# Multiple topological phase transitions in double-bridged polyacene polymers: Ab initio parametrized stacked Su-Schrieffer-Heeger model

M. Aktas, M. Mosaferi, J. Saint-Martin, A. Chin, D. Romanin

► To cite this version:

M. Aktas, M. Mosaferi, J. Saint-Martin, A. Chin, D. Romanin. Multiple topological phase transitions in double-bridged polyacene polymers: Ab initio parametrized stacked Su-Schrieffer-Heeger model. Physical Review B, 2025, 112 (4), pp.045130. 10.1103/x7mg-wvxx. hal-05175905

HAL Id: hal-05175905

https://hal.science/hal-05175905v1

Submitted on 22 Jul 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Multiple topological phase transitions in double-bridged polyacene polymers: an ab initio parametrized stacked Su-Schrieffer-Heeger model

M. Aktas,\* M. Mosaferi,† J. Saint-Martin,‡ A. W. Chin,† and D. Romanin§

(Dated: June 11, 2025)

Polyacene polymers are promising organic one-dimensional materials for optoelectronic and energy-conversion applications that can host non-trivial topologically insulating phases, without the need of spin-orbit coupling. In this work, by using ab-initio methods, we show that, in its multi-bridged configuration, it is possible to tune their topological phase by varying the bridges' number and relative position. Moreover, we show that these multi-bridged polymers can be described by a stacked Su-Schrieffer-Heeger (SSH) model, where inter-chain hoppings and next-nearest neighbour interactions play a crucial role in stabilizing multitude of non-trivial topologically insulating phases. The generality of this model is such that it can be used to describe any couple of interacting SSH chains.

## I. INTRODUCTION

Organic materials offer a combination of strong light absorption, low production costs, flexibility, and the ability to finely tune their electronic properties through chemical synthesis, which makes them ideal candidates for optoelectronic devices such as photovoltaics, light-emitting diodes and field-effect transistors [1, 2]. Moreover, recent developments in “on-surface” synthesis methods [3–5] have enabled the fabrication of such structures with atomic precision, making it also easier to alter their geometry [6–12].

One-dimensional (1D) single-bridged polyacene polymers represent a promising class of organic semiconductors, whose electronic structure can be described by the well-known topological Su-Schrieffer-Heeger (SSH) model [13–15]. By tailoring the monomer size and the chain length, signatures of a topological phase transition in the absence of spin-orbit coupling were captured both theoretically and experimentally [8, 9, 16]. These features include the change of bonding structure from aromatic/ethylynic to quinoidal/cumulenic, the closing and reopening of the band gap followed by a band parity inversion and the emergence of mid-gap edge states as a function of both the monomer size [8] and the chain length [9]. Therefore, these polymers are ideal playgrounds for exploring topological effects by simple structure modifications, which represent a novel way to tune organic materials’ electronic structure for optoelectronic applications [16–18].

<figure><img src="image_2.png" /><figcaption>FIG. 1. Stick-and-ball model of single- (left) and double- (right) bridged polyanthracene (n=3) polymer. Brown (pink) spheres are carbon (hydrogen) atoms. The periodic direction is along the chain and is denoted by black dots.</figcaption></figure>

More precisely, polyacene polymers consist of acene monomers, each characterized by a specific number of fused benzene rings (n), linked together by ethynylene bridges (Fig. 1). In Refs. [8, 9], it was observed that the topological phase transition in single-bridged polyacene polymers on Au(111) substrates occurs between polyanthracene (n = 3) and polypentacene (n = 5). By relating the electronic structures to the SSH model, they were able to rationalize the transition with a $Z_2$ invariant dependent on the relative magnitude of the intra-(inter-)monomer electron hoppings $t_1$ ($t_2$) (Fig. 2).

It is important to notice that the SSH model was initially proposed for polyacetylene, where there is a one-to-one mapping between the tight-binding (TB) model sites and the $p_z$ orbitals of the carbon atoms. Moreover, the hopping amplitudes $t_1$ and $t_2$ have a direct relation to the carbon-carbon bond lengths.

<footnote>\* melih.aktas@universite-paris-saclay.fr; Université Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies, 91120, Palaiseau, Paris, France.</footnote>

<footnote>† Sorbonne Université, CNRS, Institut des Nanosciences de Paris, UMR7588, F-75252 Paris, France.</footnote>

<footnote>‡ Université Paris Saclay, ENS Paris Saclay, CNRS, SATIE, 91190, Gif-sur-Yvette, France.</footnote>

<footnote>§ davide.romanin@universite-paris-saclay.fr; Université Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies, 91120, Palaiseau, Paris, France.</footnote>
However, recently Röntgen et al. [19] have shown that more complicated structures, even those which do not explicitly have any inversion or chiral symmetries can be effectively described with the SSH model. As a matter of fact, Cirera et al. [8] have shown that the SSH model can be employed to describe single-bridged polyacene polymers, where, in that case, a set of 8 to 10 p<sub>z</sub> orbitals constitute the TB sites (see also Sec. S1 in the Supplemental Material for the Wannier functions used in the present work). Therefore, although less direct, the hopping amplitudes can still provide physical insights on the properties of such systems.

Indeed, the gap between the highest occupied molecu-
lar orbital (HOMO) and the lowest unoccupied molecular
orbital (LUMO) of a singular acene monomer mono-
tonically decreases as we increase *n* due to delocalizing
the frontier orbitals. It is then possible to translate the
situation to polyacene polymers by relating their HOMO
(LUMO) to the (anti-)bonding SSH dimer state [8]:
the HOMO-LUMO gap will be precisely related to the
intra-monomer hopping *t*₁, while the inter-monomer
hopping *t*₂ will be linked to the ethylynic bridges
between the acene monomers. Therefore, increasing the
monomer size from *n* = 3 to *n* = 5 amounts to lowering
*t*₁ and allowing for a topological phase transition.

Although the effects of the monomer size have been
studied extensively, the effects of the bridge config-
uration between each monomer, both the number of
bridges and their positions (Fig. 1), remain unexplored
and could provide us with novel insights on achieving
topologically non-trivial polymers. In this work we
explore these degrees of freedom via both ab-initio
calculations and effective TB Hamiltonians. Moreover,
we describe stacked SSH chains, which take into account
more than 2 sites and next-nearest neighbour interac-
tions. We show how such stacked SSH model can be
used to accurately describe the topological behaviour
of double-bridged polyacene polymers, as well as any
couple of interacting SSH chains, revealing the emer-
gence of multiple topological phases. Recent advances in
on-surface synthesis techniques, which have enabled the
fabrication of complex carbon-based structures, make
the synthesis of double-bridged polyacene polymers
plausible and thereby motivate the present work.

More precisely, in Sec. II we start by discussing com-
putational details. Then, in Sec. III, we give a summary
of the topological formalism we have employed and we
provide a detailed description of our double-stacked
SSH model. In Sec. IV we discuss the structural,
electronic and topological properties of single-bridged
polyanthracene (n = 3) and polypentacene (n = 5), also
analyzing the effects of screening from the environment.
After that, in Sec. V we discuss the structural, electronic
and topological properties of double-bridged polypen-
tacene. Finally in Sec. VI we discuss our conclusions.

II. NUMERICAL METHODS

A. Geometry optimization

Density functional theory calculations are performed
both in the plane-wave basis, as implemented in Quan-
tum ESPRESSO (QE) [20, 21], and in the local Gaussian
basis, as implemented in Crystal23 (CRY) [22, 23].

Polyacene polymers are periodic along the *x* di-
rection and we added ~ 10Å of vacuum along the
non-periodic directions (*y*,*z*) in order to obtain a proper
one-dimensional system. Geometry has been relaxed
with both B3LYP and CAMB3LYP [24, 25] functionals
to compare the effects of screening. The latter choice of
exchange-correlation functional is particularly suited to
improving long-range exchange interactions [16, 26] in
vacuum.

In QE, eigenvalues and eigenstates are obtained using
a norm-conserving pseudopotential [27] with the kinetic
energy cutoffs of 60 Ry and 240 Ry for the wave functions
and the charge density, respectively. We fixed an energy
tolerance of 10<sup>-10</sup> Ry for the total energy convergence.
On the other hand, in CRY we have employed the triple-
ξ-polarized Gaussian type basis set [28] with real space
integration tolerances of 10-10-10-25-50 and an energy
tolerance of 10<sup>-12</sup> Ha for the total energy convergence.
In both cases, convergence of total energy and forces is
assured by sampling the Brillouin zone with 50 k-points
along the periodic direction.

B. Ground-state and topological invariants

Eigenvalues and eigenvectors have been obtained in
QE via the PBE exchange correlation functional on top
of both the B3LYP/CAMB3LYP optimized structures:
our goal in this work is not to give a quantitative
evaluation of the quasiparticle band gap (for which more
precise techniques are required, such as self-consistent
GW, see Ref. [16]), but to discuss the trend in the band
inversion and topological phase transitions as a function
of the electron-electron interaction.

Topological phases of the polymers have been analyzed
via numerical evaluation of the Zak phase [29] (or wind-
ing number). To achieve this, we constructed an effec-
tive TB model using Wannier interpolation [30–32] of the
QE eigenvalues and the eigenstates: after self-consistent
calculation of the equilibrium electronic density, non-self-
consistent calculations were carried on a grid of 20 × 1 × 1
k-points along the periodic direction. For each configura-
tion of monomer size and bridges’ position, we construct
a two(four)-site TB model using the maximally localized
Wannier functions (MLWFs) as a basis. We then calculate the Zak phase and winding number of each band to determine their topological phase as described in Sec. III.

### C. Numerical evaluation of Zak phases

To determine the topological phase, we discretized the Brillouin Zone onto N = 200 equally distant points and evaluated the discrete formulation of the Zak phase using the parallel-transport gauge given by ([13], chapter 3.4):

$$Z_n = - \operatorname{Im} \ln \left[ \langle u_n(k_0) | u_n(k_1) \rangle \langle u_n(k_1) | u_n(k_2) \rangle \dots \langle u_n(k_{N-1}) | e^{-iG \cdot r} | u_n(k_0) \rangle \right], \quad (1)$$

where G is the reciprocal lattice vector. The periodic gauge choice, $|u_n(k_N)\rangle = e^{-iG \cdot r}|u_n(k_0)\rangle$, was imposed to ensure smoothness at the borders of BZ. The off-diagonal terms of the position operator r have been ignored as they are three orders of magnitude smaller than the diagonal terms. The calculations were done with the tight-binding Python library PythTB [33].

### D. Edge states

Starting from Wannier interpolated Hamiltonians, we use pythTB to study open finite chains in order to observe the emergence of edge states of topologically non-trivial polymers. In order to do so we consider increasingly long chains until we notice the appearance of mid-gap states in the energy spectrum. We then use this critical length for visualizing the orbital character via DFT calculations in the CRY code. Such calculations have been performed with ~500Å of vacuum along all directions in order to have a proper finite system and by adding a hydrogen atom at the extrema.

## III. ZAK PHASE AND TOPOLOGY

In this section, we briefly discuss the connection between the Zak phase and the band topology defined on the Su-Schrieffer-Heeger (SSH) model, which has been shown to accurately describe single-bridged polyacene polymers [8]. We then generalize this formalism to a four site “double stacked” SSH model for describing double-bridged polyacene polymers.

### A. Simple Su-Schrieffer-Heeger Model

The SSH model is a one-dimensional TB model originally introduced to describe polyacetylene [14]. Due to the dimerized nature of the chain, it consists of orbitals with the same on-site energies and alternating

hopping parameters (Fig. 2).

The real space Hamiltonian is given by:

$$\hat{H} = t_1 \sum_n c_{Bn}^{\dagger} c_{An} + t_2 \sum_n c_{A(n+1)}^{\dagger} c_{Bn} + \text{H.c.}, \quad (2)$$

where $c_{An}^{\dagger}$ ($c_{An}$) and $c_{Bn}^{\dagger}$ ($c_{Bn}$) are creation (annihilation) operators of fermions on sublattices A and B of the n'th unit cell, respectively. The parameters $t_1$ and $t_2$ represent the intra-cell and inter-cell hopping amplitudes, respectively. “H.c.” stands for Hermitian conjugate. Equal on-site energies (cast in the form of an identity matrix) are neglected as they do not have any effect on physics, except for a trivial energy shift.

The topological phase of the model depends on the relative strength of $t_1$ and $t_2$: when the intra-cell hopping is stronger (weaker) than the inter-cell hopping, the system is in a topologically trivial (non-trivial) phase. This classification is due to the emergence of the zero-energy edge states on an open finite chain in the non-trivial phase due to the bulk-boundary correspondence [13, 34].

### B. $Z_2$ Invariant

The Zak phase is the phase acquired by an electron (a periodic Bloch state on a given band) as it traverses the Brillouin zone (BZ):

$$Z_n = i \int_{\text{BZ}} \langle u_n(\mathbf{k}) | \nabla_{\mathbf{k}} | u_n(\mathbf{k}) \rangle d\mathbf{k}, \quad (3)$$

where $\nabla_{\mathbf{k}}$, $|u_n(\mathbf{k})\rangle$ and n are the gradient operator, the cell-periodic Bloch function and the band index, respectively. The integral is carried out on a non-contractible loop over the BZ torus where the periodic gauge choice $|u_n(k_f)\rangle = e^{-iG \cdot r} |u_n(k_i)\rangle$ is imposed. Since the Zak phase is defined with the position operator, it is necessary to use the so-called basis II [35] to construct the Bloch functions. This basis takes the relative positions of each orbital into account.

The average position of a Wannier function in the unit cell at the origin of the crystal ($R = 0$), $\langle \mathbf{r} \rangle = \langle w_0 | \mathbf{r} | w_0 \rangle$ (Wannier center) can be cast in the following form [36] (we omit the band index without loss of generality):

$$\langle w_0 | \mathbf{r} | w_0 \rangle = i \frac{V}{(2\pi)^D} \int_{\text{BZ}} d\mathbf{k} \langle u(\mathbf{k}) | \nabla_{\mathbf{k}} | u(\mathbf{k}) \rangle, \quad (4)$$

where V and D are the volume of the unit cell and the dimensions of the system, respectively.

From Eqs. 3 and 4, it is possible to relate the Zak phase and the position of the Wannier center:
<figure><img src="image_3.png" /></figure>

$$Z_2 = 2(\langle x \rangle^t - \langle x \rangle^{t'}) = 0$$

trivial

<figure><img src="image_4.png" /></figure>

$$Z_2 = 2(\langle x \rangle^{nt} - \langle x \rangle^{t'}) = 1$$

non-trivial

<caption>FIG. 2. Schematic representation of SSH model in the trivial and nontrivial phases. Thick lines correspond to stronger bonds. Left: $t_1 > t_2$, the Wannier center is located at the center of the intra-cell bond and $Z_2 = 0$ (trivial). Right: $t_1 < t_2$, the Wannier center is located at the center of the inter-cell bond and $Z_2 = 1$ (non-trivial).</caption>

$$\langle \mathbf{r} \rangle_n = \frac{V Z_n}{(2\pi)^D}, \qquad (5)$$

and in one-dimension, taking the lattice length as unity, we get:

$$\langle x \rangle_n = \frac{Z_n}{2\pi}. \qquad (6)$$

The Wannier centers in one-dimensional crystals are usually located at the inversion centers: in the SSH model, they are located at the centers of the bonds in between the sites. Moreover, one can show that the Wannier centers are always located at the center of the intra- (inter-) cell bond in the trivial (non-trivial) phase [34, 37] (Fig. 2).

The Zak phase is continuously dependent on the Wannier center and, as a consequence, on the choice of the unit cell. Therefore, to remove this dependence on the choice of the unit cell, we make the following definition of a $Z_2$ invariant where we calculate the Zak phase with respect to the intra-cell bond center $\langle x \rangle_n^{triv}$, i.e., the Wannier center of a topologically trivial SSH chain [38]:

$$Z_2 := \frac{Z_n - 2\pi \langle x \rangle_n^{triv}}{\pi}. \qquad (7)$$

With this definition, we get exactly $Z_2 = 0$ for a trivial SSH (the Zak phase coincides with the intra-cell bond center) and $Z_2 = 1$ for a non-trivial SSH (the Wannier center shifts by half a unit cell from the trivial case).

It is important to notice that in Ref. [38], the authors do not distinguish the two dimerizations of SSH and they propose a $Z_2$ invariant defined with respect to a bond center, irrespective of whether the bond is intra- or inter-cell, which is defined modulo half a unit cell. As a result,

their $Z_2$ invariant is defined modulo 1 and it vanishes for both dimerizations. Since there is only one natural way to cut the polyacene polymers (through the bridges), there is no ambiguity in the choice of the unit cell as we draw the analogy between the infinite and the finite chains. This is why in this work, we explicitly distinguish the two dimerizations of SSH with different topological invariants, as opposed to [38].

### C. Double-Stacked Su-Schrieffer-Heeger Model

In the presence of multiple bridges, there are multiple pairs of bands appearing near the Fermi level studied. Therefore, in order to accurately capture the physics of these polymers, it becomes necessary to work with models with more than two sites. The polymers under investigation in this work consist of acene monomers connected via two ethynylene bridges, which therefore have to be modeled with (at least) four sites in order to isolate the bands closest to the Fermi level.

In this work we consider a minimal “double-stacked” SSH model (Fig. 3), which consists of two equivalent SSH chains closely separated on the (non-periodic) y axis resulting with inter-chain hoppings. The real space Hamiltonian is given by,

$$\begin{aligned} \hat{H} = & \sum_n t_v (c_{A_2 n}^\dagger c_{A_1 n} + c_{B_2 n}^\dagger c_{B_1 n}) \\ & + \sum_{n, \alpha = \{1, 2\}} t_1 c_{B_\alpha n}^\dagger c_{A_\alpha n} + \sum_{n, \alpha = \{1, 2\}} t_2 c_{A_\alpha (n+1)}^\dagger c_{B_\alpha n} \\ & + \sum_n t_{c1} (c_{B_2 n}^\dagger c_{A_1 n} + c_{B_1 n}^\dagger c_{A_2 n}) \\ & + \sum_n t_{c2} (c_{A_2 (n+1)}^\dagger c_{B_1 n} + c_{A_1 (n+1)}^\dagger c_{B_2 n}) + \text{H.c.}, \end{aligned} \qquad (8)$$
<figure><img src="image_5.png" /><figcaption>FIG. 3. Schematic representation of the double-stacked SSH model. Two closely separated equivalent SSH chains hybridize due to the inter-chain hoppings $t_{c1}$ and $t_{c2}$ to create two “sub-SSH” models, with anti-bonding (b) and bonding (c) superpositions of the vertically neighbouring A (B) sites. Consequently, the 4-site model can be treated by solving the two sub-SSH models individually.</figcaption></figure>

where $t_v$ and $t_{c1/c2}$ are the inter-chain (cross) hoppings and the subscripts on A and B denote which chain the sites belong to.

Due to the inter-chain hoppings, the chains become hybridized, and the system mediates two different pairs of SSH-like electronic bands with bonding and anti-bonding vertical superpositions and consequently possibly different topological phases while conserving the

chiral (sublattice) symmetry. Henceforth, the polymers under study can be accurately described with this 4-site model and the topological formalism of the simple SSH model still applies. As a matter of fact, in this formalism, the HOMO-LUMO gap of the single monomer unit will be related to the intra-cell hopping amplitude of the anti-bonding sub-SSH system, i.e., to $t_1 - t_{c1}$. We will show that electronic bands of the anti-bonding sub-SSH will be indeed closer to the Fermi level in most cases when the inter-cell hopping amplitude is relatively weak.

In order to classify the topology of double-bridged polymers, assuming periodic boundary conditions, we first rewrite Eq. 8 by expanding the field operators $c_{ln}$ in real space over operators in reciprocal space $c_{lk}$:

$$c_{ln}^{\dagger} = \frac{1}{\sqrt{N}} \sum_{k} e^{ikna} c_{lk}, \qquad (9)$$

where $l = A_1, A_2, B_1, B_2$ is the sublattice index, $k$ the crystal momentum, $N$ the number of unit cells, and $a$ the unit cell length. We choose units such that $\hbar = 1$. After substituting Eq. 9 in Eq. 8, we obtain:

$$H = \sum_{k} c_{lk}^{\dagger} \mathcal{H}_{lm}(k) c_{mk}, \qquad (10)$$

where momentum $k$ resides in the one-dimensional BZ $[-\pi, \pi)$ and the Einstein summation on the sublattice indices $(l, m)$ is implied.

Since we choose not to include the relative positions of the sites within each cell on the exponential factors in Eq. 9, the Bloch Hamiltonian $\mathcal{H}(k)$ has $2\pi$-periodicity. This choice of basis is the so-called basis I which is mathematically more convenient to use.

The periodic Bloch Hamiltonian has the following 4 × 4 matrix structure with the basis $\{A_1, A_2, B_1, B_2\}$ at each $k$ value (Fig. 3):

$$\mathcal{H}(k) = \begin{pmatrix} 0 & t_v & t_1 + t_2 e^{-ika} & t_{c1} + t_{c2} e^{-ika} \\ t_v & 0 & t_{c1} + t_{c2} e^{-ika} & t_1 + t_2 e^{-ika} \\ t_1 + t_2 e^{-ika} & t_{c1} + t_{c2} e^{-ika} & 0 & t_v \\ t_{c1} + t_{c2} e^{-ika} & t_1 + t_2 e^{-ika} & t_v & 0 \end{pmatrix}. \qquad (11)$$

In order to diagonalize the Hamiltonian by 2 × 2 blocks, we make a change of basis to:

$$\left\{ \frac{A_1 - A_2}{\sqrt{2}}, \frac{B_1 - B_2}{\sqrt{2}}, \frac{A_1 + A_2}{\sqrt{2}}, \frac{B_1 + B_2}{\sqrt{2}} \right\}. \qquad (12)$$

The Hamiltonian becomes:
$$ \mathcal{H}(k) = \begin{pmatrix} -t_v & (t_1 - t_{c1}) + (t_2 - t_{c2})e^{-ika} & 0 & 0 \\ (t_1 - t_{c1}) + (t_2 - t_{c2})e^{ika} & -t_v & 0 & 0 \\ 0 & 0 & t_v & (t_1 + t_{c1}) + (t_2 + t_{c2})e^{-ika} \\ 0 & 0 & (t_1 + t_{c1}) + (t_2 + t_{c2})e^{ika} & t_v \end{pmatrix}, \quad (13) \\ = \left( \frac{\mathcal{H}_{-}^{SSH}(k)}{0} \middle| \frac{\mathcal{H}_{+}^{SSH}(k)}{0} \right), $$

where the 2 × 2 matrices on the diagonals $\mathcal{H}_{\pm}^{SSH}(k)$ correspond to simple SSH Bloch Hamiltonians with hopping terms $t_1 \pm t_{c1}$ and $t_2 \pm t_{c2}$, sublattices $\frac{A_1 \pm A_2}{\sqrt{2}}$ and $\frac{B_1 \pm B_2}{\sqrt{2}}$, and on-site energies $\pm t_v$ (Fig. 3).

Each sub-Hamiltonian can be parametrized with the Pauli spin matrices $(\sigma_x, \sigma_y, \sigma_z)$, ignoring the on-site energies $\pm t_v$:

$$ \mathcal{H}_{\pm}^{SSH}(k) = \mathbf{d}_{\pm}(k) \cdot \boldsymbol{\sigma}, \quad (14) $$

$$ \mathbf{d}_{\pm}(k) = \begin{bmatrix} (t_1 \pm t_{c1}) + (t_2 \pm t_{c2}) \cos k \\ (t_2 \pm t_{c2}) \sin k \\ 0 \end{bmatrix}. \quad (15) $$

The eigenvalues at each k value are given by:

$$ E_{\pm}(k) = \{-|\mathbf{d}_{\pm}|, +|\mathbf{d}_{\pm}|\}, \quad (16) $$

for the valence and the conduction band respectively.

The eigenstates can be parametrized as well via the azimuthal angle $\phi_{\pm}$ of the Bloch sphere:

$$ |u_{\pm}^{v}(k)\rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -e^{i\phi_{\pm}(k)} \end{pmatrix}, \quad (17) $$

$$ |u_{\pm}^{c}(k)\rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ e^{i\phi_{\pm}(k)} \end{pmatrix}, \quad (18) $$

where $e^{i\phi_{\pm}(k)} = (d_{\pm}^{x} + id_{\pm}^{y})/|d_{\pm}|$.

Due to the absence of a $\sigma_z$ term in Eq. 14 (chiral symmetry), the Hamiltonian vector $\mathbf{d}_{\pm}(k)$ resides on the x-y plane, thus the tip of its norm $\hat{\mathbf{d}}_{\pm}(k) = \mathbf{d}_{\pm}(k)/|\mathbf{d}_{\pm}(k)|$ resides on the equator of the Bloch sphere. Consequently, it is possible to define a winding number $n_w$ (i.e., the number of times $\hat{\mathbf{d}}_{\pm}$ encircles the origin of the Bloch sphere in the anticlockwise orientation over the BZ) as a mapping from the BZ circle to the equator of the Bloch sphere [34].

The winding number is intimately related to the eigenstates. As a matter of fact, each winding around the origin corresponds to an orbital character exchange between the valence and the conduction band: the

coefficients of the modified B sites of each associated eigenstate exchange signs over the BZ (see Eq. 18). One can quantify the winding number by calculating the Berry connection over the BZ:

$$ \begin{aligned} \varphi_{\pm} &= i \int_{\text{BZ}} \langle u_{\pm}(k) | \partial_k | u_{\pm}(k) \rangle dk, \\ &= \begin{cases} -\pi, & (t_2 \pm t_{c2}) > (t_1 \pm t_{c1}) \\ 0, & (t_1 \pm t_{c1}) > (t_2 \pm t_{c2}) \end{cases}. \end{aligned} \quad (19) $$

If the intra-cell hopping is stronger than the inter-cell hopping, the angle $\phi_{\pm}(k)$ traces a finite segment over the BZ and ends up with the same value, thus $\varphi_{\pm} = 0$. On the other hand, if the inter-cell hopping is stronger, $\phi_{\pm}(k)$ makes a full circle over the BZ and increases by $2\pi$, thus $\varphi_{\pm} = -\pi$. Then, the winding number will be given by:

$$ n_w = -\frac{\varphi_{\pm}}{\pi}. \quad (20) $$

Note that $\varphi_{\pm}$ has the exact same structure as the Zak phase but without the periodic gauge choice. Due to the bulk-boundary correspondence, each winding creates a pair of mid-gap edge states on an open finite chain obtained by cutting the periodic infinite chain in accordance with the choice of unit cell. This is a characteristic of a topological insulator, therefore, we call $n_w = 1$ the topologically non-trivial phase and $n_w = 0$ the topologically trivial phase, which are the two possible values we can observe in polyacene polymers. In this choice of the unit cell-restricted sense, the winding number can also be recognized as a $\mathbb{Z}_2$ invariant. In the Supplemental Material (Figs. S15 and S16) we show in detail the winding number analysis of the single-(double-)bridged polymers, including the trajectories of the normalized Hamiltonian vectors and the change of the orbital parities.

As opposed to the Zak phase approach, with the winding number analysis we can have further insights on how deep each sub-system is in their respective phase by visualizing the trajectories of the Hamiltonian vector $\mathbf{d}_{\pm}(k)$. Moreover, we can understand the significance of the next-nearest neighbour (NNN) interactions better (see Fig. S20 in the Supplemental Material for a schematic representation of the second nearest neighbour
<figure><img src="image_6.png" /><figcaption>FIG. 4. Multiple topological phase transitions using a double-stacked SSH model for different sets of hopping parameters. The values of $t_{c1}$ and $t_{c2}$ are fixed with respect to $t_1$. Top panel: electronic band structures and the corresponding finite energy spectra (22 unit cells) for three different $t_2/t_1$ ratios denoted with the red A, B and C labels on the bottom scale. The green (pink) bands are of the (anti-)bonding sub-SSH (Fig. 3). Bottom panel: green (pink) vertical line corresponds to a topological phase transition of the green (pink) bands. $A \to (\mathbb{Z}_2^{(i)}, \mathbb{Z}_2^{(o)}) = (0,0)$, $B \to (\mathbb{Z}_2^{(i)}, \mathbb{Z}_2^{(o)}) = (1,0)$, $C \to (\mathbb{Z}_2^{(i)}, \mathbb{Z}_2^{(o)}) = (1,1)$.</figcaption></figure>

interactions). Indeed, the exponential factors in the periodic Bloch Hamiltonian are related to the inter-cell distance, therefore, the NNN hoppings will be present with greater exponential factors, e.g., $t_3e^{i2ka}$, $t_4e^{i3ka}$ etc. As a result, the trajectory of the corresponding Hamiltonian vector deviates from a perfect circle due to these NNN hoppings. In the Supplemental Material (Fig. S21) we show the winding number analysis of the double-bridged polypentacene both with and without including the NNN hoppings.

By modeling a 4-band SSH-like insulator with the double-stacked SSH model, we can keep track of the topological phase of each band. For instance, a material which hosts 4 non-trivial bands exhibits 2 pairs of mid-gap states, each associated to the inner bands (the bands closer to the Fermi level) and outer bands (the bands further away from the Fermi level) respectively. The simple SSH model can only show one of the pairs of SSH-like topologically non-trivial bands, failing to capture the correct structure of the bands near the Fermi level and the correct number of mid-gap states.

In Fig. 4, we can observe the evolution of the band structure of the double-stacked SSH model over three values of $t_2/t_1$ ratios with fixed $t_{c1}$ and $t_{c2}$. Each value corresponds to a distinct topological phase. On the left, both bands are trivial, i.e., $(\mathbb{Z}_2^{(i)} = 0, \mathbb{Z}_2^{(o)} = 0)$, where $\mathbb{Z}_2^{(i)}$ ($\mathbb{Z}_2^{(o)}$) is the $\mathbb{Z}_2$ invariant of the inner (outer) sub-bands, on the center, the first topological phase transition occurs where the inner bands become non-trivial, i.e., $(\mathbb{Z}_2^{(i)} = 1, \mathbb{Z}_2^{(o)} = 0)$, and on the right, the second topological phase transition occurs where both bands are now non-trivial, i.e., $(\mathbb{Z}_2^{(i)} = 1, \mathbb{Z}_2^{(o)} = 1)$.

It is important to notice that our TB model represents a toy model that can be used to describe hypothetical systems made of two coupled SSH chains with hopping parameters that can be continuously tuned. In the following we provide two examples of possible systems which can be modeled in this way. We would also like to stress that the quality of our description derives from the quality of the Wannier interpolation, as we parametrize the TB model starting from DFT wavefunctions and eigenvalues: in the Supplemental Material, for each polymer, we provide the details of the Wannier functions, as well as their localizations and the band-interpolation (see Sec. S1).

## IV. RESULTS ON SINGLE-BRIDGED POLYACENE POLYMERS

In order to validate our numerical method for the evaluation of the topological invariants in the simple SSH model, we consider single bridged polyacene polymers and we focus our analysis on polyanthracene ($n = 3$) and polypentacene ($n = 5$). As a matter of fact, these simple one-bridge systems have been previously studied in literature [8], allowing us a direct comparison of our results with the state-of-the-art.

### A. Structural properties

Single-bridged polyacene polymers are one-dimensional (1D) carbon materials (Fig. 5) consisting of acene monomers, each characterized by a specific number $n$ of benzene rings, linked together by either ethylynic ($-C \equiv C-$) or cumulenic (= C = C =)
<figure><img src="image_7.png" /><figcaption>FIG. 5. Stick-and-ball model of single-bridge a) polyanthracene (n=3) and b) polypentacene (n=5). Brown (pink) spheres are carbon (hydrogen) atoms. The 2-site SSH model constructed with MLWFs is also highlighted with red and blue boxes for A and B sites respectively. Yellow (blue) spheres are p<sub>z</sub>-like MLWFs. t<sub>1</sub> (t<sub>2</sub>) is the intra-cell (inter-cell) hopping amplitude of the simple SSH model.</figcaption></figure>

bridges. The difference between triple and single bond-lengths in the chain is an important structural parameter called bond length alternation (BLA<sub>b</sub>, subscript “b” for bridge), which is correlated to the trivial or non-trivial topological behaviour of the polymers. Moreover, each bridge configuration is associated with an aromatic or a quinoidal ring structure: the aromatic structure consists of two degenerate possibilities of alternating single and double bonds (depicted with a ring), while the quinoidal configuration has a fully conjugated cyclic structure.

<figure><img src="image_8.png" /></figure>

To classify aromatic/quinoidal rings, we also define a bond length alternation of the benzene rings connected by the bridges: BLA<sub>m</sub>, subscript “m” for monomer. The reason why we disregard the remaining benzene rings constructing the monomer is that due to their single hydrogen atom termination on both sides, they cannot be in the quinoidal form. In the Supplemental Material, we give the details of the BLA definitions and show all the bond lengths for each polymer we studied in this work (see Sec. S3). Depending on the choice of the functional, we can obtain different structures. B3LYP and CAMB3LYP optimized *n* = 3 shows similar values of BLA<sub>b</sub> with 0.1950 Å and 0.2146 Å, respectively. Moreover, for each geometry, BLA<sub>m</sub> almost vanish. These results suggest that they both have the aromatic/ethylynic configuration. For *n* = 5, the BLA<sub>b</sub> of the B3LYP optimized geometry decreases to 0.1212 Å. Moreover, the bond connected to the bridge slightly increases, 1.4600 Å, with respect to the neighbouring one constructing the benzene ring, 1.4403 Å, (see Figs. S9 and S11 in the Supplemental Material for the bond lengths of B3LYP optimized *n* = 3 and *n* = 5) both indicating a transition towards the quinoidal/cumulenic configuration. However, it is important to notice that a fully cumulenic structure with vanishing weight of the ethylynic phase would exhibit BLA<sub>b</sub>=0. In contrast to the B3LYP optimized geometry, the BLA<sub>b</sub> and BLA<sub>m</sub> of the CAMB3LYP optimized geometry stays almost unchanged. This suggests that the polymer is still estimated to be in the aromatic/ethylynic configuration. In all cases, we obtain similar lattice parameters ranging from 6.8865 Å to 6.9174 Å. The relevant geometry parameters are summarized in Tab. I. Our B3LYP (CAMB3LYP) optimized geometries match perfectly with those reported in literature [8, 16].

<table><caption>TABLE I. Bond length alternation of the bridge (connected benzene ring): BLA<sub>b</sub> (BLA<sub>m</sub>) and lattice parameters (a) for single-bridge polyacene polymers as a function of the exchange-correlation functional (XC) and number of carbon rings (n) in the acene monomer. The calculations were done with PBE on top of the geometries relaxed with the XC stated in the table.</caption><thead><tr><th>XC</th><th>n</th><th>BLA<sub>b</sub> (Å)</th><th>BLA<sub>m</sub> (Å)</th><th>a (Å)</th></tr></thead><tbody><tr><td>B3LYP</td><td>3</td><td>0.1950</td><td>0.0114</td><td>6.9070</td></tr><tr><td>CAMB3LYP</td><td>3</td><td>0.2146</td><td>0.0159</td><td>6.8865</td></tr><tr><td>B3LYP</td><td>5</td><td>0.1212</td><td>0.0198</td><td>6.9113</td></tr><tr><td>CAMB3LYP</td><td>5</td><td>0.2096</td><td>0.0257</td><td>6.9174</td></tr></tbody></table>
Notice that it was shown by Cirera et al. [8] that, for single-bridged polyacene polymers, the aromatic/ethylynic and the quinoidal/cumulenic configurations are energetically non-degenerate. Furthermore, they showed that the polymers always exhibit a combination of the two configurations with different weights, as a function of *n*: the chemical structures (and BLA) do not change sharply between two well-defined values but instead they show a tendency to go towards a quinoidal/cumulenic structure from an aromatic/ethylynic one as *n* is increased in a discrete way. This correlation between the chemical structure and the topological phase is later consolidated by Bhattacharjee et al. [39], where they analyzed 24 different polymers, 15 of which in aromatic phase and 9 in the quinoidal phase. The authors found a correlation between the chemical structure and the Zak invariant determined by the presence of edge states, where the aromatic/ethylynic (quinoidal/cumulenic) polymers exhibited $\mathbb{Z}_2 = 0$ ($\mathbb{Z}_2 = 1$)

## B. Electronic properties and topological phase transition

In Fig. 6, we show the electronic dispersion relation and the density of states (DOS) for the valence and conduction bands of polyanthracene ($n = 3$) and polypentacene ($n = 5$) projected onto the $p_z$ orbitals of C:

$$|\psi_n(\mathbf{k})\rangle = \sum_j C_j^n(\mathbf{k})|\phi_j\rangle, \quad (21)$$

where $|\psi_n(\mathbf{k})\rangle$ is the Bloch function of the $n^{th}$ band corresponding to the canonical Bloch Hamiltonian $e^{-i\mathbf{k}\cdot\mathbf{r}}He^{i\mathbf{k}\cdot\mathbf{r}}$, $|\phi_j\rangle$ are the orbitals of the system and $|C_j^n(\mathbf{k})|^2$ are the probability amplitudes.

The main orbitals responsible for the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) of the polyacene polymers are the $p_z$ orbitals on the Carbon atoms connected by the ethynylene bridges. Therefore, we use the two $p_z$ orbitals centered on these atoms as initial guesses for the Wannier interpolation (Fig. 5).

The PBE band gap of the CAMB3LYP (PBE-CAMB3LYP) optimized polyanthracene, 0.8187 eV, is relatively larger than the PBE gap of the B3LYP optimized geometry (PBE-B3LYP), 0.6535 eV. Nevertheless, for what concerns when going from $n = 3$ to $n = 5$, the PBE-B3LYP gap decreases slightly whereas it keeps diminishing to 0.1705 eV for PBE-CAMB3LYP.

We find overall lower hopping values for $n=5$ and this can be understood from the fact its bands are relatively narrower compared to those of $n=3$ (Fig. 6 and Tab. II).

<table><caption>TABLE II. Hopping amplitudes ($t_1, t_2$), band gap ($E_g$) and Zak phase ($\mathbb{Z}_2$) for single-bridge polyacene polymers as a function of the exchange-correlation functional (XC) and number of carbon rings (n) in the acene monomer. The calculations were done with PBE on top of the geometries relaxed with the XC stated in the table.</caption><thead><tr><th>XC</th><th>n</th><th>t<sub>1</sub> (eV)</th><th>t<sub>2</sub> (eV)</th><th>E<sub>g</sub> (eV)</th><th>Z<sub>2</sub></th></tr></thead><tbody><tr><td>B3LYP</td><td>3</td><td>0.9405</td><td>0.5492</td><td>0.6535</td><td>0</td></tr><tr><td>CAMB3LYP</td><td>3</td><td>0.9914</td><td>0.5197</td><td>0.8187</td><td>0</td></tr><tr><td>B3LYP</td><td>5</td><td>0.3820</td><td>0.5214</td><td>0.4713</td><td>1</td></tr><tr><td>CAMB3LYP</td><td>5</td><td>0.5202</td><td>0.3656</td><td>0.1705</td><td>0</td></tr></tbody></table>

Likewise, $n=5$ has relatively longer bond lengths compared to $n=3$ which usually precipitates lower hopping values. Moreover, in the B3LYP optimized geometry we can observe the topological phase transition from $n = 3$ to $n = 5$ with the inter-cell hopping $t_2 = 0.5214$ eV surpassing the intra-cell hopping $t_1 = 0.3820$ eV, consistent with the experimental findings of Ref. [8] on Au(111) substrates. However, we do not observe the same transition on the CAMB3LYP optimized geometry, since it is deeper in the trivial regime due to greater bridge size, consistent with results of Ref. [16] in vacuum where the phase transition happens between $n=5$ and $n=7$. These results confirm the connection between the aromatic/ethylynic (quinoidal/cumulenic) configuration and the topologically (non-)trivial phase.

Equivalently, in the insets of Fig. 6, we can observe the evolution of the corresponding Hamiltonian vectors (Eq. 15) over the BZ. They are confined in the x-y plane of the Bloch sphere due to chiral symmetry. Encircling the origin once corresponds to a winding number $n_w = 1$ which is exhibited only by the B3LYP optimized $n = 5$. Indeed, a non-zero winding number corresponds to an orbital character exchange between the bands over the BZ, a footprint of a transition to the topologically non-trivial phase.

In the Supplemental Material we show the open-finite chain (12 monomers) energy spectra (Fig. S18) obtained via direct diagonalization of our effective Wannier interpolated TB Hamiltonians, where we highlighted in red the HOMO/LUMO states and provided the associated wave functions. We can observe the emergence of quasi-double-degenerate mid-gap edge states of the B3LYP optimized polypentacene. Increasing values of $t_1$ will strengthen the coupling of the edge states with the bulk states. These couplings depend on the decay factors of the edge states towards the bulk [40, 41]:

$$a_m \sim e^{-m/\xi}, \quad (22)$$

$$\xi \equiv \frac{1}{\ln\left(\frac{t_2}{t_1}\right)}, \quad (23)$$

where *m* is the index of the unit cell increasing from the
<figure><img src="image_9.png" /><figcaption>FIG. 6. In each panel we plot the orbital-projected electronic dispersions (left), the total and orbital-projected density of states(right), the winding numbers (inset) of polyanthracene (n=3, left-side panels) and polypentacene (n=5, right-side panels). The orange disks represent the contribution coming from the p_z orbitals on the carbon atoms connected by the ethynylene bridges and the blue circles represent the ones on the Carbon atoms constructing the bridges. The size is proportional to the sum of the probability amplitudes of each set of orbitals. Since the polymers are insulating, we placed the Fermi level in the middle of the band gap. In both systems, results have been obtained via PBE on top of B3LYP (upper panels) and CAMB3LYP (lower panels) relaxed structures.</figcaption></figure>

left edge to the right. $a_m$ is the coefficient of the contribution coming from each A sublattice of the left edge state. Similarly, the right edge state has contributions only coming from the B sublattices and they decay towards the left edge. As a result, in Fig. S18 we can observe a weak hybridization between the edge states:

$$ \langle \psi_{A, \text{left}} | \hat{H} | \psi_{B, \text{right}} \rangle \sim e^{-N/\xi}, \quad (24) $$

with the anti-bonding configuration ($A - B$) being higher in energy than the bonding one ($A + B$). Greater values of $t_2$ or increasing the chain length $N$ can break the hybridization completely.

In order to validate these results and to observe the orbital character of the mid-gap states, we also calculate them via DFT, confirming the existence of the edge states as can be seen in Fig. 7: it is possible to observe the (anti-)bonding hybridization of HOMO (LUMO), consistent with the TB approach, and the spreading of the HOMO-1/LUMO+1 around the bulk as opposed to

the mid-gap states, highlighting their profound spatial difference.

### C. Effect of exchange-correlation functional

The substrate on which polymers are grown or deposited provides us with a strong control over their topological properties. Indeed, the screening of the substrate could inhibit the effects of long-range electron-electron interactions, which naturally exist for organic materials. For this reason, while a range-separated hybrid functional (such as CAMB3LYP) works better for long chains in vacuum [16], global hybrid functionals (such as B3LYP [8]) work better for modeling the polymers on a substrate: as a matter of fact a substrate screens long-range interactions and strongly modifies correlation energies, as discussed in Refs. [16, 42].

From our analysis, polypentacene is the most sen-
<figure><img src="image_10.png" /><figcaption>FIG. 7. Atomic wavefunctions with relative phases of topologically non-trivial finite-sized (12 monomers) single-bridged polypentacene (n=5) obtained via the PBE functional on top of the B3LYP optimized geometry. HOMO/LUMO represent the degenerate mid-gap edge states. We show the first (last) three monomers for the left (right) end-side and the central three monomers in order to show the spatial localization of the edge-states compared to the delocalized HOMO-1/LUMO+1. See Fig. S17 in the Supplemental Material for the corresponding energy spectrum.</figcaption></figure>
sitive case to the choice of the exchange-correlation
functional as it is very close to the topological phase
transition. Since our goal is to rationalize the physics
behind multiple topological phases as a function of the
number of bridges, in the following we will consider only
geometry optimization via CAMB3LYP as we neglect
the screening effect of the substrate.

V. RESULTS ON DOUBLE-BRIDGED POLYACENE POLYMERS

For $n = 5$ there are two symmetric possibilities for the placement of the carbon bridges (Fig. 8), which also leads to an additional SSH-like pair of bands appearing near the Fermi level: we call them external-double-bridged (Fig. 8 a) and internal-double-bridged (Fig. 8 b). Therefore, a double-bridged configuration can be seen as two interacting 1D chain models, mediated along each bridge: our double-stacked SSH model (Sec. III) can describe two sub-SSH systems characterized by different vertical superpositions of the sites (bonding and anti-bonding), resulting with detrimental or incremental contributions of the inter-chain hopping amplitudes.

In this section, we will show that the two polypentacene with different bridge configuration will be in different topological phase. However, it is important to note that the bridge configuration is not a parameter that one tunes continuously to induce a transition, unlike mechanical strain for instance [39, 43–45].

A. Structural properties

Due to symmetry along the main axis, we can divide
the polymer in two halves with the same chemical
structure. We can therefore limit our analysis to the
upper half of the polymer.

Both the external-double-bridged and internal-double-
bridged polypentacene have a similar bridge structure
with BLA<sub>b</sub> of 0.2178 Å and 0.2074 Å, respectively,
suggesting an ethylynic configuration. Moreover, for
both cases we find significantly lower values of BLA<sub>m</sub>
(External: 0.0234 Å, Internal: 0.0216 Å), hinting to an
aromatic benzene ring. Similarly, we obtain almost equal
lattice parameters for both cases (External: 6.8964 Å,
Internal: 6.8860 Å). The relevant geometry parameters
are summarized in Tab. III.

<figure><img src="image_11.png" /><figcaption>FIG. 8. Stick-and-ball model of a) external- and b) internal-double-bridged polypentacene (n=5). Brown (pink) spheres are carbon (hydrogen) atoms. The 4-site SSH model constructed with MLWFs is also highlighted with red and blue boxes for A and B sites respectively. Yellow (blue) spheres are p<sub>z</sub>-like MLWFs. t<sub>1</sub> (t<sub>2</sub>) is the intra-cell (inter-cell) hopping amplitude, t<sub>c1</sub> and t<sub>c2</sub> are the inter-chain cross hopping amplitudes and t<sub>v</sub> is the on-site energy of the stacked SSH model.</figcaption></figure>
<table><caption>TABLE III. Bond length alternation of the bridge (connected benzene ring): BLA<sub>b</sub> (BLA<sub>m</sub>) and lattice parameters (a) for external- and internal-double-bridged polypentacene polymers, computed via PBE XC on top of CAMB3LYP optimized geometry.</caption><thead><tr><th>System</th><th>BLA<sub>b</sub> (Å)</th><th>BLA<sub>m</sub> (Å)</th><th>a (Å)</th></tr></thead><tbody><tr><td>External</td><td>0.2178</td><td>0.0234</td><td>6.8964</td></tr><tr><td>Internal</td><td>0.2074</td><td>0.0216</td><td>6.8860</td></tr></tbody></table>

## B. Electronic properties and topological phase transition

In Fig. 9, we show the electronic dispersion relation and the density of states (DOS) for the HOMO/HOMO+1 and LUMO/LUMO-1 bands of a) external- and b) internal-double-bridged polypentacene (n = 5) projected onto the p<sub>z</sub> orbitals of C, both obtained with PBE on top of CAMB3LYP optimized geometries.

Both pairs of bands exhibit a similar orbital character where the p<sub>z</sub> orbitals around the Carbon atoms connected by the bridges dominate. The associated electronic wave functions of the inner bands will be more localized around the central benzene rings (see Fig. S7 in the Supplemental Material). Therefore, when we switch to the configuration with the bridges near the center, we can observe that the inner bands are driven towards the topological regime from the diminishing band gap of 0.1996 eV compared to 0.5144 eV of the external-bridged configuration. On the other hand, in both cases, the molecular orbitals of the outer bands are spread around the whole monomer except for the central benzene ring (see Fig. S7). As a result, changing the location of the bridges does not significantly impact the outer band structures.

In Sec. III we discussed how we can divide the double-stacked SSH model into two sub-SSH systems on which the topological formalism of the simple SSH model applies. Therefore, we compute the hopping parameters with Wannier interpolation, and we determine the ordered pair of Z<sub>2</sub> invariants of both bridge configurations, each one associated with the inner bands and the outer bands, respectively. We only find the inner bands of the internal-double-bridged configuration to be in the topologically non-trivial phase (Z<sub>2</sub><sup>(i)</sup> = 1, Z<sub>2</sub><sup>(o)</sup> = 0):

$$t_1 - t_{c1} < t_2 - t_{c2} \Rightarrow 0.4925 \text{ eV} < 0.5151 \text{ eV},$$

Indeed, it can be seen in Tab. IV that the intramonomer hoppings t<sub>1</sub> and t<sub>c1</sub> are almost the same for each bridge configuration as expected, whereas the inter-monomer hoppings differ. t<sub>2</sub> is stronger when the bridges are between the benzene rings adjacent to the central one since the MLWFs tend to localize around these benzene rings. This leads to an increase in the

<figure><img src="image_12.png" /><figcaption>FIG. 9. Orbital-projected electronic dispersions (left), total and orbital-projected density of states(right) and winding numbers (inset) of a) external- and b) internal-double-bridged polypentacene (n=5) polymer. The orange disks represent the contribution coming from the p<sub>z</sub> orbitals on the carbon atoms connected by the ethynylene bridges and the blue circles represent the ones on the Carbon atoms constructing the bridges. The size is proportional to the sum of the probability amplitudes of each set of orbitals. Since the polymers are insulating, we placed the Fermi level in the middle of the band gap. In both systems, results have been obtained via PBE on top of CAMB3LYP relaxed structures.</figcaption></figure>

destructive overlap between them, ultimately increasing t<sub>2</sub>. In contrast, t<sub>c2</sub> is weaker for this configuration. Similarly, this is due to the increase in the constructive overlaps of the MLWFs.

In the double-bridged configuration, the constraint on the geometry is increased with the addition of the second bridge. As a result, observing the correlation between the chemical structure and the topological phase becomes more complicated with respect to the single-bridged configuration. Indeed, contrary to Bhattacharjee et al. [39], we have shown that double-bridged polymers can host topological states while being in the aromatic/ethylynic configuration (internal-double-bridged polypentacene). This can also be understood by the fact that this polymer is quite close to the critical point of transition, according to our TB model. As a matter of fact, Mosaferi et al. [45] have recently shown that double-bridged six ring (n = 6) polyacene polymers showcase multiple topological phase
transitions due to the application of mechanical strain. Moreover, they found out that as the polymer is driven deep towards the topological regime, only then it becomes possible to observe the quiniodal/cumulenic behaviour and not around the critical point of transition.

In the insets of Fig. 9 (a), we can observe that the Hamiltonian vectors of the external-bridged configuration follow an ellipse shape rather than a circle. This deviation is due to the NNN interactions, with hopping term $(t_4 \pm t_{c4})e^{ik(3a)}$ (i.e., the inter-cell hopping between the B sites and the A sites in the third neighbouring cells). As the elongation of the circles alternates between each sub-system, it suggests that the contribution is primarily coming from an inter-chain hopping $t_{c4}$, changing sign for each sub-system. Indeed, we find that $t_{c4} = 0.0100$ eV, which is $\sim t_1/10$. In situations where the intra-cell and the nearest inter-cell hoppings are almost identical, the NNN interactions become crucial as they can be the determining factor whether the material is topologically trivial or not.

<table><caption>TABLE IV. Intra-chain ($t_1, t_2$) and inter-chain ($t_{c1}, t_{c2}$) hopping amplitudes, band gap ($E_g$) and inner/outer (i/o) Zak phases ($\mathbb{Z}_2^{(i)}, \mathbb{Z}_2^{(o)}$) for external-and internal-double-bridged polypentacene polymer computed via PBE on top of CAMB3LYP optimized geometry. All hopping amplitudes and energy gaps are in eV.</caption><thead><tr><th>System</th><th>t<sub>1</sub></th><th>t<sub>2</sub></th><th>t<sub>c1</sub></th><th>t<sub>c2</sub></th><th>E<sub>g</sub></th><th>(Z<sub>2</sub><sup>(i)</sup>, Z<sub>2</sub><sup>(o)</sup>)</th></tr></thead><tbody><tr><td>External</td><td>0.9900</td><td>0.4150</td><td>0.4402</td><td>0.1612</td><td>0.5144</td><td>(0, 0)</td></tr><tr><td>Internal</td><td>0.9622</td><td>0.5545</td><td>0.4697</td><td>0.0394</td><td>0.1996</td><td>(1, 0)</td></tr></tbody></table>

In the Supplemental Material (Fig. S19), we show the energy spectra of the open-finite chains obtained with the 4-site TB model. We can observe the emergence of a pair of mid-gap states located at the edges of the internal-bridged configuration. Moreover, the vertically anti-bonding nature of the states further consolidates our description of the material with the two sub-SSH systems, since it is consistent with the sub-basis of the inner bands we have identified. Furthermore, we can observe the (horizontally) bonding and anti-bonding hybridization of the edge states with the latter being higher in energy.

In Fig. 10, we show the atomic wavefunctions of the 4 energy states closest to the Fermi energy obtained via DFT on the non-trivial internal-bridged polypentacene (22 monomers), showing the localized nature of mid-gap edge states (HOMO/LUMO) as opposed to the delocalized trivial states (HOMO+1/LUMO-1). The geometry of double-bridged polypentacene is such that there are odd number of benzene rings between the two bridges. In such geometries, each vertically separated MLWF have

the same phase localized around their associated bridge in order to have constructive overlap around the central benzene ring (Fig. 10). Moreover, since we are not far from the topological phase transition, we can also observe (Fig. 10) the weak horizontal hybridization between the edge states on each end due to their slow spatial decay (Eq. 22).

## VI. CONCLUSIONS

In this work we have identified a minimal four bands TB model that can be described via stacked SSH chains. Due to orbital hybridization, we have shown that the problem can be decoupled into vertically bonding and anti-bonding sub-SSH models, allowing for a full solution in terms of simple SSH chains: in this description, inter-chain hoppings are added to (subtracted from) the intra-chain hoppings for the (anti-)bonding sub-SSH. Moreover, we have shown how, by tuning inter-chain and intra-chain hoppings it is possible to obtain subsequent multiple topological phase transitions.

As a practical application, we have shown that our formalism can accurately capture the physics of multi-bridged polyacene polymers. More precisely, we have studied double-bridged polypentacene, which effectively behaves as a 4-band insulator. We have shown how, by changing the relative position of the bridges, it is possible to effectively change the inter-monomer electron hopping amplitudes along the bridges ($t_2$) and across one another ($t_{c2}$), thus selectively tuning the topological phase for one of the two sub-bands.

We would like to point out that the dynamical stability of this system will be the subject of future work. In particular, the light atomic mass and low dimensionality necessitate the inclusion of quantum anharmonic effects.

In conclusion, our stacked SSH model can be used to describe any material that shares the inversion symmetries of the model and most conveniently, stacking of any material whose electronic structure can be described by the SSH, such as polyacene polymers.

## ACKNOWLEDGMENTS

D.R. and M.A. acknowledge support from the HPC resources of IDRIS, CINES, and TGCC under Allocation No. 2024-A0160914101 made by GENCI, as well as funding from the project TyLDE (Grant No. ANR-23-CE50-0001) founded by the French National Research Agency. A.W.C. and M.M. wish to acknowledge support from the ANR Project ACCEPT (Grant No. ANR-19-CE24-0028). The authors thank G. Marini for useful discussions.
[1] A. Köhler and H. Bässler, *Electronic Processes in Organic Semiconductors: An Introduction* (John Wiley & Sons, New York, 2015).

[2] H. Bronstein, C. B. Nielsen, B. C. Schroeder, and I. McCulloch, The role of chemical design in the performance of organic semiconductors, *Nature Reviews Chemistry* **4**, 66 (2020).

[3] J. Méndez, M. F. López, and J. A. Martín-Gago, On-surface synthesis of cyclic organic molecules, *Chemical Society Reviews* **40**, 4578 (2011).

[4] J. Lisiecki and P. Szabelski, Halogenated anthracenes as building blocks for the on-surface synthesis of covalent polymers: structure prediction with the lattice monte carlo method, *The Journal of Physical Chemistry C* **125**, 15934 (2021).

[5] X. Li et al., Recent progress on surface chemistry I: Assembly and reaction, *Chinese Chemical Letters* **35**, 110055 (2024).

[6] G. Allinson, R. J. Bushby, J.-L. Paillaud, and M. Thornton-Pett, Synthesis of a derivative of triangulene; the first non-kekulé polynuclear aromatic, *Journal of the Chemical Society, Perkin Transactions 1*, 385 (1995).

[7] A. Sánchez-Grande, B. de la Torre, J. Santos, B. Cirera, K. Lauwaet, T. Chutora, S. Edalatmanesh, P. Mutombo, J. Rosen, R. Zbořil, R. Miranda, J. Björk, P. Jelínek, N. Martín, and D. Écija, On-Surface Synthesis of Ethynylene-Bridged Anthracene Polymers, Angewandte Chemie International Edition **58**, 6559 (2019).

[8] B. Cirera, A. Sánchez-Grande, B. De la Torre, J. Santos, S. Edalatmanesh, E. Rodríguez-Sánchez, K. Lauwaet, B. Mallada, R. Zbořil, R. Miranda, et al., Tailoring topological order and π-conjugation to engineer quasi-metallic polymers, *Nature nanotechnology* **15**, 437 (2020).

[9] H. González-Herrero, J. I. Mendieta-Moreno, S. Edalatmanesh, J. Santos, N. Martín, D. Écija, B. de la Torre, and P. Jelinek, Atomic Scale Control and Visualization of Topological Quantum Phase Transition in Conjugated Polymers Driven by Their Length, *Advanced Materials* **33**, 2104495 (2021).

[10] J. Hieulle, S. Castro, N. Friedrich, A. Vegliante, F. R. Lara, S. Sanz, D. Rey, M. Corso, T. Frederiksen, J. I. Pascual, and D. Peña, On-Surface Synthesis and Collective Spin Excitations of a Triangulene-Based Nanostar, Angewandte Chemie International Edition **60**, 25224 (2021).

[11] S. Cheng, Z. Xue, C. Li, Y. Liu, L. Xiang, Y. Ke, K. Yan, S. Wang, and P. Yu, On-surface synthesis of triangulene trimers via dehydration reaction, *Nature Communications* **13**, 1705 (2022).

[12] A. Jiménez-Martín, Z. Sosnová, D. Soler, B. Mallada, H. González-Herrero, S. Edalatmanesh, N. Martín, D. Écija, P. Jelínek, and B. de la Torre, *Atomically Precise Control of Topological State Hybridization in Conjugated Polymers*, ACS Nano **18**, 29902 (2024).

[13] D. Vanderbilt, *Berry Phases in Electronic Structures* (Cambridge University Press, Cambridge, UK, 2018).

[14] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in polyacetylene, *Phys. Rev. Lett.* **42**, 1698 (1979).

[15] A. M. Pendás, J. Contreras-García, F. Pinilla, J. D. Mella, C. Cardenas, and F. Muñoz, A chemical theory of topological insulators, *Chemical Communications* **55**, 12281 (2019).

[16] D. Romanin, M. Calandra, and A. W. Chin, Excitonic switching across a $Z_2$ topological phase transition: From mott-wannier to frenkel excitons in organic materials, *Physical Review B* **106**, 155122 (2022).

[17] W. J. Jankowski, J. J. P. Thompson, B. Monserrat, and R.-J. Slager, Excitonic topology and quantum geometry in organic semiconductors (2024), arXiv:2406.11951 [cond-mat.mes-hall].

[18] J. J. P. Thompson, W. J. Jankowski, R.-J. Slager, and B. Monserrat, Topologically-enhanced exciton transport (2024), arXiv:2410.00967 [cond-mat.mes-hall].

[19] M. Röntgen, X. Chen, W. Gao, M. Pyzh, P. Schmelcher, V. Pagneux, V. Achilleos, and A. Coutant, Latent Schrieffer-Heeger models (2023), arXiv:2310.07619 [cond-mat].

[20] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, et al., *Quantum espresso: a modular and open-source software project for quantum simulations of materials*, *Journal of Physics: Condensed Matter* **21**, 395502 (2009).

[21] P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. Buongiorno Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, et al., *Advanced capabilities for materials modelling with quantum espresso*, *Journal of Physics: Condensed Matter* **29**, 465901 (2017).

[22] A. Erba, J. K. Desmarais, S. Casassa, B. Civalleri, L. Donà, I. J. Bush, B. Searle, L. Maschio, L. Edith-Daga, A. Cossard, C. Ribaldone, E. Ascrizzi, N. L. Marana, J.-P. Flament, and B. Kirtman, *Crystal23: A program for computational solid state physics and chemistry*, *Journal of Chemical Theory and Computation* **19**, 6891 (2023), pMID: 36502394.

[23] C. R. R. O. C. M. Z.-W. F. P. B. C. K. D. N. H. I. B. P. D. M. L. M. C. Y. N. L. M. A. E. M. R. S. C. B. S. J. D. R. Dovesi, V. R. Saunders, *Crystal23 user's manual*, University of Torino, Torino (2023).

[24] T. Yanai, D. P. Tew, and N. C. Handy, A new hybrid exchange-correlation functional using the coulomb-attenuating method (cam-b3lyp), *Chemical physics letters* **393**, 51 (2004).

[25] Y. Tawada, T. Tsuneda, S. Yanagisawa, T. Yanai, and K. Hirao, A long-range-corrected time-dependent density functional theory, *The Journal of chemical physics* **120**, 8425 (2004).

[26] D. Romanin, L. Monacelli, R. Bianco, I. Errea, F. Mauri, et al., *Dominant role of quantum anharmonicity in the stability and optical properties of infinite linear acetylenic carbon chains*, *Journal of Physical Chemistry Letters* **12**, 10339 (2021).

[27] D. R. Hamann, *Optimized norm-conserving vanderbilt pseudopotentials*, *Phys. Rev. B* **88**, 085117 (2013).

[28] D. Vilela Oliveira, J. Laun, M. F. Peintinger, and T. Breddow, *Bsse-correction scheme for consistent gaussian basis sets of double-and triple-zeta valence with polarization quality for solid-state calculations*, *Journal of Computational Chemistry* **40**, 2364 (2019).

[29] J. Zak, *Berry's phase for energy bands in solids*, *Physical Review Letters* **62**, 2747 (1989).
[30] N. Marzari and D. Vanderbilt, Maximally localized wannier functions for composite energy bands, Physical Review B **56**, 12847 (1997).

[31] I. Souza, N. Marzari, and D. Vanderbilt, Maximally localized wannier functions for entangled energy bands, Physical Review B **65**, 035109 (2001).

[32] A. A. Mostofi, J. R. Yates, G. Pizzi, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, An updated version of wannier90: A tool for obtaining maximally-localised wannier functions, Computer Physics Communications **185**, 2309 (2014).

[33] S. Coh and D. Vanderbilt, Python tight binding (pythtb). zenodo. https://doi.org/10.5281/zenodo.12721316 (2024).

[34] J. Cayssol and J. N. Fuchs, Topological and geometrical aspects of band theory, Journal of Physics: Materials **4**, 034007 (2021).

[35] C. Bena and G. Montambaux, Remarks on the tight-binding model of graphene, New Journal of Physics **11**, 095003 (2009).

[36] E. I. Blount, Formalisms of the band theory of solids, Solid State Physics **13**, 305 (1962).

[37] B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, Topological quantum chemistry, Nature **547**, 298 (2017).

[38] J.-N. Fuchs and F. Piéchon, Orbital embedding and topology of one-dimensional two-band insulators, Phys. Rev. B **104**, 235428 (2021).

[39] R. Bhattacharjee and M. Kertesz, Topological Transition in Aromatic and Quinonoid π-Conjugated Polymers Induced by Static Strain, Journal of the American Chemical Society **146**, 26497 (2024).

[40] P. Delplace, D. Ullmo, and G. Montambaux, Zak phase and the existence of edge states in graphene, Phys. Rev. B **84**, 195452 (2011).

[41] J. K. Asbóth, L. Oroszlány, and A. Pályi, *A Short Course on Topological Insulators*, Lecture Notes in Physics, Vol. 919 (Springer International Publishing, Cham, 2016).

[42] J. B. Neaton, M. S. Hybertsen, and S. G. Louie, Renormalization of molecular electronic levels at metal-molecule interfaces, Phys. Rev. Lett. **97**, 216405 (2006).

[43] R. Bhattacharjee and M. Kertesz, Continuous topological transition and bandgap tuning in ethynylene-linked acene π-conjugated polymers through mechanical strain, Chemistry of Materials **36**, 1395 (2024).

[44] Y. Li, K. Zhang, H. Lv, and X. Wu, Topological Transition in One-Dimensional Conjugated Polymers via Strain Engineering, ACS Macro Letters **14**, 329 (2025), publisher: American Chemical Society.

[45] M. Mosaferi, M. Aktas, D. Romanin, and A. W. Chin, Multiple strain-induced topological phase transitions in 1d polyacene polymers: A first-principles study, Phys. Rev. Mater. **9**, 044204 (2025).
<figure><img src="image_13.png" /><figcaption>FIG. 10. Atomic wavefunctions with relative phases of topologically non-trivial finite-sized (22 monomers) internal-double-bridged polypentacene (n=5) obtained via the PBE functional on top of the CAMB3LYP optimized geometry. HOMO/LUMO represent the degenerate mid-gap edge states. We show the first (last) three monomers for the left (right) end-side and the central three monomers in order to show the spatial localization of the edge-states compared to the delocalized HOMO-1/LUMO+1. See Fig. S17 in the Supplemental Material for the corresponding energy spectrum.</figcaption></figure>