&0&0&1&0&0\\\
0&0&1&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0\\\ 1&0&0&0&0&0&0&0\\\ 0&1&0&0&0&0&0&0\\\
0&0&0&0&0&0&e^{ik_{y}}&0\\\ 0&0&0&0&0&0&0&e^{ik_{y}}\end{pmatrix},$ (28)

where BS (GS) stands for blue (gray) sublattice. According to the Lieb’s
theorem [36], the sublattice imbalance within the unit cell
($\\#_{GS}-\\#_{BS}=4$) equals the number of zero-energy flat bands under PBC.
This is the mechanism behind the formation of highly degenerate zero-energy
flat bands that appear for a class of Lieb-like lattices studied recently
[37].

Figure 5: Schematic illustration of the result of squaring the
$\sqrt{\text{HOTI}}$ lattice (and the respective Hamiltonian below) with an
integer number of unit cells in $x$ and $y$. Up to a constant overall energy
shift $c_{1}$,the squared model is composed of two decoupled lattices, one
yielding the HOTI in the blue sublattice and the other yielding the residual
model in the gray sublattice. The meaning of the shaded areas is given in Fig.
2(a). Orange impurity sites with an on-site energy offset in relation to the
other sites of the same lattice appear at different boundaries with opposite
effects: they gap out the boundary modes in the HOTI lattice but drive the
boundary modes in the residual lattice to the same gaps of the corresponding
ones in the HOTI.

The combination of the reflection symmetries leads to a two-fold rotation
symmetry written as

$\displaystyle C_{2}$ $\displaystyle=$ $\displaystyle
M_{x}M_{y}=\begin{pmatrix}C_{2}^{BS}&0\\\ 0&C_{2}^{GS}\end{pmatrix},$ (29)
$\displaystyle C_{2}^{BS}$ $\displaystyle=$
$\displaystyle\sigma_{0}\otimes\sigma_{x},$ (30) $\displaystyle C_{2}^{GS}$
$\displaystyle=$ $\displaystyle\begin{pmatrix}0&0&0&0&1&0&0&0\\\
0&0&0&0&0&e^{-1k_{x}}&0&0\\\ 0&0&0&1&0&0&0&0\\\ 0&0&1&0&0&0&0&0\\\
1&0&0&0&0&0&0&0\\\ 0&e^{-1k_{x}}&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&e^{ik_{y}}\\\
0&0&0&0&0&0&e^{ik_{y}}&\end{pmatrix}.$ (31)

The $k_{\mu}$-dependence in $M_{\mu}$ implies that $\gamma_{\mu}^{j}$ in (6)
is not $\pi$-quantized in general [17]. In turn, neither the polarization
$P_{\mu}^{j}$ in (5) nor, consequently, the quadrupole moment $q_{xy}$ in
(18), regardless of the energy gap considered, will necessarily be quantized,
as exemplified by the polarization profile of the lowest energy band shown in
Fig. 4(c). Quantized topological indices will only be recovered at the level
of the diagonal block of the squared model corresponding to the asymmetric 2D
SSH model analyzed in the previous section. This truly establishes the model
in Fig. 4(a) as a square-root HOTI, since the usual tools for the topological
characterization of its boundary modes fail when directly applied, and must be
derived from the “well behaved” topological features of the original HOTI one
arrives at by squaring the model.

The $\mathbf{k}$-dependence of the bulk spatial symmetries [38] [see (23),
(26) and (29)], regardless of the choice of unit cell, means that these same
symmetries are broken in real-space when an integer number of unit cell is
considered, as in Fig. 4(a). In order to restore the same symmetries in real-
space, extra sites from incomplete unit cells would have to be included along
the two edges where they are missing [e.g, including the dangling gray sites
present at the right and bottom edges also along the left and top edges in
Fig. 4(a), as explained above]. The energy spectrum for the model in Fig. 4(a)
for an open square lattice of $8\times 8$ unit cells, with $t_{2x}=1$ (the
energy unit) and $t_{2y}=0.25$, keeping it, as before, as a function of a
linear variation of $\sqrt[4]{t_{1x}}=\sqrt[4]{t_{1y}}$, is shown in Fig.
4(b). The two corner bands form a finite energy chiral pair, with their states
appearing at the bottom right corner as illustrated in Fig. 4(a). Four green
(red) bands of vertical (horizontal) edge states are present until the bulk
gap closing points occurring at $\sqrt[4]{t_{1x}}\approx 0.7$
($\sqrt[4]{t_{1x}}=1$) are reached, and are located around a single edge, more
concretely the bottom (right) edge with the geometry of Fig. 4(a). Again, the
corner states evolve into horizontal edge states across the gap closing point
between pairs of horizontal edge bands occurring at
$\sqrt[4]{t_{1x}}=\sqrt[4]{0.25}\approx 0.7$ (that is, $\sqrt{t_{1x}}=0.5$, if
we use the square-root energy units of the $\sqrt{\text{HOTI}}$ model of this
section).

Squaring the $\sqrt{\text{HOTI}}$ Hamiltonian in (21) leads to

$\displaystyle H^{2}_{\sqrt{\text{HOTI}}}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\begin{pmatrix}H_{\text{par}}&0\\\
0&H_{\text{res}}\end{pmatrix},$ (32) $\displaystyle H_{\text{par}}$
$\displaystyle=$ $\displaystyle
h_{\sqrt{\text{HOTI}}}^{\dagger}h_{\sqrt{\text{HOTI}}}=c_{1}I_{4}+H_{\text{HOTI}},$
(33) $\displaystyle H_{\text{res}}$ $\displaystyle=$ $\displaystyle
h_{\sqrt{\text{HOTI}}}h_{\sqrt{\text{HOTI}}}^{\dagger}=c_{1}I_{8}+H_{\text{res}^{\prime}},$
(34)

where $H_{\text{par}}$ ($H_{\text{res}}$) is the parent (residual)
Hamiltonian, $H_{\text{HOTI}}$ is given in (1),
$c_{1}=t_{1x}+t_{2x}+t_{1y}+t_{2y}$ is a constant energy shift, $I_{n}$ is the
$n\times n$ identity matrix. The result of squaring the open lattice in Fig.
4(a) is schematically depicted in Fig. 5, where it can be seen that edge and
corner states are located around the unperturbed (perturbed) edges and
corners, respectively, for the lattice corresponding to $H_{\text{HOTI}}$
($H_{\text{res}^{\prime}}$). The different offsets in the self-energies of the
perturbed sites, all colored in orange for simplicity, are a consequence of
their lower coordination number in the $\sqrt{\text{HOTI}}$ lattice, in
relation to the other sites in the same sublattice. Due to this, when
$H_{\sqrt{\text{HOTI}}}$ is squared in real-space, and for an integer number
of unit cells, these sites acquire fewer terms in their self-energy, akin to
the edge corrections appearing in second-order perturbation theory to the
states with weight at the edge sites [39]. As such, the boundary modes coming
from $H_{\text{HOTI}}$ are topologically originated, as demonstrated in the
previous section, while the ones coming from $H_{\text{res}^{\prime}}$ are
impurity (also called Tamm [40] or Shockley [41]) states forming a degenerate
boundary spectrum with the former. The energy spectrum coming from the
diagonalization of $H^{2}_{\sqrt{\text{HOTI}}}$ with $8\times 8$ unit cells
and a global down energy shift of $c_{1}(t_{1x})=1.25+2t_{1x}$ (since
$t_{2x}=1$, $t_{2y}=0.25$, and $t_{1x}=t_{1y}$), coincides with the one in
Fig. 2(b), with all boundary curves doubly degenerate, both edge and corner
ones. From (33) and (34) one has [6]

$\displaystyle H_{\text{HOTI}}\ket{\psi^{\text{HOTI}}_{j}}$ $\displaystyle=$
$\displaystyle(\epsilon_{j}-c_{1})\ket{\psi^{\text{HOTI}}_{j}},$ (35)
$\displaystyle H_{\text{par}}\ket{\psi^{\text{HOTI}}_{j}}$ $\displaystyle=$
$\displaystyle\epsilon_{j}\ket{\psi^{\text{HOTI}}_{j}},$ (36) $\displaystyle
H_{\text{res}^{\prime}}\ket{\psi^{\text{res}^{\prime}}_{l}}$ $\displaystyle=$
$\displaystyle(\epsilon_{l}^{res}-c_{1})\ket{\psi^{\text{res}^{\prime}}_{l}},$
(37) $\displaystyle H_{\text{res}}\ket{\psi^{\text{res}^{\prime}}_{l}}$
$\displaystyle=$
$\displaystyle\epsilon_{l}^{res}\ket{\psi^{\text{res}^{\prime}}_{l}},$ (38)

where $\ket{\psi^{\text{HOTI}}_{j}}$ ($\ket{\psi^{\text{res}^{\prime}}_{l}}$)
is the $j^{\text{th}}$ ($l^{\text{th}}$) eigenstate of the parent (residual)
Hamiltonian, $\\{\epsilon_{j}\\}=\\{\epsilon_{1},\dots,\epsilon_{4N}\\}$ with
$\epsilon_{j}\geq 0$, $N$ is the total number of unit cells,
$\\{\epsilon_{l}^{\text{res}}\\}=\\{\\{\epsilon_{j}\\},\\{0\\}_{4N}\\}$ and
$\\{0\\}_{4N}=\\{0,0,\dots,0\\}$ is a vector of size $4N$ containing the
excess zero-energy levels in $H_{\text{res}}$ coming from the sublattice
imbalance in $H_{\sqrt{\text{HOTI}}}$. Focusing on the finite energy states
within the relevant set $\\{\epsilon_{j}\\}$, the eigenstates of the original
$\sqrt{\text{HOTI}}$ lattice can be written in terms of the eigenstates of the
HOTI and shifted residual lattices as

$H_{\sqrt{\text{HOTI}}}\ket{\psi_{j,\pm}}=\pm\sqrt{\epsilon_{j}}\ket{\psi_{j,\pm}})=\pm\sqrt{\epsilon_{j}}\begin{pmatrix}\ket{\psi^{\text{HOTI}}_{j}}\\\
\pm\ket{\psi^{\text{res}^{\prime}}_{j}}\end{pmatrix},$ (39)

where, due to the presence of chiral symmetry, defined in real-space as
$CH_{\sqrt{\text{HOTI}}}C^{-1}=-H_{\sqrt{\text{HOTI}}}$, with
$C=\text{diag}\big{(}I_{4N},-I_{8N}\big{)}$, all finite-energy states come in
chiral pairs $\ket{\psi_{j,\pm}}$ (note, however, that the presence of
impurity sites in the parent and residual lattices, as shown in Fig. 5, breaks
the chiral symmetry in $H_{\sqrt{\text{HOTI}}}^{2}-c_{1}I_{12}$). In
particular, both the weak edge and the higher-order corner states of the
$\sqrt{\text{HOTI}}$ lattice in Fig. 4(b) can be said to be “half”-topological
and “half”-impurity, with their topological features directly inherited from
the $\ket{\psi^{\text{HOTI}}_{j}}$ component. Thus, the model in Fig. 4(a)
constitutes an example of a square-root weak and higher-order topological
insulator.

##  IV Quartic-root of the asymmetric 2D SSH model

Here, we want to go a step further than in the previous section and construct
the quartic-root version of the asymmetric 2D SSH model, which we label
$\sqrt[4]{\text{HOTI}}$. Our starting point is the $\sqrt{\text{HOTI}}$ in
Fig. 4, with all its sites colored in blue, since we will require that the 12
sites of its unit cell become a sublattice of the $\sqrt[4]{\text{HOTI}}$.
Then, as before, we subdivide the $\sqrt{\text{HOTI}}$ by including a gray
site at the middle of each link and renormalizing the two resulting hoppings
to the adjacent blue sites as $\sqrt{t_{i\mu}}\to\sqrt[4]{t_{i\mu}}$, leading
to the unit cell depicted in Fig. 6 without the green sites at this stage.

Figure 6: Unit cell of the $\sqrt[4]{\text{HOTI}}$. Blue sites form one
sublattice and the green and gray sites form another. Open sites at the edges
belong to adjacent unit cells.

To see why this is not enough, we recall that the $\sqrt[4]{\text{HOTI}}$
model should be such that, upon squaring its bulk Hamiltonian, one gets the
$\sqrt{\text{HOTI}}$ as one of its diagonal blocks, apart from a constant
energy term, replicating for a higher root-degree the same relation
established above between the $\sqrt{\text{HOTI}}$ and the HOTI in (32).

Figure 7: Real part of the energy spectrum, with the same parameters and color
scheme for the curves as in Fig. 2(b), obtained from diagonalization of: (a)
$H_{\sqrt[4]{\text{HOTI}}}$, the Hamiltonian of the open
$\sqrt[4]{\text{HOTI}}$ lattice with $8\times 8$ unit cells plus 36 extra
sites from incomplete unit cells (16 extra gray sites with index 35 and 36,
from Fig. 6, placed on top and 16 extra gray sites with index 16 and 28 placed
at the left); (b)
$H_{\sqrt[4]{\text{HOTI}}}^{2^{\prime}}=H_{\sqrt[4]{\text{HOTI}}}^{2}-c_{2}I$,
with $c_{2}$ given in (45); (c)
$H_{\sqrt[4]{\text{HOTI}}}^{4^{\prime}}=H_{\sqrt[4]{\text{HOTI}}}^{2^{\prime}}H_{\sqrt[4]{\text{HOTI}}}^{2^{\prime}}-c_{1}I$,
with $c_{1}$ given below (34). (d)-(e) Imaginary part of the spectrum for the
corresponding cases above. The dashed orange vertical line crossing (a) and
(d) at $\sqrt[4]{t_{1x}}=0.5$ separates the non-Hermitian region to the left
from the Hermitian region to the right.

However, if one considers only the blue and gray sites in Fig. 6, the self-
energy is not the same at all the sites in the blue sublattice upon squaring
the $\sqrt[4]{\text{HOTI}}$ bulk Hamiltonian,

$\displaystyle\bra{j}H^{2}_{\sqrt[4]{\text{HOTI}}}\ket{j}$ $\displaystyle=$
$\displaystyle\sum\limits_{\begin{subarray}{c}i=1,2,\\\
\mu=x,y\end{subarray}}\sqrt{t_{i\mu}},\ \ \ j=1,2,3,4,$ (40)
$\displaystyle\bra{5}H^{2}_{\sqrt[4]{\text{HOTI}}}\ket{5}$ $\displaystyle=$
$\displaystyle\bra{9}H^{2}_{\sqrt[4]{\text{HOTI}}}\ket{9}=2\sqrt{t_{1x}},$
(41) $\displaystyle\bra{6}H^{2}_{\sqrt[4]{\text{HOTI}}}\ket{6}$
$\displaystyle=$
$\displaystyle\bra{10}H^{2}_{\sqrt[4]{\text{HOTI}}}\ket{10}=2\sqrt{t_{2x}},$
(42) $\displaystyle\bra{7}H^{2}_{\sqrt[4]{\text{HOTI}}}\ket{7}$
$\displaystyle=$
$\displaystyle\bra{8}H^{2}_{\sqrt[4]{\text{HOTI}}}\ket{8}=2\sqrt{t_{1y}},$
(43) $\displaystyle\bra{11}H^{2}_{\sqrt[4]{\text{HOTI}}}\ket{11}$
$\displaystyle=$
$\displaystyle\bra{12}H^{2}_{\sqrt[4]{\text{HOTI}}}\ket{12}=2\sqrt{t_{2y}},$
(44)

where the $\mathbf{k}=(k_{x},k_{y})$ dependence is implied in
$H^{2}_{\sqrt[4]{\text{HOTI}}}$ for convenience. In order to recover a
constant energy shift $c_{2}$ for the entire blue sublattice, we first set it
to the self-energy at sites 1 to 4 in (40),

$c_{2}=\sum\limits_{\begin{subarray}{c}i=1,2,\\\
\mu=x,y\end{subarray}}\sqrt{t_{i\mu}},$ (45)

and then add extra green sites in Fig. 6, connected to sites 5 to 12 by
hoppings chosen such that the difference between (41)-(44) and (40) is
compensated, yielding a $c_{2}$ self-energy term for all sites in the blue
sublattice. These new hopping terms can be readily found to be given by

$\sqrt[4]{v_{j\nu}}=\sqrt{\sum\limits_{\\{i,\mu\\}\neq\\{j,\nu\\}}\sqrt{t_{i,\mu}}-\sqrt{t_{j,\nu}}}.$
(46)

If we set, as in the two previous sections, $t_{2x}=1$, $t_{2y}=0.25$ and
$t_{1x}=t_{1y}$, this leads to $\sqrt[4]{v_{2x}}=\sqrt{2\sqrt{t_{1x}}-0.5}$,
which is purely imaginary and non-Hermitian for $t_{1x}<0.0625$
($\sqrt[4]{t_{1x}}<0.5$). The non-Hermiticity of $\sqrt[4]{v_{2x}}$ in this
interval can be understood by noticing that, without the extra sites, the on-
site energies at sites 6 and 10 in Fig. 6, coupled by the strong $t_{2x}$
links to the respective adjacent sites, are greater than $c_{2}$ when the
system is squared. In order to compensate for the difference to $c_{2}$, the
squared extra hopping terms $\sqrt[4]{v_{2x}}$ (light blue links coupling to
the extra sites 20 and 32) have to yield a negative contribution
($\sqrt{v_{2x}}<0$) in this $\sqrt[4]{t_{1x}}<0.5$ region, requiring
$\sqrt[4]{v_{2x}}$ to be non-Hermitian.

The real and imaginary parts of the energy spectrum of the
$\sqrt[4]{\text{HOTI}}$ are shown in Figs. 7(a) and (d), respectively,
considering an open square geometry of $8\times 8$ unit cells plus 36 extra
sites from incomplete unit cells (16 extra gray sites with index 35 and 36,
from Fig. 6, placed on top and 16 extra gray sites with index 16 and 28 placed
at the left). The role of the extra sites is to ensure a uniform $c_{2}$ on-
site energy on all sites of the relevant blue sublattice upon squaring [14],
such that the $\sqrt{\text{HOTI}}$ in Fig. 4(a) is recovered as one of the
diagonal blocks of the squared Hamiltonian 222Note that adding extra sites
from the larger sublattice has a trivial effect on the energy spectrum, namely
the addition of the same amount of zero-energy states due to sublattice
imbalance.. The dashed orange vertical line separates the non-Hermitian region
to the left from the Hermitian region to the right. Within the non-Hermitian
region, two additional subregions can be distinguished: i) a parity-time
($\mathcal{PT}$) symmetry breaking [43] subregion below the exceptional point
[44] at $\sqrt[4]{t_{1x}}\approx 0.2$, characterized by the appearance of a
set of eigenstates with imaginary energies, given by the dashed blue curves in
Fig. 7(d), and ii) a $\mathcal{PT}$-symmetry preserving subregion for
$0.2\lesssim\sqrt[4]{t_{1x}}<0.5$, characterized by a purely real spectrum
[the imaginary energies evolve into the symmetric blue dashed curves of bulk
states bifurcating from $E=0$ at $\sqrt[4]{t_{1x}}\approx 0.2$ in Fig. 7(a)].
Four bands or corner states, given by the dashed black curves, are present in
Fig. 7(a), together with a plethora of 1D vertical (green curves) and
horizontal (red curves) bands of edge states that are driven into the bulk
blue bands in the Hermitian region.

Squaring the $\sqrt[4]{\text{HOTI}}$ bulk Hamiltonian leads to

$\displaystyle H^{2}_{\sqrt[4]{\text{HOTI}}}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\begin{pmatrix}H_{\sqrt{\text{par},2}}&0\\\
0&H_{\sqrt{\text{res},2}}\end{pmatrix},$ (47) $\displaystyle
H_{\sqrt{\text{par},2}}$ $\displaystyle=$ $\displaystyle
c_{2}I_{12}+H_{\sqrt{\text{HOTI}}},$ (48) $\displaystyle
H_{\sqrt{\text{res},2}}$ $\displaystyle=$ $\displaystyle
c_{2}I_{24}+H_{\sqrt{\text{res},2^{\prime}}},$ (49)

where $c_{2}$ is given in (45), $H_{\sqrt{\text{HOTI}}}$ is the Hamiltonian of
the $\sqrt{\text{HOTI}}$ given in (21), while
$H_{\sqrt{\text{res},2^{\prime}}}$ is the Hamiltonian of a residual lattice
involving the sites in the gray and green sublattice of Fig. 6, which has the
same spectrum as the $\sqrt{\text{HOTI}}$ apart from the extra 12 bands with
$E=-c_{2}$ [zero-energy bands, if the constant shift in (47) is included]
coming from the sublattice imbalance.

The real and imaginary parts of the squared energy spectrum, with a global
downshift of $c_{2}$ given in (45), of the open $\sqrt[4]{\text{HOTI}}$
lattice of Figs. 7(a) and (d) is shown in Figs. 7(b) and (e), respectively.
The spectrum is purely real and coincides with that of the
$\sqrt{\text{HOTI}}$ lattice in Fig. 4(b), as expected, apart from the
degenerate lowest energy curve with $E=-c_{2}$. Interestingly, the imaginary
part of the spectrum vanishes everywhere in Fig. 7(e), even though the the
residual block $H_{\sqrt{\text{res},2^{\prime}}}$ in (47) remains non-
Hermitian for $\sqrt[4]{t_{1x}}<0.5$. It is easy to see that, in this region,
upon squaring the lattice in Fig. 6 one obtains a non-Hermitian $i$ phase
factor in the hopping connecting, e.g., sites 15 and 20, picked up along the
light blue $\sqrt[4]{v_{2x}}$ (recall that $H_{\sqrt{\text{res},2^{\prime}}}$
comprises the sites in the gray and green sublattice). On the other hand, we
know that the relevant blue sublattice remains Hermitian upon squaring, since
the non-Hermitian $\sqrt[4]{v_{2x}}$ couplings, for $\sqrt[4]{t_{1x}}<0.5$,
translate by design as an extra real and negative on-site contribution at
sites 6 and 10 in Fig. 6. Since, as shown in Appendix A,
$H_{\sqrt{\text{par},2}}$ shares the same finite energy spectrum as
$H_{\sqrt{\text{res},2}}$, with the extra zero-energy states coming from
sublattice imbalance, it follows that $H_{\sqrt{\text{res},2}}$ displays a
purely real spectrum even within the non-Hermitian region. This constitutes a
new mechanism that should be distinguished from that reported in recent
studies of square-root topology in non-Hermitian models [6, 7, 9], where the
same $\mathcal{PT}$-symmetry broken region with complex energy spectrum is
found in both the original and squared models.

Finally, by squaring the energy downshifted versions of
$H^{2}_{\sqrt[4]{\text{HOTI}}}$ in (47), and subsequently downshifting by
$c_{1}=t_{1x}+t_{2x}+t_{1y}+t_{2y}$ (the resulting on-site energy at the
relevant HOTI sublattice), one arrives at

$\displaystyle H^{4}_{\sqrt[4]{\text{HOTI}}^{\prime}}(\mathbf{k})$
$\displaystyle:=$
$\displaystyle\big{(}H^{2}_{\sqrt[4]{\text{HOTI}}}-c_{2}I_{36}\big{)}^{2}-c_{1}I_{36}$
(50) $\displaystyle=$ $\displaystyle\begin{pmatrix}H_{\text{par}}&0&0\\\
0&H_{\text{res}}&0\\\ 0&0&H_{\text{res},2}\end{pmatrix}-c_{1}I_{36},$
$\displaystyle=$ $\displaystyle\begin{pmatrix}H_{\text{HOTI}}&0&0\\\
0&H_{\text{res}^{\prime}}&0\\\ 0&0&H_{\text{res,2}^{\prime}}\end{pmatrix},$
$\displaystyle H_{\text{res},2}$ $\displaystyle=$ $\displaystyle
c_{1}I_{24}+H_{\text{res,2}^{\prime}},$ (51)

with $H_{\text{par}}$ and $H_{\text{res}}$ given, respectively, in (33) and
(34). The real and imaginary parts of the energy spectrum of
$H^{4}_{\sqrt[4]{\text{HOTI}}^{\prime}}$, coming from the open
$\sqrt[4]{\text{HOTI}}$ lattice of Figs. 7(a) and (d), is shown in Figs. 7(c)
and (f), respectively. Even though the decoupled lattice corresponding to
$H_{\text{res,2}}=H_{\sqrt{\text{res},2^{\prime}}}^{2}$ still has non-
Hermitian terms for $\sqrt[4]{t_{1x}}<0.5$, the energy spectrum is again
purely real, which can be demonstrated by applying the same reasoning as above
for the squared spectrum in Fig. 7(b). As indicated by the arrows, the
degenerate zero-energy curve in Fig. 7(a), originated from sublattice
imbalance in $H_{\sqrt[4]{\text{HOTI}}}$, becomes the $E=-c_{2}$ curve in Fig.
7(b) and the $E=c_{2}^{2}-c_{1}$ curve in Fig. 7(c). The degenerate zero-
energy curve in Fig. 7(b), originated both from the sublattice imbalance
within $H_{\sqrt{\text{HOTI}}}$ in (48) and from the degenerate finite energy
spectrum of $H_{\sqrt{\text{par,2}}}$ in (48) and $H_{\sqrt{\text{res,2}}}$ in
(49), becomes the $E=-c_{1}$ curve in Fig. 7(c), which delimits the lowest
bulk continuum from below.

Aside from these extra bands that are present due to successive sublattice
imbalances and degeneracies, the energy spectrum in Fig. 7(c) corresponds to a
four-fold degenerate spectrum of the HOTI lattice in Fig. 2(b), with one
degeneracy coming from $H_{\text{res}^{\prime}}$ and two others from
$H_{\text{res,2}^{\prime}}$ in (50). In particular, this four-fold degeneracy
affects the 1D edge states, both the vertical ones at the bottom edge and the
horizontal ones at the right edge (see their location in Fig. 5), and the 0D
higher-order states at the right bottom corner. Note, however, that only one
these states in each fourfold-degenerate group is topological in nature,
coming from $H_{\text{HOTI}}$, while the other three are impurity states, one
coming from $H_{\text{res}^{\prime}}$ and two from
$H_{\text{res,2}^{\prime}}$. Generalizing what was shown in the previous
section for the $\sqrt{\text{HOTI}}$ lattice, we can say that each boundary
mode of the starting $\sqrt[4]{\text{HOTI}}$ lattice in Fig. 7(a) can be
written as a linear combination of the corresponding four degenerate states in
Fig. 7(c), all with a quarter of their weight on the topological state within
the group. As such, both the weak edge and the higher-order corner states of
the $\sqrt[4]{\text{HOTI}}$ lattice can be said to be a “quarter” topological
and “three quarters” impurity, with their topological features directly
inherited from the $\ket{\psi^{\text{HOTI}}_{j}}$ component. Thus, the
$\sqrt[4]{\text{HOTI}}$ model with the unit cell of Fig. 6 constitutes an
example of a quartic-root (or 4-root) weak and higher-order topological
insulator.

It is clear at this point that this method can be repeated to find the 8-root
HOTI, 16-root HOTI, etc. As the root-degree increases so will the size of the
unit cell. Let us summarize the process for the construction of the
$2^{n}$-root HOTI, with $n\in\mathbb{N}$:

(i) One starts by identifying all sites in the $2^{n-1}$-root model as the
relevant sublattice of the $2^{n}$-root model (which we depict in blue as
exemplified in Fig. 5, where the blue sublattice in the $\sqrt{\text{HOTI}}$
corresponds to the sites of the HOTI).

(ii) One subdivides the lattice [6] by introducing an extra site (depicted
here in gray, as shown in for the $\sqrt{\text{HOTI}}$ of Fig. 4) in the
middle of every hopping link such that it splits in two, each connecting the
new site to one of the two adjacent blue sites. Together with the assumption
that the original HOTI has no mass terms, that is, no modulation on its on-
site energies, this step ensures that the $\sqrt[2^{n}]{\text{HOTI}}$ under
construction is bipartite, such that its squared Hamiltonian can be written in
a block diagonal form.

(iii) The phases and magnitudes of the new hopping parameters are renormalized
as
$\sqrt[2^{n-1}]{t_{ij}}e^{i\phi_{ij}}\to\sqrt[2^{n}]{t_{ij}}e^{i\phi_{ij}/2}$,
which keeps the same flux pattern of the $\sqrt[2^{n-1}]{\text{HOTI}}$ (no
phases have been introduced so far, but they will appear in some of the models
studied below).

(iv) One checks the squared on-sites energies of the different types of sites
in the blue sublattice, based on the number and magnitudes of their hopping
connections. If there is at least one discrepancy, one sets the squared on-
site energy of one of the types of blue sites as the uniform level and
compensates for the difference in the other types by including extra sites
(depicted here in green, as in Fig. 6) connected to them by appropriately
tuned hopping terms. Note that within the global constraint of having to
compensate for a fixed value of the energy difference, the number of extra
sites included and the amplitudes of the new hopping terms are degrees of
freedom of the method. This step is skipped whenever it is not necessary, that
is, whenever there is already a uniform squared on-site energy for all blue
sites. Typically we set the type of blue sites with highest squared on-site
energy for the entire parameter range, if there is one, as the constant energy
level, in order to avoid the inclusion of purely imaginary non-Hermitian
hoppings to the other blue sites (to shift down their higher squared on-site
energy). However, even if non-Hermitian terms are to be included, the
successive squared spectra remain purely real, due the shared finite spectrum
of the parent block containing the lower-root degree Hamiltonian (which is
real by default) and the residual block at any given level after each squaring
of the Hamiltonian (see Appendix A).

(v) Under OBC and for $n>1$ (above the square-root model), blue sites along
specific edges and corners have fewer connections than their bulk
counterparts, which leads to a negative squared on-site energy offset at these
sites, relative to the bulk ones. Extra sites from incomplete unit cells,
placed along the directions with a connectivity deficit, must be included in
the lattice. For the open $\sqrt[2^{n}]{\text{HOTI}}$ lattice, with $n>1$, the
number of included extra sites is such that it must guarantee that, after
$n-1$ successive squaring operations to the Hamiltonian followed by a
downshift of the $c_{2^{n-1}}$ term of the respective level, one arrives at
the $\sqrt{\text{HOTI}}$ as one of the diagonal blocks with uniform on-site
energy $c_{2}$.

We point out that, even though the method outlined above for constructing
$2^{n}$-root models is not unique, the required criteria underlying it are,
namely, (i) bipartition of the higher root-degree lattice, (ii) ensuring the
same flux pattern upon squaring the model, and (iii) ensuring a uniform
squared on-site energy at the relevant blue sublattice. As an example, we
refer the reader to [45], where a different approach was followed for the
construction of the square-root version of the 2D SSH model with NNN hoppings
which, however, still obeys criteria (i)-(iii).

##  V Quartic-root Chern Insulator

Here, we apply the method outlined at the end of the last section in the
construction of the 4-root version of a well-known Chern insulator (CI),
namely, the Haldane model [46], with the change in notation HOTI$\to$CI.

###  V.1 Original CI model

We take our original CI to be the Haldane model without the mass term (zero
on-site energy for both sites within a unit cell) depicted in Fig. 8(c). The
complex next-nearest-neighbor (NNN) hoppings break time-reversal symmetry,
lifting the degeneracy at the Dirac points in the bulk spectrum and opening a
gap between the two bands given by

$\Delta E=6\sqrt{3}\lambda\sin\phi.$ (52)

We consider a cylindrical geometry with zigzag edges, with an open system
along $\mathbf{a_{1}}$ and periodic along $\mathbf{a_{2}}$. The momentum
$k_{2}$ along the latter direction is a good quantum number and the energy
spectrum of the Haldane model, our original CI, has the form of Fig. 8(f) for
$t=1$ as the energy unit, $\lambda=\frac{0.2}{3\sqrt{3}}$ and
$\phi=\frac{\pi}{2}$. A chiral pair of red edge bands crossing at zero-energy
appears at the gap between the two bulk bands, which can be related to the
non-trivial Chern number $C=1$ of the lower bulk band.

Figure 8: Plaquette of the (a) quartic-root, (b) square-root, and (c) original
version of the CI, corresponding to the Haldane model without a mass term.
Numbered sites indicate the respective unit cells. Energy spectrum as a
function of $k_{2}$, with
$(t,\lambda,\phi)=(1,\frac{0.2}{3\sqrt{3}},\frac{\pi}{2})$ of: (d)
$H_{\sqrt[4]{\text{CI}}}$, (e)
$H_{\sqrt[4]{\text{CI}}}^{2^{\prime}}=H_{\sqrt[4]{\text{CI}}}^{2}-c_{2}I$,
with $c_{2}$ given below (57), and (f)
$H_{\sqrt[4]{\text{CI}}}^{4^{\prime}}=H_{\sqrt[4]{\text{CI}}}^{2^{\prime}}H_{\sqrt[4]{\text{CI}}}^{2^{\prime}}-c_{1}I$,
with $c_{1}$ given below (55) and $H_{\sqrt[4]{\text{CI}}}$ the Hamiltonian of
the $\sqrt[4]{\text{HOTI}}$ lattice open along $\mathbf{a_{1}}$ and periodic
along $\mathbf{a_{2}}$, with 24 extra sites in incomplete unit cells above the
top edge in $\mathbf{a_{1}}$, given by the 6 blue sites $\\{4,7,8,9,10,11\\}$
plus the 18 gray and green sites connected to them in (a). Blue (red) bands
correspond to bulk (edge) bands.

###  V.2 $\sqrt{\text{CI}}$ model

The $\sqrt{\text{CI}}$ has a unit cell with 11 sites, as shown in Fig. 8(b).
Its bulk Hamiltonian in the ordered $\\{\ket{j,\mathbf{k}}\\}$ basis, with
$j=1,2,\dots,11$, with $\mathbf{k}=(k_{1},k_{2})$ and assuming that
$\mathbf{a_{1}}=a_{1}\hat{e}_{x}$ and $\mathbf{a_{2}}=a_{2}\hat{e}_{y}$ are
orthogonalized by square-shaping the geometry of the model, and setting also
$a_{1}=a_{2}=1$, reads as

$\displaystyle H_{\sqrt{\text{CI}}}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0&h_{\sqrt{\text{CI}}}^{\dagger}\\\
h_{\sqrt{\text{CI}}}&0\end{pmatrix},$ (53) $\displaystyle
h_{\sqrt{\text{CI}}}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\sqrt{t}&\sqrt{t}\\\
0&\sqrt{\lambda}\big{(}e^{-i\frac{\phi}{2}}+e^{-i(k_{2}-k_{1}-\frac{\phi}{2})}\big{)}\\\
\sqrt{\lambda}\big{(}e^{i\frac{\phi}{2}}+e^{-i(\frac{\phi}{2}-k_{2})}\big{)}&0\\\
\sqrt{t}e^{ik_{2}}&\sqrt{t}\\\
\sqrt{\lambda}\big{(}e^{-i\frac{\phi}{2}}+e^{i(\frac{\phi}{2}+k_{1})}\big{)}&0\\\
0&\sqrt{\lambda}\big{(}e^{i\frac{\phi}{2}}+e^{-i(\frac{\phi}{2}-k_{1})}\big{)}\\\
0&\sqrt{\lambda}\big{(}e^{i(\frac{\phi}{2}+k_{1})}+e^{-i(\frac{\phi}{2}+k_{2}-k_{1})}\big{)}\\\
\sqrt{\lambda}\big{(}e^{-i(\frac{\phi}{2}-k_{1})}+e^{i(\frac{\phi}{2}+k_{2})}\big{)}&0\\\
\sqrt{t}e^{ik_{2}}&\sqrt{t}e^{ik_{1}}\end{pmatrix},$ (54)

yielding, when squared,

$H_{\sqrt{\text{CI}}}^{2}(\mathbf{k})=c_{1}I_{11}+\begin{pmatrix}H_{\text{CI}}(\mathbf{k})&0\\\
0&H_{\text{res}^{\prime}}(\mathbf{k})\end{pmatrix},$ (55)

where $c_{1}=3t+6\lambda$ is a constant energy shift,
$H_{\text{CI}}(\mathbf{k})=h_{\sqrt{\text{CI}}}^{\dagger}h_{\sqrt{\text{CI}}}-c_{1}I_{2}$
is the two-band bulk Hamiltonian of the CI model in Fig. 8(c), and
$H_{\text{res}^{\prime}}(\mathbf{k})=h_{\sqrt{\text{CI}}}h_{\sqrt{\text{CI}}}^{\dagger}-c_{1}I_{9}$
is the nine-band bulk Hamiltonian of a residual Hamiltonian built on the gray
sublattice of Fig. 8(b). Notice that step (iv), regarding the inclusion of
extra sites, was skipped for the construction of the $\sqrt{\text{CI}}$, since
both blue sites in the unit cell of Fig. 8(b) already have the same squared
on-site energy $c_{1}$. Also, the $\sqrt{\text{CI}}$ is bipartite and can be
subdivided in a blue and a gray sublattice, even though the CI itself is not
bipartite,

The energy spectrum of the $\sqrt{\text{CI}}$ for the same ribbon geometry as
before, with the same parameters as Fig. 8(f), has the form of Fig. 8(e). Two
symmetric pairs of edge bands are found, with the crossing between the bands
of each pair occurring at $k_{2}=\pi$ and $E=\pm\sqrt{c_{1}}\approx\pm 1.8$.
However, it should be emphasized that we included the set of six extra sites
$\\{4,7,8,9,10,11\\}$ in incomplete unit cells above the top edge already at
this $\sqrt{\text{CI}}$ stage, contrary to what was stated in step (v), which
only applies above square-root topology. These extra sites were included to
enable a direct comparison between Fig. 8(e) and Fig. 3(d) in [6], where they
are implicitly included. Without them, only one edge band would be found for
positive energy and a symmetric one with negative energy, with both lying at
the lower edge. The sevenfold-degenerate zero-energy band is a consequence of
the sublattice imbalance. If one squares this spectrum and applies a general
energy downshift of $c_{1}$ one arrives, apart form the extra degenerate
bands, to the spectrum of Fig. 8(f) with double degeneracy for all bands,
including the red edge bands, summing to a total of four, with two topological
edge bands coming from the CI lattice and two impurity edge bands coming from
the residual lattice.

###  V.3 $\sqrt[4]{\text{CI}}$ model

Starting by identifying all the sites in the $\sqrt{\text{CI}}$ in Fig. 8(b)
as the blue sublattice of the $\sqrt[4]{\text{CI}}$, we can then apply all the
steps towards the construction of this model, which leads to the
$\sqrt[4]{\text{CI}}$ depicted in Fig. 8(a). The size of the unit cell is
considerably enlarged to 38 sites, including nine extra green sites connected
to the blue sublattice by two newly defined hopping terms,

$\displaystyle\sqrt[4]{w}$ $\displaystyle=$
$\displaystyle\sqrt{3\sqrt{t}+4\sqrt{\lambda}},$ (56)
$\displaystyle\sqrt[4]{v}$ $\displaystyle=$
$\displaystyle\sqrt{\sqrt{t}+6\sqrt{\lambda}},$ (57)

introduced in order to ensure a uniform squared on-site energy for all sites
in the blue sublattice, given by $c_{2}=3\sqrt{t}+6\sqrt{\lambda}$.

In Fig. 8(d), we show the energy spectrum of the $\sqrt[4]{\text{CI}}$ in Fig.
8(a), obtained from diagonalization of $H_{\sqrt[4]{\text{CI}}}$ in a ribbon
geometry with 24 extra sites in incomplete unit cells above the top edge,
given by the 6 blue sites $\\{4,7,8,9,10,11\\}$ plus the 18 gray and green
sites connected to them, for a parameter set
$(t,\lambda,\phi)=(1,\frac{0.2}{3\sqrt{3}},\frac{\pi}{2})$. Four edge bands
are found, with the crossing between the bands of each pair occurring at
$k_{2}=\pi$ and $E=\pm\sqrt{c_{2}\pm\sqrt{c_{1}}}$. The zero-energy 16-fold-
degenerate bulk band appears due to sublattice imbalance.

In Fig. 8(e), we show the energy spectrum obtained from diagonalization of
$H^{2^{\prime}}_{\sqrt[4]{\text{CI}}}=H^{2}_{\sqrt[4]{\text{CI}}}-c_{2}I=\text{diag}(H_{\sqrt{\text{CI}}},H_{\sqrt{\text{Res},2^{\prime}}})$,
with $I$ the identity matrix of $\dim(I)=\dim(H_{\sqrt[4]{\text{CI}}})$ and
$H_{\sqrt{\text{Res},2^{\prime}}}$ the square-root of the residual lattice
built on the gray plus green sublattice of Fig. 8(a), with impurity sites
along the top and bottom edges. The global spectrum is a superposition of two
identical spectra, one corresponding to each of the diagonal blocks. In
particular, each red edge band is therefore doubly-degenerate, with one
stemming from the $\sqrt{\text{CI}}$ and the other from the square-root
residual block. The zero-energy states in Fig. 8(d) become the $E=-c_{2}$
states of $H_{\sqrt{\text{Res},2^{\prime}}}$, which fall below the energy
range of Fig. 8(e).

The energy spectrum obtained from diagonalization of

$\displaystyle H^{4^{\prime}}_{\sqrt[4]{\text{CI}}}$ $\displaystyle=$
$\displaystyle
H^{2^{\prime}}_{\sqrt[4]{\text{CI}}}H^{2^{\prime}}_{\sqrt[4]{\text{CI}}}-c_{1}I$
(58) $\displaystyle=$ $\displaystyle\begin{pmatrix}H_{\text{CI}}&&\\\
&H_{\text{Res}^{\prime}}&\\\ &&H_{\text{Res},2^{\prime}}\end{pmatrix}$

is shown in Fig. 8(f), apart from the extra states with energy $E=-c_{1}$
($E=c_{2}^{2}-c_{1}$) which fall below (above) the energy range. The global
visible spectrum consists of a superposition of four identical spectra, one
coming from $H_{\text{CI}}$, one from $H_{\text{Res}^{\prime}}$, and two from
$H_{\text{Res},2^{\prime}}$. For each fourfold-degenerate red edge band, only
one is topological in nature, stemming from the Haldane lattice, our original
CI, while the other three are impurity bands stemming from the two successive
residual chains. Furthermore, each edge state in Fig. 8(d) can be written as a
linear combination of the corresponding four degenerate states in Fig. 8(f)
and shown to have exactly one quarter of its weight on the topological state
of this set, defining the model in Fig. 8(a) as a quartic-root CI.

A subsequent application of the method outlined at the end of Sec. IV on the
$\sqrt[4]{\text{CI}}$ would yield the $\sqrt[8]{\text{CI}}$, from which its
even higher root-degree versions $\sqrt[2^{n}]{\text{CI}}$, with $n>3$, could
be successively found. This method could also be applied to find the higher-
root degree versions of a modulated Haldane model recently studied [47]
(provided the mass term is set to zero), which was shown to have higher-order
boundary modes for a region in parameter space.

##  VI Quartic-root breathing kagome lattice

Figure 9: (a) Schematic illustration of the result of squaring the
$\sqrt{\text{BK}}$ triangular lattice (and the respective Hamiltonian above),
with the unit cell shown at the lower left. Up to a constant overall energy
shift $c_{1}$, the squared model is composed of two decoupled lattices, one
yielding the BK in the blue sublattice and the other yielding the residual
model in the gray sublattice with edge (corner) impurity sites with an on-site
energy offset of $-t_{2}$ ($-2t_{2}$), in relation to the red bulk sites.
Shaded red (gray) regions indicate the localization of the edge (corner)
states. (b), (c) Energy spectrum of the bulk $\sqrt{\text{BK}}$ model, in
units of $\sqrt{t_{1}}=1$, along the high-symmetry lines delimiting the orange
region of the Brillouin zone depicted at the right for (b)
$\sqrt{t_{2}}=\sqrt{t_{1}}$ and (c) $\sqrt{t_{2}}=1.2$. High-symmetry points
of the Brillouin zone: $\Gamma=(0,0)$, $K=(\frac{4\pi}{3},0)$,
$K^{\prime}=(\frac{2\pi}{3},\frac{2\pi}{\sqrt{3}})$ and
$M=(\pi,\frac{\pi}{\sqrt{3}})$.

In this section, we will derive the quartic-root version of the breathing
kagome (BK) lattice [48, 49, 50, 51, 52, 53, 54]. We will follow the method
outlined at the end of Sec. IV, with the change in notation HOTI$\to$BK, when
finding the $\sqrt[4]{\text{BK}}$ model, while the $\sqrt{\text{BK}}$ model
was taken directly from existing literature, as explained in more detail in
Sec. VI-B.

###  VI.1 Original BK model

Let us consider the BK lattice, which is a kagome lattice with alternating
hopping terms as depicted in the middle of Fig. 9(a). This model has been
shown to host both conventional (edge) and higher-order (corner) boundary
modes for $t_{1}<t_{2}$ located, respectively, around the red and gray shaded
regions. The energy spectrum of this model for a triangular configuration, as
a function of a linear variation of $\sqrt[4]{t_{1}}$, has the form of Fig.
10(c). The spectrum displays two blue continua of bulk states related, in the
corresponding bulk spectrum under PBC, to a single bulk band for the top one
and to two bulk bands for the bottom one (one being a flat band following the
lowest blue curve), two red continua of edge states, with one visible in the
gap between the bulk continua around $E=1$ and the other one buried within the
lower bulk continuum, and a three-fold degenerate zero-energy corner band
(dashed black curve) in the $\sqrt[4]{t_{1}}<1$ region, but gapped only in the
$\sqrt[4]{t_{1}}\lesssim 0.9$ in Fig. 10(c). The corner states disappear above
the simultaneous bulk and edge band gap closing point $t_{1}=t_{2}$ (remember
that the lower continuum of edge states is buried within the lower continuum
of bulk states).

Some studies have identified these corner modes with topological higher-order
states, proposed to be protected either by (i) a generalized chiral symmetry
[51, 55] allowing for perturbations on the hopping terms between sites in
different sublattices, or (ii) a combination of the three-fold rotation
symmetry ($C_{3}$-symmetry) with its three mirror symmetries [49]. The
polarization vector $\mathbf{P}=p_{1}\mathbf{a_{1}}+p_{2}\mathbf{a_{2}}$ is
used as the $\mathbb{Z}_{3}$ topological index of the system [56], since
$p_{1}=p_{2}=0,\frac{e}{3},\frac{2e}{3}\ \mod e$. However, these conclusions
have been challenged by Miert and Ortix in a recent paper [57], where the
authors considered a specific set of perturbations that respected the
symmetries in (i) and (ii) and showed how, nevertheless, the corner states can
be gapped out without closing either the bulk or edge bands. Consequently, the
corner modes were characterized as conventional higher-order states without
topological protection. For the sake of completeness, as well as for
comparison with related works, we will still calculate the quantized
polarization of the $\sqrt{\text{BK}}$ and $\sqrt[4]{\text{BK}}$ lattices
analyzed next. On the other hand, another recent study [58] relates the zero-
energy corner states to the topology of the lower edge band, which displays a
non-trivial Zak phase. Due to this ongoing debate, we refrain from topological
considerations when discussing the different $\sqrt[2^{n}]{\text{BK}}$ models
below.

Figure 10: Energy spectrum, in units of $t_{2}=1$, as a function of a linear
variation of $\sqrt[4]{t_{1}}$, obtained from diagonalization of: (a)
$H_{\sqrt[4]{\text{BK}}}$, the Hamiltonian of the $\sqrt[4]{\text{BK}}$
triangular lattice with the unit cell of Fig. 12(a); (b)
$H_{\sqrt[4]{\text{BK}}}^{2^{\prime}}=H_{\sqrt[4]{\text{BK}}}^{2}-c_{2}I$,
with $c_{2}=3(\sqrt{t_{1}}+\sqrt{t_{2}})$; (c)
$H_{\sqrt[4]{\text{BK}}}^{4^{\prime}}=H_{\sqrt[4]{\text{BK}}}^{2^{\prime}}H_{\sqrt[4]{\text{BK}}}^{2^{\prime}}-c_{1}I$,
with $c_{1}=t_{1}+t_{2}$. The different types of states are indicated in (c).
Green labeled points along the dashed orange vertical line in (a) correspond
to the respective states depicted in Fig. 12(d).

###  VI.2 $\sqrt{\text{BK}}$ model

The $\sqrt{\text{BK}}$ model, shown in a triangular configuration at the left
of Fig. 9(a), describes a decorated honeycomb lattice, also called super-
honeycomb [59, 60, 61], edge-centered honeycomb [62] or honeycomb-kagome
lattice [63, 64], which has been experimentally realized in photonic lattices
[65, 66] and predicted to be the ground state configuration of several
monolayer materials [67, 68, 69, 70, 71, 72, 73, 74, 75, 76]. Note that this
$\sqrt{\text{BK}}$ model was not derived from our proposed scheme outlined at
the end of Sec. IV. Instead, we chose it from comparison with existing
literature [5] and for allowing it also to serve, in the limit of constant
hopping terms, as the square-root version of the topological semimetal
analyzed in the next section. However, one could have chosen to follow our
method to derive the $\sqrt{\text{BK}}$ lattice, yielding a subdivided version
of the BK lattice in the middle of Fig. 9(a), which would not change the
fundamental properties of the system (e.g., it would have the same energy
gaps, the same edge and corner energy levels, etc.).

The bulk energy spectrum of the $\sqrt{\text{BK}}$ model along the high-
symmetry lines is shown in Fig. 9(b) for $\sqrt{t_{1}}=\sqrt{t_{2}}$, where a
semimetallic behavior is found for a Fermi level set at the band touching at
the $K$ point between the top or bottom pairs of bands, and in Fig. 9(c) for
$\sqrt{t_{1}}\neq\sqrt{t_{2}}$, where the degeneracies are lifted at this
point and a gap opens at symmetric energies [5]. In both cases a spin-1 Dirac
cone is found centered at zero-energy around the $\Gamma$ point, reminiscent
of the low-energy behavior of the Lieb [77] and other kagome-type[78]
lattices.

The $\sqrt{\text{BK}}$ model has a three-fold rotation symmetry about the $z$
axis centered at site 5 defined, in the ordered $\\{\ket{j,\mathbf{k}}\\}$
basis, with $j=1,2,\dots,5$, as

$\displaystyle C_{3}:\ $ $\displaystyle R_{3}(\mathbf{k})$ $\displaystyle
H_{\sqrt{\text{BK}}}(\mathbf{k})R_{3}^{-1}(\mathbf{k})=H_{\sqrt{\text{BK}}}\bigg{(}R\Big{(}\frac{2\pi}{3}\Big{)}\mathbf{k}\bigg{)},$
(59) $\displaystyle R_{3}(\mathbf{k})$
$\displaystyle=\begin{pmatrix}0&0&1&0&0\\\ 1&0&0&0&0\\\ 0&1&0&0&0\\\
0&0&0&e^{-i\mathbf{k}\cdot\mathbf{a_{2}}}&0\\\ 0&0&0&0&1\end{pmatrix},$ (60)

with
$R(\varphi)\mathbf{k}=(\cos(\varphi)k_{x}-\sin(\varphi)k_{y},\sin(\varphi)k_{x}+\cos(\varphi)k_{y})$,
from where the polarization at a given energy gap can be determined through
[56]

$e^{-i\frac{2\pi}{e}p}=\prod\limits_{n\in``occ"}\frac{\theta_{n}(K)}{\theta_{n}(K^{\prime})},$
(61)

where $p=p_{1}=p_{2}$ as mentioned above,
$\theta(\mathbf{k})=\bra{u_{n}(\mathbf{k})}R_{3}\ket{u_{n}(\mathbf{k})}$ is
the expectation value of $R_{3}$ on band $n$ and the high-symmetry points are
given by $K=(\frac{4\pi}{3},0)$ and
$K^{\prime}=(\frac{2\pi}{3},\frac{2\pi}{\sqrt{3}})$.

Figure 11: Polarization profile, in units of $e$, along both the
$\mathbf{a_{1}}$ and $\mathbf{a_{2}}$ directions for the lowest bulk band of
the $\sqrt{\text{BK}}$ model (green dots) and of the $\sqrt[4]{\text{BK}}$
model (red squares).

The polarization profile as a function of $t_{1}$, considering that only the
lowest energy band is filled, is represented by the green dots in Fig. 11. For
$t_{1}<1$ the polarization is quantized to $p=\frac{e}{3}$, at $t_{1}=t_{2}$
we have the gap closing point and for $t_{1}>1$ the polarization is trivial.

As schematized in Fig. 9(a), the $\sqrt{\text{BK}}$ model, when squared and
downshifted by $c_{1}=t_{1}+t_{2}$ (squared on-site energy at the blue
sublattice), decouples into the BK lattice and the honeycomb lattice with
alternating on-site potentials, which are perturbed along the edges, with an
on-site energy offset of $-t_{2}$, and at the corner sites, with an on-site
energy offset of $-2t_{2}$, in both cases relative to the bulk red sites.
These impurities are the origin of the corner and edge modes of this residual
lattice, which are degenerate with the respective ones from the BK lattice.
The energy spectrum of the $\sqrt{\text{BK}}$ model, as a function of a linear
variation of $\sqrt[4]{t_{1}}$, has the form of Fig. 10(b), where it can be
seen that in the $t_{1}<1$ region there are two three-fold degenerate bands of
corner-states with finite and symmetric energies.

###  VI.3 $\sqrt[4]{\text{BK}}$ model

Figure 12: (a) Unit cell of the $\sqrt[4]{\text{BK}}$. Blue sites form one
sublattice and the green and gray sites form another. (b)-(c) Energy spectrum
of the bulk $\sqrt[4]{\text{BK}}$ model, in units of $\sqrt{t_{1}}=1$, along
the high-symmetry lines delimiting the orange region of the Brillouin zone
depicted in Fig. 9, for the same parameters as the corresponding cases there.
The four gaps closings at the Dirac $K$-points in (b) are opened in (c). (d)
Profile of the lower left corner states highlighted in Fig. 10(a) of the
$\sqrt[4]{\text{BK}}$ triangular lattice, 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. Each state has two
others degenerate in energy lying at the top and lower-right corners. The
arrows indicate the site with the same component in all four states and
corresponds, after two squaring and energy downshifting operations, to the
lower-left corner site of the BK lattice depicted in the middle of Fig. 9(a).

The application of the steps for the construction of the $\sqrt[4]{\text{BK}}$
model from the $\sqrt{\text{BK}}$ model leads to a lattice with the 16-sites
unit cell of Fig. 12(a). Note that five extra green sites were included,
connected to blue sites by the newly defined hopping parameters,
$\sqrt[4]{v_{1(2)}}=\sqrt[4]{9t_{1(2)}}$ and
$\sqrt[4]{v_{3}}=\sqrt{2(\sqrt{t_{1}}+\sqrt{t_{2}})}$, in order to ensure a
constant squared on-site energy $c_{2}=3(\sqrt{t_{1}}+\sqrt{t_{2}})$ in all
sites of the blue sublattice.

The bulk energy spectrum along the high-symmetry lines is shown in Fig. 12(b)
for $\sqrt[4]{t_{1}}=\sqrt[4]{t_{2}}$, where a semimetallic behavior is found
when the Fermi level is placed at one of the four band touchings occurring at
the $K$-point, and in Fig. 9(c) for $\sqrt[4]{t_{1}}\neq\sqrt[4]{t_{2}}$,
where the degeneracies are lifted at this point and four extra energy gaps
appear. In both cases asymmetric spin-1 Dirac cones centered at the
$\Gamma$-point and $E=\pm\sqrt{c_{2}}$ appear in the spectrum. Apart from the
six-fold degenerate flat band coming from sublattice imbalance, the squared
spectrum of Figs. 12(b) and (c), downshifted by $c_{2}$, yields the spectra in
Figs. 9(b) and (c), respectively, with a global double degeneracy due to the
diagonalization of $H_{\sqrt{\text{res},2^{\prime}}}(\mathbf{k})$, the bulk
Hamiltonian of the square-root residual block.

The $\sqrt{\text{BK}}$ model has a three-fold rotation symmetry about the $z$
axis defined, in the ordered $\\{\ket{j,\mathbf{k}}\\}$ basis, with
$j=1,2,\dots,16$, as

$\displaystyle C_{3}:\ $ $\displaystyle R_{3}(\mathbf{k})$ $\displaystyle
H_{\sqrt[4]{\text{BK}}}(\mathbf{k})R_{3}^{-1}(\mathbf{k})=H_{\sqrt[4]{\text{BK}}}\bigg{(}R\Big{(}\frac{2\pi}{3}\Big{)}\mathbf{k}\bigg{)},$
(62) $\displaystyle R_{3}(\mathbf{k})$
$\displaystyle=\setcounter{MaxMatrixCols}{16}\begin{pmatrix}0&0&1&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ 0&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&e^{-i\mathbf{k}\cdot\mathbf{a_{2}}}&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&e^{-i\mathbf{k}\cdot\mathbf{a_{2}}}&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&e^{-i\mathbf{k}\cdot\mathbf{a_{2}}}&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&e^{-i\mathbf{k}\cdot\mathbf{a_{2}}}&0&0&0&0&0&0&0\\\
0&0&0&0&0&e^{-i\mathbf{k}\cdot\mathbf{a_{2}}}&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1\\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0\\\
0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0\end{pmatrix},$ (63)

where sites 7 and 15 are assumed here collinear in the $z$-direction with
sites 4 and 5, respectively, within the unit cell of Fig. 12(a). The
polarization profile as a function of $t_{1}$, computed through (61)
considering that only the lowest energy band is occupied, is represented by
the red squares in Fig. 11, where it can be seen that it follows the same
behavior as that of the lowest band of the $\sqrt{\text{BK}}$ model.

The energy spectrum as a function of $\sqrt[4]{t_{1}}$ of the
$\sqrt[4]{\text{BK}}$ triangular lattice with the unit cell of Fig. 12(a),
obtained from the diagonalization of the Hamiltonian $H_{\sqrt[4]{\text{BK}}}$
with a maximum of 10 unit cells aligned along the base, is shown in Fig.
10(a). Four three-fold degenerate dashed black corner bands are found, along
with four continua of red edge bands appearing at different gaps, with four
more continua buried within the blue bulk continua centered around
$E=\pm\sqrt{c_{2}}$ (e.g., centered at $E=\pm\sqrt{3}\approx 1.72$ at the
atomic limit $\sqrt[4]{t_{1}}=0$). The zero-energy states come from the six-
fold degenerate bulk band that appears due to sublattice imbalance.

The energy spectrum obtained from diagonalization of
$H^{2^{\prime}}_{\sqrt[4]{\text{BK}}}=H^{2}_{\sqrt[4]{\text{BK}}}-c_{2}I=\text{diag}(H_{\sqrt{\text{BK}}},H_{\sqrt{\text{Res},2^{\prime}}})$,
with $I$ the identity matrix of $\dim(I)=\dim(H_{\sqrt[4]{\text{BK}}})$ and
$H_{\sqrt{\text{Res},2^{\prime}}}$ the square-root of the residual lattice
built on the gray plus green sublattice of Fig. 12(a), is shown in Fig. 10(b).
The global spectrum is a superposition of two identical spectra, one
corresponding to each of the diagonal blocks, such that, e.g., each of the two
dashed black curves of corner states is now six-fold degenerate. The zero-
energy states in Fig. 10(a) become the $E=-c_{2}$ states of
$H_{\sqrt{\text{Res},2^{\prime}}}$, which fall below the energy range of Fig.
10(b), except for a small segment near the lower left corner for
$\sqrt[4]{t_{1}}\lesssim 0.15$.

The energy spectrum obtained from diagonalization of

$\displaystyle H^{4^{\prime}}_{\sqrt[4]{\text{BK}}}$ $\displaystyle=$
$\displaystyle
H^{2^{\prime}}_{\sqrt[4]{\text{BK}}}H^{2^{\prime}}_{\sqrt[4]{\text{BK}}}-c_{1}I$
(64) $\displaystyle=$ $\displaystyle\begin{pmatrix}H_{\text{BK}}&&\\\
&H_{\text{Res}^{\prime}}&\\\ &&H_{\text{Res},2^{\prime}}\end{pmatrix}$

is shown in Fig. 8(f), apart from the extra states with energy $E=-c_{1}$
($E=c_{2}^{2}-c_{1}$) which fall below (above) the energy range. The global
visible spectrum consists of a superposition of four identical spectra, one
coming from $H_{\text{BK}}$ [the middle lattice in Fig. 12(a)], one from
$H_{\text{Res}^{\prime}}$ [the right lattice in Fig. 12(a)] and two from
$H_{\text{Res},2^{\prime}}$.

The four labeled lower-left corner states along the $\sqrt[4]{t_{1}}$ line in
Fig. 10(a) are represented in Fig. 12(d). For each state, two degenerate
states live on the top and lower-right corners of the triangular lattice. The
site indicated by the arrows corresponds, after two squaring and energy
downshifting operations, to the lower-left corner site of the BK lattice
depicted in the middle of Fig. 9(a). The component at this site for all four
states represents a quarter of the weight of the total state, with neglible
contributions from other sites of the BK lattice since we are near the atomic
limit with $t_{1}=0.2^{4}\approx 0$. As such, each corner state in Fig. 12(d)
can be written as a linear combination of the corresponding four degenerate
zero-energy states at the same corner in Fig. 10(c) and shown to have exactly
one quarter of its weight on the BK state of this set, defining the model with
the unit cell of Fig. 12(a) as the quartic-root version of the BK lattice.

##  VII Quartic-root topological semimetal

Figure 13: Energy spectrum, in units of $t$, as a function of $k_{2}$,
obtained from diagonalization of: (d) $H_{\sqrt[4]{\text{TS}}}$, the
Hamiltonian of the $\sqrt[4]{\text{TS}}$ lattice with the unit cell of Fig.
12(a) with $t_{1}=t_{2}=t$, open along $\mathbf{a_{1}}$ and periodic along
$\mathbf{a_{2}}$; (b)
$H_{\sqrt[4]{\text{TS}}}^{2^{\prime}}=H_{\sqrt[4]{\text{TS}}}^{2}-c_{2}I$,
with $c_{2}=6\sqrt{t}$; (f)
$H_{\sqrt[4]{\text{TS}}}^{4^{\prime}}=H_{\sqrt[4]{\text{TS}}}^{2^{\prime}}H_{\sqrt[4]{\text{TS}}}^{2^{\prime}}-c_{1}I$,
with $c_{1}=3t$. Blue (red) bands correspond to bulk (edge) bands.

In this section, we expand $2^{n}$-root topology to semimetallic systems,
using the example of the graphene ribbon as the original topological semimetal
(TS) from which we construct its higher root-degree versions. We will follow
the method outlined at the end of Section IV, with the change in notation
HOTI$\to$TS.

###  VII.1 Original TS model

We will take our original TS to be the honeycomb lattice with a uniform
hopping constant $t=1$ as the energy unit for this whole section. Considering
a cylindrical square-shaped geometry with zigzag edges, where
$\mathbf{a_{1(2)}}=a_{1(2)}\hat{e}_{x(y)}$ and $a_{1}=a_{2}=1$, with an open
system along $\mathbf{a_{1}}$ and periodic along $\mathbf{a_{2}}$, allowing us
to define the momentum $k_{2}$ along the latter direction as a good quantum
number, the energy spectrum our original TS has the form of Fig. 13(c), apart
from the lowest energy flat band states, whose origin will become clear in the
next sections. The most prominent feature of this spectrum is the appearance
of two degenerate flat red bands of topological zero-energy states connecting
the Dirac points.

The appropriate topological invariant for this model is given by the Zak’s
phase along $\mathbf{a_{1}}$ (equivalent to the $x$-direction) of the lower
band, computed at each $k_{2}$-value [32],

$\gamma_{x}(k_{2})=i\oint
dk_{1}\bra{u_{1}(\mathbf{k})}\frac{d}{dk_{1}}\ket{u_{1}(\mathbf{k})}\mod
2\pi,$ (65)

where $\ket{u_{1}(\mathbf{k})}$ is the bulk eigenstate of the lower band. The
result is given by the solid blue dots in Fig. 14, where a non-trivial
$\pi$-value is obtained for the same $k_{2}$-region where the topological
state are seen to lie in Fig. 13(c).

###  VII.2 $\sqrt{\text{TS}}$ model

The $\sqrt{\text{TS}}$ corresponds to the $\sqrt{\text{BK}}$ model with the
unit cell shown at the lower right of Fig. 9(a) in the
$\sqrt{t_{1}}=\sqrt{t_{2}}=\sqrt{t}$ regime [8] since, as shown in Fig. 9(b),
the bulk spectrum along the high symmetry lines has semimetallic behavior when
the Fermi level is set at $E=\pm\sqrt{3}$. Notice that the roles of the
squared blocks are interchanged, which is a reflection of the interchanged
roles of the sublattices of the $\sqrt{\text{TS}}$, that is, gray sites 4 and
5 form now the relevant sublattice from which the the honeycomb lattice is
obtained upon squaring [$H_{\text{res}^{\prime}}(\mathbf{k})\to
H_{\text{TS}}(\mathbf{k})$], while blue sites 1, 2 and 3 form the other
sublattice from which the residual lattice (the kagome lattice) is obtained
upon squaring [$H_{\text{BK}}(\mathbf{k})\to
H_{\text{res}^{\prime}}(\mathbf{k})$]. At the same time, it is now the
constant squared on-site energy at the gray sublattice that defines the new
energy shift, transformed according to $c_{1}=t_{1}+t_{2}\to c_{1}=3t$ (each
gray site has a coordination number of three). The squared bulk Hamiltonian of
the $\sqrt{\text{TS}}$ becomes then

$H_{\sqrt{\text{TS}}}^{2}(\mathbf{k})=c_{1}I_{5}+\begin{pmatrix}H_{\text{TS}}(\mathbf{k})&0\\\
0&H_{\text{res}^{\prime}}(\mathbf{k})\end{pmatrix}.$ (66)

Figure 14: Zak’s phase along $\mathbf{a_{1}}=\hat{e}_{x}$ as a function of
$k_{2}$ for the lowest bulk band of the TS model (solid blue dots) and of the
$\sqrt{\text{TS}}$ model (open red dots).

Considering the $\sqrt{\text{TS}}$ in the same cylindrical geometry as in the
previous subsection, its energy spectrum has the form of Fig. 13(b). Two two-
fold degenerate flat bands of edge states with symmetric finite energies are
found, connecting the pairs of Dirac points appearing at
$E=\pm\sqrt{3}=\pm\sqrt{c_{1}}$. However, if we compute the Zak’s phase of the
lower bulk band along $x$ as a function of $k_{2}$, through (65), we obtain
the curves of open red points in Fig. 14, which are not $\pi$-quantized
neither in the trivial nor in the non-trivial $k_{2}$-regions. This model (the
super-honeycomb with uniform hoppings) is thus defined as a
$\sqrt{\text{TS}}$, since a quantized topological index is only recovered at
the level of the topological block of the squared model, i.e.,
$H_{\text{TS}}(\mathbf{k})$ in (66).

###  VII.3 $\sqrt[4]{\text{TS}}$ model

The unit cell of the $\sqrt[4]{\text{TS}}$ model corresponds to the one
depicted in Fig. 12(a) with $t_{1}=t_{2}=t$. As shown in Fig. 12(b), the bulk
spectrum along the high symmetry lines has semimetallic behavior when the
Fermi level is set at either of the four energies where a band touching at the
$K$-point occurs. The squared bulk Hamiltonian of the $\sqrt[4]{\text{TS}}$ is
written as

$H_{\sqrt[4]{\text{TS}}}^{2}(\mathbf{k})=c_{2}I_{16}+\begin{pmatrix}H_{\sqrt{\text{TS}}}(\mathbf{k})&0\\\
0&H_{\sqrt{\text{res},2^{\prime}}}(\mathbf{k})\end{pmatrix},$ (67)

with $c_{2}=6\sqrt{t}$ 333In order to avoid extra definitions, we derived the
unit cell of the $\sqrt[4]{\text{TS}}$ as that of Fig. 12(a) in the
$t_{1}=t_{2}=t$ limit, which leads to a squared on-site energy shift of
$c_{2}=6\sqrt{t}$. However, the model could be simplified by suppressing sites
7 and 15 (along with their hopping connections $\sqrt[4]{v_{1}}$ and
$\sqrt[4]{v_{2}}$) and redefining $\sqrt[4]{v_{3}}=\sqrt[4]{t}$. In this case,
the only changes are trivial and consist of having $c_{2}=3\sqrt{t}$ and two
fewer zero-energy flat bands in the bulk spectrum due to a lower imbalance
between the residual and relevant sublattices, as sites 7 and 15 are removed
from the former..

Considering the $\sqrt[4]{\text{TS}}$ in the same cylindrical geometry as in
the previous subsections, its energy spectrum is shown in Fig. 13(a). Four
two-fold degenerate flat bands of edge states with symmetric finite energies
are found, connecting the pairs of Dirac points appearing at
$E=\pm\sqrt{c_{2}\pm\sqrt{c_{1}}}$. The energy spectrum obtained from
diagonalization of
$H^{2^{\prime}}_{\sqrt[4]{\text{TS}}}=H^{2}_{\sqrt[4]{\text{TS}}}-c_{2}I=\text{diag}(H_{\sqrt{\text{TS}}},H_{\sqrt{\text{Res},2^{\prime}}})$,
with $I$ the identity matrix of $\dim(I)=\dim(H_{\sqrt[4]{\text{TS}}})$ and
$H_{\sqrt{\text{Res},2^{\prime}}}$ the square-root of the residual lattice, is
shown in Fig. 13(b). The global spectrum is a superposition of two identical
spectra, one corresponding to each of the diagonal blocks, such that, e.g.,
each topological flat band is now four-fold degenerate. The zero-energy states
in Fig. 13(a) become the $E=-c_{2}$ states of
$H_{\sqrt{\text{Res},2^{\prime}}}$, which fall below the energy range of Fig.
13(b).

The energy spectrum obtained from diagonalization of

$\displaystyle H^{4^{\prime}}_{\sqrt[4]{\text{TS}}}$ $\displaystyle=$
$\displaystyle
H^{2^{\prime}}_{\sqrt[4]{\text{TS}}}H^{2^{\prime}}_{\sqrt[4]{\text{TS}}}-c_{1}I$
(68) $\displaystyle=$ $\displaystyle\begin{pmatrix}H_{\text{TS}}&&\\\
&H_{\text{Res}^{\prime}}&\\\ &&H_{\text{Res},2^{\prime}}\end{pmatrix}$

is shown in Fig. 13(f), apart from the extra states with energy
$E=c_{2}^{2}-c_{1}$ which fall above the energy range. Apart from the states
in the lowest energy flat band with $E=-c_{1}$, which stems from sublattice
imbalance at the higher root-degree levels of the model, the global visible
spectrum consists of a superposition of four identical spectra, one coming
from $H_{\text{TS}}$, one from $H_{\text{Res}^{\prime}}$ and two from
$H_{\text{Res},2^{\prime}}$. There are now eight degenerate zero-energy
topological flat bands, with only two of them coming from the topological
block $H_{\text{TS}}$ and the other six from the two successive residual
chains.

##  VIII Conclusions

A systematic approach to the construction of $2^{n}$-root versions of several
emblematic lattice models was presented here, generalizing to 2D systems the
concept of $2^{n}$-root topology previously introduced for 1D chains [14].
Specifically, we constructed the 4-root version of: (i) the asymmetric 2D SSH
model, shown to exhibit both 4-root weak and higher order topological states;
(ii) the Haldane model with zero mass, behaving as a 4-root Chern insulator;
(iii) the breathing kagome lattice in a triangular configuration, shown to
host weak and higher-order boundary modes, extending the relevance of our
method to non-topological systems also; (iv) the honeycomb lattice, which
behaves as a 4-root topological semimetal in a cylindrical geometry, where
several flat bands of topological states appear connecting different pairs of
Dirac points. The general method for the construction of these models was
outlined at the end of Section IV, such that their $2^{n}$-root versions can
be straightforwardly found from the lower $2^{n-1}$-root version.

Due to their high degree of tunability, artificial lattices are the best
candidates for the experimental realization and topological probing of the
models introduced here. The original and square-root versions of several
models analyzed here have already been implemented in acoustic [80, 81, 55,
11] and photonic [82, 83] lattices, as well as in topoelectrical circuits [84,
85, 86, 10, 87]. This latter platform seems especially suited for implementing
the $2^{n}$-root 2D topological insulators, since the hopping terms of the
tight-binding models can be easily emulated by a judicious choice of coupled
capacitors and inductors [88].

The extension of our work to higher-dimensional models, higher-order
topological superconductors and more general non-Hermitian systems is an open
subject that invites further studies. Another extension would be to explicitly
consider spinful models hosting spin-polarized edge and higher-order boundary
modes [89]. Although the method for constructing the higher-root degree
versions of these systems would remain unchanged, the introduction of spin-
orbit couplings (SOCs) would pose new and interesting questions. Concretely,
if the SOCs are understood as interlayer hoppings between layers of different
spin projections, then the construction of the square-root model will entail,
by virtue of subdivision of the lattice, the inclusion of new sites in the
middle of each SOC, which can be viewed as a new layer of sites related to a
specific spin projection (e.g., a fermionic spin-1/2 original TI with two spin
layers from which a bosonic square-root spin-1 model with three layers can be
constructed). This prospect of having $\sqrt[2^{n}]{\text{TIs}}$ where the
different root-degrees relate to different spin quantum number sectors, and in
particular the possibility of formally deriving a bosonic model from a lower
root-degree fermionic model, or vice-versa, is a very interesting one, and one
that we are currently addressing.

## 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 PTDC/FIS-
MAC/29291/2017. A.M.M. acknowledges financial support from the FCT through the
work Contract No. CDL-CTTRI-147-ARH/2018.

##  Appendix A Degenerate squared energy spectrum

In this appendix we want to show that $H_{\sqrt{\text{par},2}}$ and
$H_{\sqrt{\text{res},2}}$ in (47) share the same finite-energy spectrum. A
similar derivation can be found in the Appendix B of [6]. We start by writing
the $\sqrt[4]{\text{HOTI}}$ bulk Hamiltonian in a manifestly chiral-symmetric
form, by using the site ordering of the unit cell in Fig. 6:

$H_{\sqrt[4]{\text{HOTI}}}(\mathbf{k})=\begin{pmatrix}0&h^{R}\\\
h^{L}&0\end{pmatrix},$ (69)

with $h^{R}$ ($h^{L}$) a $12\times 24$ ($24\times 12$) matrix. Note that no
Hermiticity condition, which would read as $h^{R^{\dagger}}\equiv h^{L}$, is
imposed here. By squaring the Hamiltonian in (69), one can recast (47) as

$\displaystyle H^{2}_{\sqrt[4]{\text{HOTI}}}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\begin{pmatrix}H_{\sqrt{\text{par},2}}&0\\\
0&H_{\sqrt{\text{res},2}}\end{pmatrix},$ (70) $\displaystyle
H_{\sqrt{\text{par},2}}$ $\displaystyle=$ $\displaystyle h^{R}h^{L},$ (71)
$\displaystyle H_{\sqrt{\text{res},2}}$ $\displaystyle=$ $\displaystyle
h^{L}h^{R}.$ (72)

The Schroedinger equation applied to the square-root parent subspace is
written as

$H_{\sqrt{\text{par},2}}\ket{\psi_{\sqrt{\text{par},2}}(k)}=E_{\sqrt{\text{par},2}}(k)\ket{\psi_{\sqrt{\text{par},2}}(k)},$
(73)

where $E_{\sqrt{\text{par},2}}(k)=c_{2}+E_{\sqrt{\text{HOTI}}}(k)\neq 0$ is
assumed. Inserting (71) into (73) yields

$h^{R}h^{L}\ket{\psi_{\sqrt{\text{par},2}}(k)}=E_{\sqrt{\text{par},2}}(k)\ket{\psi_{\sqrt{\text{par},2}}(k)}.$
(74)

Multiplying by $h^{L}$ from the left yields

$h^{L}h^{R}h^{L}\ket{\psi_{\sqrt{\text{par},2}}(k)}=E_{\sqrt{\text{par},2}}(k)h^{L}\ket{\psi_{\sqrt{\text{par},2}}(k)},$
(75)

which, after defining
$h^{L}\ket{\psi_{\sqrt{\text{par},2}}(k)}:=\ket{\psi_{\sqrt{\text{res},2}}(k)}$
and using (72), leads to

$H_{\sqrt{\text{res},2}}\ket{\psi_{\sqrt{\text{res},2}}(k)}=E_{\sqrt{\text{par},2}}(k)\ket{\psi_{\sqrt{\text{res},2}}(k)},$
(76)

thus showing that $E_{\sqrt{\text{res},2}}(k)=E_{\sqrt{\text{par},2}}(k)$ with
regards to the finite energy spectrum.

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