# Restoring the topological edge states in a finite optical superlattice

A. Katsaris∗ Department of Physics, National and Kapodistrian University of
Athens, GR-15784 Athens, Greece  I. A. Englezos∗ Zentrum für Optische
Quantentechnologien, Luruper Chaussee 149, Universität Hamburg, 22761 Hamburg,
Germany  C. Weitenberg  Institut für Quantenphysik, Luruper Chaussee 149,
Universität Hamburg, 22761 Hamburg, Germany  The Hamburg Centre for Ultrafast
Imaging, University of Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
F. K. Diakonos  Department of Physics, National and Kapodistrian University of
Athens, GR-15784 Athens, Greece  P. Schmelcher  Zentrum für Optische
Quantentechnologien, Luruper Chaussee 149, Universität Hamburg, 22761 Hamburg,
Germany  The Hamburg Centre for Ultrafast Imaging, University of Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany

###### Abstract

We consider the emergence of edge states in a finite optical lattice and show
that the boundaries of the lattice play a decisive role for their location in
the corresponding energy spectrum. We introduce a simple parametrisation of
the boundaries of the optical lattice and demonstrate the existence of an
optimal choice of the values of the parameters which lead to an approximate
restoration of chiral symmetry. A crucial property of this optimization is the
suppression of tunneling between next-nearest neighboring wells of the
lattice. This in turn allows the mapping of the optical lattice set-up to a
finite SSH model. The topological character of the emerging edge states is
discussed.

**footnotetext: These authors contributed equally.

##  I Introduction

Ever since the experimental realization of Bose-Einstein condensation (BEC)
Anderson  _et al._ (1995); Bradley  _et al._ (1995); Davis  _et al._ (1995)
ultracold atomic platforms have emerged as a highly versatile test-bed for a
wide range of phenomena in atomic, molecular and condensed matter systems
Bloch  _et al._ (2008, 2012). A crucial step in that direction is the ability
to precisely control the inter-atomic interaction strengths, via Feshbach
resonances Courteille  _et al._ (1998); Inouye  _et al._ (1998); Chin  _et
al._ (2010). Modern optics can be utilized so that the shape of the external
confinement results in almost arbitrary trap geometries, including setups of
low-dimensional lattices Bakr  _et al._ (2009). This level of control has
enabled among others the realization of Hubbard models and the study of the
Superfluid-Mott insulator transition Jaksch  _et al._ (1998); Greiner  _et
al._ (2002). Moreover, it has facilitated the realization of topological
phases of matter related to the quantum Hall effect and topological insulators
such as the Su-Schrieffer-Heeger (SSH) model Cooper  _et al._ (2019);
Weitenberg and Simonet (2021).

The simplicity of the SSH model establishes it as an ideal starting point for
understanding the topological phases of matter. One of the most important
features of the SSH model is its chiral symmetry, which is strongly connected
to its topological properties. Namely, since there is a strong bulk-edge
correspondence (BDI symmetry class Ryu  _et al._ (2010)), one can predict when
topologically protected edge states will be supported by the system. Those
states are of great interest because of their robustness against certain
disorders, rendering them promising candidates for quantum information
processing Palaiodimopoulos  _et al._ (2021); D’Angelis  _et al._ (2020);
Huang  _et al._ (2022). The SSH model can be easily extended to describe more
complex systems, such as higher dimensional systems with enriched topological
phases Ma  _et al._ (2022) or including interactions Grusdt  _et al._ (2013);
Stenzel  _et al._ (2019); Zhou  _et al._ (2023).

Realizations of the SSH model in optical lattices are mainly focused on the
study of the quantized charge transport both for fermionic Nakajima  _et al._
(2016) and bosonic Lohse  _et al._ (2016) atoms. In this context, the
parameters of the discrete SSH model are derived from the continuous optical
lattice potential, by fitting their corresponding Bloch spectra primarily in
the two lowest bands, whose gap determines the topological properties of the
system Cooper  _et al._ (2019). Importantly, in principle next-nearest
neighbor (NNN) hopping is always present in optical lattices, although it is
significantly suppressed for deep potential wells and it is usually ignored in
studies of the SSH model. However, from the theoretical side, even a small
next-nearest neighbor hopping amplitude explicitly breaks the underlying
chiral symmetry of the SSH model Li  _et al._ (2014); Pérez-González  _et al._
(2019), leading to an extended SSH model (eSSH) belonging to a different
symmetry class (AI) and possessing a different topological invariant.
Additionally, edge states in 2D topological systems were observed only
recently by adding a sharp wall potential to a driven honeycomb lattice Braun
_et al._ (2023) and a rotating trap Yao  _et al._ (2023). Other observations
of edge states relied on artificial dimensions using internal states for
defining sharp edges Boada  _et al._ (2012); Mancini  _et al._ (2015); Stuhl
_et al._ (2015); Kanungo  _et al._ (2022). Edge states were also probed in
tweezer arrays with Rydberg interactions de Léséleuc  _et al._ (2019), where
the edges also occur naturally.

For the above reasons, we focus on a systematic study of the conditions under
which a finite continuous optical lattice may be mapped, within the tight-
binding approximation, to an SSH model. Specifically, we demonstrate and
analyze the problems that emerge when finite confined systems are considered,
and we provide the essential tools to control them. In particular, we
introduce a simple extension of the potential domain of the optical lattice,
and we find the optimal choice of its parameters values so that the effects of
the confinement are maximally suppressed. Finally, we address the presence of
NNN hopping when the potential wells are not deep enough, and propose criteria
so that the corresponding terms can be omitted.

This work is structured in the following way. In Section II we mention the
basic characteristics of the SSH and eSSH models and describe the optical
superlattice potential we employ to realise them. The explicit mapping from
the continuous to the discrete system is also presented. Then, in Section III
we analyze the criteria and in particular boundary conditions under which
topological edge states emerge in the finite continuous system, via directly
solving the corresponding Schrödinger equation. In Section IV we present the
results derived from the mapping to the discrete (tight-binding) system and
examine their deviations from the (extended) SSH model with the same bulk
parameters. Finally, in Section V we summarise our results and highlight
future perspectives.

##  II Tight-binding models and Optical lattice Potentials

In this section, we begin by noting the basic characteristics of the SSH model
and its extension (eSSH) when NNN hopping terms are considered (Sec. II.1). We
then proceed with a continuous setup involving a superlattice optical
potential, which enables the implementation of the SSH model in ultracold
atomic platforms (Sec. II.2). Lastly, we present the tight binding (TB)
approximation, which we employ for the detailed comparison between the
continuous and the corresponding discrete system (Sec. II.3).

###  II.1 SSH and eSSH models

The Hamiltonian of an SSH model Su  _et al._ (1979); Batra and Sheet (2020)
with $M$ unit cells can be written as:

$\mathcal{H}_{\text{SSH}}=v\sum_{m=1}^{M}a^{\dagger}_{m}b_{m}+w\sum_{m=1}^{M}a^{\dagger}_{m+1}b_{m}+h.c$
(1)

where $v,w$ are the intracell and intercell hopping amplitudes,
$\hat{a}^{\dagger}_{m}$ ($\hat{a}_{m}$) and $\hat{b}^{\dagger}_{m}$
($\hat{b}_{m}$) are the creation (annihilation) operators, creating
(annihilating) a particle in the $m$ cell on the $A/B$ sublattice
respectively. A schematic of the SSH model is provided in Fig. 1(a). We also
consider an extension of the SSH model with the addition of NNN hopping terms
of the form:

$\mathcal{H}_{\text{NNN}}=t\sum_{m=1}^{M-1}\left(\hat{a}^{\dagger}_{m+1}\hat{a}_{m}+\hat{b}^{\dagger}_{m+1}\hat{b}_{m}\right)+h.c.$
(2)

where $t$ is the NNN hopping amplitude (see Fig. 1(b)). The Hamiltonian of
this extended SSH (eSSH) is:

$\mathcal{H}_{\text{eSSH}}=\mathcal{H}_{\text{SSH}}+\mathcal{H}_{\text{NNN}}$
(3)

Figure 1: Schematics of (a) SSH and (b) eSSH models with $M=3$ cells.

These two models are of great interest because they can both support
topologically protected edge states (TESs), i.e. states that are localized on
the edges of the lattice and their energies reside in the center of the band
gap. We emphasize that in the SSH model the existence and protection of TESs
are connected to the chiral symmetry of the system (BDI symmetry class) Ryu
_et al._ (2010), while in the case of the eSSH model they are related to the
inversion symmetry (AI symmetry class) Li  _et al._ (2014); Pérez-González
_et al._ (2019). Specifically, in the SSH model when $v>w$ the system is in
the topologically trivial phase and it does not support edge states. In
contrast, for $v<w$ the system resides in the topologically non-trivial (i.e.
topological) phase where it supports two edge states. On the other hand, for
the case of eSSH, even though the bulk-edge correspondence holds, specifics of
the edge states, such as their position on the energy spectrum or their
robustness, can not be easily predicted. However, when edge states are
supported by eSSH, they are in fact of topological nature. In most of the
cases considered below the parameter $t$ will always be relatively small in
comparison to the values of the $v$ and $w$ parameters, hence the NNN terms
could be treated as a perturbation.

###  II.2 Superlattice potential

An optical lattice can be implemented experimentally by forming a standing
wave, utilizing two counter propagating laser beams. A sequence of $M$ double
wells as required to realize the SSH and eSSH models, can be achieved by
superimposing two such standing waves with different frequencies Anderlini
_et al._ (2007); Fölling  _et al._ (2007); Cooper  _et al._ (2019), leading to
the superlattice (SL) potential:

$\displaystyle
V_{SL}(x)=V_{s}\cos^{2}{\left(2k_{r}x\right)}+V_{l}\cos^{2}{\left(k_{r}x\right)}$
(4)

where $V_{s},~{}V_{l}$ are the amplitudes of the two standing waves,
$k_{r}=2\pi/\lambda_{0}$ is the single-photon recoil momentum, and
$\lambda_{0}$ is the wavelength of the lattice. For a system of $M$ cells and
total length $2L$ it holds that $\lambda_{0}=2L/M$.

Figure 2: Schematic of the SL potential for $M=3$, $V_{l}=15E_{r}$ and
$V_{s}=40E_{r}$, and $x$ in units of $k_{r}^{-1}$.

The stationary Schrödinger equation

$\hat{H}\ket{\Psi_{n}}=E_{n}\ket{\Psi_{n}},~{}\text{with}~{}~{}\hat{H}=-\frac{\hslash^{2}}{2m}\frac{\partial^{2}}{\partial
x^{2}}+\hat{V_{SL}}$ (5)

can then be solved numerically, e.g. via exact diagonalization (ED), to obtain
the complete spectrum of eigenstates and eigenenergies of the system. In the
following, our analysis will focus on the lowest band, i.e. the eigenstates
with the $2M$ lowest laying eigenvalues.

The heights of the two barriers of the SL potential shown in Fig. 2 are given
by the following expressions:

$\displaystyle V_{low}=V_{s}(1-\frac{V_{l}}{4V_{s}})^{2},\quad V_{high}$
$\displaystyle=V_{s}(1+\frac{V_{l}}{4V_{s}})^{2}$ (6)

Evidently, changing one of the amplitudes $V_{s,l}$ results in a non-linear
shift of both the relative and absolute heights of the potential barriers.
Hence, it is convenient to express our results in terms of the height of the
higher barrier $V_{high}$ and the ratio between the lower and higher barrier
$u=V_{low}/V_{high}$. Finally, for computational convenience, we recast the
Schrödinger equation in a dimensionless form by expressing the length in terms
of $k_{r}^{-1}$ and the energy in terms of the recoil energy
$E_{r}=\hbar^{2}k^{2}_{r}/2m$, where $m$ is the atomic mass and $\hbar$ the
reduced Plank constant.

###  II.3 Tight Binding Approximation

In the context of the tight binding (TB) approximation, we assume that in a
deep enough lattice, the description of the system can be accurately truncated
to the first, or first few, energy bands Marzari  _et al._ (2012); Wei  _et
al._ (2024a). For a system of $M$ double wells (cells), i.e. a system with
$\mathcal{N}=2M$ minima (sites), restricted to the lowest band, the TB
Hamiltonian takes the form:

$\hat{\mathcal{H}}_{TB}=\sum_{i=1}^{\mathcal{N}}\sum_{j=1}^{\mathcal{N}}h_{i,j}\hat{\alpha}^{\dagger}_{i}\hat{\alpha}_{j},$
(7)

where $\hat{\alpha}^{\dagger}_{i}$ ($\hat{\alpha}_{i}$) are the creation
(annihilation) operators, creating (annihilating) either a bosonic or a
fermionic particle in the lowest band at the site $i$, and the matrix elements

$h_{i,j}=\int
w^{*}_{i}(x)\Big{[}\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial
x^{2}}-\hat{V_{SL}}\Big{]}w_{j}(x)dx$ (8)

are defined in terms of the Wannier functions $w_{i}(x)$ localized at each
site $i$. In the following, we define the Wannier functions as

$w_{i}(x)=\bra{x}\ket{\chi_{i}}$ (9)

where $\ket{\chi_{i}}$ are the eigenstates of the position operator restricted
to the lowest band ($\hat{\mathcal{X}}_{band}$), which can be obtained by
solving the eigenvalue problem
$\hat{\mathcal{X}}_{band}\ket{\chi_{i}}=\chi_{i}\ket{\chi_{i}}$, with

$\hat{\mathcal{X}}_{band}=\sum_{n=1}^{\mathcal{N}}\sum_{m=1}^{\mathcal{N}}\ket{\Psi_{n}}\bra{\Psi_{n}}\hat{x}\ket{\Psi_{m}}\bra{\Psi_{m}},$
(10)

and $\ket{\Psi_{n}}$, $\ket{\Psi_{m}}$ fulfilling the eigenvalue problem given
by Eq. (5). In 1D this definition of the Wannier functions has been shown to
produce uniquely defined maximally-localized Wannier functions, even when
generalized to take into account higher bands Marzari and Vanderbilt (1997).
In the following, we aim to determine the conditions under which the TB
Hamiltonian (7), corresponding to the finite continuous system subjected to
the SL potential, can be accurately mapped to the SSH or eSSH model.

\begin{overpic}[width=416.27809pt]{Figures/Fig3_abc.pdf} \put(9.0,10.0){{(a)}}
\put(43.5,10.0){{(b)}} \put(79.0,10.0){{(c)}}
\end{overpic}\begin{overpic}[width=416.27809pt]{Figures/Fig3_def.pdf}
\put(9.0,10.0){{(d)}} \put(43.5,10.0){{(e)}} \put(81.0,10.0){{(f)}}
\end{overpic}

Figure 3: (a,b) The energy eigenvalues of the superlattice potential with
hard-wall boundary conditions for $V_{high}=40E_{R}$, $V_{low}=20E_{R}$, and
$M=9$ in the trivial and topological regime obtain via ED. (c) The non-
topologically protected edge states appearing in the presence of hard-wall
boundary conditions for the continuous system ($x$ in units of $k_{r}^{-1}$).
(d,e) The energy eigenvalues of the SSH-model in the trivial and the
topological regime. (f) The edge states of the associated SSH-model. The
relevant hopping amplitudes for the SSH model, in (d)-(f), have been obtained
from the bulk values of the TB Hamiltonian i.e. $v,w=h_{M,M\pm 1}$ and
$\mu=h_{M,M}$. The overall energy scale offset of the graphs (a,b) and (d,e)
is related to the larger on-site amplitudes of the edge sites in comparison to
the bulk sites, which in this case we have not taken into account.

##  III Impact of the Boundary Conditions on the finite continuous system

In order to obtain a uniquely defined set of eigenstates and eigenergies (up
to an overall phase), specific boundary conditions have to be implemented when
we solve the Schrödinger equation. For a periodic system, e.g. in a ring
geometry, one has to employ periodic boundary conditions (PBC). In this case,
it has been shown that the lowest band of the energy spectrum of the
continuous system subjected to the SL potential given by Eq (4), is in good
agreement with that of the corresponding discrete SSH model with PBC Cooper
_et al._ (2019); Greiner  _et al._ (2002). However, in the case of PBC the
system cannot exhibit edge states by construction. Hence, in order to directly
observe topological edges states we have to consider finite lattices with open
boundary conditions, i.e. systems with clearly defined edges.

When considering finite systems, the choice of boundary conditions is not
unique. A common choice, is the consideration of hard wall boundary conditions
(HWBC), i.e. to demand that the eigenstates are exactly zero (vanish) at the
boundaries of the potential landscape. Moreover, we expect a vanishing
probability of observing a particle in the regions where the energy lies below
the potential strength, namely in the classically forbidden regions. So, the
energetically low-lying eigenstates are expected to take values close to zero
in the vicinity of the positions of the potential’s local and global maxima.
In the context of optical lattices the boundary conditions are usually
considered at its maxima, so the eigenstates are expected to not exhibit an
abrupt transition of their profiles, even in the case of HWBC. Ultimately,
from a theoretical point of view HWBC seems at first to be a particularly
natural choice for our system, especially when the description is restricted
to the lowest band. However, in this section we illustrate that the they are
not an appropriate choice for capturing topologically protected edge states in
the continuous finite system (Sec. III.1) and we propose an alternative
boundary which instead allows for their emergence (Sec. III.2).

###  III.1 The case of hard wall boundary conditions

First, we begin our analysis with a topologically trivial setting where no
edge states emerge. Namely, we consider the continuous system with HWBC,
potential barriers of heights $V_{high}=40E_{r}$, $V_{low}=20E_{r}$ and $M=9$
cells and compare the energy spectrum in the lowest band with the one obtained
from the SSH model with open boundary conditions (OBC) as shown in Fig. 3(a)
and (d) (see also Fig. 4(a) for a schematic of the potential in the
topologically trivial regime). Evidently, we indeed find a good agreement with
the corresponding SSH model with OBC in the topologically trivial regime.

Surprisingly, the situation is profoundly different when considering the
topological phase for the same parameters and HWBC (see Fig. 4(b) for a
schematic of the potential). Specifically, the energy spectrum for the
continuous system in Fig. 3(b) exhibits a structure featuring one sub-band
with $M-1$ states and a higher sub-band with $M+1$ states, in sharp contrast
with the spectrum of the corresponding SSH model depicted in Fig. 3(e), which
features two sub-bands containing $M-1$ states and the two edge states
residing in the center of the gap. Interestingly, the states in the middle of
the spectra of both the continuous system and the SSH model ($\Psi_{M}$,
$\Psi_{M+1}$), exhibit localization at the edges as shown in Fig. 3(c) and
(f).

\begin{overpic}[width=407.60385pt]{Figures/Fig4_abc.pdf}
\put(12.5,14.25){{(a)}} \put(43.5,14.25){{(b)}} \put(71.0,14.25){{(c)}}
\end{overpic}\begin{overpic}[width=173.44534pt]{Figures/Fig4_d.pdf}
\put(87.0,30.0){{(d)}} \end{overpic}

\begin{overpic}[width=173.44534pt]{Figures/Fig4_e.pdf} \put(89.0,30.0){{(e)}}
\end{overpic}

Figure 4: The superlattice potential with hard-wall boundary conditions for
$V_{high}=40E_{r}$, $V_{low}=20E_{r}$, and $M=3$ in the topologically (a)
trivial, (b) non-trivial regime. (c) Same as (b) with the addition of the
linearly extended boundary for a slope $\alpha=7Er\cdot k_{r}$ and
$d=0.3L_{cell}$. In (a)-(c) $V(x)$ is in units of $E_{r}$ and $x$ in units of
$k_{r}^{-1}$. (d) The energy eigenvalues for $M=9$ for increasing the length
of the extended boundary $d$ and vanishing slope ($\alpha=0$). (e) Same as
(d), but for increasing slope ($\alpha$) and fixed length $d=\pi/k_{r}$.

However, the edge localized states predicted by the continuous system have
support in both odd and even sites –unlike the topologically protected states
of the SSH model– which is a strong signature of the particle-hole symmetry
breaking of the system. Moreover, we note that the edge localized states of
the continuous system are significantly less localized, and become well
separated from the bulk only in the case of very large or deep lattices (see
Appendix A for an illustrative example).

Finally, it is clear that the edge localized states in the continuous system
are not topologically protected, since they are not separated by a sub-band
gap from the bulk states. We interpret the results presented here as an
indication that the HWBC cause an offset to the potential experienced by an
atom at the edge sites $1$ or $2M$ as compared to an atom residing in the
center of the lattice. Evidently, this seems irrelevant when considering a
topologically trivial lattice, however it has a profound effect in a
topological lattice where edge states may be found.

###  III.2 Extension of the potential domain

To address the discrepancy between the continuous system and the SSH model we
introduce an extension to the SL potential, in order to simulate a more
physical boundary than the infinite wall. This extension should be simple
enough to aid the theoretical analysis and at the same time to be
experimentally feasible. So we consider a linear extension of length $d$ and
slope $\alpha$, as follows (see also Fig. 4(c)):

$V_{ext}(x)=\begin{cases}~{}V_{s}-\alpha(x+L),&-(L+d)\leq x\leq-L\\\\[5.0pt]
~{}0,&-L<0<L\\\\[5.0pt] ~{}V_{s}+\alpha(x-L),&L\leq x\leq L+d\end{cases}$ (11)

Quite surprisingly, this rather simple extension readily fixes the discrepancy
of the spectral behaviour between the continuous system and the SSH model. As
shown in Fig. 4(d) upon increasing the length $d$ of the linear extension (for
fixed $\alpha=0$) the edge state energies move towards the center of the gap
between the two sub-bands. In contrast, upon increasing the slope $\alpha$
(for fixed $d=\pi/k_{r}$), and hence recovering the effect of a sharp wall at
the edge (in the limit $\alpha\rightarrow\infty$ and $d\neq 0$), progressively
the edge state energies shift towards the higher sub-band. This is shown in
Fig. 4(e), where the logarithmic scale of the $x$-axis highlights the
relatively large values of the slope $\alpha$ required for the HWBC effect to
re-emerge. Moreover, we indeed observe topologically protected edge states,
even for smaller systems with $M=3,5$ and $M=9$ as shown in Fig. 5.

Finally, in Fig. 4(d) and (e) we observe asymptotically constant behaviour of
the eigenenergies when $d$ is increased (or $\alpha$ decreased) beyond a
certain value. Based on this behavior we can make an estimation about the
required length of the linearly extended boundary, so that the energies of the
two edge states reside in the center of the two sub-bands. For all values of
the parameters ($M$, $V_{high}$ and $V_{low}$), we find that a relatively
small length of the boundary extension is needed $d\leq 0.5\pi/k_{r}$ for the
energies of the edges states to saturate with respect to $d$, indicating that
only the local behavior of the boundary extension around the first and the
last site is relevant.

\begin{overpic}[width=416.27809pt]{Figures/Fig5.pdf} \end{overpic} Figure 5:
The topologically protected edge states appearing in the presence of the
linearly extended boundary at $d=\pi/k_{r}$ and $\alpha=0$, for systems with
$M=5,9,13$.

##  IV Tight-Binding Approximation Analysis

In this section we are focusing on the results related to the corresponding
discrete system, i.e. the form of the tight-binding Hamiltonian
($\hat{\mathcal{H}}_{TB}$) and the solution of its eigenvalue problem.
Specifically, we introduce a decomposition of the $\hat{\mathcal{H}}_{TB}$
into two terms, the $\hat{\mathcal{H}}_{0}$ for the bulk values and the
$\delta\hat{\mathcal{H}}$ that expresses the deviations due to the HWBC (Sec.
IV.1). Then, we analyse the behaviour of $\delta\hat{\mathcal{H}}$ with
respect to the parameters of the extension of the potential ($d$ and
$\alpha$), and we find the region in which it is minimised (Sec. IV.2).
Finally, we consider systems with $\delta\hat{\mathcal{H}}=0$ and we focus on
the behaviour of $\hat{\mathcal{H}}_{0}$ (Sec. IV.3).

###  IV.1 Introducing the ${\bf\it\mathcal{H}_{0}}$ and
${\bf\it\delta\mathcal{H}}$ terms

In order to make our analysis simpler, we establish specific notations for the
hopping amplitudes $h_{i,j}$, where $i$ and $j$ are the indices of the sites
of the discrete system (corresponding to the minima of the superlattice
potential). So we set the on-site ($i=j$) amplitudes as $\mu_{i}$, the nearest
neighbour ($\absolutevalue{i-j}=1$) hopping amplitudes as $J_{v_{i}}$ and
$J_{w_{i}}$ and the next-nearest neighbour hopping amplitudes
($\absolutevalue{i-j}=2$) as $J_{t_{i}}$. In particular, $J_{v_{i}}$ and
$J_{w_{i}}$ express tunneling through the odd and even barriers of the
superlattice potential respectively. Finally, we consider $h_{i,j}$ = 0 for
$\absolutevalue{i-j}>2$ because these terms are significantly suppressed, due
to the fact that they correspond to tunnelling through at least three
consecutive barriers.

In the case of periodic boundary conditions (PBC), as the potential landscape
is also periodic, it is evident that
$\mu_{i}=\mu,~{}~{}J_{v_{i}}=J_{v},~{}~{}J_{w_{i}}=J_{w}$ and
$J_{t_{i}}=J_{t}~{}~{}\forall i$. In essence, all the corresponding sites have
the same potential environment. However, we are interested in the realization
of edge states via a finite optical lattice so the latter has to possess both
a left and a right end at a finite $x$-value. This necessarily implies that
the external potential has to be confining, breaking the translation symmetry.
In our analysis the confinement comes from the hard wall boundary conditions
(HWBC) or from the boundary extension described above.

\begin{overpic}[scale={0.57}]{Figures/Fig6_a.pdf} \put(8.0,55.75){{(a)}}
\end{overpic}\begin{overpic}[scale={0.57}]{Figures/Fig6_b.pdf}
\put(-2.0,22.0){\rotatebox{90.0}{$\rho\left(\delta\mathcal{H}\right)$}}
\put(8.0,55.75){{(b)}} \end{overpic}

Figure 6: (a) The elements $\lambda_{11},\lambda_{12}$ and $\lambda_{13}$
versus the length of the linearly extended boundary $d$ and $\alpha=0$. (b)
The spectral radius of the matrix $\delta\mathcal{H}$ versus the length of the
linearly extended boundary $d$ and three cases of the slope $\alpha=0,0.1,1$
in units of $E_{r}\cdot k_{r}$.

Our mapping is mostly affected by the boundaries in the case of the
topologically non-trivial configuration of the potential landscape. In
particular, the tails of the localized states related to the edge sites are
forced to zero in a sharp (unnatural) way. Another perspective of the finite
boundaries effect, is that every site of the discrete system is mapped to a
minimum of the superlattice potential that features a different environment.
With that in mind, we define:

$h_{i,j}=\begin{cases}\mu+\delta\mu_{i}&i=j\\\ J_{v,w}+\delta
J_{v_{i},w_{i}}&\absolutevalue{i-j}=1\\\ J_{t}+\delta
J_{t_{i}}&\absolutevalue{i-j}=2\\\ \end{cases}$ (12)

where $\mu$, $J_{v}$, $J_{w}$ and $J_{t}$ are the bulk values corresponding to
the center sites ($M$ and $M\pm 1$, $M\pm 2$), while $\delta\mu_{i},~{}\delta
J_{v_{i}},~{}\delta J_{w_{i}}$ and $\delta J_{t_{i}}$ denote the deviations
from these bulk values at each site. Those deviations will be present in any
confining potential, since they express the gradual modification of the
potential environment towards the edges of the system. So, even though the
following analysis is based on the consideration of HWBC or of a linear
boundary, the techniques that we develop here can be applied to any kind of
confining potential. The Hamiltonian of the system can be written as:

$\mathcal{H}_{TB}=\mathcal{H}^{(0)}+\delta\mathcal{H}$ (13)

where $\mathcal{H}^{(0)}$ has as elements the bulk values, and
$\delta\mathcal{H}$ the deviations.

\begin{overpic}[width=424.94574pt]{Figures/Fig7_a.pdf} \put(27.0,23.25){{(a)}}
\end{overpic}\begin{overpic}[width=424.94574pt]{Figures/Fig7_b.pdf}
\put(25.5,23.25){{(b)}} \put(28.0,19.25){$\mathcal{H}^{TB}_{ij}$}
\put(59.5,19.25){$\mathcal{H}^{TB}_{ij}$}
\put(90.5,19.25){$\mathcal{H}^{TB}_{ij}$} \end{overpic}

Figure 7: (a) The $\lambda$ matrix for a system of $M=5$ cells,
$V_{high}=40E_{R}$ and $u=0.5$ for different values of the length $d$ of the
linear extension of the potential area ($\alpha=0$). (b) The
$\mathcal{H}_{TB}$ matrix for a system of $M=5$ cells, $V_{high}=40E_{R}$,
$d=0.14L_{cell}$ (and $\alpha=0$) for different values of the ratio $u$ of the
heights of the potential. In all cases, we have used color plots as a visual
tool to display the elements of the matrices. The elements
$\mathcal{H}^{TB}_{ij}$ are in units of $E_{r}$.

###  IV.2 Minimising the ${\bf\it\delta\mathcal{H}}$ term

In order to examine the behaviour of the $\delta\mathcal{H}$ term, we can
define the $\lambda$ matrix, with elements:

$\lambda_{ij}=\big{|}\delta\mathcal{H}_{ij}\big{|}/\big{|}\mathcal{H}^{(0)}_{ij}\big{|}$
(14)

expressing the relative deviation from the bulk values. As we can see in Fig.
7(a), the strongest (and only significant) deviations arise from the terms
that correspond to the endpoints, in accordance with the results of the
continuous system. Furthermore, we see that the elements of the $\lambda$
matrix depend on the parameters of the linear extension of the potential
domain. Specifically, as we can see in Fig. 6 (a), for fixed $\alpha$ and
increasing $d$, they decrease down to certain minimum values and then increase
until they stabilize and become $d$-independent. In order to determine the
critical length $d_{crit}$ for which the elements of the $\lambda$ matrix are
minimized, we employ the spectral radius Horn and Johnson (1985). Namely, we
solve the eigenvalue problem of the matrix

\begin{overpic}[scale={0.63}]{Figures/Fig8.pdf} \end{overpic} Figure 8: Color
plot of the $d_{crit}$ in units of $L_{cell}$ defined as the point of minimum
of the spectral radius versus the ratio of the heights of the potential
barriers $u$ and the slope $\alpha$. The area colored in deep red indicates
the parameter region where the spectral radius has no minimum.

$\delta\mathcal{H}$ for specific values of the parameters $d$ and $\alpha$,
and we find the eigenvalue with the maximum absolute value, which is the
spectral radius $\rho(\delta\mathcal{H}$) of the matrix. We repeat this
process for different values of the parameter $d$. Finally, we identify as
$d_{crit}$ the value of $d$ for which $\rho(\delta\mathcal{H})$ is minimized.
As shown in Fig. 6 (b) for three different cases of the slope $\alpha$ there
is indeed a $d_{crit}$. Moreover, we observe in Fig. 8 that for specific
values of the parameters of the system there is no minimum in the spectral
radius, and one should find a different way to define $d_{crit}$. For example,
in those cases $d_{crit}$ could be defined as the value of $d$ for which the
elements of the $\lambda$ matrix become $d$-independent.

###  IV.3 The behaviour of the ${\bf\it\mathcal{H}_{\text{NNN}}}$ term

In this subsection we consider systems with $d=d_{crit}$ so that we can omit
the $\delta\mathcal{H}$ term. Hence, we can write $\mathcal{H}_{TB}$ as:

$\mathcal{H}_{TB}=\mathcal{H}^{(0)}=\mathcal{H}_{\mu}+\mathcal{H}_{SSH}+\mathcal{H}_{\text{NNN}}$
(15)

where $\mathcal{H}_{\mu}$ and $\mathcal{H}_{\text{NNN}}$ express the on-site
and next-nearest neighbour hopping terms respectively. We can also neglect the
$\mathcal{H}_{\mu}$ term, since it represents an overall offset in the energy
scale that does not affect the phenomenology of our systems. So we reach to
$\mathcal{H}_{TB}=\mathcal{H}_{\text{SSH}}+\mathcal{H}_{\text{NNN}}$ which is
clearly the exact same as Eq. (3). Thus, we have ultimately obtained the usual
SSH model with the addition of NNN hopping terms (eSSH). In Fig. 7 (b) we
present how the final form of the TB Hamiltonian is affected by the ratio of
the heights of the two potential barriers $u$. For fixed $V_{high}$ and
increasing $u$ we see $J_{v}$ increasing, $J_{w}$ decreasing and $J_{t}$
slightly decreasing (see also Fig. 10 for the exact values for three different
cases of $V_{high}$).

In order to neglect the $\mathcal{H}_{\text{NNN}}$ term we have to make a
strong assumption, since we have to take into account the different scales of
magnitude of the hopping amplitudes. To this end, we define the ratios that we
present in Fig. 9. Specifically, $J_{v}/J_{w}$ and $J_{t}/J_{v}$ express
relative sizes of the weak to strong and NNN to weak hopping amplitudes,
respectively. As we increase $u$, we observe $J_{v}/J_{w}$ increasing and
$J_{t}/J_{v}$ decreasing. Therefore, there is a specific value of $u$ for
which the two ratios are crossing (orange point in the graph of Fig. 9). We
could in principle define this value as a $u_{crit}$ and omit the
$\mathcal{H}_{\text{NNN}}$ term for $u>u_{crit}$. Alternatively, we could
define $u_{crit}$ as the value of $u$ for which $J_{v}/J_{w}$ and
$(J_{t}/J_{v})/(J_{v}/J_{w})$ are crossing (red point in the graph of Fig. 9).
The ratio $(J_{t}/J_{v})/(J_{v}/J_{w})$ expresses how $J_{v}$ varies from
small (close to $J_{t}$) to large (close to $J_{w}$) orders of magnitude (for
fixed $V_{high}$ and increasing $u$). This is naturally a more strict
definition, since for the first definition we omit the $\mathcal{H}_{NNN}$
term when $J_{v}/J_{w}>J_{t}/J_{v}$ while for the second when
$J_{v}/J_{w}>\sqrt{J_{t}/J_{v}}$. We mention that for both definitions
$u_{crit}$ is $V_{high}$ dependent.

We close this subsection by noting that both SSH and eSSH support edge states.
However, when it comes to their experimental realization via an optical
lattice, the NNN terms will always be present if both wells are not deep
enough. Thus, the eSSH model is the easier to be implemented. We note that
next-neighbor tunneling terms in the SL were, e.g., considered in the context
of Thouless pumps, where they can lead to a deviation of the pumped charge
Schouten and Cheianov (2021).

\begin{overpic}[scale={0.5}]{Figures/Fig9.pdf} \end{overpic} Figure 9: The
ratios $J_{v}/J_{w}$, $J_{t}/J_{v}$ and $(J_{t}/J_{v})/(J_{v}/J_{w})$ for the
hopping amplitudes of a system with $M=5$, $V_{high}=40E_{r}$ and
$d=d_{crit}$. The horizontal dashed line shows the $0.2$ ratio value, while
the vertical dashed lines show the values of $u$ where the crossing of
$J_{v}/J_{w}$ with $J_{t}/J_{v}$ and $(J_{t}/J_{v})/(J_{v}/J_{w})$ occurs.

###  IV.4 Experimental Considerations

An extension of the boundary as proposed here is experimentally feasible by
combining an optical lattice with an arbitrary optical dipole potential
projected via a high resolution imaging system. This allows shaping the
potential at the scale of a single lattice site and also to prepare desired
lattice occupations Weitenberg  _et al._ (2011); Preiss  _et al._ (2015) and
was recently used to create sufficiently steep walls for the creation of edge
states in 2D systems Braun  _et al._ (2023); Yao  _et al._ (2023). The above
analysis shows that the correct mid-gap states are reached for a range of
parameters without fine tuning, making it experimentally feasible. The
selective preparation of the edge states could be achieved, e.g., by an
appropriate charge pump or by heating away all other atoms via an amplitude
modulation that leaves the edge state unaffected.

\begin{overpic}[scale={0.35}]{Figures/Fig10_a.pdf} \put(14.0,30.0){{(a)}}
\end{overpic}\begin{overpic}[scale={0.35}]{Figures/Fig10_b.pdf}
\put(14.0,31.5){{(b)}} \end{overpic}

\begin{overpic}[scale={0.35}]{Figures/Fig10_c.pdf} \put(14.0,29.0){{(c)}}
\end{overpic}

Figure 10: The $J_{v},J_{w}$ and $J_{t}$ hopping amplitudes values in units of
$E_{r}$ for systems with $M=5$, (a) $V_{high}=20E_{r}$ and $d=0.15L_{cell}$
(b) $V_{high}=40E_{r}$ and $d=0.14L_{cell}$ and (c) $V_{high}=60E_{r}$ and
$d=0.15L_{cell}$. In (a) the x-axis starts from $0.6$, since for $u<0.6$ not
all eigenergies of the first band lies below the lowest barrier ($V_{low}$).

##  V Summary and Perspectives

We have studied the mapping of a finite continuous optical lattice, within the
tight-binding approximation, to an SSH and an eSSH model. We revealed the
complications that emerge from the confinement of the system, when hard wall
boundary conditions are considered. Specifically, we found that especially in
the topologically non-trivial case, there is a substantial discrepancy in the
characteristics of the edge states of the system between what is expected for
the SSH model and the results provided by the ED. We found that this is
related to the fact that the potential environment is gradually changing from
the bulk towards the ends of the system, due to the confinement. We then
proceeded by providing a solution to this problem through a linear extension
of the potential area. In detail, we established and used qualitative criteria
in order to describe the minimization of the effects related to the HWBC. We
also examined the behaviour of the NNN hopping terms in system with
experimentally feasible depths of wells.

The goal of the present work was to reveal the problems occurring in the
mapping of a continuous optical lattice potential to a discrete SSH model when
finite size and open boundary conditions are considered. From a theoretical
point of view, these problems are resolved with the boundaries extension we
proposed here. Nevertheless, for the design of an optimal experimental set-up
capable to detect the associated edge states, a more advanced optimization
procedure, involving a more general functional form of the potential
extension, may be needed. This functional form should be eventually also
adapted to the experimental requirements for the specific set-up.
Additionally, the observability of the continuous formed edge states may be
influenced significantly by their dynamics. However, a corresponding dynamics
analysis goes beyond the scope of the present work and will be investigated in
the future.

The exact realization of tight-binding models for ultracold atoms was recently
also considered for tweezer arrays, including the homogeneity of the on-site
interaction strength Wei  _et al._ (2024b). The study of interacting
superlattice systems is particularly interesting, because the bosonic model
with weak interactions lacks chiral symmetry and the edge states are no longer
mid-gap Grusdt  _et al._ (2013). Further interesting extensions would be to
consider optimal parameters for mid-gap states in the bulk of the system
induced by impurities Lang and Chen (2014) or by a step in the confining
potential Grusdt  _et al._ (2013), as well as optimal boundaries when
combining superlattices with a Floquet drive to realize the AIII symmetry
class Velasco and Paredes (2017). It would also be interesting to perform a
similar anaylsis for engineered systems with indirect band gaps, where
surprisingly no localized edge states were found Malki  _et al._ (2019).

## Acknowledgments

This work (P.S. and C.W.) has been supported by the Cluster of Excellence
“Advanced Imaging of Matter” of the Deutsche Forschungsgemeinschaft (DFG)-EXC
2056, Project ID No. 390715994.

## APPENDIX

##  Appendix A Edge localized states of very large/deep lattice

In Fig. 11 we present the energy spectrum and the edge localized states of a
very (a) large and (b) deep lattice when HWBC are considered, i.e. without the
extension of the potential domain. In both cases the energies of the two edge
localized states feature an energy shift towards the upper band. This is to be
expected, since it is related to the consideration of HWBC for the system.
However, in the case of the very deep lattice we observe that the profiles of
the two edge localized states look similar to those of an eSSH model. This is
remarkable, since the discrepancy on the energies is still present.

\begin{overpic}[scale={0.425}]{Figures/Fig11_a.pdf} \put(12.0,22.25){{(a)}}
\end{overpic}\begin{overpic}[scale={0.425}]{Figures/Fig11_b.pdf}
\put(12.0,22.25){{(b)}} \end{overpic}

Figure 11: The energy spectrum and the profiles of the corresponding edge
localized states for a very (a) large and (b) deep lattice ($x$ is in units of
$k_{r}^{-1}$).

## References

  * Anderson  _et al._ (1995) M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269, 198 (1995).
  * Bradley  _et al._ (1995) C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Phys. Rev. Lett. 75, 1687 (1995).
  * Davis  _et al._ (1995) K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995).
  * Bloch  _et al._ (2008) I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008).
  * Bloch  _et al._ (2012) I. Bloch, J. Dalibard, and S. Nascimbène, Nature Phys. 8, 267 (2012).
  * Courteille  _et al._ (1998) P. Courteille, R. S. Freeland, D. J. Heinzen, F. A. van Abeelen, and B. J. Verhaar, Phys. Rev. Lett. 81, 69 (1998).
  * Inouye  _et al._ (1998) S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kurn, and W. Ketterle, Nature 392, 151 (1998).
  * Chin  _et al._ (2010) C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev. Mod. Phys. 82, 1225 (2010).
  * Bakr  _et al._ (2009) W. S. Bakr, J. I. Gillen, A. Peng, S. Fölling, and M. Greiner, Nature 462, 74 (2009).
  * Jaksch  _et al._ (1998) D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys. Rev. Lett. 81, 3108 (1998).
  * Greiner  _et al._ (2002) M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
  * Cooper  _et al._ (2019) N. R. Cooper, J. Dalibard, and I. B. Spielman, Rev. Mod. Phys. 91, 015005 (2019).
  * Weitenberg and Simonet (2021) C. Weitenberg and J. Simonet, Nature Physics 17, 1342 (2021).
  * Ryu  _et al._ (2010) S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, New J. Phys. 12, 065010 (2010).
  * Palaiodimopoulos  _et al._ (2021) N. E. Palaiodimopoulos, I. Brouzos, F. K. Diakonos, and G. Theocharis, Phys. Rev. A 103, 052409 (2021).
  * D’Angelis  _et al._ (2020) F. M. D’Angelis, F. A. Pinheiro, D. Guéry-Odelin, S. Longhi, and F. m. c. Impens, Phys. Rev. Res. 2, 033475 (2020).
  * Huang  _et al._ (2022) L. Huang, Z. Tan, H. Zhong, and B. Zhu, Phys. Rev. A 106, 022419 (2022).
  * Ma  _et al._ (2022) H. Ma, Z. Zhang, P.-H. Fu, J. Wu, and X.-L. Yu, Phys. Rev. B 106, 245109 (2022).
  * Grusdt  _et al._ (2013) F. Grusdt, M. Höning, and M. Fleischhauer, Phys. Rev. Lett. 110, 260405 (2013).
  * Stenzel  _et al._ (2019) L. Stenzel, A. L. C. Hayward, C. Hubig, U. Schollwöck, and F. Heidrich-Meisner, Phys. Rev. A 99, 053614 (2019).
  * Zhou  _et al._ (2023) X. Zhou, J.-S. Pan, and S. Jia, Phys. Rev. B 107, 054105 (2023).
  * Nakajima  _et al._ (2016) S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa, L. Wang, M. Troyer, and Y. Takahashi, Nature Physics 12, 296–300 (2016).
  * Lohse  _et al._ (2016) M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, and I. Bloch, Nature Physics 12, 350–354 (2016).
  * Li  _et al._ (2014) L. Li, Z. Xu, and S. Chen, Phys. Rev. B 89, 085111 (2014).
  * Pérez-González  _et al._ (2019) B. Pérez-González, M. Bello, A. Gómez-León, and G. Platero, Phys. Rev. B 99, 035146 (2019).
  * Braun  _et al._ (2023) C. Braun, R. Saint-Jalm, A. Hesse, J. Arceri, I. Bloch, and M. Aidelsburger, (2023), arXiv:2304.01980 .
  * Yao  _et al._ (2023) R. Yao, S. Chi, B. Mukherjee, A. Shaffer, M. Zwierlein, and R. J. Fletcher, (2023), arXiv:2304.10468 .
  * Boada  _et al._ (2012) O. Boada, A. Celi, J. I. Latorre, and M. Lewenstein, Phys. Rev. Lett. 108, 133001 (2012).
  * Mancini  _et al._ (2015) M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider, J. Catani, C. Sias, P. Zoller, M. Inguscio, M. Dalmonte, and L. Fallani, Science 349, 1510–1513 (2015).
  * Stuhl  _et al._ (2015) B. K. Stuhl, H.-I. Lu, L. M. Aycock, D. Genkina, and I. B. Spielman, Science 349, 1514–1518 (2015).
  * Kanungo  _et al._ (2022) S. K. Kanungo, J. D. Whalen, Y. Lu, M. Yuan, S. Dasgupta, F. B. Dunning, K. R. A. Hazzard, and T. C. Killian, Nature Comm. 13, 972 (2022).
  * de Léséleuc  _et al._ (2019) S. de Léséleuc, V. Lienhard, P. Scholl, D. Barredo, S. Weber, N. Lang, H. P. Büchler, T. Lahaye, and A. Browaeys, Science 365, 775 (2019).
  * Su  _et al._ (1979) W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979).
  * Batra and Sheet (2020) N. Batra and G. Sheet, Resonance 25, 765 (2020).
  * Anderlini  _et al._ (2007) M. Anderlini, P. J. Lee, B. L. Brown, J. Sebby-Strabley, W. D. Phillips, and J. V. Porto, Nature 448, 452–456 (2007).
  * Fölling  _et al._ (2007) S. Fölling, S. Trotzky, P. Cheinet, M. Feld, R. Saers, A. Widera, T. Müller, and I. Bloch, Nature 448, 1029–1032 (2007).
  * Marzari  _et al._ (2012) N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, Rev. Mod. Phys. 84, 1419 (2012).
  * Wei  _et al._ (2024a) H.-T. Wei, E. Ibarra-García-Padilla, M. L. Wall, and K. R. A. Hazzard, Phys. Rev. A 109, 013318 (2024a).
  * Marzari and Vanderbilt (1997) N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847 (1997).
  * Horn and Johnson (1985) R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press (1985).
  * Schouten and Cheianov (2021) K. J. M. Schouten and V. Cheianov, Phys. Rev. A 104, 063315 (2021).
  * Weitenberg  _et al._ (2011) C. Weitenberg, M. Endres, J. F. Sherson, M. Cheneau, P. Schauß, T. Fukuhara, I. Bloch, and S. Kuhr, Nature 471, 319–324 (2011).
  * Preiss  _et al._ (2015) P. M. Preiss, R. Ma, M. E. Tai, A. Lukin, M. Rispoli, P. Zupancic, Y. Lahini, R. Islam, and M. Greiner, Science 347, 1229–1233 (2015).
  * Wei  _et al._ (2024b) H.-T. Wei, E. Ibarra-García-Padilla, M. L. Wall, and K. R. A. Hazzard, Physical Review A 109 (2024b).
  * Lang and Chen (2014) L.-J. Lang and S. Chen, Journal of Physics B: Atomic, Molecular and Optical Physics 47, 065302 (2014).
  * Velasco and Paredes (2017) C. G. Velasco and B. Paredes, Phys. Rev. Lett. 119, 115301 (2017).
  * Malki  _et al._ (2019) M. Malki, L. Müller, and G. S. Uhrig, Phys. Rev. Res. 1, 033197 (2019).