[1]Simone Traverso

[1]Physics Department, University of Genoa, Via Dodecaneso 33, Genova, 16146,
Italia

2]CNR-SPIN, Via Dodecaneso 33, Genova, 16146, Italia

# Emerging topological bound states in Haldane model zigzag nanoribbons

<EMAIL_ADDRESS>Maura Sassetti<EMAIL_ADDRESS>Niccolò
Traverso Ziani<EMAIL_ADDRESS>*  [

###### Abstract

Zigzag nanoribbons hosting the Haldane Chern insulator model are considered.
In this context, a reentrant topological phase, characterized by the emergence
of quasi zero dimensional in-gap states, is discussed. The bound states, which
reside in the gap opened by the hybridization of the counter-propagating edge
modes of the Haldane phase, are localized at the ends of the strip and are
found to be robust against on-site disorder. These findings are supported by
the behavior of the Zak phase over the parameter space, which exhibits jumps
of $\pi$ in correspondence to the phase transitions between the trivial and
the non-trivial phases. The effective mass inversion leading to the jumps in
the Zak phase is interpreted in a low energy framework. Setups with non-
uniform parameters also show topological bound states via the Jackiw-Rebbi
mechanism. All the properties reported are shown to be extremely sensitive to
the strip width.

###### keywords:

topological insulators, Haldane model, bound states, Zak phase

##  1 Introduction

The discovery of topological insulators and topological superconductors
completely revolutionized the usual classification of the phases of matter,
shedding light on the fact that the Ginzburg-Landau classification was but a
partial description [1]. Starting from the first feasible proposal for a
topological insulator [2], in less than two decades the field has undergone
several major breakthroughs, like the classification of topological phases by
symmetry classes [3], the discovery of higher order topology [4] and, very
recently, of non-Hermitian topology [5].  In this context, the dimensionality
of the systems plays a crucial role for the definition of the topological
invariants. Indeed, these are typically defined relative to systems
compactified in all directions. The bulk boundary correspondence, then,
suggests that when the system is non-trivial, and is made semi-infinite in one
direction, metallic states appear at the boundary. These metallic states
persist as long as the uncompactified dimension of the system does not become
comparable with their decay length.

The improvement in the nanostructuration of topological phases of matter has
made it possible to realize samples in which this condition is not met [6, 7].
In this regard, a very active branch of research is nowadays related to the
study of finite size effects on topological phases [8, 9, 10, 7, 11, 6, 12,
13, 14, 15, 16, 17, 18]. Indeed, dimensional crossovers between topological
phases is a promising way to engineer novel topological systems [19, 20]. In
this context, the way has been paved by the extensive studies performed on
graphene nanoribbons (GNRs), that have revealed an extremely rich
phenomenology. For instance, it was found that these systems, depending on the
nanoribbon width and on the nature of its terminations, can host robust
topological bound states [21, 22] amenable to detection with local probes [23,
24, 25].

Taking a step back, even before the first theoretical proposal for a symmetry
protected topological insulator was conceived, many topologically non-trivial
systems had been object of study, first of all the quantum Hall system [26].
In that context, a milestone for the comprehension of topological phases had
been conceived: the Haldane model [27]. Dated back to 1988, it represented the
first theoretical proposal for a system realising a quantum anomalous Hall
phase and is now regarded as the most famous model for a Chern insulator. Even
more importantly, the time reversal doubling of the Haldane model results in
the Kane-Mele model which, describing spinful fermions on a honeycomb lattice
with strong spin-orbit interaction, is the prototypical model of a time
reversal protected topological phase [28, 29]. Although the Kane-Mele model
was originally proposed to describe the electrons in graphene, where the
predicted spin Hall phase cannot be observed since the spin-orbit coupling is
too small to yield a sizable topological gap, it has recently found a direct
experimental realization in Bismuthene [30] and Germanene [31], and is hence
receiving renewed attention.

In light of this, Haldane model nanoribbons represent a significant model for
studying the physics of these newly discovered honeycomb-based topological
materials. Even more importantly, they configure themselves as the optimal
theoretical playground for merging the physics of nanostructured topological
insulators and the one inherited from graphene nanoribbons. In this paper, we
focus on zigzag Haldane nanoribbons and assess the effects of dimensional
reduction on the topological phase of the Haldane model. We find that, for
thin enough strips, there are multiple regions of the parameter space in which
the chiral edge states gap out and, correspondingly, degenerate pairs of quasi
zero-dimensional (0D) end-states appear whose energy lies inside the gap. Such
regions however are intercalated, through topological quantum phase
transitions, to phases without bound states. We hence unveil a complex, width-
dependent, reentrant quantum phase diagram, which we characterize by
numerically computing a well established indicator for the classification of
topological phases in (effectively) one-dimensional systems, that is, the Zak
phase [32, 33, 34]. Moreover,  we explain the mechanism leading to the mass
inversion via a phenomenological low energy theory, effectively modelling the
chiral edge states coupling. Finally, we show that domain walls in the on-site
staggered potential distribution can localize quasi 0D bound states. These
Jackiw-Rebbi like states [35], ubiquitous in topological phases of matter [36,
11, 37, 38, 39], retain a fractional charge of $\pm\frac{e}{2}$ ($e$ the
electron charge) and are found to be robust against the introduction of random
on-site disorder. More generally, our results demonstrate that the physics of
coupled topological edges can be way richer than what is expected from naive
low energy theories.

##  2 Results

###  2.1 Haldane model on zigzag nanoribbons

The Haldane model [27] describes spinless fermions on a honeycomb lattice,
pierced by an orthogonal periodic magnetic field and having the full symmetry
of the lattice and zero net flux over the unit cell. Time reversal symmetry is
broken for the model, so that the transverse Hall conductance can be non-
trivially quantized [40]. We denote the two sublattices of the honeycomb
lattice as $A$ and $B$ and we choose as primitive vectors
$\mathbf{a}_{1}=a(1,0)$ and
$\mathbf{a}_{2}=a(\frac{1}{2},\frac{\sqrt{3}}{2})$. Moreover, we place the
unit cell origin on the $A$ sites, so that the basis vectors are given by
${\bm{\delta}}_{a}=(0,0)$ and
${\bm{\delta}}_{b}=a(\frac{1}{2},\frac{1}{2\sqrt{3}})$. The Hamiltonian is
thus given by

$\begin{split}\mathcal{H}&=t_{1}(\sum_{l,n}a^{\dagger}_{ln}b_{ln}+a^{\dagger}_{ln}b_{l-1n}+a^{\dagger}_{ln}b_{ln-1})+\text{h.c.}\\\
&+t_{2}\mathrm{e}^{\mathrm{i}\phi}(\sum_{l,n}a_{ln}^{\dagger}a_{l+1n}+a^{\dagger}_{ln}a_{l-1n+1}+a^{\dagger}_{ln}a_{ln-1})+\text{h.c.}\\\
&+t_{2}\mathrm{e}^{-\mathrm{i}\phi}(\sum_{l,n}b_{ln}^{\dagger}b_{l+1n}+b^{\dagger}_{ln}b_{l-1n+1}+b^{\dagger}_{ln}b_{ln-1})+\text{h.c.}\\\
&+m(\sum_{l,n}a^{\dagger}_{ln}a_{ln}-b^{\dagger}_{ln}b_{ln}),\end{split}$ (1)

where $a_{ln}$ ($b_{ln}$) destroys a fermion at
$l\mathbf{a}_{1}+n\mathbf{a}_{2}+\bm{\delta}_{a(b)}$; $t_{1}$ and $t_{2}$
parameterize the intensity of the nearest and next to nearest neighbor
hoppings respectively; $\phi$ is a phase accounting for the staggered magnetic
flux inside the unit cell and $m$ tunes a term of staggered on-site potential
breaking inversion symmetry. In the topological phase, that occurs for
$|m/t_{2}|<3\sqrt{3}\sin\phi$, the bulk bands are gapped and the Chern number
[40] has a non-trivial value ($c=\pm 1)$ [27].

Correspondingly, in a strip geometry chiral modes localized on opposite edges
and with gapless dispersion relation occur, in accordance with the bulk
boundary correspondence. This can be explicitly seen by imposing periodic
boundary conditions (PBC) in the $\mathbf{a}_{1}$ direction and going in
$k$-space. For a strip of length $L=Na$, the Fourier transformation is defined
as

$\displaystyle a_{ln}$
$\displaystyle=\dfrac{1}{\sqrt{N}}\sum_{k}\mathrm{e}^{-\mathrm{i}kx_{ln}^{a}}a_{n}(k)$
$\displaystyle x_{ln}^{a}$ $\displaystyle=la+\frac{n}{2}a,$ (2) $\displaystyle
b_{ln}$
$\displaystyle=\dfrac{1}{\sqrt{N}}\sum_{k}\mathrm{e}^{-\mathrm{i}kx_{ln}^{b}}b_{n}(k)$
$\displaystyle x_{ln}^{b}$ $\displaystyle=la+\frac{n+1}{2}a,$ (3)

with the Bloch momenta discretized as $k_{j}=\dfrac{2\pi}{Na}j,\
j=0,\ldots,N-1$. The Bloch Hamiltonian is found to be of the form (see also
[41])

$\mathcal{H}(k)=\begin{bmatrix}g(\tilde{k},\phi)+m&2t_{1}\cos(\frac{\tilde{k}}{2})&g(\frac{\tilde{k}}{2},-\phi)&0&0&0&\cdots&0\\\
2t_{1}\cos(\frac{\tilde{k}}{2})&g(\tilde{k},-\phi)-m&t_{1}&g(\frac{\tilde{k}}{2},\phi)&0&0&\cdots&0\\\
g(\frac{\tilde{k}}{2},-\phi)&t_{1}&g(\tilde{k},\phi)+m&2t_{1}\cos(\frac{\tilde{k}}{2})&g(\frac{\tilde{k}}{2},-\phi)&0&\cdots&0\\\
0&g(\frac{\tilde{k}}{2},\phi)&2t_{1}\cos(\frac{\tilde{k}}{2})&g(\tilde{k},-\phi)-m&t_{1}&g(\frac{\tilde{k}}{2},\phi)&\ddots&0\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\cdots&\vdots\end{bmatrix},$ (4)

with $g(\tilde{k},\phi)=2t_{2}\cos(\tilde{k}+\phi)$ and $\tilde{k}=ka$.

In Fig. 1a is shown a schematic representation of a strip with zigzag edges,
$N_{y}=60$ sites in the vertical direction, and PBC along the $\mathbf{a}_{1}$
direction. In Fig. 1b are reported the corresponding energy bands, obtained
via numerical tight-binding diagonalization. Inside the bulk gap are clearly
visible the two modes corresponding to the chiral edge states characterizing
the topological phase.

Figure 1: Haldane chiral modes in a wide zigzag nanoribbon. a Scheme of the
lattice with PBC along the $\mathbf{a}_{1}$ direction. The dashed red lines at
the two ends of the strip mark the sites that are connected by PBC. b Bands of
the Haldane model in a strip configuration with zigzag edges, with PBC along
the $\mathbf{a}_{1}$ direction and $N_{y}=60$ sites in the $y$ (or
$\mathbf{a}_{2}$) direction. The parameters are set as $t_{1}=1,\ t_{2}=0.3,\
m=0,\ \phi=\pi/2$.

It can be expected that by reducing the strip width, the chiral edge states
may hybridize because of spatial overlap, giving rise to a gap opening in the
edge spectrum.  The length scale at which this phenomenon becomes relevant is
defined by the decay length of the edge states, that, for the Haldane model on
zigzag nanoribbons, has been found to be [42, 43]

$\xi_{\text{loc}}\approx\dfrac{\sqrt{3}}{2}\left[\log\left\\{\sqrt{1+\left(\frac{t_{1}}{4t_{2}}\right)^{2}}+\frac{t_{1}}{4t_{2}}\right\\}\right]^{-1}.$
(5)

This kind of behavior has been explicitly proven in several contexts [8, 44].
In spite of this, an analysis of the topological character of the gapped
phases originated by finite size effects is still missing in the context of
Chern insulators. In the next section, the results of such a study are
presented.

###  2.2 Emerging quasi 1D topological phase diagram

We start by numerically computing the amplitude of the gap $\Delta$ as a
function of the staggered on-site potential $m$. We set the energy scale to
$t_{1}=1$, and we impose $t_{2}=0.3$ and $\phi=\pi/2$, so that the topological
bulk gap of the Haldane model is maximized. We perform our analysis for the
Haldane strip with PBC for the different widths $N_{y}=4,6,8,10$. The results
are reported in Fig. 2.

Figure 2: Energy gap and Zak phase. Gap width $\Delta$ (top panels) and Zak
phase $\varphi$ (bottom panels) as a function of the staggered on-site
potential $m$, computed for strips of various widths: (a, e) $N_{y}=4$; (b, f)
$N_{y}=6$; (c, g) $N_{y}=8$; (d, h) $N_{y}=10$. The model parameters are set
to $t_{1}=1$, $t_{2}=0.3$ and $\phi=\pi/2$. For $N_{y}=6$ and $N_{y}=10$ the
system is gapless at $m=0$. In correspondence of every gap closing the Zak
phase jumps of $\pi$.

By comparing the top panels (Fig. 2a-d) it appears that the number of gap
closings and reopenings grows monotonically with the strip width. This non
trivial pattern hints to the fact that a topological phase transition may be
associated with the gap closings. Interestingly, for the strips whose number
of sites in the vertical direction is given by $N_{y}=4M+2$, the edge spectrum
is gapless for $m=0$: in these cases, despite the wave functions of the chiral
modes on opposite edges are brought in close proximity, they do not hybridize
with each other. This counterintuitive behavior is proven analytically in
Supplementary Note 1.

In order to characterize the topology of the zigzag Haldane strips, we use a
well established tool for assessing the topological properties of 1D solids:
the Zak phase [32], i.e. the natural open-loop extension of the Berry phase
[45] when the parameter space is the Brillouin zone. The Zak phase associated
with an isolated band was originally defined in terms of the cell-periodic
Bloch functions $\ket{u_{k}}$ as [32]:

$\varphi=\mathrm{i}\int_{0}^{2\pi/a}\mathrm{d}k\braket{u_{k}}{\partial_{k}}{u_{k}}.$
(6)

However, the above definition cannot be applied in the present case. Indeed,
we are dealing with a multiband system in which the valence bands may cross
each other and whose Hamiltonian can only be diagonalized numerically. Thus,
we follow the prescription given in [46] for the multi-band case. Given a
discretization of the Brillouin zone $k_{j}=\dfrac{2\pi}{a}\dfrac{j}{N},\
j\in\\{0,\ldots,N-1\\}$, for each momentum $k_{j}$ we compute a basis of
eigenstates of the Hamiltonian in Eq. 4 for the occupied bands. The resulting
$N_{y}$-dimensional vectors will be denoted as
$\ket{u_{nk_{0}}},\ldots,\ket{u_{nk_{N-1}}}$, $n$ being the band index. Then
we enforce the periodic gauge, by defining [46]

$\ket{u_{nk_{N}}}_{\ell}=\mathrm{e}^{-\mathrm{i}\frac{2\pi}{a}t_{\ell}}\ket{u_{nk_{0}}}_{\ell},$
(7)

where $\ket{u_{nk_{j}}}_{\ell}$ is the $\ell$-th component of the eigenvector
$\ket{u_{nk_{j}}}$ and $t_{\ell}$ the $x$ position of the $\ell$-th site
inside the strip unit cell [47], whose origin we place on the $A$ site at the
bottom edge. The Zak phase for the occupied bands is thus defined as [46, 48,
47]

$\varphi=-\Im\log\det\prod_{j=0}^{N-1}S(k_{j},k_{j+1}),$ (8)

with the overlap matrix $S$ given by [46, 48, 47]

$S(k_{j},k_{j+1})_{mn}=\braket{u_{mk_{j}}}{u_{nk_{j+1}}}.$ (9)

This formula is unaffected by any erratic behavior of the phase randomly
appended to the eigenvectors by the numerical diagonalization routine [48]. It
must be noted that since the Hamiltonian in Eq. 4 is real, the Zak phase can
only be $0$ or $\pi$. The values obtained for the Zak phase at different
values of $m$ are reported in the bottom panels of Fig. 2, below the
corresponding plots of the energy gap. We find that the Zak phases jump of
$\pi$ at each gap closing, confirming that some kind of topological phase
transition actually occurs.

In order to gain any insight about which phase is trivial and which is
topological we need to make some more physical considerations. As a matter of
fact, the Zak phase itself does not have an absolute meaning [33, 49, 50],
since its value depends on the choice of the unit cell origin in real space.
However, the difference between the values of the Zak phases in two regions of
the parameter space is uniquely defined (modulo $2\pi$).

In the large $m$ limit, the system at half filling is expected to be a trivial
insulator: in fact, in this scenario the electrons localize on the sublattice
which is lower in energy (depending on the sign of $m$) and the hoppings
between sites are suppressed. In view of this, we expect the regions of
parameter space characterized by a value of the Zak phase differing by $\pi$
($0$) from that of the large $m$ limit to be topologically non-trivial
(trivial). It is worth pointing out that for $N_{y}=6$ and $N_{y}=10$ the
limits $m\to\pm\infty$ correspond to different values of the Zak phase. We
prove analytically in Supplementary Note 2 that this is a general fact for
$N_{y}=4M+2$. In these cases, the comparison of the Zak phases should be made
with the $m\to-\infty$ limit for negative values of $m$ and with the
$m\to\infty$ limit for positive ones. The phase diagram emerging from this
argument is depicted in Fig. 3, where for each of the widths considered a star
is drawn in every region of the parameter space which is expected to be
topologically non-trivial. By virtue of bulk boundary correspondence, quasi 0D
bound states should occur when the quasi 1D strips are considered under open
boundary conditions (OBC) along the $\mathbf{a}_{1}$ direction and the
parameters fall inside one of the topological regions depicted in Fig. 3.

Figure 3: Emerging phase diagram. Phase diagram of the model as a function of
the staggered on-site potential $m$ for different widths: (a) $N_{y}=4$; (b)
$N_{y}=6$; (c) $N_{y}=8$; (d) $N_{y}=10$. The model parameters are set to
$t_{1}=1$, $t_{2}=0.3$ and $\phi=\pi/2$. The regions of the parameter space
where a star is drawn, are those expected to be topologically non trivial. The
yellow and red dots in (d) mark the two phases inspected in Fig. 6.

To check our predictions we perform numerical diagonalization of the model in
an uncompactified geometry along the $\mathbf{a}_{1}$ direction and we inspect
the low energy spectra in the different regions of the parameter space. The
OBC strips are cropped with a rectangular geometry, with the long edges
parallel to $\mathbf{a}_{1}$. In Fig. 4, for each of the widths taken into
account, the bands in PBC geometry (Panels a-d) and the corresponding low
energy spectra in the OBC case (Panels e-h) are reported. The values of $m$ at
which the diagonalization was performed for each strip, were chosen close to
the point of the topological regions where the gap was maximum (cf. Fig.
2a-d). The spectra of the finite size systems present two degenerate
eigenvalues located inside the gap. In Panels i-l we report the 1D profile of
the probability density distributions corresponding to the eigenvalue
n${}^{\circ}10$ (marked in orange) of the spectra in Panels e-h respectively
as a function of the position, together with insets representing the
corresponding unprojected probability densities directly on the strips. From
the localization pattern of these states, we conclude that the in-gap doublets
characterizing the topological phases depicted in Fig. 3 correspond to quasi
zero-dimensional bound states exponentially localized at the two ends of the
strips. We numerically checked the robustness of these 0D bound states against
the introduction of random on-site disorder finding that, though their energy
is inevitably slightly shifted, they survive as long as the disorder strength
does not become comparable with the gap width (see Supplementary Note 3).

Figure 4: Topological quasi 0D bound states. Results from numerical
diagonalization in topologically non-trivial regions for different widths: (a,
e, i) $N_{y}=4$; (b, f, j) $N_{y}=6$; (c, g, k) $N_{y}=8$; (d, h, l)
$N_{y}=10$. For different values of $m$ for each of the widths considered
(respectively $0,\ 0.5,\ 0.8,\ 0.92$), the top row reports the bands of the
strips with PBC along $x$ ($\mathbf{a}_{1}$) and the mid row reports the low
energy spectra of the corresponding OBC counterparts, obtained for strips of
length $20a$ (e), $40a$ (f), $80a$ (g), $160a$ (h). The values of $m$ chosen
for each strip belong to the regions marked with a star in Fig. 3. A doublet
of degenerate modes (marked in blue and orange) is visible inside each gap of
the OBC spectra: these correspond to bound states localized at the ends of the
strips. The 1D probability density profiles of the states associated to the
eigenvalue n${}^{\circ}10$ (marked in orange) of each of the spectra (e-h) are
reported in the bottom row (i-l) as a function of the position for the left
half of the various strips ($x<0$). For each of these plots, the insets show
the unprojected probability densities. The model parameters are set to
$t_{1}=1$, $t_{2}=0.3$ and $\phi=\pi/2$.

The results just discussed prove that the dimensional reduction of the Haldane
model, when operated on a strip with zigzag edges, gives rise to a reentrant
topological phase diagram, characterized by the emergence of degenerate
doublets of in-gap 0D end states. Quite curiously, such phenomenology has no
counterpart in the case of armchair nanoribbons, at least in the parameter
range we numerically inspected. A qualitative motivation of this peculiar
asymmetry is given by the end of the next subsection, where we interpret our
results through an effective model.

###  2.3 Low energy theory

In order to understand our findings, we frame them into an effective low
energy theory. The minimal low energy model describing the coupling between
the two counter-propagating edge states in a zigzag Haldane nanoribbon would
be

$H_{\text{edge}}=[v_{F}^{\text{ZZ}}(k-\pi)+\tilde{m}]\tau_{z}+\mathcal{M}\tau_{x},$
(10)

where [42]

$v_{F}^{\text{ZZ}}\approx\dfrac{6t_{2}t_{1}}{\sqrt{t_{1}^{2}+16t_{2}^{2}}},\qquad\tilde{m}\approx\dfrac{mt_{1}}{\sqrt{t_{1}^{2}+16t_{2}^{2}}}.$

Unfortunately, here $\mathcal{M}$ is an effective coupling induced by the
spatial proximity between the two zigzag edges in the thin strip limit. As
such, in principle it depends in a non trivial way on all of the model
parameters, as well as on the Bloch momentum $k$. This severely hinders the
derivation of a reliable low energy theory for the present model.
Nevertheless, we can still achieve some qualitative predictions– at least for
$m=0$ –by the following hand waving argument. To effectively describe the
coupling of the chiral edge states, we consider two 1D chains, representing
the two edges of the zigzag nanoribbon. We assume that the sites of the two
chains are connected via a coupling which decreases exponentially with the
distance, as it is usually done when modelling proximity effects. However, we
add a crucial physical input: We introduce a sharp cutoff in the hopping
range, effectively coupling only sites that in the actual nanoribbon are
connected by the minimum amount of first neighbor hoppings. The motivation
behind this choice, is that the coupling between any other pair of edge sites
would represent a higher order (negligible) correction.

Omitting the details of the derivation, for which we refer the interested
reader to Supplementary Note 4, we find for the two classes considered
($N_{y}=4M$ or $N_{y}=4M+2$) and assuming $m=0$

$\displaystyle N_{y}=4M:$
$\displaystyle\mathcal{M}^{\text{teo}}(k)=\tilde{\Delta}[1+\sum_{j=1}^{M}2\cos(kj)\mathrm{e}^{-(\sqrt{w^{2}+j^{2}}-w)/\xi}],$
(11) $\displaystyle N_{y}=4M+2:$
$\displaystyle\mathcal{M}^{\text{teo}}(k)=\tilde{\Delta}\sum_{j=0}^{M}2\cos(k(j+1/2))\mathrm{e}^{-(\sqrt{w^{2}+(j+1/2)^{2}}-w)/\xi},$
(12)

where $w=\frac{\sqrt{3}}{2}\frac{N_{y}}{2}-\frac{1}{\sqrt{3}}$ is the strip
width and $\xi$ is of the order of the chiral edge states localization length
(Eq. 5).

To benchmark our results, we extract the low energy bands directly from the
exact numerical diagonalization. Let us denote by $E_{N_{y}}(m;k)$ the lowest
(positive) energy band for a strip of width $N_{y}$ (all the other parameters
of the model fixed). We can assume that, close to the Dirac point,
$E_{N_{y}}(m;k)$ is well described by the spectrum of the low energy model in
Eq. 10, with a certain unknown function $\mathcal{M}$. Thus, noting that the
hybridization between the chiral modes is exponentially suppressed with the
strip width– i.e. $\mathcal{M}\xrightarrow[]{N_{y}\to\infty}0$ –we can recover
(the modulus of) $\mathcal{M}$ for a given $N_{y}$ as

$|\mathcal{M}(m;k)|=\sqrt{E_{N_{y}}(m;k)^{2}-E_{\infty}(m;k)^{2}}.$ (13)

In Fig. 5a-d are shown the plots of $\mathcal{M}^{\text{teo}}(k)$ as a
function of $k$ for the different widths considered, with $\xi$ set to twice
the localization length of the edge states ($\xi_{\text{loc}}$) for
$t_{2}=0.3$ (assuming $t_{1}=1$). In Panels e-h instead, are reported the
corresponding plots of $|\mathcal{M}(0;k)|$, obtained numerically as described
in Eq. 13. Note that the points in $k$ space where $\mathcal{M}$ stops
oscillating and steeply goes up correspond to the points where the edge states
of the reference strip ($N_{y}\to\infty$) merge with the bulk states. Beyond
this limit, the low energy Hamiltonian in Eq. 10 is no longer valid, since it
does not account for the bulk degrees of freedom.

Figure 5: Effective toy model validation. Comparison between the effective
mass term derived analytically in Eqs. 11 and 12 (a-d) and the ones obtained
numerically according to Eq. 13, for $m=0$ (e-h) and $m=0.5$ (i-l) with the
other parameters set to $t_{1}=1,\ t_{2}=0.3,\ \phi=\frac{\pi}{2}$. The dashed
vertical line in the plots (e-l) indicates the position of the Dirac point,
which varies with $m$ according to $k_{\text{D}}=\pi-\frac{m}{6t_{2}}$ (cf.
Eq. 10). The $y$-axis in Panels (a-d) are in arbitrary units.

By comparison between the two rows of plots, one can see that the effective
mass term in Eqs. 11 and 12 correctly reproduce some qualitative features of
the one obtained numerically for $m=0$. More specifically, setting the
parameter $t_{2}$ in the range $[0.1\sim 0.3]$, we correctly recover the
number of nodes of $\mathcal{M}(0;k)$ for the different widths, the fact that
the mass term spreads in $k$-space as $t_{2}$ is increased and, crucially, the
fact that $\mathcal{M}(0;\pi)=0$ for $N_{y}=4M+2$. Further support to these
statements can be found in Supplementary Figs. 4, 5 and 6, where, for each of
the widths considered, we report the plots of $|\mathcal{M}|$ as a function of
$k$ for different values of $m$ and $t_{2}$. It is worth mentioning that the
shape of $\mathcal{M}(0;k)$ more closely resembles that of a sinusoidal-like
function for smaller values of $t_{2}$ ($t_{2}\sim 0.1-0.2$). The deviations
from the sinusoidal pattern predicted by the effective low energy theory
observed in Fig. 5e-l ($t_{2}=0.3$), are probably due to the fact that the
first has been derived taking into account first neighbor hoppings only. That
being the case, it is expected to be more reliable for smaller values of
$t_{2}$.

Furthermore, the numerical results for $\mathcal{M}$ show that, at least at a
qualitative level, for $m>0$ the mass term is shifted to the _right_ in the
Brillouin zone (see Fig. 5i-l and Supplementary Figs. 4, 5 and 6). On the
other hand, we know that for $m>0$ the Dirac point moves to the _left_ (dashed
vertical lines in Fig. 5i-l). Therefore, there must be a (set of) value(s) of
$m$ for which the nodes of $\mathcal{M}(m;k)$ coincide with the shifted Dirac
point. This mechanism qualitatively explains the mass inversion in our model
and, consequently, the reentrant topological phase diagram retrieved
numerically.

Finally, we briefly address the reason why armchair nanoribbons do not share
this kind of phenomenology. In the Haldane model on armchair nanoribbons the
Dirac point is at $k=0$, and is insensitive to variation of the staggered
mass. This can be understood by noticing that, in contrast with the zigzag
case, each armchair edge has the same number of $A$ and $B$ sites. Thus,
varying $m$ does not shift the energy of the edge states. This hints to the
fact that in the case of armchair geometry, the properties of the edge states
are less likely to be tuned by varying the staggered mass $m$. Moreover, the
edge states in zigzag Haldane nanoribbons present a peculiar property: as
shown by Eq. 5, $\xi_{\text{loc}}$ grows with $t_{2}$ in the zigzag case. On
the other hand, the Haldane edge states on armchair nanoribbons have their
localization length inversely proportional to $t_{2}$ [43]. Thus, in the
zigzag case, bigger values of $t_{2}$ lead both to a wider topological bulk
gap in the 2D limit, and to a larger localization length of the edge states.
This feature, which is not shared by armchair nanoribbons, elects thin zigzag
nanoribbons as the optimal playground to inspect the finite size effects on
the Haldane edge states.

###  2.4 Jackiw-Rebbi like bound states

Interestingly, if we consider a setup of our model in which the staggered mass
term $m$ interpolates between two values in adjacent regions of the phase
diagram with different Zak phases, we observe the occurrence of a bound state,
localized at the transition point and with energy lying inside the gap. Thus,
leaning on the results from Jackiw and Rebbi [35], we can conclude that such
bound state retains a fractional charge of $\pm\frac{e}{2}$ [36, 51].
Strikingly, the bound state is present even when the mass $m$ interpolates
between two regions that, despite having different Zak phases, do not host
bound states against the vacuum (more details in Supplementary Note 5).

In Fig. 6 we report an explicit example for a strip with $N_{y}=10$ and
$L=200a$, in which the on-site staggered potential is set to $\mp 0.3$ for the
$A$ and $B$ sites respectively on the left part and to $\pm 0.3$ on the right
part (with $m=0$ at the interface to smoothen out the jump). These two regions
of the phase diagram are marked with a yellow and red dot respectively in Fig.
3d. All the other parameters of the model are left unchanged. In Panel a is
reported a density plot of the on-site potential close to the transition
point, where the sign of $m$ switches for the two sublattices. By performing
real space numerical diagonalization we obtain the low energy spectrum and the
corresponding eigenvectors. The first is shown in Panel b, where an isolated
eigenvalue is clearly visible inside the gap: this corresponds to a bound
state localized at the point where $m$ switches its sign, as demonstrated by
the plot of its probability density in Panel c. From Fig. 2d, we see that from
$m=-0.3$ to $m=+0.3$ the Zak phase jumps of $\pi$, so that the occurrence of a
bound state at the domain wall is actually expected according to our analysis.

Figure 6: Jackiw-Rebbi bound states at domain walls. Results obtained via
numerical diagonalization for a strip with $N_{y}=10$ and $L=200a$ with
spatially varying on-site potential. According to Jackiw-Rebbi theory, a bound
state is expected in correspondence of mass inversion (which we identify as a
jump of $\pi$ in the Zak phase). The model parameters are set to $t_{1}=1$,
$t_{2}=0.3$ and $\phi=\pi/2$. a Representation of the on-site potential for
the central region of the strip. The left side of the strip ($x<0$) has
$m=-0.3$, while the right side ($x>0$) has $m=0.3$ (cf. yellow and red dot of
Fig. 3d). The Zak phases in the two regions differ of $\pi$. b Low energy
spectrum obtained via numerical diagonalization. An isolated eigenvalue is
visible inside the gap (marked in orange). c 1D profile of the probability
density of the state associated with the in-gap eigenvalue for $-50a<x<50a$,
with an inset showing the corresponding unprojected probability distribution
directly on the strip. The state appears to be exponentially localized at the
domain wall.

##  3 Discussion

In this paper we have studied the role of finite size effects on the Haldane
model. We have shown that, in the case of zigzag strip geometry, the chiral
edge modes can, as expected, hybridize and develop a gap. Surprisingly,
however, such gap can be both trivial or topological in the sense that, in the
uncompactified geometry, bound states can be present or absent depending on
the value of the trivial mass, even in the topological phase of the Haldane
model. In other words, we unveiled a phase diagram which presents a width-
dependent reentrant behavior with respect to the tuning of the on-site
staggered potential. Moreover, we have reinterpreted such reentrant structure
within an effective minimal model describing the coupling between the chiral
edge states of the Haldane model in zigzag nanoribbons.

We have then proven that, when present, the bound states are robust against
on-site random disorder. Besides, we have shown that they also occur in
correspondence of domain walls in the on-site staggered potential and,
consequently, that they bear a fractional charge of $\pm\frac{e}{2}$. The
topological nature of the bound states is witnessed by the behavior of the Zak
phase. Indeed, we can hence conclude that the mass associated to the tunneling
between the edges competes in a definitely non-trivial fashion with the other
masses of the model, generating a rich phenomenology.

The implications of our results are diverse. In the context of two-dimensional
topological insulators and Chern insulators they imply, for instance, that the
transport properties of setups where constrictions are present might be
affected by the presence of zero modes, and hence show resonances. Moreover,
given that a similar gap structure also characterizes the 2D Kitaev model [19,
52] for topological superconductivity, the impact of our results bears also
consequences on the field of Majorana zero modes [53, 54, 55] and parafermions
[56, 57, 58], paving the way to new possibilities for implementing such non-
abelian excitations.

##  4 Methods

###  4.1 Numerical diagonalization tools

The finite size model construction and the numerical diagonalization have been
performed using the package Pybinding [59].

###  4.2 Numerical computation of the Zak phase

The computation of the gaps and of the Zak phase as a function of the
staggered on-site potential have been performed with an original code and the
results for the Zak phase have been benchmarked with existing packages.

## Declarations

Data availability All data relevant to the paper are reported in the main text
and in the Supplementary Information. All the numerically generated points
reported in the plots of this paper and in the accompanying supplementary are
obtained as described in section 4. The actual codes used to produce the
results reported in this paper and in the accompanying supplementary are
available from the corresponding author upon request.

Acknowledgments N.T.Z. acknowledges the funding through the NextGenerationEu
Curiosity Driven Project “Understanding even-odd criticality”. N.T.Z. and M.S.
acknowledge the funding through the “Non-reciprocal supercurrent and
topological transitions in hybrid Nb-InSb nanoflags” project (Prot.
2022PH852L) in the framework of PRIN 2022 initiative of the Italian Ministry
of University (MUR) for the National Research Program (PNR).

Author contributions Conceptualization, M.S. and N.T.Z. Development and
preparation of the first draft S.T. All authors contributed to the
interpretation of the results and reviewed the manuscript.

Competing interests The authors declare no competing interests.

Supplementary information Details of the calculations and further numerical
results can be found in the accompanying Supplementary Information.

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## References

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  * [2] Bernevig, B. A., Hughes, T. L. & Zhang, S.-C.  Quantum spin hall effect and topological phase transition in hgte quantum wells.  _Science_ 314, 1757–1761 (2006).
  * [3] Ryu, S., Schnyder, A. P., Furusaki, A. & Ludwig, A. W. W.  Topological insulators and superconductors: tenfold way and dimensional hierarchy.  _New J. Phys._ 12, 065010 (2010).
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  * [7] Stühler, R. _et al._ Effective lifting of the topological protection of quantum spin hall edge states by edge coupling.  _Nat. Commun._ 13, 3480 (2022).
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  * [50] Cayssol, J. & Fuchs, J. N.  Topological and geometrical aspects of band theory.  _J. Phys. Mater._ 4, 034007 (2021).
  * [51] Ziani, N. T., Fleckenstein, C., Vigliotti, L., Trauzettel, B. & Sassetti, M.  From fractional solitons to majorana fermions in a paradigmatic model of topological superconductivity.  _Phys. Rev. B_ 101, 195303 (2020).
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  * [55] Mourik, V. _et al._ Signatures of majorana fermions in hybrid superconductor-semiconductor nanowire devices.  _Science_ 336, 1003–1007 (2012).
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