<EMAIL_ADDRESS>
# Tunable zero modes and quantum interferences in flat-band topological
insulators

Juan Zurita  Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco,
E-28049 Madrid, Spain  Universidad Complutense de Madrid, E-28040 Madrid,
Spain  Charles Creffield  Universidad Complutense de Madrid, E-28040 Madrid,
Spain  Gloria Platero  [ Instituto de Ciencia de Materiales de Madrid (CSIC),
Cantoblanco, E-28049 Madrid, Spain

###### Abstract

We investigate the interplay between Aharonov-Bohm (AB) caging and topological
protection in a family of quasi-one-dimensional topological insulators, which
we term CSSH ladders. Hybrids of the Creutz ladder and the SSH chain, they
present a regime with completely flat bands, and a rich topological phase
diagram, with several kinds of protected zero modes. These are reminiscent of
the Creutz ladder edge states in some cases, and of the SSH chain edge states
in others. Furthermore, their high degree of tunability, and the fact that
they remain topologically protected even in small systems in the rungless
case, due to AB caging, make them suitable for quantum information purposes.
One of the ladders can belong to the BDI, AIII and D symmetry classes
depending on its parameters, the latter being unusual in a non-superconducting
model. Two of the models can also harbor topological end modes which do not
follow the usual bulk-boundary correspondence, and are instead related to a
Chern number. Finally, we propose some experimental setups to implement the
CSSH ladders with current technology, focusing on the photonic lattice case.

##  1 Introduction

Topological materials have a gapped bulk, which can be insulating or
superconducting, and a metallic surface spectrum. For one-dimensional (1D)
materials, this implies the existence of topological zero modes, robust end
states pinned at zero energy and protected by symmetries of the Hamiltonian.
Depending on these symmetries, topological materials can be classified into
ten topological phases, five of which can show topological phases in 1D [1].

The robustness of their topological end modes can serve as a way to prevent
errors in quantum information protocols. For instance, the well-known SSH
chain end modes can be used as a way to implement a robust quantum state
transfer protocol [2, 3, 4, 5, 6], or a quantum memory resistant to off-
diagonal disorder [5]. Ladder models are quasi-1D lattice models that consist
of two chains of sites coupled by tunneling amplitudes or interactions. By
tuning the different parameters in ladder models, several BDI [7, 8, 9, 10],
AIII [11, 12, 10, 13, 14] and CII [15, 16] topological insulators (TIs) have
been found. The best-known topological ladder model is probably the Creutz
ladder [17], a BDI topological insulator that also shows flat bands in a
certain regime, where particles experience a self-localization phenomenon
termed “Aharonov-Bohm (AB) caging” due to a magnetic flux. This effect has
previously been studied in other 1D and 2D lattices, like the rhombus chain
[18, 19, 20, 21] and the $\mathcal{T}_{3}$ and $\mathcal{T}_{4}$ lattices [22,
23], and has been observed experimentally in several setups [24, 25, 26, 27,
28, 29].

Figure 1: CSSH ladder diagram in the general case. The model is determined by
its four horizontal ($J_{i}$), and four diagonal ($t_{i}$) tunneling
amplitudes, its two vertical tunneling amplitudes $m_{1},m_{2}$ (which we will
consider equal) and the magnetic flux $\phi$. A unit cell is shaded in orange.

Flat-band models have attracted a great deal of research in recent years, in
part because of the discovery of magic-angle twisted bilayer graphene [30].
The Creutz ladder and other quasi-1D flat-band models like the rhombus chain
can serve as toy models to explore the interplay of localization dynamics with
interactions or topology [20, 7, 31, 32, 33, 21]. Other studied variants of
the Creutz ladder include additional Peierls phases or on-site energies [34,
12, 13, 11], Floquet driving [35, 36], spin-$1/2$ particles [37, 35, 16] or
superconductivity [38, 32].

Figure 2: (a) The Creutz ladder, obtained from the general case (Fig. 1) when
$J_{i}=t_{i}=J$ (in dark brown), and $m_{1}=m_{2}$ (in purple). (b) The SSH
chain, with tunneling amplitudes $v$ and $w$. (c) The hourglass CSSH ladder,
with $J_{3}=J_{4}=t_{3}=t_{4}=J$, $J_{1}=J_{2}=t_{1}=t_{2}=J^{\prime}$. The
darker (lighter) links represent $J$-valued ($J^{\prime}$-valued) amplitudes.
(d) The rhomboid CSSH ladder, with $J_{2}=J_{3}=t_{2}=t_{4}=J$,
$J_{1}=J_{4}=t_{1}=t_{3}=J^{\prime}$. (e) The delta CSSH ladder, with
$J_{2}=J_{4}=t_{2}=t_{3}=J$, $J_{1}=J_{3}=t_{1}=t_{4}=J^{\prime}$. (f) The
limit of the rhomboid CSSH ladder when $J^{\prime}=0$ is the orthogonal dimer
chain, which is also a topological insulator.

In this work, we propose and investigate a family of quasi-1D topological
insulators related to the Creutz ladder and the SSH chain, which we term
Creutz-Su-Schrieffer-Heeger (CSSH) ladders [see Figs. 1 and 2]. They present a
regime where all four bands are flat due to AB caging. This phenomenon also
localizes the edge states in one or two sites whenever the vertical links of
the ladders vanish. This prevents the overlap of the left and right states,
and allows them to be degenerate at zero energy even in small systems. This
unusual interplay between quantum interference and topology, inherited from
the Creutz ladder, enhances the topological protection of the states, which is
preserved even in the one-plaquette limit of the ladders. A remarkable result
of our work is that, in the CSSH ladders, the wavefunction of these end states
can be tuned, using the parameters of the model, to take the form of almost
any state in the Bloch sphere spanned by the two end sites. This can be used
to prepare the Creutz ladder end states starting from a particle in a single
site, or to control a particle in a topologically protected way even in small
systems, something that can be useful for quantum information purposes.

Interestingly, one of the models, the hourglass CSSH ladder, can harbor
topological zero modes reminiscent of both the Creutz ladder and the SSH chain
end states. Another one of them, the delta CSSH ladder, has topological phases
in the BDI, AIII and D classes, such that these three classes can be explored
by changing the parameters of the system, without closing the gap or losing
the protection of the states. This is something remarkable, because models in
the D class in 1D are usually topological superconductors [39, 40]. In the
rhomboid and delta CSSH ladders, we also report the existence non-protected
topological end modes which do not follow the usual bulk-boundary
correspondence. They can instead be related to a Chern number, even though
they appear in a quasi-1D model. These modes are very similar to those found
in the trimer chain [41]. The mentioned features make the CSSH ladders
versatile and exotic models, which can also be realized experimentally in
several bosonic and fermionic platforms with current technology.

In Section 2, we present the Hamiltonian of the CSSH ladders. Then, in section
3, we review the topology of the SSH chain and the Creutz ladder, including an
original full description of the symmetries of the latter. In Section 4, we
discuss the topological phase diagrams of the models, their spectra,
symmetries and zero modes. In Section 5 we propose some experimental
implementations of the models, focusing on the photonic lattice case. Finally,
in Section 6, we present our conclusions.

.

##  2 The CSSH ladder models

The Creutz ladder [17, 11, 33], a quasi-1D topological insulator, is formed by
two chains of sites, connected by vertical, horizontal, and diagonal links
[see Fig. 2 (a)]. A magnetic field is applied in the perpendicular direction.
On the other hand, the Su-Schrieffer-Heeger (SSH) chain [42, 43], the simplest
1D TI, consists of a chain of sites connected by alternate hopping amplitudes
[see Fig. 2 (b)]. Both systems are chiral TIs for some values of their
parameters (see Section 3).

In this work, we consider a family of models closely related to the Creutz
ladder, which will also include the alternating hopping amplitudes of the SSH
model. Therefore, we will refer to them as CSSH ladders. To label each site,
we will use the unit cell number $n=1,\ldots,N$ and the internal coordinate
$\alpha=A,B,C,D$, as indicated in Fig. 1. We will also use the nomenclature
$\overline{A}=B$, $\overline{B}=A$, $\overline{C}=D$, $\overline{D}=C$. The
general Hamiltonian for these models is (see Fig. 1):

$\displaystyle\mathcal{H}_{C\\!S\\!S\\!H}=-\sum^{\left\lfloor
L/2\right\rfloor}_{n=1}\left(J_{1}^{+}c^{\dagger}_{n,C}c^{\vphantom{\dagger}}_{n,A}+J_{2}^{-}c^{\dagger}_{n,D}c^{\vphantom{\dagger}}_{n,B}+J_{3}^{+}c^{\dagger}_{n+1,A}c^{\vphantom{\dagger}}_{n,C}+J_{4}^{-}c^{\dagger}_{n+1,B}c^{\vphantom{\dagger}}_{n,D}+\right.$
$\displaystyle\left.+t_{1}c^{\dagger}_{n,D}c^{\vphantom{\dagger}}_{n,A}+t_{2}c^{\dagger}_{n,C}c^{\vphantom{\dagger}}_{n,B}+t_{3}c^{\dagger}_{n+1,B}c^{\vphantom{\dagger}}_{n,C}+t_{4}c^{\dagger}_{n+1,A}c^{\vphantom{\dagger}}_{n,D}+m_{2}c^{\dagger}_{n,C}c^{\vphantom{\dagger}}_{n,D}+h.c.\right)+$
$\displaystyle-\sum^{\left\lceil
L/2\right\rceil}_{n=1}\left(m_{1}c^{\dagger}_{n,A}c^{\vphantom{\dagger}}_{n,B}+h.c.\right),$
(1)

where $\lfloor\cdot\rfloor$ and $\lceil\cdot\rceil$ are the floor and ceiling
functions, and $J_{i}^{\pm}=J_{i}e^{\pm i\phi/2}$ for $i=1,2,3,4$. The
operator $c^{(\dagger)}_{n,\alpha}$ destroys (creates) a particle in site
$n,\alpha$. Our work is applicable to both bosons and spinless (or spin-
polarized) fermions. The upper limits of the sums have been chosen to take
into account the possibility of a finite system with an odd length $L$, in
which the last unit cell is not complete. When not explicitly noted otherwise,
we will assume that $L$ is even, and then $\left\lfloor
L/2\right\rfloor=\left\lceil L/2\right\rceil=N$, the total number of unit
cells. We consider the same phase shift $\phi$ in every horizontal link,
implementing a magnetic flux of $\Phi=\hbar\phi/q$ through each square
plaquette, where $q$ is the charge of the particle.

We will also fix the vertical hopping amplitudes as equal, $m_{1}=m_{2}=m$.
This leaves 10 degrees of freedom, $\\{J_{i},t_{i},\phi,m\\}_{i=1}^{4}$. Under
periodic boundary conditions, this Hamiltonian can be rewritten as
$\mathcal{H}_{C\\!S\\!S\\!H}=\sum_{k\in
BZ}\ket{k}\bra{k}\otimes\mathcal{H}(k)$, where the first operator acts on the
spatial coordinate (that is, the unit cell number $n$), while the $4\times 4$
matrix $\mathcal{H}(k)$ acts on the internal degree of freedom $\alpha$. The
quasimomentum $k$ takes values along the first Brillouin zone. The bulk
Hamiltonian has the form:

$\mathcal{H}(k)=-\begin{pmatrix}0&m&J_{1}^{-}+J_{3}^{+}e^{-ik}&t_{1}+t_{4}e^{-ik}\\\
m&0&t_{2}+t_{3}e^{-ik}&J_{2}^{+}+J_{4}^{-}e^{-ik}\\\
J_{1}^{+}+J_{3}^{-}e^{ik}&t_{2}+t_{3}e^{ik}&0&m\\\
t_{1}+t_{4}e^{ik}&J_{2}^{-}+J_{4}^{+}e^{ik}&m&0\end{pmatrix},$ (2)

and will allow us to probe the topology of the systems in Section 4.

Out of all possible systems, we analyze particular cases in which the
horizontal and diagonal hoppings, $J_{i}$ and $t_{i}$, take two different
values, $J$ and $J^{\prime}$, in the spirit of the SSH model. In addition, we
will focus our attention on cases where all energy bands are non-dispersive
due to AB caging when $\phi=\pi$ and $m=0$. The CSSH models will have, in
general, 4 different bands, given that their unit cell consists of 4 sites. In
addition to the Creutz ladder [Fig. 2 (a)], where $J_{i}=t_{i}=J$ $\forall i$,
three different kinds of CSSH ladders that exhibit this behavior are:

  * •

The hourglass
($\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{1.5}{$\Join$}}}$) CSSH
ladder, with $J_{3}=J_{4}=t_{3}=t_{4}=J$, $J_{1}=J_{2}=t_{1}=t_{2}=J^{\prime}$
[Fig. 2 (c)].

  * •

The rhomboid () CSSH ladder, with $J_{2}=J_{3}=t_{2}=t_{4}=J$,
$J_{1}=J_{4}=t_{1}=t_{3}=J^{\prime}$ [Fig. 2 (d)].

  * •

The delta ($\Delta$) CSSH ladder, with $J_{2}=J_{4}=t_{2}=t_{3}=J$,
$J_{1}=J_{3}=t_{1}=t_{4}=J^{\prime}$ [Fig. 2 (e)].

The name for each kind of ladder comes from the shape formed by the links of
the same value. We will parametrize $J$ and $J^{\prime}$ in the following way:

$J=\xi\sin^{2}\theta;\hskip 20.0ptJ^{\prime}=\xi\cos^{2}\theta,$ (3)

with $0\leq\theta<\pi/2$. Thus, the Hamiltonian for any of the CSSH ladders is
completely determined by the hopping amplitudes $\xi=J+J^{\prime}$ and $m$,
the angle $\theta$ and the magnetic flux phase $\phi$. Unless specified
otherwise111Some properties of the models, like the winding number or the
chirality of the zero modes, are periodic in $\phi$ with period $4\pi$, not
$2\pi$. This is related to the fact that the smallest closed loop in the
$m\neq 0$ model is a triangle with a flux of $\phi/2$. However, most features
explored in this work have a period of $2\pi$., all values of $\phi$ are to be
taken modulo $2\pi$. From this point on, we set $\hbar=1$ and we consider
$\xi=1$, which sets the energy scale of the system but has no other effect in
the dynamics. As we will see below, the CSSH ladders inherit some features
from the Creutz ladder, and some others from the SSH chain.

These three models have not been previously studied in the literature,
although they bear some relation to other quasi-1D insulators. The -CSSH
ladder, when either $J$ or $J^{\prime}$ are zero, reduces to the orthogonal
dimer chain [Fig. 2 (f)], a topological model that is usually studied as a
chain of interacting spins [44, 45, 46, 47, 48, 49, 50, 51, 52]. The -CSSH
ladder model connects the topological phases of the orthogonal dimer chain and
of the $\pi$-flux Creutz ladder in an adiabatic way, without losing the
topological protection of the edge states at any point222In this regard, this
model is similar to the $\alpha$-$\mathcal{T}_{3}$ lattice, a non-topological
2D model with a flat band and two different tunneling amplitudes, which
interpolates between the honeycomb lattice and the $\mathcal{T}_{3}$ lattice
[116, 117, 118]..

The spin-$1/2$ SSH chain with spin-orbit interaction [53, 54, 55, 56] is
similar to the hourglass CSSH ladder in a certain regime, except for the
distribution of magnetic fluxes. The hourglass-CSSH ladder with $\phi=0$, in
its trivial phase, also has the same geometry as certain
$\text{Cu}_{2}\text{Te}_{2}\text{O}_{5}\text{X}_{2}$ compounds with X$=$Cl, Br
[57]; and has been analyzed as a chain of interacting spins [58, 57]. Other
models similar to the CSSH ladders, but with some missing tunneling
amplitudes, have been studied, like the SSH ladder [59, 60, 9, 61, 62], the
SSH bowtie ladder [10, 63] or the chequerboard chain [64]. A thorough
exploration of other topological ladders can be found in [10].

##  3 Topology in simple models

Figure 3: Topological phase diagrams (TPDs). Some figures are striped to show
the value of two invariants, which are
$[\tilde{\mathcal{Z}}_{1},\tilde{\mathcal{Z}}_{2}]$ unless specified
otherwise. (a) TPD for all models in the Creutz limit, $\theta=\pi/4$,
represented by its Zak phase $\mathcal{Z}$, which takes the values of 0
(trivial) or $\pi$ (topological). Overlapped on the diagram, we mark the three
regimes in the $(\phi,m)$ plane where the end states can be protected, for at
least some value of $\theta$: $\phi=\pi$ (orange) and $m=0$ (yellow) in all
systems, and $\phi=0$ in the
$\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{1.5}{$\Join$}}}$-CSSH case
(blue). For other values of $\phi$, the Zak phase is quantized but the states
are not protected if $m>0$. (b)
$\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{1.5}{$\Join$}}}$-CSSH ladder
TPD when $\phi=\pi$. The phase diagram looks identical when $\phi\neq 0,\pi$,
but the states are not protected there when $m\neq 0$. (c) -CSSH ladder TPD
when $\phi=\pi$. $\tilde{\mathcal{Z}}_{1}$ is not defined (N/D), because the
bands touch [see Fig. 4 (c)], so only $\tilde{\mathcal{Z}}_{2}$ is pictured.
(d) -CSSH ladder TPD for $\phi=\pi/2$. The absence of inversion symmetry means
that the Zak phase is not quantized to only 0 or $\pi$. Additionally,
$\tilde{\mathcal{Z}}_{1}\neq\tilde{\mathcal{Z}}_{3}$ here, so we identify the
phases by
$[\tilde{\mathcal{Z}}_{1},\tilde{\mathcal{Z}}_{2},\tilde{\mathcal{Z}}_{3}]$,
see the main text for details. In this case, the diagram is striped to show
$\tilde{\mathcal{Z}}_{2}$ and $\tilde{\mathcal{Z}}_{3}$. Three phases arise,
where the invariants oscillate around a given value, something we indicate
with a $\sim$ symbol. The model belongs to a topological symmetry class (BDI)
only for $m=0$, where the second gap has protected zero modes. (e)
$\Delta$-CSSH ladder TPD for $\phi=\pi$. The first invariant has a constant
value of $3\pi/2$, except at the Creutz limit $\theta=\pi/4$, where it becomes
undefined. (f) $\Delta$-CSSH laddder TPD for $\phi=\pi/2$. It is similar to
the $\pi$-flux case, but the values of the invariants now oscillate around the
indicated values. In the $m=0$ line (AIII case), the second gap is
topological. All results are obtained from the bulk Hamiltonian
$\mathcal{H}(k)$, following the formula (8) using 100 values of $k$. All CSSH
ladders have $\xi=1$.

The three main symmetries that are usually considered in the context of
topology are particle-hole (PH) symmetry ($\mathcal{C}$) and time-reversal
(TR) symmetry ($\mathcal{T}$), which are both antiunitary, and chiral or
sublattice symmetry ($\mathcal{X}=\mathcal{C}\mathcal{T}$), which is unitary.
They satisfy:

$\displaystyle\mathcal{C}\mathcal{H}\mathcal{C}^{-1}=-\mathcal{H}$ (4)
$\displaystyle\mathcal{T}\mathcal{H}\mathcal{T}^{-1}=\mathcal{H}$ (5)
$\displaystyle\mathcal{X}\mathcal{H}\mathcal{X}^{-1}=-\mathcal{H}.$ (6)

Their presence and type define the well-known symmetry classes in the periodic
table of topological insulators [1, 65, 66], some of which can have
topological phases, depending on their spatial dimension. The five topological
classes in 1D can be consulted in Table 1.

In this paper, we follow the approach of considering the requirements (4-6) as
the defining property of the three symmetries, which is a useful point of view
for the analysis of topological insulators and has been used extensively in
the literature [39, 67, 11, 16, 68, 69]. However, it is important to note that
the particle-hole symmetries we discuss do not have the same implications as
in a topological superconductor. Specifically, it does not imply the presence
of Majorana end modes in the topological phases. The unit cells in the models
we discuss are composed of more than one particle mode, and so their end modes
are never formed by half a fermionic mode [70, 39, 67, 71].

Class |  A |   AIII |  AI |  BDI |   D |  DIII |   AII |  CII |  C |  CI
---|---|---|---|---|---|---|---|---|---|---
1D T. Inv. |  |  $\mathbb{Z}$ |  |  $\mathbb{Z}$ |  $\mathbb{Z}_{2}$ |   $\mathbb{Z}_{2}$ |  |  $2\mathbb{Z}$ |  |
Table 1: Altland-Zirnbauer symmetry classes for 1D systems and their
topological invariants [1]. The delta CSSH ladder spans three of the five
topological classes, AIII, BDI and D, while the other CSSH ladders and the
Creutz ladder can be BDI or AIII.

In this section, we will review the relevant symmetries of the SSH chain and
Creutz ladder, including the hidden symmetries in the Creutz ladder, essential
for the protection of the edge states. Then, we will describe the symmetries
and topology of the three CSSH ladder models proposed in this work. All the
cases we consider are represented in Table 2, in which the symmetry classes
and invariants of the different models are detailed.

###  3.1 The SSH chain

The SSH chain [73] [Fig. 2 (b)] is a 1D chain with two sites per unit cell,
$I$ and $I\\!I$, with a intracell tunneling amplitude of $v$, and an intercell
amplitude of $w$, in which particles move around. Its Hamiltonian is:

$\mathcal{H}_{S\\!S\\!H}\\!=\\!-\\!\sum_{x}\left(vc^{\dagger}_{x,I\\!I}c_{x,I}+wc^{\dagger}_{x+1,I}c_{x,I\\!I}+h.c.\right)\\!,$
(7)

where $c^{(\dagger)}_{x,\sigma}$ destroys (creates) a particle in unit cell
number $x=1,\ldots,L/2$ and sublattice $\sigma=I,I\\!I$.

This Hamiltonian has particle-hole
($\mathcal{C}^{(2)}_{S}=\sigma_{z}\mathcal{K}$), time-reversal
($\mathcal{T}^{(2)}_{S}=\mathcal{K}$) and chiral
($\mathcal{X}^{(2)}_{S}=\sigma_{z}$) symmetries, where $\mathcal{K}$ is
complex conjugation. The matrices act on the internal degree of freedom, the
subindex $S$ refers to the SSH chain, and the superindex indicates the
matrices are $2\times 2$. The model belongs to class BDI [42, 1], and so it
can be characterized by its Zak phase (for a more detailed discussion, please
refer to Appendix A). The Zak phase of a given band $i$ is given by333As
detailed in Appendix A, there are two slightly different quantities that have
been called “Zak phase” in the literature. We follow the convention in [42],
which is tied to the choice of the so-called Basis I for tight-binding models
[99].:

$\mathcal{Z}_{i}=i\oint_{BZ}\bra{u_{k}^{i}}\partial_{k}\ket{u_{k}^{i}}dk,$ (8)

where $\\{\ket{u_{k}^{i}}\\}$ is the $i$-th eigenstate of the bulk Hamiltonian
$\mathcal{H}(k)$ of the system [74].

The internal states of the SSH chain $\ket{I}=(1,0)$ and $\ket{I\\!I}=(0,1)$
have positive and negative chirality, respectively, corresponding to the two
sublattices. The bulk states have equal support on both sublattices, while the
topological edge states, when present, only have support on one of them. This
is a general phenomenon in chiral-symmetric topological phases. The SSH chain
is topological whenever $v<w$, giving rise to one chiral zero mode in each end
of the chain, protected by the chiral and PH symmetries [75].

###  3.2 The Creutz ladder

####  3.2.1  $\phi=\pi$

The Creutz ladder Hamiltonian can be obtained from equation (1) setting
$J_{i}=t_{i}=J$. Its unit cell is formed by the upper and lower sites, $a$ and
$b$, of each rung. The system is metallic when $m=2J$ or $\phi=0$
$\\!\\!\\!\mod 2\pi$. When $\phi=\pi$, the system belongs to class BDI, with
the symmetries $\mathcal{C}^{(2)}_{C}=\sigma_{z}\mathcal{K}$,
$\mathcal{T}^{(2)}_{C}=\sigma_{x}\mathcal{K}$ and
$\mathcal{X}^{(2)}_{C}=i\sigma_{y}$ [38, 34, 68], where the subindex $C$
refers to the Creutz ladder. This TR symmetry, as well as several others that
we describe below, is non-conventional, but it behaves exactly like a usual TR
symmetry for topological considerations, so we will consider it as such. For a
more detailed discussion, see Appendix B.

The topology of the model can be characterized by the Zak phase, which is
$\pi$ if $m<2J$ and vanishes otherwise; except if $\phi=0,\pi\mod 2\pi$, where
it is not defined because the system is metallic. These lines at constant
$\phi$ in the phase diagram separate distinct topological phases, which can be
distinguished by the chirality of their edge states or the sign of their
winding number, $\nu=\pm 1$ [8, 12]. One protected chiral edge state appears
in each end of the ladder when the system is topological.

The chiral basis is now, in the $a,b$ basis, $(1,\mp i)/\sqrt{2}$,
corresponding to the chirality of the left/right edge state. The edge states
are protected by both the chiral and PH symmetries.

A unitary transformation [34] maps the $\phi=\pi$ ladder to the SSH chain,
with $v=m$ and $w=2J$, as the topological phase transitions (TPTs) of each
model suggest. For instance, this means that the rungless case ($m=0$), when
AB caging completely localizes the topological edge states of the Creutz
ladder in the first and last rungs, corresponds to the fully dimerized limit
($v=0$) in the SSH chain, where the edge states are localized at the first and
last sites. This has important consequences, as we discuss below. The mapping
also relates the chiral bases of the models. This illustrates a key difference
between them: the two chiral states in the SSH chain coincide with its sites,
while those in the Creutz ladder do not. While one of the chiral states (and,
thus, one of the end states, if they are present) can be easily eliminated in
the SSH chain due to open boundary conditions, this is impossible in the
Creutz ladder. Thus, the topology of the Creutz ladder does not depend on the
choice of unit cell, unlike that of the SSH chain.

SSH |  BDI: $\mathbb{Z}$ [$\mathcal{C}^{(2)}_{S},\mathcal{T}^{(2)}_{S},\mathcal{X}^{(2)}_{S}$]
---|---
Creutz | $\bm{\phi=0}$ | $\bm{\phi\neq 0,\pi}$ | $\bm{\phi=\pi}$
$\bm{m\neq 0}$ | Trivial |  AI: NT [$\mathcal{T}^{(2)}_{C}$] |  BDI: $\mathbb{Z}$ [$\mathcal{C}^{(2)}_{C}\\!,\mathcal{T}^{(2)}_{C}\\!,\mathcal{X}^{(2)}_{C}$]
$\bm{m=0}$ | Trivial |  BDIa: $\bm{\mathbb{Z}}$ [$\tilde{\mathcal{C}}_{SC},\mathcal{T}^{(2)}_{C}\\!,\tilde{\mathcal{X}}_{S}$ ] |  BDI: $\mathbb{Z}$ [$\mathcal{C}^{(2)}_{C}\\!,\mathcal{T}^{(2)}_{C}\\!,\mathcal{X}^{(2)}_{C}$]
$\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{1.5}{$\Join$}}}$-CSSH | $\bm{\phi=0}$ | $\bm{\phi\neq 0,\pi}$ | $\bm{\phi=\pi}$
$\bm{m\neq 0}$ |  AIb: NT [$U$: $\mathcal{T}^{(2)}_{S}$]HHH |  HHH AI: NT [$\mathcal{T}_{C}$]HHH |  HHH BDI: $\mathbb{Z}$ [$\mathcal{C}_{C},\mathcal{T}_{C},\mathcal{X}_{C}$]
$\bm{m=0}$ |  BDI: $\mathbb{Z}$ [$U$: $\mathcal{C}^{(2)}_{C}\\!,\mathcal{T}^{(2)}_{C}\\!,\mathcal{X}^{(2)}_{C}$] |  BDI: $\mathbb{Z}$ [$\mathcal{C}_{SC},\mathcal{T}_{C},\mathcal{X}_{S}$] |  BDI: $\mathbb{Z}$ [$V$: $\mathcal{C}^{(2)}_{C}\\!,\mathcal{T}^{(2)}_{C}\\!,\mathcal{X}^{(2)}_{C}$]
-CSSH | $\bm{\phi=0}$ | $\bm{\phi\neq 0,\pi}$ | $\bm{\phi=\pi}$
$\bm{m\neq 0}$ | Trivial |  AI: NT [$\mathcal{T}^{\prime}_{\scalebox{0.5}{ \leavevmode\hbox to6.64pt{\vbox to4.49pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{ }{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{6.23701pt}{4.095pt}\pgfsys@lineto{2.45702pt}{4.095pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.78pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}$] |  BDI: $\mathbb{Z}$ [$\mathcal{C}_{C},\mathcal{T}_{\scalebox{0.5}{ \leavevmode\hbox to6.64pt{\vbox to4.49pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{ }{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{6.23701pt}{4.095pt}\pgfsys@lineto{2.45702pt}{4.095pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.78pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}},\mathcal{X}_{\scalebox{0.5}{ \leavevmode\hbox to6.64pt{\vbox to4.49pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{ }{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{6.23701pt}{4.095pt}\pgfsys@lineto{2.45702pt}{4.095pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.78pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}$]
$\bm{m=0}$ | Trivial |  BDI: $\bm{\mathbb{Z}}$ [$\mathcal{C}^{\prime}_{\scalebox{0.5}{ \leavevmode\hbox to6.64pt{\vbox to4.49pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{ }{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{6.23701pt}{4.095pt}\pgfsys@lineto{2.45702pt}{4.095pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.78pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}$, $\mathcal{T}^{\prime}_{\scalebox{0.5}{ \leavevmode\hbox to6.64pt{\vbox to4.49pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{ }{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{6.23701pt}{4.095pt}\pgfsys@lineto{2.45702pt}{4.095pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.78pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}$,$\mathcal{X}_{S}$] |  BDI: $\mathbb{Z}$ [$W$: $\mathcal{C}_{W},\mathcal{T}_{W},\mathcal{X}^{(2)}_{S}$]
$\Delta$-CSSH | $\bm{\phi=0}$ | $\bm{\phi\neq 0,\pi}$ | $\bm{\phi=\pi}$
$\bm{m\neq 0}$ | Trivial | A: NT [none] |  D: $\mathbb{Z}_{2}$ [$\mathcal{C}_{C}$]
$\bm{m=0}$ | Trivial |  AIII: $\mathbb{Z}$ [$\mathcal{X}_{S}$] |  BDI: $\mathbb{Z}$ [$\mathcal{C}_{C},\mathcal{T}_{SC},\mathcal{X}_{S}$]
Table 2: Symmetries and symmetry classes of the SSH, Creutz and CSSH
Hamiltonians. Letters $\mathcal{C}$, $\mathcal{T}$ and $\mathcal{X}$ refer to
PH, TR and chiral symmetries, respectively. Hidden symmetries are marked by a
tilde. Inversion symmetry is not included in the table, see text. All
operators are defined in the text, and listed again in Appendix B. In the CSSH
models, $J\neq J^{\prime}$ is assumed. When the Hamiltonian is block-
diagonalizable using a unitary symmetry $U$, it is indicated by $U$:, followed
by the symmetries of the resulting blocks. “NT” means the Altland-Zirnbauer
classification predicts the model is not topological in 1D. Footnotes: a The
hidden symmetries of the model change its symmetry class into a topological
one and protect the edge states. b The model only differs from the $m=0$ case
in an element proportional to $\mathbbm{1}$, which has no effect on the
topological invariant [72] nor, in this case, the protected edge states.

####  3.2.2 $\phi\neq 0,\pi$

For other values of the magnetic flux, the TR symmetry holds but the PH and
chiral symmetries break down. The edge states are only protected in the
rungless case. When $m\neq 0$, the two edge states drift away from zero
energy, given that the model belongs to class AI, with no protecting
symmetries (see Table 2). The Zak phase is still quantized due to inversion
symmetry [42], which also causes the end states to be degenerate in the
thermodynamic limit. Thus, in this regime, the system is a topological
crystalline insulator [8, 76]. However, in it, the end states are not
topologically protected against any kind of disorder, given that any disorder
will break inversion symmetry. The topological phase diagram of the Creutz
ladder as a function of $\phi$ and $m$ is depicted in Fig. 3 (a), as well as
the regions where topological protection will be found in the CSSH ladders:
whenever $\phi=\pi$ or $m=0$, like in the Creutz ladder, and, in the hourglass
case, also when $\phi=0$.

When $m=0$, the topological protection can only be explained due to the
presence of so-called “hidden symmetries”. These are symmetries that satisfy
one of the conditions (4-6), but they act in a non-trivial way on the spatial
degree of freedom, so they cannot be reduced to their effect on the inner
space Hamiltonian [77]. They can determine the topology of the system, exactly
like their conventional counterparts [78, 79, 80, 81, 82, 83, 84].

The topological protection when $m=0$ can be explained due to the presence of
the hidden chiral symmetry $\tilde{\mathcal{X}}_{S}$, which reflects the fact
that the lattice is bipartite, and the hidden PH symmetry
$\tilde{\mathcal{C}}_{SC}$. The subindex $S$ indicates this symmetry is
analogous to the chiral symmetry in the SSH chain, while $SC$ denotes that the
operator is related to $\tilde{\mathcal{X}}_{S}$ and $\mathcal{T}^{(2)}_{C}$.
In the spatial basis of the full Hilbert space, they are represented by the
operators:

$\displaystyle\tilde{\mathcal{X}}_{S}=\mathop{\mathrm{diag}}\nolimits_{2L}(1,1,-1,-1,1,1,-1,-1,\ldots)$
(9)
$\displaystyle\tilde{\mathcal{C}}_{SC}=\mathop{\mathrm{diag}}\nolimits_{L}(\sigma_{x},-\sigma_{x},\sigma_{x},-\sigma_{x},\ldots)\mathcal{K},$
(10)

where $\mathop{\mathrm{diag}}_{\ell}(a,\ldots,z)$ is the diagonal or block-
diagonal matrix formed by the $\ell$ elements $a,\ldots,z$. In general, we
will mark operators acting on the whole Hamiltonian with a tilde, and those
that act on the inner space Hamiltonian $\mathcal{H}(k)$ with no tilde. These
operators cannot be factorized to separate their effect on the inner and
spatial subspaces, as is customary. These hidden symmetries place the rungless
Creutz ladder in the BDI class (see Table 2). The operators
$\tilde{\mathcal{X}}_{S}$ and $\mathcal{T}^{(2)}_{C}$ are related to the
hidden symmetries $\mathcal{S}$ and $\mathcal{V}$ mentioned in Ref. [80] by
the inversion symmetry of the ladder.

####  3.2.3 AB caging and topological protection

We want to bring attention to an important detail at this point, which is
often overlooked. In most topological materials, if the system has a small
size, the energies of the protected end modes have a finite (albeit small)
value, due to their overlap. In the systems we study, these energies are at
least one order of magnitude smaller than the gap, and so do not break the
bulk-boundary correspondence. However, the zero modes in the rungless Creutz
ladder are completely localized to the two end sites of the ladder, and this
causes their overlap to vanish. This localization has the same nature as AB
caging, as it is caused by destructive interference of the wavefunction due to
the magnetic flux and connectivity of the ladder. Thus, the end modes are
pinned at zero energy for any length of the ladder $L$, including the one
plaquette limit, $L=2$. This interplay between AB localization and topology
protects the zero modes even better than topology alone can, and allows the
detection and use of protected states in systems with only a few sites. This
also happens in the three CSSH ladders, where the edge states present a larger
variety than in the Creutz case444The end states in the completely dimerized
SSH chain also have zero energy for any length of the chain, as a trivial
consequence of disconnecting the end sites from the rest of the chain. In the
$\pi$-flux Creutz and CSSH ladders, this phenomenon occurs while the end sites
are connected to the rest of the system..

Figure 4: Bands of the CSSH ladders with periodic boundary conditions in $k$
space as a function of $m$. The parameters used are $\xi=1$, $J=2J^{\prime}$,
and $\phi=\pi$ for the figures on the left (a,c,e), and $\phi=\pi/2$ for those
in the right (b,d,f). The former show flat bands when $m=0$ (marked in red),
and a TPT when the middle bands touch at $k=0$. (a,b)
$\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{1.5}{$\Join$}}}$-CSSH ladder
spectra. The BDI case (a) has the most symmetric spectrum, while the AI case
(b) is only symmetric in $k$, due to the TR symmetry. When $m=0$, the spectrum
(b) is BDI. (c,d) -CSSH ladder spectra. The BDI case (c) now has only one gap,
with the two upper and the two lower bands touching at $k=\pi$ for all $m$,
something that does not happen in the AI case (d). When $m=0$, the (d) case is
also BDI. (e,f) $\Delta$-CSSH ladder spectra. Now, the $\pi$-flux spectrum (e)
is only symmetric about the origin, indicating the sole presence of a PH
symmetry (D class). For a generic value of $\phi$ (f), the spectrum shows no
symmetries (A class), except for $m=0$, where only a chiral symmetry exists
(AIII class), making the spectrum symmetric only in $E$.

####  3.2.4 $\phi=0$

When $\phi=0\mod 4\pi$, the Hamiltonian has a unitary commuting symmetry
$U=\sigma_{x}$, that makes it decouple into two topologically trivial
subspaces: a dispersive band spanned by $(\ket{a}+\ket{b})/\sqrt{2}$, and a
flat band at energy $E=m$, spanned by $(\ket{a}-\ket{b})/\sqrt{2}$. When
$\phi=2\pi\mod 4\pi$, the states that form each subspace switch their roles,
and the value of their energies is reversed (for instance, the flat band is
located at $E=-m$).

##  4 Topology of the CSSH ladders

We will now analyze the three CSSH models. They have topological phases with
protected zero modes if $\phi=\pi$ ($\pi$-flux regime) or $m=0$ (rungless
regime). Additional constraints must be satisfied for each model to be
topological, something we explore below using their topological phase diagrams
(Fig. 3). The coexistence of the symmetries from the $\pi$-flux and rungless
regimes causes the $\phi=\pi,m=0$ case to be block-diagonalizable in the
$\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{1.5}{$\Join$}}}$-CSSH and
-CSSH ladders, revealing a trivial and a topological subspace. The
$\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{1.5}{$\Join$}}}$-CSSH ladder
is also topological in the $\phi=0$ case, while the other two models are not.
In the chiral models, two different nontrivial phases appear with edge modes
of opposite chirality and winding number, in the same way than in the Creutz
ladder [8, 12]. These phases are separated by two topological phase
transitions at $\phi=0,2\pi\mod 4\pi$. For more details on the relevant
topological invariants, please refer to Appendix A.

The different topological classes, symmetries and invariants of the SSH,
Creutz and CSSH systems for every combination of parameters can be found in
Table 2. The detailed list with the analytical expression of all symmetry
operators is reserved for Appendix B, although the most relevant ones will be
mentioned in the text. The bulk bands for the different models, depicted in
Fig. 4, reflect the presence of these symmetries. The spectra for the finite
ladders, with different types of topological end modes, is also studied below
(Fig. 5). Finally, the spatial distribution of some energy eigenstates of
different 32-site ladders are represented in Fig. 6. Each of the
aforementioned figures is referenced below, in the context of the different
models. In all cases, the energy scale has been fixed by setting $\xi=1$.

Figure 5: Spectra of different CSSH ladders with open boundary conditions as a
function of $\theta$ (a-h) and $m$ (i-l). Bulk states are marked in blue or
orange, protected zero modes in red, and other edge states in green. All
systems have $L=200$ and $\xi=1$, and were obtained by exact diagonalization.
The parameters and CSSH family of each subfigure are included above it. (a-d)
$\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{1.5}{$\Join$}}}$-CSSH ladder
spectra in $\theta$. (a) Flat-band regime. The band energies can be tuned
independently changing $J$ and $J^{\prime}$, and they correspond to a trivial
(orange) and a topological (blue) completely dimerized SSH system (see text).
(b) General BDI $\pi$-flux case. Zero modes and non-protected topological edge
states appear. Two TPTs occur at the points where the zero modes join the bulk
bands. (c) BDI $\pi/2$-flux rungless topological phase, with protected zero
modes. (d) Fluxless BDI case, showing the SSH-like subspace with a TPT at
$\theta=\pi/4$, and a degenerate flat band (orange). (e,f) -CSSH ladder
spectra in $\theta$. (e) Flat-band case, always topological. (f) BDI
$\pi$-flux case. The two upper and lower bands touch, so only one gap appears.
When it closes, there is a TPT. (g,h) $\Delta$-CSSH ladder spectra in
$\theta$. (g) Flat-band regime. Non-protected end states appear in addition to
the zero modes. (h) D-class $\pi$-flux case, with non-protected end states and
two TPTs. (i-l) Spectra as a function of $m$. (i) $\pi$-flux
$\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{1.5}{$\Join$}}}$-CSSH
spectrum. A TPT can be seen at $m=1$, and non-protected edge states come out
of the inner bands as $m$ increases. (j) $\pi$-flux -CSSH ladder spectrum. A
TPT occurs at the same point as in (i). The two upper and lower bands touch,
so only one gap is seen. (k) -CSSH ladder spectrum for $\phi=\pi/2$. When
$m\neq 0$, the edge states in the central gap are not degenerate, because they
are not related by inversion symmetry. The color red has been used for
clarity. The third gap undergoes a phase transition (see insert). (l)
$\pi$-flux $\Delta$-CSSH ladder spectrum. A TPT is also found at $m=1$, and
non-protected end states go into the outer bands as $m$ increases. In (i,j,l),
the flat-band regime can be seen at $m=0$.

###  4.1 The hourglass CSSH ladder

####  4.1.1 $\phi=\pi$

In this case, the
$\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{1.5}{$\Join$}}}$-CSSH ladder
[see Fig. 2 (c)] belongs to class BDI, with symmetries
$\mathcal{C}_{C}=\mathbbm{1}_{2}\otimes\sigma_{z}\mathcal{K}$,
$\mathcal{T}_{C}=\mathbbm{1}_{2}\otimes\sigma_{x}\mathcal{K}$ and
$\mathcal{X}_{C}=\mathbbm{1}_{2}\otimes(i\sigma_{y})$, which are equivalent to
the Creutz ladder symmetries in the same regime, and have dimension 4 (the
superindex $(4)$ was omitted in all matrices for clarity). This is reflected
on the symmetric form of its spectrum [see Fig. 4 (a)].

In the rungless $\pi$-flux case, the system has four flat bands and a
commuting unitary symmetry $V=\sigma_{z}\otimes(i\sigma_{y})$. This kind of
symmetry allows us to block-diagonalize the Hamiltonian, and study the
topology of the different subspaces independently. In this case, the system
can be mapped into two independent, fully dimerized SSH chains, a trivial and
a topological one, with amplitudes $v_{1}=2J^{\prime}$, $w_{1}=0$ and
$v_{2}=0$, $w_{2}=2J$. They correspond, respectively, to the flat bands with
energies $E=\pm 2J^{\prime}$ and $E=\pm 2J$ [see Fig. 5 (a)]. The
$\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{1.5}{$\Join$}}}$-CSSH ladder
end modes correspond to the edge states of the topological chain. When $m$ is
switched on, it couples the two chains and the symmetry $V$ breaks.

In the $m\neq 0$ case, the full four-band system must be considered. The
topological invariant related to the boundary states that appear in a
particular gap in the system can be obtained as the sum of the Zak phases of
all the bands below it [85],
$\tilde{\mathcal{Z}}_{i}=\sum_{j=1}^{i}\mathcal{Z}_{j}$. In a topological
symmetry class, the number of topological edge states per end in that gap will
be $\tilde{\nu}_{i}=\tilde{\mathcal{Z}}_{i}/\pi$ [86, 85, 87, 88]. When
quantized, the topological invariants associated with the first and third gaps
of all CSSH ladders have the same value,
$\tilde{\mathcal{Z}}_{1}=\tilde{\mathcal{Z}}_{3}$, so to identify different
phases we will use $[\tilde{\mathcal{Z}}_{1},\tilde{\mathcal{Z}}_{2}]$.

The topological phase diagram for this regime is depicted in Fig. 3 (b). The
first invariant, $\tilde{\mathcal{Z}}_{1}$, behaves like the one in the SSH
chain, being topological when $J>J^{\prime}$. The second one,
$\tilde{\mathcal{Z}}_{2}$, is topological when
$m<\xi\sin(2\theta)=2\sqrt{JJ^{\prime}}$. When $J=J^{\prime}$, this condition
is reduced to the one in the Creutz ladder. The states in the second (or
central) gap are protected by the chiral and PH symmetries in the topological
phase [see Fig. 5 (i)]. Those in the first and third gaps only appear when
$\tilde{\mathcal{Z}}_{1}=\pi$, and thus satisfy the bulk-boundary
correspondence. However, they are not pinned in the middle of their gaps, as
can be seen in Fig. 5 (b). This is because the symmetries of the system can
only protect states in the middle gap, given that they only guarantee a
symmetric spectrum about zero energy. This is further explained in Appendix E.
They do guarantee, however, that the states in the first gap have the opposite
energy to those in the third gap. These states can be described as
topological, but not protected. If $\pi/4<\theta<\arcsin(m/\xi)/2$, the system
has six topological edge states, two of which are protected at zero energy. In
Fig. 6 (b,c), the edge states in the first and second gap of the
$\phi=\pi,\theta=3\pi/8,m=0.5$ system are shown.

####  4.1.2 $\phi\neq 0,\pi$

For these values of the magnetic field, the TR symmetry still holds, but the
other two break down, in a similar situation to that of the Creutz ladder.
This can be appreciated in Fig. 4 (b). If $m\neq 0$, the model is a
topological crystalline insulator in class AI. Its states are not
topologically protected against disorder, but they are still degenerate in a
long enough ladder due to spatial inversion.

When $m=0$, the same chiral and PH symmetries as in the Creutz ladder appear.
However, they are not hidden anymore, because the unit cell of the system is
larger now. Their operators on the bulk Hamiltonian are
$\mathcal{C}_{SC}=\sigma_{z}\otimes\sigma_{x}\mathcal{K}$ and
$\mathcal{X}_{S}=\sigma_{z}\otimes\mathbbm{1}_{2}$. As noted earlier, the
subindex $S$ indicates this symmetry is analogous to the chiral symmetry in
the SSH chain. In this regime, $\tilde{\mathcal{Z}}_{1}=\pi$ only when
$J>J^{\prime}$, and $\tilde{\mathcal{Z}}_{2}=\pi$ always holds. Like in the
$\phi=\pi$ case, only the states in the middle gap are protected [see Fig. 5
(c)]. Their exact form is described in Appendix C. The Zak phases take the
same values as in the $\pi$-flux regime [Fig. 3 (b)].

####  4.1.3 $\phi=0$

If the magnetic field is turned off in the
$\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{1.5}{$\Join$}}}$-CSSH ladder,
the topology inherited from the Creutz ladder disappears, but not the one
generated by the staggered, SSH-like amplitudes. In this case, the bulk
Hamiltonian has a unitary commuting symmetry,
$U=\mathbbm{1}_{2}\otimes\sigma_{x}$, that generates two independent
subspaces, exactly like in the Creutz case. One of them is a topologically
trivial doubly degenerate flat band at energy $E=m$, spanned by the internal
states $(\ket{A}-\ket{B})/\sqrt{2}$ and $(\ket{C}-\ket{D})/\sqrt{2}$. This
subspace can be turned into two non-degenerate flat bands if the vertical
hopping amplitudes $m_{1},m_{2}$ are set to different values in (1).

The other subspace is now analogous to an SSH chain with amplitudes
$v=J^{\prime},w=J$, which will be topological if $J^{\prime}<J$, in contrast
to the Creutz case, where the system is metallic at $\phi=0$. Its spectrum is
pictured in Fig. 5 (d). The vertical amplitude $m$ behaves like a homogeneous
chemical potential, shifting the SSH chain spectrum downwards, but having no
other effect in the topology. The topological chiral end states are analogous
to those in the SSH chain, with the left (right) state only supported on the
odd (even) sites, as can be seen in Fig. 6 (d) for the $\theta=3\pi/8$, $m=0$
system.

###  4.2 The rhomboid CSSH ladder

####  4.2.1 $\phi=\pi$

When $\phi=\pi$, the -CSSH ladder [see Fig. 2 (d)] is in class BDI, with the
symmetries

$\displaystyle\mathcal{C}_{C}=\mathbbm{1}_{2}\otimes\sigma_{z}\mathcal{K}$
(11) $\displaystyle\mathcal{T}_{\scalebox{0.5}{ \leavevmode\hbox
to5.94pt{\vbox to4.04pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{5.54402pt}{3.64pt}\pgfsys@lineto{2.18402pt}{3.64pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.36pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}=\begin{pmatrix}-is_{\vartheta}&c_{\vartheta}&0&0\\\
c_{\vartheta}&-is_{\vartheta}&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{pmatrix}\mathcal{K}$ (12)
$\displaystyle\mathcal{X}_{\scalebox{0.5}{ \leavevmode\hbox to5.94pt{\vbox
to4.04pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{5.54402pt}{3.64pt}\pgfsys@lineto{2.18402pt}{3.64pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.36pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}=\begin{pmatrix}is_{\vartheta}&c_{\vartheta}&0&0\\\
-c_{\vartheta}&-is_{\vartheta}&0&0\\\ 0&0&0&1\\\ 0&0&-1&0\end{pmatrix}$ (13)

where

$\displaystyle s_{\vartheta}=\sin\vartheta=\frac{J^{\prime 2}-J^{2}}{J^{\prime
2}+J^{2}}$ (14) $\displaystyle
c_{\vartheta}=\cos\vartheta=\frac{2J^{\prime}J}{J^{\prime 2}+J^{2}}.$ (15)

The angles $\theta$ and $\vartheta$ are related by the expression:

$\vartheta=2\arctan(\cos 2\theta).$ (16)

The first and third gaps are closed at $k=\pi$, forming two Dirac cones [see
Fig. 4 (c)], and this renders $\tilde{\mathcal{Z}}_{1}$ and
$\tilde{\mathcal{Z}}_{3}$ not meaningful, as only the middle gap exists. The
ground state of the system lies at $k=0$, something apparent in Fig. 6 (e) for
$J=2J^{\prime}$, $m=0.5$. The relevant phase diagram is shown in Fig. 3 (c).
The system is topological when $m<\sqrt{2(J^{2}+J^{\prime
2})}=\xi\sqrt{(3+\cos 4\theta)/2}$. When that condition is met, the gap has
zero modes, protected by the chiral and PH symmetries. Their form for
$J=2J^{\prime}$, $m=0.5$ is depicted in Fig. 6 (f), and they can be seen in
its spectra as a function of $\theta$, in Fig. 5 (f), and as a function of
$m$, in Fig. 5 (j).

The chiral symmetry of the system $\mathcal{X}_{\scalebox{0.5}{
\leavevmode\hbox to5.94pt{\vbox to4.04pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{5.54402pt}{3.64pt}\pgfsys@lineto{2.18402pt}{3.64pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.36pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}$, which protects
the end modes along with $\mathcal{C}_{C}$, depends on $\theta$. It
interpolates between a Creutz-like chiral symmetry, at $\theta=\pi/4$, and its
counterpart in the orthogonal dimer chain, at $\theta=0,\pi/2$. This provides
a degree of freedom to choose the system in which the protecting symmetries
will be less broken in a particular implementation. The Creutz ladder chiral
symmetry can withstand a broader range of on-site disorder configurations
without breaking than that in the orthogonal dimer chain (see Appendix B for a
more detailed discussion). On the other hand, the orthogonal dimer chain is a
simpler model, and can therefore have less sources of disorder than the Creutz
ladder, so it might be preferable for some applications.

As can be seen in Fig. 3 (c), when $J$ or $J^{\prime}$ vanish, the system can
still be topological, unlike the other models. The resulting lattice is the
orthogonal dimer chain, depicted in Fig. 2 (f). It is topological when
$\phi=\pi$ and $m<\sqrt{2}J$ (assuming $J^{\prime}=0$). As mentioned above,
this model is usually studied in the context of interacting spins [45, 47, 48,
50, 52]. The -CSSH ladder connects the topological phases of the Creutz ladder
(obtained when $\theta=\pi/4$) and the orthogonal dimer chain
($\theta=0,\pi/2$), two previously unrelated models. The superposition of both
topological edge states in the orthogonal dimer chain with $m=0.5$ is shown in
Fig. 6 (i).

Furthermore, when $\phi=\pi$ and $m=0$ (for any value of $\theta$), the system
has a unitary commuting symmetry
$W=\mathcal{X}_{S}\mathcal{X}_{\scalebox{0.5}{ \leavevmode\hbox to5.94pt{\vbox
to4.04pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{5.54402pt}{3.64pt}\pgfsys@lineto{2.18402pt}{3.64pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.36pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}$ that creates two
subspaces that can be mapped to two fully dimerized SSH chains, with
amplitudes $v_{1}=0=w_{2},w_{1}=i\sqrt{2(J^{2}+J^{\prime 2})}=-v_{2}$. They
are topological and trivial, respectively. The complex amplitudes,
corresponding to a magnetic potential in the direction of the chain, do not
affect the topology, and only modify the form of the PH and TR operators of
the SSH chain. The topological subspace has two protected zero modes. The
upper $(+)$ and lower $(-)$ bands of the two SSH chains are degenerate, with
energies $E=\pm\sqrt{2(J^{2}+J^{\prime 2})}$. However, given that their states
never mix, there is no need to resort to the non-abelian topological invariant
[67]. The spectrum of the system is pictured in Fig. 5 (e).

####  4.2.2 $\phi\neq 0,\pi$

The system is still TR-symmetric for these values of $\phi$ [see Fig. 4 (d)],
but the form of the TR operator is now more complicated (see equation (43) in
Appendix B). When $m=0$, the chiral symmetry $\mathcal{X}_{S}$ pins the edge
states in the middle gap exactly at zero energy. The left zero mode is
depicted in Fig. 6 (g) for $J=2J^{\prime}$, $\phi=\pi/2$. A PH symmetry,
$\mathcal{C}^{\prime}_{\scalebox{0.5}{ \leavevmode\hbox to5.94pt{\vbox
to4.04pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{5.54402pt}{3.64pt}\pgfsys@lineto{2.18402pt}{3.64pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.36pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}$ (see Appendix B),
is also present there. Therefore, the $m=0$ case belongs to the BDI class,
while the $m\neq 0$ system belongs to class AI, which is not topological.

To understand the topological phase diagram, we need to keep in mind that the
Zak phase is only guaranteed to be quantized if the system with periodic
boundary conditions is symmetric under a spatial inversion operation that
leaves the point at the center of the unit cell invariant [84]. The -CSSH
ladder is symmetric under a specific spatial inversion symmetry, in which the
inversion center is located between the $C$ and $D$ sites (see Appendix B for
a matrix representation).

In this regime, the invariants associated with the first and third gaps are no
longer equal. In fact, one of them has a topological phase transition between
non-quantized values, while the other one does not. Only
$\tilde{\mathcal{Z}}_{3}$ presents a phase transition if $-\pi<\phi<\pi\mod
4\pi$, and only $\tilde{\mathcal{Z}}_{1}$ presents one otherwise. This can be
seen in Fig. 3 (d). Non-protected end states appear in the first and third
gaps, and they do not follow the usual bulk-boundary correspondence. One of
these end states is pictured in Fig. 6 (h). The topological phase transitions
are related to gap closings in the spectrum that do not change the number of
end states [see the insert in 5 (k)], but do change the end (left or right) in
which they appear. We discuss these states in depth in Section 4.4.

In the -CSSH ladder, the Zak phase is quantized whenever the system belongs to
a topological class, except for $\tilde{\mathcal{Z}}_{1}$ and
$\tilde{\mathcal{Z}}_{3}$ in the $\phi\neq\pi,m=0$ case. There,
$\tilde{\mathcal{Z}}_{2}$ is equal to $\pi$, indicating the presence of
topological zero modes. Additionally, the second gap end states in a ladder
with an even number of rungs are not related by spatial inversion, and so they
are not degenerate, even in a large system. This occurs because the open
system loses its global inversion symmetry, and its effects can be seen in the
corresponding spectrum as a function of $m$ in Fig. 5 (k).

####  4.2.3 Edge state tunability

To show an application of the high tunability of the protected edge states,
let us consider the left topological zero mode of the rungless -CSSH ladder
[see Fig. 6 (g)]:

$\ket{\mathcal{L}_{\scalebox{0.5}{ \leavevmode\hbox to5.94pt{\vbox
to4.04pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{5.54402pt}{3.64pt}\pgfsys@lineto{2.18402pt}{3.64pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.36pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}\\!=\\!\frac{J\ket{1,A}-J^{\prime}e^{i\phi/2}\ket{1,B}}{\sqrt{J^{2}+J^{\prime
2}}}.$ (17)

The dependence on $\phi$ and $\theta$ of the zero modes provides a
topologically protected method to control a particle located in the two end
sites, by changing the parameters of the system.

We can establish any state in the Bloch sphere, with $\theta$ controlling the
weight of each site and $\phi$ controlling their relative phase. To obtain a
given state $\ket{\psi}=a\ket{1,A}+be^{i\varphi}\ket{1,B}$ with
$a,b,\varphi\in\mathbb{R}$, the following parameters must be chosen:

$\displaystyle\phi_{\psi}=2(\varphi-\pi)\mod 4\pi$ (18)
$\displaystyle\theta_{\psi}=\arctan\sqrt{\frac{a}{b}},$ (19)

such that $a=J/\sqrt{J^{2}+J^{\prime 2}}$, $b=J^{\prime}/\sqrt{J^{2}+J^{\prime
2}}$. In the cases of $\varphi=0,\pi$ ($\phi=0\mod 2\pi$), the system becomes
topologically trivial. The states defined by expression (17), however, are
still eigenvalues of the model, and become part of a flat band instead of
being topological zero modes.

In particular, the system can be initiated with the particle in eigenstate
$\ket{1,A}$ in the model with $J^{\prime}=0$ and $\phi=\phi_{\psi}$, and then
turning $J^{\prime}$ on adiabatically until the condition
$\theta=\theta_{\psi}$ is reached. In particular, this allows us to prepare
the topological end modes of the Creutz ladder, by choosing $\theta=\pi/4$. As
long as the chiral symmetry of the system, $\mathcal{X}_{S}$, is not broken,
this protocol is topologically protected against disorder, even in small
systems. The left state in the $\Delta$-CSSH ladder can also be used, with the
same results. For more information, refer to Appendix C, where we discuss the
caged zero modes of all models.

Figure 6: Spatial distribution of the wavefunctions of different energy
eigenstates in different 32-site ladders with open boundary conditions, with
$\xi=1$. We identify the ladders by their symbol and parameters
$(\theta,\phi,m)$, and the states by its number in a square, by ascending
order of energy. (a) Color code used to show the phase of the wavefunction.
(b,c)
$\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{1.5}{$\Join$}}}(3\pi/8,\pi,0.5)$
ladder, edge states in the first gap (b) and in the second gap (c). (d)
$\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{1.5}{$\Join$}}}(3\pi/8,0,0.5)$
ladder, protected zero mode. Its profile is reminiscent of an SSH chain edge
state. (e,f) $\leavevmode\hbox to8.32pt{\vbox
to5.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{7.92003pt}{5.20001pt}\pgfsys@lineto{3.12003pt}{5.20001pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{4.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(0.96,\pi,0.5)$
ladder. The value of $\theta\approx 0.96$ was chosen such that
$J=2J^{\prime}$. Its ground state (e) lies at $k=0$ [see Fig. 4 (c)], and its
topological zero modes (f) are similar to those in the Creutz ladder, but have
a larger support on the upper leg. (g,h) $\leavevmode\hbox to8.32pt{\vbox
to5.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{7.92003pt}{5.20001pt}\pgfsys@lineto{3.12003pt}{5.20001pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{4.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(0.96,\pi/2,0)$ ladder
left AB-caged zero mode (g) and right non-protected state (h). The left zero
mode (g) can be tuned using $\theta$ and $\phi$ to take almost any form in the
Bloch sphere, and is degenerate with its left counterpart (see text). The
relative phase between its sites is $\phi/2=\pi/4$. (i) $\leavevmode\hbox
to8.32pt{\vbox to5.6pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{7.92003pt}{5.20001pt}\pgfsys@lineto{3.12003pt}{5.20001pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{4.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(\pi/2,\pi,0.5)$
ladder topological state. This model has $J^{\prime}=0$ and is identical to
the orthogonal dimer chain. (j) $\Delta(0.96,\pi,0)$ ladder AB-caged non-
protected state in the first gap. It is localized in the four left sites due
to AB caging. (k,l) $\Delta(0.96,\pi,0.5)$ ladder (D-class TI). The loss of
chiral and TR symmetries now shifts the ground state of the system (k) to a
nonzero value of $k$ [see Fig. 4 (e)]. This is reflected on the changing phase
of the wavefunction as a function of $r$ [compare to (e)]. The states in the
central gap (l) are now only protected by PH symmetry.

####  4.2.4 $\phi=0$

In this regime, the system is topologically trivial. When $m=0$, a doubly
degenerate flat band appears at $E=0$, and one dispersive band is located at
each side of it, touching it if $\theta=\pi/4$. When $m$ is turned on, a flat
band remains at $E=m$, spanned by $(\ket{C}-\ket{D})/\sqrt{2}$, and the other
three are dispersive.

###  4.3 The delta CSSH ladder

####  4.3.1 $\phi=\pi$

When $m\neq 0$, this $\pi$-flux model, in contrast with all the previous ones,
does not have TR and chiral symmetries inherited from the $\phi=\pi$ Creutz
ladder, and only has its PH symmetry, $\mathcal{C}_{C}$. This is related to
the fact that one of the legs is formed by strong links, and the other one by
weak ones [see Fig. 2 (e)]. Given that the TR symmetry in Creutz-related
ladders swaps the two legs, this imbalance breaks TR symmetry. The effects of
this symmetry breaking can be easily seen in its spectrum [Fig. 4 (e)], which
is symmetric about the origin $k=0,E=0$, but not in $E$ or in $k$. The first
and third gaps are narrowest at opposite values of $k$, and not in $k=\pi$ as
in the other ladders.

Therefore, the $\Delta$-CSSH ladder with $\phi=\pi$ and $m\neq 0$ belongs to
the D class. Other instances of 1D or quasi-1D topological D-class materials
are typically superconducting systems. Its topology is characterized by a
$\mathcal{Z}_{2}$ invariant, which in the 1D D class coincides with the Zak
phase555Its value for the middle gap can also be calculated using the Pfaffian
of the Hamiltonian. This has been confirmed numerically to coincide with
$\tilde{\mathcal{Z}}_{2}$ in the $\Delta$-CSSH ladder. [39, 89]. As depicted
in Fig. 3 (e), $\tilde{\mathcal{Z}}_{1}$ takes a constant value of $3\pi/2$,
except in the Creutz limit $\theta=\pi/4$, where it becomes undefined. On the
other hand, $\tilde{\mathcal{Z}}_{2}$ indicates that protected zero modes will
appear if $m<2\sqrt{JJ^{\prime}}$. These states are represented in Fig. 6
(k,l) for $J=2J^{\prime}$ and $m=0.5$. They are only protected by PH symmetry,
as there is no chiral symmetry in the system.

Additionally, non-protected edge states located in the left end of the ladder
appear in the first and third gaps. They can be seen in the spectrum in Fig. 5
(g,h,l). One state of this kind in the flat band case is depicted in Fig. 6
(j). Interestingly, these states do not follow the usual bulk-boundary
correspondence, and can be related instead to a Chern number. We discuss these
states in detail in subsection 4.4, and in Appendix D.

As usual, when $\phi=\pi$ and $m=0$, the bands flatten and the chiral and TR
symmetries $\mathcal{X}_{S}$ and $\mathcal{T}_{SC}$ appear (see Appendix B).
The model is BDI and topological ($\tilde{\mathcal{Z}}_{2}=\pi$). The end
states of the system are now totally localized to two sites due to AB caging.

####  4.3.2 $\phi\neq 0,\pi$

In the $m\neq 0$ case, the model has no relevant symmetries [see Fig. 4 (f)]
and belongs to class A, not topologically protected in 1D. The second gap end
modes are not degenerate even in the even-runged, large chain case, because
the boundaries of the system break spatial inversion. In this case,
$\tilde{\mathcal{Z}}_{1}\approx\tilde{\mathcal{Z}}_{3}$, so we only represent
$[\tilde{\mathcal{Z}}_{1},\tilde{\mathcal{Z}}_{2}]$ in the phase diagram [Fig.
3 (f)].

When $m=0$, the chiral symmetry $\mathcal{X}_{S}$ puts the system in class
AIII and makes the system topological, with $\tilde{\mathcal{Z}}_{2}=\pi$ and
$\tilde{\mathcal{Z}}_{1}=3\pi/2$, and hosts a pair of protected zero modes in
the middle gap (see Appendix C). Like their -CSSH counterparts, these states
are localized in only two sites, and they can be tuned in the same way as
them. Non-protected states of the same nature as in the $\pi$-flux can also
appear, see the next subsection for details. There are other known ways to
achieve AIII TIs with the geometry of the Creutz ladder: to include an energy
imbalance between the legs [11], or additional complex hopping amplitudes in
the diagonal or vertical tunneling amplitudes [10, 12, 13]. As we have seen,
by changing its parameters, the $\Delta$-CSSH ladder can belong to three of
the five topological classes in 1D (see Table 1). The end modes remain
protected when changing from one class to another.

The Zak phase is not always quantized in the $\Delta$-CSSH ladder because,
like the -CSSH case, it is only invariant under a specific kind of generalized
inversion symmetry. This transformation only flips the longitudinal coordinate
of the ladder (see Appendix B). The value of $\tilde{\mathcal{Z}}_{2}$ is
quantized when the system belongs to a topological symmetry class, but not
$\tilde{\mathcal{Z}}_{1}$, as we discuss in Section 4.4.

####  4.3.3 $\phi=0$

The system is topologically trivial. Like in the other models, a doubly
degenerate flat band appears in $E=0$ in the rungless regime. The other two
bands are dispersive and touch the flat band for all values of $\theta$,
making the system metallic. Like in the -CSSH case, when $m$ is switched on, a
non-degenerate flat band spanned by $(\ket{C}-\ket{D})/\sqrt{2}$ lies at
$E=m$, while the rest of the bands become dispersive.

###  4.4 Non-protected states and Chern numbers

In the rhomboid and delta CSSH ladders, a single edge state can be found in
the first or in the third gaps, and they often appear in both at the same
time. These states, when present, are both localized at the same side of the
ladder, not one at each side. They can be found at the left or right ends in
the -CSSH ladder, and only at the left end in the $\Delta$-CSSH ladder. These
states are not protected, as they are not located in the central gap (see
Appendix E for details), and they do not follow the usual bulk-boundary
correspondence, as they appear even though $\tilde{\mathcal{Z}}_{1}$ takes
intermediate values between $0$ and $\pi$ [see Fig. 3 (d,e,f)]. However, they
have a topological nature, given that they appear due to a topological
obstruction that can be understood by exploring the Wannier functions of the
model. For a detailed analysis, please refer to Appendix D.

Interestingly, this kind of edge states have been recently reported in a
recent paper [41] in trimer chain models with no inversion symmetry. In said
work, it is argued that these edge states, even though they exist in a 1D
model, are related to the 2D Chern number associated to the model changing
along a closed path in its space of parameters. This is associated with the
topological pumping that would happen if the model was physically changed in
time following the chosen path [41, 90].

In the CSSH ladders, an analogous invariant can be found, if we consider an
appropriate extension of the models. In the -CSSH ladder, the relevant path
along parameter space can be chosen as [see Fig. 7 (a)]:

$\displaystyle t_{1}=A_{0}+A\sin(-\frac{3\pi}{4}+\varphi)$ $\displaystyle
t_{2}=A_{0}+A\sin(\frac{3\pi}{4}+\varphi)$ $\displaystyle
t_{3}=A_{0}+A\sin(-\frac{\pi}{4}+\varphi)$ $\displaystyle
t_{4}=A_{0}+A\sin(\frac{\pi}{4}+\varphi)$ (20)

where $A_{0}=(J+J^{\prime})/2$, $A=(J-J^{\prime})/\sqrt{2}$, $0<\varphi<2\pi$
is the parameter along the path, $t_{i}$ are the diagonal hopping amplitudes
in Hamiltonian (1). The rest of the hopping amplitudes are constant along the
path. For the $\Delta$-CSSH ladder, the value of $t_{3}$ and $t_{4}$ must be
interchanged.

For $\varphi=0$, the system is a \- or $\Delta$-CSSH ladder as defined in
Section 2. In the opposite point of the path, $\varphi=\pi$, the system is the
result of reflecting said ladder along the horizontal axis, and thus the value
of its Zak phase will be the opposite. This system will have non-protected end
modes on the opposite side to the non-reflected version. As an example, the
different models visited along the path in the delta case are depicted in Fig.
7 (b).

Figure 7: (a) Hopping parameters along the path for the calculation of the
Chern number, for $J=0.7$ and $J^{\prime}=0.3$. (b) Scheme of the hopping
parameters for the delta case as a function of $\varphi$, where the darker
links have an amplitude of $J$ and the lighter, dotted lines take the value
$J^{\prime}$. (c) Zak phase of the first and second gaps along the path in the
$\theta=1,\phi=\pi/2,m=0$ $\Delta$-CSSH ladder. $\tilde{\mathcal{Z}}_{1}$
winds around once in the negative direction, and so $C_{1}=-1$, while
$\tilde{\mathcal{Z}}_{2}$ does not ($\tilde{C}_{2}=0$). (d) Spectrum with
periodic (light blue, in the background) and open boundary conditions (lines)
for the system in (c). The end states in the first and third gaps (green) move
around as dictated by its Chern number, while those in the central gap (red)
remain pinned at zero energy. The finite system has $L=16$.

For the $\phi=\pi$ $\Delta$-CSSH ladders, the PH symmetry must be broken at
some points to avoid band degeneracies along the path, and so in that case we
also consider a changing $\phi$, for instance:

$\phi(\varphi)=\frac{\pi}{4}[3+\cos(2\varphi)].$ (21)

Thus, $\phi(\varphi=0,\pi)=\pi$, and the degeneracies that otherwise appear at
$\varphi=\pi/2,3\pi/2$ are avoided. In the rhomboid case, the first and third
gaps of the $\pi$-flux systems close, so the invariant loses meaning there.

As the system changes along the path, the Zak phase of the first band starts
increasing or decreasing, and eventually wraps around and returns to its
initial value. This winding of the Zak phase indicates that the lowest band
has a nontrivial Chern number as it changes along the path. In the
$\Delta$-CSSH ladder, $C_{1}=\pm 1$ when $\theta\lessgtr\pi/4$. We can see
this for $\theta=1,\phi=\pi/2,m=0$ in Fig. 7 (c), where $C_{1}=-1$. The second
gap has an invariant of $\tilde{C}_{2}=C_{1}+C_{2}=0$, which predicts no edge
states of this kind there (the states in this gap are related to the usual Zak
phase instead). Equivalently, the value of $\tilde{\mathcal{Z}}_{2}$ along the
path does not wrap around, and instead remains constant [see Fig. 7 (c)]. The
value of $\tilde{C}_{3}=C_{1}+C_{2}+C_{3}$ is nontrivial, indicating the
possible existence of an edge state in the third gap.

It is important to note that invariant is associated with the whole path, not
just the particular system we are interested in. For this reason, the bulk-
boundary correspondence is only guaranteed to be satisfied when considering
the entire path, not at a single point in it. The value of the Chern number
cannot predict if the end states in the CSSH ladders will appear in the gaps
or not, it only indicates that a single state can appear in each gap. Its sign
also has a physical meaning: it indicates the direction of pumping of the end
modes if the system is changed along the path. The Zak phase itself, however,
is not without meaning. A value of $\tilde{\mathcal{Z}}_{1}$ around $\pi/2$
($3\pi/2$) indicates the edge states in the first gap will appear in the right
(left) end of the ladder, due to its relation to the Wannier centers, as
explained in Appendix D.

The situation is completely analogous to the trimer chain case [41]. In Fig. 7
(d), the spectrum with periodic (light blue background) and open boundary
conditions (lines) for the $\Delta$-CSSH ladder in subfigure (c) is
represented along the path, for $L=16$. We can appreciate the non-protected
end states, in green, moving around in their gaps, as dictated by their Chern
number. The protected end states in the central gap, in red, stay constant at
zero energy, as their Zak phase of $\pi$ dictates.

In the -CSSH ladder, $\tilde{C}_{1}$ and $\tilde{C}_{3}$ are also nontrivial,
while $\tilde{C}_{2}=0$. As discussed in Section 4.2.1, one of the two outer
gaps undergoes a topological phase transition as $m$ increases [see Fig. 3
(d)], across which the Zak phase jumps discretely but the number of end states
remains constant. The end of the ladder in which the end state is located
changes accordingly, from the right to the left end. This is accompanied by a
change in the Chern number associated with that gap. If $\theta<\pi/4$,
$\tilde{C}_{1}=-1$ if $-\pi<\phi<\pi\mod 4\pi$, where no phase transition is
present in the first gap. For other values of $\phi$, $\tilde{C}_{1}$ changes
from $-1$ to $1$ as $m$ increases and transverses its critical value. All
signs are reversed if $\theta>\pi/2$. The behavior of $\tilde{C}_{3}$ is
completely analogous, but the phase transitions are now only present if
$-\pi<\phi<\pi\mod 4\pi$, and all signs are reversed.

###  4.5 CSSH ladders with an odd number of rungs

The discussion above assumes systems with an even number of rungs, that is, an
integer number $N$ of unit cells. We now briefly discuss now the case when the
number of rungs $L$ is odd. This means half a unit cell is left unpaired,
either in the left or the right end of the ladder. If this choice of
boundaries broke the protecting symmetries, the central gap edge states would
not be protected anymore. However, all the chiral symmetries of the systems
are preserved under this choice of boundaries, implying that, when present,
the edge states in the middle gap will be chiral and protected. In the cases
where the topology is protected by $\mathcal{X}_{C}$ or
$\mathcal{X}_{\scalebox{0.5}{ \leavevmode\hbox to5.94pt{\vbox
to4.04pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{5.54402pt}{3.64pt}\pgfsys@lineto{2.18402pt}{3.64pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.36pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}$, the states appear
in the half unit cell when expected, and in the \- and $\Delta$-CSSH ladders
they take an analogous form to their counterparts on the other end, something
that is not true in the even-runged ladders. This is a consequence of the
presence of a global inversion symmetry, which will make the left and right
states degenerate in the thermodynamic limit, even in the $m\neq 0$, $\phi\neq
0,\pi$ case.

In the cases where the topology is protected by $\mathcal{X}_{S}$, the SSH-
like chiral operator, topological states appear in the half unit cell when the
Zak phase is trivial, and no states appear when it is topological. This is a
well-known phenomenon in the SSH chain [91, 43, 3], and can be easily
understood in terms of the lattice geometry. In the D-class topological
$\Delta$-CSSH, the states are protected by the PH symmetry $\mathcal{C}_{C}$,
which is not broken either in odd-runged ladders, and zero modes appear in the
half unit cells. For a discussion of the $m=0$ states in even and odd-runged
ladders, see Appendix C.

Finally, the non-protected states in the first and third gaps of the \- and
$\Delta$-CSSH ladders can also appear or disappear in the same way as SSH-type
zero modes. Depending on the choice of boundaries, both left and right non-
protected modes can be found in both ladders at the same time.

##  5 Experimental implementations

Our work considers the single-particle case only, and so it is applicable for
both bosonic and fermionic particles. As mentioned above, the Creutz ladder
has been implemented in a cold atom system using orbital-momentum coupling
[92, 93] and can be simulated in a specific regime [27] in a rhombus chain
photonic lattice [27, 28, 29]. Recently, a Creutz ladder plaquette was also
implemented in a superconducting system [94]. Some other theoretical proposals
have been made to implement the Creutz ladder in cold atom systems, either by
using two hyperfine states as the legs of the ladder [11] or two zigzag chains
[12]. A system of helical coupled-resonator waveguides [95] also produces the
Creutz ladder as an effective model. By changing the tunneling amplitudes in
the appropriate way, something particularly tunable in the mentioned systems,
these setups can be employed to implement the CSSH ladders, under certain
constraints. The easiest model to implement is probably the
$\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{1.5}{$\Join$}}}$-CSSH ladder,
given that the staggering of its amplitudes follows a simple SSH-like pattern.

Modified rhombus chains with a large chemical potential in its central sites
can be used to implement the three CSSH models, following the Creutz ladder
implementation presented in [27]. Their geometries are represented in Fig. 8.
The $\pi$-flux $m=0$ case can be obtained directly, but the $\phi$ and $m$
degrees of freedom are not independent, and that restricts the regimes that
can be obtained. For a detailed explanation and the relevant effective
Hamiltonians, refer to Appendix F. This general scheme can be used to
implement the CSSH ladders in a photonic lattice or other systems, like
ultracold atoms or superconducting circuits.

In particular, the tunneling amplitudes in cold atom implementations [12, 11,
92, 93] are related to the intensity of the lasers that induce the Raman-
assisted transitions. This provides an easy way to independently tune each
amplitude and obtain any of the CSSH ladders. In some proposals, like for
instance [12], the $\phi$ and $m$ parameters can be tuned independently, and
this would allow for the implementation of the D class TI or the rungless
topological phase with any $\phi$.

Four-site versions ($L=2$) of all $m=0$ CSSH models can be implemented in the
superconducting system in [94], by changing the appropriate tunneling
amplitudes. It is worth noting that, in this limit, the
$\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{1.5}{$\Join$}}}$-CSSH ladder
reduces to a Creutz-style plaquette (which, in turn, can be rearranged as a
square plaquette with a magnetic flux of $2\phi$), and the  and $\Delta$-CSSH
ladders become identical to each other. These systems, although small, still
have states at exactly zero energy, which are protected by the same symmetries
as those in the longer models.

The complex hopping amplitudes, which constitute one of the main difficulties
in implementing these models, a periodic on-site energy driving in a waveguide
lattice [27], and it appears naturally as a result of the two-photon
transition in the cold atom implementation [93], and of the pump that couples
the modes in the superconducting scheme [94].

Figure 8: Photonic waveguide lattice diagrams for the CSSH ladders. (a)
Photonic rhombus chain with a large energy offset in the middle sites (green).
Its low-energy effective Hamiltonian is equivalent to the Creutz ladder [27].
(b) By deforming the rhombi as depicted, the
$\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{1.5}{$\Join$}}}$-CSSH ladder
can be implemented. For $\phi=\pi$, the rungless ($m=0$) regime is realized,
while for other values of $\phi$, the vertical links are nonzero and acquire a
phase. (c) If the odd rhombi are substituted by rhomboids, the -CSSH ladder
arises. As in (b), when $\phi=\pi$ the rungless model is achieved. When $\phi$
takes another value, $m$ is turned on. (d) If the odd rhombi are modified as
pictured, a variant of the $\Delta$-CSSH ladder is obtained. It has some
additional on-site energies that can be compensated to obtain the desired
model. Like in the other ladders, the topological $\pi$-flux rungless case can
be obtained, and other values of $\phi$ cause $m\neq 0$.

Additionally, given that the fermionic Creutz model is equivalent to a
spin-$1/2$ particle hopping in a chain of sites in a specific gauge field [80,
11, 96], different fermionic implementations could also be considered for both
the Creutz and the CSSH ladders, including fermionic cold atoms,
semiconducting quantum dots or trapped ions, provided that the appropriate
gauge field and different tunneling amplitudes and spin-orbit couplings can be
established.

##  6 Conclusions

In this work, we propose and analyze a family of quasi-1D topological
insulators, the CSSH ladders, with rich topological phase diagrams and flat-
band regimes due to Aharonov-Bohm caging. Each of the models has some peculiar
properties. The hourglass CSSH ladder can exhibit Creutz-like and SSH-like
topological zero modes. The latter arise in the absence of a magnetic field,
in contrast with the Creutz ladder case, where the $\phi=0$ model is trivial.
The rhomboid CSSH ladder has a tunable chiral symmetry, which can be used to
predict which will be the best quasi-1D model to use in each experimental
setup to increase the protection against disorder. And, finally, the
$\pi$-flux delta CSSH ladder is a quasi-1D topological material in the D class
with no superconducting term, something rather unusual. By changing its
parameters, this topological phase can be adiabatically connected with a BDI
($\phi=\pi$,$m=0$) and an AIII phase ($\phi\neq 0,\pi$,$m=0$), without losing
the protection of the edge states. This can be of interest to explore the
differences between 1D topological symmetry classes in a simple model.

The wide variety of topological states presented here are also tunable by
changing the $m$, $\phi$ or $\theta$ parameters, which provides a high degree
of control. They also have the robustness implicit to topological protection,
which is still present in small, experimentally accessible systems in the
rungless case. For these reasons, they are highly suitable to be used in the
fields of quantum information and computation. For instance, they could be
used in a quantum memory, as part of a symmetry-protected qubit architecture,
or as a means of transferring particles or states between the two ends of the
ladder in a robust way. Additionally, the topological end modes in the
rungless CSSH ladders are pinned at exactly zero energy, even for ladders only
a few sites long, thanks to AB localization. This feature is inherited from
the Creutz ladder, and makes a wide variety of topologically protected zero
modes experimentally available, even in small systems. An interesting follow-
up to this research would be to explore the potential of these topological
zero modes in different quantum information tasks, and compare them to other
topological systems. Other intriguing topic is the study of the non-protected
states in the rhomboid and delta CSSH ladder, which do not follow the bulk-
boundary correspondence and are instead related to a Chern number. Previously
reported on the trimer chain, they are reminiscent of topological states in
quasicrystals. Their generalization to higher dimensions may be a promising
topic for future research. Finally, the applications of the wide variety of
topological pumps that are possible in these models, some of them present in
the Creutz ladder [12] and others in the trimer chain [90], also constitute an
interesting subject to explore further.

Finally, we propose a setup to implement the CSSH ladders in a photonic
lattice system as an effective low-energy model in three modified rhombus
chains. The large number of flat-band systems recently implemented in this
kind of environment show that these lattices can be realized with the current
technology. Given that our study applies to both bosons and fermions, the CSSH
ladders can be realized in many other platforms, such as cold atoms,
semiconducting quantum dots or superconducting circuits.

## Acknowledgements

We would like to thank Álvaro Gómez-León and Luca Leonforte for fruitful
discussions. C. E. Creffield was supported by Spain’s MINECO through Grant No.
FIS2017-84368-P. G. Platero and J. Zurita were supported by Spain’s MINECO
through Grants No. MAT2017- 86717-P and PID2020-117787GB-I00 and by CSIC
Research Platform PTI-001. J. Zurita recognizes the FPU program FPU19/03575.

##  Appendix A Topological invariants

To probe the topology of the CSSH ladders, we have used the Zak phase,
calculated numerically using bulk Hamiltonian (2). In this Appendix, we
specify some additional details about this calculation, which are not often
discussed in the literature.

In the strong topological insulator phases of the models (that is, when
$\phi=\pi\mod 2\pi$ or $m=0$), they belong to the symmetry classes BDI, AIII
and D, which can harbour topological phases in 1D. The topological invariant
in classes BDI and AIII is the winding number $\nu$, of type $\mathbb{Z}$. In
class D, the invariant, $\mathcal{Q}$, is related to the Pfaffian of the bulk
Hamiltonian, of type $\mathbb{Z}_{2}$.

However, these topological invariants satisfy the following relations to the
Zak phase $\mathcal{Z}$ [12, 10, 39, 89]:

$\displaystyle\nu=\mathcal{Z}/\pi\mod 2$ (22)
$\displaystyle\mathcal{Q}=\mathcal{Z}/\pi,$ (23)

provided spatial inversion symmetry is present and the center of the unit cell
is chosen as the origin of coordinates.

Even in the cases where inversion symmetry is not present, the Zak phase is
still a good topological invariant for protected zero modes, as we explain in
the main text, and can also give us information about the non-conventional end
states, related to a Chern number instead, that appear in the \- and
$\Delta$-CSSH ladders (see subsection 4.4 and Appendix D).

In the CSSH ladders, two distinct nontrivial phases appear, with $\nu=\pm 1$,
corresponding to $0<\phi<2\pi$ and $-2\pi<\phi<0$ respectively, taken
$\\!\\!\mod 4\pi$. These two phases cannot be distinguished with the Zak phase
alone. They can be characterized by the chirality of their edge states [8,
12], just like their counterparts in the Creutz ladder, to which they can be
connected adiabatically. Apart from this detail, our work focuses between the
difference between trivial and nontrivial phases, to which the Zak phase is
sensitive.

To calculate the Zak phase numerically, we have used the following formula,
which can be obtained from its definition (8) by discretizing momentum space
and using the Taylor formula for the logarithm [97, 98]:

$\mathcal{Z}=-\textrm{Im}\enskip\textrm{log}\prod_{k\in
BZ}\braket{u_{k}^{i}}{u^{i}_{k+\delta k}}$ (24)

where $\\{\ket{u_{k}^{i}}\\}$ is the $i$-th eigenstate of the bulk Hamiltonian
$\mathcal{H}(k)$ of the system. This expression will be valid as long as the
step $\delta k$ in momentum space is small enough. It is independent of the
gauge chosen for the eigenvectors $\\{\ket{u_{k}^{i}}\\}$, something necessary
for numerical calculations.

In the convention we use for the bulk Hamiltonian, which is sometimes called
Basis I in the literature [99, 100], the Zak phase calculated using (24) does
not depend on the positions of the orbitals inside the unit cell, given that
the unit cell number $n$ is treated as the spatial coordinate, and $\alpha$ as
an internal degree of freedom. Another convention exists, Basis II, in which
these positions would be taken into account in the bulk Hamiltonian, because
the spatial coordinate is made to depend on both $n$ and $\alpha$. The version
of the Zak phase obtained in this basis carries more information than in Basis
I, because it depends on the position of the orbitals. For example, the
Wannier centers calculated with it will be sensitive to the distribution of
the wavefunctions inside each unit cell, while the Zak phase in Basis I will
not. These distinctions have been noted in recent works, and some authors
maintain that the name “Zak phase” should be reserved for the quantity that
contains information about the positions of the orbitals [100, 101]. However,
we keep the terminology for its Basis I counterpart, in agreement with many
works in the literature [102, 103, 8, 104, 105, 72, 106, 107, 108] and the
reference text by Asbóth et al. [42]. When calculating the Zak phase in Basis
II or in the continuum, an additional non-homogeneous phase factor has to be
included in the last ket of the closed loop in expression (24) [98].

Even though the Zak phases in Bases I and II are two distinct quantities, both
are equally valid to study the topology of 1D materials. Basis I seems to be
used more often in the literature, probably because its calculations are more
straight-forward [42, 102, 103, 8, 104, 105, 72, 106, 107, 108]. Works on the
experimental detection of the Zak phase have used both Basis I [107] and Basis
II [109].

Finally, we want to note an important point. The Zak phase, in both bases,
depends on the choice of the origin of coordinates. This happens because that
choice is related to a gauge transformation on the Bloch states
$\\{\ket{u_{k}^{i}}\\}$ [98, 110]. In particular, the Bloch states at
$k=\pm\pi/2$ can have, in general, different phases. The expression we use,
equation (24), corresponds to the Zak phase calculated using a periodic gauge
for the cell-periodic Bloch states, where both states are equal. In Basis I,
the choice of coordinates $x_{n}=na_{0}$ (where $a_{0}$ is the lattice
constant, taken to be 1 in the main text), which we use, provides this type of
gauge, and also allows us to use expressions (22) and (23).

##  Appendix B Full list of symmetries

The full list of symmetries defined in the text is detailed below. The
conventional, untilded symmetries act on $\mathcal{H}(k)$, and are written in
the standard internal basis. As usual, they act on the spatial coordinate by
leaving it unchanged, if unitary, or with a complex conjugation, if
antiunitary. The hidden symmetries are represented by tilded operators, which
act on the full Hilbert space and are expressed in the standard basis.

The symmetries of the SSH chain Hamiltonian (7), expressed in the
$\\{\ket{I},\ket{I\\!I}\\}$ basis, are
$\mathcal{C}^{(2)}_{S}=\sigma_{z}\mathcal{K}$,
$\mathcal{T}^{(2)}_{S}=\mathcal{K}$ and $\mathcal{X}^{(2)}_{S}=\sigma_{z}$,
where the superindex is a reminder of the dimension of the matrices and
$\mathcal{K}$ is complex conjugation.

The $m=0$ hourglass and rhomboid CSSH ladders satisfy a chiral symmetry of
this type:

$\displaystyle\mathcal{X}_{S}=\begin{pmatrix}\mathbbm{1}_{2}&0\\\
0&-\mathbbm{1}_{2}\end{pmatrix},$ (25)

in the basis $\\{\ket{A},\ket{B},\ket{C},\ket{D}\\}$, where $\mathbbm{1}_{2}$
is the $2\times 2$ identity matrix. The $m=0$ Creutz ladder satisfies the
related hidden symmetry:

$\tilde{\mathcal{X}}_{S}=\mathop{\mathrm{diag}}\nolimits_{2L}(1,1,-1,-1,1,1,-1,-1,\ldots),$
(26)

in the basis $\\{\ket{1,a},\ket{2,b},\ldots,\ket{L,b}\\}$.

The Creutz ladder Hamiltonian with $\phi=\pi$ satisfies the symmetries
$\mathcal{C}^{(2)}_{C}=\sigma_{z}\mathcal{K}$,
$\mathcal{T}^{(2)}_{C}=\sigma_{x}\mathcal{K}$ and
$\mathcal{X}^{(2)}_{C}=i\sigma_{y}$, written in the basis
$\\{\ket{a},\ket{b}\\}$.

The symmetry $\mathcal{T}^{(2)}_{C}$ flips the upper and lower legs of the
ladder, something unexpected in the bosonic case [34]. It is, however, a more
natural TR transformation if we consider the two legs of the ladder as the up
and down components of a spin-$1/2$ particle, something we can do if we take
$c^{(\dagger)}_{n,\alpha}$ to be fermionic operators. This approach has been
explored in [38, 34, 80, 96, 111]. However, a fermionic TR symmetry usually
squares to $-\mathbbm{1}$, not $\mathbbm{1}$ as is the case, so it is still
non-conventional in that regard. In any case, $\mathcal{T}^{(2)}_{C}$ behaves
like a regular TR symmetry in all cases as far as topology is concerned, so we
keep the usual nomenclature for its operator and the associated symmetry
classes. The same can be said about many of the transformations below, which
have a non-conventional form but fulfill their role as chiral, PH or TR
symmetries.

We say then that the $\pi$-flux Creutz ladder belongs to the BDI class [7, 38,
11, 16, 68]. Other authors do not consider the mentioned TR and PH symmetries,
and thus classify the system as a member of the AIII class, with only chiral
symmetry [8, 112]. Given that the chiral symmetry is enough to protect the
topological states, both classifications are valid.

Even though $\mathcal{C}^{(2)}_{S}$ and $\mathcal{C}^{(2)}_{C}$ have the same
matrix expression, they have different generalizations in the CSSH ladders due
to the different degrees of freedom in which they act. The analogues of the
mentioned symmetries in the CSSH ladders are (see Section 4):

$\displaystyle\mathcal{C}_{C}=\begin{pmatrix}\sigma_{z}&0\\\
0&\sigma_{z}\end{pmatrix}\mathcal{K}$ (27)
$\displaystyle\mathcal{T}_{C}=\begin{pmatrix}\sigma_{x}&0\\\
0&\sigma_{x}\end{pmatrix}\mathcal{K}$ (28)
$\displaystyle\mathcal{X}_{C}=i\begin{pmatrix}\sigma_{y}&0\\\
0&\sigma_{y}\end{pmatrix}.$ (29)

There are also some composite symmetries, formed by the composition of an SSH-
type and a Creutz-type symmetry (hence the $SC$ subindex):

$\displaystyle\mathcal{C}_{SC}=\mathcal{X}_{S}\mathcal{T}_{C}^{-1}=\begin{pmatrix}\sigma_{x}&0\\\
0&-\sigma_{x}\end{pmatrix}\mathcal{K}$ (30)
$\displaystyle\mathcal{T}_{SC}=\mathcal{C}_{C}^{-1}\mathcal{X}_{S}=\begin{pmatrix}\sigma_{z}&0\\\
0&-\sigma_{z}\end{pmatrix}\mathcal{K},$ (31)

expressed in the internal CSSH ladder basis.

The $m=0$ Creutz ladder satisfies the following hidden composite PH symmetry:

$\tilde{\mathcal{C}}_{SC}=\mathop{\mathrm{diag}}\nolimits_{L}(\sigma_{x},-\sigma_{x},\sigma_{x},-\sigma_{x},\ldots)\mathcal{K},$
(32)

in the basis $\\{\ket{1,a},\ket{2,b},\ldots,\ket{L,b}\\}$.

Some symmetries of the $\pi$-flux rhomboid CSSH ladder have a unique form:

$\displaystyle\mathcal{T}_{\scalebox{0.5}{ \leavevmode\hbox to5.94pt{\vbox
to4.04pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{5.54402pt}{3.64pt}\pgfsys@lineto{2.18402pt}{3.64pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.36pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}=\begin{pmatrix}-is_{\vartheta}&c_{\vartheta}&0&0\\\
c_{\vartheta}&-is_{\vartheta}&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{pmatrix}\mathcal{K}$ (33)
$\displaystyle\mathcal{X}_{\scalebox{0.5}{ \leavevmode\hbox to5.94pt{\vbox
to4.04pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{5.54402pt}{3.64pt}\pgfsys@lineto{2.18402pt}{3.64pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.36pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}=\begin{pmatrix}is_{\vartheta}&c_{\vartheta}&0&0\\\
-c_{\vartheta}&-is_{\vartheta}&0&0\\\ 0&0&0&1\\\ 0&0&-1&0\end{pmatrix}$ (34)

where

$\displaystyle s_{\vartheta}=\sin\vartheta=\frac{J^{\prime 2}-J^{2}}{J^{\prime
2}+J^{2}}$ (35) $\displaystyle
c_{\vartheta}=\cos\vartheta=\frac{2J^{\prime}J}{J^{\prime 2}+J^{2}},$ (36)

and with $\vartheta=2\arctan(\cos 2\theta)$. These symmetries reduce to
$\mathcal{T}_{C}$ and $\mathcal{X}_{C}$ when $J=J^{\prime}(\vartheta=0)$,
because the model becomes a Creutz ladder. On the other hand, when
$\vartheta=\pm\pi/2$, which corresponds to $J=0$ and $J^{\prime}=0$
respectively, we obtain the orthogonal dimer chain, with the following
symmetries:

$\displaystyle\left.\mathcal{T}_{\scalebox{0.5}{ \leavevmode\hbox
to5.94pt{\vbox to4.04pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{5.54402pt}{3.64pt}\pgfsys@lineto{2.18402pt}{3.64pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.36pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\right|_{\vartheta=\pm\pi/2}=\begin{pmatrix}\mp
i\mathbbm{1}_{2}&0\\\ 0&\sigma_{x}\end{pmatrix}\mathcal{K}$ (37)
$\displaystyle\left.\mathcal{X}_{\scalebox{0.5}{ \leavevmode\hbox
to5.94pt{\vbox to4.04pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{5.54402pt}{3.64pt}\pgfsys@lineto{2.18402pt}{3.64pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.36pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\right|_{\vartheta=\pm\pi/2}=\begin{pmatrix}\mp
i\sigma_{z}&0\\\ 0&i\sigma_{y}\end{pmatrix},$ (38)

in addition to the PH symmetry $\mathcal{C}_{C}$, present for all -CSSH
ladders with $\phi=\pi$.

The chiral and PH symmetries protect the zero modes in these models, as long
as they are not broken. These symmetries are usually broken by on-site energy
disorder. In the Creutz limit, however, the chiral symmetry can withstand any
on-site energy configuration, as long as the energies in the upper leg of the
ladder have the opposite value as those in the lower leg. This will also make
it more robust against general on-site energy disorder, given that some of the
disordered configurations will randomly approach this case. The chiral
symmetry in the orthogonal dimer chain, however, only resists symmetric on-
site energies in the $C,D$ sites.

We now show this analytically. We can quantify the degree of symmetry breaking
in an open system with on-site energies in the odd rungs, $\epsilon_{A},B$,
with the following quantities:

$\displaystyle x=\|{\tilde{\mathcal{X}}_{\scalebox{0.5}{ \leavevmode\hbox
to5.94pt{\vbox to4.04pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\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@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{5.54402pt}{3.64pt}\pgfsys@lineto{2.18402pt}{3.64pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.36pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>\leavevmode\hbox to5.94pt{\vbox to4.04pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\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@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{5.54402pt}{3.64pt}\pgfsys@lineto{2.18402pt}{3.64pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.36pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>\leavevmode\hbox to5.94pt{\vbox to4.04pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\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@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{5.54402pt}{3.64pt}\pgfsys@lineto{2.18402pt}{3.64pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.36pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}^{\dagger}+\mathcal{H}}\|$
(39) $\displaystyle t=\|{\tilde{\mathcal{C}}_{C}\mathcal{H}_{\scalebox{0.5}{
\leavevmode\hbox to5.94pt{\vbox to4.04pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{5.54402pt}{3.64pt}\pgfsys@lineto{2.18402pt}{3.64pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.36pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\tilde{\mathcal{C}}_{C}^{-1}+\mathcal{H}}\|,$
(40)

where $\|{\cdot}\|$ is the Frobenius norm, and the Hamiltonian
$\mathcal{H}_{\scalebox{0.5}{ \leavevmode\hbox to5.94pt{\vbox
to4.04pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{5.54402pt}{3.64pt}\pgfsys@lineto{2.18402pt}{3.64pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.36pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}$, as well as the
tilded operators, acts on the full real space. The results, in the Creutz and
orthogonal dimer chain limits, are, respectively,

$\displaystyle x_{C}=2\sqrt{2}|\epsilon_{A}+\epsilon_{B}|$ (41) $\displaystyle
x_{OD}=4\sqrt{\epsilon_{A}^{2}+\epsilon_{B}^{2}}$ (42)

while $t=x_{OD}$ for all values of $\theta$. Thus, $x_{OD}>0$ for any value of
the energies, while $x_{C}$ is zero as long as $\epsilon_{A}=-\epsilon_{B}$.

The TR symmetry for the -CSSH ladder with a general value of $\phi$ has a more
complicated form:

$\displaystyle\mathcal{T}^{\prime}_{\scalebox{0.5}{ \leavevmode\hbox
to5.94pt{\vbox to4.04pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{5.54402pt}{3.64pt}\pgfsys@lineto{2.18402pt}{3.64pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.36pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}=\begin{pmatrix}S(\theta,\phi)&C(\theta,\phi)&0&0\\\
C(\theta,\phi)&S(\theta,\phi)&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{pmatrix}\mathcal{K},$ (43)

with

$\displaystyle S(\theta,\phi)=e^{i\phi/2}\frac{J^{2}-J^{\prime
2}}{J^{2}-e^{i\phi}J^{\prime 2}}$ (44) $\displaystyle
C(\theta,\phi)=(1-e^{i\phi})\frac{JJ^{\prime}}{J^{2}-e^{i\phi}J^{\prime 2}},$
(45)

which does not depend on $\xi$, only on $\theta$ (as mentioned in the text,
$J=\xi\sin^{2}\theta$ and $J^{\prime}=\xi\cos^{2}\theta$). Equation (33) is
actually a special case of this operator, with $\phi=\pi$. When $m=0$, the
system also has the chiral symmetry $\mathcal{X}_{S}$ and a PH symmetry
represented by:

$\displaystyle\mathcal{C}^{\prime}_{\scalebox{0.5}{ \leavevmode\hbox
to5.94pt{\vbox to4.04pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{5.54402pt}{3.64pt}\pgfsys@lineto{2.18402pt}{3.64pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.36pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\\!=\\!\mathcal{X}_{S}\mathcal{T}_{\scalebox{0.5}{
\leavevmode\hbox to5.94pt{\vbox to4.04pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{5.54402pt}{3.64pt}\pgfsys@lineto{2.18402pt}{3.64pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.36pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}^{\prime-1}\\!=\\!\begin{pmatrix}S(\theta,\phi)&C(\theta,\phi)&0&0\\\
C(\theta,\phi)&S(\theta,\phi)&0&0\\\ 0&0&0&-1\\\
0&0&-1&0\end{pmatrix}\mathcal{K}.$ (46)

The SSH chain with complex amplitudes that appears in one of the subspaces of
the rungless $\pi$-flux -CSSH ladder has the symmetries
$\mathcal{C}^{(2)}_{W}=\sigma_{x}\mathcal{K}$,
$\mathcal{T}^{(2)}_{W}=\sigma_{z}\mathcal{K}$ and
$\mathcal{X}^{(2)}_{W}=\sigma_{z}$.

The unitary commuting symmetries that allow us to reduce the Hilbert space
into two subspaces are:

$\displaystyle U=\begin{pmatrix}\sigma_{x}&0\\\ 0&\sigma_{x}\end{pmatrix}$
(47) $\displaystyle V=\begin{pmatrix}i\sigma_{y}&0\\\
0&-i\sigma_{y}\end{pmatrix}$ (48) $\displaystyle
W=\begin{pmatrix}is_{\vartheta}&c_{\vartheta}&0&0\\\
-c_{\vartheta}&-is_{\vartheta}&0&0\\\ 0&0&0&-1\\\ 0&0&1&0\end{pmatrix},$ (49)

present in the
$\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{1.5}{$\Join$}}}$-CSSH ladder
with $\phi=0$, in the
$\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{1.5}{$\Join$}}}$-CSSH ladder
with $\phi=\pi$ and $m=0$, and in the -CSSH ladder with $\phi=\pi$ and $m=0$,
respectively.

In the  and $\Delta$-CSSH $\phi=0$ cases, the Hamitonian acquires a TR
symmetry, $\mathcal{T}_{S}=\mathcal{K}$, and when $m=0$ it also has the PH and
chiral symmetries $\mathcal{C}_{S}=\sigma_{z}\otimes\mathbbm{1}\mathcal{K}$
and $\mathcal{X}_{S}=\sigma_{z}\otimes\mathbbm{1}$ (defined above). These are
the four-site counterparts of the SSH symmetries. However, these systems are
topologically trivial. A doubly degenerate flat band appears at zero energy
when $m=0$, and two dispersive bands are located over and under it, touching
it if $\theta=\pi/4$ in the -CSSH ladder, and for all values of $\theta$ in
the $\Delta$ ladder. When $m$ is turned on, one flat band remains at $E=m$,
and the other three bands are dispersive. Non-protected states appear in the
lower gap in the rhomboid ladder.

Finally, the inversion symmetry of the SSH, Creutz and
$\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{1.5}{$\Join$}}}$-CSSH ladder
with periodic boundary conditions can be expressed as

$\tilde{\mathcal{I}}=\mathop{\mathrm{diag}}\nolimits_{M}^{-1}(1\ldots 1),$
(50)

where $M$ is the total number of sites in the system and
$\mathcal{I}=\mathop{\mathrm{diag}}_{\ell}^{-1}(a,\ldots,z)$ is the
$\ell\times\ell$ antidiagonal matrix with elements
$\mathcal{I}_{1,\ell}=a,\ldots,\mathcal{I}_{\ell,1}=z$.

In the  and $\Delta$-CSSH ladders with periodic boundary conditions, the
inversion symmetry can be represented by the operators:

$\displaystyle\tilde{\mathcal{I}}_{\scalebox{0.5}{ \leavevmode\hbox
to5.94pt{\vbox to4.04pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{5.54402pt}{3.64pt}\pgfsys@lineto{2.18402pt}{3.64pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.36pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}=\left(\begin{array}[]{c|c}\mathcal{I}_{6}&0\\\
\hline\cr 0&\mathcal{I}_{4N-6}\end{array}\right)$ (53)
$\displaystyle\tilde{\mathcal{I}}_{\Delta}=\left(\begin{array}[]{c|c}\mathcal{J}_{6}&0\\\
\hline\cr 0&\mathcal{J}_{4N-6}\end{array}\right),$ (56)

respectively, where
$\mathcal{I}_{n}=\mathop{\mathrm{diag}}\nolimits^{-1}_{n}(1,\ldots,1)$ and
$\mathcal{J}_{n}=\mathop{\mathrm{diag}}\nolimits^{-1}_{n/2}(\sigma_{x},\ldots,\sigma_{x})$.

##  Appendix C Form of the caged topological edge states

In all CSSH ladders, the topological edge states are completely localized in
the two sites of the end rungs of the model when $m=0$ and $\phi\neq 0$. This
localization has the same nature as the AB caging of the $\pi$-flux case, but
for other values of $\phi$ it does not affect the bulk states. As discussed in
the main text, the protected modes will be pinned at exactly zero energy for
any $L>1$, making topological protection meaningful in systems with as few as
4 sites. The analytical form of the zero modes for the $m=0$ ladders with an
even number of rungs are:

$\displaystyle\ket{\mathcal{L}_{\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{0.7}{$\Join$}}}}}\\!=\\!\frac{\ket{1,A}-e^{i\phi/2}\ket{1,B}}{\sqrt{2}}$
(57) $\displaystyle\ket{\mathcal{L}_{\scalebox{0.5}{ \leavevmode\hbox
to5.94pt{\vbox to4.04pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{5.54402pt}{3.64pt}\pgfsys@lineto{2.18402pt}{3.64pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.36pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}\\!=\\!\ket{\mathcal{L}_{\Delta}}\\!=\\!\frac{J\ket{1,A}-J^{\prime}e^{i\phi/2}\ket{1,B}}{\sqrt{J^{2}+J^{\prime
2}}}$ (58)
$\displaystyle\ket{\mathcal{R}_{\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{0.7}{$\Join$}}}}}\\!=\\!\ket{\mathcal{R}_{\scalebox{0.5}{
\leavevmode\hbox to5.94pt{\vbox to4.04pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{5.54402pt}{3.64pt}\pgfsys@lineto{2.18402pt}{3.64pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.36pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}\\!=\\!\ket{\mathcal{R}_{\Delta}}\\!=\\!\frac{\ket{N,C}-e^{-i\phi/2}\ket{N,D}}{\sqrt{2}}$
(59)

When the number of rungs is odd, we find half a unit cell in one end of the
system. The edge states in that end, which we mark with a prime, are still
protected, but their form changes to one analogous to the state in the other
side, but with the opposite chirality. Their form is:

$\displaystyle\ket{\mathcal{L}^{\prime}_{\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{0.7}{$\Join$}}}}}\\!=\\!\ket{\mathcal{L}^{\prime}_{\scalebox{0.5}{
\leavevmode\hbox to5.94pt{\vbox to4.04pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{5.54402pt}{3.64pt}\pgfsys@lineto{2.18402pt}{3.64pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.36pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}\\!=\\!\ket{\mathcal{L}^{\prime}_{\Delta}}\\!=\\!\frac{\ket{1,A}-e^{i\phi/2}\ket{1,B}}{\sqrt{2}}$
(60)
$\displaystyle\ket{\mathcal{R}^{\prime}_{\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{0.7}{$\Join$}}}}}\\!=\\!\frac{\ket{N,C}-e^{-i\phi/2}\ket{N,D}}{\sqrt{2}}$
(61) $\displaystyle\ket{\mathcal{R}^{\prime}_{\scalebox{0.5}{ \leavevmode\hbox
to5.94pt{\vbox to4.04pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{5.54402pt}{3.64pt}\pgfsys@lineto{2.18402pt}{3.64pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.36pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}\\!=\\!\frac{J^{\prime}\ket{N,C}-Je^{-i\phi/2}\ket{N,D}}{\sqrt{J^{2}+J^{\prime
2}}}$ (62)
$\displaystyle\ket{\mathcal{R}^{\prime}_{\Delta}}\\!=\\!\frac{J\ket{N,C}-J^{\prime}e^{-i\phi/2}\ket{N,D}}{\sqrt{J^{2}+J^{\prime
2}}}$ (63)

We can see the states in the hourglass CSSH ladder do not change with respect
to the even-runged case.

Finally, the only non-protected edge states that can experience AB caging are
those in the $\Delta$-CSSH ladder, and only for $m=0$ and $\phi=\pi$. This is
because the other models do not have any non-protected states for that
parameter choice (see Fig. 5). One of them is represented in Fig. 6 (j). They
have the form:

$\displaystyle\ket{\ell\pm}=\frac{J^{\prime}\ket{1,A}+iJ\ket{1,B}}{\sqrt{2(J^{2}+J^{\prime
2})}}\mp\frac{i\ket{1,C}+\ket{1,D}}{2}$ (64)

##  Appendix D Non-protected end states and Wannier functions

In 1D topological insulators, the Zak phase acts like a $\mathbb{Z}_{2}$
topological invariant, which can also distinguish between the even and odd
phases of a $\mathbb{Z}$-type phase (inside each group, different phases can
be identified by their winding number). At the same time, it is related with
the 1D electric polarization. As shown by Zak in its original paper [74], the
Zak phase of a band $\mathcal{Z}_{i}$ is related to its Wannier center
$q_{i}$, by the relation $\mathcal{Z}_{i}=2\pi q_{i}$. This point is
calculated as the center of mass in the spatial coordinate of the Wannier
functions of the band.

A Zak phase of zero is associated with a Wannier function which is centered in
its unit cell. This corresponds to the trivial insulating phase. If the Zak
phase takes a value of $\pi$, the Wannier function has shifted half a unit
cell, and it is now spread between at least two contiguous unit cells. This is
directly related with the appearance of topological end modes: if open
boundary conditions are enforced, the two Wannier functions at the ends will
each be missing half of their support states, and so they will form states
outside the bands [105]. This is the situation in the SSH chain, the Creutz
ladder, the
$\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{1.5}{$\Join$}}}$-CSSH ladder
and for some parameters in the -CSSH ladder. It is most easily seen in the
flat band limit of the ladders, where an eigenbasis of Wannier functions can
be chosen, which only have support over a small number of sites (i.e., they
are compact). Each Wannier function spans four sites in these models, in a
similar way to the Creutz ladder case [7, 113]. The hourglass CSSH ladder bulk
eigenstates are:

$\displaystyle\ket{u_{\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{0.7}{$\Join$}}}n}^{\pm}}=\frac{1}{2}\left[\ket{n,A}+i\ket{n,B}\mp\left(i\ket{n,C}+\ket{n,D}\right)\right]$
$\displaystyle\ket{v_{\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{0.7}{$\Join$}}}n}^{\pm}}\\!=\\!\frac{1}{2}\left[\ket{n\\!\\!+\\!\\!1,A}\\!-\\!i\ket{n\\!\\!+\\!\\!1,B}\\!\pm\\!\left(i\ket{n,C}\\!-\\!\ket{n,D}\right)\right]$

with $n=1,\ldots,N$ the cell coordinate. The intracell (intercell) states
$\ket{u_{\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{0.7}{$\Join$}}}n}^{\pm}}$
($\ket{v_{\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{0.7}{$\Join$}}}n}^{\pm}}$)
form flat bands at
$E_{u\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{0.7}{$\Join$}}}}^{\pm}=\pm
2J^{\prime}$
($E_{v\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{0.7}{$\Join$}}}}^{\pm}=\pm
2J$). The rhomboid CSSH ladder bulk states are:

$\displaystyle\ket{u_{\scalebox{0.5}{ \leavevmode\hbox to5.94pt{\vbox
to4.04pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{5.54402pt}{3.64pt}\pgfsys@lineto{2.18402pt}{3.64pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.36pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}n}^{\pm}}=\frac{1}{\kappa}(J^{\prime}\ket{n,A}+iJ\ket{n,B})+$
$\displaystyle\mp\frac{1}{2}(i\ket{n,C}+\ket{n,D})$
$\displaystyle\ket{v_{\scalebox{0.5}{ \leavevmode\hbox to5.94pt{\vbox
to4.04pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@roundjoin\pgfsys@invoke{
}{{}}{}{}{}{}{}{}{}{}\pgfsys@moveto{5.54402pt}{3.64pt}\pgfsys@lineto{2.18402pt}{3.64pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.36pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}n}^{\pm}}=\frac{1}{\kappa}(J\ket{n+1,A}-iJ^{\prime}\ket{n+1,B})+$
$\displaystyle\pm\frac{1}{2}(i\ket{n,C}-\ket{n,D})$

with $n=1,\ldots,N$. The positive energy inter- and intracell states are
degenerate, with energy $E_{+}=\sqrt{2(J^{2}+J^{\prime 2})}$. The degenerate
negative energy states have energy $E_{-}=-\sqrt{2(J^{2}+J^{\prime 2})}$.

The intercell Wannier functions are centered on the boundary between unit
cells, and so they are responsible for the topological Zak phase and the
appearance of end modes. We demonstrate this in a schematic way in Fig. 9 (a).
In the Creutz and
$\mathbin{\rotatebox[origin={c}]{90.0}{\scalebox{1.5}{$\Join$}}}$-CSSH
ladders, the bulk-boundary correspondence always holds, and the number of end
states in a gap is predicted by the sum of the Zak phases of the bands below.

Figure 9: Wannier basis and topological edge states in the flat-band ladders.
(a) Wannier states in the Creutz, hourglass and rhomboid CSSH ladders with
periodic boundary conditions. Each pair of Wannier functions, marked with
$\pm$, span four contiguous sites. (b) In the open system, two unpaired halves
of Wannier states, one at each end, become the topological edge states. (c) In
the periodic delta CSSH ladder, each group of four Wannier states, labelled by
$A/B\pm$, span six sites in total. The whole ladder is covered by them. (d) In
the open system, three modes in total are left unpaired on the left end of the
ladder, and one on the right end. These become the two protected zero modes,
$\ket{\mathcal{L}},\ket{\mathcal{R}}$, and the two non-protected end states
$\ket{\ell\pm}$, which appear in the first and third gaps.

This is not the case for the \- and $\Delta$-CSSH ladders. An example can be
seen in the flat band $\Delta$-CSSH ladder. Its compact Wannier function
eigenbasis is:

$\displaystyle\ket{w_{\Delta
n}^{(A)\pm}}=\frac{1}{\sqrt{2}}\ket{n,A}\mp\frac{i}{\sqrt{8}}\left[\ket{n,C}-\ket{n-1,C}\right.$
$\displaystyle\left.-i\left(\ket{n,D}+\ket{n-1,D}\right)\right]$
$\displaystyle\ket{w_{\Delta
n}^{(B)\pm}}=\frac{1}{\sqrt{2}}\ket{n,B}\pm\frac{i}{\sqrt{8}}\left[\ket{n,D}-\ket{n-1,D}\right.$
$\displaystyle\left.+i\left(\ket{n,C}+\ket{n-1,C}\right)\right]$

with $n=1,\ldots,N$. The flat bands spanned by the two former states have
$E=\pm 2J^{\prime}$, and those spanned by the two latter, $E=\pm 2J$.

These Wannier states span five sites each (located on three rungs), and are
always centered on the odd rungs of the ladder, as we depict in Fig. 9 (c).
Their centers are located at a position of $3/4$ in each unit cell (with a
lattice constant of $a_{0}=1$ between equivalent sites). Given that
$\mathcal{Z}_{1}=3\pi/2$ there [see Fig. 3 (e)], the first-band Wannier center
position coincides with $\mathcal{Z}_{1}/2\pi$, as expected. In the bulk, all
states are accounted for, each $A$ or $B$ site participates in two of the
Wannier states, and each $C$ or $D$ site is part of four of them. When the
system is opened (without breaking any unit cells), a single end mode
$\ket{\mathcal{R}}$ appears on the right side, in the same way than in the
other ladders. However, in the left end, a total of 3 modes are left unpaired,
which reorganize into a protected end state $\ket{\mathcal{L}}$ in the middle
gap and two unprotected topological states $\ket{\ell\pm}$, with support on
the four first sites, in the first and third gaps [see Fig. 9 (d)]. Away from
the flat band point, the Wannier functions are no longer energy eigenstates,
but they can still cause the appearance of the end states can still appear. An
analogous situation occurrs in the -CSSH ladder, although not in its flat band
limit, so the analysis of its Wannier functions would be more involved.

By referring to the $\Delta$-CSSH flat band case again, we can conclude that
the topological states do not only appear because the compact Wannier